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RELATIVISTIC EFFECTIVE CORE POTENTIALS AND THE THEORETICAL CHEMISTRY OF THE TRANSACTINIDE ELEMENTS
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
Clinton Scott Nash, B. A., M. S.
The Ohio State University 1996
Dissertation Committee: Approved by
Dr. Bruce E. Bursten ~^^****' ^ Dr. Russell M. Pitzer Advisor Dr. Sherwin J. Singer Department of Chemistry UMI Number: 9710632
UMI Microform 9710632 Copyright 1997, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 ABSTRACT
Shape-consistent, Ab initio relativistic efifective core potentials (RECPs) and Gaussian valence basis sets were derived for Am-Element 118. These were used in conjunction with large-scale, self-consistent field and configuration interaction calculations to explore the electronic structure and, by inference, the likely chemistry of the transactinide (Z>102) elements. It is found that owing to relativity, severe shell-structure and spin-orbit effects dramatically alter the expected behavior of these elements vis'-a-vis’ their presumed placement in the periodic table.
11 DEDICATION
It is difficiilt for me to dedicate this to just one person. I have been blessed with the most loving and supportive family imaginable. I have always been allowed by the people around me to follow my muse, and this dissertation is the result. To my parents, Chuck and Rilla, my brother Cory, and my grandparents, aunts, uncles and cousins I would just say that I will be eternally grateful that God's design has put me among them. But the one person who has endured long absences, put up with my preoccupations, and generally had the most to complain about- and hasn't- is my wonderfiil wife Lisa. I don't know if she knew exactly what she was getting into when she gave me her hand but I know that without her this accomplishment would be empty. It is to her that I dedicate this dissertation, the woman who shares my name and life and who has completely captured my heart.
ui ACKNOWLEDGMENTS
There are quite a few people who have a stake in this doccument who deserve recognition. First, I thank my advisor, Bruce Bursten, for allowing me the freedom to pursue the research that interested me as well as for his personal and financial support. I'm not certain that I could have finished this process in any other kind of working environment. Second, I'd like to gratefully acknowlege Dr. Walter C. Ermler lately of the Department of Energy in Germantown Maryland who sacrificed a good deal of his spare time to teach me the ins and outs of core potential generation. Without his assistance the majority of the work reported here would not have been done. Next, I'd like to thank Dr. Russell Pitzer and members of the Pitzer group Jean Blaudeau, Ke Zhao, Nora Wallace, and Scott Brozell who have been more than helpful in teaching me how to use the relativistic Cl codes. I thank David Heisterberg for spending a good portion of his life teaching me quantum mechanics. Finally, I'd like to recognize Craig Hamilton, the man who shares an office with me. Job had nothing on this guy.
IV VITA
June 21,1968 Bom, Cleveland Ohio 1990-1996 BA. - Ohio Wesleyan University, Delaware Ohio 1990 Graduate Student, The Ohio State University 1994 M.S. - The Ohio State University
Publications: C. S. Nash, B. E. Bursten Metalloaromaticity in Metallocarbohedrenes: The Electronic Structures of C20 and TigCl2. Proceedings of the Electrochemical Society 94-24 (Recent Advances in the Chemistry and Physics of Fullerenes and Related Materials), 389-96. C. S. Nash, B. E. Bursten Comparisons Among Transition Metal, Actinide, and Transactinide Complexes: The Relativistic Electronic Structures of Cr(C0)6, W(C0)6, U(C0)6, and Sg(C0)6. New Journal of Chemistry, 19(5-6). 1995. C. S. Nash, B. E. Bursten Ab Initio Relativistic Effective Core Potentials with Spin-Orbit Operators VII. Am- Element 118: Journal of Chemical Physics, in press. TABLE OF CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGMENTS iv
VITA V LIST OF TABLES vn i
LIST OF FIGURES xi
CHAPTER 1 Introduction 1 CHAPTER 2 Relativistic Effective Core Potentials and Basis sets for the Transactinide Elements 14 CHAPTERS Transactinide Atoms, Hydrides, and Dimers 30 CHAPTER 4 Relativistic Effects in Transactinyl Fluorides 134 CHAPTERS Seaborgium Hexacarbonyl: A 6d Organotransition Metal Complex 194 LIST OF REFERENCES 212 APPENDIX A Small-Core Relativistic Effective Core Potentials for Am-Element 118 220 APPENDIX B Large-Core Relativistic Effective Core Potentials for Rf-Element 118 245 APPENDIX C Cartesian Gaussian Basis Sets Corresponding to the ECPs from Appendix A 262 APPENDIX D Cartesian Gaussian Basis Sets Corresponding to the ECPs from Appendix B 287 APPENDIX E Comparisons Among All-Electron Dirac-Fock, Small- Core REP, and Large-Core REP Atomic SCF Calculations 303 LIST OF TABLES
Table Pg. 1.1 The relationship among orbital, spin, and total angular 4 momentum quantum numbers and labels for jj spinor orbitals. 1.2 A listing and brief description of components of the 10 COLUlVffîUS su ite o f programs used for m uch o f the work reported here. 1.3 The American Chemical Society slate of proposed names 13 and atomic symbols for Elements 104*109. 2.1 f Polarization functions for use w ith large core (LC) 24 RECPs of elements Rf-Element 118. 2.2 Total Hartree-Fock energies (in Hartrees) for Am* 27 Element 118. 3.1 Computational summary for the lawrencium atom. 36 3.2 AREP SCF energies of lawrencium. 37 3.3 Computational summary for the rutherfordium atom. 41 3.4 Computational summary for element 111 atom. 45 3.5 Computational summary for element 112 atom. 48 3.6 Computational summary for element 113 atom. 52 3.7 Computational summary for element 114 atom. 54
viu 3.8 Computational summary for element 115 atom. 56 3.9 Computational summary for element 116 atom. 58 3.10 Computational summary for element 117 atom. 60 3.11 Hydrogen basis set used throughout Chapter 3. 62 3.12 Computational summary for Lr-H. 64 3.13 Computational summary for RfH. 68 3.14 Computational summary for (IIO)H. 71 3.15 Computational summary for (lll)H. 76 3.16 Computational summary for (113)H. 81 3.17 Computational summary for (114)H. 88 3.18 Computational summary for (115)H. 95 3.19 Computational summary for (116)H. 102 3.20 Computational summary for(117)H. 106 3.21 Computational summary for (111)2 112
3.22 Computational summary for (113^. 117
3.23 Computational summary for (117)2* 125 3.24 Results of augmented SOCI singles calculations for 132 atom ic elem ent 117. 4.1 Fluorine basis set used throughout this chapter. 135 4.2 Information pertaining to the spin-orbit Cl calculations. 145 4.3 Equilibrium metal-Ruorine bond lengths of several 6d 147 Hexafluorometallates calculated optimized at the SCF and SOCIS levels. 4.4 Optical transitions in NsFg with various Cl expansions. 154
IX 4.5 d-d Optical transitions in HsFg. The states marked with 158 a (?) were not unam biguously assigned.
4.6 d-d optical transitions in MtF6. 163 4.7 d-d optical transitions in (110)%. 164
4.8 d-d optical transitions in MtFg^'. 165
4.9 Computational summary for (113)F. 167 4.10 Computational summary for (114)F 172 4.11 Computational summary for (114)F2 180 4.12 Computational summary for (118)F4 187
5.1 Carbon basis set used in Sg(CO)6. 199 5.2 Oxygen basis set used in Sg(CO)6. 200
5.3 Atomic Coordinates of Seaborgium Hexacarbonyl 202 Optimized at the SCF Level of Theory. 5.4 Atomic Coordinates of Seaborgium Hexacarbonyl 203 Optimized at the MP2 Level of Theory. 5.5 Theoretical Interatomic Bond Lengths for Group VI 204 Hexacarbonyls. 5.6 Experimental and Theoretical Values of free CO bond 205 lengths and stretching frequencies. 5.7 Values of CO stretching modes calculated at the MP2 206 level of theory for Group VI transition metal carbonyls. Frequencies are in cm"l. 5.8 Theoretical Values of CO Stretching Frequencies in 207 Sg(C0)6. 5.9 Theoretical and Experimental values of CO stretching 208 frequencies for known Group VI transition metal carbonyls. Frequencies are in cm 1
X LIST OF FIGURES
Figure Pg.
31 Potential energy curve of Lr-H in its ground (0=0+)) 35 state calculated at the CISD level with and without the spin-orbit operator. 3.2 Potential energy surface for several low-lying states of 72 (IIO)H. The ground state is denoted by CISD while the first and second excited states are designated CISD 2 and CISD 3, respectively. 3.3 Potential energy curve of the ground state of of element 77 111 hydride calculated at the SCF and SOCISD levels of theory. 3.4 Three ground state potential energy surfaces of (113)H. 82 The curves labeled CISD and CISD NOSO were both calculated using the s.e. expansion and the latter was also calculated without the spin-orbit potential. The curve labeled CISD mcd was generated using the more extensive (Le.) Cl expansion. 3.5 Potential energy surface of element 113 Hydride 83 calculated at the SCF and SOCI-l e. levels. 3.6 A potential energy curve for (113)H which focuses on the 84 large expansion SOCI calculation. 3.7 Potential energy curve for the ground state of element 90 114 hydride calculated at the SCF and CISD levels with and without the spin-orbit potential
XI 3.8 Potential energy curve for several low-lying states of 93 element 114 hydride. The curve labeled CISD is the ground state. The first excited state is labeled CISD 2, the second CISD 3 etc. 3.9 Potential energy curve of element 115 hydride 98 calculated at the SCF and CISD levels. 3.10 A potential energy curve of element 115 hydride 100 calculated a t the CISD le v e l 3.11 Potential energy surface of element 116 hydride 103 calculated at the SCF, CISD, and CISD-NOSO levels. 3.12 A potential energy surface of several low-lying states of 104 element 116 hydride. 3.13 Potential energy curve of element 117 hydride 107 calculated with and without the inclusion of the spin- orbit potential 3.14 A potential energy curve for element 117 hydride 109 calculated at the SCF and large expansion CISD levels. 3.15 Periodic trends in equillibrium dissociation energies, 110 De, and bond distances. Re, among the halogen hydrides. The results for AtH and (117)H are calculated while those of the other atoms are experimental 3.16 A potential energy surface of (11 Dgcalculated at the 113 SCF and small expansion 01 levels. 3.17 A potential energy curve of (111^ calculated at the 115 large expansion CISD level. 3.18 Potential energy curve of (113)2calculated at the SCF 119 le v e l
3.19 A potential energy curve for the 0=0" ground state of 122 (lllg calculated with two different Cl expansions. The curve labeled mcd included single excitations from the set of 10 d orbitals. The other curve did not include these d-singles but did have a more extensive virtual space.
XU 3.20 Potential energy surface of the ground state of (117)2 127 calculated with (CISD A) and without (CISD-NOSO) the spin-orbit potential using the s.e. Cl expansion. 3.21 Potential energy surface of (117)2 calculated at the SCF 129 and large expansion (Le.) CISD levels. In this diagram, the latter is denoted as CISD Ag MC. 3.22 Periodic trends in values of Re and De for the dihalogen 130 molecules. The data for At2 and (117)2 is calculated while that of the others is empirical 4.1 Model of a typical octahedral transition metal complex. 138 4.2 Qualitative orbital diagram depicting the effects on a 140 free atom d orbital of an octahedral ligand field, the spin-orbit effect, and the simultaneous imposition of both. 4.3 Potential energy surface of SgFg calculated at the SCF 149 and SOCIS levels and representing the variation of total energy with the metal-fiuorine bond distance. 4.4 Potential energy surface of NsFg calculated at the SCF 152 and SOCIS levels and representing the variation of total energy with metal-fiuorine bond length. 4.5 Potential energy surface of HsFg caluclated at the SCF 156 and SOCIS levels representing the variation of total energy with metal-fiuorine bond distance. 4.6 Potential energy surface of MtFg calculated at the SCF 161 and SOCIS levels and representing the variation of total energy with metal-fiuorine bond cQstance. 4.7 Potential energy surface of the ground state of (113)F 169 calculated at the SCF and CISD levels. 4.8 Potential energy surface of element 114 fluoride 175 calculated at the SCF CISD and SOCIS levels. 4.9 Potential energy surface for the five lowest states of 177 element 114 fluoride calculated at the CISD level
xni 4.10 Potential energy surface of (114)F2calculated at the 183 CISD level depicting the variation of total energy with F-(114)-F bond angle. 4.11 Nonbonding Hartree-Fock MO's of element 118 185 tetxafluoride. Their symmetry labels are appropriate to the Dg subgroup. 4.12 Potential energy surface of element 118 tetrafluoride 189 calculated at the SCF level for the tetrahedral and square planar (flat) geometric configurations of (118)F4 4.13 Potential energy surface of element 118 tetrafluoride in 190 both its tetrahedral and square planar geometries. The calculations were done at the CISD level with and without (NOSO) the spin-orbit potential 4.14 Potential energy surface of element 118 tetrafluoride. 192 This is a reiteration of the curve in Figure 4.13. 5.1 Octahedral Transition Metal Hexacarbonyl in its 197 Cartesian orientation.
XIV CHAPTERl INTRODUCTION
From the very beginning of their education, chemists are impressed with a notion of the Periodic Table as perhaps the single most powerful tool for the prediction of atomic and molecular properties. Even in cases when such predictions are not entirely borne out, it organizes the elements into easily recognizable classes and provides a framework within which the width and breadth of chemical complexity can be understood. As each new element is discovered, interesting questions arise about how closely their properties conform to what would be expected given their presumed placement the Periodic Table. To begin to address such questions has been the motivation behind the work discussed in this dissertation. » The first synthetic transuranium elements, Np and Pu were discovered by McMillan and Seaborg in 1940.^ The intervening decades have seen the completion of the actinide series with Lawrencium and upon the addition of the as of yet unnamed element 112 have brought us to the end of the fourth (6d) transition metal series.^ Subsequent elements, if they are to be synthesized would undoubtedly begin the 7p main group. This progress might have brought about remarkable changes in synthetic chemistry and materials science if it were not for the fact that all of these elements have comparatively short half-lives, hi fact, many of them are produced only on an atom-at-a-time basis.^ Indeed, if this were not so, these elements would not be synthetic at all but would likely occur in the earth's crust in at least trace amounts. As it is, however, the heaviest element with a half-life on the order of the age of the Earth is Uranium."^ Half-lives of the most stable isotopes of elements with a greater atomic number range from 80 million years for Plutonium to several microseconds for element 1 1 1 .5 A good deal of the experimental chemistry of the middle actinides, Am-No, has been explored as many of them are sufficiently stable to allow their production in macroscopic amounts.® Of the others, however, very little experimental information is available and the limited information that is available sheds only the faintest light.7 This general lack of knowledge about some of the most fundamental chemistry of a large class of elements is intolerable and begs to be remedied. In order to do just that, we must resort to the methods that are available to us. The inherent difficulty or impossibility of performing complex chemical transformations with atoms that last for as little as a few millionths of seconds requires that we instead appeal to tried and proven theoretical methods. In the current state of affairs there is no shortage of established computational techniques which might be used. Rather, the problem is to find and use ones that adequately account for the relativistic and correlation effects, the inclusion of which is necessary for even a qualitatively correct description of the electronic structure of heavy-element systems. The practical consequences of relativity in atomic electronic structure are generally divided into two categories, shell-structure and spin-orbit effects.^ The first of these is explained that a relativistic mass increase of inner shell electrons is actually experienced in varying degrees by all electrons. This causes a direct radial contraction and energetic stabilization of all relativistic orbitals in comparison to their non-relativistic counterparts. A second, indirect shell-structure effect is seen when atomic orbitals with large probability densities near the nucleus, primarily s and p orbitals, are relativistically contracted to the point that they more effectively shield outer electrons firom the positive nuclear charge. The net result of these two sheU-structure effects is generally a radial contraction and energetic stabilization of s and p atomic orbitals and expansion/destabilization of d and f orbitals. The second relativistic effect, spin-orbit coupling, is simply the observation that 1 and s are no longer good quantum numbers; they no longer represent conserved quantities (l^, 1%; s^, Sz). They are instead replaced by a composite angular momentum, j=l+s (J= S ji), where I and s are the orbital and spin angular momenta of individual electrons. This means that the familiar s, p, d, and f, etc. spatial atomic orbitals 0=0,1,2,3,...) are transformed into combined spin-space functions indexed by j=l±l/2. So, for instance, the p orbital 0=1) is split in the relativistic case into j=l/2 (pi/2 or p*) and j=3/2 (pg/g or p) spin-orbit components. The only spatial orbital not transformed in such a way is the s orbital for which j=s=l/2. These spin-space functions are commonly referred to as spinor orbitals or simply spinors. A spin-orbit scheme such as this is known as jj coupling and is most appropriate for situations in which spin-orbit coupling is much stronger than interelectronic Coulombic repulsions. In cases where coulombic interactions dominate over spin-orbit coupling the familiar LS or Russell-Saunders (J=L+S, L=E li, S=Z Si) coupling scheme is preferred. In Russell-Saunders coupling the orbital angular momenta of all of the electrons add vectorially as do all of the spin angular momenta. These L and S vector sums are then coupled to form a composite angular momentum, J. It is important to note that these two cases of spin-orbit coupling, jj and LS, represent unrealistic extremes. The real' coupling scheme is an intermediate of these extremes and only J is conserved in nature despite our attempts to impose any particular representation.
Atomic Spinor 1 quantum j quantum k quantum Parity Orbital Chbital number number number Label Label s s 0 172 -1 + P P* 1 172 1 -
P 1 3/2 -2 - d d* 2 3/2 2 + d 2 5/2 -3 + f (* 3 5/2 3 - f 3 7/2 -4 - g g* 4 7/2 4 + g 4 9/2 -5 +
Table 1.1 The relationship among orbital, spin, and total an gu lar momentum quantum numbers and labels for jj spinor orbitals. A further complication introduced by the spin-orbit effects is the inadequaqr of normal (single) group theory for the description of the symmetry properties of atomic and molecular wavefunctions. The reason for this can be seen when one considers the transformation properties of functions characterized by half-odd integral J values. In contrast to the case where J is a whole number for which the single group is perfectly adequate, a rotation by 2jc is not equivalent to the identity but is rather the negative of it. In order to resolve this dilemma and treat both cases on an equal footing, Bethe introduced the mathematical artifice that the identity is not rotation by 2iz but is rather rotation by 4n.^ Rotation by 2 k is then found to be an additional symmetry operator and the order of the group is effectively doubled, hence the name double group'. It is not necessarily the case that the number of classes and therefore the number of irreducible representations is doubled, although it may be so. The proper calculation of spin-orbit Hamiltonian matrix elements takes place in the context of the double group and so this is a very important concept for future considerations. The chemical importance of electron correlation has been widely recognized for quite some time, but that of relativity has largely been ignored. The reason for this is simple, for the great majority of chemical species of interest to chemists, relativistic effects are negligible while correlation effects frequently are not. As a result, while there are a great many ways of including correlation, from configuration interaction and coupled cluster methods to Moeller-Plesset (MP) perturbation theory, there are fewer options if one wants to take into consideration the additional effects of relativity. The recognition of the shortcomings of the Schrœdinger equation in light of the requirements of special relativity has existed almost from its inception. The four-component Dirac equation (eq. 1.1) has answered the inadequacies of the Schrœdinger equation, at least for the hydrogen atom. ¥ (eq. 1.1) I 0 a = vO -^ J The relativistic equivalent to the many-body Schrœdinger Hamiltonian is the Dirac-Coulomb (DC) operator
^DC “ ^ ^D;i ^ (^q 1.2) i ij where ho,i is the one-electron Dirac operator from equation 1.1 applied to electron i and the second term is the normal pairwise sum of electronic repulsions. 12 While the two-electron part of the DC Hamiltonian is not Lorentz covariant, no formally correct many-body Dirac equation is known and it is found that this problem poses no real difBculties in actual calculations. 13 A common extension to the Dirac-Coulomb Hamiltonian involves the use of the Breit operator which, as the first term in a quantum electrodynamical expansion for the interelectronic interaction, gives corrections for retardation of the electromagnetic field and spin-other orbit interactions. 14 Because the effects of the Breit interaction are usually no more than about 3% of those of Dirac relativity, for our purposes, this operator may be ignored. 15 Although the DC Hamiltonian has been used in numerical all electron atomic calculations for some time, computational constraints and various practical concerns (e.g. difBculties with kinetic balance in basis set MO methods) have limited the widespread appUcation of the DC Hamiltonian to molecular problems of chemical interest. Only now are such calculations beginning to become feasible for even small molecular s y s t e m s . The DC Hamiltonian also suffers firom its spinor nature. The solutions to equation 1.1 and 1.2 are not the familiar scalar wavefimctions firom non-relativistic quantum mechanics but are instead four-component vectors' (eq. 1.3).
V, »F =a (eq. 1.3) Ws p
The multi-componential nature of relativistic wavefunctions is necessary for the proper description of spin which is intrinsic to the relativistic theory. However, it also complicates their interpretation. In actuality, the non-relativistic functions have two components, each corresponding to a particular spin state, but this fact is usually suppressed in the non- relativistic theory and spin is treated as an od hoc addition. A common approach for including relativistic effects in chemistry is through the use of a LS spin-orbit operator, a mass-velocity operator, and a Darwin correction as perturbations to non-relativistic wavefunctions.^® (The Darwin term is a purely relativistic correction which accounts for errors in the wavefunction that arise from the fact that it is not possible to simply remove two components of a four-component spinor) This treatment of relativistic effects, however, is found to be insufficient for even relatively light atoms. 1® Other techniques involve the solution of the relativistic Hartree-Fock equations, a self-consistent analog of the perturbational approach mentioned above.20 Among still others are the Cowan-GrifBn equations,21 quasirelativistic density fimctional theory,22 Dirac-Fock one center expansion methods,23 the two-component, no- pair theory of Hess et oZ.,24 and various semiempirical approaches.25 Another adaptation of non-relativistic quantum chemical methods to the problem of relativistic effects entails the use of effective core potentials. These replace the core electrons of atoms with “atom-like” potentials derived from relativistic or quasirelativistic atomic calculations.26 These potentials are usually employed in the single component, Hartree-Fock- Roothaan equations familiar from non-relativistic molecular calculations. Valence spin-orbit effects can then be calculated at a later stage employing spin-orbit potentials also derived from these atomic calculations.27 It is this last approach that was used here in order to begin the task of outlining the general chemistry of the transactinide elements. There are comparatively few theoretical studies of the chemistry of transactinide elements and none that are comprehensive surveys.28 While certainly not comprehensive, the research reported in this document is an attempt to establish the parameters of the debate, a first pass, if you will, through the transactinide elements. Its purpose is to provide a starting point for the further illumination of transactinide chemistry while providing some preliminary clues about how our notions of periodicity and chemical analogy are supported or refiited in the behavior of atoms at the frontier of the periodic table. Time constraints dictated that tough choices be made concerning which molecules will and which will not be examined here. It
8 is only hoped that the choices made establish the groundwork for future work in this area. Chapter 2 describes the derivation of the relativistic effective core potentials (RECPs) and valence basis sets for Âm-element 118 used throughout the three subsequent chapters. Each of these three deals with a different kind of chemical question. Chapter 3 addresses the most elementary questions including the importance of correlation, shell- structure, and spin-orbit effects in some transactinide atoms, simple monohydrides, and dimers. Chapter 4 deals with simple inorganic complexes, the octahedral hexafluorometaUates of the 6d transition metals as well as some low-valent, 7p-block fluorides. These complexes have well- known analogs in the lighter transition metals and it is instructive to see how things such as ionic radii and electronic spectra compare between the periods. This is especially interesting in light of the commonality of our notions of crystal and ligand held theories as they relate to the chemistry of transition meals. Chapter 5 focuses on a particular aspect of the organometallic chemistry of a 6d transition metal. Seaborgium (Sg, element 106). The molecule of interest is seaborgium hexacarbonyl, Sg(C0)6. the logical next member of the isoelectronic series including Cr(CO)e> Mo(CO)e, and W(CO)e. It remains to be seen to what degree the
structural and electronic similarities within this fam ily of molecules will be continued for superheavy elements.
There were several different suites of quantum chemical programs used to perform these calculations each with its own strengths and weaknesses and each suited to different types of problems. The core potentials and basis sets were derived using several relativistic and non- relativistic atomic programs as well as others designed specifically for this purpose. As noted earlier these will he discussed in Chapter 2. For the molecular work of Chapters 3 and 4, components of the COLUMBUS program system was used along with some adjunct programs not directly incorporated into the general COLUMBUS release.29.30 These extra codes permit the inclusion of spin-orbit effects in a (double group) configuration interaction step. Table 1.2 lists and provides a brief description of the
function of these codes.3i-36
Program Description computes molecular integrals over ARGOS an AO basis creates a PK supermatrix using the CNVRT results of ARGOS a restricted molecular SCF program SCFPQ transforms integrals firom the AO LSTRN basis of ARGOS to an MO basis using the coefficient matrix from SCFPQ generates configuration list needed CGDBG for double group Cl program a double group configuration CIDBG interaction program that includes spin-orbit matrix elements
Table L2 A listing and brief descriptioii of components of the COLUMBUS suite of programs used for much of the work reported here.
10 The procedure for the use of the spin-orbit configuration interaction codes is as follows. Once a target molecule in the desired geometry has been selected, ARGOS is used to calculate all one and two-electron integrals over the given AO basis set. This includes integrals over the core and spin- orbit potentials firom the ECP. ARGOS calculates integrals over symmetry- adapted linear combinations of atomic orbitals under D2h> DZ, and C2v symmetry and so is usefiil for any molecule belonging to one of these full point group or belonging to one that has one of these as a subgroup. The next step is to use CNVRT to convert the integrals file into a PK supermatrix format usable by the self-consistent field program SCFPQ. In this step, the spin-orbit integrals are not used. If a configuration interaction calculation is to be performed it is desirable to obtain virtual orbitals which maximize the efficiency of the (truncated) Cl expansion and this is done by again using SCFPQ in a so-called improved virtual orbital (IVO) calculation.37 The rVO procedure involves freezing the occupied molecular orbitals from a previous converged SCF run while removing one or several electrons. The program is then allowed to undergo one SCF cycle in which the virtual orbitals experience a n-1 (or several) electron Coulombic repulsion. The result is a set of filled molecular orbitals (MOs) originating from the original SCF calculation and a set of virtual orbitals originating from a molecular cation calculation. Because of the freezing of the occupied MOs in IVO calculations, orthogonality between all of the orbitals is maintained but the space of virtual orbitals is reorganized. If a full Cl calculation were to be performed or if all of the virtual orbitals in a given calculation were to be used, this procedure would probably be unnecessary.
11 But as we are limited by practical concern to reasonably small Cl expansions, it is of paramount importance to get as much as possible out of orbitals that are used. Once a set of MO coefiBcients has been obtained, LSTRN is used along with the molecular orbital coefficients file to convert the integrals over atomic orbitals firom ARGOS to integrals over molecular orbitals. It is this new integrals file that is used in the 01 calculation. The symmetry restrictions of ARGOS arise firom the original design of the code for use in GUGA driven Cl calculations for which the use of higher symmetry is d if f ic u lt .38 The spin-orbit Cl matrix is blocked according the irreducible representations of the molecular double point group. The use of the double group is a natural result of including spin in the molecular calculation and in general the spin-orbit matrix elements will be complex (as opposed to real). It is fortuitous however that the limitation to these three symmetry groups facilitates the evaluation of spin-orbit matrix in the Cl calculation by allowing all calculations be performed using real arithmetic.^® The double-group adapted configuration list specific to the desired molecular state is generated using CGDBG. The molecular integrals file from LSTRN and the configuration list from CGDBG are then used by CIDBG in the spin-orbit Cl calculation. In its current incarnation, COLUMBUS is adapted for single point MCSCF/MRCI calculations and is not particularly well suited for such things as geometry optimizations and calculation of vibrational spectra. For this reason, certain calculations were also performed using the academic version of the GAMESS program system.^® The latter also has certain advantages in that, in addition to automated geometry
12 optimizations and analytical gradients and Hessians, it permits the treatment of electron correlation at the MP2 level of theory. However GAMESS does suffer from its inability to use the spin-orbit potentials, a fact which somewhat reduces its usefulness in these systems. Finally, it is important to note that at the time of the writing the names of the transactinide elements are a source of come controversy.'^ ^
Whenever applicable, the slate of names and recommended symbols adopted by the American Chemical Society will be used.^2 These are compiled in Table 1.3. Names for the known elements 110, 111, and 112 (eka-platinum, eka-gold, eka-mercury) have not as of yet been proposed and elements 113 through 118 (eka-radon) are not yet known. These will be referred to by atomic number enclosed by parentheses so, for example, the hexafluoride of element 110 will be referred to as (110)F6 and the dimer of (113) will be indicated by (113)2.
Atomic Number______Proposed Name______Proposed Symbol 104 Rutherfordium Rf 105 Hahnium Ha 106 Seaborgium Sg 107 Nielsbohrium Ns 108 Hassium Hs 109 Meitnerium Mt
Table L3 The American Chemical Society slate of proposed names and atomic qmibols for Elements 104-109.
13 CHAPTER 2 RELATIVISTIC EFFECTIVE CORE POTENTIALS AND BASIS SETS FOR THE TRANSACTINIDE ELEMENTS
This chapter deals with the development of the relativistic effective core potentials (RECPs) and valence basis sets for the transplutonium actinide and transactinide elements which are to be used in subsequent chapters. The theory of the effective core potential (ECP) traces its origin to 1960 and the pseudopotential method of Phillips and Kleinman.^G This method depends on the chemically reasonable assumption that the atomic core orbitals in atoms are essentially unchanged upon bond formation. Although objections have been raised to the Phillips-Kleinman (PK) formalism as applied to many electron molecular calculations (among other things, it tends to underestimate the electron density in the valence region), it is this assumption that forms the basis for the two major classes of ECP methods in use today.^'^ These classes differ in their treatment of the core valence separation. The first follows the original PK formalism and explicitly orthogonalizes the valence orbitals to the core and projects the resulting orbitals onto the valence.'^S The second class makes use of so- called nodeless pseudoorbitals.**® These pseudoorbitals have the same
14 characteristics of the Hartree Fock (or Dirac-Fock) orbitals in the valence region but are allowed to go smoothly to zero at the nucleus. This second strategy avoids treatment of nodes and has distinct advantages over the first. The RECPs developed here have evolved from this second branch of the ECP tree. Effective core potentials serve several purposes which make them indispensable for use in large scale quantum chemical calculations. First, they dramatically reduce the number of electrons that need to be explicitly treated. This means that far fewer basis functions, primitive and contracted, are required because the need to represent the chemically inert core electrons is eliminated. The number of basis functions is further reduced if, as noted earlier, care is taken to construct ECPs that result in valence orbitals which have their inner nodes removed. This is the fundamental way in which the ECP method differs from a simple frozen core approximation. In the latter, the nodes of the valence orbitals remain and may require several radially contract (as opposed to diffuse) primitive or contracted functions to describe them. This means, for example, that all six radial nodes of the U 7s orbitals would have to be described by some contraction or contractions of Gaussians which would clearly require a very large number of such functions. Clearly, the frozen core approach is undesirable in such a case. Naturally, as the number of valence orbitals and the basis functions used to describe them is reduced, higher levels of post Hartree-Fock computation are possible. Another major advantage, and the one which lead to the generation and use of ECPs here, is the ability to include relativistic effects in the molecular calculation. The direct inclusion of relativistic effects at the
1 5 Hartree-Fock self-consistent field (SCF) level through the use of additional terms in the Fock operator such as mass-velocity or Darwin corrections is possible but is very computationally costly. Spin-orbit corrections to such quasirelativistdc wavefimctions require an even greater expenditure of effort because of the loss spin-space independence. The RECP on the other hand permits the inclusion of shell structure effects (direct stabilization/contraction of all orbitals, indirect destabilization/expansion of d and f orbitals) at the Hartree-Fock level without any alterations of the Fock operator beyond those needed to accommodate ECPs in the first place. These transactinide ECPs have been derived based on the second strategy, the use of so-called shape-consistent nodeless pseudospinors in the context of a two-component Dirac-spinor form alism .'^7,48 Previous tabulations of such "shape-consistent effective potentials" have included lithium through plutonium and this work represents a natural extension of this list.'^®'®'^ To the authors' knowledge, no published compilation of ECPs exists for elements with Z>103. However, Dolg et. al. have published energy-adjusted nonrelativistic and quasi-relativistic ECPs for much of the periodic table including all of the actinides.®®
Method. Relativistic effective core potentials (REP) have been generated in the form =Ul^{r) + % £ I W]|ljn)(ljn I , (eq. 2.1) i=0 = - J where the Uij(r) are obtained from the two-component spinor equation |_ i - C7 ^ y ixy = CyXy . (Oq. 2.2)
1 6 L is generally taken as one greater than the largest 1 quantum number of the core electrons and J is the corresponding value of j required to define the relativistic quantum number for the atomic shell unambiguously.'^® Zy is the atomic number minus the number of core electrons. Xlj is a pseudospinor generated from the large component of the Dirac-Fock (DF) valence spinor and £ij is the corresponding one-electron energy. Wy is the Coulomb plus exchange potential between Xy and the remaining valence pseudospinors. The pseudospinors are defined by = V'ij(r) + Fy(r) (2.3) where \|/ij(r) is the numerical radial function corresponding to the large component of the Dirac-Fock valence spinor.®® Fjj is chosen to cancel the radial oscillations in the core region by equating up to five derivatives at the matching point of Fy and \(/y, thereby ensuring that Xy goes smoothly to zero at the nucleus.'^’^ The spinor orbital, l|fy(r), results from self-consistent solution of the numerical atomic Dirac-Fock equation. The result of such a DF calculations for an atom are a set of spinor orbitals of the form
where Pnk (=\l/y(r))and Qnk represent respectively the large and small
17 component radial wavefimctions and %km and %.km are each two- component spinors representing the intrinsic coupling of spin and spatial orbital angular momenta viz.
^ (l) ® where
The quantity 1 represents orbital angular momentum in the uncoupled representation, m is the normal magnetic quantum number in the coupled representation, the first 1/2 is the familiar electron spin angular momentum quantum number, and the +1/2 and -1/2 represent the possible components of the spin angular momentum along a particular axis. The quantum number k is unique to the relativistic equation and is commonly associated with an alignment vector between j and 1. It can be seen that the terms in brackets are simply the necessary vector coupling coefficients. In addition, the two-component column f vectors and j represent the a and P (non-relativistic electron) .0. V spinors. The result of this ansatz is a four-component spinor function which is an eigenfunction of and jz- The use of only the large component of the DF spinor orbitals for the construction of the REP is justified by the fact that the small component is vanishingly small in the valence region of even very heavy atoms.
18 The 1-1/2 and 1+1/2 components of the Uy of eq. (1) are then used to form weighted average REP (ARE?) operators and (SO) operators that have the fbrms^G'57
=c7f°-(r) + £ X (r)-C7,""“’(r)]|l;n ){lm | (eq. 2.4) 1=0 m *—i where a (r) = {21+ + {l+l)a% ( r ) ] (eq. 2.5) and
=s*X[2/Cl+1)] A i7 f“’ ( r ) X XIIHI-^ I (eq 2 .6 ) 1=1 m '=— Im =— 1 where (eq. 2.7)
The details of the procedures for deriving the REPs are given elsewhere.58 The AREPs and SO operators are fit to expansions in Gaussian type functions (GTFs) of the form
jAREP e x p (-Q r ') (eq. 2.8) i where Cy and Çy are linear and non-linear expansion coefficients.
The residual potentials Ul are fitted directly. The Ui (1 U t r (r) = C7 (r) - 1; r (r) . (eq. 2.9) Similarly, the exponential factors determined in (8) are used to fit the SO operators in the form = j^A u;‘'{r) . (eq. 2.10) 19 Note that the coefficients of the expansions for Ui^O must be multiphed by 2/1 for use in computer programs based on equation 2.6.5? Results and Discussion, Relative Effective Core Potentials Gaussian expansions of AREPs and SO operators are given in Appendix A for the 24 atoms Am through element 118 (elements 113-118 remain undiscovered at the time of writing). The core/valence partitioning for these REPs is the same as that used in a previous tabulation for Fr-Pu.53 That is, 78 electrons in the subshells ls-5s, 2p-5p, 3d-5d, and 4f were taken as core. The 6s, 7s, 6p, 6d, 5f and 5g subshells comprise the space of valence orbitals in this "actinide-like" potential for all 24 atoms. Such a scheme by analogy to the earlier actinides appears to be the most appropriate for the later actinides, but it seems prudent in light of the uncertainties of the electronic structure of transactinides to include such a partitioning for them as well. At the same time, as the later actinides are known to more closely resemble the late lanthanides in the chemical inertness of the valence f electrons, it is certainly to be expected that such would be true of transactinide elements for which this 5f shell is complete, energetically stable, and radially contracted. In light of this fact, compiled in Appendix B are 15 92-electron, large-core (LC) REPS for the atoms Rf-element 118, derived such that the 5f electrons have been attributed to the core. However, in this set the 6s and 6p orbitals are retained in the valence space. Although they are energetically very stable and not likely to have a 2 0 dramatic effect on the chemical properties of transactinide atoms, their inclusion in an "outer core" serves to allow relaxation and polarization of the atomic wavefimction. Proper treatment of such effects may be of increased importance for the description of electronic spectra and bonding interactions in these elements.^7 The propriety of assigning even more of the electrons to the core remains an open question which merits further exploration. Basig-sets. Energy-optimized valence basis sets corresponding to (SC) and (LC) effective potentials are reported in Appendices C and D respectively. These basis sets were derived using ATMSCF, an atomic code associated with the COLUMBUS suite.59 In contrast to previous tabulations accompanying shape-consistent REPs, these are Cartesian Gaussians. The functions of s symmetry are taken as the x^+y^+z^ combination of the 6 Cartesian d orbitals and the p functions are taken as the appropriate linear combination of the 10 f Cartesians. The s and d functions as well as the p and f fimctdons are expanded in common sets of exponents, denoted sd' and pf respectively. In addition, for the late actinides and d-block transactinides (Am-element 112) two additional "true" p functions are provided. This is done in order to ensure that there is sufficient flexibility in the basis set to account for the relativistically enhanced importance of the 7p shell in the description of the electronic structure of these atoms. Three such functions are provided for element 113-element 118, as these are thought to be actual p-block atoms. In both cases, attention is paid to this issue because, while a common set of p and f exponents may work very well to describe the 2 1 relatively radially contracted 6p and 5f shells, it is by no means certain that the same set will adequately represent the 7p shell. This is less of a concern for the sd basis hmctions because two radially difhise sets of functions, 7s and 6d, are represented in this expansion. There is evidence that the use of radial functions with increased n values is a more natural choice for this type of core potential.®® As noted earlier, the pseudospinors used to define the REPs are constrained to go smoothly to zero at the nucleus. The 's' combination of d orbitals contains the d orbital radial factor of r^ which eliminates the need to do this by relying on the differencing of pure spherical harmonic s-like functions. Experience with these core potentials and various types and sizes of basis sets has shown that often a lower or at least comparable total valence energy is obtained using the Cartesian basis sets in Heu of the pure spherical harmonic basis sets. In addition, this lower energy is obtained with fewer primitive functions — in some cases, half as many. The sd and pf exponents for atoms (Am-No) were simultaneously optimized for the ground state of each atom under LS coupling. Unlike the case for earlier actinides (Ac-U), this ground state arises primarily from a f^d® electron configuration. The competition between d and f atomic orbitals for electron density has largely been decided in favor of the latter by the middle of the actinide series. The optimization of the sd exponents for the 7s shell, however, means that there are basis functions of d symmetry in radial and energetic proximity to the bonding region. The extra "difiuse" p primitives were optimized for the lowest state arising from a 2 2 ApO > Pi-lpl excitation while holding the sd and pf exponents fixed at their previously optimized values. The sd and pf exponents in Lr-Element 112 were optimized for the lowest LS state arising from a d^ configuration. This despite strong evidence firom various sources that the ground states of Lr, Ha, and Rf exhibit significant contributions firom the j=l/2 (7p*) spin-orbit component of the 7p shell. (The "normal" ordering seems to be recovered at Rf, element 104). This valence description, however, relies on a jj-coupled model of the atom which is incompatible with the philosophy used in these optimizations. To wit, the scalar relativistic effects (net contraction/stabilization of s and p shells, net expansion/destabilization of the d and f shells) are accounted for in the SO averaged calculation while sufficient flexibility is built into the basis sets to allow for a proper description of spin-orbit coupling at a later (Cl) stage. To this end, the diffiise p exponents were optimized for the lowest state resulting firom a d*^- > d’^'lpl excitation. All of the exponents for p-block Element 113 to Element 118 were optimized simultaneously for lowest LS state resulting from the p^ configuration. In all cases, the contraction coefficients for occupied orbitals are those corresponding to the ground atomic state. For atomic orbitals that are not occupied in the ground atomic state, 7p for Am-Element 112, 6d for Am-No, the contractions coefficients are given for the lowest LS terms arising firom 5f^p® -> 5f^-l7pl (Am-No), 5f^6d0 > 5f^ l 6dKAm-No), and 6d"7pO .> 6d"-^7p^(Lr-Element 112) excitations. It is often desirable to augment a basis set with additional primitive functions corresponding to 1 values one greater than the maximum of that 23 in the valence. These additional functions serve two purposes: (1) to allow more flexibility in the description of valence shell polarization and (2) to allow more degrees of freedom in the post-Hartree-Fock description of correlation. To this end, exponents of single f primitives for LC basis sets are provided in Table 2.1 for each of the atoms whose core potentials have been derived. These exponents were not found variationally but were chosen to maximize the radial overlap of the f primitive polarization function with the d valence orbital.® ^ The maximization of radial overlap seems to be optimal for recovering dynamical correlation. Element fmcponent Element f exponent Rf(104) 0.301566 (112) 0.725568 Ha (105) 0.369118 (113) 0.815619 Sg(106) 0.426112 (114) 0.893519 Ns (107) 0.483141 (115) 0.980506 Hs (108) 0.529726 (116) 1.053230 Mt(109) 0.577671 (117) 1.133899 (110) 0.625409 (118) 1.202255 (111) 0.674824 Table 2.1 f Polarization functions for use with large core (LC) RECPs of elements Rf-Element 118. 24 figSHltS» The degree to which these REPs successfully represent the core electrons of late actinides and transactinides is detailed in Appendix E. There are compiled the results of numerical all-electron Dirac-Fock and jj- coupled, REP-based atomic SCF calculations for Am-Element The electron configuration under jj coupUng was chosen in each case to result in a single, unambiguous electronic state. The REPs were generated by "unaveraging" the AREPs reported in the appendices using the derived spin-orbit potentials. That is, the spin-orbit operator was recombined with the l±l/2 components of the averaged potential, taking into account the proper weighting, to form a potential indexed by relativistic quantum number k rather than 1. The unaveraged potentials are related to the averaged potentials by: AC/,0 (eq. 2.11) : ; if K< 0 and C/j^=C/,4.AC/,o (eq. 2.12) if K> 0. This procedure allows a direct comparison of the results of SCF calculations using these REPs with the fully relativistic DF results. The purpose of unaveraging the averaged potential rather than simply taking the numerical REP directly from the results of the shape-consistent procedure is to provide a direct test of the tabulated potentials as opposed to comparing those from an intermediate step. 25 Because two core potentials are reported for each atom with Z >103, the first including the 5f shell (SC) in the valence space, the second (LC) relegating this shell to the core, the results of two atomic jj-SCF calculations are shown. The 1+1/2 and 1-1/2 components of the jj-coupled atomic orbitals are expanded in the same set of basis fimctions. The basis sets used in these atomic calculations are the optimized basis sets Usted in Appendices C and D. This procedure provides an indirect test of the adequacy of the basis set to replicate the spin-orbit splitting. In all cases, agreement between the results of jj-SCF calculations and the results of all electron Dirac-Fock calculations is to within 3%. This agreement may improve is the basis fimctions were actually optimized for the jj-SCF calculation, but this was not thought to be necessary. We find, therefore, that this procedure results in relativistic effective core potentials which are faithful representations of the core electrons of these superheavy element systems. 2 6 Atom large-core RECP (LC) small-core RECP (SC) Am -82.13401102 N/A Cm -94.85004552 N/A Be -108.87527459 N/A Of -125.38727042 N/A Es -142.56567706 N/A Fm -160.95546351 N/A No -202.77602258 N/A I f -225.08240853 N/A Rf -249.48573454 -47.02071422 Ha -275.34528137 -54.92203347 Sg -302.76700270 -63.87092280 Ns(lOT) -331.24682725 -73.83002694 HsdOS) -361.59345720 -84.69762565 Mt(109) -393.65978348 -96.61879400 (110) -430.61916177 -109.64280207 (111) -462.48824793 -124.02177479 (112) -499.80681297 -139.59996090 (113) -538.28217477 -156.23744264 (114) -578.74505290 -173.71912536 (115) -620.20292196 -192.59289405 (116) -665.69098333 -212.59833900 (117) -710.40717320 -233.90663502 (118) -758.92467758 -256.34148073 Table 2J2 Total Hartree-Fock energies (in Hartrees) for Am-Element 118 2 7 Snin.Orhit Brfaxntinn Riiaiw It seems appropriate at this time to mention an interesting aspect of the use of spin-orbit averaged ECPs (AREPs) versus those based on the unaveraged REPs. As was noted earlier in this chapter, the AREPs derived here and tabulated in Appendices A and B were 'unaveraged' and used in an atomic SCF calculation based on jj coupling. This was done in order to allow a direct comparison of the orbital eigenvalues from Dirac-Fock calculations with those from atomic REP-based calculations. These comparisons yielded an interesting result which in hindsight seems obvious but which is nonetheless surprising. Among the results compiled in Table 2.2, one finds that the total Hartree-Fock AREP energy of the dosed-shell No atom was calculated to be -202.77602 Hartrees. An analogous calculation using the same basis set and the unaveraged ECP (REP) obtained by recombining the No AREP with the spin-orbit potential gave a total energy of -203.05458215 Hartrees. To be sure, this is a small energy difference amounting to only 0.13% of the total-but at first glance it seems that the results should be identical given the fact that No is a closed shell atom. It is particulary surprising that the jj-coupled calculation gives a lower total energy given the fact that the basis set was optimized for the LS problem. Actually, this discrepancy can be is a simple reflection of the fact that the average coulomb repulsion of the valence electrons in an 1-indexed orbital is different than the average Coulomb repulsions of its spin-orbit components. By construction, the core potential and the spin-orbit potential reflect and reproduce the effect of the core electrons on the valence electrons. Such a core potential cannot simultaneously reflect the 2 8 difference in valence-valence interactions between the spin-orbit non spin- orbit cases. Consider the case of a valence p shell containing 6 electrons. The spin-orbit effect splits this threefold degenerate orbital into a doubly degenerate pi/2 (p*) spinor orbital and a quadruply degenerate ps/2 (p) spinor orbital. The average interelectronic repulsions between the six equivalent p electrons in the spin-orbit less case is not necessarily the same as the average repulsions among and between the two p* and four p electrons in the spin-orbit case. So, this is not an inadequacy of the ECP itself but is rather a consequence of the valence description of the atom and as such is not avoidable. It is fortunate then that this difference introduces only a very small, uniform error. 29 CHAPTERS TRANSACTINmE ATOMS, HYDRH)ES, AND DIMERS In this chapter application is made of the RECPs and basis sets derived in Chapter 2 to problems concerning relativistic effects on transactinide atoms and diatomic molecules. That these relativistic effects, shell-structure effects and spin-orbit coupling, have profound consequences in heavy element chemistry is well established and not really in question. This is certainly true of superheavy elements, so while a comparison of relativistic' element 111 hydride with the non-relativistic' hydride may serve as a useful starting point for a general discussion of relativistic effects, it is not the purpose of this chapter. Simply put, molecules such as non-relativistic (lll)H do not exist and one must routinely account for relativistic effects in the theoretical study of such species. Indeed, in many cases their electronic structures bear little resemblance to what would be expected of them given their placement in the periodic table.®^ Rather, this dissertation is an attempt to deconvolute the interrelated influence of shell, spin-orbit, and correlation effects in the discussion of the of transactinides. 30 The primary focus of this chapter is the electronic structure of the 7p- block elements and their molecules. However, some 6d transition elements are also considered, in particular some low-valent species with only one or two valence electrons (or holes). Not all of the species examined here are reasonable molecules; they have been chosen to illustrate various electronic and structural properties and begin the task of outlining the chemistry of these elements at the most basic level. The reasoning behind this approach holds that a hill appreciation of the fundamentals of bonding in more complex transactinide molecules begins with an understanding of the factors that influence the bonding in simpler molecules and groups. It is not possible to have addressed all of the pertinent questions about the nature of these interactions, but it is hoped that the selection of transactinide atoms and molecules made here constitutes a representative sample and a good start in the process of outlining their chemistry. The organization of this chapter roughly follows the order of the transactinide elements by atomic number. For the purposes of this discussion. Lawrencium, Lr-element 103, is categorized among the transactinides despite the fact that it is technically the last actinide. This is done because Lr has one electron outside completely closed 7s and 5f shells. This makes it the first open-shell, transuranium element in which the electronic structure is not characterized by partially filled f shells. The details of the calculations reported in this chapter are, for the most part, presented in tabular format so as not to disrupt too severely the flow of the discussion. Each of these tables is particular to the problem being addressed and therefore the information presented in them varies accordingly. All of the calculations reported in this chapter as well as the 31 next (Chapter 4) were done using the spin-orbit configuration interaction (SOCI) scheme described briefly in Chapter 1. The computational detail tabulations are divided into three general sections. The first section gives pertinent information about the self-consistent- field calculations which provide the one-electron basis for the SOCI calculations. Of course, these SCF calculations provide interesting and usefiil information in their own right. It includes an identification of the effective core potential and valence basis set used in the SCF step. In all cases, the metallic RECPs and basis sets are those which were described in Chapter 2 and compiled in the Appendices. The description of the contraction scheme, (6sd5pf2p)/[5sd4pf2f] for example, proceeds as follows. The figures in parentheses give the number and types of primitive Gaussian basis fimctions in each set; ’6sd' indicates that 6 sd-type primitive basis fimctions were used. The remainder of the symbols are interpreted similarly. The figures in the square brackets represent the number and types of contracted basis fimctions, that is fixed linear combinations of primitive Gaussian basis fimctions. By convention, the number of contracted fimctions of each type beyond the number of valence atomic shells of that type is equal to the number of diffuse primitives in the contraction set. So in the above example for Lr atom, the 6sd primitives are partitioned into 5 contracted basis fimctions. These sd basis fimctions are meant to describe three atomic orbitals (6s, 7s and 6d) and therefore the last two basis fimctions in the contraction set are the two most diffuse sd primitives. In several instances, a so called unbiased' (as opposed to normal') calculation is done in order to generate the one-electron basis. Such 32 calculations are used in cases where open-shell species do not have a clear-cut Hartree-Fock electron configuration. The unbiased calculation is an SCF calculation on the nearest closed shell species. In the case of the lawrencium atom where it is not immediately obvious to which atomic orbital, 6d or 7p, the outer, unpaired electron should be assigned, this closed-shell species is simply the lawrencium monocation, Lr+. Molecular orbitals generated in such a way are then used in spin-orbit Cl calculations incorporating the full complement of electrons. This procedure should be seen as a way of comparing the results of such SOCI calculations with orbitals generated in different ways. The next table section provides information pertaining to the definition of the active space and generation of the spatial configuration list for use in the SOCI calculations. The term computational double group' refers to the double group symmetry under which the calculation was performed as opposed to the actual symmetry of the molecule. As mentioned in Chapter 2, the methodology is restricted to symmetry double groups D2h. 02. and Cgv so all species examined must belong to one of these groups or have one of them as a subgroup. For atoms, the symmetry is usually chosen as D2h because this is the highest usable subgroup of spherical symmetry. For linear Coov molecules C2v is the appropriate computational double group symmetry under which the calculation is to be performed. Similarly, the term computational irrep’ refers to electronic state symmetry under the labels of the computational double group. The electronic state symmetry is reflective of the actual symmetry even though the labels are not. For instance, the 2pg/2 (J=3/2) state of the (113) atom manifests itself as two degenerate Ei/2u roots of the Cl matrix. 33 In. the definitions of the reference spaces, again in the Lr atom, the notation '6p®(singles only)5fl^7s27pl / doubles' means that double excitations are allowed firom each of the three spatial configurations corresponding to p%l, Pyl, and pz^ under the constraint that the 6p shell always contains at least 5 electrons. It is hoped that this information is all that is required to provide the reader with sufficient background for each calculation presented here. The remainder of this chapter proceeds as a series of short discussions on the properties of various transactinide atoms and molecules. The Atoms Indisputably, the key to any real understanding of molecular electronic structure begins with an understanding of the electronic structure of its constituent atoms. Such is certainly true of molecules containing transactinide atoms. In this section, the electronic structure of various low-valent transactinide atoms is examined Lawrencium, (Element 103) Lawrencium is the last member of the actinide series and, according to its traditional placement in the periodic, it should be a d-block element (7s2 5fl^ 6d^). This would also follow from the observation that, at least for the early actinides, the 6d and 5f orbitals are veiy close in energy and therefore one might expect that the filling of the f shell ought to put the next electron into a d shell. However, the question as to whether the relativistic stabilization of the 7p* spinor orbital is sufficient to make it lower in energy 34 than the (relativistically destabilized) 6d* spinor orbital bas warranted the attention of several autbors.®^ Most recently, Wiejesundere et. al., report that although single configuration Dirac-Fock calculations find the ground state to be (Ts^Tp*!), a multiconfigurational calculation finds 2D3/2 (Ts^Gd*!) to be the ground state.®^ Kaldor, et. al., on the other band, using a relativistic coupled cluster approach find that ^Pi/2 is indeed the ground state of Lawrencium and that its energy is lower than the ^ 03/2 state by 1300cm" 1.®® This suggests that Lr ought to be considered a p-block element rather than a member of a d-block transition series. We also have found that at both the Dirac-Fock self-consistent field (DF-SCF) and the spin-orbit averaged relativistic effective potential (AREP) Hartree-Fock levels of theory, the ground state electron configuration of lawrencium atom to be [Rn] 7s^ Sf^"^ 7pl. This stands in contrast to the (non-relativistic) Hartree-Fock result which has [Rn] 7s^ 6d^ as the lowest energy configuration. Only the latter configuration corresponds with our expectations given the placement of Lr within the periodic table as an analog of lutetium, the last lanthanide. 35 acE Hartree Fock Electron Configuration: normal: unbiased: 5fl^s^p®7s2 (+1 ion) ECP / Basis Set Contraction Scheme: 78-electron EOF firom Chapter 2/ (6sd5pf2p)/[5sd3pf2p] Hartree Fock State: 2p (Is, unbiased) Hartree Fock SCF Energy (Hartrees): -225.0826244 Configuration Interaction Computational Double Group Symmetry: Dg Number of Virtual Orbitals: 16 Definition of Reference Space / allowed ^citations: 6p6(singles only) 7s^ 7pl / doubles 6p®(singles only) 7s^ 6d^ / doubles Computational brrep Number of Number of Double Spatial Configurations Group Functions Ei/2 11157 103029 B oults normal: State (parity) Energy (Hartrees) AECE-Eo) (eV) Primary Character J=l/2 (-) -225.173322 — 7p*l J=3/2 (+) -225.155808 0.476 6d*l J=3/2 (-) -225.144782 0.760 7pl J=5/2 (+) -225.137695 0.968 6d^ unbiased: State (parity) Energy (Hartrees) AE(E-EO) (eV) Primary Character J=l/2 (-) -225.171013 — 7p*l J=3/2 (+) -225.155548 0.421 6d*l J=3/2 (-) -225.144782 0.714 7pl J=5/2 (+) -225.137695 0.907 6d^ Table 3.1 Computational summary for the lawrencium atom. 36 SptnOfbit averaged fiiCF Energy Lr Configuration (Hartrees) [Rn]7s2 5fl46dl -225.07544098 [Rn] 7s2 5fl4 7pl -225.08240853 Table 3^ AKEIP-SCF energies of lawrencium Of course, it is not surprising that the effects of electron correlation and spin-orbit coupling play a major role in the electronic structure of the lawrencium atom and that role is now discussed. The double group symmetry group under which the spin-orbit conhguration interaction (SOCI) calculation was performed was chosen as D2 rather than D2h* This was done because the representations, and therefore the wavefiinctions, under the latter are labeled by parity (g or u, + or -) whereas those under the former are not. As the energy differences of states with different parities are being compared, it is desirable to treat them on an equal footing by using a single configuration list for aU states of interest. The odd number of electrons in the Lr atom dictates that the total angular momentum quantum number, J, takes on half-odd integer values. This in turn requires that the symmetry of the electronic state be labeled according to one of the additional' irreducible representations (irrep) of the double g r o u p . 6 4 In D2h there is one double group irrep of even parity, Ei/2gi and one of odd parity, Ei/2u- The absence of the inversion operator in D2 means that there is only one additional' double group irrep and it is labeled simply Ei/2. Of course, the wavefunction actually retains its parity 37 as it must according to the true spherical (Kh) symmetry of the atom despite the lifting of this constraint in the Cl calculation. As noted earHer, spin-orbit configuration interaction calculations were performed using both normal orbitals generated using the neutral atomic 7p^ Hartree-Fock configuration (followed by an IVO calculation) and unbiased orbitals using the atomic monocation. The results of both calculations are substantively the same giving a ^Pi/2 (p*^) ground state and the same ordering of excited states. There is also a strong correspondence between the energies of the excited states between the two types of calculations. This is significant because it demonstrates that the choice of 7pl as the Hartree-Fock valence electron configuration in the SCF calculation does not bias the results of the SOCI calculation toward stability of the state resulting from p orbital occupation relative to that resulting from d orbital occupancy. Having said this, the normal' orbitals give a total energy which is somewhat lower in energy than the unbiased orbitals and are therefore probably more appropriate for the calculation of transition energies. It is interesting that there is an interleaving of states originating from different orbital occupations. This underscores the energetic proximity of the 6d and 7p atomic orbitals and leads to the conclusion that the standard orbital notation (s, p, d, f,...) is in this case less useful than the configurations tabulated according to jj coupling (s, p*, p, d*, d,....). The results of these atomic configuration interactions calculations are clear, the ground state of the gas-phase Lawrencium atom is ^Pl/2 (J=l/2 (-)). What is still not clear is whether Lr is appropriately classified as a p-block element. No experimental evidence exists which suggests that Lr 38 behaves more like T1 than Lu. Hofi&nan has suggested that the electronic nature of Lr may be easily perturbed in the interaction of the atom with a chemical environment such as an elution c o lu m n .6 5 Such an interaction may stabilize the 6d orbital relative to the 7p orbital and bring about a reordering of the resulting electronic states. This is a suggestion that will be examined later in this chapter. Rutherfordium, Element 104 As was the case with lawrencium, there is some question surrounding the identity of the dominant ground state electron configuration of rutherfordium, element 104. In strict analogy with group IV metals of the 1st, 2nd, and 3rd transition series, this configuration would be 6d2. However, as we saw in the case of Lr, because of the relativistic stabilization of the 7p* orbital, this analogy does not always hold and we must examine the question directly. Using a relativistic coupled cluster method (RCC) Kaldor et al, have in fact found the ground state electron configuration to be 7s^6d^.66 This is in opposition to the results of earlier multiconfigurational Dirac-Fock (MCDF) calculations that had 7s^6dl7p*l as the proper ground state configuration.^? The major reason for the difference in the results of the RCC calculation versus the MCDF study is the inclusion in the former of dynamic correlation. The multiconfigurational Dirac-Fock approach only accounts for non-dynamical correlation which is roughly defined as stemming from the inadequacy of a single or small number of determinants to properly describe the wavefunction. Kaldor concludes. 39 therefore, the inclusion of dynamic correlation is necessary for the proper description of the electronic structure of the Rf atom. The results of Hartree-Fock SCF calculations on this problem are certain to be uninformative since spin-orbit effects are so dramatic for this atom, therefore no such results are reported here. The orbitals for the spin-orbit 01 calculation used to determine the neutral atomic ground state were generated from the closed shell Rf^+ ion in a manner similar to the unbiased' Lr atom calculation described previously. The reason that the +2 ion was chosen to generate the orbitals rather than the neutral atom is that, as in the case of Lr, it seems reasonable to avoid biasing the result of the spin-orbit 01 calculation by generating the one-electron functions presuming any particular electron conGguration or HF state. And, for the same reasons as in the treatment of the Lr ground state, the computational symmetry of the atom was taken as Dg rather than Dgh-it allows a treatment on an equal footing of states with different parities. States arising from d^ (d*^, d*ldl, and d^) or p2 (p*2, p*lpl, and p2) configurations are of even parity while those arising from d^p^ configurations are odd. Unlike the case of Lr, which has an odd number of electrons, the even number of electrons of rutherfordium (104) requires the use of more than one configuration list for the calculation of the states of interest. This owes to the fact that J may only take on integer values and therefore the symmetries of the states are the normal (as opposed to additional) representations of the double group. In the case of D2 these are A, Bi, B2, and Bg. 40 SCE Hartree Fock Electron Configuration: unbiased: (+2 ion) ECP / Basis Set Contraction Scheme: 78-electron ECP from Chapter 2/ (6sd5pf2p)/[5sd3pf2p] Hartree Fock State: ^S, unbiased Hartree Fock SCF Energy (Hartrees): Not important Configuration Interaction Computational Double Group Symmetry: D 2 Number of \^rtual Orbitals: 16 Definition of Reference Space / allowed excitations: Sfl'^Bs^Bp^Ts^Tp^ / singles Bfl^ggSgpBY g2gd2 / singles Sfl'^Bs^Bp^Ts^TplBdl / singles Computational Irrep Number of Spatial Number of Double Group Configurations Functions A B309 19701 Bl(=B2=Bg) B300 19B92 Results unbiased: State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character (1=2 (+) -249.57787 --- 6d*^ J=1(-) -249.57479 0.08376 6d*l7p*l J=3 (+) -249.55369 0.65787 6d*2 Table 3 .3 Computational s u m m a r y for the rutherfordium atom 41 The configuration lists of each symmetry under Dg were generated by allowing single excitations from all 13 occupied orbitals in each of the 45 spatial reference configurations into 16 virtual orbitals. These reference configurations include the Sfl^^Gs^Gp^Ts^ active space while distributing the remaining two electrons in all possible ways among the 3 p and 5 d atomic orbitals. Despite the fact that only single excitations were allowed from the reference space, this reference space itself consisted of some double excitations which generated some triple excitations for the configuration Ust. The orbitals were optimized for the N-2 electron system so Brillouin's theorem does not hold in the N electron calculation. Still, while it is desirable to allow a greater level of excitation out of the reference space, to do so in an evenhanded way while avoiding the generation of an unmanageable number of configurations is a problem. So although the Cl expansion is not as large as might be ideal, it is still quite large and provides enough flexibility to adequately address this problem. The lowest four roots of the Hamiltonian matrix of each symmetry were found and the results indicate that the ground state of Rf is a d^ J=2 state in agreement with Kaldor's results and in opposition to MCDF studies, reinforcing the importance of dynamic correlation.^? The first excited state, however, has J=1 and arises from a p^d^ configuration and lies a mere 0.08376 eV above this ground state. The second excited state, with J=3, again arises from a d^ configuration. It is comforting that these RECP/SOCI results are in qualitative, if not quantitative, agreement with the RCC results which is quite complex and involves a much more extensive treatment of correlation. 68 This despite the fact the fact that the 42 RECP itself is derived from a Dirac-Fock calculation which gives the opposite result. In short, the results of the RECP/SOCI procedure go beyond the reproduction of the DF results and mirror the more expensive RCC results. This is suggestive that this procedure is effective in representing cases of intermediate coupling. It is apparent that although the expected d^ ground electronic configuration of the group IV metal Rf is realized, the energetic proximity of (odd) states based on partial occupation of 7p atomic orbitals means that one ignores these orbitals in molecular calculations at one's own peril. This is not surprising in light of what we have already discovered about the electronic structure of Lr. Element 111 (eka-gold) Element 111, which was discovered relatively recently, would be expected to occupy a position on the periodic table which classifies it among the coinage metals with Cu, Ag, and Au.G9 These last three are all characterized by fully filled valence d shells and a single electron in the valence s orbital. On the surface, then, the electronic structures of such metals are relatively simple, especially in comparison to those of earlier transition metals which are made more complex by the presence of open d shells. The ^8 electronic (Hartree-Fock) ground state means that a great many coinage metal compounds are univalent. The extent to which this familial characteristic is maintained at element 111 remains to be seen. At this point it should be noted that element 111 is referred to here as a coinage metal based on its presumed position in the periodic table rather than any particular chemical property. 43 Kaldor et al, conclude that the relativistic stabilization of the 7s orbital relative to the 6d orbital causes a change in their relative energetic ordering and leads to a ^Dg/2 ground atomic state.^O They also find that the first excited state, the spin-orbit counterpart of the ground state, ^D3/2, lies 2.719 eV above this ground state. It is the second excited (^Si/2) state that corresponds to the ground state of lighter coinage metals and it lies 3.006 eV above the ground state. In our calculations, the orbitals used in the Cl calculation were generated for the neutral (111) atom with a 7s^ 6d^ configuration. The computational symmetry for both the SCF and Cl calculations were chosen as D2h as rather than Ü2 which was used in similar calculations on Lr and Rf atoms. The higher group was used because unlike the earlier atoms, the parities of the ground state and lower-lying excited states are not really in question. It is expected and indeed found that the 7s and 6d shells are the most important for the proper description of these states and these are both even. An unbiased calculation was also done for (111) in order to provide another one-electron basis for the Cl calculation. These orbitals correspond to the anionic 7s26dl® electron configuration and are used in SOCI calculations just for purposes of comparison. 44 SCE Hartree Fock Electron Configuration: normal: 6s2 6p® 7s2 unbiased: (-i ion) ECP / Basis Set Contraction Scheme: 92-electron ECP finm Chapter 2/ (6sd5pii;/L6sd4plfI Hartree Fock State: (^S, unbiased) Hartree Fock SCF Energy (Hartrees): -124.02177 Cgnfigiiratiffl Intgi Computational Double Group Symmetry: D2h Number of ^rtual Orbitals: 23 Definition of Refierence Space / allowed excitations: normal: 7s^6d9 / doubles 7sl6d^0 / doubles large expansion: 6s2(singles only)6p^(singles only)7s^6d^ doubles 6s2(singles only)6p®(singles only)7sl6d^0 doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions El/2g normal: 8708 66908 large expansion: 14569 152149 Table 3.4 Computational summary for element 111 atom. 45 Table 3.4 continued R esu lts State (parity) Energy (Hartrees) AECE-Eo) (eV) Primary Character J=5/2 (+) -124.33667 — 7s26d*46d5 J=3/2 (+) -124.23969 2.6389 7s26d*^6d6 J=l/2 (+) -124.20973 3.4539 7sl6d*46d6 State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=5/2 (+) -124.32051 •— 7s26d*46d5 J=3/2 (+) -124.22394 2.6279 7s26d*^6d6 J-V2 (+) -124.20520 3.1376 7sl6d*46d6 normal (large expansion): State (parity) Energy (Hartrees) AEGE-Eq) (eV) Primary Character J=5/2 (+) -124.51719 — 7s26d*46d5 J=3/2 (+) -124.42149 2.6041 7s26d*^6d6 J=l/2 (+) -124.37134 3.9688 7sl6d*46d6 46 Separate SOCI calculations were performed which differed in the definition of the active space. The results of these calculations are essentially the same however. Unlike the case in copper, silver, or gold, the ground electronic state of element 111 is found to be ^D5/2 rather than ^Si/2. The first excited state is the other spin-orbit component of the state, 2D s/2 and the % i/2 state is the 2nd excited state. This conclusion is also in agreement with Kaldor's study and is fundamentally interesting in the sense that it represents another violation of the aufbau principle. This is clearly a consequence of the relativistic stabilization of s orbitals and destabilization of d orbitals. The effect is great enough by element 111 that it becomes energetically favorable to transfer an electron firom the 6d shell to the 7s shell. This will surely have significant consequences for the chemistry of this atom distinguishing it firom the other' coinage metals. What these particular consequences may be remain to be seen. Element 112 (eka-mercury) The salient feature of (112) is the presumed complete occupancy of its valence orbitals, apart, of course, of any possible valence involvement of the 7p orbitals. As such, it would be predicted to closely resemble Hg in its chemical and physical properties. First among these properties is its relative inertness and one of the ways to probe the degree of this inertness is to determine the energy required to promote an electron firom the closed valence shells to the low-lying p orbitals. One can consider such a promotion as a first step in any sort of chemical bond formation. For Hg, the lowest such transition energy occurs for the promotion of a 6s electron 47 to the 6p shell. This corresponds quite well to the observation that in Au the ns shell lies higher in energy than the 5d shell, so the lowest transitions in Hg are of the form Gs^GpO —> 6s ^ôp^. These transitions typically occur at energies around 4.26 to 6.76 eV, depending on the spin-orbit states involved.G iven what we have learned about (111) however, it is reasonable to assume that the corresponding transitions in (112) occur with a transfer Grom the 6d to the 7p orbitals. Any transfer of electrons Grom closed d shells, or for that matter s shells, to the p shells will result in states of odd parity. It is clear Grom the results listed in Table 3.5 that the lowest such electronic transitions occur at about 8 eV, an energy approximately double that of the lowest transition energy in Hg. Granted, these are different transitions than in Hg, d to p rather than s to p, but it illustrates the point that the closed-shell conGguration of (112) is extremely stable compared to open, ’bond-forming' conGgurations. This, in turn is indicative that element (112) might be extraordinarily inert, certainly more so than even Hg. The energy difference might be surprising considering the relativistic destabihzation of the 6d shell and the stabilization of the 7p shell. But it is perhaps more appropriate to consider this dramatic difference as the result of an extreme stabilization of the 7s orbital which swamps any other shell effects. In other words, the d to s transition in Hg may be larger than that in (112) but in Hg this is not the lowest transition while in (112) it is. 48 SCE Hartree Fock Electron Configuration: ECP / Basis Set Contraction Scheme: 92-electron ECP from Chapter 2/ (6sd5plf)/[6sd4plf| Hartree Fock State: Ig Hartree Fock SCF Energy (Hartrees): -139.59996 Configuration Interaction Computational Double Group Symmetry: D2h Number of ^rtual Orbitals: 24 ïlfffinitinn of Reference Space / allowed mccitations: Sp6(singles only)7s26dl0 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions Ag 4192 14011 Au 4176 13959 Biu(=B2u=B3u) 4176 13959 State (parity) Energy (Hartrees) AE(E-Eq) (eV) Primary Character J=0 (+) -140.09231 --- 7s^6d*46d6 J=2 (-) -139.79005 8.225 7s26d*46d57p*l J=3(-) -139.77744 8.568 7s26d*46d57p*l Table 3.5 Computational summary for element 112 atom. 49 Element 113 (eka-thallium) Consideration of Lr aside, element 113 is the first official member of the 7p block of elements, elements which should exhibit the most severe spin-orbit effects in the periodic table. Such has been demonstrated in a seminal paper by Grant et al., dealing with such atoms using MCDF calculations.*^^ Element 113 and its cations have also been studied by Kaldor et al. , using his relativistic coupled cluster m e t h o d . ^ 3 Our purpose in the examination of this atom is to lay the groundwork for a discussion of bonding in its molecules- a subject that will be dealt with later in this chapter. The Ts^Gd^Oypl (Hartree-Fock) electronic configuration of (113) might be seen to lead quite readily to the formation of chemical bonds. And, in light of what we have seen of (112) we also might expect that such chemical bonds should be primarily single bonds, the 7s orbitals just being too stable to easily participate in such bonding. Such would be a manifestation of an enhanced inert-pair' effect. This is the observation that the stability of oxidation states two less than the group number (TU in the case of (113)) is enhanced at the bottom of the periodic table. Indeed, the inert-pair effect should be enormous for (113). At any rate, the bonding in such a molecule will certainly be greatly influenced by spin-orbit coupling. From the information in Table 3.6, it is clear that spin-orbit effects are quite large, amounting to a greater than 3.5 eV energy difference between the 2pi/2 (J=l/2 (-))) and (J=3/2 (-)) spin- orbit states. This value is more than three times the experimental value 0.966 eV for the next lightest Group III element, thallium .75 The question then becomes at what point do the spin-orbit components more closely resemble different electronic shells than components of the same 50 (nonrelativistic) shell. In other words, to what extent do conGguration labels like 7pl cease to be truly descriptive of the valence electronic properties of the atom. These results suggest that such a label is not really useful at all for element 113. 51 gCE Hartree Fock Electron Configuration: 6s^p^6d^^7s^7p^ ECP / Basis Set Contraction Scheme: 92-electron ECP from Chapter 2/ (6sd6plf)/[6sd5plfl Hartree Fock State: Hartree Fock SCF Energy (Hartrees): -156^32889 Configuration Interaction Computational Double Group Symmetry: D2h Number of Virtual Orbitals: 27 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s27pl / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions Ei /2u 4720 46792 Results State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=l/2(-) -156.48141 7 s27 p*1 J=3/2 (-) -156.35127 3.541 7s27pl Table 3.6 Computational summary for (113) atom. 52 Element 114 (eka-lead) The as of yet undiscovered Element 114 occupies a unique position among transactinide elements because of the prediction that the nucleus with Z=114 and N=288 should exhibit enhanced stability owing to a double magic number' of nucleons.^G These numbers correspond to closed nuclear shells and the associated stability of such is analogous to the chemical stability of atoms with closed electronic shells. Element 114 is also interesting as an analog of Pb and ultimately of C. We saw that for element 113, spin-orbit coupling led to a 3.5 eV sphtting in the ground state of the free atom and a ground ^Pi/2 ground state. This state is best described as resulting from dominant 7p*l (Jssl/2) electronic (spinor) configuration. The additional electron of element 114 completes the filling of this 7p* shell and would be expected to result in a J=0 ground state. The data in Table 3.7 certainly bears this suggestion out with a better than 3.8 eV difference between the J=0 state and the next highest state which results from a 7p*2 -> 7p*l7pl excitation. This transition energy compares to a value of 0.969 eV for the equivalent transition in Pb so it is clear that the stability of the 7s^7p*2 configuration is greatly enhanced in (114) compared to its lighter homolog.In this regard, element 114 closely resembles a closed shell atom, an observation that has led Pitzer, among others, to predict that such an atom might actually be an inert gas.78 The only p2 configurational microstates that can contribute to the J=0 ground state are p*2 and p^(=p3/2^) so it is clear that the spin-orbit splitting of the p manifold leads to a kind of secondary period of the 7p block characterized by the complete filling of the 7p* 'subshell'. 53 gCE Hartree Fock Electron Configuration: ECP / Basis Set Contraction Scheme: 92-electron ECP firom Chapter 2/ (6sd6plf)/[6sd5plfl Hartree Fock State: 3p Hartree Fock SCF Enorgy (Hartrees): -173.718785 Configuration Interaction Computational Double Group Symmetry: D2h Number of \lrtual Orbitals: 26 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s27p2 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions Ag 10120 62722 B ig(=B 2g=B 3g) 10014 62616 Results State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=0 (+) -174.15651 — 7 s27 p*2 J=1 (+) -174.01496 3.851 7s^7p*l7pl J=2 (+) -173.99803 4.312 7s27p*l 7p l J=2 (+) -173.82485 9.024 7s27p2 Table 3.7 Computational summary for element 114 atom. 54 This situation is exemplary of the breakdown of Russell-Saunders (LS) coupling at the heavy element limit. One generally considers that the atomic terms stemming from a p2 configuration are ^P2,l,0. ^D2> and ^Sq . Of these, Hund's rules would dictate that ^Sq is the highest in energy but such a label is probably the most appropriate of these term symbols for the description of the (114) ground state. Of course, we don't really expect LS coupling to work for these atoms, but this phenomenon certainly underscores the point. Element 115 (eka>bismuth) Given what we have just seen about the stability of the 7s2?p*2 configuration of element 114 and the resulting secondary periodicity in the entire 7p shell, it stands to reason that the dominant electron configuration of element 115 should be 7s^7p*27pl The additional electron in the 7p(=7p3/2) orbital would be comparatively easily lost compared to the other two p (7p*) electrons. Such is seen in bismuth which has a well-known stable +1 oxidation state and this oxidation state in (115) should be even more stable relative to the +3 state. Such a suggestion is supported in the results listed in Table 3.8. The lowest energy electronic transition involves the promotion of an electron from the 7p* spinor orbital in the J=3/2 (-) ground state to the 7p spinor orbital. We see that this energy is quite large, more than 5 eV, an energy with suggests that while (114) is mostly inert (115) is decidedly univalent. In fact, element 115 might more closely resemble an alkali metal than a main group element. While it is difficult to directly compare the 55 SCE Hartree Fock Electron CJonfiguratioii: ECP / Basis Set Contraction Scheme: 92-electron ECP from Chapter 2/ (6sd6plf)/[6sd5plf| Hartree Fock State: ^ Hartree Fock SCF Energy (Hartrees): -194.60437 Configuration Interaction Computational Double Group Symmetry: D2h Number of Mrtual Orbitals: 16 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s27p3 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions E1/2u 2876 18056 Results State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=3/2 -193.00135 -- 7s^7p*^7pl J=l/2 -192.80666 5.298 7s27p*l7p2 J=3/2 -192.77384 6.190 7s27p*l7p2 J=5/2 -192.75137 6.802 7s^7p*l7p2 J=l/2 -192.72174 7.609 7s27p*l7p2 J=3/2 -192.63362 10.007 7s27p3 Table 3.8 Computational summary for element 115 atom. 56 electronic states of (115) with those of Bi, it is again interesting to note that there is a complete breakdown of the Russell-Saunders coupling scheme. This is seen in the fact that although both atoms have a Js3/2 ground state, the ordering of the excited states is dramatically different. For Bi the J values of the (all odd) excited states are, by increasing energy, 3/2, 5/2,1/2, and 3/2.79 The first two of these Bi J values are the spin-orbit components of the 2 d LS term while the second pair arise firom the 2p LS term. In addition, the spin-orbit splittings of these LS terms amounts to about 0.5 eV and 1.4 eV and the energy difference between these terms is around 1.7 eV. In the case of element 115 the ordering of excited states is quite different and it is not straightforward to discern the LS terms giving rise to these states. Element 116 (eka polonium) If any more evidence is required of the effects of secondary periodicity' in the 7p block, it should be satisfied with the results of atomic SOCI calculations for element 116. In the case of element 115, the lowest possible energy transition involved the promotion of an electron firom the 7p* spinor orbital to the 7p spinor orbital. For element 116, we have possible transitions (selection rules aside) among the spin-orbit states of the 7 s27 p* 2yp 2 configuration. From the information in Table 3.10 it can be seen that the J=2 ground state differs by 1 eV from its J=0 configurational counterpart. While this is still a reasonably large difference, it is quite small compared to those we saw in (114) or (115) and quite comparable to the analogous transition in Po (0.89 eV).80 These are the only two states that can arise from the 7 s27 p* 2yp 2 57 SCE Hartree Fock Electron Configuration: 6s26p®6dl^7s27p4 ECP / Basis Set Contraction Scheme: 92-electron ECP &om Chapter 2/ (6sd6plf)/[6sd5plfl Hartree Fock State: Hartree Fock SCF Energy (Hartrees): -212.59835 Configuration Interaction Computational Double Group Symmetry: Dgh Number of Mrtual Orbitals: 16 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s^7p4 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions A g 6190 36874 Big(=B2g=B3g) 6090 36774 Results State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=2 (+) -212.92518 7s^7p*27p2 J=0 (+) -212.88829 1.005 7s27p*27p2 J=1 (+) -212.71394 5.760 7s^7p*l7p^ J=2 (+) -212.71021 5.851 7s27p*l7p3 Table 3.9 Computational summare for element 116 atom 58 configuration so the next transition involves the transfer of an electron firom the 7p* spinor to 7p. This energy is much higher, about 5.7 eV, which is much greater than the second excited state of Po, which lies about 2.09 eV above the ground state. So, just as (115) is predicted to have an extraordinarily stable +1 oxidation state, (116) should have a very stable +2 oxidation state. Element 117 (eka-astatine) With element 117 we near the end of the 7p-block and have an atom whose electronic state is characterized by a single electron hole. This atom can be seen as the hole equivalent of element 113 which has a single valence electron. Element 117 would be classified among the halogens, and the spin-orbit effect in this atom should be more severe than in any other atom below the 8p block. And indeed, this is found to be the case, the spin-orbit splitting in the p manifold being greater than 7.5 eV compared to a theoretical value of 2.7 eV for At.®l This should settle once and for all the issue of whether or not there is a chemical difference between the spin-orbit components of the 7p shell. We shall explore this difference further in the next chapter with a discussion of the tetrafiuoride of element 118, which is analogous to the known compound XeF4. 59 acE Hartree Fock Electron Configuration: 6s^p^6d^^7s^7p^ ECP / Basis Set Contraction Scheme: 92-electron EOF from Chapter 2/ (6sd6p lf)/[6sd5p If] Hartree Fock State: Hartree Fock SCF Energy (Hartrees): -233.90667 Configuration Interaction ComputationalDouble Group Symmetry: Dgh Number of ^rtual Orbitals: 25 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s^7p5 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions Ei/2u 10216 101368 Results State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character J=3/2 (-) -234.31777 -- 7s27p*27p5 J=l/2 (-) -234.03534 7.685 7s^7p*l7p6 Table 3.10 Computational summary for element 117 atom. 6 0 Transactinide Hydrides In this section, we will deal with the simple monohydrides of most of the atoms described in the previous section. The study of the interactions of these atoms with hydrogen provides a first insight into the nature of bonding in transactinide molecules, despite the fact that these hydrides are not particularly interesting or realistic species. The same hydrogen basis set was used in all of the calculations and is given in Table 3.11. Of particular interest is the effect of spin-orbit coupling on the nature of the bonding interactions given the demonstrated significance of these interactions in the corresponding transactinide atoms. In order to separate the various effects of electron correlation and spin-orbit coupling, in many cases Cl calculations were performed in which the spin-orbit potential was set to zero. Such calculations correspond to the fictional systems in which there is no spin-orbit effect but the level of electron correlation should be about the same as in the true SOCI calculation. Now, it is true that is not actually possible to rigorously separate spin-orbit and correlation effects, but this procedure goes a long way toward that end. These calculations omitting the spin-orbit effect are denoted by the term NOSO. 61 s Exponents Contraction CoefBdents 12.997698 0.019061 0.0 0.0 L9612909 0.134231 0.0 0.0 0.44446653 0.474487 1.0 0.0 0.12194430 0.509091 0.0 1.0 p Exponents Contraction CoefBdents 0.33792 0.09162 0.08002 0.473193 0.02472 0.579760 Table 3.11 Hydrogen basis set used throu^out this chapter. 62 Lr-H It remains to be seen how the ’non-intuitive' ^'Py2 ground state of Lr might effect its chemistry. Using a Dirac-Fock one center expansion method, Pykko has examined the electronic structure of LrH within the context of the lanthanide contraction and the so-called actinide contraction.®^ However, because his purpose was simply to outline trends along the series, his treatment of LrH was only cursory. The study did demonstrate that although the effective radius of Ac is -15 pm larger than that of Lr, the radius of Lr was only -3 pm greater than that in Lu. This indicates that the actinide contraction is actually more severe than the well- known lanthanide contraction; this is a topic to which we will return. The main purpose of the examination of this molecule is to determine the relative importance of p versus d orbital participation in the bonding in lawrencium compounds. The SCF energy of Lr-H is mininiized at an intemuclear distance of 3.85 Bohr. At that bond length, the major atomic overlap populations of the atomic orbitals comprising the a-bonding orbital are: 0.75 Lr(s), 0.17 Lr(d), 0.09 Lr(p), and 0.98 H(s). From this we can seen that the bonding interaction occurs primarily through an sd hybrid metallic orbital with little contribution from the valence p orbital. At the SOCI equilibrium bond length, this primarily sigma bonding orbital lies about 0.16 Hartrees below a filled primarily non-bonding molecular orbital 63 gCE Hartree Fock EtectnmConfiguratioi]: ECP / Basis Set ContcBction Scheme: Lr: 78-electron ECP 6rom Chapter 2/ (6sd5pf2p)/[5sd4pf2p] H: No ECP/ basis set from Table 3.11 Hartree Fock State: Re(SCF) (Bohr)= 3.85 Configiiratignlt Computational Double Group Symmetry: C2v Number of Virtual Orbitals: 15 DedBnition of Reference Space / allowed excitations: 6(sp)8(singles only)5fl^(singles only) 7 s2o2 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions A i 6556 22621 Comments: Re(CISD) (Bobr)= 3.82 Important Cl Coefficients: Reference Configuration at Rg: Co=0.94735 Co(NOSO)=0.96 Next Most Important Configuration: C(7s2 ~> p7c2)=0.0784 Table 3.12 Computational summary for Lr-H. 6 4 Lr-H -225.73 225.74 ECCISD) E(CISD) NOSO -225.75 5-225.76 'S-225.77 225.78 -225.79 225.80 -225.81 R(Lr-H) (a^) Figure 3.1 Potential energy curve of Lr-H in its ground (lz+ (0=0+)) state calculated at the CISD level with and without the spin-orbit operator. 65 which is composed of mostly Lr 7s, 6d, and 7p AOs. It is this mostly non bonding HOMO for which the 7p plays a more significant role. According to a MuUiken population analysis, the hydrogen is assigned a -0.18 charge while Lr is +0.18. This analysis also has the AO population of the Lr 6d as 0.52 and the 7p as 0.42 but as has already been mentioned, the bonding interaction occurs primarily through an sd hybrid orbital. Of course, the SCF calculation does not take spin-orbit effects into account although relativistic scalar effects are included through the use of the AREP. Still, as was seen in the earlier case of the Lr atom, the preference for p orbital occupancy over d orbital occupancy in the Lr valence is reproduced at the AREP-SCF level without the inclusion of the spin-orbit effects. In any case, the molecular SCF results indicate that Hoffinan's conjecture holds, that interaction with other atoms proceeds primarily through d orbitals. This interpretation is not at aU changed by the inclusion of spin-orbit and correlation effects in SOCI calculations. The coefficient of the reference configuration in the Cl expansion is nearly 0.95 indicating that the GHartree-Fock) reference configuration is a fairly good representation of the molecular wavefiinction. The next most important spatial configuration involves a double excitation from the mostly nonbonding HOMO of the reference configuration into each of two orbitals which aure best described as 7px, but these excitations have coefficients of about 0.079 and so do not significantly alter the bonding picture in this molecule. The bond length is only marginally shorter in the SOCI calculation than in the SCF calculation. In addition, the NOSO-CI potential energy curve closely mirrors the SOCI curve demonstrating a similarly small spin-orbit effect on the bond length. As we shall see later in 66 this chapter in the discussion of (113)H, transactinide monohydrides in which there is significant p character in the bonding orbitals tend to exhibit dramatic bond length changes upon the inclusion of spin-orbit effects. We can therefore conclude fairly confidently that despite the fact that in the Lr atom the p orbitals determine the ground electronic state, the d orbitals are of prime importance in bond formation. At first glance, the most obvious difference between RfH and LrH is presence in the former of an additional electron residing in a x or 5 atomic orbital. This is the defining electronic characteristic of this molecule and it leads to a fairly complex electronic spectrum. Unlike the case for most of the other molecules in this chapter, the equilibrium bond distance was not optimized at the SOCI level of theory. The bond length was instead chosen as 3.45 Bohr which corresponds to the SCF optimized geometry. A single point spin-orbit Cl calculation was then performed at this geometry. Under C2v. the computational symmetry of the molecule, molecular orbitals which are tc and S to with respect to the intemuclear axis span the same irreducible representations. Also, the open-shell coupling coefficients used in SCFPQ for each of these states are the same. This means that the calculation itself does not presume any particular A character. The SCF wavefiinction converges to a state but ultimately this is unimportant. The 2n(A=l, 2=1/2) and ^A(A=2, 2=1/2) states wül extensively mix under the influence of spin-orbit coupling and the state labels here under A2 coupling 67 SCE Hartree Fock Electron Configuration: 1 ECP /Basis Set Contraction Scheme: Rf: 92-electron ECP from Chapter 2/ (6sd5plf)/[5sd4pli] H: No ECP/ basis set from Table 3.11 ^rtree Fock State: 2n(2A) Bond Length: Taken as 3.45 Bohr Hartree Fock SCF Energy (Hartrees): -47.59804 Configuration Interaction Computational Double Group Symmetry: C2v Number of Virtual Orbitals: 16 Definition of Reference Space / allowed excitations: 6p®(singles onlylTs^n^jcl / doubles 6p6(singles only)?s2(y2cy*l / doubles Gp^Csingles oulylTs^a^ô^ / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions E i/2 6182 62222 Table 3.13. Computational summary for RfH. 68 Table 3.13. continued State Energy AE(E-Eo) (eV) Primary (Hartrees) Character 0=3/2 -47.76137 a2ôl(~60%), a^TC^ (-20%) 0=1/2 -47.75614 0.142 cZjcl (-12%); cy2a*l (-68%) 0=3/2 -47.73137 0.816 a2j[l (-60%); c25k~20%). 0=5/2 -47.72956 0.866 cy2sk ~ 80 %). 0=1/2 -47.72097 1.099 a2a*l (-68%); a2jtl (-12%) 69 are as inappropriate as those under LS coupling in transactinide atomic cases. The extensive miiring of II and A states is evident in the results tabulated above. The states with 0=3/2 arise hum coupling of the projection along the intemuclear axis of the spin of the open-shell electron (S) with the states of both II and A orbital angular momentum projections. Conversely, the states with 0=1/2 and 0=5/2 arise solely &om spin coupling to II and A A-states, respectively. They do mix with other configurations,however, but not with each other. These results seem to indicate that even though there is a great deal of II-A mixing, that the ground state of RfH is based primarily on the Il-type orbital function. If nothing else, however, the mixing of states in this molecule demonstrates that O is the only important conserved quantity in these superheavy hydrides. a i s m If RfH is an example of a diatomic transactinide molecule with one electron outside closed shells, then (IIO)H is the corresponding molecule with one hole' outside closed shells. And, as was the case in Rfil, the use of C2v as the computational symmetry does not designate the location of the hole to be in a 7C or 5 molecular orbital for the Haurtree-Fock calculation. Unlike in the calculations on RfH, the metal-hydrogen bond length was optimized at both the SCF and SOCI levels of theory. 70 SCE Hartree Fock Electron Configuration: ECP /Basis Set Contraction Scheme: (110): 92-electron ECP from Chapter 2/ (6sd5p lf)/[5sd4p If] H: No ECP/ basis set from Table 3.11 Hartree Fock State: 2a (2n) Re(SCF) (Bohr)=2.95 Configuration Interaction Computational Double Group Symmetry: C2v Number of Mrtual Orbitals: 16 Definition of Reference Space / allowed excitations: 7s26d536dji^a2 / doubles ReCClSD) (Bobr): 2.93 Computational Irrep Number of Spatial Number of Double Group Configurations Functions Ei /2 6870 54630 Results (at R=Rg(CISD)) State (parity) Energy (Hartrees) AE(E-Eo) (eV) Primary Character 0=5/2 -110.37538 --- 7s2(y2j[4g3(_g2%), 0=3/2 -110.35185 0.5913 7s2a2jc354(~50%); 7 s2ct2tc453(~32%) 0=1/2 -110.30154 2.009 7s2(j2;j3g4(^61%): 7slc2n4g4(_i6%). Table 3.14 Computational summary for (IIO)H. 71 (IIO)-H •109.90 ■E(SCF) E(CISD) - 110.00 ■=“E(CISD)"2 g110.10î I 12= 1/2 * 110.20 Q=3/2 ■110.30 0=5/2 ■ -110.40 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 R(dlO)-H) (^) Figure 3.2 Potential energy surface for several low-lying states of (IIO)H. The ground state is denoted by CISD while the first and second excited states are designated CISD 2 and CISD 3, respectively. 72 Comparison of the results of this calculation with the previous results on RfH as well as with recent results on the monohydride of platinum (Ft is valence isoelectronic with (110)) reveals some interesting results. In RfH, the fl=3/2 ground state was weighted in favor of contributions from 11 terms but still included significant contributions from A terms, the case in (110)H is quite different. The ground state in the latter, 0=5/2, arises solely from the coupling projections of orbital and spin angular momentum of the 5 hole. The first excited state of (IIO)H is a 0=3/2 state with contributions from both H and A terms, but this lies nearly 0.6 eV above the ground state— not really that much in absolute terms but quite a bit more than the analogous energy gap in RfH. It is the second excited state, 0=1/2, that is identifiable as arising solely from spin-coupling of the n terms. As for comparisons of the electronic structure and geometry of (IIO)H with that of its fighter cousin, PtH, there are several significant differences.®^ The ground state in both cases is 0=5/2 but in the case of PtH the first excited state is 0=1/2— the state arising solely from the localization of the hole on the dx molecular orbitals. This first excited state lies a mere 0.146 eV above the ground state- an energy gap comparable to that seen in RfH. It is the second excited state of PtH that has 0=3/2 and it lies more than 0.5 eV from the ground state. It seems reasonable to conclude that the enormous spin-orbit coupling in (IIO)H causes the II and A terms to mix to the extent that this 0=3/2 state is lowered in energy relative to the 0=1/2 state for the heavier molecule. Optimized bond lengths in the two molecules are very similar with the 2.93 Bohr of (IIO)H comparing with the 73 2.84 Bohr separation in PtH. It is interesting that the SCF optimized bond length of (IIO)H differs from the SOCI result by only 0.02 Bohr but this coincidence prohahly results from cancellation of spin-orbit and correlation effects. a iD H If the electronic structure of (110)H is defined by its valence hole, that of (lll)H is marked by the filling of this hole to give a nominally closed-shell molecule. The fact that element 111 is characterized by a filled valence s orbital and valence d hole along with the large energy difference between the associated states may have profound consequences for the chemistry of this atom, especially in comparison to others of its family The normal model for bonding in univalent coinage metal compounds involves the ionic loss or covalent sharing of the unpaired valence s electron with another atom or small group. It is not inappropriate, at least to a first approximation, to consider the this bonding to be similar to bonding in simple hydrogen-containing molecules. Of course, the chemistry of coinage metals is much more complex than that of hydrogen - the d orbitals cannot simply be ignored - but the point stands. For element 111, however, the situation changes. The open 6d orbital which would be energetically most amenable for bonding interactions is radially shielded from the chemical environment by the more stable, and less corruptible, 7s shell. 74 This is visible in the orbital plots for element 111 given in Appendices C and D in conjunction with the derived Gaussian basis sets. In a sense, but perhaps to a lesser degree and for different reasons, this situation is similar to the case in lanthanide and actinide molecules. In these, the f orbitals are partially filled and are energetically of valence type but are so radially contract compared to the s shell that they play only a very small role in determiningchemical behavior. This is the reason that the lanthanides, and to a lesser extent the later actinides, are renowned for their uniformity in properties along the period. Almost certainly, the chemical role of the valence d orbitals of element 111 is greater for that atom than that of the f orbitals of lanthanides, but the extent to which this is true is not clear. It is clear, however, that bonding in molecules containing element 111 must involve a good deal of sd hybridization. While not an interesting molecule from a chemical point of view, the study of (lll)H serves to illustrate the extent to which such hybridization plays a role in the bonding mechanisms in element 111. It begins the task or sorting out the result of the competition of the tendency for the open d shell to participate in bonding interactions with Pauli repulsions due to the closed s shell which discourage such interactions. In other words, will the energy required for sd hybridization be recovered by bond formation? Seth et al, have also studied (lll)H and where appropriate our results will be compared with theirs.84 75 gCE Hartree Fock Election Ginfiguration: 7s2(A^4 ECP/Basis Set Contraction Scheme: (111): 92-electron ECP firom Chapter 2/ (6sd5plf)/[5sd4plfl H: No ECP/ basis set firom Table 3.10 Hartree Fock State: Re(SCF) (Bohr). 2.89 Configuration Ii Computati Computational Irrep Number of Spatial Number of Double Group Configurations Functions A i 6573 38913 B e (C I S D ) (Bohr)=2.94 Ground State: 0+(lZ+) Major Configurations: Co= 0.91917 C(7i4->7t3cy*) = .1385 Table 3.15. Computational s u m m a r y for (lll)H. 76 -124.35 -124.40 -124.45 g-124.50 '1-124.55 124.60 -124.65 -124.70 -124.75 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Figure 3.3. Potential energy curve of the ground state of of element 111 hydride calculated at the SCF and SOCISD levels of theory. 77 At the self-consistent field level, it would seem that concerns that the magnitude of sd hybridization energy would be too large to be recovered by bond formation seem unfounded. The SCF dissociation energy is around 6.5 eV which more than compensates for any hybridization energy. With the inclusion of spin-orbit and correlation effects in the calculation however, this value drops to about 3.2 eV which compares very well with the value of 3.05 eV obtained by Seth et al., in their spin-orbit pseudopotential/coupled cluster work on this molecule. The methods are somewhat different, however, which may account for some of this discrepancy. The equilibrium bond length at the SOCI level of 2.94 Bohr also agrees with the other group’s finding of 2.84 Bohr. Again the difference in the method probably accounts for the discrepancy. It is not clear which method is more appropriate for this kind of problem. While the coupled-cluster method in conjunction with the use of their relativistic (energy-adjusted) pseudopotentials does a fine job of including the effects of dynamic correlation, the spin-orbit Cl technique seem to do a better job of treating correlation and spin-orbit effects on a par.85 Overall, the SOCI results reported here give properties that are more similar to those of AuH than do the results of the paper referenced above (for AuH, De=3.25 eV, Re=3.12 B o h r ) .8 6 In both cases, however, the values of De and Re for (lll)H and are much more similar to each other than they are to the hydrides of the other coinage metals. It is also interesting that both give bond lengths which are dramatically shorter than that in AuH. This is perhaps indicative of the central role that the 6d orbitals play in bonding in (lll)H whereas in AuH they are of only secondary importance. Naturally this is a result of the anomalous nature of the (111) ground state. 78 The fact that very different techniques yield qualitatively similar results is encouraging. Overall, however, it is found that the hydride of element 111 is very similar to that of AuH. (113)H The bonding in (113)H seems like it should be fairly straightforward, the single (113) 7p valence orbital would overlap with the hydrogen Is orbital and form a more stable sigma bonding molecular orbital. And, this is, in fact, the most appropriate description of this molecule at the SCF level, which gives an equilibrium bond length of 3.79 Bohr. Inclusion of correlation and spin-orbit effects however seems to complicate the matter. Figure 3.4 is a comparison of three 01 potential energy surfaces for (113)H differing by the sizes of the Cl expansion and the omission in one of them of the spin-orbit potential. The curves labeled CISD and CISD NOSO were done using a reasonably small configuration list (s.e. from Table 3.16) but the important feature is how they relate to each other. The third curve labeled CISD mcd (an arbitrary label) was done using a more extensive Cl expansion Q.e. in Table 3.16). The SCF bond length of (113)H is found to contract by about 0.11 Bohr upon the inclusion of correlation without the presence of the spin-orbit operator. However, the analogous calculation including the spin-orbit operator shortens the bond length from the SCF level by 0.42 Bohr. So, inasmuch as we can distinguish between these interrelated effects, we can seen that the spin-orbit is responsible for an approximate 0.31 Bohr bond length contraction compared to the calculation including the same degree of correlation but without the spin-orbit operator. This observation makes 79 sense in light of the contracted nature of the 7p* spinor orbital as opposed the 7p*/7p average. This was not seen in Lr, the other 7 p l’ element because the Cl mixed in states of the relativistically' expanded 6d orbital and the net effect was no dramatic spin-orbit shortening of the bond. In (113) all of the 6d states are filled and only the p orbitals make any significant contribution, thus the effect that we see. Having demonstrated that spin-orbit coupling in this atom leads to a severe bond length contraction, the potential energy curve of (113)H was recalculated using the so-called large expansion which, in addition to the ground-state Hartree-Fock configuration, also included doubles from the excited «ylxl HF configurations. The d orbitals in element 113 are still close enough both radially and energetically to the valence orbitals that the effects of core-valence polarization involving them should not be disregarded.®® Especially, perhaps, in light of the known importance of relativistic indirect' d-shell expansion. A further 0.06 Bohr bond shortening is seen to result from the inclusion of excitations from the configurations— the larger SOCI expansion leads to fiirther refinements in the treatment of correlation and especially spin-orbit effects. In both Figure 3.4 and 3.5 these refinements are seen to flatten out the potential energy surface and weaken the bond. 8 0 gCE Hartree Fock Electron Configuratioii: ECP/Basis Set Conbracti Conficurationli: Computational Double Group Sÿnunetry: C2v Number of ^%tual Orbitals: 18 Definition of Reférenœ Space / allowed excitations: small expansion: Bd^^Csingles only)7s^o^ / doubles large expansion: 6dl0(singles only)7s2a2 / doubles 6dl0(singles only)7s2(rl%l /doubles Computational Lnep Number of Spatial Number of Double Group Configurations Functions small expansion(s,e.) Al 2350 7399 large expansion(l.e.) Al 6214 36829 Ground State: 0+(lz+) Be(CISD-s.e.) (Bohr) = 3.37 Re(CISD-s.e.NOSO) (Bohr) = 3.68 Re(CISD>largee^ansion) (Bohr) = 3.31 Major Configurations: Co= 0.71607 C(a2->07t) = 0.36099 Table 3.16. Computational summary for (113)BL 81 (113)H -156.75 E(CISD) I -156.80 E(GiSDrN0S(b— E CISD mcd @156.85 IM 56.90 g l 56.95 -157.00 -157.05 2 3 4 5 6 7 8 9 10 Figure 3.4 Three ground state potential energy surfaces of (113)H. The curves labeled CISD and CISD NOSO were both calculated using the s.e. eq)ansion and the latter was also calculated without the spin-orbit potential The curve labeled CISD mcd was generated using the more extensive (Le.) Cl expansion. 82 (113)H -156.65 I I I I I f r I I I I * r I I I I I 1 1 I 1 I 1 I I I I I 1 , 1 I 1 -156.70 ^156.75 §156.80 1 156.85 S i 56.90 -156.95 -157.00 -157.05 Figure 3.5 Potential energy surface of element 113 Hydride calculated at the SCF and SOCI-Le. levels. 83 (113)H -156.92 -156.94 g156.96 Ml 56.98 4— E(CISp) mdd -157.00 -157.02 4 5 6 7 8 9 10 R((113)-H) (^) Figure 3.6 A potential energy curve for (113)H which focuses on the large expansion SOCI calculation. 84 The experimental value of Re for TIH of 3.53 Bohr compares to the calculated value of 3.31 (SOCI-l.e.) for (113)H and the experimental value for De of 2.06 eV for TIH compares to 2.4 eV (also SOCI-l.e.) for (113)H.®^ This second value compares to a De of at least 3.5 eV for (113)H at the SCF level. We have already touched on the reason for the bond length shortening- severe spin-orbit (SO) effects. Because SO coupling is so much stronger in (113) than in Tl, it stands to reason that the bond should shrink more in the former than in the latter. The explanation of the bond weakening also hinges on the spin-orbit effect, but in a way not yet discussed. This explanation also has consequences beyond (113)H so it is worthwhile to discuss it now using this molecule as an example. In this next discussion I borrow liberally from a paper by Christiansen and apply it myself to the hydride of element 113.88 If we for a moment consider a scalar version of (113)H. we can see that the 7p orbitals of the heavy atom lead to two angular momentum projections along the intemuclear axis, IXI =0 (c) and 1X. I =1 (x). We can imagine that the singly degenerate a projection comes from the po (complex) atomic orbital while the doubly degenerate n projection comes from the equivalent p±i orbitals. In the case of a spin-orbit atom, the p* (=Pl/2) and p (=p3/2) spinors are linear combinations of all three of these complex spatial orbitals (along with appropriate spin functions and coupling coefficients). So, as these atomic spinors are resolved in an axial field, they retain neither pure sigma or pi character but are rather a combination of both. The p* spinor for instance has only an |o)|=l/2 angular momentum projection along the intemuclear axis. This spinor is therefore only singly degenerate, aside from the intrinsic Kramers' degeneracy, and 85 must exhibit a 2/3 x to 1/3 a spatial character in order to reflect the proper ratio of spatial functions in the linear combination. The p (p3/2) spinor, on the other hand, has two possible projections of angular momentum along the intemuclear axis, |ca|=l/2 and |o)|=3/2. Now, the |o)|=3/2 component can have no a spatial character whatsoever (because the combination X=0 and 0=1/2 can n ot give |o)|=3/2) so it must possess a purely x spatial character. Therefore the |m|=l/2 projection of the p3/2 spinor must possess 1/3 x and 2/3 <5 spatial character in order to preserve the overall 2:1 x to a ratio in the nonrelativistic limit. The rationale for the bond weakening in (113)H with more complete treatment of spin-orbit effects follows directly from this argument. The p* spinor orbital provides the major bonding interaction in the monohydride of element 113 as well as thallium. Because this spinor has only 1/3 a character, it is not well suited for interaction with a a-only ligand like hydrogen. For element 113, the energy gap between the spinors is simply too great to facilitate an interaction which would include the I co I =1/2 component of the ps/2 spinor and allow the recovery of full c character. So, as the treatment of spin-orbit effects is improved, the bond is weakened. It is somewhat curious, then, that the value of De for (113)H is higher than that in TIH. After all, in the case of Tl, the |o)|=l/2 (recall: 2/3 o, 1/3 x) component of the p3/2 spinor is close enough in energy to the pi/2 spinor that linear combinations of the two adlow for a better c-bonding interaction than would be the case for (113)H. So, although there is a spin-orbit weakening of the sigma bond in both (113)H and TIH, the larger spin-orbit effect in (113) relative to Tl does not result in a weaker bond. Clearly there are other factors involved. We shall see other examples of anomalously 8 6 high bond dissociation energies and possible explanations for these will be discussed at the end of the chapter. The SCF potential energy curve of (114)H exhibits a clear and well defined minimum at 3.69 Bohr. With the inclusion of electron correlation at the 01 level of theory excluding the spin-orbit effect, this bond is lengthened somewhat to 3.73 Bohr, a not too terribly large effect. As seen in Figure 3.7, the spin-orbit less Cl curve is somewhat flatter and shallower than the SCF curve indicating a lowering of the calculated dissociation energy from about 4.6 eV to about 2.7 eV. This can be seen purely as an electron correlation effect. It is dramatic then with the inclusion of both spin-orbit and correlation effect that the potential well for (114)H should become so very shallow and wide. This calculated dissociation energy hovers around 0.8 eV and the minimum energy on the potential surface (inasmuch as there is one) occurs at 3.93 Bohr. These values are in stark contrast to those of PbH for which Re is calculated at about 3.5 Bohr and De at about 1.55 eV.89 Actually, the explanation for this effect is really quite simple. As we say in the case of the (114) atom, a very substantial amount of energy is required to promote an electron from the 7p* spinor orbital to the 7p spinor. Such a promotion would be necessary in order to allow opening of the closed 7p* shell' and the formation of a bond between (114) and any other atom. In the absence of the spin-orbit effect, there is no such closed shell to be opened. This is yet another manifestation of the secondary periodicity mentioned earlier in this chapter. 87 gCE HartreeFock Electron Configuration: ECP /Basis Set Contraction Scheme: (114): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plfl H: No ECP/ basis set from Table 3.10 HartreeFock State: Re(SCF) (Bohr)=3.69 Configuration Interaction Computational Double Group Symmetry: C2v Number of Mrtual Orbitals: 17 Definition of Refierence Space / allowed excitations: 6dl0(singles onlylTs^a^Ttl / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions E i/2 5401 54568 Ground State: 0=1/2 ReCCISD) (Bohr): 3.94 Re(CISD-NOSO) (Bohr): 3.73 Re(CISD2) (Bohr): 3.40 Re(CISD3) (Bohr): 4.66 Table 3 .1 7 . Computational s u m m a r y for element 1 1 4 hydride. 8 8 Table 3.17. continued State Energy (Hartrees) AE(E-Eo) (eV) Primary Character 0=1/2 -174.44145 7s2a2%l(_60%). 7s2(yl(y*l(-21%); 7s2(jl;t2(-ll%) 0=3/2 -174.35974 2273 7s2c2tc1(~57%), 7s2a^Tc2 (-23%) 0=1/2 -174.34370 2.660 7g2(ylK2(-26%) 0=1/2 -174.28347 4299 0=3/2 -174.28027 4.386 89 -174.10 -174.15 >--174.25 1-174.30 — ETSCF).. -174.35 - -E(CISD) ■E(CISD)NOSO -174.40 -174.45 2 3.0 4.0 5.0 6.0 7.0 8.0 9.0 Figure 3.7 Potential energy curve for the ground state of element 114 hydride calculated at the SCF and CISD levels with and without the spin* orint potential 90 Further evidence supporting this interpretation is found in Figure 3.8 which is a potential energy curve showing several low-lying states of (114)H. The ground Q=l/2 ground state is only very weakly bound and corresponds to the interaction of the hydrogen atom and its unpaired electron with the (114) atom in its dosed-shell 7p*2 (JsO) ground state configuration. The fimdamental weakness of this interaction owes to the stability of this closed 7p* shell which we have already seen in the case of the (114) atom. The next two excited states having £2=3/2 and £2=1/2 respectively both exhibit relatively deep potential wells, especially in comparison to that in the ground state. These two states correspond to the hydrogen atom interaction with the (114) atom in excited states characterized by a 7p*^7pl electron configuration. Two other states are depicted in Figure 3.8 which are both unbound. All of these excited states correspond to separated atoms limits corresponding to (114) in excited electronic states. The potential energy minima of the two bound excited states occur at very different values of intemuclear separation with that of the narrower £2=3/2 state occurring at a radius more than a full Bohr shorter than that of the broader £2=1/2 state. The former state is easily interpreted as the interaction of the hydrogen atom with the 7p* spinor to form a sigma bond leaving the unpaired electron in the 7p(=7pg/2) spinor while the latter involves sigma bond formation of the hydrogen with the 7p spinor leaving the extra electron in 7p*. Of course, this interpretation is not strict, since the p and p* spinor orbitals both have |co|=l/2 projections along the intemuclear axis so they do interact with one another. But it does explain why there is such a large 91 difiference in the Re values of these two states; bond formation with the radially more compact p* spinor leads to a shorter bond and bond formation with the more expansive p spinor leads to a longer bond. The narrowness of the well for 0=3/2 is indicative that the interaction of hydrogen with the p* spinor is a shorter range interaction compared to that with the p spinor, a notion which also jibes with our understanding of their differing radial character. The depths of these wells are very similar amounting to about 1.6 eV, values very similar to De for the ground state of PbH. And in (114)H just as in the case of (113)H, these values are lower than the corresponding values in the spin-orbit less calculation owing to the spin-orbit ct/tc admixtures. To wit: the ct bond in the D=3/2 state involves the interaction of the hydrogen s with the (114) p* (|oj|=l/2) spinor component which we have already seen to possess only 1/3 a character. The a bond in the £2=1/2 state is generated from overlap of the hydrogen s with the (114) p3/2 (|o)|=l/2) spinor component which has 2/3 a character. The energy difference between this interaction and the single electron in the lower energy 7p* (|o)|=l/2) spinor means that they do not interact significantly and the a bonding character of 92 E(CISD) 174.15 .J!L.E(-CISD) 2 -E(CISD) 3 -174.20 —.T.T.T.T.E(CISD ) 4 0^372 E(CISD) 5 §174.25 0 = 1/2 (114) (J=2,l,0) +H?S i /2) IM 74.30 Ml 74.35 0 = 1/2 0 = 3/2 -174.40 (114I.CJ*0.)...± H^Sj/2 ) 0 ^1/2 174.45 2.0 3.0 4.0 5.0 6.0 7.0 8.Ô o7o R((114)-H) (^) Figure 3.8 Potential energy curve for several low-lying states of element 114 hydride. The curve labeled CISD is the ground states. The first excited state is labeled CISD 2, the second CISD 3 etc. 93 the bonding is not substantively enhanced. In both cases, the c character of the (114) spinor orbital is reduced compared to the same problem in the absence of the spin-orbit effect. (115)-H The closing of the p* shell at element 114 means that the additional electron in (115) resides in a the 7p3/2 spinor orbital. The bonding interaction of this atom with hydrogen atom then will occur by overlap of the |o)|=l/2 component of this spinor with the s orbital of hydrogen. The large energy gap between the 7p* and 7p shell along with the fact that the former is filled in (115) means that this bonding proceeds almost exclusively through the ps/2 (|0)|=l/2) spinor. As we have seen, this spinor orbital contains 2/3 a character, more than the in corresponding pi/2 spinor orbital for (113)H but still not enough to recover the pure sigma bonding character seen in the spin-orbit less case. Therefore, contributions from the p* (|co|=l/2) spinor that would allow the full recovery of a character will be minimal. This, in part, is the reason for the decrease in De from the SCF of 5.1 eV to the SOCI value of about 2.4 eV. 94 acE HartreeFock Electron Configuration: 6dl07 g2q2;[2 ECP /Basis Set Contraction Scheme: (114): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] H: No ECP/ basis set from Table 3.10 HartreeFock State: ReCSCF) (Bohr)=3.57 Configuration Interaction Computational Double Group Symmetry: Cgv Number of Mrtual Orbitals: 16 Definition of Reference Space / allowed excitations: small expansion: 6dl0(singles only)7s^c^TC^ / doubles large expansion: 6dl0(singles only)7s2a2jc2 / doubles Gd^^Csingles only)7G^j[4 / singles Computational Irrep Number of Spatial Number of Double Group Configurations Functions small expansionCs.e.) A i 4718 33734 B l 4697 33713 large expansionde.) A i 8798 46454 A2 8681 33734 Bi (=B2) 8681 33734 Table 3.18 Computational summary for (115)H. 95 Table 3.18 continued Ground State: 0=0+ ReCCIS) (Bohr): 3.75 Re(CISD*s.e.) (Bohr): 3.87 Results at RsRo and with large expansion State Energy (Hartrees) AE(E-Eo) (eV) Primaiy Character 0 =0+ -193.36545 7s2(j2TCxljiyl(-45%) 7 s2a27rx^/JCy2(~28%); 7s2ala*l7Cx^7tv^ (-11%) 0=1 -193.29502 1.917 7s2CT2jîxlTCyl(-47%); 7s2a^7t3(«.21%) 0 =0- -193.28165 2.280 7s2a27Cxl7iyl(~33%) 7 s2ct1ti3(-29%) 7s2al(T*lTCx^7tv^ (-9%) 96 In the discussion of (113)H, the bond length contraction upon the inclusion of spin-orbit effects was attributed to the difference between the interaction of the averaged p orbital with the hydrogen s orbital and the particular interaction between this s orbital and the 7p spin-orbit components. Because the bonding occurs primarily through the 7p* (|cd|=l/2) component the bond length is shorter. An extension of this argument to element 115 would have the bond length increasing with the inclusion of spin-orbit effects because the interaction is with the 7p (|û )|=1/2) spinor which is expanded relative to the average. This expectation is home out in these calculations. The SCF optimized bond length of 3.57 Bohr is seen to increase to 3.87 at the CISD level. The intemuclear distance was also optimized at the SOCI singles (SOCIS) level which also gave an increased bond length compared to the SCF result of 3.75 Bohr. This SOCIS calculation can be considered to include a portion of the spin-orbit effect but without incorporating any electron correlation. This is because unlike a usual Cl calculation, in the spin-orbit Cl scheme, the single excitations do mix with the Hartree-Fock ground state configuration. This Hartree-Fock determinant is optimized according to an SCF potential that does not include any portion of the spin-orbit operator. Of course in the SOCI step, the spin-orbit term is included and the SCF orbitals are not stationary with respect to this new operator. The portion of the Cl matrix that involves matrix elements of single excitations with the reference determinant are therefore not necessarily zero. Correlation effects, on the other hand, are only indirectly included at this level because the (rij)'l terms in matrix elements involving the reference are still 97 identically zero. That is not to say that the whole of the spin-orbit effect is included in such a way; this is just a first-order inclusion of them. At any rate, the SOCIS calculation demonstrates that the better part of the bond length expansion is directly attributable to the spin-orbit effect. -192.90 -193.00 g-193.10 E(SCF) E(CISD) ffl-193.20 -193.30 -193.40 4.0 5.06.0 7.0 8.0 9.0 Figure 3.9 Potential energy curve ofelement 115 hydride calculated at the SCT and CISD levels. 96 At the SOCI optiinized bond length the 0=0+ ground state, which corresponds to the closed (115) 7p* shell and the a interaction of the 7p (|©|=l/2) spinor with hydrogen, lies nearly 2 eV lower in energy than the lowest excited which has 0=1. This state results from a promotion of one of the 7p* electrons. The analogous transition Grom the 0=0+ ground state in BiH occurs at about 0.61 eV indicating that the tendency for closed p* shells in these isoelectronic molecules is much stronger in (115) than in Bi.90 This only makes sense in light of the relative spin-orbit effects in these two atoms. All of the methods that include spin-orbit effects give bond lengths substantially greater than the theoretical value for Re in BiH of 3.5 Bohr- again partially attributable to the increase in their magnitude. 99 -193.26 E(CISD) -193.28 @193.30 @193.32 -193.34 -193.36 3 4.0 5.0 6.0 8.0 9.07.0 Figure 3.10. A potential energy curve of element 115 hydride calculated at the CISD level (116)-H If element 114 hydride represents the case of one electron outside a closed molecular orbital shell then element 116 hydride represents the opposite, one hole outside a closed molecular orbital shell. Our model for 1 0 0 (115)H consists of a closed 7p* shell and a sigma orbital formed by the overlap of the 7p3/2 (|(a|=l/2) spinor and the hydrogen s orbital. The additional electron in (116)H then occupies the 7p3/2 (Iwl =3/2) spinor orbital resulting in a doubly degenerate £2=3/2 ground state. The equibbrium intemuclear distance in this ground state is lengthened relative to that in both the SCF and CISD-NOSO results. The interpretation for this finding is the same as for (115)H; the 7p3/2 spinors, through which the interaction occurs, are radially more expansive than the average, although the difiference is not as large as it is between 7p* and the average. The £2=1/2 excited state results firom the promotion of an electron firom the s bonding molecular orbital to the non-bonding 7p (|co|=3/2) spinor. This closes the 7p (|oj|=3/2) spinor shell but opens up a hole in the a molecular orbital resulting in a decreased bond order and gives a |co|=l/2 state. The dissociation limits of all three states depicted in Figure 3.12 all correspond to the hydrogen in its ground state and (116) in various states resulting from 7p*^7p2 configurations— again reinforcing the fact of the closed 7p* shell and the non-equivalence of the orbitals in the p shell. The value of De for (116)H is also lowered relative to that in the SCF case for all of the same reasons outlined earlier. The decrease amounts to 4.6 eV, from about 7.9 eV to 3.3 eV. The dissociation energy of the first excited state is very small supporting the assignment of this state as resulting firom the transfer of an electron firom a bonding orbital to a non bonding orbital. The second excited state is purely dissociative indicating the presence of an electron in the antibonding a* molecular orbital 1 0 1 SCE HartreeFock Electron Configuration: ECP/Basis Set Contraction Scheme: (116): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] H: No ECP/ basis set from Table 3.10 HartreeFock State: % Re(SCF)(Bohr)=3.49 Configiiration Interaction Computational Double Group Symmetry: C2v Number o f ~\^rtual Orbitals: 16 Definition of Reference Space / allowed mccitations: 6dl0(singles onlylTa^a^icS / doubles 6dl0(singles onlylTa^a^Ji^ / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions El/2 9192 88488 Ground State: 0=3/2 ReCCISD) (Bohr): 3.68 Re(CISD-NOSO) (Bohr): 3.53 Re(CISD2) (Bohr): 4.05 Re(CISD 3) (Bohr): 4.52 Results at R=R^(C1SD) State Energy (Hartrees) AE(E-Eq) (eV) Primary Character 0=3/2 -193.36545 7s2a27i3(~69%) 0=1/2 -193.29502 2.647 0=1/2 (?) -193.28165 4.536 Table 3.19. Computational summary for (116)H. 1 0 2 -212.60 E(SCF) -212.80 — JECCISDXNOSO. - E(CISD) « £-213.00 I M-213.20 -213.40 -213.60 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 R((116)-H) (sQ Figure 3.11. Potential energy surface of element 116 hydride calculated at the SCF, CISD, and CISD-NOSO levels. 103 -213.10 -213.15 ■E(GISD) E(CISD) 2 ^-213.20 "=""-~ErCrSDT3 g-213.25 1-213.30 Q = l/2 S-213.35 -213.40 -213.45 n=3/2 -213.50 0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 R((116)-H) (^) Figure 3.12. A potential energy surface of several low-lying states of element 116 hydride. 104 (117)-H The family of hydrogen halides are among the most studied class of molecules and (117)H represents its latest member. The valence hole in the ground state of (116)H is filled in (117)H and therefore the ground state is totally symmetric. For this reason, we shall not consider excited states of this molecule as we have for some other 7p block hydrides. It is interesting to compare the properties of this molecule with the others of its class especially in light of the extraordinary spin-orbit effects expected. Indeed, we have already seen that such dominate the electronic structure of the earher 7p-block hydrides. However, unlike those other 7p-hydride species which might be prepared and spectroscopically studied only in the gas phase, its closed-shell nature means that (117)H is a reasonable molecule which might be prepared (provided the problem of extreme nuclear instability could be solved). For purposes of comparison, the intemuclear distance was optimized with and without the spin-orbit operator using the small (I.e.) Cl expansion described in Table 3.20. Once again, we see that the inclusion of spin-orbit effects increases Re- This distance actually decreases a very small amount from the SCF value of 3.44 Bohr to 3.42 Bohr when correlation effects are included without the spin-orbit terms in the Hamiltonian. When these terms are included, however. Re increases to 3.58 Bohr. The bond length was also optimized with a much longer (I.e.) Cl expansion which gave a Re only 0.03 Bohr greater than the smaller expansion. All of these values are substantially longer than the 3.27 Bohr calculated for AtH.^2 This agrees with the trend in bond lengths in actinide monohaUdes which increase firom HF to AtH as seen in Fig. 3.15.^^ 105 SCE HartreeFock Electron Configuration: ECP/Basis Set Contraction Scheme: (114): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd4plf] H: No ECP/ basis set from Table 3.10 HartreeFock State: ReCSCF) (Bolir)= 3.44 Configuration Interaction Computational Double Group Synunetiy: C2v Number of Virtual Orbitals: 16 Definition of Reference Space / allowed excitations: small expansion: Gd^^Csingles onlylTs^a^x^ / doubles large expansion: 6dlO(singles only)7s2(y2;i4 / doubles 6d^0(singles onlylTs^crlx^ ;[*!/ doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions small expansionCs.e.) AI 4225 27105 large expansionCl.e.) AI 7945 63978 Ground State: 0=0+ Re(CISD-s.e.) (Bobr)= 3.58 Re(CISD-s.e.-NOSO) (Bobr)= 3.42 ReCCISDLe.) (Bobr)= 3.61 Cl Coefficients at Re(CISD*s.e.): Co(CISD-s.e.)=0.81646 Co(CISD-s.e. NOSO)=.962484 Co2(CISD-l.e.)=.654019 S c 2(g17C*1)=.156454 Table 3.20. Computational summary forC 117)BL 106 -234.45 -234.50 9 -234.55 à -234.60 E(CISD) E(CISD)NOSO -234.65 -234.70 2.5 3.0 3.5 4.0 5.0 5.54.5 Figure 3.13 Potential energy curve of element 117 hydride calculated with and without the inclusion of the spin-orbit potential 107 And once again we see that the bond dissociation energies decreases from the SCF value of about 7.1 eV to about 3.5 eV at the CISD Q.e.) level. It is striking that this 3.5 eV is more than a full eV higher than the calculated value of 2.19 eV for AtH.®^ This result is in opposition to the trend in bond dissociation energies of halogen monohydrides, also depicted in Figure 3.15, which fall from HF (6.13 eV) to AtH (2.19-calculated). The reason for this anomaly is not entirely clear but will be addressed in the discussion of a similar finding in the dissociation energy of (117)2- 1 0 8 -234.20 ■E(SCF) E(CISD) Ag MC •234.30 2-234.40 t *234.50 -234.60 -234.70 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 R((117-H) (^) Figure 3.14. A potential energy curve for element 117 hydride calculated at the SCF and large expansion CISD levels. 109 • De(eV) 2.00 o Re(Bohr) 7.00 1.80 6.00 u 1 60 -o- 5.00 (g A M 1.40 -O- 4.00 < ^ 1.20 3.00 1.00 o 0.80 2.00 HF HCl HBr HI HAt H(117) Figure 3.15. Periodic trends in equilibrium dissociation energies, De, and bond distances. Be, among the halogen hydrides. The results for AtH and (117)H are calculated while those of the other atoms are experimental 110 Dimers In the first two chapters we have examined the effects of spin-orbit coupling on the natures of the transactinide atoms and on the interaction of these atoms with hydrogen. Now we address the question of the interaction of transactinide atoms with themselves. Although only three dimers are examined in this section, they all have familial analogs among the ranks of the periodic table. (111)2 Perhaps as well as any transition metal, coinage metals are recognized for their propensity to dimerize in the gas phase.^5 This is most certainly due to the fact that Cu, Ag, and Au resemble hydrogen in that for the ground state atom, a single electron resides in the valence s orbital and all other orbitals are completely filled. As we have seen, however, such is not the case for the (111) atomic ground state. Still, the energetic proximity of the d and s valence shells along with the valence hole in element 111 should lead to a bonding interaction akin to those seen in earlier coinage metal dimers. There is a distinct minimum in the potential energy curve of element 111 dimer occurring at an intemuclear separation of about 4.66 Bohr. This value remains relatively constant with the differing levels of calculation, self-consistent field, small expansion SOCI, and large expansion SOCI. This is virtually identical to the experimental value of Re for Au2 of 4.67 Bohr.9® Similarly, the dissociation energy of element 111 dimer is calculated at about 2.3 eV, a value equal to the experimental De of Aug. I l l SGE Hartree Fock Electron Configuration: TA[*4g4g*4q:2 ECP /Basis Set Contraction Scheme: (111): 92-electron ECP from Chapter 2/ (6sd5plf)/[5sd4plf] Hartree Fock State: 1%+ Re(SCF)(Bohr)=4.65 Configuration Interaction Computational Double Group Symmetry: D2h Number of ^rtual Orbitals: 23 Definition of Reference Space / allowed excitations: small expansion: x47[*4g4g*4g2 / doubles lai^e expansion: ;j4jf * 4 5 4 5* 4^ 2 / doubles jj37t*4g4§*4^y2(j* 1 / doubles Computational Irrep Number of Spatial Number of Double Group Configurations Functions small expansion(s.e.) Ag 3031 10258 large expansiond.e.) Ag 9590 64538 Ground State: 0=0+ Be(CISD-s.e.) (Bobr): 4.67 Re(CISDLe.)(Bobr): 4.66 Cl Coefficients at Re(CISD-s.e.): Co2(CISD-l.e.)=0.7831106 Table 3.21. Computational summary for (111)2. 1 1 2 (111) -247.95 -248.00 « 2248.05 I E(SCF) M248.10 _T_E(CISD)_Ag -248.15 -248.20 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 R ((lll)-(lll))/2 (Sg) Figure 3.16. A potential energy surface of element 111 dimer calculated at the SCF and small expansion Cl levels. 113 This similarity of properties between (111)2 and Au2 is especially compelling in light of their very different atomic properties. Recall that the ground state of atomic gold in the gas phase is ^Si/2 resulting primarily from a 5d^®6sl electron configuration. As we have seen earlier in this chapter, the relativistic stabilization of the 7s shell and destabilization of the 6d shell of element 111 results in the filling of the former and the presence of a valence hole in the latter. This in turn leads to a ^D5/2 atomic ground state which is more than 3 eV lower in energy than the presumably more bonding-firiendly ^ 81/2 state in (111). It is also true that the closed 7s shell is more radially difiuse than the d shell and this might lead to concerns that the valence hole will be shielded from bonding by this stable s shell. Clearly this concern seems to be u n m e t in these results. Apparently enough sd hybridization occurs to allow the formation of a very strong bond. Clearly also, the mechanism of bonding in (111)2 versus Au2 is very different. In the latter, the bonding is very much like that in hydrogen, resulting from the overlap of two singly occupied s orbitals to form a a bonding molecular orbital and a a* antibonding orbital. The d orbitals in this case are relegated to only secondary importance and this is reflected in its potential energy curve. Das and Balasubramanian have calculated the potential energy surface for several low-lying states of Au2 and their results clearly indicate the dominance s orbital interaction for bonding in the ground electronic state.97 The separated atom limit of this state corresponds to the Au atoms in their ^Sl/2 ground state. Reassuringly, the separated atom limit in (111)2 has both in their ^D5/2 ground state underscoring the importance of valence d-d interactions in this molecule vis'-a-vis' Au2- 114 (Ill) -248.16 -248.18 CISD Le. -248.24 -248.26 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Figure 3.17. A potential energy curve of (111)2 calculated at the large expansion CISD level 115 We have not concerned ourselves here with problem of the various excited states of element 111 dimer. The purpose here has been simply to get a rough estimate of the equilibrium intemuclear bond distance and dissociation energy of this molecule compared to Au2. In further work, in will be interesting to compare the excited states of (111)2 with the theoretical results of Das and Balasubramanian for Au2 in order to gauge the degree to which spin-orbit coupling dominates the electronic structure in the (111) dimer. (113)2 Intuitively, the dimer of element 113 would seem to be among the simplest and most easily understood, as would that of TI 2. The simpHstic picture has the 7p (6p in the case of Tl) electron on each of the atom pairing with its counterpart on the other to form a single a bond and leading to a Q=Og+ (IZo with no spin-orbit eflfect) ground state. This picture is seen as inadequate, however, even in the case of TI 2 which has been shown by Strassinger and Bondybey to possess a very weakly bound ground state of Ou" symmetry. 98 The assignment of this Ou" ground state is supportive of calculations on several lower lying states of TI 2 by Christiansen.99 This state does not arise from a molecular orbital configuration but rather a configuration in which one electron has been excited from this c (|o)|=l/2) orbital to a x orbital (|o)|=l/2). The œ-O) coupling in the molecule then has the ground state being 0=0 but with the (-) reflection. This non-intuitive electronic nature of TI2 is also seen in (113)2. 116 SEE Hartree Fock Electron Ck>nfiguration: (o^o*^:Ac*4g4g*4)(y2(y*2(jl;[l ECP /Basis Set Contraction Scheme: (113): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] Hartree Fock State: Re(SCF) (Bohr)= 628 Configuration Interaction Computational Double Group Symmetry: D2h Number of \lrtual OAitals: 22 Definition of Refierence Space / allowed excitations: small expansion: fj2^2(y2 / doubles jj2(j*2q1;i1 / doubles / doubles çy2(j=t=2jt2 / doubles / doubles large expansion: d[(a2o*2j:4jc*4545*4)(singles only)] / doubles d[(G2o*2j^ji[*4§4§*4)(gingles only)] / doubles d[(a2a*2;:4jj*4545*4)(sxngles only)] / doubles d[(a 2o* 27i4jj* 454§*4)(siugles only)] a2o*2crl7c*l / doubles d[(a2c*2j[4j;*4§4g*4)(gingieg only)] a 2a* 27i2 / doubles d[(a 2o* 27t4jj* 4545* 4)(gingies only)] a2a*2jt*2 / doubles Table 3^2. Computational summary for (113)2. 117 Table 3.22 continued Computational Irrep Number of Spatial Number of Double Group Configurations Functions small expansion(s.e.) Au 5996 28826 large expansiond.e.) Au 6375 37160 Ground State: G=0* Re(CISD-s.e.) (Bohr): dissociative Re(CISDLe.)(Bohr): 6.78 Cl Coefficients at Re(CISD-s.e.): Co2(CISD-l.e.)=0.7831106 118 (113) -312.48 E(SCF) 0- -312.50 @312.54 -312.56 -312.58 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Figure 3.18. Potential energy curve of (113)2 calculated at the SCF level 119 The SCF ground state (113)2 is found to arise from a Hartree- Fock molecular orbital configuration and exhibits a profound bonding interaction with a D$ of about 2.2 eV and R@ of 6.28 Bohr. At the SOCI level, however, this potential well becomes very shallow indicating a dramatic relativistic (spin-orbit) weakening of this bond. Indeed, the equilibrium dissociation energy of the thallium dimer is calculated to be 0.16 eV and so one might not expect that the well for (113)2 to be as deep as it is at the SCF level.^9 The electronic properties of element 113 dimer were first discussed by Wood and Pyper using an REP based SCF procedure. They find a weak butnon-vanishing bonding interaction in accord with the expected similarity of Tl and element 113. This calculation was comparatively primitive, however, and the problem deserves to be addressed at higher levels of theory. Two different types of expansions were used to calculate the SOCI potential energy curves of (113)2 depicted in Figure 3.19. These differ mainly by the inclusion in one of single excitations from d-based molecular orbitals. It was found that without the inclusion of these d excitations, the (113)2 potential energy curve is entirely dissociative. With the inclusion of these excitations, however, a shallow potential minima of 0.19 eV at 6.78 Bohr is found. In their discussion of TI 2, Christiansen and Ermler discuss the need to include d-orbital excitations in their calculation and note that the lack of these in their calculations leads to potential energy curves which are probably a bit too repulsive. 101 The other substantive way in which these SOCI calculations differ is the number of restrictions placed on the occupancy of certain virtual orbitals. These additional restrictions were necessary when the d-singles were included in the calculation so that the 1 2 0 CI expansion did not unmanageably large. An interesting result of this procedure, however, is the observation that at long bond lengths, the energy of the s.e. SOCI potential energy curve is lower than that under the large expansion (including d-singles). Clearly, the importance of these d-singles decreases dramatically at these long bond lengths and the broader range of excitations from p and s based virtual orbitals dominates. No doubt that this owes, at least in part, to the fact that a single Hartree-Fock determinant is not a very good representation of the ground state wavefunction and with increased flexibility in the reference space this problem is mitigated. At any rate, these calculated values of De and Re imply that the electronic structure of (113)2 is very similar indeed to that of Tl2. The question remains about why the bonding in such pi systems should be so very weak. Actually, the answer to this has been touched on earlier in the context of (113)H. In the discussion of the bonding in that molecule, we attributed the weakening of the bond to the fact that the spinor character of the 7p* ( I w I =1/2) spinor orbital consisted of 1/3 a character 2/3 K character. Because the hydrogen is non-bonding with respect to k interactions, the reduced a character of the spinor reduces the bonding capabilities of the atoms. Further, the 7p3/2 (co=l/2) spinor component, which consists of 2/3 c and 1/3 x character, lies too high in energy to contribute significantly to the ground state and thereby increase the c character of the bonding molecular orbital. 12 1 (113) -312.65 -312.65 ^312.66 Eccism_Oz___. CO — — -E(CISD) mcd @312.67 U k312.67 @312.68 -312.68 -312.69 -312.69 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 R((113)-(113))/2 (a^) Figure 3.19. A potential energy curve for the Q=0* ground state of (111)2 calculated with two different Cl expansions. The curve labeled mcd included single excitations horn the set of 10 d orbitals. The other curve did not include these d sin^es but did have a more extensive virtual space. 1 2 2 The situation in (113)2 is quite similar. Note that the gerade combination of 7p* ( I o) I =1/2) spinor orbitals results in a molecular orbital which is 1/3 a bonding and 2/3 iz antibonding. The ungerade combination on the other hand is 1/3 a antihonding and 2/3 k bonding leading to a bond order of approximately 1/3. And in this case as well, contributions &om the atomic 7p (o)=l/2) spinor orbital in order to increase the a bonding capabilities of the p-shell are very unfavorable owing to the same large energy gap. In hghter atoms, of course, this is not a problem because of the energetic proximity of the 7p* and 7p atomic orbitals. This is essentially the same argument given by Pitzer et al., to explain the shallow potential well of Tl2 and it appUes just as well to this situation. ^02 Having said this, it is again prudent to point out that there are other factors involved in determining these bond strengths besides spin- orbit coupling. We have seen the values of De in both (113)H and (113)2 are higher than in the analogous thallium compounds. Earlier in this chapter this questions was considered in discussion of (113)H and now the answer for this is apparent in Figure 3.19. The inclusion of single excitations from the d manifold of orbitals introduced a shallow potential well into an otherwise totally repusive energy surface. The relativistic destabilization and expansion of the 6d orbitals of (113) relative to the 5d orbitals of Tl means that they play a much more active role in bonding compounds (113) than in those of Tl. Therefore, it can be seen that d-orbital polarization is responsible, in whole or in part, for overcoming the increased spin-orbit bond weakening of compounds containing element 113 and thereby leading to unexpectedly large De values. 1 2 3 (117)2 Just as (117)H was introduced as the heaviest hydrogen halide, so is (117)2 the heaviest halogen molecule. We saw in the case of element 117 hydride that the bond dissociation energy is anomolously high compared to that predicted by extrapolation of the known values for halogen monohydrides. It remains to be seen if the bond dissociation energy of this superheavy diatomic halogen is similarly rebellious. All of the known dihalogen molecules have distinctive and well-recognized colors, the molecules absorb at lower and lower wavelengths from ?2 to Ig and therefore their color changes from pale yellow for F2 to deep purple for I 2. It will be interesting to seen if the trend continues for (117)2, or for that matter even for At2. In the bond length optimization we see once again that there is a substantial increase in Re upon the inclusion of spin-orbit effects. The SCF and spin-orbit less 01 (small expansion) results give very similar values of 5.77 and 5.85 Bohr, respectively. The same s.e. 01 expansion including the spin-orbit operator gives an equilibrium bond length of 6.20 Bohr, a 0.35 Bohr increase over the NOSO-CI result that we can attribute directly to spin-orbit effects. An even greater increase to 6.27 Bohr is found with the longer configuration list in the SOOI calculation. This compares to intemuclear separations of 3.75 Bohr for OI 2 , 4.31 Bohr for Br2, 5.04 Bohr for I2, and 5.75 Bohr (calculated) for At2-^®^ These values are plotted for the halogens (disregarding F 2 which is known to be unusual) in Figure 3.22. 124 SEE Hartree Fock Electron Configuration: (a 2o* 27j4jjH545*4) Computational Irrep Number of Spatial Number of Double Configurations Group Functions small expansion(s.e.) Ag 2476 14545 large expansionQ.e.) Ag 6655 28168 Au 6585 27921 Blu 6585 27921 B2u(=BSu) 6585 27921 Table 3.23. Computational summary for (117)2. 125 Table 3.23. continued Ground State: O=0g+ Be(CISD-s.e.) (Bohr): 6J20 Re(CISD«e.NOSO)(Bohr): 5.85 ReCCISDLe.) (Bohr): 6.27 Cl (Coefficients at Re(CISD-s.e.): Cq 2(CISD-NOSO)=0.894880 Co2(CISD)=0.507085 ZC(o27t37c*4o*i)=o.234260 8 other spatial configurations with coefficients of about 0.10 R esults a t B^((C1S1 State Energy AE(E-Eo) (eV) Prim ary (Hartrees) A ssignm ent Og+ -468.1990176 c2o*2j[4j[*4(y2 ground state Ou'*’ -468.1469107 1.4179 K*—> a* Ou -468.1466164 1.4259 jc* —> a* lu -468.2564114 1.5618 K*—> a* Ou" -468.2802171 2.2096 a —> a* lu ■468.2824110 2.2693 (T—> a* Og+ -468.2910652 2.5048 K —> 126 (117) -468.08 -467.85 E(CISD) NOSO -468.10 I ^-468.12 ICR -467.90 O t -468.14 0 M-468.16 6 o cc O -468.18 -467.95 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Figure 3.20 Potential energy surface of the ground state of (117)2 calculated with (CISD A) and without (CISD-NOSO) the spin-orbit potential using the s.e. Cl expansion. 127 Also plotted in Figure 3.22 are the values of De for the dihalogens. As with the case of the halogen monohydrides, the dissociation energy of about 1.8 eV for (117)2 is anomolously large according to the periodic trends in the other dihalogens. Of course, this is greatly reduced from the SCF value of 3.5 eV but is still much higher than would be expected given the calculated value of 0.63 eV for At2; in fact this 1.8 eV is actually greater than the De of It is strange that the equilibrium bond length of (117)2 is predicted quite well by periodic trends among the halogens while the bond dissociation energy is not. We will return to this question at the end of the chapter. The original motivation behind the examination of this molecule was to predict its color. The electronic absorption bands that give the halogens their characteristic hues are generally assigned as a -> a* and k* > a* transitions. In the non-relativistic picture this amounts to a HOMO-LUMO electron excitation. As one proceeds down the periodic table from Cl2 to Br2 and then to l2, this transition occurs at lower and lower energies and hence the observed color of the dihalogen moves to shorter and shorter wavelengths. Several calculated lower-energy transitions for (117)2 are listed in Table 3.23 along with primary assignments. These results show two major bands of transitions, one centered around 1.5 eV and the other around 2.3 eV. The absorbed radiation corresponding to these transitions would then have wavelengths centered around 830 nm for first band and around 540 nm for the second. Transitions into this first band lead to colorless absorbtions but those into the second absorb greenish-yellow light which results in a purple-red appearance. It would probably be very much like l2 in this regard. 1 2 8 (117) -467.68 -468.13 -467.70 -468.14 E(SCF) |467.72 468.15 W467.74 O -468.16 S-467.76 k 468.17 «■467.78 468.18 ^ 4 6 7 .8 0 E(CISD) Ag MC -467.82 -468.19 -467.84 468.20 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 R((117)-(117))/2 (Sg) Figure 3.21 Potential energy surface of (117)2 calculated at the SCFand large expansion (Le.) CISD levels. In this diagram, the latter is denoted as CISDAgMC. 129 o De (eV) • Re (aO) 6.5 1 1 1 r I .... 3.0 4 > 6.0 ------cp ------2.5 4 » - 5.5 - - ...... c 2.0 O - (D 5.0 4b g e - C) 1.5 ^ 4.5 - - - 4 ► - 4.0 1.0 4 » 3.5 __i— i_i_ i_ 1 'III 1 1 1-1 0.50 CI2 Brg Ig Atg (1 1 7 i Figure 3.22 Periodic 130 We have seen that both (117)H and (117)2 exhibit bond dissociation energies which buck the trends of decreasing values of De as one descends the periodic table. We have justified the destabilization of bonds at the bottom of the periodic table by noting that spin-orbit splitting generates orbitals that are admixtures of a and x character. Pure c and x bonding occurs when these admixtures can themselves mix, a prospect which becomes less likely as the spin-orbit components of various spatial orbitals become energetically far removed fiom one another. Certainly this is not less of a difficulty for element 117 than it for astatine or other lighter halogens, so what can be the source of the enhanced dissociation energies in these two molecules? This question has prompted the réévaluation of the electronic structure of the (117) atom, as well as the other 7p-block atoms. Using a reference space consisting of single excitations into 25 virtual orbitals from the 10 6d electrons in configurations resulting from 7 more electrons distributed in all possible ways among the 7s, 7p, and 8s orbitals, a SOCI calculation was performed. States of both even and odd parity were calculated and the configuration fists consisted of 1369 and 1323 spatial configurations, respectively (7729 and 7971 double group functions). The somewhat surprising results are summarized in Table 3.24. We have already established earlier in this chapter the fact that there is an extraordinary difference in energy between the spin-orbit components of the 7p shell in element 117. We have used this difference to posit the argument that the 7p* and 7p shells are more appropriately considered independently and not as more or less equivalent as we do for fighter elements. What was not considered then and what presents itself now is 131 State (parity) Energy (Hartrees) AE(E-Eo) (eV) Prim ary C haracter Js3/2 (-) -234.1607344 — 7p*Z7pii Js5/2 (+) -233.9823179 4.854 7p*27p28sl J=3/2 (+) -233.9460126 5.840 7p*27p28sl J=l/2 (+) -233.9126181 6.752 7p*27p2ssl J a l/2 (-) -233.8836564 7.539 7p*i7p4 Table 3^4 Results of augmented SOCI singes calculatioiis for atomic element 117. the fact that the states resulting from the promotion of an electron from the 7p (7p3/2) spinor orbital to the virtual 8s orbital Ue lower in energy than the spin-orbit counterpart to the atomic J=3/2 (-) ground state. This finding argues compellingly for the possibihty that these low-lying, relativistically stabilized 8s orbitals participate to some degree in the bonding of these superheavy halogen elements. They are certainly more important for molecules containing element 117 than the 7s orbitals are for astatine containing molecules. Indeed, a reexamination of the sigma molecular orbitals with large coefficients in the Cl expansions shows that they do contain a significant amount of 8s character. The 8s orbitals are much closer to the valence than are the (full) 7s orbitals and it seems reasonable to propose that at least part of the enhanced bond dissociation energies is attributable to them. This is a relativistic shell-structure eflfect and so it should manifest itself in the non-spin-orbit calculations as well. Indeed, 132 the sigma molecular orbitals containing significant s character are found to be the most important configurations in those Cl expansions as well. This proposition merits and will be given further attention in future research.  similar reexamination of states of the other 7p-block hydrides demonstrates the increasing importance of the 8s orbitals as one moves fi*om left to right along the 7p series. In element 113, the J=l/2 (+) (2S1/2) arising firom the 7s^7p*l > 7s^ Ss^ transition lies 6.93 eV above the ground state. In (114), the J=0 (-) and J=1 (-) states arising fi-om the 7s^ 7pl Ss^ configuration lie 8.31 eV and 8.42 eV above the J=0 (+) ground state. This is still above all of the states arising firom 7s^7p*l7pl configurations but below states arising firom the 7s^7p2 configuration, like the second J=2 (+) state in Table 3.7. Similarly in (115), the J=l/2 (+) state arising firom 7s^7p*%s^ lies above all of the states arising firom both 7s^7p*^7p^ and 7s^7p*l7p^ but below that arising firom 7s27p^. A similar result is seen in (116). It is seen that states arising form the occupancy of the 8s orbital are interspersed with those arising from various occupancies of spinors of the 7p manifold for all such elements except (113). This is yet another example of the need to treat 7p* and 7p spinor orbitals as if they were separate shells entirely. This is the first example, however, of such an effect resulting not only from relativistic spin-orbit effects but relativistic shell-structure effects as well. 1 3 3 CHAPTER 4 RELATTVISTIC EFFECTS IN TRANSACTINYL FLUORIDES In the previous chapter, we examined the effects of relativity as it affects the electronic structure of atoms and the geometry of simple diatomic hydrides and dimers. In this chapter, this theme is extended to species which have fluorine as a common ligand. Fluorine compounds are among the most common and most widely studied. As a result, it is interesting to compare the electronic structure of the superheavy fluorides selected in this chapter with the more familiar molecules of the lighter elements. All of the calculations reported in this chapter were done with essentially the same fluorine basis set which is given in Table 4.1.104 134 sEzponents Contcaction CoeflBcientB 52.19 -0.009723 0.0 9.339 -0.1336174 0.0 1.182 0.6065861 0.0 0.3626 0.5077537 1.0 pFxpnnents Contraction Coefficients 22.73 0.0448314 0.0 4.986 0.2355939 0.0 1.182 0.5089400 0.0 0.3626 0.4578976 1.0 dEzponent Coefficient 1.666 1.0 Table 4.1. Fluorine basis set used throu^out this chapter. 135 fïH aliPdrgl WggafliinmniAfailliites of d-Blodc Tm ngflrtim d*» Atoms Almost any course in inorganic or even general chemistry includes a discussion of crystal field theory. Essentially, this theory attempts to explain features in the electronic absorption spectra and magnetic susceptibility of metal complexes, usually of high symmetry, by considering what effect the electrostatic field created by the ligands has on the (open shell) metallic orbitals. Although the crystal field model is simplistic, ascribing only a point charge to each ligand and ignoring other properties such as polarizability or x-donation, it does form the foundation of more complete theories. Ligand field theory combines the quaUtative and more chemically intuitive crystal field model with the more flexible and quantitative molecular orbital theory. It seems appropriate to examine the effect of such a ligand field on the electronic properties of d-block transactinides given the great effort given to understand such properties for complexes of lighter transition metals. The primary objective of this section is to get an idea of the magnitude of properties like the crystal field stabilization energy'. A, for d-d optical transition energies for some simple transactinide complexes which have well known analogs in the 3d, 4d, and 5d transition metals. It remains to be seen how these d-d optical spectra as well as ligand-to-metal charge transfer transitions are affected by the simultaneous action of ligand field and spin-orbit effects. Which, if either, dominates? and how do these two effects combine and interact? Another, perhaps more basic but no less important goal is to get some idea of the ionic radii of these atoms. Certainly, we can find the radial maxima of the various atomic shells from 136 an atomic calculation but it is not certain how these translate into actual ionic radii. The particular molecules of interest are octahedral hexafluorometallates (Figure 4.1) of some 6d transactinide metals, and among them are both closed-shell and open-shell species. The selection of fluorine as the single ligand common to all of these metals was made in order to minimize any covalent interactions between the metals and the ligand environment. Fluoride is a very hard ligand, it is quite non- polarizable, and as such more closely approximates a point charge than any other single atom. In addition, the electronegativity of fluorine (F is the most electronegative element) minimizes any ambiguities surrounding the assignment of formal oxidation states to the atoms in the molecules. One can say with some authority that the oxidation state of tungsten in WFq is +6 and that its d electron count is 0 (d^). The use of only one single anion in a variety of metal complexes allows a more direct analysis of how the ligand field and spin-orbit effects change molecular properties across the fourth transition metals series. 137 Figure 4.1 Model of a typical octahedral transition metal complex. 138 üieoretical considerations In the traditional analysis of the spectra of octahedral complexes the focus has been the splitting of the metal d orbitals with the imposition of an octahedral ligand field. In the heavier (5d) transition metals and surely in the 6d transition metals this ligand field splitting is complicated by spin- orbit coupling which further splits the d manifold. Part of this added complexity is a result of the necessary use of the octahedral double group in the symmetry analysis of electronic structure. There are two limiting cases which can be considered in such an analysis. In the first, spin-orbit coupling is disregarded and the five degenerate metal d orbitals are split only by the ligand field into two sets, a doubly degenerate set spanning the Eg and a triply degenerate set spanning the T2g representations of the double group. The octahedral field ensure that the latter wül be lower in energy than the former. In this case, explicit treatment of spin is unnecessary and the we must take care only to avoid violating the Pauli exclusion principle in occupying the molecular orbitals. The second limiting case takes the form of an atom in an infinitesimal octahedral field but experiencing the spin-orbit effect. In such a case, the ligand field splitting of the metallic d orbitals is nil and the non-degeneracy of the Eg and T2g sets of d orbitals is purely formal. The spin-orbit effect splits the d G.=2) atomic level into two spinor orbitals, the quadruply degenerate d3/2 (=d*, J=3/2=l-l/2) and hextuply degenerate ds/2 (=d, J=5/2=l+l/2). Under the labels of the octahedral double point group, the d3/2 spinor spans G3/2g while the ds/2 spinor is formally split into levels spanning G3/2g and E5/2g. As long as the field remains infinitesimal, these last two are 'accidentally' degenerate. As the ligand field is turned 139 / Ÿ 63/2+ A2 i "• f®5/2+ free (scalar) atom ^ ^“’*^3/2^ *2g free (scalar) atom 6 3 /2+ atom + spin-orbit atom + ligand field atom + 0]j + spin-orbit ligand field ê Figure 4.2 Qualitative orbital diagram depicting the effects on a free atom d orbital of an octahedral ligand field, the spin-orbit effect, and the simultaneous imposition of both. on', however, the G3/2g level of the dg/2 spinor can interact with that arising from the ds/2 spinor and the accidental degeneracy is lost. All of this is represented qualitatively in Figure 4.2. Finally, it should be noted that although all of the metal hexafluoride complexes studied in this section are taken to be strictly octahedral, they may in fact be subject to a Jahn-Teller d isto rtio n . ^^5 The high symmetry of the octahedron coupled with the fact that many of the molecules have open shells, and therefore degenerate electronic states, virtually ensures this. However, such possible distortions are not dealt with here and it is assumed that any changes in molecular geometry are small compared to vibrational amplitudes. We are encouraged in this assumption by the observation that various studies have demonstrated that Jahn-Teller effects in, for example, d^ ReFg and d^ OsFg are minimal compared to the energies of the electronic transitions.^®® Of course, future more detailed work in this subject will treat distortions in a discussion of vibronic coupling in 6d hexafluorides but for this preliminary work it is sufficient to disregard them. Computational Details In this chapter, as in the last, the calculations were performed using the self-consistent field program SCFPQ to generate the molecular orbitals used in the double-group, spin-orbit configuration interaction code CIDBG. The symmetry restrictions the SOCI implementation means that we cannot take advantage of the octahedral double group but must instead resort to the use of its D2h subgroup. The essential procedure for doing such a calculation is summarized in the introduction and so will not be repeated 141 here. The metallic relativistic effective core potentials used in the calculations of all hexafluorides are the 92-core electron, large core (LC), veiriety found in Appendix B. The metallic basis sets are Grom Appendix D and all use a (6sd5plf)/[6sd4plf) contraction scheme. The fluorine basis set from Table 4.1 is used for all MFgfl" molecules with the exception that the single d primitive has been omitted. The purpose of this omission is simply to save time in computation, and these functions should not be terribly important in the description of these highly ionic molecules. The geometry of each of the hexafluorometallates was optimized at the SCF level by minimizing its total energy as a function of M-F bond length. As noted earUer, the molecule is assumed to be octahedral so all bond lengths are equal and all M-F-M bond lengths are fixed at 90.00°. For open-shell molecules, the total energy is dependent on the molecular term and ultimately the number of electrons in the (non-spin orbit) t2g molecular orbital. For this reason, the term with the highest multiplicity in each (t2g)^ case was chosen as the ground state of the molecule. This SCF optimization procedure, of course, neglects the possible effects on bond lengths of both electron correlation and spin-orbit coupHng. Of these, the former is known to have only a minor effect on the geometry of complexes of this type. 107 Using the GAMESS quantum chemistry program ,which is used extensively in the next chapter, the bond length of SgFe was found to be only about 0.03  (or about 1.5%) longer at the MP2 level than at the SCF optimized geometry. This difference is too small to trifle with especially in the absence of experimental evidence. Although this calculation included correlation at the simplest possible level, spin- orbit effects were not included and thus they remain obscure. We have seen 142 in the previous chapter that the spin-orbit efifect can have dramatic efifects on geometric properties (and we shall see more evidence of this later in this chapter) so it is pertinent to consider them here. In order to determine the possible effects that spin-orbit coupling may have on molecular geometries, the bond lengths were also energy optimized within at the spin-orbit Cl singles level. Recall from our discussion in Chapter 3 that single excitations do in fact miv with the ground state Hartree-Fock reference determinant in spin-orbit configuration interaction calculations. Briefly, this exception to Brillouin's theorem results from the fact that the SCF orbitals are optimized with respect to a Hamiltonian which does not include the spin-orbit operator. Because the spin-orbit operator is a term in the SOCI Hamiltonian, these orbitals are no longer stationary to single excitations and can therefore contribute to the ground state wavefimction. This does not necessarily mean that all of the spin-orbit effect are included in such a calculation. A good portion of them are included, however, and will provide insight about the consequences of the spin-orbit operator in such problems. The SOCI-CIS optimizations were performed with the same molecular orbitals and reference spaces as were used in the calculation of optical spectra at the equilibrium geometries. The reference spaces for all d-d transition calculations were very similar. The active spaces consisted of single excitations from the 12 highest non-d(t2g) molecular orbitals in the SCF calculation in addition to double excitations from the (t2g)^ =(d^) manifold into 23 virtual orbitals. These 12 MOs include (mostly) non-bonding F px as well as F tlu pa orbitals. An exception to this convention is found in the case NsFg for which several different configuration fists were used. These differed in the 143 number of virtual orbitals and number of excitations allowed from the F based orbitals. When doubles were allowed frnm the F px and per orbitals, the lengths of the Cl expansions grew dramatically. Even in this case of NsFg where there is only 1 t2g electron to be considered, the problems can become unmanageably large. The other exception to the reference space convention involved allowing double excitations from the F-based levels of (d^) SgFg in order to allow the calculation of the energy of some ligand to metal charge transfer transitions. Except for these last two cases, single excitations only were allowed from the primarily fluorine orbitals in order to keep the configuration lists to manageable lengths. This restriction applied equally to all molecules provides a consistent framework and allows them to be treated on an equal footing. Table 4.2 summarizes relevant information about these Cl expansions. So, we have examples of the same molecule treated with differing Cl expansions as well as different molecules treated with equivalent expansions. 144 n Computational # o f # o f #ofDouUe Lrrep(aUare Spatial Configs Determinants Group Functions teerade) 1 Ei/2 1633 6223 6223 1-big Ei/2 10239 116727 116727 2 A 4536 31557 15786 Bi (=B2=B3) 4521 31542 15771 3 Ei/2 8461 74686 74686 4 A 9814 97733 48886 Bi (=B2=B3) 9775 97694 48847 5 Ei/2 8307 73503 73503 for S ^ 6 MLCT calculation n Computational # o f # o f fofD ouU e Irrep Spatial Confies Determinants Group Functions 0 Ag 4087 30277 15229 Big(=B2g=B3g) 4626 30096 14626 Au 4734 30204 15102 Biu(=B2u=B3u) 4734 30204 15012 Table 4^ Tnfnrmarinti pgrfaiining fai thp apin^rhit flT ralfriilatinns- 145 Results The SCF and SOCI-CIS optimized M-F equilibrium bond lengths are bsted in Table 4.3. What is striking about these results is the fact that spin- orbit coupling seems to make no difference whatsoever in determining these bond lengths. Indeed, the 1.5% difference firom correlation between the MP2 and SCF M-F bond lengths swamps any difference caused by spin- orbit effects. This finding should probably not be terribly surprising. The M-F bonds are highly ionic and the oxidation states of the metals are clearly all +6. The parts of the wavefunctions that describe the interaction of the fluorines and the 6d metal (as opposed to those parts that describe the metal 6s or 6p core' electrons) are dominated by fluorine orbitals for which the spin-orbit effect is minimal. If there are electrons remaining in the t2g manifold, the es/2 spinor orbital and lower gg/2 spinor, then they are predominantly metal localized and not greatly affected by changes in the M- F bond distance. Another striking feature of this data is the relative constancy of the M-F bond distance among the neutral complexes. Again, this should not be surprising. The previous argument about the lack of impact of spin-orbit coupling on the M-F distance relied on a picture of bonding in these complexes as primarily electrostatic. The additional d electrons in the t2g levels of these neutral molecules, or its spin-orbit components, simply serve to counterbalance the increased nuclear charge of the metal. This electrostatic balance does not, indeed cannot, apply to the two anionic complexes. The extra charge results in increased interelectronic repulsions between the metal and the fluoride ligands and ultimately to 146 longer bond lengths. Disregarding the spin-orbit efifect, the metal-based t2g orbitals are only weakly jc-antibonding with respect to fluorine (owing to its hardness) and the weakness of this antibonding interaction is also evidenced by this constancy. In addition, this means that spin-orbit efifects within the t2g manifold are, for the most part, not propagated to the ligands and thus their efifect on the intemuclear distances will be mitigated. We now proceed to discuss the various molecules according to their formal d electron count. Complex foimaldcount ReCSCF) (Â) Re(CIS) (A) HaFe- dO 1.97 1.97 S ^6 dO 1.88 1.88 NsFe dl s[(t2g)l] 1.88 1.88 HsFe d2 =[(t2g)2] 1.88 1.87 MtFe d3 =[(t2g)3] 1.88 1.88 MtF6-2 d5 M!(t2g)5] 2.01 (110)F6 d4 = Kt2g)4] 1.89 1.89 Table 43 Equilibrium metal-fluoiiiie bond lengths of several 6d Hexafluorometallates calculated optimized at the SCF and SOCIS levels. 147 d% HaFg"andS^6 These molecules are formally meaning that they have no electrons in the t2g (e5/2 + g3/2) orbital manifold. Because hahnium is a Group V metal, and therefore has a maximum +5 oxidation state, an additional negative charge is required in the hexafluoride to fill the F-based levels. This, of course, means that there is no d-d optical spectrum to be calculated. It also means, however, that there wül be no Jahn-Teller distortions and they should be strictly octahedral complexes.  metal- ligand charge transfer spectrum was calculated for SgFg and it will be discussed presently. In both of these complexes, the SCF and SOCI-CIS optimized M-F bond lengths agree to within 0.01 Â, 1.88  for SgFg and 1.97  for HaFg-. The variation of total energy with M-F bond distance is depicted in Figure 4.3. As noted above, it is not at all surprising that the anionic HaFg" complex should have a much larger bond length than SgFg. What is interesting is that in both molecules the metal fluorine bond distances are both only 0.05  longer than in the corresponding compounds of 5d transition metals. The importance of this difference should not be overestimated, it is quite common for SCF or even correlated calculations on metal-fluorides to differ from experiment by this m u c h . 108 What is relevant is the fact that the bond distances, and therefore the ionic radii, are remarkably similar for the 6d and 5d transition metal complexes. We shall numerous examples of this in Chapter 4. 148 SgF -207.83 E(SCF) -207.83 -#—"Ê(GIS) • M-207.85 -207.85 -207.86 3.45 3.50 3.55 3.60 3.65 3.70 R(Sg-F) (ag) Figure 4.3 Potential energy surface of SgFg calculated at the SCF and SOCIS levels and representing the variation of total energy with the metal fluorine bond distance. 149 The ligand-to-metal charge transfer (LMCT) transitions possible in SgFg result firom excitations firom the fluorine px and pa orbitals to the d- based t2g level. The lowest energy transition fi-om the filled primarily F pa (tlu) HOMO are seen to occur no lower than 10.6 eV. In fact, there were 12 electronic states calculated between 10.6 eV and 11.2 eV all of which must be assigned as charge transfer bands. These compare to the lowest such transitions in WFg which occur between 7.25 eV and 9.85 eV.109 both compounds these LMCT bands are best described as excitations firom the tlu M-F a orbital to the metal t2g ligand field orbital. The fact that these transitions occur about 3 eV higher in energy for SgFg than in WFg results firom the increased relativistic stabilization of the 7p orbital of Sg relative to the 6p orbitals of W. This energetic stabilization allows for a better interaction with F a orbitals and lowering the energy of the tlu HOMO. Some of this energy difference can also be attributed to the metallic d orbital expansion. This allows better overlap of these d orbitals with the F x orbitals thereby slightly stabihzing them and destabilizing the d-based t2g level. Still these x orbitals remain primarily non-bonding. The net result of the small destabilization of the weakly antibonding d-based t2g level and the stabilization of the F pa tiu level is an increase in the energy of the ligand to metal charge transfer. dl; NsFg This molecule gives perhaps the clearest picture of the simultaneous efifects of spin-orbit and ligand field efifects in transactinide hexafluorides. This is because as a Group VII transition metal analogous to rhenium, the atom in its +6 oxidation state has one electron remaining in the d-based t2g 1 5 0 ligand field level. The fact that there is only one 'optically active'd electron means that the double group symmetry of the electronic states are identical to the symmetry of the singly occupied spinor orbital. As a result, this problem represents the most pure example of d-d optical transitions that will be seen. The states involved in the electronic transitions are unambiguously determined by the symmetry of the singly occupied symmetry orbital. As with the case of HaFe* and SgFg, the optimized M-F bond length in NsFg under octahedral symmetry is the same at both the SCF and SOCI- CIS levels. Unlike the case in these molecules, NsFg would certainly be subject to a Jahn-Teller distortion just as is ReFg. But as mentioned earlier, this distortion is small and not considered here. The equilibrium intemuclear distance of NsFg is calculated to be 1.88  which compares to an experimental value of 1.83  for ReFg.^^O Again, we see a 0.05  difference between the theoretical result for the 6d transition metal hexafluoride and the corresponding experimental value in 5d fluorides. Compiled in Table 4.4 are Ai and A2 transition energies (defined in Figure 4.2) calculated for NsFg calculated with varying sizes of Cl expansions. Note that g3/2' refers to the higher level of that symmetry in Figure 4.2 which corresponds to the eg ligand field orbitals. Several features in this calculated spectrum present themselves. First, comparison of Ai and A2 between NsFg and ReFg indicate that the transitions in the Ns complex occur at much higher energies, about twice 151 -217.44 -217.46 -217.48 g -217.52 *-217.54 -217.56 -217.58 3.30 3.40 3.50 3.60 3.70 3.80 R(Ns-F) (a^) Figure 4.4 Potential energy surfEioeof NsFg calculated at the SCF anH SOCIS levels and representing the variation of total energy with metal- fluorine bond length. 152 as high. For ReFg, occurs at about 0.64 eV and A2 at about 4.03 eV compared to 1.35 eV and 8.07 eV for NsFg.m The ground state of NsFg, and for that matter of ReFg, is a G 3/2 state arising from the single occupancy of the g3/2 spinor orbital. This spinor orbital is derived from the spin-orbit splitting of the t2g molecular orbital. The increase in Ai is not at all surprising, since this is a transition between spin-orbit components of this t2g ligand field orbital. The spin-orbit splitting in d orbitals is about twice as great in Ns than in Re so the doubling of Ai between the two might fairly be expected on this basis alone. This interpretation is reinforced in the results of the calculated spin-orbit less transitions shown in Table 4.4. In the absence of the spin-orbit operator, Ai disappears entirely— as is expected given its nature. It is somewhat more surprising, however, that A 2 increases so dramatically from ReFg to NsFg. It is tempting to rationalize this observation by noting that because the lower g3/2 level and the upper g3/2’ level are of the same symmetry under spin-orbit coupling, they may strongly interact so as to dramatically lower the energy of g3/2 and raise that of g3/2'. This might explain the sharp increase in the gap between them. This is not found to be the case however, as is again seen in the spin- orbit less calculation. The magnitude of A2 is not diminished by the absence of spin-orbit coupling but is actually increased. We can see, then, that the doubling of the gap between the two G 3/2 states is due entirely to ligand field effects. The actual reason for the large A2 revolves around the nature of the g3/2 spinor orbital. Recall that it arises from the upper, d-based, eg, ligand field orbital which is itself the o-antibonding component of low-lying M-F a- 153 bonding orbitals. These a-bonding orbitals result from the overlap of fluorine pa atomic orbitals with the metallic da dx^-y^) orbitals. As it turns out, an analysis of the SCF molecular orbitals reveals that the relativistic expansion of the Ns d orbitals increases its overlap with the F pa orbitals and leads to a very strong bonding interaction relative to the non- relativistic case. As a natural result of this, the antibonding orbitals associated with this strongly bonding interaction are substantially raised in energy. The eg (d) to F pa interaction is strong enough that at the SCF level these bonding molecular orbitals lie a full 3.8 eV below the mostly F pa tlu HOMO. In octahedral complexes of lighter metals, there is a much smaller energy diflerence between this a-bonding molecular orbitals and the tlu orbital thus accounting for the difference seen in the calculated spectrum. # spatial # double Al(g3/2 -> 65/2) A2(g3/2 > g3/2’) functions groiq> functions 1633 6223 1.35 8.066 No spin-orbit i 1633 6223 N/A 8.606 1963 7363 1.35 8.078 douMes from F-based orbitals included i 10303 98323 1.32 8.807 9814 101884 1.32 8.743 10239 116727 1.32 8.612 Table 4.4 Optical transitions in NsFg with various Cl expansions. 154 The assignment and interpretations of these d-d transitions do not change substantially as the reference space is enlarged to include double excitations from the F-based orbitals. The value of Ai changes only marginally, lowered by about 0.03 eV, with the inclusion of double excitations from F-based orbitals. This merely reflects a more thorough evaluation of the spin-orbit operator with the inclusion of a great many more terms in the Cl expansion. The value of Ag does change somewhat more significantly, varying from about 8.8 eV to 8.6 eV however, but is still more than twice the value for the same transition in ReFg. At the highest level of calculation Ag is 8.612 eV which is probably the most reliable of the calculated results. Granted, this differs by 6.7% from the value calculated with the small 01 expansion, but this variation is small given the uncertainties in the calculation. In fact, it is satisfying that the small expansion does so well and differs from the much larger expansion by less than 10%. d2; HsFg Once again, for HsFg as for the other problems discussed in this chapter, we see that the SCF and SOCI-CIS optimized M-F bond distances agree to within 0.01  and measure 1.88 Â. And again we see that this bond distance is approximately 0.05  longer than in the corresponding 5d metal complex, in this case OsFg.^lO Unlike the previous problems examined, however, the ground state of the molecule is not uniquely determined by the electron configuration. Rather, the double occupancy of the d-based tgg HOMO in the SCF case or of the gg/g spinor orbital in the 155 SOCI-CIS case leads to a number of states. For the SCF optimization, a ^T2g ground state was assumed, with some justification, while Eg was chosen as the double group symmetry of state under the CIS optimization. These choices were found to correspond to the lowest states in both cases. HsF -228.20 E(SCF) 3T1 E(CIS) 2A1 (E) -228.25 -228.35 -228.40 3.30 3.40 3.50 3.60 3.70 3.80 R(Hs-F) (aj Figure 4.5 Potential energy surfimeofHsFgcaludated at the SCF and SOCIS levels representing the variation of total energy with metal-fluoiide bond distance. 156 The multiplicity of states arising from the ground-state electron configuration naturally has consequences for the electronic spectrum of this molecule. What is to be expected, and what is actually found is that there are two distinct groups of state arising from d-d electronic transitions in HsPg. The first set, dubbed the lower manifold' in Table 4.5, involves transitions between states arising from spin-orbit components of the (t2g)^ electronic configuration. This includes transitions between different states arising from the spinor configuration (g3/2)^ as well as those arising from the (g3/2)^‘°(e5/2)*^ configurations. It does not include, however, transitions between the spin-orbit components of (t2g)^"^ eg^ configurations; these are called upper manifold' transitions. The calculated transitions are presented in Table 4.5. 157 State AE(Ei-Eo> Assignm ent Lower ManifoldI1 Eg 0.0 (g 3/2)2 ground state T2g 0.037 (g3/2)2 Aig 1.099 (g3/2)2 Tig 1.378 (g3/2)l(e5/2)^ T2g 2.103 (g3/2)^(eS/2)^ Eg 2.192 (g3/2) 1(65/2)1 Aig 4.072 (65/2)2 Upper Manifold A(?)g 7.259 (g3/2)l(g3/2)'l T(?)g 7.410 (g3/2)l(g3/2)'l T(?)g 7.642 (g3/2)l(g3/2)’l Eg 7.665 (g3/2)l(g3/2)’l T(?)g 7.699 (g3/2)l(g3/2)’l + CT Eg 7.742 (65/2)l(g3/2)'l + CT T(?)g 7.822 (g3/2)l(g3/2)'l + ^T T(?)g 7.836 (65/2)l(g3/2)’l + ^T T(?)g 7.836 (65/2)l(g3/2)'l + ^T A(?)g 7.859 (g3/2)l(g3/2)’l + ^T Table 4.5 d*d Optical transitions in The states marked with a (?) were not unambiguously assigned. Those labeled with a + CT included configurations corresponding to metal-to-ligand charge transfers. 158 The first thing that presents itself in the results of Table 4.5 is the near-degeneracy of the Eg ground state (under double group labeling) with the T2g first excited state. The same near-degeneracy is seen in the electronic spectrum of OsFg in which the T2g state lies between 100 cm"^ and 250 cm"^ above the Eg s t a t e . ^^3 The energy difference in HsFg is closer to 300 cm'l but this feature is remarkably similar to both systems. In both cases, OsFg and HsFg, these states arise from the Cg3/2)^ spinor orbital configuration. There is a spin-orbit mediated difference between the next state in these molecules however. In HsFg, the second excited lies about 1.1 eV above the ground state and is of Aig symmetry. This is the last state arising from the (g3/2)^ spinor-orbital configuration. In OsFg on the other hand, the 2nd excited state is of Tig symmetry meaning that the magnitude of Ai was not sufficient to overcome the Coulombic repulsions of the electrons crammed into g3/2- The lower manifold transitions of HsFfi span 1-4 eV while the same transitions in OsFg range between about 0.5 eV and 2.15 eV.ll3 This again reflects the difference in the spin-orbit coupling of d- based orbitals between 5d and 6d transition metals. Upper manifold transitions in HsFg are similar in magnitude to Ai in NsFg - certainly for the same reasons. We see a large number of transitions spanning a range of 7.2-7.S eV and there are certainly quite a few more that have not been calculated. There is some question surrounding the assignment of upper manifold transitions in OsFg. Some authors attribute such transitions around 4.42 eV and 5.05 eV to d-d transitions while others assign them as LMCT bands. In any case, these transitions are still only about half of the upper manifold transitions in HsFg, an observation that agrees with what we have already seen in NsFg. 159 We do see a number of states which are mixtures of both charge transfer (F(jc) -> d(t2g)) and upper manifold transitions and those are indicated by + CT in Table 4.5. d3/d4/d5: MtF& (llO lFft MtFs^- The same factors that affect the geometry and electronic spectra of SgFe, NsFe, and HsFg also affect these metallic hexafluorides. We shall not go into the same detail for these as we have for the previous cases except to mention to make a few points. The calculated equilibrium M-F bond lengths for these three complexes are 1.88 Â, 1.89 Â, and 2.01 Â, respectively for MtFg, (llOlFg, and MtFg^". These compare to experimental values of 1.83, 1.85 Â, and 1.93  for their isoelectronic analogs IrFg, PtFg, and IrFs^-.ilo 160 MtF -239.95 -240.00 1-240.10 -240.15 -240.20 3.40 3.50 3.60 3.70 3.80 3.90 R(Mt-F) (ag) Figure 4.6 Potential energy sur£ace of MtFg calculated at the SCF and SOCIS levels and representing the variation of total energy with metal* fluorine bond distance. 161 The hexafluoride of Meitnerium, MtFg is a complex analogous to IrFg. Its G 3/2 ground state of arises from the (g3/2)3 spinor orbital configuration and lies nearly 1.5 eV below the first excited state which arises from the promotion of one electron from the gg/2 to the eg/2 spinor orbital. Several states arise from this excited (g3/2)^(eS/2)^ configuration, all lying between 1.5 and 3.1 eV from the ground state. The next state arises from a double promotion from g3/2 to es/2 and lies more than 4.0 eV above the ground state. These comprise the lower manifold. In IrFg, the ground state is also G 3/2 and the lower manifold transitions occur between 0.77 eV and 1.88 eV.H4 Again, we see an approximate doubling in magnitude of these transition between the 5d and 6d metals energies owing to spin-orbit effects. The upper manifold transitions in MtFg occur between 6.8 and 7.1 eV but the corresponding transitions in IrFg remain obscure. This is because in IrFg, charge transfer bands are thought mix with these levels and even with lower manifold transitions. In fact, some assignments have all transitions in IrFg above 3.7 eV as such charge transfer bands. This means that deconvolution of these transitions is difficult. An analysis of the Cl vectors for the upper manifold transitions in MtFg clearly indicate that these are not primarily charge transfer bands but rather d-d transitions. It has already been shown that such CT bands seem to occur at higher energies in transactinide hexafluorides compared to those of the 3rd transition series. For our purposes, it is sufficient to note that the calculated transitions and their assignments in MtFg are in qualitative agreement with what we have seen earlier in this chapter for other 6d complexes. 162 State AE(Ei-Eo) Assignment Lower Manifold G3/2 0.0 (g3/2)^ ground state G3/2 1.496 (g3/2)^ -> (g3/2)2(e5/2)l G3/2 1.999 (g3/2)3 -> (g3/2)2(65/2)l E5/2 2.125 (g3/2)2 -> (g3/2)2(e5/2)l E5/2 3.087 (g3/2)^ -> (g3/2)2(eS/2)^ G3/2 4.042 (g3/2)3 -> (g3/2)l(e5/2)2 Upper Manifold E(?)g 6.871 (g3/2)2 -> (g3/2)2(g3/2)’l G3/2 7.005 (g3/2)^ -> (g3/2)^(g3/2)’^ G3/2 7.036 (g3/2)^ -> (g3/2)2(g3/2)’^ Figure 46 d-d optical transitions in MiFg. 163 The hexafluoride of element 110, which is an analog of platinum, would be a d^ complex with an Ag ground state. This ground state arises from the complete occupancy of the g3/2 spinor orbital and should be Jahn- Teller inactive. The Ai transition energies resulting from the promotion of one electron from this g3/2 spinor to the eg/2 spinor orbital and their magnitudes are on a par with what we have seen in the other 6d hexafluorometallates. Two more groups of states are seen in the calculated spectrum, one which arises from the promotion of two electrons from g3/2 to e5/2 and another which corresponds to the promotion of one electron from g3/2 to g3/2'. The latter are upper manifold transitions. Lower Manifold Apr 0.0 (g3/2)^ ground state T(?)g 1.336 (g3/2)^ -> (g3/2)^(e5/2)^ T(?)g 1.788 (g3/2)^ -> (g3/2)^(e5/2)^ Eg 1.893 (gS/2)'^ -> (g3/2)^(e5/2)^ T2g 3.470 (g3/2)^ -> (g3/2)2(e5/2)^ Eg 3.628 (g3/2)^ -> (g3/2)2(es/2)2 A lg 4.955 (gS/2)'^ -> (g3/2)^(e5/2)^ Upper Manifold T(?)g 6.539 (g3/2)^ -> (g3/2)^(g3/2)’^ Eg 6.596 (gS/2)'^ -> (g3/2)^(g3/2)’^ T(?)g 6.605 (g3/2)^ -> (g3/2)^(g3/2)’^ A(?)g 6.633 (g3/2)^ -> (g3/2)3(g3/2)’^ T(?)g 7.247 (g3/2)^ -> (g3/2)^(g3/2)’^ Table 4.7 d-d optical transitions in (llOlFg. 164 Unlike what we see for (110)Fg, the ground state ofPtFg does not arise from a (g3/2)^ spinor orbital configuration but rather from a mixed ( (g3/2)^^e5/2W lower manifold configuration. As a result of this mixed configuration, the spectrum of PtFg exhibits three intraconfigurational bands between 0.37 eV and 0.74 eV. This compares to a lowest energy transition of more than 1.3 eV for (110)F6, a value well more than the rule- of-thumb double of the lowest such transition in PtFg. Clearly this is a result of the stability of the closed g3/2 spinor. The strength of spin-orbit coupling is great enough in (llOlFg to overcome the coulomb repulsions of the filled shell whereas in PtFg it is not. As a result of the difference in their ground states, it is difficult to directly compare the spectra of these molecules. State AE(Ex*E)o) Assignment Lower Manifold E5/2 0.0 (g3/2)^^es/2)^ ground state G3/2 1.915 (g3/2)'^(e5/2)^ -> (g3/2)2(e5/2)2 Upper Manifold G3/2 4.048 (g3/2)'^(eS/2)^ -> (g3/2)'^(g3/2)’^ E(?) 4.984 (g3/2)'*(e5/2)l -> (g3/2)3(e5/2)l(gS/2)’^ E(?) 5.049 (g3/2)'^(eS/2)^ -> (g3/2)3(e5/2)^(g3/2)’^ Table 4.8 d-d optical transitions in MtFg^-. 165 Finally, in Table 4.8 we present some the calculated transitions in MtFe^'. The table should be self-explanatory and is presented only to indicate the similarity in electronic structure and spectra among the 6d hexafluorometallates. Simple 7p-Block Fluorides: (113)F. (114)F. and (114)Fg. In Chapter 3 we examined the electronic structures of various 7p- block monohydrides. We now take what we have learned there and apply it to the monofluorides of element 113 and 114 and the difluoride of element 114. Recall that the major findings of Chapter 3 held that the severe differences in energetic and radial properties between the 7pi/2 (7p*) and 7p3/2 (7p) atomic spinors lead to the conclusion that any theoretical treatment of 7p molecules which neglects the effects of spin-orbit coupling also neglects the governing factor in their electronic structure. Electrons in 7p* and 7p spinors are wholly inequivalent and these shells must be treated as separate entities. The same factors that instructed our interpretation of the electronic and geometric structures of the 7p monohydrides also do so in our discussion of these monofluorides. 166 SGE HartxeeFoc^EleclroiiiConfiguration: 6d^^s^sa^F7C^G^ ECP /Basis Set Contractioii Scheme: (113): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] F: 2e- ECP from Reference 104 / basis set from Table 3.10 HaitreeFock State: lz+ Re(SCF)(Bohr)s4.29 ConfiguratioDh • I'.y Computational Double Group Symmetry: Cgy Number of Mrtual Orbitals: 12 Definition of Reference Space / allowed excitations: 6dl0(singles only)7s^F7i4(y2 / doubles 6dl0(singles only)?s^Fjc^gljg* 1 / doubles Computational Lrrep Number of Spatial Number of Double Group Configurations Functions A i 6567 44283 Ground State: 0=0+ ReCCISD) (Bohr); 4.12 Co2=0.799647 Table 4.9 Computational summary for (113)F. 167 There are two major differences between the monohydrides and monofluorides. The first is the much higher electronegativity of fluorine compared to hydrogen. This difference makes it much more likely that an electron will be transferred or shared between the 7p-block element and the ligand. The second difference is that in fluorine there is a valence pTC orbital whereas in H there is not. We saw in Chapter 3 that the spin-orbit mediated reduction of a character and increase in 71 character in bonds dominated by pi/2 spinor orbitals firequently resulted in M-H bond weakening. This weakening will be mitigated to some extent by participation in bonding by the F px orbitals. One can imagine a bonding orbital between (113) and F which is neither entirely x or c but has characteristics of both. At the same time, the orbitals that corresponds to the non-bonding F px in the non-relativistic case are, in the spin-orbit case, nonbonding orbitals which themselves are combinations of a and x. In (113)F there are two valence holes, one fi*om (113) corresponding to the vacancy in the 7p* shell, and one firom F corresponding to the open 2p shell. These holes migrate to the antibonding orbitals in (113)F and we have a stable single bonding orbital which is not well described as a or x but as a doubly occupied I co I =1/2 spinor orbital. This is the reason that there is not as severe a decrease in the value of De between the SCF and SOCI calculations for (113)F as there is for (113)H. This is apparent in Figure 4.7. 168 (113)-F -180.10 e— E(SCF) #— E(CISD) -180.15 t - 1 80.20 -180.25 -180.30 3.50 4.00 4.50 5.00 5.50 6.00 6.50 R((113)-F) (^) Figure 4.7 Potential energy surface of the ground state of (113)F calculated at the SCF and CISD levels. 169 From Figure 4.7 the value of De calculated at the SOCI level is estimated to be around 2.2 eV or 210 kJ/mol. This is somewhat lower that the estimated SCF value of 3.0 eV (290 kJ/mol) and compares to an experimental value of 444 kJ/mole for TTF.ffS The dissociation energy lowering between the SCF and SOCI levels is justified by reiterating that spin-orbit coupling introduces n character into the bonding orbital and that 7C interactions are intrinsically weaker than a interactions. However, because the dissociation energy is lower for (113)F in both the SCF and CISD cases, we are loathe to attribute this difference entirely to spin-orbit effects. Surely the relativistic contraction/stabilization of the whole p manifold decreases their overlap with the F p (114)F We saw in Chapter 3 that the electronic structure of element 114 is dominated by the closed 7p* shell. This domination manifests itself by a very large promotion energy from the closed-sheU J=0 state to states of higher angular momentum arising from 7p*l7pl configurations. We also saw a dramatic weakening of the bond in (114)H as a result of this shell 170 closing which we likened to a hydrogen atom interacting with an inert gas. For the same reasons that we saw in (113)F, the situation is somewhat different for (114)F. In (113)F we described the bonding as occurring by an interaction of metallic orbitals with F p orbitals which contained spin-orbit admixtures of ff and iz character. The major bonding interaction in (114)F is the same except there is an extra electron in an antibonding/non-bonding molecular orbital outside the closed I a> I =1/2 bonding spinor orbital. 171 SCE H aitreeFock Electron Ck>nfiguration: ECP / Basis Set Contraction Scheme: (114): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] F: 2e- ECP from Reference fiF/ basis set from Table 3.10 HaitreeFock State: Re(SCF) (Bohr)=4.13 Configuration Interaction Computational Double Group Symmetry: C2v Number of Mrtual Orbitals: 11 Definition of Reference Space / allowed excitations: small expansion: Gd^^Ts^Fsa^Fx^a^Tc*! / doubles large expansion: Gd^^Ts^Fsa^FTt^a^j^l / doubles 6d l 07 g 2pg(y 2Fj[4 (yl ;[^*2 / doubles Gd^Oy s2Fsc^Fx4(ylj^y*2 / doubles Computational Irrep Number of Spatial Number of Double Configurations Group Functions small expansion:* Ei/2 7178 120035 large expansion: Ei /2 9018 154G14 * used to calculate potential energy curves in Figures 4.8 and 4.9 Ground State: (1=1/2 ReCCISD) (Bohr): 4.24 ReCCISD 3) (Bohr): -5.5 ReCCISD 5) (Bohr): 4.47 Table 4.10 Computational summary for (114)F 172 Table 4.10 continued State Energy AEOS-Eo) (eV) Primary (Hartrees) Character 0=1/2 -197.82819 7s2Fs(y2Fjt4a2jtl • (-73%); next most important: 7s2Fso2F7c4(y2(y*l 0=3/2 -197.71212 3.158 (114)3=0 + FC2P1/2) 0=1/2 -197.710097 3.2139 0=1/2 -197.695122 3.6209 0=3/2 -197.682240 3.9715 7s2alx2 (-26%) 173 From Figure 4.8 it can be seen that there is a definite decrease in the value of De, firom about 4.4 eV at the SCF level to 2.4 eV and 1.6 eV at the SOCI-CIS and SOCI levels respectively. There is at least a decrease by half upon inclusion of spin-orbit effects, a phenomenon that was not seen in a similar progression for (113)F. This is certainly due in part to the continued stability of the 7p*^ configuration. It is interesting that there is a further approximate 0.8 eV drop in De with the insertion of double excitations into the Cl expansion. This is probably attributable to a combination of correlation effects and a more complete evaluation of the spin-orbit operator upon the inclusion of doubles. The equilibrium bond length is actually marginally shorter at the SCF level than at the either of the Cl levels. This is an effect which again points to the weakening of the (114)-F bond by spin-orbit and correlation effects. Figure 4.9 is a potential energy surface depicting the five lowest states SOCI states of (114)F. In it there are several striking feature which bear mention here. Upon comparison of Figure 4.9 with Balasubramanian's calculated PES of PbF, it is clear that despite their expected similarities, they are remarkably different. The assignment of the ground state as 0=1/2 is the same in both cases, with the equilibrium intemuclear distance in PbF calculated to be about 3.9 Bohr. When compared to the 4.29 Bohr ground-state Re of (114)F we see that this is the largest bond length discrepancy between compounds of 6th and 7th row elements that we have yet encountered. The depth of this ground state well 174 (114)-F -197.50 -197.55 _-E(CIS}_ - E(CISD) k-l 97.65 ••“••E ("S GF ) • 1-197.70 -197.75 -197.80 -197.85 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 R((114)-F) (^) Figure 4.8 Potential energy surface of element 114 fluoride calculated at the SCF CISD and SOCIS levels. 175 is also very different amounting to about 3.8 eV for PbF compared to the previously mentioned value of 1.6 eV in (114)F. In both molecules as well the first excited state is assigned as £2=3/2 but while in PbF this state is bound, in (114)F this state is found to be dissociative. Both the difference in the depth of the ground state wells and the difference in boundedness of the first excited states are understandable in light of factors we have already seen. The bonding in PbF is stronger because the smaller energy gap between the 6p* and 6p spinors more easily allows mixtures of them to form hybrids optimal for bonding. The much larger gap in (114)F means that such mixing is not favorable and the bonding is correspondingly weaker for all of the reasons we have already discussed. This larger gap in (114)F also points to the difference in the £2=3/2 first excited states. At the long Re limit, this state is dominated by F in a 2pg/2 its state and (114) in its closed shell J=0 configuration. At short bond distances however, the open-shell electron resides on (114) and only spinor orbitals arising firom 7p*l7pl configurations can generate an £2=3/2 projection. This open-shell configuration is of course very unstable in (114) and thus the state is dissociative. In both PbF and (114)F too, the second excited state has £2=1/2 and is the spin-orhit counterpart of the second excited state. And this second excited state is bound for PbF and dissociative for (114)F just as is the first excited state in either case. The dissociated atom limit of all three of these lowest states corresponds to the metal in a J=0 state and F in its 2pg/2 or 2Pi/2 state. Of course, the spin-orbit spHtting is so small that this difference is not noticeable in Figure 4.9. 176 E(CISD) ('Î'ÎA\ 17 “ E(CISD) 2 U 14M < E(CISD) 3 -197.55 — — - E(CISD) 4 -197.60 E(CISD) 5 0=3/2 (114) (J=2,1,0) + F(2P) S-197.65 0 = 1/2 6-197.70 -197.75 (114) (J=0) + F(2p) -197.80 0 = 1/2 -197.85 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 R((114)-F) (^) Figure 4.9 Potential energy surface for the five lowets states of element 114 fluoride calculated at to CISD leveL 177 The dissociated atom limit for both of the last two (third and fourth) calculated excited states of (114)F corresponds to (114) in an excited 7p*l 7p^ configuration and F in its ground state. This is why both of these states are bound with respect to the dissociated atom limit. The lower of these has 0=1/2 and the higher has 0=3/2. These were assigned on the basis of two features of the potential energy surface. The first of these is the presence of avoided crossings of the third excited state with the previously assigned, dissociative 0=1/2 state (dissociated atom limit: (114) J=0). These avoided crossings are especially clear in the short bond length limit of Figure 4.9. The second feature leading to these assignments are the locations of the potential minima. The bonding in the 0=1/2 state results firom the interaction of the I to I =1/2 component of the p3/2 spinor orbital with the F pa orbital. Because this atomic spinor has a radial maximum at a much greater distance firom the (114) nucleus than do spinor orbitals arising firom the pi/2 atomic spinor, the bond length in this state is correspondingly longer. The open-shell electron then resides predominantly in the I to I =1/2 component of the 7pi/2 atomic spinor orbital giving rise to the £2=1/2 state. The fourth, £2=3/2, excited state has a much shorter range interaction leading to its interpretation as the interaction of the 7pi/2 ( IÛ) I =1/2 ) spinor with the F pa orbital. Then the extra electron would reside in the I œ I =3/2 component of the 7p3/2 spinor orbital thus leading to the £2=3/2 state. These interpretations are reinforced by noting that the potential well of the £2=3/2 state which corresponds to the interaction of the 7p* orbital with the fluorine atom is much shallower than the £2=1/2 state for which this bonding interaction occurs through the 7p orbital. We know that a-bonding is the primary mode of bonding in (114)F 178 and have already seen that a-bonding is favored for interaction with ps/2 over pi/2- (114)F2 Having explored the electronic structure of 7p molecules having one bound ligand, it behooves us to examine one with multiple ligands. The difluoride of element 114 is strictly analogous to PbFg and provides us with an excellent example of such a molecule. PbF2 is known to adopt a C2v arrangement in its ^Ai ground state with a F-Pb-F bond angle of 95.6° and a Pb-F bond length of 2.129 Â.^18 The fact of this rather severe bond angle is commonly attributed to the inert pair effect, a nonbonding pair of electrons simply takes up a great deal of room. To see if the same sort of effect is seen in (114)F2, its geometry was optimized in a MP2 calculation using the GAMESS quantum chemistry package. At this level of theory, the (114)-F bond length is calculated to be 2.19  and the (114)-F-(114) bond angle fovmd to measure 94.6°. This similarity in the properties of these two atoms leads to the conclusion that the same factors that affect these properties in Pb also are in effect for (114). This MP2 optimization, however, does not account for spin-orbit effects which we have already seen have a profound effect in molecules of 7p-block elements. 179 SEE HartreeFock Electron Configuration: (alAbl)^ ECP/Basis Set Contraction Scheme: (114): 92-electron ECP from Chapter 2/ (6sd6plf)/[5sd5plf] F: 2e- ECP from Reference 104 / basis set from Table 3.10 HaitreeFock State: ^Ai MP2 Geometry: C2v R((114)-F) (Bohr)=4.155 0(F.(114).F) (degre%)=94.61 Configuration Interaction Computational Double Group Symmetry: C2v Number of Virtual Orbitals: 12 Definition of Reference Space / allowed «citations: (114)6dl07s2Fpjc8o(ai)2o(bi)2 / doubles Computational Lrrep Number of Spatial Number of Double Group Configurations Functions A l 6229 19297 A2 6084 19152 B l 6084 19152 B2 6084 19152 Table 4.11 Computational summary for (114)F2. 1 8 0 Table 4.11 continued. Ground State: Âi State Energy (Hartrees) AE(E-Eo) (eV) Primary Character Al 1. -221.71514500 --- ground state: 7s2(ai)2(bi)2 2. -221.54273742 4.69145 7s2(ai)l(bi)2(b2)l A2 1. -221.54566219 4.61197 7s2(ai)l(bi)2(b2)l 2. -221.52826765 5.08531 7s2(ai)2(bi)l(b2)l Bl 1. -221.53252965 4.96933 7s2(ai)l(bi)2(b2)l 2. -221.52402864 5.20066 7s2(ai)2(bi)l(b2)l B2 1. -221.54263103 4.69445 7s2(ai)l(bi)2(b2)l 2. -221.52824644 5.08587 7s2(ai)2(bi)l(b2)l 181 In order to address the possible effects of spin-orbit coupling on the geometry of (114)F2 we bave calculated at the SOCI level the total energy of the Al (double group label) ground state as a ffmction of F-(114)-F bond angle. This was done while bolding the (114)-F bond distance constant at the MP2 optimized value. The result is seen in Figure 4.10 is in quabtative agreement with the MF2 optimization with the minimum energy of the ground state occurring at a bond angle of about 92°. Naturally, the bond length will probably vary with the inclusion of spin-orbit effects but the point of this exercise was just to determine if any changes in the shape' of the molecule would result from these effects. It appears as though they do not. The last thing that will be mentioned about this molecules is that there is a large, 4.61 eV, energy gap between the closed-sbell Ai ground state of (114)F2 and the first excited state (which corresponds to a non- relativistic triplet state). This compares to an analogous energy gap of about 4.08 eV in PbF2.^^® In (114)F2 there are seven excited electronic spin- orbit states between 4.61 eV and 5.29 eV above the ground state while in PbF2 these states range between 4.08 eV and 6.35 eV above the ground state. Although this range is slightly broader in PbF these calculations demonstrate that unlike the relationship between PbF and (114)F, the properties of PbF2 and (114)F2 at first glance are very similar. The reasons for this will be explored in greater detail when time permits. 182 (114)F -221.50 E(CISD) -221.55 -221.70 -221.75 80.0 100.0 120.0 140.0 160.0 180.0 200.0 0 Figure 4.10 Potential energy surface of (114)F2 calculated at the CISD level depicting the variation of total energy with F-(114)-F bond angle. 183 XeF^ vs. (118)F^ Despite their name, the heavier inert gasses, especially Xe and Rn, have been shown to participate in molecule formation with highly electronegative fluorine. In doing so they provide us with a pedagogical example of the usefulness of the valence shell electron pair repulsion (VSEPR) model for the prediction of the geometry of XeF 4 . Briefly, VSEPR theory holds that xenon tetrafluoride adopts a square planar geometry owing to the presence of 6 valence electron pairs about the central Xe atom four of which are bonding pairs and two of which are lone pairs. Eight of these electrons come from the Xe valence s and p orbitals and four come from the F pa orbitals. According to the precepts of VSEPR theory, the template of bonding for six electron pairs is octahedral and furthermore, the lone pairs are arranged so as to minimize their angular overlap and hence their mutual repulsions. These two factors together dictate the square planar arrangement. This is illustrated in Figure 4.11. So, an interesting question arises upon consideration of the possible geometry of the tetrafluoride of element 118 which is isoelectronic with Xe. Naturally, the VSEPR model does not take into account possible relativistic effects and such effects will certainly have profound consequences for the chemistry of element 118. There is good reason to believe this, after all the 7p shell is more severely affected by spin-orbit coupling than any other valence shell to that point. Dirac-Fock SCF calculations indicate that there is a 0.44 Hartree difference in energy and a 0.9 Bohr difference in the radial expectation value between the spin-orbit components of the 7p s h e l l . ^ 2 0 184 a i #/ Figure 4.11 Nonbonding Hartree-Fock MO's of element 118 tetrafluoride. Their symmetry labels are appropriate to the Dg subgroup. 185 Recall also that the SOCI splitting of the 7p orbital shell in element 117 is nearly 8 eV. So given this substantial energy difference it might well be the case that the spin-orbit components will behave as different shells entirely. Indeed, there have already been numerous examples of the inequivalence of the 6 7p electrons and the spin-orbit 7p* 'shell dosing'. At the same time, the 7s shell should be dramatically contracted and stabilized relative to the valence region as compared to the 5s orbital of Xe owing to relativity. These questions were addressed through the use of spin-orbit configuration interaction calculations applied to (118)F4. Such calculations were performed at each of two major molecular configurations, the first having square planar symmetry and the second having a tetrahedral arrangement. Both of these geometries have Dg as a common subgroup so this was the symmetry under which the spin-orbit Cl calculations were performed. The SCF orbitals for the square planar (flat) geometry were chosen as they would be in XeF^. Specifically, four (118)-F sigma orbitals spanning Ai, Bi, B2, and B2 along with two non-bonding orbitals spanning Ai and B2 were used. For the tetrahedral configuration, however, the inadequacy of the SCF procedure was evident. The four (118)F o orbitals still span the same representation as in the flat geometry but the two non-bonding pairs present a problem. One can imagine a distortion of the flat structure in Figure 4.11 into a tetrahedron. Such a distortion would serve to delocalize the non-bonding electrons throughout the structure. Under this reasoning, the a i non-bonding orbital was retained and delocalized as Rydberg s-like orbital over the entire structure. Similarly, the h i non bonding orbital was a Rydberg p-like delocalized molecular orbital. The 186 required open-shell coefficient needed to describe such a state exactly correspond to those used for an atomic calculation in a p2 (3p) configuration. 121 Of course, such a single configuration technique is not really appropriate for such a case, all the more reason that the final results of the calculation are surprising. SCE Hartree Fock Electron Configuration: see text ECP /Basis Set Contraction Scheme: (118): 92-electron ECP firom Chapter 2/ (6sd6plf)/[5sd5plf] F: 2e- ECP firom Reference 104 / basis set from Table 3.10 Hartree Fock State: Configuration Interaction Computational Double Group Symmetry: D 2 Number of Virtual Orbitals: 7 Definition of Reference Space / allowed excitations: see text Computational brep Number of Spatial Number of Double Group Configurations Functions smaller expansion: Al 8092 43066 large expansion: Al 16908 118590 Ground State: Ai Table 4.12 Computational summary for (IISIF^. 187 The configurations chosen for the Cl expansion result from double excitations of 26 electrons firom 6 reference configurations into 7 virtual orbitals. The reference consists of all possible arrangements of the two p- like 'non-bonding' electrons in the three 'nonbonding' p-like bl, b2, and b3 orbitals. In such a way, all configurations important for both the square planar and tetrahedral geometries are represented. Despite the rather limited active space, more than 40,000 double group functions were generated. A larger configuration hst was also generated which included more virtual orbitals but also entailed more restrictions on the occupancy of filled molecular orbitals. Figure 4.12 depicts the effect on the total energy of (118)F4 of varying the (118)-F bond distance at the SCF level for each of these two configurations. At all bond lengths, the square planar geometry is seen to be much more stable than the open-shell tetrahedral configuration. This is exactly what we would expect for this under the assumptions of the VSEPR model and in analogy with XeF 4 . This same result is seen at the spin-orbit less Cl level as weU, as represented in Figure 4.13 (these are labeled as NOSO). With the inclusion of the spin-orbit effect, however, we see that the energies of the two configurations become virtually identical. That these energies should be so similar is surprising, especially given the fact that the orbitals in the Td case are not really optimal for treating this case and were generated in a somewhat ad hoc fashion. Notice too that the double group (spin-orbit) symmetry of the state is Ai, certainly not a ground state well represented by spatial orbitals resulting from 'p2-Uke' molecular orbital configuration. 188 (118)F -348.50 E(SCF) Td -348.60 - -E(SCF) flat -348.90 -349.00 3.5 4.54.0 R((118)-F) (^) Figure 4 .1 2 Potential energy surface of element 1 1 8 tetrafluoride calculated at the SCF level for the tetrahedral and square planar (flat) geometric configurations of (1 1 8 )F 4 . 189 (118)F 4 -348.50 E(CISD) Td -348.60 V -ElCISDrflat ^348.70 .-ECCISDJ-TdNOSO © E(CISD) flat NOSO g348.80 ^348.90 @349.00 H -349.10 -349.20 -349.30 3.5 4.0 4.5 R((118)-F) Figure 4.13 Potential energy surface of element 118 tetrafluoride in both its tetrahedral and square planar geometries. The calculations were done at the CISD level with and without (NOSO) the spin-orbit potentiaL 190 Still, not only are these energies quite comparable but as seen in Figure 4.14, a close-up of the view of the SOCI potential energy curve , that of the tetrahedral arrangement is actually somewhat more stable. The actual energy difference is calculated to be only about 0.003 Hartrees at the SOCI optimized Re of each, -349.22573 for the Td geometry and -349.22346 for the flat geometry. This is a small difference to be sure but the real surprise is that they are comparable at all. These Cl potential energy curves were all calculated with the smaller expansion described above. Even with this smaller expansion, the generation of these curves took an enormous amount of time. In order to gauge the results of increasing the Cl expansion length, single point calculations were done at the equilibrium bond lengths for each of the geometries using the larger expansion. This larger calculation has a more liberal active space and should capture even more of the spin-orbit effect than seen in the small expansion. At this higher level of calculation, the tetrahedral geometry is even lower in total energy than the flat geometry, -349.273228 Hartrees versus -349.261342 Hartrees, a difference of .012 Hartrees. To be sure, this is still a very small difference but still it is surprising. At the very least, (118)F4 should be considered a stereochemically non-rigid molecule even if it is not purely tetrahedral. It will be necessary to improve the molecular orbital basis set and use an even larger Cl expansion to finally pin down the actual energy difference. It may even be that at this higher level of calculation, the flat geometry regains it primacy but these results clearly indicate that the spin-orbit effect has profound consequences for the geometry of this molecule. 191 (118)F •349.00 -349.05 E(CISD) Td E(CISD) flat -349.20 -349.25 3.5 4.0 4.5 R((118)-F) Figure 414 Potential energy surface of element 118 tetrafluoride. This is a reiteration of the curve in Figure 413. 192 To explain this we must simply point out that it has been shown time and time again that the 7p* shell in these superheavy elements is best regarded independently from the 7p shell. Recall that in the case of element 117 excited states resulting from the promotion of one electron from a 7p spinor to the 8s orbital are closer in energy to the 2pg/2 ground state than its spin-orbit counterpart. In such a picture, element 118 is best regarded as having four valence electrons outside very stable closed shells. In short, (118)F4 is tetravalent. Certainly, this picture is not definitive or else the difference between the tetrahedral and the flat geometries would be still greater. There are other factors involved which must be explored with a more complete level of theory, it may even be the case that the molecule adopts a configuration not yet considered. But this finding has opened up a can of worms that shall be interesting to pursue. 193 C H A P T E R S SEABORGIUM HEXACARBONYL: A 6d ORGANOTRANSmON METAL COMPLEX To be sure, the subject of this chapter deals with aspects of chemistry that may never be experimentally realized. The purpose of this work is to draw comparisons of the structure and some properties of a hypothetical transactinide organometallic compound with its analogs among the lighter transition metals. This compound is seaborgium hexacarbonyl, Sg(C0)6, a species which has well-studied cousins in carbonyls of the hghter transition metals. As a class, transition metal carbonyls occupy a central position in organometallic chemistry and often form the beginning of any introductory discussion of the subject. Arguably, the most well-known example of such a compound is chromium hexacarbonyl, Cr(C0)6. This molecule is quite often presented as the prototypical 18-electron complex in which the octahedral symmetry guarantees a large separation between filled and unfilled molecular orbitals. It is also the first member of the triad of isoelectronic and isostructural d® carbonyls, including Mo(CO)6 and 194 W(C0)6, which constitutes the only complete class of stable transition metal carbonyls. By contrast, the valence isoelectronic actinide analog of these molecules, uranium hexacarbonyl, U(C0)6, is known to exist only in low- temperature matrices. ^2 the only isolable uranium carbonyl complex is one in which the metal is in the +3 oxidation state. The next logical addition to this list of Group VI transition metal hexacarbonyls is the 6d transactinide species seaborgium hexacarbonyl. It is not a priori obvious that the electronic structure of seaborgium hexacarbonyl is more similar to the hexacarbonyls of chromium, molybdenum, and tungsten than it is to that of uranium, although it does seem reasonable that this should be so. Nevertheless, this is an assumption that was tested with the use of the density functional method. The results of this study were reported in the New Journal of Chemistry (New J. Chem. 1995,19 (5-6), 669-75) and this paper appears as Appendix F of this d issertatio n . ^24 The major conclusion of this article holds that the instabiKty of U(C0)6 relative to W(C0)6 and Cr(C0)6 owes to a high density of f-based states, some dissociative, near the ground state. This high density of states results from a poor overlap of U f orbitals with CO iz* orbitals as well as to competition between the 5f and 6d orbitals for electron density. The study further concludes that Sg(C0)6 would not suffer from these difficulties due to the simple fact that the f shell is completely full, and it will instead behave like a more traditional Group VI transition metal carbonyl. Given common understanding of trends in the periodic table, it would indeed have only been alarming if this had not been found to be the case. These conclusions were made based on the results of electron density and orbital population 195 analyses rather than explicit calculations of state and bond energies. While such an analysis was sufficient for the purposes of the New Journal of Ohemistrv paper, there are a great many pertinent questions yet to be addressed and to do so we must appeal to other methods. Having demonstrated that, in fact, the characteristics of Sg(C0)6 are more like those of transition metal carbonyls than those of actinides, it is now pertinent to consider particular properties of seaborgium hexacarbonyl vis -a-vis' its lighter homologs. The stability of the 18-electron transition metal carbonyls in general and the Group VI transition metal carbonyls in particular has long been attributed the phenomenon of x-backbonding. This stabihzing mechanism, which involves the interaction of transition metal orbitals with CO x*, orbitals results in a net transfer of electron density from the metal to the ligands. In octahedral carbonyls, x backbonding occurs through the T2g set of metallic d orbitals (the d%z, dyz, dxy according to the Cartesian coordinate system represented in Figure 5.1) with a properly symmetrized set of CO x* orbitals. The backbonding occurs in concert with a donation of the set of CO "lone pair " electrons to the metal valence ns (Aig), (n-l)dz2 and (n-l)dx2-y2 (Eg) orbitals, and np(Tiu) atomic orbitals. Such a transfer of electron density to x-antibonding orbitals reduces the CO bond order and manifests itself as a reduction in the frequency of the stretching modes of that ligand. The degree of this reduction is indicative of the level of such backdonation; the greater the reduction of the frequency, the better the dx-CO x* overlap and hence the greater the interaction. This picture certainly holds for the known Group VI carbonyls so the question of interest then is to what degree does it also hold for Sg. Given the proven significance of relativistic effects for the 196 actinides and transactinides and even some late transition metals, it stands to reason that it is necessary to properly account for these effects here. Figure 5.1 Octahedral Transition Metal Hexacarbonyl in its Cartesian orientation. One question remains unanswered, why seaborgium hexacarbonyl? There are several reasons for the focus on SgfCOlg. Chapter 4 dealt with several examples of hexafluorometallates of 6d transition metals and so we have already seen examples of the chemistry of superheavy transition elements in high (+6) oxidation states. This molecule complements this previous work, by dealing with a 6d transition metal in a low (0) oxidation 197 State. In addition, the transition metal carbonyls are a large class of molecules and form one of the pillars of organometallic chemistry. It seems very relevant in light of the ubiquity of such molecules to discuss the hypothetical behavior of seaborgium hexacarbonyl. Computatinnfll TVtoils The approach and computational methodology used for the work in this chapter is somewhat different than that in Chapters 3 and 4 of this dissertation. There, Hartree-Fock wavefiinctions in conjunction with large scale spin-orbit Cl calculations were used to probe the detailed electronic structure of reasonably simple transactinide and transactinide molecules. The work reported in this chapter, however, was done exclusively using the GAMESS quantum chemistry suite of programs for which no provision is made for the inclusion of relativistic spin-orbit effects.^25 However, the incorporation of scalar relativistic effects (shell effects) is accomplished through the use of averaged relativistic effective potentials such as those derived in Chapter 2. GAMESS was used in order to allow a more direct comparison of the results of calculations on superheavy organometallics with the large body of work dealing with hghter analogs using similar programs and methods. This choice does leave open the question of possible spin-orbit effects on the electronic or vibrational structure of this molecule, but this question is left for a later time. The core electrons of seaborgium were replaced with the 92-electron relativistic effective core potential derived in Chapter 2. The seaborgium basis set is from Chapter 2 and corresponds to this core potential. A 198 (6sd5p)/[5sd4p] contraction scheme was used according to the conventions outlined in Chapter 3. The carbon and oxygen basis sets, both using (4s4p)/[2s2p] contractions, are taken whole doth firom a paper by Blaudeau, Wallace, and Pitzer under contraction schemes shown below. ^26 These light element basis sets are assodated with 2-electron, shape-consistent RECPs published by Ermler et. al. and were published as alternatives to those included in this paper. Exponent Contraction Coefficients s C l C2 25.04 -0.0107539 0.0 3.358 -0.1374153 0.0 0.4836 0.5764856 0.0 0.1519 0.5356444 1.0 P C l C2 9.430 0.0381521 0.0 2.001 0.2094554 0.0 0.5451 0.5089665 0.0 0.1516 0.4683789 10 Table 5.1 Caibon basis set used in Sg(C0)6 199 Exponent Contraction Coefficients C l 41.04 -0.0095512 0.0 7.161 -0.1334986 0.0 0.9074 0.5985184 0.0 0.5985184 1.0 C l C2 17.72 0.0430232 0.0 3.857 0.2287623 0.0 1.046 0.5090576 0.0 0.4604005 1.0 Table 5.2 (bqrgen basis setnsedin Sg(CO)6 Because previous theoretical investigations on the properties of transition metal carbonyls have demonstrated the need for inclusion of correlation effects in order to reproduce their various properties, calculations at both the self-consistent field (SCF) and Moeller-Plesset 2nd order perturbation (MP2) levels of theory were p e r f o r m e d . ^ 2 8 An MP2 calculation in conjunction with the use of a RECP is sometimes referred to as relativistic MP2.' In this context, this terminology can be misleading, however, in that only scalar relativistic effects are taken into account. 200 At the SCF level, the geometry of Sg(C0)6 was optimized within the constraints of octahedral symmetry. The known Group VI transition metal carbonyls, Cr(C0)6, Mo(CO)6, and W(C0)6, are invariably octahedral so it is reasonable to assume that this assumption does not introduce any unwarranted restrictions on the wavefunction.l29 This supposition is somewhat vindicated by observation that the MP2 optimized geometry retains Oh symmetry despite the fact that the calculation itself was constrained to only D2h- The subgroup symmetry was used in the MP2 calculation because the efficiency of the integral transformation algorithm is increased with the use of Abelian groups. B o u lts Tables 5.1 and 5.2 respectively contain the SC F and MP2 optimized atomic coordinates for Sg(C 0)6. The results indicate a 0.05  C -0 bond length increase and a 0.02  M-C bond length decrease at the MP2 level relative to the SC F geometry. These differences are on a par with what is seen in the results of MP2 vs. SCF calculations for the other Group VI hexacarbonyls. ^30 Table 5.3 summarizes theoretical and experimental bond lengths for Group VI transition metal carbonyls including these results for SgCCOlg. Essentially, they indicate that the seaborgium molecule is remarkably similar in geometry to the other species. The metal to carbon bond length is only slightly longer in S g(C 0)6 than in W (C 0)6, there is more of a difference between the M-C distances of C r(C 0)6 and Mo(CO)6.^^^ This approximate 0.05  difference in the metal to ligand bond distances between Sg(C 0)6 and W(CO)e is about the same difference that 201 was seen in Chapter 4 for the same metal-fluorine distances. The most striking difference in the geometry of seaborgium hexacarbonyl compared to the others is the difference in C-0 bond lengths. For Cr(C0)6, Mo(CO)6, and W(C0)6, this distance remains fairly constant, varying from 1.165  to 1.168  at the MP2 optimized geometries and 1.140  to 1.148  experimentally. On the other hand, the MP2 Sg(C0)6 carbonyls bond distance is calculated to be 1.197 Â, perhaps 1.17  accounting for the empirical differences between the experimental and MP2 results for the other M(C0)6 systems. 132 COORDINATES OF SYMMETRY UNIQUE ATOMS (ANGS) ATOM CHARGE X Y z SEABORGIUM106.0 0.0000000000 0.0000000000 0.0000000000 CARBONl 6.0 2.1190401402 0.0000000000 0.0000000000 OXYGENl 8.0 3.2602926037 0.0000000000 0.0000000000 COORDINATES OF ALL ATOMS ARE (ANGS) ATOM CHARGE X Y z Sg 106.0 0.0000000000 0.0000000000 0.0000000000 C 6.0 -2.1190401402 0.0000000000 0.0000000000 C 6.0 0.0000000000 2.1190401402 0.0000000000 C 6.0 0.0000000000 -2.1190401402 0.0000000000 C 6.0 0.0000000000 0.0000000000 2.1190401402 C 6.0 0.0000000000 0.0000000000 -2.1190401402 C 6.0 2.1190401402 0.0000000000 0.0000000000 0 8.0 -3.2602926037 0.0000000000 0.0000000000 0 8.0 0.0000000000 3.2602926037 0.0000000000 0 8.0 0.0000000000 -3.2602926037 0.0000000000 0 8.0 0.0000000000 0.0000000000 3.2602926037 0 8.0 0.0000000000 0.0000000000 -3.2602926037 0 8.0 3.2602926037 0.0000000000 0.0000000000 Table 5.3 Atomic Coordinates of Seaborgium Hexacarbonyl Optimized at the SCF Level of Tbeoiy. 202 COORDINATES OF SYMMETRY UNIQUE ATOMS (ANGS) ATOM CHARGE X Y z Sg 106.0 0.0000000000 0.0000000000 0.0000000000 C 6.0 2.1018485913 0.0000000000 0.0000000000 0 8.0 3.2992065001 0.0000000000 0.0000000000 COORDINATES OF ALL ATOMS ARE (ANGS) ATOM CHARGE XY z Sg 106.0 0.0000000000 0.0000000000 0.0000000000 C 6.0 -2.1018485913 0.0000000000 0.0000000000 c 6.0 0.0000000000 2.1018485913 0.0000000000 c 6.0 0.0000000000 -2.1018485913 0.0000000000 c 6.0 0.0000000000 0.0000000000 2.1018485913 c 6.0 0.0000000000 0.0000000000 -2.1018485913 c 6.0 2.1018485913 0 .0000000000 0.0000000000 0 8.0 -3.2992065001 0.0000000000 0.0000000000 0 8.0 0.0000000000 3.2992065001 0.0000000000 0 8.0 0.0000000000 -3.2992065001 0.0000000000 0 8.0 0.0000000000 0 .0000000000 3.2992065001 0 8.0 0.0000000000 0.0000000000 -3.2992065001 0 8.0 3.2992065001 0.0000000000 0.0000000000 Table 5.4 Atomic Coordinates of Seaborgium Hexacarbonyl Optimized at the MP2 Level of Theory. 203 Sg(CO)6 RCM-C) A R(C-O) A SCF 2.119 1.141 MP2 2.102 1.197 W(CO)6 R(M-C) A R(C-0)A SCF 2.115 1.122 MP2 2.064 1.166 Expt 2.058 1.148 Mo(CO)6 R(M-C) A RCC-OA SCF 2.121 1.121 MP2 2.049 1.165 Espt 2.063 1.145 Cr(C0)6 R(M-C) A R(C-0)A SCF 2.005 1.120 MP2 1.868 1.168 Expt 1.914 1.140 Table 5.5 Theoretical Interatomic Bond Lengths for Group VI Hexacarbonyls. If, for the hexacarbonyls of Cr, Mo and W, the ligated CO bond length increase relative to that in free CO (see Table 5.4) is attributable to k- backbonding, then it is clear from these results that the d orbitals of Sg have the best x-donating characteristics in Group VI. This is also evidenced by the calculated CO stretching frequencies in Sg(C0)6 relative to its lighter co n gen ers. 204 FStpmm«»nt SCF MP2 r (cm))A 1.128 1.130 1.154 v(CO) (cm 1) 2143 2264 2129 Table 5.6 E ^ e rimental and Theoretical Vaincs nffrmm CO tinnii tengths ^iiH stretching frequencies. The carbonyl stretches in the octahedral Group VI hexacarbonyls are resolved into three symmetry adapted vibrational modes spanning Ag, Eg, and Tiu- Of these, the first two are IR inactive and Raman active while the third is IR active and Raman inactive. The frequencies of all of these stretches for C r(C 0)6, Mo(CO)6, and W (C 0)6, occur at wavelengths below the corresponding frequency of firee (unHgated CO), a characteristic which is a hallmark of x-backbonding. At both the SCF and MP2 levels of calculation, the CO stretches in S g (C 0 )6 occur at wavelengths even lower than those in the lighter hexacarbonyls, these results are summarized in Tables 5.5, 5.6, and 5.7.133 In all cases where data is available, the MP2 calculations underestimate the vibrational fi-equencies by between 30 to 40 cm 1.134 Even so, they represent a substantial improvement over the SCF calculated frequencies which differ from experiment by hundreds of wavenumbers. In any case, the MP2 calculated CO stretching frequencies in Sg(C0)6, 1903 cm‘1 for the Aig mode, 1793 cm‘1 for the Eg mode, and 1789 cm 'l for the Tiu mode are dramatically lower than even the values 205 in W(C0)6- Even if we add back the 35 or so wavenumbers in order to compensate for empirical error the Tiu frequency, for example, is still only around 1820 cm-1. This is a value usually associated with doubly bonded ketone stretches rather than triply bonded carbonyl stretches. This, along with the substantial increase in C-0 bond length, indicates that k- backbonding in SgCCOg is more extensive than in any of the other ceirbonyls of Group VI. Mode Mo(CO)6 W(CO)6 Sg(CO)6 A ig 2086 2088 1903.1 % 1996 1988 1793.7 T iu 1964 1967 1789.0 Table 5.7 Values of CO stretching modes calculated at the MP2 level of theory for Group VI transition metal carbonyls. Frequencies are in cm 1. The explanation for this enhanced level of backbonding in seaborgium complexes compared to those of similar 3d, 4d, and 5d transition metal is essentially the same as for the anomalously large A values in transactinide octahedral hexafluorometallates from Chapter 4. Relativity causes the expansion and destabilization of the 6d orbitals to such an extent that the overlap of dx orbitals with the CO k* orbitals is 206 much larger for Sg than any other Group VI metal. This accounts nicely for the increased lengthening of the CO bond in the Sg complex relative to W(C0)6; more backdonation means more electron density in the carbonyl it* system and a consequential decrease in 0-0 bond order. At the same time, the relativistic contraction and energetic stabilization of the valence 7s and 7p orbitals, through which the CO a-donation is accomplished, leads to a M- C bond length which is remarkably similar to that in W(C0)6. Relativity is seen to counteract any shell-structure effects which might be expected to result in metal to carbon distances on the order of 2.5 Â. In short, relativity leads to a Sg(C0)6 geometry which is more like that of W(C0)6 than it would be otherwise and vibrational frequencies that are very much lower than they would be otherwise. Mode A ig % T iu SCF 2248.5 2137.6 2093.4 MP2 1903.1 1793.7 1789.0 Table 5.8 Theoretical Values of CO Stretching Frequencies in Sg(CO)g. 207 W(C0)6 Mode A le % T iu SCF 2418 2325 2292 MP2 2088 1988 1967 Expt. 2126.2 2021.1 1997.6 Mb(CO)6 Mode A ig % Tiu SCF 2418 2336 2310 MP2 2086 1996 1964 Expt 2120.7 2024.8 2003.0 Cr(CO)6 Mode A ig % Tiu SCF 2417 2322 2324 MP2 N/A N/A N/A Expt 2118.7 2026.7 2000.4 Table 5.9 Theoretical and Ehq)eriniental values o f CO stretching frequencies for known Group VI transition metal carbonyls. Frequencies are in cm'l 208 Finally, we shall consider the carhonyl dissociation energy of Sg(C0)6 in the following reaction. Sg(C0)6 ------> S g(C 0)5 + CO (eq. 5.1) This energy required to break the M-CO bond is of course a measure of the M-C bond strength and it is interesting to compare its value in S g(C 0)6 with those of its cousins. Having optimized both the SCF and MP2 geometries and obtained the corresponding total energies of Sg(C 0)6 and CO, it was necessary to do so for Sg(C 0)5. The metal fragment products of the analogous reactions of the lighter group VI transition metals are known to have a C4v arrangement very similar to the octahedral complex with one CO removed. 135 It is reasonable to assume that the same would be true of S g(C 0)5 but nevertheless, its geometry was optimized at the SC F level under the constraints of both C4v symmetry and Dsh symmetry. The latter corresponds to the trigonal bipyramidal geometry of, for example, F e(C 0)5. The C4v optimized geometry of Sg(C 0)5 was found to lie somewhat lower in energy than the Dsh geometry in agreement with experimental results for the other pentacarbonyls. The geometry of Sg(C0)5 was then optimized at the MP2 level and the total energy so obtained was added to the MP2 energy of CO and this sum was subtracted from the MP2 energy of Sg(C0)6. This gives a first bond dissociation energy (FBDE) for the reaction above of 2.56 eV or 59.1 kcal/mol. At the SCF level, the FBDE is 1.64 eV or 37.8 kcal/mol. These compare to experimental values of 36.8 kcal/mol, 40.5 kcal/mol, and 46.0 kcal/mol for Cr(C0)6, Mo(CO)6, and W(C0)6, respectively. 136 209 Having said this, it should be noted that the agreement between MP2 and experimental FBDEs for the Group VI hexacarbonyls are rather suspect, the MF2 values overestimating it by about 5 10 kcal/mol. This suggests that a more realistic estimate of the carhonyl dissociation energy would be about 50-55 kcal/mol. This still implies a substantially stronger M-CO bond for Sg than for W, Mo, or Cr and reinforces the interpretation of increased 7C-backdonation in the hypothetical seaborgium complex relative to the others.137 These results are all preliminary and intended as an inital foray into further research into the field. The high degree of x-backbonding and accompanying C-0 bond reduction seen in Sg(C0)6 seems to support the intuitive picture of relativistic shell-structure effects in which d orbitals are expanded and s and p orbitals are contracted. On the other hand, a good deal remains to be done. We have seen elsewhere in this dissertation that spin-orbit effects are quite often relevant to any discussion of bonding in transactinide molecules and these have not been treated here. It is somewhat reassuring to note, however, that such effects should be much less important for 6d orbitals than 7p orbitals, a result of the 1"3 dependence of the spin-orbit operator. Still, in a complete treatment of the subject they should be treated on an equal footing with correlation. Naturally, the study of the organometallic chemistry of the transactinide elements is somewhat of an academic exercise. The hope is that by doing so, we will gain insight into the nature of this chemistry in the traditional transition metals. After all, relativistic trends in transactinide molecules are simply exaggerations of those already present in 3d, 4d, and especially 5d species. It is hoped that the work presented in this chapter 210 and in the balance of this dissertation has begun the task of outlining the chemistry of the transactinides and has shed new light on the more established branches of the field. 211 LIST OF REFERENCES 1. Seaborg, G. T., McMillan, E. M., Kennedy, J. W., Wahl, A. C.; Phys. Rev 6 9 , 366 (1948). 2. Hoffinan, S., Ninov, V., Hessberger, F. P., Armburster, P., Folger, H., Miinzenberger, G., Schott, H, J., Popeko, A. G., Yeremin, A. V., Saro, S„ Janik, R., Leino, M.; Z, Phys. A. 354, 229 (1996). 3. Hoffinan, D. C,; Radiochim. Acta 61,123 (1993). 4. Holden, N. E., Pure Appl. Chem. 6 1 , 1483 (1989). 5. Hoffinan, S., Ninov, V., Hessberger, F. P., Armburster, P., Folger, H., Miinzenberger, G., Schott, H. J., Popeko, A. G., Yeremin, A. N., Andreyev, A. N., Saro, S., Janik, R., Leino, M.; Z. Phys'A. 350, 277 (1995). 6. Katz, J. J., Seaborg, G. T., Morss, L. R., Eds. The Chemistry of the Actinide Elements Vol 1,2 (1986). 7. Kratz, J. V.; J. Alloys Compd. 213/214 20 (1994). 8. Pyykko, P.; Chem, Rev 88, 563 (1988). 9. (a) Bethe, H.; Ann. Physik, 3 , 133 (1929). (b) Hamermesh, M.; Group Theorv and its Application to Phvsical Problems. (1964). (c) Landau, L. D., Lifshitz, E. M.; Quantum Mechanics: N-Relativistic Theorv. (1977). 10. Wilson, S. Ed.; An Introduction to the Methods and Relativistic Molecular Quantum Mechanics. (1996). 11. Schrodinger, E. Ann. der Phys., 79, 361 (1926). 12. Dirac, P. A. M.; Proc. Roy. Soc. London A. 1 2 3 , 714 (1929). 13. (a) Grant, I. P.; J. Phys. B: At. Mol. Opt. Phys. 19, 3187 (1986). (b) Grant, I. P., (b) Quiney, H. M. J. Phys.B.: At. Mol. Opt. Phys. 22, 15 (1989). 212 14. (a) Grant, I. P.; Adv. Phys. 1 9 , 747 (1970). (b) Breit, G. E.; Phys. Rev. 3 5 , 554. 15. Pyykko, P; Adv. Quantum Chem. 11, 353 (1978). 16. see, for example: Dyall, K G., Malli, 6., Ishikawa, Y., Faegri, K., Mohanly, A., Matsuoka, O., Aerts, P. C. J., Visscher, L., Nieuwport, W. C.,... 17. Rose. M. E.: Relativistic Electron Theorv. (1961). 18. Lo, B., Saxena, K. M. S., Fraga, S.; Theor. Chim. Acta 2 4 , 300 (1972). 19. (a) Ratkowitz, F., Marian, C. W.; Chem. Phys. Lett, 2 5 7 , 105 (1996). (b) Pyper, N. C., Marketos, P.; J. Phys. B 1 4 , 1387 (1981). 20. Wood, J. H.; Boring, A. W. Phys. Rev. B. 1 9 7 8 , 18, 2701. 21. Cowan, R. D.; Griffin, D. C. J. Ope. Soc. Am. 1 9 7 6 , 66,1010 22. Snijders, J. G.; Baerends, E. J. Mol. Phys. 1 9 7 8 , 36, 1789. 23. Desclaux, J. P.: Relativistic Effects in Atoms. Molecules, and Solids. 1st ed. (1983). 24. Hess, B. Phys. Rev. A. 1986, 33, 3742. 25. Cory, M. G.; Koestimeier, S.: Kotzian, M.; Roesch, N.: Zerner, M. J. Chem. Phys. 1994^ 100, 1353. 26. Wahlgren, U.; Lect. Notes Chem. 1 9 9 2 , 58, 413. 27. Ermler, W. C., Ross, R. B., Christiansen, P. A.; Int. J. Quantum Chem. 40, 829 (1991). 28. Hoffinan, D. C. The Transuranium Elements: From Neptunium and Plutonium to Element 112. in press. 29. Shepherd, R.; Shavitt, I., Pitzer, R. M., Comeau, D. C., Pepper, M.; Lischka, H., Szalay, P. G., AWrichs, R., Brown, F. B., Zhao, J. (3.; k t. J. Quantum Chem. Quantum Chem. Symp. 2 2 , 149 (1988). 30. CNVRT: Written by Y. Yamaguchi and B. Brooks; modified by F. B. Brown and R. M. Pitzer. 31. SCFPQ: See Hsu, S. L., Pitzer, R.M., Davidson, E. R.; J. Chem. Phys., 65, 609 (1976). with modifications made at Ohio State University, Iowa State University, and Argonne National Laboratory. 213 32. LSTRN: Modified by R. M. Pitzer fi*om CITRAN (F.B. Brown, Ph. D. Thesis, Ohio State University, 1982). 33. CGDBG: Modified by R. M. Pitzer and N. W. Winter firom MQM23 written at the CaUfomia Institute of Technology by B.D. Olafson and R. C. Ladner. 34. CIDBG: Written by R. M. Pitzer and N. W. Winter using section firom programs developed at the California Istitute of Technology, Los Alamos National Laboratory, Lawrence Livermore National Laboratory, and Ohio State University. See R.C. Ladner, Ph. D. Thesis, California Institute of Technology, 1974; R. M. Pitzer and N. W. Winter, J. Phys. Chem., 92, 3061 (1988), and I. Shavitt, NASA-Ames Report (1977, unpublished). 35. ARGOS: R, M. Pitzer (principal author). This program is part of the COLUMBUS suite of programs. See R. Shepherd, I. Shavitt., D. C. Comeau., M. Pepper, H. Lischka, P. G. Szalay, R. Ahlrichs, F. B. Brown, and J. G. Zhao, Int. J. Quantum Chem. Symp. 22,149 (1988). 36. The descriptions of the COLUMBUS codes in references 30-35 are adapted firom the Ph. D. Thesis of J. P. Blaudeau (The Ohio State University, 1993). 37. (a) Goddard, W. A.; Phys. Rev. 1 5 7 , 73 (1967). (b) Goddard, W. A.; Phys. Rev. 1 5 7 , 81 (1967). (c) Goddard, W. A.; J. Chem. Phys. 4 8 , 450 (1968). (d) Goddard, W. A.; J. Chem. Phys. 4 8 , 1008 (1968). 38. Lischka, H., Shepherd, R., Brown, F. B., Shavitt, I.; Int. J. Quantum Chem. Quantum Chem. Symp. 1 5 , 91 (1981). 39. Pitzer, R. M., Winter, N. W.; J. Phys. Chem. 92,3061 (1988). 40. Schmidt, M. W., Baldridge, K K., Boatz, J. A., Elbert, S. T., Gordon, M. S., Jensen, J. H., Kosek, S., Matsunaga, N., Nguyen, K A., Su, S. J., Windus, T. L., Dupuis, M., Montgomery, J. A.; J. Comput. Chem. 1 4 , 1347 (1993). 41. Anon.; Pure Appl. Chem. 66, 2419 (1994). 42. American Chemical Society recommended slate of names 43. Phillips, J. C., Kleinman, L.; Phys. Rev. llfi 287 (1959). 44. Szadz, L., Brown, L.; J. Chem. Phys. 6 5 , 1393 (1976). 45. Bonifadc, V., Huzinaga, S., J. Chem. Phys 6 0 , 2779 (1974). 46. MeUus, C. F., Goddard, W. A.; Phys. Rev. A 1 0 , 1528 (1974). 214 47. Christiansen, P. A., Lee, Y. S., Pitzer, K. S.; J. Chem. Phys. 71, 4445 (1979). 48. Lee, Y. S., Ermler, W. C., Pitzer, K S.; J. Chem. Phys. 6 7 , 5861 (1979). 49. Pacios, L. F., Christiansen, P. A.; J. Chem. Phys. 82, 2664 (1985). 50. Hurley, M. M., Pacios, L. F., Christiansen, P. A., Ross, R. B., Ermler, W. C.; J. Chem. Phys. 84, 6840 (1986). 51. LaJohn, L. A., Christiansen, P. A., Ross, R. B., Atashroo, T., Ermler, W. C.; J. Chem. Phys. 87, 2812 (1987). 52. Ross, R. B., Powers, J. M., Atashroo T., Ermler, W. C., LaJohn, L. A., Christiansen, P. A.; J. Chem. Phys. 9 3 , 6654 (1990). 53. Ermler, W. C., Ross, R. B., Christiansen, P. A.; Int. J. Quantum Chem. 40, 829 (1991). 54. Ross, R. B., Gayen, S., Ermler, W. C., J. Chem. Phys. 1 0 0 , 8145 (1994). 55. Kuechle, W., Dolg, M., Stoll, H., Preuss, H.; J. Chem. Phys. 100, 7535 (1994). 56. Ermler, W. C., Lee, Y. S., Christiansen, P. A., Pitzer, K. S.; Chem. Phys. Lett. 8 1 , 70 (1981). 57. Pitzer, R. M., Winter, N. W.; Int. J. Quantum Chem. 40, 773 (1991). 58. Ermler, W. C., Ross, R. B., Christiansen, P. A.; Adv. Quantum Chem. 19, 139 (1988). 59. (a) Roothaan, C. C. J., Bagus, P. S.; Meth. Comp. Phys. 2, 47 (1963). (b) Roos, B., Salez, C., Veillard, A., Clementi, E.; IBM Research Report RJ518 (1968). (c) Pitzer, R. M.; QCPE, 1 0 , 587 (1990). 60. Wallace N. M., Blaudeau, J. P.; Pitzer, R. M.; Int. J. Quantum Chem. 40, 789 (1991). 61. (a) Dunning, T. H.,; J. Chem. Phys. 9 0 , 1007 (1989). (b) Woon, D. E., Dunning, T. H.; J. Chem. Phys. 9 8 , 1358 (1993). 62. Hoffinan, D. C.; Chem. Eng. News 7 4 , 24 (1994). 63. (a) Nugent, J. L., Vander Sluis, Fricke, B., Mann, J. B.; Phys Rev. A 9 , 2270 (1974). (b) Desclaux, J. P., Fricke, B.; J. Phys. 4 1 , 943 (1980). 215 64. (a) Wijesimdera, W. P., Vosko, S. H., Parpia, F. A.; Phys. Rev. A 51, 278 (1995). (b) Eliav, E., Kaldor, U., Ishikawa, Y.; Phys. Rev. A 51, 225 (1995). 65. Ho&nan, D. C.; Transuranium Elements: A Half Centurv. Morss, L. R., Fuger, J. Eds. (1992). 66. Eliav, E., Klador, U., Ishidawa, Y.; Phys. Rev. Lett. 7 4 , 1079 (1995). 67. (a) Glehov, V. A., Kasztura, L., Nefedov, V. S., Zhuikov, B. L.; Radiochim. Acta 46,117 (1989). (b) Johnson, E., Fricke, B., Keller, 0. L., Nestor, C. W., Tucker, T. C.; J. Chem. Phys. 93,8041 (1990). 68. Kaldor, U.; Recent Progress in Manv-Bodv Theories. Schachinger, E. et a/., Eds. 135(1995). 69. Hoflman, S., Ninov, V., Hessberger, F. P., Armburster, P., Folger, H., Miinzenberger, G., Schott, H. J., Popeko, A. G., Yeremin, A. N., Andreyev, A. N., Saro, S., Janik, R., Leino, M.; Z. Phys A. 3M, 277 (1995). 70. Eliav, E., Kaldor, U., Schwerdtfeger, P., Hess, B. A., Ishikawa, Y.; Phys. Rev. Lett. 73, 3203 (1994). 71. Eliav, E., Kaldor, U., Ishikawa, Y; Phys. Rev. A 52, 2765 (1995). 72. Pyper, N. C., Grant, I. P.; Proc. Roy. Soc. London A 376, 483 (1981). 73. EHav, E., Kaldor, U., Ishikawa, Y., Seth, M., Pyykko, P.; Phys. Rev. A 53, 3926 (1996). 74. Hancock, R. D., Shaikjee, M. S., Dobson, S. M., Boeyens, J. C. R.; Inorg. Chim. Acta 154, 229 (1988) 75. Ross, R. B., Ermler, W. C., Christiansen, P. A.; J. Chem. Phys. 84, 3297 (1986). 76. (a) Herrman, G., Angew. Chem. Int. Ed. Engl. 34, 1713 (1995). (b) Beiner, M., Flocard, H., Veneroni, M., Quentin. P.; Phys. Scr. IDA, 84 Q974). 77. Brewer, L., Chang, C.; J. Chem. Phys. 56, 1728 (1972). 78. Pitzer, K S.; J. Chem. Phys 6 3 ,1032 (1975). 79,80, 81. Kiichle, W., Dolg, M., Stoll, H., Preuss, H.; Mol. Phys. 74, 1245 (1991). 82. Pyykko, P.; Phys Scr. 20, 647 (1979). 2 1 6 83. Manan, C., Wahlgren, U.; Chem. Phys. Lett. 251, 357 (1996). 84. Seth, M., Schwerdtfeger, P., Dolg, M., Faegri, K , Hess, B. A., Kaldor, U.; Chem. Phys. Lett. 250,461 (1996). 85. Huber, K. P.; Herzberg, G., Molecular Spectra and Molecular Structure Constants of Diatomic Molecules. (1979). 86. Balasubramanian, K., Tao, J. X.; J. Chem. Phys. 94, 3000 (1991). 87. Kiichle, W., Dolg, M., Stoll, H., Preuss, H.; Mol. Phys. 74,1265 (1991). 88. Christiansen, P. A., Ermler, W. C., Pitzer, K. S.; Ann. Rev. Phys. Chem. 36, 407 (1985). 89. Kiichle, W., Dolg, M., Stoll, H., Preuss, H.; Mol. Phys. 74, 1245 (1991). 90. Akekseyev, A. B., Buenker, R. J., Libermann, H. P., Hirsh, G.; J. Chem. Phys. 100, 2989 (1994). 91. Kiichle, W., Dolg, M., Stoll, H., Preuss, H.; Mol. Phys. 74, 1245 (1991). 92. Dolg, M.; Mol. Phys. 88,1645 (1996). 93. Visscher, L., Styszynske, J., Niewpoort, W. C.; J. Chem. Phys. 105, 1987 (1996). 94. Kiichle, W., Dolg, M., Stoll, H., Preuss, H.; Mol. Phys. 74, 1245 (1991). 95. Lian, L., Hackett, P. A., Rayner, D. M.; J. Chem. Phys. 99, 2583 (1993). 96. Hilpert, K , Gingerich, K. A.; Ber. Bunsenget. Phys. Chem. 84, 739 (1980). 97. Balasubramanian, K., Das, K. K.; J. Mol. Spec. 140, 280 (1990). 98. Stangassinger, A., Bondybey, V. A.; J. Chem. Phys. 103, 10804 (1995). 99. Christiansen, P. A.; J. Chem. Phys. 79, 2928 (1983). 100. Wood, C. P.; Pyper, N. C.; Chem. Phys. Lett. 84, 614 (1981). 101. Ermler, W. C., Ross, R. B.; Christiansen, P. A.; Adv. Quantum Chem. 1 9 ,139 (1988). 102. Christiansen, P. A.; Ermler, W. C., Pitzer, K. S.; Ann. Rev. Phys. Chem. 36, 407 (1985). 217 103. Huber, K. P.; Herzberg, G., Molecular Spectra and Molecular Structure Constants of Diatomic Molecules. (1979). 104. Wallace N. M., Blaudeau, J. P.; Pitzer, R. M.; Bit. J. Quantum Chem. 40, 789 (1991). 105. Kiseljov, A. A.; J. Phys. B 2 ,270 (1969). 106. Ischenko, A. A., Ogurtsov, I. Ya., Kazantseva, L. A., Spiridonov, V. P., Deyanov, R. Z.; J. Mol. Struct. 298, 103 (1993). 107. Msscher, L., Nieuwpoort, W. C.; Theor. Chim. Acta 88, 447 (1994). 108. Marx, R., Seppelt, K., Ibberson, R. M.; J. Chem. Phys. 104, 7658 (1996). 109. Rianda, R., Frueholz, R. P., Kupperman, A.; J. Chem. Phys. 70, 1056 (1979). 110. Holloway, J. H., Stanger, G., Hope, E. G., Levason, W., Ogden, J. S.; J. Chem. Soc. Dalton Trans. (1988). 111. Bernstein, E. R., Meredith, G. R.; J. Chem. Phys. 64, 375 (1978). 112. Rianda, R., Frueholz, R. P., Kupperman, A.; J. Chem. Phys. 70, 1056 (1979). 113. (a) Michalopoulos, D. L., Bernstein, E. R.; Mol. Phys. 47, 1 (1982). (b) Chakravarty, A. S.; Indian, J. Phys. 63A, 115 (1989). 114. (a) Webb, J. D., Bernstein, E. R.; Mol Phys. 37, 1509 (1979). (b) Webb, J. D., Bernstein, E. R.; Mol. Phys. 37, 203 (1979). 115. Schwerdtfeger, P., Ischtwan, J.; J. Comp. Chem. 14, 913 (1993). 116. Ogilvie, J. F., Liao, S. C., Uehara, H., Horiai, H.; Can. J. Phys. 72, 930 (1994). 117. Balasubramanian, K.; J. Chem. Phys. 83, 2311 (1985). 118. Dai, D., Al-Zahrani, A. A., Balasubramanian, K; J. Phys. Chem. 98, 9233 (1994). 119. (a) Classen, H. H., Sehg, H., Malm, J. G.; J. Am. Chem. Soc. 84, 3593 (1962). (b) Pitzer, K S.; J. Chem. Soc. Chem Comm. 760 (1975). (c) Styszynske, J., Malli, G.; Int. J. Quantum Chem. 55, 227 (1995). 120. Desclaux, J. P.; At. Data Nucl. Data Tab. 12, 311 (1973). 218 121. Pitzer, R. M.; O.S.U.-T.C.G. Report 101 (1995), unpublished 122. Slater, J. L., Sheline, R. K , Lin, K. C„ Weltner, W.; J. Chem. Phys. 55, 5129 (1971). 123. (a) Brennan, J. G., Andersen, R. A., Robbins, J. L.; J. Am Chem. Soc. 1 0 8 , 335 (1987). G)) Bursten, B. E., Strittmatter, R. J.; J. Am. Chem Soc. 1 0 9 , 6606. 124. Nash, C. S., Bursten, B. E.; New J. Chem. 1 9 , 669 (1995). 125. Schmidt, M, W., Baldridge, K. K., Boatz, J. A., Elbert, S. T., Gordon, M. S., Jensen, J. H,, Kosek, S., Matsunaga, N., Nguyen, K A., Su, S. J., Windus, T. L., Dupuis, M., Montgomery, J, A.; J. Comput. Chem. 1 4 , 1347 (1993). 126. Wallace N. M., Blaudeau, J. P.; Pitzer, R. M.; Int. J. Quantum Chem. 40, 789 (1991). 127. Pacios, L. F., Christiansen, P. A.; J. Chem. Phys. 82, 2664 (1985). 128. Ehlers, A. W., Frenking, G.; Organometallics 14, 423 (1995). 129. (a) Wemer, H.; Angew. Chem. 1 0 2 , 1109 (1990). (b) Davidson, E. R., Kunze, F. B.; Machado, F. B.C., Chakravoty, S. J.; Acc. Chem. Res. 26, 628 (1993). 130. Ehlers, A. W., Frenking, G.; J. Am. Chem. Soc. 116 1514 (1994). 131. Jonas, V., Thiel, W.; J. Chem. Phys. 102, 8474 (8474). (b) 132. Van Wüllen, C.; Int. J. Quantum Chem. 58, 147 (1996). 133. The data in these tables for ail molecules but Sg(C0)6 come from ref. 130. 134. Observed from the trends in the data from ref. 130. 135. (a) Hay, P. J.; J. Am. Chem. Soc. 1 0 0 , 2411 (1978). (b) Demuynck, J., Kochanski, E., Veillard, A.; J. Am. Chem. Soc. 1 0 1 3467 (1979). 136. These data also come from ref. 130. 137. This is the primary conclusion of this chapter. 219 APPENDIX A SMALL CORE (SC) RELATIVISTIC EFFECTIVE POTENTIALS FOR Am-ELEMENT 118 220 (78 core/17 valence) RECP- valence 6s,p,d/7s/5f n Exponent______C(ABBE)------ClSÛl 8 s-g Potential 2 1.097300 -28.391754 2 1.294700 84.976964 2 1.801300 -171.024010 2 2.787600 357.064278 2 4.609000 -387.687859 2 8.164900 441.841574 1 26.838200 35.836269 0 21.214000 15.823783 p-g Potential 2 1.702600 64.265412 -42.453794 2 1.948800 -210.603808 105.165300 2 2.545400 442.712799 -98.901415 2 3.655100 -492.188685 34.796778 2 5.550200 437.844181 31.290680 2 8.034300 -168.011901 -42.906345 1 14.145600 70.560542 -2.422367 0 48.913500 6.548605 -1.280931 d-g Potential 2 0.925800 28.796524 -8.203218 2 1.116300 -105.516646 24.895429 2 1.497100 221.985760 -34.534530 2 2.198500 -264.265018 29.223866 2 3.372200 261.342802 -12.717561 2 5.198500 -110.510983 0.857126 1 7.962100 57.149078 0.848736 0 24.138200 7.614054 -0.039528 £-g Potential 1.249300 -2.476704 -0.080903 2.418300 37.726698 1.218867 3.249900 -114.312323 -4.186270 4.978400 236.725719 9.474528 8.136800 -310.354382 -11.741854 14.510400 466.908721 6.140497 44.864000 30.389703 -0.469460 38.131700 9.769735 0.019426 g Potential 1.447000 -1.178665 -0.034326 3.218700 -13.270015 -0.016079 7.599100 -39.103179 -0.783682 22.651400 -155.020566 -0.890858 71.293000 -354.333006 0.951217 220.402300 -61.318104 3.279723 221 Cujrium (78 c o re /18 valence) RECP- valence 6s,p,d/7s/5f ______0______Exponent ____ C(MIEP)______CtSfll 8 s-g Potential 2 1.155600 -26.337600 2 1.374300 85.682174 2 1.882000 -174.588183 2 2.895200 355.670975 2 4.740200 -370.721483 2 8.270900 423.926355 1 25.204200 35.043504 0 21.014600 15.924326 p-g Potential 2 1.124400 -35.039644 35.297041 2 1.297300 113.442334 -100.844871 2 1.686700 -223.715259 147.946981 2 2.419600 424.266954 -151.882114 2 3.682500 -484.904342 116.857108 2 5.545800 375.647735 -57.592092 1 17.957000 33.733576 -0 .477872 0 16.023600 11.893398 -1.496981 d-g Potential 2 1.004600 39.770015 -0.995880 2 1.187100 -126.651226 -0.690327 2 1 ..609700 255.058819 11.288113 2 2,.361800 -305.242517 -23.692497 2 3,,642100 300.052796 19.913111 2 5,.577900 -129.087605 -9.296907 1 8.676800 60.350841 0.408854 0 27.028400 7.491302 -0.049207 f-g Potential 2 1.306500 -2.706424 -0.093696 2 2.459500 37.519790 1.271638 2 3.306000 -111.372108 -4.194348 2 5.066900 228.429256 9.191463 2 8.235000 -297.128424 -10.822294 2 14.630000 458.379593 5.300582 1 44.623300 30.037240 -0.346644 0 37.612400 9.801611 0.013686 g Potential 2 1.548800 -1.270895 -0.037525 2 3.419100 -13.837428 -0.006941 2 8.070500 -40.887652 -0.632202 2 23.859700 -160.034801 -0.375652 2 75.013900 -364.425146 1.084822 1 231.420300 -61.785508 3.512688 222 Berkelium (78 core/19 valence) RECP- valence 6s,p,d/7s/5f ______0______Exponent ______CiABEEJ------ClSÛl 8 s-g Potential 2 1.790500 52.192068 2 2.205400 -186.360344 2 3.218100 92.782986 2 3.219000 334.955213 2 5.333700 -465.587654 2 10.145100 553.616567 1 23.302000 29.158753 0 31.064600 16.747137 p-g Potential 2 1.840900 51.229143 -83.464270 2 2.153100 -181.650865 229.116958 2 2.897600 443.109062 -292.075057 2 4.391900 -565.480871 262.838527 2 7.070500 596.354205 -156.308213 2 11.386100 -236.826528 32.701280 1 19.072900 111.001433 -6.319147 0 78.963600 9.823661 -1.515725 d-g Potential 2 1.072500 38.537797 -9.887181 2 1.275200 -128.371939 27.437703 2 1.725100 265.806633 -35.789880 2 2.523900 -312.576826 28.118190 2 3.890900 306.932510 -9.085215 2 5.892800 -127.939905 -1.886613 1 9.335500 61.205029 1.092306 0 28.418100 7.491891 -0.054453 f-g Potential 2 1.332100 -2.381826 0.283867 2 2.638200 37.631024 -9.853449 2 3.530200 -112.931082 28.677570 2 5.436100 235.850068 -45.750624 2 8.959600 -315.544566 49.038174 2 15.724700 493.098579 -17.012609 1 45.859700 30.159934 0.073132 0 41.152600 9.813851 0.043038 g Potential 2 1.774000 -1.910335 -0.048427 2 4.029400 -18.140817 0.016109 2 9.973300 -49.252275 -0.824661 2 28.408500 -186.211075 0.701747 2 94.635800 -442.212739 1.261434 1 300.058200 -64.786573 4.683575 223 Californium (78 core/20 valence) RECP- valence 6s,p,d/7s/5f ______a______Exponent ______GfAPBPI------G (SOI 8 S - g Potential 2 0.086600 -0.023150 2 0.212100 -0.244230 2 0.564600 -2.083392 2 2.450800 -112.744095 2 3.329600 361.115670 2 5.103100 -304.971728 1 7.639000 91.317723 0 45.903400 9.390824 p-g Potential 0.173300 -0.249646 -0.006092 0.720100 -4.339745 1.731808 0.942700 0.844048 -5.061527 1.734400 -4.787827 18.687476 2.798300 77.995743 -28.765700 4.644600 -16.039077 18.670466 9.477700 46.878098 0.309112 22.035200 5.985687 0.051835 d-g Potential 2 0.048300 0.006838 0.006411 2 0.574100 -1.264230 0.339470 2 1.294900 23.252780 -0.217478 2 2.131500 -61.921888 1.040052 2 3.010900 115.666878 -3.366419 2 5.096500 -23.182711 3.755714 1 7 .415600 48.154512 -0.690020 0 19.884300 8.243530 0.058987 f-g Potential 0.090600 -0.029296 .000050 0.687600 -3.009714 .010319 0.246700 -0.382043 .001317 1.963800 -8.691500 .166135 5.772700 86.565616 .620219 8.674800 -115.703917 -5.056839 16.138200 34.373082 0.234350 9.274000 9.235198 -0.011676 g Potential 0 .112400 0.063969 0.021198 0.334100 0.720262 0.138355 0 .868700 3.603620 1.908495 35.572300 -331.191754 -8.320818 5.749800 -49.507135 -1.981396 173.791700 -64.195923 3.889005 224 Einsteinium (78 core/21 valence) RECP- valence 6s,p,d/7s/5f ______n______Exponent CfAREP)------CIS2I 8 S-g Potential 2 0.904200 -14.499788 2 0.217700 -0.089871 2 1.251100 46.887831 2 1.933300 -134.738524 2 3.123600 332.500499 2 5.087700 -312.705486 1 8.040700 120.462492 0 51.814600 13.290569 p-g Potential 2 0.212800 -0 .081618 -0.015075 2 0.766700 -40.432905 27.275182 2 0.788600 42.010816 -30.202247 2 1.873400 -96.411406 43.139799 2 2.265400 169.882874 -50.353294 2 4.711100 -79.355184 24.469501 1 5.985400 57.575254 -4.215818 0 24.309300 7.183643 -0.892178 d-g Potential 2 0.094000 0.016277 0.010199 2 0.916200 -4.064139 0.626402 2 1.605400 122.576201 1.648516 2 1.998600 -208.420513 -4.192756 2 2.801600 160.102601 1.389661 2 5.722700 -78.404778 1.145532 1 5.635600 59.691111 -0.368194 0 21.450400 7 .673575 0.022983 f-g Potential 2 0.172900 -0.032916 -0.000174 2 0.434000 -0.605185 -0.006629 2 1.329800 -17.507505 -0.105784 2 1.737200 37.310971 0.256591 2 2.515800 -64.487781 -0.824368 2 3.584000 64.953050 1.368011 1 17.794800 34.131750 -1.001365 0 15.361600 9.334680 0.082032 g Potential 2 0.280200 0.220054 0.056998 2 0.845200 2.600895 0.799558 2 4.926900 -31.240355 -1.083576 2 15.657200 -69.072082 -2.128629 2 33.651100 -153.302277 2.887205 1 84.974300 -49.514857 1.007187 225 Fezmium {78 core/22 valence) RECP- valence 6s,p,d/7s/5f ______n — EgpQneat------CtAREP)______C i s a i 8 S-g Potential 2 1.120900 -34.449874 2 0.273000 -0.053970 2 1.384800 93.452520 2 1.968400 -166.937064 2 3.185900 315.457378 2 5.448900 -287.312680 1 8.250500 121.748657 0 53.435700 13.414754 p-g Potential 2 0.290500 -0.071368 -0.022101 2 0.893900 -4.895700 2.152718 2 1.551900 109.294828 -58.932739 2 1.848200 -264.049489 114.294492 2 2.597100 416.310951 -89.859740 2 3.555200 -281.262873 36.002226 1 5.338100 56.756920 -3.740923 0 23.670400 7.069498 -1.088468 d-g Potential 2 0.150900 0.029051 0.014327 2 1.389900 -66.326009 0.257370 2 1.653700 222.195547 5.893223 2 2.245700 -283.739365 -13.247011 2 3.425600 282.015119 9.957154 2 5.163100 -125.962372 -3.900062 1 7.542100 57.053576 -0 .017579 0 24.026000 7.323862 -0.010036 f-g Potential 2 0.858300 -2.390307 -0.017956 2 0.296800 -0.096259 -0.003072 2 3.459200 -38.288905 -1.287947 2 5.040700 111.312686 4.731229 2 7.663300 -75.484584 -4.029523 1 17.767600 88.395055 -0.037676 0 81.463400 8.397624 0.005319 g Potential 2 0.374900 0.209722 0.064147 2 1.003300 2.387164 0.607340 2 4.664200 -23.885361 -1.672557 2 11.071900 -41.272843 1.051893 2 30.679300 -195.209479 -0.688478 1 94.678700 -52.429053 1.566428 226 Mendeleevium (78 core/23 valence) RECP- valence 6s,p,d/7s/5f ______a______E x p o n e n t ______C(ftRBg)------£1SQL 8 S-g Potential 2 0.755600 10.108781 2 0.904100 -36.098217 2 1.269200 74.370778 2 1.960400 -152.264872 2 3.351100 334.368796 2 6.066600 -349.597904 1 9.846900 141.494938 0 76.166000 13.195672 p-g Potential 2 0.612000 -0.651938 0.116639 2 1.760800 27.558510 -22.748683 2 2.163700 -140.451828 61.969387 2 2.948200 354.183858 -44.969809 2 4.364400 -382.050962 -21.731287 2 6.090400 271.133015 36.886812 1 20.193600 65.555243 -1.501719 0 59.766100 5.579841 0 .204611 d-g Potential 2 0.220900 0.049197 0 .015009 2 1.467900 -64.929224 12.831409 2 1.758200 223.800372 -29.085527 2 2.409600 -295.631655 28 .645890 2 3.595400 295.395665 -12.390702 2 5.423100 -127.921715 0.912056 1 7.747400 57.743547 0.620858 0 24.391700 7.828492 -0.024061 f-g Potential 2 2.432000 13.131205 -3.928180 2 0.441600 -0.213042 -0.017897 2 1.289200 -5.233049 0.323111 2 4.005400 -81.413386 19.598520 2 5.935500 203.750815 -36.775307 2 9.763300 -217.600708 34.900743 1 15.588000 98.324105 -0 .775404 0 89.995600 8.201972 0 .082970 g Potential 2 0.430700 0.159764 0 .057029 2 1.116000 2.233934 0.524753 2 4.738200 -20.636832 -1.966528 2 9.466000 -32.380397 1.472259 2 28.086600 -187.271015 -0.227218 1 91.218400 -53.306916 1.544094 227 Nobélium (78 core/24 valence) RECP- valence 6s,p,d/7s/5f ______n______E x p o n e n t ------CfflBEfJ------£ISÛ1 s-g Potential 2 0.948000 -5.247927 2 1.452300 53.062085 2 1.936300 -117.289078 2 3.563800 306.650545 2 6.037800 -331.410566 1 9.500100 128.607860 0 49.944500 12.216962 p-g Potential 2 1.487300 -38.581799 85.354236 2 1.714000 127.712443 -244.852413 2 2.216300 -256.025556 355.294367 2 3.189200 525.354948 -371.593410 2 4.906800 -633.991086 295.243710 2 7.498100 508.197819 -135.280057 1 24.339100 36.281852 1.579205 0 21.459200 11.820445 -1.590535 d-g Potential 2 1.214600 42.612485 3.251585 2 1.450100 -141.191107 -12.826586 2 1.971400 286.287439 32.066290 2 2.940400 -338.348570 -50.302988 2 4.582700 343.325895 42.178249 2 7.181600 -140.282365 -20.337716 1 11.164000 66.823763 0.737634 0 34.784100 7.409760 -0.084415 F-g Potential 2 0.000000 0.000000 0.000000 2 1.181100 0.178466 0.049503 2 0.783500 -0.330725 -0.052698 2 5.195300 -67 .099483 -3.827856 2 6.897200 216.271345 11.645897 2 11.001500 -237.395913 -12.303372 1 17.620800 104.372606 0.708011 0 102.361900 8.085295 -0.049681 g Potential 2 2.027900 -1.759599 -0.080942 2 4.571700 -19.194930 0.115094 2 11.586500 -50.647092 0.211626 2 33.051600 -200.095089 0.834956 2 110.282900 -486.486179 4.496621 1 356.881800 -67.628304 5.694578 228 Lawrencium (78 core/25 valence) RECP- valence 6s,p,d/7s/5f JL_ ____ B x p g n e n t — - - .WlAKafJ------ S-g Potential 2 2.167200 49.137263 2 2.724600 -191.042990 2 4.010800 209.446511 2 4.015100 260.470903 2 6.699600 -488.956803 2 13.217900 658.068399 1 36.089200 43.209839 0 40.458300 15.955562 p-g Potential 2 1.347400 -7.355266 34.389889 2 1.553700 28 .438324 -92.556703 2 2.026900 -66.700230 118.908838 2 3.073900 186.593175 -104.380659 2 5.023700 -142.150133 69.631695 2 8.878700 190.023133 -27.202488 1 20.101500 43.405351 -2.282971 0 27.586900 7.593093 -1.009920 d-g Potential 2 1.195600 35.041400 -3.793784 2 1.423900 -113.568256 10.387445 2 1.939900 235.133881 -8.749707 2 2.838800 -271.483486 -3.117346 2 4.375800 282.181501 16.700629 2 6.633300 -105.983269 -14.341218 1 10.278200 59.506987 1.428240 0 28.457500 7.747695 -0.094302 f-g Potential 2 2.031000 -9.851342 -1.311274 2 2.986900 60.027373 7.111847 2 4.020200 -125.038772 -12.334065 2 6.562100 224.626196 16.106974 2 11.027500 -251.349898 -14.756406 1 17.351700 106.518512 0.849994 0 109.062500 8 .207049 -0.059082 g Potential 2 2.028500 -1.441703 -0.059397 2 4.393000 -15.039198 0.116671 2 9.838100 -39.659553 0.100483 2 28.350200 -161.562671 0.829881 2 82.611400 -360.188035 1.602020 1 250.899200 -62.816846 4.129600 229 Rutherfordium (78 core/26 valence) RECP- valence 6s,p,d/7s/5f ______a______E x p o n e n t ______CfaREf)------ 8 S-g Potential 2 2.058500 27.637497 2 2.815700 -159.928651 2 4.149500 464.697474 2 6.797900 -257.038009 2 6.965700 -254.119544 2 12.647800 645.653928 1 35.790200 48.396421 0 37.883000 15.586774 p-g Potential 2 1.501000 -37.393924 37.973993 2 1.727300 124.291869 -106.351762 2 2.224000 -244.904095 142.201725 2 3.164100 457.411490 -104.271484 2 4.760700 -468.097681 20 .069494 2 7.098900 358.423906 22.864369 1 19.363500 37.594209 -3.464915 0 20.903000 10.341765 0.322723 d-g Potential 2 1.311800 46.797161 -6.295990 2 1.561200 -150.722763 17.272744 2 2.132900 307.926906 -17.003536 2 3.174700 -363.821267 1.625200 2 5.000200 368.262951 18 .183437 2 7.880100 -149.462674 -18.122157 1 12.478100 70.914630 1.836045 0 40.450500 7.344888 -0 .119670 f-g Potential 2 2.285000 -17.428778 -2.819184 2 3.027700 62.139782 9.436844 2 4.400700 -133.331364 -16.395297 2 7.095800 279.473551 24.788712 2 11.784200 -341.902618 -25.703118 2 22.133600 612.217065 13.770114 1 57.551500 31.497567 -1.020299 0 58.816700 9.879494 0.040206 g Potential 2 2.438900 -2.846479 -0.082669 2 5.655300 -24.321307 0.267437 2 14.867600 -62.584277 -0.202859 2 39.853600 -222.374617 2.245994 2 134.056100 -559.875997 4.453153 1 442.392500 -71.088272 7.285639 230 (78 core/27 valence) RECP- valence 6s,Pfd/7s/5f ______0______Exponent______ClMfifJ------CISÎ2I 8 S-g Potential 2 0.806100 11.459021 2 0.988600 -38.271740 2 1.411700 79.461242 2 2.226600 -156.071270 2 3.911200 363.050096 2 7.362300 -387.526781 1 12.347600 157.080308 0 97.268700 13.126674 p-g Potential 2 1.607500 -17.352870 59.765580 2 1.876700 59.346162 -174.755978 2 2.485400 -127.779262 263.752850 2 3.770500 367.828490 -306.077982 2 6.152000 -461.057966 303.433764 2 10.164300 455.068917 -197.171465 1 32.596100 39.796318 6.541495 0 28.605000 9.117669 -3.170430 d-g Potential 2 1.339900 36.486124 -4.561019 2 1.617900 -127.748917 13.466262 2 2.213100 280.031451 -13.316194 2 3.252200 -318.986978 0.471811 2 5.101500 323.854003 15.668940 2 8.009600 -120.241587 -14.930963 1 12.690000 68 .860421 1.529534 0 38.712500 7.470073 -0.090903 f-g Potential 2 2.412500 -20.218714 1.150193 2 3.161900 74.460296 -9.101466 2 4.395100 -137 .020865 22.971021 2 7.217800 257.446343 -39.380821 2 11.784300 -273.330229 41.580562 1 19.006300 108 .773237 -0.836603 0 112.815500 8.119303 0.083482 g Potential 2 2.069400 -1.072854 -0.054496 2 4.515300 -14.627154 0.118561 2 10.553700 -44.910669 0.040125 2 31.709400 -184.397107 0.997713 2 100.320200 -437.046272 3.459612 1 319.042200 -66.318773 5.251987 231 Seaborgium (78 core/28 valence) RECP- valence 6s,p,d/7s/5f ______Q ______E x p o n e n t C( a r e p i ------CCSQi 8 S-g Potential 2 0.863000 12.059602 2 1.054200 -40.361789 2 1.494100 83.712676 2 2.329600 -160.982037 2 4.069400 368.868633 2 7.561700 -378.962570 1 12.481400 154.195371 0 92.559400 13.154970 p-g Potential 2 1.403800 -26.910976 22.395341 2 1.678900 104.842428 -73.054256 2 2.213400 -225.371662 109.119620 2 3.241700 430.559561 -82.565455 2 5.055200 -450.615717 17.743796 2 8.128800 441.563947 24.150231 1 23.101000 36.736255 -5.562491 0 21.409900 15.164863 0.463670 d-g Potential 2 1.424000 46.346517 4.773688 2 1.703400 -151.616339 -17.041877 2 2.339900 313.824316 40.616536 2 3.504700 -357.892485 -62.318665 2 5.581700 368.826484 53.281998 2 8.842900 -140.974769 -27.674417 1 14.356300 74.653018 0.982915 0 46.220700 7.297409 -0.098267 f-g Potential 2 0.200900 0 . 000781 -0.001399 2 2.505800 -24. 035559 1.547908 2 3.131000 67. 894449 -8.252546 2 4.527500 -121. 398830 21.256668 2 7.317100 232. 516207 -36.366929 2 12.348200 -253. 158704 39.502521 1 19.371100 110. 254096 -0.944473 0 119.084600 8. 231390 0.091297 g Potential 2 2.643200 -2.864170 -0.087343 2 6.071400 -24.832216 0.314629 2 15.741200 -65.208793 -0.417390 2 42.458000 -230.830314 2.887456 2 142.643100 -581.962395 4.611663 1 474.112400 -72.601275 7.950557 232 Kielabohrium (E107) (78 core/29 valence) RECP- valence 6s,p,d/7s/5f ______n E x p o n e n t ______C(ftBBg)------C .tSOI 8 s-g Potential 2 0.923900 12.892830 2 1.121200 -42.521161 2 1.580400 87.594939 2 2.448100 -168.265216 2 4.228300 378.642333 2 7.743800 -370.940538 1 12.615700 151.038240 0 87.547500 13.160754 p-g Potential 2 1.611900 -40.425377 35.752217 2 1.870800 132.456765 -99.535269 •2 2.450800 -259.200135 131.788026 2 3.572200 499.212255 -86.700331 2 5.535600 -512.780569 -6.314924 2 8.694300 414.409638 42.860230 1 25.463100 37.000409 -6.030902 0 23.464800 10.310912 0.448379 d-g Potential 2 1.506300 52.805720 -7.644621 2 1.790900 -166.949591 19.780401 2 2.466700 342.949734 -16.713511 2 3.711900 -397.512244 -3.697762 2 5.950300 402.933539 25.134549 2 9.738000 -159.585431 -23.142811 1 15.475800 79.989260 2.145572 0 53.966000 7.180165 -0.133277 f-g Potential 2 1.589000 3.474602 0.194063 2 2.227700 -30.920197 -4.369500 2 2.803200 52.616790 8.048470 2 4.850400 -101.109937 -11.362056 2 7.852300 242.440802 18.827858 2 13.851500 -290.073278 -19.092606 1 21.300700 118.626655 1.161712 0 142.670700 8.214596 -0.074111 g Potential 2 2.895600 -3.352908 -0.379876 2 6.992500 -31.312096 0 .814223 2 19.708100 -84.269482 -0.416957 2 50.646800 -250.994557 3.375802 2 169.912600 -663.907311 7.122246 1 579.140100 -76.542887 9.685665 233 Bassium (E108) (78 core/30 valence) RECP- valence 6s,p,d/7s/5f ______0______Exponent------CfMEP)------C1S21 8 s-g Potential 2 2.545400 51.801708 2 3.234800 -201.126572 2 4.847000 233.902811 2 4.852100 279.005579 2 8.135600 -496.250421 2 16.228200 661.434313 1 29.026200 71.455378 0 70.567200 15.947768 p-g Potential 1.830600 -46.538950 88.660302 2.105100 151.838787 -250.016844 2.721400 -298.162104 353.078417 3.911000 594.441252 -342.322606 6.008100 -663.107218 240.005711 9.250100 540.663495 -101.067073 27 .790000 37.351300 -1.127433 24.850900 11.900684 -1.419209 d-g Potential 2 1.561200 49.726137 -6.646368 2 1.858100 -159.462565 17.205356 2 2.548300 328.190867 -13.188035 2 3.806800 -367.870306 -6.217672 2 6.024200 377.359830 25.031248 2 9.587800 -138.683147 -21.237088 1 15.387800 76.322813 1.872121 0 49.033900 7.306534 -0.112472 f-g Potential 2 2.827700 -59.201590 -20.922512 2 3.132600 107.364218 34.310187 2 4.176400 -87.966726 -19.006835 2 8.282800 188.385017 17.211366 2 14.056600 -246.597673 -18.882652 1 21.788100 113.654027 1.104875 0 122.951200 8.019008 -0.068631 g Potential 3.048600 -3.521262 -0.357355 7.267500 -31.555325 0.861710 20.270600 -85.944269 -0.753940 52.336100 -256.313672 4.242553 175.599500 -678.365025 6.595322 601.332900 -77.539123 10.188585 234 Heltneriun (E109) (78 core/31 valence) RECP- valence 6s,p,d/7s/5f S-g Potential 2 0.935700 10.828010 2 1.138600 -36.034994 2 1.605900 73.919805 2 2.501300 -141.544461 2 4.341400 317.196819 2 7.945800 -265.940804 1 12.756700 143.721629 0 75.800600 13.339084 p-g Potential 2 1.752600 -44.133453 95.167301 2 2.039700 146.130034 -271.748796 2 2.677000 -286.204196 382.088421 2 3.944700 569.665584 -365.762877 2 6.254700 -628.399344 232.347211 2 10.451100 599.156132 -47.864464 1 32.067000 36.188430 -5.891108 0 26.348200 15.100821 0.499980 d-g Potential 2 1.612700 45.779565 -5.742294 2 1.923400 -149.023125 14.877522 2 2.634200 310.817246 -10.222090 2 3.898500 -333.047219 -7.642159 2 6.144500 345.933606 23.886140 2 9.605900 -117.377245 -19.546173 1 15.467400 73.738391 1.696060 0 45.960100 7.424849 -0 .098284 f-g Potential 2 1.154200 0.277785 0.047903 2 2.660600 -26.518504 -7.287898 2 3.304100 67.399693 16.190276 2 4.860600 -114.036208 -17.968523 2 8.082800 229.090344 20.979349 2 14.022000 -250.872236 -18.566846 1 22.290000 115.670703 0.984108 0 135.269500 8.227838 -0.060709 g Potential 2 3.219900 -3.785209 -0.348048 2 7.606300 -32.178836 0.930546 2 21.081100 -88.775589 -1.085671 2 54.509500 -262.261008 5.173478 2 182.558800 -695.461128 6.103801 1 627.899800 -78.668186 10.768032 235 Bleneot 110 (78 core/32 valence) RECP- valence 6s,p,d/7s/5f Exponent CfSO) s-g Potential 2 0.542900 -3.671164 2 0.692000 15.773653 2 0.960500 -34.350249 2 1.514900 66.841212 2 2.505800 -135.146742 2 4.567700 332.310546 2 8.925800 -306.756772 1 15.317600 163.754476 0 103.569300 13.132844 p-g Potential 2 0.641500 0.630285 -0.547001 2 1.265000 -33.085644 22.120759 2 1.612400 106.268191 -60.511193 2 2.335300 -213.291076 83.663177 2 3.750100 441.619350 -60.102264 2 6.327500 -460.578188 8.527055 2 11.840800 612.622770 34 .440507 1 43.857800 173.169588 -5.398954 0 166.251100 12.702933 0.456996 d-g Potential 2 0.642600 13.040438 -4.889224 2 0.780000 -43.325722 15.670597 2 1.092100 88 .980822 -28.924443 2 1.671000 -167 .504735 42.964233 2 2.785600 345.467130 -49.699498 2 4.888200 -402.272115 38.384338 2 9.541800 522.266521 5.676984 1 37.528500 45.726112 -2.950775 0 28.530300 12.897008 0.280182 f-g Potential 2 1.118000 12.718983 8.798987 2 0.743200 -0.779139 -0.415542 2 1.192800 -13.603269 -9.354084 2 4.168200 25.850713 18.899172 2 6.460300 -130.019386 -75.742620 2 10.289600 366.759520 146.673006 2 18.962100 -398.088804 -151.317868 1 31.528500 112.131203 -14.153835 0 201.218800 5.497999 -2.101956 g Potential 2 2.322800 -0.942699 -0.055862 2 5.565600 -16.794738 0.135595 2 12.392600 -48 .719661 0.134394 2 35.593200 -178.705263 0.825326 2 95.570400 -340.505431 6.992716 2 288.021800 -925 .312329 20.703276 1 1028.635200 -91.493930 17.037421 236 Element 111 (78 core/33 valence) RECP- valence 6s,p,d/7s/5f ______Q______E x p o n e n t ------QlASEB)------ClfiQI 8 3-g Potential 2 0.947000 6.978591 2 1.266900 -38.521601 2 1.753200 91.575333 2 2.743400 -169.090588 2 4.932300 401.814637 2 9.393200 -395.126026 1 15.881300 165.672380 0 108.679600 13.034783 p-g Potential 2 1.853100 -40.920194 82.053680 2 2.146800 132.246506 -228.242338 2 2.814500 -257.940022 313.504322 2 4.130700 516.623718 -284.203226 2 6.498300 -530.770406 160.618202 2 10.800000 532.586958 -16.076057 1 29.744900 35.666329 -6.362357 0 25.982900 15.211856 0.529237 d-g Potential 2 1.762600 52.904518 -6.705513 2 2.108100 -169.846825 17.866982 2 2.926900 357.297674 -14.640181 2 4.441300 -389.268333 -3.982257 2 7.191700 402.688391 24.313139 2 12.012700 -139.892663 -22.925138 1 19.264300 85.852344 2.136049 0 65.398400 7.152899 -0.113469 f-g Potential 2 2.489000 -2.640088 2.137267 2 3.845600 21.827728 -15.550036 2 5.366800 -73.400872 32.899682 2 8.972900 207.930518 -46.356061 2 15.688100 -239.916586 51.412269 1 25.051200 119.507803 -1.532009 0 149.488500 8-248963 0.121567 g Potential 2 3.587100 -3.966707 0.178914 2 8.367400 -35.014687 -0.276850 2 23.631600 -99.423949 0.552620 2 60.879200 -277.577194 3.523668 2 202.942700 -748.039776 12.925321 1 708.654700 -81.904162 12.002298 237 Element 112 (78 core/34 valence) RECP- valence 6s,p,d/7s/5f ______0______E x p o n e n t ______CfAREP)______CI5Q1 8 S-g Potential 2 0.940100 9.043886 2 1.174000 -31.959011 2 1.702200 70.196782 2 2.712300 -140.042079 2 4.877300 338.350651 2 9.395700 -292.859149 1 15.812700 161.048172 0 99.461400 13.202310 p-g Potential 2 1.578500 -35.738085 20.698316 2 1.889500 118.124270 -58.135269 2 2.606300 -239.417852 76.623275 2 4.053300 500.687990 -30.705496 2 6.753600 -547.915100 -53.265618 2 12.190700 637.375252 85.030885 1 39.639500 42.199140 -16.028854 0 32.614100 14.913436 0.923367 d-g Potential 2 1.822200 50.669762 -5.739149 2 2.178900 -162.967629 15.175244 2 3.020100 343.422914 -10.751567 2 4.555400 -361.293529 -7.111646 2 7.312100 381.121767 25.443610 2 11.928100 -121.967828 -22.131963 1 19.346500 83.479885 1.951785 0 61.645200 7.245259 -0.098837 f-g Potential 2 1.108400 -0.048820 0.054363 2 3.579900 1.697479 -1.805359 2 6.235300 -54.912952 14.804185 2 9.394800 210.221175 -34.692485 2 15.827800 -234.009489 42.335855 1 25.449000 118.589466 -0.860615 0 146.297200 8.245070 0.080197 g Potential 2 3.861600 -4.680760 0.166602 2 8.955400 -36.714009 -0.279290 2 25.270400 -106.208694 0.738999 2 64.780700 -285.430144 3.620953 2 214.204000 -773.820148 14.703091 1 751.535800 -83.561491 12.766648 238 Element 113 (78 core/35 valence) RECP- valence 5f,6s,p,d/7s,p ______n Exponent ______CfARBP)------CiSfil S-g Potential 2 1.611000 -33.131224 2 1.951900 100.803694 2 2.718000 -152.838238 2 5.001400 313.083659 2 9.716100 -248.092289 1 15.756300 150.037428 0 76.443200 12.667375 p-g Potential 2 1.897000 -48.996717 79.180793 2 2.218500 154.241544 -216.802091 2 2.986200 -306.370257 302.980184 2 4.482300 633.817706 -274.055932 2 7.266700 -709.170181 131.646100 2 12.480800 707.803100 22.661287 1 37.935300 38.965510 -14.618927 0 32.576600 15.029538 0.944662 d-g Potential 2 1.350000 -24.979057 0.845374 2 1.567000 86.707104 -0 .588288 2 2.059200 -178.467921 -6.211673 2 3.027100 352.840535 30.067048 2 4.723700 -371.167613 -52.750791 2 7.580200 339.475827 28.620523 1 23.912900 39.630388 -3.088462 0 22.998600 8.521371 0.163666 f-g Potential 2 1.896500 3.339513 -0.047593 2 1.769400 -2.683407 -0.004639 2 6.424100 -55.695049 -2.162127 2 9.958900 229.926801 11.327006 2 17.352700 -264.309566 -16.273096 1 28.383400 125.842273 0.887588 0 168.699600 8.177499 -0 .051675 g Potential 3.864300 -3.574774 -0.064818 9.266600 -39.994405 0.322920 27.415600 -116.860782 -0.084291 70.062600 -294.715866 5.623714 229.321100 -808.954668 13.824438 810.892500 -85.761360 13.970285 239 Blenent 114 (78 core/36 valence) RECP- valence 5f,6s,p,d/7s,p ______Q ______E x p o n e n t ______CiABEE^------QlSfil 8 s-g Potential 2 1.806300 -21.749950 2 2.279100 80.691011 2 3.354300 -80.035980 2 3.358700 -109.489436 2 5.649500 454.508048 2 11.589500 -457.978678 1 20.567200 194.288589 0 154.044700 12.678037 p-g Potential 2 1.505900 -3.713278 6.149087 2 2.463500 107.572355 -117.610861 2 3.005800 -253.914841 214.242993 2 4.725100 583.203912 -168.996010 2 7.373300 -671.680583 55.543976 2 12.682800 662.527443 69.794390 1 31.637600 44.697962 -16.747832 0 36.313000 14.938110 1.162434 d-g Potential 2 1.408200 -25.007865 13.139031 2 1.638300 87.663653 -37 .332113 2 2.158300 -182.293636 53.032999 2 3.183000 364.756096 -49.119374 2 5.007900 -380.770243 27.644585 2 8.118900 352.239144 -4.225558 1 25.685900 39.896763 -0.340525 0 24.120300 8.493500 0.052927 f-g Potential 2.319400 0.344388 -0.044913 6.677000 -53.560658 -2.701937 10.376300 240.964826 13.734170 17.217500 -272.165732 -18.163931 28.362100 124.263801 0.904586 160.970000 8.140298 -0.051590 g Potential 2 4.193000 -4.394264 -0.085876 2 9.843800 -41.135044 0.385691 2 28.901400 -122.727375 -0.121400 2 73.821000 -302.046733 6.150971 2 239.654800 -830.232192 14.943788 1 849.802000 -87.274843 14.713798 240 Blenent 115 (78 core/37 valence) RECP- valence 5f,6s,p,d/7s,p s-g Potential 2 1.076500 12.176418 2 1.337500 -41.231584 2 1.947700 87.550237 2 3.133400 -173.784545 2 5.712900 437.019279 2 11.346700 -422.076712 1 20.097100 183.295885 0 141.716700 13.074354 p-g Potential 2 2.015200 -44.678330 67.816217 2 2.361400 147.662921 -191.989010 2 3.132800 -289.689444 260.597574 2 4.690200 588.555701 -208.886126 2 7.524800 -613.505606 73.203188 2 12.836100 635.526854 45.476597 1 36.371100 37.297922 -13.335247 0 31.097400 15.155248 0.855191 d-g Potential 2 1.474300 -20.463395 11.638450 2 1.729000 79.253118 -35.099290 2 2.283800 -186.148480 54.203359 2 3.292400 375.899259 -51.033192 2 5.116300 -374.101544 27.475848 2 8.189000 343.774023 -3.966477 1 24.265600 41.621361 -0.784000 0 25.101600 8.493645 0.077283 f-g Potential 2 2.124500 5.998849 -13.183520 2 2.662700 -41.257119 91.909894 2 2.923400 40.404178 -90.974466 2 7.772800 -89.230206 93.626989 2 12.195400 356.130540 -210.710218 2 21.503800 -384.506582 215.080488 1 35.188300 119.362804 15.384129 0 219.728400 6.380794 2.080640 g Potential 2 4.573100 -5.451290 -0.112555 2 10.595500 -43.235141 0.609430 2 30.686800 -127.688550 -1.269419 2 77.318100 -306.261366 8.884558 2 248.006300 -844.509256 12.298827 1 879.541500 -88.477449 15.475510 241 Element 116 (78 core/38 valence) RECP- valence 5f,6s,p,d/7s,p n Exponent ______CfAREP}------CiSOJ. S-g Potential 2 0.663400 -5.548366 2 0.812800 18.412328 2 1.162600 -38.570839 2 1.814500 74.893012 2 3.021400 -144.812601 2 5.642100 366.231559 2 11.330000 -305.136480 1 19.895900 177.393551 0 126.626800 13.160643 p-g Potential 2 1.126900 20.224048 -11.318234 2 1.374400 -80.976157 40.815203 2 1.813600 156.702938 -64.625174 2 2.722800 -260.494312 64.776514 2 4.526700 544.743383 -4.155176 2 7.801700 -603.111682 -81.989493 2 15.096700 791.544316 89.882446 1 58 .452800 47.493730 -17.573394 0 39.531200 14.734463 0.668895 d-g Potential 2 0.816700 17.485620 -5.728665 2 0.974300 -56.001109 17.662494 2 1.335300 110.422249 -30.875773 2 2.000700 -200.911164 43.001463 2 3.269300 403.913609 -43.809588 2 5.624100 -470.120904 28.307252 2 10.589700 581.414071 -1.624791 1 43.657800 161.859015 -0 .613379 0 178.882000 11.150152 0.089240 f-g Potential 2 2.184900 9.444849 0.562252 2 2.764700 -32.695644 -1.979789 2 4.085900 68.622295 3.516815 2 6.723100 -149.085986 -6.444158 2 12.549700 420.618631 20.041761 2 26.038300 -503.514509 -39.819964 1 49.048000 116.737883 2.733584 0 488.526000 4.036544 -0.134865 g Potential 2 4.771700 -6.152120 -0.120844 2 11.230900 -44.798807 0.604031 2 30.794600 -113.975858 -0.767129 2 71.736400 -275.717847 7.125340 2 219.141200 -742.860812 8.704667 1 742.878200 -83.786401 13.296633 242 Element 117 (78 core/39 valence) RECP- valence 5f,6s,p,d/7s,p n______E x p o n e n t ______CfftRBg)------ClSfll 8 s-g Potential 2 2.027200 -30.515921 2 2.395700 -30.850335 2 2.396100 115.253743 2 3.439200 -150.098595 2 5.893100 336.092130 2 12.968900 -236.629685 1 20.244400 156.190344 0 62.572100 11.087171 p-g Potential 2 2.033700 -50.503029 60.142996 2 2.412900 165.091892 -169.118166 2 3.289400 -332.095814 232.895499 2 5.054400 699.308470 -176.103090 2 8.393100 -782.489388 21.104051 2 15.122400 834.195433 102.917844 1 48.490600 41.992650 -24.645208 0 39.998300 14.940799 1.318851 d-g Potential 2 1.533700 -28.714778 13.333700 2 1.798800 99.224392 -38.112279 2 2.402700 -204.887399 54.744150 2 3.609800 417.571683 -50.494050 2 5.811500 -434.896265 27.083135 2 9.737900 410.814634 -2.088785 1 31.635000 41.612894 -0.279540 0 27.814900 8.398550 0.050615 f-g Potential 2 1.837900 -0.429642 -0.400892 2 4.168400 8.167726 6.594189 2 7.767700 -100.227218 -49.616787 2 12.507800 367.704019 126.964598 2 24.219200 -400.856900 -158.303060 1 41.788100 120.328244 -17.037417 0 229.116500 5.348353 -1.953045 g Potential 5.212300 -7.446204 -0.148822 11.843300 -42.979261 0.698954 27.590100 -76.961212 -0.749655 61.842800 -268.866350 5.691358 206.113700 -744.908991 9.678120 702.411100 -81.817939 12.629015 243 Element 118 (78 core/40 valence) RECP- valence 5f,6s,p,d/7s,p ______D______Exponent ------G(ARBP)------GiSÛI 8 a-g Potential 2 1.601800 -11.460676 2 2.233900 59.044952 2 3.336800 -144.236014 2 6.148700 102.835760 2 6.150000 283.381075 2 13.086200 -328.943814 1 23.847200 172.123267 0 50.199900 8.991834 p-g Potential 2 2.045000 -38.008706 43.313147 2 2.471300 139.492515 -133.464654 2 3.369600 -302.084728 192.642940 2 5.194600 669.513767 -132.302850 2 8.528800 -734.128480 -15.703016 2 15.357400 794.957025 120.445524 1 45.696800 41.396905 -24.302174 0 40.021700 15.011317 1.339439 d-g Potential 2 1.522300 -28.540864 17.755676 2 1.798000 97.736303 -52.888223 2 2.433200 -202.570643 84.975874 2 3.707100 421.559962 -107.543939 2 6.051000 -452.875512 119.043351 2 10 .422700 479.521252 -93.539422 1 34.521400 44.074201 2.535648 0 29.525200 10.242098 -2.034669 f-g Potential 2 1.225900 -3.612687 -0.177583 2 1.501700 11.499070 0.561774 2 2.185000 -24.057349 -1.136952 2 3.446900 47.093881 1.885108 2 5.882900 -101.671395 -4.244293 2 10.891000 237.376453 12.885803 1 19.049900 -19.678320 -4.059337 0 38.957700 8.136366 0.246189 g Potential 3.047300 -0.018051 -0.037442 5.837500 -10.535128 0.003400 14.460500 -60.171433 0.442714 46.559400 -230.609888 1.765122 150.948900 -580.812361 9.378669 514.004300 -76.589117 9.190921 244 APPENDIX B LARGE CORE (LC) RELATIVISTIC EFFECTIVE POTENTIALS FOR Rf-ELEMENT 118 245 Rutherfordium (92 core/12 valence) RECP- valence 6s,p,d/7s ______n Exponent ______CfAREP)------£XS2I 8 s-g Potential 2 0.326600 -0.066280 2 1.227200 -17.689108 2 1.582600 70.772370 2 2.275100 -149.709878 2 3.779900 334.757358 2 6.594600 -317.540178 1 10.360100 134.175938 0 66.568800 13.305340 p-g Potential 2 0.025100 -0.002087 -0.001121 2 1.208600 2.655954 -1.445351 2 0.249700 -0.084976 0.027300 2 2.473100 -109.662388 32.574896 2 3.070400 291.623746 -36.376653 2 4.645700 -204.213180 0.277107 1 6.865400 75.008296 3.649848 0 32.442600 9.111850 -0.328723 d-g Potential 2 0.258900 -0.056172 0.007123 2 1.282700 46.696933 -11.292423 2 1.513400 -138.692929 28.646209 2 2.085200 274.697533 -32.380877 2 3.116000 -324.345066 22.343668 2 4.608800 261.811484 -5.640252 1 16.743000 32.166998 -0.063979 0 13.113200 8.764485 0.033389 f-g Potential 2 0.779900 79.362849 4.250349 2 0.905100 -205.718978 -14.231759 2 1.181900 270.892036 26.732862 2 1.643900 -225.022074 -30.942978 2 2.273200 145.163590 17.114868 2 6.229200 -22.459674 -7.546390 1 6.195800 58.644307 1.603937 0 18.502900 10.057546 -0.121622 g Potential 2 0.649800 -0.537347 -0.017555 2 1.719100 -8.067266 -0.055957 2 4.403000 -34.812390 0.209642 2 12.458000 -91.223993 0.077891 2 39.952200 -314.497607 0.550754 1 130.945900 -66.699209 2.167874 246 (92 core/13 valence) RECP- valence 6s,p,d/7s ______n______Exponent CtAREP)------CfSQl S-g Potential 2 1.173900 -34.701799 2 1.395700 76.959986 2 2.262100 -141.277010 2 3.859000 348.224227 2 7.210000 -362.542829 1 11.802800 143.069393 0 55.645800 11.258032 p-g Potential 2 0.320000 -0.102359 0.048284 2 0.038800 -0.001960 -0.001226 2 1.704500 14.734363 -14.696224 2 2.360900 -74.837257 63.486100 2 3.602200 313.126429 -117.075032 2 5.378300 -301.738298 109.766914 1 8.454800 85.846117 -18.565703 0 47.011800 7.742675 -1.878463 d-g Potential 2 0.288700 -0.054147 0 .008010 2 1.358100 40.557596 -9.878520 2 1.634200 -147.524846 29.546427 2 2.176500 300.052629 -35.553390 2 3.174100 -325.154401 23.382892 2 4.659600 253.428544 -6.027344 1 16.086400 69.351500 0.256868 0 44.618200 7.488787 0.012028 f-g Potential 2 0.760200 -46.106319 6.718892 2 0.867700 150.971250 -15.373507 2 1.104500 -244.644845 12.488236 2 1.531500 300.793130 1.352291 2 2.195400 -237.390179 -10.999295 2 3.152700 163.818076 7.443970 1 8.634100 55.569211 -0.345991 0 20.935600 10.163735 0.071181 g Potential 2 0.776000 -0.654714 -0.019782 2 2.026300 -9.763248 -0.046334 2 5.149300 -40.440588 0.234108 2 14.694200 -104.137252 0.029130 2 45.587000 -338.957500 0.953390 1 149.363300 -68.107993 2.468706 247 Seaborgium (92 core/14 valence) RECP- valence 63,pfd/7s ______n Exponent ______C(ARBP1------CISÛI 8 S-g Potential 2 0.877600 17.245318 2 1.017800 -43.637585 2 1.491800 81.757202 2 2.334300 -159.667408 2 4.065500 366.794508 2 7.559300 -377.865437 1 12.478200 154.138867 0 93.082900 13.194971 p-g Potential 2 0.320600 -0.090314 0.018952 2 1.958800 56.947141 -20.955441 2 2.363300 -180.849462 51.724950 2 3.341300 391.773371 -33.487904 2 5.169800 -419.339889 -15.116163 2 8.406100 433.023200 38 .005182 1 24.642700 36.613498 -6.943512 0 21.328800 15.146234 0 .536513 d-g Potential 2 0.936600 -21.794298 -0.245139 2 1.120400 74.874522 1.846925 2 1.501400 -153.153866 -7.550535 2 2.185900 285.913962 23.492993 2 3.331500 -311.648532 -37.084218 2 5.061200 255.272218 20.198055 1 15.040000 41.123720 -1.823123 0 19.948000 8.448281 0.120194 f-g Potential 2 0.906400 -52.109659 11.398188 2 1.040800 161.168919 -25.866594 2 1.360200 -250.287529 24.000036 2 1.920900 301.221247 -7.572543 2 2.853000 -220.960963 -6.642072 2 4.270400 176.554209 7.007899 1 10.591300 52.234596 -0.502310 0 21.022200 10.222618 0.078109 g Potential 2 0.796700 -0.474569 -0.015147 2 2.038600 -8.393960 -0 .054591 2 5.018100 -37.517650 0.222965 2 13.732500 -95.686334 0.058861 2 43.016700 -324.340102 0.694314 1 139.777900 -67.139846 2.349722 248 Nielsbohrium (E107) (92 core/15 valence) RECP- valence 6s,p,d/7s ______n______E x p o n e n t ------C(flBBf)------CISS2I 8 s-g Potential 2 0.913300 12.407078 2 1.116800 -43.630049 2 1.576200 90.794800 2 2.447900 -168.454363 2 4.225800 376.052764 2 7.737300 -367.844442 1 12.608200 150.678919 0 87.176100 13.176981 p-g Potential 2 0.369500 -0.085521 0.033364 2 2.371300 77.271760 -72.781498 2 2.824400 -248.700434 193.688060 2 3.916500 568.454026 -221.371199 2 5.984300 -663.798612 160.323809 2 9.352900 552.648334 -67.160887 1 29.155900 38.022460 -3.842963 0 25.508700 11.823279 -1.259232 d-g Potential 1.014200 -22.360015 5.853364 1.220900 85.300815 -20.979410 1.599100 -170.912516 35.058815 2.327300 306.437647 -36.420870 3.559400 -334.478158 24.671605 5.363200 273.660979 -6.064105 15.804700 42.245577 0.013683 21.233800 8.404568 0.039030 f-g Potential 2 1.028700 -48.467243 -3.400494 2 1.203100 159.866621 10.871675 2 1.595500 -242.649482 -15.143455 2 2.326800 290.391760 12.140014 2 3.549700 -196.077561 -6.356186 2 5.711500 191.153052 1.890529 1 13.519600 50.542484 -0.084732 0 22.610700 10.228659 0.004700 g Potential 2 0.983800 -0.801000 -0.023218 2 2.560600 -12.557979 -0.032929 2 6.488900 -49.942100 0.279001 2 18.936500 -129.571983 -0.018719 2 56.692800 -379.929822 1.765045 1 184.422200 -70.426221 3.065091 249 Bassium (E108) (92 core/16 valence) RECP- valence 6s,p,d/7s ______n______E x p o n e n t ------GfABEPJ------ClSai 8 s-g Potential 2 0.978900 14.474160 2 1.187900 -48.334099 2 1.667200 96.427503 2 2.569400 -174.883379 2 4.396300 382.985107 2 7.940700 -356.319795 1 12.777000 147.680222 0 82.824000 13.202260 p-g Potential 2 1.337400 -32.199752 14.656580 2 1.634700 108.107352 -44.837602 2 2.275600 -219.733332 66.985084 2 3.506800 448.845821 -40.952381 2 5.725600 -495.563898 -21.382212 2 9.826000 531.298585 54.438802 1 30.717100 39.341564 -11.138900 0 26.002000 14.986587 0.751662 d-g Potential 2 1.094000 -28.149186 7.520194 2 1..278000 89.615942 -22.419193 2 1..684500 -172.129627 35.103332 2 2,.420500 311.192736 -36.505894 2 3..634400 -326.208291 24.354420 2 5..395500 262.402293 -6.067903 1 15.188700 47.869972 0.155462 0 24.633600 8.246173 0.028026 f-g Potential 1.148400 -49.935591 -4.092185 1.346300 166.810730 12.891365 1.792200 -251.105340 -17.840795 2.620200 302.569609 14.458227 4.012400 -202.464012 -7.769137 6.455100 207.007100 2.474619 14.895300 48.090596 -0.136885 22.672400 10.327009 0.008516 g Potential 2 0.896900 -0.407071 -0.013601 2 2.313100 -8.493902 -0.058113 2 5.563600 -39.606961 0.246517 2 14.845800 -98.784565 0.039451 2 45.650800 -330.969087 0 .841044 1 147.204900 -67.526649 2.511505 250 Meitnerium (E109) (92 core/17 valence) RECP- valence 6s,p,d/7s ______n Exponent ______CfAREf)------CIS2I 8 s-g Potential 2 1.333000 -7.345620 2 2.025600 102.149702 2 2.440800 -97.023030 2 2.435100 -70.052877 2 4.658200 319.740107 2 9.052300 -335.572136 1 15.964300 172.152461 0 94.162700 11.662644 p-g Potential 2 1.500300 -26.101995 30.007907 2 1.862900 100.266320 -108.475750 2 2.558900 -222.647215 193.052577 2 3.878200 484.859225 -207.572613 2 6.264100 -535.240566 130.629773 2 10.696800 555.704365 -5.960897 1 31.225900 36.368158 -9.153359 0 26.583900 15.108423 0.701121 d-g Potential 2 1.153500 -28.653041 9.267752 2 1.368300 101.301461 -29.844699 2 1.809300 -208.376801 48 .817355 2 2.597900 363.425936 -49.280988 2 4.116300 -395.470908 32.634512 2 6.276800 332.531695 -9.120658 1 19.169200 46.906845 0.029198 0 26.246300 8.156343 0.040856 f-g Potential 2 1.270600 -45.764927 -3.957684 2 1.508900 164.030990 13.313771 2 2.031400 -253.098527 -19.559019 2 2.974600 300.892263 15.979785 2 4.753400 -192.398110 -8.649543 2 7.748500 237.146634 3.012361 1 18.346000 45.002365 -0.166199 0 23.064900 10.400747 0.009686 g Potential 2 1.095000 -0.742978 -0.022638 2 2.911000 -13.419464 -0.029112 2 7.324400 -54.972223 0.293300 2 21.521100 -143.766397 0.031556 2 64.152800 -406.553469 2.254429 1 208.512700 -71.896533 3.516351 251 Element 110 (92 core/18 valence) RECP- valence 6s,p,d/7s ______Q______Exponent CtABEE) ------Clfifll s-g Potential 2 0.912400 9.980010 2 1.162800 -38.319728 2 1.684100 86.869598 2 2.642200 -165.242650 2 4.749200 394.741663 2 9.126300 -401.456920 1 15.644100 167.145769 0 108.734200 12.862905 p-g Potential 2 1.446400 -39.120633 19.972857 2 1.751800 127.034669 -57.560144 2 2.446200 -251.352570 80.561328 2 3.843400 520.133890 -45.929702 2 6.475800 -596.807328 -34.088064 2 11.817500 673.248348 79.696649 1 40.545300 44.027724 -18.009417 0 33.485600 14.789047 1.047542 d-g Potential 2 1.571600 45.618640 27.196202 2 1.910100 -139.636335 -106.573629 2 3.021500 176.867732 -1878.870535 2 2.875600 203.081477 1761.562528 2 4.593700 -484.807182 258.919224 2 8.548100 507.970236 -72.267298 1 25.925400 51.644028 5.626461 0 36.390600 13.122716 -0.350560 f-g Potential 2 0.963300 33.983672 2.649885 2 1.114200 -106.386919 -8.299491 2 1.463200 202.399256 15.463798 2 2.085400 -263.342294 -19.649194 2 3.152300 300.345378 15.363792 2 4.696000 -148.455132 -6.628823 1 6.998600 68.818669 0.213336 0 25.595600 9.565679 -0.021264 g Potential 2 0.970400 -0.384888 -0.013079 2 2.637300 -9.264927 -0.060888 2 6.385200 -45.043155 0.291615 2 17.339300 -112.137002 -0.001503 2 52.767500 -361.939884 1.402824 1 171.289300 -69.285626 2.941767 252 Element 111 (92 core/19 valence) RECP- valence 6s,p,d/7s ______n______Exponent ______C(AREf)______CiSfll 8 S-g Potential 2 1.021500 14.501743 2 1.251200 -46.825810 2 1.784900 92.411256 2 2.807200 -172.521912 2 4.953100 406.452216 2 9.416900 -394.516748 1 15.883100 165.660574 0 110.133200 13.126524 p-g P o te n tia l 1.525700 -39 .417971 20.585826 1.835600 127 .470835 -58.207692 2.545100 -251 .712506 78.981138 3.966200 517 .517795 -39.191612 6.626200 -577 .285863 -41.937102 11.937500 652 .118962 80.429086 39.444000 42 730942 -16.341724 32.649600 14 866936 0.956148 d-g Potential 1.184000 -30.463471 15.521654 1.408200 100.255110 -47.530527 1.917200 -199.131780 79.854651 2.920300 397.951099 -107.759358 4.751600 -467.314845 124.114885 8.011400 445.343874 -92.481769 29.171700 43.254037 3.015247 25.303100 10.117769 -2.084648 f-g Potential 1.054700 35.776519 2.682551 1.229900 -114.930555 -8.729249 1.624600 219.445705 16.683464 2.360400 -282.551814 -22.129326 3.613100 321.357754 18.076024 5.565400 -152.144018 -8.240077 8.375800 73.913091 0.327714 30.584100 9.391479 -0.031080 g Potential 2 0.953100 -0.322363 -0.011069 2 2.711200 -8.950009 -0 .065001 2 6.667400 -47.344681 0.305624 2 18.549600 -120.621835 0.008247 2 56.972600 -381.879045 1.738959 1 186.632000 -70.458294 3.219319 253 Element 112 (92 core/20 valence) RECP- valence 6s,p,d/7s n E x c o n e n t CfAREP) CfSO) S-g Potential 2 1.786700 -49.131731 2 1.872200 65.119586 2 3.293200 -102.822690 2 5.163300 327.320932 2 9.976600 -278.885705 1 15.744700 148.844407 0 56.915400 11.232328 p-g Potential 2 1.563400 -36.843418 18 .586845 2 1.879400 120.385363 -52.550204 2 2.602400 -239.351888 71.094839 2 4.062800 497.398012 -33.766145 2 6.783900 -547.437549 -31.943190 2 12.240700 657.425030 50.288334 1 40.532500 45.181137 -11.659535 0 33.590000 14.859858 0.625276 d-g Potential 2 1.119500 -26.867996 8.597164 2 1.337600 85.132922 -24.540205 2 1.872300 -174.761622 38.991398 2 2.893000 358.177678 -44.075232 2 4.801500 -414.632875 33.951504 2 8.526400 468.036802 -8.991855 1 29.184300 43.069294 -0.504987 0 25.792200 13.081729 0 .077288 f-g Potential 2 1.251200 48.663568 -39.111299 2 1.474600 -156.475074 113.918586 2 1.999200 321.751371 -174.534844 2 2.983700 -431.872759 189.578896 2 4.775300 473.966786 -121.708646 2 7.786800 -225.622418 39.466344 1 11.990000 90.020987 0.842651 0 47.785600 8.876081 -0.013314 g Potential 2 0.843100 -0.184495 -0.006251 2 2.472300 -5.997739 -0 .070949 2 5.966100 -38.916524 0.246081 2 15.688000 -102.479730 0.120752 2 49.701500 -347.998677 0 .957817 1 161.479000 -68.488891 2.855113 254 Element 113 (92 core/21 valence) RECP- valence 6s,p,d/7s,p JL Exponent CfftREBL CtSQl s-g Potential 2 0.524600 -8.713354 2 0.540500 9.877405 2 1.321700 -21.819222 2 1.812600 64.814361 2 2.860300 -133.792822 2 5.087400 335.692480 2 9.620300 -273.981703 1 15.983500 156.727933 0 92.817600 13.211569 p-g Potential 2 0.930900 1.445750 -3.291370 2 1.397400 -13.785656 26.700101 2 2.164500 102.823104 -146.810358 2 2.944900 -291.154807 295.830072 2 4.386900 600.885879 -284.578860 2 7.197900 -620.629639 145.572858 2 13.013700 674.373285 3.904541 1 45.735700 171.243660 -5.215479 0 169.984300 12.678698 0 .504128 d-g Potential 2 1.375400 -36.798525 10-586301 2 1.601800 115.171563 -29 .664855 2 2.132200 -221.514595 42.677857 2 3.140900 423.052420 -36.939331 2 4.946500 -460.611015 10 .399268 2 8.243700 465.858011 18 .418527 2 14.459100 -165.126711 -22.673768 1 23.078400 96.465345 2.289332 0 86.474900 6.971832 -0.117467 f-g Potential 2 1.102500 38.350332 1.474179 2 1.282000 -120.871743 -5.348755 2 1.695100 230.473679 10 .872199 2 2.438100 -299.488235 -13.196703 2 3.743600 353.542628 8 .408723 2 5.942600 -208.641139 -2.732204 2 10.246200 283.832657 -0.153918 1 28.946400 105.780659 0 .069026 0 91.195300 8.860997 -0 .007067 g Potential 2 1.123500 -0.370921 -0.022738 2 2.933900 -8.814242 -0 .178245 2 6.909300 -42.939888 0 .591586 2 16.852300 -95.364114 -0.463326 2 46.568300 -284.667751 3.062026 2 142.420500 -641.743702 3.979159 1 447.962600 -82.607383 7.957377 255 Blenent 114 (92 core/22 valence) RECP- valence 6s,p,d/7s,p ______n Exponent ------C(ARBP)------&ISÛI 9-g Potential 2 0.602100 -16.451971 2 0.663100 ^ 33.124992 2 0.818100 -22.996162 2 1.964800 53.474577 2 2.989700 -147.877120 2 5.494300 405.930114 2 11.072500 -413.647317 1 19.807300 183.073057 0 133.678600 12.675604 p-g Potential 2 1.133300 21.042551 -29.685693 2 1.317600 -62.411422 82.073313 2 1.760500 113.422447 -123.587213 2 2.589900 -201.146311 147.864959 2 4.142300 408.057686 -111.855760 2 6.942000 -373.061612 26.808276 2 13.246900 534.605188 69.759274 1 41.694000 161.959458 -9.077900 0 146.890700 12.760735 0.838691 d-g Potential 2 0.797500 23.097251 0.453961 2 0.922200 -63.868541 -1.630085 2 1.253200 113.153229 4.571378 2 1.857300 -194.741893 -12.491259 2 3.025500 374.281244 36.270120 2 5.205300 -432.293262 -64.199344 2 9.800800 529.602043 45.865368 1 42.536100 49.038774 -7.296428 0 27.080400 12.666754 0 .287771 f-g Potential 2 1.190000 38.578574 1.650299 2 1.404400 -129.950725 -6.377919 2 1.866800 252.969453 13.360423 2 2.734600 -321.001604 -17.199529 2 4.273000 373.651962 12.534949 2 6.942800 -202.087813 -5.590807 2 12.338600 314.605535 1.296538 1 33.950600 112.113197 -0.026444 0 108.482300 8 .812570 -0.001462 g Potential 2 0.961700 -0.183383 -0.016651 2 2.632000 -5.718898 -0.152260 2 6.058800 -33.331167 0.381569 2 14.089600 -82.594524 0.014859 2 39.390500 -234.911442 1.393751 2 110.790100 -516.632964 3.379520 1 338.012200 -77.893944 6.018678 256 115 (92 core/23 valence) RECP- valence 6s,p,d/7s,p ______p Exponent CfAREP)------C .tS0 1 S-g Potential 2 0.613400 -4.281742 2 0.767200 15.861090 2 1.092100 -33.914680 2 1.722100 66.604889 2 2.883600 -133.200555 2 5.429900 352.125238 2 11.020700 -309.860070 1 19.705500 179.749067 0 131.031600 13.127737 p-g Potential 2 1.074100 21.446902 -11.028594 2 1.283700 -68.365716 32.426561 2 1.780000 140.410431 -53.764527 2 2.663000 -260.709247 62.723167 2 4.400200 543.202659 -10.811591 2 7.676500 -610.929910 -71.374716 2 14.890800 802.351385 87.056796 1 55.976400 47.713983 -19.048466 0 40.888100 14.750978 0.812638 d-g Potential 2 0.789800 15.721649 -5.390283 2 0.945800 -52.311383 16.950024 2 1.295500 105.274293 -29.762698 2 1.947900 -193.765976 41.456605 2 3.187100 393.060288 -42.244937 2 5.507400 -462.188065 26.814270 2 10.407200 565.236390 -0.579152 1 39.147900 46.743504 -2.405594 0 31.470100 12.914608 0.168120 f-g Potential 2 1.298000 45.808988 -15.298969 2 1.508500 -138.868436 41.650874 2 2.013300 260 .212299 -58.436246 2 2.941500 -327.438939 58.875036 2 4.572000 382.295075 -33.800062 2 7.347500 -196.396643 7.527880 2 13.310300 325.727930 4.429276 1 36.957200 118.554681 -0.127891 0 123.865100 8.775046 0.039255 g Potential 2 1.036300 -0.237171 -0.026051 2 2.890100 -6.687313 -0.138313 2 6.725800 -38.343498 0.448732 2 16.035500 -92.207138 -0.130929 2 45.674900 -277.988049 2.269835 2 139.173900 -631.095316 5.427263 1 440.178300 -82.666795 7.845117 257 Element 116 (92 core/24 valence) RECP- valence 6s,p,d/7s,p ------CfSOl S-g Potential 2 0.638600 -10.540090 2 0.730500 28.556432 2 0.886200 -25.292601 2 1.970200 49.599223 2 3.029500 -127.413218 2 5.683700 350.951917 2 11.288200 -293.060032 1 19.754200 174.239346 0 119.141000 13.047712 p-g Potential 2 1.124700 21.168308 -10.305705 2 1.351700 -71.959427 32.219675 2 1.839700 144.238978 -52.287396 2 2.754100 -260.100570 57.215820 2 4.534300 541.997505 -0.286464 2 7.843600 -596.333504 -82.737214 2 14.997200 782.450339 90.631775 1 53.474000 46.665035 -18.296333 0 40.443800 14.831286 0.813169 d-g Potential 2 0.830800 16.373691 -5.636108 2 0.982700 -49.793531 16.251354 2 1.368900 104.080269 -29.255910 2 2.027200 -197.515233 41.964497 2 3.283900 390 .685092 -41.302668 2 5.642200 -442.165596 25.168201 2 10.506900 542.019035 0.243737 1 34.192200 98.687111 -1.571796 0 66.914000 11.409051 0.161791 f-g Potential 2 1.498700 57.578070 -43.685990 2 1.765600 -183.660703 126.882973 2 2.394400 374.725340 -196.408871 2 3.552800 -490.566053 215.180483 2 5.671600 547.351964 -139.029881 2 9.343700 -282.563621 46.437530 2 17.163900 423.432387 4.792467 1 46.384800 125.180074 -0.268339 0 150.758800 8.725048 0.088968 g Potential 2 0.930300 -0.150500 -0.023242 2 2.676800 -4.737331 -0.121014 2 6.254400 -32.934742 0.330224 2 14.830000 -88.566498 0.115762 2 43.390200 -265.300756 1.563245 2 132.259000 -614.827583 6.221305 1 422.113100 -82.201451 7.514156 258 Element 117 (92 core/25 valence) RECP- valence 6s,p,d/7s,p J i___ E X PPneas------s-g Potential 2 0.548900 -6.255240 2 0.571300 7.355998 2 1.442300 -21.351014 2 1.988400 63.489637 2 3.215600 -134.176859 2 5.908400 361.621988 2 11.643800 -282.312197 1 19.945400 171.549893 0 115.805100 13.149717 p-g Potential 2 1.151800 21.185531 -18.120848 2 1.366800 -67.266229 53.675907 2 1.849800 126.328966 -83.157838 2 2.781700 -224.593512 94.572680 2 4.586500 472.641403 -39.623186 2 7.894100 -462.935538 -64.594340 2 15.394400 646.974245 164.078876 1 42.373900 34.980950 -33.716100 0 36.065900 15.357383 1.909562 d-g Potential 2 0.895300 17.424506 -6.413385 2 1.060600 -56.321808 19.634753 2 1.436900 111.494261 -33.763052 2 2.128000 -202.389104 46.058114 2 3.425800 401.896703 -45.679077 2 5.802200 -447.328672 29.610635 2 10.641700 547.998889 -3.988301 1 41.715400 156.444585 -0.356236 0 164.296500 11.153515 0.067454 f-g Potential 2 0.970500 -26.669952 10.124866 2 1.154500 88.514297 -31.174079 2 1.573500 -176.857399 52.616104 2 2.388400 341.516030 -70.847915 2 3.882800 -443.154700 72.650530 2 6.816200 505.219202 -38.816026 2 13.281000 -203.255220 7.167172 1 21.792300 116.876526 1.324268 0 94.373400 8.441382 -0.064561 g Potential 2 0.832800 -0.097043 -0.020749 2 2.487500 -3.422337 -0.106557 2 5.942600 -29.063526 0.247433 2 14.114300 -86.280193 0.258458 2 41.714400 -251.829294 1.160610 2 125.295100 -590.415770 6.230920 1 400.272500 -81.464127 7 .154981 259 Element 118 (92 core/26 valence) RECP- valence 6s,p,d/7s,p ______Exponent------CCMBEj------CiSflL S-g Potential 2 0.627900 -6.011416 2 0.760300 17.849394 2 1.103500 -33.594082 2 1.783200 65.429942 2 3.060900 -131.570034 2 6.023300 370.429204 2 12.886100 -324.954043 1 24.748900 201.721801 0 176.926200 13.128312 p-g Potential 2 1.240600 22.772751 -24.561063 2 1.472500 -73.744930 74.721688 2 1.988400 144.491558 -122.282756 2 2.941600 -258.592453 148.229325 2 4.774500 525.101455 -101.259969 2 8.140900 -524.478587 9.389697 2 15.700400 746.173651 71.131419 1 56.060700 193.233837 -10.176438 0 214.514800 12.739912 0.738252 d-g Potential 2 0.816900 16.294555 -4.837793 2 0.971900 -46.406032 13.176850 2 1.421300 104.539241 -25.555132 2 2.125600 -205.747489 38.619938 2 3.573200 425.202000 -37.848300 2 6.352300 -497.616756 21.510490 2 12.739300 660.944717 5.469798 1 51.271500 45.717474 -3.506810 0 36.309700 12.926904 0.176594 f-g Potential 2 0.978400 -24.515935 -1.867017 2 1.183600 87.944303 6.322383 2 1.612500 -175.530007 -12.079622 2 2.494000 336.897392 22.458803 2 4.119100 -433.326472 -33.364817 2 7.392600 497.843431 28.853337 2 15.549300 -170.563471 -18.981566 1 26.090500 125.270252 1.408628 0 115.401800 8.370827 -0.093403 g Potential 2 1.037400 -0.211666 -0.040341 2 3.109900 -6.001543 -0.082457 2 7.376300 -40.273155 0.370865 2 17.723600 -99.483130 0.025018 2 50.987300 -300 .288073 2.565311 2 158.767900 -698.911850 8.053164 1 514.739100 -86.382457 9.281211 260 261 APPENDIX C CARTESIAN GAUSSIAN BASIS SETS FOR Am-ELEMENT 118 CORRESPONDING TO THE SMALL CORE (SC) ECPS LISTED IN APPENDIX A. ALSO GIVEN ARE RADIAL PLOTS OF FILLED ATOMIC ORBITALS. 262 Am (®S) n ClGa) Cf7al C(6d) -2.2294504 -0.1880269 -0.3091971 sd 11.5158890 -0.0096236 0.0036780 -0.0034666 sd 1.9585446 -0.3135430 0.0969918 0.0122432 sd 1.3284298 0.8419227 -0.3092847 -0.0480467 sd 0.5282669 0.4885885 -0.0833127 0.5419619 sd 0.1531474 0.0363567 0.5425990 0.5883328 sd 0.0509499 -0.0056110 0.5942378 0.0633350 a Cf&pl C(7o> CfSfl -1.0558310 -0.1429350 -0.3765679 P 0.0433588 -0.0016612 0.7944826 P 0.0152267 0.0024485 0.2609128 pf 5.3774034 0.0175529 -0.0086293 0.1680031 Pf 2.4281957 -0.0075860 -0.0039696 0.3811474 pf 1.1237942 0.5127832 -0.1570190 0.3792905 pf 0.4971558 0.5009627 -0.1433581 0.2636056 pf 0.1964535 0.1332448 0.0441445 0.1169825 Am OJ V 04 R(aO) Figure C.l Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 263 Cm (?F) n cna) Cf6d) -2.3162397 -0.1908768 -0.3216563 sd 11.0265250 -0.0095322 0.0036975 -0.0038025 sd 2.3919811 -0.1833158 0.0545297 0.0060346 sd 1.3199383 0.7215014 -0.2685333 -0 .0408193 sd 0.5460793 0.4763731 -0.0742669 0.5559123 sd 0.1573956 0.0350389 0.5430889 0.5794595 sd 0.0522126 -0.0054981 0.5925290 0.0571118 n Cf6o) GfSfl -1.0895168 -0.1431574 -0.3779473 P 0.0426398 -0.0043735 0.7649497 P 0.0154094 0.0030976 0.2683084 pf 5.3398412 0.0098818 -0.0066589 0.1923966 pf 2.4037829 0.0181693 -0.0104238 0.3959676 pf 1.1019327 0.5439779 -0.1627232 0.3717785 pf 0.4826264 0.4735516 -0.1262516 0.2474920 pf 0.1879217 0.1124313 0.0720665 0.1029956 Cm V •1 0 1 2 3 4 5 R(aO) Figure C.2 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 264 Bk (6h ) n rf7al Cf6dV -2.3950358 -0.1931011 -0.4361929 sd 16.7548840 -0.0070607 0.0026848 -0.0023595 sd 2.2932401 -0.1437765 0.0344357 -0.0202350 sd 1.2577831 0.7700722 -0.2717306 0.0392519 sd 0.5201191 0.4017908 -0.0379623 0.6423980 sd 0.1554146 0.0254802 0.5556445 0.4706534 sd 0.0518245 -0.0038095 0.5727354 0.0043160 n Cfgpl Cf7pl CfSfî -1.1205361 -0.1457129 -0.3910423 P 0.0439084 -0.0036685 0.7496755 P 0.0156209 0.0029465 0.2995332 pf 5.7033444 0.0112916 -0.0066267 0.1866324 pf 2.5741569 0.0033256 -0.0049125 0.3987952 pf 1.1748984 0.5375529 -0.1486193 0.3743857 pf 0.5133695 0 .4844449 -0.1274688 0.2463364 pf 0.2000785 0.1193465 0.0571317 0.1034655 Bk V R(aO) Figure C.3 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 265 Cf (5l) n CfGsl Cf6d) -2.4824550 -0.1964053 -0.0985377 sd 20.7339090 -0.0036152 0.0013633 -0.0007206 sd 2.0937082 -0.4186404 0.1262541 -0.0030299 sd 1.5363060 0.9555671 -0.3336794 -0.0119006 sd 0.5954388 0.4827480 -0.0762605 0.1918297 sd 0.1692088 0.0349620 0.5384102 -0.4587176 sd 0.0554399 -0.0054718 0.5985638 0.5713996 n Cf6t>) Cf7pl C(Sf) -1.1229543 -0.1466235 -0.4272813 P 0.0482689 0.0011066 0.7926703 P 0.0163072 0.0018595 0.2896516 pf 6.0047068 0.0174384 -0.0087503 0.2041382 pf 2.7325189 -0.0118319 0.0000439 0.3844418 pf 1.3102299 0.4812847 -0.1355602 0.3605868 pf 0.5833715 0.5167104 -0.1423371 0.2484040 pf 0.2315253 0.1555935 0.0108988 0.1069444 Cf 13 V 03 R(aO) Figure C.4 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 266 E s (*J1 n Cf6s) cna) Cffidl -2.5618336 -0.1992953 -0.1421825 sd 14.9607040 -0.0094584 0.0034030 -0.0018058 sd 2.1642731 -0.3658535 0.1070205 0.0070943 sd 1.5640496 0.9205368 -0.3186549 -0.0257642 sd 0.6108950 0.4701972 -0.0694535 0.2623020 sd 0.1746284 0.0335079 0.5371425 0.5680298 sd 0.0569639 -0.0051325 0.5993230 0.4045873 n Cffiol cno) Cf5f) -1.1425210 -0.1475369 -0.4594158 P 0.0489932 0.0018114 0.7908259 P 0.0167286 0.0015721 0.2890171 pf 6.1699240 0.0122558 -0.0073972 0.2036136 pf 2.8361203 -0.0005612 -0.0027584 0.4014019 pf 1.3227024 0.4947964 -0.1363785 0.3642366 pf 0.5862411 0.5034383 -0.1346088 0.2336491 pf 0 .2348108 0.1483779 0.0129503 0.0965151 Es V R(aO) Figure C.5 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 267 Fm (3 l) n Cf6s1 Cf7s) C(6d) -2.6507983 -0.2019686 -0.1991112 sd 14.3727930 -0.0097520 0.0035206 -0 .0029291 sd 2.6279910 -0.1959512 0.0553928 0 .0094204 sd 1.5581072 0.7532510 -0.2668019 -0.0273166 sd 0.6341563 0.4653611 -0.0637449 0.3484136 sd 0.1793011 0.0333260 0.5370152 0.6310416 sd 0.0583133 -0.0052477 0.5986519 0.2474989 a Cf7o1 CfSfî -1.1708556 -0.1491063 -0 .4595589 P 0.0486447 0.0007528 0.7955248 P 0.0166478 0.0017394 0.2790171 pf 6.3362744 0.0257996 -0.0100900 0 .2141369 pf 2.8970511 -0.0030421 -0.0015009 0 .4060790 pf 1.3538754 0.5043659 -0.1336665 0.3576435 pf 0.5971338 0.4978951 -0.1232402 0 .2298934 pf 0.2371621 0.1433459 0.0164490 0 .0926955 Fm V R(aO) Figure C.6 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 268 Md (2 p ) n Cf 6aî Cf7a) Cf6d) -2.7217029 -0.2035649 -0.0945696 sd 22.0297190 -0.0068539 0.0024001 -0.0006096 sd 2.1642731 -0.3109555 0.0772378 0.0192519 sd 1.5673249 0.9310615 -0.3028350 -0.0512221 sd 0.6122855 0.4164343 -0.0430873 0.1772045 sd 0.1797246 0.0246946 0.5431619 0.4324920 sd 0.0584055 -0.0033160 0.5892551 0.6107423 n Cfgp) Cf7o) CfSfl -1.1935648 -0.1511375 -0.4919348 P 0 .0516183 0.0034629 0.7979947 P 0.0169365 0.0010369 0.2927572 pf 6.8582197 0.0238412 -0.0091076 0.1971442 pf 3.1413322 -0.0118515 -0.0002158 0.4130824 pf 1.4417434 0.4885636 -0.1245121 0.3668453 pf 0.6370828 0.5086035 -0.1264078 0.2272576 pf 0.2554965 0.1554167 -0.0001561 0.0934810 Md V Oj R(aO) Figure C .l Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 269 Mo (IS) n Cf6aî cn»\ CfSdl -2.8070750 -0.2073756 -0.1525306 sd 20.2777110 -0.0075236 0.0025980 -0.0012645 sd 2.1438433 -0-8133190 0.2326793 0.0573288 sd 1.8043526 1.3869298 -0.4408257 -0.0996052 sd 0.6688283 0.4555298 -0.0653627 0.3243834 sd 0.1909842 0.0310391 0.5360384 0.5564642 sd 0.0614098 -0.0046163 0.6021956 0.3819373 n Cffiol Cf7c1 CfSfl -1.2133093 -0.1422563 -0.5130335 P 0.0501206 0.0034139 0.8554108 P 0.0163136 0.0007449 0.1988928 pf 7.3730206 0.0148056 -0.0070180 0.1936398 pf 3.2225813 0.0170689 -0.0130742 0.4479412 pf 1.3980402 0.5170097 -0.1513817 0.3718620 pf 0.6139671 0.4818328 -0.1412416 0.2031199 pf 0.2492495 0.1345196 0.0356792 0.0826214 No V R(aO) Figure C.8 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set 270 L r ( 2 d ) n Cf6sl Cf7s) Cf6d) -3.0832067 -0.2426139 -0.1461748 sd 18.9171780 -0.0085509 0.0034270 -0.0011250 sd 2.5830972 -0.2986066 0.0824662 -0.0000830 sd 1.7557520 0.8993388 -0.3204553 -0.0281607 sd 0.6854340 0.4280307 -0.0664847 0.2805867 sd 0.2187084 0.0224782 0.5665836 0.5081690 sd 0.0697719 -0.0013445 0.5796018 0.4551770 n. Cffipl CO b ) CfSfl -1.3952127 -0.1437422 -0.7488966 P 0.0563985 0.0078379 0.8317910 P 0.0187504 -0.0004666 0.2546086 pf 7.5803955 0.0129924 -0.0066507 0.1967634 pf 3.4452695 -0.0084411 -0.0054326 0.4331510 p f 1.5715476 0.4897929 -0.1345607 0.3676817 pf 0.7121435 0.4987381 -0.1562686 0.2062702 p f 0.2980248 0.1550924 -0.0068684 0.0737730 Lr V ■OS -1 0 1 2 3 4 S R(aO) Figure C.9 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 271 Rf (3P) n Cf7a) Cf6d) -3.3490319 -0.2666231 -0.2101925 sd 17.2947070 -0.0094684 0.0042577 -0.0017744 sd 3.0860378 -0.1846579 0.0501892 0.0028623 sd 1.7603053 0 .8066281 -0.3129865 -0.0310379 sd 0.7093796 0.4047424 -0.0588430 0.3319306 sd 0.2453532 0.0175019 0.5650892 0.5355440 sd 0.0801662 0.0004944 0.5788189 0.3618838 n Cf6p1 cno) CfSf) -1.5493881 -0.1637377 -0.9870628 P 0.0594585 0.0076593 0.9088743 P 0.0173357 -0.0006208 0.1350060 pf 7.6187816 0.0084469 -0.0063643 0.2187136 pf 3.4633023 0.0098015 -0.0121487 0.4494106 pf 1.5783319 0.5173363 -0.1689422 0.3593007 pf 0.7232524 0.4720575 -0.1654860 0 .1804459 pf 0.3096631 0.1361044 0.0440299 0 .0538388 Rf -1 0 1 2 3 4 5 R(aO) Figure C.IO Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 272 Ha (< F ) n Cf 6sî Cf7s) Cf6d) -3.6177311 -0.2955359 -0.2181467 sd 27.4749390 -0.0065065 0.0029803 -0.0008820 sd 2.2734328 -0.9234427 0.2660744 -0.0416169 sd 1.9468378 1.5866999 -0.5599307 0.0216459 sd 0.7132286 0.3736248 -0.0431050 0.3655745 sd 0.2616137 0.0102982 0.5966263 0.5141667 sd 0.0863187 0.0019195 0.5424579 0.3354079 n Cf6t>ï Cf7o1 CfSfl -1.7148477 -0.1808088 -1.2505403 P 0.0636712 0.0082945 0.9391056 P 0.0160124 -0.0007394 0.0803765 pf 8 .1473376 0.0295028 -0.0129674 0.2100744 pf 3.7233188 -0.0193023 -0.0049723 0.4555010 pf 1.7109658 0.5149009 -0.1804400 0.'3630618 pf 0.7896551 0.4811628 -0.1788051 0.1746405 pf 0.3402368 0.1406695 0.0676564 0.0459146 Ha V Oj •1 0 1 2 3 4 S R(aO) Figure C .ll Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 273 sg (SD) n Cf7a) Cffidl -3.8833484 -0.3099646 -0.3001709 sd 25.4217990 -0.0072783 0.0035287 -0.0012225 sd 2.5987626 -0.5952257 0.1809421 -0.0088082 sd 2.0560785 1.2375087 -0.4797507 -0.0219554 sd 0.7775353 0.3873694 -0.0491834 0.3959649 sd 0.2905247 0.0129585 0.5791945 0.5345658 sd 0.0975429 0.0019599 0.5597535 0.2755108 n CfGo) Cf7p) CfSfl -1.8536465 -0.1980793 -1.4888329 P 0.0670770 0.0067602 0.9042778 P 0.0175343 -0.0005630 0.0848274 pf 8.1803385 -0.0073258 -0.0015702 0.2325852 pf 3.7089209 0.0325383 -0.0224179 0.4789429 pf 1.6650488 0.5496780 -0.2035960 0.3529889 pf 0.7708857 0 .4422707 -0.1597585 0.1410556 pf 0.3359438 0.1132807 0.1153716 0.0331126 sg V •1 0 1 2 3 4 5 R(aO) Figure C.12 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 274 NS (®S) n CfSs) Cf7s) CfSd) -4.1576334 -0.3286966 -0.3494503 sd 23.2879010 -0.0081070 0.0041663 -0.0017316 sd 2.9548861 -0.4214287 0.1347480 0.0038486 sd 2.1450303 1.0516080 -0.4422197 -0.0378840 sd 0.8329362 0.3942674 -0.0486414 0.4176870 sd 0.3153532 0.0135999 0.5776463 0.5362151 sd 0.1067331 0.0022291 0.5606683 0.2475756 n Cf6o1 Cf7o) CfSf) -2.0021949 -0.1926330 -1.7262139 P 0.0734333 0.0079585 0.8918283 P 0.0206490 -0 .0008385 0.1219603 pf 9.0385688 0.0154737 -0.0087012 0.2089035 pf 4.0558872 0.0042116 -0 .0137344 0.4897858 pf 1.8173171 0.5343428 -0.1985941 0.3609208 pf 0.8515955 0.4588743 -0.1634918 0.1399007 pf 0.3726824 0.1246171 0.0917953 0.0331984 Ns 2JS V B( b O) Figure C.13 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 275 HS (5D) n Cf7aî CfGdl -4.4668819 -0.3542071 -0 .3618368 sd 21.9203490 -0.0088519 0.0048855 -0.0016268 sd 3.4526239 -0.2775820 0.0891436 -0.0093037 sd 2.1942232 0.9073770 -0.4098645 -0.0291670 sd 0.882277B 0.3899946 -0.0393898 0.4335844 sd 0.3368128 0.0130283 0.5862175 0.5254827 sd 0.1147642 0.0024868 0.5491470 0.2413366 n CfGpl Cf7p) C(5f) -2.1722599 -0.1929305 -2.0120317 P 0.0769110 0.0059981 0.8591336 P 0.0226494 -0.0003974 0.1471806 pf 9.0496185 0.0107794 -0.0074263 0 .2283671 pf 4.1141947 0.0069355 -0.0144968 0.4918683 pf 1.8708722 0.5560975 -0 .2084999 0.3492774 pf 0.8714431 0.4473290 -0.1492658 0.1289483 pf 0.3779315 0.1114880 0.1072196 0.0251598 Hs 6s V R(aO) Figure C.14 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 276 Mt (*F) n C(6a^ rr7a> CfGdl -4.7786517 -0.3793439 -0.3860927 sd 16.9907230 -0.0123253 0.0074228 -0.0018001 sd 4.2645591 -0.1776778 0.0592194 -0.0103127 sd 2.2388334 0.7989371 -0.3964769 -0.0399022 sd 0.9508741 0.3902671 -0.0297307 0.4471858 sd 0.3635673 0.0177238 0.5890365 0.5232073 sd 0.1242963 0.0016935 0.5440694 0.2348771 n CfSol cnx>) Cf5£V -2.3346497 -0.1718330 -2.3026483 P 0.0783220 0.0038601 0.7921254 P 0.0249269 0.0000341 0.2217348 pf 9.1139125 0.0063689 -0.0058669 0.2470154 pf 4.1561459 0.0222246 -0.0190875 0.5020261 pf 1.8778761 0 .5868716 -0.2098840 0.3385913 pf 0.8651224 0.4228727 -0.1179213 0.1097074 pf 0.3714973 0.0926932 0.1062869 0.0183220 Mt 2J ès V R(aO) Figure C.15 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 277 Element 110 (3P) n cnm'i Cfgdl -5.1153931 -0.3961001 -0.4306061 sd 18.5614790 -0.0116504 0.0073688 -0.0050254 sd 4.2815409 -0.1376744 0.0366179 0.0230777 sd 2.2388334 0.7767568 -0.4041836 -0.1315039 sd 1.0755149 0.3533851 0.0015523 0.5239005 sd 0.3881528 0.0471294 0.5793733 0.5263976 sd 0.1326909 -0.0062906 0.5456437 0.2071495 n Cf6o1 Cf7t»1 CfSfl -2.5183220 -0.1705091 -2.6508087 P 0.1277398 0.0368147 0.6995712 P 0.0386306 -0 .0056803 0.4414086 pf 18.2388140 -0.0076594 -0.0013407 0.0724698 pf 6.8469230 0.0254525 -0.0179609 0.3588522 pf 3.7247118 -0.0098324 0.0035300 0.3765663 pf 1.8829343 0.6927158 -0.2782637 0.3183215 pf 0.7479886 0 .3866956 -0.0834892 0.0942456 Element 110 V Oj R(aO) Figure C.16 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 278 Element 111 (=0) -5.4062595 -0.4292062 -0.4409458 sd 28.9821980 -0.0076983 0.0046436 -0.0014923 sd 3.1520539 -1.5250903 0.5940494 -0.0096309 sd 2.7941365 2.1533160 -0.9485364 -0.0370095 sd 1.0430570 0.3872604 -0.0287443 0.4708011 sd 0.4083108 0.0156980 0.6100351 0.5081483 sd 0.1409560 0.0020620 0.5194302 0.2166593 n Cfgp) Cf7pl CfSfl -2.6795254 -0.1915391 -2.8860903 P 0.0902311 0.0039960 0.7751754 P 0.0290759 0.0000182 0.2447750 pf 9.9164969 -0.0020078 -0.0033004 0.2443478 p f 4.5599515 0.0176207 -0.0183517 0.5117339 p f 2.0820072 0.5856956 -0.2115921 0.3322471 p f 0.9727928 0.4250254 -0.1175264 0.1003968 p f 0.4197411 0 .0939279 0.1033941 0.0160408 Element ill V R(aO) Figure C.17 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 279 Element 112 (iS) n Cf6a1 Cf7a1 Cf6d) -5.7474777 -0.4530455 -0.4759295 sd 21.7119070 -0.0117747 0.0075186 -0.0018715 sd 4.0519063 -0.3501435 0.1300922 -0.0118821 sd 2.7224689 0.9633368 -0.4931527 -0.0473827 sd 1.1230624 0.3958727 -0.0259788 0.4768381 sd 0.4401965 0.0183443 0.6051899 0.5116797 sd 0.1523479 0.0019877 0.5249203 0.2124878 n Cf6o1 Cf7o1 CfSfl -2.8539318 -0.1980740 -3.1885189 P 0.0903696 0.0012573 0.7328491 P 0.0303724 0.0005968 0.2548009 pf 9.9055975 0.0083678 -0.0061138 0.2688755 p f 4.5408196 0.0394776 -0.0284981 0.5218401 pf 2.0485838 0.6269515 -0.2216032 0.3159730 pf 0.9323977 0.3864127 -0.0827334 0.0816303 pf 0.3925141 0.0697300 0.1350055 0.0106859 Element 112 Gd V Oj ■OS .1 0 1 2 3 4 5 R(aO) Figure C.18 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 280 Element 113 (2P) Cffis) C (6d) -6.2765010 -0.6008987 -0.6673128 sd 18.0441130 -0.0149534 0.0104708 -0.0039344 sd 4.1716298 -0.4412975 0.1820409 0.0222837 sd 2.9605510 1.0291651 -0.5574745 -0.0905477 sd 1.2231770 0.4162210 -0.0502859 0.4886564 sd 0.4914624 0.0206770 0.6248411 0.5197603 sd 0.1779609 0.0019305 0.5092242 0.1874097 a Cf6p) C(7t>) CfSfl -3.1945381 -0.1710262 -3.6699982 P 2.2429884 -0.0493773 0.0011555 P 0.0873824 0.0024441 -0.6956211 P 0.0237909 -0.0003154 -0.3416566 pf 10.9730000 0.0260541 0.0057268 0.0002356 pf 4.9855571 0.0340186 0.0175298 0.2353514 pf 2.2437546 0.6376294 0.1938103 0.5345413 pf 1.0307168 0.4100723 0.0951994 0.3266730 pf 0.4416325 0.0738487 -0.1269522 0.0944255 Element 113 --5I V •1 0 1 2 3 4 S R(aO) Figure C.19 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 281 Element 114 (3P) n EzB. CffiaL C(7 aL ■CLfidL -6.7786351 -0.7330663 -0.8395136 sd 39.9757620 -0.0063174 0.0045824 -0.0013781 sd 4.1097837 -0.2705511 0.0708712 -0.0092541 sd 2.7224689 0.9564804 -0.5267574 -0.0545975 sd 1.2100945 0.3172862 0.0521564 0.5513093 sd 0.4868278 0.0282116 0.6594168 0.4882353 sd 0.1817576 -0.0029294 0.4294648 0.1367065 n Cfgp) Cf7pl CfSf) -3.5272412 -0.2347786 -4.1572595 P 1.7935886 -0.0589893 -0.0131234 P 0.1433561 0.0033620 0.5712058 P 0.0512023 0.0001646 0.4781007 pf 11.0616840 0.0205419 -0.0047315 0.2496453 pf 5.1017154 0.0288886 -0.0200513 0.5266160 pf 2.3508303 0.6491614 -0.2318639 0.3197556 pf 1.1103357 0.4088821 -0.1173115 0.0831312 pf 0.4948663 0.0754787 0.0833920 0.0106347 Element 114 V R(aO) Figure C.20 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 282 Element 115 ( n. cna\ C(6d1 -7.3282088 -0.8498889 -0.9936953 sd 36.4438300 -0.0072536 0.0055731 -0.0014468 sd 4.3686812 -0.3159259 0.1032190 -0.0164593 sd 2.9362844 0.9846712 -0.5688409 -0.0547179 sd 1.2906059 0.3332858 0.0451151 0.5579176 sd 0.5328963 0.0232578 0.6678157 0.4853572 sd 0.2012507 -0.0016368 0.4239077 0.1273314 n Cffitsl Cf7o1 CfSfl -3.8479056 -0.2880350 -4.6050717 P 4.6364816 -0.0054445 -0.0127743 P 0.1597537 0.0332072 0.8768418 P 0.0468109 -0.0080145 0.2406001 pf 17.3247370 -0.0055768 0.0028949 0.0906138 pf 7.4743802 0.0386083 -0.0158459 0.4127940 pf 4.2628352 -0.0305519 -0.0046763 0.3152939 pf 2.4147049 0.6843563 -0.3135246 0.2888529 pf 1.0384794 0.3960650 -0.0993365 0.0826323 Element 115 Z5 V Figure C.21 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 283 Element 116 (3p) n Cffisl Cf7st CfSdl -7.8850375 -0.9887110 -1.1680915 sd 13.1511810 -0.0199376 0.0211636 -0.3402583 sd 5.2514419 -0.2048551 0.0561259 -0.0850615 sd 3.1100370 0.8109171 -0.5182894 -0.0207079 sd 1.5109247 0.3852593 -0.0188690 0.0819218 sd 0.6164518 0.0391145 0.6663561 -0.1957913 sd 0.2333849 -0.0035421 0.4597625 0.5714190 n Cfgol Cf7p1 C(5f1 -4.1852530 -0.3037875 -5.1452805 P 7.6428803 0.0039528 0.3771878 P 0.1813751 0.0285263 -0.0111824 Pf 0.0523035 -0.0058491 0.8826613 0.5220167 Pf 30.5204540 -0.0096676 0.2378591 0.1358131 pf 8.7320871 0.0534175 0.0044069 0.0827166 pf 4.7704078 -0.0319697 -0.0220658 0.3650974 pf 2.4780453 0.6929553 -0.0217457 0.4017765 Element 116 6 s ZS V R(aO) Figure C.22 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 284 Element 117 (2P) n rf7al Cf6dl -8.5086355 -1.1421470 -1.3491963 sd 22.4731400 -0.0101151 0.0105213 -0 .0046980 sd 7.1088634 -0.1153737 0.0405580 0.0091572 sd 2.9362844 0.8073178 -0.5946812 -0.1207304 sd 1.4785317 0.2859659 0.1434622 0.6207979 sd 0.6049914 0.0385157 0.6630447 0.4774583 sd 0.2390290 -0.0066453 0.3973327 0.1025891 a Cfgpl Cf7t>1 CfSfl -4.5388444 -0.3390333 -5.6343767 P 5.6847500 0.0000108 -0.0153859 P 0.2191043 0.0401292 0.8547123 P 0.0649429 -0.0064428 0.2939070 pf 23.4594860 -0.0124442 0.0050168 0.0732167 pf 9.2513080 0.0909913 -0.0364324 0.2863169 pf 5.5849091 -0.1109226 0.0203243 0.4068732 pf 2.8239680 0.6679633 -0.3616582 0.3362860 pf 1.2453231 0.4355963 -0.1397700 0.0862475 Elem ent 117 2J V 1 0 1 2 3 4 5 R(aO) Figure C.23 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 285 Element 118 (iS) n Cf7s1 Cf6d) -9.0884997 -1.3182007 -1.5483069 sd 37.0793660 -0.0087095 0.0079529 -0.0023385 sd 5.9976251 -0.1068773 0.0084643 -0.0079938 sd 2.9362844 0.8639953 -0.6204248 -0.0999716 sd 1.4975514 0.2268528 0.2093401 0.6473299 sd 0.6199286 0.0433008 0.6751575 0.4520697 sd 0.2486012 -0.0092151 0.3519038 0.0855188 n Cf6p1 cnp^ C(S£) -4.9137784 -0.3971641 -6.2234527 P 7.5067182 0.0025730 -0 .0160400 P 0.3084952 0.0532085 0.6416476 P 0.1046683 -0.0064566 0.5427009 pf 33.7170100 -0.0085131 0.0036635 0.0595834 pf 10.8165580 0.0390918 -0.0140288 0.2748253 pf 5.6934640 -0.0316913 -0.0292684 0.4841995 pf 2.8023531 0.6581538 -0.3750578 0.3024417 pf 1.2949519 0.3898609 -0.1598939 0.0753072 Element 118 Z 5 V R(aO) Figure C.24 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 286 APPENDIX D CARTESIAN GAUSSIAN BASIS SETS FOR RF-ELEMENT 118 CORRESPONDING TO THE LARGE CORE (LC) ECPS LISTED IN APPENDIX B. ALSO GIVEN ARE RADIAL PLOTS OF FILLED ATOMIC ORBITALS. 287 Rf (3P) n Cfgat Cf7a1 GfGdl -3.3263886 -0.2776094 -0.2003171 sd 17.1892480 0.0096481 -0.0044613 0.0019079 sd 2.4670676 0.9877880 -0.3490293 -0.0515029 sd 2.0884922 -1.5398775 0.6010914 0.0874475 sd 0.7664974 -0.4656545 0.0996382 -0.2951129 sd 0.2643783 -0.027 6267 -0.5688005 -0.5356461 sd 0.0848622 0.0004839 -0.5882937 -0.4039300 n Cf6p> Cf7t>) -1.5321399 -0.1581284 P 2.1546351 0.3849158 0.1167226 P 1.1974361 -0.6005009 -0.1776238 P 0.5257430 -0.5507484 -0.3541921 P 0.2784190 -0.1607841 0.1673399 P 0.0535180 -0.0157432 0.9778547 Rf •OJ •0.6 V 04 R(aO) Figure D .l Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 2 8 8 Ha (*F) a C(Cn\ Cf7al Cf6d) -3.5763372 -0.3004716 -0.2493722 sd 27.7442420 0.0062388 -0 .0030350 0.0008338 sd 2.4401074 0.4282833 -0.1123751 0.0243849 sd 1.8570274 -1.0940188 0.4227428 -0.0013570 sd 0.7145120 -0.3686588 0 .0336055 -0.3627134 sd 0.2654334 -0.0131839 -0.5981277 -0.5255698 sd 0.0888054 -0.0014006 -0.5354365 -0.3204753 n Cffipî Cf7p) -1.6797476 -0.1756352 P 2.0763235 0.3974513 0.1397244 P 1.1049945 -0.7955771 -0.3143258 P 0.4215962 -0.5274197 -0.2523849 P 0.1420926 -0.0510010 0.4285273 P 0.0448986 0.0002771 0 .7419423 Ha - ■ - 6d V R(aO) Figure D.2 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 289 S g (5 d ) n Cffia) rn») Cf6dl -3.8521780 -0.3194117 -0.2915552 sd 25.3638650 0.0072531 -0.0036940 0.0011939 sd 2.4552504 1.2280842 -0.4182264 -0.0020300 sd 2.1501839 -1.8581202 0.7271898 0.0358644 sd 0.7875148 -0.3975423 0.0510321 -0.3795960 sd 0.2964259 -0.0149771 -0.5882156 -0.5359924 sd 0.0994537 -0.0018645 -0.5512169 -0.2930947 n Cfgpl Cf7o) -1.8339557 -0.1911900 P 2.6999857 0.2801284 0.0988299 P 1.0950253 -0.6423904 -0.2446019 P 0.5040857 -0.4629218 -0.3494522 P 0.2620271 -0.1272591 0.2988096 P 0.0605913 -0.0096935 0.9187690 Sg V R(«0) Figure D.3 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 290 Ms (69) n Cf6a) r.na^ C(6d) -4.1337016 -0.3383607 -0.3412950 sd 23.4502300 0.0079556 -0.0043077 0.0014018 sd 2.9260999 0.4826934 -0.1647716 0.0000724 sd 2.1824614 -1.1001554 0.4825160 0.0425585 sd 0.8461273 -0.4039382 0.0489690 -0.4025190 sd 0.3220941 -0.0161185 -0.5836418 -0.5399914 sd 0.1090658 -0.0020559 -0.5544869 -0.2633308 n Cf6ot CH b I -1.9828864 -0.1851930 p 2.3630949 0.4310609 0.1521188 P 1.3200435 -0.7306819 -0.2783311 P 0.5568690 -0.5465631 -0.3333627 p 0.2504231 -0.1143814 0.2931068 P 0 .0606258 -0.0058075 0.8980086 Ns V ■0^ 09 2 R(80) Figure D.4 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 291 HS (50) n R«p C f 7 s 1 C ( 6 d l -4.4382837 -0.3651814 -0.3569391 sd 4.8877446 0.1215378 -0.0443353 0.0084776 sd 2.0752004 -0.7126109 0.3749589 0.0522393 sd 0.9641488 -0.3945236 0.0493450 -0.3675674 sd 0.4159815 -0.0400987 -0.4026420 -0.4622793 sd 0.1794204 -0.0009860 -0.5393341 -0.3010416 sd 0.0729336 0.0000719 -0.2270933 -0.0929895 n Cffioî Cf7t>i -2.1565861 -0.1872555 P 2.9629645 0.2598683 0.0937120 P 1.1018751 -0.7423971 -0.3238372 P 0.4751743 -0.4292591 -0.1939540 P 0.1814785 -0.0478921 0.4204824 P 0.0531101 -0.0005235 0.7592657 Hs -13 ¥ 03 R(aO) Figure D.5 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 292 Mt (*F) Ezp CLIfiaJ____ C(^a^ ■ C (6 d l -4.7248881 -0.3909025 -0.3821215 sd 3.6153759 0.3034625 -0.1091838 -0.0052237 sd 2.3768882 -0.8735436 0.4328342 0.0638334 sd 1.0251458 -0.4241671 0.0840072 -0.3735115 sd 0.4494950 -0.0347849 -0.4230876 -0.4632696 sd 0.1923798 -0.0042266 -0.5364695 -0.2973840 sd 0.0781970 0.0009908 -0.2212765 -0.0865266 Exp CÏGp) -2.3240103 -0.1675872 P 2.8069182 0.3456398 0.1181503 P 1.2996156 -0.7275511 -0.2857981 P 0.5658901 -0.4989085 -0.2508210 P 0.2295408 -0.0741067 0.3404931 P 0.0573721 -0.0015807 0.8455882 Mt V R(«0) Figure D.6 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 293 Element 110 (3F) n Cfg*1 Cf7al CfSd) -5.0514135 -0.4152627 -0.4063354 sd 6.9517373 0.0474840 -0.0120427 0.0052081 sd 1.9731912 -0.7914402 0.4523921 0.0337925 sd 0.9096853 -0.2630906 -0.1292185 -0.5060014 sd 0.3627890 -0.0287092 -0.5146771 -0.4442345 sd 0.1578573 0.0056117 -0.4408872 -0.1997790 sd 0.0691259 -0.0019974 -0.1243834 -0.0493378 n CfGol C(7o1 -2.4936650 -0.1753513 P 3.0620756 0.2715184 0.0924047 P 1.1870026 -0.8028607 -0.3394025 P 0.4956792 -0.3981053 -0.1230301 P 0.1687397 -0.0320002 0.4519872 P 0.0495026 0.0005655 0.7065991 Element 110 —*-6p R(80) Figure D.7 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 294 Element 111 (=0) n Cffial C(6d1 -5.3907170 -0.4402323 -0.4386647 sd 3.9517373 0.3664467 -0.1463136 -0.0175290 sd 2.6558543 -0.9431405 0.4933454 0.0820092 sd 1.1170412 -0.4186176 0.0629157 -0.4170500 sd 0.4880082 -0.0255347 -0.4613041 -0.4608012 sd 0.2099120 -0.0059659 -0.5119409 -0.2700874 sd 0.0867262 0.0015550 -0.1985607 -0.0705152 n Cf7ol -2.6690778 -0.1888060 P 3.2064431 0.2857565 0.0971265 P 1.2656554 -0.8016241 -0.3375003 P 0.5340928 -0.4059000 -0.1328456 P 0.1887285 -0.0350759 0.4269841 P 0.0541501 0.0002885 0.7351326 Element 111 •*6p — 6d V R(aO) Figure D.8 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 295 Element 112 (iS) n Cl6a\ r n a i Cf6d) -5.7329677 -0.4678673 -0.4737629 sd 8.1025671 0.0741373 -0.0301965 0.0090932 sd 2.2633356 -0.7614641 0.4693978 0.0579351 sd 1.0862309 -0.2982425 -0.1023401 -0.4988302 sd 0.4452846 -0.0347271 -0.4824433 -0.4495048 sd 0.1982186 0.0045260 -0.4662832 -0.2082785 sd 0 .0858068 -0.0015663 -0.1503145 -0.0556196 n Cf6o1 Cf7tj) -2.8445375 -0.1523161 P 3.3937630 0.2918043 0.0790929 P 1.3291110 -0 .8040308 -0.2630544 P 0.5644730 -0.4042241 -0.1316953 P 0.2033490 -0.0356860 0.3365871 P 0.0488708 0.0000958 0.8216107 Element 112 7 s V R(aO) Figure D.9 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 296 Element 113 (2p) n Cffial c a s ) CfSd) -6.2591025 -0.6048418 -0.6597650 sd 7.7356052 0.0933185 -0.0417133 0.0036141 sd 2.4642980 -0.7296300 0.4843289 0.0827091 sd 1.2391599 -0.3321236 -0.0566764 -0.4817275 sd 0.5375619 -0.0463988 -0.4404229 -0.4557102 sd 0.2512345 0.0045789 -0.5048285 -0 .2217909 sd 0.1066765 -0.0014021 -0.1750251 -0.0540187 n Cffipl c a o ) -3.1804400 -0.1737494 P 14.8116840 0.0098839 0.0041199 P 2.3115885 2.0813521 0.6992586 P 1.9776265 -2.4727975 -0.8762331 P 0.6627581 -0.5403209 -0.1428489 P 0.1649483 -0.0313092 0.4780372 P 0.0470971 0.0039046 0.6682258 Element 113 V Oj •1 0 1 2 3 4 5 R(aO) Figure D.IO Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 297 Element 114 (3P) n GfGs) cna) CfSd) -6.7685943 -0.7382074 -0.8265182 sd 3.8609610 0.6431991 -0.2489287 -0 .0026460 sd 2.9975893 -1.2925136 0.6812417 0.0574884 sd 1.1991657 -0.3608350 -0.0092066 -0.5199981 sd 0.5281082 -0.0089095 -0.5835410 -0.4505987 sd 0.2329544 -0.0056690 -0.4312795 -0.1763078 sd 0.1005854 0.0017255 -0.1057559 -0.0220048 a CfSpI Cf7o) -3.5194471 -0.2364267 P 11.3938790 0.0099186 0.0055529 P 2.7286805 0.9191454 0.3533907 P 1.9543278 -1.2879534 -0.5581423 P 0.7235397 -0.5512618 -0.1715783 P 0.1907241 -0.0334332 0.5495958 P 0.0602871 0.0042251 0.6016527 Element 114 1 V Oj •1 0 1 2 3 4 5 R(aO) Figure D .ll Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 298 Element 115 ( n fiZB------____ c t/a i------7.3055672 -0.8726522 -1.0018347 sd 15.8685160 0.0306140 -0.0145493 0.0068175 sd 2.3716175 -0.8185072 0.6203447 0.0819324 sd 1.2677254 -0.1861190 -0.2660248 -0.5934673 sd 0.5281082 -0.0584757 -0.5404211 -0.4535007 sd 0.2329544 0.0180275 -0.4331953 -0.1279408 sd 0.1005854 -0.0055942 -0.0548952 -0.0154700 n Cf6tJl Cf7t>1 -3.8387906 -0.2987631 P 11.3938790 0.0180206 0.0110423 P 2.7286805 0.8477806 0.3491423 P 1.9543278 -1.2746015 -0.6145741 P 0.7235397 -0.5233516 -0.1499160 P 0.1907241 -0.0144256 0.7020109 P 0.0602871 -0.0004676 0.4505160 Element 115 V •03 R(aO) Figure D.12 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 299 Element 116 (3P) n Ciea) Cf7sl Cf6d) -7.9089968 -1.0180667 -1.1881140 sd 8.9519690 0.0818140 -0.0372191 0.0103677 sd 2.7099956 -0.7916089 0.5927275 0.0661399 sd 1.3025451 -0.2810283 -0.2016561 -0.6083778 sd 0.5516349 -0.0173445 -0.5777326 -0.4315829 sd 0.2650537 0.0006976 -0.3643429 -0.1051792 sd 0.1242436 0 .0000724 -0.0874743 -0.0216865 n Cf 6pl Cf7o1 -4.1904306 -0.3175373 P 4.0597392 0.2870462 0.1270073 P 1.5438164 -0.8489233 -0.5063916 P 0.7018190 -0.3558393 -0.0254519 P 0.2347880 -0 .0245579 0.5431958 P 0.0978325 0.0042505 0.4752023 P 0.0422867 -0.0012982 0.1221240 Element 116 V R(aO) Figure D.13 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 300 Element 117 (2p) n EzB------CfGsl C(7a) C(6d) -8.5167889 -1.1702278 -1.3725761 sd 8.7681738 0.1036527 -0.0512182 0.0124895 sd 2.9214280 -0.7862632 0.6068075 0.0764945 sd 1.3989310 -0.3056471 -0.1799697 -0.6105323 sd 0.6058613 -0.0106083 -0.5885010 -0.4331283 sd 0.2917136 -0 .0031148 -0.3672873 -0.1057427 sd 0.1351845 0 .0013284 -0.0841367 -0.0194713 n Cf7o1 -4.5430396 -0.3528591 P 4.0248567 0.3465081 0.1687702 P 1.7593670 -0 .8015467 -0.5241224 P 0.8429019 -0.4423392 -0.0910831 P 0.2814498 -0.0422151 0.5134449 P 0.1224921 0 .0091933 0.4879869 P 0.0532600 -0.0027082 0.1616669 Element 117 Zb V R(aO) Figure D.14 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 301 Element 118 (iS) n CfSal Cf7a) Cf6dl -9.0875779 -1.3236684 -1.5435156 sd 7.1866114 0.1120459 -0.0384426 -0.0016299 sd 3,0370662 -0.8269604 0.6296703 0.0839500 sd 1.4183281 -0.2879650 -0.2238956 -0.6578379 sd 0.5967078 -0.0063335 -0.6617554 -0.4204588 sd 0.2597307 -0.0019357 -0.3122770 -0.0792021 sd 0.1068344 0.0008407 -0.0310738 -0.0065432 n Cf6p1 Cf7p) -4.9112411 -0.3993040 P 4.2348018 0.3603265 0.1908903 P 1.9149393 -0.7381018 -0.5401508 P 0.9745660 -0.4998483 -0.1254787 P 0.3250890 -0.0618169 0.4825714 P 0.1514189 0.0160100 0.4882241 P 0.0663907 -0 .0044071 0.2004374 Element 118 è s -6d ¥ •OJ R(80) Figure D.15 Radial plot of the atomic pseudoorbitals corresponding to the valence orbitals in the above basis set. 302 APPENDIX E COMPARISONS AMONG ALL-ELECTRON DIRAC-FOCK (ACRV)., SMALL CORE (SC) REP, AND LARGE CORE (REP) ATOMIC SCF CALCULATIONS. ALL QUANTITIES ARE NEGATIVE AND IN UNITS OF HARTREES. THE ELECTRON CONFIGURATIONS OF THE EXAMINED ATOMS OR IONS WERE CHOSEN TO RESULT IN SINGLE, UNAMBIGUOUS ELECTRONIC STATES. 303 Americium (Z=95) [Rn] 7s2 SP® 5fl ACRV SC St 0.35339 0.34489 5P 0.23933 0.23292 6b 2.24464 2.24000 6p* L41753 L41240 0.93255 0.92968 6d* 6d 7s 0.19114 0.19043 7p^ Tp Curium + (Z=96) 7s2 5f*6 g fl ACRV SC 5P 0.84505 0.84041 St 0.72167 0.71803 6s 2.75385 2.75476 6p* _ 1.87369 1.87221 Gp 1.33391 1.33507 6d* 6d 7s 0.44453 0.44547 7lP Tp BerkeHum+2(Z=97) 7s2 5f*6 5fl ACRV SC 5P 1.42843 1.42770 St 129288 1.29120 6s 3.34541 3.34451 ep* 2.40836 2.40487 Gp 180511 1.81225 6d* 6d 7s 0.74813 .74842 7p* 7p 304 Californium +1 (Z=98) 7s2 sf'l 5^ ACRV SC 5P 0.88010 0.90403 5£ 0.88127 0.91487 6s 2.94346 2.94297 ep* 1.91840 1.88871 ep 1.41568 1.40512 6d* 6d 7s 0.45775 0.45942 7 l^ 7p Einsteinium +2 (Z=99) 7s2 5t*2 5f8 ACRVSC 5f^ 1.47776 1.51358 5£ 1.47354 1.52323 Gs 3.55449 3.56966 ep* 2.47145 2.44562 ep 1.89809 1.88707 ed* 6d 7s 0.77023 0.77533 7p* Tp Fermium+1 (Z=100) 5f*6 5f7 ACRV SC 5P 0.51813 0.51512 5f 0.42445 0.43117 6s 2.66888 2.66821 6p* 1.68012 1.66882 ep 1.09489 1.07750 6d* 6d 7s 7p* Tp 305 Mendeleevium (Z=101) 7s^ 5f^ ACRV SC 5P 0.53937 0.54691 a t 0.43888 0.45097 68 2.72509 2.72176 ep* L68198 1.67359 4 ) 1.03470 1.02194 6d* 6d 7s 0.20729 0.20603 7P* TP Nobelium (Z=102) 7s25 f*6 5^ ACRV SC 5 f 0.56461 0.56612 5f 0.46710 0.47724 Gs 2.80908 2.80847 6p* 1.72804 1.71323 %) 1.04765 1.06437 6d* 6d 7s .21006 0.21000 7p* TP Lawrencium (Z=103) 7s2 5f*® 5f8 7p*l ACRV SC 5f^ 0.81135 0.88924 5f 0.79135 0.78683 6& 3.17189 3.16951 Gp* 2.03449 2.03174 ep 1.27227 1.27532 6d* 6d 7s 7p* 0.16601 0.16644 TP 306 Rutherfordium. + (Z=104) 7s2 5f*6 5^ 6d*^ ACRV SC LC 5P L37214 1.36782 5f L25049 1.24829 6s 3.66426 3.66525 3.64454 2.48439 2.48451 2.45589 Gp L62528 1.62896 L61243 6d* 0.50769 0.50892 0.49916 6d 7s 0.52617 0.52612 0.53963 7p^ Hahnium (Z=105) 7s2 5f*6 5fS 6d*3 ACRVSC LC 5P 1.29534 1.30069 5f 1.15500 1.15218 Gs 3.59984 3.58344 3.55957 6p* 2.39829 2.39205 2.33250 Gp 1.42331 1.42306 1.40534 6d* 0.24931 0.24764 0.23868 Gd 7s 0.28933 0.28740 0.29874 7p* Tp Seaborgium (Z=106) 7s2 5f*6 5f8 ed*4 ACRV SC LC 5P 1.54831 1.56069 5f 1.39031 1.38767 66 3.86652 3.85774 3.82775 _6p* 2.62958 2.63646 2.58661 Gp 1.53610 1.53321 1.51556 6d* 0.29032 0.28771 0.27836 Gd 7s 0.30883 0.30694 0.31645 7p* 7p 307 Nielsbohiium (Z=107) 7s2 6d*'^ 6dl ACRVSC LC 5 P L82514 1.81270 Sf L65409 1.64340 Gs 4.15775 4.14791 4.12456 Gp* 2.86356 2.85520 2.81016 Gp L67712 1.67363 L65724 6d* 0.33655 0.33258 0.32468 Gd 0.25804 0.25499 0.24821 7s 0.34407 0.33255 0.34203 7p* Tp Hassium+ (Z=108) 7s2 6d*4 6dl ACRV SC LC 5P 2.47621 2.46872 5f 2.28865 2.28276 es 4.82311 4.82212 4.79283 Gp* 3.47434 3.47056 3.41797 Gp 2.16371 2.16701 2.14221 6d* 0.73007 0.73199 0.72335 Gd 0.63615 0.63722 0.62923 7s 0.65679 0.65740 0.66764 7p* Tp M eitnerium+2 (Z=109) 7s2 6d*4 6dl ACRV SCLC 5P 3.21566 3.20680 Sf 3.01100 3.00375 es 5.57510 5.57443 5.51768 Gp* 4.16772 4.14489 4.10815 Gp 2.72208 2.72052 2.70447 Gd* 1.19046 1.19164 1.18201 Gd 1.07807 1.07784 1.06991 7s 1.03421 1.03642 1.04840 Tp^ T p.. 308 Element 110 +3 7s2 6d*4 6dl ACRV SCLC 5P 4.02759 4.113704 5£ 3.80517 3.858894 G b 6.39829 6.430118 6.35476 4.92904 4.95627 4.85571 % 3.33730 3.36535 3.30846 6d* L70566 L74314 1.69328 Gd L57318 L60790 156281 7s 1.45809 1.46487 1.47626 7p* 5» Element 111 7s2 6d*4 6d5 ACRV SC LC 5P 3.01004 3.02250 5f 2.78320 2.76350 6k 5.40834 5.39627 5.38906 3.86082 3.86396 3.78745 %> 2.25867 2.25659 2.24262 Gd* 0.51413 0.51191 0.50719 6d 0.40311 0.39949 0.40142 7s 0.43374 0.43314 0.44453 Tp Element 112 7s2 6d*^ 6d® ACRV SC LC SP 3.32644 3.34626 Sf 3.08430 3.07206 6k 5.74770 5.74540 5.73843 Gp* 4.13161 4.13752 4.04748 » 2.40977 2.40327 2.39675 Gd* 0.55890 0.55742 0.55551 6d 0.43886 0.43611 0.43584 7s 0.46041 0.45983 0.47471 7p^ TP 309 Element 113 7s2 6d*4 6d6 7p*l ACRV SCLC 5P 3.77358 3.75863 Sf 3.51246 3.50030 6s 6.22045 6.21488 6.20139 4.53928 4.50277 4.47865 Gp 6.65453 2.65181 2.64976 6d* 0.71039 0.70930 0.70641 Gd 0.56444 0.56208 0.56107 7s 0.56660 0.57074 0.57782 7p* 0.27399 0.27101 0.27014 Tp. ... Element 114 7s2 6d*4 ed® 7p*2 ACRV SC LC 5P 4.23425 4.22133 5f 3.95298 3.93257 6b 6.71069 6.67176 6.66153 ep* 4.96346 4.92972 4.91486 Gp 2.90028 2.89834 2.90238 6d* 0.86522 0.86079 0.86041 Gd 0.69014 0.68564 0.67866 7s 0.67747 0.68371 0.68951 7p* 0.31275 0.31208 0.31161 Element 115 7s2 6d*4 ed^ 7p*2 7pl ACRV SC LC 5P 4.78422 4.71401 5f 4.48222 4.46919 6^ 7.29707 7.28416 7.24147 Gp* 5.47869 5.47412 5.41703 GP 3.22778 3.24378 3.19604 Gd* 1.09107 1.09825 1.07763 Gd 0.89315 0.90027 0.88328 7s 0.84138 0.85398 0.85227 7p* 0.42287 0.41436 0.41465 Tp 0.17953 0.18132 0.17748 310 Element 116 + 7s2 6d*4 6d6 7p*2 7pl ACRV SCLC 5P 5.61302 5.61126 a 5.28927 5.28200 6b 8.16816 8.15164 8.16739 ep* 6.27683 6.27396 6.14164 €P 3.82220 3.83620 3.80843 6d* 1.58407 1.59169 158989 6 i 1.36102 137140 136836 7s 126168 1.27817 1.28620 7p* 0.78843 0.77704 0.75498 TP 0.46975 0.47580 0.47364 Elem ent 117 7s2 6d*4 ed^ 7p*2 ?p3 ACRV SC LC 5P 5.90874 5.87436 5£ 5.56268 5.52871 Gs 8.51658 8.51651 8.50632 ep* 6.54670 6.50292 6.40979 » 3.87507 3.88138 3.86017 6d* 153154 1.53337 1.52571 6d 128648 1.29037 128363 7s 116382 1.18169 1.18492 7P* 0.63433 0.63017 0.60428 TP 0.26430 0.26534 0.26542 Elem ent 118 7s2 6d*4 ed® 7p*2 ?p4 ACRV SC LC 5f* 6.49078 6.44321 Sf 6.12131 6.05996 • 6s 9.16017 9.08017 9.10480 6p* 7.11018 7.03870 6.98967 GP 4.20343 4.17863 4.19653 6d* 175571 1.72346 1.73464 6d 1.48581 1.46200 1.47104 7s 1.33230 1.33739 1.35401 7p* 0.74550 0.73594 0.71389 Tp 0.30324 0.29237 0.30570 311