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The graphical unitary group approach to configuration interaction calculations: An application to the dipole moment and potential energy surface of the water molecule
Kedziora, Gary Steven, Jr., Ph.D.
The Ohio State University, 1994
UMI 300 N. Zeeb RA Ann Aibor, MI 48106
The Graphical Unitary Group Approach to Configuration Interaction Calculations: an Application to the Dipole Moment and Potential Energy Surface of the Water Molecule.
Dissertation
Presented in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Gary S. Kedziora Jr.
The Ohio State University
1994
Dissertation Committee: Approved by
Professor Isaiah Shavitt
Professor Russell M. Pitzer
Professor Sherwin Singer Advisor Department of Chemistry DEDICATION
To my dear Barbara ACKNOWLEDGMENTS
I would like to thank Professor Isaiah Shavitt for teaching me much of what
went into this dissertation and pushing me to think carefully and clearly. He has
been a superb example of a scientist and will continue to inspire my career. Steven
Parker provided frequent and gracious help in preparing this document. I would also like to thank Lars Ojamae and Jean Blaudeau for interesting and fruitful scientific discussions, and my fellow Shavitt group members, Eric Stahlberg, Robert Zellmer, and Galen Gawboy for their help with the COLUMBUS codes.
Ill VITA
September 22, 1965 ...... Bom - Oscoda, Michigan
1987 B. S., - University of Minnesota
1987-1994 ...... Teaching Associate - The Ohio State University
Publications
P.S. Portoghese, G.S. Kedziora, D.L. Larson, B.K. Bernard, and R.L. Hall, “Reactivity of Glutothione with a,/?-Unsaturated Keytone Flavoring Substances,” Food and Chemical Toxicology, 27, 773 (1989).
FIELDS OF STUDY
MAJOR FELD: Chemistry
Theoretical Chemistry, Professor Isaiah Shavitt
IV TABLE OF CONTENTS DEDICATION...... ii ACKNOWLEDGMENTS...... iii VITA ...... iv LIST OF TABLES...... vi LIST OF FIGURES ...... ix
CHAPTER PAGE
I. Introduction and T heory ...... 1 A. Introduction ...... 1 B. Electronic Structure M ethods ...... 3 C. Spin Symmetry of Many-Eiectron Wave Functions ...... 14 D. GUGA...... 20 E. The COLUMBUS Programs: A GUGA Implementation ...... 28 F. S um m ary ...... 32 G. Tables and Figures ...... 33
II. Triple- and Quadruple-Excitation Correlation Corrections ...... 42 A. Corrections and complexity ...... 42 B. Triple- and Quadruple-Excitation Loop Extensions ...... 47 0 . S um m ary ...... 53 D. Tables and Figures ...... 54
III. Ground-State Potential Energy and Dipole Moment Surface of the Water Molecule ...... 61 A. Introduction ...... 61 B. M e th o d s ...... 65 1. Basis S e t s ...... 65 2. Reference Space ...... 70 0. Results and Discussion ...... 72 E. Sum m ary : ...... 81 F. Tables and Figures ...... 82 APPENDIX ...... 99 BIBLIOGRAPHY...... 134 LIST OF TABLES
TABLES p a g e
Table I Partitions of the integer 5 and classes of Sjsj ...... 33
Table 2 Step numbers defined in terms of changes in the Paldus a, b, and c values, intermediate occupation and spin ...... 36
Table 3 The distinct row table (DRT) for the basis in Figure 3. See text for definitions ...... 37
Table 4 The output generated by the depth-first tree-search program, corresponding to the depth-first tree given in Figure 9. See the caption of Figure 9 for an explanation of some of the data. Wo and Wj are the segment values corresponding to Equation (42) of Chapter I. In the cases where the segment value lies between the Wq and W; columns, it is a contribution from a one-body segment value, W. Aq and A] are the internal contributions to the two terms (x = 0 , 1) of the coupling coefficient. In this table the total contributions are Ai n Wa; (H Wo = 0 in these cases). Each p or q level index may be selected from a range of valid level indices defining a single loop extension (see text for examples).. . 56
Table 5 For each loop type and number of external indices, the number of loop extensions starting at each set of boundary vertices is shown next to the boundary vertex pairs. If there is no valid loop extension of a given type for a pair of boundary vertices, they are left out of the table ...... 57
Table 6 The primitives and contractions of the aug-cc-pVQZ-d(f,g) basis set. See text for the origin of the basis set ...... 82
Table 7 The primitives and contractions of the ave-ANO+(dg,spf) basis. See text for the origin of the basis set ...... 84 vi Table 8 Comparison of the two final basis set choices for the surface. The definition of each is given in the text. All calculations were done at the equilibrium geometiy R = 1.811 3o and 6 = 104.48°, with the 6 CAS reference space. 8 6
Table 9 Comparison of the choices for the reference space at various levels of correlation using the aug-cc-pVQZ-d(g,f) basis set. The final four Cl entries for each reference space include the quadruples corrections defined in Chapter 1 ...... 87
Table 10 The dimension of the MR-SDCI spaces for the cases considered in Table 9 in Cs symmetry. See text for a full definition of the spaces ...... 8 8
Table 11 Comparison of anharmonic frequencies from ab initio surfaces with experiment. The first two columns are from this work ...... 89
Table 12 Comparison with experiment of the anharmonic frequencies calculated in this work with the various energy surfaces using different correlation methods ...... 90
Table 13 Comparison of internal-coordinate force constants ...... 91
Table 14 Comparison of spectroscopic constants from the different surfaces of this work with experimental spectroscopic constants. All values but the geometry constants are in wavenumbers...... 92
Table 15 Comparison of ab initio reduced normal-coordinate force constants with experimental force constants ...... 93
Table 16 Comparison of dipole moment expansion coefficients in terms of reduced normal coordinates with other ab initio coefficients ...... 94
Table 17 Comparison with experiment of dipole derivatives with respect to reduced normal coordinates ...... 96
Table 18 Comparison of the vibrationally averaged dipole moment from perturbation theory with experiment ...... 97
Table 19 Calculated ab initio energy points (atomic units). The internal coordinates are given exactly to machine precision. . . . 1 0 0 vii Table 20 Dipole moments from the MR-SDCI wavefunction. The dipole moment data corresponds to the internal coordinates exactly as given to machine precision. The y-z coordinate frame has the HOH angle bisected by the z axis, and the rj-(l coordinates are the Eckart coordinates. The angle of rotation between the z axis and the ( axis is given in radians by 0 . See Figure 10 ant text in Results section of Chapter m. . Ill
VII) LIST OF FIGURES
FIGURES PAGE
Figure 1 The Young shapes for S 5 with corresponding irrep labels. . 33
Figure 2 The Young tableaux for the (3,2) irrep and (4,1) irrep of S 5 34
Figure 3 The Weyl tableaux for the (2, I, 0^) irrep of (7(4) are shown in the first line. This corresponds to the basis set for 5 = N = Z, and n = 4. The subduction process is shown. The lines show the correspondence between the subduced irreps at each level. This figure is based on Figure 7 of Reference 23 ...... 35
Figure 4 The Shavitt graph that corresponds to the DRT in Table 3 and the Weyl tableaux in Figure 3 ...... 38
Figure 5 An example of a two-body loop which gives a nonzero coupling coefficient for eij,ki- The solid and dashed line walks correspond to the ket and bra state functions |m) and (m'|, respectively. The loop is divided into segments by the dashed line at each level, and the segment types are shown to the right of each segment. The indices that correspond to the indices in the two-body operator are shown at the left of the relevant levels...... 39
Figure 6 The distinct loop types that contribute to the upper triangle of the Hamiltonian matrix. The spacing between the bra (dashed line) and the ket (full line) shows the relative occupancy at each level of the loop. The segment types that define the loop type are shown at the right of each different level type of the loop. The indices at the left of the figure show the position of the operator and integral indices for relevant segments of each loop type. The indices in this figure satisfy i < j < k < I. The operator and indices that go along with each loop are shown at the bottom of the loop. This figure is reproduced from Reference 27 ...... 40 ix Figure 7 A graphical representation of a DRT from the COLUMBUS cidrt program. The reference space is shown in thick lines and consists of three doubly occupied levels and a complete active space (full Cl) with 4 electrons in 4 orbitals. The thinner lines show the MR-SDCI space that results from this reference space. The external space is typically much larger than can be shown in this figure ...... 41
Figure 8 The exteraal-space Shavitt graph for excitation through quadruples is shown. The a,b values at the top of the extemal- space graph are the boundary vertices. They are labeled by R—V for the triples and quadruples. The singles and doubles boundary vertices have the traditional designations W—Z. 54
Figure 9 This is an example of a depth-first search tree for loop type (3a) with two-extemal indices. The search starts at the UX boundary vertex pair. At each node is shown the index, the segment shape, the step numbers of the bra and ket of the segment, and the vertex labels. The vertex labels implicitly define the a,b values at that level...... 55
Figure 10 The relation between the coordinate system in which the ab initio calculations were performed (the y-z coordinates) and the Eckart coordinates (the i]-( coordinates). The z axis bisects the bond angle, and the dipole moment is shown with respect to the y-z origin at the oxygen nucleus. The 77-C origin is at the center of mass of the .molecule ...... 98
Figure 11 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.471 ao (spline fits used whenever more than two points are available) ...... 119
Figure 12 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.587 a^ (spline fits used whenever more than two points are available) ...... 119
Figure 13 CISD energies as a function of 0 for various values of R 2 at a fixed R] = 1.591 a^, (spline fits used whenever more than two points are available) ...... 1 2 0
Figure 14 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.689 ao (spline fits used whenever more than two points are available) ...... 1 2 0 Figure 15 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.781 ao (spline fits used whenever more than two points are available) ...... 121
Figure 16 CISD energies as a function of 0 for various values of R 2 at a fixed Ri = 1.811 ao (spline fits used whenever more than two points are available) ...... 121
Figure 17 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.903 ao (spline fits used whenever more than two points are available) ...... 1 2 2
Figure 18 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 1.997 ao (spline fits used whenever more than two points are available) ...... 1 2 2
Figure 19 CISD energies as a function of 6 for various values of R 2 at a fixed R] = 2 .1 0 0 ao (spline fits used whenever more than two points are available) ...... 123
Figure 20 CISD energies as a function of 6 for various values of R 2 at a fixed Ri = 2.218 ao (spline fits used whenever more than two points are available) ...... 123
Figure 21 CISD z-component of the dipole moment as a function of 6 for various values of R 2 at a fixed R] = 1.471 ao 124
Figure 22 CISD z-component of the dipole moment as a function of 6 for various values of R 2 at a fixed Rj = 1.587 ao 124
Figure 23 CISD z-component of the dipole moment as a function of 9 for various values of R 2 at a fixed Rj = 1.591 ao 125
Figure 24 CISD z-component of the dipole moment as a function of 6 for various values of R 2 at a fixed Ri = 1.689 ao 125
Figure 25 CISD z-component of the dipole moment as a function of 0 for various values of R 2 at a fixed Ri = 1.781 ao 126
Figure 26 CISD z-component of the dipole moment as a function of 0 for various values of R 2 at a fixed R] = 1.811 ao 126
Figure 27 CISD z-component of the dipole moment as a function of 0 for various values of R 2 at a fixed R; = 1.903 ao 127 xi Figure 28 CISD z-component of the dipole moment as a function of 6 for various values of Rg at a fixed Ri = 1.997 a*, ...... 127
Figure 29 CISD z-component of the dipole moment as a function of 6 for various values of Rg at a fixed R] = 2.100 ao 128
Figure 30 CISD z-component of the dipole moment as a function of 6 for various values of Rg at a fixed Ri = 2.218 ao 128
Figure 31 CISD y-component of the dipole moment as a function of 6 for various values of Rg at a fixed R] = 1.471 ao 129
Figure 32 CISD y-component of the dipole moment as a function of 6 for various values of Rg at a fixed R; = 1.587 ao 129
Figure 33 CISD y-component of the dipole moment as a function of 6 for various values of Rg at a fixed Ri = 1.591 ao 130
Figure 34 CISD y-component of the dipole moment as a function of 6 for various values of Rg at a fixed R| = 1.689 ao 130
Figure 35 CISD y-component of the dipole moment as a function of 6 for various values of Rg at a fixed R] = 1.781 ao 131
Figure 36 CISD y-component of the dipole moment as a function of 6 for various values of Ra at a fixed Ri = 1.811 ao 131
Figure 37 CISD y-component of the dipole moment as a function of 6 for various values of Ra at a fixed R] = 1.903 ao 132
Figure 38 CISD y-component of the dipole moment as a function of 6 for various values of Ra at a fixed Rj = 1.997 ao 132
Figure 39 CISD y-component of the dipole moment as a function of 6 for various values of Ra at a fixed R% = 2.100 ao 133
Figure 40 CISD y-component of the dipole moment as a function of 6 for various values of Ra at a fixed R; = 2.218 ao 133
XU CHAPTER I Introduction and Theory
A Introduction
Highly accurate potential energy surfaces are useful tools for chemists for the investigation of many different chemical phenomena. The structures of molecules can be determined by locating the global minimum of the potential energy surface, transition states can be determined by locating saddle points on the surface, and other useful information like bond dissociation energies can be calculated if the dissociation limit energy is calculated as well. Also, dynamical information like scattering cross sections and rotation-vibration spectra can be calculated from a surface if a large enough section of the surface is calculated. However, routine calculation of highly accurate surfaces which give dynamical information that compares well with experiment continues to be a challenge.
Accurate surfaces require large basis sets and multireference correlation methods.
Large surfaces of this type require thousands of hours of computer time even for triatomic molecules with a small number of electrons. And if a property surface such as a dipole moment surface is desired, the demands are even greater. The Shavitt group is interested in calculating accurate surfaces, so we require efficient electronic structure methods and codes that run on state-of-the-art computers and are capable of producing accurate results. Much work has gone into producing such codes and 2
methods. The COLUMBUS codes* are a series of efficient programs that can be used
to calculate energy and property surfaces. A brief description of the methods used in this dissertation, including the graphical unitary group approach^ (GUGA), is given in Chapter I.
The COLUMBUS codes include multireference correlation methods based on
GUGA, including multireference singles and doubles Cl (MR-SDCI) and multiref erence perturbation theory^ through third order (MR-PT3). MR-PT3 is not as ac curate for small molecules as MR-SDCI, but it does scale properly with size. The
COLUMBUS codes also have approximate size-consistency corrections like the David son correction'*’ ^ and the Pople correction^. However, these corrections sometimes are not adequate for highly accurate work or fail to adequately correct for size consis tency in difficult cases. An investigation into a better correction for size consistency using perturbation theory with triple and quadruple excitations was undertaken. Such a calculation would theoretically scale as m^n^, where m is the number of occupied
(or internal) orbitals and n is the number of unoccupied (external) orbitals. For accu rate work on small molecules, typical sizes for m and n are 1 0 and 1 0 0 , respectively, so a calculation using triple and quadruple excitations would use on the order of at least 10*^ floating point computations. Thus, a very efficient algorithm must be found for practical calculations using a perturbative correction for size consistency. This is difficult in the context of GUGA as it is implemented in the COLUMBUS programs.
A discussion of the problems is given in Chapter H. Also described in Chapter II is a computer program that generates information for computation of coupling coefficients for Hamiltonian matrix elements between triple-excitation and quadruple-excitation 3
configurations and between single-excitation and double-excitation configurations.
The water molecule is a good molecule for testing accurate ab initio energy
and property surfaces because there have been many experimental studies and other
accurate ab initio studies to compare with. A potential energy surface and dipole
moment surface of the water molecule over a wide range of geometries suitable for
calculation of the rotation-vibration spectrum, including intensities, is presented in the
Appendix. The spectrum will be calculated with the computer program of Tennyson
and coworkers,^ which uses the exact rotation-vibration Hamiltonian and a variational
solution of the Schrbdinger equation. This spectrum will be the most accurate first- principles calculation of the rotation-vibration spectrum of the water molecule and will be useful in helping experimentalists observe highly excited rotation-vibration bands.
In Chapter El, the ab initio energy and dipole moment surface of the water molecule is discussed. Two accurate basis sets are compared, and it is shown that the generally-contracted averaged ANO basis sets of Widmark et al.^ give good results for both the energy and the dipole moment with a relatively small set of contracted basis functions. Two smaller reference spaces are compared to the complete active space reference space for 8 electrons in 8 orbitals. The smaller reference spaces are shown to give roughly the same results with much smaller Cl spaces. Finally, the accuracy of the surfaces are assessed based on fittings of the surface and spectroscopic data derived from perturbation they. Good agreement with experiment in anharmonic frequencies and in the dipole moment function is observed.
B Electronic Structure Methods
The main goal in electronic structure theory is to solve the stationary-state elec- tronic Schrôdinger differential equation,
( 1)
for atoms and molecules using different levels of approximation. Here we are
interested in molecules consisting of first- and second-row atoms, for which relativistic
effects are usually not important and total electron spin provides a good quantum
number. We also assume that the nuclei are stationary point charges. The electronic
Hamiltonian in first-quantized form in the nonrelativistic approximation is written in
atomic units (h = 1, rue = 1 and e = 1 ) as
, yV N Tlatoma r 7 ' ^ 1 ^atoms y
where Vp = Vp‘V? = is the kinetic energy operator for electron p, Za
is the nuclear charge of nucleus a, Tpa is the distance between nucleus a and electron p, rpq is the distance between electrons p and q, and N is the number of electrons. It is useful to separate the Hamiltonian into a one-electron part and a two-electron part and to drop the purely nuclear repulsion part when solving the electronic Schrôdinger equation, because electronic wave function does not depend on the nuclear coordinates.
The one-electron Hamiltonian for electron p is
1 Tlatom s y '‘.=-îv?-Er^. P) ^ 0=1 and the two-electron Hamiltonian for electrons p and q is just ^(p, q) = The
Hamiltonian is now simply written as
N N He = Y^hp+ ^ g{p,q). (4) p=l p
In many-electron atoms and molecules, the exact solution of Equation (1) cannot be
obtained. However, we can get as close as we want to the exact energy and wave
function provided we have enough computing power, but for most molecular systems
we must rely on approximations. Much of the work on electronic structure theory
methods is to design computer programs that give the best possible approximation to
Equation (1) for atoms and molecules while using the computing resources efficiently.
This chapter describes the electronic structure methods used in the work of this
dissertation and some of the background information.
Since the hydrogen atom Schrôdinger equation is exactly solvable and we know
the exact functional form for the orbitals, it provides a good starting point for basis
sets of all atoms and molecules. A many-electron atomic orbital is constructed from
a linear combination of functions similar to hydrogen-like orbitals, and many-electron
molecular orbitals are constructed from linear combinations of atomic orbitals. The
general atomic orbital basis functions need not be constructed from hydrogen-like orbitals, but we use what we know about the hydrogen-like orbitals as a guide to the functional form for an atomic basis. This form is
X,(r, 0, ip) = (5) where r, 9, and y are the usual spherical coordinates, and Q are parameters that are optimized for a given basis set. The angular prefactor //„,(^, (p) is proportional to a spherical harmonic function and contains a -dependant normalization factor.
Functions of the type in Equation (5) with the characteristic factor are called
Slater type functions. Because of the difficulty in calculating integrals over functions on different atomic centers, it is more convenient to use Gaussian functions, where 6
the exponential factor is replaced by e"®*’’*. This type of function does not display
the correct cusp (undefined derivative) at r = 0 or the correct asymptotic form of the
Slater functions, so it may take more Gaussian functions than Slater functions to get
a comparable solution to the Schrôdinger equation. However, because r is squared in
the exponential, the basis functions can be factored in terms of the Cartesian variables,
and this leads to simplifications of the two-electron integrals. Also, the angular part of the basis function is expressed in terms of Cartesian (x, y, z) coordinates rather than spherical polar coordinates for computational reasons. See Reference 9 for a history of Gaussian basis functions and references to review articles about basis sets.
All traditional molecular electronic structure calculations use methods that are built upon molecular orbitals, which are linear combinations of the atomic basis functions. The Hartree-Fock (HP) method is the most common way to generate molecular orbitals, and it provides a good first approximation to the exact solution of the Schrôdinger equation for many atomic and molecular systems. The Hartree-Fock method assumes that the wave function for a many-electron system can be represented by an antisymmetric product of occupied spin-orbitals. A spin-orbital is a product of a one-electron spatial orbital and a one-electron spin function. The one-electron spatial function is expanded in terms of the basis functions with linear expansion coefficients,
n 4 ( z , y,z) = ' ^ xi{x, y, z)cik, (6) 1=1 and the one-electron spin function, cr^ is either a (with S = ^, ms = ^) or ^ (with S = i, m s = —\)- A spin-orbital, then, depends on the three Cartesian degrees of freedom and a fourth spin degree of freedom; together, for electron p, the four coordinates are denoted by qp, and the spin-orbital is denoted by 0jt(qp) =
meaning that if two of the electron indices are switched, the wave function changes
sign. The easiest way to achieve this is to use a Slater determinant.
V’i(qi) ^2(qi) ••• ^jv(qi) 1 ^ i( q 2 ) V’2 (q 2 ) ••• # ( q 2 ) (7) vW I 0 i(qjv) V"2 (q # ) # ( q ^ )
A variational treatment minimizing the energy expectation value, E = ($|/f|$), leads
to a coupled system of eigenvalue equations.
(8)
for each spin-orbital k = 1,..., ra, where the Fock operator F is a one-electron operator
which depends on the orbitals (see for example. Reference (10)). The expectation
value of F with spin-orbital k is
N N N (9) p=i p=i p=i
with
{^k\Jp\^k) = V’it(qi)V’p(q2){ ^ifc(qi) 0 p(q 2 ) (10) r i 2
and
{AU The operator J = J^Jp is called the Coulomb operator and K = Y!> Fp is called the p p exchange operator, where the sum over p includes all occupied spin-orbitals. Equation (8 ) must be solved iteratively for each k (since the operator F depends on the solutions) by varying the linear coefficients c,jt until the change in each coefficient between iterations becomes small enough. When the spin-orbitals do not change significantly 8 in an iteration, the spin-orbitals are said to be self consistent. Because of this the Hartree-Fock method is often called the self-consistent field method (or SCF method). Roothaan developed a matrix method for solving the set of Equations ( 8 ).** See also Reference (10) for a detailed description of the SCF method and many of its variations. The SCF method is valid for a limited number of molecular states and geometries. It is well suited for closed-shell molecules (no unpaired electrons) in their ground state near the equilibrium geometry. Unrestricted Hartree-Fock (UHF), where each spin-orbital may have a different spatial part, works for some open-shell molecules (some of the electrons are unpaired), but the disadvantage is that the wave function is not an eigenfunction of the total electron spin operator. The restricted Hartree-Fock (RHF) method, on the other hand, has the orbitals paired so that two spin-orbitals, one with an a spin function and the other with a spin function, have the same spatial function. The RHF method is used mostly for closed-shell molecules. For most closed-shell molecules near equilibrium, the UHF and RHF methods are equivalent. Open-shell RHF (ROHF) works well for some open-shell molecules and produces an eigenfunction of the total spin operator but is not feasible for some molecules. Also, none of the Hartree-Fock methods work well when a bond is stretched substantially beyond its equilibrium bond length. For these more general situations and for more accurate work, many-determinant or many-configuration wave functions are used. Since the ^ term cannot be separated into a part that depends on only electron p and a part that depends only on electron q, a single-determinant wave function takes account only of the average interaction (through the Fock operator F). The instantaneous electron-electron interaction beyond the HF level is called electron 9 correlation. To account for electron correlation, the wave function is expanded in terms of an N-electron basis, where the N-electron basis functions are antisymmetric products of orbitals, including orbitals that are occupied in the SCF procedure as well as orbitals that are unoccupied in the SCF procedure (called virtual or external orbitals). The simplest basis is formed by replacing occupied spin-orbitals in Equation (7) by virtual spin-orbitals in all possible ways. However, these basis functions are not eigenfunctions of the total electron spin operator (they are not spin adapted) and are often not symmetry adapted. See Section C for ways of making the AT-electron basis spin adapted. Symmetry- and spin-adapted iV-electron basis functions are often called configurations, configuration functions, or configuration state functions. The wave function for the molecule using this basis is found by using the configuration interaction method.^^ The trial wave function is expanded in terms of the configuration functions as ^ ^ ( 1 2 ) m and then the Schrôdinger equation (1) is solved by minimizing the energy functional, with respect to the linear coefficients, Cm- This is equivalent to solving the matrix eigenvalue equation H e = E c, (14) where Hmn = assuming an orthonormal basis, (#m|^n) = ^mn- If the basis {$m} is obtained by replacing occupied spin-orbitals in allpossible ways with virtual orbitals, then the resulting Cl method is called/«// CL If the given 10 one-electron basis is complete, this gives the exact A^-electron wave function and gives the exact nonrelativistic energy, but in order to have a complete one-electron basis, the expansion must be infinite, so of course, we must limit the basis to a reasonable finite length. Even with a finite basis of n orbitals, with N electrons the resulting full Cl basis will have on the order o in ^ configurations, so one can see that a full Cl calculation is possible for small basis sets and for molecules with only a small number of electrons. The 7V-electron space is often limited by replacing only one or two of the occupied orbitals in the HF configuration by virtual orbitals in all possible combinations. This is called singles and doubles Cl (SDCI). A truncated Cl basis can be constructed with triple, quadruple, etc. excitations but these A^-electron bases are too large for all but very small one-electron bases. SDCI is by far the most commonly used. Truncated Cl expansions are dependent on the quality of the molecular orbitals obtained from the SCF procedure, whereas full Cl wave functions are invariant to a unitary transformation of the molecular orbitals. The atomic basis could just as well be used in a full Cl calculation instead the HF molecular orbitals. Truncated Cl wave functions such as SDCI will not compensate for an already poor HF wave function, and as already mentioned, the HF approximation is not satisfactory for many types of molecular situations. These situations include stretched bonds, excited states, and many open-shell molecules. These cases, which involve near degeneracies, rearrange ment of electron spin couplings, and many open-shell molecules, need the inclusion of nondynamical correlation to give a good first approximation to the wave function. This means that a certain number of configurations are required to give a satisfactory 11 first approximation. The additional correlation that improves the approximation of the instantaneous electron-electron correlation beyond the nondynamical level is called dynamical correlation. In cases in which nondynamical correlation is important, the Cl coefficients of those configurations, as well as the orbital coefficients, are varied simultaneously to minimize the energy. This is called the multiconfigurational self-consistent-field (MCSCF) method. The minimization of the both the orbital coefficients and the Cl coefficients is a nonlinear problem and will not be dealt with here; for a comparison of different minimization techniques see Reference (13). For a review of many aspects of the method including the optimization problem and configuration selection, see Reference (14). Even though the MCSCF method unlike SCF is not limited to certain types of molecular situations, it is not as simple to use as the SCF method. For example, it is not always clear which configurations will be most important. In some cases this can be determined a priori, but in other cases it must be determined by trying various configurations and analyzing the results. And even when a suitable set of configurations has been found, often a good set of starting orbitals is needed to converge to the proper wave function. Although these complications exist, the advantages of the method make it very useful and powerful. Potential energy surfaces require an MCSCF initial approximation if the displacement of the nuclei is very far from equilibrium. Selection of a set of configurations for the potential energy surface of the ground state of water is discussed in Chapter 4. In an MCSCF wave function there are three different classifications of orbitals. 12 There are now two types of occupied orbitals: doubly occupied or inactive orbitals and active orbitals. The doubly occupied orbitals are doubly occupied in all MCSCF configurations, and the active orbitals have variable occupation in the MCSCF con figurations but must be included in at least one configuration. The virtual or external orbitals are not occupied in any MCSCF configuration, just as in the SCF case. A Cl calculation in which one or two virtual orbitals replace the active and doubly occupied orbitals in all possible ways is called multireference SDCI (MR-SDCI) calculation. In this context the MCSCF configurations are called reference configurations. The MR-SDCI method accounts for much of the dynamical correlation as well as the nondynamical correlation, and it can be used on a wide variety of molecules because it uses an MCSCF reference function. However, the SDCI energy does not scale properly with size. A method is said to be size consistent if the energy or properties of a set of noninteracting molecules is the same whether calculated for the set as a whole or separately.*^ The SDCI method is not size consistent, but appropriate single-reference perturbation methods are size consistent. There has been much recent interest in multireference Rayleigh-Schrodinger perturbation theory (MR-PT) using an MCSCF reference function,'®’ and such a procedure has been implemented in the COLUMBUS programs.^ MR-PT has the advantage that it is nearly size consistent, and it takes much less time for calculating the third-order energy than it does to calculate the MR-SDCI energy. The third-order energy calculation requires as much time as one iteration of corresponding Cl calculation, and it typically takes 7 to 15 iterations to adequately converge the Cl energy for one state. Third-order MR-PT also gives nearly the same amount of correlation energy as MR-SDCI with a good 13 enough reference space. Simple formulae that approximately correct the SDCI correlation energy for the size consistency error have been derived using the linked cluster expansion*®’ and Cl treatments of a system of noninteracting helium atoms by Davidson^**’ ^*’ ^ and by Pople.^ These corrections are obtained with essentially no extra cost than the SDCI calculation itself. The COLUMBUS programs have three variants of the Davidson correction and the Pople correction implemented. The original Davidson correction^® is E d v i = (l - Cq )AE s d c i , (15) where Cq is the sum of the squares of the reference Cl coefficients, and AE s d c i is the SDCI correlation energy. The normalized version of this is 1 — E d v 2 = — ^ 2 ° A E s d c i- (16) Davidson and Silver used a collection of noninteracting helium atoms to refine the original Davidson correction and obtained the approximation E o v i = ^ 2 ^ - ^ ^E^s d c i - (17) Pople etal. also used a system of noninteracting helium atoms to obtain the correction® (N^ + 2NiBx?2e)'^ - N ^ = ------Ï ( i c 2 è - 1 )------ where cos# = C q. See Chapter II for a further discussion of size-consistency corrections to Cl. 14 C Spin Symmetry of Many-Electron Wave Functions All chemists know how important point groups are for a deeper understanding of bonding and spectra of molecules. For example, the irreducible representations (irreps) of the point group of a molecule are used to label the electronic states of the molecule, selection rules can be derived, and Hamiltonians can be block diagonalized with knowledge about the irreps of the group. The group of spherical symmetry is also well known for labeling the states of atoms and describing atomic orbitals. The mathematics of the group representations help characterize the atom or molecule and help derive other useful information about it. Another group that is very important in chemistry is the group of permutations of N indistinguishable objects, Sn , the symmetric group of order N. Since the electrons in a molecule are indistinguishable, any permutation of electron labels in the Hamiltonian or wave function results.in the same energy. The Hamiltonian commutes with all the elements of Sm, and the irreps of Sn label many-electron spin functions. Powerful techniques using the theory of the symmetric group derived from graphical objects called Young shapes and Young tableaux,^^’ can be used to generate functions for use in molecular calculations. Young shapes (see Figure 1) are used to label irreps of Sn - These Young shapes are generated by taking advantage of the one-to-one correspondence of the classes of Sn and the irreps of Sn - In fact, there is an obvious one-to-one correspondence between the Young shapes and the conjugate classes of Sn- To see what the conjugate classes of the symmetric group are and how they are used to label Young shapes, we first need to define the notion of cyclic permutations. Any element of the symmetric group can 15 be represented by products of cyclic permutations. For example, (123) represents the permutation in which object 1 is replaced by object 2, object 2 is replaced by object 3, and object 3 is replaced by object 1. Objects 4, 5 , N are not permuted. We also have (12)(23) = (123) and (23)(12) = (132). If the cyclic permutations are disjoint, then the cyclic structure persists in the products. For example, (12)(23)(45) = (123)(45). Notice that (45) does not contain 1,2, or 3, so that cycle did not get combined with the other cycles, but the first two cycles are not disjoint and form a larger cycle. For any disjoint product of cycles, P — P\F 2 '--Pu where / is the number of disjoint cycles, if T is an arbitrary element of Sn, then rp p T -i ^ tP iT ~ '^T P 2 T-'^ ■ • • T P iT ~ \ (19) preserving the cyclic structure. For example, let P = (123)(45), and T = (234). Then, T-^ = (243) and T P T -'^ = (234)(123)(45)(243) = (134)(25). (20) Another example of this is (123)(24516)(321) = (34526) which is still a 5-cycle. We can see then that the conjugate classes of Sn are the sets of products of disjoint cyclic permutations. Each permutation with the same cyclic structure belongs to the same class. The classes can be generated by finding all the integer partitions of N, (Ai, A2, .. ., A/), where I is the number of partitions (cycles) such that Ai + A2 -f • • • -b A/ = iV. (21) If N = 5, then the partitions and classes are shown in Table 1. The Young shapes for each irrep of S5 along with a label (A) = (Ai, A2,..., A/) are given in Figure 1. The 16 class (1^) corresponds to the completely antisymmetric irrep, while (5) corresponds to the completely symmetric irrep. Young tableaux are obtained by placing each of the indices from 1 to iV in the boxes of the Young shapes in all possible ways without repetition and with the restriction that the numbers are nondecreasing from left to right in the rows and nondecreasing down the columns. The tableau generated in this way are in a one-to-one correspondence with basis functions carrying the irreps of Sn . The Young tableau for the (3, 2) irrep and the (4,1) irrep are given in Figure 2. The tableaux with only two rows are important in chemistry because they are in one-to-one correspondence with a basis of iV-electron spin functions with spin S. The (3,2) tableau corresponds to all the doublet spin functions (with M s = ?) obtained from five electrons, and the (4,1) tableau corresponds to all the quartet spin functions (with = |) obtained from five electrons. For the two-row tableau, S = |(A i - Ag). An explicit basis for the N-electron spin functions can be obtained by allocating a- spin functions to all electrons with indices in the first row and ^-spin functions to electrons with indices in the second row of a Young tableau T. The product of these one-electron spin functions is then operated on by the Young operator'® for the given Young tableau T, y ( A ) .T ^ ^W.Tg(X).T^ (22) where = Âi---Âx, and = S i-■■ Si. (23) 17 Âi is the sum of all permutation operators, multiplied by their parity factor, that permute the numbers in column / of the given tableau, and Si is the sum of all permutation operators of row i, without a parity factor. For the third (3,2) Young tableau in Figure 2, ^(3.2),3 = (12)][1 - (3 5 )], (24) 5(3,3),3 = [1 + (13) + (14) + (34) + (134) + (143)] [1 + (25)], (25) and 0(3,2),3 ^ i(3.3).35(3,2),3^(j)^(2)a(8)a(4)/3(5) = = 12[a(l),8(2)-a(2)^(l)][a(3);8(5)-a(5)^(3)]a(4). (26) Notice that if the Young tableau had more than two rows, the N-electron spin functions would be zero, because it is not possible to have an antisymmetric product in which two of the one-electron spin functions are the same. Electron correlation calculations do not require the spin functions explicitly be cause the Hamiltonian does not depend on spin. But they require the use of spatial functions that have the correct permutation symmetry which, when combined with the spin permutation symmetry produce an antisymmetric wave function. These spa tial functions should carry irreps that are dual or conjugate to the spin irreps so that the overall N-electron function is antisymmetric. The Young shape dual to (3,2) is (2^,1) and the Young shape dual to (4,1) is (2,1^). Thus, the spin function products that were symmetric with respect to electron permutations should be multiplied by 18 spatial function products that are antisymmetric with respect to the same electron per mutations, while the spin function products that were antisymmetric with respect to electron permutations should be multiplied by products of spatial functions that are symmetric with respect to the same electron permutations. This requires using Young shapes with no more than two columns for generating spatial functions that display the correct permutation symmetry for electrons. When creating a basis for N-electron spatial functions we start with a 1-electron basis where n > N. Usually n is several times greater than N to get adequate results. We generate Weyl tableaux^^ by placing the indices 1 to n in the two- column Young shapes in all possible ways with the restriction again that the numbers are nondecreasing across rows and down columns and with the further restriction that no index appears more than twice, because only two electrons can be in any spatial orbital. These Weyl tableaux are in one-to-one correspondence with a basis that spans an irreducible tensor space with respect to G U ,nf^ and the unitary subgroups of GL(n). GL(n) is the group of nonsingular matrices of order n. The unitary subgroup of GL(n) is the group of unitary matrices of order «, U(n). This is the group of n x n matrices such that U^U = 1. The irreps of U(n) are the same as the irreps of GL(n), but there are advantages in using the unitary group that make the development easier.^^’ It should be explained what is meant by saying that a space is an irreducible tensor space with respect to U(n). This means that the space spans an irrep of U(n) and is formed from tensors with N indices, such that the indices have a characteristic permutation symmetry of an irrep of 5^?. A reducible basis could be formed by taking 19 as its basis functions all products K (ri)fe(r2) • • • (27) although such a basis is not suitable for fermions. The representation of that reducible space is U®U g) • • • ® U (N matrices in the product), with the transformation law • • • ^'tN = • • • ^ i N j N • (28) Elements P of Sn commute with U®U g • • • g) U, so the above space can be reduced into subspaces with index transformations that belong to irreps of Sn- The irreps for electrons have the two-column Young shapes mentioned above, and the dimension of these spaces is given by^^ Din,N,S) = l ) ( i r - 5 ) ' This is the dimension of the full Cl space, which grows factorially with n and is therefore often limited by excluding certain 7V-electron basis functions. The basis functions used in electron correlation calculations are called configuration state func tions (CSFs). Explicit CSFs could be generated with a procedure using Weyl tableaux analogous to the procedure for generating spin functions using Young tableaux, but we do something better. The beauty of the unitary group approach (UGA)^^ to quantum chemistry, and the graphical unitary group approach (GUGA) developed by Shavitt,^^’ ^ is that the CSFs do not have to be generated explicitly. The nonzero Hamiltonian matrix elements can be calculated directly from linear combinations of integrals using techniques based on the unitary group representation theory and Lie algebra, and information about the CSF space can be stored in well-organized and compact data structures. 2 0 D GUGA The N-electron basis used in UGA^^ and GUGA^ is the Gel’fand-Tsetlin basis. There are many ways to represent the basis,^ but here we leave out the Gel’fand representation and get to the Paldus tableau and step vector representation directly. These latter objects lead directly to the objects of GUGA, and it is the main goal of this section to describe these objects along with their relation to calculating Hamiltonian matrix elements and organizing Cl calculations. This section is a nutshell description of the work by Shavitt^^’ ^ and Paldus,^^ and those references should be consulted for a detailed description. The Gel’fand-Tsetlin basis uses the chain of subgroups^^= t/(n)DC/(n-l)D--OU(l). (30) Each irrep of C/(n) can be subduced into a representation of the subgroup U{n- 1). The unitary group is simply reducible, which means that the subduced representation is a direct sum of the irreps of U{n — 1), in which each of the irreps of U{n— 1) is contained at most once. In other words, the tensor space carrying the irrep of U{n) is decomposed into a direct sum of smaller spaces in the A’-index irreducible tensor space (and subspaces) of U{n — \) The simple reducibility property leads to the unique labeling of the Gel’fand-Tsetlin basis.^^' The subduction procedure is applied to the Weyl tableaux by eliminating the boxes with numbers n for all the Weyl tableaux for a given basis to go from U(n) to U{n — 1). To go from U{n — 1 ) to U{n — 2) the boxes with « - 1 are eliminated. This procedure is repeated until only boxes with 1 remain or no boxes remain. This is most easily explained by an example. Figure 3 lists all the Weyl tableaux for 21 5 = ^, = 3, n = 4. At the top level are all the Gel’fand-Tsetlin basis functions for this space—an irrep of U{n)—labeled with the Young shape (2,1,0^). The zeros are important in the Paldus tableaux, as we shall see, so now they are included explicitly in the permutational symmetry labels. The next lower level has the subduced irreps (basis functions) of (7(3), etc., until (7(1). Each column of the figure corresponds to one CSF and uniquely defines that CSF. Paldus devised a table for representing the basis sets of electronic structure calculations based on the subduction chain of Equation (30). For a given CSF there is an n +1 row table. Each row, k, corresponding to the irrep of U{k) for that particular table (orbital), is assigned three numbers ot, bk, and c^. The number ajt represents the number of twos in the Young-shape label for the Weyl tableau at level k of the subduction, bk represents the number of ones, and ck represents the number of zeros. The top row n is the same for all basis functions and represents the permutational symmetry of the electronic state, corresponding to the irrep label for the Young shape. The bottom row, row zero, is 0, 0, 0 in all cases. The Paldus tableaux for the first and last CSF in Figure 3 are 1 1 2 ‘ "1 1 2 ' 1 1 1 0 1 2 1 1 0 and 0 0 2 (31) 1 0 0 0 0 1 0 0 0 . 0 0 0 . respectively. For each row, k, of the Paldus tableau we have the following relations ak + bk + ck = k, Sk = ^bk, Nk = 2ak + bk. (32) 22 The Paldus tableau represents the successive spin-coupling of electrons in the various rows (orbitals), beginning with row 1 and ending with row N. Here Sk is the intermediate spin at row k, and Nk is the number of electrons at row k {Le., in orbitals 1—k); N = Nn and S = Sn- From the subduction process, we see that there are only four ways to reduce a Weyl tableau from U{k) to U{k - 1). They are 1) nothing gets removed (c* - cjt_i = 1); 2) a box with k gets removed from the left column (bk - 6jt-i = 1); 3) a box with k gets removed from the right column (a^ - ot-i = - bk-i = —1, - Ck-i = 1); and 4) two boxes with k, one from each column, get removed (at — Cfc-i = 1). These results, along with the step number dk defining the particular subduction case, are given in Table 2. We are now ready to define Shavitt’s distinct row table (DRT),^® which is the data structure that is used to define, enumerate, and order the CSF basis in GUGA electronic structure calculations. Table 3 is the DRT corresponding to the basis in Figure 3. Reorganization of the data in Figure 3 to eliminate repetition of Weyl subtableaux (or Paldus subtableaux) leads to the basic structure of the DRT. The distinct rows of the DRT are labeled by a running index r. Each row contains the a-b-c numbers of Paldus; downward chaining indices, and counting indices, y dr and Xr. The downward chaining index Vdr specifies the row at level k - 1 which is connected to row r in level k by step number d. If the connection is not defined for the current basis, the chaining index is zero. The counting indices are used to order the CSF basis and allows arbitrary organization of Hamiltonian matrix element calculation because the index of the CSF is easily calculated from the DRT information. The counting 23 index, Xr, also called the row weight, is equal to the number of subtableaux with row r at the top. The counting indices, y dr, called arc weights, are equal to the number of subtableaux with the top row r at level k preceding the subtableaux with the same row r at level k and with row r* at level k - \ . With xq = 1 these numbers are defined as Î/Or = 0 Vdr ~ yd—\,r ^rd-i,Ti 3,4) — %/4,r, (33) where Xr<(_i,r = 0 for any missing chaining index The lexical index m for a CSF given by a step vector d is n m{d)==l + Y^yd^rf (34) t = l Many of these concepts are most easily understood by looking at a Shavitt graph,^^ Figure 4. This figure contains the same information as the DRT in Table 3 and the Weyl tableaux in Figure 3. Each vertex on the graph corresponds to a row of the DRT and each arc corresponds to a chaining index with the correct step number. The vertices are labeled with the row number to make the correspondence between the graph and the DRT easy to understand. The graph is laid out on a regular grid with the level indices on the ordinate and the a and b numbers partitioning the abscissa, such that the grid increment for the a value must be wide enough to accommodate the maximum b value of the CSF basis plus one or more. On this grid, different slopes of the arcs correspond to different step numbers d. A CSF in the basis corresponds to a distinct path from the tail of the graph to the head. It is easy to calculate the 24 counting indices with the aid of the graph; they are included in Figure 4. The row weights, also known as vertex weights, are just to the left of each vertex, and the arc weights are to the right of each arc. Comparing the Shavitt graph in Figure 4 with the tree in Figure 3, one can appreciate the compactness of the Shavitt graph and the DRT. The Shavitt graph has 27 arcs, while the tree in the Weyl tableaux figure has 35 arcs. The Shavitt graph has 14 vertices, while the Weyl tableaux have 36 vertices. These differences are actually fairly modest, but the differences grow exponentially as the size of the basis set increases. Notice also that the lexical ordering defined in Equation (34) is the same as the order of the Weyl tableaux in Figure 3, but this order is difficult to define until the subduction process is performed. In the unitary group approach we use the Hamiltonian in second-quantized form. In terms of a spin-orbital one-electron basis, the TV-electron tensor product basis spans the totally antisymmetric irrep (1^) of U{2n). This irrep has the correct antisymmetric property for each A^-electron basis function but does not necessarily have the correct permutational symmetry for the total electron spin. The second-quantized Hamiltonian for this basis is given as the familiar H = ' ^ {i\h\j) + ^ [ir,kl] (35) i,j=l i,j,k,l=l ii,v=a,P where [ij]kl] = ^i(l)/:(2)|^|;(l)^(2)y If we sum a\^aj^ over the spin functions a and /3, we get Eij = (36) which are unitary group generators of U(n). They are generators in the sense that any n x n unitary matrix can be represented by U = exp I - i ^ uijEij j . The Gel’fand 25 Tsetlin basis is not created by application of the generators to a reference vacuum function like the spin-orbital creation and annihilation operators. Rather, the Gel’fand Tsetlin basis is created from the subduction process. The one-body operators, Eij, are classified as weight generators if i = j, raising generators if i < j , and lowering generators if i > j. To derive the two-body operator in the unitary group form, we use the anticom mutation relations and get II,V which after summing over fi and i/ gives the two-body unitary operator, ^ij,kl = EijEki - hjEii- (38) So, the Hamiltonian in terms of the unitary group generators is {i\h\j)Eij + l [Ü; kl]eij^kl, (39) i,j=l i,j,k,l=l and matrix elements are given by Tl ^ {m'\H\m) = {i\h\j){m'\Eij\m) + - ^ [Ü; (40) i,j=l i,j,k,l=l The coupling coefficients, {m'\Eij\m) and (j7î'|e,y,)t/|m), are real numbers that can be derived from the shapes of the loops formed between walks for |m) and (m'| on the Shavitt graph. An example of a two-body coupling coefficient loop, (m'|e,y^fc/|m) is given in Figure 5. The orbital indices i, j, k, and I are shown at the left of the loop. The solid line (walk) corresponds to the ket |m), and the dashed line (walk) corresponds to the bra (m'|. Where the bra and ket overlap, only the solid line is showing; this part does not make a contribution to the coupling coefficient loop value. 26 The loop is broken up into loop segments at each level as shown by the dotted lines in Figure 5. Each segment at level p is defined by its shape Tp, consisting of the segment type, the step numbers d' and d of the bra and ket respectively, and by the difference, 6 — b', of the b values of the ket and bra. For the loop shown in Figure 5, the shape types are shown at the right of each segment. This particular loops is a simple product of two one-body loops. The two-body operator that it represents is eij,ki = EijEki, where j < i < k < I, so the first one-body generator is a lowering generator and the second is a raising generator. This is reflected in the notation of the segment shapes. An R segment corresponds to a raising operator, an R corresponds to a raising generator in which the bra and ket walks are joined at the top of the segment, and R corresponds to a raising generator in which the bra and ket lines of the segment are joined at the bottom. The notation for the lowering one-body generator segments, L, L, L, is analogous. The two-body segments are named by a combination of two one-body segment shape names. References 27and 2 provide detailed explanations of the notation for segment types and a complete list of shapes along with tables of segment values. One-body loop values or coupling coefficients are obtained from the segment values, W{Tp,bp), by {m'\Eij\m) = W{Tp,bp). (41) i = J ] W {T „ h ,) ] ] %% W ,{T „ b ,] , (42) pgSi 1=0,1 where S 2 is the set of level indices in the intersection between the intervals (i,j) and (k,l), i.e. (S2 = {i,j) n {k, /)), and Si = (i, I) — 8 2 . The subscript x denotes the type 27 of intermediate coupling (singlet or triplet) in the derivation of the loop values (see References 27 and 29 for details). The two-body loop in Figure 5 does not contain an index set S2 , and the loop value is simply a product of two one-body loops. For the most general case when S2 is not empty, the loop value is a sum of two terms. In the practical use of the theory, the one-body loops are treated as a special case of the two-body loops. , Shavitt has compiled a list of all the distinct loop types that contribute to the upper triangle of the Hamiltonian matrix.^^ These are given in Figure 6 along with the two- body generators and integrals that go along with them. The loop types in Figure 6 show the relative intermediate occupancy (see Equations (32)) of the bra (dashed line) and ket (solid line) at each level of the operator indices and at each of the intermediate levels. The width of the top portion of loop type (la), for example, has an occupation difference of one electron. In the middle of loop types (3a) and (3b) there is a two electron difference between the bra and the ket. Where the bra and ket lines are close together, the occupation is the same, but the intermediate spin may be different. Next to the loop types at each level are the segment types that comprise the loop. The loop types are organized so that coulomb and exchange pairs go together. This can be seen by looking at the two-electron integrals that go along with each loop type. For example the integral \jk; il] that goes along with loop type (lb) is the corresponding exchange integral of [ij; &1], the integral that goes along with loop type (la). The loop types are used as templates for generation of all the possible loops on a graph as well as a tool to help organize computer programs based on GUGA. This will be discussed in more detail in the next section and in Chapter 3. In general. 28 for each of the loop types in Figure 6 there are many different cases of loops with various starting vertices and different combinations of step numbers for the bra and the ket walks. There are however restrictions on loop types ( 8 b), (9), and (11b). They must be restricted so that the first place the bra and ket lines are not overlapping, the bra line is to the left of the ket line. This ensures that they only contribute to the upper triangle of the Hamiltonian matrix. Alternatively one may drop (9) and compute (8 b) without restriction, interchanging the bra and ket arc weights whenever the bra weight is greater than the ket weight.^^ Also, note that the one-electron operator loop types (12c) and (14b) are included with the two-electron operators, and these are processed in a similar manner as the two-electron operators. They can be thought of as a subset of the two-electron loops in many discussions of the loops types and coupling coefficients. The COLUMBUS Cl programs and most other Cl programs allow only single and double excitations of electrons from orbitals in the reference space to orbitals in the external or virtual space. A typical Shavitt graph for such a space is shown in Figure 7. The external levels have a simple structure, and only a limited number of loop shapes or parts of loops can be made on this part of the graph. This was first noticed by Siegbahn.^®’ In Reference 32, Shavitt compiled a table of all the possible loops and loop extensions in the single- and double-excitation Cl external space. E The COLUMBUS Programs: A GUGA Implementation The COLUMBUS program system consists of a family of programs to perform MR-SDCI calculation and related correlation methods.* Included are an integral program, SCF program, several programs for doing an MCSCF calculation, and 29 several programs for MR-SDCI, MR-PT and related correlation methods. In addition, there are utility programs used to analyze the results and give one-electron properties. The programs are very efficient and are suitable for doing highly accurate ab initio calculations. They utilize abelian point-group symmetry to save storage and CPU time for symmetric molecules, but the implementation of the symmetry aspects will be ignored here. Since the most time consuming step is the Cl calculation, much effort has been put into optimizing this section of the code, and GUGA is the basis for the ability to organize this code to make it work so efficiently. In this section some of the organization of the Cl program is highlighted in terms of GUGA. References 2and 27 give a more detailed discussion of the organization of Cl programs based on GUGA. Most of the effort in the Cl procedure is spent in constructing the matrix-vector product w = Hv. The COLUMBUS Cl diagonalization program uses the direct Cl method, meaning that the matrix H is not stored. Instead, contributions to matrix elements of H are created in an order that makes the calculation as efficient as possible. As soon as a matrix element contribution is applied in the calculation of w, it is discarded. One or a few eigenvalues of the matrix are calculated iteratively by the Davidson method,^ which requires saving one or more trial vectors and their corresponding matrix-vector products. We can routinely solve the Cl eigenvalue problem for vectors of length on the order 10^ to 10’, but the vectors and matrix- vector products cannot always be stored in main memory, so they are divided up into segments. Pairs of segments of the current trial vector and matrix-vector product are brought into main memory from disk, and the contributions of the current trial vector segments are added to the matrix-vector product segments at that time. The 30 matrix-vector product depends on the current trial vector elements, the integrals, and the coupling coefficients via Wfji — ^ ^ Hmn^n — ^ ^ ^ (43) n n where the sum over integrals typically includes only one or two integrals for a given pair m, n. In GUGA calculations the coupling coefficient is interpreted as a loop value, where the loop is formed between the bra (m| walk and the ket |n) walk. Only nonzero loops, and thus nonzero matrix elements, are generated and calculated. This is a key advantage of the GUGA method and saves considerable time in the calculation of the matrix-vector product. The sum in Equation (43) is done in an order that maximizes the efficiency. Efficient calculation of loop values and the corresponding matrix elements requires having the correct integrals in memory when a given pair of segments are in main memory, and most importantly depends on taking advantage of the simplicity of the external space to compute the matrix-vector product with dense-matrix kernels that are easily vectorizable. The DRT for the internal space is created for each new internal space, while the external space structure, including the extemal-space DRT loop extensions are built into the program. The walks in the internal space are numbered in reverse lexical order, while the walks in the external space are numbered in lexical order. Reverse lexical order simply means that the recursive calculation of arc weights and row weights using Equations (33) begins at the head of the graph rather than the tail. This insures that the order of all the possible walks connecting a loop head to the graph head is continuous.^^ The complete set of CSFs are actually grouped by the boundary 31 vertex—the vertex at the boundary between the external and internal space—so that CSFs are contiguous in the list if they have the same boundary vertex. This is useful because the extemal-space loop extensions depend on the vertices from which they start. Loop extensions are created in the internal space by a tree-search method (similar to the one described in Chapter II, Section B for extemal-space loop extensions) prior to execution of the diagonalization program, and are stored in a formula file. The intemal-space loop extensions are organized first by the number of internal indices of the two-electron operator associated with each loop type (see Figure 6). Then the internal space indices are varied in a systematic way, and the intemal-space loop part appropriate for those indices are read from the formula file. The integral blocks with the corresponding internal indices are also read in. The intemal-space loop extensions are combined with the extemal-space loop extensions and the integrals, and the contributions to all matrix elements associated with a given loop are accounted for by looping over all walks from the head of the loop to the head of the graph and from the tail of the loop to the tail of the graph. The matrix elements and matrix-vector products are created in vectorizable dense-matrix kemels in which the dimensions of the matrices depend on the number of extemal orbitals. A clear outline cf this overall stmcture was given in Reference (33) and is repeated here: Loop over pairs of Cl vector segments. Loop over types of indices (0-4 intemal). Loop over intemal indices. Loop over formulas for a given set of intemal orbitals. Loop over upper walks. Dense matrix kernel calculations. 32 The integrals are sorted by the number of internal indices, so that most of the necessary integrals can be in memoiy during processing of the inner loops. The loop construction in the intemal space is done in an order that corresponds to fixed internal indices in the integrals, and when a given set of internal space loops are ready to be used, the integrals that go along with the loops are in memory and in an appropriate order suitable for use in the dense matrix kemels. F Summary The development of GUGA has allowed the use of general MR-SDCI to be used for highly accurate calculations on small molecules. The multireference methods provide the flexibility to perform calculations of potential energy surfaces and excited states in which the optimization of the orbitals requires nondynamical correlation by using the MCSCF procedure. In the calculation of the HgO potential energy and dipole moment surface discussed in Chapter ÜI, an MR-SDCI wave function was used with 3.6 million CSFs. The Cl diagonalization program, which took 60% to 70% of the total time, ran at sustained speeds of 213 MFLOPS on a single processor of the Cray YMP-8 , which has a single-processor theoretical maximum speed of 330 MFLOPS. 33 G Tables and Figures Table 1. Partitions of the integer 5 and classes of Sn - Partitions of N=5 Classes of S5 1 + 1 + 1 + 1 + 1 (I,l,l,l,l) = (m 2 + 1 + 1 + 1 (2,1^) 3 + 1 + 1 (3,l2) 4 + 1 (4.1) 2 + 2+1 (2\ 1) 3 + 2 (3.2) 5 (5) (4,1) (2^.1) (3,2) (1®) Figure 1. The Young shapes for S5 with corresponding irrep labels. 34 1 2 3 1 2 4 1 3 4 1 2 5 1 3 5 4 5 3 5 2 5 3 4 2 4 ’ 1 2 3 5 1 2 3 5 1 2 4 5 1 3 4 5 5 4 3 2 • Figure 2. The Young tableaux for the (3,2) irrep and (4,1) irrep of S5 U{A) ( 2. 1, 0, 0 ) 1 i| 1 1 2| 2 2| 1 3| 1 3| 2 3| 1 1| 1 2| 2 2| 1 3| 2 3| 3 3| 1 4| 1 4| 2 '*1 1 4| 2 4| 3 4| 2 2 3 3 2 3 3 4 4 4 4 4 4 2 3 3 4 4 4 U{3) ( 2, 1, 0 )' ( 2, 0, 0 ) ( 1. 1, 0 ) ( 1. 0, 0 ) 1 2 2| 1 3| 1 3| 2 il3| h h lhlal lalairnn |2|3| lahl [71 m_i_ [T] [T] [J] [3 ] 2 WWW 2 3 ^ 2 3 2 f/(2) / ( 2, 0 ) (1, 1) ( 1, 0 ) (2,0) (1,0) (0,0) (1,1) (1,0) (1,0) (0,0) Q S HKIQ 3 um □ . .13 X H] H] 13 ID X \ C/(l) (2) (1) (2) (i) (0) (1) (1) (0) (2) (1) (1) (0) (0) (1) (1) (0) (1) (0) (0) [333 X 13 0 X 030 X0XX00X0XX Figure 3. The Weyl tableaux for the (2, 1, 0^) irrep of (7(4) are shown in the first line. This corresponds to the basis set for 5 = 1, N = 3, and n = 4. The subduction process is shown. The lines show the correspondence between the subduced irreps at each level. This figure is based on Figure 7 of Reference 23. W U\ 36 Table 2. Step numbers defined in terms of changes in the Paldus a, b, and c values, intermediate occupation and spin. 4 Aok Abk Ack ANk ASk 0 0 0 1 0 0 1 0 1 0 1 1/2 2 1 -1 1 1 -1/2 3 1 0 0 2 0 37 Table 3. The distinct row table (DRT) for the basis in Figure 3. See text for definitions. level row (k) (r) Or br Cr f \ r f2r Dr }'lr y2r ysr Xr 4 1 1 1 2 2 3 4 5 8 14 17 20 3 2 1 1 1 6 7 8 9 2 5 6 8 3 1 0 2 7 9 10 3 3 5 6 4 0 2 1 8 9 1 3 3 3 5 0 1 2 9 10 2 3 3 3 2 6 1 1 0 11 12 0 1 3 3 7 1 0 1 11 12 13 1 1 2 3 8 0 2 0 12 0 1 1 1 9 0 1 1 12 13 1 2 2 2 10 0 0 2 13 1 1 1 1 1 11 1 0 0 14 0 0 0 1 12 0 1 0 14 0 1 1 1 13 0 0 1 14 1 1 1 1 0 14 0 0 0 1 38 a = 1 1 0 0 0 level b = ^ 0 2 1 0 20 ('1J 0 \ 8 17 3(4) 3(5) 0 \ 5 \ 6 X 3 \ 5 H8; 2(9.) mo) 1 (1 2 ) 1(13 1 (1 4 ) Figure 4. The Shavitt graph that corresponds to the DRT in Table 3 the Weyl tableaux in Figure 3. 39 Grapjt Head 1 Loop Head V \ \ Loop Tail Graph Tail Figure 5. An example of a two-body loop which gives a nonzero coupling coefficient for The solid and dashed line walks correspond to the ket and bra state functions |m ) and {m'\, respectively. The loop is divided into segments by the dashed line at each level, and the segment types are shown to the right of each segment. The indices that correspond to the indices in the two-body operator are shown at the left of the relevant levels. 40 (la) (1b) (2a) (2b) (3a) (3b) \ I \ \ R RE r r \ r r Rt \ RL RL RR RR RL EL ER RE R L B L ®iÿ.H ‘jU k ' ‘ il.kj mm vm lik:m m n (4a) (4b) (6a) (6b) (7) N, R h R R \ R 1 ' \ R R E RE RW ER EL RL V \ w \ EL s ,kl ^ ‘ ii,ki ' ‘ il,kk ‘ H ik ‘ klM mm lkk;il\ lik;kl] lik:kn , (8a) (11b) \ W RE RL «J-ll ‘ iUi ' ‘ iUI ' ‘ ii.ll ‘ii.ir ( (12a) (12b) (12c) (13) _ (14a) (14b) R R N R N RR WW \ W 1 1 R 1 R R 1 RR I, I ^ . EW E E BE ‘iUi ‘ IUI 4 r ‘ il,il i ‘ll,ll 4 , [K•i7] um (iIa IO im um ((Ia | i) Figure 6. The distinct loop types that contribute to the upper triangle of the Hamilton ian matrix. The spacing between the bra (dashed line) and the ket (full line) shows the relative occupancy at each level of the loop. The segment types that define the loop type are shown at the right of each different level type of the loop. The indices at the left of the figure show the position of the operator and integral indices for relevant segments of each loop type. The indices in this figure satisfy i < j < k < I. The operator and indices that go along with each loop are shown at the bottom of the loop. This figure is reproduced from Reference 27. 41 a; 5 4 3 2 1 0 b: 04321 04 321043210432104321 0 active levels doubly occupied levels external levels Figure 7. A graphical representation of a DRT from the COLUMBUS cidrt program. The reference space is shown in thick lines and consists of three doubly occupied levels and a complete active space (full Cl) with 4 electrons in 4 orbitals. The thinner lines show the MR-SDCI space that results from this reference space. The external space is typically much larger than can be shown in this figure. CHAPTER II Triple- and Quadruple-Excitation Correiation Corrections A Corrections and complexity Correlation methods that have CSFs with electron excitations through double ex citations from the reference function sometimes do not give sufficiently quantitative results. This is certainly true in many cases of single-reference methods. Multirefer ence correlation methods with single and double excitations include the most important triple-excitation CSFs for cases like these. However, even multireference correlation methods can be inadequate. For example, the van der Waals Beg potential energy curve calculated with third-order MR-PT using a reference space with 60 CSFs is not even qualitatively correct^: there is a spurious local maximum at about 6 Bohr. It would be interesting to calculate the energy curve with fourth-order MR-PT to see whether this method improves the situation, but such a calculation requires triple- and quadruple-excitation CSFs. The variational MR-SDCI curve for Bez is qualita tively correct, but needs improvement. The most important way to improve the Cl method is by correcting for the lack of size consistency. Methods that attempt to correct the Cl wavefunction for size consistency by approximations like the Davidson correction^'’ do better in the Beg example, but these do not always give adequate results. We would like to make better perturbative corrections to the Cl energy for size 43 consistency using triple and quadruple excitations, as well as to be able to calculate the fourth-order MR-PT energy. The Cl wave function for any given A^-electron basis can be represented by an infinite-order Rayleigh-Schrodinger perturbation theory (RSPT) expansion.*® If single- and double-excitation CSFs only are included in the basis, then an infinite-order RSPT expansion limited to that basis will be equivalent to the Cl wavefunction, provided the series converges. The perturbation expansion in this case will not be size extensive by the linked-diagram theorem.*® Missing are the disconnected quadruple excitations that cancel with terms in the fourth-order renormalization term, which requires only single and double excitations. To make the Cl energy size extensive up to fourth order in RSPT, single, double, triple, and quadruple excitations need to be included in the A^’-electron basis. But the size of such an N-electron basis (unless based on a very small one-electron basis set) is too large to use for a Cl calculation. We would like to calculate or approximate the contribution to the energy from the triple and quadruple excitations that make the Cl wavefunction size-extensive up to fourth order in RS perturbation theory. The most obvious way to approximate this contribution is to use some type of perturbation approach. If the perturbation approaches are of the RSPT type in a Cl program setting, as opposed to the linked- diagram approaches of MBPT, there are serious difficulties due to computer storage and organization that make the calculation nearly as difficult as the Cl calculation with quadruple excitations itself. It is not clear how to solve this problem in a CSF- based Cl-type approach. Spin-adapted multireference MBPT approaches are difficult to formulate, but this type of approach may help in organizing a fourth-order calculation 44 and help in reducing the amount of intermediate data that needs to be stored. The Ak method^'* is the simplest choice for a perturbation correction to the Cl energy. It can derived using Lowdin’s matrix partitioning theory.^^ Let H jpj denote the Cl matrix block including excitations through doubles and H q q denote the block containing only triple and quadruple excitations. Then H d q = are the off- diagonal blocks. The total matrix is written as ( Z Z ) First H dd is diagonalized to obtain = E qcq . The orthogonal complement, the space spanned by the rest of the eigenvectors o f H jjj), can be neglected for the Ak approximation, since it will not contribute, or it can be included with the H qq block for more general perturbation expansions like MR-PT. Putting H o o = C g H ij ij c o = Eq and H qq = H q ^j c û , we solve (: by eliminating c q to obtain Hoo + H qq (E — H q q ) ^ H q o = E. (46) With E = Eo + AE, the energy correction becomes AE = H oq {E - H qq )~ ^H qq . The inverse matrix can be factored and expanded by the binomial theorem as {B - Hag)-' = f ; [ [E o - - A e)] n=0 (47) where H qq is the diagonal part of the quadruples block and H qq is the off-diagonal part of the quadruples block. To zeroth order in the off-diagonal part we get an Ak 45 energy correction. 2 E ^QD<^D j £ g o f _ _ a e = Y " ______(48) ^ E q - H q q ^ E q - H q q where the sums over D and Q are over the individual single and double excitations and the individual triple and quadruple excitations, respectively. The choice of Hqq in the denominator gives a second-order Epstein-Nesbet RSPT energy. If H o o = E q already contains the most important contribution to the energy, and H q q is diagonally dominant, which are often the case, then the energy (48) should be a good approximation to the exact energy obtained from the infinite-order expansion. The Epstein-Nesbet second-order energy is size consistent, but what we really want is a size-consistency correction to SDCI, and how well Equation (48) does that depends on how well it approximates the unlinked fourth-order quadruple excitation terms that cancel the renormalization term, -E (^)($i|$i). So, rather than H q q in the denominator, it may be that a different choice will more completely cancel the unlinked terms from the renormalization term. A better correction would be to calculate everything from the fourth-order Mpller- Plesset-like RSPT energy which, apart from the renormalization term, can be written^^ ______Vq d W d q W q d 'Vd 'q______ ^ Q ( 4 ° ^--4'^)(4^-4')’ where V = H — H q, W — V — and H q is chosen to give Mpller-Plesset-like partitioning.^ This expression has the unlinked quadruple excitation terms that cancel with the renormalization term present in the SDCI energy plus the linked quadruple excitations that scale properly with size. If this expression can be calculated, then the full fourth-order MR-PT energy would also be available. 46 We can obtain an estimate of the amount of work it will take theoretically to cal culate the energy correction via Equation (48) by considering a common contribution to written in the following way, with a, b, c, and d denoting external orbitals and i, j, k, and I denoting reference occupied (internal) orbitals: where c“y is the cq coefficient for the ij to ah excitation . Since, for a nonzero contribution, there can be at most two non-coincidences between electrons in orbitals of the bra CSF and orbitals of the ket CSF, matrix elements of the type written in Equation (50) will be the most numerous. If m is the number of orbitals in the internal orbital space and n is the number of orbitals in the external orbital space then there will be on the order of contributions of the type in Equation (50). Typical values for m and « are 10 and 100 respectively, so there would be on the order of 10^2 contributions. Equation (48) can be rewritten as The calculation of the matrix elements involves linear combinations of a few integrals with the coupling coefficients as the linear coefficients. Neglecting the one-electron integrals, the most numerous contributions to the numerator in Equation (51) would be of the form, plus the appropriate exchange contribution. One way to realize the summation over the terms would be to go through all the in turn and sum over the double 47 pairs of indices that contribute in Equation (52), divide by the appropriate denominator, and add to à.E . In a GUGA Cl context, however, the calculation is loop driven and organized to take advantage of the simplicity of the external space for factorization. As a result, for a given 0^)^, the contributions due to the integrals are spread out over the entire integral list, and the interacting singles and doubles CSFs are correspondingly spread throughout the CSF list. B Triple- and Quadruple-Excitation Loop Extensions Being optimistic that a suitable algorithm would be derived for the use of triple and quadruple excitations, I developed the extemal-space GUGA loop extensions for these CSFs interacting with single- and double-excitation CSFs. These are likely to be useful in the future, so I am documenting here the procedure used to generate the loop extensions and some statistics and examples. Shavitt has generated the loop extensions for a single- and double-excitation external space “by hand.”^^ The procedure used here is an extension of what Shavitt did, but using a computer program. The extemal-space graph for CSFs through quadruple excitations is shown in Figure 8. There are five more vertices at the boundary between the external space and internal space compared to the graph for single and double excitations. The new vertices are labeled by R, S, T, U, and V as shown in Figure 8. Even though the quadruples extemal-space graph is more complicated than the singles and doubles extemal-space graph, it is not too complicated to enumerate all the possible loop extensions, particularly with the help of a computer. 48 The basic idea is to use Figure 6 as a template for all the loop types and to generate all the actual loop extensions in the external space by using a depth-first tree search (DPS), where the nodes of the tree are pairs of extemal-space vertices and the arcs are GUGA segments. Recall the definition of segments given in Chapter 1, Section D. The levels of the tree are defined by the two-body unitary group generators (or the indices of the two-electron integrals). The two-body operator indices have the restriction that i < j < k < I, just like the spatial arrangement in Figure 6, where the indices increase from the bottom to the top of each loop. The additional intermediate levels are labeled by p, q, and r with i levels do not depend on the two-electron operator indices, but define a unique loop shape depending on the bra and ket that form the loop. The intermediate indices can therefore be repeated as many times as possible with various step values as long as the step values for the bra and ket are allowed for the segment shape that goes along with the intermediate index. In the DFS, the extemal-space graph that the loops are constructed on is of sufficient length so that all of the multiple occurrences of the intermediate levels can be accommodated; it can be thought of as an infinitely long extemal-space graph. A loop-type data structure or loop-type object is defined by the level indices i, j, k, I, p, q, and r; the segment shapes that go with each level index; the number of levels; the two-body operator index sequence; the loop-type name; and the exchange symmetry type (a or b). For example, the loop-type data structure for loop type (la) is • Level index and segment shape pairs: (/,R), (r,R), (&,R), (?,-), (i,R), (p,R), (z,R) • Number of level types = 7 • Loop type name = '(la)' 49 and the loop-type data structure for loop type (66) is • Level index and segment shape pairs: (/,R), (r,R), (fc,RR), (?,R), (i,R) • Number of level types = 5 • Loop type name = ‘(66)’ • Operator index sequence = 'fdik' • Exchange symmetry type = ‘6’. The term “loop-type data structure” is used rather loosely here, because the program was written in Fortran 77, and F o r t r a n n does not provide for arbitrary data structures. In reality, the data structures are just collections of arrays and variables, but it is useful in this discussion to think in terms of data structures. The loop-type data structure is passed to subroutines that make decisions about how to proceed based on the contents of the data structure. The overall organization of the loop generation program is given below. Loop over loop types. Loop over the number of external space indices. Loop over bra boundary vertex (R, S, T, U, V) and ket boundary vertex {W. X, Y, Z) pairs. Enumerate all possible loop extensions on the quadruples external- space graph by a depth-first tree search. All the loop types of Figure 6 are taken in turn and the information for the loop-type data structure is created and passed on to a subroutine that loops over extemal-space indices. The number of extemal-space indices is either 1, 2, 3, or 4. All cases are not possible for each loop type. For example, a loop type with an operator index sequence ’«//’ cannot have Just one index in the extemal space. These cases are checked for in the loop over external indices. In the loop over boundary vertices, the program checks to see if the first segment of the loop type is defined on those boundary vertices. If it 50 to see if the first segment of the loop type is defined on those boundary vertices. If it is, then the subroutine godown is called. This subroutine gets its name from its bias to add segments going down the graph first before replacing segments at the same level or a higher level, i.e. a DFS. The subroutine godown does not save the tree as it performs the DFS. Rather, it saves all the necessary data on stacks as it goes down the graph. In addition to valid segments, level indices, bra and ket vertex labels, and segment values, pointers are saved on stacks that point to the current segment for a given segment shape, and the total number of segments for that given segment shape is also saved. When the program is working its way back up the branch of the tree, it uses these pointers to see which segment to try next. If the next segment fits on the graph, then it is pushed onto the stack and the algorithm proceeds down to the next level. When a loop extension is complete all the appropriate information for that loop extension is written to the output file, and the program goes up a level and tries a new segment of the appropriate type. If that segment is valid, then the program works its way down the graph again trying the first segment of the appropriate segment shape at each level. If that segment is not valid, then it tries the next one, and so on. An example of the output is given in Table 4 for loop type (3a) with two external indices starting at the boundary vertices U for the bra and X for the ket. The corresponding tree is given in Figure 9. The loop extensions are given in Table 4 in the order in which they are created in the DFS. Where there is an intermediate index labeled by p or g that has nonzero step numbers associated with it, there is some flexibility in the number it can take. For example, the first entry in Table 4 51 has two q levels with the step number pairs 1,1. These qs can take any of the values between j and Ux, where rix is the number of extemal space levels. Also, none of the . ^s in this entry need to have an extemal space level value if all the RR segments are in the intemal space. In general, each loop extension in the table can contribute to several actual loops. Notice also that the coupling coefficient is given in terms of the extemal-space part times the intemal-space part, where the internal-space contribution is labeled by A;. The reason for this notation has to do with the exchange symmetry of the coupling coefficients.^^ The two-body operator matrix element is broken up into two contributions, (m'|e,j;/fc|m) = 4- {m'\e]j.ik\m), (53) where the first term on the right-hand side represents the z = 0 contribution to the matrix element of Equation (42), and the second term represents the a; = 1 contribution. The exchange symmetry relations are^^ where i , j < k ,l. The notation used in the tables of loop extensions'^ denotes the internal-space contribution by of a loop types (Coulomb like) by A q or A i for the X = 0 or the a: = 1 contribution respectively. The intemal-space contributions to 6-type two-body coupling coefficients (exchange like) are analogously named by B q or B i. If a loop type satisfies one of the exchange symmetries (54) then both the a- and 6-type coupling coefficients can be written with common factors from the internal 52 space. For example, the two-external (3b) loops with boundary vertices U and X have the same shape as the (3a) loops in Table 4, and their total coupling coefficients are just the negative of the coupling coefficients given in Table 4, because they satisfy the first of Equations (54). So, the first (3b) coupling coefficient corresponding to the first entry in Table 4 is —A \ and the second is A\, etc. This is useful in reducing the amount of work in calculating the coupling coefficients. The loop extension generator program recognizes the exchange symmetry and defines the 6-type intemal space segment value in terms of a-type segment value products where ever possible. A summary of all the loop extensions is given in Table 5. The bra walk has either one, two, three or four electrons in the extemal space, while the ket has zero, one, or two electrons in the extemal space. These loop extensions would contribute to Equations (48) or (49). For each loop type. Table 5 shows the number of distinct loop extensions possible for a given number of extemal indices and for a give pair of bra and ket boundary vertices. If there are no loop extensions for a given set of extemal indices and boundary vertices, the entry is left blank. Loop extensions with an R, S, T, U, or V for the bra boundary vertex are the new loop extensions. Ones with W, X, or Y are singles and doubles loop extensions. Loop types (14a) and (14b) were left out of the search procedure, since these are trivial to generate. There is too much data to reproduce all the loop extensions here. However, the loop extensions exist in a text database suitable for manipulation by text processing programs. The all-purpose script language, peri,^^ was used to filter out loops extensions with zero values and to organize and count the various loop extensions. Such a language could be used to organize the data in a form that can be used by a 53 program that uses the loop extensions. For example, loop extensions with the same shape that are Coulomb and exchange pairs need to be combined, and loops with repeated indices that are special cases of larger loop extensions, should be organized in a manner where this is clear, so it can be taken advantage of in a computer program. C Summary Since the approximate quadruples corrections like the Davidson correction and Pople correction do not always provide adequate results, it is desirable to calculate a better correction to Cl for size consistency. However, it proved to be quite difficult to find a practical algorithm for such a correction in the GUGA Cl framework as it is implemented in the COLUMBUS programs. Even though a practical algorithm was not devised, some insight into the problem was gained. Also, the program to generate extemal-space loop extensions was a useful by-product of the investigation. This program provides electronic versions of loop extensions between any pair of boundary vertices through quadruple excitations and could be extended to higher excitations. In the future, when the computing power is adequate or possibly when new algorithms are discovered which could use higher excitation loop extensions, this program will be very useful. 54 D Tables and Figures a = 2 b = 0 2 1 0 4 3 21 0 vertex label: R S TV Figure 8. The extemal-space Shavitt graph for excitation through quadruples is shown. The a,b values at the top of the extemal-space graph are the boundary vertices. They are labeled by R—V for the triples and quadmples. The singles and doubles boundary vertices have the traditional designations W—Z. 55 q. RR. 00. UX I g, RR.ll.VY q. RR. 00. VY p. R . 00. VX I.RR, ll.xzj i.RR.10.XY P .R .ll.X Y i.R . lO.XX ! P.R.OO.XY p.R.OO.XY Î.RR.10.YZ P .R .ll.Y Z i.R .OO.YY P.R.11.YZ i.R.lO.YY p.R.OO.YZ p.R.OO.YZ P.R.OO.YZ i.R.OO.ZZ i.R.OO.ZZ Figure 9. This is an example of a depth-first search tree for loop type (3a) with two-external indices. The search starts at the UX boundary vertex pair. At each node is shown the index, the segment shape, the step numbers of the bra and ket of the segment, and the vertex labels. The vertex labels implicitly define the a,b values at that level. 56 Table 4. The output generated by the depth-first tree-search program, corresponding to the depth-first tree given in Figure 9. See the caption of Figure 9 for an explanation of some of the data. Wq and W% are the segment values corresponding to Equation (42) of Chapter I. In the cases where the segment value lies between the Wq and Wj columns, it is a contribution from a one-body segment value, W. Aq and A; are the internal contributions to the two terms (z = 0,1) of the coupling coefficient. In this table the total contributions are A^ (II Wo = 0 in these cases). Each p or q level index may be selected from a range of valid level indices defining a single loop extension (see text for examples). vertex vertex level shape d'd labels Wo W, level shape d'd labels Wo W, RR 00 u x 0 q RR 00 u x 0 1 RR 11 VY 0 q RR 11 VY 0 1 RR 00 VY 0 q RR 00 VY 0 1 RR 11 x z 0 j RR 10 XY 0 1 RR 00 x z 0 p R 00 XY 1 RR 10 YZ 0 p R 11 YZ - 1 R oo YZ p R 00 YZ 1 R 10 ZZ i R 10 ZZ 1 Coupling coefficient = Ai Coupling coefficient = -A, vertex vertex level shape d'd labels Wo W, level shape d'd labels Wo W, q RR OO u x 0 1 q RR 00 UX 0 1 q RR 11 VY 0 1 j RR 10 VX 0 1 q RR OO VY 0 1 p R 00 VX 1 j RR 10 XY 0 1 p R 11 XY -1 p R 00 XY 1 p R 00 XY 1 1 R 10 YY 1 p R 11 YZ -1 p R 00 YZ 1 i R 10 ZZ 1 Coupling coefficient = A, Coupling coefficient = Ai vertex vertex level shape d'd labels Wo W, level shape d'd labels Wo w , q RR 00 UX 0 1 q RR 00 UX 0 1 j RR 10 VX 0 1 j RR 10 VX 0 1 P R OO VX 1 P R 00 VX 1 P R 11 XY -1 i R 10 XX 1 P R 00 XY 1 i R 10 YY 1 Coupling coefficient = -A, Coupling coefficient = A, 57 Table 5. For each loop type and number of extemal indices, the number of loop extensions starting at each set of boundary vertices is shown next to the boundary vertex pairs. If there is no valid loop extension of a given type for a pair of boundary vertices, they are left out of the table. Loop Type Integral One External Two Extemal Three External Four External (la) [ijM TW 8 WW 5 TW 10 WW 1 TX 7 XX 3 TX 9 XX 1 VX 3 YY 1 VX 4 WY 3 WY 1 XY 2 XY 1 YZ 1 (lb) WM TW 8 WW 5 TW 12 WW 1 TX 7 WX 4 TX 9 XX 1 VX 3 XW 4 VX 4 WY 3 XX 3 WY 1 XY 2 YY 1 XY 1 YZ 1 (2a) WM YW 3 WW 5 TW 11 WW 1 YX 2 XX 3 TX 9 XX 1 YY 1 VX 4 WY 1 XY 1 (2b) [ik;j[\ YW 3 WW 5 TW 12 WW 1 YX 2 WX 1 TX 9 XX 1 XW 4 VX 4 XX 3 WY 1 YY 1 XY 1 (3a) [ik;jl] TW 8 RW 19 TW 10 TX 7 RX 17 TX 10 WW 1 VX 3 SW 21 VX 4 XX 1 WY 3 SX 23 WY 1 XY 2 TY 8 XY 1 YZ 1 UX 6 VY 3 WZ 1 58 Table 5. (Continued), Loop Type Integral One Extemal Two Extemal Three Extemal Four External XZ 1 (3b) \jX-il\ TW 8 RW 19 TW 12 WW 1 TX 7 RX 17 TX 9 XX I VX 3 SW 21 VX 4 WY 3 SX 23 WY 1 XY 2 TY 8 XY 1 YZ 1 UX 6 VY 3 WZ I XZ 1 (4a) [if.t/] WW 3 TW 6 WW 1 XX 2 TX 5 XX 1 YY 1 VX 3 WY 1 XY 1 (4b) [ik;i[] WW 3 TW 8 WW 1 WX 2 TX 6 XX 1 XW 2 VX 3 XX 2 WY 1 YY 1 XY 1 (5) [ik;i[\ RW 4 TW 4 WW 1 SX 3 TX 3 TY 2 WY 1 WZ 1 (6a) [kk;il] TW 8 TW 6 WW 1 TX 7 TX 7 XX 1 VX 3 VX 3 WY 3 WY 1 XY 2 XY 1 YZ 1 (6b) [ik M TW 8 TW 6 WW 1 TX 7 TX 4 XX 1 VX 3 VX 3 WY 3 WY 1 59 Table 5. (Continued), Loop Type Integral One External Two External Three Extemal Four External XY 2 XY 1 YZ 1 (7) [ik;kl] YW 3 TW 4 WW 1 YX 2 TX 3 WY 1 (8a) [yV//] TW 8 WW 5 WW 1 TX 7 XX 3 XX 1 VX 3 YY 1 WY 3 XY 2 YZ 1 (8b) [il;j[] TW 8 WW 5 WW I TX 7 WX 4 XX 1 VX 3 XW 4 WY 3 XX 3 XY 2 YY 1 YZ 1 .. (9) um YW 3 WW 5 WW 1 YX 2 WX 1 XX 1 XW 4 XX 3 YY 1 ( 10) um TW 8 RW 19 WW 1 TX 7 RX 17 VX 3 SW 21 WY 3 SX 23 XY 2 TY 8 YZ I UX 6 VY 3 WZ 1 XZ 1 ( 11a) um WW 3 WW 1 XX 2 XX 1 YY 1 60 Table 5. (Continued), Loop Type Integral One External Two External Three External Four External (11b) [/7;/7] WW 3 WW 1 WX 2 XX 1 XW 2 XX 2 YY 1 (12a) u m TW 2 WW 1 TX 2 WY 1 (12b) [//,•//] TW 8 WW 1 TX 7 VX 3 WY 3 XY 2 YZ 1 (12c) (:|A|j> TW 8 WW 4 TX 7 XX 2 VX 3 YY 1 WY 3 XY 2 YZ 1 (13) [il;i[] RW 4 WW 1 SX 3 TY 2 WZ 1 CHAPTER !l! Ground-State Potential Energy and Dipole Moment Surface of the Water Molecule A Introduction With the continuing growth in the speed with which computers can perform floating-point operations, it is becoming possible to make accurate predictions of high- energy rotational vibrational spectra in small molecules using ab initio potential energy surfaces and dipole moment surfaces. Such calculations require large basis sets with high angular momentum functions and diffuse functions that cover a wide spatial area. These calculations also require multireference correlation methods so that calculated surfaces closely parallel the true surfaces. Very accurate surfaces are now achievable for small molecules with about ten electrons and a few nuclei, and with such small molecules the rotation-vibration spectrum can be calculated variationally with high accuracy as well. The water molecule provides an excellent testing ground for accurate ab initio po tential surfaces. There are many experimental transitions including intensities available to compare with.^®’ Also, there are many unobserved high-energy vibration-rotation bands of water for which theoretical data can be used to help experimentalists locate them. The wealth of data for the water molecule allows high-quality empirical sur faces to be calculated. See Jensen’s work^^ for a high-quality empirical potential 61 62 energy surface and a list of references for experimental transition frequencies. Femley et compared recent empirical energy surfaces derived using variational meth ods to solve the vibration-rotation Schrodinger equation and found Jensen’s surface^® reproduced experimental transitions best. Variational methods for solving the nu clear motion Schrodinger equation are more accurate, and can reproduce high-energy rotation-vibration transition energies better, than perturbation methods, but there has been much earlier work using perturbation methods to derive empirical potential en ergy surfaces.'^^'^ For an overview of perturbation methods see Reference43. Although it is well known how to derive accurate energy surfaces from experi mental data and that these surfaces could be used to predict higher energy transitions, these methods alone do not give the full picture. For one thing, even though the em pirical surfaces give good agreement with experimental transition frequencies, we can not be sure that the shape of the surfaces match the true Bom-Oppenheimer potential energy surface. Furthermore, the surface must be extrapolated to high energies to make predictions about the higher energy rotation-vibration spectrum, so the potential energy function must be physically correct in the extrapolated region. High quality ab initio methods can reproduce the correct shape of the surface over the entire range necessary. Also, energy surfaces alone do not help predict the intensities of the transi tions. For this we need dipole moment surfaces, which are relatively easy to calculate with ab initio methods but are difficult to calculate with empirical methods.'^^'^^ Jonathan Tennyson suggested we calculate a highly accurate energy and dipole moment surface of the water molecule, so we could make a first-principles calculation of the rotation-vibration spectrum of water. Tennyson wrote a program^ that uses 63 a variational method and a potential energy surface to calculate rotation-vibration transition energies and uses a dipole moment surface to calculate the transition intensities. Tennyson provided us with a list of 312 geometry points covering an energy range of 25,000 cm“^ (0.1139 E^, where Eh denotes Hartrees) with bond lengths varying from 0.770  to 1.48  and the bond angles from 41° to 169° to be used for generating such surfaces. There has been recent work on using ab initio dipole moment surfaces for calculating rotation-vibration intensities,^^’ but none has been as accurate as the work undertaken here. Jprgensen and Jensen'*® recently calculated the intensities of 80,000 rotation-vibration transitions with energies below 30,000 cm"*. In their paper, they do not report any data easily compared to our data, except intensities, and we have to wait until Tennyson completes the calculation of the vibration-rotation spectrum from our surface to be able to compare these intensities. Nevertheless, we can still compare the level of theory used in each case. Jprgensen and Jensen used Jensen’s experimentally derived energy surface®® and an ab initio dipole moment surface covering an even larger range of geometries than our surface but with only 241 geometry points. The energy surface of Jensen, while reproducing the rotationai- vibrational spectrum of water better than any published surface,'*® is an empirical surface rather than an ab initio surface, so it is not a first-principles calculation. Also, their dipole moment surface was produced with a smaller basis set than was used in this work. Jprgensen and Jensen used a basis set consisting of 72 functions contracted from 95 primitive Gaussian functions that included only d polarization functions on oxygen and p polarization functions on hydrogen. In this work, the basis set consists of 132 64 functions contracted from 174 primitives and includes angular momentum functions through g on oxygen and/on hydrogen. J 0 rgensen and Jensen’s Cl expansion included only 24 reference configurations, while ours includes 95 reference configurations. The dipole moment surface produced here should be more reliable than that of Jprgensen and Jensen. Other recent work in this area includes the ab initio energy and dipole surface of Culot and Lievin,'*^ but they used an even smaller basis set than J 0 rgensen and Jensen with single reference Cl, and the range of the surface was much smaller, using the grid of Rosenberg, Ermler, and Shavitt.^® The energy surface of Bartlett et al.^^ uses the highest level of correlation treatment in CCSDT-1, but it is a single-reference calculation and uses the small grid and basis set of Rosenberg et al. Nevertheless, the surface gives good agreement with experimental band origins for low vibrational levels.^^ In this dissertation our potential energy surface and dipole surface are presented. It is shown that the averaged atomic natural orbital basis sets of Widmark et al.^ reproduce the dipole moment and energy of a large primitive set with a relatively small set of contracted functions. The savings one obtains from this basis set in comparison with other high-quality basis sets are very useful for potential energy calculations. Two smaller reference spaces are compared with a full valence Cl reference space (8 electron — 8 orbital CAS). It is seen that the much smaller reference spaces give nearly the same results as the large CAS space, but the smaller reference wave functions can be symmetry broken at certain geometries. Finally, the quality and consistency of the surfaces are assessed by looking at plots and comparing spectroscopic information 65 derived from our surfaces using a perturbation approach.^^’ B Methods Basis Sets It is well known that the calculation of accurate dipole moments with ab initio methods requires a good basis set with polarization functions and diffuse functions. Since we are calculating the dipole moment surface as well as the energy surface, we need to use a large basis set with diffuse functions. Two types of valence- correlation Gaussian basis sets (with modifications) were chosen for comparison: the augmented correlation-consistent basis sets of Dunning and coworkers^^’ and the averaged atomic natural orbital (ANO) basis set of Widmark et al} The correlation consistent basis set of Dunning^^ was designed to provide good valence electron correlation with Raffenetti contractions^^ and an economical choice of primitives. The primitive basis sets for the occupied shells are from van Duijneveldt,^^ and the primitives for the polarization functions (unoccupied shells) were optimized in SDCI calculations on the atoms (for hydrogen H 2 was used instead). The lowest angular momentum polarization function primitive exponents were optimized first, then the next higher angular momentum shell exponents were optimized, and so on. The polarization functions are left uncontracted, but for the occupied shells, the primitive set is contracted by using the Hartree-Fock orbital coefficients. Additional functions for the occupied shells are well-chosen primitives left uncontracted. The uncontracted primitives of the occupied shells are usually the most diffuse primitives of the atomic shell, but Dunning has shown that in some cases the optimal choice of uncontracted primitives may involve uncontracting less diffuse functions. Furthermore, the largest 6 6 primitive set does not necessarily provide the most correlation energy lowering for small contracted sets The correlation-consistent basis chosen for consideration as the basic basis set for the present work was the cc-pVQZ basis, which stands for correlation- consistent polarized valence-quadruple-zeta. The contraction scheme for this basis is {\2s6pid2f\g)l{6s'ip2d\f)^[5s^p?>d2f\g'\l[As?>p2d\f\. This basis set does not have diffuse functions, which are needed to give quantitatively accurate dipole moments and other properties. Notice the regular sequence of the number of each type of an gular momentum functions in the contraction scheme. This results from the definition of the correlation-consistent sets, which requires that all basis functions which lower the CISD energy by the same order of magnitude are added to the basis set together. This is similar to adding one shell at a time, for example adding 4s-4/ to the basis set together. Additional diffuse primitives were derived to augment Dunning’s correlation consistent basis sets.^^ These are single primitive gaussians with exponents chosen to optimize the energy of the singly charged anion of the atoms. Interpolation and extrapolation procedures were used to choose exponents for atoms for which the additional electron for the negative ion is not bound. The resulting aug-cc-pVXZ bases (where X is D, T, or Q) have an additional diffuse function for each shell. These diffuse functions are necessary for properties such as electron affinity, dipole moment, and polarizability, that depend on the electron distribution in the outer regions of molecules. The contraction scheme for the full aug-cc-pVQZ basis is [()s5pAd2)f2g\l[5sAp2)d2f\, which gives a total of 172 basis functions for water. This basis set is too large to use 67 for a large surface using a large reference-space wavefunction, so we truncated it by eliminating the diffuse g functions on oxygen and the diffuse/ functions on hydrogen. The resulting basis set has a total of 149 functions. This basis set is denoted by aug-cc-pVQZ-d(gt/) where the designation means that from the original aug- cc-pVQZ one diffuse g function was removed from the oxygen basis and one diffuse f function was removed from each hydrogen basis. This basis set is shown in Table 6 . The averaged ANO contracted functions of Widmark et al.^ were obtained by spherically averaging the frozen-core SDCI density matrices for the atom over various states including ionic states and low-lying states important in bonding. Also included in the averaging was the density matrix for the atom polarized by an electric field, and the H and Li basis sets used Ha and Lia density matrices, respectively. The contraction coefficients for each atomic shell are the natural orbitals that diagonalize the averaged density matrix for that shell. The primitive basis sets for the occupied-shell basis functions were taken from van Duijneveldt.^® The 7s set was used for hydrogen and the 13f8p set was used for the first-row atoms. The polarization exponents were obtained from an even-tempered optimization of the frozen-core SDCI energy of the atomic ground state (for hydrogen and lithium H 2 and Li2 were used, respectively). Diffuse primitives were added to the s, p, d, and / sets for the heavy atoms and to the s, p, and d sets for hydrogen. The full primitive sets are 14s9p4d3/for the heavy atoms and 8s4p3d for hydrogen. The number of contracted functions used depends on how much of the primitive set one wants to reproduce. Widmark et al. found that ANOs having the same number of nodal surfaces, e.g. 4s-4f, have roughly the same occupation number. Almlof 68 and Taylor^^ also documented this behavior in their original paper on ANOs, and it is related to Dunning’s definition of “correlation consistent.” Even though Widmark et al. noticed this behavior, their standard basis sets are not correlation consistent. Rather, the contractions they use are [6s5p3d2f\ for lithium through neon and [As3p2d\ for hydrogen. However, basis sets obtained from Widmark through electronic mail contained additional contractions for each primitive set. For highly accurate work, g functions are needed for first row atoms and/functions should be added for hydrogen, so we added Dunning’s cc-pVQZ oxygen g function to the oxygen basis and hydrogen / to the hydrogen basis. An additional d contracted set was added to the standard oxygen contracted set, and an additional s and p contracted set was added to the standard hydrogen contracted set considered for the water surface. So, the ANO set used here is [6s5pAd2f\g]l[5s^p2d\f\, for a total of 132 basis functions for the water molecule. This basis set will be denoted by ave-ANO+(c/g, jp /, meaning the averaged ANO basis set of Widmark et al. plus an extra / and g on oxygen and an extra s, p, and / on hydrogen. This basis set is given in Table 7. A comparison of energies and dipole moments for the water molecule at the experimental equilibrium geometry, R=1.811 Oo (0.9583 Â) and 0=104.48°, for the two basis sets is given in Table 8 . The smaller ave-ANO+(dg, jp/) basis gives a Hartree- Fock energy lower by 1.1 mE*, and is within 1 mEh of the estimated Hartree-Fock limit. Feller^® estimated the Hartree-Fock limit to be -76.0679 E/, for the geometry R=0.9572  and 0=104.52° by fitting a sequence of aug-cc-pVXZ SCF energies with X up to 5 to an exponential function and extrapolating X to oo. The correlation energy using the SDCI wavefunction with a CAS reference space of 6 electrons in 6 orbitals 69 is -0.2903 Eh for the aug-cc-pVQZ-d(g^^ basis and -0.2907 Eh for the smaller ave- A^iO+(dg,spf) basis. The number of CSFs for these wave functions is 4.62 million and 3.59 million, respectively (see Table 10). The smaller basis set gives slightly better correlation energies even though the ^-electron basis is smaller by 22%. The dipole moments are essentially the same at each level of approximation for the two basis sets. One reason for the better results with the smaller basis set is that the averaged ANO basis set has a larger number of primitives, and the contractions in this basis set give much of the correlation energy and properties of the original primitive set with a relatively small number of contracted functions. Comparison of the basis sets given in Tables 6 and 7, shows that for oxygen, the ave-ANO+(dg,spfi basis set has one more s primitive and two more p primitives than the aug-cc-pVQZ^(gJ) basis. The averaged ANO basis set in each of these s and p sets of primitives covers a wider spatial range, with tighter primitives to describe the cusp at the nucleus (giving a better energy), as well as slightly more diffuse primitives. The diffuse primitives are much closer in the two basis sets than the cusp primitives, which may account for why the dipole moment is very similar in the two cases. The hydrogen s primitive set is larger by one in the ANO basis, and again the tightest primitive is tighter in the ANO case, but the diffuse primitive is slightly more diffuse in the correlation consistent basis set. One other comparison to note about the two basis sets is that in the polarization functions, for which the number of primitives is the same in the two bases, the correlation consistent basis covers a slightly wider spatial range. 70 Reference Space The reference space for a surface must include all the configurations needed to model all the sampled geometries equally well, and the energy and properties surfaces should be as parallel to the true surfaces as possible. There are two ways of choosing configurations to meet these criteria, an empirical selection method or an a priori selection method. For a review of various selection methods see Reference 14. Briefly, an empirical selection method involves sampling all the important regions of molecular geometries and choosing configurations at each geometry based on an energy or a configuration weight threshold.^' The final set of configurations is the union of the sets at each geometry. The a priori selection methods determine which configurations to include based on knowledge of the bonding and electronic structure of the molecule. A priori methods are simpler to determine if the electronic structure is well known. While empirical methods involve more effort to determine the set of CSFs, they give lower energies per number of CSFs. The strategy used for determining the reference space of the water molecule surface was to approximate a high quality but costly reference space with a less costly one. The full valence complete active space (CAS) is a full Cl expansion with all valence electrons in as many valence orbitals. It was chosen as the best a priori active space that could realistically be used in the test calculations and was mentioned by Sexton and Handy to be a good active space for a rotation-vibration energy surface.®^ For water, this space is an 8 electron in 8 orbital CAS ( 8 CAS) and has 924 CSFs in Cs symmetry. Two other spaces with appreciably smaller MR-SDCI expansions and nearly the same quality as the 8 CAS space were chosen. The first one is a 6 electrons 71 in 6 orbitals CAS ( 6 CAS), in which the oxygen 2s orbital was moved to the doubly occupied space and the corresponding correlating orbital was moved from the active space to the external space. The other alternative considered to the 8 CAS space was the 8 orbital restricted-CI generalized valence bond space (RCI-GVB 8 ) space,''^’ ^ which is a direct product of 2 electrons in 2 orbitals full Cl spaces, one for each pair of orbitals, with all allowed spin couplings among the 2 by 2 CAS spaces. Each pair of orbitals is an occupied orbital with a corresponding correlating orbital. The results of test calculations for the three spaces at three different geometries are given in Table 9. All orbitals were optimized at the MCSCF level using the aug- cc-pVQZ-d(gt/) basis set, and all of the MCSCF configurations were included in the Cl reference space. Some of the calculations were done in Q symmetry, while others were done in C2v symmetry. The C2 v calculations included A] andfi 2 reference CSFs, to match the Cs A' reference CSFs. The Cl calculation was an MR-SDCI calculation with the oxygen core orbital uncorrelated (frozen) at the Cl level, but the core orbital was optimized at the MCSCF level. All possible spin couplings were allowed in the reference space and no interacting space restrictions were imposed. The results of these calculations show that the 6 CAS space and RCI-GVB 8 spaces give roughly the same quality results as the 8 CAS space, with the RCI-GVB 8 space giving the best overall agreement with experiment for the dissociation energies and dipole moments. The dimension of each of the spaces is given in Table 10. Because of the large number of geometry points requested for the surface (312), we chose the smaller 6 CAS reference space for the surface production. One Cl iteration takes roughly twice as long for the RCI-GVB 8 space as it does for the 6 CAS space, and 72 each CI calculation takes 60% to 70% of the total time for one full calculation at each point. Thus, with a limited amount of computer time the 6 CAS space seemed to be the most judicious choice. However, that space turned out to be symmetry broken at linear geometries. Plots of the dipole moment components for symmetric geometries showed that no discernible effects of the symmetry breaking problem occurred until bond angles as large as 175° were reached. Fortunately no points with such a high bond angle were required for the surface calculation of the vibration-rotation spectrum for which the surface was to be used. C Results and Discussion The ab initio energy data points and dipole moment data points are given in the Appendix. Along with the MR-SDCI energies are given the second- and third- order perturbation theory energies and the Cl energies with the various quadruples corrections (see the end of Chapter I). The values of the internal coordinates given in the tables of the Appendix are exact to machine precision. The dipole moment length and components computed as the expectation value of the dipole moment operator with respect to the MR-SDCI wavefunction are also given in the Appendix. The coordinates for the dipole moment table are given in two coordinate systems. The first set of coordinates is the y-z coordinates with the oxygen at the origin, the hydrogens in the third and fourth quadrants, and the z axis bisecting the bond angle. The coordinates are the Eckart coordinates^^ which are explained below in the discussion of the dipole moment function. Plots of the Cl energy are given in Figures 11-20 in the Appendix, and plots of the dipole moment components are shown in Figures 21-40 of the Appendix. The 73 geometry points (as requested by Tennyson) were given in planes in which R\ was fixed while R 2 and 9 varied, so it was only possible to plot the data versus^ to test the smoothness of the results and to check whether there were any outlying points. These plots indicate that the ab initio data are consistent but do not by themselves confirm the quality of the data. To assess the accuracy and validity of the ab initio energy and dipole moment points of the surface, the data were fit using the program survibtm^'^ and spectroscopic data were obtained to compare with experimental values and other data derived from ab initio surfaces. The program survibtm fits ab initio energy or property points of a surface to a variety of functional forms, does a normal coordinate analysis, and then uses a second-order perturbation approach^^ to calculate anharmonic frequencies and spectroscopic constants. If a property surface is fit in addition to an energy surface, then vibrational averages of the property are also calculated. The surface was fit with two different internal coordinate expansions: a Dunham expansion to eighth order in the internal displacement coordinates and an eighth order expansion in Simons-Parr-Finlan (SPF) coordinates.®^ The internal displacement coordinates are ARi = Ri — Rg, AR 2 = R 2 — Re, and A6 = 9 — 9g, where Re and 9g are the equilibrium internal coordinates determined from an initial fit of the surface with a guess for the equilibrium geometry. The surface is then refit in terms of the computed equilibrium geometry. The SPF coordinates are (A9 is not changed), and are known to have better convergence properties than the Dunham expansions. However, the results obtained here with the high-order expansions show that both expansions give nearly identical results. The root-mean-square deviations 74 were 3.2x 10"^ Eh (7.0 cm”') for the Dunham fit and 3.4x 10 Eh (7.4 cm”') for the SPF fit, while the maximum deviations were -4.8x 10”^ Eh (105 cm”') and -S.Ox 10"^ Eh (110 cm”') for the Dunham and SPF fits, respectively. The largest deviations in both fits were for veiy high energy points with small angles and asymmetric bond lengths. The equilibrium geometries of the two fits agree to 0.00002  and 0.0003° for the bond lengths and bond angle, respectively, and the harmonic frequencies match to 0.5 cm”' . The anharmonic frequencies of the two fits agree to within 3 cm”' (see Table 11) and the dipole derivatives in terms of normal coordinate match to roughly two significant figures. The best available data that we have to assess the ab initio surface by comparison with experiment are the anharmonic frequencies for the low vibrational levels obtained from second-order perturbation theory. Since the perturbation method is less accurate for high vibrational levels, this comparison is most valid in assessing the quality of the lower-energy parts of the energy surface. The perturbation theory frequencies do not account for Fermi resonance, so they are compared to experimental frequencies that had the Fermi resonance effects removed (deperturbed).®’”’^ Table 12 gives a comparison of the anharmonic frequencies obtained from the different energy surfaces. All of these values were obtained from an SPF fit surface. The vi,v2 , and % quantum numbers are in the standard order and represent the symmetric stretch, bend, and asymmetric stretch, respectively. One can clearly see that the MR-SDCI surface gives the best agreement with experiment. The MR-PT2 surface gives poor agreement with experiment especially in the asymmetric stretching frequencies. The MR-PT3 results are much better than the MR-PT2 results, except that the asymmetric stretching 75 frequencies are still too large. All of the surfaces with quadruples corrections give frequencies that are systematically too low compared to experiment, while the MR- SDCI results are randomly distributed about the experimental values. It is interesting to note that in the earlier calculation of Rosenberg et al?'^ with a 39-STO basis set, frequencies obtained from an SDCI SPF-fit surface are too large and the Davidson correction brings them into closer agreement to experiment. Handy and Sexton^^ report fundamental frequencies of water calculated using a variational procedure from various surfaces created using small- to medium-sized basis sets and Cl wavefunctions. All of the frequencies show a behavior similar to the Rosenberg et al. frequencies, but the effect was smaller with larger MR-SDCI expansions (on the order of 10^ CSFs). In Table 11 the MR-SDCI anharmonic frequencies from the Dunham and SPF fits are compared with the frequencies obtained from the Bartlett et al. CCSDT-1 surface^* and the experimental frequencies. The frequencies derived from the Bartlett et al. surface were produced in two different ways. The first used an SPF expansion and second-order perturbation method as in this work, and the second used a variational method with the MORBID Hamiltonian of Jensen.^^ The Dunham-fit and SPF-fit anharmonic frequencies match the deperturbed ex perimental frequencies about equally well. The maximum deviation is +12 cm"' and +11 cm"' for the Dunham and SPF fits, respectively. The standard deviations are 6.1 cm"' and 6.9 cm"'. This agreement is to be expected because the ab initio points extend to energies sufficiently high that the extrapolation of the fit does not affect these frequencies very much. The Bartlett et al. frequencies compared with the deperturbed experimental frequencies have a standard deviation of 26.3 cm"' and a 76 maximum deviation o f+61 cm"\ The Jensen frequencies compared to the experimen tal band origins give a standard deviation of 17.1 cm"' and a maximum deviation of 31 cm"'. The Jensen results show the superiority of the variationally calculated an harmonic frequencies over the anharmonic frequencies obtained for the same surface from perturbation theory. The equilibrium coordinates and the force constants in terms of the internal displacement coordinates are given in Table 13, along with experimentally derived internal-coordinate force constants for comparison. The force constants are derivatives of the energy function in terms of the internal displacement coordinates evaluated at the equilibrium geometry without any additional factors. There is substantial disparity between the different experimental force constants, especially at third and fourth order. However, the ab initio surface of this work matches well with the ab initio surface of Bartlett et a/.^' Jensen’s force constants^^ were obtained by taking derivatives of the potential energy in the functional form of the MORBID approach with respect to internal displacement coordinates. The experimental force constants of Hoy, Mills, and Strey (HMS)'^^ and the refinements of the HMS surface due to Mills'^^ and to Hoy and Bunker''^ are all based on fitting to experimental frequencies with parameters from a Hamiltonian obtained from perturbation theory. These methods have difficulties with providing enough data to determine all the parameters. This makes the uncertainty in the force constants high and requires some of the parameters to be left out of the fitting. The functional form of the potential energy in the MORBID approach matches more closely the correct physical form with a relatively small number of parameters, and Jensen used a large compilation of experimental data to fit his surface. 77 so the uncertainty in his data is relatively low. Because of the difference in the functional forms used and in the methods of fitting the data, it is understandable that the experimental force constants in Table 13 differ as much as they do. The ab initio force constants are within the error limits or are close to the error limits of the perturbation-based experimental force constants. However, some of the third- and fourth-order force constants are well outside the error limits of Jensen’s surface. The spectroscopic constants and reduced normal coordinate force constants ob tained from the various surfaces in this work and from experiment are given in Tables 14 and 15, respectively. There is generally pretty good agreement with experiment in the spectroscopic constants. It is difficult to make any judgement about the ab ini tio spectroscopic constants based on this data. The reduced normal coordinate force constants do not match the experimental values veiy well, but the two experimental determinations left some of the force constants out of the fitting, and do not agree well with each other. The agreement between the sets of ab initio normal coordinate force constants is much better than the agreement for the internal coordinate force constants. To be able to compare the dipole moment surface with experiment, the axes for asymmetric geometries had to be rotated to satisfy the Eckart conditions.^^’ Figure 10 shows the relation between the coordinate system in which the ab initio calculations were performed (the y-z coordinates) and the coordinate system satisfying the Eckart conditions (the 77-C coordinates). In the original coordinates, the z axis bisects the bond angle, while in the Eckart coordinates, for an asymmetric geometry, the bisector of the angle is rotated off the C axis to eliminate rotational angular momentum contributions from the asymmetric vibrations. See Reference 74 for a 78 good discussion of the Eckart conditions and the relation to normal coordinates. The rotated dipole moment components were fit with survibtm and vibrational averages were obtained with perturbation theory methods.^^’ Table 16 gives the coefficients of the dipole moment function in terms of a fourth- degree polynomial in reduced normal coordinates and compares them with coefficients from other ab initio calculations. The numbers in the table are the derivatives of the indicated dipole-moment component with respect to the reduced normal coordinates evaluated at the equilibrium geometry, incorporating the Taylor expansion 1 /n! factor (where n is the order of the derivative). For example, for the z component, where q\ is the reduced normal coordinate for the symmetric stretch and 92 is the reduced normal coordinate for the bend. All the other ab initio calculations used much smaller basis sets and single-reference correlation methods, so they are not expected to be as accurate as the results from this work. Table 17 compares the dipole moment derivatives with experimental values. The middle two columns are from Shostak and Muenter.'*® They generated vibrational wavefunctions variationally using the Hoy, Mills, and Strey (HMS) force constants"^^ and the Kuchitsu and Morino (KM) force constants^* to obtain matrix elements of the form {a\qi\b) and {a\qiqj\b), where g, is the ith reduced normal coordinate and a and b denote vibrational states. They used this data to fit experimental transition dipole moments and experimental vibrational dipole moments to an expansion of the form {a\fl\b) = HeSab + ^ + E 79 The last column of Table 17 gives dipole derivatives obtained from intensity measure ments of Camy-Peyret and coworkers’^"^^ and the equilibrium dipole moment from Clough et al.^^ The dipole derivatives obtained in this work agree best with the data in the last column. The dipole derivatives from Shostak et al. were obtained by a complicated procedure that involves approximations that could introduce inaccuracies. For example, the experimental force constants they used are designed to reproduce the vibrational spectrum well, but they may not necessarily give the correct shape of the surface, especially since some of the force constants were left out of the fittings. This could lead to errors in the vibrationally averaged reduced normal coordinates. Also, the transition dipole moment data they used had the effects of Fermi resonance removed by perturbation theory, so they had to reintroduce the effects of Fermi reso nance. It appears that the ab initio dipole derivatives are more reliable than the ones obtained by Shostak et al. Table 18 compares the experimental rotationless vibrationally averaged 6 -component of the dipole moment of Shostak et al.^^’^^ in the first few vi brational states with the corresponding ab initio vibrationally averaged dipole moments obtained from perturbation theory. The equilibrium dipole moment is also reported in each case. The difference between the ab initio and experimental dipole moments is 0.006 Debeye for the (0,0,0), (1,0,0), and (0,0,1) bands but 0.008 Debeye for the equilibrium dipole moment and 0.005 Debeye for the (0,1,0) band. One would expect the differences A/i^ between the vibrational dipole moments and the equilibrium dipole moments to be similar in the experimental and ab initio cases, but this is only approximately observed. One problem is that the dipole moment function 80 is dependent on the equilibrium geometry determined from the energy surface, which may not be accurate. Another problem is the way Shostak and Muenter determined the equilibrium dipole moment. Shostak and Muenter determined the equilibrium dipole moment by fitting their vibrational dipole moments determined from Stark measurements^® to the perturbation expression y-viV2Vi = /^e + ^ ( ^ i + 2^ ^ + 2 ) 2) ’ but since they only had four data points, they had to use the approximation that all the Bij are zero. From that fitting they obtained'^ f^viV2V3 = —1.8570 — 0.0051 + 0.0317^ — 0.0225. (58) The Ai values they obtained agree well with the differences between the ab initio vibrational dipole moments and the equilibrium dipole moment, and they also agree reasonably well with the ab initio A, values, which are -0.0061 Debye, 0.0292 Debye, and -0.0240 Debye for the vi, V2 , and V3 vibrational states, respectively. The fit of Shostak and Muenter was redone using the ab initio B ij values computed here and using Shostak and Muenter’s rotationless vibrational dipole moments to get new Ai values and a new estimate of the equilibrium dipole moment. The resulting dipole moment function to first order in the vibrational quanta is fJ‘ViV2V3 — —1.8563 — 0.0058 ^ + 0.0307 4- —^ — 0.0233 U3 + ^ ) . (59) This provides a somewhat better agreement with the ab initio Ai values and equilibrium dipole moment, and this estimate for the equilibrium dipole moment should be closer to the true value. 81 E Summary The energy and dipole moment surface of the ground state of the water molecule produced in this work are the most accurate to date. Over 1000 hours of Cray YMP -8 CPU time were used to produce the ab initio surfaces. These surfaces are intended for the simulation of the rotation-vibration spectrum of the water molecule from first principles including intensities. This work will be done with the exact Hamiltonian using a variational method^’ and will provide the best data for comparison with experiment. A preliminary check on the quality of the data was done with spectro scopic data obtained from perturbation theory/'^' and obtained better agreement with experiment than any other ab initio surface for low vibrational states. 82 F Tables and Figures Table 6. The primitives and contractions of the aug-cc-pVQZ-d(^g) basis set. See text for the origin of the basis set. Atomic Orbital Exponent Contraction Coefficients 0(J) 61420.0 0.000090 -0.000020 9199.0 0.000698 -0.000159 2091.0 0.003664 -0.000829 590.9 0.015218 -0.003508 192.3 0.052423 -0.012156 69.32 0.145921 -0.036261 26.97 0.305258 -0.082992 11.10 0.398508 -0.152090 4.682 0.216980 -0.115331 1.428 0.017594 0.288979 1.000000 0.5547 -0.002502 0.586128 1.000000 0.2067 0.000954 0.277624 1.000000 0.06959 1.000000 0 (p) 63.42 0.006044 14.66 0.041799 4.459 0.161143 1.531 0.356731 1.000000 0.5302 0.448309 1.000000 0.1750 0.24494 1.000000 0.05348 1.000000 0 {d) 3.775 1.000000 1.300 1.000000 0.444 1.000000 0.154 1.000000 0(/) 2.666 1.000000 0.859 1.000000 0.324 1.000000 0(g) 1.846 1.000000 83 Table 6. (Continued), Atomic Orbital Exponent Contraction Coefficients H(s) 82.64 0.002006 12.41 0.015343 2.824 0.075579 0.7977 0.256875 1.000000 0.2581 0.497368 1.000000 0.08989 0.296133 1.000000 0.02363 1.000000 H(p) 2.292 1.000000 0.838 1.000000 0.292 1.000000 0.0848 1.000000 m 2.062 1.000000 0.662 1.000000 0.190 1.000000 HW 1.397 1.000000 84 Table 7. The primitives and contractions of the a\e-ANO+{dg,spf) basis. See text for the origin of the basis set. Atomic Or bital Exponent Contraction Coefficients 0(s) 105374.95 0.00004590 -0.00001050 0.00000896 -0.00001090 0.0000 1811 -0.00002240 15679.24 0.00036065 -0.00008250 0.00007048 -0.00008220 0.0001 2131 -0.00002240 3534.5447 0.00191977 0.00044120 0.00037567 -0.00046750 0.00080932 -0.00089120 987.36516 0.00820666 -0.00188640 0.00161462 -0.00184650 0.00257695 -0.00469920 315.97875 0.02972570 -0.00695400 0.00593400 -0.00755850 0.01377944 -0.01381340 111.65428 0.09045579 -0.0217208 0.01878662 -0.02108680 0.02811115 -0.06289440 42.699451 0.21740537 -0.05685130 0.04946829 -0.06675110 0.13747540 0.12855560 17.395596 0.36876567 -0.11396350 0.10303987 -0.10936730 0.12206903 -0.46171980 7.438309 0.33727977 -0.16202010 0.16205865 -0.27314310 0.73734928 -0.49460860 3.222862 0.09675046 -0.03338000 0.00093665 0.20971367 -1.70575100 3.79097000 1.253877 0.00256736 0.36550685 -0.82242510 1.20348070 -0.35040600 -5.90661100 0.495155 0.00137461 0.55200311 -0.10179020 -0.67746940 3.14299460 5.49744820 0.191665 -0.00014100 0.22363927 0.42539393 -1.42988400 -3.34748100 -3.29851000 0.067083 0.00006829 0.00657453 0.68770275 1.48910680 1.40123750 1.04571810 0(p) 200.00000 0.00089331 -0.00083840 0.00126180 -0.00195280 0.00307588 46.533367 0.00736901 -0.00684910 0.01116281 -0.02434040 0.03750571 14.621809 0.03493921 -0.03285050 0.05183165 -0.09447540 0.18584407 5.313064 0.11529855 -0.11000600 0.19788446 -0.54898560 1.02706510 2.102525 0.25832314 -0.31352630 -0.17892910 -0.34954760 -1.56754100 0.850223 0.36962312 -0.31960110 -0.17892910 1.46590890 0.38159176 0.337597 0.32387894 0.22172426 -0.89820770 -0.75718940 1.14666480 0.128892 0.14679893 0.56226160 0.26666430 -0.59056730 -1.66270400 0.045112 0.03361269 0.30132513 0.62589942 0.79593212 0.97284427 0(d) 3.750000 0.12849338 -0.21820550 0.62420931 -1.21783900 1.312500 0.52118843 -0.48176950 0.24030630 1.70381520 0.459375 0.43457843 0.13575954 -1.18364200 -1.36241400 0.160781 0.14574094 0.82977340 0.92087218 0.61572759 0(f) 2.35 0.36341106 -0.88354060 0.94 0.56215546 0.22624078 0.376 0.26352789 0.67223250 0(g) 1.846 1.00000000 85 Table 7. (Continued), Atomic Or bital Exponent Contraction Coefficients HW 188.61445 0.00096385 -0.00131190 0.00242240 -0.01157010 0.01478099 28.276596 0.00749196 -0.01034510 0.02033817 -0.08371540 0.09403187 6.42483 0.03759541 -0.05049530 0.08963935 -0.44516630 0.53618016 1.815041 0.14339498 -0.20738550 0.44229071 -1.14627100 -0.60896390 0.591063 0.34863630 -0.43508850 0.57571439 2.50318710 -1.11488900 0.212149 0.43829736 -0.02472970 -0.98028900 -1.58284900 3.48208120 0.079891 0.16510661 0.32252599 -0.67215380 0.03096569 -3.76253900 0.027962 0.02102287 0.70727538 1.14176850 0.30862864 1.67669320 H(p) 2.3050 0.11279019 -0.21086880 0.75995011 -1.44274200 0.80675 0.41850753 -0.59437960 0.16461590 2.34899140 0.28236 0.47000773 0.08968888 -1.37101400 -1.99115200 0.098827 0.18262603 0.86116340 1.05931550 0.90505601 H(d) 1.819 0.27051341 -0.79380350 0.7276 0.55101250 -0.09142520 0.29104 0.33108664 0.86200334 H(/) 1.397 1.00000000 86 Table 8 . Comparison of the two final basis set choices for the surface. The definition of each is given in the text. All calculations were done at the equilibrium geometry R = 1.811 Û0 and B = 104.48°, with the 6 CAS reference space. aug-cc-pVQZ - d W ave-ANO + (dg.spf) Dipole Dipole Wave function Energy Moment Energy Moment SCF -76.065894 1.9829 -76.067000 1.9828 MCSCF -76.158373 1.8081 -76.159518 1.8086 MR-CISD -76.356217 1.8434 -76.357653 1.8434 87 Table 9. Comparison of the choices for the reference space at various levels of correlation using the aug-cc-pVQZr-d(g^ basis set. The final four Cl entries for each reference space include the quadruples corrections defined in Chapter 1. Energies Dipole Moments H :0 ‘ OH^ + H 2H+0 D./ D j /THjO /iOH Method (Eh) (Eh) (Eh) (eV) (eV) (Debeyes) (Debeyes) 6 Orbital-6 Electron Complete Active Space MCSCF -76.158373 -75.972950 -75.823207 5.05 9.12 1.8081 1.6061 MRPT2 -76.342911 -76.152348 -75.989355 5.19 9.62 MRPT3 -76.353647 -76.155928 -75.989341 5.38 9.91 Cl -76.356217 -76.158378 -75.990279 5.38 9.96 1.8434 1.6360 CI+DVl -76.363971 -76.165647 -75.996407 5.40 10.00 C1+DV2 -76.364286 -76.165942 -75.996639 5.40 10.00 C1+DV3 -76.364629 -76.166263 -75.996890 5.40 10.01 CI+POPLE -76.362393 -76.164167 -75.995140 5.39 9.99 8 Orbital-8 Electron Complete Active Space MCSCF -76.196715 -76.001949 -75.834754 5.30 9.85 1.9028 1.6872 MRPT2 -76.347249 -76.153382 -75.981474 5.28 9.95 MRPT3 -76.356144 -76.156852 -75.983070 5.42 10.15 Cl -76.358382 -76.160122 -75.984819 5.39 10.17 1.8401 1.6472 Cl+DVl -76.363213 -76.165333 -75.989355 5.38 10.17 C1+DV2 -76.363361 -76.165509 -75.989496 5.38 10.17 C1+DV3 -76.363518 -76.165698 -75.989646 5.38 10.17 Cl+POPLE -76.362173 -76.164231 -75.988381 5.39 10.17 8 Orbital RCI-GVB Space MCSCF -76.189099 -75.997689 -75.843863 5.21 9.39 1.9226 1.6842 MRPT2 -76.345454 -76.152399 -75.989401 5.25 9.69 MRPT3 -76.354449 -76.155943 -75.988567 5.40 9.96 Cl -76.357312 -76.159191 -75.991276 5.39 9.96 1.8631 1.6499 Cl+DVl -76.362781 -76.164662 -75.996003 5.39 9.98 C1+DV2 -76.362964 -76.164853 -75.996159 5.39 9.98 C1+DV3 -76.363159 -76.165057 -75.996326 5.39 9.98 CI+POPLE -76.361622 -76.163512 -75.994999 5.39 9.98 Experiment 5.44 10.07 1.8570' 1.649880 ‘The geometiy was R=1.811 Bohr and cos(d) = -0.249. '’The experimental bond length used was R=0.9697 A from Reference 81. ‘This is the dissociation energy for the reaction, H20(^Ai) -» H(*S) -f OH(^n). The experimental value is from Reference 82. ''Dissociation energy for the reaction OH(*H) —» 0(®P) -h 2H(*S) plus Dc.[. The experimental value for the OH dissociation energy is 4.63 eV, from Reference 83. 'From Reference 46 'From Reference 80. 88 Table 10. The dimension of the MR-SDCI spaces for the cases considered in Table 9 in Q symmetry. See text for a full definition of the spaces. Number of MR-SDCI CSFs aug-cc-pVQZ- Number of Ref. d(gj) ave-ANO+(dg,sp/) Reference Space CSFs basis basis 6 CAS 95 4,620,284 3,586,230 8 CAS 924 13,341,724 10,316,376 RCI-GVB8 150 • 6,677,110 5,177,856 89 Table 11. Comparison of anharmonic frequencies from ab initio surfaces with exper iment. The first two columns are from this work. Ab Initio Experimental® Bartlett, («1,^2, U3 ) SPF = Dunham** et aP Jensen** Deperturbed*^ Original (1,0,0) 3650 3651 3652 3656 3652 3657 (0,1,0) 1601 1601 1625 1614 1595 1595 (0,0,1) 3754 3755 3747 3751 3756 3756 (2,0,0) 7215 7217 7218 7192 7211 7202 (0,2,0) 3167 3168 3217 3164 3156 3152 (0,0,2) 7420 7423 7397 7442 7425 7445 (1,1,0) 5233 5234 5262 5264 5227 5235 (1,0,1) 7236 7239 7232 7235 7244 7250 (0,1,1) 5331 5333 5351 5357 5331 5331 (1,1,1) 8796 8799 8821 8838 8807 8807 “T he ab intio points were fit with Simons-Parr-Finlan expansion. •’T he ab intio points were fit with a Dunham expansion. “Reference 51. They used the same perturbation theory method to calculate the vibrational levels as in this work, which does not include Fermi resonance. ‘‘Reference 52, using the ab initio surface of Bartlett et al., but with a MORBID fitting and a variational calculation of the vibrational levels. This variational treatment automatically accounts for Fermi resonance, so these frequencies can be compared directly to the original freqeuncies. “References 84, 85. ^Deperturbed to remove Fermi resonance effects (References 67-72). These are the appropriate values for comparison of the ab intitio frequencies except those of Jensen. Table 12. Comparison with experiment of the anharmonic frequencies calculated in this work with the various energy surfaces using different correlation methods. («1,^2, us) MR-SDCI MR-PT2 MR-PT3 CI+DVl CI+DV2 CI+DV3 CI+POP Expmt.® (1,0,0) 3650 3606 3649 3641 3640 3639 3642 3652b (0,1,0) 1601 1582 1596 1594 1594 1593 1595 1595 (0,0,1) 3754 3675 3745 3746 3745 3744 3747 3756 (2,0,0) 7215 7125 7211 7196 7194 7192 7199 721 lb (0,2,0) 3167 3136 3160 3153 3152 3151 3155 3156b (0,0,2) 7420 7197 7388 7405 7404 7402 7407 7 4 2 5 b (1,1.0) 5233 5171 5227 5218 5216 5215 5220 5 2 2 7 b (1,0,1) 7236 7110 7227 7218 7216 7214 7220 7 2 4 4 b (0,1,1) 5331 5233 5318 5317 5316 5315 5319 5331 .. (li.ll_.. 8796 8651 8781 8771 8770 8767 8775 8807 “References 84, 85. ’’Deperturbed values from References 67-72. g 91 Table 13. Comparison of internal-coordinate force constants. This Work” Bartlett et at" Jensen' Mills'’ HMS' HE* R./ 0.9591 0.9591 0.9584 0.957 0.9572 0.9578 ejàtg. 104.09 104.45 104.44 104.55 104.52 104.48 /RR/mdÂ'' 8.43405(1)* 8.4430(16) 8.43938(19) 8.454(1) 8.454(1) 8.454(1) fQQimAk 0.710481(1) 0.7921(3) 0.70700(12) 0.697(1) 0.697(1) 0.697(1) /^ ,/m d A -‘ -0.099062(12) -0.1000(11) -0.10515(16) -0.101(1) -0.101(1) -0.101(1) y i m à 0.260079(2) 0.2743(3) 0.30641(23) 0.219(2) 0.219(1) 0.2371(11) /nRR/mdA'^ -58.48198(33) -56.400(60) -55.40(33) -58.2(23) -59.4(30) -59.4(30) 4^,g/radA-‘ -0.524856(44) -0.505(15) -0.447(25) -0.6(2) -0.40(28) -0.40(28) /Rw'/mdA' -0.05413(16) -0.076(16) -0.318(20) -0.8(3) 0.2(15) 0.2(15) 4gg/m d -0.326256(14) -0.3210(20) -0.3383(62) -0.2(1) -0.2(1) -0.2(1) f^ g lm A k ' -0.107042(46) -0.084(18) -0.252(55) 0.4(2) 0.4(2) -0.002(10) fggglm Ak -0.730814(9) -0.7482(30) -0.7332(70) -0.9(1) -0.9(1) -0.67881(36) 361.777(11) 338.(17) 306.0(47) 367.(50) 384.(62) 384.(62) f^ e g lm A k ' -0.22224(18) -0.28(88) -0.950(51) -2.(1) -1.4(1) -1.495(57) -0.5099(38) -0.3(32) 2.57(40) 7.(3) ■5.(18) -5.(18) f^ 'g g lm A k ^ 0.61797(19) 0.62(78) 0.1150(62) 0.9(3) 0.6(5) 0.6(5) 0.6727(33) 0.5(27) 1.93(15) 6.(2) 0.6(73) 0.6(73) y@g6g/md -0.569090(76) -0.74(24) -0.238(19) -0.1(2) -0.07(20) -0.778(13) -1.49044(94) -1.2(78) -6.14(91) 0.0’’ 0.0“ 0.0“ 0.38273(44) 0.2(24) -3.22(32) 0.0“ 0.0“ 0.0“ /oflflfl/md 0.740188(71) 0.648(126) 0.87(13) 0.0“ 0.0“ 0.0“ “The values were obtained from an eighth order Taylor’s expansion fit. '’Reference 51. 'Reference 39. ‘‘Reference 43. 'Reference 42. 'Reference 44. ^Values in parentheses are quoted uncertainties in units of the last digits. The uncertainties for this work are three times the standard deviations of the fitting. ‘‘Constrained to zero in the fit. 92 Table 14. Comparison of spectroscopic constants from the different surfaces of this work with experimental spectroscopic constants. All values but the geometry constants are in wavenumbers. MR- MR- Cl Cl Cl Cl CCSDT- Const. Cl PT2 PT3 +DV1 +DV2 +DV3 +POP P Expt.b / ? c ( A ) 0.9591 0.9610 0.9589 0.9598 0.9599 0.9600 0.9597 0.9591 0.9578' © e 104.09 103.21 103.87 104.17 104.18 104.18 104.16 104.45 104.48": (deg) W | 3828.8 3787.3 3828.8 3819.3 3818.4 3817.4 3820.7 3829.8 3832.2 Ui2 1655.7 1632.1 1649.2 1649.7 1649.3 1648.8 1650.7 1677.0 1648.5 W 3 3937.0 3927.3 3943.4 3928.3 3927.6 3926.7 3929.6 3939.5 3942.5 •*11 -42.9 -43.4 -43.4 -42.7 -42.7 -42.7 -42.7 -43.3 -42.6 ■*22 -17.2 -14.3 -16.0 -17.7 -17.7 -17.7 -17.6 -16.8 -16.8 ■*33 -43.7 -77.0 -51.2 -43.3 -43.2 -43.2 -43.3 -48.9 -47.6 -*12 -17.7 -17.5 -18.4 -17.4 -17.3 -17.3 -17.4 -14.9 -15.9 ■*13 -167.9 -171.1 -167.5 -168.9 -168.9 -169.0 -168.7 -167.4 -165.8 -*23 -23.4 -24.5 -23.7 -23.1 -23.0 -23.0 -23.1 -21.4 -20.3 c 2.5 -4.8 1.0 2.6 2.6 2.6 2.6 2.5 Go 4635.0 4581.6 4631.6 4623.0 4622.0 4620.8 4624.9 4647.5 4634 4 e 27.063 26.438 26.942 27.071 27.070 27.070 27.068 27.285 27.379 B e 14.623 14.743 14.673 14.585 14.582 14.578 14.591 14.553 14.584 C e 9.493 9.465 9.500 9.478 9.477 9.475 9.480 9.491 9.526 0 | A 0.686 0.741 0.725 0.682 0.681 0.681 0.682 0.704 0.750 <*2A -2.570 -2.396 -2.505 -2.606 -2.609 -2.611 -2.600 -2.516 -2.941 <*3A 1.151 1.238 1.197 1.149 1.149 1.149 1.149 1.155 1.253 “IB 0.221 0.201 0.207 0.224 0.225 0.225 0.224 0.218 0.238 “ 28 -0.154 -0.182 -0.167 -0.149 -0.148 -0.148 -0.150 -0.168 -0.160 “ 3B 0.097 0.073 0.085 0.099 0.099 0.099 0.099 0.101 0.078 “ 1C 0.178 0.179 0.178 0.179 0.179 0.179 0.179 0.178 0.202 “ 2C 0.148 0.149 0.149 0.147 0.147 0.147 0.147 0.149 0.139 “ 3C 0.143 0.150 0.145 0.143 0.143 0.143 0.143 0.142 0.144 “From Reference 51 •’From Reference 86 'From Reference 44 Table 15. Comparison of ab initio reduced normal-coordinate force constants with experimental force constants. E x p erim en t MR. MR- Cl Cl Cl Cl CCSDT- Const. Cl PT2 PT3 +DV1 +DV2 +DV3 +POP 1= HMS*’ KMC &ni -303.7 -300.5 -303.0 -304.1 -304.1 -304.1 -304.0 -304.0 -302.5 -319.4 &H2 12.7 17.6 15.8 12.1 12.0 12.0 12.2 13.7 17.7 13.2 &122 50.5 47.0 48.6 51.2 51.2 51.3 51.0 49.7 55.8 85.1 *133 -303.6 -303.8 -304.5 -303.3 -303.3 -303.3 -303.4 -303.9 -309.3 -307.2 *222 -46.7 -39.9 -43.1 -48.1 -48.2 -48.3 -47.9 -42.0 -63.6 -61.9 *233 45.1 52.3 48.4 44.5 44.5 44.4 44.6 45.3 46.3 49.1 *1111 31.6 30.7 31.0 32.0 32.1 32.1 32.0 31.8 31.9 38.5 *1112 -2.6 -3.6 -3.2 -2.5 -2.5 -2.5 -2.5 -2.8 *1122 -12.6 -11.8 -12.2 -12.8 -12.8 -12.8 -12.7 -12.5 -14.3 -23.4 *1133 31.7 31.4 32.0 31.6 31.6 31.6 31.6 31.9 33.6 34.9 *1222 6.6 5.8 6.3 6.8 6.8 6.8 6.8 6.1 *1233 -5.0 -6.3 -5.7 -4.9 -4.9 -4.9 -4.9 -4.9 *2222 -2.2 -1.9 -2.4 -2.2 -2.1 -2.1 -2.2 -2.5 2.1 11.2 *2233 -15.3 -15.2 -15.1 -15.3 -15.3 -15.3 -15.3 -15.2 -16.9 -20.4 *3333 31.9 10.7 27.3 32.2 32.3 32.3 32.2 32.1 35.4 35.0 “From reference 51. '’From reference 42. 'From reference 41. ÎS 94 Table 16. Comparison of dipole moment expansion coefficients in terms of reduced normal coordinates with other ab initio coefficients. Dipole TZ-f2P 6-31G** 39STO Derivative This work SDCP MP2b SDCF m jo -1.84855294 -1.909 - 1 . 9 0 7 m ^ i -0.02756172 -0.03497 -0.03100 - 0 .0 5 2 0 0 m ;2 0 .1 8 5 7 5 2 6 0 0 .1 8 8 2 4 0 .1 7 7 0 0 0 . 2 0 6 8 0 m . |i 0.00473964 0.00328 0 . 0 0 4 0 0 0 .0 0 1 5 0 m z i i -0.00392752 - 0 . 0 0 4 1 9 -0.00400 -0.00379 mz22 0 . 0 1 2 3 9 6 6 1 0 . 0 1 3 0 7 0.01350 0.00995 mz33 -0.00227600 -0.00267 -0.00400 -0.00400 m - i i i 0 .0 0 0 4 9 1 4 8 0 .0 0 0 2 1 0.00017 0.00041 m z i n 0 .0 0 0 3 7 3 5 0 0 . 0 0 0 2 0 0.00033 0.00012 m z m -0.00044499 -0.00033 - 0 . 0 0 0 1 7 - 0 . 0 0 0 2 4 "1:133 0 . 0 0 0 3 6 8 6 2 0 .0 0 0 2 6 0.00000 0.00035 mz222 0 .0 0 1 2 7 2 0 7 0 .0 0 1 3 3 0.00150 0.00051 "1:233 0 . 0 0 0 2 3 5 4 2 0.00009 0.00000 0.00006 "1:1111 -0.00005574 "1:1112 -0.00002815 "1:1122 0 . 0 0 0 0 7 3 3 2 "1:1133 0 .0 0 0 0 3 9 4 2 "1:1222 -0.00014660 "1:1233 -0.00005458 "1:2222 0 .0 0 0 0 4 2 7 2 "1:2233 0 .0 0 0 1 3 4 5 1 "1:3333 -0.00003630 "1 ,3 0 . 0 9 2 1 7 9 4 7 0 .0 9 0 8 8 0.09890 -0.11000 "1,13 -0.00504472 - 0 .0 0 4 1 1 - 0 .0 0 2 5 0 - 0 . 0 0 2 5 0 "1,23 0 . 0 2 8 2 1 3 7 2 0.02068 0.02812 -0.01300 "1,113 -0.00158645 - 0 . 0 0 0 3 3 -0.00042 0.00017 "1,123 -0.00032875 - 0 . 0 0 0 4 6 - 0 .0 0 0 6 5 0 . 0 0 0 0 0 "1,223 0.00030378 -0.00064 0.00026 -0.00100 "1,333 -0.01558064 0 .0 0 0 2 1 -0.00003 -0.00033 "1,1113 0 .0 0 0 6 5 4 4 1 95 Table 16. (Continued), Dipole TZ+2P 6-3IG** 39STO Derivative This work SDCI® MP2'’ SDCr m y i m -0.00019111 n i y i z a -0.00004845 r r ty u n -0.00080880 i r i y z m -0.00001528 0 .0 0 0 2 7 2 4 7 •From Refeience 49 '’From Reference 87 'From Reference 50. 96 Table 17. Comparison with experiment of dipole derivatives with respect to reduced normal coordinates. This work HMS^ KM'’ Literature fie -1.8486 -1.8567 -1.8574 -1.8473(10)' d fijd q i -0.0276 -0.033(7) -0.027(3) -0.0217(10)' dfJtz/dq2 0.1858 0.141(12) 0.147(4) 0.1795(20)' g g dfiyldqi 0.0922 0.0971(18)' d'^fiz/dql 0.0047 0.0073(13) 0.0045(5) d'^fizldql 0.0124 0.0082(15) 0.0100(5) 0.013(2)'* d‘^fi^ldqidq2 -0.0079 -0.0053(21) -0.0068(3) -0.0084^ 0.00069(25) g d^fizldql 0.0013 d^fiz!dq\ 0.000043 0.00010(6) 0.000144 “From the vibrational analysis of Shostak et using the Hoy-Mills-Strey"*^ force constants. '’From the vibrational analysis of Shostak et using the Kuchitsu and Morino'*' force constants. 'From Reference 45. ‘'From Reference 75. 'From Reference 76. 'From Reference 77. BSet to zero in the fit. 97 Table 18. Comparison of the vibrationally averaged dipole moment from perturbation theory with experiment. Experimental® Ab Initio State fJ-zKjy) A / i z W HzlÇD) A ;z/(D )'’ Equilibrium -1.8570 -1.8486 (0,0,0) -1.8549 0.0021 -1.8485 0.0001 (1,0,0) -1.8601 -0.0031 -1.8538 -0.0053 (0,1,0) -1.8233 0.0337 -1.8184 0.0302 (0,0,1) -1.8774 -0.0204 -1.8716 -0.0231 “From References 78, 46. '’Difference between vibrational state dipole moment and equilibrium dipole moment. 98 Figure 10. The relation between the coordinate system in which the ab initio calculations were performed (the y-z coordinates) and the Eckart coordinates (the 77-C coordinates). The z axis bisects the bond angle, and the dipole moment is shown with respect to the y-z origin at the oxygen nucleus. The origin is at the center of mass of the molecule. Appendix 99 Table 19. Calculated ab initio energy points (atomic units). The internal coordinates are given exactly to machine precision. Energy Ri R2 cos(9) MR-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1.456 1.584 -0.735 -76.26078104 -76.27133237 -76.27263160 -76.27983206 -76.28010005 -76.28038876 -76.27833761 1.471 1.471 -0.249 -76.24572278 -76.25705523 -76.25933098 -76.26653443 -76.26680178 -76.26708974 -76.26503849 1.471 1.607 0.155 -76.25592609 -76.26634434 -76.26839932 -76.27562916 -76.27589947 -76.27619078 -76.27413003 1.471 1.607 -0.249 -76.28008802 -76.29140407 -76.29373293 -76.30104378 -76.30132014 -76.30161821 -76.29953134 1.471 1.811 0.057 -76.28338112 -76.29372472 -76.29586445 -76.30325351 -76.30353725 -76.30384364 -76.30173001 1.471 1.811 0.448 -76.23482989 -76.24344844 -76.24474089 -76.25201888 -76.25229363 -76.25258994 -76.25051282 1.471 1.811 -0.249 -76.29490401 -76.30580795 -76.30815103 -76.31560046 -76.31588875 -76.31620025 -76.31406710 1.471 1.811 -0.556 -76.29075558 -76.30140362 -76.30297172 -76.31028857 -76.31056729 -76.31086809 -76.30877736 1.471 1.811 -0.890 -76.26527648 -76.27585542 -76.27747609 -76.28492752 -76.28521506 -76.28552568 -76.28339279 1.471 2.016 0.117 -76.27407995 -76.28376030 -76.28579320 -76.29331318 -76.29360802 -76.29392693 -76.29176973 1.471 2.016 -0.249 -76.28596471 -76.29642573 -76.29880956 -76.30640534 -76.30670618 -76.30703184 -76.30484983 1.471 2.016 -0.616 -76.27594579 -76.28620373 -76.28784316 -76.29530309 -76.29559371 -76.29590788 -76.29376979 1.471 2.260 0.353 -76.23569297 -76.24395687 -76.24555603 -76.25319358 -76.25349809 -76.25382789 -76.25163185 1.471 2.260 -0.249 -76.26184891 -76.27173385 -76.27417257 -76.28195324 -76.28226989 -76.28261342 -76.28036970 1.587 1.587 -0.496 -76.30556664 -76.31709137 -76.31952644 -76.32703098 -76.32732205 -76.32763662 -76.32548703 1.587 1.587 -0.749 -76.29235492 -76.30389378 -76.30624580 -76.31392757 -76.31423046 -76.31455821 -76.31235287 1.587 1.690 -0.445 -76.31951260 -76.33101117 -76.33351914 -76.34108429 -76.34138109 -76.34170214 -76.33953178 1.587 1.690 -0.644 -76.31124506 -76.32273988 -76.32522125 -76.33290584 -76.33321074 -76.33354084 -76.33133278 1.587 1.690 -0.871 -76.29416215 -76.30483315 •76.30657224 -76.31427636 -76.31458164 -76.31491212 -76.31269884 1.591 1.591 0.276 -76.27249982 -76.28250911 -76.28442113 -76.29172230 -76.29199859 -76.29229662 -76.29021222 1.591 1.591 -0.009 -76.30015786 -76.31105706 -76.31332103 -76.32067986 -76.32096080 -76.32126405 -76.31916070 1.591 1.591 -0.249 -76.30901785 -76.32032110 -76.32272040 -76.33013544 -76.33042051 -76.33072837 -76.32860697 1.591 1.692 0.157 -76.30183222 -76.31222341 -76.31435036 -76.32175432 -76.32203932 -76.32234713 -76.32022855 1.591 1.692 0.442 -76.26131209 -76.27045243 -76.27205898 -76.27939073 -76.27966976 -76.27997086 -76.27787615 1.591 1.692 -0.056 -76.31659029 -76.32757097 -76.32992068 -76.33736769 -76.33765613 -76.33796783 -76.33583512 1.591 1.692 -0.249 -76.32185408 -76.33318482 -76.33565078 -76.34314659 -76.34343864 -76.34375438 -76.34160596 1.591 1.811 0.123 -76.31173162 -76.32204708 -76.32422244 -76.33171987 -76.33201292 -76.33232980 -76.33017999 1.591 1.811 0.362 -76.28508120 -76.29445711 -76.29622813 -76.30367003 -76.30395835 -76.30426990 -76.30213859 8 Table 19. (Continued), Eneigy R i R î cos(0) MR-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1.591 1.811 -0.070 -76.32272341 -76.33354643 -76.33592784 -76.34346585 -76.34376218 -76.34408277 -76.34191960 1.591 1.811 -0.249 -76.32639400 -76.33753612 -76.34003902 -76.34762420 -76.34792408 -76.34824865 -76.34607024 1.591 1.811 -0.429 -76.32442320 -76.33575486 -76.33832347 -76.34597663 -76.34628134 -76.34661132 -76.34441122 1.591 1.811 -0.612 -76.31676895 -76.32813405 -76.33070320 -76.33846062 -76.33877252 -76.33911054 -76.33687740 1.591 1.811 -0.819 -76.30189163 -76.31260242 -76.31446819 -76.32222002 -76.32253039 -76.32286664 -76.32063642 1.591 1.931 0.144 -76.30919396 -76.31919132 -76.32132823 -76.32890802 -76.32920819 -76.32953311 -76.32735574 1.591 1.931 0.395 -76.28155271 -76.29057468 -76.29226298 -76.29978356 -76.30007854 -76.30039761 -76.29824011 1.591 1.931 -0.059 -76.32005381 -76.33059814 -76.33297759 -76.34060068 -76.34090446 -76.34123346 -76.33904173 1.591 1.931 -0.249 -76.32311488 -76.33401144 -76.33654126 -76.34421412 -76.34452172 -76.34485501 -76.34264712 1.591 1.931 -0.440 -76.31997877 -76.33109260 -76.33371130 -76.34145752 -76.34177041 -76.34210963 -76.33987825 1.591 1.931 -0.638 -76.31024067 -76.32138640 -76.32401263 -76.33187300 -76.33219381 -76.33254192 -76.33027430 1.591 1.931 -0.871 -76.29039691 -76.30085906 -76.30275950 -76.31061328 -76.31093227 -76.31127826 -76.30901412 1.591 2.058 0.213 -76.29758685 -76.30706000 -76.30907506 -76.31673161 -76.31703838 -76.31737075 -76.31516773 1.591 2.058 0.529 -76.25494403 -76.26324299 -76.26469347 -76.27227805 -76.27257826 -76.27290321 -76.27072461 1.591 2.058 -0.026 -76.31120628 -76.32136248 -76.32370490 -76.33141256 -76.33172376 -76.33206115 -76.32984101 1.591 2.058 -0.249 -76.31422471 -76.32481500 -76.32736228 -76.33512809 -76.33544387 -76.33578641 -76.33354726 1.591 2.058 -0.473 -76.30892614 -76.31978030 -76.32245280 -76.33030787 -76.33063016 -76.33098004 -76.32871222 1.591 2.058 -0.714 -76.29379385 -76.30458230 -76.30722503 -76.31522735 -76.31555983 -76.31592113 -76.31360659 1.591 2.206 0.044 -76.29583609 -76.30541808 -76.30766266 -76.31545947 -76.31577849 -76.31612473 -76.31387463 1.591 2.206 0.368 -76.27204598 -76.28063289 -76.28237350 -76.29010338 -76.29041620 -76.29075541 -76.28852813 1.591 2.206 -0.249 -76.29979264 -76.30997234 -76.31252480 -76.32039774 -76.32072295 -76.32107618 -76.31880099 1.591 2.206 -0.543 -76.28975382 -76.30027639 -76.30301985 -76.31102100 -76.31135567 -76.31171956 -76.30940308 1.591 2.206 -0.892 -76.25813106 -76.26797495 -76.27002083 -76.27806746 -76.27840339 -76.27876860 -76.27643960 1.591 2.402 0.209 -76.26844138 -76.27703772 -76.27905726 -76.28694831 -76.28727551 -76.28763102 -76.28534932 1.591 2.402 -0.249 -76.27763710 -76.28718731 -76.28972457 -76.29772969 -76.29806660 -76.29843313 -76.29611337 1.591 2.402 -0.707 -76.25307291 -76.26308133 -76.26594352 -76.27420545 -76.27456151 -76.27494964 -76.27254695 1.591 2.707 -0.249 -76.24268739 -76.25108760 -76.25360772 -76.26176428 -76.26211509 -76.26249744 -76.26012612 1.689 1.689 0.077 -76.32264277 -76.33313578 -76.33535564 -76.34285085 -76.34314371 -76.34346038 -76.34131131 1.689 1.689 0.274 -76.30396770 -76.31382411 -76.31575479 -76.32320833 -76.32349766 -76.32381037 -76.32167515 1.689 1.689 0 5 5 4 -76.25174207 -76.26044070 -76.26180700 -76.26918397 -76.26946693 -76.26977246 -76.26766260 Table 19. (Continued), Energy Ri cos(O) Rz M R-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople I.6S9 1.689 -0.091 -76.33135470 -76.34222948 -76.34461152 -76.35214317 -76.35243893 -76.35275886 -76.35059785 1.689 1.689 -0.407 -76.33325983 -76.34458420 -76.34713337 -76.35476961 -76.35507295 -76.35540141 -76.35320686 1.689 1.689 -0.567 -76.32757115 -76.33901831 -76.34159847 -76.34932594 -76.34963671 -76.34997236 -76.34774852 1.689 1.689 -0.741 -76.31616488 -76.32765884 -76.33022767 -76.33810982 -76.33843022 -76.33877778 -76.33650519 1.689 1.689 -0.949 -76.29470848 -76.30598285 -76.30833370 -76.31655794 -76.31690221 -76.31727657 -76.31489522 1.689 1.713 0.067 -76.32544932 -76.33594737 -76.33818577 -76.34570136 -76.34599599 -76.34631466 -76.34415876 1.689 1.713 0.255 -76.30865690 -76.31855982 -76.32052798 -76.32800364 -76.32829489 -76.32860976 -76.32646714 1.689 1.713 0.513 -76.26546312 -76.27428931 -76.27574792 -76.28315490 -76.28344035 -76.28374869 -76.28162890 1.689 1.713 -0.095 -76.33339578 -76.34427559 -76.34667467 -76.35422592 -76.35452339 -76.35484526 -76.35267767 1.689 1.713 -0.249 -76.33618583 -76.34739656 -76.34991287 -76.35750747 -76.35780819 -76.35813370 -76.35595211 1.689 1.713 -0.403 -76.33492050 -76.34627666 -76.34884907 -76.35650364 -76.35680865 -76.35713896 -76.35493822 1.689 1.713 -0.559 -76.32946888 -76.34090085 -76.34349876 -76.35124218 -76.35155340 -76.35189068 -76.34966168 1.689 1.713 -0.728 -76.31858755 -76.33006601 -76.33266016 -76.34055049 -76.34087190 -76.34122062 -76.33894498 1.689 1.713 -0.928 -76.29947868 -76.31018522 -76.31215483 -76.32022199 -76.32055505 -76.32091681 -76.31858566 1.689 1.811 0.051 -76.33079025 -76.34123343 -76.34351779 -76.35110998 -76.35141133 -76.35173759 -76.34955596 1.689 1.811 0.223 -76.31755529 -76.32745681 -76.32949701 -76.33705240 -76.33735060 -76.33767330 -76.33550395 1.689 1.811 0.442 -76.28798063 -76.29698800 -76.29861819 -76.30611964 -76.30641312 -76.30673050 -76.30457925 1.689 1.811 -0.102 -76.33717981 -76.34799434 -76.35043771 -76.35806491 -76.35836908 -76.35869852 -76.35650538 1.689 1.811 -0.249 -76.33921018 -76.35029892 -76.35285368 -76.36052319 -76.36083057 -76.36116362 -76.35895678 1.689 1.811 -0.397 -76.33749744 -76.34877677 -76.35140697 -76.35913555 -76.35944722 -76.35978509 -76.35755930 1.689 1.811 -0.547 -76.33197898 -76.34336075 -76.34603288 -76.35384682 -76.35416454 -76.35450920 -76.35225618 1.689 1.811 -0.709 -76.32152306 -76.33292144 -76.33560159 -76.34355423 -76.34388171 -76.34423732 -76.34194013 1.689 1.811 -0.901 -76.30332745 -76.31404177 -76.31613572 -76.32423698 -76.32457399 -76.32494026 -76.32259667 1.689 1.909 0.062 -76.32932703 -76.33955306 -76.34182593 -76.34948757 -76.34979506 -76.35012826 -76.34792325 1.689 1.909 0.238 -76.31630330 -76.32597202 -76.32798054 -76.33560405 -76.33590819 -76.33623762 -76.33404542 1.689 1.909 0.463 -76.28638298 -76.29516283 -76.29674915 -76.30431849 -76.30461777 -76.30494169 -76.30276780 1.689 1.909 -0.097 -76.33546224 -76.34607595 -76.34852740 -76.35622585 -76.35653637 -76.35687299 -76.35465581 1.689 1.909 -0.249 -76.33703157 -76.34793081 -76.35050985 -76.35825262 -76.35856655 -76.35890701 -76.35667543 1.689 1.909 -0.402 -76.33466338 -76.34577295 -76.34844430 -76.35624921 -76.35656772 -76.35691332 -76.35466174 1.689 1.909 -0.560 -76.32807714 -76.33932007 -76.34205205 -76.34994950 -76.35027465 -76.35062772 -76.34834653 S Table 19. (Continued), Energy R i Rz cos(0) M R-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1.689 1.909 -0.733 -76.31572290 -76.32698710 -76.32973868 -76.33779362 -76.33812996 -76.33849562 -76.33616415 1.689 1.909 -0.946 -76.29293167 -76.30339205 -76.30549295 -76.31368846 -76.31403363 -76.31440916 -76.31203399 1.689 2.011 0.094 -76.32294812 -76.33285078 -76.33507278 -76.34280132 -76.34311472 -76.34345462 -76.34122709 1.689 2.011 0.292 -76.30751862 -76.31677793 -76.31867779 -76.32636328 -76.32667279 -76.32700827 -76.32479532 1.689 2.011 0.557 -76.26657724 -76.27488612 -76.27632369 -76.28394937 -76.28425331 -76.28458248 -76.28238995 1.689 2.011 -0.080 -76.32962382 -76.33996252 -76.34239958 -76.35016817 -76.35048496 -76.35082869 -76.34858782 1689 2.011 -0.249 -76.33094127 -76.34160625 -76.34420145 -76.35201946 -76.35234013 -76.35268823 -76.35043122 1.689 2.011 -0.419 -76.32748620 -76.33839359 -76.34110592 -76.34899438 -76.34932029 -76.34967429 -76.34739455 1.689 2.011 -0.596 -76.31871808 -76.32978232 -76.33257619 -76.34057549 -76.34090944 -76.34127248 -76.33895719 1.689 2.011 -0.800 -76.30136351 -76.31239899 -76.31520417 -76.32341990 -76.32376939 -76.32414993 -76.32176536 1.689 2.124 0.158 -76.31127883 -76.32071130 -76.32282379 -76.33061893 -76.33093819 -76.33128472 -76.32903479 1.689 2.124 0.403 -76.28848227 -76.29710129 -76.29878917 -76.30653320 -76.30684758 -76.30718857 -76.30495621 1.689 2.124 -0.049 -76.31997077 -76.32994149 -76.33233625 -76.34017804 -76.34050140 -76.34085257 -76.33858695 1.689 2.124 -0.249 -76.32120442 -76.33157174 -76.33417397 -76.34207433 -76.34240238 -76.34275886 -76.34047400 1.689 2.124 -0.450 -76.31589963 -76.32655911 -76.32931717 -76.33730490 -76.33763953 -76.33800342 -76.33569029 1.689 2.124 -0.667 -76.30242893 -76.31327210 -76.31614384 -76.32428649 -76.32463245 -76.32500912 -76.32264619 1.689 2.124 -0.940 -76.27249792 -76.28249859 -76.28469843 -76.29302560 -76.29338330 -76.29377310 -76.29135264 1.689 2.258 0.009 -76.30565669 -76.31510285 -76.31740683 -76.32532640 -76.32565675 -76.32601585 -76.32372392 1.689 2.258 0.280 -76.29127310 -76.29999091 -76.30189509 -76.30975850 -76.31008361 -76.31043675 -76.30816402 1.689 2.258 0.646 -76.23707533 -76.24484244 -76.24624214 -76.25405406 -76.25437340 -76.25471997 -76.25246581 1.689 2.258 -0.249 -76.30725760 -76.31722364 -76.31981981 -76.32781494 -76.32815150 -76.32851763 -76.32620071 1.689 2.258 -0.508 -76.29823231 -76.30857271 -76.31139227 -76.31951096 -76.31985693 -76.32023371 -76.31787668 1.689 2.258 -0.809 -76.27235335 -76.28286381 -76.28583826 -76.29424780 -76.29461527 -76.29501632 -76.29256553 1.689 2.430 0.131 -76.28372044 -76.29234707 -76.29447244 -76.30247098 -76.30280844 -76.30317562 -76.30085694 1.689 2.430 0.555 -76.24533673 -76.25291340 -76.25445434 -76.26240374 -76.26273529 -76.26309571 -76.26079517 1.689 2.430 -0.249 -76.28767125 -76.29704974 -76.29962064 -76.30772641 -76.30807298 -76.30845052 -76.30609606 1.689 2.430 -0.630 -76.26897631 -76.27887478 -76.28180662 -76.29013572 -76.29049953 -76.29089656 -76.28846937 1.689 2.683 0.441 -76.23931420 -76.24653509 -76.24838989 -76.25648331 -76.25682870 -76.25720488 -76.25485469 1.689 2.683 -0.249 -76.25888319 -76.26727208 -76.26981123 -76.27803855 -76.27839550 -76.27878702 -76.27639093 1.781 1.781 -0.249 -76.34366464 -76.35456044 -76.35711300 -76.36483266 -76.36514457 -76.36548275 -76.36325892 S Table 19. (Continued), Energy Ri R : cos(9) MR-PT2 MR-PT3 MR-CISD DVl DV2 DV3 Pople 1.781 1.811 0.030 -76.33765448 -76.34796817 -76.35027270 -76.35793978 -76.35824781 -76.35858163 -76.35637473 1.781 1.811 0.185 -76.32753682 -76.33740958 -76.33949493 -76.34712824 -76.34743335 -76.34776388 -76.34556827 1.781 1.811 0.369 -76.30744770 -76.31666597 -76.31841225 -76.32600285 -76.32630413 -76.32663033 -76.32444915 1.781 1.811 0.634 -76.25016397 -76.25851487 -76.25983354 -76.26735715 -76.26765257 -76.26797214 -76.26581345 1.781 1.811 -0.112 -76.34259208 -76.35323973 -76.35570020 -76.36340152 -76.36371234 -76.36404929 -76.36183111 1.781 1.811 -0.174 -76.34357190 -76.35435591 -76.35687553 -76.36459422 -76.36490639 -76.36524488 -76.36302103 1.781 1.811 -0.249 -76.34384681 -76.35479504 -76.35737992 -76.36512265 -76.36543665 -76.36577719 -76,36354555 1.781 1.811 -0.309 -76.34338408 -76.35444826 -76.35707750 -76.36484251 -76.36515816 -76.36550056 -76.36326174 1.781 1.811 -0.387 -76.34194658 -76.35308896 -76.35575757 -76.36355708 -76.36387526 -76.36422051 -76.36197059 1.781 1.811 -0.528 -76.33675686 -76.34797597 -76.35069279 -76.35857314 -76.35889715 -76.35924895 -76.35697315 1.781 1.811 -0.680 -76.32712527 -76.33842652 -76.34118443 -76.34919406 -76.34952732 -76.34988951 -76.34757237 1.781 1.811 -0.856 -76.31014389 -76.32146662 -76.32424434 -76.33251749 -76.33286979 -76.33325344 -76.33085182 1.781 1.841 -0.174 -76.34343002 -76.35419577 -76.35672966 -76.36447098 -76.36478519 -76.36512599 -76.36289448 1.781 1.841 -0.249 -76.34364550 -76.35455900 -76.35715615 -76.36492171 -76.36523778 -76.36558066 -76.36334128 1.781 1.841 -0.309 -76.34314125 -76.35415539 -76.35679564 -76.36458364 -76.36490139 -76.36524616 -76.36299953 1.781 1.903 0.038 -76.33638486 -76.34655377 -76.34886513 -76.35659805 -76.35691200 -76.35725252 -76.35502335 1.781 1.903 0.196 -76.32654304 -76.33624460 -76.33832081 -76.34601892 -76.34632980 -76.34666684 -76.34444939 1.781 1.903 0.383 -76.30668773 -76.31571456 -76.31744685 -76.32510214 -76.32540909 -76.32574167 -76.32353879 1.781 1.903 0.652 -76.24877689 -76.25701014 -76.25834728 -76.26593972 -76.26624096 -76.26656709 -76.26438551 1.781 1.903 -0.107 -76.34101519 -76.35153465 -76.35401633 -76.36178476 -76.36210164 -76.36244548 -76.36020456 1.781 1.903 -0.249 -76.34194308 -76.35274127 -76.35535423 -76.36316641 -76.36348670 -76.36383439 -76.36157919 1.781 1.903 -0.392 -76.33948087 -76.35049213 -76.35320561 -76.36107759 -76.36140235 -76.36175505 -76.35948055 1.781 1.903 -0.538 -76.33344274 -76.34459733 -76.34738454 -76.35534372 -76.35567485 -76.35603472 -76.35373221 1.781 1.903 -0.699 -76.32234769 -76.33358038 -76.33642293 -76.34452844 -76.34487017 -76.34524197 -76.34289253 1.781 1.903 -0.890 -76.30211744 -76.31329506 -76.31615916 -76.32459956 -76.32496590 -76.32536547 -76.32290824 1.781 1.998 0.063 -76.33101944 -76.34091596 -76.34318921 -76.35098513 -76.35130475 . -76.35165170 -76.34940121 1.781 1.998 0.236 -76.31990680 -76.32928177 -76.33127962 -76.33903756 -76.33935373 -76.33969677 -76.33745916 1.781 1.998 0.446 -76.29569567 -76.30432554 -76.30593303 -76.31364452 -76.31395627 -76.31429428 -76.31207264 1.781 1.998 -0.094 -76.33594135 -76.34622155 -76.34869387 -76.35652819 -76J5685I07 -76.35720171 -76.35493842 1.781 1.998 -0.249 -76.33660333 -76.34718916 -76.34981747 -76.35770005 -76.35802675 -76.35838171 -76.35610261 2 Table 19. (Continued), Energy Ri R2 cos(6) M R-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1781 1.998 -0.404 -76.33332148 -76.34414598 -76.34689652 -76.35484575 -76.35517748 -76.35553811 -76.35323744 1.781 1.998 -0.566 -76.32559379 -76.33659662 -76.33944520 -76.34749657 -76.34783584 -76.34820496 -76.34587139 1.781 1.998 -0.749 -76.31100122 -76.32210547 -76.32503627 -76.33327967 -76.33363303 -76.33401805 -76.33162269 1.781 1.998 -0.983 -76.28210075 -76.29227847 -76.29457239 -76.30315528 -76.30353270 -76.30394483 -76.30144109 1.781 2.101 0.111 -76.32161594 -76.33112435 -76.33331325 -76.34117142 -76.34149664 -76.34184994 -76.33957838 1.781 2.101 0.314 -76.30686361 -76.31574758 -76.31759128 -76,32540652 -76.32572765 -76.32607631 -76.32381951 1.781 2.101 0.584 -76.26669592 -76.27477086 -76.27618074 -76,28394609 -76.28426210 -76.28460493 -76.28236565 1.781 2.101 -0.071 -76.32769450 -76.33765773 -76.34009717 -76.34799907 -76.34832810 -76.34868574 -76.34639949 1.781 2.101 -0.249 -76.32816439 -76.33848415 -76.34111886 -76.34907658 -76.34941012 -76.34977284 -76.34746823 1.781 2.101 -0.428 -76.32348667 -76.33408689 -76.33687759 -76.34491564 -76.34525532 -76.34562497 -76.34329427 1.781 2.101 -0.620 -76.31244904 -76.32326597 -76.32618962 -76.33436290 -76.33471266 -76.33509369 -76.33271932 1.781 2.101 -0.849 -76.28950434 -76.30040877 -76.30345183 -76.31195218 -76.31232637 -76.31273501 -76.31025494 1.781 2.220 0.197 -76.30675459 -76.31570370 -76.31773749 -76.32565907 -76.32598991 -76.32634958 -76.32405665 1.781 2.220 0.466 -76.28096361 -76.28911394 -76.29069203 -76.29856652 -76.29889240 -76.29924641 -76.29697016 1.781 2.220 -0.029 -76.31579054 -76.32531615 -76.32768447 -76.33565734 -76.33599287 -76.33635789 -76.33404751 1.781 2.220 -0.249 -76.31626786 -76.32623833 -76.32886765 -76.33690982 -76.33725106 -76.33762254 -76.33528923 1.781 2.220 -0.470 -76.30904909 -76.31935567 -76.32220604 -76.33035476 -76.33070423 -76.33108503 -76.32871701 1.781 2.220 -0.718 -76.29072172 -76.30131348 -76.30435209 -76.31271793 -76.31308378 -76.31348310 -76.31104475 1.781 2.367 0.051 -76.29872599 -76.30760880 -76.30984151 -76.31788763 -76.31822989 -76.31860257 -76.31626724 1.781 2.367 0.365 -76.28069370 -76.28878855 -76.29054811 -76.29853818 -76.29887473 -76.29924087 -76.29692521 1.781 2.367 -0.249 -76.29993856 -76.30941924 -76.31202574 -76.32016497 -76.32051511 -76.32089672 -76.31853037 1.781 2.367 -0.550 -76.28715223 -76.29708856 -76.30000706 -76.30831291 -76.30867609 -76.30907248 -76.30665167 1.781 2.367 -0.926 -76.24455958 -76.25477194 -76.25800251 -76.26690752 -76.26731679 -76.26776550 -76.26514957 1.781 2.568 0.228 -76.27113160 -76.27897943 -76.28097740 -76.28909600 -76.28944488 -76.28982509 -76.28746511 1.781 2.568 -0.249 -76.27693539 -76.28565239 -76.28821615 -76.29646571 -76.29682616 -76.29721955 -76.29481541 1.781 2.568 -0.727 -76.24741762 -76.25678515 -76.25988925 -76.26849981 -76.26888884 -76.26931468 -76.26679218 1.811 1.811 0.029 -76.33836786 -76.34861572 -76.35091876 -76.35860845 -76.35891852 -76.35925464 -76.35704009 1.811 1.811 0.181 -76.32876127 -76.33858813 -76.34067478 -76.34833115 -76.34863832 -76.34897118 -76.34676779 1.811 1.811 0.362 -76.30967994 -76.31888071 -76.32063079 -76.32824548 -76.32854890 -76.32887749 -76.32668821 1.811 1.811 0.617 -76.25792556 -76.26632471 -76.26760860 -76.27516014 -76.27545794 -76.27578019 -76.27361220 S Table 19. (Continued), Energy Ri R i cos(S) MR-PT2 MR-PT3 MR-CISD D Vl DV2 DV3 Pople 1.811 1.811 -0.112 -76.34310126 -76.35366043 -76.35611708 -76.36384111 -76.36415398 -76.36449327 -76.36226739 1.811 1.811 -0.174 -76.34403622 -76.35471189 -76.35722433 -76.36496591 -76.36528016 -76.36562099 -76.36338938 1.811 1.811 -0.249 -76.34427908 -76.35508098 -76.35765268 -76.36541855 -76.36573464 -76.36607756 -76.36383807 1.811 1.811 -0.309 -76.34378813 -76.35468024 -76.35729347 -76.36508182 -76.36539960 -76.36574441 -76.36349766 1.811 1.811 -0.387 -76.34224750 -76.35324351 -76.35590369 -76.36372685 -76.36404720 -76.36439489 -76.36213694 1.811 1.811 -0.527 -76.33689395 -76.34804055 -76.35076775 -75.35867182 -76.35899802 • -76.35935231 -76.35706841 1.811 1.811 -0.679 -76.32711741 -76.33837518 -76.34115458 -76.34918913 -76.34952471 -76.34988954 -76.34756388 1.811 1.811 -0.856 -76.30982532 -76.32112645 -76.32394162 -76.33224723 -76.33260251 -76.33298954 -76.33057688 1.811 1.841 -0.174 -76.34386696 -76.35454868 -76.35708200 -76.36484625 -76.36516255 -76.36550572 -76.36326642 1.811 1.841 -0.249 -76.34398526 -76.35483226 -76.35743482 -76.36522357 -76.36554175 -76.36588704 -76.36363978 1.811 1.841 -0.309 -76.34340240 -76.35436860 -76.35701928 -76.36483072 -76.36515060 -76.36549780 -76.36324321 1.811 1.903 0.036 -76.33712648 -76.34724201 -76.34955529 -76.35731100 -76.35762703 -76.35796990 -76.35573299 1.811 1.903 0.192 -76.32773619 -76.33739646 -76.33947510 -76.34719621 -76.34750917 -76.34784858 -76.34562332 1.811 1.903 0.376 -76.30885932 -76.31787327 -76.31961326 -76.32729262 -76.32760172 -76.32793674 -76.32572572 1.811 1.903 0.634 -76.25684952 -76.26508801 -76.26643089 -76.27405136 -76.27435501 -76.27468387 -76.27249292 1.811 1.903 -0,108 -76.34151194 -76.35197750 -76.35446401 -76.36225543 -76.36257443 -76.36292066 -76.36067191 1.811 1.903 -0.249 -76.34227387 -76.35302342 -76.35564535 -76.36348080 -76.36380324 -76.36415335 -76.36189023 1.811 1.903 -0.391 -76.33971395 -76.35067960 -76.35340582 -76.36130126 -76.36162820 -76.36198338 -76.35970087 1.811 1.903 -0.537 -76.33358664 -76.34469333 -76.34749676 -76.35548023 -76.35581363 -76.35617609 -76.35386526 1.811 1.903 -0.697 -76.32246249 -76.33365278 -76.33651815 -76.34464874 -76.34499285 -76.34536738 -76.34300931 1.811 1.903 -0.889 -76.30189176 -76.31307752 -76.31599777 -76.32447692 -76.32484683 -76.32525049 -76.32278003 1.811 1.997 0.061 -76.33183003 -76.34168354 -76.34395977 -76.35177755 -76.35209920 -76.35244846 -76.35019050 1.811 1.997 0.231 -76.32126754 -76.33061452 -76.33261800 -76.34039828 -75.34071651 -76.34106188 -76.33881667 1.811 1.997 0.437 -76.29838775 -76.30701776 -76.30863772 -76.31637302 -76.31668692 -76.31702737 -76.31479766 1.811 1.997 0.751 -76.21169011 -76.21994889 -76.22125561 -76.22892530 -76.22923305 -76.22956654 -76.22735927 1.811 1.997 -0.095 -76.33651309 -76.34674783 -76.34922555 -76.35708201 -76.35740695 -76.35775993 -76.35548908 1.811 1.997 -0.249 -76.33702218 -76.34756351 -76.35020060 -76.35810573 -76.35843455 -76.35879190 -76.35650509 1.811 1.997 -0.403 -76.33365511 -76.34443778 -76.34720120 -76.35517336 -76.35550726 -76.35587035 -76.35356181 1.811 1.997 -0.565 -76.32583502 -76.33680131 -76.33966860 -76.34774413 -76.34808569 -76.34845742 -76.34611555 1.811 1.997 -0.747 -76.31121409 -76.32229448 -76.32525463 -76.33352443 -76.33388033 -76.33426823 -76.33186377 o Os Table 19. (Continued), Energy Ri R2 cos(0) MR-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1.811 1.997 -0.980 -76.28214507 -76.29234829 -76.29472815 -76.30340840 -76.30379403 -76.30421552 -76.30167934 1.811 2.100 0.108 -76.32254467 -76.33201532 -76.33420757 -76.34208749 -76.34241476 -76.34277038 -76.34049137 1.811 2.100 0.308 -76.30848603 -76.31735008 -76.31920032 -76.32703791 -76.32736113 -76.32771215 -76.32544771 1.811 2.100 0.569 -76.27154934 -76.27963383 -76.28105579 -76.28884611 -76.28916440 -76.28950980 -76.28726205 1.811 2.100 -0.072 -76.32831389 -76.33823289 -76.34067681 -76.34860059 -76.34893170 -76.34929169 -76.34699793 1.811 2.100 -0.249 -76.32862768 -76.33890330 -76.34154629 -76.34952649 -76.34986217 -76.35022733 -76.34791499 1.811 2.100 -0.427 -76.32386903 -76.33442808 -76.33723196 -76.34529316 -76.34563506 -76.34600724 -76.34366856 1.811 2.100 -0.617 -76.31288310 -76.32366564 -76.32660927 -76.33480585 -76.33515789 -76.33554153 -76.33315908 1.811 2.100 -0.845 -76.28997172 -76.30088074 -76.30397145 -76.31250122 -76.31287831 -76.31329029 -76.31079998 1.811 2.218 0.191 -76.30807514 -76.31699968 -76.31903945 -76.32698225 -76.32731512 -76.32767711 -76.32537687 1.811 2.218 0.454 -76.28391461 -76.29207018 -76.29366117 -76.30155819 -76.30188620 -76.30224264 -76.29995865 1.811 2.218 -0.031 -76.31658646 -76.32607319 -76.32844562 -76.33643952 -76.33677708 -76.33714441 -76.33482677 1.811 2.218 -0.249 -76.31688038 -76.32680937 -76.32944619 -76.33750954 -76.33785284 -76.33822668 -76.33588604 1.811 2.218 -0.468 -76.30963946 -76.31991673 -76.32276983 -76.33094062 -76.33129227 -76.33167555 -76.32929986 1.811 2.218 -0.713 -76.29154588 -76.30211067 -76.30517253 -76.31356010 -76.31392820 -76.31433008 -76.31188407 1.811 2.364 0.047 -76.29977468 -76.30862594 -76.31086216 -76.31892855 -76.31927282 -76.31964778 -76.31730540 1.811 2.364 0.354 -76.28289942 -76.29098921 -76.29275745 -76.30076824 -76.30110682 -76.30147529 -76.29915244 1.811 2.364 -0.249 -76.30071766 -76.31015934 -76.31277213 -76.32093204 -76.32128424 -76.32166821 -76.31929465 1.811 2.364 -0.546 -76.28806971 -76.29796784 -76.30089834 -76.30922449 -76.30958976 -76.30998854 -76.30756058 1.811 2.364 -0.914 -76.24683641 -76.25713520 -76.26046953 -76.26939602 -76.26980795 -76.27025974 -76.26763577 1.811 2.561 0.216 -76.27316546 -76.28100966 -76.28301175 -76.29114912 -76.29149990 -76.29188229 -76.28951573 1.811 2.561 -0.249 -76.27821711 -76.28690877 -76.28947677 -76.29774491 -76.29810727 -76.29850285 -76.29609217 1.811 2.561 -0.715 -76.24986629 -76.25920271 -76.26231236 -76.27093084 -76.27132107 -76.27174830 -76.26922261 1.841 1.841 -0.174 -76.34384532 -76.35441734 -76.35694328 -76.36473022 -76.36504859 -76.36539411 -76.36314710 1.841 1.841 -0.249 -76.34393294 -76.35463378 -76.35722293 -76.36503461 -76.36535489 -76.36570255 -76.36344750 1.841 1.841 -0.309 -76.34332542 -76.35411889 -76.35675281 -76.36458737 -76.36490938 -76.36525898 -76.36299653 1.903 1.903 0.044 -76.33600942 -76.34587157 -76.34815923 -76.35598029 -76.35630231 -76.35665199 -76.35439285 1.903 1.903 0.203 -76.32680365 -76.33622641 -76.33826321 -76.34604864 -76.34636744 -76.34671346 -76.34446639 1.903 1.903 0.390 -76.30805136 -76.31686789 -76.31854298 -76.32628709 -76.32660195 -76.32694349 -76.32471064 1.903 1.903 0.651 -76.25553164 -76.26375072 -76.26500417 -76.27269461 -76.27300426 -76.27333990 -76.27112558 3 Table 19. (Continued), Energy R i Ri cos(6) MR-PT2 MR-PT3 MR-CISD DVl DV2 DV3 Pople 1.903 1.903 -0.103 -76.34016254 -76.35035932 -76.35283173 -76.36069006 -76.36101523 -76.36136847 -76.35909693 1.903 1.903 -0.249 -76.34060618 -76.35107619 -76.35369549 -76.36160069 -76.36192959 -76.36228703 -76.36000012 1.903 1.903 -0.395 -76.33754946 -76.34824191 -76.35097916 -76.35894808 -76.35928181 -76.35964472 -76.35733715 1.903 1.903 -0.547 -76.33057224 -76.34144849 -76.34428571 -76.35234986 -76.35269066 -76.35306155 -76.35072326 1.903 1.903 -0.716 -76.31781793 -76.32884438 -76.33177848 -76.34001227 -76.34036561 -76.34075063 -76.33835767 1.903 1.903 -0.924 -76.29309641 -76.30421884 -76.30729731 -76.31601071 -76.31640042 -76.31682662 -76.31427807 1.903 2.000 0.071 -76.33045445 -76.34008785 -76.34234177 -76.35022577 -76.35055359 -76.35090984 -76.34862927 1.903 2.000 0.245 -76.31995856 -76.32908530 -76.33104733 -76.33889284 -76.33921705 -76.33956922 -76.33730181 1.903 2.000 0.455 -76.29689728 -76.30534182 -76.30691569 -76.31471810 -76.31503799 -76.31538523 -76.31313286 1.903 2.000 -0.090 -76.33494513 -76.34497662 -76.34745354 -76.35537850 -76.35570985 -76.35607013 -76.35377583 1.903 2.000 -0.249 -76.33507599 -76.34543793 -76.34809570 -76.35607269 -76.35640824 -76.35677327 -76.35446185 1.903 2.000 -0.409 -76.33110262 -76.34172554 -76.34452980 -76.35257935 -76.35292049 -76.35329182 -76.35095679 1.903 2.000 -0.577 -76.32234306 -76.33315606 -76.33608262 -76.34424570 -76.34459537 -76.34497634 -76.34260458 1.903 2.000 -0.771 -76.30546606 -76.31641968 -76.31947663 -76.32787409 -76.32824136 -76.32864222 -76.32619462 1.903 2.106 0.121 -76.32080280 -76.33004944 -76.33220977 -76.34015611 -76.34048964 -76.34085239 -76.33855060 1.903 2.106 0.328 -76.30642601 -76.31506353 -76.31685844 -76.32476206 -76.32509136 -76.32544930 -76.32316236 1.903 2.106 0.599 -76.26720481 -76.27516018 -76.27654097 -76.28440399 -76.28472866 -76.28508128 -76.28280909 1.903 2.106 -0.065 -76.32647000 -76.33618018 -76.33861707 -76.34660979 -76.34694746 -76.34731493 -76.34499746 1.903 2.106 -0.249 -76.32641176 -76.33649977 -76.33916114 -76.34721486 -76.34755754 -76.34793068 -76.34559305 1.903 2.106 -0.434 -76.32094220 -76.33133530 -76.33418246 -76.34232448 -76.34267404 -76.34305497 -76.34068851 1.903 2.106 -0.635 -76.30840284 -76.31904900 -76.32206876 -76.33036513 -76.33072642 -76.33112061 -76.32870407 1.903 2.106 -0.882 -76.28092065 -76.29176919 -76.29503562 -76.30378679 -76.30418312 -76.30461705 -76.30205238 1.903 2.229 0.214 -76.30531252 -76.31397878 -76.31596055 -76.32396981 -76.32430900 -76.32467819 -76.32235509 1.903 2.229 0.491 -76.27908777 -76.28701404 -76.28853102 -76.29649958 -76.29683403 -76.29719778 •76.29488957 1.903 2.229 -0.019 -76.31418142 -76.32343152 -76.32578143 -76.33384435 -76.33418859 -76.33456354 -76.33222204 1.903 2.229 -0.249 -76.31413470 -76.32385576 -76.32650625 -76.33464547 -76.33499611 -76.33537834 -76.33301146 1.903 2.229 -0.480 -76.30579914 -76.31589820 -76.31880103 -76.32706046 -76.32742059 -76.32781355 -76.32540732 1.903 2.229 -0.743 -76.28460698 -76.29504906 -76.29822241 -76.30674651 -76.30712706 -76.30754316 -76.30505070 1.903 2.383 0.069 -76.29630998 -76.30487362 -76.30706038 -76.31519506 -76.31554605 -76.31592870 -76.31356259 1.903 2.383 0.400 -76.27716773 -76.28495473 -76.28662671 -76.29470935 -76.29505459 -76.29543062 -76.29308325 g Table 19. (Continued), Eneigy Ri Ra c o s (6 ) MR-PT2 MR-PT3 MR-CISD D V l DV2 DV3 Pople 1.903 2.383 -0.249 -76.29709720 -76.30629187 -76.30891004 -76.31714838 -76.31750832 -76.31790115 -76.31550031 1.903 2.383 -0.568 -76.28235181 -76.29204573 -76.29504056 -76.30347186 -76.30384719 -76.30425750 -76.30179327 1.903 2.383 -0.979 -76.22848654 -76.23851913 -76.24192834 -76.25136384 -76.25181956 -76.25232152 -76.24952684 1.903 2.59S 0.270 -76.26662685 -76.27408455 -76.27599046 -76.28419736 -76.28455494 -76.28494509 -76.28255442 1.903 2.595 -0.249 -76.27310480 -76.28146704 -76.28402864 -76.29237589 -76.29274627 -76.29315105 -76.29071263 1.903 2.595 -0.768 -76.23783912 -76.24692497 -76.25016863 -76.25896813 -76.25937505 -76.25982144 -76.25723391 1.997 1.997 0.098 -76.32541725 -76.33473304 -76.33692338 -76.34486508 -76.34519828 -76.34556067 -76.34326036 1.997 1.997 0.288 -76.31332487 -76.32211116 -76.32396417 -76.33186498 -76.33219422 -76.33255208 -76.33026591 1.997 1.997 0.526 -76.28406150 -76.29220702 -76.29361304 -76.30147202 -76.30179673 -76.30214942 -76.29987819 1.997 1.997 -0.076 -76.33035886 -76.34008568 -76.34252890 -76.35051501 -76.35085214 -76.35121899 -76.34890371 1.997 1.997 -0.249 -76.33027773 -76.34034277 -76.34299044 -76.35103399 -76.35137582 -76.35174800 -76.34941374 1.997 1.997 -0.422 -76.32542674 -76.33576883 -76.33858497 -76.34671070 -76.34705894 -76.34743835 -76.34507728 1.997 1.997 -0.609 -76.31449091 -76.32507476 -76.32804818 -76.33631110 -76.33666975 -76.33706096 -76.33465537 1.997 1.997 -0.831 -76.29211666 -76.30292801 -76.30611128 -76.31471509 -76.31509983 -76.31552060 -76.31300417 1.997 2.113 0.160 -76.31440313 -76.32331799 -76.32539278 -76.33339981 -76.33373907 -76.33410834 -76.33178573 1.997 2.113 0.392 -76.29663807 -76.30490097 -76.30655873 -76.31452271 -76.31485740 -76.31522146 -76.31291413 1.997 2.113 0.719 -76.23388420 -76.24193082 -76.24323505 -76.25116996 -76.25150048 -76.25185972 -76.24956383 1.997 2.113 -0.045 -76.32101035 -76.33043805 -76.33283778 -76.34089546 -76.34123938 -76.34161396 -76.33927409 1.997 2.113 -0.249 -76.32072889 -76.33058478 -76.33325395 -76.34138060 -76.34173028 -76.34211140 -76.33974862 1.997 2.113 -0.453 -76.31394057 -76.32413788 -76.32703130 -76.33526135 -76.33561919 -76.33600956 -76.33361291 1.997 2.113 -0.680 -76.29790192 -76.30838470 -76.31149906 -76.31992869 -76.32030191 -76.32070971 -76.31824807 1.997 2.113 -0.980 -76.25611676 -76.26690778 -76.27055215 -76.28005635 -76.28051670 -76.28102391 -76.27820751 1.997 2.249 0.013 -76.30716170 -76.31606568 -76.31834882 -76.32647945 -76.32683031 -76.32721281 -76.32484784 1.997 2.249 0.280 -76.29564944 -76.30389561 -76.30573616 -76.31380984 -76.31415504 -76.31453106 -76.31218594 1.997 2.249 0.617 -76.25541911 -76.26306053 -76.26443986 -76.27249247 -76.27283375 -76.27320524 -76.27086931 1.997 2.249 -0.249 -76.30705956 -76.31650156 -76.31915248 -76.32737214 -76.32773062 -76.32812180 -76.32572706 1.997 2.249 -0.512 -76.29633499 -76.30621066 -76.30917691 -76.31754476 -76.31791509 -76.31831971 -76.31587627 1.997 2.249 -0.826 -76.26596616 -76.27629792 -76.27967838 -76.28846977 -76.28887345 -76.28931598 -76.28673381 1.997 2.424 0.136 -76.28583285 -76.29390793 -76.29596150 -76.30416448 -76.30452221 -76.30491255 -76.30252270 1.997 2.424 0.549 -76.25420557 -76.26152026 -76.26298939 -76.27117001 -76.27152328 -76.27190844 -76.26952867 Table 19. (Continued), Energy Ri R î cos{fl) MR-PT2 MR-PT3 MR-CISD DVl DV2 DV3 Pople 1.997 2.424 -0.249 -76.28763956 -76.29646138 -76.29906539 -76.30739227 -76.30776099 -76.30816388 -76.30573219 1.997 2.424 -0.635 -76.26679029 -76.27620951 -76.27931894 -76.28791456 -76.28830503 -76.28873266 -76.28621235 1.997 2.681 0.449 -76.24313161 -76.24987955 -76.25156810 -76.25988090 -76.26024769 -76.26064834 -76.25822211 1.997 2.681 -0.249 -76.25907299 -76.26683506 -76.26936541 -76.27780456 -76.27818444 -76.27860012 -76.27612929 2.100 2.100 0.216 -76.30559586 -76.31411638 -76.31606528 -76.32412097 -76.32446476 -76.32483920 -76.32249995 2.100 2.100 0.492 -76.28021290 -76.28808244 -76.28953518 -76.29755298 -76.29789211 -76.29826120 -76.29593606 2.100 2.100 -0.018 -76.31415744 -76.32324068 -76.32557682 -76.33368735 -76.33403638 -76.33441681 -76.33205867 2.100 2.100 -0.249 -76.31384110 -76.32338723 -76.32603844 -76.33422866 -76.33458446 -76.33497257 -76.33258784 2.100 2.100 -0.481 -76.30519912 -76.31512354 -76.31804120 -76.32635633 -76.32672209 -76.32712150 -76.32469575 2.100 2.100 -0.748 -76.28313614 -76.29342958 -76.29664927 -76.30524988 -76.30563792 -76.30606263 -76.30354359 2.100 2.254 0.061 -76.29820971 -76.30672622 -76.30890596 -76.31709359 -76.31745008 -76.31783903 -76.31545423 2.100 2.254 0.381 -76.28108629 -76.28885833 -76.29050122 -76.29863468 -76.29898514 -76.29936715 -76.29700189 2.100 2.254 -0.249 -76.29848710 -76.30763869 -76.31027986 -76.31857332 -76.31893906 -76.31933855 -76.31691816 2.100 2.254 -0.560 -76.28401878 -76.29367839 -76.29672387 -76.30521037 -76.30559171 -76.30600895 -76.30352492 2.100 2.254 -0.956 -76.23310377 -76.24355191 -76.24751627 -76.25708744 -76.25755841 -76.25807814 -76.25523418 2.100 2.461 0.245 -76.27060031 -76.27808665 -76.27993453 -76.28820243 -76.28856637 -76.28896384 -76.28655159 2.100 2.461 -0.249 -76.27559355 -76.28399115 -76.28656539 -76.29497880 -76.29535624 -76.29576915 -76.29330715 2.100 2.461 -0.744 -76.24269181 -76.25185116 -76.25516566 -76.26401142 -76.26442433 -76.26487767 -76.26227256 2.100 2.791 -0.249 -76.23996398 -76.24693751 -76.24942166 -76.25794668 -76.25833585 -76.25876225 -76.25626067 2.218 2.218 0.112 -76.28986172 -76.29800520 -76.30007422 -76.30829987 -76.30866010 -76.30905333 -76.30665532 2.218 2.218 0.493 -76.26417828 -76.27159802 -76.27304536 -76.28123496 -76.28158984 -76.28197688 -76.27959324 2.218 2.218 -0.249 -76.29097636 -76.29983110 -76.30244749 -76.31079492 -76.31116601 -76.31157163 -76.30913244 2.218 2.218 -0.610 -76.27220809 -76.28163734 -76.28474217 -76.29333524 -76.29372633 -76.29415472 -76.29163440 2.218 2.459 0.373 -76.25364635 -76.26063682 -76.26226475 -76.27059884 -76.27096856 -76.27137259 -76.26893811 2.218 2.459 -0.249 -76.26468205 -76.27271986 -76.27525935 -76.28374267 -76.28412730 -76.28454846 -76.28206184 2.364 2.364 0.443 -76.24425657 -76.25105799 -76.25257092 -76.26094818 -76.26132136 -76.26172935 -76.25928065 2.364 2.364 -0.249 -76.25975803 -76.26761493 -76.27013411 -76.27864970 -76.27903768 -76.27946271 -76.27696468 I l l Table 20. Dipole moments from the MR-SDCI wavefunction. The dipole moment data corresponds to the internal coordinates exactly as given to machine precision. The y-z coordinate frame has the HOH angle bisected by the z axis, and the tj-C coordinates are the Eckart coordinates. The angle of rotation between the z axis and the ( axis is given in radians by See Figure 10 and text in Results section of Chapter IE. R i/O o R z/O o cos(0) H y lD I tz I D Im I/d ij)/Tad. P q / D 1 .4 5 6 1 .5 8 4 - 0 .7 3 5 0 .1 0 3 9 3 0 -1.201534 1.206020 0.010003 0.115944 -1.200434 1 .4 7 1 1 .4 7 1 - 0 .2 4 9 0 .0 0 0 0 0 0 -1.740253 1.740253 0.000000 0.000000 -1.740253 1.471 1.607 0.155 0.025916 -2.050017 2.050181 -0.011630 0.002073 -2.050180 1 .4 7 1 1 .6 0 7 - 0 .2 4 9 0 . 0 6 0 6 7 7 -1.764946 1.765988 -0.002377 0.056481 -1.765085 1 .4 7 1 1 .8 1 1 0 .0 5 7 0 . 0 7 8 6 1 7 - 2 .0 2 1 3 2 3 2.028847 -0.021870 0.034263 -2.028557 1.471 1.811 0.448 0 . 0 0 3 9 7 9 - 2 .2 7 6 1 6 6 2.276169 -0.045222 -0.098922 - 2 .2 7 4 0 1 9 1 .4 7 1 1 .8 1 1 - 0 .2 4 9 0.147681 -1.802557 1.808596 -0.005574 0.137632 -1.803352 1.471 1.811 -0.556 0 . 2 2 8 2 0 7 -1J06085 1523276 0.011987 0.246244 -1.503241 1.471 1.811 -0.890 0.366847 -0.846130 0.922232 0.040052 0.400433 -0.830762 1.471 2.016 0.117 0.094274 -2.101762 2.103876 -0.037933 0.014498 -2.103825 1 .4 7 1 2 .0 1 6 - 0 .2 4 9 0.225130 -1.836413 1 .8 5 0 1 6 1 - 0 .0 0 8 4 0 9 0.209680 -1.838241 1 .4 7 1 2 .0 1 6 -0.616 0.394838 -1.456121 1.508703 0.023977 0.429635 -1.446236 1 .4 7 1 2.260 0.353 0.007973 - 2 .2 6 9 2 8 6 2.269300 -0.079422 -0.172095 -2.262765 1 .4 7 1 2 . 2 6 0 - 0 .2 4 9 0.294718 -1.864250 1.887403 -0.011378 0 .2 7 3 4 8 9 - 1 .8 6 7 4 8 2 1 .5 8 7 1.587 -0.496 0.000000 -1.543165 1.543165 0.000000 0.000000 -1.543165 1 .5 8 7 1 .5 8 7 - 0 .7 4 9 0 .0 0 0 0 0 0 - 1 .1 7 2 4 9 9 1 .1 7 2 4 9 9 0.000000 0.000000 -1.172499 1 .5 8 7 1.690 -0.445 0 .0 6 0 6 7 4 -1.613499 1.614640 0 .0 0 1 6 0 5 0.063263 -1.613400 1 .5 8 7 1 .6 9 0 - 0 .6 4 4 0 .0 8 0 1 5 1 -1.363799 1.366152 0.005404 0.087520 - 1 .3 6 3 3 4 6 1 .5 8 7 1.690 -0.871 0 .1 0 1 9 5 7 -0.900218 0.905974 0 .0 1 1 4 3 9 0.112247 -0.898993 1 .5 9 1 1 .5 9 1 0 .2 7 6 0.000000 -2.147484 2.147484 0.000000 0.000000 -2.147484 1.591 1J91 -0.009 0.000000 -1.960973 1.960973 0.000000 0.000000 -1.960973 1.591 1.591 -0.249 0.000000 -1.781232 1.781232 0.000000 0.000000 -1.781232 1 .5 9 1 1.692 0.157 0.017206 -2.090851 2.090922 -0.008131 0.000204 - 2 .0 9 0 9 2 2 1 .5 9 1 1.692 0.442 0.001810 -2.270724 2.270725 -0.013315 -0.028424 - 2 .2 7 0 5 4 7 1 .5 9 1 1 .6 9 2 -0.056 0.030451 -1.946360 1 .9 4 6 5 9 8 - 0 .0 0 4 6 9 9 0.021305 -1.946482 1 .5 9 1 1 .6 9 2 -0.249 0.043859 - 1 .7 9 7 7 9 5 1.798330 -0.001655 0.040883 -1.797865 1 .5 9 1 1 .8 1 1 0 .1 2 3 0 .0 3 9 0 9 0 - 2 .0 9 0 0 0 2 2.090368 -0.015908 0.005838 -2.090359 1 .5 9 1 1 .8 1 1 0 .3 6 2 0.009817 -2.242124 2.242146 -0.024690 -0.045538 - 2 .2 4 1 6 8 3 1 .5 9 1 1 .8 1 1 -0.070 0.065673 -1.956815 1.957917 -0.009413 0.047251 - 1 .9 5 7 3 4 6 1 .5 9 1 1 .8 1 1 - 0 .2 4 9 0 .0 9 3 1 1 9 - 1 .8 1 6 9 2 8 1 .8 1 9 3 1 2 - 0 .0 0 3 4 7 9 0.086797 -1.817241 1 .5 9 1 1 .8 1 1 - 0 .4 2 9 0.124632 -1.648207 1.652913 0.002725 0.129123 -1.647861 1 .5 9 1 1.811 -0.612 0.163017 -1.425940 1.435228 0.009755 0.176918 -1.424282 1 .5 9 1 1 .8 1 1 - 0 .8 1 9 0 .2 1 2 9 5 4 - 1 .0 5 6 5 7 2 1.077819 0.020020 0.234062 - 1 .0 5 2 0 9 7 1 .5 9 1 1 .9 3 1 0 .1 4 4 0.051334 -2.123123 2 .1 2 3 7 4 4 - 0 .0 2 4 8 3 3 - 0 .0 0 1 3 9 9 - 2 .1 2 3 7 4 3 1 .5 9 1 1.931 0.395 0.004783 -2.280403 2.280408 -0.038831 -0.083748 -2.278870 1 .5 9 1 1.931 -0.059 0.094128 -1.983888 1.986120 -0.014595 0.065165 -1.985050 1 .5 9 1 1.931 -0.249 0 .1 3 9 0 0 4 -1.834709 1.839968 -0.005194 0.129473 -1.835406 1 .5 9 1 1 .9 3 1 -0.440 0.191367 -1.652724 1 .6 6 3 7 6 6 0 .0 0 4 6 5 9 0.199065 -1.651815 112 Table 20. (Continued), R i/f lo Rz/a, cos(fl) H y lD f i / D | p |/ D V>/rad. P q / D P(/D 1 J 9 1 1 .9 3 1 - 0 .6 3 8 0.258466 -1.402063 1 .4 2 5 6 8 8 0 . 0 1 6 2 0 9 0 .2 8 1 1 5 8 - 1 .3 9 7 6 8 9 1 3 9 1 1 .9 3 1 - 0 .8 7 1 0.364159 -0.923039 0 . 9 9 2 2 7 7 0 .0 3 5 1 1 9 0 . 3 9 6 3 4 4 - 0 .9 0 9 6 8 4 1 .5 9 1 2.058 0.213 0.044319 - 2 .1 8 4 0 1 8 2 . 1 8 4 4 6 8 -0.037741 -0.038120 - 2 .1 8 4 1 3 5 1 .5 9 1 2 .0 5 8 0 .5 2 9 -0.028704 -2.373085 2 .3 7 3 2 5 9 -0.063051 -0.178174 - 2 .3 6 6 5 6 1 1 3 9 1 2.058 -0.026 0.111091 - 2 .0 2 4 9 0 0 2 .0 2 7 9 4 5 -0.021514 0.067504 - 2 .0 2 6 8 2 1 1 3 9 1 2 .0 5 8 - 0 .2 4 9 0.182046 -1.850940 1 .8 5 9 8 7 1 -0.006886 0.169297 - 1 .8 5 2 1 5 0 1 .5 9 1 2.058 -0.473 0.267580 -1.630764 1.652571 0.008558 0.281526 - 1 .6 2 8 4 1 4 1 .5 9 1 2 .0 5 8 - 0 .7 1 4 0 .3 9 1 1 2 2 -1.288805 1.346846 0.028346 0.427492 - 1 .2 7 7 2 0 2 1 3 9 1 2 .2 0 6 0 . 0 4 4 0.105553 -2.088280 2 . 0 9 0 9 4 6 -0.033088 0.036411 - 2 .0 9 0 6 2 9 1 3 9 1 2 . 2 0 6 0 .3 6 8 -0.004502 -2.289342 2 .2 8 9 3 4 6 -0.062370 -0.147186 - 2 .2 8 4 6 1 0 1 3 9 1 2 .2 0 6 - 0 .2 4 9 0.222815 -1.864709 1.877974 -0.008714 0.206557 - 1 .8 6 6 5 8 0 1 3 9 1 2.206 -0.543 0.373968 -1.557694 1.601956 0.017465 0.401115 - 1 3 5 0 9 2 5 1 .5 9 1 2 .2 0 6 - 0 .8 9 2 0.684597 -0.874263 1.110410 0.062981 0.738265 - 0 .8 2 9 4 4 2 1391 2.402 0.209 0.038094 -2.200048 2 .2 0 0 3 7 8 -0.059402 -0.092584 - 2 .1 9 8 4 2 9 1 3 9 1 2 .4 0 2 - 0 .2 4 9 0 .2 5 6 4 5 2 - 1 .8 7 0 9 9 1 1.888485 -0.010927 0.235992 - 1 .8 7 3 6 8 2 1 .5 9 1 2 .4 0 2 - 0 .7 0 7 0J92876 -1.311927 1.439671 0.043910 0.649892 -1.284638 1 .5 9 1 2 .7 0 7 - 0 .2 4 9 0 .2 4 7 6 4 4 -1.844091 1.860645 - 0 .0 1 3 9 7 0 0.221859 -1.847370 1 .6 8 9 1 .6 8 9 0 . 0 7 7 0.000000 -2.054580 2.054580 0.000000 0.000000 - 2 .0 5 4 5 8 0 1 .6 8 9 1 .6 8 9 0 . 2 7 4 0 .0 0 0 0 0 0 - 2 .1 8 2 8 7 1 2.182871 0.000000 0.000000 - 2 .1 8 2 8 7 1 1 .6 8 9 1 .6 8 9 0.554 0.000000 -2.353377 2.353377 0.000000 0.000000 - 2 .3 5 3 3 7 7 1 .6 8 9 1 .6 8 9 - 0 .0 9 1 0.000000 -1.936253 1.936253 0.000000 0.000000 - 1 .9 3 6 2 5 3 1 .6 8 9 1 .6 8 9 - 0 .4 0 7 0.000000 -1.665500 1 .6 6 5 5 0 0 0.000000 0.000000 - 1 .6 6 5 5 0 0 1 .6 8 9 1 .6 8 9 - 0 .5 6 7 0.000000 -1.481909 1 .4 8 1 9 0 9 0.000000 0.000000 - 1 .4 8 1 9 0 9 1 .6 8 9 1 .6 8 9 - 0 .7 4 1 0.000000 -1.207655 1 .2 0 7 6 5 5 0.000000 0.000000 - 1 .2 0 7 6 5 5 1.689 1.689 -0.949 0.000000 -0.573652 0.573652 0.000000 0 .0 0 0 0 0 0 - 0 .5 7 3 6 5 2 1 .6 8 9 1 .7 1 3 0.067 0.005060 - 2 .0 5 1 8 9 7 2.051903 -0.001527 0 .0 0 1 9 2 8 - 2 .0 5 1 9 0 2 1 .6 8 9 1 .7 1 3 0 .2 5 5 0.002431 -2.174908 2 . 1 7 4 9 0 9 -0.002248 -0.002458 -2.174908 1 .6 8 9 1 .7 1 3 0.513 -0.000603 -2.333354 2.333354 -0.003398 -0.008533 - 2 .3 3 3 3 3 8 1 .6 8 9 1 .7 1 3 - 0 .0 9 5 0.007547 -1.937101 1 .9 3 7 1 1 6 -0.000937 0.005733 - 1 .9 3 7 1 0 7 1.689 1.713 -0.249 0.010154 -1.815096 1 .8 1 5 1 2 4 -0.000380 0.009465 - 1 .8 1 5 1 0 0 1 .6 8 9 1 .7 1 3 - 0 .4 0 3 0.013084 -1.672658 1 .6 7 2 7 0 9 0.000196 0.013412 - 1 .6 7 2 6 5 5 1.689 1.713 -0.559 0.016493 -1.494911 1.495001 0.000829 0 .0 1 7 7 3 3 - 1 .4 9 4 8 9 7 1 .6 8 9 1 .7 1 3 -0.728 0.021045 - 1 .2 3 4 5 4 0 1 .2 3 4 7 1 9 0.001638 0.023067 - 1 .2 3 4 5 0 4 1.689 1.713 -0.928 0.018304 -0.694933 0 .6 9 5 1 7 4 0.003108 0.020463 - 0 .6 9 4 8 7 3 1 .6 8 9 1 .8 1 1 0 .0 5 1 0.025493 -2.056942 2.057100 -0.007251 0.010578 - 2 .0 5 7 0 7 3 1 .6 8 9 1 .8 1 1 0 .2 2 3 0 .0 1 3 1 1 3 - 2 .1 7 0 7 2 9 2 . 1 7 0 7 6 9 -0.010478 -0.009633 - 2 .1 7 0 7 4 7 1 .6 8 9 1 .8 1 1 0.442 -0.000613 - 2 .3 0 6 3 6 1 2 .3 0 6 3 6 1 -0.015086 -0.035406 - 2 .3 0 6 0 8 9 1 .6 8 9 1 .8 1 1 - 0 .1 0 2 0.037538 -1.946946 1 .9 4 7 3 0 7 -0.004503 0.028770 - 1 .9 4 7 0 9 5 1 .6 8 9 1.811 -0.249 0 .0 5 0 2 4 1 - 1 .8 2 8 8 6 2 1.829552 -0.001875 0 . 0 4 6 8 1 1 - 1 .8 2 8 9 5 3 1 .6 8 9 1 .8 1 1 -0.397 0.064590 - 1 .6 9 1 1 2 0 1.692353 0.000856 0 .0 6 6 0 3 7 - 1 .6 9 1 0 6 4 1 .6 8 9 1 .8 1 1 -0.547 0.081406 - 1 .5 2 0 9 9 0 1.523167 0.003843 0 .0 8 7 2 5 1 - 1 3 2 0 6 6 6 1 .6 8 9 1.811 -0.709 0.103719 - 1 .2 7 8 2 1 3 1 .2 8 2 4 1 4 0.007592 0.113420 - 1 .2 7 7 3 8 9 1 .6 8 9 1 .8 1 1 - 0 .9 0 1 0.130836 -0.811267 0 . 8 2 1 7 4 9 0.013980 0.142165 - 0 .8 0 9 3 5 9 1 .6 8 9 1 .9 0 9 0 .0 6 2 0.041627 -2.079078 2.079494 -0.013070 0.014451 - 2 .0 7 9 4 4 4 113 Table 20. (Continued), R i/flo R tJOo c o s ( # ) f t y / D f t z I D |p | / D rl)/ta d . 1 .6 8 9 1 .9 0 9 0.238 0.019062 -2.194831 2.194914 -0.018894 - 0 .0 2 2 4 0 8 - 2 .1 9 4 7 9 9 1 .6 8 9 1 .9 0 9 0 .4 6 3 -0.005662 -2.331946 2.331953 -0.027318 -0.069357 -2.330921 1 .6 8 9 1 .9 0 9 -0.097 0.064014 -1.964687 1.965729 -0.008056 0.048185 - 1 .9 6 5 1 3 9 1 .6 8 9 1 .9 0 9 - 0 .2 4 9 0.087618 -1.841754 1.843837 -0.003290 0.081558 -1.842032 1 .6 8 9 1 .9 0 9 - 0 .4 0 2 0.114490 -1.697561 1.701418 0.001668 0.117322 -1.697368 1 .6 8 9 1 .9 0 9 -0J60 0.147096 -1314220 1.521348 0.007225 0.158032 -1513118 1 .6 8 9 1 .9 0 9 - 0 .7 3 3 0.192402 -1.242227 1 .2 5 7 0 3 9 0.014431 0.210308 - 1 .2 3 9 3 2 1 1 .6 8 9 1 .9 0 9 -0.946 0.266610 - 0 .6 1 7 8 9 1 0.672956 0.028834 0.284313 -0.609948 1 .6 8 9 2 .0 1 1 0.094 0.049994 -2.114014 2.114605 -0.020066 0.007566 -2.114592 1 .6 8 9 2 .0 1 1 0.292 0.014128 -2.241206 2.241251 -0.029580 -0.052164 - 2 .2 4 0 6 4 3 1 .6 8 9 2 .0 1 1 0.557 -0.023743 -2.394892 2.395010 -0.044721 -0.130786 - 2 .3 9 1 4 3 6 1 .6 8 9 2.011 -0.080 0.085028 -1.990340 1.992155 -0.012222 0.060696 - 1 .9 9 1 2 3 1 1 .6 8 9 2 .0 1 1 - 0 .2 4 9 0.122903 -1.853543 1.857614 -0.004682 0.114223 -1.854098 1 .6 8 9 2.011 -0.419 0.166731 -1.690086 1.698290 0.003186 0.172115 -1.689546 1 .6 8 9 2 .0 1 1 -0.596 0.222335 - 1 .4 7 4 1 7 1 1.490843 0.012235 0 . 2 4 0 3 5 4 - 1 .4 7 1 3 4 0 1 .6 8 9 2 .0 1 1 -0.800 0.310404 -1.107187 1.149876 0 . 0 2 5 4 0 4 0 . 3 3 8 4 2 7 - 1 .0 9 8 9 4 5 1 .6 8 9 2 .1 2 4 0 .1 5 8 0.043933 -2.167783 2.168228 -0.030206 -0.021557 -2.168121 1 .6 8 9 2 .1 2 4 0 .4 0 3 - 0 .0 1 1 5 3 9 -2.317126 2.317154 -0.046458 -0.119136 -2.314090 1 .6 8 9 2 .1 2 4 - 0 .0 4 9 0.097831 -2.024928 2 . 0 2 7 2 8 9 - 0 .0 1 7 8 3 2 0 . 0 6 1 7 0 9 - 2 .0 2 6 3 5 0 1 .6 8 9 2.124 -0.249 0.156699 -1.863908 1.870483 -0.006138 0.145255 -1.864835 1 .6 8 9 2 .1 2 4 -0.450 0.226853 -1.664913 1.680297 0.006144 0.237078 -1.663488 1 .6 8 9 2.124 -0.667 0.326179 -1.373234 1.411440 0.021407 0.355498 -1.365937 1 .6 8 9 2.124 -0.940 0.536505 -0.660532 0.850964 0.052513 0570436 -0.631461 1 .6 8 9 2 .2 5 8 0 .0 0 9 0.093721 -2.075727 2.077841 -0.026834 0.037994 -2.077494 1 .6 8 9 2 .2 5 8 0.280 0.008366 -2.251146 2.251162 -0.047972 -0.099594 - 2 .2 4 8 9 5 7 1 .6 8 9 2 .2 5 8 0 .6 4 6 -0.072484 -2.448887 2.449960 - 0 .0 8 4 4 0 9 -0.278688 -2.434057 1 .6 8 9 2 .2 5 8 - 0 .2 4 9 0.187970 -1.871374 1.880790 -0.007756 0.173450 -1.872776 1 .6 8 9 2 .2 5 8 -0.508 0.306564 -1.601672 1 .6 3 0 7 4 6 0.012552 0.326643 -1597698 1 .6 8 9 2 .2 5 8 - 0 .8 0 9 0.526646 -1.094431 1.214552 0.043251 0573474 -1.070637 1 .6 8 9 2.430 0.131 0.047349 -2.159663 2 .1 6 0 1 8 2 - 0 .0 4 4 9 9 9 -0.049848 -2.159607 1 .6 8 9 2 .4 3 0 0 .5 5 5 - 0 .0 9 5 1 0 3 - 2 .3 9 5 7 1 9 2.397606 -0.091974 -0.314734 - 2 .3 7 6 8 5 9 1 .6 8 9 2 .4 3 0 - 0 .2 4 9 0 .2 1 0 8 1 9 -1.870992 1.882831 - 0 .0 0 9 6 7 9 0 . 1 9 2 7 0 0 - 1 .8 7 2 9 4 5 1 .6 8 9 2 .4 3 0 -0.630 0.443090 -1.428190 1.495345 0.029245 0.484662 -1.414623 1 .6 8 9 2 .6 8 3 0.441 -0.120576 -2.305304 2.308455 - 0 .0 9 7 9 4 3 -0.345426 -2.282465 1 .6 8 9 2.683 -0.249 0.200626 -1.844942 1.855818 -0.012232 0.178044 -1.847258 1 .7 8 1 1.781 -0.249 0.000000 -1.836260 1.836260 0.000000 0.000000 -1.836260 1 .7 8 1 1 .8 1 1 0 . 0 3 0 0.006271 -2.055892 2.055901 -0.001646 0.002887 -2.055900 1 .7 8 1 1 .8 1 1 0 .1 8 5 0.003495 -2.160362 2.160365 -0.002335 - 0 .0 0 1 5 4 9 - 2 .1 6 0 3 6 4 1.781 1.811 0.369 0.000548 -2.275959 2 . 2 7 5 9 5 9 -0.003225 -0.006793 - 2 .2 7 5 9 4 9 1 .7 8 1 1 .8 1 1 0.634 -0.002350 -2.421455 2.421456 -0.004815 -0.014008 -2.421416 1 .7 8 1 1 .8 1 1 -0.112 0.009041 - 1 .9 5 1 7 3 9 1.951760 -0.001036 0.007018 -1.951747 1 .7 8 1 1 .8 1 1 -0.174 0.010333 -1.902770 1.902798 -0.000772 0.008865 -1.902777 1 .7 8 1 1 .8 1 1 - 0 .2 4 9 0.011974 -1.839885 1 .8 3 9 9 2 4 - 0 .0 0 0 4 4 9 0.011147 -1.839890 1 .7 8 1 1.811 -0.309 0 .0 1 3 3 6 2 -1.786129 1.786179 -0.000188 0.013026 -1.786131 114 Table 20. (Continued), R i / û o R 2/ 4 , c o s ( 0 ) l l y l D l i z I D MID i/tlra d . /‘ f j/D #*< /D 1 .7 8 1 1.811 -0.387 0.015287 -1.710696 1 .7 1 0 7 6 4 0 .0 0 0 1 6 0 0.015560 -1.710694 1.781 1.811 -0.528 0 .0 1 9 1 3 7 -1.552564 L552682 0.000826 0 .0 2 0 4 1 9 - 1 5 5 2 5 4 8 1 .7 8 1 1 .8 1 1 - 0 .6 8 0 0.024211 -1.334318 1.334538 0.001644 0 .0 2 6 4 0 4 - 1 .3 3 4 2 7 6 1 .7 8 1 1 .8 1 1 - 0 .8 5 6 0.032062 -0.949451 0 . 9 4 9 9 9 2 0 .0 0 2 9 0 0 0.034815 -0.949354 1 .7 8 1 1 .8 4 1 - 0 .1 7 4 0 . 0 2 0 4 0 7 -1.906528 1.906637 -0.001531 0 .0 1 7 4 8 9 - 1 .9 0 6 5 5 7 1 .7 8 1 1 .8 4 1 - 0 .2 4 9 0 .0 2 3 6 8 3 -1.843425 1.843577 -0.000891 0 . 0 2 2 0 4 0 - 1 .8 4 3 4 4 5 1 .7 8 1 1 .8 4 1 -0.309 0.026449 -1.789479 1.789675 - 0 .0 0 0 3 7 3 0 . 0 2 5 7 8 2 - 1 .7 8 9 4 8 9 1 .7 8 1 1 .9 0 3 0 .0 3 8 0 .0 2 3 2 8 1 - 2 .0 7 3 9 3 8 2 ,0 7 4 0 6 9 - 0 .0 0 6 6 6 4 0.009460 -2.074047 1 .7 8 1 1 .9 0 3 0.196 0.011915 - 2 .1 8 0 2 2 7 2.180260 -0.009458 -0.008707 -2.180242 1 .7 8 1 1 .9 0 3 0.383 0.000009 - 2 .2 9 6 4 7 8 2 .2 9 6 4 7 8 - 0 .0 1 3 0 7 7 - 0 .0 3 0 0 2 0 - 2 .2 9 6 2 8 2 1 .7 8 1 1 .9 0 3 0 .6 5 2 -0.010978 -2.437466 2 .4 3 7 4 9 1 - 0 .0 1 9 6 1 0 -0.058771 -2.436782 1 .7 8 1 1 .9 0 3 - 0 .1 0 7 0.034674 -1.967240 1.967545 -0.004194 0.026423 -1.967368 1 .7 8 1 1 .9 0 3 - 0 .2 4 9 0 . 0 4 6 9 3 6 - 1 .8 5 0 3 8 4 1 .8 5 0 9 7 9 - 0 .0 0 1 7 8 2 0 .0 4 3 6 3 9 - 1 .8 5 0 4 6 5 1 .7 8 1 1 .9 0 3 - 0 .3 9 2 0 .0 6 0 8 2 5 -1.714639 1.715718 0.000723 0.062065 -1.714595 1 .7 8 1 1 .9 0 3 -0.538 0.077352 -1J47592 1J49524 0.003472 0.082724 -1547314 1.781 1.903 -0.699 0.100081 -1.307703 1.311527 0.006969 0.109192 -1.306974 1 .7 8 1 1 .9 0 3 -0.890 0.138321 -0.844101 0.855359 0.012809 0 . 1 4 9 1 2 2 - 0 .8 4 2 2 6 0 1 .7 8 1 1 .9 9 8 0 .0 6 3 0.034867 -2.102611 2 . 1 0 2 9 0 0 - 0 .0 1 2 3 0 4 0.008994 -2.102881 1 .7 8 1 1 .9 9 8 0.236 0.013419 - 2 .2 1 7 0 4 1 2.217082 -0.017679 -0.025777 -2.216932 1 .7 8 1 1 .9 9 8 0 .4 4 6 -0.008761 -2.343200 2.343216 -0.025002 -0.067336 -2.342249 1 .7 8 1 1 .9 9 8 - 0 .0 9 4 0.056312 -1.987873 1 .9 8 8 6 7 1 - 0 .0 0 7 6 5 4 0 .0 4 1 0 9 6 - 1 .9 8 8 2 4 6 1.781 1.998 -0.249 0.079764 -1.859835 1.861544 - 0 .0 0 3 0 9 0 0.074018 -1.860073 1 .7 8 1 1 .9 9 8 -0.404 0.106503 -1.710140 1.713453 0.001629 0 .1 0 9 2 8 9 - 1 .7 0 9 9 6 4 1 .7 8 1 1 .9 9 8 -0.566 0.139967 -1.517380 1523821 0.006997 0.150580 -1.516364 1.781 1.998 -0.749 0.189962 -1.218238 1.232959 0 .0 1 4 2 7 7 0 . 2 0 7 3 3 4 - 1 .2 1 5 4 0 2 1.781 1.998 -0.983 0.287464 -0.355546 0.457218 0.032373 0.298821 -0.346055 1 .7 8 1 2 .1 0 1 0 .1 1 1 0 . 0 3 6 9 2 7 -2.145211 2.145528 - 0 .0 1 9 7 5 0 - 0 .0 0 5 4 4 6 - 2 .1 4 5 5 2 2 1 .7 8 1 2 .1 0 1 0 .3 1 4 0.001967 -2.274384 2.274385 -0.029084 -0.064173 -2.273479 1 .7 8 1 2.101 0.584 -0.031858 -2.422425 2.422634 - 0 .0 4 4 0 8 8 -0.138592 -2.418667 1 .7 8 1 2 .1 0 1 - 0 .0 7 1 0.072180 -2.014851 2 . 0 1 6 1 4 4 - 0 .0 1 1 9 5 6 0 .0 4 8 0 8 5 - 2 .0 1 5 5 7 0 1 .7 8 1 2 .1 0 1 - 0 .2 4 9 0.110856 -1.867912 1.871199 -0.004435 0 . 1 0 2 5 7 1 - 1 .8 6 8 3 8 5 1 .7 8 1 2.101 -0.428 0.156254 -1.690646 1.697852 0 .0 0 3 4 2 8 0 . 1 6 2 0 4 9 - 1 .6 9 0 1 0 0 1 .7 8 1 2.101 -0.620 0.217720 -1.445156 1.461464 0.012862 0 .2 3 6 2 8 9 - 1 .4 4 2 2 3 6 1 .7 8 1 2 .1 0 1 - 0 .8 4 9 0.329565 -0.981648 1.035493 0.027995 0 .3 5 6 9 1 3 - 0 .9 7 2 0 3 8 1 .7 8 1 2 . 2 2 0 0 .1 9 7 0.020965 -2.208771 2.208870 -0.031389 -0.048366 - 2 .2 0 8 3 4 1 1.781 2.220 0.466 - 0 .0 3 4 3 9 8 - 2 .3 6 6 1 7 5 2.366425 -0.049217 - 0 .1 5 0 7 6 6 - 2 .3 6 1 6 1 7 1 .7 8 1 2 . 2 2 0 - 0 .0 2 9 0 . 0 7 7 2 0 7 - 2 .0 5 3 7 5 9 2.055210 -0.018277 0.039660 -2.054827 1 .7 8 1 2 .2 2 0 - 0 .2 4 9 0 .1 3 9 8 4 1 -1.873630 1.878842 - 0 .0 0 5 9 0 3 0 . 1 2 8 7 7 8 - 1 .8 7 4 4 2 3 1 .7 8 1 2 .2 2 0 - 0 .4 7 0 0 .2 1 6 2 9 1 - 1 .6 4 6 1 1 8 1.660267 0.007150 0 . 2 2 8 0 5 4 - 1 .6 4 4 5 3 0 1 .7 8 1 2 .2 2 0 - 0 .7 1 8 0 .3 3 8 0 3 2 - 1 .2 8 2 4 3 1 1.326233 0.024633 0 . 3 6 9 5 1 7 - 1 .2 7 3 7 1 6 1 .7 8 1 2 .3 6 7 0 .0 5 1 0 . 0 5 8 2 4 6 - 2 .1 1 4 0 5 3 2.114855 -0.029378 - 0 .0 0 3 8 7 7 - 2 .1 1 4 8 5 2 1 .7 8 1 2 .3 6 7 0 .3 6 5 -0.034012 -2.309675 2 . 3 0 9 9 2 6 -0.054156 -0.158984 -2.304448 1 .7 8 1 2 .3 6 7 - 0 .2 4 9 0 .1 6 3 3 2 7 - 1 .8 7 3 8 8 1 1.880985 -0.007601 0 .1 4 9 0 7 9 - 1 .8 7 5 0 6 8 1 .7 8 1 2 .3 6 7 - 0 .5 5 0 0.301665 -1.541232 1 . 5 7 0 4 7 7 0 .0 1 5 8 3 1 0 . 3 2 6 0 2 6 - 1 .5 3 6 2 6 3 115 Table 20. (Continued), Ri/ûo Rî/Oo cos(S) HylD HzlD M io V>/rad. Pq/D P(/D 1.781 2.367 -0.926 0.675682 -0.704711 0.976301 0.061703 0.717852 -0.661704 1.781 2J 68 0.228 -0.023157 -2.217713 2.217834 -0.054851 -0.144706 -2.213108 1.781 2.568 -0.249 0.168974 -1.859358 1.867020 -0.009736 0.150863 -1.860915 1.781 2.568 -0.727 0.474870 -1.231028 1.319443 0.041850 0.525957 -1.210083 1.811 1.811 0.029 0.000000 -2.059383 2.059383 0.000000 0.000000 -2.059383 1.811 1.811 0.181 0.000000 -2.162118 2.162118 0.000000 0 .0 0 0 0 0 0 -2.162118 1.811 1.811 0.362 0.000000 -2.275949 2.275949 0.000000 0.000000 -2.275949 1.811 1.811 0.617 0.000000 -2.416852 2.416852 0.000000 0.000000 -2.416852 1.811 1.811 - 0.112 0.000000 -1.955619 1.955619 0.000000 0.000000 -1.955619 1.811 1.811 -0.174 0.000000 -1.906475 1.906475 0.000000 0.000000 -1.906475 1.811 1.811 -0.249 0.000000 -1.843354 1.843354 0.000000 0.000000 -1.843354 1.811 1.811 -0.309 0.000000 -1.789392 1.789392 0.000000 0.000000 -1.789392 1.811 1.811 -0.387 0.000000 -1.713671 1.713671 0.000000 0.000000 -1.713671 1.811 1.811 -0.527 0.000000 -1.556253 1.556253 0.000000 0.000000 -1.556253 1.811 1.811 -0.679 0.000000 -1.337907 1.337907 0.000000 0.000000 -1.337907 1.811 1.811 -0.856 0.000000 -0.950791 0.950791 0.000000 0.000000 -0.950791 1.811 1.841 -0.174 0.010057 -1.910066 1.910093 -0,000759 0.008607 -1.910073 1.811 1.841 -0.249 0.011697 -1.846713 1.846750 -0.000442 0.010881 -1.846718 1.811 1.841 -0.309 0.013084 -1.792551 1.792599 -0.000185 0.012753 -1.792553 1.811 1.903 0.036 0.017225 -2.076340 2.076411 -0.004959 0.006928 -2.076400 1.811 1.903 0.192 0.008773 -2.181671 2.181688 -0.007020 -0.006543 -2.181679 1.811 1.903 0.376 -0.000059 -2.296229 2.296229 -0.009674 -0.022271 -2.296121 1.811 1.903 0.634 -0.008213 -2.432584 2.432598 -0.014279 -0.042945 -2.432219 1.811 1.903 -0.108 0.025747 -1.969885 1.970053 -0.003124 0.019592 -1.969956 1.811 1.903 -0.249 0.034923 -1.853309 1.853638 -0.001333 0.032453 -1.853354 1.811 1.903 -0.391 0.045317 -1.718026 1.718624 0.000527 0.046223 -1.718002 1.811 1.903 -0.537 0.057780 -L550610 1.551686 0.002582 0.061784 -1.550456 1.811 1.903 -0.697 0.074853 -1.312351 1.314484 0.005177 0.081646 -1.311946 1.811 1.903 -0.889 0.104150 -0.848157 0.854528 0.009551 0.112246 -0.847124 1.811 1.997 0.061 0.029179 -2.104588 2.104790 -0.010415 0.007258 -2.104778 1.811 1.997 0.231 0.011156 -2.217520 2.217548 -0.014901 -0.021888 -2.217440 1.811 1.997 0.437 -0.007485 -2.341522 2.341534 -0.020977 -0.056599 -2.340850 1.811 1.997 0.751 -0.017925 -2.467049 2.467114 -0.033717 -0.101082 -2.465043 1.811 1.997 -0.095 0.047403 -1.989993 1.990558 -0.006485 0.034496 -1.990259 1.811 1.997 -0.249 0.067335 -1.862124 1.863341 -0.002628 0.062441 -1.862295 1.811 1.997 -0.403 0.090067 -1.712801 1.715168 0.001359 0.092395 -1.712677 1.811 1.997 -0.565 0.118729 -1.519585 1.524217 0.005921 0.127725 -1.518855 1.811 1.997 -0.747 0.161428 -1.222409 1.233022 0.012066 0.176166 -1.220372 1.811 1.997 -0.980 0.242949 -0.383917 0.454331 0.027037 0.253239 -0.377209 1.811 2.100 0.108 0.032318 -2.146307 2.146551 -0.017587 -0.005433 -2.146543 1.811 2.100 0.308 0.001324 -2.274193 2.274194 -0.025811 -0.057370 -2.273470 1.811 2.100 0.569 -0.028401 -2.417732 2.417898 -0.038660 -0.121825 -2.414828 1.811 2.100 -0.072 0.063700 -2.016599 2.017605 -0.010680 0.042159 -2.017164 1.811 2.100 -0.249 0.098323 -1.869635 1.872218 -0.003976 0.090889 -1.870011 116 Table 20. (Continued), Ri/flo Rî/Oo cos(6) fly/D ItilD ImI/d rpfiad. M17/D P(/D 1.811 2.100 -0.427 0.138974 -1.692566 1.698262 0.003032 0.144105 -1.692137 1.811 2.100 -0.617 0.193730 -1.449475 1.462365 0.011386 0.210220 -1.447175 1.811 2.100 -0.845 0.294069 -0.992072 1.034739 0.024787 0.318567 -0.984479 1.811 2.218 0.191 0.018597 -2.207907 2.207985 -0.028567 -0.044475 -2.207537 1.811 2.218 0.454 -0.031481 -2.362792 2.363002 -0.044502 -0.136564 -2.359052 1.811 2.218 -0.031 0.069618 -2.054424 2.055604 -0.016723 0.035254 -2.055301 1.811 2.218 -0.249 0.126896 -1.874622 1.878912 -0.005435 0.116705 -1.875284 1.811 2.218 -0.468 0.196839 -1.648287 1.659998 0.006468 0.207495 -1.646979 1.811 2.218 -0.713 0.307532 -1.290160 1.326306 0.022300 0.336224 -1.282982 1.811 2.364 0.047 0.052788 -2.113317 2.113976 -0.027269 -0.004852 -2.113971 1.811 2.364 0.354 -0.031936 -2.306666 2.306887 -0.049890 -0.146929 -2.302203 1.811 2.364 -0.249 0.150148 -1.874314 1.880319 -0.007126 0.136787 -1.875336 1.811 2.364 -0.546 0.277991 -1.545093 1.569901 0.014523 0.300400 -1.540893 1.811 2.364 -0.914 0.612581 -0.752679 0.970455 0.055467 0.653367 -0.717561 1.811 2J61 0.216 -0.021849 -2.213559 2.213667 -0.050855 -0.134343 -2.209587 1.811 2.561 -0.249 0.155984 -1.859648 1.866178 -0.009230 0.138814 -1.861008 1.811 2561 -0.715 0.434969 -1.250767 1.324242 0.038115 0.482315 -1.233284 1.841 1.841 -0.174 0.000000 -1.913510 1.913510 0.000000 0.000000 -1.913510 1.841 1.841 -0.249 0.000000 -1.849915 1.849915 0.000000 0.000000 -1.849915 1.841 1.841 -0.309 0.000000 -1.795543 1.795543 0.000000 0.000000 -1.795543 1.903 1.903 0.044 0.000000 -2.092886 2.092886 0.000000 0.000000 -2.092886 1.903 1.903 0.203 0.000000 -2.200262 2.200262 0.000000 0.000000 -2.200262 1.903 1.903 0.390 0.000000 -2.315507 2.315507 0.000000 0.000000 -2.315507 1.903 1.903 0.651 0.000000 -2.445295 2.445295 0.000000 0.000000 -2.445295 1.903 1.903 -0.103 0.000000 -1.983602 1.983602 0.000000 0.000000 -1.983602 1.903 1.903 -0.249 0.000000 -1.861653 1.861653 0.000000 0.000000 -1.861653 1.903 1.903 -0.395 0.000000 -1.720475 1.720475 0.000000 0.000000 -1.720475 1.903 1.903 -0.547 0.000000 -1.542448 1.542448 0.000000 0.000000 -1.542448 1.903 1.903 -0.716 0.000000 -1.280919 1.280919 0.000000 0.000000 -1.280919 1.903 1.903 -0.924 0.000000 -0.711141 0.711141 0.000000 0.000000 -0.711141 1.903 2.000 0.071 0.012984 -2.121754 2.121793 -0.005430 0.001463 -2.121793 1.903 2.000 0.245 0.003598 -2.237293 2.237295 -0.007779 -0.013805 -2.237253 1.903 2.000 0.455 -0.005798 -2.361578 2.361585 -0.010972 -0.031708 -2.361372 1.903 2.000 -0.090 0.022618 -2.002666 2.002794 -0.003363 0.015882 -2.002731 1.903 2.000 -0.249 0.033214 -1.868996 1.869291 -0.001337 0.030715 -1.869039 1.903 2.000 -0.409 0.045469 -1.711042 1.711646 0.000773 0.046792 -1.711006 1.903 2.000 -0.577 0.061120 -1.505378 1.506619 0.003197 0.065933 -1.505175 1.903 2.000 -0.771 0.086115 -1.172200 1.175358 0.006628 0.093882 -1.171603 1.903 2.106 0.121 0.017480 -2.164644 2.164715 -0.012403 -0.009369 -2.164694 1.903 2.106 0.328 -0.004192 -2.296447 2.296451 -0.018291 -0.046192 -2.295986 1.903 2.106 0.599 -0.022765 -2.437614 2.437720 -0.027702 -0.090274 -2.436048 1.903 2.106 -0.065 0.039697 -2.029687 2.030075 -0.007500 0.024473 -2.029928 1.903 2.106 -0.249 0.064499 -1.874770 1.875879 -0.002724 0.059391 -1.874939 1.903 2.106 -0.434 0.093895 -1.686885 1.689496 0.002274 0.097732 ■1.686667 117 Table 20, (Continued), Ri/flo Rz/Oo cos(ô) llylü Pi/D Ifl/D ÿ/rad. /‘c/D 1.903 2.106 -0.635 0.135230 -1.420879 1.427300 0.008402 0.147163 -1.419693 1.903 2.106 ■0.882 0.219824 -0.873852 0.901077 0.019084 0.236460 -0.869498 1.903 2.229 0.214 0.004683 -2.231581 2.231586 -0.023317 -0.047348 -2.231084 1.903 2.229 0.491 -0.033685 -2.390835 2.391072 -0.036760 -0.121530 -2.387982 1.903 2.229 -0.019 0.045985 -2.070068 2.070579 -0.013548 0.017936 -2.070501 1.903 2.229 -0.249 0.093046 -1.877583 1.879887 -0.004245 0.085075 -1.877961 1.903 2.229 -0.480 0.151353 -1.632406 1.639407 0.005591 0.160478 -1.631534 1.903 2.229 -0.743 0.250377 -1.225935 1.251241 0.019238 0.273913 -1.220892 1.903 2.383 0.069 0.027981 -2.134269 2.134452 -0.024346 -0.023984 -2.134318 1.903 2.383 0.400 -0.045549 -2.340122 2.340565 -0.045394 -0.151694 -2.335644 1.903 2.383 -0.249 0.114961 -1.873976 1.877499 -0.006026 0.103667 -1.874635 1.903 2.383 -0.568 0.232028 -1.506771 1.524531 0.013783 0.252773 -1303430 1.903 2.383 -0.979 0.648249 -0.366904 0.744880 0.061566 0.669595 -0.326324 1.903 2.595 0.270 -0.054681 -2.248849 2.249514 -0.050305 -0.167692 -2.243255 1.903 2.595 -0.249 0.115036 -1.852786 1.856354 -0.008277 0.099696 -1.853675 1.903 2.595 -0.768 0.402926 -1.120267 1.190525 0.040621 0.448087 -1.102980 1.997 1.997 0.098 0.000000 -2.148955 2.148955 0.000000 0.000000 -2.148955 1.997 1.997 0.288 0.000000 ■2.272956 2.272956 0.000000 0.000000 -2.272956 1.997 1.997 0.576 0.000000 -2.405813 2.405813 0.000000 0.000000 -2.405813 1.997 1.997 -0.076 0.000000 -2.020802 2.020802 0.000000 0.000000 -2.020802 1.997 1.997 -0.249 0.000000 -1.874225 1.874225 0.000000 0.000000 -1.874225 1.997 1.997 -0.422 0.000000 -1.699839 1.699839 0.000000 0.000000 -1.699839 1.997 1.997 -0.609 0.000000 -1.459971 1.459971 0.000000 0.000000 -1.459971 1.997 1.997 -0.831 0.000000 -1.026187 1.026187 0.000000 0.000000 -1.026187 1.997 2.113 0.160 0.005248 -2.199234 2.199241 -0.007506 -0.011259 -2.199211 1.997 2.113 0.392 -0.007492 -2.342310 2.342322 -0.011305 -0.033970 -2.342076 1.997 2.113 0.719 -0.012982 -2.468410 2.468444 -0.018520 -0.058692 -2.467746 1.997 2.113 -0.045 0.018653 -2.051726 2.051811 -0.004470 0.009482 -2.051789 1.997 2.113 -0.249 0.033887 -1.878468 1.878773 -0.001519 0.031034 -1.878517 1.997 2.113 -0.453 0.052235 -1.665382 1.666201 0.001568 0.054846 -1.665298 1.997 2.113 -0.680 0.080203 -1.343558 1.345950 0.005554 0.087664 -1.343092 1.997 2.113 -0.980 0.159348 -0.359477 0.393211 0.015625 0.164945 -0.356943 1.997 2.249 0.013 0.024280 -2.099708 2.099849 -0.011172 0.000820 -2.099848 1.997 2.249 0.280 -0.009788 -2.280720 2.280741 -0.019762 -0.054856 -2.230081 1.997 2.249 0.617 -0.034485 -2.451757 2.451999 -0.033345 -0.116203 -2.449244 1.997 2.249 -0.249 0.063848 -1.878863 1.879947 -0.003193 0.057848 -1.879057 1.997 2.249 -0.512 0.114511 -1.587838 1.591962 0.005307 0.122936 -1.587208 1.997 2.249 -0.826 0.220154 -1.025631 1.048994 0.018775 0.239370 -1.021317 1.997 2.424 0.136 -0.002675 -2.184341 2.184343 -0.024430 -0.056032 -2.183624 1.997 2.424 0.549 -0.066031 -2.419783 2.420683 -0.049042 -0.184574 -2.413637 1.997 2.424 -0.249 0.083976 -1.870132 1.872016 -0.005197 0.074257 -1.870543 1.997 2.424 -0.635 0.209455 -1.390210 1.405900 0.016025 0.231705 -1.386675 1.997 2.681 0.449 -0.118467 -2.340911 2.343906 -0.063881 -0.267662 -2.328574 1.997 2.681 -0.249 0.068451 -1.833865 1.835143 -0.007867 0.054022 -1.834347 118 Table 20. (Continued), Ri/Oo Rî/Oo cos(9) /‘s/D /ii/D ll'I/D l&Aad. P(/D 2.100 2.100 0.216 0.000000 -2.242006 2.242006 0.000000 0.000000 -2.242006 2.100 2.100 0.492 0.000000 -2.401108 2.401108 0.000000 0.000000 -2.401108 2.100 2.100 41018 0.000000 -2.076950 2.076950 0.000000 0.000000 -2.076950 2.100 2.100 -0.249 0.000000 -1.880220 1.880220 0.000000 0.000000 -1.880220 2.100 2.100 -0.481 0.000000 -1.629820 1.629820 0.000000 0.000000 -1.629820 2.100 2.100 -0.748 0.000000 -1.209792 1.209792 0.000000 0.000000 -1.209792 2.100 2.254 0.061 0.006896 -2.139489 2.139500 -0.007542 -0.009240 -2.139480 2.100 2.254 0.381 -0.015353 -2.346541 2.346592 -0.013922 -0.048019 -2.346100 2.100 2.254 -0.249 0.033951 -1.878199 1.878506 -0.001903 0.030377 -1.878260 2.100 2.254 -1560 0.070339 -1.515195 1.516827 0.004179 0.076671 -1.514888 2.100 2.254 -0.956 0.189179 -0.521776 0J55013 0.017388 0.198223 -0.518408 2.100 2.461 0.245 -0.030921 -2.258585 2.258797 -0.024768 -0.086847 -2.257126 2.100 2.461 -0.249 0.053602 -1.863127 1.863897 -0.004259 0.045667 -1.863338 2.100 2.461 -0.744 0.196139 -1.167841 1.184197 0.019362 0.218712 -1.163825 2.100 2.791 -0.249 0.010220 -1.801998 1.802027 -0.007601 -0.003478 -1.802024 2.218 2.218 0.112 0.000000 -2.178505 2.178505 0.000000 0.000000 -2.178505 2.218 2.218 0.493 0.000000 -2.408952 2.408952 0.000000 0.000000 -2.408952 2.218 2.218 -0.249 0.000000 -1.876211 1.876211 0.000000 0.000000 -1.876211 2.218 2.218 -0.610 0.000000 -1.432246 1.432246 0.000000 0.000000 -1.432246 2.218 2.459 0.373 -0.037227 -2.339887 2.340183 -0.020025 -0.084072 -2.338672 2.218 2.459 -0.249 0.025601 -1.856498 1.856675 -0.002772 0.020454 -1.856562 2.364 2.364 0.443 0.000000 -2.383237 2.383237 0.000000 0.000000 -2.383237 2.364 2.364 -0.249 0.000000 -1.854556 1.854556 0.000000 0.000000 -1.854556 119 B, = 1.471 -0.20 R ,= 1.471 o Rg = 1.607 Rg = 1.811 Rg = 2.016 -X— Ro = 2.260 ^ ' •0.24 •0.26 £ -0.28 I iS -0.30 -0.32 -0.34 -0.36 L _ 40.00 60.00 80.00 100.00 120.00 140.00160.00 180.00 e/(degrees) Figure 11. CISD energies as a function of 6 for various values of Rg at a fixed R, 1.471 flo (spline fits used whenever more than two points are available). R. = 1.587 -0.20 R , = 1.587 R , = 1.690 -0.22 •0.24 g #• -0.28 I iS -0.30 -0.32 -0.34 -0.36 L - 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 12. CISD energies as a function of 9 for various values of R% at a fixed Rj 1.587 Oo (spline fits used whenever more than two points are available). 120 R, = 1.591 -0.20 Rg = 1.591 ^ R2 —1692 " -0.22 Rg — 1.811 -&■- R )= 1.931 “X*— Rg = 2.058 -A-*- Rg = 2.206 -0.24 Rg = 2.402 Rg = 2.707 A •0.26 cI £ -0.28 e -0.30 -0.32 -0.34 -0.36 •— 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degress) Figure 13. CISD energies as a function of 0 for various values of R 2 at a fixed R] = 1.591 Oo (spline fits used whenever more than two points are available). R, = 1.689 -0.20 Rg = 1.689 •• Rg —1.713 •+••• •0.22 Rg — 1.811 O' •• Rg= 1.909 - X - - Rg = 2.011 Rg = 2.124 -*— •0.24 Rg = 2258 Rg — 2.430 -A— Rg = 2.683 s •0.26 £ -0.28 S' i5 -0.30 -0.32 -0.34 -0.36 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 14. CISD energies as a function of 0 for various values of Rg at a fixed R, = 1.689 Qo (spline fits used whenever more than two points are available). 121 R. = 1.781 -0.20 R , = 1.781 * R , = 1.811 -f- ■0.22 R j= 1.841 0 - R , = 1.903 -X - R , = 1.998 - t r Rg = 2.101 -0.24 R , = 2.220 R , = 2.367 -A- R , = 2.568 -0.26 E -0.28 -0.32 -0.34 -0.36 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 ©/(degrees) Figure 15. CISD energies as a function of 9 for various values of Rg at a fixed R] = 1.781 Oo (spline fits used whenever more than two points are available). R^ = 1.811 -0.20 Rg = 1.471 -o - Rg “ 1.591 -0.22 Rg = 1.689 -B— Rg = 1.781 X Rg —1.811 A ' Rg = 1.841 -0.24 - Rg s= 1.903 -*■— Rg = 1.997 Rg = 2.100 Rg = 2.218 •+••• -0.26 - Rg = 2.364 Rg = 2.561 -X— & 60.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 16. CISD energies as a function of 6 for various values of R; at a fixed R, = 1.811 Oo (spline fits used whenever more than two points are available). 122 R, = 1.903 •0.20 = 1.781 ^ ^2 = 1.811 •+••• -0.22 ^2 — 1.903 4D" " R2 ~ 2.000 -X» — R9 —2.106 "A— R2 = 2.229 •0.24 Rg = 2.383 ' R2 = 2.595 -A— "CI 1 -0.28 m -0.30 -0.32 .X- •0.34 -0.36 — 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e /(degrees) Figure 17. CISD energies as a function of 6 for various values of R 2 at a fixed R] 1.903 Oo (spline fits used whenever more than two points are available). R, = 1.997 -0.20 R ,= 1.811 Rg = 1.997 H— •0.22 R2 — 2.113 -Ej—* Rg = 2.249 -K— Rg = 2.424 Rg = 2.681 ~M— •0.24 •0.26 I £ . -0.28 & -0.30 V\ -0.32 -0.34 -0.36 I— 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 18. CISD energies as a function of6 for various values of R2 at a fixed Ri = 1.997 Oo (spline fits used whenever more than two points are available). 123 R, =2.100 -0.20 Rg = 1.811 “©■— Rg = 2.100 "i ' '■ •0.22 Rg = 2.254 Rg = 2.461 -X— Rg = 2.791 4 •0.24 S a I m •0.30 •0.32 •0.34 •0.36 60.00 80.00 100.00 120.00 140.00 160.00 180.0040.00 e/(degrees) Figure 19. CISD energies as a function of 0 for various values of R] at a fixed R] 2.100 Oo (spline fits used whenever more than two points are available). R. =2.218 •0.20 Pg = 1.611 o Pg = 2.218 —H— •0.22 Pg = 2.459 -a— •0.24 -0.26 s I •0.28 •0.32 •0.34 •0.36 60.0040.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 20. CISD energies as a function of 6 for various values of R2 at a fixed R, 2.218 Oo (spline fits used whenever more than two points are available). 124 R, = 1.471 -0.10 -0.20 -0.30 Rg —1.471 0 Rg = 1.607 H— Rg “ 1.811 -EJ" Rg = 2.016 -X — -0.40 Rg = 2.260 -A™ -0.50 a. -0.60 ■0.70 -0.80 ■0.90 -1.00 — 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 0 /(degrees) Figure 21. CISD z-component of the dipole moment as a function of 0 for various values of R 2 at a fixed Ri = 1.471 Qq . R- = 1.587 •0.10 •0.20 -0.30 Rg — 1.690 -H— •0.40 -0.50 a- -0.60 -0.70 -0.80 -0.90 -1.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 8 /(degrees) Figure 22. CISD z-component of the dipole moment as a function of 0 for various values of R2 at a fixed R] = 1.587 Oq . 125 R. = 1.591 •0.10 •0.20 •0.30 R , = 1.591 Rg = 1.811 •□• Rg = 1.931 -K- •0.40 Rg = 2.058 •A-- Rg = 2.206 -»•• Rg = 2.402 •«■ •0.50 Rg = 2.707 •*•■ 3 S. •0.70 •0.80 •0.90 -1.00 40.00 80.00 100.00 120.00 140.00 160.00 180.0060.00 e/(degrees) Figure 23. CISD z-component of the dipole moment as a function of 6 for various values of R 2 at a fixed Ri = 1.591 Qq. R, = 1.689 -0.10 •0.20 •0.30 Rg = 1.689 Rg = 1.713 “+• Rg = 1.811 •□• Rg = 1.909 - X - •0.40 Rg = 2.011 - 4-. Rg = 2.124 Rg = 2.258 •■• •0.50 Rg = 2.430 -4. = Rg = 2.683 » •0.60 •0.70 •0.80 •0.90 -1.00 — 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 24. CISD z-component of the dipole moment as a function of 0 for various values of R2 at a fixed Ri = 1.689 Oo- 126 R, = 1.781 -0.10 •0.20 -0.30 Rg = 1.781 -O' - Rg = 1.811 -4— Rg = 1.841 Rg = 1.903 -X— -0.40 Rg = 1.998 Rg = 2.101 -**■— Rg = 2.220 -0.50 Rg = 2.367 -o— Rg = 2.568 -o-" a -0.70 -0.90 -1.00 40.0060.00 80.00 100.00 120.00 140.00 160.00 180.00 e /(degrees) Figure 25. CISD z-component of the dipole moment as a function of 6 for various values of R z at a fixed R i = 1 .7 8 1 flo - R, = 1.811 Rg = 1.471 Rg= 1.591 Rg = 1.689 Rg = 1.781 Rg = 1.811 Rg = 1.841 Rg = 1.903 Rg = 1.997 Rg = 2.100 Rg = 2.218 Rg = 2.364 Rg = 2.561 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 26. CISD z-component of the dipole moment as a function of 6 for various values of R2 at a fixed R| = 1.811 flo. 127 R, = 1.903 ■0.10 •0.20 -0.30 Ro “ 1.781 -o— Ro = 1.811 - 4— Rg = 1.903 •□•. Rg — 2.000 X— -0.40 Rg = 2.106 -A— Rg = 2.229 -*■'- Rg ~ 2.383 ■*. -0.50 Rg = 2.595 * -0.70 -0.80 -0.90 -1.00 •— 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 0 /(degrees) Figure 27. CISD z-component of the dipole moment as a function of 0 for various values of R 2 at a fixed Ri = 1.903 Uq- R. = 1.997 -0.10 -0.20 -0.30 Rg = 1.811 0 Rg = 1.997 - 4— Rg = 2.113 Rg = 2.249 - X - -0.40 Rg = 2.424 ^ Rg = 2.681 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00 40.00 60.00 100.00 120.00 140.00 160.00 180.0080.00 e/(degrees) Figure 28. CISD z-component of the dipole moment as a function of 6 for various values of R2 at a fixed R; = 1.997 Oq- 128 R, = 2.100 -0.10 -0.20 -0.30 R? = 1.811 0 Rg = 2.100 —i— Rg = 2.254 -Q— Rg = 2.461 -X— -0.40 Rg = 2.791 -*■— -0.50 a- -0.60 -0.70 -0.80 -0.90 -1.00 <— 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e /(degrees) Figure 29. CISD z-component of the dipole moment as a function of 9 for various values of Rg at a fixed R] =2.100 Oq . R, =2.218 -0.10 -0.20 -0.30 Rg — 1.811 ■ o ' Rg = 2.216 - 4— Rg = 2.459 -B" -0.40 -0.50 a- -0.60 -0.70 -0.80 -0.90 -1.00 L- 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 ©/(degrees) Figure 30. CISD z-component of the dipole moment as a function of 6 for various values of Rg at a fixed R| = 2.218üq . 129 R, = 1.471 0.30 0.25 0.20 Rg = 1.471 0 Rg —1.607 Rg = 1.811 ♦Q** Rg = 2.016 ~K— 0.15 Rg = 2.260 0.10 I 0.05 .Q' 0.00 •0.05 -0.10 -0.15 *— 40.00 60.00 80.00 100.00 120.00 140.00 180.00160.00 6 /(degrees) Figure 31. CISD );-component of the dipole moment as a function of B for various values of Rg at a fixed Ri = 1.471 Oq . R, = 1.587 0.30 0.25 0.20 R , = 1.587 -f r- Ro “ 1.690 0.15 0.10 I ^ 0.05 0.00 -0.05 -0.10 -0.15 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 32. CISD y-component of the dipole moment as a function of B for various values of Rg at a fixed R; = 1.587 Oq- 130 R, = 1.591 0.30 0.25 0.20 R , = 1.591 Ro ~ 1.692 —H— R o s 1.811 - a - R2 — 1.931 "X—- 0.15 R2 = 2.058 «A— R2 = 2.206 Hfr*- R2 ~ 2.402 *■ 0.10 Ro = 2.707 a « G" ^ 0.05 0.00 •0.05 -0.10 -0.15 •— 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 ©/(degress) Figure 33. CISD y-component of the dipole moment as a function of 6 for various values of R 2 at a fixed Ri = 1.591 ûq- 0.30 0.25 0.20 R, = 1.689 R, = 1.713 R, = 1.811 R, = 1.909 0.15 R ,= 2.011 R, =2.124 R, = 2.258 T 0.10 R, = 2.430 I R, = 2.683 0.05 ■ Q—' 0.00 -0.05 -0.10 -0.15 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 ©/(degrees) Figure 34. CISD y-component of the dipole moment as a function of 0 for various values of R2 at a fixed Rj = 1.689 ûq- 131 R , =1.781 0.30 0.25 - 0.20 Rg — 1.781 0 Rg = 1.811 •+—• Rg = 1.641 Rg = 1.903 0.15 Rg = 1.998 -A-.- Rg = 2.101 R g s 2.220 •«••• — 0.10 Rg s 2.367 •A»— I Rg = 2.568 •• ^ 0.05 0.00 g - " g. # -0.05 -0.10 -0.15 _ 1_ _L_ 40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 35. CISD y-component of the dipole moment as a function of 9 for various values of R; at a fixed Ri = 1.781 Qq- R, = 1.811 R, = 1.471 R , = 1.591 Rj = 1.689 Rj = 1.781 Rj = 1.811 R; = 1.841 R2 = 1.903 Rj = 1.997 R; = 2.100 R: = 2.218 R, = 2.364 Rg = 2.561 ■■•o.... 40.00 100.00 120.00 140.00 160.00 180.00 e/(degrees) Figure 36. CISD ^-component of the dipole moment as a function of 6 for various values of Rg at a fixed R] = 1.811 Aq- 132 R, = 1.903 0.30 0.25 0.20 R, = 1.781 Ro “ 1.811 “+—• R g = 1.903 *E3** Rg = 2.000 -X — 0.15 Rg = 2.106 Rg = 2.229 H»... Rg = 2.383 « g 0.10 Rg = 2.595 •■A— =f" 0.05 0.00 -0.05 - 0.10 60.00 80.00100.00 120.00 140.00 160.00 180.00 0 /(degrees) Figure 37. CISD ^-component of the dipole moment as a function of 6 for various values of R 2 at a fixed R; = 1.903 Oq . R. = 1.997 0.30 0.25 0.20 R , = 1.811 R , = 1.997 R2 = 2.113 R j = 2.249 0.15 R j = 2.424 R , = 2.681 T 0.10 4 ^ 0.05 0.00 —g: -0.05 - 0.10 -0.15 60.0040.00 80.00 100.00 120.00 140.00 160.00 180.00 0 /(degrees) Figure 38. CISD ^/-component of the dipole moment as a function of 6 for various values of R% at a fixed R, = 1.997 Oq . 133 R, =2.100 0.30 0.25 R ,= 1.811 R, = 2.100 R j = 2.254 R , = 2.461 0.15 R , = 2.791 ^ 0.10 I 0.05 0.00 -0.05 •0.10 -0.15 40.00 60.00 100.00 120.00 140.00 160.00 180.0080.00 0 /(d e g ree s) Figure 39. CISD j-component of the dipole moment as a function of 9 for various values of R; at a fixed Rj =2.100 ag. 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