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Ab initio investigations of the sulfur dioxide electronic spectrum in the 3900—1700 A region
Zellmer, Robert J., Ph.D.
The Ohio State University, 1992
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
Ab Initio Investigations of the Sulfur Dioxide Electronic Spectrum in the 3900 — 1700 A Region
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Robert J. Zellmer, M.S.
* * * * *
The Ohio State University
1992
Dissertation Committee: Approved by Isaiah Shavitt
Russell M. Pitzer / s ~ - ; __ C. Weldon Mathews Isaiah Shavitt Department of Chemistry To Debbie, Courtney, Amanda, Rose, Mickey, Bruce, and Denise
ii ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to Dr. Isaiah Shavitt for his guidance and undying patience throughout the long years taken to complete of this research.
I am very appreciative of him not giving up on me and allowing me to continue my studies. I would also like to thank the other professors in the Physical Division, especially those on my committee, who have put up with me and helped me during this research. I would like to thank Drs. Frank Brown and Don Comeau for insights provided through discussions during the early part of this research. I also appreciate the programming efforts of Drs. Melanie Pepper and Kyungsun Kim which made much of the research possible. I owe a great deal of graditude to Dr.
Eric Stahlberg for all his programming support and the many helpful discussions we had. I would also like to thank Mr. Gary Kedziora and Galen Gawboy for
their ongoing programming support and helpful insights. The Ohio Supercomputer
Center provided the computer time necessary to complete this research.
I wish to thank my family for providing years of encouragement and support, especially my mother, who never let me give up. I would also like to thank my father for his support, especially in regards to employment during the last couple
of years. Finally, I would like to express my utmost appreciation and indebtedness to my wife and children for their ceaseless support. I can never begin to repay them for the many years they had to put up with my absences and for turning their world upsidedown. They gave up so very much so that I could fulfill my dreams. I will never forget what they did for me. VITA
March 6, 1955 ...... Born—Charleston, S.C.
May 1976 ...... B. S. Department of Chemistry Wright State Uni verity Dayton, Ohio
March 1981 ...... M. S. Department of Chemistry The Ohio State Uni verity Columbus, Ohio
Sept. 1977-Aug. 1982 Graduate Teaching Associate Department of Chemistry The Ohio State University Columbus, Ohio
Jan. 1983-Dec. 1984. Graduate Research Associate Department of Chemistry The Ohio State University Columbus, Ohio
Sept. 1990-Dec. 1990 Instructor Department of Chemistry The Ohio State University Columbus, Ohio PUBLICATIONS
Martin L. Meyers, Robert J. Zellmer, R. Kent Sorrell and Paul G. Seybold, ’’Room-Temperature Phosphorescence of Anilinonaphthalenesulfonate ’’Fluores cence Probe” Compounds”, J. Luminescence , 20, 215 (1975).
Donald C. Comeau, Robert J. Zellmer and I. Shavitt, “The Location and Char acterization of Stationary Points on Molecular Potential Energy Surfaces,” in Geometrical Derivatives of Energy Surfaces and Molecular Properties, eds. P. J0rgensen and J. Simons, 1986, pp. 243-251.
FIELDS OF STUDY
Major Field: Chemistry Theoretical Chemistry. The Ohio State University. Professor Isaiah Shavitt, Adviser. Table of Contents
D E D IC A T IO N ...... ii ACKNOWLEDGEMENTS...... iii VITA ...... v LIST OF FIGURES ...... xi LIST OF TABLES ...... xviii
CHAPTER PAGE I Introduction ...... 1
II Overview of Electronic Structure Theory ...... 10 2.1 The Hartree-Fock SCF Approximation ...... 12 2.2 Basis Sets ...... 13 2.3 Electron Correlation ...... 15 2.3.1 Non-Dynamical Correlation E nergy ...... 16 2.3.2 Dynamical Correlation Effects ...... 17 2.4 Applications ...... 20
III Literature Review of Experimental and Theoretical Work on Sulfur Dioxide ...... 21 3.1 Introduction ...... 21 3.2 Introduction to S02 Spectroscopy ...... 26 3.3 The Geometry and Absolute Energy of the JAi Ground State of S02 ...... 27 3.3.1 Experimental Investigations ...... 27 3.3.2 Theoretically Determined Geometries and En ergies ...... 30 3.3.3 Possible Isomers and a Double Minimum Po tential Surface of S 02 ...... 43 3.3.4 Summary ...... 48 3.4 The 2400-1700 A Absorption Region — The C1B2 (bV) Excited S tate ...... 49
vii 3.4.1 Experimental Investigations ...... 49 3.4.2 Theoretical Investigations ...... 58 3.4.3 Summary ...... _...... 65 3.5 The 3400-2500 A Absorption Region — The A lA2 and B xBx Excited States ...... 65 3.5.1 Experimental Investigations ...... 67 3.5.2 Theoretical Investigations ...... 88 3.5.3 Summary ...... 97 3.6 The 3900-3400 A Absorption Region — The a3Bi, b3A2 and 3B2 Excited States ...... 98 3.6.1 Experimental Investigations ...... 99 3.6.2 Theoretical Investigations ...... 115 3.6.3 Summary ...... 126 3.7 Summary ...... 127
IV Discussion of the Basis Set Employed in the Present Calcu lations ...... 130 4.1 Introduction ...... _...... 130 4.2 Comparison of Basis Sets for the X lAi Ground State, at Single Point Geometries ...... 135 4.2.1 Results at the X lAi Experimental Geometry. . 136 4.2.2 Results at the EBP SCF X xAi Optimized Ge ometry...... 143 4.2.3 Summary of Basis Set Comparisons for the X xAi State at Single Point Geometries ...... 143 4.3 Comparison of Basis Sets for the C XB2 Excited State. 149 4.4 Comparison of Properties Calculated at the [6,4,2/4,2,1] and [7,5,2/5,3,1] SCF Optimized Geometries ...... 154 4.4.1 Results at the X xAi Optimized Geometries. . . 154 4.4.2 Results at the C XB2 Optimized Geometries. . . 159 4.5 Summary ...... 167
V Methods Employed in the Present Calculations ...... 169 5.1 Introduction ...... 169 5.2 MCSCF Methods used for the Potential Surface Studies 174 5.2.1 Discussion of the MCSCFl Active Space .... 177 5.2.2 Discussion of MCSCF2 Active S p ace ...... 1S2 5.3 Cl Methods used in the Present Study ...... 188 5.3.1 Discussion of SRCI Configuration Spaces. . . . 188 5.3.2 Discussion of MRCI Configuration Spaces. . . . 191
viii VI Discussion of Potential Surface Calculations ...... 209 6.1 Introduction ...... 209 6.2 Geometry Optimization Procedures ...... 209 6.2.1 Minimizations without Gradients ...... 212 6.2.2 Minimizations with Gradients ...... 214 6.3 SCF Geometry Optimizations ...... 216 6.4 MCSCF Geometry Optimizations ...... 219 6.5 Summary of Geometry Optimizations ...... 220 6.6 Transition Energies ...... 226 6.6.1 Adiabatic Transition Energies, Te ...... 227 6.6.2 Vertical Excitation Energies, A Ev e r t...... 241 6.6.3 Emission Energies, AFem,-4 ...... 260 6.7 Approximate MCSCF2 Harmonic Frequencies ...... 270 6.8 Summary ...... 274
VII Crossing of the 3A2—3Bi and IA 2—*Bi Potential Surfaces . . . 277 7.1 Method used to Determine the Minimum Crossing Point of Potential S u r fa c e s ...... 277 7.2 MINCROV: Program for Determining Critical Points of Crossing Surfaces ...... 281 7.3 Test of MINCROV Program ...... 282 7.4 MCSCF Minimum Crossing Points ...... 283 7.5 Summary ...... 287
VIII Summary and Conclusions ...... 289 8.1 Summary ...... 289 8.2 Conclusion ...... 294
APPENDICES PAGE A Definitions of Abbreviations for Basis Sets and Methods . . 296 A.l Abbreviations for Basis Sets ...... 297 A.1.1 Generic Basis Set Abbreviations ...... 297 A.1.2 Specific Basis Set Abbreviations ...... 298 A.2 Abbreviations for Ab Initio M eth o d s ...... 299 A.3 Abbreviations for Non-46 Initio M eth od s ...... 300
B Definitions and Explanations of Properties, Symbols and Units 301 B.l Properties ...... 302 C Mulliken Population Analysis ...... 307
REFERENCES...... 310
x List of Figures
FIGURE PAGE
1.1 The spatial orientation of the sulfur dioxide m o lecu le ...... 3
3.1 Schematic representation of the excitation energies for a diatomic m olecule ...... 23
3.2 Schematic representation of the double-well potential of the C 1B2 (*A') state along its v3 mode...... 57
3.3 Schematic representation of the experimental excitation energies of the states responsible for the S02 spectrum in the 3900-1700 A region. 128
8.1 Schematic representation of the experimental and MRCI-DAV2S ex citation energies of the states responsible for the S02 spectrum in the 3900-1700 A region. The solid lines are the experimental val ues and the dashed lines are the calculated values. The calculated A E vert values were determined at the X *Ai state’s experimental ge ometry and are shown in the center of the figure. The calculated Te values were determined at the respective MCSCF2 optimized ge ometries of the individual states and are shown on either side of the AEveri values...... 291 List of Tables
TABLE PAGE
3.1 Experimentally determined geometries for the xAi ground state. . . 29
3.2 Results of various ab initio SCF studies for the ground state (’Ai) of SO 2 at experimental geometries ...... 31
3.3 Geometries of the ground state (*Ai) of SO 2 determined by non -ab initio methods ...... 35
3.4 Optimized geometries for the ground state (aAi) of S02 from various ab initio SCF studies...... 36
3.5 Results of various studies on the ground state (*Ai) of SO 2 using correlated methods ...... 40
3.6 Geometries and energies of the isomers of SO 2 ...... 45
3.7 Experimentally determined geometries and excitations energies for the C 1B 2 (A') excited state ...... 51
3.8 Calculated geometries and excitations energies for the 1B 2 excited state...... 59
3.9 Experimentally determined geometries and excitation energies for the jA 2 and xBi excited states ...... 68
3.10 Calculated geometries and excitation energies for the xA 2 excited state...... 89
3.11 Calculated geometries and excitation energies for the xBi excited state...... 90
xii 3.12 Experimentally determined geometries and excitations energies for the triplet excited states ...... 101
3.13 Calculated geometries and excitation energies for the 3BX excited state...... 116
3.14 Calculated geometries and excitation energies for the 3A2 excited state...... 117
3.15 Calculated geometries and excitation energies for the 3B2 excited state...... 118
4.1 Contracted [6,4,2/4,2,1] Gaussian basis set for (9s5pld) oxygen and (Ils7p2d) sulfur primitive basis sets ...... 133
4.2 The experimental and [7,5,2/5,3,1] SCF optimized geometries for the X *Ai ground state of S02 ...... 136
4.3 Comparison of S 02 X XAX SCF orbital energies calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X XAX ex perimental geometry ...... 137
4.4 Comparison of S 02 XxAx SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the experimental X xAi ge ometry ...... 138
4.5 Comparison of S02 X XAX SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X XAX experimental geometry ...... 140
4.6 The experimental properites for the X XAX state of S02...... 142
4.7 Comparison of S 0 2 X XAX SCF orbital energies calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X XAX [7,5,2/5,3,1] SCF optimized geometry ...... 144
4.8 Comparison of S 02 XxAx SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X xAx [7,5,2/5,3,1] SCF optimized geometry ...... 145
4.9 Comparison of S02 X XAX SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X XAX [7,5,2/5,3,1] SCF optimized geometry ...... 147 4.10 Comparison of SO 2 Cl B2 SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the C1B2 [7,5,2/5,3,1] SCF optimized geometry ...... 150
4.11 Comparison of S02 C 1B2 SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the C *B2 [7,5,2/5,3,1] SCF optimized geometry...... 152
4.12 Comparison of S02 XMi SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective X *Ai SCF optimized geometries ...... 155
4.13 Comparison of S02 X *Ai SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective X *Ai SCF optimized geometries ...... 157
4.14 Comparison of S02 X *Ai SCF orbital energies calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the respective X aAj SCF optimized geometries ...... 158
4.15 Comparison of S02 X*Ai DZP and EBP SCF changes in the orbital energies when changing from the experimental to the respective op timized geometries ...... 160
4.16 Comparison of S02 X *Ai DZP and EBP SCF properties changes when changing from the experimental to the respective optimized geometries ...... 161
4.17 Comparison of S02 X *Ai DZP and EBP SCF population analy ses changes when changing from the experimental to the respective optimized geometries ...... 163
4.18 Comparison of S 02 C1B2 SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective C 1B2 SCF optimized geometries ...... 164
4.19 Comparison of S02 C1B2 SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective C 1B2 SCF optimized geometries ...... 166
5.1 SCF description (in C2v symmetry) of the orbitals based on the Mulliken population analysis of the X *Ai state at its experimental geometry ...... 176
xiv 5.2 The number of CSF’s for all states for the various calculation meth ods used in the present study...... 178
5.3 The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X*Ai experimental geometry from MCSCF1 calculations. 179
5.4 The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X *Ai MCSCFl optimized geometry from MCSCF1 calculations ...... 180
5.5 The coefficient of the SCF CSF, number of CSF’s, diagonal ele ments of the Fock matrix and natural orbital occupation numbers for all states at their respective MCSCFl optimized geometries from MCSCFl calculations ...... 181
5.6 The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X*Ai experimental geometry from MCSCF2 calculations. 184
5.7 The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X XAX MCSCF2 optimized geometry from MCSCF2 calculations ...... 185
5.8 The coefficient of the SCF CSF, number of CSF’s, diagonal ele ments of the Fock matrix and natural orbital occupation numbers for all states at their respective MCSCF2 optimized geometries from MCSCF2 calculations ...... 186
5.9 The absolute value of the coefficients for the SCF CSF’s for all states for various calculation methods at the X JAj experimental geometry. 190
5.10 The number of CSF’s with coefficients > 0.03 in which excitations from the SCF CSF occur into or out of these orbitals as determined from SRFCC1 calculations at the X*Ai experimental geometry. . . . 193
5.11 MCSCF3 CSF’s with coefficients > 0.1 for all states determined at the X JAi experimental geometry ...... 200
5.12 MCSCF3 CSF’s with coefficients > 0.1 for all states determined at the respective MCSCF2 optimized geometries of each state ...... 201
xv 5.13 MRCI CSF’s with coefficients > 0.1 for all states determined at the X experimental geometry ...... 205
5.14 MRCI CSF’s with coefficients > 0.1 for all states determined at the respective MCSCF2 optimized geometries of each state ...... 206
5.15 MCSCF3 CSF’s with coefficients > 0.1 for the X aAj state deter mined at the respective MCSCF2 optimized geometries of each state. 207
5.16 MRCI CSF’s with coefficients > 0.1 for the X*Aj state determined at the respective MCSCF2 optimized geometries of each state. . . . 208
6.1 Comparison of MINPT fits to SCF data for all states ...... 217
6.2 Comparison of the experimental and ab initio geometries and changes in the geometries on going from the *Ai ground state to the various excited states for S02 ...... 221
6.3 Total energies (in hartrees) of the XW i, a 3Bj, b 3A2, A 1A2, and C 1B2 states of S02 at their respective experimental geometries. . . 228
6.4 Total energies (in hartrees) of all states of S 0 2 at their respective MCSCF2 optimized geometries ...... 229
6.5 Correlation energies, ECOTT (in hartrees), of the X 1A1, a 3Ba, b 3A2, A jA2, and C 1B2 states of S02 at their respective experimental geometries ...... 231
6.6 Correlation energies, EC0Tr (in hartrees), of all states of S02 at their respective MCSCF2 optimized geometries ...... 232
6.7 Adiabatic transition energies, Te (in eV), of the a 3Bi, b 3A2, A*A2, and C *B2 states of S 02 at their respective experimental geometries. 235
6.8 Adiabatic transition energies, Te (in eV), of the excited states of S02 at their respective MCSCF2 optimized geometries ...... 236
6.9 Relative ordering of the X a 3Bi, b 3A2, A XA2, and C 1B2 states of S02 at their respective experimental geometries ...... 237
6.10 Relative ordering of the excited states of S02 at their respective MCSCF2 optimized geometries ...... 238 6.11 Total energies (in hartrees) of all states of SO 2 at the X xAi experi mental geometry ...... 243
6.12 Total energies (in hartrees) of all states of S 0 2 at the X SCF2 optimized geometry...... 244
6.13 Total energies (in hartrees) of all states of SO 2 at the XxAi MCSCF2 optimized geometry...... 245
6.14 Correlation energies, Ecorr (in hartrees), of all states of S 0 2 at the X xAi experimental geometry ...... 246
6.15 Correlation energies, Ecorr (in hartrees), of all states of SO 2 at the X xAj SCF2 optimized geometry...... 247
6.16 Correlation energies, Ecorr (in hartrees), of all states of SO 2 at the X xAa MCSCF2 optimized geometry ...... 247
6.17 Vertical excitation energies, A E vert (in eV), of S 0 2 calculated at the X xAi experimental geometry ...... 249
6.18 Vertical excitation energies, A Evert (in eV), of SO 2 calculated at the X xAj SCF optimized geometry ...... 250
6.19 Vertical excitation energies, A Evert (in eV), of S 0 2 calculated at the X xAj MCSCF2 optimized geometry ...... 251
6.20 Relative ordering of the excited states of SO 2 at the X xAj experi mental geometry ...... 256
6.21 Relative ordering of the excited states of S02 at XxAi SCF optimized geometry ...... 257
6.22 Relative ordering of the excited states of S02 at X xAj MCSCF2 optimized geometry ...... 258
6.23 Total energies (in hartrees) of the X*Ai state of S02 at the respective experimental geometries of the excited states ...... 261
6.24 Total energies (in hartrees) of the XxAi state of S02 at the respective MCSCF2 optimized geometries of the excited states ...... 262
xvii 6.25 Correlation energies, E corr (in hartrees), of the X 1A1 state of S02 at the respective experimental geometries of the excited states. . . . 263
6.26 Correlation energies, Ecorr (in hartrees), of the X *AX state of S02 at the the respective MCSCF2 optimized geometries of the excited states...... 264
6.27 Emission energies, A Eem{3 (in eV), of S02 calculated at the respec tive experimental geometries of the excited states ...... 266
6.28 Emission energies, Ai?em,a (in eV), of S02 calculated at the respec tive MCSCF2 optimized geometries of the excited states ...... 267
6.29 Relative ordering of A Eemi3 of S 02 calculated at the respective ex perimental geometries of the excited states ...... 268
6.30 Relative ordering of A E emi3 of S02 calculated at the respective MC- SCF2 optimized geometries of the excited states ...... 269
6.31 Harmonic frequencies (cm-1) for the symmetric stretching (u;x) and bending (u;2) modes calculated using the MCSCF2 potential surfaces.272
7.1 SCF minimum crossing points of the 3A2-3B! and 1A2- 1B1 pairs of states of S02 used to test the MINCROV program ...... 284
7.2 The minimum crossing points of the 3A2-3BX and ’A ^ B i pairs of states of S02 at the SCF and MCSCF levels ...... 286
xviii C H A P T E R I
Introduction
There are several reasons for performing theoretical studies on chemical systems
(atoms and molecules), not the least of which are:
1. To test new theories and methods
2. Help in the confirmation and interpretation of experimental results
3. To guide the design of experiments of previously unexplored chemical sys
tems.
The last two are as important as the first and are after all the ultimate goal of theoretical chemistry. A good example of the second reason would be how theoretical studies helped in the determination of the bond angle in the ground state (3Bi) and the singlet-triplet energy gap of methylene (CH 2 ) (for a good review of the methylene problem see the review article by I. Shavitt1). Another molecule which has presented some difficulty in the interpretation of experimental results
(especially in its spectra) is sulfur dioxide (SO 2 ). 2
The study of sulfur dioxide is of great interest since it plays an important role as a pollutant in the Earth’s atmosphere.2-6 It has also been found in the atmospheres of Venus7-9 and Io,10-18 one of Jupiters satellites. Thus, it is very desirable to understand the mechanisms of its reactions in these atmospheres, especially our own. It is believed that many of the Earth’s atmospheric reactions take place with SO 2 in an excited state.19-21 Therefore, a thorough understanding of the photochemistry of SO 2 is necessary.
Sulfur dioxide’s ultraviolet absorption spectrum consists of three main regions:
3900-3400 A, 3400-2500 A, and 2400-1700 A. The first region is attributed to absorption from the singlet ground state ^A,) to a triplet state (3Bj) and is thought to be perturbed by other low-lying triplet states. The second region is considered to be due to excitation to more than one excited singlet state (in particular, to a XA2 with perturbations from a nearby or to a singlet state which is a combination of these two states). The third region is also a singlet to singlet transition. All of these states are believed to be those generated by excitations from the highest three occupied orbitals in the ground state to the first unoccupied orbital. For an
SO2 molecule lying in the yz plane as in Figure 1.1, the ground state electronic configuration (for C 2 U symmetry) is
(la,)2 (lb2)2 (2a,)2 (3a,)2 (2b2)2 (lb,)2 (4a,)2 (5a,)2 (3b2)2 (6a,)2
(4b2)2 (7a,)2 (2b,)2 (5b2)2 (la2)2 (8a,)2 (3b,)° (9a,)° (6b2)°. 3
z
Figure 1.1. The sulfur dioxide molecule is situated in the yz-plane with the sulfur nucleus at the origin and the oxygen nuclei in the fourth and third quadrants. Ri and R2 are the S-Oi and S-02 bond lengths and are assumed to be equal in this study (C2„ symmetry). 0 is the 0 r S-02 bond angle. The principal l70 field gradient axes are indicated by b and c, and the angle cr(170) measures the deviation from the S-’'0 bond direction. (There is some question as to the ordering of the 4b2 and 7a! orbitals.) Thus, the excited states in question would be 1,3Bi ( 8ai —► 3bx ), 1,3B2 ( la2 —► 3b} ), and
1-3A2 ( 5b2 -+ 3b, ).
A large volume of work, both experimental and theoretical, has been done on these states and the spectra produced by transitions from the ground state to the excited states. Even so, a number of questions still remain and a theoretical study of S02 was undertaken in this laboratory.
There are many different theoretical methods which can be used in performing calculations and these can be grouped into two categories, semi-empirical and ab initio. The semi-empirical methods, such as CNDO, INDO, and MINDO, make approximations to certain types of integrals of the electronic Hamiltonian. In these cases all or some of the integrals are neglected and the core-Hamiltonian integrals are approximated by using experimental data or by taking some of them as semi- empirical parameters adjusted to agree with either experimental data or minimal basis set ab initio SCF calculations or both. These methods can give good results
(at least for geometries and some properties) but the results will depend on the proper choice of the parameters.
The main advantages of these methods are the ease and speed at which the calculations can be performed. The main disadvantage is that the approximations in the models are not generally valid. Another disadvantage is that experimental data for the chemical system being studied or from similar chemical systems is needed to fit the parameters. This means that if the experimental data is not of good quality then the calculations won’t be. Also, the same problem can occur when one tries to fit parameters for a particular system to those found for other similar systems. Just because it appears that two chemical systems are similar and should behave the same doesn’t mean that they will.
These types of problems do not occur in ab initio methods since they are based on first principles and integrals are not neglected. These methods include self-consistent field (SCF), multi-configurational SCF (MCSCF), configuration in teraction (Cl), and many-body perturbation theory (MBPT - MP2, MP3, MP4).
Of course, these methods are not without there own problems. In general, ab initio calculations take longer, but this problem is being conquered with the ad vent of larger and more powerful computers (especially supercomputers). Another problem is in the choice of basis sets. Minimal basis set SCF calculations can give fairly good optimized geometries for molecules (the bond distances tend to be too short) and even fair electronic excitation energies in many cases. However, many times properties calculated with these basis sets are of poor quality. For small to medium sized molecules, extended basis sets of at least double-zeta quality can be used, so in these cases basis set size does not have to be a limiting factor, at least for SCF calculations.
One can improve upon the SCF calculations by using methods which include electron correlation. One such method is MCSCF, which includes more than one configuration of the same symmetry in a calculation. These configurations are generated by allowing excitations of electrons from occupied orbitals into a number
(usually limited to a few) of virtual (unoccupied) orbitals. The orbitals in which the electron population can be less than two per orbital are called the active space.
In this procedure both the coefficients of the orbitals and those of configurations are optimized. This method can recover a portion of the correlation energy but tends to give bond distances that are too long.
Another electron correlation method is Cl. In this method one starts with a reference configuration (or several reference configurations), the reference space, and then “excites” electrons from the occupied orbitals of the reference configu ration^) to the virtual orbitals. This usually generates a large total number of configurations. Like SCF and MCSCF this is a variational procedure. However, in this case only the coefficients of the configurations are optimized and not those of the orbitals.
There are various levels of Cl, depending on the number of reference configu rations and the types of excitations allowed from them. The most widely used is single reference SD-CI, in which there is a single reference configuration and only single and double excitations from the occupied to the virtual orbitals are allowed.
This will recover a large portion of the correlation energy (usually much more than
MCSCF). The geometries and excitation energies are usually quite good and are usually in close agreement with experiment. Cl is of course not without its own problems. If it is not a full Cl then it is not size consistent. Therefore, when doing dynamic problems (such as computing dissociation energies or potential energy curves out to dissociation) one does not always obtain the proper results. Like SCF and MCSCF, the results can depend on the choice of basis set. Also, the choice of the reference space can effect the results.
Even considering the problems associated with ab initio methods they are still the method of choice for theoretical calculations. In most cases they produce reasonably accurate results for energies, geometries, and properties. Also, one doesn’t need any experimental results in order to carry out the calculations.
With all of this in mind it was decided that ab initio calculations on S02 would be performed. The original work was performed by D. D. Lindley.22 In that investigation basis set studies at the SCF level were carried out for the ground state. The optimized geometries and absolute energies for the ground state and the six excited states, given above, were obtained. Vertical and adiabatic excitation energies and properties were also obtained. I continued and extended this work for my master’s thesis.23 I also used the SCF method, but concentrated on the excited states. In particular the crossing of the potential energy curves of the 3A2
- 3Bj and *A2 - 1Bi systems were studied in hopes of shedding some light on what is occurring in the first and second regions of the UV spectrum. However, some problems inherent to SCF (particularly symmetry breaking) were encountered. From those results it was concluded that methods that include more than one configuration and electron correlation (MCSCF or Cl) should be used.
In this paper the results of ab initio calculations using SCF, MCSCF, single reference Cl (SRCI), and multireference Cl (MRCI) methods are reported. Opti mized geometries, absolute energies, vertical and adiabatic excitation and emission energies and properties were obtained. The crossing regions of the 3A2 - 3Bi and
*A 2 - 1 B1 systems were also reinvestigated using the MCSCF method.
Chapter II is a review of the methods to be used in this dissertation. These will only be briefly discussed since there are m any books and reviews covering these subjects.
Chapter III is a review of the literature dealing with previous experimental and theoretical work done on SC> 2 -
Chapter IV discusses the basis set used in the calculations performed in the present study. The results are compared to the results obtained using other basis sets in previous studies in this lab.
Chapter V discusses the various MCSCF active spaces and MRCI configuration spaces used. The MCSCF a n d MRCI configuration spaces for all the states are analyzed at the various geometries studied.
Chapter VI presents the results of the geometry optimizations, the equilibrium energies and the excitation and emission energies. The results of the determination of the MCSCF minimum crossing points of the the 3A 2 - 3Bi and *A 2 - 1 B1 systems are presented in Chapter VII.
The final chapter, Chapter VIII, contains the summary and the conclusions to be drawn from this investigation.
The appendices contain definitions of abbreviations and properties used in this dissertation. C H A P T E R II
Overview of Electronic Structure Theory
The basic underlying premise of quantum chemistry is the Schrodinger equation.
Its non-relativistic, time-independent form is,
H $ = EV, (2 .1 ) where H is the Hamiltonian operator, which describes all the interactions of the system, is the wavefunction, and E is the total energy. For atoms and molecules the Hamiltonian, including nuclear motion, can be written in atomic units as
ft = - E 2^-vi-iEv?-Ep + E -+ E § ^ , (2.2) A 2 M a 2 . i A r,A r j j Ai, l/r,j and 1 1R a b terms are the Coulomb energies between the electrons and nuclei,
10 electrons and electrons, and nuclei and nuclei. The last four terms represent the electronic energy which, under the Born-Oppenheimer approximation (see below), is a parametric function of the nuclear coordinates, E (R a b )• This potential energy surface can be used to determine the equilibrium geometry, vibrational frequencies, and as the potential energy functions for the nuclear Hamiltonian describing the motion of the nuclei.
The Schrodinger equation (Eq. 2.1) can be solved exactly only for a few simple cases, such as any one-electron atom or diatomic. For molecular systems the basic approximation which is employed to obtain solutions to this equation is the Born-
Oppenheimer approximation . 24 This approximation is based on the fact that, since the nuclei are much heavier than electrons, the nuclear displacements are very slow compared to the motion of the electrons. This allows the wavefunction to be written as a product of nuclear and electronic wavefunctions which allows
Equation 2.2 to be separated into nuclear and electronic parts.
Various approximation methods (using various wavefunction representations) have been developed to solve the many-electron problem. These are briefly de scribed in this chapter. 2.1 The Hartree-Fock SCF Approximation
The Hartree-Fock (HF) self-consistent field (SCF) approximation 2 5 ,2 6 expresses the wavefunction as a single Slater determinant , 27
| $ s c f ) = IX1X2X3 • • • Xiv), (2.3)
which is an antisymmetrized product of spin-adapted one-electron orbitals (spin- orbitals). The spin-orbitals, Xi can be written as,
Xi = 0i(r)w, (2-4) where V’t'(r) is a spatial function and u represents the a or /3 spin of the electron. In restricted HF (RHF) the spatial orbitals are identical for and spin functions.
The HF SCF method also replaces the many-electron problem by coupled one- electron problems in which the electron-electron repulsion is treated in an average way. Each electron is subjected to an average potential due to the remaining N — I electrons. This one-electron approximate potential is iterated for each electron until the orbitals and the potential which they generate are consistent.
The introduction of the linear variational approach by C. C. J. Rootliaan 28 using a basis set formulation greatly reduced the amount of work required to solve the HF equations. The equations could now be solved analytically rather than numerically. This was extended to molecular systems through the use of the linear combination of atomic orbitals (LCAO) method. This allowed the molecular 13 orbitals to be expressed as linear combinations of nuclear centered atomic orbitals.
The best wavefunction is the one which gives the lowest possible energy, in a variational sense,
E s c f = ($scf|H |$scf) > E0, (2.5) where E0 is the true total energy of the system. More detailed reviews of the
Hartree-Fock SCF method can be found elsewhere . 29,30
2.2 Basis Sets
The LCAO method is general in the sense that any type of basis function can be used. However, the basis functions are normally chosen to be similar to hydrogen like atomic orbitals since one is, after all, trying to describe atomic and molecular systems. A desirable form for these basis functions is given by Slater-type orbitals
(STO), which have the principal radial factor e-(*r, since they have the proper asymptotic behavior for large R and the cusp at the origin. However, integrals over
Slater functions on different centers are difficult to solve. Instead, the primitive basis set consists of nuclear-centered Cartesian Gaussian-type orbitals (GTO),
(Ji = N xlymzne-<'T\ (2.6) where N is the normalization constant and depends on /, m, and n. The angular dependence of the atomic orbital (AO) functions is introduced by the product of the
Cartesian coordinates to various powers, xlymzn. For instance an s-type function 14
would occur when I = m = n = 0 and a px-type function would have I = 1 and
m = n = 0. Even though the GTO’s do not have the correct long range description or the proper cusp behavior at the origin, their computational advantages outway
this concern.
The problems associated with the deficiencies in using GTO’s is addressed by taking linear combinations of the primitive functions. Selected primitive basis functions are then grouped together (contracted) and treated as one function.
These contractions are given by:
X i = C iid ii (2-7) j where gj is the original primitive basis set, is the contracted basis set, and the
Cji are the contraction coefficients which relate the two sets. The exponents and contraction coefficients are variationally optimized to produce the lowest possible
HF energy.
At the present time there are two basic contraction schemes. The segmented contraction scheme 31 uses a different set of exponents for each linear combination.
Often in this scheme, at least one GTO with the same exponent but a different coefficient is used in two separate contractions of different orbitals. Most often the
GTO with one of the smaller exponents in one contracted functions is used as the largest exponent in another contracted function of the same orbital type. Thus in this scheme integrals over primitive functions are often repeated. The generalized contraction scheme 32 employs a common set of exponents which are used to form different linear combinations. This scheme is readily becoming the one of choice since it offers the greatest flexibility in the contractions and primitive function integrals are not repeated.
In general, for small to medium sized atomic and molecular systems, basis sets of at least double-zeta quality (each orbital is described by two contracted functions) in the valence shell are used. In addition, for molecules, polarization functions (angular momenta higher than the valence orbitals) are needed to obtain proper description of the bonding regions. In SO2 for example, d-type polariza tion functions are necessary to obtain a proper description of the bonding, prop erties, and excitation energies. Also, additional polarization functions and diffuse functions (functions with smaller exponents than in the valence orbitals, used to describe regions of space farther from the nucleus than those covered by the basic functions) are often essential in obtaining good property values.
2.3 Electron Correlation
In the IIF SCF procedure the electronic interactions are dealt with in an ap proximate manner and the instantaneous correlated motion of the electrons is not accounted for. Even the IIF limit, in which the basis set approaches complete ness and is the best possible SCF description, does not account for the electron correlation. The energy obtained beyond the HF description is referred to as the 16 correlation energy and is given by,
Ecott = E ~ Es c f - (*2.8)
There are two types of correlation energy: ( 1 ) nondynamical correlation due to near degeneracies or other deficiencies in the HF wavefunction and ( 2 ) dynamical correlation due to the instantaneous correlated motion of the electrons.
2.3.1 Non-Dynamical Correlation Energy
There are many times when a single-configuration RHF wavefunction does not ade quately describe the physical nature of the system. This includes breaking of bonds
(improper dissociation), near-degeneracies of states, and symmetry breaking. For these cases, a multiconfigurational description is necessary. The multiconfigura- tional SCF (MCSCF) method uses a few important configurations to obtain a better description of the wavefunction. In this method, the wavefunction
I'EVcscf) = (2.9) i is variationally optimized over both the coefficients, C{, and the orbital coefficients.
One particular MCSCF procedure, complete active-space MCSCF (or CASSCF), attempts to produce an unbiased wave function by allowing all possible config urations which can be generated by rearranging N electrons in M orbitals (the active space consisting of a few SCF occupied and virtual orbitals). Of course the resulting wavefunction is only truly unbiased if the proper orbitals were chosen for 17 the active space. The MCSCF method is the most widely accepted method in use for accounting for nondynamical correlation effects.
2.3.2 Dynamical Correlation Effects
The method used in the present study to account for the dynamical correlation effects is configuration interaction (Cl). The best possible Cl calculation is full Cl
(FCI) in which the wavefunction is expressed as the sum of the reference function and all possible excited configurations,
l*)=^l'M + E $ijk.'.'.i ^e subscripts indicate internal orbitals from which excitations have occurred and the a,b, c ,... superscripts indicate the virtual orbitals which replace them. (Excitations into partly occupied orbitals should also be included.) The coefficients in Equation 2 . 1 0 are determined by variationally minimizing the expression, F 19 111 (tfjtf) “ EfCI 18 The full Cl answer serves as an upper bound for the energy but it is impossible to obtain at the present time, except for small systems, because of the length of the expansion 2.10. Instead an approximation, truncated Cl, is often used. Typically the Cl wavefunction is truncated after the double excitations to pro duce what is referred to as singles and doubles Cl (SDCI or CISD). The CISD wavefunction can be written as, |*CKD> = Co|*o) + E + (2-12) i,a «' This is a reasonable approximation since H has no more than two-electron oper ators . 3 3 The coefficients, Co, c“ and c^, and the CISD energy, are obtained by minimizing the following expression with respect to the coefficients: 17 _ 17 ,1 7 {^S d \E \9 Sd ) , o\ E c is d - E s c f + Ecorr(c,SD) - - {y sD^ sD) • (2-13) The CISD energy is an upper bound to the FCI and true energies. If the ref erence function, I 'f 'o ) ) is the SCF function, I ' I ' s c f ) , then this is referred to as single-reference Cl (SRCI). If an MCSCF wavefunction, I^a/oscf), is used as the reference function then the calculation is referred to as multi reference Cl (MRCI). Even though, in most cases, truncated Cl gives good reasonable answers, it is not without its problems. It is not size extensive, i.e., for large N the calculated total energy of a system of N non-interacting subsystems does not scale linearly with N as it should, E(N ■ A) f N ■ E{A). (2.14) Also, CISD is not size-consistent , 34 which requires the total energy for a non interacting system of particles to be separably additive, E{A + B) = B(A) + E(B). (2.15) Except for a system of two particles, for which CISD is size-consistent, the energy scales as N 2 for large system size N. There are several approximations that have been applied to CISD results to compensate for these problems. The Davidson corrections 3 5 ,3 6 add the effects of the major quadruple excitations contributions. The correlation energy with the unnormalized Davidson correction is, E dav \ = E cisd + ( 1 - cD&E cisd (2.16) and that with the normalized corrections is, E dav2 = E cisd + AECisd, (2-17) where Cq is the coefficient of th e reference configuration (for SRCI) in the final wavefunction and A E C IS d is the CISD correlation energy (A E c i s d = ECI -f E0). For MRCI Cq is the sum of the cf of the reference configurations. In order to make the correction vanish for two-electron systems a scaling factor of (N — 2)/A, where N is the number of correlated electrons, can be included to give scaled Davidson corrections, 20 and f N — 2 \ (\ - c 3\ A E dav 2-s = E cisd + y—^ —J ( ^ 2 ) &E cisd , (2.19) There is one other widely used correction to CISD, proposed by Pople ,3 7 to correct for the size extensivity problems. This correction is given by, (N 2 -f 2 AT tan 2 20)*^ — N E p o flb = ERe, + ------^ 2(, —i)------A E c r s n , (2.20) where cq = cos 9. Finally, performing an MRCI calculation can also recover some of the dynamical correlation that is absent in a SRCI calculation. An MRCI expansion will often include what amounts to triple and quadruple excitations from what would be the SCF wavefunction. 2.4 Applications The above methods, among others, are implemented in several computer programs that were used to perform the calculations for this study. The integral program ARGOS 3 8 was used to calculate the integrals. The SCF calculations were per formed using the SCFPQ program . 3 9 Finally, the MCSCF and Cl calculations were performed using the CO LU M B U S 40 system of programs. CHAPTER III Literature Review of Experimental and Theoretical Work on Sulfur Dioxide 3.1 Introduction As mentioned in Chapter I, much work has been done on SO 2 . A complete review of the literature is beyond the scope of this dissertation. The review presented in this chapter is primarily limited to those things which are directly related to the present work. There are two main topics concerning sulfur dioxide that are discussed. These involve the ground state (xAj) and the first six excited states (1 ,3A2, 1 ,3 B2), which are generated by exciting an electron from one of the highest three occupied orbitals into the lowest unoccupied (virtual) orbital (as shown in Chapter I). The first topic deals with the equilibrium geometries and absolute energies (in the case of theoretical studies) of the various states. The second topic covers the absorption and emission spectra, as they apply to excitation (transition) energies and crossings of the 1 ,3A2 - 1,3Bi potential surfaces. 21 22 There are two types of excitation energies which can be considered, vertical and adiabatic. These are shown for a diatomic molecule in Figure 3.1. A vertical excitation (absorption) energy is the energy difference between two electronic states (usually the ground state and an excited state) calculated at the equilibrium internuclear geometry of the lower state. Experimentally, the reported vertical transition is from the zeroth vibrational level of the lower state to an ex cited vibrational level of the upper state, and is most commonly reported if one cannot observe the (OOO)-(OOO) transition, discussed below. One can also consider a vertical emission energy which is analogous to the vertical excitation energy, except that the energy difference is taken between energies found at the equilib rium internuclear geometry of the upper state. This, of course, would produce a vibrationally excited lower state. According to the Franck-Condon principle, these transitions would appear as the most intense bands in the UV absorption and emission spectra, respectively. Both of these vertical transition energies are quite easy to obtain theoretically (vertical excitation energies being the most commonly calculated). The minimum or adiabatic excitation energy, Te, is the difference in energy between two states which are each at their equilibrium internuclear geometry (dif ferences in the energies at the bottoms of the potential wells). Experimentally, this can be obtained from the (OOO)-(OOO) transition of the spectrum, if one knows the zero point vibrational energies (height of the zeroth vibrational level ( 0 0 0 ) above molecule Figure 3.1. Schematic representation of t|ic excitation (!iiergics f°r a diatomic diatomic a f°r (!iiergics excitation t|ic of representation Schematic 3.1. Figure E(R) oo oo R AZPE 23 24 the bottom of the potential well) for both of the states. This relationship is given by Te = T0 — A ZPE (3.1) where To is the (OOO)-(OOO) transition energy A ZP E = G'0go — Gqoo an Gooo and Gooo are the zero point vibrational energies of the lower and upper states, respectively and can be seen in Figure 3.1. The Gq 0 0 or G ^ are usually determined from the infrared spectrum or a vibrational analysis of the absorption or emission spectra. They are determined by using the general equation for the vibrational energies (assuming nondegenerate energy levels and ignoring higher-order terms) in cm-1, Gv!V2V3 = 52(vi + -)u>i -f- ^2 + o)(vj + y)xij (3-2) *=1 1 .=1 j> i 1 z where the a;,- are the zero-order harmonic frequencies and the are the anhar- monic terms. One can rewrite this in terms of the difference between any vibra tional level and the zero point vibrational level, 3 3 3 y A*/ = ( GVIV2V3 - Gooo) = £ ViOi + X) Y } vivj + «(»>.• + Wj)]*y (3-3) ,=i «=i j>i L Experimentally, this is the equation that is used, since it is the differences in the vibrational levels that are determined from the spectra. The to, and are then 25 determined by fitting the observed At/ to this equation. Once this is done, the zero point vibrational energy can be calculated by using the equation for GV1V2V3 with vi = V2 = v3 = 0. Of course, one may not always have good enough data to obtain all of these terms (especially the i,j). In this case, if one does have at least the fundamental frequencies, then Gooo can be approximated by using the harmonic terms with the u\ set equal to the fundamental frequencies. In the following discussions, if the zero point energies are not given explicitly by any investigators, then the above methods will be used to at least obtain a rough estimate of their value. This is easier than obtaining To from ab initio methods. To determine T 0 from ab initio calculations Te values have to first be calculated and then the harmonic vibrational frequencies have to be calculated from the theoretically determined potential energy surfaces. It would require a large number of calculations to determine Te (as described below) and a good potential energy surface. Furthermore, to obtain a fairly accurate value for T 0 the anharmonic constants, would also need to be determined. This would require an even greater number of calculations since a larger region of the potential surface would need to be determined. It requires more work to theoretically obtain Te than to obtain the vertical excitation energy. This is because to obtain Te the absolute energies for each state must be calculated at their respective equilibrium geometries. This can be done by performing the calculations at the respective experimentally determined geome 26 tries. Ideally though, it is most desirable to calculate the equilibrium geometries for the two states and then use the energies obtained at these “optimized” geome tries to calculate Te. This can require a very large number of calculations and a great deal of time. Consequently, Te is calculated less frequently. After a brief introduction to the spectroscopy of SO 2 , each of the two topics above is discussed as part of separate sections, which deal with the different regions of the SO 2 spectrum. Each of these sections is further divided into sections covering experimental and theoretical investigations. 3.2 Introduction to SO 2 Spectroscopy Sulfur dioxide has three main absorption regions in the near ultraviolet: 3900- 3400 A, 3400-2500 A, and 2400-1750 A. The first region (3900-3400 A) is a t tributed to an a 3Bi <— X xAj electronic transition,, with perturbations caused by other states, possibly other low-lying triplets ( 3A2 or 3 B2). The second region (3400-2500 A) is made up of bands of medium intensity with a continuous back ground of weak bands. The strong bands of this region are now generally assigned to A *A 2 <— X ^ 1 , while the weak bands are assigned to B 1 B1 <— X aAi. The third region (2400-1750 A), the most intense group of bands, was thought to consist of more than one electronic transition, one of which was C 1 B2 <— XhV], More recent studies have been able to explain all the lines in the spectrum of this region 27 by postulating the transition to be C 4A' <- X *Ai, in which the upper state has unequal bond lengths. 3.3 The Geometry and Absolute Energy of the *Ai Ground State of SO 2 The ground state of SO 2 has been of great interest to both experimentalists and theoreticians and the investigations pertaining to it are, by far, the most prodi gious .22,41"94 It is a small triatomic molecule and being the ground state its vi brational and rotational spectra show few perturbations, at least for the first few vibrational levels. This makes it relatively easy for experimentalists to obtain the molecular constants (such as the moments of inertia) and therefore the geome try. Also, having a relatively small number of electrons (32) and multiple bonding (involving d orbitals) has made it a good candidate for theoretical studies. This section is divided into subsections which report on the experimentally and theoretically obtained geometries for the ground state of SC> 2 - Also reported are the absolute energies obtained in the theoretical studies, both at the experimental and optimized geometries. 3.3.1 Experimental Investigations Many different types of experiments have been performed on the 4Ai ground state of SO 2 , in both its gaseous and solid phases. The aim of much of this work has been to obtain the molecular constants (vibrational frequencies and rotational constants), geometry, and properties. T he experimentally determined geometries from several of these studies are given in Table 3.1.41-53 As can be seen in the table, all of the experiments give bond distances around 1.43 A and all but two give a bond angle of between H9 .3 0 and 119.5°. Most of these investigations have involved analyses of microwave or infrared data. One also notices that all but one of the microwave or infrared analyses since about 1960 report a bond distance of 1.431 A and a bond angle of 119.33°. T h e only two experiments that did not give a bond angle between 119.3° and 119.5° were b°th electron diffraction experiments on gaseous S0 2 . 5 1 ,5 2 It should be pointed out however, that the values that were obtained had rather large un certainties and the upper limits place both of them in the proper range. Also, a m ore recent electron diffraction experiment by Holder and Fink 5 3 gives a bond distance and angle which is in bettor agreement with those obtained from the other methods. It appears fr0m all of this data th at the geometry of the ground state is well established. The m ost likely values for the S - 0 bond distances and O S O bond angle are 1.431 A and 119.33 , respectively. \Vhile not to be discussed jn detail here, the zero-point vibrational energy (G«oo) of the X b\j ground state is im portant, as it will be used in later sections about the excited states. It will be used, along with th e zero-point energy of the excited state (G ^), to obtain the minimum (adiabatic) transition energy, Te, from 29 Table 3.1. Experimentally determined geometries for the JAj ground state." Geometry Method Ref. Rso (A) Qoso (deg) 1.433 119.5 Microwave analysis 416 1.4321 119.035 Microwave analysis 426 1.432 119.53 Microwave analysis 436 1.4321 119.536 Microwave analysis 44 6 1.4308 ± 0.0002 119.32 ± 0.03 Microwave analysis 45c 1.43076 ± 0.00013 119.33 ± 0.012 Microwave analysis 46c 1.43498 ± 0.00015 119.346 ± 0.024 Microwave analysis 47 6 1.43074 ± 0.00007 119.333 ± 0.003 Microwave analysis 48° 1.43076 119.330 Infrared analysis 49c 1.430 ± 0.015 119.5 ± 1.5 X-ray diffraction 50 1.4309 db 0.001 118.98 ± 0.5 Electron diffraction 51c 1.427 ± 0.002 118.5 ± 1.0 Electron diffraction 52c 1.4313 ± 0.0006 119.5 ± 0.3 Electron diffraction 53c "All studies, with the exception of the X-ray diffraction, are of gaseous SO 2 . JThe R so reported is Ro- cThe R so reported is Re. 30 the the zero-zero transition energy, To- The zero-point vibrational energy of the X *At state has been obtained by many investigators .5 4 -6 0 The value of Shelton et al. , 5 4 ,5 5 approximately 1530 cm - 1 (0.190 eV), is the accepted experimental value. They, along with several of the other investigators, also obtained values for the harmonic and anharmonic constants (of Equations 3.2 and 3.3) and higher order potential (force) constants. 3.3.2 Theoretically Determined Geometries and Energies Theoretical studies of the ground state of SO 2 have been done for several reasons. The wealth of good experimental data gives theoreticians something with which to compare the calculated results. Also, it has been used extensively to test the need for polarization functions (in particular d-type functions) in molecular basis sets. All types of theoretical calculations, from semi-empirical to correlated ab initio, have been performed to obtain absolute energies, optimized geometries, and properties . 2 2 '6 1 -9 4 Tables 3.2 through 3.5 show the results of several levels of calculations. In order to facilitate the discussion of the tables and theoretical work, definitions for the abbreviations used for the basis sets and methods are given in Appendix A. Table 3.2 shows the results of self-consistent-field (SCF) calculations at several experimental geometries. Since these geometries are essentially the same, this table can be used to see the effects of various basis sets on the total energy calculated 31 Table 3.2. Results of various ab initio SCF° studies for the ground state ^Ai) of SO2 at experimental geometries. G eom etry (A, deg) Energy (a-u.) Basis Set6 Contraction Scheme Ref. R s o e o s o 1.43° 120 -540.6481 STO-3G (9,6/6,3) —»[3,2/2,1] 61,62 1.4321c 119.5 -540.60254 STO-3G (9,6/6,3)—* [3,2/2,1] 63 1.4321c 119.5 -543.79626 3G-STO-MBS" (9,6/6,3) — (3,2/2,1] 63 1.43° 120 -541.0076 STO-3G* (9,6,3/6,3) — [3,2,1 /2,1] 61,62 1.43c 120 -541.0495 ST 0-3G . (‘sp lit d ’)d (9,6,3/6,3) —► [3,2,1/2,1] 62 1.4321c 119.5 -544.84792 ST O -nG *' (13,8,3/12,4) — [3,2,1/2,1] 64 1.4308’ 119.3 -545.2777 DZV-STO-nG* (14,9/11,6) —► [4,3/3,2] 65 1.4308* 119.3 -545.5325 DZVBF-STO-nG’ 65 1.4308' 119.3 -545.5794 DZVBF-STO-nG>’« 66 1.4321c 120.00 -541.7576 DZV-ST0-3G* (12,9,6/9,6) — [4,3,2/3,2] 67 1.4308’ 119.3 -547.23386 e m b s p (11,7,2/95,1) — [4,3,l/3,2,l]r 68 1.4308’ 119.34 -543.1985 3G -ST0-D Zm (18,12/12,6) —> [6,4/4,2] 69 1.432° 119.54 -546.629 DZ (10,6/7,3)-[6,4/4,2] 70 1.4321c 119.536 -546.9512 DZ (12,9/10,5) — (6,4/4,2] 71 1.4316 119.0 -543.340 DZP (6,4,2/4,2) 72 1.4321° 119.536 -546.786 DZP (12,9,1/95) -(6,4,1/4,2] 73,74 1.432c 119.54 -546.831 DZP (10,6,1/75,1)-[6,4,1/4,2,1] 70 1.4321° 119.536 -547.2089 DZP° (12,9,1/10,5,1)-(6,4,1/4,2,1] 71 1.4321° 119.536 -547.018 DZP (12,9,2/95,1) - [6,4,2/4,2,l] 74 1.432° 119.54 -547.231466 DZP' (11,7,2/95,1) -[6,4,2/4,2,1] 22k 1.4321° 119.536 -546.837 EBP (13,10,2/10,6,1) — [7,5,2/5,3,1]‘ 73,74 1.4321° 119.536 -547.233 EBP (12,9,1/11,7,1)— [7,5,1/5,3,1] 74 1.432° 119.54 -547.191349 EBP' (11,7,1/95,1) — [7,5,1/5,3,1] 22 1.432° 119.54 -547.236964 EBP' (11,7,2/95,1) — [7,5,2/5,3,1] 22 1.432° 119.5 -547.224711 EBP (12,8,3/10,6,2) — [7,6,3/5,4,2]° 75 1.432° 119.5 -547.248465 EBP (12,9,3/9,5,2) — [6,5,3/5,3,2]p 76 “See methods definitions in Section A.2. ^See basis set definitions in Section A.I. 'Experimental geometry from Kivelson.44 dThree 3d G T F’s split into 1 GTF and a group of 2 GTF's (same exponents for each). 'Six d-type polarization functions (gives 5 3d-typc and a 3-s-type functions). ■^For S: n = 5 G TF’s for Is STO, n = 4 GTF’s for 2s STO, etc. (5,4,4,4,4,3/6,6,4). h Experimental geometry from Haase and Winnewisser.M 'Experimental geometry from Morino et al,45 1 Partial optimization of the exponents of the BF's and their position along the S-O bond. ibThis reference contains energies calculated with several other primitive and contracted basis sets. *Same as the author’s DZP basis with diffuse s, aIK) d-type functions added (and not contracted). m3-GTF expansion to DZ-STO basis set of Clementi et a/.95"97 n3-GTF expansion to SZ-STO basis of Clementi et al.s5~9T (Different exponents for ns and np functions.) °(ll>7,3/9,5,2) — [6,5,3/4,3,2] diffuse s and p functions on S and O. (Basis uses 6 d-type functions.) pBasis has a larger number of primitives and a larger number of contracted functions than previous basis set (before the addition of the diffuse s and p functions and only 5 d-type functions). 7Each Is function was fitted with five Gaussians and all other functions with three- 'Using general contraction method of Raffcnctti.32 for S 0 2. It is fairly obvious from this table that minimal basis sets give energies which are ~ 3.3 hartree or more above the near Hartree-Fock (HF) value. The least effective of these is the ST0-3G basis set. An improvement over the minimal basis sets are the split-valence basis sets (double zeta for the valence orbitals - DZV). The best calculation with this type of basis set is that by Burton et al.65 who used an STO-nG basis with n = 5 for the Is orbitals and n = 3 for all other functions and obtained an energy of —545.2777 hartree. This energy is still higher than the near Hartree-Fock (HF) value. From the table it can be seen that basis sets of double-zeta (DZ) quality are usually required to obtain energies below —546 hartree. These basis sets are usually made up of primitive Gaussian- type orbitals (GTO) which are then contracted in various ways into a smaller set of contracted Gaussian-type orbitals (CGTO) (see various papers related to this topic32,98-102). The quality of these basis sets will depend, at least to some extent, on the number of primitive GTO’s and how they are contracted to CGTO’s. This can be seen by comparing the results for the DZ basis sets of Rothenberg and Schaefer 71 (E = —546.9512 hartree) to that of Roos and Siegbahn 70 (E = —546.629 hartree), the latter using a smaller set of primitives. The next step in the construction of a complete basis set is the addition of one or more sets of polarization functions (d-type GTO’s) on sulfur or on both sulfur and oxygen. Addition of these functions lowers the energy in all cases (even for MBS calculations), the greatest effect occurring for the ST0-3G* basis sets (energy lowered by as much as 0.4 hartree62). The effect of adding d-type GTO’s to DZ basis sets is smaller, which is due to the fact that these are more complete basis sets to begin with. While the effects of adding polarization functions appears to be small, compared to going from a minimal to double-zeta basis, their importance becomes even clearer for geometry optimization and property calculations, as will be discussed below. Their importance is even seen for a single point calculation, as evidenced by the calculations of Lindley . 22 If one compares the result from his DZP calculation with two d-type functions on sulfur (and one on oxygen) to that from his EBP calculation with only one d-type function on sulfur (EBP1) and his EBP calculation with two d-type functions (EBP2), it is apparent th at the extra polarization function helps to lower the energy. The energy from the DZP calculation is 0.040 hartree lower than that from the EBP1 calculation. Part of this energy lowering (0.04 hartree) could be due to the fact th at he used six d-type functions (which give five pure d’s and one 3s), thus adding to the completeness of the s orbital space. However, the EBP2 energy Js only 0.006 hartree lower than the DZP energy but 0.046 hartree lower than the EBP1 energy, thus about 0.04 hartree is due to the addition of the second d-type function on sulfur. The best basis sets used to date are the extended basis with polarization (EBP). The two best calculations using these types of basis sets are those of Lindley 22 (E = —547.236964 hartree) and Bacskay et al.7e (£ — —547.248465 hartree). The latter authors also calculated many properties, with several basis sets, at the 34 SCF and correlated (Cl and CPP) levels. It should be pointed out that Dunning and Raffenetti 68 , | 0 alm ost as well w ith an EMBSP basis set, using the general contraction method of Raffenetti .32 The rest of this section concerns the optimized geometry and corresponding energy of the groun empirical to correlated, using a variety of basis sets. The results from the semi- empirical calculate , , s 7 7 ' 8 1 are given in Table 3.3. Most of these calculations tend to give bond distances which are too long and bond angles which are too small. The best of these is th at of R. 0. Jones, who performed extensive geometry opti mizations for theground state and a few excited states using the density functional method . 8 1 His geometry is cl°se to the experimental geometry (with both his bond length and angle being slightly too large). Table 3.4 sho\Vs the optimized geometries and the corresponding energies for the ground state from various SCF studies .22,69*33-82-92 The first thing to note is that MBS calCu]atjons do a very poor job of describing both the optimized geometry and enCrgy T h ey tend to give bond lengths which are too long and bond angles which are too small (by as much as 0.23 A and 14°). The energies can also be too high by as rnuch as ^ 5 5 hartree. As was the case for the basis set studies discussed above, th e worst basis set is the STO-3G. The results gencraJly improve on going to DZV and DZ basis sets. However, the bond distances are stjll too long and the angle to small. Even the EB calculation Table 3.3. Geometries of the ground state (xAi) of SO 2 determined by non-a 6 initio methods. Geometry Method® Basis Set6 Reference R so (A) Ooso (deg) 1.4321° 1 1 1 CNDO/2 MBS-STO 77 1.4321° 116.9 CNDO/2 MBS-STO* 77 1.517 115.6 CNDO/BW MBS-STO 78d 1.59 119.5 INDO-CP MBS-STO* 79 1.432' 119.536 x„sw 80g 1.455 119.4 DFC-LSD ? 81 “See the definitions for methods in Section A.3. JSee the definitions for basis sets in Section A.I. °Experimental bond distance from Kivelson.44 ^Result from author’s parameter set I. eFor the Cl calculation only the doubly-occupied double-excitations to all virtual orbitals were allowed. •^Experimental geometry from Kivelson.4'1 s C alculated energy of —546.278 hartree. 36 T ab le 3.4. Optimized geometries for the ground state (xAi) of S 0 2 from various ab initio SCF studies. G eom etry (A, degl Energy (a.u.) Basis Set® Contraction Scheme Ref. H s o Voso 1.628 105.5 -540.73480 STO-3G (MBS) (9,6/6,3) —♦ (3,2/2,1] 82 1.504 121.0 -541.06681 STO-3G* ( ‘sp lit d ’)6 (9,6,3/6,3) - [3,2,1/2,1] 82 1.416 117.6 -540.97909 MBSBF-STO-3G 83 1.423 117.6 -546.59497 DZVBF-44-31G ... 83 1.562 109.19 -540.64868 STO-3G (MBS) See Sec. A .l 84 1.531 115.05 -546.37328 44-31G (DZV) 84 1.446 119.88 -540.91012 STO-3G* (MBSP) 84 1.526 114.0 3-21G (DZV) 85 1.419 118.7 -544.50373 3.21G* (D Z V P ) 85 1.414 118.8 -547.16901 6-31G* (DZVP) 86 1.431“ 122.0 3G-STO-MBSd (9,6/6,3) — [3,2/2,1] 69 1.530 119.45 STO-3G (MBS) (9,6/6,3) — [3,2/2,1] 63 1.445 112.5 3G -STO -M B S“ (9,6/6,3) - [3,2/2,1] 63 1.659 108.6 -544.03184 MINI-1 (MBS) See Sec. A.l 87 1.523 115.9 -544.25957 MIDI-1 (DZV) 87 1.527 114.1 -546.36440 MIDI-4 (DZV) 87 1.437 118.1 -544.49831 Mi d i - i * ( d z v p ) 87 1.438* 118.1 -546.57233 Mi d i -4* ( d z v p ) 87 1.543 113.32 ECP-21G (DZV)9 88h 1.426 118.31 ... ECP-21G* (DZVP)* 88* 1.404 118.8 -547.239412 EBP (11,7,2/9,5,1)-*[7,5,2/5,3,1] 22 1.404* 118.8 -547.240598 EBP' (12,8,2/9,5,1) —* [8,6,2/5,3,1] 22 1.552 112.4 -546.99113 DZ (12,9/9,5)-*[6,4/4,2] 89 1.539 112.7 -547.04144 MP-DZ”* 89 1.423 118.4 -547.22464 DZP" (12,9,1/9,5,1) —► [6,4,1/4,2,1] 89 1.421 118.4 -547.26796 M P -D Z P " 89 1.526 114.6 -5 4 6 .3 7 7 7 EB (10,6/7,3) -*[7,4/5,3] 90 1.434 118.2 -5 4 6 .5 2 7 5 EBP (10,6,1/7,3,1) —► [7,4,1/5,3,1] 90 1.417 118.3 -547.208431 e b p ° (U ,8,1/9,6,1) —* [6,5,1/4,3,1] 91 1.416 118.2 -547.20769 DZP (U,7,1/9,5,1)-*[6,4,1/4,2,1] 92 “ See basis set definitions in Section A.I. ^ T h ree 3 d GTF’s split into 1 GTF and a group of 2 GTF’s (same exponents for each). “Experimental bond distance from Morino et a l* b d3-GTF expansion to SZ-STO basis set of C l e m e n t i et al.95' 97 “3-GTF expansion to SZ-STO basis set of C l e m e n t i et al.9s~97 (Different exponents for ns and np functions.) ■t Geometry optimized at the MIDI-3* level. 9 Effective-core-potential (ECP) for inner orbitals with double zeta for the valence orbitals for sulfur and 6-31G for oxygen. h Calculated Hartree-Fock (HF) valence energy is — 159.3d 169 hartree. ‘ ECP-21G with a single set of polarization functions (5 d-type) on sulfur. 1 Calculated Hartree-Fock (HF) valence energy is —159.51436 hartree. * Optimized geometry using the smaller basis set. 'Previous basis plus Rydberg orbitals (5 and j>) on sulfur. m Model potential used for 6 inner s and 3 inner P on sulfur and 3 inner s on oxygen (( 6,6/ 6,5) -* [5,4/3,2]). n A single set of 6 d-lype functions added on both sulfur and oxygen. “ Double zeta basis with 1 diffuse p function and one d function on both sulfur and oxygen. 37 of Schmiedekamp et al.90 gives this same result. It is fairly obvious that basis sets without polarization functions ju st are not adequate for obtaining accurate geometries. The absolute energies generally improve as the basis sets improve from minimal to extended. From the table, it is readily apparent that polarization functions are necessary to get a good description of the geometry. The improvement can even be seen in the STO-3G* calculations. The geometry seems to stabilize at around 1.42 A and 118.5° once one goes to a DZVP basis set. Even the small 3-21G* basis set of Pietro et al.85 gives a good geometry ( Rso = 1.419 A and Qoso = 118.7°). However, very good energies usually are not obtained unless one uses a DZP or better basis set. This can be seen by looking at the calculations of Huzinaga and Yoshimine .89 They used a good DZ basis set and obtained an energy lowering of « 0.22 hartree upon adding polarization functions to both sulfur and oxygen. Also, although not shown here, the polarization functions are extremely important in calculating good properties (dipole and higher moments, polarizability, and dissociation energies). The best explanation for the effect on the bonding upon adding polarization functions can be seen by noting the change in character of the outer orbitals when this occurs. In a MBS description, the sulfur s and p are not of the correct symmetry to participate in the la 2 MO. Thus the l a 2 ^ orbital is non-bonding (lone-pair on oxygen). 38 Upon addition of d-type functions on sulfur the la 2 orbital takes on S-0 dir - pn bonding character and slight 0 - 0 antibonding character (while still being mostly lone-pair on oxygen). This is also evident in the overlap population, which in creases by about tenfold (of which about 13 % is due to the dir - p^r backbond- ing ) . 71’70’67’65,22,82 The d-type functions also change the 8 ai orbital, causing it to take on more of a lone-pair character on sulfur and become less S-0 antibonding. Roos and Siegbahn 7 0 show an electron density difference map obtained when the polarization functions are added to their basis set. It clearly shows both of the effects discussed above. The best calculations to date use DZP or EMP basis sets . 2 2 ,8 9 "9 2 The lowest all electron calculation energy (E = —547.239412 hartree) was obtained by Lindley, using a EBP basis set (2 d’s on S and 1 on O ) . 22 He also used this same basis with a set of Rydberg orbitals (s and p) on sulfur, which lowered the energy by « 0 . 0 0 1 hartree. It should also be noted that he obtained the shortest bond distance (Rso = 1.404 A) and largest bond angle (0Oso = 118.8°) for these types of basis sets. Two other things should be noted about the results presented in Table 3.4. First, there are two studies which were not strictly SCF. These were the effective- core-potential calculations of Janszky et al.s 8 and the model potential calculations of Huzinaga and Yoshimine . 89 Both of these methods used a potential function to describe the inner shells. While these are approximate SCF methods, they both obtained geometries similar to the SCF methods. Huzinaga and Yoshimine’s MP- 39 DZP calculation gives a valence energy of —41.51115 hartrees. Their total energy of —547.26796 hartrees is obtained by adding the free atom core energies of the sulfur and oxygen atoms to the MP-SCF valence energy. Secondly, SCF calculations almost always give bond angles which are too small, the exceptions being those obtained by some studies using ST0-3G or ST0-3G* basis sets. The bond distances vary in a more uneven pattern, but those obtained using DZVP or better basis sets tend to give bond distances which are too short. The next step in a series of calculations would be to add the effects of electron correlation. The results from various studies using correlated methods ,68,91,92,76 both at experimental and optimized geometries, are given in Table 3.5. Two of these studies were done at the experimental geometry of S0 2 .6 8 ,7 6 Dun ning and Raffenetti 6 8 performed SCF, GVB, and various GVB-CI calculations using an extended minimal basis sets with polarization functions on both sulfur and oxygen. Their SCF results are given in Table 3.2. The generalized valence bond (GVB) method is a type of MCSCF and gives an energy approximately 0.06 hartree lower than the SCF energy. They then did two types of GVB 2R-CI cal culations. The first was what they label GVB-CI, which is a 2R-CI only within the valence orbitals. This gave an energy which was 0.06 hartree below that of the GVB calculation and a total correlation energy, A E corr ( E c v b - ci — E s c f), °f 0.11 hartree. The other Cl calculation was termed a POL-C1, which is a 2R-CI with the restriction that no more than one electron occupy a virtual orbital outside 40 Table 3.5. Results of various studies on the ground state (*Ai) of SO2 using correlated methods. Geometry Energy (a.u.) Method" Basis Set6 Ref. Rso (A) O q s o (deg) 1.4308° 119.3 -547.28972 GVB DZVPd 68 1.4308° 119.3 -547.34521 GVB-CI6 DZVPd 68 1.4308° 119.3 -547.49588 POL-Cl* DZVPd 68 1.45 120 -547.34671 GVB-CI6 DZVPd 68 1.46 120 -547.49843 POL-CI* DZVPd 68 1.45 120 -547.64468 GVB+l+2r® DZVPd 68 1.416* 118.2 -547.64831 SD-CI DZP* 92 1.416* 118.2 -547.65476 2R-CI DZP* 92 1.444 118.1 -547.656835 SD-CI EBP-’ 91 1.444* 118.1 -547.714762 SAC EBP* 91 1.432' 119.5 -547.277532 CASSCF-2 EBPm 76” 1.432' 119.5 -547.742395 SD-CP EBPm 76 1.432' 119.5 -547.803142 SD-CI/Dav° EBPm 76 1.432' 119.5 -547.816329 CPF0 EBP"* 76 1.432' 119.5 -547.836291 SD-CP EBPm 76 1.432' 119.5 -547.851354 CPFP EBPm 76 “See methods definitions in Section A.2. lSee basis set definitions in Section A.l. 'Experimental geometry from Morino et al.45 ^Contraction scheme is (11,7,2/9,5,1) —*■ [4,3,1/3,2,1]. 'GVB-2R-CI, excitations only within the valence orbitals. 1GVB-2R-CI, excitations with the restriction that no more than one electron occupy a virtual orbital. JGVB-2R-CI, excitations with the restriction that no more than one electron be excited from the ns-like orbitals (lai, 2ai, lb 2). '‘SCF optimized geometry with same basis set. ‘Contraction scheme is (11,7,1/9,5,1) —*■ [6,4,1/4,2,1]. ■’Double zeta basis with 1 diffuse p function and one d function on both sulfur and oxygen ((11,8,1/9,6,1) -[6,5,1/4,3,1]). *SD-CI optimized geometry. Experimental geometry from Kivelson.44 "•Contraction scheme is (12,9,3/9,5,2) —► [6,5,3/5,3,2]. "Includes the calculation of many properties with multiple basis sets. °The sulfur Is, 2s, 2p and oxygen Is are frozen (i.e., no excitations from these orbitals are allowed). pOnly the Is on sulfur and oxygen are frozen. 41 the valence space. This calculation gave an energy of -547.49588 hartree, which is ~ 0.15 hartree below the GVB-CI energy. This gives an overall correlation energy, A E corr ( Epol-ci ~ Escf), of 0.26 hartree. The best work to date is that by Bacskay et al.76 who did SCF, CASSCF- 2 , SD-CI, and CPF calculations at the experimental geometry. They used an extended basis set with three d-type functions on sulfur and two on oxygen. Their SCF energy is —547.248465 hartree. They then did a two-configuration MCSCF calculation, which lowered the energy by 0.03 hartree. In addition, two types of SD-CI and C PF calculations were performed. In the first set of SD-CI and CPF calculations, the inner shell core orbitals on sulfur (Is, 2 s, 2 p) and oxygen (Is) were frozen (i.e. no excitations allowed from these orbitals). Their SD-CI energy is 0.494 hartree (0.555 hartree with the Davidson correction) below that of the SCF value, while the CPF value is 0.568 hartree lower than SCF. In the second set of calculations, only the Is on sulfur and oxygen were frozen. The SD-CI energy in this case is —547.836291 (A Ecorr = 0.588 hartree) and the CPF energy is —547.851354 (A Ecorr = 0.603 hartree). It can also be seen, by comparing these values to those from the previous SD-CI and CPF calculations, that Ri 0.035 hartree (5.81 %) of the correlation energy can be recovered by allowing excitations from the n = 2 inner shell on sulfur and oxygen. The rest of the results presented in Table 3.5 deal with geometry optimiza tion .68,91'92 Murrell et al.92 performed SD-CI and 2R-CI calculations, using a DZP 42 basis set, at their SCF optimized geometry (Rso = 1.416 A and 6oso — 118.1°). Their correlation energy, from the 2R-CI calculation, is 0.447 hartree. Only two studies did actual geometry optimizations at the correlated level. Dunning and Raffenetti 6 8 did three types of GVB-CI calculations similar to those discussed above for the experimental geometry. Their optimized geometries, from all three types of calculations, were essentially the same. Their best calculation was a GVB 2R-CI, with the restriction that no more than one electron be excited from the ns-like orbitals (laj, 2 a1? lb 2) (GVB+l+ 2 r). Their optimized geometry is Rso = 1-45 A and 0oso = 1 2 0 ° and the energy is —547.64468 hartree. This geometry is extremely close to the experimental geometry, with an error of about 1.4 % and 0.6 % for the bond distance and angle, respectively. The only other study to optimize the geometry at the correlated level was that of Hirao . 91 He used an EBP basis set and optimized the geometry at the SD-CI level, obtaining a bond distance of 1.444 A and a bond angle of 118.1° (error relative to experiment being 0.91 % and 1.2 %, respectively). This was an improvement over his SCF optimized bond distance (1.417 A) but not his SCF optimized bond angle (11S.30). The energy at this geometry is —547.656835 hartree (AEcorr = 0.45 hartree, relative to the energy at the SCF optimized geometry — see Table 3.4). This is the lowest energy obtained to date for an optimized geometry at the correlated level. Hirao also calculated the energy at the SD-CI 43 optimized geometry using the symmetry-adapted cluster (SAC) method, obtaining —547.714762 hartree (A E corr — 0.51 hartree). 3.3.3 Possible Isomers and a Double Minimum Potential Surface of SO 2 There has been some spectroscopic evidence for an isomeric form of SO ^ 10 3 *110 These isomers were proposed to explain a strong continuous absorption which gradually disappeared with the appearance of the S0 2 discrete bands in the com bustion of H 2 S, CS2, and COS . 1 0 3 -1 0 5 This same type of behavior was observed by Norrish and Oldershaw in their flash photolysis study of S0 2 . 10 6 This isomer was postulated to be either a superoxide form (SOO) or a closed ring form with a bond angle around 70°, much like that postulated for ozone. Brown and Burns studied the effect of temperature on the absorption spectrum and noted that as the temperature is raised the discrete bands in the far u.v. are replaced by contin uous bands . 1 0 7 They also obtained an enthalpy change of 4.1 ± 0 . 4 kcal/mole for the isomerization, although they could make no conclusions on the nature of the isomer. It should be noted however, that there are other interpretations of this same data. Norrish and Oldershaw themselves have said that it could be due to a sulfur monoxide dimer (S 2 0 2)> an S20 molecule, or even to an excited triplet state of S0 2 . 1 0 6 Herman et al.108 and Gaydon et al.109 concluded that the continuum is due to an increase in the complexity of the rotational structure at high temper ature. The latter investigators also suggested that the mechanisms proposed by 44 Norrish and Oldershaw to explain the reactions that take place during the flash photolysis of S0 2 would not be altered that much if th eir isomer was regarded as an excited triplet state ( 3 Bi). Finally, Giguere and Savoie 110 , who studied the effects of temperature on the IR spectra, concluded that their data could not fit an SOO isomer. Instead, they attribute the few weak bands that appear in the continuum to “hot” bands of an electronically excited state of S0 2 and/or to S 2 0 . A number of theoretical studies have also looked into the possibility of isomers of S0 2 and a double minimum in the *Ai potential surface .22,68-69'79.81.92-94 The results of these studies are shown in Table 3.6. Two of these studies were semi-empirical studies , 7 9 ,9 3 which found a crossing of the 5b 2 (occupied) and 3bx (virtual) M O’s. These results are in agreement with the Walsh diagram for S 0 2 -94 Hayes and Pfeiffer 9 3 calculated the orbital energies as a function of th e 0S0 bond angle (at a fixed S - 0 bond distance of 1.434 A) using an extended Huckel model. They found the orbital crossing occurred at a bond angle of about 72°. Chung 7 9 performed INDO-CI calculations on S 0 2 a n d also found a crossing of the 5b 2 and 3bx orbitals at around 75°. They looked at the *A, state with the 3bj as the doubly occupied orbital instead of the 5 b2. T he optimized equilibrium geometry for this excited state had a bond angle of 6 2 ° and a bond length of 1.64 A. The calculated adiabatic excitation energy from the normal ground state was 2.47 eV. 45 Table 3.6. Geometries and energies of the isomers of SO 2 . Isomer Geometry (A, d e g ) Energy (a.u.) Isomerization M ethod0 Basis set6 Ref. (ABC) R - a b R b c L a b c Energy (eV)c oso 1.431-1 1.431 119 -543.1985 0.0 SCF 3G-STO-DZ9 69 1.45 1.45 120 -547.34671 0.0 GVB-CI* D ZV P9 68 1.46 1.46 120 -547.49843 0.0 POL-CI* DZVP9 68 1.45 1.45 12 0 -547.64468 0.0 GVB+l+2r‘ DZVP9 68 1.46 1.46 119 ... 0.0 DFC-LSD ? 81 1.416 1.416 1 1 8 .2 -547.20769 0.0 SCF D Z P° 92 1.416P 1.416 1 1 8 .2 -547.65476 0.0 2R -C I9 D Z P" 92 SOO 1.69 1.27 1 2 6 -543.1642 0.93 SCF 3G-STO-DZ9 69 1 .67 1.33 120 -547.20065 3.97 GVB-CI* DZVP9 68 1.6 7 1.33 120 -547.33943 4.33 POL-CI* DZVP9 68 1.568 1.236 123.2 -547.00938 5.40 SCF D Z P° 92 1.568P 1.236 123.2 -547.49417 4.370 2R -C I9 D Z P° 92 OSO 1.434d 1.434 72 E-Huckel MBS-STO*' 93 (ring) 1.64 1.64 62 2.47 INDO-CI MBS-STO*' 79h 1 .6 8 1.69 60-3 -547.20230 3.93 GVB-CI* D ZV P9 68 1.6 7 1.67 6 0-0 -547.34685 4.12 POL-CI* D ZV P9 68 1 .66 1.68 60-7 -547.48391 4.37 G V B + 1+ 2,.1 D ZV P9 68 1 .76 1.76 49 4.1m DFC-LSD ? 81 1.660" 1.660 6 0 7 -547.03191 4.78 SCF D Z P° 92 1 .660P 1.660 60-7 -547.50769 4.002 MR-CI9 D Z P° 92 aSee methods definitions in Sections A.2 and A.3. ^See basis s e t definitions in S e c t i o n A .l. cAdiabatic e n e rg y differences (Bi jom cr ~~ E s o 3 )• dFixcd experim ental bond distance from Morino el al.45 'Minimal basis set of Slater-type orbitals with a d-type STO on sulfur (See Table 3.1). ■^Experimental geometry from Kivelson44 sThe authors’ 3-GTF expansion to DZ-STO basis set of Clementi et al.95~97 ^ The ’Aj state doubly excited relative to the ground state (3bi doubly occupied instead of the 5b2). 'GVB-211-Cl. excitations only w ithin the valence orbitals. J Contraction scheme is (11,7,2/9,5,1) -* [4,3,1/3,2,1]. ^GVB-2R-Cl, excitations with the restriction that no more than one electron occupy a virtual orbital. *GVB-2R-Cl> excitations wit|, the restriction that no more than one electron be excited from the ns-like orbitah (la t, 2al* Ito). mCalcnlated activation barrier between the two minima is ss 5.4 eV. n Geometry taken from Dunning a,,d Raffenetti.68 "Contraction scheme is (11,7,1/9,5,1)-► [6,4,1/4,2,1]. pGcometry optimization done a t the SCF level. 5 Excitations from SCF occupied-valence orbitals into the low-lying virtual orbitals. 46 Farantos et al.69 carried out an SCF calculation using a DZ ST0-3G type basis set without polarization functions. They found the superoxide isomer (SOO) to have an S - 0 bond distance of 1.75 A, an 0 - 0 bond distance of 1.27 A and an SOO bond angle of 126°. The isomerization energy ( E i , omer — E so 2) °f this structure was 0.93 eV above their calculated energy for SO 2 . The SCF study of Lindley , 22 using a larger EBP basis (see Table 3.4), did not show any crossing of the 5b 2 and 3bj orbitals between the OSO bond angles of 60° and 180°, at a fixed optimized S - 0 bond distance of 1.404 A (see his Figure 2 on p. 70). He also performed calculations on the doubly excited xAi state at three bond angles between 60° and 90°. At none of these angles was this state’s energy lower than that of the usual ground state at the same geometry. Therefore, this SCF study did not indicate a double minimum in the potential surface for the ground state. However, it does appear in his Figure 2, th at the 5b 2 and 3 bi will cross at some angle less than 60°. So, it is possible that a second minimum may have existed at an angle of less than 60°. However, this would most likely be a local minimum. There have been a few more recent calculations concerning the isomers of S0 2 and the possibility of a double minimum in the potential surface .68,81'9 2 Dunning and RafFenetti 68 performed Generalized valence bond (GVB) and GVB-CI calcu lations on the open isomer of S02, the ring isomer, and the superoxide isomer (SOO). They did three levels of Cl calculations (discussed previously) for the open 47 and ring isomers, but only two for the SOO isomer. The best calculation common to all three isomers is their POL-CI. The open isomer of S 0 2 has an optimized geometry of 1.46 A and 1 2 0 °. The ring isomer has a geometry with S-0 and 0 -0 bond distances of 1.67 A and an OSO angle of 60.0° and it lies 4.12 eV above the ground state. (A slightly better Cl calculation on this isomer gave S-0 and 0 -0 bond distances of 1 . 6 6 A and 1 . 6 8 A, respectively, an OSO bond angle of 60.7° and an isomerization energy of 4.37 eV.) The superoxide isomer is calculated to have S - 0 and 0 -0 bond distances of 1.67 A and 1.33 A, respectively and an SOO bond angle of 120°. It is calculated to have an isomerization energy of 4.33 eV, just above that for the ring isomer. Since the energies of both of these isomers are so much higher than that of the ground state, these authors suggest that the “isomer” in the combustion reactions is vibrationally-rotationally excited S02, as suggested by Herman et a/ . 108 and Gaydon et al..109 Murrell et al.92 performed fairly extensive calculations on S 0 2 and its two possible isomers. They used a DZP basis set and optimized the geometries of the open and superoxide isomers. (For the ring isomer they used the geometry obtained by Dunning and Raffenetti.68) They obtained isomerization energies similar to those of Dunning and Raffenetti. Jones 81 has performed density functional calculations on S02. His calculations show a crossing of the 5b 2 and 3bj orbitals at around 60°. He also obtained an optimized geometry for the second JAi state (doubly excited relative to the ground 48 state) with an SO bond distance of 1.76 A and OSO angle of 49°. This state has an adiabatic energy difference of 4.1 eV from the ground state. The activation barrier between the two minima was calculated to be ss 5.4 eV. From these more recent studies, especially those at the correlated level, it seems probable th at stable isomers of SO 2 do exist. It is also likely that the potential surface of SO 2 does have a double minimum. However, the isomers would lie at substantially higher energies than X 3.3.4 Summary From the discussion above two things are very clear. A basis set of at least double- zeta quality, including polarization functions (usually d-type) on both sulfur and oxygen, is necessary to obtain a good description of the equilibrium geometry and minimum energy. The size of the uncontracted basis set and the contraction scheme can also have a sizable effect. The best contraction schemes tend to be the segmented or general contractions, the latter of which seem to be better. In addition, the method of calculation is important. Methods which include electron correlation greatly improve the optimized equilibrium geometry (com pared to SCF) and the total energy. The energy due to including valence electron correlation amounts to as much as « 0.6 hartree (« 16 eV) of the total energy. Thus electron correlation can have a large effect on the energy and, as will be discussed in the following sections, the transition energies to the various excited 49 states. Also, both the basis set selection and calculation method can determine whether secondary minima on the potential surface can be located (metastable transition states or isomers). 3.4 The 2400-1700 A Absorption Region — The C ll$2 (lA') Excited State This region of the SO 2 spectrum exhibits the most intense bands. It is due to an ►tt* (3bj <— la 2 ) transition. This state is of interest for two reasons. First, it is possible that SO 2 predissociates via it . 1 1 1 -1 1 5 Second, it is very probable that this excited state exhibits a double-minimum potential in its antisymmetrical ( 1/3 ) mode of vibration, making it quite different from the ground state and the other excited states discussed in this paper. There have been many experimental studies , 1 1 4 -1 4 2 most aimed at answering the question as to whether this region of absorption is due to just one or several excited states. Most recent studies actually postulate that there is only one excited state involved and that it has an asymmetrical (Cs) structure. Theoretical studies of this state 82’'9’143,22,81 have been mostly limited to the calculation of the equilibrium C21, geometry and excitation energy. 3.4.1 Experimental Investigations The experimentally determined excitation energies (vertical and (OOO)-(OOO) tran sition or band origin, T0) and equilibrium geometries for the 1 B2 state are given 50 in Table 3.7. Some of these values were obtained assuming that the excited state had Civ structure (and there was possibly more than one excited state) and others assumed a C« structure. While this does affect the geometry it can be seen from the table that it has little effect on the excitation energy. The earliest studies seemed to give conflicting results as to the number and type of excited states responsible for this region of absorption . 1 1 6 ’11 7 Chow 1 1 6 attributed this region to more than one electronic system, with a vertical transition energy of about 6.24 eV. However, Price and Simpson 1 1 7 postulated that only one upper electronic state was responsible for the absorption, with a band origin around 5.34 eV and a vertical transition range of 6.14-6.33 eV. Up to the late 1960’s it was “generally” accepted that more than one electronic upper state existed in this region . 1 1 8 -1 2 4 Duchesne and Rosen proposed that this region is formed by at least two electronic systems and probably a third as well . 118 Dubois actually assigned this region to a 1Bi <— JAi and xAj <— xAj transition . 120-1 2 3 He also obtained an observed (OOO)-(OOO) transition energy, To, of 42589.65 cm 1 (5.280 eV). Dubois also obtained the geometry for the upper state from both rotational and vibrational analyses. The more accurate results from the rotational analysis give a bond distance of 1.52 A and bond angle of 109.35°. In the early 1970’s this view began to change. Brand and Srikameswaran 125 performed a rotational analysis at 20 °C and —80 °C and determined the structure to be that of an A-type transition of an asymmetric rotor. They therefore assigned 51 Table 3.7. Experimentally determined geometries and excitations energies for the C jB 2 (W ) excited state. Geometry0 A E (eV) Method* Ref. Band R s o (A )“ Ooso (deg) Vertical Origin (To)d . . . 6.14-6.33 5.344 UV 117 5.228“ Vibrational 118 5.235' Vibrational 119 1.55 (0.12) 109.78 (-9.55) 5.280 Vibrational 123s 1.520 (0.089) 109.35 (-9.98) Rotational 123s 1.560 (0.129) 104.3 (-15.0) 5.2784 Rotational 125 5.414 V ib.— Rot.'* 126 1.525 (0.094)' 105.15 (-14-2) 5.279 Vibrational 127,144 1.560 (0.129) 104.3 (-15.0) 5.2785 V ib.—Rot. 114 1.6337 (0.203) 104.3 (-15.0) V ib.— Rot. 128 1.4863 (0.055) 1.6394 (0.208)* 103.75 (-15.58) V ib.— Rot. 129 1.4908 (0.060) 1.639 (0.208) 103.75 (-15.58) V ib.— Rot. 130 1.490 (0.059) 6.1 Electron Impact1 136 6.2 Electron Impact 138,142' 6.3 Electron Impact 137,139-141"* “Quantities in parentheses represent the change compared to X *Ai ( R s o = 1-431 A and BoSO — 119.33°). 6The vibrational and rotational analyses are of the ultraviolet (UV) spectra, unless otherwise noted. “Entries with more than one bond distance are for the proposed C, structure ('A' state). Origin of the (OOO)-(OOO) transition. See Section3.4.1 for zero-point energy corrections and minimum excitation energy, Xe. “The band origin of the authors’ orj band system. ' The authors’ revised band origin of Duchesne and Rosen’s a i band system.118 9 Assigned by the author to 1 B i. '“Vibrational and rotational analysis of the fluorescence spectrum. The A Evert corresponds to the most intense line in the spectrum. 'An alternative structure of R s o = 1-526 A and B q s o = 108.4°is also given. 1 The C2v structure has R s 0 = 1.5525 A (A R = 0.1215 A). Also referred to as electron energy loss spectrum (EELS). * Reference 138 gives an energy loss range of 5.3-7.3 eV. m Reference 141 studied solid S 02. ” Reference 140 gives an energy loss range (Franck-Condon limits) of 5.5-7.1 eV. 52 the transition to a 1 B2 «— They also could not explain one band in the spectrum in terms of iq and 1/2 progressions alone. A vibrational analysis of the resonance fluorescence spectrum by Brand et al.126 supported this assignment and also showed the presence of v3 progressions. This provided strong evidence that the upper state could have an asymmetrical (Ca) structure. Coon and coworkers 127 studied this region in both the vapor phase and in solid rare gas matrices at temperatures near 4.2 K. Their results also showed progressions in vz and a large interaction between and 2 ^3 . They associated this with the anharmonicity of a double-minimum potential in the antisymmetrical (i/3) mode of vibration. By the late seventies the electronic origin for this band system had been fairly well determined to be 5.2785 eV for 7b ( vheadfl-o = 5.279 eV). Brand et al.126 had also determined the vertical emission energy to be 5.414 eV. Also, it was becoming apparent that the C ^ 2 state was probably unsymmetrical and should be considered to have Cs symmetry ( 1A'). Most of the work was then aimed at finding the asymmetrical structure and the height of the saddle point (C 2,, structure) above the absolute minima (two equivalent Cs structures ).114,128-130 Brand et al.114 rotationally analyzed the transition as a C 4 B2 <— X ^ 1 . How ever, to account for the vibrational and rotational band structure, they hypothe sized a double minimum potential in the antisymmetric stretching coordinate q3 and found that the rotational structure fitted that of A-type bands of a prolate asymmetric rotor. The difference in S-0 bond distances, estimated from the res onance fluorescence work of Brand et al.,126 was approximately 0 . 1 2 A. They also obtained a preliminary value of fa 1 0 0 cm- 1 for the height of the saddle point at the C2,, configuration above the absolute Cs minimum. Brand and Rao 128 were the first to analyze the vibrational-rotational struc ture completely in terms of a double-minimum potential in v3. They obtained an expression for the vibration-rotation Hamiltonian in terms of two curvilinear stretching coordinates pl and pz (where 2 px = R x + R 2 and 2 p3 = R 1—R 2) and one rectilinear bending coordinate. Using this, along with potential constants adjusted to fit the term values for certain vibrational levels found by Brand et a/ . , 1 14 they obtained a geometry of R} = 1.6337 A, R2 = 1.4863 A and 0Oso = 104.3°. Their barrier height was calculated to be 90 cm-1. Brand and Hoy 129 also analyze this region using a vibrational Hamiltonian based on a double minimum potential in the antisymmetrical stretching coordinate <73 and strong anharmonic coupling of q3 with the symmetric stretch qi (where R\ — R2 = 0.1047<73, -f R2 — 2R e = 0.0789<7j, and Re is the average equilibrium C2u bond distance). Their average equilibrium C2„ geometry is Re = 1.5525 A and Ooso = 103.75°. The <71 — 0 section of their potential surface contains the saddle point corresponding to the C2u structure, but the absolute minima occur for small positive values of <71 • They calculate a barrier height of 141 ± 20 cm - 1 at <71 = 0.32 and q3 = 1.419. This corresponds to the bond distances R x = 1.639 A and R2 = 1.491 A. Also, the zeroth vibrational level is 43 ± 2 0 cm- 1 below the C2u 54 saddle point, corresponding to an asymmetric distortion in the lowest vibrational level. Mezey and Rao 1 3 0 performed a similar analysis to Brand and Rao 1 28 using the C2u geometry of Brand and Hoy . 129 They obtained a geometry for the asymmetrical structure with Ri = 1.639 A and R2 = 1.490 A and a barrier height of 140 cm-1, in good agreement with the previous analyses. There have been a few more recent investigations of this state . 1 1 5 ’1 3 1 -1 3 5 JafFe132 performed a nonnormal mode vibrational analysis of the excited state using a double-minimum potential. He assigns the two fundamental frequencies, 560 cm - 1 and 960 cm-1, to the long and short S-0 bonds, respectively. The two-photon excitation experiments of Vasudev and McClain 115 support the assignment of the excited state as XB 2 ^A') and that there is a double-minimum in the potential. However, they approximate a barrier height of 1000 ± 150 cm-1. They also con clude that photodissociation of the C 1 B2 state proceeds through unsymmetrical stretching of the S-0 bonds. Ivanco 134 performed laser induced fluorescence mea surements and has concluded that the barrier height is only 49 cm-1, which is below the zero point vibrational energy. He thus constructs a one dimensional potential in the <73 coordinate. He also finds a coupling of this state with higher vibrational levels of the ground state and possibly one or more triplet states. This is supported by several other investigators .131'133,135 55 The preceding discussion details the optical experiments. There have also been several electron impact excitation or electron energy loss (EEL) experiments .136' 142 As seen in Table 3.7 all of these experiments obtained a vertical excitation energy of around 6.2 eV. The electron energy loss range or Franck-Condon limits for this region is 5.3-7.3 eV . 138,140 These results agree very well with the vertical excitation energies obtained from optical experiments. In conclusion, it seems that the majority of the experiments point to a double minimum potential in the antisymmetrical stretching mode of the excited state (Cs structure). The absolute minima occur for S - 0 bond lengths of about i?i = 1.639 A and R.2 = 1.490 A and a OSO bond angle of 103.75°. The shorter bond distance is close the values of 1.5002 A and 1.493 A found in the b 1£ + and a 1A excited states of SO, respectively . 145 ,1 4 6 The SO ground state bond distance is 1.48108 A, which corresponds to a pure S-0 double bond . 146 The longer bond is comparable to a single S-0 bond, as determined in SO 3 polymers . 1 4 7 Also, using the Schomaker- Stevenson rule , 1 4 8 Powers and Olson 149 calculated the S-0 single bond length to be 1.69 A and Gordy 150 calculated the S - 0 double bond length to be 1.49 A, both of which are remarkably close to those found above. The barrier to interconversion between the two structures has a height of ap proximately 140 cm-1. The top of the barrier corresponds to the C 2 ,, structure with a S - 0 bond distance of about 1.5525 A and OSO angle of 103.75°, the same as for the Cs structure. The change in these values from the ground state geometry 56 (Rso = 1.431 A and doso = 119.33°, see Section 3.3.1) are 0 . 1 2 2 A and —15.58° for the S-0 bond distance and OSO angle, respectively. The relationship between the two C3 structures and the C2v structure along the potential curve is shown schematically in Figure 3.2. This conclusion seems to be supported by similar behavior of the excited states of several other molecular systems; the 2B 2 state of NO 2 , 151,152 the 1 B2 state of Se0 2 , 15 3 and the 2A2 state of CIO2 . 1 5 4 ’1 5 5 All of these transitions involve the exci tation of an electron into an antibonding bx orbital. Mulliken 15 6 has suggested that the antibonding effect of this orbital is expected to be smaller for an unsymmetrical structure than for a symmetrical one. A rough estimate for the minimum excitation energy, Te, can be obtained using the above information and that in Section 3.3.1 on the XAX ground state. It has been established that To is 5.2785 eV. The zero point energy for the XAX ground state is approximately 1530 cm - 1 (0.190 eV). The zero point energy of the XB 2 state is not easily obtained because of its double minimum potential in v3. If this is ignored and only the zero point energy of the ground state is used one can obtain a maximum to Te (TCmai) of 5.47 eV. An approximation to the *B 2 zero point energy can be obtained from the data of Brand and Hoy . 129 The give harmonic frequencies for the symmetric stretching ( qx) and bending (q2) coordinates of 935.2 and 384.9 cm-1, respectively. The contribution to the zero-point energy of the XB 2 state due to these degrees of freedom is « 660.05 cm_1(0.082 eV). The contribution to the 57 Figure 3.2. Schematic representation of the double-well potential of the C 1 B2 ('A') state along its inode. 58 zero-point energy due to the asymmetric stretching component of the potential surface (q3) can be determined by using Brand and Hoy’s analysis for the double minimum in this coordinate, as discussed above. They obtained a barrier height above the absolute minimum of ss 141 cm-1and place the zeroth vibrational level a t ~ 43 cm-1below the top of the barrier. This gives a contribution to the zero- point energy of ss 98 cm- 1 (0.012 eV). Thus the total zero-point energy of the 1 B2 state is 0.094 eV. This can be subtracted from TemoI calculated using only the X *Ai state’s zero-point energy to obtain a Te of approximately 5.37 eV for the 1 B2 state. 3.4.2 Theoretical Investigations As mentioned previously only a few studies have been performed on the excited states of sulfur dioxide .22,23’64,79’81'82*143’157,158 Most of the calculations on this state were performed by investigators who also performed calculations on the ground state, as discussed in Section 3.3.2. Since the methods and basis sets used were discussed in that section, they will not be discussed in detail here. Table 3.8 contains the absolute and excitation energies and optimized geometries for the 1 B2 state from the theoretical studies. Two of the studies were semi-empirical . 79,81 Chung 7 9 performed an INDO-CI calculation for both the ground state and 1 B2 states. However, the Cl for the ex cited states was different than that for the ground state. For the ground state only 59 Table 3.8. Calculated geometries and excitations energies for the 1 B2 excited state.0,6 Geometry 0 AE {eV)d Method Ref. Rso (A) Ooso (deg) Vertical Minimum (Te) 1-573 (0.069) 98.1 (-22.9) 8 . 0 SCFe 82' 1-595 (0.091) 107.6 (-13.4) 5.3 S-CP S2f 6 .933* S-CI‘,J 143 1-69 (0.10) 104 (-15.5) 5.18 3.84 INDO/S-CIfc 79 1-609 (0.154) 106 (-13) 5.9 4.2 DFC-LSD 81 1-524 (0.120)' 101.3 (-17.5) 7.095m 6.054" SCF° 2 2 7.498p SCF0 2 2 °See Tables 3.4 and 3.3 for geometries and energies of the xAi ground state. 6Unless otherwise noted the basis sets are the same as those used for the xAi ground state reported in Section 3.3.2. cQuantities in parentheses represent the change compared to the ground state. dUnless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. eVirtual orbital approximation applied to the ground-state wavefunction. ; Using ST0-3G* (‘split’ d) basis. 9 All single excitations from the seven valence MO’s to all virtual MO’s. 6Geometry for xAi not given. Assumed to be experimental geometry, although the energy is incorrectly reported as -654.78725 a.u. ‘Using a 4-31G basis set. •'All single excitations from the three highest occupied MO’s to the three lowest virtual MO’s. kSingle excitations from doubly-occupied orbitals to the two singly-occupied virtual orbitals in the SCF description (la^SbJ). 'Geometry is not fully optimized. mCalculated at the xAi experimental geometry of Kivelson 44 Absolute energy of —546.976230 hartree. "The absolute energy is -547.016935 hartree. "Using their [7 ,5,2/5,3,1] contracted basis. pCalculated at the *Ai optimized geometry. Absolute energy o f —546.963873 hartree. 60 double-excitations from doubly-occupied orbitals to all virtual orbitals were con sidered, while for the excited states only single-excitations of doubly-occupied or bitals to the two singly-occupied virtual ground state orbitals were allowed. Jones 81 performed density functional calculations for several states and obtained slightly better results than those of Chung’s. As can be seen from the table, both of these studies produce bond distances which are too long and excitation energies which are much too low, when compared to the experimental values. The most comprehensive study was by Lindley . 22 He performed SCF calcula tions using an extended basis set with polarization functions. He obtained absolute energies, vertical and minimum excitation energies, and properties for the excited states. Some of the results for the XB 2 state are shown in Table 3.8. The geometry of this state was not optimized to the same degree as the other states studied, but it is still very good (error from experimental values is 1 . 8 % for Rso and 2.4 % for Ooso)- The bond distance and angle are both smaller than the experimental C/2 v values, as is the usual case in SCF calculations. Also, the change in geometry on going from the ground to excited state agrees quite well with the experimental changes (see Section 3.4.1). The relative error between the experimental and calcu lated changes are 1 . 2 % for A Rso and 1 2 % for A Ooso- As it turns out, this study produced the best results for the geometry and the change in geometry between the xAi and XB 2 states. However, the results for the excitation energies are not as good. The vertical and minimum excitation energies are both about 0.7 eV above 61 the experimental values (with the vertical excitation energy at the ground state experimental geometry being 0.4 eV lower than that at the optimized geometry). Bendazzoli and Palmieri 143 calculated the vertical excitation energies using sin gle reference Cl for the excited states, and a small 4-31G basis set. Their Cl was very limited, allowing only single excitations from the three highest occupied MO’s (5b2, la 2, 8 a!) to the three lowest-lying virtual MO’s (3b!, 9 ai, 6 b2). Their Cl expansion for the C 1 B2 state has three important contributions, described by one- electron excitations as (la 2 —>3bi), (5b 2 —>9 ai), and ( 8 ax —> 6 b2). Their vertical excitation energy is still at least 0.6 eV higher than the experimental value, not much better than the SCF value of Lindley. There may be two reasons for this. First, they do not give the exact geometry at which the calculations are done. It appears to be the X xAi experimental geometry. A different geometry could give a very different result. Second and most importantly, they appear to have calculated the energy for the X 4Ai state at the SCF level only. If reason two is true, then they are not taking energy differences between absolute energies calculated using equal treatments of both the ground and excited states. It might be found that other configurations (not just the SCF one) are as important for a good description of the *Ai state, as they were for the 1 B2. A Cl calculation on the ground state could have a large effect on its energy, thus affecting the A E (it is not possible to say for sure in which direction this would shift). Hillier and Saunders 8 2 performed virtual orbital and Cl calculations to obtain optimized geometries and minimum excitation energies. They used a minimal basis set with “split” d polarization functions to allow some optimization of the 3d orbital (see Table 3.2). As can be seen in the table the virtual orbital approximation results are very poor, especially for the bond angle and excitation energy. They improved upon this by doing a single-reference Cl in which all single excitations from the seven valence MO’s to all the virtual MO’s were allowed. They found the MO configurations of greatest importance are those arising out of single excitations from the 5 b 2 , la 2 , and 8 ai occupied MO’s to the 3bi virtual MO. Thus the XB 2 state’s most important configuration, in their calculation, is the SCF configuration arising from the transition, la 2 —>3^. Their result for the geometry is fairly good, being too large in both the bond distance and angle (by about the same amounts as Lindley’s are too small). Their result for the minimum excitation energy is very good. However, this could be fortuitous since it seems that their energy difference is between the SCF energy of the xAi state and the Cl energy of the XB 2 state. Also, they used a very small basis set which does not produce very good absolute geometries and energies. Therefore, they may have just happened to get a cancellation of errors to give the proper Te Also, their A R$o and A Ooso are the differences between the SCF optimized equilibrium geometry for the xAj ground state and the Cl optimized equilibrium geometry for the *B 2 excited state. These would undoubtedly change if a Cl optimization were also performed for the *Ai state. There are two studies which do not appear in Table 3.7. Zellmer 23 did SCF calculations, using the [7, 5 ,'2 / 5 ,3,1] contracted basis set of Lindley , 2 2 near the op timized geometry of the >A 2 a n d 3A2 states (Rs0 = 1-503 A , 0OSO ~ 92.65°). The energy differences between the JAi ground state a t its optimized geometry and the and 1 B2 states at the 'A, optimized geometry are 1.95 and 6.13 eV , respectively. The 1 B2 state is still the highest excited state, at this geometry. Phillips and Davidson 158 performed SCF and C l (single and doubles, SD-CI, and SD-CI w ith quadruples’ corrections, SD(Q)-CI) calculations on several of the excited states at the experimental geometry of th e 3Bi excited state. They used a DZP basis set based on the same Dunning contractions 9 8 -102 used by Lindley 22 in some of their calculations ( (11,7,1/9,5,1) _> [ 6 ,4,1/4,2,1] ). The only difference is in the exponents of the d polarization functions- For their Cl calculations they used either a single or multiconfiguration reference space, depending on the state. They found th at for the X state two reference configurations were needed: the SCF configuration (lcore]:la^ 3 b?) and the doubly excited configuration relative to the SCF one ([core]:la.°3bf). The SCF, SD-CI, and SD(Q)-CI energies at this geometry are, -547.188, - 5 4 7 -5 4 3 , and -547.699 hartree, respectively. As can be seen the correlation energy is quite large (AEcorr = 9 -7 a°d 13.9 eV for SD-Cl and SD(Q)-CI, respectively). For the C !B 2 state they used a three configuration reference space: two singly-excited configurations, the SCF one (la 2 —>3bx) and the one given by the 5b 2 —► 9ai excitation, and a doubly-excited configuration ([core]:2b}5b 2 3b}9aJ). Their calculated SCF, SD-CI, and SD(Q)-CI energies at this geometry are, —546.951, —547.298, and —547.481 hartree, respectively. The SCF value is close to that of Lindley’s calculated using a bigger basis set and a slightly different geometry ( Rso = 1.51 A, Ooso = 118.8°, and E = -546.995 hartree). They also calculated energy differences between the various states. For the C 1 B 2 <_-X 1Aj transition they obtained A Escf> A E sdci , and A E sd{Q )ci values of 6.45, 6.67, and 5.93 eV, respectively. These are in agreement with the values reported in the previous investigations discussed above, keeping in mind of course, th a t this calculation was performed at a very different geometry. Two things should be made clear about all of the calculations just discussed concerning this state. All were carried out with the *B 2 state having a C2u sym metry. To my knowledge no calculation to date has been done on this state using C5 symmetry. This should be done since it has been established experimentally that this state has an asymmetrical structure. Also, in at least two of the investi gations 1 4 3 ,1 5 8 it was determined that this state is best described by more than one configuration. Thus, any good calculation would have to start with a multirefer ence space, meaning SCF is definitely not an adequate level at which to perform the calculation. 65 3.4.3 Summary The most recent experimental analyses conclude that this state is the C 1 B2 and most probably has an asymmetrical structure ( 1A/). Also, the zeroth vibrational level is probably below the symmetric saddle point (barrier to linearity). The ge ometries for the C xB 2 state from the theoretical investigations agree fairly well with the experimental results for the C2u structure. The changes in the geometry on going from the xAi ground state to the XB 2 state from both experiment and the ory also agree quite well. Also, the vertical and minimum ( Te) excitation energies from the ab initio studies agree fairly well with those from experiment, especially when considering the level at which the calculations were performed. They defi nitely fall within the energy range obtained from the electron impact studies (see Section 3.4.1). In summary, both experimental and theoretical results lead one to conclude, with a great deal of confidence, that this region of the spectrum is due to excitation to a C XB 2 OA') excited state. 3.5 The 3400-2500 A Absorption Region — The A A 2 and B *Bi Excited States This region of the S0 2 spectrum exhibits bands of medium intensity, overlaying a continuous background of weak bands. It is severely perturbed and has been the most difficult region to analyze. Thus, this region and the excited states responsible for it have been the subject of many investigations. It has been agreed 66 that these excited states are singlets, although the exact nature of these states has been difficult to determine. These singlet states are very important in the photochemistry of sulfur diox ide. To some small degree, one or more of them directly participates in the photochemical processes of SO 2 . 1 5 9 -1 6 8 However, most of the important pho tochemical reactions of SO 2 take place with the molecule in its excited triplet state(s) (a 3Bi and/or b 3A2)109-159_165>169-195 Several investigations have shown that the triplet state(s) can be populated by excitation to the singlet state(s) in this region (fluorescence and phosphorescence both occur on absorption in this region ) . 1 5 9 -1 6 5 ’1 6 9 -1 7 6 ’1 7 9 ’181’1 8 5 ’190’1 9 3 ,1 9 6 -2 1 2 Also, the major sunlight absorption of sulfur dioxide occurs within this region . 1 7 9 Thus, these states are a major con tributor to the formation of the triplet state(s) in the atmosphere . 1 6 4 ,1 7 9 The extent to which the singlet state(s) can form the triplet state(s) will be determined by the different energy degradation reactions in which the singlet(s) can partici pate .163-165,174'179,198-204 This will largely be controlled by the extent of mixing of the excited singlet states with the neighboring singlet and triplet states. Therefore, it is very important to understand the nature of these excited singlet states and their interaction with the other states of SO 2 . The main question concerning this region of the spectrum has been whether it is due to excitations from the xAi ground state to ( 1 ) only one excited state, the XA 2 or 4 Bi, (2) both the XA 2 and xBi states, (3) an electronic state which is a combina 67 tion of the two states (XA"), or (4) a combination of all of these. Theoretical studies have obtained excitation energies, for both of these states, which lie within this region . 2 2 ,2 3 ’7 9 ,8 1 ’8 2 ,1 4 3 ’1 5 7 Formally, only the B 1 Bi <— X xAi transition ( 8 ai ->3bi) is allowed. However, the electronically forbidden n* —► tt* A XA2 <— X xAi transi tion (5b 2 —»3bi) is a vibronically allowed C-type 1 B1 <— lA* transition, through the z/ 3 (b2) vibration mode. Also, it appears that there is Renner-Teller coupling between the 1Bi and xAi states, Jahn-Teller interaction between the 1 B 1 and XA 2 states, and spin-orbit coupling between these states and the triplet states ( 3 Bi, 3 B2, and 3A2). All of this will be discussed more fully in the following sections. 3.5.1 Experimental Investigations The experimentally determined excitation energies and equilibrium geometries for the XA 2 and xBj states are given in Table 3.9. As can be seen from the table, the geometry information from experiments is very sparse and the agreement in excitation energies is only fair. The earliest studies 213-215,222 all give essentially the same vertical transition energies in absorption, 4.2 eV. However, their T 0 values are not reliable since the full nature of the transitions in this region had not been determined. In fact, Watson and Parker 213 and Clements 214 considered this band system and the one due to the triplet states (3900-3400 A) to be part of one system. Metropolis ’215 To value falls almost midway between the presently accepted values. All of these 68 Table 3 .9 . Experimentally determined geometries and excitation energies for the XA2 and xBj excited states. State Geometry® AE (eV) Method** Ref. B and R so (A) 0OSO (deg) Vertical Origin (T0)c A, Bd 3.84 Vibrational 213 4.2° 3.96* Vibrational 214 100 (-19.3) 4.21 3.67 Vibrational 215 4.28 Ultraviolet 216 3.44 Fluorescence 190,199s 3.65* ... Fluorescence 197 4.25 Electron Impact1 136 4.39 Electron Impact 137 4.31 Electron Impact 13& 4.5 Electron Impact 139-141*’ 4.3 Electron Impact 142 A jA2 4.22 3.50 Vibrational 217m 1.50 (0.07) 112 (-7.3) 3.6 R o tatio n al 218” 1.53 (0.10) 99 (-20.3) ... 3.463 Vib.—Rot. 219,220° B 1B1 3.85 Fluorescence 20p 3.96 V ib.— R ot. 220« 3.873r Vibrational* 221 “Quantities in parentheses represent the change compared to X *Ai (Rso = 1.431 A and 0Oso = 119.33°). *The vibrational and rotational analyses are of the ultraviolet (uv) spectra, unless otherwise noted. cOrigin0f the (OOO)-(OOO) transition. See Section 3.5.1 for zero-point energy corrections and minimum excitation energy, y e. ^The excited state was not known for sure or the transition may be to more than one singlet state. 'Most intense band of the author’s intense letter bands from 4.01-4.48 eV. ■^This is the energy of the author's A band (31945 cm-1 ), which is now believed to be near the 1B1 origin. ^Maximum of total fluoresoence emission. Reference 199 used an excitation wavelength of around 3000 A. h Maximum of to tal fluorescence emission. Strongest “resonance” band, which occurs for an exciting wavelength of 3020 A, has an energy of 3.96 eV. 'Also referred to as electron energy loss spectrum (EELS). 3 Reference 138 gives an energy loss range of 3.75-5.3 eV. * Reference 141 studied solid SC> 2. Reference 140 gives an energy loss range (Franck-Condon limits) of 3.9-5.5 eV. mThe authors actually attributed this transition to a 1B] state (based on analysis of the supposed (040)-(000) band at 2 9 4 5 2cm -1 (3.65 eV)), feeling the band system as a whole was too intense to be due to *A 2 «— 'A i . "The authors based their analysis on the supposed (010)-(000) band. °Thc authors based their analysis on the (031)-(000) band and extrapolated back to get T0. These are the presently accepted values for the geometry and To. p Based on fluorescence lifetime and bimolecular quenching rate data showing two distinct regions. ?The midpoint ° f the range, 3100-3160 A in which the authors say the origin occurs. rThis is the presently accepted value for To. 3 Vibrational analysis of the fluorescence spectrum. 69 studies showed relatively long progressions in both vx and v2, indicating that both the bond distance and angle of the excited state are quite different than those of ground state. Very early investigations of the emission spectra showed anomalously long ra diative lifetimes.196-199,223,224 Several of these investigations also place the vertical emission at between 3.44 and 3.65 eV (maximum of total fluorescence emission). Douglas 223 was the first to explain this as being due to the emitting state being perturbed by interelectronic level mixing with the vibrational and/or rotational levels of another state (either the ground state or a lower-lying excited state). Mettee197 attributed the interaction as being due to the lower lying a 3Bx state, since absorption in this region not only caused fluorescence but also phosphores cence. Strickler and Howell 199 supported this conclusion in part, but said that the main interaction was with higher vibrational levels of the ground state. They also attribute the change in the absorption band, from a relatively uniform progression to a complex series of peaks at about 2800 A, to possible strong interactions with the expected XA 2 state. None of these studies actually confirmed the symmetry of the upper state, but most believed it to be a xBx. In fact Gardner 2 2 5 used this fact to explain the fluorescence lifetime behavior as being due to a Renner-Teller effect involving the coupling of the IB 1 and xAx states, which correlate with the xAa state of the linear molecule. The results of these studies seemed to be supported by the early photochemical kinetic studies of several investigators .159-164,170’171,181’190,200’201’205,226 Sidebottom et al.226 also observed a non-exponential decay in the fluorescence spectrum for exciting wavelengths below 2860 A, and attributed this, at least in part, to the 1 B1 crossing over to a non-emitting near-lying excited state (possibly the ^ 2 ). The singlet decay was exponential for incident wavelengths above 2980 A and was at tributed to low-lying vibrational levels of 1 B1. Heicklen and coworkers 159-162,170,171 also interpreted their results in terms of two reactive emitting states ( 1 B 1 and and several reactive non-emitting states (!A2, 3A2, and 3 B2), with the singlets being strongly coupled. The results of Calvert and coworkers 1 6 3 ,1 6 4 ’181,183 confirmed the observations of Heicklen and coworkers, but they favored only one reactive singlet and triplet state, with involvement of some undefined intermediate excited state (possibly the \A2) or unstable isomer as a new source of phosphorescent triplet at high pressures. Again, none of these studies could conclusively determine the nature of the excited singlet state(s) (or triplet states). Other studies, mostly optical, were also taking place at this time in hopes of elucidating the exact nature of the singlet excited state .19,20’217-220’227-229 In experi ments dealing with the fluorescence lifetime in this region, Brus and McDonald 19,20 found biexponential decay and unexpectedly long lifetimes (that of the long-lived component being 2-12 times longer than the short-lived component). They con cluded that the long-lived component, which is strongly observed in excitation to 71 the blue of 3220 A, is a very nearly linear zeroth-order Born-Oppenheimer (XAS) state, which is strongly coupled to the \Aj (XAS) ground state through a Renner-Teller interaction. This was the same conclusion reached earlier by G ard ner . 225 The short-lived component, mainly observed by excitation to the r e d of 3220 A, is attributed to moderately perturbed bi levels of the XA 2 state with in tensity borrowed from the B xBi <— X xAx transition. Calvert 2 0 6 was also able to reinterpret some of the earlier data from his group in terms of short- and long-lived components in the fluorescence decay. Hardwick and Eberhardt 2 2 8 interpret the results of their magnetic rotation experiments and those of Kusch and Loomis , 2 2 7 in the region 2940-3170 A, in terms of Renner-Teller coupling between the xAx ground state and a nearly linear xBi ( 1 Ag) state with a bond distance close to that for the Wj. They could also not entirely rule out mixing of the xBi with a triplet state that would have a similar geometry and lie at a lower energy. Brand and Nanes ,217 however, suggested th a t the upper state was a xBi with a nonlinear geometry, but a significantly larger bond angle than the ground state. They also suggested that the band origin (!T0) Gf this system is 28238 cm - 1 (3.50 eV), but is obscured by the triplet-singlet transition. The early single vibronic level (SVL) fluorescence spectra studies of Shaw and coworkers 2 3 0 seemed to also suggest an excited state with a quasi-linear structure. Caton and Gangadharan 211 combined their results with those of Sidebottom a/ . 226 and suggested that the region of maximum interaction between the singlet 72 and triplet manifolds, or betw<.(.„ these two manifolds and some other manifold, occurs around 3000 A. This h a p p e n s to be near the maximum intensity in the magnetic rotation spectrum observed by Hardwick a n d Eberhardt. Other investigators218-220.^ fc|t that this transition occurred through a Bi vibronic state of the ^ 2 (a ^(bj) vibrational mode) state, with its intensity being “borrowed” by vibronic interaction w ith a nearby B <— X XAX transition. Dixon and Halle218 rotationally ana|ym j th e bands in t j,c 3 1 50-3400 A region as type c bands and arrived at a value 0f 3.6 e V for r 0. Hamada and Merer219’220 perforrned a very extensive rotational analysis of the bands in the 3000-3400 A regjon T h e y concluded th a t all the analyzable bands (the weak bands in the region 3250—3400 A) are highly perturbed type C bands, which appear by means of a vibronic coupling (Herzberg-Teller mixing) through the ^3 (b2 species) vibrational levels of th e A*A2 electronic state (a “forbidden” transition) to give Ba vibronic states- The geometry f° r this state, R s o = 1-53 A and O oso = 99°, is based on tlle relatively unperturb ed band at 3395 A (031-000 transition). They also placed tllc o r i g i n f°r the ]A2 s ta te at 27930 cm -1 (3.46 eV). The B xBi state appears to lUuv been reduced to a weak underlying continuum due to perturbations caused by stro n g Renner-Teller coupling w ith the higher vibrational levels of the 1A, ground state. Tllere is indirect evidence that the origin for this state lies between 3 1 ^ 0 and 3160 A, which is where the Clements letter bands begin (strong bands between 2750 and 3130 A). Below 3200 A the vibrational bands begin to overlap and a full rotational analysis was impossible due to the perturbations. It is also here (3220 A) that Brus and McDonald 20 observed that the long lived component in the fluorescence spectrum becomes important. The shortest wavelength band in which they were able to make a rotational analysis was at 3001 A (4.13 eV), the Clements “G” band. They also suggested that the anomalous vibrational intensity distribution in the Clements bands was not in accord with a Herzberg-Teller vibronic coupling but was likely due to strong Born-Oppenheimer (nuclear momentum) coupling between the XA 2 and 1Bi states, where they are effectively degenerate. (The rotational constants of these hybrid levels, which are still dominated by the XA 2 levels, correspond to a geometric structure part way between that of the XA 2 and that expected for the 1 Bi, which could result from the mixing of xBi character into the XA 2 levels.) This implies that the higher vibrational levels of the A state are influenced the most by the lowest levels of the B state (i.e., if a crossing occurs it is close to the bottom of the potential well for the xBi but higher up the potential for the xA2). Finally, they also suggest that the magnetic effects exhibited by some of the bands and the perturbations up to 3220 A are, for the most part, due to a transfer of the Renner-Teller coupling from the xBi state to the XA 2 state, via the Born-Oppenheimer coupling between the two excited states. At wavelengths greater than 3220 A the perturbations are caused only by spin-orbit coupling with lower triplet states, or possibly from direct b-axis electronic Coriolis coupling with the ground state. These latter suggestions were supported by the Zeeman studies of Brand et al. in the 3250-2950 A region. They concluded that the nature and strength of the Zeeman and the magnetic rotation effects could only b e accounted for by both Renner-Teller coupling of the three singlet states ( 1A1, xB i, and XA2) and spin-orbit coupling of the A XA 2 state with the higher vibrational levels of the 3 B2 and/or 3Bi states. They also speculate that higher levels of the 3B i state could be quite extensively mixed with the A XA2 ♦- X ‘A, system through the coupling of spin with the orbital angular momentum of the Renner-Teller interaction, which would mix triplet character into the important background levels of the ground state. In the years immediately following the work of these authors, the hypothesis that two coupled excited states exist in this region was confirmed by many in vestigators .221,230'235 Su et al, 2 3 1 studied the kinetics of the fluorescence decay of S0 2, excited in the 2662-3273 A region, and found it to be biexponential, con firming the results of Brus and McDonald .19 They favor the designation of the short- and long-lived states as A XA2 and B XB1? respectively. They also found that when the relatively unperturbed band near 3 2 2 6 A, attributed to the mainly pure A xA 2 *— X xAi transition, was excited they obtained long-lived fluorescence emission, which seemed to support xBi as the em itter. This led them to believe that the two states are coupled and there is some sort of conversion between the two, i.e., they do not decay independently. Also, from their data they suggested that the origin of the B 1 B 1 «— XxAj transition is at a wavelength longer than 3273 A (energy < 3.79 eV). These results are supported by the work of Rudolph and Strickler 2 3 6 who also found biexponential decay. They suggested that the 1 B1 and XA 2 states might be so thoroughly mixed that the absorption should be rein terpreted in terms of two new states composed of the xBi and XA 2 states. Simons and coworkers 1 8 5 ,2 1 2 studied the intersystem crossing in S02, which showed that virtually all the fluorescence is associated with radiative decay from 1 Bi, even though the excitation is mainly into the XA 2 state. Their results also show that the net triplet yield passed through a second broad maximum, centered at ap proximately 3100 A, which parallels the onset of the strong Zeeman effect 2 2 9 and the development of a magnetic rotation spectrum 227,228 in the A XA2 <— X xAi band system. They suggest that intersystem crossing contributes to the magnetic be havior. These results supported those of Caton and Gangadharan 211 and imply the existence of more than one singlet state in this region. Glinski et a/ . 193 studied the chemiluminescence resulting from the reaction of ozone with methyl mercaptan. At a pressure of about 0.015 torr, they found that most of the fluorescence was from the *A 2 state (with a lifetime of ~ 2 0 fis) and that as the fluorescence yield passed a maximum the phosphorescence began to appear. They concluded that the xA 2 state is responsible for the production of the phosphorescent triplet state through intersystem crossing. Shaw and coworkers 221,230,232,233 performed extensive single vibronic level (SVL) fluorescence studies in region 2940-3280 A. The Franck-Condon emission patterns suggested that the OSO angle changes very little and the bond length shows only a moderate change in going from the ground state to the excited state. However, the rotational analysis of the absorption spectra by Hamada and Merer 219*22 0 gave an excited stated with a large change (ss 20°) in the OSO angle and a medium change in the S-0 bond length from the ground state. This result, along with those from previous emission studies, indicated that the bulk of the emission is from an excited state with a geometry different than that found for the excited state from rotational analysis, thus indicating that two excited states exist in this region. They interpreted their results in terms of the levels of an unperturbed 1Bi electronic state (with T 0 « 31240 cm-1, i/x « 690 cm-1, and v2 « 460 cm-1) that are very strongly vibronically coupled to many more XA 2 levels (with a lower To, i/i ~ 800 cm-1, and v2 ~ 300 cm-1). Except at the lowest energies, there are no pure xBi and XA 2 states, but rather large admixtures of the zero-order states. The bulk of the emission is what would be expected from the unperturbed xBi levels spread among XA 2 levels, whereas the absorption is more characteristic of a A XA2 <— X xAi transition. The also suggested that the presence of i / 3 progressions in emission are a result of Coriolis coupling between the rotational levels of the zero-order XA 2 and XBX states, involving a b 2 rotation (about the c axis). This 77 would make it possible for absorption to take place to levels with A 2 vibronic character that can then emit to B 2 levels of the ground state. Other schemes have also been suggested to explain the absorption and emis sion spectra. Henneker et al.234 interpreted the anomalously long bending-mode progression observed in the A XA 2 «— X xAi transition (longer than what would be expected for a change of 20°) as a non-Condon effect due to a larger change in the OSO angle between the coupled A and B states than between the X and A states. This implies that the */2 potentials in A and B are not parallel but shifted by an appreciable amount relative to each other (the adiabatic potentials will tend to cross along the z / 2 coordinate). Therefore, they cannot be taken as harmonic, but will show avoided crossings. Several investigators 235,237,238 used infrared multiphoton excitation (IRMPE) to study the fluorescence of S 0 2 in this region. They concluded that the emission is made possible through inverse electronic relaxation (IRE), in which excited vi bronic states are populated by nonadiabatic coupling with high-lying vibrational levels of the ground state. Again, this shows the interaction of the XA 2 and xBx states with the JAi ground state. Ultraviolet multiphoton laser excitation experiments 1 6 6 -1 6 8 at 2 4 8 0 A, leading to photofragmentation, have been performed on sulfur dioxide. They have shown, among other things, that absorption of one photon at this wavelength is predomi nantly into the !Bi state. The maximum of the emission envelope occurs between 3100 A and 3200 A and the fluorescence lifetime is anomalously long (ss 40//s). This supports the previous observations of other studies on the fluorescence life times.19-20'196’197,199’224’231 An important method, extensively used since about 1980, is single rotational (rovibronic) level (SRL or SRVL) fluorescence, usually of rotationally cooled S02. In these studies, single rotational levels of the upper state are excited using lasers with very narrow bandwidths (< 0 . 2 cm-1) and then the rotationally-resolved flu orescence emission is observed. Since the conditions are usually such that there are few collisions, the fluorescence emission hopefully occurs from these same ro tational levels to various rovibronic levels of the ground state. This has been used to study supposedly pure (unperturbed) levels of the XA 2 state, look for pure levels of the xBi state, study fluorescence lifetimes, and discern which types of pertur bations are present by combining the technique with Zeeman studies. The most extensive band studied in this region, using this method, has been the Clements “E” band near 3043 A (specifically the transition at 32813.20 cm-1) because it is believed to contain many relatively unperturbed levels of the XA 2 state. The hope is of course that this technique can give a better understanding of the nature of the interactions occurring in this region of the spectrum. Several groups have done extensive studies using this method . 2 3 9 -2 5 3 Lee and coworkers 2 3 9 -2 4 6 have used this method to study the fluorescence life times of single rotational levels in the 3000-3400 A region and the mechanisms of collision-induced electronic and rotational relaxation. They studied samples at room temperature (~ 296 K) and cooled to about 1 K and used a laser with a bandwidth of ~ 0.2 cm-1. Their early studies centered around the supposedly unperturbed levels of the “E” band. In particular they used the pP 7 (7 ) transi tion at 32813.20 cm - 1 to selectively excite the K' = 6 , J ' = 6 level. The most intense vibronic emission band terminated on the ( 1 ,0 ,0 ) vibrational level of the ground state. It was found that the PP?(7) fluorescence emission line exhibited single exponential decay with a “collision-free” zero-pressure lifetime of « 13.4 (is. This was much shorter than that of the short-lived component found earlier by Brus and McDonald 19,20 and Su et a/ . , 23 1 but still longer than their estimated value of 3 (is for the A 1A2 —* X:Ax transition. They also found that the entire unresolved emission profile gave a double exponential decay consistent with the previous studies . 1 9 ,2 0 ’231 They then extended this work to cover 41 SRL’s in 7 different vibronic levels from 3274 A (30552 cm-1), the (141) *A 2 vibration band, to 3001 A (33331 cm-1), the “G” band. Overall, their SRL spectra were found to be very similar to those of Shaw et al.221 The ratio of continuum to discrete emission was observed to increase with increasing excitation energy and pressure. They also found that the zero-pressure lifetimes showed a systematic decrease with increasing excitation energy, decreasing from « 50 (is for the “unperturbed” (141) vibrational level to ps 15 (is for the “E” band (those for the “G” band increased slightly to ~ 20 (is). What was more interesting was that there seemed to be two lifetime regions, with the lifetimes between 30552 cm 1 and 31235 cm 1 (3274- 3210 A) being 2-3 times longer than those between 32187 cm - 1 and 33517 cm - 1 (3107-3001 A). This dividing line falls close to the electronic origin of the 1 B1 state (31240 cm"1) determined by Shaw et al. 221 They interpret this in the follow ing way. There are two types of xBi vibronic levels, those that are relatively free of xAi contamination (very short lifetimes of ~ 1 fis) and those that are strongly mixed with the xAi by Renner-Teller coupling. Above the 1 BJ origin there should be greater 1A2- 1 Bi mixing, thus imparting greater xBi character to the XA 2 state. Since they feel that the XA 2 levels being excited are mixed with non-Renner-Teller coupled xBi states, the “pure” xBi character results in shorter lifetimes. Also, the geometry difference between XA2 and xBi would result in stronger coupling between the two states at higher energies and therefore greater xBi character and shorter lifetimes. Overall, the spectrum can therefore be interpreted in terms of a mixed vibronic state involving the zero-order XA 2 state and xBi state perturbed to varying degrees by the Renner-Teller interaction with the xAj ground state. One last thing that should be mentioned is that these workers also noticed a Zeeman effect in some of the SRL’s of the “E” band . 243,245 They interpret this, along with their lifetime data, to favor spin-orbit interaction between XA 2 and high vibrational levels of a low-lying 3 B2 state, since this would affect all XA 2 levels and not just those involving xBi Renner-Teller perturbed levels. This coupling would at least help to explain the anomalous lifetime of the “unperturbed” XA 2 levels. Watanabe and coworkers 2 4 7 -2 5 2 also used this method to study SO 2 rotationally cooled to between 5 and 10 K in a supersonic jet. They studied SRL’s in the same regions looked at by Lee and coworker s . 2 3 9 -2 4 6 The main differences in their meth ods is that Watanabe and coworkers used lower source-gas pressures (100-700 torr compared to 500-5300 torr) and a narrower laser bandwidth (0.02 cm - 1 compared to 0.2 cm-1). Under these conditions they obtained biexponential decay from all rovibronic levels having a short lifetime of 3-5 fis and a long one of 15-30 fis, which did not agree with the results of Lee and coworkers . 2 3 9 -2 4 6 However, their short lifetime did agree with the lifetime of ~ 3 fis calculated for a pure A 1A2 —*• X4Ai transition. Lee and coworkers suggested that the biexponential decay and the low values for the lifetimes were due to collisions under the conditions employed by Watanabe and coworkers. Watanabe and coworkers claimed that the behavior observed by Lee and coworkers was due to the fact that the bandwidth used by the latter could not resolve each rotational line. Therefore, the appearance of the fast-decay component may be interpreted as an indication of more contributions of the fluorescence from particular SRVL’s in which the short-lived component pre dominates. From their own data, Watanabe and coworkers concluded that when a single 4A2 rovibronic level which is vibronically coupled with the zero-order 1 B 1 state is excited, the decay is biexponential because the initially prepared *A 2 level effectively couples with a small number of levels of the ground and/or triplet states. The degree of coupling is controlled by accidental approach of the levels. When, 82 by chance, the coupling is very weak, the fluorescence decays single-exponentially with an intrinsic *A 2 lifetime. The lifetimes can increase slightly f°r levels of the xA 2 state with a weaker vibronic coupling to th e 1Bj state. They also state th a t this may be why Lee and coworkers obtained a lifetime of « 13 fis. Watanabe and coworkers 2 4 9 -2 5 2 also performed Zeeman quantum beat studies on SRL’s in several vibronic bands. They found that almost all rotational levels of upper vi bronic states of various bands (D,F,L, and N) possess sizable m agnetic moments, while only a limited number of rovibronic lines of the less perturbed bands (E and G) possess a magnetic moment and show beating fluorescence decays. They also found that biexponential decay and longer lifetimes were associated with larger g values, with nearly all the rotational levels of th e “D” (32624.93-32632.29 cm-1) and “F” (33106.34-33109.69 cm-1) bands showing this behavior. They concluded that spin-orbit coupling is the dominant intramolecular mechanism responsible for these effects (especially in the J' = 1, K' = 0 level of the “E ” band, where a Renner-Teller perturbation is not expected). Suzuki et al.253 used SRVL fluorescence spectra to study th e appearance of odd quanta of the ground state antisymmetric stretching mode (^ 3 ), which is a forbidden transition in C2u symmetry. They concluded that this could be due to two types of Coriolis coupling. The first possibility is vibrational Coriolis coupling in the ^ state (between the ai and b 2 vibrations) made possible by rotation about the c-axis (b 2 x-axis). The second possibility is electronic-rotational Coriolis 83 coupling of the 1 B1 and *A 2 electronic states, again made possible by rotation about the c-axis. Unfortunately, based on their d ata, they could not discern which of the two couplings actually takes place. Another very promising method, sub-Doppler laser spectroscopy, has been re cently developed. In this method the absorption and fluorescence spectra of rota- tionally cooled (Tro, < 1()K) molecules can be measured w ith a spectral resolution of less than 0 . 0 0 1 cm-1. T his allows the study of transitions between single rota tional levels. In areas of a spectrum which are due to transitions to more than one excited state, the levels due to each state can be picked out. Of course, since there are now hundreds or thousands of individual lines, this can be a very tedious task. Demtroder and coworkers 2 5 4 " 2 5 8 have used this method to study the absorption and fluorescence spectra in the 30300-33000 cm - 1 (3030-330 0 A) region. They have also used the method in combination with magnetic fields to study the Zeeman effect on individual rovibronic levels. Their first observation was that rather than being an apparent continuum, the absorption or fluorescence consists of hundreds of lines at an average density of 300 lines/cm-1. They found many new vibronic bands and found that many bands, especially the Clements letter bands, actually consist of several vibrational levels (more than one vibronic band). They were also able to fit most of the intense lines in the “E” and 3226 A bands, as well as many lines in a few other bands, and redetermined the rotational constants. Their 84 results agree very well with those of Hamada and Merer . 2 1 9 ’2 20 However, they had not as yet attributed any of the bands to pure 1 B1 levels. They found that the bands above the origin of the xBi state (31240 cm-1), especially the Clements letter bands, show many, sometimes large, perturbations. For many of the bands, they have proposed a multilevel vibronic coupling scheme, where many asymmetric vibrational levels of the XA 2 state are coupled to only a few symmetric vibrational levels of the 1Bi state. In particular, they suggest three ai vibrational levels of the B xBj state are coupled to fourteen b2 vibrational levels of the A xA 2 state to give seventeen (B xBj ai + A XA 2 b2) “hybrid” levels of Bi vibronic symmetry. Using this scheme they could explain many of their observed band intensities. Also, since there are many more XA 2 levels, it explains why the rotational constants are closer to those of the XA 2 state than to those of the xBi state. These hybrid levels can then also be perturbed by Renner-Teller coupling or by spin-orbit coupling with triplets. Unfortunately, their model didn’t work for all the bands (particularly the Clements “A” bands). They suggested that other perturbations, due to singlet-triplet mixing or Fermi-resonances between vibrational levels of the XA 2 state, are present in these cases. Another drawback to their model is that it incorrectly describes the isotope shifts of the vibronic origins. For the bands below the XB! origin, the 3226 A band, large perturbations due to direct interaction with xBj levels can be excluded. They suggested therefore 85 that local perturbations could not be due to Renner-Teller coupling but should instead be due to singlet-triplet interactions. From their Zeeman studies, they concluded that for “pure” JA 2 states (3226 A band) the main mechanism which causes the magnetic moment and Zeeman split tings, and the perturbations when there is no field, is spin-orbit coupling between the singlet state and several vibronic triplet states. They also concluded that for higher vibronic singlet levels (hybrid states in the E band) this is the dominant perturbation mechanism, with Renner-Teller coupling or Coriolis coupling of the 1 B1 ai vibronic part of the hybrid state with high-lying levels of the X 4Ai state playing only a small role. Finally, from their fluorescence lifetime studies on selectively excited rovibronic levels, they found single exponential decays. The lifetimes of nearly “unperturbed” levels decrease from « 30 /as in the 3226 A band to w 8 /is in the E band. These results and their interpretation by Demtroder and coworkers agree with those of Lee and coworkers .2 3 9 -2 4 6 For some of the perturbed levels they suggested several perturbation schemes. Some perturbations are caused by other radiating (B ^j ai + A *A 2 b2) hybrid levels. They also suggested that radiating hybrid states of the type (B 1 B 1 b 2 + A lA2 ax) interact with the (B ^ ax -f A 4A2 b2) hybrid states via a “vibrational” Coriolis coupling, as suggested by Suzuki et al.253 Finally, they suggested that there are differing degrees of perturbations due to couplings with 86 non-radiating levels, producing a variable lengthening of lifetimes depending on the coupling strength. In conclusion, the experimental results indicate that this region of the spec trum is due to transitions from the X xAi ground electronic state to a complicated interacting system of excited states composed, to varying degrees, of the XA 2 and xBi electronic states. The vertical absorption energy from optical experiments is about 4.2 eV. This is supported by the vertical transition energies of between 4.25 and 4.5 eV obtained from electron impact experiments . 1 3 6 -1 4 2 The low-energy part of this region is due to a relatively pure AxA 2 state, with an origin, T0, of approximately 3.46 eV (27930 cm-1). The best geometry for this state is that of Hamada and Merer 219 (RSo = 1.53 A and 90so = 99°). A rough estimate of Te can be obtained from the zero-point energies (G ooo’s) for the X xAj and A XA 2 states. For the X xAi state Gooo = 1530 cm- 1 (see Section 3.4.1). The estimated zero-point energy of the XA 2 state, using only the three harmonic frequencies (coi = 788 cm-1, uj2 = 306 cm-1, and w3 = 607 Cm-1 ),217,219,220,238 is 851 cm-1. Using these zero-point energies, one obtains a T e Ri 3.55 eV (28609 cm-1). The BXB! state has its origin ( T0) around 3.87 eV (31240 cm-1). The estimated zero-order frequencies for this state are 690, 460, and 900 cm - 1 for u >l5 u;2, and u>3, respectively (the latter value taken from the 3 B! state ) .221,234,238 These give an estimated zero-point energy, G 0 0 o> of 1025 cm - 1 and a Te of approximately 3.94 eV (31745 cm-1). No experimental geometry has been determined for this 87 state, although from the emission spectra it is presumed to have a geometry much closer to that of the ground state than is that of the *A 2 state. Above the 1Bi origin there are (B *Bi ax + A *A 2 b2) and possibly (B XBX b 2 -f AxA 2 ax) hybrid states (the latter hybrids perturbing the previous ones) consisting, at least at first, of only a few *BX levels coupled with several xA 2 levels. The degree of mixing of the 1 BX state increases with increasing energy- Presumably as one goes to the highest energies of this region the excited state(s) should be predominately 1 BX, although this has not been experimentally confirmed. There are also several types of interactions with other states which cause per turbations of the levels of the excited singlet states. Below 31240 cm - 1 (“pure” A *A 2 state) the perturbations are mainly due to spin-orbit coupling of the *A 2 with one or more triplet states. This interaction is also expected to increase with increasing energy. Above 31240 cm - 1 (especially the Clements letter bands) a Renner-Teller coupling of the B *BX and X *AX states causes perturbations. The perturbations are so great for the !BX state that its absorption appears as a contin uum in low resolution spectra. The A *A 2 <— X *AX transition borrows its intensity through coupling of the lA 2 b 2 vibrational states with the 1 BX ax vibrational levels and also undergoes a Renner-Teller perturbation due to this coupling. The very long fluorescence lifetimes are probably due to this Renner-Teller coupling of the B JBX and X *AX states. This perturbation also increases with increasing energy (past the “hybrid” states responsible for the Clements letter bands there is very 88 little discernible vibrational structure in the absorption spectrum). There are also smaller perturbations caused by various types of Coriolis couplings. One such in teraction is probably responsible for transitions in emission to 1/3 levels of the X xAi ground state. It appears that there is still much experimental work that needs to be done. The origins and geometries of the two states, especially 1 Bi, need to be more accurately determined. The same can be said for the regions where the A XA 2 and B xBi states first begin to interact with each other and with the other states. With the advent of the latest techniques for studying individual rovibronic levels in transitions, there is hope that this can be done soon. 3.5.2 Theoretical Investigations Most of the theoretical work done on these states 2 2 ,2 3 ,7 9 ,8 1 ,8 2 ’1 4 3 ’157 was done by investigators who also performed calculations on the xAx ground state and XB 2 excited state. For a major discussion of the details of these calculations the reader is referred to Sections 3.3.2 and 3.4.2. The results of calculations on the XA 2 and xBi excited states can be found in Tables 3.10 and 3.11, respectively. For the XA 2 state all the geometries agree fairly well with the experimental geometry given by Hamada and Merer . 2 1 9 The bond length calculated by Lindley 22 is the closest to the experimental value. However, his value for the OSO bond angle is much smaller than the experimental value and even the values from the other 89 Table 3.10. Calculated geometries and excitation energies for the *A 2 excited state . 0,6 Geometryc AE (eV)d Method Ref. R so (^) Ooso (cleg) Vertical Minimum ( Te) 1.587 (0.083) 100.5 (-20.5) 4.0 SCFe 82f 1.585 (0.081) 100.3 (-20.7) 3.3 S-CF 82* 4.766/l S-CI*-J 143 1 . 6 8 (0.09) 103 (-16.5) 2.61 1.78 INDO/S-CI* 79 5.0 DFC-LSD 81 1.503 (0.099) 92.56 (-26.24) 4.605' 3.451m SCF” 2 2 4.985° SCF” 2 2 1.503 92.56 (1.501)p SCF 23 "See Tables 3.4 and 3.3 for geometries and energies of the 'A; ground state. ’’Unless otherwise noted the basis sets are the same as those used for the lAi ground state reported in Section 3.3.2. 'Quantities in parentheses represent the change compared to the ground state. dUnless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. 'Virtual orbital approximation applied to the ground-state wavefunction. f Using STO-3G* (‘split’ d) basis. 9 All single excitations from the seven valence MO’s to all virtual MO’s. hGeometry for lAi not given. Assumed to be experimental geometry, although the energy is incorrectly reported as —654.78725 a.u. 'Using a 4-31G basis set. J All single excitations from the three highest occupied MO’s to the three lowest-lying virtual M O ’s. ^Single excitations from doubly-occupied orbitals to the two singly-occupied orbitals in the SCF description (5b23b}). 'Calculated at the ’Ai experimental geometry of Kivelson.44 Absolute energy of —547.067728 hartree. mThe absolute energy is —547.112585 hartree. "Using their [7,5,2/5,3,1] contracted basis. "Calculated at the :Ai optimized geometry. Absolute energy of —547.056215 hartree. p Vertical emission energy calculated at the geometry of Lindley, using his basis set. 90 Table 3.11. Calculated geometries and excitation energies for the 1 B1 excited state . 0 ’6 Geometry 0 AE (eV)d Method Ref. Rso (A) Ooso (deg) Vertical Minimum (Te) 1.568 (0.064) 125.2 (4.2) 4.0 SCF® 82/ 1.564 (0.060) 125.0 (4.0) 3.4 S-CI® 82' 3.699* 143 1.63 (0.04) 126 (6.5) 2.13 2.03 INDO/S-CI* 79 1.535 (0.100) 124.5 (5.5) 3.9 3.5 DFC-LSD 81 1.495 (0.090) 117.3 (-1.5) 4.627' 4.470m SCF" 2 2 4.980° SCF" 2 2 1.541 140 CNDO/2-SCP 157 °See Tables 3.4 and 3.3 for geometries and energies of the lAi ground state. ^Unless otherwise noted the basis sets are the same as those used for the ground state reported in Section 3.3.2. 'Quantities in parentheses represent the change compared to the ground state. dUnless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. 'Virtual orbital approximation applied to the ground-state wavefunction. 1 Using STO-3G* (‘split’ d) basis. 5All single excitations from the seven valence MO’s to all virtual MO’s. hGeometry for *Aj not given. Assumed to be experimental geometry, although the energy is incorrectly reported as —654.78725 a.u. ‘Using a 4-31G basis set. ■' All single excitations from the three highest occupied MO’s to the three lowest virtual MO’s. ^Single excitations from doubly-occupied orbitals to the two singly-occupied virtual orbitals in the SCF description (8a}3b}). Calculated at the 'Ai experimental geometry of Kivelson.44 Absolute energy of —547.066928 hartree. ’"The absolute energy is —547.075142 hartree. "Using their [7,5,2/5,3,1] contracted basis. "Calculated at the JAi optimized geometry. Absolute energy of —547.056409 hartree. PA maximum of 25 configurations are used. 91 calculations. Since the experimental geometry was obtained by using the rotational constants determined for the (031) vibrational level, it is possible that the angle for the ( 0 0 0 ) vibrational level will be less than that estimated by Hamada and Merer.219 If so, this would improve the agreement between the experimental bond angle and that calculated by Lindley . 22 More importantly, the calculated changes in the geometry on going from the to \A 2 state agree very well with those predicted by Hamada and Merer 219 and qualitatively agree with the experimental Franck-Condon absorption patterns. The Te values from the ab initio calculations agree fairly well with each other and with the experimental Te of 3.55 eV, estimated in Section 3-5.1, both the C l value of Hillier and Saunders 8 2 and the SCF value of Lindley 2 2 being somewhat lower than the experimental value. The ab initio vertical absorption energies are all higher than the experimental values. The agreement between the different cal culations is not as good in this case as it was for Te (at least for the ab initio calculations) and seems to depend to a great degree on the geometry at which the Albert's were determined. Looking at the SCF calculation of Lindley ,22 one finds a difference of 0.38 eV between the AFuert’s determined at the experimental and optimized equilibrium ground state geometries (with that at the experimental geometry being lower). Also, a vertical emission energy of 1.501 eV was calculated by Zellmer 2 3 at the optimized XA 2 equilibrium geometry Qf Lindley . 22 This value is much lower than that determined experimentally. F in ally, it should be men 92 tioned that the primary configuration in the Cl wavefunctions determined by both Hillier and Saunders 82 and Bendazzoli and Palmieri 143 is the SCF configuration ([core]:5b23bJ). For the xBi state there is no experimentally determined geometry to compare with the calculated values. The variation in the geometries between the various calculations is about the same as it was for the 1A2 state. The bond distance is slightly shorter than that of the lA 2 state and the bond angle is about 25° larger than in the XA 2 state (when comparing similar calculations). Again, the bond distance and angle from the SCF calculation of Lindley 22 are less than for the other calculations. Also, this calculation is the only one to give a bond angle which is smaller than that in the xAj ground state. The other calculations give a bond angle close to the experimental and theoretical values for the a 3Bj excited state (see Section 3.6). The main thing to notice is that the change (from the ground state) in bond distance is not as great as for the XA 2 state and that the change in bond angle is small (i.e., the bond angle for the 1 B 1 is nearly the same as for the xAi ground state). This is what was deduced from the experimental Franck-Condon emission patterns (a moderate progression in t/\, slightly shorter than for the XA2, and a short progression in 1/2 )- The transition energies agree fairly well, except that the values from the semi- empirical calculation of Chung 79 are too low, as they were for the XA2. All the other calculations give Aj Evert > 3.7 eV and Te > 3.4 eV. As for the XA 2 state, 93 this value for Te is lower than the experimental value. Also, the SCF values of Lindley 22 are not as close to the other ab initio calculations as they were for the xA 2 state. This seems a little strange, since the primary configuration for the Cl wavefunctions was again just the SCF configuration ([core]:8a}3b } ) . 82,143 As for the ^ 2 state, Lindley’s 22 SCF calculation shows a difference of « 0.35 eV between the A£„ert’s calculated at the two different geometries. Also, a vertical emission energy (A Eemis) can be approximated for the xBi state from the SCF data found in Appendix D of Lindley’s thesis .22 It contains energies and A£?’s (relative to the energy of X xAi at its SCF optimized geometry) at various geometries. The energy and A E (Te in this case) at the 1 B1 optimized geometry (R s o = 1.495 A and Ooso = 117.3°) are —547.075142 hartree and 4.470 eV, respec tively. At a geometry with Rso = 1.504 A and O oso = 118.0°, the energy and A E are —547.074915 hartree and 4.476 eV, respectively. These values are very close to those at the xBi optimized geometry. For the ground state, at Rso = 1.510 A and Ooso — 118.8°, the energy and A E are —547.210516 hartree and 0.786 eV, respectively. Also, for both of these states, the energies vary very little for small changes in geometry in the region around these points. Using these last two sets of values for A E of the xBi and xAi states the vertical emission energy is estimated to be 3.7 eV, which is very close to the experimental values given in Table 3.9. This AJSemts is very close to Te for the state. This is another indicator that the equilibrium geometries of the xBi and xAi states are very similar. This is similar 94 to what occurs in the transitions to and from the 3Bi state, which has a geometry close to that of the xAj ground state (see Section 3.6). As with the geometries, perhaps the most important thing to look at is how the experimental and calculated Ai^’s for the 1 Bi and XA 2 states compare. The Te values for the XA 2 state, from all calculations, are lower than those for the xBt state. This is what has been experimentally observed. The opposite is generally true for the A Evert. The only exception is Lindley’s 2 2 SCF calculation at the experimental xAi ground state geometry. In this case the vertical transition energy for xBi is higher than th at for XA2. However, the difference between the two at this geometry is only 0.022 eV, the two states being nearly degenerate. As a matter of fact, his values for the A_Euert’s at the optimized equilibrium xAi geometry are, for all intents and purposes, degenerate (difference of only 0.005 eV). This is very interesting since the perturbations in many of the absorption and emission bands have been postulated as being due to interactions between the XA 2 and xBx states, where they are accidentally degenerate (due to a crossing of their potential surfaces). A calculation, not discussed in detail above, by Zellmer ,23 attempted to answer the question about the intersection between the XA 2 and xBi potential energy sur faces. He performed SCF calculations using the [7,5,2/5,3,1] contracted basis set of Lindley and Shavitt . 2 2 His Figures 1 and 2 (pages 65 and 6 6 ), reproduced from the work of Lindley , 22 show the variation of the energy of the seven lowest-lying states of S0 2 with respect to bond length at the optimized bond angle, and with respect to bond angle at the *AX optimized bond length, respectively. The *A 2 and 1 Bi excited states essentially cross at the ^ optimized equilibrium geometry, as discussed above. These figures also show that the crossing occurs very near the optimized bond lengths and angles of the xBi state. His Figures 3 and 4 (pages 67 and 6 8 ) are similar to Figures 1 and 2, except the energies plotted were deter mined using the optimized bond angle and length of the *A 2 and 3A2 states. As seen in Figure 3 (energy versus bond angle at constant bond length) the JA 2 and 1 B1 potential curves cross at around 122.65°, which is again near the minimum of the v -2 potential of the 1 B 1 state. Figure 4 (energy versus bond length at constant bond angle) shows that the 1A 2 and 1 B1 potentials are nearly parallel, at that bond angle. This is w hat was suggested by Henneker et a / . 234 to explain the ob served intensity distributions in the absorption spectrum (non-Condon effect). The minimum C2„ crossing point occurs at R so = 1.487 A and 0Oso = 122.35°, very near the 1 B1 optimized equilibrium geometry. The A E vert at this geometry, using C2„ symmetry restrictions, is 4.495 eV, which is very close to Lindley’s 2 2 Te value of 4.47 eV. He also found that another solution existed if the calculation was performed at this geometry using Cs symmetry restrictions (“symmetry-broken” solution). In C 3 symmetry the ]A 2 and 1 B1 states combine to form a *A" state. The energy for this 1A" state is 1.309 eV lower than the energies of the *A 2 and 1 B1 states. Thus AEvert is 3.18G eV and is less than the Te’s of the *A 2 and 1 Bx states. He also varied the bond lengths (at the minimum C2u crossing angle of 122.35°) using C 3 symmetry and found the absolute minimum in the asymmetric potential ( 1/3 ) curve occurred at approximately R\ = 1.57 A and R2 = 1.40 A. The A E vert at this point is 2.784 eV, which is approximately 0.4 eV lower than the Te of the *A 2 state. This symmetry breaking may be ascribed to an artifact of the SCF treatment. Otherwise, it would seem to imply that one or both of the \A 2 or 1 B1 states should be considered to have Cs symmetry (unsymmetrical equilibrium geometry), in which they could have a lower optimized energy than they have under C2v symmetry. Another interesting point is that the work of Lindley 22 and Zellmer 2 3 shows that the potential surfaces of the lA 2 and 1Bi states cross those of the 3A2, 3 Bi, and 3 B2 states (see Figures 1-4 of Zellmer). The potential curves of the 3Bi and 3 B 2 states cross that of the XA 2 very near its minimum, while the potential curves of the 3A2 and 1Bi cross near the minimum of the IBi state. Spectroscopists have suggested that perturbations in the A *A 2 <— X *Aj absorption spectrum, especially near the origin of the transition, are due to spin-orbit coupling with one or more triplet states. They have also suggested that the phosphorescent a 3Bi state can be populated by intersystem crossing from the B 1 Bi- 97 3.5.3 Summary In summary, both the experimental and theoretical investigations give the same geometry for the XA 2 state (around R s o ~ 1-53 A and Ooso ~ 99°). Also, both show that the XBX state has a geometry with a bond angle close to that of the XAX ground state and a bond length close to that of the XA 2 state. Thus a reasonable approximate geometry for the XBX state is; Rso ^ 1 . 5 A and Ooso & 1 2 0 °. Both theory and experiment show that the Tc (and T0) for th e XA 2 state is lower than that for the XBX state. Also, theoretically calculated values for A Evert for excitation to the xA 2 state are closer to experimental values, while those for emission from the XBX state are closer to the experimental values from the emission spectrum. This supports the spectroscopists’ interpretation that absorption appears to be to the xA 2 state, while emission appears to occur from the XBX state. Also, the SCF calculations of Lindley 22 and Zellmer 2 3 support the spectroscopists’ conclusions that the potential curves of the XA 2 and XBX states cross near the minimum of the xBi state and further up the potential curve of the XA 2 state. Thus, as suggested by Demtroder and coworkers , 254' 2 5 8 there would be several vibronic levels of the xA 2 state coupled to only a few vibronic levels of the xBi state, starting near the origin of the XBX. Finally, the theoretical calculations support the spectroscopists’ views that some of the perturbations in this region of the S0 2 spectrum could be due to interactions of the excited singlet states with excited triplet states. 98 Thus, with some confidence, it can be reasoned that the 2500-3400 A region of the spectrum is due to excitation to two excited states, the XA 2 and 1 B1. These states interact with each other and all the resulting singlet states are perturbed by interactions with the XxAj ground state and the three low-lying excited triplet states, 3A2, 3 Bj, and 3 B2. 3.6 The 3900—3400 A Absorption Region — The a 3Bi, b 3A 2 and 3B 2 Excited States This region of the S0 2 spectrum is very weak and has been attributed to a a 3Bi <— X*Ai transition. It exhibits fairly strong perturbations, but these are not as severe as in the region due to absorption to A XA 2 and B xBi excited states (the first few vibrational levels are unperturbed or only slightly per turbed). A great deal of work has been done to determine the exact nature of the triplet states in this region and how they take part in the reactions of g Q ^ 22,23,64,79,81,82,109,138-144,156,158-165,169-212,216,236,259-289 As mentioned in Section 3.5, the triplet state(s) is very important in the photochemical processes of S02, both as a reactant 109,159' 165,169-188 and a prod uct.172,188-195 Also mentioned in that section is the fact that the triplet state(s) can be populated by intersystem crossing from the XA 2 and/or 1 B1 states.159-165,169-176,179,181,185,190,193,196-212 Many of these photochemical experiments have invoked more than one triplet state as the reactive species to explain the kinetics of the reactions of S0 2 with various reactants and the kinetics of S 0 2 quenching by unreactive gases. While theoretical calcu lations 22 , 23, 64, 79 '8 i . 8 2 , place all three of these triplet states in this region (some actually place the 3 B 2 state as the lowest of the three), only the a 3Bi and possibly the b 3A2 have been detected by experimental means .138-142,216,267-281 The only evidence for either of the other two states are the perturbations which begin in the ( 1 0 0 ) and ( 1 1 0 ) vi brational levels of the 3Bi state. These findings bring up several questions about the nature of the excited states in this region. First, why do the 3A2 and 3 B2 states not appear in the absorption or emission spectra? If 3 B2 is lower than the 3Bi state, as many calculations have predicted, then why does phosphorescence from it not appear in the emission spectrum? Another question is, are the perturbations in this band system due to interactions of the 3Bi state with 3A2 and/or the 3 E$ 2 or are they due to some other type of interaction (i.e., with neighboring singlet states or between its own overlapping vibrational and rotational levels)? The following sections discuss the many studies that have been done in an attempt to answ er these questions. 3.6.1 Experimental Investigations Many experiments 1 3 8 -1 4 2 ,14 4 ,1 5 6 ,1 5 9 - 165,1 70,171,1 8 1 - 1 8 7 ,207-2 1 2 ,216,2 3 6 ,2 5 9 -2 63,266'29i haVe been performed in the hopes of elucidating the nature of the states responsible for this region of the spectrum. The experimentally determined excitation energies and 100 equilibrium geometries for the triplet states, from many of these studies, are given in Table 3.12. T he early studies of the absorption and emission spectra by Gay- don 2 6 7 and Metropolis and Beutler 2 6 8 both placed the (OOO)-(OOO) transition at about 3.19 eV. These spectra also showed that the Franck-Condon maxima (verti cal transitions) occur very near this transition, indicating fairly small differences in the equilibrium geometries of the 4Ai ground state and the excited state. Neither study was able to determine the actual nature (symmetry) of the excited state. A magnetic field study of the absorption spectra by Douglas 269 showed that this region displays a pronounced Zeeman effect and he therefore postulated that the excited state was a triplet. These early studies were followed by several vibrational and rotational studies of the absorption and emission spectra .144,156,216-270-285-290 In their medium-resolution vibrational and rotational analyses of the absorption spectra, Russell et al.270~272 assigned the transition to an excited state with 3Bi symmetry and obtained the geometry shown in Table 3.12. They also found that the transitions to v3 levels (3A2 vibronic symmetry) appear more prominently than would normally be expected. Coon 282 and Mulliken 156 use(j this fact, and the fact that there seemed to be a large positive anharmonicity (# 3 3 ), to postulate that the upper state might have a slightly asymmetrical geometry (C* symmetry). Merer216,274 attempted to answer some of these questions by performing a ro tational analysis of some of the bands using a higher resolution than the previous 101 Table 3.12. Experimentally determined geometries and excitations energies for the triplet excited states. S tate Geometry" AB (eV) M e th o d 6 R ef. B and R so (A) Ooso (deg) Vertical Origin (To)c a, bd (2.98)* 3.192 Vibrational^ 267 3.31 3.196 Vibrational 2 6 8 3.40 ... Electron Impact 9 I 38* 3.45-3.50 Electron Impact 139—142' a 3Bi 3.36 3.19 UV A b s o rp tio n 179,199,291 (2.98)* 3.19 Phosphorescence 190,193,196,197,199,211'' 1.491 (0.060) 124.82 (5.49)- F-C Analysis' 144 1.491 (0.060) 126.1 ( 6.8) 3.35m 3.19 F-C Analysis' 278 1.490 (0.059) 124.95 (5.62)- • - F-C Analysis' 283 1.489 (0.058) 125.97 (6.64)- .. 3.196 Rotational 271,272 1.494 (0.063) 126.08 (6.75)-•• 3.1947 Rotational 216 1.4926 (0.062) 126.22 (6.89)-•• 3.1946 R o ta tio n a l 275,276 1.4930 (0.062) 126.23 (6.90). •• 3.1946 Rotational" 279 1.472 (0.042)° 1 26.7(7.7) 3.27p Vibrational 273 b 3A2 ...... 3.31-3.35 Rotational 9 277 ... 3.26r Phosphorescence 289 1.55 (0.12) 97 (-22.33) 3.23 Rotational* 281,292 “Q uantities in parentheses represent the change compared to X ([is o = 1.431 A and 6qs o = 119.33°). 6The vibrational and rotational analyses are of the ultraviolet (UV) spectra, unless otherwise noted. "Origin of the (OOO)-(OOO) transition. See Section 3.5.1 for zero-point energy corrections and m inim um excitation energy, Te- dThe excited state was not known for sure or the transition may be to more than one singlet state. "Maximum of the emission spectrum. Range of maximum absorption (Franck-Condon limits) of 2-92-3.13 eV. J Vibrational analysis of the phosphorescence spectrum. s Also referred to as electron energy loss spectrum (EELS). h Reference 138 gives an energy loss range of 3.05-3.75 eV. ‘Reference 141 studied solid SO 2. J Reference 140 gives an energy loss range (Franck-Condon limits) cf 3.1 -3 .9 eV . * Reference 196 studied microcrystalline solid SO 2 a t 77 K. 1A Franck-Condon analysis of the intensities of the vibrational bands is used to determine the geometry. "*Tlie maximum of phosphorescence is 2.99 eV. " A n a l y z e d partially rotationally cooled S0 2 (-78 °C ). “Geometry difference is computed using the X ‘At geometry in solid S0 2 ( R s o = 1-430 A and O oso = 119°). p A nalyzed solid S 0 2 (4.2K). To for the emission occurred 80 ctp-t to the red of the absorption (3.26 eV) — see te x t. 1 The To was inferred from perturbations in the rotational structyre Qf t||e (lOO) and (110) vibrational bands of the a3Bi X 'At. "Analyzed solid S 0 2 (4.2l<). To for the excitation occurred 83 cm - l t o th e blue of the emission (3.26 eV). A ttributed the excitation at (3.27 eV) to th e a 3B j s ta te an d the emission to the b 3A2 state. * Rotational analysis of the perturbing levels of the a 3Bi X ]A, transition. 102 investigators. He found the bands to be of perpendicular type and assigned the upper state to a 3 B 1 with an origin at 25766.88 cm - 1 (3.1947 eV). The geometry he obtained (R s o = 1-494 A and Ooso = 126.08°) is similar to that from earlier analyses .144’270-272 As in the study of Russell et a / . , 2 7 0 -2 7 2 he observed A 2 vibronic bands (due to the antisymmetric stretching frequency mode), a doubling of some sub-bands, and a subsidiary intensity maximum, approximately 2 0 cm- 1 to the violet of the main maximum. He proposed that the appearance of ^ 3 (3A2 vibronic levels) is due to spin-orbit mixing of the 3A2 levels with higher singlet levels, allow ing the ( 2 0 1 ) level to appear nearly as strong as some of the 3 BX levels. Since the 3A 2 bands also appear as perpendicular bands, they may be mixed with a 4 B2 state, while the 3 BX levels are mixed with the JAX ground state. Merer also observed two types of fairly strong perturbations in the rotational levels of the (HO)-(OOO) band. He proposed that these are Coriolis perturbations of type A K = 0 and A K = ± 1 . He states that the first could be caused by an interaction of a Bx vibronic level with an Ax vibronic level due to an antisymmetric level of a 3 B2 state lying under the (110) level. This state could also account for some unexplained perturbations in higher vibrational levels. The second type of perturbation (occurring in two different rotational levels, K' ~ 3 and 1 2 ) could be due to interaction with A ,2 or B 2 vibronic states. He states that if the same state causes both perturbations then it would have an origin 1272 cm " 1 (0.158 eV) above the (000) level of the 3 BX state (a T0 » 3.35 eV). 103 Brand et al.275~2 7 7 performed an extensive vibrational and rotational analysis of the first few vibrational bands of this system. Their T0 of 25765.701 cm - 1 (3.1946 eV) and geometry (R so = 1.4926 A and Ooso = 126.22°) are the best values to date for gas phase data. They found that the only two unperturbed bands are the (000)-(000) and (010)-(000) transitions. From the analysis of the 0,0 band they found C-type rotational structure and assigned the system as a 3Bi <— X They also found that the (010), (100), and (110) bands have rotational structure like that of the (000) band. The (100) band is the first level which is perturbed, with the perturbations increasing with increasing energy. They determined that the coupling scheme is close to the Hund’s case (b) limit and that the intensity of the transition is due to spin-orbit coupling of the 3Bi state mainly with XB 2 states and to a lesser degree with xAi states. They explain the perturbations in the following way. The large negative anharmonicity in higher bands and the N- independent perturbations in the ( 1 1 0 ) band are probably due to the interaction of 3Bi with Bt vibronic levels of a 3A2 state. A weak perturbation of the K ' — 17 subband of the (100) band may also be due to this interaction. They state that if this interaction is due to near degeneracy of these 3Bj levels with the ( 0 0 1 ) level of the 3A2 state then the upper limit of this 3A2 vibrational level would fall between « 27024 cm- 1 (the i/0 for the 3Bj (110) band) and the K' = 17 subband of the (100) band (vQ = 26672.0r cm-1). Therefore the energy of the (001) level of the 3A2 state would be rs 27000 cm - 1 (3.35 eV) above the xAi ground state. The N- 104 dependent perturbations in the ( 1 0 0 ) and ( 1 1 0 ) bands are attributed to a 6 -axis rotational-electronic coupling with B 2 vibronic levels of a 3 B2 state (although they could not entirely rule out a c-axis Coriolis interaction between the (110) and (011) bands or a rotational-electronic coupling with A 2 vibronic levels of the 3A 2 state). All of their results point to the existence of at least two other triplet states whose origins lie very close to th a t of the a 3Bi state. Hallin et a/ . 279’280 performed a rotational analysis on cooled S 0 2 (-78 °C) in order to determine the electron-spin fine-structure constants. Their results for T0 and the geometry are shown in Table 3.12 and are the same as those from Brand et al 2 7 5 -2 7 7 Their molecular constants are slightly better than those of th e previous authors, th e only major discrepancy being for the spin-spin interaction parameter (/?). They also state th a t the small perturbations in the low-lying levels of the a3Bx state appear to be spin-orbit perturbations caused by the unseen b 3A2 state, the two states being degenerate to within a few hundred wavenumbers. Hallin et alm have also partially characterized the b 3A2 state. They found that all vibrational levels of the a ^ state, except the ( 0 0 0 ) level, suffer at least one local rotational perturbation. They found similar perturbations occurring at approximately the same energy in the spectra of S l 6 0 2 and S 18 0 2, which they explained in terms of a single perturbing electronic state. This state lies only a few hundred cm - 1 above th e b 3A2 state. They rotationally analyzed some of the perturbations and found they correspond to vibronic 3Bi perturbing levels. In 105 addition, they found that the rotational constants A — B and B for the perturbing state are very similar to those of the AXA 2 state. These results lead them to propose that the perturbing levels are vibrational b 2 levels of the b 3A2 state. There have also been references by several investigators 141,260,261,266,289 to unpublished results and to private communications with Merer concerning the b 3A2 and 3 B2 states. According to these references, from a detailed rovibronic analysis Merer has placed the b 3A2 state at 300 db 100 cm - 1 above the origin of the a 3 Bx state (at sa 3.23 ± 0.01 eV). However he has found no evidence for a 3 B2 state in this region. The results of these absorption studies received support from other types of studies on the absorption or emission spectra .144,278,283,290 Wampler et al.290 used laser-induced phosphorescence to study several low-lying vibrational bands. They found perturbations in the (110)-(000) band from K' = 15 upwards, just as had Merer 21 6 ,2 7 4 and Brand et al277 Several investigators 144,278,283 used a Franck- Condon analysis to obtain the geometry of the 3 BX state. In this analysis, the geometry is varied until a good fit is obtained between the calculated and ob served relative intensities. The results are shown in Table 3.12. The geometry of Vikesland and Strickler, who used both the absorption and emission spectra and allowed for a variation in the transition moment with angle, agrees very well with the geometries from the rotational analyses. They also state that the differences between the observed and calculated relative intensities in the regions 3.37-3.43 eV and 3.49-3.53 eV indicate the presence of other electronic transitions. All of the above studies were on gaseous S 0 2. There have been several studies of SO2 phosphorescence in matrices ( 2 0 K and 77 K ) 2 8 4 -2 8 6 and of solid S0 2.273,287-289 The phosphorescence spectra studies of matrix isolated S0 2 of Phillips et al.285 and Lalo and Vermeil 286 show similar results. In both cases To is shifted to lower energies than in the gas phase (by an average of « 450 cm-1) and the ground state frequencies are also shifted. As in the gas phase, no transitions in the antisymmet- rical stretching-vibration mode (^3) are observed. However, unlike the gas phase, the anharmonicity seems to be small. Also, excitation to both the A and C singlet regions leads to phosphorescence and the intensity is greater for excitation into the first singlet than into the second. The studies of solid S 0 2 all show similar results .273,287,289 Hochstrasser and Marchetti 2 7 3 performed a vibrational analysis of both the absorption and emission and made several interesting observations. The (OOO)-(OOO) transition in both absorption and emission is ~ 600 cm - 1 higher in energy than in the gas phase. In both absorption and emission, there are strong progressions in and i/2, but no evidence is found for the appearance of 1/ 3 . In the absorption spectrum a large anharmonicity in the excited state potential curves for both v[ and u'2 is indicated. There seem to be large perturbations in the bands around 3400 A (near the start of the singlet transitions). Also, the bond lengths for both the a 3 Bx and XlAi states decrease on going from the gas to solid phase (with the greatest change, —0.032 A, occurring for 3 BX). They also made two other very interesting observations. The 107 appearance of the absorption spectra differs between the gas and solid phases, especially in the complex region between 3590 and 3625 A . In the gas phase this region has very definite twin-band structure, while in the solid there is a series of sharp single peaks. They suggest that the difference in the complexities between the gas and solid phases are caused by interactions between two electronic states having different relative energies in the two phases. They also observe that the 0 , 0 band origin in emission is 80 cm - 1 lower in energy than in absorption (26305 and 26385 cm-1, respectively). They attribute this gap to the formation of focal structural deformations of the lattice in the neighborhood of the excited molecule. The results of Tinti’s 2 8 7 analysis of the phosphorescence from solid SO 2 at 4.2 K paralleled those of Hochstrasser and Marchetti. However, at 1.4K, he did observe weak emission involving 1/3 (for excitation at 3131 and 3650 A ). He suggests that the observed emission and the 80 cm - 1 difference in the 0 , 0 bands are due to site defects or “traps”. The most recent study of solid S 0 2 (4.2 K) was done by Snow et al. 2S9 As with the two previous studies, they observe a gap of nz 83 cm - 1 between the phosphores cence excitation and emission origins (26387 and 26304 cm-1, respectively). They also found a very weak peak (150 times weaker than the a3!^ origin) in the excita tion spectrum at exactly 26304 cm-1. They performed site-selection experiments and determined that the gap is not due to defect sites in the crystal. They propose that their results indicate the existence of a new electronic state whose origin in solid SO 2 is « 83 cm- 1 lower in energy than that of the 3 BX state. The 3Bi state is seen in absorption, while the phosphorescence occurs from this “new” triplet. They suggest that this is the 3A2 state, citing several reasons. First, the weak peak seen in absorption would be expected for the spatially forbidden b 3A2 <— X xAi. In the crystal it could borrow its entire transition intensity from the a 3 Bx <— X xAi. This could be why the emission appears as if it is from a 3Bj state rather than 3A2. They also cite the unpublished work of Merer and coworkers, discussed above, which places the 3A2 state just above that of the a 3 Bx state. They reason that the large blue shift in the transition energy of the 3Bi state on going from gas to solid is due to a probable decrease in the dipole moment in 3 BX compared to the ground state. They also suggest that if the dipole moment of the 3A2 state was larger than the ground state (to be expected for the more strongly bent excited state), then this would result in a red shift in energy of 3A2 on going from gas to solid. There fore it would be possible for the ordering of the 3A2 and 3 BX states to be reversed in solid S02. It should be pointed out that the SCF calculations of Lindley 22 do show a decrease in dipole moment for 3 BX and an increase for 3A2 (compared to the XAX ground state). This lends some support to their line of reasoning. If correct, this would be the first direct observation of b 3A2 <— X *Ai. There have also been several non-optical studies performed on S0 2 in this re gion .266,138~142 The most important and interesting of these is the electron-impact or electron energy-loss spectra (EELS ) . 1 3 8 -1 4 2 All of these studies give a maximum transition of « 3.45 eV, in fairly good agreement with the vertical transition en ergies (for absorption) from optical experiments. Also, all studies have found that the transition is spin forbidden and they attribute it to excitation to the a 3Bi state. None of these studies have found any evidence for other triplet states be low 3 eV. Vuskovic and Trajmar 139,140 have found evidence for a triplet state near 4.9 eV (in the energy range of the first singlet states). In addition, they observed that the cross section decreases toward low scattering angles, which is character istic of parity-forbidden transitions. They tentatively rationalize this behavior in terms of an additional transition to a state with A 2 symmetry. The fairly recent work of Abuain et al.142 also confirms the presence of additional excited states in this region. They claim that all three triplet states lie in the 3.1-3.9 eV range. They feel this is indicated for several reasons. First, the maximum energy l°ss occurs between 3.4 and 3.5 eV, which is « 0.1 eV higher than the optical values. Secondly, the EELS band extends to higher energies than does the optical a 3Bi band (3.1-3.9 eV compared to 3.2-3.65 eV). From these facts and the fact that they could not locate any energy loss th at could be attributed to the 3 B2 state, they feel that all three triplets lie in this region of the spectrum. From the results of these last two EELS experiments and the preceding opti cal experiments it seems plausible that at least the a 3Bi and b 3A2 states have been located in this region of the spectrum. The existence of the 3 B2 s ta te is still in question. Many studies have postulated the existence of all three of these close-lying triplet states to explain the results of photophysical and photochem ical experiments . 159- 165’170’171-181- 185'187’207-211'2 1 2 -2 3 6 *259- 2 6 1 ’291 Heicklen and cowork- ers159"162’170’171 have studied the photolysis of SO 2 when irradiated at 3130 A in the presence of foreign gases. They have invoked the participation of three triplet states (one emitting and two non-emitting) to explain the kinetics of the reactions. It should be pointed out, however, that Su and coworkers 184,208’209 have questioned this conclusion. Instead, they claim their data, and that from the previous investi gators, can be fit just as well by a one-reactive-state model incorporating pressure saturation of SC> 2 (3 Bi) physical quenching. This model depends on the idea that 3Bi will be “diluted” by a rovibronic quasicontinuum of 4Ai ground state levels that can provide a pathway for irreversible intersystem crossing (via intramolecular spin-orbit coupling). Strickler and coworkers 236,259-261 have invoked two models as possible explanations for their results from lifetime and quenching studies. Their kinetic model invokes the participation of at least the 3A2 state and possibly the 3 B2 state. Their radiationless transition model proposes that there is an interac tion between individual rovibronic levels of the 3Bj state and nearly isoenergetic high-lying levels of the JAi ground state. They favor the second model but are not able to conclusively choose between the two. However, calculations of intersystem crossing rates by Takahashi 2 6 2 ,2 6 3 have shown that the 3 B, state is not diluted to any appreciable extent by !Ai, so that the above models incorporating this fact I ll may be in error. (He does suggest that the *A 2 and 'Bj states are well diluted by triplet states.) Two of these types of studies seem to be a little more important than the others in implying the existence and the relative energies of these triplet states. Simons and coworkers 185,212 studied the chemical and physical quenching of S0 2 and the S 0 2 photosensitized phosphorescence of biacetyl. They present evidence against 3 B2 being the lowest triplet state. No phosphorescence or singlet-triplet absorption bands other than those associated with 3Bj have been observed. The data are consistent with the direct population of the 3Bj state through a Renner- Teller coupling mechanism in the region 2950-3200 A (first excited singlet region). At shorter wavelengths they observe an “excess” triplet yield and propose that this may be due to additional non-emitting triplet states ( 3A2 and/or 3 B2) being populated. The study suggests that the lowest triplet state energy is greater than 20.000 cm - 1 since it quenches triplet biacetyl (with an energy level at 19,200 cm-1) very inefficiently. (Some theoretical calculations 2 2,82 place the 3 B2 origin below 20.000 cm-1.) Also, direct measurement of the rate of appearance of triplet biacetyl through energy transfer from the triplet states of S 0 2 populated at 2950-3000A gives a rate coefficient equal to that obtained for energy transfer from 3Bi alone. Therefore, the authors postulate that it is unlikely that the 3 B2 state’s energy would lie below the 3Bi level at 25,765 cm In addition, to explain their quenching data, they postulate that the a3Bi state is populated by a cascading mechanism in 112 which the 1 B 1 state is excited and populates the 3 B2 (which lies at a slightly higher energy than the 3 Bj) by intersystem crossing which in turn populates the a 3 BX via a collisional process. The a3Bi state can then phosphoresce or be quenched by collisions to populate the 3A2 state which would lie at a lower energy (which then reacts with another S O 2 molecule to produce SO and S O 3 ) - The other study was by Wampler et al.,207 who invoStigated the tem perature dependence and mechanism of S0 2 (3 Bi) quenching reactions. They obtained acti vation energies of between « 2.4 and 3.2 kcal/mole ( 8 7 4 and 1119 cm-1) for quench ing by various atmospheric gases and fairly chemically unreactive molecules. The energies of the 3 Bi(1 0 0 ) and the 3 Bi(001) vibrational levels at w 906 cm 1 (r/i = 905.5 cm_1and u>i = 990.80 cm - 1 ) 2 6 8 ,2 7 0 and « 940 Cn r ' ( u 3 = 940.30 cm - 1 ) , 270 respectively. They suggest that it is not just a coincidence that the activation energy for quenching is the same as the energy of the (lOO) level which seem s to be perturbed by neighboring triplet states. They propose th a t physical quenching of the 3Bj state occurs when S0 2 molecules are promoted to this region of 3 BX which then undergo intersystem crossing to the potential energy surface of th e 3A2 and/or 3 B 2 states. The new state must be either short-lived and nonradiative (be cause it couples very efficiently with the vibrational levels of the ground state and undergoes internal conversion) or long-lived (so that vibrational relaxation makes it impossible to return to 3 Bj) with a radiative lifetime longer than that of 3 BX and a very low emission intensity. 113 From these photophysical and photochemical studies it seems plausible that there is more than one triplet state present in this energy region. It remains to be seen if further work of this type can help in getting a better idea of what states actually do exist and their relative energy ordering. In conclusion, the a 3Bi state has been positively identified experimentally within this region of the spectrum. Its origin ( T0) lies at 25765.7 cm- 1 (3.1946 eV) and is the lowest experimentally determined excited state (at least in the gas phase). It has an S-0 bond length of 1.4930 A and an OSO bond angle of 126.23°. Of all the excited states whose geometries have been experimentally determined, that of 3Bi is closest to the that of the xAj ground state. (The B xBj state is predicted to have a similar structure.) Using the the zero-point energies of the ground state (Gq 0 0 = 1530 cm-1) and the 3Bj state {G'0 0 0 ~ 1158 cm-1, determined from the three harmonic frequencies given by Russell , 2 7 0 tui = 990.80, ct >2 = 385.15, and u >3 = 940.30 cm-1) the minimum excitation energy (Te) f°r the a 3 B! <— X xAj is approximately 26138 cm- 1 (3.24 eV). The b 3A2 state’s origin (To), in the gas phase, has been tentatively placed just above that of the a 3Bj state at 26066 cm- 1 (3.23 eV ) . 281 It has also been sug gested that in the solid phase it is actually the lowest excited state, with an origin of « 26304 cm- 1 (3.26 eV) compared to the 3Bi state’s origin of « 26387 cm- 1 (3.27 eV) . 289 The approximate geometry for this state is, R so = 1.55A and Ooso _ 9 7 °.28X This is remarkably close to that of the A XA 2 state and the difference in geometries between the a 3Bj and b 3A2 is very similar to that predicted for the corresponding singlet states. Only a rough estimate of Te can be obtained since no experimental zero-point energy or harmonic frequencies have been determined for this state. Since the equilibrium geometries of the *A 2 and 3A2 states are very similar, their harmonic frequencies and zero-point energies will be similar. This is born out by the fact that the harmonic frequencies (for and u>2) of these two states calculated in the present study are very similar (see Section 6.7). Based on this assumption, the zero-point energy of 3A2 is approximately 851 cm - 1 (0.11 eV) (see Section 3.5.1) and A ZPE is « —679 cm- 1 (0.08 eV). This gives an estimated Te of approximately 26745 cm - 1 (3.3 eV). The 3 B2 state still has not been found. Its existence has only been inferred by the perturbations in the ( 1 0 0 ) and ( 1 1 0 ) vibrational levels of the a 3 B! <— X'Aj. The interactions between the states in this region are similar to those found for the A *A 2 and B *Bi states (see Section 3.5.1). except that the perturbations produced are not quite as severe. As discussed above and in Section 3.5.1, the triplet states are spin-orbit coupled to the singlet states (a 3 B1 <— X borrowing its transition intensity from the coupling to the ’B 2 and/or *Aj states and being manifested as strong perturbations in the transitions to and from the A b \ 2 and B *Bi states). Also, the 3Bi state may be involved in a Renner-Teller mecha nism with the X *Aj state. Finally, there are several possible perturbations of the a 3Bi <— X ‘A, transition (starting in the (100) level and perhaps to a small extent 115 in the (010) level), caused by the b 3A2 and/or the 3 B2 state(s). These include: ( 1 ) vibronic coupling with Bi vibronic levels of the b 3A2 state, (2 ) 6 -axis rotational- electronic coupling with B 2 vibronic levels of the 3 B2 state, (3) small spin-orbit perturbations in the low-lying levels of the a 3Bi state caused by the b 3A2 state, and (4) less likely, but still possible, a c-axis Coriolis interaction between the (110) and (Oil) bands or rotational-electronic coupling with A 2 vibronic levels of the b 3A2 state. More experimental work still needs to be done to determine the exact nature and extent of the interactions of the three triplet states. Also, as discussed below, many theoretical calculations predict that the 3 B2 state is the lowest triplet state. If this is true, it remains to be seen why this state can not be seen in either absorption or emission. 3.6.2 Theoretical Investigations The results, from several calculations ,22,23>64>79>81'82 ’143,158 for the three excited triplet states can be found in Tables 3.13-3.15. The details of the calculations can be found in Sections 3.3.2, 3.4.2, and 3.5.2. For the a 3Bi state all the geometries agree fairly well with the experimental geometries shown in Table 3.12. All of the calculations produced bond lengths and angles which are slightly larger than the experimental values, except for the SCF bond length of Lindley 22 which is slightly shorter (as is expected for SCF). The 116 Table 3-13. Calculated geometries and excitation energies for the 3Bi excited state." '6 Geometry0 A E (eV)d Method Ref. "so (A) Ooso (deg) Vertical Minimum ( Te) 1.548 (0.044) 131.9 (10.9) 2.6 SCFe 82' 1.547 (0.043) 129.2 (8.2) 2.1 S-CP 82/ 2.523fe s -c r-j 143 1.61 (0.02) 127 (7.5) 1.70 1.57 INDO/S-CIfc 79 1.529 (0.074) 126.5 (7.5) 3.2 2.8 DFC-LSD 81 1.450 (0.046) 127.0 (8.2) 3.128' 3.099m SCFn 22 3.353° SCF" 22 2.3863p SCF9 64 (2.39)r SCF 158 (2.83)r SD-CI 158 (2.83)r SD(Q)-CI 158 “See Tables 3.4 and 3.3 for geometries and energies of the xAj ground state. ‘Unless otherwise noted the basis sets are the same as those used for the *Ai ground state reported in Section 3.3.2. 'Q uantities in parentheses represent t h e change compared to the ground state. ^Unless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. 'Virtual orbital approximation applied to the ground-state wavefunction. f Using STO-3G* (‘split’ d) basis. sAll single excitations from the the seven valence MO’s to all virtual MO’s. ‘Geometry for X 'Aj not given. Assumed to be experimental geometry, although the energy is incorrectly reported as —654.78725 a.u. ‘Using a 4-31G basis set. J All single excitations from the three highest occupied MO’s to the three lowest virtual MO’s. ‘Single excitations from doubly-occupied orbitals to the two singly-occupied virtual orbitals in the SCF description (8aj3b{). 'Calculated at the X ‘Ai experimental geometry of Kivelson.44 Absolute energy of —547.122007 hartree. mThe absolute energy is —547.125513 hartree. "Using their [7,5,2/5,3,1] contracted basis. 'Calculated at the X ‘Ai optimized geometry. Absolute energy of -547.116189 hartree. pC o m p u t e d at the X *Ai state’s experimental geometry (Rso = 1 -4321 A and Ooso = 119.5°). ’The 3 Table 3.14. Calculated geometries and excitation energies for the 3A2 excited state.0’6 Geometry 0 AE (eV)d Method Ref. Rso (A) Ooso (deg) Vertical Minimum (Te) 1.590 (0.086) 101.3 (-19.7) 3.4 SCFe S2f 1.586 (0.082) 1 0 0 . 8 (-2 0 .2 ) 2.5 S-CP 82* 4.2086 S-CP’-' 143 1.63 (0.04) 104 (-15.5) 2.14 1.58 INDO/S-CP 79 4.6 DFC-LSD 81 1.503 (0.099) 92.56 (-26.24) 4.313' 3.184171 SCFn 2 2 4.690° SCF" 2 2 2.7454p SCF? 64 1.503 92.56 (1.234)r SCF 23 aSee Tables 3.4 and 3.3 for geometries and energies of the lAi ground state. 6Unless otherwise noted the basis sets are the same as those used for the *Ai ground state reported in Section 3.3.2. cQuantities in parentheses represent the change compared to the ground state. dUnless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. eVirtual orbital approximation applied to the ground-state wavefunction. f Using ST 0-3G * (‘sp lit’ d) basis. 3 All single excitations from the seven valence MO’s to all virtual MO’s. hGeometry for *Ai not given. Assumed to be experimental geometry, although the energy is incorrectly reported as -654.78725 a.u. ’Using a 4-31G basis set. J All single excitations from the three highest occupied M O ’s to the three lowest-lying virtual M O’s. kSingle excitations from doubly-occupied orbitals to the two singly-occupied virtual orbitals in the SCF description (Sb^bJ). 'Calculated at the *Ai experimental geometry of Kivelson.44 Absolute energy of -547.078478 hartree. mThe absolute energy is -547.122404 hartree. "Using their [7,5,2/5,3,1] contracted basis. "Calculated at the *Ai optimized geometry. Absolute energy of -547.067046 hartree. pComputed at the X *Ar state’s experimental geometry (R-so - 1.4321 A and Ooso = 119.5°). ’ T he 3 Table 3.15. Calculated geometries and excitation energies for the 3B2 excited state.®’*’ Geometry 0 AE (eV)d Method Ref. Rso (A) Ooso (deg) Vertical Minimum ( Te) 1.622 (0.118) 104.8 (-16.2) 2 . 1 SCFe 82/ 1.616 (0 .1 1 2 ) 104.5 (-16.5) 1.3 S-CP 82/ 2.540*1 S-CI*’J 143 1.69 (0.10) 108 (-11.5) 1.15 INDO/S-CIfc 79 1.609 (0.154) 107 (-12.0) 4.8 3.5 DFC-LSD 81 1.561 (0.157) 103.5 (-15.3) 3.020' 1.894"1 SCF" 2 2 3.552° SCF" 2 2 2.2530p SCF? 64 “See Tables 3.4 and 3.3 for geometries and energies of the *Ai ground state. ‘ Unless otherwise noted the b asis sets are the same as those used for th e *Ai ground state reported in Section 3.3.2. 'Quantities in parentheses represent the change compared to the ground state. ‘'Unless otherwise noted, the A E are obtained using the energies computed at the optimized equilibrium geometries. 'Virtual orbital approximation applied to the ground-state wavefunction. 1 Using STO-3G* (‘split’ d) basis. s All single excitations from the seven valence MO’s to all virtual MO’s. ‘ Geometry for X 'Ai not given. Assumed to be experimental geometry, although the reported energy is —654.78725 a.u. ‘Using a 4-31G basis set. > All single excitations from the three highest occupied MO’s to the three lowest-lying virtual M O’s. ‘Single excitations from doubly-occupied orbitals to the two singly-occupied virtual orbitals in the SCF description (la^b]). 'Calculated at the X 'A! experimental geometry of Kivelson.44 Absolute energy of -547.125964 hartree. mThe absolute energy is —547.169792 hartree. "Using their [7,5,2/5,3,1] contracted basis. °C alculated at the X ‘Ai optim ized geometry. A bsolute energy of -547.108862 hartree. pComputed at the X 'At state’s experimental geometry ( Rso = 1-4321 A and 0Oso = 119.5°). ’The 3d-orbital exponent is optimized for both the X *Ai (1.53) and the a 3Bi (1.67) states. 119 change in the geometry on going from the XxAi to the a3Bi state from experiment and theory compare well. Except for the calculation of Jones ,81 the change in the calculated S-0 bond length is smaller than that found experimentally. All the values calculated for the change in the 0S0 bond angle are slightly larger than the experimental value. The agreement between Te values from experiment and theory is not as good as for the geometries. The only calculation which gives a Te value that is close to the approximate experimental value of 3.24 eV (To= 3.1946 eV) is the SCF calculation of Lindley 22 (Te= 3.099 eV). The other calculations, even the Cl calculation of Hillier and Saunders 8 2 (Te= 2.1 eV), give values much less than the experimental one. The vertical excitation energies calculated by Lindley 22 and Jones 81 agree fairly well with the experimental value of fa 3.4 eV. Again the other calculations give much lower values. It should be pointed out that the SCF A EveTt values of Keeton and Santry were calculated using different 3d orbital exponents for each of the states they studied. The vertical emission energies calculated by Phillips and Davidson 158 agree fairly well with that from experiment (the SD(Q)-CI value is 2.83 eV compared to the experimental emission maximum of 2.98 eV). The absolute energies from their calculations, at the SCF, SD-CI, and SD(Q)-CI levels, are given in a footnote to Table 3.12. Two things about their calculations should be pointed out. First, their most important Cl configuration is the one resulting from excitation from the 8 ai to 3bi orbital (SCF configuration), as is the case for 120 the other Cl calculations 8 2 ,1 4 3 listed in the table. This is used as their reference configuration for the CL Their second, and most important, discovery is that two distinct solutions exist for the 3Bj state. There is the symmetry restricted solution (C2v) with an energy of —547.0992 hartree, and the symmetry broken solution with an energy of —547.1115 hartree. From MCSCF geometry optimizations, they find that this symmetry breaking appears to be an artifact of the SCF (RHF) method. This could mean that other SCF calculations suffered from this same problem- The results of calculations for the b 3A2 state are given in Table 3.14. The geometries from all the calculations agree fairly well, again with the exception of the SCF calculation of Lindley , 2 2 whose values for the bond length and angle are quite a bit lower than those from the other calculations. The calculated geometries are in good agreement with the tentative experimental geometry for this state (R so = 1.55 A and Ooso = 97°), given by Merer et al.281 Interestingly, the geometries calculated for the 3A2 state are practically identical to those for the 4A2 state (those of Lindley 2 2 are identical). This result also agrees with the experimental findings of Merer and coworkers 2 8 1 ’2 1 9 ’2 2 0 which show that the geometry for the b 3A2 state is very close to that for the A !A 2 state (Rso = 1-53 A and Ooso = 99°). The calculated changes in geometry on going from the !Ai ground state to this state agree fairly well with those from experiment The Te values calculated for the 3A2 state show the same pattern as for the 3Bi state. In this case the SCF results of both Lindley 22 and Hillier and Saunders 8 2 121 are a bit higher than the Cl value of Hillier and Saunders . 8 2 The SCF values for Te (3.482 and 3.184 eV22) are in fairly good agreement w ith the approximate To value of 3.23 eV 281 from experiment. For the most part the vertical excitation energies are in agreement with each other but quite a bit higher than the maximum absorption energy of fa 3.4 eV. T h e vertical emission energy calculated by Zellmer 23 is 1.234 eV. This is much less than the experimental value of 2.98 eV. It should also be pointed out that Phillips and Davidson158 found that two configurations were needed in the Cl reference space. These were the singly excited SCF configuration ([core]: 5 b 2 3 b}) and the doubly excited configuration ([core]: 7 a|9 a}5 b 2 3 bJ), relative to the ground state. This is in contrast to the other two Cl studies which found that the only important contribution to the Cl wave- function was from the SCF configuration. Table 3.15 shows the results for various theoretical studies on the 3 B2 state. Since there is absolutely no direct experimental data on this state, the theoretical calculations are very important- The agreement between the calculated geome tries is about the same as it was for the other states. The main difference between the various studies is in the calculated change in bond length- The SCF calcu lation of Lindley 2 2 and the density functional calculation of Jones 81 both give a A R so ~ 0.155 A- The other calculations obtained a ARso « 0 . 1 1 A. This is the biggest discrepancy in A Rso found for the triplet states, at least for the ab initio calculations. 122 The Te values calculated by ab initio methods are in fairly good agreement and are very small. They are much lower than the respective values for the a 3Bi and b 3A2 states. The spread in calculated vertical excitation energies is about the same as for the 3Bj state and much worse than that for the 3A2 state. The calculated A Evert is within the range for the triplet states found from experiment. Unfortunately, a A Eemi, could not be calculated as it was for the xBi state in Section 3.5.2. It would be helpful to have this value. As for the two other triplet states above, it should be noted that all the C l studies show that the single most important configuration for the 3 B2 state is the SCF configuration ([core]:la^3bJ). This is to be contrasted to the XB 2 state for which three configurations are important, according to Bendazzoli and Palmieri 1 4 3 and Phillips and Davidson 158 (although they only agree on two out of the three). Since experiments have raised some questions as to the ordering of these triplet states and their interactions, it would be instructive to take a closer look at what the theoretical calculations show. Concerning the ordering of the states, only two things can be said for sure. The A Evert and Te values for the 3A2 state are higher than those for the 3Bi and 3 B2 states and all three triplet states have transition energies which are nearly degenerate. In fact, the Te values for the 3Bi and 3A2 states differ by at most 0.4 eV and the A Evert values for the 3Bi and 3 B2 states differ by no more than 0.2 eV (for ab initio calculations). Unfortunately, the issue on which of the 3Bj and 3 B2 states is the lowest is somewhat clouded. Most of the calculations (all ab initio methods) give much lower Te values for the 3 B2 state than for 3 Bi. The only one in which the reverse is true is the DFC calculation of Jones . 81 For the vertical excitation energies the matter of which state is lowest depends a great deal on the method and geometry used. For instance if one looks at the SCF calculations of Lindley , 2 2 it can be seen that there is a large difference in the A Evert values calculated at the experimental and SCF optimized X xAi geometries (as was the case for the excited singlet states). In fact, this difference is large enough in the 3Bi and 3 B 2 cases that the 3Bi state is the lowest triplet at the X xAj SCF optimized geometry, while the order is reversed at the X XAX experimental geometry. Phillips and Davidson 1 5 8 performed calculations on all three of these triplet states, at the experimental geometry of the 3 BX. They give the SCF, SD-CI, and SD(Q)-CI energies and energy differences, at this geometry, for the XAX, 3 Bi, 3 B2, 3A2 , and XB 2 states (see Tables 3 and 4 of their paper). At the SCF level the 3 B2 state is lower than the 3 BX state (although, if the symmetry broken solution is used for the 3 BX, then the two are very nearly degenerate). At the Cl levels this ordering is reversed and the 3 BX is « 0.62 eV (SD(Q)-CI result) lower than the 3 B2 state. Since the SD(Q)-CI results should be fairly reliable, it is probably safe to say that, at the experimental geometry of the 3 BX state, the 3 BX is the lowest triplet state. As can be seen from these results, theoretical calculations have not conclusively shown which of the two 3B states is the lowest. The interactions of the triplet states with each other, and with the singlet states, have also been studied using theoretical methods .23,143,158 Bendazzoli and Palmieri143 and Phillips and Davidson 1 5 8 computed the spin-orbit matrix elements, oscillator strengths of transitions from X XAX state to the excited states, and the radiative lifetime of the a 3 BX state. They found that the 3Bi state is spin-orbit coupled to the XAX and and several XB 2 states (the largest contribution coming from X xAx and C 1 B2) and that the X xAx state is spin-orbit coupled to several 3A2 states (the largest contribution coming from b 3A2). Their results support the findings of the experiments. In particular, Phillips and Davidson 158 computed a radiative lifetime of 8.1 ± 2.5 msec, which agrees well with that from experiment and seems to support the claims of some spectroscopists 208,210,265 that the principal route for depopulating the a 3 BX state is through radiative decay. Zellmer performed calculations, similar to those done for the A XA 2 and B 1 BX states (discussed in Section 3.5.2), on the a 3 BX and b 3A2 states. 23 The results are similar to those for the analogous singlet states. The i / 2 potential curves of the 3 BX and 3A2 states, calculated near the X XAX optimized bond length, cross near their minima (the 3 BX curve crossing closer to the minimum of 3A2). The 3 B2 state v-i potential curve crosses near the minimum of both the 3 BX and 3A2 potentials. For the i/x potentials, near the XxAx optimized bond angle, the 3 BX and 3A2 states cross, but not as near to the minimum of either state. The 3 B2 state crosses almost right at the minimum of 3 BX. For calculations done near the SCF optimized geometric parameters of the 3A2 state, it is seen that the v2 potentials of the 3Bi and 3A2 states cross near their minima, while the V\ potentials are parallel. This is analogous to the situation in the singlet states of the same symmetry. Also, the 3 B2 state crosses near the minimum of the 3A2 state in both the i/x and 1/2 potentials (especially v2). For the a 3Bi and b 3A 2 states, the minimum C2v crossing point occurs at R$o = 1.487 A and O q s o — 108.65°. This bond length is close to Lindley’s 2 2 SCF optimized value for the 3A 2 state (1.503 A). The bond angle is midway between his values for the 3 BX and 3A 2 states (average bond angle is 109.78°). The A E vert at this geometry, using C2v symmetry restrictions, is 3.510 eV. This is close to Lindley’s 22 Te values of 3.100 and 3.184 eV for the 3 BX and 3A2 states, respectively. As was the case for the analogous singlet states, there is a “symmetry broken” solution at the C2„ minimum crossing point. In C, symmetry the 3Bi and 3A2 states form the 3A" state. At this geometry the energy for the 3A" state is 1.002 eV lower than the energies of the 3 BX and 3A 2 states. Thus A E vert is 2.508 eV for the “symmetry broken” solution and is lower than the Te’s of either of the C2„ triplet states. The absolute minimum, near the crossing point, occurs in the i / 3 potential surface. The minimum geometry and energy (at the minimum C2u crossing angle of 108.65°) are approximately, R\ = 1.56 A and R 2 = 1.41 A and 2.200 eV, respectively. This is almost as low as the minimum excitation energy for the 3 B2 state (Te = 1.894 eV). If this situation is not an artifact, it could be possible that the absolute physical minimum in the j / 3 potential near the crossing 126 point of the 3Bi and 3A2 states is lower than the minimum of the 3 B2 state. This would help to explain why the 3 B2 state is not seen in absorption or emission. In addition to the above interaction between the triplet states, this study also indicates that the X *AX state’s potential surface crosses near the minimum of the b 3A2 and 3 B2 potential surfaces. As mentioned above the ground state should interact with the 3A2 state. In, conclusion the theoretical studies do show that all three triplet states fall within the range of this transition and that they may interact with each other and with the singlet states. However, they have not been successful in giving a definite answer as to which of the a 3 Bx or 3 B2 states is the lowest. More accurate calculations, including some form of electron correlation, will be necessary to get a definitive answer to this problem. 3.6.3 Summary Both experiment and theory show that the a 3 BX, b 3A2, and 3 B2 states are nearly degenerate. The geometry for the a 3 Bx state, from experimental investigations, is approximately Rso — 1.493 A and Ooso = 126.23°. This geometry is supported by the theoretical calculations. What is more important is that both experiment and theory show essentially the same changes in geometry on going from the X *AX state to the a 3 Bx state. For the b 3A2, the results of experimental and theoretical studies are also remarkably similar. Both give geometries with Rso ^ 1-55 A and 127 Ooso ft* 1 0 0 °. They both also predict that the structures of the b 3A2 and A IA 2 states are almost identical. Thus in this case, where the geometry of the b 3A2 state was hard to determine experimentally, theory has given much support for the experimental value. The SCF calculations of Lindley 22 and Zellmer 2 3 support the spectroscopist’s view that the a 3 Bx and b 3A2 states interact near their minimum (the ( 0 0 1 ) vibrational level of b 3A2 interacting with the ( 1 0 0 ) and ( 1 1 0 ) levels of a 3 Bx). Although experiment and theory have given a fairly good picture of what is occurring in th e 3400-3900 A region, there are still many questions that need a more thorough answer. The exact nature of where and how the triplet states interact still needs to be answered, as does the question of where exactly is the 3 B2 state and why it cannot be found experimentally. 3.7 Summary Figure 3.3 presents a schematic energy level diagram for the six states responsible for the absorption in the 3900-1700 A region of the SO 2 spectrum. The best present values for the geometries, T0, Te, and A Evert are presented in the figure. The figure makes it plain just how close the T0 and Te values are for the 3 BX and 3A2 states. Also, it is seen that the vertical excitation energy to the 3 BX state is very similar to its To and Te values (according to spectroscopists, the other triplet states’ vertical excitation energies are also supposed to appear in the same region 128 AEvert— 6.2 ‘B, ‘B2 (R= 1.553,0= 103.8) 'A'/ R , = 1.639, R2= 1.491 A V 0 = 103.8 / -5 T0 = 5.28, Te = 5.37 e ! (eV) AEven- 4.2 »A, *B, ~4 ’B, T0 = 3.87, Te = 3.94 T0 = 3.23, % = 3.32 »A2 (R = 1.53,0 = 99) AEvcrt= 3.35 j A2 (R= 1-55,9 = 97) T0 = 3.46, % = 3.55 Bi 3B, ^ B t (R = 1.493, 0= 126.23) T0 = 3.19, T; = 3.24 - 3 *A,(R= 1.431, 0=119.33) Figure 3.3. Schematic representation of the experimental excitation energies of the states responsible for the S 0 2 spectrum in the 3900-1700 A region. as the 3Bi state). This signifies that the geometry of the 3Bi state should be similar to the ground state, as it turns out to be. It is also very plain that the 'B 2 state is much higher in energy than the other states found in the 3900-1700 A region. C H A P T E R IV Discussion of the Basis Set Employed in the Present Calculations. 4.1 Introduction This chapter discusses the basis set used for the calculations reported in this disser tation, the results of which are discussed in the following chapters. As mentioned in Chapter I and Section 3.3.2, an in-depth basis set study had been performed in this laboratory by D. D. Lindley ,22 and therefore no basis set optimization was done in the present study. The basis set used in the present study is slightly smaller than the one used in our previous studies . 2 2 ,2 3 The details of this basis and comparisons to the SCF results obtained in the previous studies (using a larger basis) are discussed below. Lindley performed SCF calculations on the X *Ai ground state (at its experi mental geometry) using several different primitive basis sets of G T O ’s contracted in different ways to give several different CGTO basis sets (see Chapter 3 of his thesis22). 130 The primary CGTO basis set chosen for subsequent use in geometry optimiza tions and property calculations for the XxAi ground state and the lowest six excited states (1 ,3A2 , 1 ,3 Bi, and 1 ,3 B2 ) was a [7,5,2/5,3,1] contraction of an (11,7,2/9,5,1) primitive basis (an EBP type basis set). Lindley also performed calculations on these states, at the X xAi SCF optimized geometry, using a slightly larger basis (a set of s and p Rydberg type orbitals were included on the sulfur). The effect of us ing this larger basis set on the total energy, orbital energies and selected properties was small. The total energy and orbital energies were lowered by no more than 1 . 2 and 2.4 mhartree, respectively. The effect of this basis set on the dipole and sec ond moments on sulfur, for the ground and excited states, while more pronounced than that on the energies, was also fairly small (the largest change occurred for the dipole moment and was ~ 0.0056 a.u.). The actual calculated values for this basis can be found in Tables 3, 11, and 22 of Lindley’s thesis . 22 I also used the [7,5,2/5,3,1] basis set to determine the minimum crossing points of the potential surfaces of the 1,3A2 - 1,3Bi pairs of states and to optimize the geometries of the corresponding 1 ,3A" states in the neighborhood of the crossings . 23 The basis set used in the present study is Lindley’s smaller DZP basis set which has two sets of 6 d-type orbitals on sulfur and one set on oxygen. This basis is a [6 ,4,2/4,2,1] contraction of the same (11,7,2/9,5,1) primitive basis used in the previous studies performed in this laboratory. The exponents of the primitive 132 basis functions and the contraction coefficients for the [ 6 ,4,2/4,2,1] CGTO basis are given in Table 4.1. The 6 d-type orbitals used as polarization functions contain the standard five spherical harmonic linear combinations of dxi, Chapter V) total energies (calculated at the X experimental geometry) are lowered by fa 0 . 0 0 2 hartree upon the addition of the 3 s orbitals to both sulfur and oxygen. The smaller contracted basis set was chosen for several reasons. Optimized geometries of the seven states of SO 2 discussed in this dissertation and potential energy curve crossings of the 1 ,3A2 - 1,3Bi states were to b e determined, at least at the MCSCF level. This requires the calculation 0f the energy of a particular state at several (sometimes a few dozen) geometries. Each energy calculation requires a minimum of two steps: ( 1 ) an integration step and (2) an SCF, MCSCF, or Cl step. The integration time is reduced when going from the EBP to the DZP basis (the number of primitive function integrals is th e same, but the number of 133 Table 4.1. Contracted [ 6 ,4,2/4,2,1] Gaussian basis set for (9s5pld) oxygen and (\\s7p2d) sulfur primitive basis sets.“ Oxygen s sets Sulfur s sets Exponents Contraction [4s] Exponents Contraction [ 6 s] 7816.5400 0.002031 35710.0000 0.002565 1175.8200 0.015436 5397.0000 0.019405 273.1880 0.073771 1250.0000 0.095595 81.1696 0.247606 359.9000 0.345793 27.1836 0.611832 119.2000 0.635794 3.4136 0.241205 119.2000 0.130096 9.5322 1.000000 43.9800 0.651301 0.9398 1.000000 17.6300 0.271955 0.2846 1.000000 5.4200 1.000000 2.0740 1.000000 0.4264 1.000000 0.1519 1.000000 Oxygen p sets Sulfur p sets Exponents Contraction [2p] Exponents Contraction [4p| 35.1832 0.019580 212.9000 0.014091 7.9040 0.124189 49.6000 0.096685 2.3051 0.394727 15.5200 0.323874 0.7171 0.627375 5.4760 0.691756 0.2137 1.000000 5.4760 -0.626737 2.0440 1.377051 0.5218 1.000000 0.1506 1.000000 Oxygen d set Sulfur d sets Exponents Contraction [Id] Exponents Contraction [2d] 0.7250 1.000000 1.0823 1.000000 0.3206 1.000000 “Taken from D. D. Lindley.22 contracted function integrals is reduced since the number of contracted basis func tions is reduced from 74 to 62). The tim e needed for the SCF or MCSCF step is also reduced. With the smaller basis set there are fewer orbitals which need to be variationally optimized and fewer integrals to be read in and processed. This can be very important for an MCSCF calculation which, especially for a large number of configurations, takes considerably more time than an SCF calculation (the co efficients of the configurations must also be variationally determined). Finally, for Cl calculations the smaller basis set reduces the size of the virtual orbital space, thus substantially reducing the number of configurations that can be generated. All the previous results concerning the SCF optimized geometries, excitation energies, properties, and the crossing points were obtained with the [7,5,2/5,3,1] basis . 2 2 ,2 3 One could ask how these entities are affected upon going to the smaller DZP basis set (i.e., how “good” is this basis compared to the larger EBP basis?). This can be investigated by: • Comparing energies (orbital, total, and vertical excitation) and properties (one-electron properties and Mulliken population analysis) calculated with each basis for Qne or more of the states being studied, at single point geome tries (such as the X xAi experimental geometry). 135 • Obtaining new SCF optimized geometries and crossing points using the present basis DZP set and comparing them to those obtained using the pre vious EBP basis. The first alternative is by far the easiest, while the second would be far more difficult and time consuming. Hopefully, if the first set of calculations showed that the two basis sets produce similar results, then it could be assumed that the SCF optimized geometries, minimum excitation energies (Te), and crossing points obtained from both basis sets would also be similar. Lindley did compare the energies and some selected property values (includ ing Mulliken population analysis) calculated with both basis sets, at the X *AX experimental geometry of Kivelson 4 4 (Rso = 2.7063 bohr and 0osO = 119.536°). Lindley’s results can be found in Tables 3, 4c, and 5 of his thesis. I have recalcu lated most of these same entities, as well as some additional ones, for the X *Aj ground state and the C 1 B2 excited state, the results of which are discussed below. 4.2 Comparison of Basis Sets for the X !A i Ground State, at Single Point Geometries. Basis set studies for the X 'Ai ground state were performed at t wo molecular geometries, the experimental geometry of Kivelson 44 and the EBP SCF optimized ground state geometry of Lindley , 2 2 as given in Table 4.2. The orbital and total energies, one-electron properties, and Mulliken population analysis computed using 136 Table 4.2. The experimental and [7,5,2/5,3,1] SCF optimized geometries for the X *Ax ground state of SO 2 . Experimental Geometry Rso 1-4321 A (2.7063 bohr) Ooso 119.536° [7,5,2/5,3,1] Optimized Geometry R,w 1-404 A (2.653 bohr) Ooso 118.8° both the DZP and EBP basis sets, at these geometries, are compared below. The definitions of the properties and population analysis are given in Appendices B and C, respectively. 4.2.1 Results at the X *Ai Experimental Geometry. The results for the various properties calculated at the X*Ai experimental geometry are given in Tables 4.3, 4.4 and 4.5. The absolute differences between the DZP and EBP results are calculated as ( Pd zp — Re b p )- The DZP values obtained here are not identical to those of Lindley, using the same [ 6 ,4 ,2/4,2,1] basis set. This is most likely due to the fact that the geometry used here was slightly different than Lindley’s. He rounded off his y and z coordinate values so that his calculations were actually done at Rso = 2.70630376 bohr and Ooso = 119.53604410°, while the present geometry is Rso = 2.70630001 bohr and 0OSo = 119.53599994°. This difference in geometry produces only an insignificant difference in property values 137 Table 4 .3 . Comparison of SO 2 X 'Aj SCF orbital energies calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X *AX experimen tal geometry.0’6 MO 16,4,2/4,2,11 [7,5,2/5,3,1]° Difference lax -92.185406 -92.182989 -0.002417 162 -20.620489 -20.625337 0.004848 2 a x -20.620459 -20.625307 0.004848 3a 1 -9.165709 -9.164036 -0.001673 262 -6.845S34 -6.844047 -0.001787 I61 -6.844S35 -6.843253 -0.001582 4ax -6.842S96 -6.841450 -0.001446 5a j -1.509545 -1.508330 -0.001215 362 -1.405182 -1.404114 -0.001068 6 a j -0.869319 -0.868681 -0.000638 462 -0.700653 -0.699444 -0.001209 7«i -0.695310 -0.693553 -0.001757 26j -0.665615 -0.664037 -0.001578 5 63 -0.542255 -0.540749 -0.001506 la2 -0.520908 -0.519097 -0.001811 8 a j -0.495079 -0.494057 -0 . 0 0 1 0 2 2 “Rso = 1.4321 A. (2.7063 bolir) and Ooso = 119.5360°. Energ|0S m j,ivrtrces cTaken from T a b le 10 of Lindley.22 138 Table 4.4. Comparison of S02 X SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the experimental X XAX geometry.0-6 P rop erty [6,4,2/4,2,1] [7,5,2/5,3,l]c D ifference Total energy -547.231493 -547.236964 0.005471 Dipole moment 0 .8 2 8 9 3 9 0.826281 0.002658 Second moments xx(S) -1 5 .3 2 2 9 4 -15 .2 69 897 -0 .0 5 3 0 4 yy(S) -2 0 .7 3 1 2 7 -20.666908 -0 .0 6 4 3 6 zz(S) -1 8 .0 8 8 2 4 -18.037508 -0 .050 73 b b (0 ) -2 1 .4 0 5 0 3 -21.339067 -0 .065 96 c c (0 ) -1 5 .1 5 5 4 1 -15 .1 13 519 -0 .0 4 1 8 9 r2(S) -5 4 .1 4 2 4 5 -53 .9 74 313 -0 .1 6 8 1 4 Third moments zzz(S) 3 5 .1 7 0 4 8 34.983680 0 .18 68 0 xxz(S) 1 0 .7 986 9 10.723620 0 .0 7 5 0 7 yyz(S) 1 7 .2 0 0 0 6 17.108549 0.09151 Potential at nucleus *(S) -5 8 .9 8 4 1 1 -58.986618 0 .00251 4>(0) -2 2 .2 8 7 3 4 -22.280822 -0 .0 0 6 5 2 Electric field gradient at nucleus Qxx( S) -1 .9 9 7 5 4 7 -2.119418 0.121871 9yy(S) -0 .0 4 1 7 7 7 -0.044878 0.003101 M S) 2 .0 3 9 3 2 4 2 .164296 -0.1 2 4 9 7 2 9aa(0) -0 .8 4 5 1 0 2 -0 .855898 0.0 1 0 7 9 6 766(0) -0 .5 9 7 9 6 5 -0.581491 -0.0 1 6 4 7 4 7cc( 0 ) 1 .4 4 3 0 6 7 1.437389 0.0 056 78 Table 4.4: (continued) Property [6,4,2/4,2,1] [7,5,2/5,3,1] Difference Asymmetry parameter ?/(33S) -0.959028 -0.958529 -0.000499 v(170) -0.171259 -0.190907 0-019648 Electronic second moments xxe((CM) -15.32294 -15.269897 -0.05304 yye/(CM) -108.2081 -108.144020 -0-0641 ZZe;(CM) -31.81245 -31.765365 -0.04709 r2(CM) -155.3435 -155.179282 -0.1642 Electronic third moments ZZZe;(CM) -0.664562 -0.751360 0.086798 xxze;(CM) 0.356053 0.317123 0.038930 yyze,(CM) 62.65431 62.606894 0.04742 Second moments xx(CM) -15.32294 -15.269897 -0.05304 yy(CM) -20.73127 -20.666907 -0.06436 zz(CM) -16.95839 -16.911280 -0.04711 r2(CM) -53.01260 -52.847084 -0.16552 Third moments zzz(CM) -0.656132 -0.742933 0-086801 xxz(CM) 0.356053 0.317123 0.038930 yyz(CM) 3.071622 3.023967 0.047655 aRso = 1 4321 A (2.7063 bohr) and 60so = 119.5360°. b Properties given in atomic units. cTaken from Tables 3, 4c, and 15 of Lindley.22 140 Table 4.5. Comparison of SO 2 X *Ax SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X *Ai experimental geometry.0,6 [6,4,2/4,2,1] [7,5,2/5,3,l]c Difference Total s 0 3.775 3.785 -0.010 s 5.696 5.688 0.008 Total p 0 4.670 4.680 -0.010 s 8.545 8.528 0.017 Total d 0 0.092 0 . 1 0 1 -0.009 s 0.687 0.672 0.015 Gross atomic population 0 8.536 8.566 -0.030 s 14.928 14.869 0.059 Overlap O-S 0.9062 0.8678 0.0384 0 - 0 -0.0802 -0.0825 0.0023 aRs o = 1.4321 A (2.7063 bohr) and 0Oso = 11 9 .5360°. 6Properties given in atomic units. cTaken from Tables 13 and 14 of Lindley.22 141 calculated for this basis (the total energy differs by ~ 27 /xhartree, while most properties agree to better than ±0.001 a.u.). As one can see from the tables, the agreement in the energies and proper ties calculated using the two different basis sets is fairly good. The orbital ener gies agree to better than 0.35%, with all the DZP values (execpt for the oxygen Is orbitals 162 and 2cti) being lower than the EBP values. The total energies agree to 5.5 mhartree or 0.001% (the relative difference is calculated using \(Pdzp — P e b p )/P ebp \ X 100). The other properties and population analysis re sults generally show differences in the hundredth decimal place and, except for a few cases, agree to better than 1% (the largest relative difference is less than 13%). The few major discrepancies which show percentage differences of greater than 1% occur in the cases in which the differences are taken between two small numbers (such as for the electric field gradient at the nucleus, asymmetry parame ter for oxygen> electronic and total third moments at the center of mass, and some population analysis results). The results using both basis sets can also be compared to the experimental values, where available. These values are given in Table 4.6. The values calculated with the larger basis set are, for the most part, closer to the experimental ones. However, the differences between the experimental and the EBP SCF values are much greater than those between the two basis sets. Therefore, the DZP basis set can be considered to do essentially as well as the EBP basis set in reproducing 142 Table 4.6. The experimental properites for the X *Ai state of S02. Property Experimental value experimental value (atomic units) (cgs units) Dipole moment (debyes) Px 0.639999 1.62673° Electric field a t nucleus fl0~ 8 dvne«-e*u—M8 E*(S) (0.00) (0.00) Ey(O) (0.00) (0.00) E * (0 ) (0.00) (0.00) Electric field gradient at nucleus (1016 esu/cm 3) Asymmetry param eter (dimensionless) „ (33S) 0.87° Q uadrupole coupling constants (MHzV* eqxxQ(M S) 0.102143 24.0 ±0.2° eqyyQ(33 S) 0.007235 1.7 ± 0 .2 ° eqxxQ(33 S) -0.109378 -25.7 ±0.2° Electronic second moments flO-16 ™ ! l xxe|(CM ) -14.998476 4.2 ± 0.2/ yyei(CM ) -107.131968 30.0 ± 0.2/ zze|(CM ) -31.068271 8.7 ± 0 .2 / r «i(CM) -153.198714 42.9 ± 0.6/ Quadrupole moment tensor (10~26 esu-cm2) ©xx (CM) 2.973874 4.0 ± 0.6/ ©yy(CM) -3.940384 - 5 .3 ± 0 .4 / ©xx(CM) 0.966509 1.3 ± 0.3/ Diamagnetic susceptibility (10-6 erg/G2-mole) X i i (CM) -138.296415 -1 6 4 .3 ± 0 .5 / Xyy(CM) -46.211036 -5 4 .9 ± 0.3/ Xrz(CM) -122.135179 -145.1 ±0.3/ Xou(CM) -102.214210 -121.4 ±0.4/ “Reference 293. 6Heilman-Feynman forces, F(A), which depend on E(A), are also zero by definition. “Calculated from the quadrupole coupling constants of Reference 294. dQ(33S) = —0.065 barns. “R eference 294. ■f Reference 295. The signs of the properties depend on the components of the molecular g values along the axes all being negative. 143 experimental property values (at the experimental ground state geometry) at the SCF level. 4.2.2 Results at the EBP SCF X A i Optimized Geometry. The same comparisons are also made at the EBP SCF optimized X geometry and are shown in Tables 4.7, 4.8 and 4.9. Although some of the properties (the dipole moment, electric field gradient at the nucleus, asymmetry parameter for oxygen, electronic second and third moments and total third moment at the center of mass, and some population analysis results) show large changes due to the change in geometry, the agreement between the DZP and EBP energies and properties is very similar to that obtained at the experimental geometry. 4.2.3 Summary of Basis Set Comparisons for the X *Ai State at Single point Geometries. Several comments can be made about the previous results for the X JAx ground state at the two geometries considered. In general, the DZP and EBP properties for this state are in good agreement at both geometries. The largest difference is about 16% for the xxz c o m p o n e n t of the electronic and total third moments at the center of mass, calculated at the EBP SCF optimized geometry. The only discrepancy in the orbital energy ordering occurs for the 4 b2 and 7fli orbitals. The ordering of these orbitals is 462 7fli using both basis sets at the experimental geometry and using the EBP basis at the EBP optimized geometry. 144 Table 4.7. Comparison of S02 X 'A, SCF orbital energies calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X *Ax [7,5,2/5,3,1] SCF optimized geometry.0’6 MO [6,4,2/4,2,1] [7,5,2/5,3,l]c Difference la i -92.176131 -92.173751 -0.002380 152 -20-614900 -20.620192 0.005292 2ai -20.614868 -20.620160 0.005292 3ax -9.158419 -9.156691 -0.001728 262 -6.839002 -6.837158 -0.001844 I6i -6.837434 -6.835796 -0.001638 4ai -6.835638 -6.834135 -0.001503 5aj -1.527633 -1.526266 -0.001367 362 -1.417117 -1.415930 -0.001187 6ax -0.862142 -0.861474 -0.000668 462 -0.704250 -0.702989 -0.001261 7ax -0.704451 -0.702566 -0.001885 26, -0.676101 -0.674431 -0.001670 562 -0.544878 -0.543326 -0.001552 lfl2 -0.527591 -0.525739 -0.001852 Sax -0.494255 -0.493230 -0.001025 0Rso = 1 .4 0 4 A (2.653 bolu) and Ooso ~ 118.80°. 6Energies in hartrees. Taken from Table 11 of Lindley.22 145 Table 4.8. Comparison of SO 2 X xAi SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the X xAi [7,5,2/5,3,1] SCF optimized geometry.0’6 Property [6,4,2/4,2,1] [7,5,2/5,3,l]c Difference Total energy -547.233835 -547.239412 0.005577 Dipole moment Vz 0.804090 0.802226 0.001864 Second moments xx(S) -15.22752 -15.172016 -0.05550 yy(S) -20.50558 -20.440482 -0.06510 zz(S) -18.00678 -17.954485 -0.05230 r*( S) -53.73987 -53.566983 -0.17289 Third moments zzz(S) 34.47845 34.285521 0.19293 xxz(S) 10.59431 10.515427 0.07888 yyz(S) 16.83259 16.745275 0.08732 Potential at nucleus *(S) -58.99314 -58.995588 0.00245 $ (0 ) -22.29292 -22.285901 -0.00702 Electric field gradient at nucleus -2.090240 -2.220434 0.130194 9w(S) 0.030037 0.034526 -0.004489 Qzz{ S) 2.060203 2.185908 -0.125705 qaa(O) -0.868881 -0.885245 0.016364 Qbb(0) -0.482930 -0.451020 -0.031910 9cc(0) 1.351811 1.336265 0.015546 Table 4.8: (continued) P rop erty [6,4,2/4,2,1] [7,5,2/5,3,1] D ifference Asymmetry parameter i?(33S) -1.029159 -1.031590 0.002431 *?(170) -0.285507 -0.324954 0.039447 Electronic second moments x x ej(C M ) -15.22752 -15.172016 -0.05550 y y e/(C M ) -103.9276 -103.862550 -0.0651 zzej(C M ) -31.51660 -31.466824 -0.04978 r2el( CM) -150.6717 -150.501390 -0.1703 Electronic third moments ZZZe/(C M ) -0.923001 -1.012498 0.089497 XXZe;(C M ) 0.307152 0.265767 0.041385 y y z e/(C M ) 59.30538 59.262048 0.04333 Second moments x x (C M ) -15.22752 -15.172016 -0.05550 y y (C M ) -20.50558 -20.440486 -0.06509 zz(C M ) -16.92035 -16.870575 -0.04978 r 2(C M ) -52.65344 -52.483077 -0.17036 Third moments zzz(C M ) -0.914790 -1.004288 0.089498 x x z(C M ) 0.307152 0.265767 0.041385 y y z(C M ) 2.979773 2.936430 0.043343 aR so = 1-404 A (2.653 bohr) and Ooso — 1 18.8°. ^Properties given in atomic units. cTaken from Tables 3 and 18 of Lindley.22 147 Table 4.9. Comparison of SO2 X *Ai SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1 ] contracted basis sets at the X *Ai [7,5,2/5,3,1] SCF optimized geometry. °,b [6,4,2/4,2,1] [7,5,2/5,3,l]c Difference Total s 0 3.757 3.770 -0.013 s 5.696 5.664 0.032 Total p 0 4.680 4.691 -0.011 s 8.525 8.507 0.018 Total d 0 0.096 0.105 -0.009 s 0.712 0.696 0.016 Gross atomic population 0 8.534 8.567 -0.033 s 14.933 14.867 0.066 Overlap o-s 0.9249 0.8765 0.0484 0-0 -0.0913 -0.0926 0.0013 aRso = 1-404 A (2.653 bohr) and 0Oso = 118.8°. bProperties given in atomic units. 'Taken from Tables 16 and 17 of Lindley.22 148 The ordering of the these orbitals using the DZP basis set is reversed from that using the EBP basis at the EBP optimized geometry. However, it should be pointed out that these orbitals are very nearly degenerate using both basis sets at both geometries. The total energies from the two basis sets differ by about 0.006 mhartree at both geometries. The differences between the individual DZP and EBP properties at the two geometries are of the same order of magnitude and have the same signs. The only exceptions to this last statement occur for the yy component of the electric field gradient and asymmetry parameter calculated at sulfur (< 7vy(S) and r/(33S), respectively). The value for qyy(S) and the differences for this property, for both basis sets, changes from a very small negative number to a very small positive number when the geometry changes. This component of electric field gradient is obtained by adding two small numbers which are almost equal in magnitude but opposite in sign. Small changes in the geometry or basis set will have a much greater effect on this property than on other properties (for example, the dipole and second moments ). In addition, the differences are taken between two very small numbers which have relatively large errors, as explained above. Thus, the change in the sign of this property and the differences between the DZP and EBP values for this property, when changing the geometry, could be considered to be small. (The asymmetry parameter is calculated from the electric field gradient, so that errors in the latter are likely propagated to the former.) Finally, one notices 149 that, in general, the properties that show the largest deviation upon changing the basis set are also those that show the greatest change when the geometry at which they are calculated changes (for the same basis set). Considering just the results for the X *Ai state, one could conclude that the DZP basis does almost as well as the EBP basis in determining the total energy and properties at the SCF level. However, since excited states are also being studied it would be instructive to see how these properties compare for the two basis sets when determined for an excited state. 4.3 Comparison of Basis Sets for the C *B 2 Excited State. The C ^ 2 state was the excited state chosen for the basis set comparison because, as will be discussed below, it was the state which seemed to show the largest change in its SCF optimized geometry when going from the EBP to DZP basis. Therefore, it was felt that this state might be the most sensitive to a change in the basis set. The geometry at which this comparison is made is the EBP “optimized” geometry calculated by Lindley22 ( Rso = 1.524 A and Ooso = 101.5°). (As he states, this geometry was not optimized as well as that for the other states studied.) The comparisons of the properties and population analysis results, calculated using the two basis sets, are given in Tables 4.10 and 4.11. It should be pointed out that Lindley’s values for the electric field gradient at the oxygen nucleus are those 150 Table 4.10. Comparison of SO 2 C 1B2 SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the C *B2 [7,5,2/5,3,1] SCF optimized geometry.0’6 Property [6,4,2/4,2,1] [7,5,2/5,3,1]c Difference Total energy -547.011080 -547.016935 0.005855 Dipole moment A4* 1.002201 1.022937 -0.020736 Second moments xx(S) -15.764100 -15.727276 -0.036824 yy(S) -20.159880 -20.144736 -0.015144 zz(S) -19.389530 -19.393443 0.003913 r 2(S) -55.313510 -55.265455 -0.048055 Third moments zzz(S) 50.011070 50.412542 -0.401472 xxz(S) 13.509040 13.629022 -0.119982 yyz(S) 22.527320 22.637658 -0.110338 Potential at nucleus *(S) -59.002020 -59.001920 -0.000100 $ (0 ) -22.278620 -22.272903 -0.005717 Electric field gradient at nucleus Q'xr(S) 0.128353 0.157871 -0.029518 -1.332502 -1.433254 0.100752 MS) 1.204148 1.275382 -0.071234 M ° ) e -1.148905 -1.086057 -0.062848 Qyy(O) 0.440407 0.399586 0.040821 M 0) 0.708498 0.686471 0.022027 151 Table 4.10: (continued) Property [6,4,2/4,2,l] [7,5,2/5,3,l] Difference Asymmetry parameter V ^S) 1.213186 1.247567 -0.034381 v(17o y -2.243213 -2.164174 -0.079039 Electronic second moments xxe/(CM) -15.764100 -15.727276 -0.036824 yye/(CM) -99.744170 -100.326660 0.582490 ZZei(CM) -44.125880 -44.561828 0.435948 r2(CM) -159.634100 -160.615764 0.981664 Electronic third moments zzze/(CM) -0.523813 -0.518856 -0.004957 xxze/(CM) -0.857549 -0.834756 -0.022793 yyze/(CM) 76.643430 77.810686 -1.167256 Second moments xx(CM) -15.764100 -15.727276 -0.036824 yy(CM) -20.159880 -20.144736 -0.015144 zz(CM) -17.562820 -17.511931 -0.050889 r2(CM) -53.486800 -53.383943 -0.102857 Third moments zzz(CM) -0.503654 -0.498163 -0.005491 xxz(CM) -0.857549 -0.834756 -0.022793 yyz(CM) 4.154649 4.111310 0.043339 a R s o = 1 -524 A (2.880 bohr) and 0Oso — 1 01.5°. h Properties given in atomic units. cTaken from Tables 23 and 43 of Lindley.22 e Lindley’s values for oxygen are actually those for the global coordinate system. The present values in the principal axis system of oxygen are, qxx( 0) = qaa(0) = -1.148905, qtb(O) = -0.858843, and qcc(0 ) = 2.007748. / As for the electric field gradient, Lindley’s value is for the global coordinate system. The present value in the principal axis system of oxygen is -0.144471. 152 Table 4.11. Comparison of SO 2 C1B2 SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the C XB 2 [7,5,2/5,3,1] SCF optimized geometry.0’6 [6,4,2/4,2,1] [7,5,2/5,3,l]c Difference Total s O 3.827 3.836 -0.009 s 5.712 5.699 0.013 Total p O 4.593 4.602 -0.009 s 8.744 8.729 0.015 Total d 0 0.077 0.085 -0.008 s 0.549 0.527 0.022 Gross atomic population 0 8.497 8.522 -0.025 s 15.006 14.955 0.051 Overlap O-S 0.4654 0.4444 0.0210 0 - 0 -0.0934 -0.0918 -0.0016 aRso = 1.524 A (2.880 bohr) and Ooso = 1 0 1 .5 °. ‘'Properties given in atomic units. cTaken from Tables 41 and 42 of Lindley.22 153 calculated in the global coordinate system and not the principal axis system of oxygen. As can be seen in the table, the agreement in the properties is essentially the same as obtained for the X :Aj ground state discussed above. In fact, there are almost as many instances where the agreement in the properties calculated from the two basis sets is better for the MI 2 state as there are where it is better for the JAi state. The energy agrees to « 5 . 9 mhartree, as was th e case above for th e X*Ai state. The largest percent difference, of « 19%, occurs for the xx component of the electric field gradient at the sulfur nucleus. As for for the X ’Ai state, the value for the qxx(S) component for th e *B 2 state is obtained by subtracting two numbers of almost equal magnitude. Thus, one might expect the difference for this property to be greater than for some of the other properties. These results support the preliminary conclusion, made above f0r the X *Ai state, that this basis set does nearly as well as the larger basis set in describing the energies and wave functions (as determined by the properties and population analysis) at the SCF level. One additional check is to compare the properties calculated at the DZP SCF optimized geometry to those calculated at th e EBP SCF optimized geometry, for one or m ore of the states studied. 154 4.4 Comparison of Properties Calculated at the [6,4,2/4,2,1] and [7,5,2/5, 3 , 1 ] SCF Optimized Geometries. Extensive SCF reoptimizations for all of the states were not carried out, as one aim of this work was to use the MCSCF method to obtain optimized geometries, excitation energies, and the crossing points of the 1 ,3A2 - li3Bi pairs of states. However, this optimization was done for the X *Ai ground state and the C 1 B 2 state, the r e s u l t s of which are discussed below and in Chapter VI. 4.4.1 Results at the X *Ai Optimized Geometries. The “new” DZP SCF optimized geometry, energy, properties, population analysis, and orbital energies and the “old” EBP SCF values are given in Tables 4.12, 4.13, and 4 .1 4 As one can see from the tables, the DZP optimized geometry is very similar to th e EBP optimized geometry (the difference being only 0.07% and 0.02% in the S-O bond distance and OSO angle, respectively). The results for the comparison of th e properties are very similar to those discussed in Sections 4.2.1 and 4.2.2. The difference in the total energy is again less than 6 mhartree, as in the cases discussed previously. The largest percent difference (as 20.4%) again occurs for the yy component of the electric field gradient at the sulfur nucleus (^(S)). A comparison of the orbital energies also shows similar behavior as in the previous cases. In this case the ordering of the 462 and 7a! orbitals is the same for 155 Table 4.12. Comparison of SO 2 X xAi SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective X SCF optimized geometries.0 Property [6,4,2/4,2,1] [7,5,2/5,3,l]6 Difference Optimized geometry Rso 2 . 6 5 4 8 8 0 2 . 6 5 3 0 0 0 0 . 0 0 1 8 8 0 0oso 1 1 8 . 8 2 0 1 0 0 1 1 8 . 8 0 0 0 0 0 0 . 0 2 0 1 0 0 Total energy -547.233839 -547.239412 0 . 0 0 5 5 7 3 Dipole moment 0.804828 0.802226 0 . 0 0 2 6 0 2 Second moments xx(S) - 1 5 . 2 3 0 9 9 0 - 1 5 . 1 7 2 0 1 6 - 0 . 0 5 8 9 7 4 yy(s) - 2 0 . 5 1 3 7 3 0 - 2 0 . 4 4 0 4 8 2 - 0 . 0 7 3 2 4 8 zz(S) - 1 8 . 0 0 9 4 5 0 - 1 7 . 9 5 4 4 8 5 - 0 . 0 5 4 9 6 5 r 2 (S) - 5 3 . 7 5 4 1 7 0 - 5 3 . 5 6 6 9 8 3 - 0 . 1 8 7 1 8 7 Third moments zzz(S) 3 4 . 4 9 9 3 3 0 3 4 . 2 8 5 5 2 1 0 . 2 1 3 8 0 9 xxz(S) 1 0 . 6 0 0 4 8 0 1 0 . 5 1 5 4 2 7 0 . 0 8 5 0 5 3 yyz(S) 1 6 . 8 4 4 3 3 0 1 6 . 7 4 5 2 7 5 0 . 0 9 9 0 5 5 Potential at. nucleus *(S) - 5 8 . 9 9 2 8 0 0 - 5 8 . 9 9 5 5 8 8 0 . 0 0 2 7 8 8 * ( 0 ) - 2 2 . 2 9 2 7 1 0 - 2 2 . 2 8 5 9 0 1 - 0 . 0 0 6 8 0 9 Electric field gradient at nucleus < 7 x x (S ) - 2 . 0 8 6 7 9 4 - 2 . 2 2 0 4 3 4 0 . 1 3 3 6 4 0 9y«(S) 0 . 0 2 7 4 7 8 0 . 0 3 4 5 2 6 - 0 . 0 0 7 0 4 8 t? Z z (S ) 2 . 0 5 9 3 1 6 2 . 1 8 5 9 0 8 - 0 . 1 2 6 5 9 2 Qaai 0 ) - 0 . 8 6 8 0 5 4 - 0 . 8 8 5 2 4 5 0 . 0 1 7 1 9 1 9 6 6 ( 0 ) - 0 . 4 8 6 8 8 9 - 0 . 4 5 1 0 2 0 - 0 . 0 3 5 8 6 9 9 c c ( 0 ) 1 . 3 5 4 9 4 2 1 . 3 3 6 2 6 5 0 . 0 1 8 6 7 7 Table 4.12: (continued) Property [ 6 ,4,2/4,2,1] [7,5,2/5,3,1] Difference Asymmetry parameter t?(3 3 S) -1.026689 -1.031590 0.004901 77(1 7 0) -0.281315 -0.324954 0.043639 Electronic second moments xxe/(CM) -15.230990 -15.172016 -0.058974 yye,(CM) -104.082900 -103.862550 -0.220350 ZZej(C M ) -31.524330 -31.466824 -0.057506 re/(CM) -150.838200 -150.501390 -0.336810 Electronic third moments zzze/(CM) -0.913935 -1.012498 0.098563 xxze/(CM) 0.308764 0.265767 0.042997 yyzez(CM) 59.420080 59.262048 0.158032 Second moments xx(CM) -15.230990 -15.172016 -0.058974 yy(CM) -20.513730 -20.440486 -0.073244 zz(CM) -16.921800 -16.870575 -0.051225 r 2 (CM) -52.666510 -52.483077 -0.183433 Third moments zzz(CM) -0.905718 -1.004288 0.098570 xxz(CM) 0.308764 0.265767 0.042997 yyz(CM) 2.983015 2.936430 0.046585 Properties given in atomic units. Taken from Tables 3 and 18 of Lindley.22 157 Table 4.13. Comparison of SO 2 X *Ai SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective XlAi SCF optimized geometries.0 [6 ,4,2/4,2,1] [7,5,2/5,3,l ]6 Difference Total s 0 3.758 3.770 -0 . 0 1 2 s 5.696 5.664 0.032 Total p 0 4.680 4.691 -0 . 0 1 1 s 8.526 8.507 0.019 Total d 0 0.096 0.105 -0.009 s 0.711 0.696 0.015 Gross atomic population 0 8.534 8.567 -0.033 s 14.933 14.867 0.066 Overlap 0-S 0.9242 0.8765 0.0477 0 - 0 -0.0908 -0.0926 0.0018 “Properties given in atomic units. '’Taken from Tables 16 and 17 of Lindley.22 158 Table 4.14. Comparison of S02 X *Ax SCF orbital energies calculated using the [6,4 ,2/4,2,1] and [7,5,2/5,3,1] contracted basis sets at the respective X *AX SCF optimized geometries.0 MO [6 ,4,2/4,2,1]6 [7,5,2/5,3,l]c Difference laj -92.176474 -92.173751 -0.002723 1 ^ 2 -20.615109 -20.620192 0.005083 2 ax -20.615077 -20.620160 0.005083 3ax -9.158684 -9.156691 -0.001993 2 6 2 -6.839251 -6.837158 -0.002093 I 61 -6.837704 -6.835796 -0.001908 4ax -6.835902 -6.834135 -0.001767 5ax -1.526984 -1.526266 -0.000718 3&2 -1.416701 -1.415930 -0.000771 . 6 ax -0.862394 -0.861474 -0.000920 4 6 2 -0.704137 -0.702989 -0.001148 7ax -0.704121 -0.702566 -0.001555 2 &x -0.675724 -0.674431 -0.001293 5 6 2 -0.544795 -0.543326 -0.001469 la 2 -0.527358 -0.525739 -0.001619 8 ax -0.494278 -0.493230 -0.001048 “Energies in hartrees hR s o = 1-40490 A (2.65488 bohr) and 0Os o = 118.8201°. “Taken from Table 11 of Lindley.22 Rs o = 1.4Q4 A (2.653 bohr) and O oso = 118.80°. 159 both basis sets (although these orbitals are even more nearly degenerate for the DZP basis than in the previous cases discussed above). Finally, one can also compare how the orbital energies and properties change, within each basis set, when changing from the experimental to the respective op timized geometries. The results for the X xAi state are given in Tables 4.15, 4.16, and 4.17- As the geometry changes from the experimental to the SCF optimized geometry, when using a particular basis set, the changes in the properties and orbital energies are very similar. The changes in each property, within each basis set, all have the same sign (i.e., the change in /z2 is « +0.024 a.u., the changes in the second moments are all negative, etc.). T hese results show that, at least for the X xAx state, changes in properties at the SCF level as the geometry is changed can be fairly well reproduced using either basis set. 4.4.2 Results at the C !B 2 Optimized Geometries. Unlike th e X xAx state, the DZP results at the DZP optimized geometry for the XB2 state do not agree as well with those for the EBP results at the EBP optimized geometry (Tables 4 + 8 and 4.19). The error is as large as 57% (for qyy(0)). The poorer agreement for this state is most likely due to the fact that the DZP optimized geometry is not as close to the EBP optimized geometry as it was for the xAi state. The bond distance differs by only 0.01 A, but the bond angle is 2.7° 160 Table 4.15. Comparison of SO 2 X'Aj DZP and EBP SCF changes in the or bital energies when changing from the experimental to the respective optimized geometries .a MO [6 ,4,2/4,2,1] [7,5,2/5,3,l]fr Differences Differences lai -0.008932 -0.009238 I 62 -0.005380 -0.005145 2 ai -0.005382 -0.005147 3flj -0.007025 -0.007345 262 -0.006583 -0.006889 lfci -0.007131 -0.007457 4ai -0.006994 -0.007315 5ai 0.0174 39 0.017936 362 O.OU5 1 9 0.011816 6 ai -0.006925 -0.007207 4 6 2 0.003484 0.003545 7ai 0.008811 0.009013 2bi 0.010109 0.010394 5 6 2 0.002540 0.002577 la 2 0.006450 0.006642 8 ai -0 . 0 0 0 8 0 1 -0.000827 a Energies given in hartrees. determined from results given in Tables 10 and 11 °f Lindley. 161 Table 4.16. Comparison of SO 2 X *Ai DZP and EBP SCF properties changes when changing from the experimental to the respective optimized geometries.0 Property [6,4,2/4,2,l] [7,5,2/5,3,1]6 Differences Differences Geometry Rso 0 . 0 5 1 4 2 0 0 . 0 5 3 3 0 0 6 0 so 0 . 7 1 5 9 0 0 0 . 7 3 6 0 0 0 Total energy 0 . 0 0 2 3 4 6 0 . 0 0 2 4 4 8 Dipole moment 0 . 0 2 4 1 1 1 0 . 0 2 4 0 5 5 Second moments xx(S) - 0 . 0 9 1 9 5 0 - 0 . 0 9 7 8 8 1 yy(S) - 0 . 2 1 7 5 4 0 - 0 . 2 2 6 4 2 6 zz(S) - 0 . 0 7 8 7 9 0 - 0 . 0 8 3 0 2 3 r 2 (S) - 0 . 3 8 8 2 8 0 - 0 . 4 0 7 3 3 0 Third moments zzz(S) 0 . 6 7 1 1 5 0 0 . 6 9 8 1 5 9 xxz(S) 0 . 1 9 8 2 1 0 0 . 2 0 8 1 9 3 yyz(S) 0 . 3 5 5 7 3 0 0 . 3 6 3 2 7 4 Potential at, nucleus *(S) 0 . 0 0 8 6 9 0 0 . 0 0 8 9 7 0 $ ( 0 ) 0 . 0 0 5 3 7 0 0 . 0 0 5 0 7 9 Electric field gradient at nucleus 0 . 0 8 9 2 4 7 0 . 1 0 1 0 1 6 < 7 y y (S ) - 0 . 0 6 9 2 5 5 - 0 . 0 7 9 4 0 4 Qzz{ S) - 0 . 0 1 9 9 9 2 - 0 . 0 2 1 6 1 2 < / a a ( 0 ) 0 . 0 2 2 9 5 2 0 . 0 2 9 3 4 7 9 6 6 ( 0 ) - 0 . 1 1 1 0 7 6 - 0 . 1 3 0 4 7 1 < 7 c c ( 0 ) 0 . 0 8 8 1 2 5 0 . 1 0 1 1 2 4 Table 4.16: (continued) Property [6,4,2/4,2,11 17,5,2/5,3,1] Differences Differences Asymmetry parameter r/(33S) 0.067661 0.073061 v(17 o) 0.110056 0.134047 F.lectronic second moments X X e l ( C M ) -0.091950 -0.097881 yye/(CM) -4.125200 -4.281470 zze/(CM) -0.288120 -0.298541 re((CM) -4.505300 -4.677892 F.lectronic third moments Z Z Z e l ( C M ) 0.249373 0.261138 XXZe/(CM) 0.047290 0.051356 yyzei(CM) 3.234230 3.344846 Spcond moments xx(CM) -0.091950 -0.097881 yy(CM) -0.217540 -0.226421 zz(CM) -0.036590 -0.040705 r2(CM) -0.346090 -0.364007 Third moments zzz(CM) 0.249587 0.261355 xxz(CM) 0.047290 0.051356 yyz(CM) 0.088607 0.087537 Properties given in atomic units. Determined from results in Tables 3, 4c, 15, and 18 of Lindley. 163 Table 4.17. Comparison of S02 X xAi DZP and EBP SCF population anal yses changes when changing from the experimental to the respective optimized geometries.® Property [6, 4,2/4,2,1] [7,5,2/5,3,l]6 Differences Differences Total s 0 0.017 0.015 s 0.000 0.024 Total p 0 -0.010 -0.011 s 0.019 0.021 Total d 0 -0.004 -0.004 s -0.024 -0.024 Gross atomic population 0 0.002 -0.001 s -0.005 0.002 Overlap 0-S -0.0180 -0.0087 0-0 0.0106 0.0101 “Properties given in atomic units. ‘D eterm ined from results in Tables 13, 14, 16, and 17 of Lindley. 164 Table 4.18. Comparison of S02 C *B 2 SCF properties calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective C 1B2 SCF optimized geometries.® Property [6,4,2/4,2,1] [7,5,2/5,3,l]6 Difference Optimized geometry Rso 2 . 8 7 2 0 0 0 2 . 8 8 0 0 0 0 - 0 . 0 0 8 0 0 0 #o so 9 8 . 8 0 0 0 0 0 1 0 1 . 5 0 0 0 0 0 - 2 . 7 0 0 0 0 0 Total energy -547.011643 -547.016935 0 . 0 0 5 2 9 2 Dipole moment Vz 1.043454 1.022937 0 . 0 2 0 5 1 7 Second moments xx(S) - 1 5 . 7 6 1 2 3 0 - 1 5 . 7 2 7 2 7 6 - 0 . 0 3 3 9 5 4 yy(S) - 2 0 . 0 2 0 8 8 0 - 2 0 . 1 4 4 7 3 6 0 . 1 2 3 8 5 6 zz(S) - 1 9 . 4 9 0 2 3 0 - 1 9 . 3 9 3 4 4 3 - 0 . 0 9 6 7 8 7 r2(S) - 5 5 . 2 7 2 3 4 0 - 5 5 . 2 6 5 4 5 5 - 0 . 0 0 6 8 8 5 Third moments zzz(S) 5 1 . 4 1 8 1 8 0 5 0 . 4 1 2 5 4 2 1 . 0 0 5 6 3 8 xxz(S) 1 3 . 8 6 6 1 4 0 1 3 . 6 2 9 0 2 2 0 . 2 3 7 1 1 8 yyz(S) 2 2 . 8 7 0 4 0 0 2 2 . 6 3 7 6 5 8 0 . 2 3 2 7 4 2 Potential at nucleus *(S) - 5 9 . 0 0 2 5 0 0 - 5 9 . 0 0 1 9 2 0 - 0 . 0 0 0 5 8 0 $(0) - 2 2 . 2 8 1 1 9 0 - 2 2 . 2 7 2 9 0 3 - 0 . 0 0 8 2 8 7 Electric field gradient at nucleus Qxx ( S ) 0 . 1 0 7 1 6 4 0 . 1 5 7 8 7 1 - 0 . 0 5 0 7 0 7 %y(S) - 1 . 3 4 4 7 9 2 - 1 . 4 3 3 2 5 4 0 . 0 8 8 4 6 2 Qzz{ S) 1 . 2 3 7 6 2 9 1 . 2 7 5 3 8 2 - 0 . 0 3 7 7 5 3 ?xx(0)c - 1 . 1 9 1 4 8 3 - 1 . 0 8 6 0 5 7 - 0 . 1 0 5 4 2 6 9 s/!/(0) 0 . 6 2 7 6 3 0 0 . 3 9 9 5 8 6 0 . 2 2 8 0 4 4 Qzz( 0) 0 . 5 6 3 8 5 3 0 . 6 8 6 4 7 1 - 0 . 1 2 2 6 1 8 165 Table 4.18: (continued) Property [6,4,2/4,2,!] [7,5,2/5,3,l] Difference Asymmetry parameter rtf33S) 1.173175 1.247567 -0.074392 T](v 0 ) d -3.226216 -2.164174 -1.062042 Electronic- second moments xxe,(cM) -15.761230 -15.727276 -0.033954 yye/(CM*) -96.103030 -100.326660 4.223630 zzei(C M ) -45.485450 -44.561828 -0.923622 re/(CM) -157.349700 -160.615764 3-266064 BliEtroniC-lh ili moments zzze|(C M f) -0.525005 -0.518856 -0-006149 xxze/(C M ) -0.867007 -0.834756 -0-032251 yyze,(C N 0 75.235390 77.810686 -2-575296 Second rrioments xx(CM) -15.761230 -15.727276 -0-033954 yy(CM) -20.020880 -20.144736 0-123856 zz(CM) -17.539450 -17.511931 -0-027519 r2(CM) -53.321560 -53.383943 0-062383 Ihild m oments zzz(CM ) -0.503250 -0.498163 -0-005087 xxz(CM ) -0.867007 -0.834756 -0-032251 yyz(CM ) 4.155456 4.111310 0-044146 % adrui?ole moment tensor GQ c M ) 3.01893 1 3.101057 -0-082126 O^CM) -3.370533 -3~.525133 0-154600 0 « (C M ) 0.351602* 0.424075 -0-072473 a Properties given *n atomic units. 6Taken from Tables 23 and 43 of Lindley.22 cLindley’s va|ues f°r oxygen are actually those for the global coordinate system . The present v a lu es in the p r in c ip a l axis system o f oxygen are, qaa( 0 ) = qxx(0) = -1.191483, qbh(0) = -2.050047, and q c d O ) = -0.8585634- dA s for the electric 9e*4 gradient, fvindley’s value is for the global coordinate system. The present value in the principal axis system °f oxygen is —1.000001. >. 166 Table 4.19. Comparison of SO 2 C aB2 SCF population analysis calculated using the [6,4,2/4,2,1] and [7,5,2/5,3,1] basis sets at the respective C 1!^ SCF optimized geometries.® Property [6,4,2/4,2,1] [7,5,2/5,3,l]6 Difference Total s 0 3.828 3.836 -0.008 s 5.709 5.699 0 . 0 1 0 Total p 0 4.593 4.602 -0.009 s 8.746 8.729 0.017 Total d 0 0.078 0.085 -0.007 s 0.555 0.527 0.028 Gross atomic population 0 8.495 8.522 -0.027 s 15.010 14.955 0.055 Overlap 0-S 0.4788 0.4444 0.0344 0 - 0 -0.1159 -0.0918 -0.0241 Properties given in atomic units. ^Taken from Tables 41 and 42 of Lindley.22 167 smaller for the DZP basis. Part, if not most, of this large change in the optimized geometry is due to the fact th at Lindley did not optimize the geometry for this state as well as that for the *Ai state. Taking into account the larger change in the optimized geometry between the two basis sets (as compared to the geometry change for the *Ar state) the difference between the DZP and EBP properties is reasonable. For the majority of the properties the relative error between the DZP and EBP results is still less than 5%. Also, the difference in the total energy is between 5-6 mhartree, as it was at the EBP optimized geometry (see Table 4.10). Thus, it is reasonable to say that the DZP basis produces results for the 1B2 state at the SCF level comparable to those determined with the EBP basis. 4.5 Summary From the results presented above it can be seen that, at least for the X !Aj and C ^B2 states, the SCF results obtained with the DZP basis set are comparable to those obtained with the EBP basis set. For the \Aj state, at its experimental geometry, the DZP basis predicts the experimental properties nearly as well as does the EBP basis. The optimized geometries and properties predicted using the DZP basis are very similar to those determined with the EBP basis. Also, Te for the C *B2 <— X *Aj calculated using the DZP and EBP basis sets, at their respective optimized geometries, are within 0.3 mhartree (0.002 eV) of each other. Therefore, it appears that the DZP basis set is more than adequate for the purposes of determining the SCF optimized geometries, excitation energies and properties for the states of interest in the present study. C H A P T E R V Methods Employed in the Present Calculations. 5.1 Introduction As mentioned in Chapter I, some problems were encountered with symmetry break ing of the SCF wavefunction at the C2„ geometry crossing points.23 At the C2„ geometry crossing points there are two Gs symmetry-restricted solutions which are degenerate and should have energies equal to the C2t/ solution. However, the Cs energies for the 3A" and 1A" states were lower than the C2„ energies for the 1,3A2 and 1,3Bx states by 0.036826 hartrees (1.002 eV) and 0.048087 hartrees (1.309 eV), respectively. Phillips and Davidson have reported symmetry breaking for the 3Bi excited state (although the problem disappeared upon performing MCSCF calculations).158 This type of symmetry breaking is an artifact of the SCF approxi mation and has no physical significance. It is due to the fact that the wavefunction had constraints placed on it by using C2„ symmetry restrictions. Problems with symmetry breaking for a host of systems are well documented and discussed rather extensively elsewhere.296-300 One way of dealing with this problem is to perform all 170 calculations using Cs symmetry restrictions. Another way of solving the symme try breaking described here would be to perform a multi-configuration calculation, such as MCSCF or MRCI, with a large enough configuration space th a t includes enough configuration state functions (CSF’s) to correct the problem. Usually this does not require an extremely large number of CSF’s. For example, for th e problem of the discontinuity in the potential curve discussed above, the simplesf solution might be to use a linear combination of th e two Cs solutions (a t w o - configuration MCSCF calculation) to obtain the wavefunctions for the A2 and Bx states at the C2u geometries. MCSCF is easier and computationally cheaper than MRCI. Also, th e reference space for MRCI is generally generated by performing an MCSCF calculation and then using the resulting configurations as the MRCI reference space- Thus, the choice of the configurations for an MCSCF calculation is very important. The MCSCF configurations are in turn generated by choosing an active space with the correct orbitals necessary to describe the desired system. In general, choosing an active space is often a bit more of an art than a science. Most of the tim e it is just an intelligent trial and error procedure. The best MCSCF calculation is a complete-active-space MCSCF (CASSCF) in which all possible electronic configurations (excitations) within a set of orbitals (a select number of SCF occupied and virtual orbitals) are allowed. The best C ASSCP would be the case in which all SCF occupied and virtual orbitals are included in 171 the active space. Of course this would be equivalent to a full Cl calculation. This is impossible to perform computationally except for a few extremely small systems. The next best type of MCSCF is a CAS which excludes the atomic core orbitals (usually these are still optimized) and includes the same number of virtual orbitals as there are occupied orbitals in the active space. This type of CASSCF is of ten referred to as a Fully Optimized Reaction Space (FORS) MCSCF301-304 since it contains all the atomic valence orbitals and the chemically important virtual orbitals and allows all CSF’s which can be generated by excitations within these orbitals. This cuts down on the number of CSF’s with little effect on chemically important results. While this is the best type of MCSCF calculation which can be performed computationally, it quickly becomes prohibitive for even medium sized chemical systems with DZP basis sets, especially if a potential surface study is being performed. Practicality usually calls for a choice to be made on which orbitals to leave out of the CAS. Usually one starts by leaving out “inner” orbitals (non-core orbitals with the lowest SCF orbital energies) and including only the first few (lowest) virtual orbitals. This is done until a manageable size active space (number of CSF’s) is obtained. This type of MCSCF is best referred to as an N x M CASSCF, where N is the number of electrons and M is the number of orbitals, with M usually equal to or somewhat greater than N. An example of this type of CASSCF would be a case with 5 SCF occupied orbitals and 3 SCF virtual orbitals with 10 electrons 172 (a 10 x 8 CASSCF). The maximum excitation level in this case would be 6 (6 electrons from three occupied orbitals are excited to the 3 empty virtual orbitals). The other type of MCSCF not yet discussed is a non-CAS MCSCF. This type of active space can still contain the same number of orbitals as a CASSCF but the excitation level is restricted so that not all possible CSF’s within the active space are generated. An example of this type of MCSCF would be the case of using the same 8 active space orbitals as above but allowing only single and double excitations from the 5 occupied orbitals to the 3 virtual orbitals (no more than two electrons in the virtual orbitals in any CSF). Most often this type of MCSCF is used for large systems in which a CASSCF would be impractical, or as preparation for an MRCI calculation (creation of the reference space). Of course MCSCF calculations are not without their problems. For instance, choosing an active space is more difficult if potential surface studies, studies of several states (electronic spectra), or a non-CAS MCSCF are being performed. Choosing active spaces for a potential surface study are difficult because the or dering of the orbitals can change as one moves along the surface. All orbitals which are important at any one point on the surface need to be included at all points to obtain a smooth potential surface. For the study of electronic spectra (several states), different orbitals can be more important to one state than another. One needs to be sure that orbitals 173 needed to describe all states being studied are included in the active spaces for all states. A non-CAS MCSCF active space must be chosen with care to ensure that the most important configurations are included in the wavefunction. Choosing a non- CAS active space for several states is more difficult since one wants to be sure of treating all states equivalently. This may entail allowing different excitation levels or patterns for the different states. Also, since the first few v ir tu a l orbitals are often antibonding in nature, the MCSCF calculation can be biased (the wavefunction has too much antibonding character). MCSCF bond distances are often too long due to this fact (angles are normally good). Even with all these problems, multireferenoe methods (MCSCF and MRCI) are extremely important. There are many chemical systems, such as radicals, excited states, and symmetry-broken solutions, which are poorly described by single refer ence methods (SCF). Even performing SRCI on these types of systems most often does not recover what is lost by starting with the poor single-reference description as the reference space for the Cl. Furthermore, a single-reference description is not equally good in different regions of a potential energy surface, and b eco m es progressively inadequate as bonds are stretched and broken. Keeping the above discussion in mind, three calculation methods were employed in this study: SCF, MCSCF (both CAS and non-CAS), and Cl (SRCI and MRCI). 174 A general review of the methods was presented in Chapter II. Three different types of MCSCF treatments (labeled throughout the dissertation as MCSCF1, MCSCF2, and MCSCF3), which differ in the type of active space used, were applied in the present study. Two of these, MCSCF1 and MCSCF2, were CASSCF calculations and were used to perform geometry optimizations. The MCSCF2 treatment was also used to obtain excitation and emission energies and properties. Single- and multi-reference Cl calculations (using the non-CAS MCSCF3 CSF’s as the refer ence space) were also performed at single points to obtain excitation and emission energies. The details of the three types of MCSCF and Cl calculations performed (which orbitals are used in the active space, the number of configurations generated for each state, why three types of calculations were performed, etc.) are discussed in the remainder of this chapter. 5.2 MCSCF Methods used for the Potential Surface Stud ies Two types of MCSCF complete active spaces were used to perform the potential surface studies (discussed in Chapters VI and VII). The SCF configuration of the X JAi state, [core]: (5ax)2 (362)2 (6a!)2 (4 b2)2 (7ax)2 {2bl)2 (5b2)2 (1 a2)2 (Sax)2 (36x)° (9ax)° (662)°, 175 was used as the base configuration in determining which orbitals would be included in the active spaces for all the states. This ordering of the molecular orbitals corresponds to the X *Ai experimental geometry (C 2 W symmetry). The lax orbital is lower than the 4 & 2 at the ‘A! SCF optimized geometry. A description of the SCF occupied orbitals is given in Table 5.1. Since it was desired to perform complete-active-space MCSCF (CASSCF) cal culations, the number of orbitals allowed in the active space needed to be kept to the minimum necessary to give an adequate description of the states under consideration. The \at~4ai, 162- 2 6 2 , and 16j orbitals are the atomic core orbitals (inner-shell orbitals on sulfur and oxygen). The highest core orbital, 4a1? has an orbital energy ^ 5 3 hartree lower than the first valence orbital, 5ai- There is a definite cutoff point between the core and valence orbitals. Therefore, the core orbitals were not included in the active space in any MCSCF calculations. The 5a 1 and 362 sigma bonding orbitals are separated from the other valence orbitals by « 0.7 hartree and the 6 ax lone pair orbital is « 0.17 hartree below the 462. The other valence orbitals, 4 6 2 - 8 0 !, are separated by only « 0.2 hartree. Thus, it was believed that the 5a x, 362, and 6 ai orbitals would participate only to a very small extent in a niulticonfiguration description of the states of interest and were considered “core” orbitals (i.e., remained doubly occupied) for both MCSCFl and MCSCF2. Orbitals 7 a i~ 6 6 2 were part of the active spaces and the maximum exci tation level was 6 for both types 0f MCSCF calcnlations. Thus, the only difference 176 Table 5.1. SCF description (in C2u symmetry) of the orbitals based on the Mul- liken population analysis of the X 'Aj state at its experimental geometry.0 MO Orbital Type C om m ents lai sulfur Is core li>2 oxygen Is core 2ai oxygen Is core 3ai sulfur 2s core 262 sulfur 2p core l k sulfur 2p core 4ai sulfur 2p core 5ai S -0 bonding doubly occupied in MCSCF 1 & MCSCF2 362 S -0 bonding doubly occupied in MCSCF 1 & MCSCF2 6ai sulfur lone pair doubly occupied in MCSCF 1 & MCSCF2 462 weak S-0 bonding doubly occupied in M CSCF1 7ai S -O p 26i S-O pTr - pir bonding active space orbital 562 oxygen lone pair active space orbital la2 oxygen lone pair active space orbital 8a 1 sulfur lone pair, active space orbital S-0 antibonding 3 k S-0 antibonding6 active space orbital 9ai active space orbital 662 active space orbital aR so = 1-4321 A (2.7063 bohr) and 0Oso ~ 119.5360°. 6Based on excited states. 177 in the two MCSCF treatments was how the 462 orbital was treated. Also, for both MCSCF1 and MCSCF2 all orbitals, including the core, were optimized. 5.2.1 Discussion of the MCSCF 1 Active Space Initially the 462 orbital was not included as part of the active space (kept doubly occupied). Even though this orbital’s energy was very close to those of the active space orbitals, it was originally believed, from the population analysis, that this orbital might also have little importance in a multiconfiguration description of the states being studied. Thus, the active space consisted of 10 electrons in 8 orbitals, 7ai~6b2. This MCSCF treatment is referred to throughout this dissertation as MC- SCF1. The number of configurations for each electronic state for this active space is given in Table 5.2, along with those for the other MCSCF and Cl calculations. Towards the end of the MCSCF1 geometry optimizations a possible problem was discovered with the active spaces. Tables 5.3, 5.4, and 5-5 contain information about the active spaces for all the states at the X experimental and MCSCF1 optimized geometries and the respective MCSCF1 optimized geometries for all the states. This includes the coefficient for the SCF configuration state function (CSF), the number of CSF’s with coefficients larger than thresholds of 0.10, 0.05, and 0.02, and the diagonal elements of the Fock matrix (Fock orbital energies) and natural orbital occupation numbers for all orbitals except the core. In these tables the 462 orbital is included with the active space orbitals, even though it remained 178 Table 5.2. The number of CSF’s for all states for the various calculation methods used in the present study. Methodology xAi xa 2 % xb 2 3B, 3b 2 3a2 SCF 1 1 1 1 1 1 1 MCSCF1 318 282 288 288 384 384 378 MCSCF2 666 614 614 626 855 861 855 MCSCF3 38 56 57 48 76 63 75 SRCI 73535 172164 172025 174220 298475 303445 298869 SRFCCP 23029 51490 51427 52720 87252 89988 87416 MRCI 1362324 2103030 2134049 1849948 3938503 3397354 3879051 “Frozen-core SRCI; the number of CSF’s with only the Is orbitals frozen is 47758 for lAi. Table 5.3. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X experimental geometry from MCSCF1 calculations.0’6 STATEC SCF CSF No. of C SF’s with Coeff. coefficients > O.IO 0.05 0.02 5aj 3b? 6aj 7ai 8a j 9aj ld2 24] 35] 452 5 62 652 *Ai 0.964 3 9 21 -2.593 -2.572 -1.651 -1.493 -1.391 -0.065 -1.069 -1.326 -0.113 -1.262 -1.485 -0.032 2.000 2.000 2.000 1.980 1.989 0.040 1.937 1.979 0.084 2.000 1.970 0.021 3B, 0.957 2 10 29 -2.937 -2.762 -1.567 -1.598 -0.620 -0.017 -1.133 -1.363 -0.400 -1.246 -1.252 -0.049 (8a, - 36]) 1.981 1.025 0.014 1.974 1.992 1.033 2.000 1.938 0.043 3b 2 0.973 1 8 21 -2.638 -2.606 -1.680 -1.670 -1.211 -0.054 -0.737 -1.227 -0.445 -1.343 -1.515 -0.027 (la2 —> 36]) 1.990 1.981 0.038 1.020 1.975 1.004 2.000 1.972 0.019 3a 2 0.949 5 11 19 -2.828 -2.840 -1.735 -1.412 -1.099 •0.046 -1.179 -1.381 -0.381 -1.452 -0.718 -0.025 (5b2 - 36]) 1.911 1.982 0.044 1.962 1.994 1.042 2.000 1.041 0.024 'a 2 0.943 5 11 16 -2.810 -2.838 -1.722 -1.440 -1.096 -0.047 -1.178 -1.376 -0.382 -1.448 -0.710 -0.024 (5b2 - 3b]) 1.897 1.984 0.047 1.954 1.993 1.051 2.000 1.051 0.022 ’B] 0.936 4 10 27 -2.983 -2.796 -1.558 -1.574 -0.611 -0.027 -1.144 -1.369 -0.386 -1.256 -1.220 -0.046 (8a] - 3bj) 1.972 1.053 0.026 1.947 1.983 1.064 2.000 1.910 0.046 ] b 2 0.921 7 10 28 -2.772 -2.797 -1.685 -1.616 -1.088 -0.070 -0.721 -1.216 -0.366 -1.417 -1.172 -0.033 (la2 —> 3bj) 1.987 1.949 0.109 1.118 1.937 0.946 2.000 1.905 0.049 aRso = 1-4321 A (2.7063 bohr) and dOSo = 119.5360°. 410 electrons in 8 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. I Table 5.4. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X*Ai MCSCF1 optimized geometry from MCSCF1 calculations.0’6 STATE SCF CSF No. of C SF’s with Coeff. coefficients > •A CO 0.10 0.05 0.02 5ai 362 6ai 7«i 8ai 9ai l a 2 26x H 4 i2 562 662 ’A, 0.962 3 9 23 -2.600 -2.571 -1.655 -1.355 -1.495 -0.067 -1.060 -1.313 -0.119 -1.253 -1.474 -0.034 2.000 2.000 2.000 1.989 1.979 0.042 1.932 1.978 0.090 2.000 1.968 0.023 3B, 0.952 2 10 29 -2.920 -2.753 -1.564 -1.591 -0.625 -0.016 -1.128 -1.351 -0.406 -1.239 -1.244 -0.050 (8 3b 2 0.973 1 9 24 -2.634 -2.589 -1.678 -1.652 -1.207 -0.056 -0.734 -1.216 -0.450 -1.322 -1.527 -0.028 (la 2 —> 360 1.990 1.980 0.039 1.020 1.975 1.005 2.000 1.971 0.020 3a 2 0.946 5 11 20 -2.808 -2.824 -1.738 -1.399 -1.099 -0.047 -1.170 -1.367 -0.389 -1.444 -0.716 -0.026 (562 —> 360 1.907 1.981 0.046 1.960 1.994 1.045 2.000 1.043 0.025 *a2 0.940 5 11 17 -2.791 -2.821 -1.724 -1.427 -1.096 -0.049 -1.170 -1.363 -0.390 -1.440 -0.709 -0.024 (562 - 360 1.892 1.983 0.050 1.952 1.992 1.054 2.000 1.053 0.023 ‘ Bi 0.932 4 10 27 -2.963 -2.785 -1.557 -1.570 -0.614 -0.027 -1.137 -1.356 -0.395 -1.251 -1.210 -0.047 (8aj —* 360 1.972 1.056 0.026 1.943 1.983 1.068 2.000 1.905 0.048 ■b 2 0.917 7 10 29 -2.765 -2.782 -1.684 -1.595 -1.085 -0.075 -0.720 -1.202 -0.373 -1.406 -1.163 -0.034 (la 2 —> 360 1.985 1.950 0.117 1.122 1.934 0.944 2.000 1.899 0.048 aRso = 1-4511 A (2.7422 bohr) and 0Oso = 119.4877°. b 10 electrons in 8 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. oo o Table 5.5. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and nat ural orbital occupation numbers for all states at their respective MCSCF1 optimized geometries from MCSCF1 calculations.1,6 STATEC SCF CSF No. of CSF’s with Coeff. coefficients > 0.10 0.05 0.02 5a, 3i>2 6a, 7a, 8a, 9a, la 2 26, 35, 462 562 662 'A! 0.962 3 9 23 -2.600 -2.571 -1.655 -1.355 -1.495 -0.067 -1.060 -1.313 -0.119 -1.253 -1.474 -0.034 2.000 2.000 2.000 1.989 1.979 0.042 1.932 1.978 0.090 2.000 1.968 0.023 3B, 0.956 2 10 25 -2.879 -2.727 -1.555 -1.588 -0.624 -0.016 -1.112 -1.322 -0.424 -1.221 -1.216 -0.056 (8a, - 3fci) 1.981 1.035 0.014 1.961 1.990 1.047 2.000 1.919 0.052 3b 2 0.973 1 7 31 -2.602 -2.488 -1.667 -1.364 -1.376 -0.062 -0.697 -1.137 -0.494 -1.138 -1.506 -0.047 ( la 2 —► 36,) 1.993 1.965 0.044 1.015 1.975 1.008 2.000 1.963 0.035 3a 2 0.945 3 6 28 -2.729 -2.694 -1.767 -1.138 -1.353 -0.040 -1.090 -1.296 -0.441 -1.358 -0.687 -0.032 (562 — 3f>i) 1.895 1.977 0.038 1.942 1.993 1.064 2.000 1.061 0.031 >a 2 0.938 3 7 24 -2.717 -2.692 -1.755 -1.136 -1.365 -0.041 -1.089 -1.301 -0.433 -1.355 -0.682 -0.030 (562 36]) 1.878 1.978 0.040 1.930 1.991 1.077 2.000 1.076 0.029 *B, 0.917 3 7 28 -2.883 -2.718 -1.571 -1.524 -0.661 -0.027 -1.093 -1.298 -0.447 -1.234 -1.125 -0.058 (Sai —• 3bi ) 1.973 1.090 0.025 1.909 1.979 1.107 2.000 1.855 0.064 'B j 0.919 5 11 28 -2.673 -2.632 -1.698 -1.513 -1.058 -0.092 -0.691 -1.147 -0.390 -1.274 -1.143 -0.047 (lo2 — 36)) 1.980 1.941 0.117 1.119 1.922 0.959 2.000 1.902 0.060 “See Chapter VI. HO electrons in 8 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. 182 doubly occupied, for ease of comparison to the 5 b 2 and 6 b 2 orbitals and to similar tables for the M CSCF2 case (Section 5.2.2). As can be seen in the tables, there are several instances in which an active orbital has a lower Fock orbital energy than a doubly occupied orbital of the same symmetry. For the 3Bi and *Bi states the 7ai active orbital has a slightly lower Fock orbital energy than the corresponding 6a! doubly occupied orbital. This also occurs for the 462 and 5 b 2 orbitals for the !Ai, 3Bi, and 3B2 states. It is especially noticeable for the *Ai and 3B2 states, in which the 5 b 2 Fock orbital energy is « 0.2 hartrees below that of the 462 at the xAi experimental and MCSCF1 optimized geometries. This is even worse for the 3B2 state at its MCSCF1 optimized geom etry, where the energy difference is ~ 0.37 hartrees. The best and easiest way to investigate this possible problem would be to include the 462 orbital in the active space. 5.2.2 Discussion of MCSCF2 Active Space The MCSCF2 active space differs from that of the MCSCF1 only by the addition of the 462 orbital to the active space. This active space consists of 12 electrons in 9 orbitals, Ab 2~Qb2. As seen in Table 5.2, the number of configurations increases by slightly more than a factor of two. This increased the time required to perform the MCSCF calculations, but it was still reasonable. Tables 5.6, 5.7, and 5.8 contain 183 information about the active spaces for the MCSCF2 calculations corresponding to that presented for the MCSCF1 calculations. As can be seen from these tables, the problems discussed above for MCSCF1 concerning active space orbitals having lower Fock orbital energies than doubly occupied orbitals do not exist for this active space. Adding the 4b 2 orbital to the active space not only corrected the problems with the 462 and 5 b 2 orbitals but also those with the 6ax and 7ax orbitals. Also, at first glance it appears that the 7ax and 8ax orbitals for the 1,3A 2 states have switched at the X XAX experimental and MCSCF2 optimized geometries. The Fock orbital energy for the 8ax orbital is lower than that for the 7ax orbital, while the natural orbital occupation number for the 8ax orbital is larger than that for the 7ax orbital. To investigate this problem a program was written to calculate the overlaps between any two sets of molecular orbitals which were determined at the same point on the potential surfaces. The overlap between the MCSCF and SCF wavefunction was determined and the results indicated that there is considerable mixing of the 7ax and 8ax orbitals. However, the MCSCF “8ax” orbital is most like the SCF 7ax orbital and the MCSCF “7ax” orbital is most like the SCF 8ax orbital. Thus, the orbitals have not actually switched but just the orbital labels. There are two important reasons for the inclusion of the 4b 2 orbital in the active space. As mentioned in Chapter IV, the 4 b 2 and 7ax orbitals are essentially energetically degenerate. Also, in Cs symmetry these two orbitals both have a' Table 5.6. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the X experimental geometry from MCSCF2 calculations.a,i> STATE0 SCF CSF No. of CSF's with Coeff. coefficients > 0.10 0.05 0.02 5a, 362 6a, 7a, 8a, 9a, l a 2 26, 3b, 4 b2 5b2 6b2 'A, 0.960 3 10 19 -2.636 -2.463 -1.662 -1.659 -1.166 -0.068 -1.074 -1.327 -0.119 -1.550 -1.322 -0.048 2.000 2.000 2.000 1.987 1.977 0.045 1.934 1.978 0.089 1.990 1.969 0.032 3B, 0.959 2 8 37 -3.008 -2.594 -1.669 -1.421 -0.614 -0.049 -1.141 -1.363 -0.398 -1.543 -1.139 -0.044 (8ai -> 3b,) 1.983 1.034 0.041 1.972 1.991 1.036 1.975 1.932 0.038 3b 2 0.974 2 7 23 -2.679 -2.459 -1.688 -1.640 -1.185 -0.058 -0.742 -1.265 -0.412 -1.632 -1.382 -0.040 (la 2 —> 3b,) 1.989 1.978 0.042 1.025 1.968 1.006 1.991 1.971 0.030 3a 2 0.953 3 5 31 -2.753 -2.475 -1.710 -1.201 -1.408 -0.059 -1.177 -1.380 -0.387 -1.824 -0.718 -0.038 (5b2 -» 3b,) 1.917 1.980 0.047 1.960 1.992 1.046 1.985 1.045 0.028 ' a 2 0.946 3 6 27 -2.742 -2.479 -1.696 -1.187 -1.440 -0.060 -1.177 -1.384 -0.380 -1.814 -0.712 -0.036 (5b2 —* 3 b ,) 1.902 1.981 0.049 1.950 1.991 1.057 1.985 1.058 0.026 !B, 0.946 3 6 25 -2.738 -2.604 -1.674 -1.669 -0.636 -0.055 -1.156 -1.371 -0.372 -1.550 -1.149 -0.040 (8ai - 3b,) 1.988 1.059 0.040 1.947 1.990 1.061 1.973 1.906 0.035 ! b 2 0.920 5 10 28 -2.775 -2.498 -1.685 -1.642 -1.051 -0.078 -0.720 -1.268 -0.318 -1.729 -1.167 -0.048 ( la 2 3 b ,) 1.985 1.946 0.110 1.121 1.936 0.945 1.989 1.906 0.061 aRso = 1-4321 A (2.7063 bohr) and 60so = 119.5360°. {12 electrons in 9 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. T able 5.7. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and natural orbital occupation numbers for all states at the XhAi MCSCF2 optimized geometry from MCSCF2 calculations.0’1’ STATEC SCF CSF No. of CSF's with Coeff. coefficients > 0.10 0.05 0.02 5a, 362 6a, 7a, 8a, 9a, la2 26, 36, 462 5 b2 662 JAi 0.957 3 10 21 -2.639 -2.472 -1.667 -1.612 -1.179 -0.071 -1.064 -1.312 -0.125 -1.513 -1.321 -0.050 2.000 2.000 2.000 1.986 1.976 0.047 1.928 1.976 0.096 1.989 1.967 0.034 3b , 0.956 2 8 36 -2.973 -2.593 -1.697 -1.399 -0.620 -0.053 -1.136 -1.350 -0.406 -1.522 -1.134 -0.045 (8n, — 35,) 1.983 1.037 0.044 1.968 1.990 1.040 1.972 1.927 0.038 3B2 0.973 2 8 23 -2.675 -2.466 -1.690 -1.595 -1.200 -0.060 -0.739 -1.249 -0.421 -1.593 -1.381 -0.041 ( la 2 — 36,) 1.988 1.977 0.044 1.026 1.967 1.007 1.990 1.969 0.031 3a 2 0.950 3 5 33 -2.741 -2.484 -1.713 -1.189 -1.396 -0.061 -1.167 -1.364 -0.396 -1.788 -0.716 -0.039 (5i>2 - 36,) 1.913 1.978 0.049 1.957 1.992 1.050 1.984 1.048 0.029 ‘a 2 0.9-13 3 7 29 -2.730 -2.487 -1.700 -1.176 -1.427 -0.063 -1.167 -1.368 -0.389 -1.779 -0.711 -0.037 (562 - 36,) 1.897 1.980 0.052 1.947 1.990 1.061 1.984 1.061 0.028 *B, 0.943 3 7 26 -2.721 -2.601 -1.675 -1.655 -0.644 -0.058 -1.148 -1.357 -0.383 -1.533 -1.137 -0.042 (8a, - 36,) 1.987 1.064 0.043 1.943 1.990 1.066 1.971 1.900 0.037 ‘ B j 0.916 5 10 28 -2.765 -2.504 -1.685 -1.617 -1.049 -0.082 -0.719 -1.252 -0.326 -1.693 -1.156 -0.050 (la 2 —* 36,) 1.984 1.946 0.117 1.125 1.933 0.943 1.989 1.900 0.062 aRSo = 1.4511 A (2.7422 bohr) and 0Oso = 119.4877°. 412 electrons in 9 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. Table 5.8. The coefficient of the SCF CSF, number of CSF’s, diagonal elements of the Fock matrix and nat ural orbital occupation numbers for all states at their respective MCSCF2 optimized geometries from MCSCF2 calculations.2’4 STATE' SCF CSF No. of CSF's with Coeff. coefficients > 0.10 0.05 0.02 5 a. 362 6a 1 7ai 8aj 9a 1 102 2&i 36i 462 562 66 2 ’A , 0.957 3 10 21 -2.639 -2.472 -1.667 -1.612 -1.179 -0.071 -1.064 -1.312 -0.125 -1.513 -1.321 -0.050 2.000 2.000 2.000 1.986 1.976 0.047 1.928 1.976 0.096 1.989 1.967 0.034 3Bi' 0.940 2 12 41 -2.704 -2.593 -1.716 -1.506 -0.646 -0.079 -1.112 -1.287 -0.446 -1.443 -1.092 -0.049 ( 8 qi - 36i) 1.980 1.053 0.065 1.950 1.988 1.060 1.958 1.904 0.042 ^B2 0.966 1 7 37 -2.619 -2.487 -1.717 -1.398 -1.265 -0.070 -0.706 -1.148 -0.486 -1.408 -1.249 -0.057 (la2 —> 3&i) 1.988 1.965 0.054 1.024 1.965 1.010 1.962 1.986 0.045 3a 2 0.932 2 8 45 -2.679 -2.487 -1.752 -1.412 -1.105 -0.061 -1.075 -1.272 -0.467 -1.527 -0.681 -0.060 (562 -* 36]) 1.965 1.894 0.048 1.931 1.987 1.081 1.969 1.077 0.049 ‘Aj 0.922 3 10 39 -2.671 -2.485 -1.743 -1.421 -1.100 -0.061 -1.072 -1.276 -0.459 -1.524 -0.680 -0.058 (562 -> 360 1.966 1.874 0.049 1.913 1.986 1.099 1.968 1.098 0.047 ‘ Bi 0.906 3 10 37 -2.650 -2.555 -1.724 -1.498 -0.701 -0.073 -1.086 -1.263 -0.464 -1.439 -1.049 -0.057 (8ai —f 36i) 1.977 1.116 0.060 1.892 1.983 1.122 1.958 1.840 0.052 “B2 0.901 5 13 39 -2.665 -2.532 -1.720 -1.491 -1.037 -0.095 -0.697 -1.149 -0.389 -1.384 -1.111 -0.080 (la2 —> 36i) 1.978 1.915 0.118 1.139 1.913 0.950 1.977 1.905 0.105 aSee Chapter VI. 412 electrons in 9 active orbitals ; core orbitals not included. The first line for each state gives diagonal Fock-matrix elements, the second line natural orbital occupation numbers. CSCF description of the excited state. 187 symmetry. Furthermore, using the above-mentioned overlap program showed that there is a great deal of mixing of the 4 b 2 and 562 orbitals compared to the SCF wavefunction (the coefficients of the overlap matrix for these orbitals are almost equal). Since these two orbitals mix to such a great degree, they should both be included in the active space. After looking at the MCSCF2 results in the tables it appears that the 6a! and 7ai orbitals have very close Fock orbital energies for the xAi, 3B2, 1B1, and XB2 states at the xAi experimental and MCSCF2 optimized geometries (particularly the experimental geometry). This could signify that the 6ai orbital should be included in the active space. However, it was still left out for several reasons. First, as mentioned above, at the SCF level this orbital was « 0.2 hartree below the active space orbitals (while they were all within ~ 0.2 hartree of each other). Secondly, at the time the calculations were begun, adding this orbital would have made the active spaces too large to permit an extended study of the potential surfaces. Finally, the overlap program shows that the 6ai and 7a! orbitals do not mix to as great an extent as do the 4 b 2 and 5&2 orbitals. Thus, it is still believed that this orbital is not critical to the description of the states examined in the present study. Finally, since potential surfaces were calculated, one needs to be concerned with how the active space behaves for the entire part of the surface being studied. As can be seen in the tables, the coefficients of the SCF CSF’s change only slightly 188 as the geometry changes from the xAj experimental geometry to the respective MCSCF2 geometry of each state. All the coefficients remain greater than 0.9, with the largest change (~ 0.04) occurring for the xBi state. Also, th e fact that most of the coefficients at the respective MCSCF2 optimized geometry (Table 5.8) are less than 0.95 (« 0.90 for the xBi and *B 2 states) indicates that a multireference description of the states is necessary. This MCSCF active space was the one chosen to perform the precise geometry optimizations of the ground and excited states. It was also used to calculate excitation and emission energies and properties. 5.3 Cl Methods used in the Present Study Single and multi reference SDCI (SRCI and MRCI) calculations of excitation and emission energies and selected properties were performed in this study. The se lection of the reference space is the most important aspect of performing Cl cal culations. A Cl calculation is not much better than SCF if an incorrect reference space is used. This section discusses the Cl reference spaces and the resulting configuration spaces chosen for the calculations in the present study. 5.3.1 Discussion of SRCI Configuration Spaces. The SRCI calculations are very straightforward. The reference space for each of the states is simply the SCF configuration. The CSF’s are generated by allowing single and double excitations from the internal orbitals to all external orbitals. Two types of SRCI calculations were performed, differing in which orbitals were correlated. The case in which all SCF occupied orbitals, including the atomic core orbitals, were included in the internal space is referred to simply as SRCI. This case involved 32 electrons in 16 internal orbitals for the xAi ground state (17 orbitals for the excited states) being excited into 46 external orbitals (45 for the excited states). The other case, in which the atomic core orbitals (lai-4a l7 1 6 2 - 2 6 2 , and I 6 j) are frozen (no excitations from these orbitals were allowed), is referred to as SRFCCI. This case involved 18 electrons in 9 internal orbitals for the JAi ground state and 1 0 internal orbitals for the excited states being excited into the same number of external orbitals as for the SRCI case (no external orbitals were frozen). The number of CSF’s for both of these cases for all states is given in Table 5.2. The absolute value of the coefficients of the SCF CSF for each state at the X *Ai experimental geometry for the SRCI and SRFCCI cases, along with the values for the other methods used, are given in Table 5.9. The coefficients for the SRCI case are slightly larger than those for the SRFCCI case. A calculation for the xAi state with only the Is orbitals on sulfur and oxygen frozen (SRFCCI2) was also performed and produced 47758 CSF’s and a SCF CSF coefficient of 0.937419 (intermediate between those for SRCI and SRFCCI). The smallest SCF CSF co efficient is 0.927932 (SRFCCI) for the XB 2 state. The next largest CSF (CSF2) coefficient for any of the states is less than ~ 0.077, again for the 1 B2 state. 190 Table 5.9. The absolute value of the coefficients for the SCF CSF’s for all states for various calculation methods at the X xAi experimental geometry.0 Methodology % 3Bj 3b 2 3a 2 1a 2 % xb 2 SCF 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 MCSCF 1 0.964270 0.956530 0.973260 0.948980 0.943200 0.935590 0.920540 MCSCF2 0.959590 0.959460 0.973780 0.952570 0.945990 0.946420 0.919980 MCSCF3 0.755621 0.765055 0.933961 0.878858 0.870705 0.897670 0.860133 SRCI 0.938929 0.939269 0.941085 0.937326 0.937040 0.936046 0.932893 SRFCCI6 0.935888 0.935978 0.937320 0.933699 0.933167 0.932280 0.927932 MRCI 0.721988 0.739871 0.874390 0.835274 0.826101 0.843327 0.819441 aRSo = 1.4321 A (2.7063 bohr) and 6OSo = 119.5360°. ‘The xAi SCF CSF’s coefficient with only the Is orbitals frozen is 0.937419. 191 As one can see, the SRCI and SRFCCI descriptions of the states stays fairly consistent over the part of the potential surfaces studied. The SCF CSF coeffi cients decrease only slightly as the geometry changes to the respective MCSCF2 optimized geometry of each state. The largest change (ss 0.008) occurs for the 1Bi state and the smallest SCF CSF coefficient at the MCSCF2 optimized geometry is that for the 1B2 state. 5.3.2 Discussion of MRCI Configuration Spaces. Multireference Cl calculations are not as straightforward as SRCI. One needs to choose a reference space which includes, at the least, the most important config urations. Doing this for a potential surface study or for single point calculations at very different geometries is even more difficult since the most important con figurations can change. In this case one would need to include all the important configurations that occur at any point on the surface and use them at every point. For calculations of several states it is also desired to treat each state as equiva lently as possible, and this usually entails choosing “equivalent” reference spaces. Thus, the choice of the reference space (MCSCF configurations or active space) is of utmost importance. The MCSCF active space selection for the calculations per formed for the determination of the reference space for the MRCI and the MRCI configuration space are discussed below. 192 MRCI MCSCF active space selection The CASSCF spaces used for the MCSCF1 and MCSCF2 calculations were too large to be used a a reference space for M RCI calculations. A preliminary calcu lation for the X XAX state, using the MCSCF2-based CSF’s as the reference space, indicated there would be over 32 million C SF’s for the MRCI calculation. There fore, it was decided that a non-CAS MCSCF would be used to generate the MRCI reference space. This MCSCF treatment is referred to as MCSCF3 throughout this dissertation. This requires two steps: (1) deciding which orbitals should be included in the active space and ( 2 ) which excitations into the virtual orbitals would be allowed. Choice of active space orbitals for MCSCF3: The initial decision on which orbitals would be included in the MCSCF3 active space was based on the MCSCF 2 results. It was decided that including the orbitals present in the MCSCF2 active space ( 4 6 3 - 6 6 2 ) was necessary to get a good description of all the states over the geometries studied. It was also decided to include the 6 ai orbital since the 6 a x and 7ax orbitals have very close Fock orbital energies and there is a fairly large degree of mixing of these orbitals for several of th e states. This choice was supported by looking a t the SRFCCI results for all states at the X xAx experimental geometry. Table 5.3.2 gives the number of CSF’s with coefficients greater than 0.03 in which excitations from the SCF CSF occur into Table 5.10. The number of CSF’s with coefficients > 0.03 in which excitations from the SCF CSF occur into out of these orbitals as determined from SRFCCI calculations at the X 'Ai experimental geometry.*1 Num ber S tate of C S F ’s 6ai 7ai 8ai 9ai 12ai 13ai la 2 2a2 3a2 26i 36i 66i 462 562 662 762 ‘Ai 9 0 0 1 1 0 0 6 1 1 4 8 1 0 1 1 0 3Bi 4 0 0 2 0 0 0 2 0 0 1 3 0 1 1 0 0 (8ai -+ 36i) lg 9 2 1 4 0 1 0 2 0 0 3 5 0 2 2 2 0 (8fll - 3bv) 3b2 5 0 0 1 1 0 0 3 1 0 4 2 0 0 1 1 0 (la2 - 36x) % 12 0 1 4 3 1 1 8 0 0 8 3 1 0 4 3 1 (la2 - 360 3a 2 8 0 5 2 1 0 1 2 1 0 1 4 0 0 4 1 0 (5b2 —- 36i) jA 2 9 0 5 3 2 1 1 2 0 0 1 3 0 0 3 1 0 (5b2 - 360 aR s o = 1.4321 A (2.7063 bohr) and 9OSO = 119.5360°. or out of the orbitals listed in the tables. The orbitals in the table are the only ones which are involved in these CSF’s. For example, for the *Ai state, a 1 in the column with heading 8ai means that there is one CSF with a coefficient > 0.03 in which at least one electron has been excited from this SCF occupied orbital to one of the SCF virtual (external) orbitals. Likewise, a 1 in the 9ai column means there is one CSF with a coefficient > 0.03 in which at least one electron has been excited into this SCF virtual orbital. For the excited states, a one in the 3bi column (which has an orbital occupancy of one for the SCF CSF) means that there is a CSF in which an electron has been excited from this orbital (leaving it empty) or into this orbital (making it doubly occupied). Looking at the table one sees that all the orbitals chosen for the MCSCF3 active space are involved in an excitation in at least one CSF for the majority of states. The only exception is the 6ai orbital which is involved in two CSF’s only for the 1B1 state. If one considers CSF’s with coefficients > 0.02 then the 6ai orbital is involved in excitations for several CSF’s for several of the states. Also, looking at the overlap of the SRFCCI natural orbitals with the SCF orbitals for the xAi state, th e re is a large mixing of the 6a! and 7aj orbitals (the coefficient of these orbitals are almost equal), as there was for the MCSCF3 case. For this reason and since it seemed fairly important for the 1Bi state, the 6ai orbital was left in the MCSCF3 active space. 195 One might ask why the 5ax and 3 6 2 occupied orbitals were not included in the MCSCF3 active space. The overlap program results for the MCSCF2 and SCF show that, in general, the 5ax and 362 orbitals mix to a much smaller extent with the 6ai-8ai and 4 6 2 - 5 6 2 orbitals, respectively, than those orbitals mix among themselves. Also, the SRFCCI results indicate that the 5ai and 362 orbitals are not involved in any excitations for CSF’s with coefficients > 0.02 (except for the 1B2 state in which the 362 orbital is involved in one CSF with a coefficient as 0.026). Thus, these two orbitals were considered to be relatively unimportant to a multiconfiguration description of the states. One might also ask why the 2a2 virtual orbital was not included in the active space, since this orbital would seem to “complete” the active space (there would then be one virtual orbital of each symmetry type). The SRFCCI results show that even for CSF’s with coefficients > 0.02 this orbital is involved only a few times (no more than three CSF’s) for any of the states. As a matter of fact, there are many other virtual orbitals which are involved in as many CSF’s as is the 2a2 orbital. Finally, it should be kept in mind that the purpose of the MCSCF3 calculations was to produce a reference space for MRCI calculations. Since the 5ai, 362, and 2a2 orbitals seemed to be of very little importance in the SRFCCI calculations, it was felt that any configuration arising from excitations involving these orbitals would be of little importance in the reference space. Also, these configurations would appear as CSF’s in the MRCI descriptions. 196 E x citatio n s allowed for M CSCF3 calculations: Once the active space or bitals ( 6 0 1 - 6 6 2 ) were chosen, the next step was to decide which excitations would be allowed from the occupied or partially occupied orbitals (SCF CSF) into or out of the partially occupied and into the SCF virtual orbitals (i.e. which MRCI reference CSF’s would be generated). Again, the MCSCF 2 and SRFCCI results were used to determine which CSF’s seemed to be most important. For all of the states, at both the XJAj experimental geometry and the respective MCSCF2 optimized geometries, the MCSCF2 CSF’s with coefficients > 0.05 were studied to see what were the orbital occupations of the 36j, 9ai, and 6 6 2 orbitals. There were several instances in which the 36x orbital was doubly occupied, either by itself or with one electron also in the 9 ai or 6 6 2 orbitals (36J, 3 b\§a\, or 3 b] 6 b\). Also, the CSF with the 36i orbital doubly occupied (36J) usually has the second largest coefficient (> 0.10). The configurations which were most prevalent in num ber were those in which there was one electron in the 36j orbital and one electron in either the 9aj or 6 6 2 orbitals (36}9aj or 3 b\ 6 b\). These CSF’s also had the third largest coefficient, often > 0 . 1 (second largest for the 3 B2 state, with a coefficient « 0.10). There were only a few cases in which the 9ai or 6 6 2 orbitals were doubly occupied (9a3, 6 6 2 , 36}9a3, or 36J 6 6 | ) and in those cases the CSF coefficients were less that 0.07. This occurred only once or twice for a few of the states ^B ^ 3 B2, 1 Bt, and 3 B! in particular) at their MCSCF2 optimized geometries. 197 The xB 2 state seemed to be a little unusual compared to the other states. Its CSF’s with the second largest coefficient (> 0.18) and either the third or fourth largest coefficients > 0.14 involved an empty 3&x orbital ([ ]5b\lal3b°9a\ and [ ] 8 a]ld 2 3 6 j 6 6 2 )- Both of these CSF’s involved an excitation into the other orbital involved in the SCF CSF excitation (la2) and one of the other two SCF virtual orbitals (9ax or 6 6 2 )- The SRFCCI results shown in Table 5.3.2 indicate that the most important CSF’s are those in which the 36i orbital is involved. The XAX, 3 BX, 3 B2, XBX, and XA2 states’ second most important and the 3A2 state’s third most important CSF involves a doubly occupied 3 bx orbital. There are no cases for these CSF’s in which either the 9ax or 6 6 2 orbitals are doubly occupied but there are cases in which the 3bi and either the 9 a x or 6 b 2 orbitals are singly occupied (36}9a} or 36j662)- The xB 2 state’s most important CSF’s involve singly occupied 9ax or 6 b 2 orbitals. From the MCSCF 2 and SRFCCI results discussed above, it was decided that the MCSCF3 active space would involve only single and double excitations from the SCF CSF into the 3&i, 9ax, and 6 6 2 orbitals. Any CSF’s so produced would allow the 36x orbital to be doubly occupied but either the 9ax or 6 b 2 orbitals would only be singly occupied (but not both at the same time). The desired CSF’s would thus have the following orbital occupancies for the 36X, 9ax, and 6 6 2 orbitals: 3&i9a^66° (h \x SCF CSF), 3 6 }9 a ? 6 6 2 (excited state SCF CSF’s), 3bi9a\6b%, 36?9a?66j, 36}9aj66^, 3 b\ 9 a^6 b\, 36^9a?662, 36j9a}6h3, 3&x9a?66j. 198 To accomplish this for all the states required different excitations for the *Ai state and the excited states. For the XAX state no more than two electrons were allowed in the 36i, 9ai and 662 orbitals with the added stipulation that only one electron was allowed in the 9 ai and 6 6 2 orbitals (i.e. < 2 e~ in [36i9ai662] with < le- in [9 0 1 6 6 2 ])- For the excited states, three electrons were allowed in the 36l5 9ax and 6 6 2 orbitals, again with the stipulation that only one electron was allowed in the 9 0 4 and 662 orbitals (i.e. < 3e- in [36i9ax662] with < \e~ in [9ai662]). It is felt that the MCSCF3 active spaces treat all the states equivalently and produce the proper CSF’s to use as the reference space for the MRCI calculations. The number of MCSCF3 CSF’s and the coefficients of the SCF CSF are given in Tables 5.2 and 5.9, respectively. Finally, the atomic core orbitals were frozen at the SCF level while all other orbitals were MCSCF optimized. Generation of MRCI configuration spaces The MRCI calculations used the MCSCF3 CSF’s as the reference space for SDCI calculations. To get the MRCI CSF’s the 5a! and 362 orbitals were added to the MCSCF3 active space orbitals so that the MRCI internal orbitals consisted of the 12 orbitals, 5a!-662. Single and double excitations were then allowed out of these orbitals of the multi reference configurations to all the external orbitals (and to the MCSCF partially occupied orbitals). The number of CSF’s and SCF CSF coefficients are given in Tables 5.2 and 5.9, respectively. As one can see, the 199 smallest number of CSF’s is 1,362,334 for the XAX state and the largest number is 3,938,503 for the 3Bi state. It is felt that these MRCI configuration spaces are large enough to encompass all of the most important CSF’s for the states studied. Summary of MCSCF3 and MRCI configuration space selection As one can see from Table 5.9 the MCSCF3 SCF CSF coefficients for most of the states are considerably less than 0.95. The MCSCF3 CSF’s with coefficients >0.1 for all states, at the X XAX experimental and respective MCSCF2 optimized geometries are shown in Tables 5.11 and 5.12. As seen from the tables, the number of MCSCF3 CSF’s generally increases when going from the X XAX experimental geometry to the respective MCSCF2 optimized geometries. This is especially true for the 1,3B X states. Also, except for XBX and XB 2 , the CSF’s with coefficients > 0.1 and their relative ordering are fairly constant. As the geometry changes to the respective MCSCF2 optimized geometries the contribution to the wavefunction of the SCF CSF decreases for every state and that of the second most important CSF (CSF2) increases, except for the X-3A2 states in which the importance of several non-SCF CSF’s increases. The xBi state seems to exhibit the most dramatic differences at the different geometries. The contribution to the wavefunction of the SCF CSF decreases from ^ 81% at the XAX experimental geometry to sa 41% at the XBX MCSCF2 optimized geometry. The contribution to the wavefunction of CSF2 increases from « 7% to 200 Table 5.11* MCSCF3 C S F ’s with coefficients > 0 .1 for all states determined at the X *Ai experimental geometry.0 S ta te CSF# Coef. 6aj 462 7ai 26i 562 la 2 8a i 36i 9ai 662 »Ai 1 0.756 + - + - + - + - + - + - + - 11 0.525 + - + - + - + + - + - + ~ — 2 0.215 + - + - + - + - + - + - + + - - 7 0.179 + - + - + - + - + + - + - + - + - — 6 -0.137 + - + - + - + - + - + - + - 26 0.116 + - + + - + - + - + + - —- 3B i 1 -0.765 + - + - + - + - + - + - + + (8a i — 3fci) 24 0.393 + - + - + + - 4 ~ + - + - + 15 0.286 + - + - + - + + - + - + + - 8 0.237 + - + - + - + - + + - + + — 10 0.131 + - + - + - + - + + + - + - 59 -0.124 + + - + - -1— + - + - + + - ‘ B i 1 0.898 + - + - + - + - + - + - + — (8ai — 36l) 7 0.267 + - + - + - + - + + - + -- 8 -0.185 + - + - + - + - + - + - + - 22 -0.156 + - + - + + - + - + - + —- 13 0.116 + - + - + - + + - + - — + - 37 0.102 + - + + - + - + + - + - —- 3b 2 1 0.934 + - + - + - + - + - + + - + ( l a 2 3 6 i) 5 0.205 + - + - + - + - + - + + + - 10 -0.137 + - + - + - + - + + + - + - 3 0.114 + - + - + - H— + - - + + + 47 -0.104 + - + + - + + - + - + - + — >b 2 1 0.860 + - + - + - + - + - + + - — ( l a 2 —► 3bi) 4 -0.214 + - + - + - + - + - + + -- 6 0.207 + - + - + - H— + + - + - - 18 -0.196 + - + - + - + + + - + - -- 31 0.138 + - + + - + - + - + + - -- 2 -0.113 + - + - + - + - + - + - + - 27 -0 .1 1 0 + - + - + + + - + - + - -— 22 0.104 + - + - + + - + - + + - -- 14 0.103 + - + - + - + + - + - + -- 20 -0.102 + - + - + + - + - + - + - - 3A2 1 0.879 + - + - + - + - + + - + - + (5b2 —* 36l) 5 0.313 + - + - + - + - + + - + + - 15 -0.161 + - + - + - + + + - + - + - 10 0.155 + - + - + - H— + - + + + - 4 -0.126 + - + - + - + - + + - + - + 38 0.116 + - + + - + - + - + - + - + 41 -0.103 + - + + - + - + + - + - + - *a 2 1 0.871 + - + - + - + - + + - + - — (562 — 36i) 4 -0.342 + - + - + - + - + + - + -- 8 -0.183 + - + - + - + - + - + - + - 30 0.149 + - + + - + - + - + - + - - 13 0.149 + - + - + - + - + - + - + - 32 0.109 + - + + - + - + + - + - — - “The + and — entries indicate orbital occupancies and spin couplings. 201 Table 5.12. MCSCF3 CSF’s with coefficients > 0.1 for all states determined at the respective MCSCF 2 optimized geometries of each state. S ta te CSF# Coef. 6 a j 4 ^ 7aj 26! 5&2 1^2 8 oj 36j 9 a j 662 * A i 1 0.743 + - + - + - + - + - + - + - 11 0.536 + - + - + - + + - + - + - _ 2 0.216 + - + - + - + - + - + - + — 7 •0.179 + - + - + - + - + + - + - — 6 •0.143 + - + - + - + - + - + - + - 26 0.121 + - + + - + - + - + + - — — 16 0.102 + " + - + - + - + - + - + - 3 B l 1 0.726 + - + - + - + - + “ + “ + + (8 a , — 3 b ,) 24 -0.398 + - + - + + - + - + - + - + 15 •0.300 + - + “ + - + + - + - + + - 8 -0.242 + - + - + - + - + + - + + — 10 -0.171 + - + - + - + - + + + - + - 27 0.129 + - + - + + - + - + - + + — 35 •0.116 + - + - + + + - + - + - + - 59 0.116 + + - + - + - + - + - + + — 48 -0.112 + - + + - + - + + - + - + ~ _ l B l x 1 0.640 + - + - + - + - + - + - + (80 , — 3 b ,) 20 •0.459 + - + - + + - + “ + - + - _ 13 0.320 + - + “ + “ + + - + - — + - 7 0.281 + - + “ + - + + - + — 8 -0.264 + - + - + - + - + — + - + - 27 •0.178 + - + - + — + - + - + - + - 26 0.138 + - + - + + - + + - + - — 37 0.119 + - + + - + - + + - + - _— 45 •0.111 + + - + - + - + - + “ + - , 3 b 2 1 0.920 + - + - + - + - + - + + - + ( lo 2 — 3b , ) 5 0.210 + - + - + “ + - + - + + + — 10 ■0.175 + - + - + - + - + + - + — 3 0.110 + - + - + - + - + - — + + + ‘ b 2 1 0.828 + - + - + - + - + - + + - _ ( 1«2 — 3 b i) 22 -0.226 + - + - + + - + - + + - _ — 31 0.193 + - + + - + - + “ + + - _ — 14 0.193 + - + - + - + + - + - + _— 18 •0.186 + - + - + - + + + - + - _— 2 •0.168 + - + - + - + - + - + - + — 6 0.157 + - + - + - + - + + “ + - — 8 0.141 + - + - + - + - + + + - _— 12 •0.111 + - + - + - + + - + - — + — 36 •0.103 + + - + + - + - + - — — 21 •0.102 + - + - + + - + ~ “* + - + 1 0.849 + - + - + - + + - + - / 3A2 * + - + (562 — 3 bj) 5 0.245 + - + - + - + - + + - + + — 15 -0.222 + - + - + - + + + - + ~ + - 10 0.199 + - + - + - + - + - + + + - 38 0.167 + - + + - + ~ + “ + - + - + 41 -0.159 + - + + - + - + + “ + - + — 39 -0.111 + - + + - + - — + - + - + + 1 a 2 ^ 1 0.819 + - + - + “ + - + + - + - (5b2 — 3fri) 4 •0.265 + - + - + - + - + + - + _ — 30 0.259 + - + + - + - + - + - + - _ 8 •0.235 + - + - + - + - + - + — + - 13 0.211 + - + - + - + — + - + - + - 32 0.175 + - + + - + - + + - + - _ — 23 -0.102 + - + - + + - + “ + - + — — 202 « 21% at the *Bi optimized geometry. More importantly, the second C S F at the 1Bj optimized geometry has a coefficient of only 0.0496 (contribution of & 0.2%) at the ’Ai experimental geometry. Furthermore, there are only two C S F ’s with coefficients > 0 . 2 at the *AX experimental geometry while there are five of these CSF’s at the 1 B1 MCSCF2 optimized geometry. Finally, the CSF’s with coefficients > 0.1 contribute a minimum of 91% to the wavefunction (for the 3 BX state) at either geometry. The CSF’s for all states, except 3Bi and 1 B2, at the xAj experimental geometry and for the singlet states at the respective MCSCF2 optimized geometries account for at least 95% °f ^ e wavefunction. The MRCI CSF information is given in Tables 5.13 and 5.14. The CSF indices given in these tables are actually the corresponding MCSCF3 CSF indices rather than the MRCI indices since it makes comparisons easier. One can readily see that for the most part the MRCI CSF’s with coefficients > 0.1 are the same as the corresponding MCSCF3 CSF’s (the top 3 or 4 CSF’s are the same except for some minor changes in ordering). For all states, all of these CSF’s are MCSCF3 reference CSF’s. In addition, the first 11-23 CSF’s (depending on which state is considered) are reference CSF’s. As the geometry changes, the results are similar to the MCSCF3 results, al though the increase in numbers of CSF’s with coefficient > 0.1 is not as great. 203 As for the MCSCF3 calculations, t h e greatest changes occur for the xBi and XB 2 states. In addition, the contribution of C S F ’s with coefficients >0.1 to th e wavefunc tion varies from a minimum of 81% (3Bi and l Ba) to a maximum of 8 6 % (XA2), at both geometries considered. M ore importantly, th e MCSCF3 reference CSF’s account for 88-89% of the MRCI wavefunction. This implies that for these states the reference space adequately covers the most im portant configurations at the XA, experimental and the respective MCSCF2 optimized geometries. Also, the MRCI wavefunctions would seem to be good enough to give fairly accurate vertical and adiabatic excitation energies. However, since MRCI emission energies were also calculated at the excited states’ MCSCF2 optimized geometries it is important to look at the configuration spaces of the X xAi state at these geometries. The MCSCF3 and MRCI CSF’s for the *Ai state at these geometries a re given in Tables 5.15 and 5.16, respectively. In general, for both MCSCP 3 an(j MRCI CSF’s the CSF’s with coefficients > 0 . 1 are the same at all the excited state geometries. T he ordering 0f the CSF’s is also more or less the same at the various geometries (at least for the first few CSF’s). Also, for the most part, the MRCI CSF’s with coefficients > 0 . 1 are the same as the MCSCF3 reference CSF’s jn all cases the CSF’s labeled 1 (SCF CSF) and 11 (CSF with a 261 to 36j excitation) have nearly equal coefficients. One thing deserves special attention. The MCSCF3 SCF CSF’s at the 1,3A2 and 1B2 states’ MCSCF2 optimized geometries are not the primary CSF’s (CSF 11 has switched with the SCF CSF). However, in these cases the coefficients are essen tially equal. Also, the SCF CSF switches back to the primary CSF for the MRCI wavefunction. Furthermore, it is the MRCI results and not the MCSCF3 results that are of primary interest. Thus this “problem” with the MCSCF3 calculations was deemed to be of little importance. In summary, it is felt that the MRCI configurations spaces are large enough to produce a good description of the wavefunctions. Therefore, the MRCI wavefunc- tions should give fairly reliable excitation and emission energies and properties. 205 Table 5-13. MRCI CSF’s with coefficients > 0.1 for all states determined at the X *Ai experimental geometry. State C S F # ° Coef. 5ai 362 6a i 4!^ 7ai 2 bi 562 l a 2 8ai 36i 9a i 662 'A i 1 0.722 4— + - + - + - + - + - + - 4 ~ 4— 11 0.480 4— 4— + - + - + - + + - 4— 4— - 2 0.213 4 — 4— + - + - + - + - + - 4 ~ 4- - 7 0.185 4— 4— + - + - + - + - + 4 ~ 4— - 6 0.123 4— + - + - + - + - + - + - 4 ~ 4 ~ 3B, 1 0.740 4— + - + - + - + - + - + - 4— 4- 4- (8a i — 3 6 i) 15 0.277 4— + - + - + - + - + + - 4— 4- 4— 24 0.239 4 ~ + - + - + - + + - + - 4— 4 ~ 4- 56 0.237 4— + - + + - + - + - + - 4— 4— 4- 41 0.159 4— + - + - + + - + - + - 4— 4- 4- - 27 0.156 4 ~ + - H—■ + - + + - + - 4— 4- 4- - 8 0.155 4— + - + - + - + - + - + 4— 4- 4- - 25 0.104 4 ~ + - + - + - + + - + - 4— — 4- 4" JB i 1 0.843 4— + - + - + - + - 4— + - 4— 4- _ (8ai —► 3bi) 7 0.240 4 ~ + - + - + - + - + - + 4— 4- -- 8 0.158 4— + - + - + - + - + - + — 4— 4— 22 0.154 4 — + - + - + - + + - + - 4— 4- -- 13 0.115 4— + - + - + - + - + + - 4— - 4— 43 0.114 4 ~ + - + + - + - + - + - 4— 4 ~ — 3B 2 1 0.874 4— + - + - + - + - + - + - 4- 4 ~ 4- ( la 2 —«• 3i>i) 5 0.205 4— + - + - + - + - + - + - 4- 4- 4- - 10 0.144 4— + - + - + - + - + - + 4- 4— 4- - 3 0 .1 1 2 4— + - + - + - + - + - + - - 4- 4- 4* “Bz 1 0.819 4 ~ + - + - + - + - + - + - 4- 4 ~ — ( la 2 —>• 3 6 i) 4 0 .2 0 2 4— + - + - + - + - + - + - 4- 4- -- 6 0.185 4— + - +-* + - + - + - + 4— 4 ~ — 18 0.156 4— + - + - + - + - + + + - 4— -- 31 0.132 4— + - + - + + - H— + - 4- 4— -- 2 0 .111 4 ~ + - H— + - + - + - 4— 4— + - 20 0.101 4— + - + - + - + + - 4— 4— 4— - 3A2 1 0.835 4— + - + - + - + - + - 4- 4— 4— 4- (562 — 3 6 i) 5 0.278 4— + - + - + - + - + - 4- 4— 4- 4- - 15 0.155 4— + - + - + - + - + 4- 4 ~ 4— 4— 10 0.127 4— + - + - + - + - H— 4— 4- 4- 4— 38 0.121 4— + - + - + + - H— 4— 4— 4— 4- 4 0.109 4— + - + - + - + - + - 4- 4— 4- - 4- •a 2 1 0.826 4 ~ + - + - + - + - + - 4- 4— 4— _ (562 — 3 6 i) 4 0.298 4— + - + - + - + - + - 4- 4— 4- -- 8 0 .1 6 8 4 ~ + - + - + - + - + - 4 ~ 4- - 4— 30 0.157 4— + - + - + + - + - 4— 4— 4— - 13 0.143 4— + - + - + - + - + - 4— 4— 4— 32 0.106 + - + - + - + + - H— 4- 4— 4— - - “The C SF numbers refer to the MCSCF3 CSF numbers and not those generated in the Cl calculation. 206 T able 5.14. MRCI CSF’s with coefficients > 0.1 for all states determined at the respective MCSCF2 optimized geometries of each state. State CSF#“ Coef. 5ai 362 6ai 462 7 a i 261 562 l a 2 8 a 1 36j 9a 1 6bi ‘A, 1 0.711 4— 4— + - + - + - 4 ~ 4-- 4— 4 ~ 11 0.489 4— 4— + - + - H— 4- 4 - 4— 4 ~ - 2 0.214 4— + - + - + - + - 4— 4 ~ 4— 4- - 7 0.186 4— 4— +- +- + - 4— 4- 4— 4 ~ — 6 0.129 4 ~ 4— + - + - + - 4— 4 ~ 4 ~ 4— 3B> 1 0.705 4— 4— + - + - 4— 4— 4-- 4 ~ 4- 4- (8 0! - 360 24 0.334 4 ~ 4 ~ + - + - + 4— + - 4— 4— 4- 15 0.292 4— 4— + - + - + - 4- 4 ~ 4— 4- 4— 8 0.229 4— 4— + - + - 4— 4 ~ 4- 4— 4- 4- — 27 0.132 4— 4— + - + - + 4 ~ 4-- 4— 4- 4- - 10 0.131 4— 4— + - + - + - 4 ~ 4- 4- 4 ~ 4-- 59 0.120 4 ~ 4— + + - + - 4— 4 - 4— 4- 4- - 35 0.101 4 ~ 4 ~ + - + - + 4- 4-- 4— 4 ~ 4— ‘Bi 1 0.619 4— 4— + - + - + - 4 ~ 4— 4— + - (8oi - 3 6 j ) 20 0.414 4— 4— + - + - + 4— 4 - 4— 4 ~ - 13 0.313 4— + - + - + - + - 4- 4-- 4 ~ — 4— 7 0.261 4— + - + - + - + - 4— 4- 4— 4- - — 8 0.233 4 ~ + - + - + - 4— 4— 4- _ 4— 4"- 27 0.157 4— + - + - + - + - +~ 4— 4— 4— 26 0.115 4 ~ + - + - + - 4" 4— + 4— 4— -— 45 0.114 4— + - + + - + - 4— 4-— 4— 4- — - 3b2 1 0.850 + - + - + - + - + - 4— 4 - 4- 4— 4- (lo 2 -» 3 6i ) 5 0.209 + - + - + - + - + - 4— 4 ~ 4- 4- 4- - 10 0.183 4 ~ + - + - + - + - 4— 4- 4- 4— 4- — 3 0.107 4— + - + - + - 4— 4— 4-- - 4- 4- 4- ‘b2 1 0.787 4 ~ + - + - + - 4— 4— 4-- 4- 4— - (lo 2 - 3 6 ! ) 22 0.205 4 ~ + - + - + - 4- 4— 4-- 4- 4 ~ -- 31 0.182 4— + - + - + 4 ~ 4— 4 ~ + 4— -— 14 0.166 4— + - + - + - 4— 4- + - + - 4- - — 2 0.164 4 ~ + - + - + - 4— 4 ~ 4— + - 4- — 18 0.150 4 ~ + - + - + - 4— 4- 4- 4— 4 ~ -- 8 0.143 4 ~ + - + - + - 4— 4— 4- 4- 4 ~ -— 6 0.141 4— + - + - + - 4— 4 ~ 4- 4— 4 ~ - 12 0.105 4— + - + - + - 4— 4- 4— 4— — 4- — 3a 2 1 0.804 4 ~ + - + - H— 4— 4— 4- 4— 4 ~ 4- (562 —*■ 36l ) 5 0.214 4— + - + - H— 4— 4 ~ 4- 4— 4- 4- - 15 0.208 4 ~ + - +- +- 4— 4- 4- 4— 4 ~ 4— 38 0.167 4 ~ + - +- + 4— 4— 4— 4 -- 4— 4- 10 0.162 4 ~ + - + - + - 4— 4— + ~ + 4- 4— 41 0.156 4— + - +- + 4— 4— + 4— 4— 4- - 39 0.107 +- + - +- + 4— 4— - 4— 4— 4- 4- — *a2 1 0.773 4 ~ + - + - + - 4— 4— 4- 4— 4 ~ (562 - 36i) 30 0.259 4— + - + - + 4— 4— 4-- 4— 4— - 4 0.224 4— + - + - 4— 4— 4— 4- 4— 4- - - 8 0.213 4— + - + - + - 4— 4 ~ 4-- 4- - 4— 13 0.197 4— + - + - + - 4— 4- - 4— 4— 4 ~ 32 0.172 + - + - + - + 4— 4— 4- 4— 4— — — “The CSF numbers refer to the MCSCF3 CSF numbers and not those generated in the Cl calculation. 207 Table 5.15. MCSCF3 CSF’s with coefficients > 0.1 for the state determined at the respective MCSCF2 optimized geometries of each state. G eom etry C S F # Coef. 6 a j 462 7®l 2 6 , 5 6j l a 2 8 a , 3 6 , 9 a , 662 1a 1 0.743 + - 4 ~ 4 - 4 - 4 - 4 - 4 “ 11 0.536 4 - 4 “ 4 - 4 4 - 4 - 4 ~ — 2 0.216 4 - 4 “ 4 - 4 - 4 - 4 - 4 — 7 -0.179 + - 4 “ 4 - 4 - 4 4 - 4 - — 6 -0.143 4 - H ~ 4 - 4 - 4 - 4 - 4 - 26 0.121 4 “ 4 4 - 4 - 4 - 4 4 - — — 16 0.102 4 - + - 4 - 4 — 4 - 4 - 4 - 1 0.677 4 - + - 4 - 4 - 4 - 4 - 4 ~ 11 0.582 + - + - 4 — 4 4 — 4 - 4 - — 2 -0.224 + - + - 4 - 4 - 4 — 4 - 4 — 6 •0.174 + - + - 4- 4 - 4 - 4 - 4 - 7 -0.169 4 - + - 4 - 4 — 4 4 — 4 - — 16 0.137 4 - + - 4 - 4 - 4 - 4 - 4 - 26 -0.136 4 - + 4 - 4 - 4 - 4 4 - —— 13 0.128 4 - + - 4 — 4 4 — 4 - 4 —— 20 -0.104 4 - 4 - 4 4 - 4 — 4 4 - - — 1 0.622 + - 4 - 4 - 4 - 4 - 4 - 4 - 11 0.617 4 - 4 - 4 - 4 4 - 4 - 4 “ — 6 -0.216 4 - 4 - 4 - 4 - 4 - 4 - 4 - 7 -0.195 + - 4 - 4 - 4 - 4 4 - 4 - — 2 -0.184 + - 4 - 4 - 4 - 4 - 4 - 4 — 16 0.179 4 - 4 - 4 - 4 - 4 - 4 - 4 - 26 0.130 4 - 4 4 - 4 - 4 — 4 4 - —— 13 0.115 4 - 4 - 4 - 4 4 - 4 - 4 —— 20 0.115 + - 4 - 4 4 - 4 - 4 4 - —— 15 0.111 + - 4 - 4 - 4 4 4 - 4 - —— 11 0.618 + - 4 - 4 - 4 4 - 4 - 4 - 1 0.618 + - 4 - 4 - 4 - 4 - 4- 4 - 6 •0.224 + - 4 - 4 - 4 — 4 - 4 - 4 - 7 -0.213 + - 4 - 4 - 4 — 4 4 - 4 - — 16 0.188 + - 4 - 4 - 4 - 4 - 4 - 4 - 2 •0.161 + - 4 - 4 - 4 - 4 - 4 - 4 — 15 0.124 + - 4 - 4 - 4 4 4 - 4 - — — 26 0.117 + - 4 4 - 4 - 4 - 4 4 - —— 20 -0.106 + - 4 - 4 4 - 4 - 4 4 - — — 11 0.619 + - 4 - 4 - 4 4 - 4 - 4 - _ 1 0.617 + - 4 - 4 - 4 — 4 - 4 - 4 - 6 -0.224 + - 4 - 4 - 4 - 4 - 4 - 4 - 7 -0.213 + - 4 - 4 - 4 - 4 4 - 4 - — 16 0.188 + - 4 - 4 - 4 - 4 - 4 - 4 - 2 • 0.161 + - 4 - 4 - 4 - 4 - 4 - 4 — 15 0.123 + - 4 - 4 - 4 4 4- 4 - — — 26 0.117 + - 4 4 - 4 - 4- 4 4- —— 20 -0.107 + - 4 - 4 4- 4- 4 4- -— 1 0.640 + - 4 - 4 — 4- 4- 4- 4- 11 0.606 + - 4 - 4 — 4 4 - 4- 4 - — 2 -0.204 + - 4 - 4 - 4 - 4 “ 4- 4 — 6 -0.200 + - 4 - 4 - 4 - 4- 4 - 4 - 7 •0.181 + - 4 - 4 - 4 - 4 4~ 4 - — 16 0.163 + - 4 - 4 - 4 — 4- 4- 4- 26 -0.136 4- 4 4- 4- 4- 4 4- —— 13 0.126 4- 4 - 4- 4 4 ~ 4- 4 —— 20 0.112 4- 4- 4 4 — 4- 4 4- — 11 0.619 4 - 4 - 4- 4 4- 4- 4- _ 1 0.617 4 - 4 — 4 — 4- 4- 4- 4- 6 -0.218 4- 4 - 4 - 4- 4- 4- 4- 7 -0.192 4 - 4 - 4 - 4 - 4 4 - 4 - — 2 -0.187 4 - 4 - 4 - 4 - 4- 4 - 4 — 16 0.181 4 - 4 - 4 - 4 - 4 — 4 - 4 - 26 0.132 4 - 4 4 - 4 - 4- 4 4 - —— 13 0.120 4 - 4 - 4 - 4 4- 4- 4 — — 20 0.115 4- 4 - 4 4- 4- 4 4- — — 15 0.111 4- 4- 4- 4 4 4- 4- — — 208 Table 5.16. MRCI CSF’s with coefficients > 0.1 for the X *Ai state determined at the respective MCSCF2 optimized geometries of each state. G eom etry C S F # ° Coef. 5 a j 362 6 a j 4 6 j 7aj 26j 562 la 2 Ba] 3 6j 9aj 662 'A, 1 0.711 + - + - + - + - + - + - + - + - + - 11 0.489 +- +- + - + - + - + + - + - + - — 2 0.214 + - + - + - + - + - + - + - + - + — 7 0.186 + - + - + - + - + - + - + + - + - — 6 0.129 + - +- + - + - + - + - + - + - + - 3Bj 1 0.651 + - +- + - + - + - + - + - + - + - 11 0.529 +- +- + - + - + - + + - + - + - — 2 0.224 + - + - + - + - + - + - + - + - + — 7 0.173 + - + - + - + - + - + - + + - + - — 6 0.156 + - + - + - + - + - + - + - + - + - 13 0.120 + - + - + - + - + - + + - + - + — — 16 0.117 + - + - + - + - + - + - + - + - + - 3b 2 1 0.601 + - + - + - + - + - + - + - + - + - 11 0.564 + - + - + - + - + - + + - + - + - — 7 0.204 + - + - + - + - + - + - + + - + - — 6 0.191 + - + - + - + - + - + - + - + - + - 2 0.183 + - + - + - + - + - + - + - + - + — 16 0.156 + - + - + - + - + - + - + - + - + - 15 0.111 + - + - + - + - + - + + + - + - — — 13 0.106 + - + - + - + - + - + + - + - + — - 3a2 1 0.597 + - + - + - + - + - + - + - + - + - 11 0.567 + - + - + - + - + - + + - + - + - — 7 0.223 + - + - + - + - + - + - + + - + - — 6 0.197 + - + - + - + - + - + - + - + - + - 16 0.164 + - + - + - + - + - + - + - + - + ~ 2 0.158 + - + - + - + - + - + - + - + - + — 15 0.123 + - + - + - + - + - + + + - + - — - *a 2 1 0.596 + - + - + - + - + - + - + - + - + - 11 0.567 + - + - + - + - + - + + - + - + - — 7 0.223 + - + - + - + - + - + - + + - + - — 6 0.197 + - + - + - + - + - + - + - + - + - 16 0.164 + - + - + - + - + - + - + - + - + - 2 0.158 + - + - + - + - + - + - + - + - + — 15 0.123 + - + - + - + - + - + + + - + - ~ ^ l 1 0.617 + - + - + - + - + - + - + - + - + - 11 0.553 + - + - + - + - + - + + - + - + - — 2 0.204 + - + - + - + - + - + - + - + - + — 7 0.188 + - + - + - + - + - + - + + - + - — 6 0.177 + - + - + - + - + - + - + - + - + - 16 0.141 + - + - + - + - + - + - + - + - + - 13 0.117 + - + - + - + - + - + + - + - + —- 1b 2 1 0.596 + - + - + - 4- 4- 4- 4- 4- 4- 11 0.566 + - + - + - 4- 4- 4 4- 4- 4- — 7 0.200 + - + - + - 4- 4 - 4- 4 4- 4- — 6 0.192 + - + - + - 4 — 4- 4- 4- 4- 4- 2 0.186 + - + - + - 4- 4- 4- 4- 4- 4 — 16 0.158 + - + ~ + - 4- 4- 4- 4- 4- 4- 13 0.110 + - + - + - 4- 4- 4 4- 4- 4 — _ 15 0.110 + ~ + - 4 — 4- 4- 4 4 4- 4- - - “The CSF numbers refer to the MCSCF3 CSF numbers and not those generated in the Cl calculation. C H A P T E R V I Discussion of Potential Surface Calculations 6.1 Introduction This chapter presents the results for the geometry optimizations at the SCF and MCSCF levels. The adiabatic and vertical excitation energies, Te and A Evert, re spectively, and the emission energies at the experimental and optimized geometries calculated at the SCF, MCSCF, SRCI, and MRCI levels are also reported. Finally, approximate harmonic frequencies calculated at the MCSCF level are presented. 6.2 Geometry Optimization Procedures The geometry optimizations performed for this dissertation were done using the MINPT program developed by the Shavitt research group.305-308 This program uses the computed values of the energy and energy derivatives (if available from a gradient program) and the corresponding geometries (internal or Cartesian coordi nates) and performs a least-squares fit of the data to a polynomial. The resulting polynomial fit of the potential energy surface is then used to locate an appropri 209 210 ate stationary point (in this case the minimum energy point of the surface). The polynomial can also be used to determine the Hessian matrix elements which can then be used to find the vibrational frequencies and the location of reaction paths on the potential surface. This will not be discussed since the main interest is to determine the potential surfaces and optimized geometries. The procedure for finding the minimum of a potential surface (optimized ge ometry) using MINPT is an iterative process consisting of several steps: 1. Calculate enough energy and analytical gradient points to fit a polynomial (usually a quadratic) to the desired degrees of freedom (in the case of C 2 V symmetry, two degrees of freedom). This first set of points would normally include an experimental or previously optimized geometry, if available. 2. Locate the minimum of the polynomial. This is done by using MINPT to approximate the potential energy surface with a polynomial and obtain the minimum of the surface. Also, the Hessian eigenvalues are checked to insure that this point is in the region of a minimum of the surface (all positive eigenvalues). 3. Calculate the energy and gradients (if being used) at the minimum. 4. Repeat this process with new data points (most often using the latest point, if close to the minimum) until the desired precision of the stationary point 211 is obtained, or until the analytical energy gradients, if calculated, become sufficiently small, i.e., • A E= 1.0 x 10“ 10 hartrees, • A x = 1.0 x 10~8 bohrs, • analytical gradients = 1.0 x 10-6 hartrees/bohr. In general, using energies only, the minimum number of points needed to fit a general order polynomial to a general number of degrees of freedom is given by (nr) <“> where n is the number of degrees of freedom and m is the order of the polynomial. For example, for S02 in C 2 v symmetry there are two degrees of freedom and to fit a second order polynomial (quadratic) requires 6 points, a third order polynomial (cubic) requires 10 points, etc. The number of points needed to fit a polynomial (and find the minimum in the potential surface) can be reduced dramatically by including analytical energy derivatives, especially when there are many degrees of freedom. For cases in which there are few degrees of freedom, such as for SO 2 in C2„ symmetry, the Value of calculating the analytical gradients is reduced. In fact 212 in terms of computer time, it can be less costly to calculate only the energies at more points than to calculate the energy and gradients at fewer points. For S 02, using C2u symmetry restrictions, there are only two independent de grees of freedom which need to be minimized. For most of the calculations, the minimization was done in terms of the internal degrees of freedom, Rso and 60so- This was done for two reasons. Primarily, the present MCSCF gradient program (in which case Cartesian coordinates must be used) was not available when the calculations were started. Secondly, the gradient calculations, especially at the MCSCF level, take considerably more computer time to run than do the integral and energy calculations. As explained below, this may not be a sufficient reason not to calculate the energy gradients, as it can come down to the question of whose time is more valuable, the computer’s or the investigator’s. 6.2.1 Minimizations without Gradients For the minimizations performed, in which energy gradients were not calculated, the procedure was as follows. A single point is chosen, the experimental geometry or a partially optimized geometry, and several other points around this point are generated by varying the bond distance and angle by some amounts, which decrease in size as the central point gets closer to the optimized geometry. This requires a minimum of six points for each ’’pass” in this iterative scheme, the central point, two points with R = ± A R, two points with 6 = ± A 0, and one point in which both the bond distance and angle change by some amount from the central point. These points are then used in MINPT to fit the energy to a quadratic in R and 0- The calculated minimum from MINPT is usually then used as the central point for the next set of points. However, the accuracy of the prediction depends on how well the covered region of the potential energy surface is represented by a quadratic polynomial. This problem can generally be taken care of by removing previous points which are relatively far from the predicted minimum, which produces a better fit to the data points close to the new minimum. Another way to take care of of these problems, at least until one has enough points very close to the desired minimum, is to use a higher order polynomial (such as a cubic or quartic) to fit the data. There have been cases where this actually seemed to work as well or better than throwing out a lot of the points (see Section 6.3). Another problem that one encounters as the minimum is approached and the points used are rather close together, is the precision limits of the calculation. This can limit th e precision that one can obtain in the optimized geometry to only a few decimal places. This is especially true in cases in which the potential energy surface is very flat, as it is for some of the excited states of SO 2 (especially in th e angular degree of freedom). In these cases, even moderate changes in the geometry often p r o d u c e only small changes in the total energy. Many times these changes in the energy are barely within the precision of the calculations (about 12 decimal places). In fact as will be pointed out below, the predicted minimum point may actually have a higher 214 calculated energy than a neighboring point, and the only way to know which point is actually the minimum is by calculating the energy gradient. In these cases the MINPT program’s precision in fitting the energy data is limited and one has to place m ore importance on the fit to the gradients. The precision of the optimized geometries using this method is usually within ±0.005 bohr for the bond distance and ±0.01° for the angle. The energy gradients at the minimum points, determined by this procedure, are on the order of « 1 x 10-4 hartree/bohr (for Cartesian gradients) or smaller. 6.2.2 Minimizations with Gradients The procedure for minimizations with energy gradients is similar to the above but there are certain aspects of it that warrant further discussion. At first glance, since SO2 has only two degrees of freedom (C 2 U symmetry) which need to be optimized, it would seem that the calculation of energy gradients would not be warranted. However, for a minimization with gradients for SO 2 in C 21, symmetry, the minimum number of points needed to fit a quadratic is reduced to three. More importantly, using gradients provides a better fit and an accurate min imum can be obtained more quickly. Also, points that are not very close to the minimum can be retained and a higher order polynomial (than a quadratic) can be used. The higher order polynomials assure that the computed surface will fit the data points very well, especially near the minimum (errors between the observed 215 and calculated energies on the order of 10"9 hartree or better). The polynomial can be forced to fit the points near the minimum better than those farther away by giving them larger energy and gradient weights. By using gradients and the highest order polynomial possible the cyclic proce dure discussed above of choosing a central point and several points surrounding it can quickly be abandoned. The first spread of points needs to be just large enough to cover the region of the potential surface where the minimum is located. If this produces a very good quadratic or moderately good higher order polynomial then one just needs to start adding the successive minima determined by MINPT to those already calculated, raising the order of the polynomial whenever possible, until the desired precision in the geometry is obtained. It was found during the MCSCF2 minimizations performed for SO 2 that, as soon as enough points were obtained to fit a cubic and then a quartic, the minimum (with gradients on the order of 10-9 hartree/bohr or better) could be found using only 7 or 8 data points. This means that the cyclic procedure of choosing a central point and surrounding points only consisted of one step. Of course the optimizations were started with experimental or previously obtained optimized geometries so another cycle may have been necessary if the optimization had been started very far from the minimum. As mentioned above, using gradients for S 0 2 may seem a little more costly than just using the non-gradient procedure (since gradient calculations require consid 216 erably more time than energy calculations). However, the number of data points needed to obtain a very accurate optimized geometry can be reduced significantly and the total computer time and cost may be only slightly higher than not using gradients. Also, this can significantly cut down the time the investigator spends on the optimizations, which is more important than the extra computer time required. Finally, for minimizations with gradients for small molecules with only a few de grees of freedom, very accurate optimized geometries can be obtained. Optimized geometries with gradients on the order of 10~9 hartree/bohr are easily obtained, giving about 7 or 8 digits of precision for the geometries. This gives Cartesian coordinates to ~ ±10~7 bohrs, which translates to « ±10-7 A for bond distances and between ±10-4 and dhlO-5 degrees for the angle (depending on the size of the angle). 6.3 SCF Geometry Optimizations Full gradient optimizations at the SCF level were only performed for the xAi and xB2 states. By this it is meant that several SCF geometry points with gradients were used to fit the surface to polynomials of various order (2nd, 3rd and 4t/l) using MINPT. The final Cartesian gradients for the xAi and 1B2 states were less than 3.6 x 10~6 and 1.4 x 10~4 hartrees/bohr, respectively. The results of these fits and the SCF optimized geometries are given in Table 6-1. As can be seen, the cubic and quartic fits fairly accurately predict the actual calculated optimized 217 Table 6.1. Comparison of MINPT fits to SCF data for all states. State Present Calculations Previous Calculations0 No. of Rso Ooso E + 547 T t Rso Ooso E -f 547 T e Terms4 (bohr) (deg) (hartree) (eV) (bohr) (deg) (hartree) (eV) xAr calc. 2.6549 118.8201 -0.233839 0.000 2.653 118.79 -0.239412 0.000 6 2.6532 118.6938 -0.233840 0.000 10 2.6547 118.8541 -0.233839 0.000 15 2.6549 118.8269 -0.233839 0.000 3 b 2 calc. 2.9580 103.8700 -0.163744 1.907 2.955 103.50 -0.169792 1.894 6 2.9460 104.9679 -0.163817 1.905 10 2.9580 103.8507 -0.163743 1.907 15 2.9580 103.8741 -0.163744 1.907 3 B i calc. 2.7400 127.2860 -0.120255 3.091 2.740 127.00 -0.125513 3.099 6 2.7312 127.2832 -0.120682 3.079 10 2.7394 127.4203 -0.120252 3.091 15 2.7401 127.2863 -0.120255 3.091 3a 2 calc. 2.8340 93.1510 -0.116572 3.191 2.840 92.65 -0.122404 3.184 6 2.8276 91.8843 -0.116560 3.191 10 2.8334 93.1500 -0.116577 3.191 15 2.8342 93.1506 -0.116572 3.191 xa 2 calc. 2.8360 92.9440 -0.106729 3.459 2.840 92.65 -0.112585 3.451 6 2.8271 91.5210 -0.106822 3.456 10 2.8353 92.8726 -0.106728 3.459 15 2.8359 92.9440 -0.106729 3.459 XB! calc. 2.8250 117.8340 -0.069497 4.472 2.825 117.30 -0.075142 4.470 6 2.8138 116.9199 -0.069547 4.471 10 2.8243 117.7746 -0.069498 4.472 15 2.8246 117.8336 -0.069500 4.472 xb 2 calc. 2.8720 98.8000 -0.011643 6.046 2.880 101.50 -0.016935 6.054 6 2.8719 98.6781 -0.011643 6.046 10 2.8717 98.7880 -0.011643 6.046 15 2.8717 98.7877 -0.011643 6.046 ° Results of Lindley 22 6The first entry for each state ( “calc.”) represents the results at the most accurate minimum. The following entries represent the predictions of quadratic, cubic, and quartic fits, in order. 218 geometries. The norm of the residuals is « 10-9. If these calculated optimized geometries are added to the fits then the new quartic predicted geometries, total energies, changes in y and z Cartesian coordinates and change in energy for the xAi state are 2.6548761 bohrs, 118.81986°, —547.2338386019 hartrees, « 10~6 bohrs and ~ 10-11 hartrees. Likewise, for the XB2 state these values are 2.871712 bohrs, 98.7892°, —547.0116425876 hartrees, ps 10-4 bohrs and « 10-8 hartrees. The other states were not optimized in the same manner as the xAi and XB2 states. Instead only the SCF energies calculated at the MCSCF2 geometry opti mization points were fit by MINPT. Polynomial fits up through quartic order were used to estimate the optimized geometries. The predicted optimized geometry from the quartic fit was then use to perform one additional calculation and obtain the total energy and gradients. These results, along with the MINPT predicted geometries and energies for the various states, are given in Table 6.1. The calcu lated Cartesian gradients at the optimized geometries range between 3.2 x 10~4 (for the 1B1 state) and 6.1 X 10-6 hartree/bohr (for the XA2 state). As can be seen, the quartic fits using this method predicted the optimized geometries and energies pretty well and about as well as that used for the xAi and XB2 states. The cubic fits do about as well as the quartic fits in predicting the bond distances but not as well in predicting the bond angles. This supports the contention presented in Section 6.2 that the highest order polynomial that the data allows should be 219 used to fit the data even if it means keeping data points that might otherwise be discarded when fitting a quadratic. These results can be compared to the previous results of Lindley22 which are also presented in Table 6.1. As can be seen, the basis set used in this study produces optimized geometries that are very close to those obtained using the larger basis set. The new bond distances are within 0.01 bohr of the previous bond distances. The largest differences are for the angles of the 3A2 and 1B2 states, 0.5°and 2.7°, respectively. As discussed in Chapter IV, Lindley did not fully optimize the *B2 sta te so the large difference in angle between his result and the current values is m ore likely due to this fact rather than the change in basis set. Also given in the table are the adiabatic excitation energies, Te, from both studies. The new Te values are within about 0.01 eV of Lindley’s results. These results, along with those for the geometries, indicate that this basis set adequately predicts SCF optimized geometries and adiabatic excitation energies. 6.4 MCSCF Geometry Optimizations As mentioned in Chapter V two types of MCSCF active spaces were employed for the MCSCF geometry optimizations. The first set of geometry optimizations, MCSCF1, were not gradient optimized. The gradients were calculated only at the final optimized geometries and were on the order of 1.5 x 10~4 to 6.0 x 10~6 hartree/bohr. This gives geometries precise to about 4 or 5 digits (about 1 or 220 2 decimal places for the angle and 3 or 4 decimal places for the bonds). The MCSCF1 optimized geometries are given in Table 6.2, along with the experimental geometries, SCF1, SCF2, and MCSCF2 optimized geometries. The MCSCF2 geometries shown in Table 6.2 were gradient optimized until the Cartesian gradients were < 2.2 x 10-9 hartree/bohr. This took about 6 to 8 steps (geometry points) because the optimizations were begun using the non-gradient method mentioned in Section 6.2.1. The optimized geometries are precise to at least 4 decimal places (actually closer to 6 or 7) and the energy is converged past the precision of the calculations (14 or 15 digits, which for SO 2 translates to about 11 decimal places). As a m atter of fact, as mentioned in Section 6.2.1, there were cases in which the energy at a previously calculated point was slightly lower than the energy at the optimized geometry. This simply means that for practical purposes, for any calculations involving the minimum energies (such as excitation energies), these energies can be obtained well before full optimization is achieved. The total energies are given in Section 6.6.1, which discusses adiabatic excitation energies. 6.5 Summary of Geometry Optimizations It is instructive to make a comparison of the various ab initio optimized geometries to each other and to the experimental values. Also, as with energies, it is often just as important to compare the changes in geometry on going from the ground state 221 Table 6.2. Comparison of the experimental and ab initio geometries and changes in the geometries on going from the *Ai ground state to the various excited states for S 0 2.a-6 % 3Bi 3b 2 3a 2 xa 2 xb 2 EX E0 R so 1.4321 1.4930 1.55 1.53 1.5525 A# 0.0609 0 .1 2 0 . 1 0 0.1204 Ooso 119.5360 126.23 97 99 103.75 A 0 6.69 -23 - 2 1 -15.79 S C F ld R so 1.4039 1.4499 1.5637 1.5029 1.5029 1.4949 1.5240 &R 0.0460 0.1598 0.0990 0.0990 0.0910 0 .1 2 0 1 Ooso 118.7867 127.0000 103.5000 92.6500 92.6500 117.3000 101.5000 AO 8.2133 -15.2867 -26.1367 -26.1367 -1.4867 -17.2867 SOF2 R so 1.4049 1.4499 1.5653 1.4997 1.5007 1.4949 1.5198 A R 0.0450 0.1604 0.0948 0.0958 0.0900 0.1149 Ooso 118.8201 127.2860 103.8700 93.1510 92.9440 117.8340 98.8000 AO 8.4659 -14.9501 -25.6691 -25.8761 -0.9861 - 2 0 .0 2 0 1 M CSCF 1 R so 1.4489 1.4949 1.5953 1.5413 1.5426 1.5351 1.5844 A R 0.0460 0.1464 0.0924 0.0937 0.0863 0.1355 Ooso 119.7420 119.1850 104.9800 99.1141 99.5700 113.0500 109.8300 A 0 -0.5570 -14.7620 -20.6279 -20.1720 -6.6920 -9.9120 M CSCF2 R so 1.4511 1.5446 1.5912 1.5714 1.5732 1.5841 1.6004 A R 0.0935 0.1401 0.1203 0 .1 2 2 1 0.1330 0.1493 Ooso 119.4877 122.3170 106.1522 96.2304 96.5360 114.5770 107.7199 A 0 2.8293 -13.3356 -23.2573 -22.9517 -4.9107 -11.7678 “B on d d istan ces in A and bond angles in degrees. 4The calculated values for each state at each ab initio level are precise to the number of digits shown. The calculated values are accurate to about 0.05 A for the bond distances and 0.5°, at the MCSCF2 level. 'See Chapter III for references. dResults of Lindley.22 222 to the excited states (A R and AO) predicted by the various ab initio methods to those determined from experiment. Table 6.2 also contains these changes in geometry determined from experiment and the various ab initio methods. This is especially useful for assessing the accuracy of predictions based on the ab initio results for excited states for which there are no experimental results. Comparing the calculated geometries to the experimental geometries one sees, as expected, that the SCF bond distances are too short while the MCSCF values are too long. While the MCSCF calculations are necessary in many cases to obtain accurate excitation energies and even some properties, the virtual orbitals that are usually included in the active space tend to be non-bonding or anti-bonding in character. Thus MCSCF calculations tend to reduce the bonding character of the bonds, which causes them to be longer than the SCF values. Interestingly, the MC- SCF1 optimized bond distances and A il’s are actually closer to the experimental values than are the MCSCF2 values (although this is not necessarily true for the angles). It is felt that this may be due to the fact that the 4 b2 orbital, which is weakly S-0 bonding, remains doubly occupied in the MCSCF1 active space. Fewer bonding electrons are being excited to the anti-bonding virtual orbitals of th e active space, thus reducing their anti-bonding effect on the bond lengths. This causes the resulting optimized bond distances to be slightly shorter than MCSCF2 values, for which the 4 b2 orbital is in the active space. This doesn’t mean that the MCSCFl calculations are better than the MCSCF2 calculations. Quite the con 223 trary, as discussed in Section 5.2, it is felt that the MCSCF2 active space is better and more reliable than that used for MCSCF1. In general, MCSCF optimized angles tend to be better than those obtained at the SCF level. Also, the MCSCF2 angles appear to be better than the MCSCFl angles. Furthermore, in the cases in which the SCF and MCSCF optimized geometries show large deviations from each other, both SCF calculations, as well as both MCSCF calculations, give similar results. Looking at specific states, one sees that the MCSCF2 xAi optimized geometry is quite close to the experimental geometry, much closer than the result of either SCF calculation. The bond distances are within 0.02 A and bond angles are within 0.05° of each other (errors of 1.3% and .04%, respectively). The present results are also very close to previously calculated values (see Sections 3.4 and 3.5). For the 3A 2 and XA 2 states, the experimental geometries have not been deter mined very accurately. As mentioned in Chapter III, the experimental geometries for these states were not determined from the (000) vibrational level of the excited states. Therefore, the geometries are not what they would be if determined at the equilibrium positions (at the minimum of the potential surface). In particu lar the reported experimental value for the OSO angle is larger than it would be otherwise. If the geometries were determined based on data obtained for the (000) vibrational level the angles for both states would be smaller. Even though the experimental geometries of these two states are not known that accurately, most experiments support the idea that the geometries of these states are very similar (see Tables 3.9 and 3.12). The MCSCF angles are quite close to the experimental values, especially considering the discussion above. More importantly, at all lev els of calculation these two states have extremely similar geometries. This agrees with previous ab initio results (Tables 3.10 and 3.14) and supports the experimen tal contention that the geometries of the two states are very similar. The changes in geometry, A R and A 0, for these two states are fairly consistent for all levels of calculation and experiment. The two MCSCF calculations give results closest to the experimental results and are also close to the results of previous calculations (Tables 3.10 and 3.14). The 3B2 and 1B2 states also have very similar MCSCF2 optimized geometries (within 0.01 A and 1.6°) and A ii’s and A#’s. The SCF and MCSCF results for the XB2 state differ the most from each other. The differences between the SCF2 and MCSCF2 results are approximately 0.08 A and 9°. The 1B2 A R and A 0 values calculated at the SCF and MCSCF levels also show very large differences, unlike the 3A2, xA2, and 3B2 states (A R decreases by fa 0.04 A and A0 decreases by « 8.2°). The present MCSCF2 results do agree with previous ab initio results and which also show that the 3B2 and *B2 states have similar optimized geometries (see Tables 3.15 and 3.8). The 3Bi and xBi states do not show the same general behavior as the other pairs of excited states. T heir geometries are fairly dissimilar, especially for the bond angles. Like the 1B2 state, the differences between the SCF and MCSCF results are fairly large for these two states. The difference between the MCSCF2 and SCF2 bond distances is ft* 0.09 A for both states and that for the bond angle is ft* 5.4° and ~ 3.2° for the 3Bi and 1Bi states, respectively. For both states, the A R and A 0 values also change considerably depending on the level of calculation performed. The MCSCF2 results for the bond distances for both states agree within 0.02 A to those determined by previous calculations (see Tables 3.13 and 3-11). However, the MCSCF2 bond angles differ considerably from those determined by the previous calculations, which give angles of 126°-129° for the 3Bj state and ft* 125° for the ^ state. There are several possible explanations for this. The previous calculations used basis sets that were smaller and of poorer quality (STO-3G* and 4-31G) than that used in the present study. Also, while the previous studies involved Cl calculations, they included only single excitations from a single reference configuration. The present MCSCF2 calculations used a fairly large complete active space and a good basis set. It is felt that while the MCSCF2 optimized geometries for the 3Bi and ^ states are not perfect, they are more reasonable than those determined by the previous calculations. Also, the regions of the surface in which the other calculations predict the minimum geometries to be located were covered by the calculations performed in this study and were found to have higher energies than at the MCSCF2 optimized geometries. Finally, for each of the states at all levels of calculation (except MCSCF1 for the 3Bj state) the A R and AO show the same general trends. For all excited states the bond distances increase over that of the ground state. Also, all but one excited state exhibit decreases in bond angles compared to the ground state. The only state to exhibit an increase in the bond angle is the 3BX state. However, for the MCSCF2 calculations the change is very small. The only anomaly is the MCSCF1 bond angle and A0 for the 3Bi state, which is actually smaller than the ground- state bond angle. Since the MCSCF1 active space was deemed to be inadequate for several of the states, this result is likely incorrect. The 3A2 and XA2 states show the largest change in angle, while the 3Bi and 1Bi states show the smallest change. Also, the 3B2 state exhibits fairly consistent AW s and A0’s for all level of calculations. This can also be said for A 9 for the 3A2 and *A2 states. 6 .6 Transition Energies This section presents the results for excitation and emission energies determined at various geometries. The total energies used to determine these quantities are also presented for completeness. The relative orderings of the states at the various points on the potential surfaces used to calculate these quantities are also discussed. Several methods, SCF, MCSCF, SRCI, SRFCCI, and MRCI, were used to determine these quantities. The MCSCF active spaces and the Cl reference spaces and configuration spaces were discussed in Chapter V. The Cl energies corrected 227 by various estimates for the quadruples contribution to the energies (described in Chapter II) are also presented. The SRFCCI and MRCI calculations were both performed with the atomic core orbitals, Is, 2s, and 2 p on sulfur and Is on oxygen, frozen (i.e., no excitations from these orbitals to the external orbitals were allowed). Also, for the MRCI calculations these orbitals were not optimized at the MCSCF level (i.e., they were frozen at their SCF level of optimization). 6.6.1 Adiabatic Transition Energies, Te The adiabatic transition energies, Te, were calculated at both the experimental geometries and the SCF and MCSCF2 optimized geometries. The total energies and Te values calculated at the SCF optimized geometries were presented in Ta ble 6.1. The total energies at the experimental and MCSCF2 optimized geometries are given in Tables 6.3 and 6.4. As expected, the total energies calculated at all levels of SRCI are lower than the corresponding SRFCCI values. Also, the MRCI total energies calculated at the MCSCF2 optimized geometries are lower than the SRFCCI values. The results for the *Ai state can be compared to those of previous studies (see Table 3.5). The present MCSCF2 *Aj energy is lower than the GVB and GVB-CI results of Dunning and RafFenetti,68 who used a slightly smaller ba sis set than used in the present study. The present MCSCF3 results are also in pretty good agreement. The Cl results are in very good agreement with those of previous studies. The SRCI and SRFCCI values are slightly higher (by about 0.06 228 Table 6.3. Total energies (in hartrees) of the X ^ , a 3Bj, b 3A2, A XA2, and C XB2 states of S02 at their respective experimental geometries.0 Method 6 *Ai 3Bi 3a 2 xa 2 xb 2 SCF -547.231493 -547.116564 -547.110614 -547.102967 -547.007983 MCSCF2 -547.359497 -547.222684 -547.235262 -547.225613 -547.147269 SRCI -547.770642 -547.656374 -547.652857 -547.641222 -547.556521 SRCI-DAV1 -547.834484 -547.722922 -547.722240 -547.709561 -547.630216 SRCI-DAVlS -547.830493 -547.718763 -547.717904 -547.705289 -547.625610 SRCI-DAV2 -547.843059 -547.732280 -547.732421 -547.719499 -547.641653 SRCI-DAV2S -547.838532 -547.727536 -547.727448 -547.714607 -547.636333 SRCI-POPLE -547.847470 -547.737398 -547.738294 -547.725170 -547.648700 SRFCCI -547.678695 -547.565192 -547.562080 -547.550347 -547.466482 SRFCCI-DAV1 -547.734199 -547.623622 -547.623421 -547.610785 -547.532535 SRFCCI-DAVIS -547.728032 -547.617130 -547.616605 -547.604069 -547.525196 SRFCCI-DAV2 -547.742064 -547.632371 -547.633066 -547.620225 -547.543653 SRFCCI-DAV2S -547.735023 -547.624907 -547.625179 -547.612461 -547.535078 SRFCCI-POPLE -547.741842 -547.632588 -547.633744 -547.620830 -547.545120 “See Table 6.2 for the experimental geometries. 6See Chapter II and Section A.2 for definitions of the methods. Table 6.4. Total energies (in hartrees) of all states of SO 2 at their respective MCSCF2 optimized geometries.® Method 3B, 5B2 3a2 *A2 !B, 'Bj SCF -547.227479 -547.104352 -547.162460 -547.106383 -547.096273 -547.056864 -546.996454 MCSCF26 -547.360307 -547.226030 -547.243303 -547.235831 -547.228323 -547.206333 -547.150147 MCSCF3 -54 7.327551 -547.204649 -547.206969 -547.210090 -547.202401 -547.180753 -547.126455 SRCI -547.769506 -547.650577 -547.661840 -547.650765 -547.638831 -547.611638 -547.551015 SRCI-DAV1 -547.834628 -547.720486 -547.722372 -547.721320 -547.709531 -547.687077 -547.628104 SRCI-DAV1S -547.830558 -547.716116 -547.718589 -547.716910 -547.705113 -547.682363 -547.623286 SRCI-DAV2 -547.843520 -547.730746 -547.730721 -547.731826 -547.720124 -547.698950 -547.640550 SRCI-DAV2S -547.838894 -547.725735 -547.726416 -547.726760 -547.715044 -547.693493 -547.634954 SRCI-POPLE -547.848200 -547.736667 -547.735174 -547.737995 -547.726391 -547.706381 -547.648561 SRFCCI -547.677768 -547.559919 -547.570400 -547.560188 -547.548363 -547.521774 -547.461535 SRFCCI-DAV1 -547.734543 -547.621750 -547.623509 -547.622741 -547.611247 -547.589443 -547.531092 SRFCCI-DAV1S -547.728235 -547.614880 -547.617608 -547.615791 -547.604260 -547.581924 -547.523363 SRFCCI-DA V2 -547.742734 -547.631459 -547.631457 -547.632742 -547.621408 -547.600970 -547.543324 SRFCCI-DAV2S -547.735516 •-547.623511 -547.624673 -547.624681 -547.613292 -547.592170 -547.534236 SRFCC1-POPLE -547.742646 -547.632131 -547.631651 -547.633599 -547.622375 -547.602615 -547.545413 MRCl-REFc -547.324592 -547.201078 -547.204081 -547.207623 -547.199888 -547.177913 -547.124176 MRCI -547.717456 -547.603010 -547.598991 -547.604200 -547.594309 -547.575288 -547.521803 MRCI-DAV1 -547.761415 -547.651883 -547.646803 -547.651965 -547.641285 -547.623575 -547.570118 MRCI-DAVIS -547.756531 -547.646453 -547.641491 -547.646657 -547.636065 -547.618210 -547.564750 M RCI-DAV2 -547.766953 -547.658648 -547.653389 -547.658505 -547.647636 -547.630254 -547.576801 MRCI-DAV2S -547.761454 -547.652466 -547.647345 -547.652471 -547.641711 -547.624147 -547.570690 “MCSCF2 optimized geometries given in Table 6.2. 6MCSCF1 results at the respective MCSCF1 optimized geometries are: -547.345118, —547.196339, -547.231497, -547.206274, -547.198630, -547.178352, and -547.129788. 'Energy of the MCSCF reference space CSF’s using the coefficients for these CSF’s as determined by the Cl calculation. 230 hartrees) than those determined by Bacskay et al.,309 who used a larger basis set (an extra set of p- and d-orbitals on both sulfur and oxygen and an e x t r a s-orbital on oxygen). The correlation energies ( E corr) at the experimental and MCSCF2 optimized geometries are given in Tables 6.5 and 6.6. The MCSCF2 correlation energies for the 3A2, and XA2 states at the experimental geometries are fairly constant. The MCSCF2 correlation energies for the 3Bj and XB2 states differ by % ±0.02 hartrees from those for the other states. The XB2 state appears to benefit the m ost and the 3Bi state the least from a multiconfigurational description of the wavefunction. As expected, the various Cl correlation energies for each state increase when the esti mates for the quadruples are included. Unlike the MCSCF2 correlation energies, the Cl values at a particular level stays fairly constant for all states. T h is seems to indicate that dynamical correlation is pretty consistent for the various states, at least for SRC I. The £ corr’s at the MCSCF2 optimized geometries also stay fairly constant (within 0.8 eV) except for the 3B2 state, which exhibits the smallest E cor r less than that of the next smallest £corr, for the 3Bi state). The xBi and !B2 states seem to benefit the most from using a multiconfigurational description of the wave function. The 3B2 state does not benefit nearly as much as th e other states from an MCSCF treatment. Also, the £ corr(MCSCF2) ’s at the MCSCF2 optimized geometries are greater (lower) than at the experimental geometries by 231 Table 6.5. Correlation energies, Ecorr (in hartrees), of the X XA1} a 3Bl5 b 3A2, A XA2, and C 1B2 states of S 02 at their respective experimental geometries.® Method 3Bi 3a 2 xa 2 * b 2 MCSCF2 -0.128003 -0.106120 -0.124648 -0.122645 -0.139286 SRCI -0.539148 -0.539810 -0.542243 -0.538254 -0.548538 SRCI-DAV1 -0.602990 -0.606359 -0.611627 -0.606593 -0.622233 SRCI-DAV1S -0.599000 -0.602199 -0.607290 -0.602322 -0.617627 SRCI-DAV2 -0.611565 -0.615717 -0.621808 -0.616532 -0.633670 SRCI-DAV2S -0.607039 -0.610972 -0.616835 -0.611639 -0.628349 SRCI-POPLE -0.615976 -0.620835 -0.627681 -0.622203 -0.640717 SRFCCI -0.447202 -0.448629 -0.451466 -0.447380 -0.458499 SRFCCI-DAV1 -0.502705 -0.507058 -0.512808 -0.507818 -0.524552 SRFCCI-DAV1S -0.496538 -0.500566 -0.505992 -0.501102 -0.517213 SRFCCI-DAV2 -0.510570 -0.515808 -0.522453 -0.517258 -0.535669 SRFCCI-DAV2S -0.503529 -0.508343 -0.514565 -0.509493 -0.527095 SRFCCI-POPLE -0.510348 -0.516024 -0.523131 -0.517863 -0.537137 “See Table 6.2 for the experimental geometries. 232 Table 6.6. Correlation energies, Ecorr (in hartrees), of all states of SO 2 at their respective MCSCF2 optimized geometries.0 M ethod 3Bi 3B 2 3a 2 'A 2 * b 2 M CSCF26 -0.132828 -0.121678 -0.080843 -0.129448 -0.132050 -0.149469 -0.153692 MCSCF3 -0.100072 -0.100296 0.044508 -0.103707 -0.1O6129 -0.123889 -0.130000 SRCI -0.542027 -0.546224 -0.499380 -0.544382 -0.542559 -0.554774 -0.554561 SRCI-DAV1 -0.607149 -0.616133 -0.559912 -0.614937 -0.613258 -0.630213 -0.631649 SRCI-DAV1S -0.603079 -0.611764 -0.556129 -0.610527 -0.608840 -0.625498 -0.626831 SRCI-DAV2 -0.616042 -0.626394 -0.568261 -0.625443 -0.623852 -0.642086 -0.644096 SRCI-DAV2S -0.611416 -0.621383 -0.563956 -0.620377 -0.618771 -0.636629 -0.638500 SRCI-POPLE -0.620721 -0.632315 -0.572714 -0.631612 -0.63O118 -0.649517 -0.652107 SRFCCI -0.450289 -0.455567 -0.407940 -0.453805 .0.452090 -0.464909 -0.465081 SRFCCI-DAV1 -0.507064 -0.517398 -0.461048 -0.516358 -0.514975 -0.532578 -0.534637 SRFCCI-DAVIS -0.500756 -0.510528 -0.455147 -0.509408 -0.507987 -0.525060 -0.526909 SRFCCI-DAV2 -0.515255 -0.527107 -0.468997 -0.526360 -0.525135 -0.544106 -0.546869 SRFCCI-DAV2S -0.508037 -0.519158 -0.462213 -0.518298 -0.517019 -0.535306 -0.537782 SRFCCI-POPLE -0.515167 -0.527779 -0.469190 -0.527216 -0.526102 -0.545751 -0.548959 MRCI-REF -0.097113 -0.096726 -0.041621 -0.101240 -0.103615 -0.121048 -0.127721 MRCI -0.489977 -0.498658 -0.436531 -0.497817 -0.498036 -0.518424 -0.525349 MRCI-DAV1 -0.533936 -0.547531 -0.484343 -0.545582 -0.545012 -0.566711 -0.573664 MRCI-DAV1S -0.529052 -0.542100 -0.479O31 -0.540275 -0.539792 -0.561345 -0.568296 MRCI-DAV2 -0.539474 -0.554296 -0.490929 -0.552122 -0.551363 -0.573390 -0.580347 MRCI-DAV2S -0.533975 -0.548114 -0.484885 -0.546088 -0.545438 -0.567282 -0.574236 “MCSCF2 optimized geometries given in Table 6.2. JMCSCF1 results at the respective MCSCF1 optimized geometries are: —0.117639, —0.091987, -0.069036, -0.099891, -0.102357, -0.121488, and -0.133334. as little as 0.13 eV (for the XAX and 3A2 states) and by as much as 0.42 eV (for the 3BX state). This indicates that using MCSCF description of the wavefunction is more important at the MCSCF2 optimized geometries. One can also look at the difference in the MCSCF correlation energy between the CAS MCSCF2 and non- CAS MCSCF3 calculations. The MCSCF2 calculations recover about the same amount of correlation energy (« 0.025 hartrees) over the MCSCF3 calculations for most of the states. However, for the xAi and 3B2 states the MCSCF2 treatment has a greater energy advantage (« 0.035 hartrees) over MCSCF3. This may indicate that the MCSCF3 active space does not treat these two states as well as it treats the other states. The E corr’s for the various SRCI methods at the MCSCF2 optimized geometries show the same general behavior as exhibited at the experimental geometries. As for the Ecorr(MCSCF2) energies, SRCI and SRFCCI E c o r r for the 3B 2 state are considerably smaller than those of the other states (by about 1.2 eV). The XB, and XB2 states again show the largest E c o r r . The MRCI E corr follow the same trend as the SRCI calculations. As for SRCI, the E Co r r for the MRCI calculation for the 3B2 state is considerably less than that for the other states (by about 1.4 eV). Finally, at all levels of calculation the 3A2 and XA2 states show remarkably similar results for E c o r r , just as they do for their geometries. The differences in the MRCI correlation energies between the various states are mostly due to the multireference treatment. Looking at the incremental MRCI correlation energies, 234 defined as E m r c i —E m r c i - r e f , one sees that they are fairly constant for all states. Thus, the differences in the correlation energies between the states at the MRCI level is mosly due to non-dynamical correlation. This points out the necessity of using a multireferenoe Cl treatment. The Te values at the experimental and MCSCF2 optimized geometries are given in Tables 6.7 and 6.8. The relative orderings of the states at these geometries are given in Tables 6.9 and 6.10. As one can see from these tables, the MCSCF2 Te results are comparable at the experimental and MCSCF2 optimized geometries (within ~ 0.07 eV). The experimental To values for the 3A 2 and XA2 states were determined from vibrational levels other than the (000) level and were extrapolated back to obtain the To- Thus the error in the value for T0 for these two states may be larger than that for the other states. This larger error will also apply to Te. The various SRFC and SRFCCI results at both geometries for each state are also fairly constant, differing at most by about 0.18 eV. However, this difference is reduced to about 0.04 eV when the DAV2, DAV2S, and Pople quadruple corrections are included. Since the experimental geometries of the 3B2 and *Bi states are unknown, it is more instructive to study the results for the various states at the optimized geometries. There are considerable differences between the MCSCF and SCF Te values, especially for the 3B2 state. The MCSCF2 Te values are greater than the MCSCF3 results, except for the 3B2 state. The 3B2 state exhibits the largest 235 T able 6.7. Adiabatic transition energies, Te (in eV), of the a 3Bi, b 3A2, A *A 2 , and C xB2 states of S02 at their respective experimental geometries.0 Method 3Bi 3A2 *a 2 1b 2 EXP6 3.24 3.30 3.55 5.37 SCF 3.127471 3.289383 3.497459 6.082157 MCSCF2 3.722950 3.380687 3.643255 5.775137 SRCI 3.109464 3.205164 3.521783 5.826644 SRCI-DAV1 3.035802 3.054360 3.399404 5.558525 SRCI-DAV1S 3.040406 3.063785 3.407053 5.575283 SRCI-DAV2 3.014498 3.010665 3.362303 5.480638 SRCI-DAV2S 3.020434 3.022821 3.372271 5.502264 SRCI-POPLE 2.995265 2.970891 3.328027 5.408928 SRFCCI 3.088642 3.173339 3.492614 5.774732 SRFCCI-DAVl 3.009016 3.014478 3.358341 5.487666 SRFCCI-DAV1S 3.017863 3.032129 3.373260 5.519562 SRFCCI-DAV2 2.984944 2.966035 3.315478 5.399158 SRFCCI-DAV2S 2.996466 2.989069 3.335159 5.440889 SRFCCI-POPLE 2.973012 2.941546 3.292957 5.353183 “See Table 6.2 for the experimental geometries. '’Estim ated T e values. To values are 3.19, 3.26, 3.46, and 5.28, respectively (see Chapter III). 236 Table 6.8. Adiabatic transition energies, Te (in eV), of the excited states of S02 at their respective MCSCF2 optimized geometries.® Method 3B i 3 b 2 3a 2 xa 2 'B i xb 2 EXP6 3.24 3.30 3.55 3.94 5.37 SCF 3.350524 1.769286 3.295263 3.570376 4.642766 6.286638 MCSCF2C 3.653949 3.183901 3.387230 3.591550 4.189934 5.718875 MCSCF3 3.344409 3.281277 3.196348 3.405560 3.994658 5.472222 SRCI 3.236300 2.929799 3.231178 3.555909 4.295889 5.945565 SRCI-DAV1 3.106041 3.054707 3.083337 3.404128 4.015142 5.619932 SRCI-DAV1S 3.114183 3.046900 3.092577 3.413614 4.032688 5.640284 SRCI-DAV2 3.068815 3.069486 3.039432 3.357851 3.934042 5.523233 SRCI-DAV2S 3.079283 3.060756 3.051416 3.370230 3.956657 5.549629 SRCI-POPLE 3.035020 3.075641 2.998892 3.314664 3.859160 5.432558 SRFCCI 3.206903 2.921698 3.199595 3.521384 4.244927 5.884130 SRFCCI-DAV1 3.069324 3.021468 3.042345 3.355116 3.948472 5.536319 SRFCCI-DAV1S 3.084610 3.010383 3.059817 3.373590 3.981412 5.574964 SRFCCI-DAV2 3.028006 3.028064 2.993096 3.301529 3.857693 5.426360 SRFCCI-DAV2S 3.047884 3.016246 3.016041 3.325957 3.900719 5.477224 SRFCCI-POPLE 3.007347 3.020412 2.967403 3.272811 3.810527 5.367105 MRCI-REF 3.361068 3.279349 3.182971 3.393439 3.991438 5.453729 MRCI 3.114319 3.223666 3.081924 3.351082 3.868687 5.324120 MRCI-DAV1 2.980586 3.118805 2.978360 3.268984 3.750905 5.205563 MRCI-DAV1S 2.995445 3.130456 2.989867 3.278106 3.763992 5.218736 MRCI-DAV2 2.947196 3.090296 2.951087 3.246864 3.719860 5.174420 MRCI-DAV2S 2.965765 3.105115 2.965625 3.258444 3.736396 5.191053 “MCSCF2 optimized geometries given in Table 6.2. ^Estimated T e values. To values are 3.19, 3.26, 3.46, 3.87, and 5.28, respectively (see Chap ter III). cMCSCFl results at the respective MCSCF 1 optimized geometries are: 4.048587, 3.091866, 3.778224, 3.986239, 4.538030, and 5.859557. 237 Table 6.9. Relative ordering of the X ’Ai, a 3Bi, b3A2, A xA2, and C XB2 states of S02 at their respective experimental geometries.0 Method 3Bx 3a 2 xa 2 xb 2 EXP 1 2 4 6 SCF 1 2 4 6 MCSCF2 2 1 4 6 SRCI 1 2 4 6 SRCI-DAV1 1 2 4 6 SRCI-DAV1S 1 2 4 6 SRCI-DAV2 2 1 4 6 SRCI-DAV2S 1 2 4 6 SRCI-POPLE 2 1 4 6 SRFCCI 1 2 4 6 SRFCCI-DAV1 1 2 4 6 SRFCCI-DAV1S 1 2 4 6 SRFCCI-DAV2 2 1 4 6 SRFCCI-DAV2S 2 1 4 6 SRFCCI-POPLE 2 1 4 6 “See Table 6.2 for the experimental geometries. 238 Table 6.10. Relative ordering of the excited states 0f S 02 at their respective MCSCF2 optimized geometries.0 Method 3B, 3b 2 3a 2 *a2 *8; EXP 1 2 4 5 6 SCFfc 3 1 2 4 5 6 MCSCF2c 4 1 2 3 5 6 MCSCF3 3 2 1 4 5 6 SRCI 3 1 2 4 5 6 SRCI-DAVl 3 1 2 4 5 6 SRCI-DAV1S 3 1 2 4 5 6 SRCI-DAV2 2 3 1 4 5 6 SRCI-DAV2S 3 2 1 4 5 6 SRCI-POPLE 2 3 1 4 5 6 SRFCCI 3 1 2 4 5 6 SRFCCI-DAV1 3 1 2 4 5 6 SRFCCI-DAV1S 3 1 2 4 5 6 SRFCCI-DAV2 2 3 1 4 5 6 SRFCCI-DAV2S 3 2 1 4 5 6 SRFCCI-POPLE 2 3 1 4 5 6 m r c i-r e f 3 2 1 4 5 6 MRCI 2 3 1 4 5 6 MRCI-DAV1 2 3 1 4 5 6 MRCI-DAV1S 2 3 1 4 5 6 MRCI-DAV2 1 3 2 4 5 6 MRCI-DAV2S 2 3 1 4 5 6 aMCSCF2 optimized geometries given in Table 6.2- ‘Ordering at the respective SCF optimized geometries is 2, 1, 3 4 i 5 , 6 (see Table 6 . 1). cMCSCFl results at the respective MCSCF1 optimized geometries are: 2, 1, 3, 4, 5 _ g. 239 correlation effect on the Te values at the MCSCF level. The relative ordering of the first three excited states changes at each of these levels of calculation. The 1Bj and !B 2 states have the highest Te values and have the same relative order at all levels of calculation. The MCSCF Te values are still too large but are much closer to the experimental results than are the SCF results (the MCSCF3 re sults are within 0.05 eV). The SRCI and SRFCCI calculations produce comparable results for these two states. While the SRCI and SRFCCI Te values themselves are still too large, the DAV2, DAV2S, and Pople results are just slightly less than the experimental values for these two states. The MRCI Te values are further reduced from the SRFCCI results but still within about 0.28 eV of experiment. The SCF and MCSCF2 Te for the VS .2 state are very close to each other and to experiment, while the MCSCF3 Te is about 0.15 eV smaller than experiment. All calculated Cl Te values are smaller than the experimental values, with the MRCI-DAV2 and MRCI-DAV2S about 0.3 eV below the experimental results. The present results for the 1B2 state agree fairly well with the S-CI calculations of Hillier and Saun ders82 (within about ±0.1 eV), although the DFC-LSD result of Jones81 is about 1.1 eV lower (see Table 3.8). The present result for the ^ 2 state agrees well with previous results (Table 3.10). The present Cl results corrected for quadruple con tributions are slightly smaller than the S-CI value of Hillier and Saunders82 (they agree to within sa 0.05 eV). The present MRCI results for the 1B1 state are about 0.2-0.5 eV higher than the results of Hillier and Saunders82 and Jones81 (see Ta 240 ble 3.11). Their results also indicate that the 1Bi state is nearly degenerate with the XA2 state (within 0.1 eV). This was not found to be the case in this study, which places the 1BJ minimum about 0.5 eV above that of XA2. The triplet states exhibit the most interesting changes resulting from correlation effects. The 3B2 state, which was the lowest excited state at the SCF level (about 1.2 eV below the 3A2 and 3Bi states), is essentially degenerate with the other two triplet states, especially at the Cl levels. All three states are within ~ 0.3 eV of each other at all Cl levels, the difference decreasing to less than about 0.14 eV at the MRCI level. At the highest level of calculation, MRCI-DAV2 and MRCI-DAV2S, the three states differ by only 0.04 eV and the 3A2 and 3Bi states are degenerate (T^s agree to 0.0001 eV at the MRCI-DAV2S level) and lower than 3B2. This also turns out to be the case for the 3A2 and 3Bi states at their experimental geometries (their SRCI and SRFCCI Te’s with DAV2, DAV2S, and Pople corrections are within about 0.03 eV of each other). This agrees with the experimental results which show the minima of the 3A2 and 3Bi states to be within « 0.1 eV of each other. Previous theoretical studies also exhibit the reordering of these states depending on the level of calculation used (see Tables 3.13-3.15). The S-CI results of Hillier and Saunders82 show the ordering of these states to be 3B2, 3Bi, and 3A2 (1.3, 2.1, and 2.5 eV, respectively). The DFC-LSD results of Jones81 give Te values for the 3Bi and 3B2 states of 2.8 and 3.5 eV, respectively. These are not nearly 241 as degenerate as the present calculations suggest, but th e states are in the same order. In summary, for all states except 3B 2 , including correlation lowers the Te values. The 3B 2 state exhibits the largest correlation effect upon Te, actually increasing it by about 1.3-1.5 eV, depending on the level of correlated calculation. Also, the SRCI and SRFCCI Te for 3B2 actually increase when the corrections for the effects of the most important quadruple excitations are added (although this trend is reversed for the MRCI calculations). Perhaps the easiest and only definite thing that can be said about th e excited state minima is that the ordering of the singlet states is *A2, 1B1, and 1B 2 and all singlet minima are above the triplet minima. Also, the difference between the Te values for the singlet states is about the same at all Cl levels and about the same as experiment (the *A2 state is « 0.05 and 2 eV below the 1B1 and 1B 2 states, respectively). For the triplet states, based on the Cl results (especially when quadruple cor rections are included), the 3A2 and 3Bi states are, for all intents and purposes, degenerate. Also, the 3B2 state lies just slightly above the 3A2 and 3Bi minima. 6.6.2 Vertical Excitation Energies, &EVert The vertical excitation energies, A Evert, were calculated at the experimental and the SCF2 and MCSCF2 optimized geometries of the jOAi state. The total energies 242 at these geometries are given in Tables 6.11-6.13, respectively. The only surprise is the SCF total energy for the 3BX state. For this state, E s c f at the X exper imental geometry is lower than E s c f at the 3Bi experimental geometry. However, the MCSCF and C l calculations reverse this (the MCSCF1 energy for 3Bi at its experimental geometry is 6 mhartrees lower). The correlation energies ( Ecorr) at these geometries are given in Tables 6.14- 6.16. The correlation energies differ slightly at the three different geometries. The E cott at the JAi SCF optimized geometry is less than that at the *Ai experimental energy, at all levels of calculation. The difference in F corr(MCSCF2) is about 0.006 hartree for all states except 3B 2 , for which it is 0.002 hartree. The differences in F corr(SRCI) and Z£COrr(SRFCCI) at these two geometries is 0.001-0.005 hartree but increases to the same levels as for the MCSCF2 calculations when quadruple corrections are included. The correlation energies at the *Ax MCSCF1, and MCSCF2 optimized ge ometries are greater in magnitude than those at the xAx experimental geometry. However, the differences between the correlation energies at the experimental and MCSCF geometries are much less than those between the experimental and SCF optimized geometries. This is understandable since the *Ai MCSCF2 optimized geometry is closer to the *Ai experimental geometry than is the SCF optimized geometry. Table 6.11. Total energies (in hartrees) of all states of SO 2 at the X experimental geometry.11 Method 3Bi 3b 2 3a 2 ja 2 'B i % SCF -547.231493 -547.116660 -547.119389 -547.072406 -547.061651 -547.061135 -546.969817 MCSCF1 -547.344458 -547.188872 -547.180116 -547.163207 -547.155850 -547.158766 -547.087134 MCSCF2 -547.359497 -547.206396 -547.193183 -547.181231 -547.173916 -547.172399 -547.100258 MCSCF3 -547.327911 -547.192245 -547.166255 -547.164391 -547.157310 -547.157520 -547.084195 SRCI -547.770642 -547.647388 -547.618387 -547.604419 -547.592330 -547.603872 -547.512803 SRCI-DAV1 -547.834484 -547.709893 -547.675452 -547.669016 -547.657049 -547.671072 -547.583234 SRCI-DAVIS -547.830493 -547.705986 -547.671886 -547.664979 -547.653004 -547.666872 -547.578833 SRCI-DAV2 -547.843059 -547.718237 -547.682821 -547.677944 -547.666038 -547.680568 -547.593732 SRCI-DAV2S -547.838532 -547.713809 -547.678794 -547.673348 -547.661431 -547.675775 -547.588674 SRCI-POPLE -547.847470 -547.722493 -547.686404 -547.682717 -547.670876 -547.685795 -547.599902 SRFCCI -547.678695 -547.555563 -547.526056 -547.512824 -547.500829 -547.512475 -547.422168 SRFCCI-DAV1 -547.734199 -547.609964 -547.575438 -547.569288 -547.557570 -547.571534 -547.485019 SRFCCI-DAVIS -547.728032 -547.603919 -547.569951 -547.563014 -547.551265 -547.564972 -547.478036 SRFCCI-DAV2 -5 4 7.742 064 -547.617661 -547.582263 -547.577591 -547.565988 -547.580426 -547.495161 SRFCCI-DAV2S -547.735023 -547.610761 -547.576018 -547.570395 -547.558748 -547.572875 -547.487051 SRFCCI-POPLE -547.741842 -547.617432 -547.581904 -547.577654 -547.566123 -547.580691 -547.496115 MRCI-REF -547.325042 -547.189110 -547.163379 -547.162197 -547.154979 -547.153667 -547.082088 MRCI -547.716508 -547.587665 -547.550721 -547.548346 -547.538135 -547.551514 -547.472461 MRCI-DAV1 -547.759796 -547.633604 -547.594456 -547.591573 -547.580259 -547.597699 -547.516780 MRCI-DAV1S -547.754986 -547.628499 -547.589597 -547.586770 -547.575579 -547.592568 -547.511855 MRCI-DAV2 -547.765178 -547.639589 -547.600023 -547.597022 -547.585462 -547.603765 -547.522456 MRC1-DAV2S -547.759770 -547.633819 -547.594545 -547.591613 -547.580204 -547.597959 -547.516901 243 aR S0 = 1-4321 A (2.7063 bohr) and 6OSo = 119.53600.44 Table 6.12. Total energies (in hartrees) of all states of S02 at the X lA 1 SCF2 optimized geometry.0 Method 3Bi 3b 2 3a 2 *a 2 'B j jb 2 SCF -547.233839 -547.111074 -547.102855 -547.061312 -547.050494 -547.050993 -546.957806 MCSCF2 -547.355065 -547.194596 -547.174434 -547.164510 -547.157031 -547.155713 -547.081645 SRCI -547.768756 -547.637510 -547.601102 -547.590163 -547.578035 -547.590918 -547.497241 SRCI-DAV1 -547.830776 -547.698216 -547.657266 -547.653201 -547.641183 -547.656611 -547.565940 SRCI-DAVIS -547.826900 -547.694422 -547.653756 -547.649261 -547.637236 -547.652505 -547.561647 SRCI-DAV2 -547.838910 -547.706129 -547.664402 -547.661732 -547.649770 -547.665711 -547.575966 SRCI-DAV2S -547.834525 -547.701841 -547.660445 -547.657259 -547.645287 -547.661036 -547.571046 SRCI-POPLE -547.842955 -547.710030 -547.667788 -547.666166 -547.654263 -547.670592 -547.581712 SRFCCI -547.676512 -547.545405 -547.508565 -547.498301 -547.486263 -547.499248 -547.406323 SRFCCI-DAV1 -547.730217 -547.598039 -547.557004 -547.553196 -547.541418 -547.556783 -547.467423 SRFCCI-DAVIS -547.724250 -547.592191 -547.551622 -547.547097 -547.535289 -547.550390 -547.460634 SRFCCI-DAV2 -547.737632 -547.605297 -547.563571 -547.561083 -547.549410 -547.565256 -547.477059 SRFCCI-DAV2S -547.730841 -547.598642 -547.557459 -547.554107 -547.542394 -547.557921 -547.469199 S'RTCCI-’POPIT -547.737234 -547.604898 -547.5630G9 -547.560066 -547 .549358 -547.565m -547.477763 aRso = 1.405 A (2.655 bohr) and 8o so = 118.82°. 4 4 2 Table 6.13. Total energies (in hartrees) of all states of S02 at the X*Ai MCSCF2 optimized geometry.0 Method 3Bi 3b 2 3a 2 ja 2 'B i 1b 3 SCF -547.227479 -547.117835 -547.129055 -547.078726 -547.068049 -547.065605 -546.976879 MCSCF26 -547.360307 -547.212202 -547.204214 -547.191303 -547.184075 -547.181528 -547.111328 SRCI -547.769506 -547.651441 -547.628449 -547.612809 -547.600769 -547.610130 -547.521979 SRCI-DAV1 -547.834628 -547.715208 -547.686090 -547.678452 -547.666535 -547.678375 -547.593474 SRCI-DAV1S -547.830558 -547.711222 -547.682488 -547.674349 -547.662425 -547.674110 -547.589006 SRCI-DAV2 -547.843520 -547.723862 -547.693612 -547.687650 -547.675798 -547.688154 -547.604267 SRCI-DAV2S -547.838894 -547.719336 -547.689539 -547.682972 -547.671109 -547.683278 -547.599124 SRCI-POPLE -547.848200 -547.728378 -547.697325 -547.692658 -547.680874 -547.693632 -547.610708 SRFCCI -547.677768 -547.559812 -547.536252 -547.521393 -547.509449 -547.518925 -547.431519 SRFCCI-DAV1 -547.734543 -547.615458 -547.586244 -547.578917 -547.567257 -547.579051 -547.495445 SRFCCI-DAVIS -547.728235 -547.609276 -547.580689 -547.572526 -547.560833 -547.572371 -547.488342 SRFCCI-DAV2 -547.742734 -547.623474 -547.593240 -547.587509 -547.575968 -547.588246 -547.505904 SRFCCI-DAV2S -547.735516 -547.616400 -547.586908 -547.580163 -547.568577 -547.580544 -547.497639 SRFCCI-POPLE -547.742646 -547.623375 -547.592958 -547.587700 -547.576236 -547.588655 -547.507019 aR so = 1-4511 A (2.7422 bohr) and 9OSo = 119.4877°. lMCSCFl results at the X tAi state MCSCF1 optimized geometry are: —547.345118, -547.192552, -547.189637, -547.170854, -547.163623, -547.164851, and -547.096180. 246 Table 6.14. Correlation energies, E COTT (in hartrees), of all states of SO 2 at the X *Ai experimental geometry.0 M ethod ‘Ai *B, 3b 2 3a 2 ‘A* 1B , ‘ b 2 MCSCF1 -0.112964 -0.072213 -0.060727 -0.090801 -0.094198 -0.097631 -0.117317 MCSCF2 -0.128003 -0.089736 -0.073793 -0.108825 -0.112264 -0.111264 -0.130441 MCSCF3 -0.096417 -0.075585 -0.046865 -0.091985 -0.095659 -0.096385 -0.114378 SRCI -0.539148 -0.530728 -0.498998 -0.532013 -0.530678 -0.542737 -0.542986 SRCI-DAVl -0.602990 -0.593234 -0.556063 -0.596610 -0.595398 -0.609937 -0.613417 SRCI-DAV1S -0.599000 -0.589327 -0.552496 -0.592573 -0.591353 -0.605737 -0.609015 SRCI-DAV2 -0.611565 -0.601578 -0.563431 -0.605537 -0.604387 -0.619433 -0.623915 SRCI-DAV2S -0.607039 -0.597150 -0.559404 -0.600942 -0.599780 -0.614640 -0.618856 SRCI-POPLE -0.615976 -0.605834 -0.567014 -0.610311 -0.609225 -0.624660 -0.630085 SRFCCI -0.447202 -0.4389(M -0.406666 -0.440418 -0.439177 -0.451340 -0.452351 SRFCCI-DAVl -0.502705 -0.493304 -0.456049 -0.496882 -0.495918 -0.510399 -0.515202 SRFCCI-DAV1S -0.496538 -0.487260 -0.450562 -0.490608 -0.489614 -0.503837 -0.508218 SRFCCI-DAV2 -0.510570 -0.501001 -0.462874 -0.505185 -0.504337 -0.519291 -0.525344 SRFCCI-DAV2S -0.503529 -0.494101 -0.456629 -0.497989 -0.497097 -0.511740 -0.517233 SRFCCI-POPLE -0.510348 -0.500772 -0.462514 -0.505248 -0.504472 -0.519556 -0.526297 MRCI-REF -0.093548 -0.072450 -0.043989 -0.089791 -0.093328 -0.092532 -0.112271 MRCI -0.485014 -0.471005 -0.431332 -0.475940 -0.476484 -0.490379 -0.502643 MRCI-DAV1 -0.528302 -0.516944 -0.475067 -0.519167 -0.518608 -0.536564 -0.546963 MRCI-DAV1S -0.523493 -0.511840 -0.470207 -0.514364 -0.513927 -0.531433 -0.542038 MRCI-DAV2 -0.533684 -0.522929 -0.480633 -0.524616 -0.523811 -0.542630 -0.552639 MRCI-DAV2S -0.528276 -0.517160 -0.475155 -0.519207 -0.518552 -0.536824 -0.547084 aRSO = 1.4321 A (2.7063 bohr) and 0Oso = 119.53600.44 247 Table 6.15. Correlation energies, E corr (in hartrees), of all states of SO 2 at the X SCF2 optimized geometry.® M ethod ]A, 3 B , 3B 2 3a 2 * a 2 *8, 3b 2 MCSCF2 -0.121227 -0.083522 -0.071579 -0.103199 -0.106537 -0.104720 -0.123838 SRCI -0.534917 -0.526436 -0.498247 -0.528851 -0.527541 -0.539925 -0.539435 SRCI-DAV1 -0.596937 -0.587142 -0.554412 -0.591889 -0.590689 -0.605618 -0.608134 SRCI-DAV1S -0.593061 -0.583348 -0.550901 -0.587949 -0.586743 -0.601512 -0.603840 SRCI-DAV2 -0.605071 -0.595055 -0.561547 -0.600420 -0.599276 -0.614718 -0.618160 SRCI-DAV2S -0.600686 -0.590767 -0.557591 -0.595947 -0.594793 -0.610043 -0.613240 SRCI-POPLE -0.609117 -0.598956 -0.564933 -0.604854 -0.603769 -0.619598 -0.623906 SRFCCI -0.442674 -0.434332 -0.405711 -0.436989 -0.435769 -0.448255 -0.448516 SRFCCI-DAV1 -0.496379 -0.486965 -0.454149 -0.491884 -0.490924 -0.505790 -0.509616 SRFCCI-DAV1S -0.490412 -0.481117 -0.448767 -0.485785 -0.484796 -0.499397 -0.502827 SRFCCI-DAV2 -0.503791 -0.494223 -0.460716 -0.499771 -0.498916 -0.514263 -0.519252 SRFCCI-DAV2S -0.497003 -0.487568 -0.454605 -0.492795 -0.491900 -0.506928 -0.511393 SRFCCI-POPLE -0.503396 -0.493824 -0.460245 -0.499654 -0.498864 -0.514338 -0.519957 aRso = 1-405 A (2.655 bohr) and &oso = 1 18.82°. Table 6.16. Correlation energies, Ecorr (in hartrees), of all states of SO 2 at the X MCSCF2 optimized geometry.® M ethod 'A, 3 B i 3B2 3a 2 *a2 j b 2 M CSCF26 -0.132828 .0.094367 -0.075159 -0.112578 -0.116026 -0.115923 -0.134448 SRCI -0.542027 -0.533607 -0.499393 -0.534083 -0.532720 -0.544524 -0.545100 SRCI-DAV1 -0.607149 -0.597373 -0.557035 -0.599726 -0.598486 -0.612770 -0.616595 SRCI-DAV1S -0.603079 .0.593387 -0.553432 -0.595623 -0.594376 -0.608505 -0.612126 SRCI-DAV2 -0.616042 .0.606027 -0.564556 -0.608924 -0.607749 -0.622549 -0.627388 SRCI-DAV2S -0.611416 -0.601501 -0.560483 -0.604247 -0.603060 -0.617672 -0.622245 SRCI-POPLE -0.620721 -0.610543 -0.568270 -0.613933 -0.612826 -0.628027 -0.633829 SRFCCI -0.450289 -0.441977 -0.407197 -0.442667 -0.441400 -0.453319 -0.454640 SRFCCI-DAV1 -0.507064 -0.497624 -0.457188 -0.500191 -0.499208 -0.513446 -0.518566 SRFCCI-DAVlS -0.500756 -0.491441 -0.451634 -0.493800 -0.492785 -0.506765 -0.511463 SRFCCI-DAV2 -0.515255 -0.505639 -0.464185 -0.508783 -0.507919 -0.522641 -0.529025 SRFCCI-DAV2S -0.508037 -0.498565 -0.457853 -0.501437 -0.500528 -0.514938 -0.520760 SRFCCI-POPLE -0.515167 -0.505540 -0.463902 -0.508975 -0.508187 -0.523050 -0.530140 aRso = 1.4511 A (2.7422 bolir) and Oq s o — 119.4877°. 4MCSCF1 results at the X ‘Ai state MCSCF1 optimized geometry are: -0.117639, —0.074717, -0.060581, -0.092128, -0.095574, -0.099245, and -0.119300. 248 Comparing the results at these three geometries to those at the experimental and MCSCF2 optimized geometries of the respective states (Tables 6.5 and 6.6) one sees that the Ecorr at the latter geometries are greater in magnitude. The general trend for Ecorr for each state at the different points on the potential surface are similar. The *Ai, 1Bi, and 1B2 states have the largest correlation energies. The 3B2 state has the smallest Ecorr, for both MCSCF and Cl. Also, the 3A2 and XA2 states show very similar results to each other, as they did at the previous geometries. Finally, the difference in E corr between the MCSCF2 and MCSCF3 results at these geometries are slightly less than at the respective experimental and MCSCF2 optimized geometries. However, the trends are the same in th a t the MCSCF2 and MCSCF3 differences are greatest for the xAi and 3B2 states. The vertical excitation energies, A E vert, at the X xAi experimental, SCF, and MCSCF2 optimized geometries are given in Tables 6.17-6.19, respectively. Except for the 3Ba SCF A Evert, all AEvert are greater than the corresponding Te’s. The 3Bj SCF A Evert at the xAi experimental geometry is slightly less than its Te at its experimental geometry (by only about 0.003 eV). The reasons for this were discussed above for the total energies. For all states, all A Evert at the xAi SCF optimized geometry are greater than at the xAi experimental geometry, while at the xAi MCSCF2 geometry they are less than at the xAt experimental geometry. The results for the XB2 state are similar to those obtained for Te, The A E vert at the three geometries are all greater than the experimental value. The A E vert 249 T able 6.17. Vertical excitation energies, A Evert (in eV), of S02 calculated at the X *Ai experimental geometry? M ethod 3B X 3b 2 3a 2 1a 2 % *B2 E X P 6 3.4 4.3 6.2 SCF 3.124862 3.050576 4.329087 4.621743 4.635796 7.120735 M CSCF1 4.233782 4.472059 4.932199 5.132393 5.053044 7.002294 M CSCF2 4.166189 4.525739 4.850974 5-050026 5.091299 7.054409 M CSCF3 3.691744 4.398987 4.449710 4.642376 4.636687 6.631998 SRCI 3.353995 4.143158 4.523248 4.852227 4.538154 7.016301 SRCI-DAV1 3.390355 4.327563 4.502695 4.828349 4.446762 6.836989 SRCI-DAV1S 3.388083 4.316038 4.503980 4.829841 4.452474 6.848196 SRCI-DAV2 3-396636 4.360392 4.493109 4.817080 4.421690 6.784681 SRCI-DAV2S 3.393971 4.346815 4.494992 4.819276 4.428969 6.799157 SRCI-POPLE 3.400861 4.382936 4.483261 4.805465 4.399494 6.736809 SRFCCI 3.350661 4.153621 4.513688 4.840103 4.523193 6.980612 SRFCCI-DAV1 3.380680 4.320199 4.487553 4.806430 4.426434 6.780675 SRFCCI-DAV1S 3.377345 4.301690 4.490457 4.810171 4.437185 6.802890 SRFCCI-DAV2 3.385256 4.348486 4.475615 4.791364 4.398494 6.718713 SRFCCI-DAV2S 3.381412 4.326834 4.479845 4.796779 4.412349 6.747813 SRFCCI-POPLE 3.385440 4.352227 4.467864 4.781658 4.385235 6.686727 MRCI-REF 3.698971 4.399168 4.431331 4.627740 4.663447 6.611249 MRCI 3.506076 4.511382 4.576024 4.853872 4.489810 6.641010 M RCI-DAV1 3.433936 4.499223 4.577687 4.885546 4.410967 6.612949 MRCI-DAV1S 3.441952 4.500574 4.577502 4.882027 4.419727 6.616067 MRCI-DAV2 3.417524 4.494196 4.575862 4.890410 4.392357 6.604946 MRCI-DAV2S 3.427363 4.496105 4.575880 4.886351 4.403186 6.608953 aR s o = 1 4321 A (2.7063 bohr) and 0o so = 119.5360°,44 ‘See Chapter III. 250 Table 6.18. Vertical excitation energies, A E veTt (in eV), of SO 2 calculated at the X JAi SCF optimized geometry.0 Method 3B i 3b 2 3a 2 *a 2 xB i % SCF 3.340671 3.564334 4.694800 4.989180 4.975593 7.511393 MCSCF2 4.366689 4.915332 5.185373 5.388903 5.424777 7.440318 SRCI 3.571459 4.562201 4.859882 5.189908 4.839321 7.388453 SRCI-DAV1 3.607206 4.721541 4.832171 5.159198 4.739380 7.206704 SRCI-DAVIS 3.604972 4.711582 4.833903 5.161117 4.745627 7.218063 SRCI-DAV2 3.613220 4.748710 4.821365 5.146867 4.713089 7.155218 SRCI-DAV2S 3.610610 4.737053 4.823773 5.149557 4.720978 7.169795 SRCI-POPLE 3.617158 4.766666 4.810792 5.134703 4.690368 7.108962 SRFCCI 3.567678 4.570169 4.849496 5.177067 4.823718 7.352399 SRFCCI-DAV1 3.596842 4.713487 4.817103 5.137619 4.719489 7.151172 SRFCCI-DAV1S 3.593602 4.697563 4.820702 5.142002 4.731070 7.173530 SRFCCI-DAV2 3.601123 4.736560 4.804266 5.121908 4.690718 7.090733 SRFCCI-DAV2S 3.597407 4.718072 4.809291 5.128037 4.705496 7.119807 SRFCCI-POPLE 3.601145 4.738559 4.796612 5.112495 4.677846 7.060738 a R s o = 1.405 A (2.655 bohr) and O q so = 118.82°. 251 Table 6.19. Vertical excitation energies, A Evert (in eV), of SO 2 calculated at the X aAx MCSCF2 optimized geometry . 0 Method 3Bi 3b2 3a2 *A2 'Bi 1B2 SCF 2.983632 2.678296 4.047870 4.338408 4.404902 6.819313 MCSCF26 4.030238 4.247597 4.598919 4.795619 4.864926 6.775223 SRCI 3.212767 3.838446 4.264031 4.591665 4.336945 6.735696 SRCI-DAV1 3.249662 4.042005 4.249871 4.574135 4.251947 6.562272 SRCI-DAVIS 3.247356 4.029282 4.250756 4.575231 4.257259 6.573111 SRCI-DAV2 3.256144 4.079317 4.241545 4.564064 4.227827 6.510559 SRCI-DAV2S 3.253433 4.064263 4.242951 4.565789 4.234647 6.524630 SRCI-POPLE 3.260578 4.105587 4.232582 4.553250 4.206082 6.462617 SRFCCI 3.209821 3.850934 4.255278 4.580305 4.322449 6.700920 SRFCCI-DAV1 3.240526 4.035521 4.234890 4.552197 4.231233 6.506323 SRFCCI-DAV1S 3.237115 4.015011 4.237155 4.555320 4.241368 6.527945 SRFCCI-DAV2 3.245317 4.068035 4.223995 4.538035 4.203931 6.444610 SRFCCI-DAV2S 3.241373 4.043913 4.227471 4.542732 4.217100 6.473089 SRFCCI-POPLE 3.245613 4.073322 4.216383 4.528347 4.190408 6.411880 aRso = 1-4511 A (2.7422 bohr) and 0OSO = 119.4877°. ‘MCSCFl results at the X *Ai state MCSCF1 optimized geometry are: 4.151640, 4.230954, 4.742090, 4.938851, 4.905436, and 6.774116. decreases as higher levels of correlation are included. The Afv„ert(SCF) at the ’Aj experimental and at the SCF optimized geometry are 7.12 and 7.51 eV, re spectively, compared to the experimental value of 6.2 eV. The A invert (MCSCF2) at the *Ai experimental and MCSCF2 optimized geometries are 7.05 and 6.78, respectively (the MCSCF1 values are 7.00 and 6.77 eV). The A-F„ert(SRFCCI- POPLE) at the 'A, experimental a n d SCF and MCSCF2 optimized geometries are 6.69, 7.06, and 6.41 eV, respectively. At the 4Ai experimental geometry the A £ uer<(MRCI-DAV2 S) is 6.61 eV. x h e present results agree fairly well with previ ous results (see Table 3-8). The S-CI A E vert value of Bendazzoli and Palimeri 143 is 6.93 eV and the D F C -L S D result of Jones 81 is 5.9 eV. For the ^ state all A F wri(MCSCF) at all three geometries are greater than the SCF value (by about 0.5 eV). However, the Cl A E vert decrease compared to SCF. At all three georTietries, A£„ert(SRFCCI-POpLE) — AF„ert(SCF) is between —0 . 2 and —0.3 eV and at the X 1Al experimental geometry A£t(ert(MRCI-DAV2S) — AFJ„ert(SCF) = -0-23 eV. The Ai%>ert(SCF) at the JAj SCF optimized geome try is 4.98 eV and the A F„ert(MCSCF2) at the xAj MCSCF2 optimized geometry is 4.40 eV (the MCSCFI result is 4 .9 I eV). The AFuert(SRFCCI-POPLE) at the xAi experimental and SC F and MCSCF2 optimized geometries are 4.39, 4.68, and 4.19 eV, respectively T h e AFver((MRCI-DAV2S) a t the JAi experimental geome try is 4.40 eV. This does not agree w ith the result of Bendazzoli and Palimeri , 143 253 3.70 eV, or the value of 3.9 eV of Jones 81 at the xAj experimental geometry (see Table 3.11). The present result is « 0.5 eV higher. The vertical excitation energy of the 1A2 state increases when correlation effects are included, compared to the A E vert{SCF). The MRCI values also increase com pared to the MCSCF3 and MRCl-REF values. In general, the correlation effects on A Fvert are less for this state than for the other states. At the xAi experimen tal geometry the SCF, MCSCF1, MCSCF2, SRFCCI-POPLE, and MRCI-DAV2S A E v e r t are 4.62, 5.13, 5.05, 4.78, and 4.89 eV, respectively. The SCF, MCSCF2, and SRFCCI-POPLE A Evert at the SCF optimized geometry are 4.98, 5.42, and 5.11 eV and at the MCSCF2 optimized geometry they are 4.34, 4.80, and 4.53 eV. The Cl values at the 4Ai experimental and MCSCF2 optimized geometries are within 0.6 eV of the experimental value of 4.3 eV. Also, the present results agree well with the results of Bendazzoli and Palimeri 143 and Jones , 81 which are 4.77 and 5.0 eV, respectively. The A Evert for the 3A2 state also increases compared to AEuert(SCF) when correlation effects are added. The SCF, MCSF1, MCSCF2, SRFCCI-POPLE, and MRCI-DAV2S A E Vert at the !Ai experimental geometry are 4.33, 4.93, 4.85, 4.47, and 4.58 eV, respectively. At the SCF and MCSCF2 optimized geometries the SCF, MCSCF2, and SRFCCI-POPLE are 4.69, 4.92, 4.80 and 4.05, 4.60, 4.22 eV, respectively. The A ^ er<(SRFCCl-POPLE) and A£„ert(MRCI-DAV 2 S) agree well 254 with previous results (see Table 3.14). Bendazzoli and Palimeri 143 obtained a value of 4.21 eV and Jones’ value 81 is 4.6 eV. The 3 B2 state exhibits the greatest correlation effect on A Evert, increasing by about 1.5 eV relative to SCF for A £;„crt(MRCI-DAV2S) and 1.3 eV for A.E„er*(SRFCCI- POPLE). The SCF, MCSF 1 , MCSCF2, SRFCCI-POPLE, and MRCI-DAV2S A E vert at the JAi experimental geometry are 3.05, 4.47, 4.53, 4.35, and 4.50 eV, respec tively. The SCF, MCSCF 2 , and SRFCCI-POPLE A E vert at the SCF and MCSCF 2 optimized geometries are 3.56, 4.92, 4.74 eV and 2.68, 4.25, 4.07 eV. The previous theoretical results for this state are in Table 3.15. The DFC-LSD result of Jones , 81 4.8 eV, agrees fairly well with the present results. However, the SCF value of 2.25 eV of Keeton and Santry 64 and the S-CI value of 2.54 eV obtained by Ben dazzoli and Palimeri 1 4 3 are much lower than the respective values obtained in the present study. The results for the 3Bi state at all levels and all three geometries are within 0.8 eV of the experimental result of 3.4 eV. At the JAj experimental geometry the A Evert at the SCF, MCSFl, MCSCF 2 , SRFCCI-POPLE, and MRCI-DAV2S levels of calculation are 3.12, 4.23, 4.17, 3.39, and 3.43 eV. The Cl values agree very well with the experimental result. The SCF, MCSCF2, and SRFCCI-POPLE AEvert at the SCF optimized geometry are 3.34, 4.37, and 3.60 eV, respectively. The corresponding values at the MCSCF2 optimized geometry are 2.98, 4.03, and 3.25 eV. Even at these geometries the A Fuert(SRFCCI-POPLE) is within 0.2 eV of 255 the experimental value. The previously calculated results exhibit large variations (Table 3.13). The DFC-LSD result of 3.2 eV obtained by Jones 81 agrees well with the present result. However, the S-CI result of 2.54 eV obtained by Bendazzoli and Palimeri 143 is markedly differenet. Since the A E vert show such variability, it is helpful to look at the relative orderings of the excited states at the three X xAi geometries as given in Tables 6.20- 6 .2 2 . At the xAi experimental and MCSCF 2 optimized geometries the SCF ordering of the excited states is 3 B2, 3 B i, 3A2, 1A2, 1 Bi, and 1 B2. At the xAa SCF optimized geometry the SCF order of the 3Bj and 3 B2 states is reversed, as is the order of the XA 2 and 1 Bi states. The MCSCF2 ordering at the three xAi geometries is: 3 B i, 3 B2, 3A2, xA2, 1 B1, and 1 B2. The MCSCFl and MCSCF3 ordering of the states at the XAX experimental geometry is the same except for the XA 2 and 1 B1 states. At the xAi experimental geometry, the XA 2 and xBi states have nearly degenerate SCF and MCSCF 2 energies. The Cl calculations show very interesting results for the ordering of the states. As mentioned above, the A Everl of the *Bi state drops while that for the 3 B2 state increases as higher levels of correlation are included. The SRCI and SRFCCI ordering at the xAi experimental and MCSCF2 optimized geometries is: 3 Bj, 3 B2, 3A2, XBi, jA2, and *B2. As corrections for quadruples are included in the SRCI and SRFCCI calculations the XB, state becomes the third lowest state (exchanging with the 3A2 state). At the lAj SCF optimized geometry, the SRCI and SRFCCI 256 Table 6.20. Relative ordering of the excited states of SO 2 at the X Wi experi mental geometry.0 Method 3Bj 3b 2 3a 2 ja 2 aBi % SCF 2 1 3 4 5 6 MCSCF1 1 2 3 5 4 6 MCSCF2 1 2 3 4 5 6 MCSCF3 1 2 3 5 4 6 SRCI 1 2 3 5 4 6 SRCI-DAV1 1 2 4 5 3 6 SRCI-DAV1S 1 2 4 5 3 6 SRCI-DAV2 1 2 4 5 3 6 SRCI-DAV2S 1 2 4 5 3 6 SRCI-POPLE 1 2 4 5 3 6 SRFCCI 1 2 3 5 4 6 SRFCCI-DAV1 1 2 4 5 3 6 SRFCCI-DAV1S 1 2 4 5 3 6 SRFCCI-DAV2 1 2 4 5 3 6 SRFCCI-DAV2S 1 2 4 5 3 6 SRFCCI-POPLE 1 2 4 5 3 6 MRCI-REF 1 2 3 4 5 6 MRCI 1 3 4 5 2 6 MRCI-DAV1 1 3 4 5 2 6 MRCI-DAV1S 1 3 4 5 2 6 MRCI-DAV2 1 3 4 5 2 6 MRCI-DAV2S 1 3 4 5 2 6 a R s o = 1-4321 A (2.7063 bohr) and 0O So = 119.5360°.44 257 Table 6.21. Relative ordering of the excited states of SO 2 at X*Ax SCF optimized geometry.0 Method 3 BX 3 B2 3a 2 * a 2 *BX >b 2 SCF 1 2 3 5 4 6 MCSCF2 1 2 3 4 5 6 SRCI 1 2 4 5 3 6 SRCI-DAV1 1 2 4 5 3 6 SRCI-DAV1 S 1 2 4 5 3 6 SRCI-DAV2 1 3 4 5 2 6 SRCI-DAV2 S 1 3 4 5 2 6 SRCI-POPLE 1 3 4 5 2 6 SRFCCI 1 2 4 5 3 6 SRFCCI-DAV1 1 2 4 5 3 6 SRFCCI-DAV1S 1 2 4 5 3 6 SRFCCI-DAV2 1 3 4 5 2 6 SRFCCI-DAV2S 1 3 4 5 2 6 SRFCCI-POPLE 1 3 4 5 2 6 a R s o — 1.405 A (2.655 bohr) and Oq s o = 118.82°. 258 Table 6.22. Relative ordering of the excited states of SO 2 at X xAj MCSCF2 optimized geometry.® Method 3Br 3 b 2 3a 2 xa 2 'B r xb 2 SCF 2 1 3 4 5 6 MCSCF2 6 1 2 3 4 5 6 SRCI 1 2 3 5 4 6 SRCI-DAVl 1 2 3 5 4 6 SRCI-DAV1 S 1 2 3 5 4 6 SRCI-DAV2 1 2 4 5 3 6 SRCI-DAV2S 1 2 4 5 3 6 SRCI-POPLE 1 2 4 5 3 6 SRFCCI 1 2 3 5 4 6 SRFCCI-DAV1 1 2 4 5 3 6 SRFCCI-DAV1S 1 2 3 5 4 6 SRFCCI-DAV2 1 2 4 5 3 6 SRFCCI-DAV2S 1 2 4 5 3 6 SRFCCI-POPLE 1 2 4 5 3 6 aRso = 1.4511 A (2.7422 bohr) and 0OSo = 119.4877°. 6MCSCF1 results at the X JAi state MCSCF1 optimized geometry are: 1, 2, 3, 5, 4, 6. ordering of the states is: 3 Bi, 3 B2, 1 B1, 3A2, 1A2, and 1 B2. As the quadruple corrections are included the xBi state A Evert falls below the 3 B2 state, making it the second lowest state. Finally, the ordering of the states at all MRCI levels is: 3 Bi, 1 Bi, 3 B2, 3A2, xA2, and XB2. As one can see, the orderings of the xBi, 3 B2, and 3A2 states is quite variable. These three states are within at most 0.18 eV of each other and the xBi and 3 B2 states are within 0.09 eV at the MRCI-DAV2S level. The ordering of the XA 2 and XB 2 states is the same as that obtained in previous studies. The DFC-LSD calculations of Jones 81 give the ordering of the other states as 3 Bi, xBx, 3A 2 and 3 B2. The placement of the xBi as the second lowest excited state in Jones’ calculation may just be coincidental. Jones’ A E vert is within 0.7 eV of the 3Bj value, while the present value is 1.03 eV higher. Also, his A Evert for the 3 B2 state is 0.3 eV higher than the present value. The Bendazzoli and Palimeri 143 S-CI ordering is: 3 Bi, 3 B2, 1 Bi, 3A2, XA2, and XB2. However, their A E vert for the 3 B2 state (2.54 eV) is calculated to be nearly degenerate with that for the 3Bi state (2.52 eV), both of which are considerably below the present calculations. In summary, looking at the MRCI results and the Davidson and Pople quadru ple corrected SRCI values, the likely ordering of the excited states is: 3 Bj, xBi, 3 B2, 3A2j xA2, and XB2. It is fairly certain that the 3 Bi, XA2, and XB 2 states are in the proper order. Also, it is probably fairly certain that the xBi and 3 B2 states are very nearly degenerate and lie just below the 3A2 state. 260 6.6.3 Emission Energies, AEcrnis Vertical emission energies, A E e m is, were calculated at the excited states’ experi mental geometries (where available) and at their MCSCF2 optimized geometries. The total energies and correlation energies at the various geometries are given in Tables 6.23-6.26. The X xA j SCF total energies show a great deal of variability between the experimental and MCSCF2 optimized geometries. However, the MC- SCF2 and various Cl energies at all of the experimental and MCSCF2 optimized geometries are fairly close. The correlation energies are greater in magnitude at the excited states MC- S C F 2 optimized geometries than at their experimental geometries. The M C S C F 2 correlation energies of the 3A2, XA2, and JBi states at their experimental geometries are very similar. The same is true at the M C S C F 2 optimized geometries. At both the experimental and M C S C F 2 optimized geometries, the E c o r r of the 3 B 2 and *B2 states are larger than th e other states, while that of the xAr state is smaller. The M C S C F 3 correlation energies at the M C S C F 2 optimized geometries follow the same pattern. The Cl correlation energies are all approximately the same at all the excited state geometries. The vertical emission energies, A Eemia, calculated at the excited states’ ex perimental and MCSCF2 optimized geometries are given in Tables 6.27 and 6.28. The relative ordering of the emission energies (position of the xAi potential surface with respect to that of the respective excited state) at these geometries are given 261 Table 6.23. Total energies (in hartrees) of the X^Aj state of SO 2 at the respective experimental geometries of the excited states.® Method 3Bi 3a2 % 1b2 SCF -547.208303 -547.158345 -547.173916 -547.172325 MCSCF2 -547.354054 -547.318195 -547.327183 -547.332203 SRCI -547.757795 -547.714670 -547.727088 -547.728516 SRCI-DAV1 -547.826168 -547.787034 -547.797842 -547.800720 SRCI-DAV1S -547.821895 -547.782511 -547.793420 -547.796207 SRCI-DAV2 -547.835885 -547.797854 -547.808219 -547.811492 SRCI-DAV2S -547.831004 -547.792655 -547.803148 -547.806306 SRCI-POPLE -547.841272 -547.804239 -547.814202 -547.817831 SRFCCI -547.666553 -547.624144 -547.636300 -547.637962 SRFCCI-DAV1 -547.726563 -547.688299 -547.698810 -547.701917 SRFCCI-DAV1S -547.719895 -547.681171 -547.691865 -547.694811 SRFCCI-DAV2 -547.735605 -547.698547 -547.708582 -547.712100 SRFCCI-DAV2S -547.727933 -547.690280 -547.700551 -547.703862 SRFCCI-POPLE -547.735883 -547.699415 -547.709216 -547.712933 “See Table 6.2 for the excited state experimental geometries. “See Table 6.3 for energies of the excited states. 262 Table 6.24. Total energies (in hartrees) of the X xAi state of SO 2 at the respective MCSCF2 optimized geometries of the excited states.0 M ethod 3Bi 3 b 2 3a 2 *Aa ‘ B j »b 2 SCF -547.185061 -547.152531 -547.145341 -547.145215 -547.162393 -547.148153 MCSCF2 -547.344544 -547.324242 -547.311718 -547.312099 -547.332384 -547.322737 MCSCF3 -547.304965 -547.282984 -547.272682 -547.272911 -547.290812 -547.280609 SRCI -547.741413 -547.714068 -547.704599 -547.704688 -547.723372 -547.710957 SRCI-DAV1 -547.813246 -547.789015 -547.778494 -547.778691 -547.797816 -547.786553 SRCI-DAV1S -547.808757 -547.784331 -547.773875 -547.774066 -547.793164 -547.781828 SRCI-DAV2 -547.823896 -547.800558 -547.789744 -547.789972 -547.809207 -547.798282 SRCI-DAV2S -547.818741 -547.795153 -547.784422 -547.784642 -547.803842 -547.792824 SRCI-POPLE -547.830117 -547.807609 -547.796524 -547.796781 -547.816112 -547.805507 SRFCCI -547.650747 -547.623953 -547.614323 -547.614429 -547.633154 -547.620943 SRFCCI-DAVI -547.714256 -547.690695 -547.680040 -547.680256 -547.699348 -547.688344 SRFCCI-DAV1S -547.707199 -547.683279 -547.672738 -547.672942 -547.691994 -547.680855 SRFCCI-DAV2 -547.724285 -547.701702 -547.690750 -547.690998 -547.710179 -547.699549 SRFCCI-DAV2S -547.716114 -547.693063 -547.682258 -547.682490 -547.701621 -547.690815 SRFCCI-POPLE -547.725029 -547.702955 -547.691853 -547.692117 -547.711333 -547.700906 MRCI-REF -547.301492 -547.279544 -547.269297 -547.269524 -547.287302 -547.277128 MRCI -547.701055 -547.680289 -547.669123 -547.669417 -547.688274 -547.678423 M RCI-DAVl -547.748457 -547.728783 -547.717040 -547.717381 -547.736727 -547.727239 MRCI-DAV1S -547.743190 -547.723394 -547.711716 -547.712052 -547.731344 -547.721 815 MRCI-DAV2 -547.754837 -547.735459 -547.723564 -547.723918 -547.743387 -547.734000 MRCI-DAV2S -547.748861 -547.729329 -547.717515 -547.717863 -547.737264 -547.727 824 “See Table 6.2 for the excited state MCSCF2 optimized geometries. 4See Table 6.4 for energies of the excited states. 263 Table 6.25. Correlation energies, Ecorr (in hartrees), of the X xAi state of S02 at the respective experimental geometries of the excited states.® Method 3B, 3a 2 xa 2 xb 2 MCSCF2 -0.145751 -0.159850 -0.153267 -0.159878 SRCI -0.549491 -0.556325 -0.553173 -0.556191 SRCI-DAV1 -0.617865 -0.628689 -0.623926 -0.628395 SRCI-DAV1S -0.613591 -0.624166 -0.619504 -0.623882 SRCI-DAV2 -0.627581 -0.639509 -0.634303 -0.639167 SRCI-DAV2S -0.622701 -0.634310 -0.629233 -0.633981 SRCI-POPLE -0.632969 -0.645894 -0.640286 -0.645506 SRFCCI -0.458250 -0.465799 -0.462385 -0.465637 SRFCCI-DAV1 -0.518259 -0.529954 -0.524895 -0.529592 SRFCCl-DAVlS -0.511592 -0.522826 -0.517949 -0.522486 SRFCCI-DAV2 -0.527302 -0.540202 -0.534666 -0.539775 SRFCCI-DAV2S -0.519629 -0.531935 -0.526635 -0.531537 SRFCCI-POPLE -0.527579 -0.541070 -0.535301 -0.540608 “See T a b le 6.2 for the e x c i t e d state experimental geometries. 264 Table 6.26. Correlation energies, Ecorr (in hartrees), of the X *Ai state of S 0 2 at the the respective MCSCF2 optimized geometries of the excited states.® Method 3B, 3b 2 3a 2 % *Bi 1B 2 MCSCF2 -0.159483 -0.171712 -0.166377 -0.166884 -0.169990 -0.174584 MCSCF3 -0.119904 -0.130453 -0.127341 -0.127696 -0.128419 -0.132455 SRCI -0.556353 -0.561538 -0.559259 -0.559473 -0.560978 -0.562804 SRCI-DAV1 -0.628185 -0.636484 -0.633153 -0.633476 -0.635423 -0.638399 SRCI-DAV1S -0.623696 -0.631800 -0.628535 -0.628851 -0.630770 -0.633675 SRCI-DAV2 -0.638835 -0.648028 -0.644403 -0.644757 -0.646814 -0.650129 SRCI-DAV2S -0.633680 -0.642622 -0.639082 -0.639427 -0.641449 -0.644671 SRCI-POPLE -0.645056 -0.655079 -0.651184 -0.651566 -0.653719 -0.657353 SRFCCI -0.465686 -0.471423 -0.468982 -0.469214 -0.470761 -0.472790 SRFCCI-DAV1 -0.529195 -0.538164 -0.534700 -0.535041 -0.536955 -0.540190 SRFCCI-DAV1S -0.522138 -0.530749 -0.527398 -0.527727 -0.529600 -0.532701 SRFCCI-DAV2 -0.539224 -0.549172 -0.545409 -0.545783 -0.547785 -0.551396 SRFCCI-DAV2S -0.531053 -0.540533 -0.536917 -0.537275 -0.539227 -0.542662 SRFCCI-POPLE -0.539968 -0.550424 -0.546512 -0.546902 -0.548940 -0.552753 MRCI-REF -0.116431 -0.127014 -0.123956 -0.124309 -0.124908 -0.128975 MRCI -0.515994 -0.527759 -0.523783 -0.524202 -0.525881 -0.530270 MRCI-DAV1 -0.563396 -0.576252 -0.571699 -0.572166 -0.574334 -0.579086 MRCI-DAV1S -0.558129 -0.570864 -0.566375 -0.566837 -0.568950 -0.573662 MRCI-DAV2 -0.569776 -0.582928 -0.578224 -0.578703 -0.580994 -0.585846 MRCI-DAV2S -0.563800 -0.576798 -0.572175 -0.572648 -0.574870 -0.579671 “See Table 6.2 for the excited state MCSCF2 optimized geometries. 265 in Tables 6 . 2 9 and 6 . 3 0 . As can be seen in the tables, A Eemia values for all states are smaller at the M C S C F 2 optimized geometries than at the experimental geome tries. The *A 2 state’s A E em ,-s changes the most. One note of interest concerns the Ai?em.s(SCF) of the 3B 2 state at its M C S C F 2 optimized geometry. A £emts(SCF) is actually negative, which means that at the S C F level the 3Bi state is lower than the xAx state at the same geometry. However, this does not occur once correlation is added. The M C S C F 2 and M C S C F 3 AFemis are greater than the S C F values. The C l emission energies are also greater than the S C F values but, in general, are smaller than the the M C S C F 2 values. The only exception is the M R C I value for the 3B2 state, which increases slightly. All M R C I emission energies axe larger than the M C S C F 3 values, except for the 3Bi state. The 3B 2 state’s A F emis is affected most by correlation. The calculated values of A E e m is for the 3Bi, xBi and XB 2 states are all lower than the experimental values but, except for XB2, are in fairly good agreement. For the 3Bx state at its experimental geometry, the SRCI-POPLE and SRFCCI- POPLE AEemis values of 2.83 and 2.81 eV, respectively, agree very well with the SRCI-SD(Q) value of 2.83 eV calculated by Phillips and Davidson.158 The SCF values for the 3A 2 and XA2 states agree fairly well with the values of 1.23 and 1.50 eV, respectively, calculated from results of Lindley.22 The relative orderings of A E em is at each level of calculation are the same at the excited states’ experimental and MCSCF2 optimized geometries. However, as 266 Table 6.27. Emission energies, A E emia (in eV), of S 02 calculated at the respective experimental geometries of the excited states.? Method 3Bi 3A2 *A2 xb 2 EXP 2.99b 5.41c SCF 2.496425 1.298869 1.930653 4.472064 MCSCF2 3.574844 2.256788 2.763939 5.032428 SRCI 2.759866 1.682047 2.336603 4.680318 SRCI-DAV1 2.809523 1.763164 2.402314 4.639751 SRCI-DAV1S 2.806420 1.758094 2.398207 4.642287 SRCI-DAV2 2.819291 1.780569 2.414251 4.621643 SRCI-DAV2S 2.815577 1.774411 2.409398 4.625310 SRCI-POPLE 2.826607 1.794500 2.422740 4.602391 SRFCCI 2.758243 1.688906 2.338969 4.666298 SRFCCI-DAV1 2.801224 1.765462 2.395350 4.609210 SRFCCI-DAV1S 2.796448 1.756956 2.389086 4.615553 SRFCCI-DAV2 2.809199 1.781854 2.404378 4.583778 SRFCCI-DAV2S 2.803537 1.771527 2.397110 4.592947 SRFCCI-POPLE 2.810852 1.787035 2.405161 4.566508 “See Table 6.2 for the excited state experimental geometries. 6 References 190,193,196,197,199,211,278. 'Reference 126. 267 Table 6.28. Emission energies, A E e m ia (in eV), of S02 calculated at the respective MCSCF2 optimized geometries of the excited states.0 Method 3B i 3b 2 3a 2 *A2 'B i % EXP 2.996 3.4-3.7 C 5.41d SCF 2.196248 -0.270207 1.060119 1.331814 2.871663 4.128031 MCSCF2 3.225002 2.202511 2.065035 2.279731 3.430085 4.696527 MCSCF3 2.729808 2.068517 1.703261 1.918710 2.994936 4.194837 SRCI 2.471851 1.421233 1.464944 1.792077 3.040495 4.352342 SRCI-DAV1 2.524209 1.813483 1.555815 1.881977 3.013429 4.311709 SRCI-DAV1S 2.520937 1.788968 1.550135 1.876358 3.015121 4.314248 SRCI-DAV2 2.534798 1.900402 1.576063 1.900692 3.000311 4.292207 SRCI-DAV2S 2.530864 1.870454 1.569118 1.893903 3.002822 4.295965 SRCI-POPLE 2.542955 1.971100 1.592704 1.915461 2.986003 4.270803 SRFCCI 2.471602 1.457285 1.473127 1.797799 3.030899 4.337813 SRFCCI-DAV1 2.517270 1.828273 1.559223 1.877857 2.990759 4.279141 SRFCCI-DAV1S 2.512196 1.787052 1.549657 1.868961 2.995219 4.285660 SRFCCI-DAV2 2.525958 1.911500 1.578501 1.893680 2.971797 4.251219 SRFCCI-DAV2S 2.519918 1.861032 1.566792 1.883027 2.978364 4.260841 SRFCCI-POPLE 2.527958 1.940325 1.585215 1.897821 2.958444 4.231269 MRCI-REF 2.732464 2.053516 1.678276 1.894929 2.976701 4.162153 MRCI 2.668005 2.212274 1.766690 2.043836 3.074586 4.261956 MRCI-DAV1 2.627969 2.230816 1.770830 2.070741 3.079110 4.275568 MRCI-DAV1S 2.632417 2.228756 1.770370 2.067752 3.078608 4.274055 MRCI-DAV2 2.617493 2.233264 1.770389 2.075798 3.078583 4.277685 MRCI-DAV2S 2.623105 2.230932 1.769978 2.072246 3.078139 4.275937 °See Table 6.2 for the excited state MCSCF2 optimized geometries. 4 References 190,193,196,197,199,211,278. ‘Could not be specifically attributed to either the 1Bi or *A2 state. See references 190,197,199. ^Reference 126. 268 Table 6.29. Relative ordering of A Eemis of SO 2 calculated at the respective experimental geometries of the excited states.® Method 3Bi 3a 2 *A2 % SCF 4 2 3 6 MCSCF2 4 1 3 6 SRCI 4 2 3 6 SRCI-D AVI 4 1 3 6 SRCI-DAVlS 4 1 3 6 SRCI-DAV2 4 1 3 6 SRCI-DAV2S 4 1 3 6 SRCI-POPLE 4 1 2 6 SRFCCI 4 2 3 6 SRFCCI-DAV1 4 1 3 6 SRFCCI-DAV1S 4 1 3 6 SRFCCI-DAV2 4 1 2 6 SRFCCI-DAV2S 4 1 3 6 SRFCCI-POPLE 4 1 2 6 “See Table 6.2 for the excited state experimental geometries. 269 Table 6.30. Relative ordering of A Eemia of S02 calculated at the respective MCSCF2 optimized geometries of the excited states.® Method 3Bx 3b2 3a 2 *A2 % *B2 SCF 4 1 2 3 5 6 MCSCF2 4 2 1 3 5 6 MCSCF3 4 3 1 2 5 6 SRCI 4 1 2 3 5 6 SRCI-DAV1 4 2 1 3 5 6 SRCI-DAV1S 4 2 1 3 5 6 SRCI-DAV2 4 2 1 3 5 6 SRCI-DAV2S 4 2 1 3 5 6 SRCI-POPLE 4 3 1 2 5 6 SRFCCI 4 1 2 3 5 6 SRFCCI-DAV1 4 2 1 3 5 6 SRFCCI-DAV1S 4 2 1 3 5 6 SRFCCI-DAV2 4 3 1 2 5 6 SRFCCI-DAV2S 4 2 1 3 5 6 SRFCCI-POPLE 4 3 1 2 5 6 MRCI-REF 4 3 1 2 5 6 MRCI 4 3 1 2 5 6 MRCI-DAV1 4 3 1 2 5 6 MRCI-DAV1S 4 3 1 2 5 6 MRCI-DAV2 4 3 1 2 5 6 MRCI-DAV2S 4 3 1 2 5 6 “See Table 6.2 for the excited state MCSCF2 optimized geometries. for A Evert., the orderings change as the level of calculation changes. The ordering at the MCSCF2 level is 3A2, 3B2, XA2, 3Bi, 1B1, and 1B2- The ordering of the 3Bt, % , and xB2 states stays the same at all levels of calculation. The 3A2 state’s AEemia is lowest at all MRCI levels and for SRCI when estimates for quadruple excitations are included. The ordering of the 3B2 and XA2 states changes depending on the level of calculation, but seems to settle on the XA2 A E emis being lower for the MRCI calculations and once the quadruple corrections are included in the SRCI results. Thus, the final MRCI-DAV2S ordering of the vertical emission energies is: 3A2, xA2, 3B2, 3Bx, xBi, and XB2. 6.7 Approximate MCSCF 2 Harmonic Frequencies Calculating the harmonic frequencies and comparing them to any experimental values can provide information on how well the calculated potential energy surface represents the experimental surface in the neighborhood of the minimum. Frequen cies can be obtained by using the program FREQ developed by M. Pepper.307 This program uses the Hessian (second derivatives) produced in the MINPT program to produce harmonic frequencies. The normal procedure to obtain good frequencies would be first to find the minimum point on the potential surface and then calcu late several points near the bottom of the well. These points would be chosen such that small incremental steps, both in the positive and negative directions, were taken for each degree of freedom. These points and the minimum point would 271 then be used with MINPT to obtain a good fit near the minimum and the result ing Hessian values would then be used as input to FREQ to obtain the harmonic frequencies. However, it was not a major objective of the present study to produce good harmonic frequencies. In order to do this many more calculations would have been needed. Also, no asymmetrical (Cs) geometries were considered in this study, so the asymmetric stretching frequency could not be calculated. Therefore, only approximate frequencies were obtained for the symmetric stretching and bending vibrational modes. This was done by using the quartic fits generated by MINPT using the same points which were used to obtain the optimized geometries and the final minimum point (including the gradients a t those points). The resulting Hessian values were then used in FREQ to obtain the harmonic frequencies given, along with the experimental harmonic frequencies, in Table 6.31. The MCSCF2 frequencies calculated for the X *Ai state agree quite well with the experimental values. The calculated value for experimental value, while the calculated is only about 2 cm-1 higher than the experimental value. This is understandable since the MCSCF2 optimized geometry is quite close to the experimental geometry. This also means that the shape of the MCSCF2 potential surface may closely mimic the actual experimental shape near the minimum. 272 Table 6.31. Harmonic frequencies (cm 1) for the symmetric stretching (u>i) and bending (ui2) modes calculated using the MCSCF2 potential surfaces.0 State Experimental6 Calculated xAx 1167.6 1116 526.3 528 3Bx 990.8 799 385.2 346 690 779 460 373 3A2 . . . 906 374 xa 2 788 896 306 371 3b 2 . . . 866 400 xb 2 935 814 382 292 “The masses of the sulfur and oxygen atoms used to obtain the frequencies are 31.972070 and 15.994915, respectively. ^Harmonic frequencies (see Chapter III). For the other states th e experimental and calculated values do not agree as well. In general, the calculated values for u)2 agree with th e experimental values better than do those for oJ\. This is consistent with the fact that MCSCF2 bond distances tend to have greater error than do bond angles, when compared to ex perimental values. The m ore interesting thing to notice is how close th e values are for the singlet-triplet pairs of states of the same spatial symmetry. T he cor responding calculated harmonic frequencies for the 3B, and states are both within about 30 cm '1 of each other, while the corresponding experimental values differ widely. Assuming th a t this behavior would hold tru e for more accurately calculated frequencies, it appears likely that the experimental frequencies for one or both states are in error- The 3B2 and ^ states also have very similar symmetric stretching mode fre quencies. However, their bending frequencies are different. The calculated fre quencies for the xB2 state agree about as well with the experimental values as do those for the other states. Of course, the calculated values for the ^ state may not be very reliable, since this state most likely actually has Cs symmetry (1A/). The calculated frequencies for the JA2 state are both higher than the experimen tal values. More important, however, [s the fact that the calculated frequencies for the 3A2 and *A2 states are essentially the same. This means that the shapes of their potential energy surfaces near the minima are very similar. Also, this supports the 274 use of the experimental harmonic frequencies of the *A2 state for estimating Te from the experimental To value for the 3A2 state. 6.8 Summary It is felt that the MCSCF2 optimized geometries and potential surface shapes arc fairly reliable, especially for the bond angles and the differences in the bond d is t a n c e s and angles between the X *AX state and excited states. Based on the SRCI and SRFCCI calculations including corrections for quadru ple excitations and on the MRCI calculations, some statements can be made con cerning T e, AEverh and A Eemi, and the relative ordering of the various states. The calculated Te valnes at both the experimental and MCSCF2 optimized geometries indicate that the ordering of the excited state minima is: 3A2, 3BX, 3B2, XA2, JBX, 1B2. T h is result is quite different than those obtained at the SCF and MCSCF2 levels. Furthermore, the triplet states are all very nearly degenerate, especially the 3A2 and 3BX states. Also, the adiabatic transition energy of the *A2 state lies very close to those of the triplet states. The calculated A Evert values at these Cl levels give a relative ordering of the states of 3Bi, 'B,, 3B2, 3A2, *A2, and 1B2, at the experimental and SCF opti mized geometry of the X *AX state. The ordering at the X *AX MCSCF2 optimized geometry reverses the positions of the 3B2 and 'Bj states. 275 The ordering of the emission energies, & Eemis, is: 3A2, XA2, 3B2, 3BX, 1B1, and 1B2. This lends support to the experimental results of Snow et al.,289 which indicate that the phosphorescence may be due to the 3A2 state, using borrowed transition intensity from the 3BX state, thus giving the appearance that the emission is due to the 3BX state. Also, the calculated A Eemia for the *Bi state is much closer to the experimental value than is that for the XA2 state. These results indicate several things. First, the *B2 state is undoubtedly re sponsible for the 2400-1700 A (5.17-7.3 eV) region of the S 0 2 spectrum. Second, the results lend support to the experimental conclusion that there are at least two, probably three, low-lying triplet states in the 3900-3400 A region of the S02 spectrum. The results also indicate that the minima of the JA2 and 1BX states may very well also fall in this region of the spectrum. This would agree with experimental work which indicates that one or both of the singlet states may be causing perturbations in this region of the spectrum. Finally, the vertical excitation energies indicate that the most intense transi tions to the 3B2 and 3A2 states may actually be found in the 3400-2500 A region, which is usually attributed to the *A2 and XBX states. This lends support to the contention of many spectroscopists that many of the perturbations found in this region may be due to interactions with some of the triplet states. Also, the results for the A Eemis of the 1B1 state support the contention of many spectroscopists that, while the excitation in this region is mainly to the XA2 state, the emission is from the 1Bi state. CHAPTER VII Crossing of the 3A2-3 Bi and 1A2~1Bi Potential Surfaces As mentioned in Chapter III, the regions of the spectrum due to the 3A2-3 Bi and 1A2-1B1 states exhibit many perturbations. It has been postulated that the potential surfaces of these states cross in the regions where these perturbations first appear and are the strongest. The crossings of these surfaces were studied previously at the SCF level by Zellmer23 using the EBP basis set of Lindley22 (see Chapter IV). Part of the objective of the present study was to reinvestigate the crossing of these pairs of potential surfaces at the MCSCF level of theory. This chapter presents the results of the MCSCF calculations of the minimum crossing points of the 3A2-3 Bj and 1A2-1BX potential surfaces. 7.1 Method used to Determine the Minimum Crossing Point of Potential Surfaces There are several ways to determine the minimum crossing of two potential sur faces. By using the MINPT program an analytical expression for the surfaces in a 277 278 particular region can be obtained. The total energies of the two two-dimensional surfaces in this region can be expressed as quadratic equations by E i(x,y) - ax + bix + cxy + dxxy + exx2 + f xy2 (7.1) E 2( x , y) = a2 + b2x + c2y + d2xy + e2x2 -f f 2y2. (7.2) In general, the crossing of two potential surfaces is of dimensionality one less than that of the individual surfaces. Thus, the crossing of two-dimensional surfaces occurs along a one-dimensional surfaces, i.e., along a curve. Where the surfaces cross the energies at these points will be equal, and one can set equations 7.1 and 7.2 equal to each other, ElfiriXcriVcr) = E 2 ,cr(^cr, 1/cr)- (^-3) Two equations can be generated from the sum and difference of equations 7.1 and 7.2. A “crossing function” defined using the average of equations 7.1 and 7.2, is given by, Ecr{xcr,17 yCT) \ = ------E i cr(xcr,ycr)-^-E 2 cr(,XCriycr') (7.4)(n A or Ecr(xCT,y cr) = A + B x cr + Cy„ + D xcryc r + E x2cr + F y2cr (7.5) where A = (ai + d2) /2 279 B = (b\ + &2)/2 F = ( / i + / a)/2 Another equation can be obtained by subtracting equations 7.1 and 7.2 to give Lcr{Xcr, D c t ) = A' + B'x„ + C't/cr + D'X„y„ + E'x^. + F ' y = 0 (7.6) where A' = (ai - a2) B' = (6 x -6 2) F' = ( / i - / 2) A solution of equation 7.6 defines the relationship between x„ and y„ along the crossing curve. But if we are interested in the lowest crossing point we need to minimize the energy along the crossing curve. This can be achieved by minimizing E„ , Eq. 7.4, subject to the constraint, Lcr= 0, using the method of Lagrange multipliers. A third equation can be formed from equations 7.5 and 7.6, ■^*cr(®crj Vcr) ~ FCT(,Xcr? Vcr) 4" AZ/cr (*^cr5 J/cr) (7-7) This equation is then minimized, VFcr(xcr,j/cr, A) = VEcr(xcr,ycr) + V[ALcr(xcr,?/cr)] = 0 (7.8) 280 with the constraint given by equation 7.6. The simplest way to solve this is to use Newton’s method. This is an iterative procedure which makes use of the first and second derivatives. Successive solutions to {xcr^jcr) are found by solving the following equation, (7.9) or H(£f)3+i = i > 0 (7.10) where Si+i = corrected solution of Xi Zi+i = £,+i — a:,-, the (j -f l)tfccorrection to x g(xi) = vector of first derivatives of F„ H (* ) = Hessian matrix of Fcr. Since in this case a critical point (minimum) is desired, one wants g(i£i) = 0. Therefore, one needs to find x, such that, (7.11) or since dei[H(x,)] 7^ 0 , ^ +1 = 0 . Essentially, one chooses some inital values for x cr •> Vcn and A and iterates until the correction vector equals zero (or some small tolerance), in which case the first derivatives will be zero. 281 Since analytical expressions for E i Ecrt Lcr, and F„ are also analytical expressions. Therefore, the first and second derivatives (Hessian) can be computed analytically. It is a very simple matter to write a computer program taking advantage of this method to quickly find the critical points of the crossing surface. 7.2 MINCROV: Program for Determining Critical Points of Crossing Surfaces A computer program was written to make use of results from MINPT and the method discussed in the previous section. The program presently makes use of only quadratic and cubic equations. The general outline of how the program works and the procedures used to determine the critical points of the crossing surface is as follows: 1. Use MINPT to calculate the polynomial expressions, E i(x,y) and E2(x,y) for the two surfaces that cross. 2. Read in inital guesses for the minimum crossing point, a;crio and ycr,o, and a tolerance used to check for completion of the iterative procedure (i.e. to check when the corrections to X{ (^+i) are small enough). 3. The program forms ECT(xCT,ycr) and Lcr(xcr,ycr). 282 4. An initial value for the Lagrange muliplier, A, is computed using the first element of g(x{) (i.e., A0 = -(d E cr/d x cr)/(d L cr/d x cr) ). 5. Solve equation 7.10 iteratively by, (a) using (x^., 0 , j/CTi0, A0) to get H(x0) and g(x0) (b) solving H(x0)ii = ~g(xo) to get zx (c) computing corrected x , x\ = z\ + xo (d) iterating this loop for succesive values of x until all elements of z are less than the tolerance which was read in. 6 . Printing the final values for x/, Ecr and L cr at the critical point. This procedure only gives the critical points. For minimizations which do not involve a Lagrange multiplier, if the diagonal elements of the Hessian and the determinant of the Hessian relative to x and y are both positive, then the critical point is a relative minimum. Whether the critical point is a maxium, minium, or neither is usually fairly easy to determine from the physical situation. T.3 Test of MINCROV Program Since the aim of this study was to produce MCSCF minimum crossing points, no geometries specific to the SCF crossing region were studied. Instead, the SCF 283 results at the points used to perform the MCSCF crossing studies were used with MINPT to obtain SCF fits. The resulting SCF energy functions were then used in MINCROV to determine the SCF minimum crossing points. The results of this procedure using quadratic and cubic fits are given in Table 7.1, along with previous SCF results23 (which had been obtained with the larger Lindley basis set22). As one can see, the estimated minimum crossing points using both the quadratic and cubic fits are in excellent agreement with the previous SCF results. The excitation energy at the minimum crossing point, A Ecr, calculated by taking the difference between the minimum crossing energy and the X aAi minimum energy, are also in excellent agreement. These results indicate that the MINCROV program does a very good job at finding the minimum crossing points of these potential surfaces. 7.4 MCSCF Minimum Crossing Points MCSCF minimum crossing points were determined using both the MCSCF1 and MCSCF2 methods. This was done in a manner similar to that used for finding the optimized geometries. Initially the geometry points used to determine the opti mized geometries were used to fit the surfaces to quadratic and cubic polynomials using MINPT, which were then used in MINCROV. The resulting estimated min imum crossing point was then used along with geometries close to it to repeat the procedure using MINPT and MINCROV. This was continued until the energies for each of the 3A2- 3 B1 and 1A2 - 1 Bi pairs of states were equal to six or more decimal Table 7.1. SCF minimum crossing points of the 3A2~3 Bi and 1A2- 1 Bi pairs of states of SO 2 used to test the MINCROV program . 0 States Present Calculations Previous Calculations 6 No. of Rso Qoso Ecr+ 547 A E ct Rso 9oso ECt~\' 547 A E„ Terms' (bohr) (deg) (hartree) (eV) (bohr) (deg) (hartree) (eV) 3A2- 3 B! 6 2.8220 108.8410 -0.103803 3.539 2.810 108.65 -0.110420 3.510 1 0 2.8130 108.5688 -0.104637 3.516 1A2 - ’Bi G 2.8140 122.3510 -0.068564 4.497 2.810 122.35 -0.074240 4.495 1 0 2.8110 122.2570 -0.068667 4.495 aAEcr calculated using the energy of the X*Ai at its SCF optimized geometries (see Table 6.1). ‘Results of Zellmer,23 using Lindley’s basis.22 'Quadratic fits used 6 terms, cubic 10 terms. 285 places. The results of the MCSCF1 and MCSCF2 calculations for the minimum crossing points are given in in Table 7.2, along with the SCF results. As discussed in the previous section, the calculated SCF 1 and estimated SCF2 results are very similar. The results for the MCSCF 1 and MCSCF2 minimum crossing bond an gles, Ooso,cti are also in very good agreement. The MCSCF Ooso,ct for the triplet states are about 6 ° larger than the SCF angles. Also, the A 0oso,Cr calculated at the MCSCF level is about half the size of that calculated at the SCF level. The &oso,cr f°r the singlet states stays fairly constant at all levels, increasing by about 2.7° on going from the SCF to the MCSCF2 level of calculation. The crossing bond angles for the 3A2- 3 B1 and 1A2- 1 B1 pairs differ at all levels of calculation. Both the SCF and MCSCF &oso,cr f°r the triplet states are smaller than the ground state angle, while for the singlet states they are greater. There is considerable differences in the various minimum crossing bond dis tances, R so ,c r > at the various levels. However, at each level the Rso,cr for both pairs of states are very similar. The largest differences in Rso,cr between the two pairs of states is 0.016 A for the MCSCF 1 calculations. The transition energies to the crossing points, A Ecr, are fairly constant at the SCF, MCSCF1, and MCSCF2 levels of calculation. For the 3A2- 3 Bi sur faces A/?Cr(MCSCF2) is 3.71 eV (about 0 . 2 eV higher than the SCF value). The Af?cr(MCSCF2) for the singlet surfaces is 4.29 eV and is about 0.2 eV lower than the SCF value. The MCSCF1 values for A Ecr seem to be a little too large. 286 Table 7.2. The minimum crossing points of the 3A2-3B! and 1A2- 1 Bi pairs of states of S02 at the SCF and MCSCF levels.0 3a 2- 3b 1 1a 2- 1b 1 SCF1 R so 1.404 1.49 1.49 A R 0.083 0.083 Ooso 118.7867 108.65 122.35 A 0 -10.14 3.56 E c r + 547 -0.239412 -0.110423 -0.074242 A E cr 3.51013 4.49466 SCF2c R so 1.40490 1.488576 1.487517 A R 0.083674 0.082615 Ooso 118.8201 108.569 122.257 AO -10.251 3.437 E c r + 547 -0.23383860 -0.104637 -0.068667 A E Cr 3.51583 4.49465 MCSCF1 Rso 1.4489 1.5124 1.496036 A R 0.06355 0.04715 Ooso 119.742 114.46 123.67639 A0 -5.28 3.934 E c r + 547 -0.34511816 -0.195270 -0.17238940 A E cr 4.07767 4.70029485 MCSCF2 Rso 1.4510917 1.56225 1.56486 A R 0.11116 0.11377 Ooso 119.48774 114.506 125.0537 AO -4.982 5.5660 E c r + 547 -0.36030679 -0.224150 -0.2026097 A E c r 3.70510 4.291253 “Bond distances in A, bond angles in degrees, total energies in hartrees and excitation energies in eV. bE cr for 'Ai is actually the total energy of this state at its optimized geometry. “Estimated values using cubic fits in the MINCROV program (see Section 7.3). 287 7.5 Summary These crossing results can be compared to the optimized geometries, Te’s, and A-fi^ert’s for the 3A 2 , 3Bi, XA2 , and XBX states. For both the triplet and singlet surfaces, the surface crossings occur nearer the 3BX and XBX equilibrium geometries, respectively, especially for the bond angles (see Table 6.2). The A Ecr for the singlet states is closer to Te and A E vert of the XBX states. This result, along with the geometry results, seems to support the experimental contentions, presented in Section 3.5, that the interactions of the XA2 and 1B1 states begin to occur close to the XBX minimum and further up the potential surface of the jA 2 state. This is further supported by the fact that Te for the XA2 state lies below that for the XBX state and its A Evert is higher than that of the XBX state. This means that the XA 2 potential surface is steeper than the XBX potential surface (a fact supported by the experimental and calculated vibrational frequencies, presented in Table 6.31). As for the case of the singlet states, A Ecr for the triplet states is very similar to A E vert for the 3BX state and the crossing geometry is closer to the 3BX state minimum than to that of the 3A2 state. Also, the 3A2 state potential surface is steeper than that for the 3BX state. This fact is also supported by the fact that the calculated vibrational frequencies for the 3A2 state are larger than those for the 3BX state (Table 6.31). These results are consistent with the experimental conclusion that the only vibrational level of the 3B! state that is completely free from any perturbations is the (000) level. Also, it has been postulated that the interactions of the 3A 2 state begin near its (001) vibrational level, which is much closer to the minimum of this state than was the case for the XA2 state. This is consistent with the fact that the triplet state potential surface minimum crossing point is closer to the 3A 2 minimum then was the case for the XA2 state and the singlet minimum crossing point. CHAPTER VIII Summary and Conclusions 8.1 Summary To the best of my knowledge, the calculations performed in this study represent the most detailed theoretical study of the absorption and emission energies in volving the ground states and first six excited states of S02. The CAS-MCSCF potential surface studies use fairly large configuration spaces and produce good results, considering the inherent problems associated with MCSCF studies. The non-CAS-MCSCF configuration space probably could be improved, but it is felt that it adequately provides a description of the most important CSF’s and a good reference space for MRCI which was its purpose. The MRCI calculations deal with between about 1.4 and 4 million CSF’s for the various states studied. It is felt that these MRCI configuration spaces are large enough and include all of the most important CSF’s to provide good results for the problems considered in this study. 289 The results of the MRCI-DAV2S calculations for Te and A Evert are presented in Figure 8.1, along with the experimental results that were previously presented in Figure 3.3. The experimental results are shown as solid lines and the calculated results are shown as dashed lines. The calculated Te values are those determined at the respective MCSCF2 optimized geometries for the states considered. The calculated A Evert values are those determined at the X xAx state’s experimental geometry. These will be approximately 0.14-0.20 eV too high for the 3B, and 1Bi states and approximately 0.25-0.29 eV too high for the other states, compared to what would be obtained at the X xAi state’s MCSCF2 optimized geometry (based on the differences obtained at the SRFCCI level). As can be seen in Figure 8.1, the calculated A E vert values for all states are higher than the experimental val ues. However, this might not be the case if the A Evert values were determined at the X xAi MCSCF2 optimized geometry (based on the discussion above). The calculated Te values for all states are lower than the experimental values. The results indicate that the 3900-3400 A region of the spectrum is due to the 3B i, 3A2, and 3B2 states, which all have Te’s that are very close in energy. In fact, the MRCI results suggest that the 3Bx and 3A2 states are essentially degenerate. The MCSCF optimized geometries for these states indicate that the geometry f°r the 3Bi and 3A2 state agree well with the experimental results. Also, the calculated geometry for the 3Bx state is very close to that of the X xAi state, as predicted by experiment. In addition, the 3Bi state A Evert calculated at the Cl levels is 291 6.61 b 2 - 6.2 ‘b 2 -6 % -5.37 5.19 >A2------4.89 3a2 ------4.58 3B2------4.50 E l *B, 4.40 (eV) — i i------4.2 ‘A2 % *B -3.94 XB 3.74 -3.55 3B,- 3.43 * 2 3.32 3.35 3B, 3A2 3B2 *a 2 3.26 3b , 3.24 % 3.11 2.97 3a 2 3b , >A,- Figure 8.1. Schematic representation of the experimental and MRCI-DAV2S excitation energies of the states responsible for the S02 spectrum in the 3900- 1700 A region. The solid lines are the experimental values and the dashed lines are the calculated values. The calculated A Evert values were determined at the X *Ai state’s experimental geometry and are shown in the center of the figure. The calculated Te values were determined at the respective MCSCF2 optimized geometries of the individual states and are shown on either side of the A Evcrt values. 292 very similar to the experimental value and just slightly higher than its Te. These results support the experimental facts that the (OOO)-(OOO) transition in this region is very near the Franck-Condon maximum (vertical transition). Furthermore, the calculated minimum crossing point of the 3A2 and 3Bi potential energy surfaces occurs near the minima of these two states. This agrees with experimental results that indicate the interactions of these states begin just above the (000) vibrational level of the 3Bj state and just slightly higher on the 3A2 potential surface (near its (001) vibrational level). Finally, the results for A Eemis support the view of many spectroscopists th at phosphorescence probably does not occur from the same state to which the excitation occurs and may be due to the 3A2 state. The emission energy for the 3B2 state also is lower than that for the ^ state. The results for the IA2 and 1Bi states indicate that they are the states likely responsible for most of the absorption in the 3400-2500 A region of the spectrum. As for the 3Bj state, the 1B1 state calculated Te and A Evert are quite similar. Also, the MCSCF optimized geometry for the state is much closer to that of the X*Ai state than is that for the *A2 state. These results indicate that the 1B1 origin and its Franck-Condon maximum should be very close, and near the lower energy part of the experimental spectrum. Also, the maximum in the *Bi state absorption should occur below that of the 3A2 and 3B2 states (or at least in the same region). This supports the spectroscopists’ views that the absorptions to the triplet states may also be perturbed by singlet states and vice versa. Also, the origin of the XA2 state actually occurs in the 3900-3400 A region, and is much lower than its vertical transition (by about 1.6 eV). The calculated minimum crossing point of the XA2 and 1Bi potential energy surfaces occurs near the minimum of the 1B1 state and much further up the XA2 surface. This agrees with the experimental results which indicate that the interactions of these states begin near the xBj minimum. Demtroder et a/.254-258 have postulated that this region is due to “hybrid” levels consisting of many vibrational levels of the XA2 state and only a few vibrational levels of the xBi state. This would be understandable if, as indicated by the present calculations, the crossing of these potential surfaces occur near the xBi origin and further up the XA2 states potential surface. Finally, the MCSCF2 emission energy for the xBj state is calculated to fall in the range of the experimental results for this region of the spectrum, although the Cl values are at least 0.4 eV lower. The results for A Eemis of the XA2 state fall well below the experimental fluorescence range (by about 1.5 eV). This supports the conclusions that fluorescence is mainly due to the XB! state. There seems to be little doubt that the bulk of the transitions to the 2400- 1700 A region of the S02 spectrum is due to the XB2 (XA') state. The calculated C2„ geometries are close to those from experiment. Also, Te, A E vert, and A Eem{s are all very close to the experimental values. The one remaining question is whether this state may actually be unsymmetrical and have C* symmetry. 294 8.2 Conclusion It is felt that the present calculations provide useful information concerning the SO2 spectrum and the states responsible for it. However, there are still several questions that merit further theoretical studies. These are: 1. Better geometry optimizations using MP2, MR-MP, or MRCI should be done. This would probably shorten the bond distances to somewhere between those obtained at the SCF and MCSCF levels and bring them closer to the exper imental results. This would help in determining the relative ordering of the minima and vertical excitations, since these differ slightly at the experimen tal and SCF and MCSCF optimized geometries. Also, if better geometries were obtained it may be possible to determine the harmonic frequencies more accurately than was done in this study. 2. More accurate studies of the triplet state minimum energy differences (Te’s) should be performed. It appears that these energy differences, especially for the 3BX and 3A 2 states, are very small and very reliable results can only be obtained if very accurate calculations are performed. These should be done using MRCI, MR-MP2, or another multi-reference correlated method (such as ACPF). Also, a bigger and better reference space should probably be used. 3. Further, in connection with the last item, it would be very interesting to see exactly where A E veri for the 1Bi state lies. The final MRCI results of this 295 study indicate that it may lie below that for the 3A2 and 3B2 states. However, as for the triplet state Te values, this result varies depending on whether SRCI or MRCI calculations are performed and whether the experimental or MCSCF2 optimized geometries are used to perform the calculations. Again, very accurate multireference calculations may be necessary to determine the placement of A Evert for this state relative to those for the other states. 4. The 1B2 state should be studied for the possibility of an asymmetrical equi librium geometry (a C s symmetry W state). This would include obtaining accurate harmonic frequencies and the height of the C2t) saddle point relative to the Ca minimum. If higher level calculations are performed, then a larger basis set should be employed. Larger basis sets can now be handled more easily than when this study was started. It is not unusual now to see basis sets with 100 or more basis functions for molecules the size Qf SC> 2 - ^ would probably be possible to use a basis set of triple-zeta quality with additional polarization functions. Good candidates for correlated calculations are th e recently developed correlation-consistent polarized DZV, TZV, and Q ZV basis sets of Dunning et a/.310>3u These basis sets were optimized using correlated calculations on atoms and ions and are meant to be used in correlated molecular calculations. A ppendix A Definitions of Abbreviations for Basis Sets and M eth od s The abbreviations for the types of basis sets and methods used in the theoretical calculations referred to throughout this work are defined in this chapter. I have attempted to adhere to those abbreviations most commonly in use. Many of these are taken from the CH 2 review paper by I. Shavitt.1 Others are taken from the papers in which the respective information appeared. Finally, in cases when there is no commonly used abbreviation, I have tried to make as descriptive an abbreviation as possible using the name of the basis set or method. Any basis set or method which needs further explanation than that given here is discussed in the section or table in which it appears. The appendix is divided into sections containing the definitions for the basis sets, ab initio methods, and non-a6 initio methods. 296 297 A .l Abbreviations for Basis Sets. This section contains the abbreviations used for the various basis sets referred to in this work. Those for the generic basis sets are given first followed by those used for more specific basis sets. A. 1.1 Generic Basis Set Abbreviations STO = Slater type orbitals or a basis set consisting of such; GTO (GTF) = Gaussian type orbitals (functions) or a basis set consisting of such; CGTO = A linear combination of G TO’s used to describe an atomic orbital (AO) basis; MBS (SZ) = minimal basis set (1 basis function/atomic orbital - single zeta), (3s,2p) for sulfur, (2s,lp) for oxygen; EMBS = extended minimal basis set, (4s,3p;3s,2p); DZV = double zeta (2 basis functions/atomic orbital) on valence orbitals only, (4s,3p;3s,2p); DZVP = DZV basis with polarization functions, (4s,3p,2d;3s,2p,ld); DZ = double zeta basis set, (6s,4p;4s,2p); EB = extended basis, without polarization functions, (I0s,7p;7s,4p); MBSP = MBS with polarization functions, (3s,2p,ld;2s,lp,ld); EMBSP = EMBS with polarization functions, (4s,3p,ld;3s,2p,ld); DZP = DZ basis with polarization functions, (6s,4p,2d;4s,2p,2d); EBP = EB with polarization functions, (7s,6p,3d;5s,4p,2d); MBSBF = MBS with bond functions (Is and 2 p G T F ’s positioned along the bond between the atoms) as polarization functions; DZVBF = DZV with bond functions; STO-nG = MBS of STO’s represented by a linear combination of n GTO’s as determined by a least squares fit. The best known of these is the STO-3G basis of Hehre, Pople, and coworkers312-314; STO-nG* = STO-nG basis with polarization functions. 298 A. 1.2 Specific Basis Set Abbreviations. The next few abbreviations are for the basis sets developed by Hehre, Pople, and coworkers.84-86’312-318 As for the STO-nG basis sets, if polarization functions are included, the basis set abbreviation is followed by an asterisk. 44-31G = DZV basis with 4 GTO’s in each inner shell CGTO and a double zeta contraction ((4)/[2]) in the valence shell; 6-31G = DZV basis with 6 GTO’s in each inner shell CGTO and a double zeta contraction ((4)/[2]) in the valence shell; 3-21G = DZV basis with 3 GTO’s in each inner shell CGTO and a double zeta contraction ((3)/[2]) in the valence shell (made from their 6-21G basis sets). The following abbreviations are for the basis sets developed by Huzinaga et al 63,87,319-321 Again, if an asterisk appears after the abbreviation it signifies that polarization functions have been added. MINI-1 = MBS with 3 GTO’s for each orbital (i.e., (3s,3s,3s/3p,3p) for sulfur and (3s,3s/3p) for oxygen); MIDI-1 = DZV basis set. Same as MINI-1 except split valence (double zeta on valence), (3s,3s,2s,ls/3p,2p,lp) on sulfur and (3s,2s,ls/2p,lp) on oxygen; MIDI-4 = Same as MIDI-1 except 4 GTO’s for inner most s and p orbitals (i.e., (4s,3s,2s,ls/4p,2p,lp) for sulfur). 299 A.2 Abbreviations for Ab Initio M eth od s SCF == self-consistent field; 2C-SCF = two-configuration SCF; MCSCF = multiconfiguration SCF; CASSCF = complete active space MCSCF; CASSCF-2 == two-configuration CASSCF; Cl configuration interaction; S C I single excitations Cl, single reference; SD-CI = single and double excitations Cl, single reference; CISD = SD-CI; SD(Q)-CI = SD-CI with quadruples correction (usually Davidson’s correction), single reference; SR-CI EE single-reference SD-CI (also SRCI); 2R-CI = two-reference SD-CI; MR-CI multireference SD-CI (also MRCI); DAV1 unnormalized Davidson correction for quadruples (usually used as a qualifier to SRCI or MRCI, i.e., SRCI-DAV2) - see Chapter II; DAV2 normalized Davidson correction for quadruples (i.e., SRCI-DAV2) - see Chapter II; DAVIS = scaled unnormalized Davidson correction for quadruples (i.e., SRCI-DAV1S) - see Chapter II; DAV2S = scaled normalized Davidson correction for quadruples (i.e., SRCI-DAV2S) - see Chapter II; POPLE = POPLE correction for quadruples (usually used as a qualifier to SRCI or MRCI, i.e., SRCI-POPLE) - see Chapter II; CPF EE coupled pair functional; ACPF =E averaged coupled pair functional; SAC EE symmetry-adapted cluster; MP = M0ller-Plesset pertrubation theory (usually MP2, MP3, and MP4); MR-MP multireference Mpller-Plesset pertrubation theory; GVB generalized valence bond. 300 A.3 Abbreviations for No n-Ab Initio M eth o d s This section contains the definitions of the abbreviations used for the non-ai initio or semiempirical methods. CNDO complete neglect of differential overlap; CNDO/2 CNDO with improved parameterization; CNDO/BW Boyd and Whitehead78,322,323 version of CNDO (parameters selected to give improved molecular structures and force constants); INDO intermediate neglect of differential overlap; x„sw XQ scattered-wave; DFC-LSD density functional calculation using the local spin density approximation for the exchange and correlation; E-Hiickel extended Huckel. A ppendix B Definitions and Explanations of Properties, Symbols and Units The definitions and explanations of the one-electron properties, symbols and units used in this study are presented in this appendix. The definitions of the properties were taken from several sources.324-327 The values of the fundamental constants used to determine the conversion factors from atomic units (a.u.) to SI and cgs units were taken from the IUPAC manual ’’Quantities, Units and Symbols in Phys ical Chemistry328 published in 1988. The definitions for the properties given in this appendix use the following symbols: 1. denotes a normalized wavefunction (SCF, MCSCF, or Cl); 2. the symbols used in the property operators (except the energy operator) denote a sum of N one-electron operators, (i.e. ('I’l q;|'fr)); 3. p, r, and A denote the x, y, or z components in the Cartesian coordinate system defined in Figure 1.1. They also denote the x, y, or 2 position oper ators; 301 302 4. A deontes the position of the sulfur and oxygen nuclei; 5. the index K runs over the nuclei with charges Zk and position vectors r^. A prime on a summation over the nuclei indicates that the term with K = A is omitted; 6- rKp, Ap, and rp denote the x , y, or z components of the rK, A, and r position vectors. Vertical bars denote distances (i.e. \rj< — A\, |r — A|). 7. the subscript letters a, 6, and c label the principal oxygen electric field gra dient axes. The angle a (170 ), in degrees, measures the deviation of the principal axis system from the S-0 bond direction (see Figure 1.1). B.l Properties 1. Energy (1 a.u. = 1 hartree (h) = 27.2113962 eV = 219474.628 c m '1) (a) Total Energy, E E = {iB\Hd \9) (B.l) (Hel = Tei A V = electronic Hamiltonian operator) (b) Virial Ratio, —V/2T _ v i2 T = = (V) (B.2) 2('I>|Te/|vI>) 2{Tel) (c) Correlation Energy, E COrr ECott — E c i — ESc f (B.3) 303 (d) Orbital Energies, e,- e« = {V iW W i) (B.4) ( 2. Dipole Moment, p(A) (1 a.u. = 2.54174777 debye or D) = T .Zk(tk, - A,) - <«|(r, - yl,)|*) (B.5) K (independent of A for a neutral molecule) 3. Second Moment Tensor, pT (A ) and r2,(>l) (1 a.u. = 4.48655414 x 10 40 C-m2 = 1.34503509 x 10~26 esu-cm2 = 0.28002856 A2) Pr iA ) = S z n (rKP ~ Ap){rKr - A r) K -m rp-A p)(vT-A rm (B.6) r2(A) = x2(A) + y2(A) -f z2(A) (B.7) rei(A) = <*||r- A|2|tf) = (r2)e/ (B.8) 4. Third Moment Tensor, p t A(A) ( 1 a.u. = 2.37418238 x 10~5° C-m3 = 7.11761971 x 10“35 esu-cm3 - 1.48184744 x 10"25 cm3) prA(A) = ^2 ZK(rKP- A P)(rKT-A T)(rKx-Ax) K -(® l(rp - Ap)irr ~ Ar){r\ ~ Aa)|«) (B.9) 304 5. Quadrupole Moment Tensor, &{A) (1 a.u. = 4.48655414 x lO-40 C-m2 = 1.34503509 x 10-26 esu-cm2) ©pr = \ [3/>r(A) - 6pTr2{A)] (B.10) 6. Octupole Moment Tensor, J?(j4) (1 a.u. = 2.37418238 x 10-5° C-m3 = 7.11761971 x 10-35 esu-cm3) ttprA = \ | 5prA(i4) - 8r\ Pjj(A) - 8pX £ Tjj(A) - 8pr ^ A j j (A) { i j j (B.ll) 7. Electrostatic Potential, 4>(A) (1 a.u. = 27.2113962 V = 9.07674474 x 10-2 esu-cm-1 = 17.7504539 ppm) $(A) =Y,'2K\rK - Ak \-' - {1 /r A) (B.12) K where (l/r^ ) = ( $ ||r - > l |-1|^ ) (B.13) 8. Electric Field, E(,4) (1 a.u. = 5.14220825 x 1011 V-m-1 = 1.71525604 x 107 dynes-esu-1) EP{A) = - J2'ZK(rKp - Ap)!\r - A|3 + |(r, - A p)/\r - ,4|3|tf) (B.14) K 9. Hellman-Feynman Forces, F(A) (1 a.u. =8.23873186 millidyne) Fp = Za E p(A) (B.15) 10. Electric Field Gradient, q{A) (1 a.u. = 9.71736456x 1021 V-m-2 = 3.24136392x 1015 esu-cm-3) 9pt(A) = - Y2'Zk [3(rK> ~ Ap){rKr — AT) ~ 6pT\rK — A|2] / | — A\5 K + (* | [3(r, - A,)(rT - AT) - S„|r - A\2} /|r - ^ |5|« ) (B.16) 11. Asymmetry Parameter, rj(A) (dimensionless) » r s ) = (f e (S) - q„(S))/q„(S) (B.17) >)(‘70 ) = (fa(O ) - 12. Quadrupole Coupling Tensor, eqQ(A) (1 a.u. = 1.55689413 x 103 J-m-2 = 234.96474140 MHz/barn x Q) N<5]Pr (4) = eqpT(A)Q(A) (B.19) where q(A) = electric field gradient tensor, Q(A) = quadrupole moment of nucleus A, <5(33S) = —0.065 barn and Q(170 ) = —0.0256 barn71 306 13. Diamagnetic Shielding Tensor, = - ^ r 1!*) = j(l h'A)el = - ’i* e,(A) (B.21) 3 14. Diamagnetic Susceptibility Tensor, X d(A) (1 a.u. = 1.18802296 x 10'6 cm3- mole-1 ) xir(A) = ( = pret - Sprrl(A) (B.22) XavC'4) = ~ [Xjr(^ ) + Xyy{A) -f X zr(^)] = A\2\ = ~r <*,(.*) ( B .2 3 ) A p p en d ix C Mulliken Population Analysis The population analysis gives a rough idea about the charge distribution in the orbitals (orbital population) and on the atoms in a molecule. It also gives infor mation about the overlap between two atoms in an orbital and the total overlap between two atoms. This appendix discusses the population analysis developed by Mulliken.329’330 A general molecular orbital (MO) can be written in terms of a nuclear-centered AO basis set as V>i = EECfr,Xr. (C.l) k r where q>i denotes the ith MO and Xrk denotes the rth AO basis set function on atom k. Then the charge distribution due to orbital ifi can be written as NiWi? = Ni'£T.T,Y.C>r,C<,,S,k„ (C.2) fc T I S where STkS, denotes the overlap integral between the AO’s Xrk and and N{ is the occupation number of MO N{ = 0. 307 308 The atomic orbital population N (i : r*.) of AO Xrk >n Pi is given by N ( i : rk) = iV, E E CinCiaiS rkai (C.3) I a The partial gross atomic population N (i : k) on atom k in MO pi (i.e. the total charge population on atom k due to MO Pi) is given by N(i:k) = J2N(i:rk) (C.4) r The gross atomic population N(k) on atom k (i.e. the total electronic charge on atom k due to all the MO’s) is given by JV(*) = £ £ * ( • : '• * ) (C.5) * r The gross atomic populations satisfy Af=£AT(t) (C.6) k where N is the total number of electrons in the system. the orbital overlap population P(i : rk, si) for the AO’s Xr* and Xst in MO 93 ,• is P(i '■ rk,st) = (C.7) The partial gross atomic overlap population P(i : k, I) (i.e. the overlap between atoms k and I in MO ,) is given by P(t:t,l) = T,T,P(i--n,s,) (C.8) 309 Finally, the gross atomic overlap population P (k, /), which represents the total overlap between atoms k and / due to all the MO’s, is given by P(k,l) = '£/P(i: k,l) (C.9) R eferences [1] I. Shavitt, Tetrahedron 1985, 4U 1531. [2] W. W. Kellogg, R. D. Cadle, E. R. Allen, A. L. Lazrus, and E. A. Martell, Science 1972, 175, 587. [3] Air: II. control of NOx and SOx emissions, volume 71 of AIChE Symposium Series, No. 148, edited by C. Rai and R. D. Siegel, American Institute fo Chemical Engineers, New York, N. Y., 1975. American Institute of Chemical Engineers. [4] A. C. Stern, R. W. Boubel, D. B. Turner, and D. L. Fox, Fundamentals of Air Pollution, Academic Press, Inc., New York, N. Y., 1984, 2nd edition. [5] T. Godfish, Air Quality , Lewis Publishers, Inc., Chelsa, Michigan, 1985. [6] Proceedings of the International Symposium on Sulfur in the Atmosphere, volume 12 of Atmospheric Environment, No. 1-3, edited by D. J. Moore, J. P. Lodge, Jr., W. Klug, and R. B. Husar, Pergamon Press, New York, N. Y., 1978. [7] A. I. Stewart, J. D. E. Anderson, L. W. Esposito, and C. A. Barth, Science 1979, 203, 777. [8] E. S. Barker, Geophys. Res. Lett. 1979, 6, 117. [9] R. R. Conway, R. P. McCoy, and C. A. Barth, Geophys. Res. Lett. 1979, 6, 629. [10] J. Pearl, R. Hanel, V. Kunde, W. Maguire, K. Fox, S. Gupta, C. ponnam- peruma, and F. Raulin, Nature 1979, 280, 755. [11] B. A. Smith, E. M. Shoemaker, S. W. Kieffer, and A. F. Cook II, Nature 1979, 280, 738. [12] T. V. Johnson, A. F. Cook II, C. Sagan, and L. A. Soderblom, Nature 1979, 280, 746. [13] S. Kumar, Nature 1979, 280, 758. 310 311 [14] F. P. Fanale, R- H. Brown, D. P. Cruikshank, and R. N. Clake, Nature 1979 280, 761. [15] D. B. Nash and R. M. Nelson, Nature 1979, 280, 763. [16] W. D. Smythe, R. M. Nelson, and D. B. Nash, Nature 1979, 280, 766. [17] A. F. Cheng, Astrophys. J. 1980, 242, 812. [18] D. E. Shemansky, Astrophys. J. 1980, 242, 1266. [19] L. E. Brus and J. R. McDonald, Chem. Phys. Lett. 1973, 21, 283. [20] L. E. Brus and J. R. McDonald, J. Chem. Phys. 1974, 61, 97. [21] J- G- Calvert, F. Su, J. W. Bottonheim, and O. P. Strausz, Atmospheric Environment 1978, 12(1-3), 197. [22] D. D. Lindley, Master’s thesis, Department of Chemistry, the Ohio State University 1976. [23] R- J. Zellmer, Master’s thesis, Department of Chemistry, The Ohio State University 1981. [24] M. Born and J- R- Oppenheimer, Ann. Physik 1957, 84, 457. [25] D. R. Hartree, Proc. Cambridge Phil. Soc. 1928, 24- [26] V. Fock, Z. Physik 1930, 61, 126. [27] J. C. Slater, Phys. Rev. 1929, 34, 1293. [28] C. C. J. Roothaan, Rev. Mod. Phys. 1951, 23(2), 69. [29] S. M. Blinder, Am . J. Phys. 1965, 33(6), 431. [30] A. Szabo and N- S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill, Inc., New York, 1982. [3 1 ] E. Clementi and D. R. Davis, J. Comput. Phys. 1966, 7(2), 223. [32] R. C. Raffcnetti, J ■ Chem. Phys. 1973, 55(10), 4452. [33] I- Shavitt, in Methods of Electronic Structure Theory, edited by H. F. Schae fer III, volume 3 of Modern Theoretical Chemistry, Plenum Press, New York, N.Y., 1977. [34] J- A. Pople J. S. Binkley, and R. Seeger, Int. J. Quantum Chem. Symp. 1976, 10, 1. [35] S. R. Langhoff and E. R. Davidson, Int. J. Quantum Chem. 1974, 5(1), 61. 312 [36] E. R. Davidson and D. W. Silver, Chem. Phys. Lett. 1977, 52(3), 403. [37] J. A. Pople, R. Seeger, and R. Krishnan, Int. J. Quantum Chem. Symp. 1977, 11, 149. [38] R. M. Pitzer, J. Chem. Phys. 1973, 58, 3111. [39] S. Auster, P. Bertoncini, D. C. Comeau, E. R. Davidson, R. N. Diffenderfer, D. Hood, H.-L. Hsu, K. Kaiser, R. Land, M. J. M. Pepper, R. M. Pitzer, R. C. Raffenetti, K. Ruedenberg, H. F. Schaefer III, I. Shavitt, A. Wahl, and R. J. Zellmer, (Unpublished). [40] R. Shepard, I. Shavitt, R. M. Pitzer, D. C. Comeau, M. Pepper, H. Lischka, P. G. Szalay, R. Ahlrichs, F. B. Brown, and J.-G. Zhao, Int. J. Quantum Chem. Symp. 1988, 22, 149. [41] B. P. Dailey, S. Golden, and E. B. Wilson, Jr., Phys. Rev. 1947, 12, 871. [42] G. F. Crable and W. V. Smith, J. Chem. Phys. 1951, 19, 502. [43] M. H. Sirvetz, J. Chem. Phys. 1951, 19, 938. [44] D. Kivelson, J. Chem. Phys. 1954, 22, 904. [45] Y. Morino, Y. Kikuchi, S. Saito, and E. Hirota, J. Mol. Spectrosc. 1964, 13, 95. [46] S. Saito, J. Mol. Spectrosc. 1969, 30, 1. [47] W. H. KirchhofF, J. Mol. Spectrosc. 1972, Jf, 333. [48] J. K. G. Watson, J. Mol. Spectrosc. 1973, 479. [49] A. S. Pine, G. Dresselhaus, B. Palm, R. W. Davies, and S. A. Clough, J. Mol. Spectrosc. 1977, 67, 386. [50] B. Post, R. S. Schwartz, and I. Fankuchen, Acta Cryst. 1952, 5, 372. [51] J. Haase and M. Winnewisser, Z. Naturforsch. 1968, 23A, 61. [52] A. H. Clark and B. Beagley, Trans. Faraday Soc. 1971, 67, 2216. [53] J. C. II. Holder and M. Fink, J. Chem. Phys. 1981, 75, 5323. [54] R. D. Shelton, A. H. Nielsen, and W. H. Fletcher, J. Chem. Phys. 1953, 21, 2178. [55] R. D. Shelton, A. H. Nielsen, and W. H. Fletcher, J. Chem. Phys. 1954, 22, 1791. [56] R. J. Whitehead and N. C. Handy, J. Mol. Spectrosc. 1975, 55, 356. 313 [57] N. C. Handy, S. M. Colwell, and W. H. Miller, Disc. Faraday Soc. 1977, 62, 29. [58] S. M. Colwell and N. C. Handy, Mol. Phys. 1978, 35, 1183. [59] A. D. Isaacson, D. G. Truhlar, K. Scanlon, and J. Overend, J. Chem. Phys. 1981, 75, 3017. [60] C. W. Eaker and G. C. Schatz, J. Chem. Phys. 1984, 81, 2394. [61] I. H. Hillier and V. R. Saunders, Chem. Phys. Lett. 1969, 4, 163. [62] I. H. Hillier and V. R. Saunders, Trans. Faraday Soc. 1970, 66, 1544. [63] A. N. Tavouktsoglou and S. Huzinaga, J. Chem. Phys. 1980, 72, 1385. [64] M. Keeton and D. P. Santry, Chem. Phys. Lett. 1970, 7, 105. [65] P. G. Burton, N. R. Carlsen, and E. A. Magnusson, Mol. Phys. 1976, 32, 1687. [66] P. G. Burton and N. R. Carlsen, Chem. Phys. Lett. 1977, 46, 48. [67] M. F. Guest, I. H. Hiller, and V. R. Saunders, J. Chem. Soc. Faraday Trans. 7/1972, 68, 114. [68] T. H. Dunning, Jr. and R. C. Raffenetti, J. Phys. Chem. 1981, 85, 1350. [69] S. Farantos, E. C. Leisegang, J. N. Murrell, K. Sorbie, J. J. C. Texeira-Dias, and A. J. C. Varandas, Mol. Phys. 1977, 34, 947. [70] B. Roos and P. Siegbahn, Theoret. Chim. Acta 1971, 21, 368. [71] S. Rothenberg and H- F. Schaefer III, J. Chem. Phys. 1970, 53, 3014. [72] P. D. Dacre and M. Elder, Theoret. Chim. Acta 1972, 25, 254. [73] L. S. Cederbaum, W. Domcke, and W. von Niessen, Mol. Phys. 1977, 33, 1399. [74] L. S. Cederbaum, W- Domcke, W. von Niessen, and W. P. Kraemer, Mol. Phys. 1977, 34, 381. [75] G. B. Bacskay, J. Chem. Phys. 1983, 79, 2090. [76] G. B. Bacskay, A. P. L. Rendell, and N. S. Hush, J. Chem. Phys. 1988, 89, 5721. [77] D. P. Santry and G. A. Segal, J. Chem. Phys. 1967, J7, 158. [78] R. J. Boyd and M. A. Whitehead, J. Chem. Soc., Dalton Trans. 1972, page 81. 314 [79] K. Chung, Ph.D. Dissertation, Department of Chemistry, The Ohio State University, Columbus, Ohio, 1974. [80 L. Noodleman and K. A. R. Mitchell, Inorg. Chem. 1978, 17, 2709. [81 R. 0. Jones, J. Chem. Phys. 1985, 82, 325. [82 I. H. Hillier and V. R. Saunders, Mol. Phys. 1971, 22, 193. [83 N. R. Carlsen, Chem. Phys. Lett. 1977, ^7, 203. [84 J. B. Collins, P. v. R. Schleyer, J. S. Binkley, and J. A. Pople, J. Chem. Phys. 1976, 64, 5142. [85 W. J. Pietro, M. M. Francl, W. J. Hehre, D. J. DeFrees, J. A. Pople, and J. S. Binkley, J. Am. Chem. Soc. 1982, 104, 5039. [86 M. M. Francl, W. J. Pietro, W. J. Hehre, J. S. Binkley, M. S. Gordon, D. J. DeFrees, and J. A. Pople, J. Chem. Phys. 1982, 77, 3654. [87 Y. Sakai, H. Tatewaki, and S. Huzinaga, J. Comput. Chem. 1981, 2, 100. [88 J. Janszky, R. H. Bartram, A. R. Rossi, and G. Corradi, Chem. Phys. Lett. 1986, 124, 26. [89 S. Huzinaga and M. Yoshimine, J. Chem. Phys. 1978, 68, 4486. [90 A. Schmiedekamp, D. W. J. Cruickshank, S. Skaarup, P. Pulay, I. Hargittai, and J. E. Boggs, J. Am. Chem. Soc. 1979, 101, 2002. [91 K. Hirao, J. Chem. Phys. 1985, 83, 1433. [92 J. N. Murrell, W. Craven, M. Vincent, and Z. H. Zhu, Mol. Phys. 1985, 56, 839. [93 E. H. Hayes and G. U. Pfeiffer, J. Am. Chem. Soc. 1968, 90, 4773. [94 A. D. Walsh, J. Chem. Soc. 1953, page 2266. [95 E. Clementi and D. L. Raimondi, J. Chem. Phys. 1963, 38, 2686. [96 E. Clementi, J. Chem. Phys. 1964, 40, 1944. [97 E. Clementi and C. Roetti, At. Data Nucl. Data Tables 1974, 14, 177. [98 T. H. Dunning, Jr., J. Chem. Phys. 1970, 53, 2823. [99 A. Veillard, Theoret. Chim. Acta 1968, 12, 405. [100 J. L. Whitten, J. Chem. Phys. 1966, 44, 359. [101 S. Huzinaga, J. Chem. Phys. 1965, 4 2, 1293. 315 102] T. H. Dunning, Jr. and P- J- Hay, in Methods of Electronic Structure Theory, edited by H. F. Schaefer III, volume 3 of Modern Theoretical Chemistry, Plenum Press, New York, NY, 1977, Chapter 1, pages 1-27. 103] F. R. G. W. Norrish and A. P. Zeelenberg, Proc. R. Soc. London 1957, A240, 293. 104] A. L. Myerson, F. R. Taylor, and P. L. Hanst, J. Chem. Phys. 1957, 26, 1309. 105] J. J. McGarvey and W- D- McGrath, Proc. R. Soc. London 1964, A278, 490. 106] F. R. G. W. Norrish and G. A. Oldershaw, Proc. R. Soc. London 1959, A249, 498. 107] J. Brown and G. Burns, Con. J. Chem. 1969, ^7, 4290. 108] L. Herman, J. Akrische, and H. Grenat, J ■ Quant. Spectrosc. Radiat. Trans fer 1962, 2, 225. 109] F. A. G. Gaydon, G. H. Kimbell, and H. B. Palmer, Proc. R. Soc. London 1963, A276, 461. 110] P. A. Giguere and R. Savoie, Can. J. Chem. 1965, 43, 2357. 111] P. Warneck, F. F. Marmo, and J. O. Sullivan, J. Chem. Phys. 1963, 40, 1132. 112] H. Okabe, J. Am. Chem. Soc. 1971, 93, 7095. 113] M. Hui and S. A. Rice, Chem. Phys. Lett. 1972, 17, 474. 114] J. C. D- Brand, p. H. C hiu, A. R. Hoy, and H. D. Bist, J. Mol. Spectrosc. 1976, 60, 43. 115] R. Vasudev and W. M. McClain, J. Mol. Spectrosc. 1981, 89, 125. 116] T. C. Chow, Phys. Rev. 1933, 4 4 , 638. 117] W. C- Price and D. M. Simpson, Proc. R- Soc. London 1938, A165, 272. 118] J. Duchesne and B. Rosen, J ■ Chem. Phys. 1947, 15, 631. 119] I. Dubois and B. Rosen, Disc. Faraday Soc. 1963, 35, 124. 120] I. Dubois, Bull. Soc. R oy■ Sci. Liege 1963, 32{ 10), 777. 121] I. Dubois, Bull. Soc. Roy. Sci. Liege 1963, 32(11-12), 899. 122] I. Dubois, Bull. Soc. Roy. Sci. Liege 1964, 33(3-4), 190. 316 [123] I. Dubois, J. Mol. Structure 1969, 3, 269. [124] I. Dubois and A. Grotsch, Bull. Soc. Roy. Sci. Liege 1964, 53(12), 833. [125] J. C. Brand and K. Srikameswaran, Chem. Phys. Lett. 1972, 15, 130- [126] J. C. D. Brand, D. R. Humphrey, A. E. Douglas an d I. Zanon, Can. J. Phys. 1973, 51, 530. ’ [127] V. T. Jones and J. B. Coon, J. Mol. Spectrosc. 1973, 47, 45, and references therein. [128] J. C. D. Brand and Ch. V. S. Ramachandra Rao J Mol. Spectrosc. 1976, 61, 360. [129] A. R. Hoy and J. C. D. Brand, Mol. Phys. 1978, 36, 1409. [130] P. G. Mezey and Ch. V. S. Ramachandra Rao, J Chem. Phys. 1980, 72, 121. [131] W. Sharfin, M. Ivanco, and S. C. Wallace, J. Chem. Phys. 1982, 76, 2095. [132] C. Jaffe, J. Chem. Phys. 1984, 81, 616. [133] M. Ivanco, J. Hager, W. Sharfin, and S. C. Wallace J. Chem. Phys. 1983, 78, 6531. [134] M. J. Ivanco, Ph.D. Dissertation, University of Toronto (Canada), Toronto, Canada, 1985. [135] H. Kanamori, J. E. Butler, K. Kawaguchi, C. Yamada, and E. Hirota, J. Chem. Phys. 1985, 83, 611. [136] V. Y. Foo, C. E. Brion, and J. B. Hasted, Proc. R Soc. London 1971, A322, 535. [137] J. S. Bulger and J. M. Goodings, Can. J. Phys. 1971, 43, 1437. [138] W. M. Flicker, 0. A. Mosher, and A. Kuppermann J ■ Chem. Phys. 1978, 69, 3910. [139] L. Vuskovic and S. Trajmar, J. Raman Spectrosc. 1981, 10, 136. [140] L. Vuskovic and S. Trajmar, J. Chem. Phys. 1982, 77, 5436. [141] P. Avouris, J. E. Demuth, D. Schmeisser, and S. D Colson, J. Chem Phys. 1982, 77, 1062. [142] T. Abuain, I. C. Walker, and D. F. Dance, J. Chem Soc., Faraday Trans. 2 1985, 81, 1061. [143] G. L. Bendazzoli and P. Palmieri, Int. J. Quantum Chem. 1975, 5, 537. 317 [144] J. B. Coon, R. E. DeWames, and C. M. Loyd, J. Mol. Spectrosc. 1962, 8 285. 145] R. Colin, Can. J. Phys. 1968, 46, 1539. 146] R. Colin, Can. J. Phys. 1969, 47, 979. 147] R. Westrick and C. H. MacGillavary, Acta. Crystallog. 1954, 7, 764. 148] V. Schomaker and D. P. Stevenson, J. Am. Chem. Soc. 1941, 63, 37. 149] D. Powers and H. G. Olson, J. Chem. Phys. 1980, 73, 2271. 150] W. Gordy, J. Chem. Phys. 1947, 15, 81. 151] J. B. Coon;, F. A. Cesani, and F. P. Huberman, J. Chem. Phys. 1970, 52, 1647. 152] R. J. Pirkle and V. T. Jones, J. Mol. Spectrosc. 1975, 54, 375. 153] G. W. King and P. R. McLean, J. Mol. Spectrosc. 1974, 51, 363. 154] J. B. Coon, F. A. Cesani, and C. M. Loyd, Disc. Faraday Soc. 1963, 35, 118. 155] Y. Hamada, A. J. Merer, S. Michielsen, and S. A. Rice, J. Mol. Spectrosc. 1981, 86, 499. 156] R. S. Mulliken, Can. J. Chem. 1958, 36, 10. 157] P. Zahradnik and J. Leska, Collection Czech. Chem. Commun. 1977, 42, 2060. 158] P. Phillips and E. R. Davidson, J. Comput. Chem. 1983, 4, 337. 159] E. Cehelnik, C. W. Spicer, and J. Heicklen, J. Am. Chem. Soc. 1971, 93, 5371. 160] S. E. Braslavsky and J. Heicklen, J. Am. Chem. Soc. 1972, 94, 4864. 161] I. L. Stockburger, S. Braslavsky, and J. Heicklen, J. Photochem. 1973, 2, 15. 162] E. Cehelnik, J. Heicklen, S. Braslavsky, L. Stockburger, and E. Mathias, J. Photochem. 1973, 2, 31. 163] F. B. Wampler, A. Horowitz, and J. G. Calvert, J. Am. Chem. Soc. 1972, 94, 5523. 164] F. B. Wampler, J. G. Calvert, and E. K. Damon, Int. J. Chem. Kinet. 1973, 5, 107. 318 [165] K. Chung, J. G. Calvert, and J. W. Bottenheim, Int. J. Chem. Kinet. 1975, 7, 161. 166 C. Fotakis, Chem. Phys. Lett. 1981, 82, 68. 167 M. W. Wilson, M. Rothschild, D. F. Muller, and C. K. Rhodes, J. Chem. Phys. 1982, 77, 1837. 168 C. Fotakis, A. Torre, and R. J. Donovan, J. Photochem. 1983, 23, 97. 169 R. B. Timmons, Photochem. and Photobiol. 1970, 12, 219. 170 A. M. Fatta, E. Mathias, J. Heicklen, I. L. Stockburger, and S. Braslavsky, J. Photochem. 1973, 2, 119. 171 N. Kelly, J. F. Meagher, and J. Heicklen, J. Photochem. 1976, 5, 355. 172 N. Basco and R. D. Morse, Chem. Phys. Lett. 1973, 20, 557. 173 R. A. Cox, J. Phys. Chem. 1972, 76, 814. 174 S. Okuda, T. N. Rao, D. H. Slater, and J. G. Calvert, J. Phys. Chem. 1969, 73, 4412. 175 G. E. Jackson a n d J. G. Calvert, J. Am. Chem. Soc. 1971, 93, 2593. 176 K. Otsuka and J. G. Calvert, J. Am. Chem. Soc. 1971, 93, 2581. 177 C. C. Badcock, H. W. Sidebottom, J. G- Calvert, G. W. Reinhardt, and E. K. Damon, J. Am. Chem. Soc. 1971, 93, 3115. 178 H. W. Sidebottom, C. C. Badcock, J. G. Calvert, R. R. Rabe, and E. K. Damon, J. Am. Chem. Soc. 1971, 93, 3121. 179 H. W. Sidebottom, C. C. Badcock, G. E. Jackson, J. G. Calvert, G. W. Reinhardt, and E- K. Damon, Environ. Sci. Technol. 1972, 6, 73. 180 A. Horowitz and J. G. Calvert, Int. J. Chem. Kinet. 1972, 4, 207. 181 A. Horowitz and J- G. Calvert, Int. J. Chem. Kinet. 1973, 5, 243. 182 K. L. Demerjian, J- G. Calvert, and D. L. Thorsell, Int. J. Chem. Kinet. 1974, 6, 829. 183 K. L. Demerjian and J. G. Calvert, Int. J. Chem. Kinet. 1975, 7, 45. 184 F. Su and J. G. Calvert, Chem. Phys. Lett. 1977, 52, 572. 185 F. C. James J. A. Kerr, and J. P. Simons, Chem. Phys. Lett. 1974, 25, 431. [186] F. B. Wampler, Int. J. Chem. Kinet. 1976, 8, 687. 319 [187] F. B. Wampler, T. C. Oldenberg, W. W. Rice, and J. R. R. Karl, J. Chem. Phys. 1978, 69, 2569. [188] C. Lalo, J. Masanet, F. Lempereur, and J. T. de Maleissye, Chem. Phys. Lett. 1986, 124, 447. [189] C. J. Halstead and B. A. Thrush, Nature 1964, 204, 992. [190] C. J. Halstead and B. A. Thrush, Proc. R. Soc. London 1966, A295, 380. [191] A. McKenzie and B. A. Thrush, Proc. R. Soc. London 1968, A308, 133. [192] J. S. Robertshaw and I. W. M. Smith, Int. J. Chem. Kinet. 1980, 12, 729. [193] R. J. Glinski, J. A. Sedarski, and D. A. Dixon, J. Phys. Chem. 1981, 85, 2440. [194] R. J. Glinski, J. A. Sedarski, and D. A. Dixon, J. Am. Chem. Soc. 1982, 104, H26. [195] R. J. Glinski and D. A. Dixon, J. Phys. Chem. 1985, 89, 33. [196] K. F. Greenough and A. B. F. Duncan, J. Am- Chem. Soc. 1961, 83, 555. [197] H. D. Mettee, J. Chem. Phys. 1968, 42, 1784. [198] H. D. Mettee, J. Phys. Chem. 1969, 73, 1071- [199] S. J. Strickler and D. B. Howell, J. Chem. Phys. 1968, 49, 1947. [200] T. N. Rao, S. S. Collier and J. G. Calvert, J. Am. Chem. Soc. 1969, 91, 1609. [201] T. N. Rao, S. S. Collier, and J. G. Calvert, J. Am. Chem. Soc. 1969, 91, 1616. [202] T. N. Rao and J. G. C alvert, J. Phys. Chem. 1970, 74, 681. [203] S. S. Collier, A. Morikawa, D. H. Slater, J. G. Calvert, G. Reinhardt, and E. Damon, J. Am. Chem. ’Soc. 1970, 92, 217. [204] A. Horowitz and J. G. Calvert, Int. J. Chem. Kinet. 1972, 4, 175. [205] A. Horowitz and J. G. Calvert, Int. J. Chem. Kinet. 1972, 4, 191. [206] J. G. Calvert, Chem. Phys. Lett. 1973, 20, 484. [207] F. B. Wampler, K. Otsuka, J. G. Calvert, and E. K. Damon, Int. J. Chem. Kinet. 1973, 5, 669. [208] F. Su, J. W. Bottenheim, D. L. Thorsell, J. G. Calvert, and E. K. Damon, Chem. Phys. Lett. 1977,' 4 9, 305. 320 [209] F. Su, F. B. Wampler, J. W. Bottenheim, D. L. Thorsell, J. G. Calvert, and E. K. Damon, Chem. Phys. Lett. 1978, 51, 150. 210 J. P. Briggs, R. B. Caton, and M. J. Smith, Can. J. Chem. 1975 , 53, 2133. 211 R. B. Caton and A. R. Gangadharan, Can. J. Chem. 1974, 52, 2389. 212 F. Ahmed, K. F. Langley, and J. P. Simons, J. Chem. Soc., Faraday Trans. II 1977, 73, 1659. 213 W. W. Watson and A. E. Parker, Phys. Rev. 1931, 37, 1484. 214 J. H. Clements, Phys. Rev. 1935, ^7, 224. 215 N. Metropolis, Phys. Rev. 1941, 60, 295. 216 A. J. Merer, Disc. Faraday Soc. 1963, 35, 127. 217 J. C. D. Brand and R. Nanes, J. Mol. Spectrosc. 1973, 46, 194. 218 R. N. Dixon and M. Halle, Chem. Phys. Lett. 1973, 22, 450. 219 Y. Hamada and A. J. Merer, Can. J. Phys. 1974, 52, 1443. 220 Y. Hamada and A. J. Merer, Can. J. Phys. 1975, 53, 2555. 221 R. J. Shaw, J. E. Kent, and M. F. O’Dwyer, J. Mol. Spectrosc. 1980, 82, 1. 222 T. C. Chow and H. D. Smyth, Phys. Rev. 1931, 38, 838. 223 A. E. Douglas, J. Chem. Phys. 1966, 46, 1007. 224 B. P. Levitt and D. B. Sheen, Trans. Faraday Soc. 1967, 63, 540. 225 P. J. Gardner, Chem. Phys. Lett. 1969, 4, 167. 226 H. W. Sidebottom, K. Otsuka, A. Horowitz, J. G. Calvert, B. R. Rabe, and E. K. Damon, Chem. Phys. Lett. 1972, 13, 337. 227 P. Kusch and F. W. Loomis, Phys. Rev. 1939, 55, 850. 228 J. L. Hardwick and W. H. Eberhardt, The Magnetic Rotation Spectrum of Sulfur Dioxide in the 3000 A Region , 27th Symposium on Molecular Struc ture and Spectroscopy, The Ohio State University, Columbus, Ohio 1972. 229 J. C. D. Brand, J. L. Hardwick, D. R. Humphrey, Y. Hamada, and A. J. Merer, Can. J. Phys. 1976, 54, 186. 230 J. E. Kent, M. F. O’Dwyer, and R. J. Shaw, Chem. Phys. Lett. 1974, 24, 221. 321 [231] F. Su, J. W. Bottenheim, H. W. Sidebottom, J. G. Calvert, and E. K. Damon, Int. J. Chem. Kinet. 1978, 10, 125. [232 R. J. Shaw, J. E. Kent, and M. F. O’Dwyer, Chem. Phys. 1976, 18, 155. [233 R. J. Shaw, J. E. Kent, and M. F. O’Dwyer, Chem. Phys. 1976, 18, 165. [234 W. H. Henneker, W. Siebrand, and M. Z. Zgierski, Chem. Phys. Lett. 1979, 68, 5. [235 G. L- Wolk, J. R. E. Weston, and G. W. Flynn, J. Chem. Phys. 1980, 73, 1649. [236 R. N- Rudolph, Ph.D. Dissertation, University of Colorado at Boulder, Boul der, Colorado, 1977. [237 T. B- Simpson and N. Bloembergen, Chem. Phys. Lett. 1983, 100, 325. [238 S. Kimel, D. Feldmann, J. Laukemper, and K. H. Welge, J. Chem. Phys. 1982, 76, 4893. [239 D. L. Holtermann, E. K. C. Lee, and R. Nanes, Chem. Phys. Lett. 1980, 75, 91. [240 D. L- Holtermann, E. K. C. Lee, and R. Nanes, Chem. Phys. Lett. 1980, 75, 249. [241 E. K- C. Lee, B. G. MacDonald, D. L. Holtermann, and R. Nanes, J. Pho tochem. 1981, 17, 495. [242 D. L. Holtermann, E. K. C. Lee, and R. Nanes, J. Chem. Phys. 1982, 76, 3341- [243 D. L. Holtermann, E. K. C. Lee, and R. Nanes, J. Chem. Phys. 1982, 77, 5327. [244 B. G- MacDonald and E. K. C. Lee, J. Phys. Chem. 1982, 86, 323. [245 B. G. MacDonald and E. K. C. Lee, J. Phys. Chem. 1982 , 86, 4275. [246 D. L- Holtermann, E. K. C. Lee, and R. Nanes, J. Phys. Chem. 1983, 87, 3926. [247 II. Watanabe, Y. Hyodo, S. Tsuchiya, and S. Koda, Chem. Phys. Lett. 1981, 81, 439. [248 H. Watanabe, S. Tsuchiya, and S. Koda, J. Phys. Chem. 1982, 86, 4274. [249 II. Watanabe, Y. Hyodo, S. Tsuchiya, and S. Koda, J. Phys. Chem. 1982, 86, 685. 322 [250] H. Watanabe, S. Tsuchiya, and S. Koda, Faraday Discuss. Chem. Soc. 1983, 75, 365. [251 H. Watanabe, S. Tsuchiya, and S. Koda, J. Phys. Chem. 1983, 87, 906. [252 H. Watanabe, S. Tsuchiya, and S. Koda, J. Chem. Phys. 1985, 82, 5310. [253 T. Suzuki, T. Ebata, M. Ito, and N. Mikami, Chem. Phys. Lett. 1985, 116, 268. [254 A. Fischer, R. Kullmer, and W. Demtroder, Chem. Phys. 1984, 83, 415. [255 R. Kullmer and W. Demtroder, J. Chem. Phys. 1984, 81, 2919. [256 R. Kullmer and W. Demtroder, Chem. Phys. 1985 , 92, 423. [257 R. Kullmer and W. Demtroder, J. Chem. Phys. 1985, 83, 2712. [258 R. Kullmer and W. Demtroder, J. Chem. Phys. 1986, 84, 3672. [259 R. N. Rudolph and S. J. Strickler, J. Am. Chem. Soc. 1977, 99, 3871. [260 S. J. Strickler and R. N. Rudolph, J. Am. Chem. Soc. 1978, 100, 3326. [261 S. J. Strickler and R. D. Ito, J. Phys. Chem. 1985, 89, 2366. [262 K. Takahashi, Chem. Phys. Lett. 1979, 67, 304. [263 K. Takahashi, J. Chem. Phys. 1980, 73, 309. [264 B. P. Levitt and D. B. Sheen, Trans. Faraday Soc. 1965, 61, 2404. [265 R. B. Caton and A. B. F. Duncan, J. Am. Chem. Soc. 1968, 90, 1945. [266 S. D. Colson, W. Y. Cheung, J. H. Glownia, and S. J. Riley, Chem. Phys. Lett. 1980, 76, 515. [267 A. G. Gaydon, Proc. R. Soc. London 1934, A146, 901. [268 N. Metropolis and H. Beutler, Phys. Rev. 1940, 57, 1078A. [269 A. E. Douglas, Can. J. Phys. 1958, 36, 147. [270 R. K. Russell, B. L. Landrum, and E. E. Vezey, ASTIA document no. AD81060, 1956, as referenced by Merer216 and Coon et. al.144 [271 R. K. Russell, ASTIA Document No. AD152030, 1958, as referenced by Merer216 and Coon et. al.144 [272 R. K. Russell, as referenced by Coon et. al.144 [273 R. M. Hochstrasser and A. P. Marchetti, J. Mol. Spectrosc. 1970, 35, 335. 323 [274] A. J. Merer, Disc. Faraday Soc. 1963, 35, 230. [275] J. C. D. Brand, C. DiLauro, and V. T. Jones, J. Am. Chem. Soc. 1970, 92, 6095. [276] J. C. D. Brand, V. T. Jones, and C. DiLauro, J. Mol. Spectrosc. 1971, 40, 616. [277] J. C. D. Brand, V. T. Jones, and C. DiLauro, J. Mol. Spectrosc. 1973, 45, 404. [278] J. P. Vikesland and S. J. Strickler, J. Chem. Phys. 1974, 60, 660. [279] K.-E. J. Hallin, Y. Hamada, and A. J. Merer, Can. J. Phys. 1976, 54, 2118. [280] K.-E. J. Hallin and A. J. Merer, J. Mol. Spectrosc. 1977, 65, 163. [281] K.-E. J. Hallin, Y. Hamada, and A. J. Merer, Characterization of the b ZA2 State of S02, paper WC7, 31st Symposium on Molecular Spectroscopy, The Ohio State University, Columbus, Ohio, 1976. [282] J. B. Coon, Bull. Am. Phys. Soc. 1957, 2, 100. [283] E. V. Doktorov, I. A. Malkin, and V. I. Man’ko, J. Mol. Spectrosc. 1975, 56, 1. [284] B. Meyer, L. F. Phillips, and J. J. Smith, Proc. Natl. Acad. Sci. U.S.A. 1968, 61, 7. [285] L. F. Phillips, J. J. Smith, and B. Meyer, J. Mol. Spectrosc. 1969, 29, 230. [286] C. Lalo and C. Vermeil, J. Chim. Phys. 1976, 73, 25. [287] D. S. Tinti, Chem. Phys. Lett. 1971, 12, 169. [288] G. E. Nelson and R. F. Borkman, J. Chem. Phys. 1975, 63, 208. [289] J. B. Snow, D. C. Hovde, and S. D. Colson, J. Chem. Phys. 1982, 76, 3956. [290] F. B. Wampler, R. C. Oldenberg, and W. W. Rice, J. Appl. Phys. 1979, 50, 6117. [291] H. W. Sidebottom, C. C. Badcock, J. G. Calvert, G. W. Reinhardt, B. R. Rabe, and E. K. Damon, J. Am. Chem. Soc. 1971, 93, 2587. [292] A. J. Merer, as referenced by several other authors.289,141,260’261’266 [293] D. Patel, D. Margolese, and T. T. Dyke, J. Chem. Phys. 1979, 70, 2740. [294] G. R. Bird and C. H. Townes, Phys. Rev. 1954, 94, 1203. 324 [295] J. M. Pocham, R. G. Stone, and W. H. Flygare, J. Chem. Phys. 1969, 51, 4278. [296] C. F. Jackels and E. R. Davidson, J. Chem. Phys. 1976, 64, 2908. [297] R. Manne, Mol. Phys. 1972, 24, 935. [298] J. Paldus and A. Veillard, Chem. Phys. Lett. 1977, 50, 6. [299] J. Paldus and A. Veillard, Mol. Phys. 1978, 35, 445. [300] D. T. Grow and R. M. Pitzer, J. Chem. Phys. 1977, 67, 4019. [301] K. Ruedenberg and K. R. Sundberg, in Quantum Science, edited by J. L. Calais, O. Goscinski, J. Linderberg, and Y. Ohm, Plenum Press, New York, 1976, page 505. [302] K. Ruedenberg, M. W. Schmidt, M. M. Gilbert, and S. T. Elbert, Chem. Phys. 1982, 71, 41. [303] K. Ruedenberg, M. W. Schmidt, M. M. Gilbert, and S. T. Elbert, Chem. Phys. 1982, 71, 51. [304] K. Ruedenberg, M. W. Schmidt, M. M. Gilbert, and S. T. Elbert, Chem. Phys. 1982, 71, 65. [305] D. C. Comeau, R. J. Zellmer, and I. Shavitt, in Geometrical Derivatives of Energy Surfaces and Molecular Properties , edited by P. Jprgensen and J. Simons, volume 166 of NATO ASI Series C: Mathematical and Physical Sciences, D. Reidel Publishing Co., Dordrecht, 1986, pages 243-251. [306] M. J. M. Pepper, D. C. Comeau, and I. Shavitt, Unpublished, 1988. [307] M. J. M. Pepper, Ph.D. Dissertation, Department of Chemistry, The Ohio State University, Columbus, Ohio, 1989. [308] D. C. Comeau, Ph.D. Dissertation, Department of Chemistry, The Ohio State University, Columbus, Ohio, 1990. [309] G. B. Bacskay, A. P. L. Rendell, and N. S. Hush, J. Chem. Phys. 1988, 89(9), 5721. [310] T. H. Dunning, Jr., J. Chem. Phys. 1989, 90, 1007. [311] R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 1992, 96(9), 6796. [312] W. J. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys. 1969, 51, 2657. 325 [313] W. J. Hehre, R. Ditchfield, R- F. Stewart, and J. A. Pople, J. Chem. Phys. 1970, 52, 2769. [314] W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, Ab Initio Molec ular Orbital Theory , John W iley and Sons, Inc., 1986. [315] R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys. 1971, 54, 724. [316] W. J. Hehre and W. A. Lathan, J. Chem. Phys. 1972, 56, 5255. [317] W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 1972 , 56, 2257. [318] M. S. Gordon, J. S. Binkley, J • A. Pople, W. J. Pietro, and W. J. Hehre, J. Am. Chem. Soc. 1982, 1 0 4 , 2797. [319] H. Tatewaki and S. H uzinaga, J ■ Chem. Phys. 1979, 71, 4339. [320] H. T atew aki and S. H uzinaga, J • Comput. Chem. 1980, 1 , 205. [321] Y. Sakai, H. Tatewaki, and S- Huzinaga, J. Comput. Chem. 1981, 2, 108. [322] R- J- Boyd and M. A. Whitehead, J. Chem. Soc., Dalton Trans. 1972, page 73. [323] R- J- Boyd and M. A. Whitehead, J. Chem. Soc., Dalton Trans. 1972, page 78. [324] D. W. Davies, The Theory of the Electric and Magnetic Properties of Molecules, John Wiley and Sons, London, 1967. [325] J- F. Harrison, J. Chem. Phys■ 1967, ^7(8), 2990. [326] D. Newmann and J. W. Moskowitz, J. Chem. Phys. 1968, 49(5), 2056. [327] S. Fraga and G. Malli, in Many-electron Systems: Properties and Interac tions, edited by R. Stevenson and M. A. Whitehead, volume 7 of Studies in Physics and Chemistry, W. B- Saunders Co., Philadelphia, 1968. [328] I- Mills, T. Cvitas, K. Homann, N. Kallay, and K. Kuchitsu, (eds.), Quanti ties, Units and Symbols in physical Chemistry, IUPAC, Blackwell Scientific Publications, Alden Press, Oxford, G. B., 19SS- [329] R- S. Mulliken, J. Chem. Phys. 1955, 23(10), 1833. [330] R- S. Mulliken, J. Chem. Phys. 1955, 53(10), 1841.