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Spin-Orbit Configuration Interaction Calculations of Actinide and Lanthanide Systems

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Scott Raymond BrozeU, M. S.

*****

The Ohio State University

1999

Dissertation Committee: Approved by

Professor Russell M. Pitzer, Adviser Professor Eric Herbst Adviser Professor Emeritus Isaiah Shavitt Graduate Program in Chemical Physics UMI Number: 9951<^6

UMI MScroform 9951636 Copyright 2000 by Bell & Howell Wbrmadon and Learning Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ABSTRACT

Spectroscopic properties of actinide and lanthanide systems were calculated via spin-orbit multireference configuration interaction (SOMRCI) methods using rela- tivistic efiective potentials (REPs) and correlation consistent (cc) basis sets.

A program for generating electron configuration lists was written in C + + using the standard template library. Based on an earlier program, this rewrite has no restrictions on the number of basis functions or the number of electrons and has an improved user interface.

Uranium REP core sizes were studied. A 78-electron core size, valence space of

6s~6p^5f^6d^7s^, leads to energy levels in mediocre agreement with experiment and to uranyl ion (UO 2 bond lengths that are a few picometers too short. Including the electrons in the valence space corrects these problems. Including, in addition, the electrons in the valence space yields worse results because relativistic effects are large for this subshell.

cc-pVDZ and cc-pVTZ basis sets for actinides and lanthanides have been de­ veloped. In a study on trends in correlation and SO energies are similar to those found in Dunning’s original work on the correlation energy in the oxygen atom.

Exponents of Gaussian functions optimized for both correlation and SO energy are

u nearly identical to those optimized for correlation energy only. Contraction coeffi­

cients for Gaussian basis sets for use with REPs are best determined from atomic

natural orbitals, as indicated by a study on the oxygen atom.

The ground and excited states of AmOg AmOg and CmOg ^‘'"were calculated.

The ground state of the americyl cation is ‘^^ 3 / 2 u- The first ligand to metal charge

transfer (LMCT) state is The ground and first LMCT states of the

isoelectronic dioxoamericum(V) and curyl cations are and

^Ilo+g. Agreement with experimental excitation energies is good. Other electronic

/ <— / transitions have been tentatively assigned.

Electronic spectra of various cations of uranium, plutonium, americium, curium, praseodymium, and europium, were calculated. For and Pr^^, the agreement is within 20% of experiment for SOMRCI singles and doubles (SOh'IRCISD) using 78 electron core REPs for U and a 46 electron core REP for Pr and cc-pVDZ basis sets.

For U, either using a cc-pVTZ basis set or using a 68 electron core REP improves the energy levels to within 10%.

All of these cations have been more extensively studied experimentally as dopants in solid crystalline materials. A model of crystalline calcium fluoride was developed and electronic spectra of these cations as dopants in CaFg were calculated. Generally, the agreement is within 25% of experiment. However, the ground state crystal field splittings of the p ions Am^"^, Cm^"^, and Eu^'*' are about a factor of 20 smaller than the corresponding experimental values, which are less than 50 cm~^.

ui DEDICATION

Shall I compare thee to a summer’s day? Thou art more lovely and more temperate. Rough winds do shake the darling buds of May, And su m m er’s lease hath all too short a date. Sometime too hot the eye of heaven shines, And often is his gold complexion dimm’d; And every fair from fair sometime declines. By chance or nature’s changing course untrinun’d; But thy eternal summer shall not fade Nor lose possession of that fair thou ow’st: Nor shall Death brag thou wander’st in his shade. When in eternal lines to time thou grow’st: So long as men can breathe or eyes can see. So long lives this, and this gives life to thee.

William Shakespeare

IV ACKNOWLEDGMENTS

I thank Dr. Russell M. Pitzer for his thoughtful guidance, boundless patience, and professional deportment. He is an exemplary scientist, a comprehensive scholar, and a quintessential advisor.

I thank many colleagues for their Mendship and assistance, especially Lars Ojamae,

Wolfgang Kiichle, Jo Ann Balozy, Phillip Christiansen, Clinton Nash, Michael Seth,

Bert de Jong, Steve Parker, Gary Kedziora, and former and current Pitzer group students: Ke Zhao, Zhiyong Zhang, Jean-Phillipe Blaudeau, and Spiridoula Matsika. VITA

1989 ...... B .S . — Case Western Reserve Univer­ sity, Cleveland, U.S.A. 1993 ...... M. S. — John Carroll University, Cleveland, U.S.A. 1993-1999 ...... Graduate School Fellowship, Teach­ ing Associate and Research Associate, Ohio State University

PUBLICATIONS

“Atomic Orbital Basis Sets for Use with Efiective Core Potentials,” J.-P. Blaudeau, S. R. Brozell, S. Matsika, Z. Zhang, and R. M. Pitzer, Int. J. Quantum Chem. Submitted.

FIELDS OF STUDY

Major Field: Chemical Physics

Studies in Theoretical Chemistry: Professor Russell M. Pitzer

VI TABLE OF CONTENTS

Page

Abstract...... ii

Dedication...... iv

Acknowledgments ...... v

V i t a ...... vi

List of Tables...... x

List of Figures ...... xiv

Chapters:

1. Introduction ...... 1

2. Configuration Generation Softw are ...... 9

2.1 Introduction ...... 9 2.2 Multireference Configuration E xp losion ...... 10 2.3 The Original CGDBG Program ...... 15 2.4 Design of the New CGDBG Program ...... 19 2.4.1 First Implementation...... 19 2.4.2 Final Implementation ...... 21 2.4.3 User In terface...... 24 2.4.4 Program Notes ...... 24 2.5 Profiles of CGDBG Programs ...... 25 2.6 Conclusions ...... 27

vu 3. Relativistic Effective Potential Development...... 28

3.1 Introduction ...... 28 3.2 B ackground...... 29 3.2.1 Relativistic Quantum Physics ...... 29 3.2.1-1 Special Relativity ...... 30 3.2.1.2 The Klein-Gordon Equation ...... 32 3.2.1.3 The Dirac Equation ...... 33 3.2.1.4 Many-EIectron Hamiltonians ...... 36 3.2.1.5 Two-Component Hamiltonians ...... 38 3.2.1.6 Electron Correlation Methods ...... 41 3.2.2 Relativistic Quantum Chemistry ...... 42 3.2.3 Relativistic Effective Potentials ...... 45 3.2.3.1 Historical Development...... 46 3.2.3 2 Selected Results...... 57 3.2.4 Spin-Orbit Coupling ...... 59 3.3 M e th o d s ...... 62 3.4 Results and Discussion...... 63 3.4.1 Uranium V ...... 63 3.4.2 Uranyl Cation ...... 88 3.5 Conclusions and Future W o r k ...... 94

4. Basis Set Development ...... 96

4.1 Introduction ...... 96 4.2 B ackground ...... 96 4.2.1 Correlation Consistent Basis Sets ...... 98 4.2.2 Cl Basis Sets Requirements ...... 101 4.2.3 Effective Core Potential BasisSets ...... 102 4.3 M e th o d s ...... 104 4.4 ECP Basis S e t s ...... 104 4.4.1 First- and Second-Row A to m s ...... 107 4.4.2 Actinides and Lanthanides...... 107 4.5 Uranium V Correlation and Spin-Orbit Energies ...... 110 4.5.1 Uranium REP cc-pVTZ Basis Sets ...... 118 4.6 Conclusions and Future W o r k ...... 124

5. Actinyl Ions ...... 126

5.1 Introduction ...... 126 5.2 Background ...... 128

vm 5.3 M eth o d s ...... 131 5.4 Results and Discussion ...... 137 5.4.1 A m 02^ + ...... 137 5.4.2 AmOz ^ and CmOg ...... 139 5.5 Conclusions and Future W o r k ...... 142

6. Lanthanide and Actinide Ions: Free and Doped into Crystalline Calcium Fluoride ...... 147

6.1 Introduction ...... 147 6.2 B ackground ...... 148 6.3 M eth o d s ...... 152 6.3.1 Crystal M o d el ...... 158 6.4 Free Ion Energy L e v e ls ...... 165 6.5 Crystal Doped Ion Energy Levels ...... 172

Bibliography ...... 176

IX LIST OF TABLES

Table Page

2.1 Mumber of Electron. Configurations ...... 11

2.2 Timings of Configuration Generation Programs ...... 26

3.1 SOMRCISD/cc-pVDZ Program Timings for 29 Roots .... 74

3.2 SOMRCISD/cc-pVDZ Energy Levels ...... 75

3.3 SOMRCISD/cc-pVDZ Energy Level Relative E r ro r s ...... 76

3.4 E78 SOMRCISD Energy Levels ...... 78

3.5 E78 SOMRCISD Energy Level Relative E rro rs ...... 79

3.6 SOMRCIS/cc-pVDZ Energy Levels ...... 80

3.7 U'*'*' SOMRCIS/cc-pVDZ Energy Level Relative Errors ...... 81

3.8 U'*"’" SOMRCIS/cc-pVDZatom Energy Levels ...... 82

3.9 SOMRCIS/cc-pVDZatom Energy Level Relative Errors ...... 83

3.10 U^"'’ MRCISD/cc-pVDZ Energy L e v e ls ...... 85

3.11 MRCIS/cc-pVDZ Energy Levels ...... 86

3.12 SOMRCIS/cc-pVDZ E78 AREP Energy Levels with SO Operators from C68 87

3.13 ^H4 Energies from SOMRCIS with C62 AREP and SO Operators. 89

X 3.14 Terms from MRCIS with C62 AREP and SO Operators and Small Basis Sets...... 90

3.15 Terms from MRCIS with 062 AREP and SO Operators and Large Basis Sets...... 91

3.16 SOMRCIS/cc-pVDZ Energy Levels with C62 AREP and SO Op­ erators...... 92

3.17 SOMRCIS/ cc-pVDZ Energy Level Relative Errors with C62 .AREP and SO Operators...... 93

3.18 UOa^"^ Bond Lengths in B o h r ...... 94

4.1 Oxygen 2s^2p^ ^P Correlation Energies in MiUihartrees [ 1 ] ...... 100

4.2 K-type Open-Shell Energy Coefficients for Average of All States of s, p, d, and / Configurations Where Jaao = 0...... 105

4.3 K-type Open-Shell Energy Coefficients for Average of AR States of / and g Configurations Where Jaao = 0 106

4.4 6s^6p^5/^ / Optimized Exponents and Total Energies in H artrees...... 112

4.5 6s^6p®5/^ p Optimized Exponents and Total Energies in H artrees...... 113

4.6 6s^6p®5/^ s Optimized Exponents and Total Energies in H artrees...... 114

4.7 6s^6p®5/^ ^H / Correlation and Spin-Orbit Energy Lowerings in M iUihartrees...... 115

4.8 6s^6p®5/^ ^H p Correlation and Spin-Orbit Energy Lowerings in M iUihartrees...... 116

4.9 6s^6p®5/^ ^H s Correlation and Spin-Orbit Energy Lowerings in MiUihartrees...... 117

XI 4.10 6s^6p®5/^ Total Energies in Hartrees from MRSOCISD with Ennler [2] R E P ...... 119

4.11 Energy Levels from MRSOCISD with Ermler [2] R E P ... 122

4.12 Energy Level Relative Errors from MRSOCISD with Ermler [2] REP ...... 123

5.1 Ground State SCF/(cc-pVDZ Atom) Population A n a ly se s ...... 138

5.2 AmOg SCF/ (cc-pVDZ Atom) Electronic Spectrum ...... 138

5.3 Symmetric Stretch Frequencies ...... 142

6.1 CaFa Cluster Size ...... 163

6.2 U®"*" Energy Levels from SOCISD/ cc-pVDZ with E78 R E P ... 166

6.3 U®'*’ Energy Levels from SOMRCISD/cc-pVDZ with E78 REP and a (5/, 6d, 7s, 7p)^ Reference S p a c e...... 166

6.4 Energy Levels from 5/^ Reference Space SOMRCISD/cc-pVDZ with E78 REP ...... 167

6.5 Low Lying Energy Levels from 5/^ Reference Space SOVIRCIS/cc- pVDZ with E78 REP ...... 167

6.6 Pr'^'*' Energy Levels from SOMRCISD/cc-pVDZ with Ermler 46 Elec­ tron REP and a (4/, 5d, 6s, 6p)^ Reference S p a ce 168

6.7 Pr^"*' Energy Levels from 4/^ Reference Space SOMRCISD/cc-pVDZ with Ermler 46 Electron R E P ...... 168

6.8 Pr^""" Low Lying Energy Levels from 4/^ Reference Space SOMRCIS/cc- pVDZ with Ermler 46 Electron R E P ...... 169

6.9 Eu^"*" Low Lying Energy Levels from, 4/® Reference Space, 5s^5p® Frozen Core, SOMRCISD with cc-pVDZ Basis Set and Ermler 46 Elec­ tron REP...... 170

XU 6.10 Low Lying Energy Levels from, 5/® Reference Space, 6s^6p® Frozen Core, SOMRCISD with cc-pVDZ Basis Set and Nash 78 Elec­ tron REP...... 171

6.11 in CaFg Low Lying Energy L e v els ...... 173

6.12 Pr^+ in CaFg Low Lying Energy Levels ...... 174

6.13 Ground State Splitting Energies of f~ Ions in CaFa...... 175

xni LIST OF FIGURES

Figure Page

3.1 E78 cc-pVDZ Basis S e t ...... 64

3.2 U E78 cc-pVDZ Atom Basis Set ...... 65

3.3 C78 cc-pVDZ Basis S e t ...... 66

3.4 C68 cc-pVDZ Basis S e t ...... 67

3.5 C62 cc-pVDZ Basis S e t ...... 68

3.6 C60 cc-pVDZ Basis S e t ...... 69

3.7 E78 cc-pVTZ Basis S e t ...... 70

3.8 U Uncontracted Stuttgart [3] Basis Se t ...... 71

3.9 U Uncontracted 9sd 7p 8f 2g Ih Basis S e t ...... 72

3.10 O cc-pVDZ Basis Set ...... 73

4.1 U‘‘+ E78 cc-pVTfZ Basis Set ...... 120

4.2 E78 cc-pVTspfZ Basis Set ...... 121

5.1 Am^"*" cc-pVDZ Basis Set ...... 132

5.2 Am cc-pVDZ Atom Basis S e t ...... 133

5.3 Cm cc-pVDZ Atom Basis Set ...... 134

XIV 5.4 O Augmented cc-pVDZ Basis S e t ...... 135

5.5 ArnOo^"*" MRSOCIS/cc-pVDZ Electronic Spectrum ...... 140

5.6 AmOa^"^ MRSOCISD/cc-pVDZ Groimd State Potential Energy . . . 141

5.7 AmÛ2 MRSOCIS/cc-pVDZ Electronic Spectrum ...... 143

5.8 CmOg^"^ MRSOCIS/cc-pVDZ Electronic Spectrum ...... 144

5.9 CmOa^'*'MRSOCISD/cc-pVDZ Ground State Potential Energy . . . 145

6.1 Pu^'*’ cc-pVDZ Basis S e t ...... 154

6.2 Am^"*" Stuttgart Pseudopotential [3] cc-pVDZ Basis S e t ...... 155

6.3 Eu cc-pVDZ Atom Basis Set ...... 156

6.4 Pr^"^ cc-pVDZ Basis Set ...... 157

6.5 Crystalline CaF 2 Cluster Model: Inner Two Layers [4] ...... 160

6.6 Ca^"*" Stuttgart 10 Electron Pseudopotential pVSZ BasisS e t ...... 161

6.7 F~ Stuttgart 2 Electron Pseudopotential aug-VDZ Basis S e t ...... 162

6.8 Crystalline CaF 2 Cluster Model: Ca7gFfOT [4] 164

XV C H A P T E R 1

Introduction.

Quantum chemistry, in particular, the ab initio calculation of chemical and spec­

troscopic properties, has come of age. At the end of the tumultuous decade in which

quantum mechanics was formulated, Dirac wrote [5] “The underlying physical laws

necessary for the mathematical theory of a large part of physics and the whole of

chemistry are thus completely known, and the difficulty is only that the exact appli­

cation of these laws leads to equations much too complicated to be soluble.” Seventy years later as a new millennium is approached, desk-top computers are enabling the numerical solution of the above equations [6]. Quantum chemical calculations now routinely complement experimental spectroscopy. Ab initio computations are an inte­ gral component of chemical dynamics vis-à-vis potential energy surfaces. Applications range from drug design to astrophysics to nuclear waste disposal. This introduction describes the state of the art and enough background to read the ensuing chapters independently.

Independent-particle models are applicable to huge systems, those containing sev­ eral hundred atoms. In molecular orbital theorj' approximate wavefimctions for a molecule consist of antisynunetrized products (Slater determinants) of one-electron functions called spin orbitals. Each of these is a product of a spatial function and a

1 spin function. In practical automated calculations of solutions to the many-electron

time-independent Schrodinger equation, Htl; = Eij;, these spatial functions, termed

molecular orbitals (MOs), are represented as linear combinations of K atomic orbital­ like one-electron basis functions ipi = Collectively the basis functions ((ifi are called the basis set, and the are the MO expansion coefficients. Basis set de­ velopment is discussed in Chapter 4. To complete the specification of a wavefimction, these unknown coefficients are determined via the variation method and Hartree-

Fock (HP) theory or some extension of it. The variation principle states that the energy expectation value of an approximate wavefimction is always greater than the energy of the exact ground-state wavefimction, E = {r};\H\ilj) > Eexact- In restricted

HP theory electrons whose spins are paired occupy the same spatial orbital. The

HP wavefimction always corresponds to a single electron configuration and is usually one Slater determinant. In the restricted open-shell HP method, however,the wave- function may be multideterminantal [7-9]. In the Bom-Oppenheimer approximation, where the motions of the nuclei and the electrons in a molecule are calculated sep­ arately, the nonrelativistic electronic Hamiltonian operator for a molecular system consisting of N electrons and M nuclei in atomic units is

N -1 iV Af 7 N'—1 H 1 g = + E E ( 1.1) 1=1 ^ t=l A=1 ' 1=1 /=t+l ' U

In HP theory a mean-field approximation replaces the nonseparable two-electron

Coulomb term, with a one-electron operator, 6t(r), representing the average repulsion of all the other electrons on electron i. The MO expansion coefficients axe varied self-consistently until the energetically best wavefimction is obtained. Thus, this implementation of HP theory is referred to as self-consistent field (SCF) theory. One advantage of HF theory as well as density-functional theory is that the result­ ing molecular orbitals can be easily interpreted with chemical intuition. The wide availability and black-box simplicity of ab initio packages, such as GAUSSIAN, have provided organic and inorganic chemists with a valuable tool to elucidate electronic structure. Recent Nobel Laureates Pople and Kohn are largely responsible for the routine use of ab initio programs. A disadvantage of HF theory is its inaccuracy.

In order to attain chemical accuracy, a maximum error of 1 kcal/mol, electron correlation must be achieved. The correlation energy is the difference between the exact nonrelativistic energy and the HF energy, F'corr = ■Ê'eiact — Egp- Electron cor­ relation comes in two overlapping varieties: dynamical and nondynamical. The HF mean-held approximation exphcitly neglects the correlated motions of the negatively charged electrons; this is dynamical correlation. The single conhguration HF wave- function explicitly ignores the contributions of other Slater determinants, and may fail because of degeneracies or near-degeneracies between electron configurations; this is nondynamical correlation. Three post-HF methods are popular [10-13]: configura­ tion interaction (Cl), many-body perturbation theory (MBPT), and coupled-clusters

(CC).

Cl is the simplest and most flexible method. A natural way to improve the HF single-configuration wavefimction is to include Slater determinants in which electrons are excited relative to the reference HF configuration; the excitations may be into partially occupied orbitals in the reference, essential for nondynamical correlation, or into unoccupied (virtual) orbitals or both. The Cl wavefunction is

= + Ecg^®ÿ + "-. (1.2) I,a »

Singles Cl (CIS), including all single excitations only, does not lower the HF energy because of Brillouin’s theorem which states that singly excited determinants do not interact directly with a reference HF determinant [12, pages 128-129}. Doubles Cl

(CID), including all double excitations only, accounts for the bulk of the dynamical electron correlation. Quadruple excitations are the next most important [10, page

85]. Singles and triples contribute about equally. Because the number of configura­ tions increases rapidly with excitation number, singles and doubles Cl (CISD) is the practical limit for medium and larger systems. A major disadvantage of truncated

Cl is that it does not scale properly with system size — it is not size consistent.

Two advantages of Cl are that it is tractable for excited states and that it trivially generalizes to multireference approaches, that is, where $o is a linear combination of

Slater determinants. Shavitt [14] has recently reviewed the history of Cl, and David­ son [15] has summarized the nature of Cl wavefimctions. C l methodology is discussed in Chapter 2.

Second-order MpUer-Plesset perturbation (MP2) theory, second-order MBPT with a Hartree-Fock unperturbed wavefimction, is economical, usually recovers a large percentage of the correlation energy, and enables easy calculation of one-electron properties, vibrational firequencies, and energy gradients for geometry optimizations.

CC with single, double, and noniteratively computed triple excitations (CCSD(T)) is the method of choice for the ground states of closed-shell systems. In combina­ tion with basis sets designed for electron correlation, such as correlation consistent basis sets, better than chemical accuracy can be obtained, however, only for medium and smaller systems because the calculations are expensive. A/IBPT and CC are size consistent but not variational and are difficult to generalize to multireference ap­ proaches. Bartlett [16] has reviewed recent developments in CC theory, including open-sheU applications and excited states. Roos et al. [17] have recently reviewed multiconfigurational PT and its application to electronic spectra.

Systems containing actinides, lanthanides, or transition metals are computation­ ally demanding [18-20]. Open / shells or open d shells or both spawn a multitude of nearly degenerate electronic states. In addition to this nondynamical correlation, which necessitates multireference methods, the dynamical correlation from closely spaced open-shell electrons is substantial. Relativistic effects are important for these heavy elements. In particular, spin-orbit (SO) coupling determines both the sym­ metry and the good quantum numbers of such systems. A spin-orbit term, such as

YliLi 2 eV ^ ■ Si, an approximate simple one-electron operator where 1% and s, are the orbital and spin angular momentum operators of electron i, must be included in the electronic Hamiltonian because it is energetically on a par with the two-electron

Coulomb operator. The resulting intermediate coupling of such systems may aggra­ vate the need for multireference treatments.

In 1929, Dirac stated [5] that relativistic effects would be “of no importance in the consideration of atomic and molecular structures and ordinary chemical reactions.” In fact, to reach chemical accuracy for some systems containing only first-row elements requires the inclusion of one-electron relativistic effects; in these systems the scalar relativistic perturbations are of the same order of magnitude as core-valence and core-core correlation [21,22]. In order to attain spectroscopic accuracy, a maximum error of 1 cm“^, SO effects must be treated. For example, the fine structure of the sodium D line is caused by the 17 cm~^ SO splitting of the ls^2s^2p®3p^ term into the ^Pi/ 2 ground level and the ^Pa/a level. In 1979, two landmark papers, one by K. S. Pitzer [23] and the other by P. Pyykko and J. P. Desclaux [24], indicated the importance of relativity in chemistry. Among the dramatic phenomena in which special relativity plays a significant role are the color of gold, the room temperature liquid state of mercury, the unusually low valence of thallium and lead, and the lanthanide contraction. These and other phenomena are rationalized by the interplay of three relativistic effects on atomic orbitals: (1) the direct effect which contracts and stabilizes orbitals, (2) the indirect effect which expands and destabilizes orbitals, and (3) the SO effect which splits orbitals with angular momentum quantum number

Z > 0 into orbitals with total angular momentum quantum numbers I the I — \ orbitals are contracted and stabilized; the orbitals are expanded and destabilized.

For example, the unusual color of gold, with respect to an extrapolation of the colors of its congeners, copper and silver, is explained by the direct effect on the valence 6s orbital — its stabilization leads to an anomalously low s <— d transition.

Methods for the treatment of relativistic effects are classified as either all-electron or valence-only. All-electron approaches are based on a relativistic approximate

Hamiltonian (the multielectron relativistic exact Hamiltonian is not known.) These methods are indispensable for creating valence-only potentials and for benchmark­ ing. They are presently restricted to small systems, but as computational power increases so will their purview. Valence-only methods employ a frozen-core approxi­

mation replacing the core electrons with a relativistically derived effective potential

or pseudopotential. They customarily treat valence electrons nonrelativistically with

traditional techniques. This tradeoff usually gives the best of both worlds: the rel­

ativistic effects are captured for the fastest electrons, the core electrons, while their

removal reduces the calculation size, so that the more difficult electron correlation

problem can be handled. Both all-electron and valence-only approaches are amend­

able with correlation methods in principle, but all-electron methods pay a price in

software development as well as in computing resources. Relativistic and electron cor­

relation effects are coupled but can be crudely approximated as additive. Relativistic

effects and methods are discussed in Chapter 3.

Two methods stand out from the Department of Energy Computational Grand

Challenge Application: Relativistic Quantum Chemistry of Actinides [25] initiative

to characterize and process nuclear waste: density functional theory (DFT) and SO

multireference Cl (SOMRCI). Both methods can treat relativity and correlation.

DFT is computationally inexpensive, thus applicable to large systems, and excels at

calculating geometries and vibrational frequencies in good agreement with experi­

ment. SOMRCI is computationally expensive, but applicable to medium systems,

and excels at calculating excited states in agreement with experiment. In the latter

approach traditional SCF methods generate MOs for a reference space consisting of significant Slater determinants. Excitations from this multireference space define a Cl expansion. Relativistic effects are incorporated with Relativistic Effective Potentials

(REPs) at both the SCF and Cl levels of theory and with SO operators included in the CI Haxniltonian. Various properties, such as magnetic moments, can be calculated from the resulting Cl wavefunction.

Two serial Cl implementations were employed: a determinant based program [26,

27], which explicitly creates the Hamiltonian matrix and is limited to Cl expansions of size about half a million, and a Graphical Unitary Group Approach (GUGA) pro­ gram [28] based on the COLUMBUS suite [29], which is a direct Cl, wherein the

Hamiltonian matrix is not created, and is limited to Cl expansions of size about ten million. Each program has its advantages and disadvantages. Using REPs and cor­ relation consistent basis sets, SOMRCI calculations of electronic spectra of various cations of u ran iu m , plutonium, americium, curium, praseodymium, europium, and gadolinium are compared with available experimental spectra in Chapter 6. All of these cations have been more extensively studied experimentally as dopants in solid crystalHne materials. A model of crystalline calcium fluoride is developed and elec­ tronic spectra of these cations as dopants in CaF2 are calculated with the method above and contrasted with experiment in Chapter 6. The ground and excited states of AmOg .Am02 and Cm02 which are interesting analogs of the important uranium ion uranyl, UO 2 are calculated using SOMRCI. The results and a com­ parison with available experimental data are presented in Chapter 5.

8 C H A P T E R 2

Configuration Generation Software

2.1 Introduction

The spin-orbit package in the COLUMBUS suite consists of six programs [27,29].

ARGOS [30] evaluates integrals over symmetry-adapted linear combinations of con­ tracted Gaussian atomic orbitals (AOs). CNVRT [31] converts the above symmetry- orbital integrals into supermatrix form [8]. SCFPQ [7,32] iteratively solves the re­ stricted Hartree-Fock (RHF) equations using the supermatrix integrals and producing molecular orbitals (MOs). TRAN [33] transforms the AO integrals into MO integrals.

CGDBG [34] generates a list of electron configurations. CIDBG [35] performs con­ figuration interaction (Cl) calculations, in which relativistic effects are treated with relativistic effective potentials (REPs) and spin-orbit (SO) operators, using the MO integrals and configuration list. Configuration generation is the focus of this chapter.

In the next section the rapid growth of configuration lists is quantified. Section 2.3 details the shortcomings of the original CGDBG program. The goals and design of a new configuration generation program are described in Section 2.4. Section 2.5 presents timing data on the new program. The last section contains concluding re­ marks. 2.2 Multireference Configuration Explosion

The spin-orbit configuration interaction program CIDBG requires a list of electron

configurations in the form of a sequence of occupation numbers of real spatial orbitals.

Each spatial orbital may contain zero, one, or two electrons. The sequence is usually

partitioned by the irreducible representations (irreps) of the point group symmetry

of the system. Each irrep subsequence is usually in ascending order of orbital energy.

When a distinction between the conventional form of an electron configuration, e.g.,

6s^6p®5/°, and the specialized form, e.g., 22220000000, is necessary the specialized

form will be denoted a one-dimensional electron configuration.

The electron configurations in a Cl calculation are classified as singles, doubles,

etc. based on the number, one, two, etc., of electrons excited from nonempty orbitals

to nonfull orbitals. The reference space is the set of electron configurations on which

excitations are performed. The most common type of Cl calculation uses some subset

of all single and double excitations.

the number of one-dimensional electron configurations corresponding to

the atomic electron configuration P , where 0 < n < 41 -f- 2, is

where the summation is over the number of electron pairs. The first binomial coeffi­

cient is the number of combinations of impaired electrons, and the second is for paired

electrons. Binomial coefficients are defined for all pairs of real numbers using limits of gamma functions [36-39]. Table 2.1 fists the number of one-dimensional electron con­ figurations for some atomic electron configurations. The number of one-dimensional

1 0 Configuration Number Configuration Number

1 9^ 1

s’- 1 9^ 9

1 9^ 45 p’- 3 9^ 156

p2 6 / 414 p3 7 9'^ 882

æ 1 9^ 1554 d} 5 / 2304 (f 15 9^ 2907 æ 30 9^ 3139

45 hP 1

51 1 1

/° 1 hP 6 6 r 7 hP 275 p 28 880 p 77 2277 p 161 hP 4917 p 266 hP 9042 p 357 hP 14355 p 393 hP 19855 24068 25653

Table 2.1: Number of Electron Configurations

1 1 electron configurations for a shell more than half full is equal to the value for because of the equivalence of electrons and holes [40, pages 131-132][41].

The Cl calculation of electronic energy levels, as well as most other properties, necessitates a balanced reference space. This means that ail of theone-dimensional electron configurations that correspond to a reference atomic electronic configura­ tion are included in the reference space. The number of one-dimensional electron configurations is informally denoted as the number of references. For a system con­ taining non-interacting atomic electron configurations, Ç* , where i is an element of a nonempty index set, X, the number of references is

(2 -2 ) ier a consequence of probability theory for independent events [42]. For example, a uranium valence space of 6s^6p^5f^6d^7s^ requires 1-1-77-5 -1== 385 references for a balanced treatment. Many systems are interacting, however, such as through bond formation, which may result in a closed-sheU configuration. The latter is adequately described by a single reference. However, if the Hamiltonian includes a spin-orbit operator, which may mix states of different spatial and spin symmetries, then, even when some or all of the atomic degeneracy is quenched by bond formation, additional references may be required. In practice that is firequently the case for molecules containing heavy elements, but for atomic calculations, the references necessary for a balanced treatment of the atomic degeneracy usually already include those that would mix via SO coupling.

The total number of one-dimensional electron configurations in a multireference calculation is approximately directly proportional to the number of references. This is Shavitt’s rule of thumb [14, page 10]. Assuming that the number of references is

1 2 much less than the total number of configurations in the excitation space, then the justification of the rule is that most configurations are excitations from the common

closed-shell part of the references to the common virtual part. In practical cases where

the number of singly occupied orbitals is small, usually single digits, the rule fails as

any of these three limits are approached: no common doubly occupied orbitals, no

common virtual orbitals, or the excitation number is that of a frill Cl.

Consider a single-reference one-dimensional electron configuration with no restric­

tions on excitations. If the reference is a closed shell then the number of configurations generated by all single excitations is the product of the number of doubly occupied orbitals, d, times the number of empty orbitals, v. In the general case, where the reference contains s singly occupied orbitals, the number of configurations generated by all single excitations, is

M[S) = (d -f- s) (s + u) — s. (2-3)

Intuitively, this is the product of the number of nonempty orbitals times the number of nonfrill orbitals minus a correction for self-excitation.

Double excitations are somewhat more complicated to count. If the reference is a closed shell then the number of configurations generated by all double excitations is

E E Ü- (2-4) I < i < d l ^ < o

This is based on the corresponding formula for single excitations and two nested iterations over the doubly occupied and empty orbitals. Another approach is to partition the possible double excitations [43]:

''"+''(2) + ( 2)"+(2) 0 '

13 where the terms represent the number of ways to excite from one occupied orbital into

one virtual orbital, one occupied into two virtuals, two occupied into one virtual, and

two occupied into two virtuals, respectively. These formulae are equal [37, sections

1.2.3 and 1.2.6], which is readily demonstrated by reducing each to:

^d-h

. 2 H r y

In the general case, the number of configurations generated by all double excitations,

AfiD), is

M{D) =

dv -i- (gin - h S dv + d\

4- 4 - 2 ( “ ) dn

(2.6)

The terms represent the number of ways to excite from doubly occupied orbitals into virtual orbitals, doubly into one singly and one virtual, doubly into virtual and singly into virtual, doubly into singly, singly into virtual, singly into singly and doubly into virtual, singly into singly and doubly into singly, singly into singly and singly into virtual, and singly into singly, respectively. The presence of singly occupied orbitals complicates the problem of counting the same excitation twice [44, section 5.2].

The combination of equations 2.1, 2.3, 2.6, and to a lesser extent in practice

2 . 2 leads to an explosive growth of the size of a multireference singles and doubles configuration interaction calculation. For example, the americium(II) cation has a ground state electron configuration of [Xe]4/^^5d^°6s^6p®5/^; the number of reference

14 one-dimensional electron configurations is 393. With a large 78 electron REP and a

small correlation consistent basis set representing the 6 s, 6 p, 5/, and 6

total number of configurations is 2 140 828-

In s u m m ary, a multireference singles and doubles configuration interaction calcu­

lation on a system consisting of a few atoms where each atom is represented with a

modest basis set requires the generation of a list of on the order of one hundred thou­

sand configurations. Larger calculations, such as on systems containing actinides,

lanthanides, or transition metals, can easily reach into the m illions of configurations.

2.3 The Original CGDBG Program

The original configuration generation program, CGDBG [34], has three major

shortcomings: ( 1 ) outdated limitations on the size of a configuration list, ( 2 ) an

awkward user interface, and (3) unreadable source code. The first deficiency is the

most serious; the second is the most annoying; the third is leading to the program’s

ultimate demise.

The original configuration generation program is usually adequate for a system

with less than approximately 40 basis functions. The limit of the program is 72 ba­

sis functions on machines with an integer size of four bytes. A Cray Yh/IP version

was modified to handle up to approximately 106 basis functions using the integer=64

Fortran compiler directive [45]. For systems with more than approximately 40 basis

functions, the run time can be prohibitive — hours or days on either a Sun Ultra 1

Sparc or a Cray YMP. If the input configurations are reordered so that the doubly occupied orbitals precede the singly occupied orbitals which precede the virtual or­

bitals then the run time is reduced by an order of magnitude [46]. This reordering is

15 tedious. As computational power has increased over time, the size of desirable calcu­ lations has exceeded the CGDBG program’s limit on basis functions and its ability to generate lists in a reasonable time.

The original program requires formatted input, wherein each inputed quantity has a contiguous range of character positions to contain its value. The program’s docu­ mentation [34] indicates that it was written not later than 1972. In that era of punched card input decks [47, page 650], the format is understandable. In todays era of screen based input, that format is still workable for small numbers of basis functions. Be­ yond 80 basis functions, however, the format requirements are error prone, tedious, and thwart sophisticated features of editors that could ease the pain, such as, column cutting, column pasting, and line based substitution commands. Furthermore, the user interface lacks abstraction in its representation of electron configurations. The ideal interface should accept the same form of electron configurations that humans normally use. For example, a calculation on with a reference space of 6 s^ 6 p® 5/^ should ideally require no more than that simple sequence of orbital designations and populations. Instead it requires two times twenty-eight sequences of one-dimensional electron configurations. For a [12slIpl0d8/lp]/(12sllpl0dS/lp) uranium basis set each one-dimensional electron configuration consists of two adjacent eighty character sequences. If the basis set is contracted to (6s4p4d2/lp) then the previous input is useless and must be totally rewritten. If symbolic electron configurations are used then only the number of orbitals of each irrep needs to be changed.

The original program is mainly written in Fortran 6 6 . It is a small program of 930 lines. More than half of its 240 comments are empty lines. There are 7 subroutines and 1 function. Almost all the variables are declared implicitly and have no comment

16 to decipher their at most 6 character long name. Many statements have labels, and there are 40 go to’s. These statistics are mentioned for reference. The essential problem with the program is that one cannot understand its structure and control flow by casual reading.

Software engineering has advanced significantly in the past thirty years. Dijkstra’s

1968 note entitled “Go to statement considered harmful” [48] started the structured programming revolution. Despite beautifiil expositions on the subject in both the classic paper by Wirth [49] and the classic text by Dahl, Dijkstra, and Hoare [50] many programmers myopically focused on the removal of go to statements instead of on the far greater matter of writing well structured, easily readable and understand­ able programs, as pointed out by Knuth in the 1974 paper “Structured Programming with go to Statements”. (Incidentally that 1972 classic text also contains the prin­ ciples of the data abstraction and the object oriented paradigms.) By restricting a programmer to use the five structured programming constructs: statement sequence, if then else, while loop, do until loop, and case [51, pages 129—138] the reader of a program can understand the structure at a particular level of abstraction without knowing the details at a lower level of abstraction. Structured programming officially made its way into Fortran with the 1977 standard. Of course structured source code is not enough. Programming style directly impacts program readability [52,53]. Style is acquired through experience: “Carefiil study and imitation of good programs leads to better writing” [54]. The “Numerical Recipes” books provide examples in scien­ tific computing [55-57]. Project management may determine the success or failure of large scale software engineering [58]. Structured design [59] is the logical evolu­ tion of structured programming. The decomposition of a program into subroutines

17 or modules is best done so that the material within a module is highly interelated and the communication between modules is minimized [51]. This is related to the principle of information hiding which states that implementation details should be hidden to achieve data abstraction [60]. Data abstraction and modularprogramming officially made its way into Fortran with the 1990/95 standard [61]. Object oriented programming supports a hierarchal structuring of information [62]. Object oriented features are slated to enter into Fortran with the 2002 standard [63,64].

The main algorithm of the original CGDBG program lacks structured constructs, lacks style, and is not adequately decomposed into subroutines. For example, an electron configuration is stored in a packed form. No doubt this was an essential storage saving technique in an era where physical memory was on the order of kilo­ bytes, not gigabytes as it is today. But there are no modules which perform packing and unpacking. It is not possible to differentiate the packing and unpacking code from the configuration generation code without understanding the complete main al­ gorithm. Because this program cannot be easily understood, the effort of the original programmer cannot be reused. Frankly, the decision to rewrite is obvious. It is com­ mon software engineering industry practice to rewrite [65]. Legacy systems are the exception which prove the rule. It is ironic in this year of Y2K [ 6 6 - 6 8 ] frenzy that the responsible legacy code is frequently being patched and not discarded [69,70].

18 2.4 Design of the New CGDBG Program 2.4.1 First Implementation

The primary goal of the first implementation was to determine the feasibility of a configuration generator for systems with large numbers of basis functions and elec­ trons. Because the algorithm of the original program could not be easily deciphered, it was not known whether the generation of configurations became intractable as the system size increased. The secondciry goals were to create the program quickly, to use the program to produce configuration lists for various eagerly anticipated CIs, and to verify the output of the new spin-orbit Graphical Unitary Group Approach (GUGA)

CIDRT program [28,71].

The choice of programming language can determine the outcome of a software de­ velopment project. Fortran is the de facto standard for scientific computing. It is well known for creating eflScient executables. Many optimizing compilers and fine-tuned numerical libraries exist. However, no numerical computation is needed for configu­ ration generation. Algorithms and data structures for the manipulation of large lists are required. This is the bread and butter of traditional computer science. General purpose languages which come to mind are Algol [72, references therein], Lisp [73],

Pascal [74], C [75], Ada [76], C-4—I- [77], and Modula-3 (successor of Modula-2) [78].

Fortran 77 is completely inappropriate as it lacks container libraries, parametrized types, support for data abstraction, dynamic memory, and even pointers; some of these latter tools are available in the recently released Fortran 90 [61]. In fact, some assembly languages, such as, PDP - 1 1 [79], are better suited to this type of program­ ming than Fortran. Among the languages listed above, C and C+-f- are the most familiar to scientific programmers and are the only ones besides Fortran available on

19 our Sun Ultra 1 Spares. C-M- was the obvious choice because of its many container library implementations and because it is a better C [72|.

The initial algorithm to generate recitations of order n on a list of references was to perform single excitations on the list, storing the resultant configurations in the list, and to repeat for n iterations. Then configurations not allowed by symmetry require­ ments, user defined restriction sets, and CIDBG limitations on the maximum number of singly occupied orbitals were removed. Single excitations on a configuration were obtained by attempting all movements of all electrons. This simple procedure was straightforwardly and quickly implemented in C-f—i- using the Rogue Wave container library [80] which was freely available on our Sun Ultra 1 Spares. The container used to represent the list of configurations was a hash set.

This program accomplished its goals. Lists as large as a million configurations could be generated in a few hours on a Sun Ultra I. There were no explicit restric­ tions on the number of basis functions, electrons, or configurations. The program was written in a few days and was used by members of the group to create lists for Cl calculations on actinyl ions. Only one bug was discovered [46]. However, the program also had several flaws. A practical limit on the number of basis functions or total con­ figurations or both existed because after real memory was consumed virtual memory paging slowed the program to a crawl. Several features of the original program were not implemented, such as restriction sets, retaining the order of an insert list, and different excitation numbers for different reference configurations. Finally, the user interface was no better than the original program’s and was different enough to be a nuisance.

2 0 2.4.2 Final Implementation

The new cgdbg program implements all the commonly used features of the original

program. There are no hard coded restrictions on the number of basis functions, electrons, or configurations. A practical limit on the number of basis functions or total configurations or both still exists because after real memory is consumed virtual memory paging stalls the program. This practical limit is now, however, significantly beyond the size of any Cl program including the pareillel CIDBG program [81]. There are two user interfaces. One differs from the original interface only in that whitespace is used to delimit tokens, that is, the number of blanks between values is not relevant as long as there is at least one blank. Basically it is free format or list-directed input in Fortran terminology. This interface is convenient for using old cgdbgin files.

The other user interface attempts to attain the ideal discussed in Section 2.3. The programming of this interface is presented in Section 2.4.3.

The configuration generation algorithm for a single reference is based on the count­ ing formula for double excitations from a closed-shell configuration. Formula 2.4: for i = first filled orbital to last nonempty orbital

for j = first nonfull orbital to last empty orbital

move an electron from orbital i to orbital j

if the configuration is valid then store it in the hash set

call this algorithm with the configuration and one less excitation

end for end for

2 1 This recursive algorithm is optimal in hash set storage attempts for a closed-shell

configuration. Duplicate configurations are generated only if there are singly occupied

orbitals. is the number of times the nested loop body is executed for single

excitations.

= (d-hs)(s-+-u), (2.7)

where d is the number of doubly occupied orbitals, s is the number of singly occupied

orbitals, and v is the number of empty orbitals, as in Section 2 .2 . For single and

double excitations

M {SD)= Y . Z y- ( 2 -8 ) l < i < i + s l

validation, the hash function computation, and the copying of an inserted configura­

tion all iterate over the orbitals; if the hash table is sufficiently large then the insert operation does not increase the overall time complexity [82, section 6.4]. Therefore,

the configuration generation algorithm is of order

'J'C{S') — (d -l- s -(- 'y)(d -j- s)(s + 1/) (2.9) in execution time for single excitations only and

rC{SD) = {d^s^v){d + sf{s + VŸ (2.10) in execution time for single and double excitations only. Formulae for triples, quadru­ ples, etc. have not been elucidated.

If the reference space has more than one configuration then this algorithm is called for each one. Obviously, the overall time complexity of the configuration generation step is a sum, over references, of the above single reference time complexities. For

2 2 practical calculations, as mentioned above regcirding Shavitt's rule of thumb, an upper bound on the time complexity is the number of references times the appropriate above formula (2.9, 2.10, etc.) for a reference configuration that has a maximal number of singly occupied orbitals. The program may be viewed as comprising three steps: input, configuration generatiou, and output. For large lists the execution time may be dominated by the output step.

The container for the electron configurations must provide fast storage and re­ trieval. Elements in the container must be unique. A hash set [82, section 6.4] is a good representation of the list of configurations. The hash set from the Stan­ dard Template Library (STL) of C+- 1- is employed [83]. The STL provides container classes, algorithms, and iterators to represent the basic algorithms and data struc­ tures of computer science [84]. The library is generic and based on the template facility of C-t—I- [77]. The STL is part of the recent C-i—I- standard. A freely avail­ able implementation of the STL from Silicon Graphics Computer Systems, Inc. is used [85].

The space complexity of the configuration generation step, SC is straightforward and unsurprising:

SC = basis(references 4- 2 • inserted + 2 • deleted 4 - excited), (2.11) where basis is the number of basis functions, references is the number of references, inserted is the number of configurations to insert at the top of the list, deleted is the number of configurations to remove from the list, and excited is the number of configurations generated by excitation, inserted and deleted arise because of the program’s ability to tailor the configuration list by explicitly inserting and deleting configurations [34]. The factors of two suggest potential targets for optimization, but

23 in practice inserted and deleted are usually insignificant — only the references are inserted and deletions are only occasionally necessary. A more detailed expression is useful to gauge actual memory requirements. Because the details are intimately dependent on the implementation of the hash set and other data structures, no deriva­ tion is presented:

basis S A = 4.\7 + j (references + 2 - inserted 4- 2 - deleted 4- excited) 4- 10240 4 is the actual space, in bytes, using a Sun Ultra I C4-4- compiler.

2.4.3 User Interface

The plan is to support input and output suitable for human consumption. Fortu­ nately, the output of the original CGDBG program was an ASCII file. The essential idea is that the input to a program can be viewed as a httle language [8 6 ]. Yacc (Yet

Another Compiler-Compiler) and Lex (a lexical analyzer generator) were employed both to read the original Fortran input format and to parse the new input language.

The classic reference is the dragon book [87]. The apphcation of yacc to the develop­ ment of a small interpreted programming language is described in detail in Kemighan and Pike [8 8 ].

2.4.4 Program Notes

Using the data abstraction paradigm the design of the new program naturally decomposes into the design of four types or classes: ElectronConfiguration, Symmetry,

RestrictionSet, and ElectronConfigurationList. The object-oriented paradigm is used to model symmetries: Symmetry has two derived classes: D_2h and C_ 2 v. Excellent references on C4 -P are Stroustrup’s text [77] and Meyer’s collection of tips [89,90].

24 Even tiny, one programmer software projects can benefit firom source code control systems. At least three are freely available on UNIX machines: Source Code Control

System (SCCS) [ 8 8 ], Revision Control System (RCS) [91], and Conclurent Versions

System [92]. Initially SCCS, available on the Sun Ultra 1 SunOS 5jc computer, was used, but conversion to the GNU CVS was simple and worthwhile.

Two approaches to bypass the real memory restrictions of the current version are disk based merge sorting [82, Chapter 5] and paxallelization [93]. For multirefer­ ence configuration generation, task distribution on a parallel computing system via references would be trivial to implement; however, either merging of the local con­ figuration lists or use of a global array [94] would be required to produce a final list without repetitions. Clearly, parallelization with global arrays is the simplest method to program.

2.5 Profiles of CGDBG Programs

Profilers monitor the execution of a computer program counting the number of times a statement is executed and measuring the amount of time spent in subroutines.

Two profilers are available in the UNIX operating system: prof and gprof. The use of program profiling was described by Knuth in the classic paper “Empirical Study of

Fortran Programs” [95]. Many of the speedup techniques discussed in that paper and in later works [96], such as inlining, loop transformations (collapsing, fusion, inter­ change, and unrolling), and arithmetical expression reformulation are now routinely available from optimizing compilers [97]. Changes in algorithms and data structures may reap much greater performance improvements [86,98].

25 Table 2.2: Timings of Configuration Generation. Programs. Version Max Mem Jser CPU System CPU Wall Time 78REP, cc-pVDZ, 28 refs, SD =+ 126588 configs

Original 62 Mb 17160 sec 2 . 2 19003

Reordered 62 Mb 1 2 2 2 1.3 1235 V0.90 gprof 27 14S 4.5 276 V0.90 27 108 4.0 230 V0.90 opts 27 50 4.0 194 78REP, cc-pVDZ, 28 refs, SOT =+ 2727444 configs

Original < 600 1204000 — 1470000 Reordered < 600 701817 37 960177

V0.95 opts 250 8 8 8 8 6 995

U^'*' 78REP, cc-pVDZ, 1 ref, SDTQ =+ 4816401 configs

V0.95 opts 4SS 1276 2 2 2 6051

The various CGDBG program s were profiled because of the known computational

demands of configuration generators. Some conclusions can be drawn: no routine

accounts for more than ten percent of the cpu time, the output of the configuration

list accounts for more than half of the wall time. This is probably evidence that the

character based C + + output is poorly optimized; no attempt at improvement has

been made [97, page 174]. Table 2.2 displays some execution times for various versions

of cgdbg programs. Because other jobs were sometimes executing, wall times are not

significant. The reordered entries are statistics for the original program with the

reference configurations in descending order of orbital occupation. The table shows the dramatic improvement of the new program for this 6s^6p®5/^ uranium cation

26 with, a large 78 electron REP and a small correlation consistent basis set representing the 6 s, 6 p, 5/, and 6 d orbitals (47 basis functions). The substantial amelioration from compiler optimization of the program is foimd with larger calculations.

2.6 Conclusions

Bom of necessity, later work on this program was justified by the parallelization of the CIDBG program [81]. Unfortunately, the parallel CIDBG program could not be applied to the calculations in Chapters 6 and 5, despite many attempts. The future of that program is unclear. The existence of a GUGA based Cl method does not eliminate the need for even a serial CIDBG because of the latter’s flexibility, more detailed wavefunction output, and ability to converge many roots quickly. Hopefully this new CGDBG program will suit user needs. The user interface probably requires at least fine tuning. Readable software has an important pedagogical role. Future programmers should be more fam iliar with structured programming. Fortran is slowly but steadily providing more powerful programming paradigms. Likewise C-t—I- has caught up to and in a few cases exceeded Fortran’s performance in numerically inten­ sive scientific computing [99]. In part this is because of C-I—t-’s support for powerful programming paradigms and its design philosophy that unused features do not iucur runtime expense [72]. Some of these numerical techniques have been packaged in the

Blitz library [100].

27 CHAPTER 3

Relativistic Effective Potential Development

3.1 Introduction

Poor excitation energies for uranium V, and other actinide cations as well

as an anomalous bond length for the uranyl cation, UOa^'*', reported at the Second

Conference of the Department of Energy Computational Grand Challenge Applica­

tion: Relativistic Quantum Chemistry of Actinides [25], motivated an investigation of

uranium relativistic effective potentials (REPs). The tradeoff between the incorpora­

tion of relativistic effects, which stem from the inclusion of an atomic subshell in the

REP, and both the accommodation of orbital polarization and the explicit treatment

of electron correlation, which are permitted by the inclusion of an atomic subshell in

the valence space, is examined at the self-consistent field (SCF) and the configuration

interaction (Cl), both with and without spin-orbit (SO) coupling, levels of theory us­

ing several new U REPs, with decreasing core sizes, developed by P. A. Christiansen.

In the next section relevant background information is presented on relativistic quan­

tum physics, relativistic quantum chemistry, REPs and SO coupling. Methodology is summarized in Section 3.3. excitation energies and UO 2 bond lengths, from calculations with the original U REP of Ermler et al. [2] and with the new REPs of

28 various core sizes, are presented and discussed in Sections 3.4.1 and 3.4.2, respectively.

Conclusions and suggestions for future work are stated in Section 3.5.

3.2 Background

The section on relativistic quantum physics is a pragmatic overview focusing on

Hamiltonians for atomic and molecular ab initio methods. Section 3.2.2 on rela­

tivistic quantum chemistry elaborates on the introductory material in Chapter 1 and

provides example applications to topics of relevance in later Chapters. The devel­

opment of REPs from the original nonrelativistic pseudopotentials to the present

shape-consistent formalism is chronicled in Section 3.2.3, including some early REP

results with bearing on core sizes. The last section discusses the treatment of SO

coupling and its impact on the calculation of excitation energies.

3.2.1 Relativistic Quantum Physics

This section is principally based on the following references. Because the material

is of a general nature, detailed citations are usually omitted. Numerous physics texts

present special relativity [101—103]. The Klein-Gordon amd the Dirac equations and solutions are discussed in Bethe [40]. Pyykko [104] outlines the physics as histori­ cally applied to chemistry. Dolg and Stoll [105] review many-electron Hamiltonians and valence-only methods including some of the fundamental physics. Balasubra- manian [106] presents the basic physics for an audience of chemists with emphasis on the REP methods developed by K. S. Pitzer and coworkers. Hefi, Marian, and

Peyerimhoff [107] discuss the physics in order to derive the no-pair Hamiltonian.

29 Additional information is available firom many sources. Bethe and Salpeter [108] give a complete but dated treatment of the whole subject applied to one- and two- electron atoms. Grant [109], concentrating on atoms, begins at a level where this overview ends. Wilson [110] introduces all-electron ab initio molecular structure methods. Moss [1 1 1 ] treats thoroughly the whole subject applied to molecular sys­ tems. Dyall and Faegri Jr. [112] are writing an introductory graduate level text on the subject.

3.2.1.1 Special Relativity

The postulates of Special Relativity are:

• Relativity Postulate: The laws of physics are valid in aU inertial reference

firames.

• Light Postulate: The speed of light is the same in all inertial reference firames.

In other words, there is no preferred inertial firame of reference, and the speed of light is independent of the motion of its source. With these postulates one can derive how to transform the equations of physics firom one inertial reference firame to another, the Lorentz transformation of relativistic mechanics:

y' = y

where the primed and the unprimed firames coincide at = t = 0 and the former is moving in the x direction with velocity v. The major differences with the Galilean

30 transformation of Newtonian mechanics, which these equations reduce to in the limit c —>■ oo, are that time is not absolute and that the temporal and spatial coordinates are treated equivalently. The latter homogeneity is a natural consequence of the invariance of time and space under a Lorentz transformation. Charge and spin are also

Lorentz invariants. Lorentz covariance is the property that an equation is invariant in form under the Lorentz transformation, as exemplified by Maxwell’s equations.

An exact relativistic equation is Lorentz covariant. Four-vector notation, where ct is the first component and the spatial coordinates are the next components, simplifies equations in relativity.

The strange behavior of space and time in special relativity can be derived firom the Lorentz transformation: the rest duration is the shortest, the rest length is the longest, and the rest mass is the lightest. Relativistic equations for the mass, the momentum, and the energy, all of which depend on the inertial reference frame, are:

- = p = m v = y (3.2)

E = m(? (3.3)

— p^(? = TMqC^ (3.4) where mo is the rest mass. The final relation is a fundamental identity between momentum and energy. It is Lorentz covariant. Furthermore, the righthand side is a constant and thus invariant to the firame of reference. Consequently, so is the lefthand side.

31 3.2.1.2 The Klein—Gordon. Equation

A recipe to convert from classical observables to linear Hennitian quantum me­ chanical operators is:

Classical Quantum

E

p

Applying these to the nonrelativistic energy expression for a free particle one obtains the nonrelativistic time-dependent Schrodinger equation:

s = ^ 2 m

d t 2 m

Because the temporal variable appears as a first derivative and the spatial variables as second derivatives, this equation is not Lorentz covariant. It has a plane wave solution: exp (p - r — Et)^. Applying the recipe to the relativistic energy expression for a free particle one obtains the Klein-Gordon equation:

= p^(? 4- m^c^

As one might expect from the similar treatment of temporal and spatial variables this equation is Lorentz covariant. Its solution is a plane wave in analogy to the nonrelativistic free particle wavefunction.

^ = e x p |^ (p -r —

where E = ± ^ ^ ^ 2 4 -

32 Thus the Klein-Gordoa equation admits solutions of negative energy! This leads to another perceived difficulty of the equation. The probability density p = can be negative. These apparent problems led Dirac to consider a different form for a relativistic wave equation. However, it was later shown that the Klein-Gordon equation is valid for particles with zero spin.

3.2.1.3 The Dirac Equation.

Dirac saw that the fundamental problem with the Klein-Gordon equation could be solved if only first derivatives in time of the wavefunction are allowed. The Dirac equation for a free particle is:

-I -a - = 0 c a t h where the operators at and /), arising from the intrinsic spin of the free particle, have several possible matrix representations, such as

/ / 0 0 0 1 \ r 0 0 0 - i \ f ^ 0 1 0 \ 0 0 10 0 0 i 0 0 0 0 - 1 a 0 10 0 0 - 2 0 0 1 0 0 0 \V 1 0 0 0 j \ i 0 0 0 J l o - 1 0 0 J) and / 1 0 0 0 \ 0 10 0 /) = 0 0-10 V 0 0 0 - I J Four items are noteworthy. First, this equation is Lorentz covariant. Second, it is not just a vector equation but a vector matrix equation. Thus a solution of the

Dirac equation consists of four components. Third, the equation admits negative energy solutions just like the Klein-Gordon equation. At the time this was consid­ ered a problem, but later the hypothesis that negative energy solutions corresponded

33 to positively charged electrons was confirmed by the experimental detection of the positron. Finally, the probability density for the Dirac equation is p = and can­ not be negative. Thus it has the same interpretation as in nonrelativistic quantum mechanics as the probability of the particle being at a position r at time t .

The four linearly independent free particle solutions of the Dirac equation are of the form ^(r, t) = N'ipÇr) exp (p - r — Et)^ , where the normalization constant.

1 0

0 1 0 +/s(r) = c (P x -iP v ) E^+moc^ E++moc^

c{Px-HVv) — Cpz \ E++ma<^ / \ E+ + m oc^ /

/\ f - c ( P x - iP v ) —E—+moc^ —E-+moc^ —c{p x+ ip v) £E=__ _ —E—+mot^ —E—^moc^ 1 0

\ 0 V 1 / ip+a and are degenerate with eigenvalue E+; ip^a and are degenerate with eigenvalue E-. The physical interpretation of these four-component plane wave so­ lutions is that two of the components correspond to positive energy solutions, the other two to negative energy solutions, and each positive or negative solution is com­ posed of two components because of the intrinsic one-half spin of the electron and the positron. This is seen in the solutions above because one distinct component prédominants in each wavefrmction. For example, the dominant component of the positive energy spin up solution, V’+a> is the first component; it is called the positive

34 energy spin up component, see below. For each solution two of the components go to zero in the nonrelativistic limit. For electronic solutions the negative energy compo­ nents go to zero as c —> oo and are called the small components; the positive energy components are called the large components. Thus in the spectrum from relativistic calculations to nonrelativistic calculations three types of wavefunction are possible, four-component, two-component, and one-component:

^ 4>+q. \

V J

Large — 4>+0

^Non—Rel — ( )

Note that the dominant component of the positive energy spin up solution, tp+a, is the first component, labeled

(f)-a and are neglected, perhaps after being decoupled from the large (see the

Foldy-Wouthuysen transformation below), then a two-component wavefunction re­ sults. Further, if all of the one-electron wavefunctions are chosen such that one of their components is zero then a one-component wavefunction is obtained. This is the situation when spin is not included in the Hamiltonian and is only present as an angular momentum operator, S, whose magnitude has a complete set of orthonormal eigenfunctions a and /3 that are used to construct spin orbitals, as in conventional nonrelativistic methods.

35 3.2.1.4 Many-Electron Hamiltonians

A time-mdependent relativistic electronic Hamiltonian in atomic units for a many- electron system is:

Hoc = ho{i) ^ — i < j where

hoij) = cof - Pi + f3iC^ - ^ ^ 7 ^ 7* and 7 is a nucleus index. This is the Dirac-Coulomb Hamiltonian. The one-electron operator is the exact relativistic Dirac Hamiltonian for an electron in an external

Coulombic field. Nuclei are assumed to be point charges. This assumption and the neglect of other nuclear effects, such as those stemming from nuclear spin, are usually valid in most chemical applications; notable exceptions are NMR and Mossbauer spec­ troscopy. Under these assumptions the nuclei only appear in the nucleus-electron ra­ dial vector terms, Tyi, and the corresponding summations will be omitted from future electronic Hamiltonians. Operators for nuclear effects and the nuclear Hamiltonian are presented in Moss [111, Chapter 10]. The two-electron Coulomb repulsion term above is not Lorentz covariant in the wave equation, Hdc "^ = E%1). In fact, no exact closed-form relativistic many-electron Hamiltonian exists because of complications in solving the field equations from quantum electrodynamics.

Approximate treatments using perturbation theory lead to an expansion in powers of the fine structure constant a. The first relativistic correction to the Coulomb interaction is the Breit interaction:

gfl = E L . a, - I 5 I 2rÿ _ r?- . j

36 The first term is the magnetic interaction between electrons, and the second term occurs because of the retardation of the Coulomb interaction due to the finite velocity of light. The Dirac-Coulomb-Breit Hamiltonian, H oc + H b , is correct to order

Energetically the retardation term usually contributes about one-tenth as much as the magnetic term [105, page 24]. Furthermore, it requires two-electron integrals which are more complicated than those of the Coulomb interaction and is frequently neglected, changing the Breit operator to the Gaunt operator:

Ha = i§ < j { - ^ } Higher order corrections to the Coulomb interaction as well the Breit interaction are usually cited as not chemically important.

The Dirac-Coulomb, Dirac-Coulomb-Gaunt, and Dirac-Coulomb-Breit Hamil­ tonians are the most widely used in four-component methods. Analytic solutions are not attainable and the equations are solved via the self-consistent field approach.

Dirac-Hartree-Fock (DHF) theory is analogous to HF theory. The one-electron fimc- tions, however, are four-component spinors, and the Hamiltonian is spin dependent.

DHF methods are complicated because of the spin dependence and computationally expensive because both the large and the small components require basis sets. In fact, to prevent variational collapse in DHF calculations because of the presence of positronic states, special kinetically balanced basis sets are constructed. Because of the relativistic kinetic energy operator, the small component, which is dominant for positronic solutions, requires approximately twice as many basis functions as the more important, for electronic solutions, large component. The small component can have significant magnitude in the core region and is a measure of the relativistic effect.

Because the number of integrals scales formally as the fourth power of the number

37 of basis functions, DHF requires roughly 81 times the computational effort of HF.

Numerical DHF methods, which avoid basis set problems, have been [113] and are increasingly popular.

3.2.1.5 Two-Component Hamiltonians

The complication and expense of four-component approaches has led to the use of two-component methods. The small components are decoupled from the large via transformations such as the Foldy-Wouthuysen or the Douglas—KroU. In the presence of an external electric field, such as that of a nucleus, the separation of the large and small components can only be achieved approximately to some order of o:.

Nevertheless, the resulting two-component methods restore the traditional picture of molecular electronic structure, where all interactions are between electrons and nuclei

— no positrons.

The Foldy—Wbuthuysen unitary transformation of the Dirac-Coulomb Hamilto­ nian yields the approximate relativistic Pauli Hamiltonian:

H p = H q -\- H D a rw in + H m V + H s o

Kj

H m v = 8 ^ E p ? (3.7)

B s o - 2^ E (3.3) i n 38 Pt) ' (St + 2Sj) « Jr« It can. also be obtained directly from perturbation theory by eliminating the small

components via expansions in Hq is the nonrelativistic Hamiltonian. HDarwin

has no classical analog; it is due to electron Zitterbewegung or trembling motion, in

effect the electron charge is smeared in relativistic quantum mechanics. This term is

frequently written as:

However, in this form the singularities at the nucleus are not obvious, and there is

some ambiguity in the apphcation of the Laplacian operator. Hmv is the mass-

velocity operator arising from the dependence of the electron’s mass on its velocity.

Hso is the spin-orbit coupling Hamiltonian; the one-electron term is caused by the

interaction of the electron’s spin magnetic moment with the magnetic field created

by the electron’s motion in the nuclear electric field; the two analogous two-electron

terms are: the spin-own-orbit, where the magnetic field is produced by the electron’s own motion in the electric field of the other electron (the first two-electron term in Hso, i.e., the one with Sf) and the spin-other-orbit, where the magnetic field is produced by the other electron’s motion (the second two-electron term in Hso, i.e., the one with 2Sj). Note that different sources have different signs and expressions for the two-electron terms [114, pages] [111, page 176] [107, page 159] [115, page

384] [116, page ]; Hso, Equation 3.8, for the case of two electrons is consistent with the SO operator for He in Bethe and Salpeter [108, page 181]. Application of the

Foldy-Wouthuysen transformation to the Dirac-Coulomb-Breit Hamiltonian leads to the Breit—Pauli Hamiltonian. Both these Hamiltonians suffer from singularities at

39 the nucleus and possible divergence of the mass-velocity correction. They are not appropriate for variational methods and are usually applied perturbatively to first- order using HF wavefunctions. Cowan and GriflSn [117], however, have implemented the Pauli Hamiltonian in the variational HF approach by neglecting the SO term, restricting the systems to be atomic, and solving the SCF equations numerically.

Another transformation to decouple the small component from the large com­ ponent is the Douglas-KroU transformation. This is an expansion in the external potential V, as opposed to a. The Douglas-KroU Hamiltonian is:

Hdk = 53 ^DK{i) + 5 3 — i i<3 where

hoKij) — Ei -f- Ai (yi 4- riViRt) A

+ \m ,W i\ + WiEiWi)

Ei + c^

Wi = linear operator in Vi

It is correct to order V^. The Hamiltonian is nonsingular and suitable for variational methods [105, page 29]. Similarly derived no-pair HamUtonians, so denoted because

40 the operator for particle-antiparticle pair creation has been removed during the proper

quantum electrodynamics formulation, have been applied extensively by Hefi and

coworkers [107]. A spin-orbit operator has been derived and applied using correlated

wavefunctions, both perturbatively and in a variation-perturbation approach wherein

the SO coupling matrix is diagonalized in a small Cl space [107, Sections 2.5 and

2.6]. A SOCI method, wherein SO coupling and electron correlation are treated

equivalently by including the SO operator in the, consequently two-component. Cl

Hamiltonian, was near completion in 1995 [107, page 270].

3.2.1.6 Electron. Correlation Methods

The traditional, i.e., one-component, approaches for including electron correla­

tion are adaptable to two-component and four-component methods. Quiney [118] in

1988 reviewed relativistic many-body perturbation theory. Malli [119] contains pa­ pers on both relativistic and electron correlation effects. Until recently the increased complexity and computational demands of four-component calculations have been prohibitive. Two examples of the significant developments in this area are (1) Dirac-

Coulomb-Breit Hamiltonian based coupled-cluster singles and doubles calculations on various atomic systems yielding impressive agreement with experimental excitation energies [120-127], and (2) Dirac-Coulomb-Gaunt Hamiltonian based configuration interaction singles and doubles and coupled-cluster singles and doubles with a per- turbative triples correction calculations on various atomic, molecular, and solid state systems yielding good agreement with corresponding experimental results [128] — this is an advancement of the MOLecular Fock DHlac programs (MOLFDER) [129].

Two-component Cl programs, using the REP method to obtain SO operators, were the first to be developed and applied [26,130,131]; Recent significant contributions

41 in this area are (1) a determinantal approach direct SOCI applicable to one- and

two-electron SO operators [132j, and (2) a graphical unitary group approach direct

SOCI using the REP one-electron SO operators [28,71].

3.2.2 Relativistic Quantum Chemistry

Innumerable reviews exist; an incomplete list since the prime year 1979 follows.

The ground-breaking papers by K. S. Pitzer [23] and by P. Pyykko and J. P. De-

sclaux [24] focus on chemical anomalies, at least in part attributed to special relativ­

ity, and are aimed at the general chemistry community. Pyykko’s earlier review [104]

is a concise summary of relativistic calculations for theoretical chemists. The 1981

NATO Advanced Study Institute on “Relativistic Effects in Atoms, Molecules and

Solids” produced a volume [133] containing papers mostly on theoretical develop­

ments. Results using the REP methods developed by K. S. Pitzer and coworkers,

which enabled the first truly ab initio study of molecular systems, are presented

in several reviews: emphasizing potential energy curves and spectroscopic proper­

ties derived from those curves [134], concentrating on the REP method and general

relativistic effects [135], covering ab initio methods broadly [136], emphasizing SO

effects and the consequent intermediate coupling [114], and surveying Cl calcula­

tions on bonding in heavy element containing systems [137]. Pjykko’s next reviews

continue the exploration of structural and periodic trends in Light of relativistic ef­

fects [138] while describing his semiempirical relativistic method [139]. In contrast to

the previous work, mainly on tiny systems and primarily not containing /-elements,

M. Pepper and B. E. Bursten [140] review methods for and results from electronic structure calculations on actinide-containing molecules from tiny hydrides to medium

42 actinocenes- Ab initio approaches to systems with /-elements became practical in

the 1990’s; K. Balasubramanian [141] surveyed approximate and ab initio methods

and results on lanthanide and actinide containing molecules in 1994; M. Dolg and

H. Stoll [105] reviewed methods and results for lanthanides in 1996. Section 3.2.4

contains references to reviews aimed exclusively at SO methods and effects.

In chemical systems three important effects on orbitals due to special relativity

are [104]: (1) the direct effect, (2) the indirect effect, and (3) the SO effect. The

direct effect is the contraction and stabilization of orbitals caused by the relativistic

velocities of electrons. An approximate velocity of an electron in a hydrogenlike Is

orbital is Z in atomic units. The speed of light is c = ^ = 137. Thus by the sixth

period of the periodic table a Is electron has attained a significant fraction of the

speed of light. According to Equation 3.1 the mass of such an electron has increased

to six-fifths its rest value. The Bohr radius of this orbital is oq = So the Is

orbital contracts by 20 % and is correspondingly stabilized. By orthogonality, even orbitals which do not have substantial amplitude near the nucleus are contracted and stabilized. The indirect effect is the expansion and destabilization of orbitals caused

by the increased nuclear shielding from relativisticly shrunken orbitals. The SO effect splits orbitals via the coupling of the electron’s orbital and spin angular momenta, j = 1 4- s. The resultant total angular momentum, j, quantum numbers designate orbitals: Si/2 ,pi/2 ,P3 / 2 ,<^3 /2 ,c^5 / 2 ,/ 5 /2 > and /t/2 - The I — | orbitals are contracted and stabilized relative to the pure I orbitals; the ( -t- ^ orbitals are expanded and destabilized. For heavy element systems the SO operator may exceed the Coulomb repulsion operator energetically. Jcommutes with an atomic Hamiltonian contain­ ing a SO term; however, neither L nor S do. Nevertheless, coupling intermediate

43 between j —j and L—S is found in the known atoms. This nondynamical correlation

is a major complication in theoretical calculations necessitating multiconfigurational

treatments. The energetics and intermediate coupling ste m m in g from the SO effect

have been known since the dawn of quantum mechanics [142]. However, the angular

properties of j-orbitals are different from those of 1-orbitals and can play a significant

role in bond formation as documented half a century later by K. S. Pitzer [143]. Ex­

planations of relativistic phenomena in chemistry are based on the interplay of these

three relativistic effects on orbitals.

The lanthanide contraction proper is generally defined as the decrease in the

atomic or ionic radii of the lanthanide metals as the series is crossed from left to

right. For the trivalent cations, from eight-coordinated crystal data, the decrease

is 18.3 pm [138, page 573]. The contraction stems from the incomplete shielding

by the filling 4 / subsheU of the increasing nuclear charge. The similar chemistry of

the lanthanides is rationalized by the nature of the 4 / orbitals and the lanthanide

contraction. It also has a significant effect on the chemistry of the latter elements

as well; for example, the Ad and 5d transition metals are anomalously comparable in

size. The usual explanation is the lanthanide contraction. However, special relativity,

in terms of all three effects above but usually the direct and indirect, plays a role as

demonstrated by Bagus, Lee, and Pitzer [144] using a novel approach wherein calcula­

tions are performed on pseudo-atoms. A 5d pseudo-atom is a fictitious atom in which

the 4/^^ subshell has been removed from its real counterpart and the nuclear charge

reduced by 14. This enables at least an approximate differentiation of lanthanide

contraction and relativistic effects, which may not necessarily be additive. Based on

SCF orbital energies the two causes have equal effects. From SCF radial expectation

44 values either effect may dominate depending on the particular 5d element. Other

work, described in various sources [24,105,138,145], ascribes various fractions of im­

portance to the lanthanide contraction cause and the relativistic cause depending on

what element is under consideration and which property is being measured. The

pseudo-atom technique has been applied to the 3d contraction [146] and the actinide

contraction [147]. The actinide contraction proper is 30 pm [138, page 576]. Seth

et al. [147] have shown that there is a delicate balance between the three relativistic

effects and the shell structure contraction effect, although for the lanthanides and

the 5d transition metals the shell structure contraction usually dominates and for the

actinides and the 6d transition metals the relativistic effects usually dominate. Given

the recent fashion to calculate chemical properties of superheavy elements [148—150]

no doubt the 5g contraction wiU be studied soon.

3.2.3 Relativistic Effective Potentials

Valence electrons determine chemistry. This fundamental concept underlies the periodic table of the elements and the idea of pseudopotentials — which later de­ veloped into effective potentials (EPs), model potentials, effective core potentials

(ECPs), and relativistic effective potentials (REPs). Model potentials were initially empirically adjusted, but in work by Bonifacio and Huzinaga have been based on ab initio calculations. Model potentials are not discussed below; they are reviewed by [151]. Immediately following is historical and introductory material based on the text by Szasz [152] and the review article by Krauss and Stevens [153], which contain references to the original works that are omitted herein. Next highlights from the

45 initial, pioneering relativistic ECP method of K. S. Pitzer and coworkers are pre­

sented. Subsequently their improved, currently in use, shape-consistent REP method

is described.

3.2.3.1 Historical Development

With the valence electron concept and the computational advantage of radically reducing the number of electrons in systems containing atoms as motivation, HeU- mann in 1935 showed that the requirement that a valence orbital be orthogonal to a core orbital, which is a consequence of the Pauli exclusion principle, could be met by a modification of the Hamiltonian. The altered Hamiltonian contains a pseudopotential to replace the orthogonality constraint. For a one valence electron atom Hellmann’s time-independent Schrodinger equation is:

Ip = E'lp 2 r T where A and k are such that the potential energy curve has the classic shape. The solutions, "0, are pseudoorbitals. They have no nodes that correspond to core orbitals because of the pseudopotential that keeps the valence electron out of the core. In particular, the lowest energy pseudoorbital is nodeless, but, in principle, otherwise similar to the true valence orbital. Assuming that the addition of a second valence electron does not change the core orbitals, i.e., the frozen core approximation, Hell- mann was able to build up to a many-valence-electron atom.

In 1940 Fock and coworkers justified the frozen core approximation for HF wave­ functions. If a Hartree-Fock wavefunction is a product of core and valence compo­ nents:

^ = A{

46 where A is the antisymmetrizer, {0,- : i = 1,. - -, c} is the set of core orbitals, and is the valence wavefunction representing v electrons, then the energy can be reduced to a sum of core and valence energies:

-E' = Ecore -h ^^Val

If the core orbitals are known then the valence only Hamiltonian:

^Vai = ~ ^ ^ ^Core = ^ ^ Cf 4" Vcore i= l \ ^ y i = l where U is the two-electron Coulomb repulsion or its one-electron approximation and

Vcore is the core-valence interaction, can be solved variationaUy in the Schrodinger equation, Hvai^Vai = Evai^vai- The solution requires that ^vai is orthogonal to each core orbital and that Vcore can be evaluated at each iteration. Thus given an invariant core wavefunction a valence wavefunction can be determined.

In 1959 Phillips and Kleinman provided a rigorous foundation for Hellmann’s pseudopotential approach by deriving, for the case of a single valence electron, an exact pseudopotential, for inclusion in the Hamiltonian, that replaces the core-valence orthogonality requirement. With voL =

H v a l< f> V a l = E v a l< f> V a l becomes

[H voI + Ppic) Xval = EvalXVal where the Phillips-Kleinman pseudopotential is

^PK = y^jEvai — E^\(f>i){(f)i\ i = l

47 the pseudoorbital is

C XVVaL + ^ (3-9) £=L

Hvaii for 2 = 1 to c, and the coefficients avai and a, are arbitrary except for

normalization. The pseudoorbital, Xvai^ is not necessarily orthogonal to the core. In

1968 Weeks and Rice generalized the Phillips-Kleinman derivation for many valence

electrons.

Significant simplification and computational savings can be achieved, if the core-

valence interaction, Vcore, and the core-valence orthogonality replacement, VpK, are combined into a single operator, Vpp. This effective potential (EP) keeps the valence electrons out of the core, without explicit orthogonalization, and reproduces the actual core potential felt by the valence electrons. In 1968 Goddard proposed a general method for creating EPs. For the single valence electron case, using the independent- particle model Goddard transformed the valence Schrodinger equation to:

(h + Vkp,i) XVal^i = EiXVal,i where

KEP,(r) +

The Xvai,i are non-orthogonal, smooth, and nodeless pseudoorbitals that provide a unique definition of the local, angular momentum dependent (1-dependent) EP. In

1974 Mehus and Goddard showed that HF orbitals could be used in place of the

Xvai,i to generate the EPs. Because the localization of the EP freezes in valence orbital exchange interactions, Goddard’s local EP may lack the flexibility needed for use in molecular calculations [153, pages 364—365].

48 In 1976 Kahn, Baybutt, and Truhlar [154] presented a popular method for devel­ oping ab initio EPs from many valence electron numerical HF atomic wavefunctions.

Using a modified valence Hamiltonian similar to that of Goddard but with an ef­ fective nuclear charge, Z^ff — Z — 2c, and a valence potential, Vvai-, evaluated over pseudoorbitals, instead of the original valence orbitals, their EP, called an effective core potential (ECP), is:

E [+

Each valence orbital of a given angular momentum. I, generates a unique ECP. If an

I core orbital exists then the ECP is repulsive to mimic orthogonality. If no I core orbital exists then the ECP is attractive to reproduce the unshielded effective nuclear charge. The ECP contains core-valence exchange interactions. The total atomic ECP is a sum over angular momenta:

yECP ^ (lm)(lm| 1=0 m=—l

Let L be one greater than the highest I quantum number of the core electrons. Because core-valence exchange is not significantly I dependent when I is greater than or equal to L, for I > L are approximately equal [154, pages 3830-3832]. Thus, in this approximation:

y EC P _ lA/CPjr) + £ \im) {lm\ (3.10) 1=0 m=—l

is called the residual potential. Note that ECPs are nonlocal, but that the nonlocality is restricted to the angular momentum dependence [154, pages 3830—

3832]. Their Phillips-Kleinman pseudoorbitals, see Equation 3.9, are determined via three auxihary conditions: pseudoorbitals are radially nodeless, pseudoorbitals are

49 smooth functions, and pseudoorbitals reproduce the original HF valence orbitals in

the valence region. The first condition is necessary to obtain the lowest energy eigen­

functions; the second is necessary for basis set economy; and the third is necessary

to produce a reasonable ECP. Algorithmically, the Kahn, Baybutt, and Truhlar pro­

cedure is (1) calculate the HF orbitals numerically, (2) transform the HF orbitals

into pseudoorbitals, and (3) determine the ECPs firom the pseudoorbitals using the

modified valence Hamiltonian.

In 1977 Lee, Ermler, and Pitzer [155] modified the ECP methods of Melius and

Goddard and of Kahn, Baybutt, and Truhlar [154] to generate ab initio effective core

potentials which include relativistic effects via the Dirac-Coulomb Hamiltonian from

numerical DHF calculations (REPs). In their procedure four-component valence or­

bitals are transformed into two-component valence pseudoorbitals through neglect of

the small component. Their Phillips-Kleinman pseudoorbitals have the same auxil­

iary conditions as those of Kahn, Baybutt, and Truhlar (note the proviso that the

pseudoorbitals reproduce the DHF valence orbitals). The details of the determination

of the coefficients in Equation 3.9, however, differ [155, page 5866]. For the case of

one valence electron, their modified valence Hamiltonian is:

where includes the valence relativistic interactions as well as the usual ECP

terms (the core-valence interactions, exchange and otherwise, and the core-valence

orthogonality replacement, V p k ). As indicated above, for all the EP methods, the

many valence electron case includes expressions for the Coulomb and the exchange interactions of the valence electrons; details of which are given in the original pub­

lication and elsewhere [114,135,155]. The REPs obtained from the two-component

50 Schrodinger equation, îtvaixïj^^ = are I and j dependent:

- W

The total atomic REP is an infinite sum over angular momenta. Using the approxi­ mation &om the ECPs, Equation 3.10, namely that for L equal to one greater than the largest I quantum number of the core electrons, the for I > L are equal, L-l C+l/2 + E E W - Off Wj E lijmWiM (3.11) f= 0 j=\l—l/2{ m=—j

For REPs this approximation also entails that relativistic effects are approximately equal for electrons in orbitals with I > L. This is reasonable given that these highly excited electrons spend less time near the nucleus and bom out by tests on Ne, Xe, and An [155, pages 5868-5869]. Direct application of requires two-component programs and j —j coupled basis sets.

To make use of ubiquitous one-component programs and prosaic l—s coupled basis sets, the j dependence of is averaged out to yield a spin-independent averaged relativistic effective potential (AREP) [156]:

-k ^ ^ \lm) {lm\ (3.12) 1=0 m = —l where the weighted average is:

^ + (i + l)£l,:gf/2(r)] (3.13)

The AREP, published in 1978, contains all the relativistic effects included by the

Dirac-Coulomb Hamiltonian except the SO coupling. A spin-orbit operator is im­ plicitly defined by and [157]:

Hso = (3.14)

51 L-1 (+1/2

(=0 j=|f-l/2(

m = —J As I increases the difference between the REPs for j = Z — 1/2 and j = 1 + 1/2

decreases rapidly [155, pages 5868-5869]. Thus Z/£f^(r) — U ^ ^ ^ (r) can usually be

neglected. Substituting U/^^^{r) from Equation 3.13:

Hso = (3.15)

(=1 1 + 1 1 1 ~ 0TZ\ S KX - m=-i+l/2 ^ ^ where

AC/;^^(r) = [Z]^f}2(r) - C/gf^zCr)

This form of the SO operator, published in 1981 by Ermler, Lee, Christiansen, and

Pitzer [157], enabled the first ab initio Cl nonperturbative treatment of SO coupfing.

(Hafner, Schwarz, and coworkers [158,159] had defined a similar SO operator and applied it in SCF calculations.) In a procedure published in 1982 by Christiansen,

Balasubramanian, and Pitzer and applied to TLH [130], atomic orbital (AO) integrals, including AREP and SO, are computed; a one-component SCF calculation produces molecular orbitals (MOs) for a transformation of the non-SO AO integrals into MO integrals (this avoids complex arithmetic in the transformation); the SO matrix ele­ ments, which may be complex, and the usual real Cl matrix elements are evaluated and combined; the resulting SOCI matrix is diagonalized using iterative techniques, which require trivial modifications to accommodate complex arithmetic, in partic­ ular, the Davidson method [160]. Because Hso, Equation 3.15, is a one electron

52 operator, the number of complex SOCI matrix elements is relatively small. This pro­

cedure handles SO coupling and electron correlation equitably. In practice, complex

arithmetic introduces only minor complications and SOCI complex matrix diagonal-

ization requires about twice the time for that of a real Cl matrix. The method has

been applied extensively, principally by Balasubramanian and Pitzer, to primarily

heavy p-block element containing diatomics [161—167] and third row transition metal

diatomics [168,169]. Early [134] and later [136] work has been reviewed.

Published in 1988 by R. M. Pitzer and Winter [26] (but formulated as early as

1984 by R. M. Pitzer [135, page 419]) an alternative equivalent form of Hso is:

H s o = s - E ^ (3.16) 1=1 - I - i m ' = - l m = - l In this form the SO integrals can be stored as commonplace spatial functions as

opposed to the spinor format required by the Hso of Equation 3.15. Techniques

to compute these SO integrals [170] and the AREP integrals [170,171] have been

implemented, in 1983 by R. M. Pitzer, in the ARGOS [30] program of the COLUM­

BUS suite of programs [29]. In conjunction with symmetry restrictions to guarantee

pure real or pure imaginary SO matrix elements and with methods to ensure real arithmetic, as published by R. M. Pitzer and Winter in 1991 [170] but developed as early as 1984 [135, page 430], the procedure, using program CIDBG [35], out­ lined at the start of Chapter 2 has been used extensively, principally by R. M. Pitzer and coworkers on primarily polyatomics: actinocenes [27], fuUerenes [172], transition metal systems [173—175], and alkali ion—rare gas dimers [176]. The significant realiza­ tion of the method of R. M. Pitzer and Winter is that, while a transformation from four-components to two-components loses information, a transformation from two- components to one does not. The choice between two- and one-component methods

53 is then pragmatic. If the highest affordable method is SCF then for heavy element containing systems a j - j coupled SCF might be a wise choice. However, in the usual situation, where the more difficult problem of electron correlation will be treated, one-component methods are the obvious choice for many reasons, most importantly economy. Incidentally, program CIDBG iteratively diagonalizes the real SOCI matrix via the block method of optimal relaxation (MGR) [177,178].

The SOCI method of Christiansen, Balasubramanian, and Pitzer [130] was gener­ alized for polyatomics by Balasubramanian in 1988 [131]. The procedure starts with

ARGOS to evaluate integrals; multiconfigurational SCF (MCSCF, see Chapter 4) of the complete active space variety (CASSCF) generates orbitals for use in large scale one-component Cl calculations; integrals are transformed over the resulting natural orbitals and used in a two-component Cl based on the SO operators of REPs; the latter SOCI step is analogous to that of R. M. Pitzer’s CIDBG, but retains the com­ plex arithmetic of the original diatomic program. Balasubramanian and coworkers have applied the method extensively [137,141,179,180].

In 1979 Christiansen, Lee, and K. S. Pitzer [181] made an important contribu­ tion to EP development. They showed that Phillips-Kleinman pseudoorbitals, see

Equation 3.9, may transfer electron density from the valence to the core region, re­ sulting in molecular potential energy curves that are too low at short intemuclear distances. This in turn leads to abnormally large dissociation energies and anoma­ lously short equilibrium bond lengths as obtained by several workers. They proposed a pseudoorbital of the form:

XVal = (f>Val + fval (3.17)

54 where fvaiir) is zero in the valence region, r > Tmatch, and cancels the radial oscilla­ tions in the core region, r < rmatch.- Thus they explicitly guarantee that pseudoorbitals reproduce the original HF valence orbitals in the valence region, replacing Kahn, Bay­ butt, and Truhlar’s third auxiliary condition with the objective of close agreement with all electron molecular calculations. The determination of fvaiif) and rmatch are described in the original work [181, page 4447] and revised as in [182, page 5161]

— essentially Xvaiif) is constrained to be smooth in the core region by equating up to five derivatives of (j>vai{f’) and fva[{f) at rmatch- Pseudoorbitals of this form had been proposed earlier [181, references 19 and 20] and are called norm conserving or shape consistent. Improved ab initio ECPs based on these pseudoorbitals were tested on Fa, CI2 , and LiCl [181], Ara, Ar^, Kra, Kr^, Xea, and XeJ [183], and the static electric dipole polarizabihties of Rb and Cs [184]. Agreement with experiment was significantly ameliorated. Initial application to the generation of REPs was for Tl, which were used in the first ab initio MCSCF a/-w coupling, i.e., two-component, calculations [182,185,186]. All relevant publications of K. S. Pitzer and coworkers after 1980 incorporated this improvement. Starting in 1985 Christiansen, Ermler, and coworkers developed ab initio AREPs, Equation 3.12, and SO operators. Equa­ tion 3.15, using the relativistic formalism of Lee, Ermler, and K. S.Pitzer [155] and the shape consistent pseudoorbitals of Christiansen, Lee, and K. S. Pitzer [181]. Minor modifications to the shape consistent method were employed to obtain compact ana­ lytic potentials [187]. The L — 1 summation upper bounds in Equations 3.15 and 3.16 were incremented to L to increase their accuracy for light elements. To facilitate use with omnipresent generic integral programs the potentials are fit to Gaussian

55 expansions [154,155,187]:

U ^ ^ i r ) = r"2 Ç Car"" exp(-Oir^) (3.18) i

The corresponding l SO operator:

U f ° i x ) = (3-19)

is fit to a Gaussian expansion using exponential factors, Çu, determined by the AREP

fit. AU known elements have been tabulated: atomic numbers from 3 to 18 [187],

from 19 to 36 [188], from 37 to 54 [189], from 55 to 57 and from 72 to 86 [190—192],

from 58 to 71 [193], from 87 to 94 [2], and from 95 to 118 [194]. Ccurefiil attention

was paid to core size for the non / elements; two REPs are presented for the p and

d block elements. Because of the known importance of treating core-valence corre­

lation for the alkaliand alkaline earth elements [195], only onecore size is provided

— the (n — l)s(n — l)p subshells are in the valence space. For the 3d transition metals, interconfiguration excitations, that is excitations from the lowest energy level of one electron configuration to the lowest energy level of another configuration, using large core REPs, 3d4s valence space were in error by upto 0.4 eV as compared to all-electron Cowan-Grifl&n calculations [196]. A significant part of the error stemmed from core-valence partitioning of the 4s pseudoorbital; moving the match point closer to the nucleus allowed the 4s pseudo orbital to properly relax upon excitation of an electron from that orbital. For the sixth-row p block elements recent calculations [192] have shown that accuracy requires including the 5d^° electrons in the valence space.

Furthermore, core-valence partitioning in the creation of the pseudoorbitals may be

56 critical to bond length predictions. In particular, a radial inflection in the large com­

ponent of the 5 / 5 / 2 spinors for the sixth-row p block elements lead to pseudospinor-

spinor discrepancies out to three bohr, clearly in the valence region. Early work on

core-valence polarization and correlation [195] has been continued [197].

3.2.3.2 Selected Results

The initial test systems of Lee, Ermler, and Pitzer [155] were xenon and gold.

Xe HF excitation energies are within —1.6 % to 2.5 % relative error after an ad­

justment for electron correlation [155, pages 5871-5872]. The adjustment is not de­

scribed, so its accuracy is not clear. Core size was and is an enigma. Five core sizes

were studied for Au: [Kr]4d^°, [Kr]4d^°4/^'*, [Kr]4d^°4/^^5s^, [Kr]4d^°4/^'^5s^5p®, and

[Kr]4d^°4/^^5s^5p®5d^° with the number of valence electrons being 33, 19, 17, 11,

and 1 [155, pages 5872-5874]. Based on orbital energies, radial expectation values

(both (r) and and the outermost node of the 6s orbital, it was concluded

that the 4 / and 5s could reasonably be in the core, that the 5p could somewhat

marginally be in the core, and that the 5d and 6s should clearly be in the valence

region. Au HF excitation energies are within —23 % to 24 % relative error.

A later HF study [156], the second paper in a series of five, on Xe 2 , XeJ, and

Xe^ indicated that the scalar relativistic effects from the mass-velocity and Darwin

terms are not important for spectroscopic properties. Furthermore, inclusion of SO

coupling effects by an empirical procedure, where experimental SO splittings of the

atoms are used to determine the SO matrix elements, is reasonable because of the

lack of significant configuration mixing. Scalar relativistic effects were adequately

treated by ECPs, with regard to potential energy curves and the resulting spectro­ scopic properties, for the ground and excited states of Au 2 from HF, MCSCF, and

57 CI calculations with. SO coupling included by the same empirical procedure in the

fourth paper [198].

Core sizes for T1 and Pb were examined in paper five [199]. The same two core

sizes were studied for each: [Xe]4/’-'^ and [Xe]4/^‘^5d^° with valence electrons num­

bering 13 and 3. Using the averaged ECPs, one-component SCF ionization energies

of T1 have relative errors of —11.0 % and —12.4 %, respectively; excitation energies

of Pb do not change with core size and have relative errors of —38.8 % for ^P <—

and —7.3 % for <— ^S; ionization energies of Pb have relative errors of —16.4 %

and —16.8 %, respectively. Thus, the electrons are not too important in these

processes. However, core size does have a substantial eSect on the equilibrium bond

length and dissociation energy of TIH. In two-component SCF calculations in­

creases 0.05 A with inclusion of the electrons in the valence space, bringing it

closer to experiment. In nonrelativistic SCF calculations the increase is 0.06 A but

agreement is worse because Re is longer than experiment. Later MRSOCISD cal­

culations with a shape-consistent 13 valence electron REP and a double zeta basis

set for T1 with three references, 6s^6p^, and the electrons firozen gave a ^P SO

splitting of 0.92 eV and a relative error of 5.2 % [130]. In similar calculations with

shape consistent 14 valence electron REPs and a double zeta basis sets for Sn and

Pb with excitations from the ns^np^ reference spaces, ^P^ and ^P2 excitation energies

are 1970 cm~^ and 3950 cm“^ for Sn and 7725 cm"^ and 11740 cm~^ for Pb; relative errors are 16.4 %, 15.2 %, —1.2 %, and 10.2 %, respectively.

58 3.2.4 Spin—Orbit Coupling

Condoa and Shortley [200] is the classic work on atomic systems containing the complete theory including SO coupling- Slater [201] is an updated account of the subject. Several reviews concentrating on molecular systems treated perturbatively employing all-electron ab initio wavefimctions are available. Langhoff and Kem [115] examine molecular fine structure by applying the Breit-Pauli Hamiltonian. Richards.

Trivedi, and Cooper [116] is an introductory graduate level text on the subject. Hefi,

Marian and Peyerimhoff [107] examine the calculation of both SO coupling and elec­ tron correlation using a Douglas-KroU derived, variationally stable, one- and two- electron SO Hamiltonian in perturbative and variation-perturbative methods.

SO coupling in systems containing only light elements is adequately treated per­ turbatively via empirical, semi-empirical, or ab initio methods. Applications to di­ atomic molecules, for example, abound [202-205]. Stevens and Krauss were the first to use the REP SO operator [157] perturbatively. Calculations on carbon, silicon, and some diatomics containing those elements and oxygen and hydrogen indicated that the SO operators were transferable between valencies and accurate except for lacking the Breit corrections [206,207]. In heavy element containing systems, however, the

SO interaction is as significant (sometimes more so) than the interelectron repulsion interaction and cannot be satisfactorily accounted for via perturbation theory. The lanthanides are generally too heavy. Calculated atomic SO coupling constants using

HF wavefunctions are consistently greater than experimental values by as much as

20 % for lanthanides but are within ±5 % for 4d transition metals [208].

The emphasis of K. S. Pitzer and coworkers was on molecular methods. However, in all the REP development papers except the last two, atomic SO splittings were

59 calculated [2,187—190,193]. For various neutral atoms and cations three SCF results

are compared: (AREP) first order Hsoi Equation 3.15, corrections using the HF

numerical AREP, Gaussian basis set wavefunctions, (REP) energy difference between

the appropriate J states from two-component numerical REP Gaussian basis set HF

calculations, and (DF) numerical Dirac-Fock, excluding Breit corrections, with the

appropriate core orbitals frozen. The REP splittings are consistently within 10 % of

the DF values except for some light elements where basis set augmentation improves

the results to within 10 % and for the lanthanides where the relative difference is as

large as 53 % and is probably due to the unusually large core size (valence space is

4/5d6s). This indicates that the REP formalism reliably reproduces the relativistic

effects of the core orbitals on the valence orbitals. Comparison of the AREP and

R E P splittings gauge the relativistic orbital relaxation; this is a sensitive function of element and orbital — percent relaxations ranging from 0 to 126 %. Comparison of SCF sphttings to experimental values yields relative errors of about ± 5 0 %. An

important point that is always stated is that the errors due to the REP approximations are smaller than the errors due to neglect of relativity in aU-electron calculations even for the lightest elements. In 1986 Ross, Ermler, and Christiansen [209] explicitly studied SO coupling in the Group 13 and 17 atoms, using the methods above as well as including the Breit corrections in DF calculations and nonrelativistic HF splittings. With better than triple zeta basis sets the REP-DF error is within 4 % and the AREP-DF within 7 %. For REPs where the residual SO operator, that for the maximum I, L'm Equation 3.15, is responsible for the splitting, errors are larger because of the difficulty of representing the l/r^ behavior of that operator in both the SO expansion fitting and the orbital Gaussian basis set. In general, for heavy

60 elements, perturbation theory using nonrelativistic wavefunctions was inadequate,

perturbation theory using AREP wavefunctions was adequate, DF was within 5 % of

experiment, and for nonfirst-row elements the Breit correction contributes about 5 %

of DF.

Excitation energies from the lowest energy level of one electron conhguration to

the lowest energy level of another configuration have been studied for transition met­

als [188-190], lanthanides [210], and actinides [211]. Relative errors range from 0 %

to 100 %; absolute errors range from 0 eV to 1.5 eV. In the lanthanide and actinide

studies pseudopotential and density functional calculations had sim ilar errors. Ab ini­

tio computation including electron correlation of all atomic excitation energies within

a configuration containing more than two energy levels has only been performed by

a few workers. Jankowski and Sokolowski [212] studied electron correlation in the

4/^ configuration of Fr^"*" using Cl on multiconfigurational HF orbitals; however,

SO coupling was not treated. Cai, Umar, and Froese Fischer [213] performed Cl on

multiconfigurational DF wavefunctions for Pr^'*' o b ta in in g agreement with experiment

within ten relative percent except for one level, ^Ig- Rakowitz, Casarrubios, Seijo, and

Marian [214] used the spin-free-state-shifting technique, wherein a one-component

multireference CISD is followed by a SOCl containing an operator to shift the spin- free MRCISD states to their appropriate energies (as determined either empirically or by the MRCISD), on Ir"^ yielding agreement usually within ten relative percent of experiment. (Other recent work, on sixth-row p block hydrides [215], has also shown that SO coupling and electron correlation energies are separable.) Sanoyama,

Kobayashi, and Yabushita [216] studied the transition energies of Pr^'*', Pm^^, and

Tm^"^ with MRSOCISD. By comparison with other groups REPs [217,218], the 4 /

61 only valence space of Ermler and coworkers [193] was found to be a major source of

the up to 28 % relative errors obtained. The best results are by Kaldor and coworkers

using a no-pair Dirac-Coulomb—Breit Hamiltonian, correlated via a coupled-cluster

singles and doubles method with large basis sets [126, references therein]. Results for

are in excellent agreement with experiment: 114 cm~^ average error, within 2 %

relative error except for the ^Sq level which has a 5.6 % relative error [122]. Slightly

worse errors were found for Pr^"*'. They found that inclusion of the Breit interaction

leads to significant improvements.

3.3 Methods

The Ermler [2] uranium REP, henceforth denoted E78, has a 78 electron core of

[Kr]4d^°4/'^‘‘5s^5p®5d^° leaving 14 valence electrons in the neutral: 6s^6p®5/^6d^7s^.

It was developed using the cation. Its SO operator was fit to a Gaussian ex­ pansion, Equation 3.18, using exponential factors, Qi, determined by the AREP fit.

P. A. Christiansen created four new U ECPs of core sizes 78, 68, 62, and 60, hence­ forth denoted: C78, 068, C62, and C60. The valence space of C78 equals that of

E78; the valence space of C68 contains the 5d^° electrons as well; C62 contains the

5p® electrons; and C60 the 5s^ electrons. They were also developed using the cation. Their SO operators were fit to a Gaussian expansion. Equation 3.18, using exponential factors, Cu, determined to best fit each particular SO operator.

Existing cc-pVDZ basis sets for the E78 REP were produced by Z. Zhang and

R. M. Pitzer [71,219]. cc-pVDZ basis sets for the new U REPs and a cc-pVTZ basis set for the E78 REP were developed, Figures 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7. The

(12s lip lOd 8f lg)/[12s lip lOd 8f Ig] basis set is from the Stuttgart group [3],

62 Figure 3.8. The (9sd 7p 8f 2g lh)/[9sd 7p 8f 2g Ih] set. Figure 3.9 is based on the

C68 cc-pVDZ basis; the polarization functions are from E78 cc-pVTZ. All basis sets were created using the methods described in Chapter 4. The valence space of these basis sets is 6s, 6p, 5 /, and 6d. SubsheUs previously in the 78 electron REPs are not in the valence space of the basis set, and are only single-zeta. To facilitate comparison of results from different REPs and to simplify basis set construction, uncontracted basis sets were employed in calculational series which attempted to reach the basis set limit. For the uranyl calculations an oxygen REP with a Is^ core [187] and a cc-pVDZ basis set [219] as discussed in Chapter 4, Figure 3.10, were employed.

AJl SOCI calculations were performed using the procedure described in Chapters 1 and 2. Multireference calculations in Section 3.4.1 include the complete 5/^ reference space: Q) -h = 28 spatial (one-dimensional) configurations.

3.4 Results and Discussion 3.4.1 Uranium V

The experimental values in this section are from Wyart, Kaufinan, and Sugar [220] and Van Deurzen, Rajnak, and Conway [221]. The former observed and identified aU but the ^Sq level of uranium V, from sliding spark spectra. The latter measured the ^Sq level after a semi-empirical calculation narrowed down its position.

Table 3.2 displays all 13 energy levels, 91 wavefunctions, from the 5/^ open-shell electron configuration of These SOMRCI singles and doubles (SOMRCISD) calculations with cc-pVDZ basis sets are all of the same size, 126 588 configurations and 971 733 double-group-adapted functions, because the n = 5 subshells not in the

REP for C68, C62, and C60 were frozen in the CL In general, four calculations are

63 The Uranium +2 Ion Core pVDZ Correlation Consistent Set

Basis Set: (4sd4p4flg)/[3sd2p2f1gJ

Core Potential: W.C. Ermler. R.B. Ross, & P.A. Christiansen, Ini J. Quantum Chem. 40, 829-846 (1991).

State: 6s(2)6p(6)6d(2)5f(2), av. of conRg.

Hartree-Fbck Calculations Total HF Energy -49.85442599

(HF+1+2) Calculations (J=4) Total Cl Energy HF Coefficient Spin-orfoit Splitting -xxjoooooooc Ojoooooooc xjoooooooc

4 3 3 / INo. primitives, PQN, No. contracted fhs (U sd set) 2.168000 -0.1289505 -0.0195499 0.0 1.009000 0.7955080 -0.0090364 0.0 0.402500 0.3649706 0.5279641 0.0 0.139800 0.0020985 0.5899125 1.0 4 2 2 / !No. primitives, PQN, No. contracted fhs (U p set) 6.728000 -0.0033035 0.0 1.419000 -0.3142991 0.0 0.619900 0.7755420 0.0 0244500 0.4902717 1.0 4 4 2 / INo. primitives, PQN, No. contracted fhs (U f set) 4.436000 0.1957684 0.0 1.860000 0.4559656 0.0 0.755200 0.4265113 0.0 0277000 0.1970811 1.0 1 5 1 / INo. primitives, PQN, No. contracted fns (U g set) 1.690000 1.0000000 I I The atomic set was oijtained by optimizing the (p,sd.f)-exponents in I Hartree-Fock calculations on the 6d(2)5f(2) av. of config. The I polarization set was obtained by optimizing the exponent of a single I g primitive in HF+1 +2 calculations on the lowest state of this I configuration. I December1997

Figure 3.1: E78 cc-pVDZ Basis Set

64 The Uranium Atom Cora pVDZ Corralation Consistent Set

Basis Set: (5sd6p4f1g)/[4sd4p2f1g]

Core Potential: W.C. Ermler. R.B. Ross, & P.A. Christiansen. Int J. Quantum Chem. 40,829-846 (1991).

State: 6s(2)6p(6)6d(2)5f(2), av. of config.

Hartree-Fock Calculations Total HF Energy -49.85550776 (uncontracted) -49 joooooooc (contracted)

5 3 4 / INo. primitives. PQN, No. contracted fhs (U sd set) 2.168 -0.1287677 0.0389513 -0.0170584 0.0 1.009 0.7990894 -0.3648756 -0.0085024 0.0 0.4025 0.3613013 0.0173240 0.4548192 0.0 0.1398 0.0005438 0.6476391 0.5427471 1.0 0.04898 0.0030768 0.4543083 0.1932255 0.0 2 4 / INo. primitives, PQN, No. contracted fhs (U p set) 6.728 -0.0032503 0.0014651 0.0 0.0 1.419 -0.3132348 0.0875586 0.0 0.0 0.6199 0.7729305 -0.2529905 0.0 0.0 0.2445 0.4835471 -0.1615202 1.0 0.0 0.05899 0.0197168 0S393910 0.0 0.0 0.02096 -0.0022506 0.5898553 0.0 1.0 4 2 / INo. primitives, PQN. No. contracted fhs (U f set) 4.436 0.1946438 0.0 1.860 0.4550805 0.0 0.7552 0.4246053 0.0 0.2770 0.2029756 1.0 1 5 1 / INo. primitives, PQN, No. contracted fhs (U g set) 1.690 1.0000000 I I The atomic set was obtained by optimizing the (sd,p,f)-exponents in I Hartree-Fock calculatidns on the 6d(2)5f(2) av. of config., with the I idea of obtaining an f basis d ose to optimum for an f(2) U(IV) or I higher oxidation-state ion. Trying to add another sd primitive and I trying to separate the sd basis sets in this caiculatidn both led to I intractable exponent-collapse problems. The fifth sd primitive was I chosen to optimze the energy of 5f(2)6d(2)7s(2) av. of config. The I outer two p primitives were chosen to optimize the energy of 5f(2) 16d(2)7s(1 )7p(1 ) av. of config. The polarization set was obtained by I optimizing the exponent of a single g primitive in HF+1+2 I calculations on the lowest state of 5f(2)6d(2), allowing for I conelation of the 5f shell only. I 1...... November 1997

Figure 3.2: U E78 cc-pVDZ Atom Basis Set

65 The Uranium-I-2 Ion Core Potentiel pVOZ Corralation Consistent S et

Basis Set: (4sd4p4flg)/[3sd2p2f1gi

Core Potential: 78 electron core. P. A. Christiansen Unpublished, Wed Dec 16 16:13 EST 1998.

State: 6s(2)6p(6}5f(2)6d(2), av. of config.

Hartree-Fock Calculations Total HF Energy State -49.77930313 6s(2)6p(6)5f(2)6d(2), av. of config.

(HF+1+2) Calculations (J = 6 ground state) Total Cl Energy HF Coefficient Description 5f(2)6d(2) Ref.. 5f to 5g

4 3 3 / INo. primitives. PQN. No. contracted fhs (U sd set) 2.274000 -0.1308037 -0.0180790 0.0 1.011000 0.7861748 -0.0153744 0.0 0.405900 0.3711698 0.5295897 0.0 0.140700 0.0031309 0.5915341 1.0 4 2 2 / INo. primitives. PON. No. contracted fhs (U p set) 8.241000 -0.0012864 0.0 1.411000 -0.3380539 0.0 0.632800 0.7809159 0.0 0.247300 0.5034316 1.0 4 4 2 / INo. primitives, PQN, No. contracted fhs (U f set) 6.493000 0.0952453 0.0 2.443000 0.4183949 0.0 0.955000 0.4939684 0.0 0.331700 0.2704160 1.0 1 5 1 / INo. primitives. PQN. No. contracted fhs (U g set) 1.690000 1.0000000 I I The atomic se t was obtained by optimizing the (4sd,4p.4f)-exponents In I Hartree-Fock calculations on the 5f(2)6d(2) av. of config.. with the I idea of obtaning a d and an f basis dose to optimum for a U(li) I or higher oxidation-state ion. I The first and second sd. the first p. and the first f contractions, I which represent the 6s. 6d. 6p. and 5f orbitals respectively. I were obtained from a single Hartree-Fock calculation on the above I optimized exponents. I The third sd, second p. and second f contractions, which represent the 16d, 6p, and 5f orbitals respectively, were obtained by freeing the most I diffuse exponent of the respective symmetry. I Thus, the valence space of 6d, 6p, and 5f is double zeta and I the core space of 6s Is single zeta. I The polarization set was usurped from an earlier U liasis set. I December 1998

Figure 3.3: C78 cc-pVDZ Basis Set

66 ! The Uranium-t-2 Ion Cora PotontialpVDZConolatlonConsistem Set 1 ! Basis Set: (5sd4p4f1g)/[4sd2p2f1gl ! I C ore Potential: ! 68 electron core. ! P. A. Christiansen ! Unpublished. Mon Nov 9 13:50 EST 1998.

State: 5d(10)6s(2)6p{6)5f(2)6d(2), av. ofconHg.

Hartree-Pbdc Calculations Total HF Energy State -214.13612675 5a(10)6s(2)6p(6)5f(2)6d(2). av. of config.

(HF+1+2) Calculations (J = 6 ground state) Total Cl Energy HF Coefffcfent Description -214.30206505 0.998474 5f(2)6d(2) Ref.. 5f to 5g

5 3 4 /INo. . primitives, PON. No. contracted fhs (U sd set) 16.040000 -0.0220342 0.0024798 0.0067835 0.0 2.688000 0.7135959 -0.1074646 -0.3001008 0.0 1.071000 0.3561567 0.6544832 -0.0538307 0.0 0.467800 0.0172843 0.4660201 0.4690028 0.0 0.155600 0.0052717 0.0179823 0.6717449 1.0 4 2 2 /INo.. primitives. PON. No. contracted fhs (U pset) 7.579000 -0.0022712 0.0 1.391000 -0.3322258 0.0 0.623700 0.7830963 0.0 0 2 45300 0.4975686 1.0 4 4 2 / INo. primitives. PON. No. contracted fhs (U f set) 6.539000 0.0944153 0.0 2.443000 0.4174352 0.0 0.952000 0.4949070 0.0 0.331400 02717946 1.0 1 5 1 / !No. primitives. PQN, No. contracted fhs (U g set) 1.645000 1.0 ! I The atomic set was obtained by optimizing the (5sd.4p,4f)-exponents in ! Hartree-Fbck calculations on the 5f(2)6d(2) av. of config.. with the ! idea of obtaining a d and an f basis dose to optimum for a U(ll) I or higher oxidation-state ion. 1 An additional f. p. or sd primitive lowered the energy by 0.0023,0.0015, I or 0.0117. respectively, resulting in exponent-collapse forp orsd. I The first, second, and third sd. the first p. and the first f contractions. I which represent the 5d, 6s. 6d, 6p, and 5f orixtals respectively. I were obtained from a single Hartree-Fock calculation on the above optimized ! exponents. I The fourth sd. second p. and second f contractions, which represent the 16d, 6p, and 5f orbitals respectively, were obtained by freeing the most I diffuse exponent of the respective symmetry. ! Thus, the valence space of 6d, 6p, and 5f is double zeta and I the core space of 5d and 6s is single zeta. I The polarization set was obtained by optimizing the exponent of ! a single g primitive in HF+1 +2 calculations on the lowest state of 15f(2)6d(2), allowing for correlation of the 5f shell only. I ! November 1998

Figure 3.4: C68 cc-pVDZ Basis Set

67 Tiw Uranfum * 2 km Com Potontial pVDZ Correiation Consistent Set

Bas» Set: (6sdSp4flg)/[4sd3p2f1gl

Core Potential: 6 2 electron c o re . P . A. C tiristiansen Unpublistied, Mon Feb 15 11:35 EST 1999.

State: 5p(6)5d{10)6s(2)6p(6)5f(2)6d(2). av. of config.

Hartree-Fbck Calculations Total HF Energy State -394.30215631 5p(6)5d(10)6s(2)6p(6)5f(2)6d(2), av. of config.

6 3 4 / INo. primitives, PQN, No. contracted fns (U sd set) 11.560000 -0.0546415 0.0013659 0.0186980 0.0 3.032000 0.5790943 -0.1258706 -0.2450997 0.0 1.463000 0.3994789 0.1713492 -0.1279283 0.0 0.820900 0.1155683 0.7083269 0.1082917 0.0 0.346800 0.0195902 0.2618109 0.5515968 0.0 0.126700 -0.0017781 -0.0009742 0.5108995 1.0 5 2 3 / INo. primitives. PQN. No. contracted fhs (U p set) 7.617000 -0.3434578 0.2044875 0.0 3.171000 0.7908208 -0.6438652 0.0 1.530000 0.4452836 -0.1251881 0.0 0.545200 0.0320231 0.7751910 0.0 0.225800 -0.0030549 0.3972886 1.0 4 4 2 / INo. primitives. PQN. No. contracted fhs (U f set) 4.724000 0.1638100 0.0 1.993000 0.4399246 0.0 0.802300 0.4498583 0.0 0.288500 0.2240230 1.0 1 5 1 / INo. primitives. PQN. No. contracted fhs (U g set) 1.690000 1.0 I 1 The atomic set was obtained by opta'miang the (6sd,5p.4f)-exponents in I Hartree-Fock calculations on the 5f(2)6d(2) av. of config.. with the I idea of obtaining a d and an f basis dose to optimum for a U(ll) I or higher oxidation-state ion. I The orbital energies are: I 6 s 5p 6p 5d 6d 5f I -2.67 -9.32 -1.66 -4.78 -0.681 -0.938 I An additional p primitive lowered the energy by 0.0213. I resulting in exponent-collapse. I The first, second, and third sd. the first and second p. I and the first f contractions, I which represent the 5d, 6s. 6d. 5p. 6p. and 5f orbitals respectiveiy. I were obtained from a single Hartree-Fock calculation on the above I optimized exponents. I The fourth sd, third p. and second f contractions, which represent the 16d. 6p. and 5f orbitals respectively, were obtained by freeing the most I diffuse exponent of the respective symmetry. I Thus, the valence space of 6d. 6p, and 5f is double zeta and I the core space of 5p. 5d and 6s is single zeta. I The polarization set was usurped from an earlier U basis s e t I

Figure 3.5: C62 cc-pVDZ Basis Set

68 The Uranium 4^2 Ion Cora Potential pVOZConeletlon Consistent Set

Basis S e t C7sd5p4f1g)/[5sd3p2f1g]

Core Potential: 60 electron core. P. A. Christiansen Unpublished, Mon Feb 15 11.-35 EST 1999.

State: 5s(2)5p(6)5d(10)6s(2)6p(6)5f(2)6d(2), av. of config.

Hartree-Fock Calculations Total HF Energy State -472.96690642 5s(2)5p(6)5d(10)6s(2)6p(6)5f(2)6d(2), av. of config.

7 3 5 / INo. primitives. PQN, No. contracted fhs (U sd set) 12.170000 -0.1672025 -0.0440428 0.1056774 0.0180599 0.0 5.764000 0.4245170 -0.0551898 -0.3800659 0.0084656 0.0 3.352000 0.5827382 0.5320343 -0.2855268 -0.2069748 0.0 1.614000 0.1590751 0.4774210 02685059 -0.1816746 0.0 0.777800 -0.0053722 0.1373928 0.7149104 0.1533703 0.0 0.319100 0.0027233 0.0097956 02111060 0.5601716 0.0 0.120800 -0.0007071 -0.0002898 -0.0055511 0.4670242 1.0 5 2 3 / INo. primitives, PON, No. contracted fhs (U p set) 7.651000 -0.3204507 0.1798474 0.0 2845000 0.9575219 -0.7207867 0.0 1.125000 02976132 0.0521028 0.0 0.485900 -0.0210693 0.7903827 0.0 0205700 0.0071736 0.3116052 1.0 4 4 2 / INo. primitives, PQN, No. contracted fhs (U f set) 4.720000 0.1642768 0.0 1.989000 0.4402481 0.0 0.799900 0.4498254 0.0 0287000 02239136 1.0 1 5 1 / INo. primitives, PQN, No. contracted fhs (U g set) 1.690000 1.0 I I The atomic set was obtained by optimizing the (7sd,5p,4f)-exponents in I Hartree-Fock calculations on the 5f(2)6d(2) av. of config., with the I idea of obtaining a d and an f basis dose to optimum fora U(ll) I or higher oxidation-state ion. I The orbital energies are: I 5s 6s 5p 6p 5d 6d 5f I -1325 -2.74 -9.32 -1.66 -4.78 -0.681 -0.936 I An additional sd primitive lowered the energy by 0.00246, I resulting in exponent-collapse. I The first, second, third, and fourth sd, the first and second p, I and the first f contractions, I which represent the 5s, 5d, 6s, 6d, 5p, 6p, and 5f orbitals respectively, I were obtained from a single Hartree-Fock cdculation on the above optimized I exponents. I The fifth sd, third p, and second f contractions, which represent the 16d, 6p, and SforlAals respectively, were obtained by freeing the most I diffuse exponent of the respective symmetry. I Thus, the valence space of 6d, 6p, and 5f is double zeta and I the core space of 5s, 5p, 5d and 6s is single zeta. I The polarization set was usurped from an earlier U basis set. I July 1999

Figure 3.6: C60 cc-pVDZ Basis Set

69 Uranium Ion E78 cc>pVTZ

4 3 2 /Usd(4sd7pf2d8f2g1h)/[2sd3p2d4f2g1hl 2.168000 -0.1289505 -0.0195499 1.009000 0.7955080 -0.0090364 0.402500 0.3649706 0.5279641 0.139800 0.0020985 0.5899125 5 4 1 /Up(4sd7ped8f2g1h)/[2scl3p2d4eg1h] 3.225000 0.0122353 1.022000 0.5305372 0.505600 0.4559062 0.254700 0.1110168 0.086710 0.0010245 1 4 1 /Up(4sd7p(2d8eg1h)42sd3p2d4Qg1hI 0.483000 1.0 1 4 1 /Up(4sd7pf2d8f2g1h)/I2sd3p2d4eg1h] 1.131000 1.0 1 3 1 /Ud(4sd7ped8«2g1h)/[2sd3p2d4<2g1h] 0.540000 1.0 1 3 1 /Ud(4sd7pQd8eg1h)/[2sd3p2d4f2g1hI 0.810000 1.0 1 4 1 /Uf(4sd7ped8f2g1h)/[2sd3p2d4eg1h] 0.614000 1.0 5 4 1 / U f (4sd7pl2d8eg1 h)/[2sd3p2d4Kg1 h] 4.528000 0.1823236 1.983000 0.4125621 0.891100 0.3902867 0.397800 0.2300015 0.169600 0.0671855 1 4 1 /Uf(4sd7pf2d8f2g1h)/[2sd3p2d4fêg1h] 0.367000 1.0 1 4 1 /Uf(4sd7pI2d8f2g1h)/[2sd3p2d4eg1h] 1.190000 1.0 1 5 1 /U g (4sd7pBd8Kg1h)/[2sd3p2d412g1h] 1.050000 1.0 1 5 1 /Ug(4sd7pt2d8f2g1h)/[2sd3p2d4fêg1h] 3.330000 1.0 1 6 1 /Uh(4sd7pt2d8f2g1hV[2sd3p2d4f2g1hI 1.728000 1.0

Figure 3.7: E78 cc-pVTZ Basis Set

70 12 112 /U s StuttgartECP60MWB 1534.933600 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 227.748380 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 30.696831 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18.170626 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10.813537 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.733298 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.431498 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.615298 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.286639 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.071170 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.030539 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.005000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 11 2 1 1 /U p StuttgartECP60MW B 553.345250 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 109.255010 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 23.476030 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.794472 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.432319 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 2.702169 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.493857 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.792817 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.351542 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.143962 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.005000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 10 3 10 /U d StuttgartECP60MWB 81.202858 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 18.325575 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10.454699 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.666312 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.923349 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.989638 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.495346 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.204455 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.073273 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.005000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 8 4 8 /U f StuttgartECP60MWB 55.334525 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 16.588649 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.757518 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.387550 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.130195 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.489535 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.181420 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.005000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1 5 1 /Ug(6sd7p4flg)/[4sd7p2f1g] zz,rmp 1.690000 1.0

Figure 3.8: U Uncontracted Stuttgart [3] Basis Set

71 9 3 9 /Usd(9sd7p8f2g1h)/[9sd7p8f2g1h]srb,nnp 12.170000 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.764000 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.352000 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.614000 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.777800 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.319100 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.120800 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.060000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.030000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 7 2 7 /Up(9sd7p8fôg1h)/I9sd7p8f2g1h]sfb,nnp 7.651000 1.0 0.0 0.0 0.0 0.0 0.0 0.0 2.845000 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.125000 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.485900 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.205700 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0 .1 0 0 0 0 0 0 .0 0 .0 0 .0 0.0 0 .0 1.0 0 .0 0.050000 0.0 0.0 0.0 0.0 0.0 0.0 1.0 8 4 8 / U f (9sd7p8f2g1h)/l9sd7p8f2g1h]srb,nnp 22.360000 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.368000 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.310000 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.682000 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.857000 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.430100 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.214000 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.104500 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1 5 1 /U g (9scf7p8fâg1h)/[9scl7p8f2g1hl ccoptfor5f^25g 1.060000 1.0000000 1 5 1 /U g (9sd7p8f2g1h)/[9sd7p8f2g1h] ccoptfor5f^25g 3.350000 1.0000000 1 6 1 /U h (9sd7p8f2g1h)/[9sd7p8f2g1h] ccoptfbr5f''26h 1.726000 1.0000000

Figure 3.9: U Uncontracted 9sd 7p 8f 2g Ih Basis Set

72 The Oxygen Atom Cora pVOZComlatron Consistent Set

Basis Set: (4s4p1d)/[2s2p1d]

Core Potential: L.F. Paoos & PJL Christiansen. J . C hem . Phys- 8 2 ,266 4 (1985).

State: 2s(2)2p(4).(3)P

Hartree-Fock Calculations Total HF Energy -15.65532836 (uncontracted) -15.65487713 (contracted)

(HF+1+2) Calculations (J=2,0) Total Cl Energy Lead Coefficient Spin-orbit Splitting -15.78244491 0.98433572 0.00094250

4 1 2 / INo. primitives, PQN, No. contracted fns (O s set) 41.04 -0.0097241 0.0222003 7.161 -0.1318703 0.1265661 0.9074 0.5903463 -1.6261307 0.2807 0.5169632 1.5531546 4 2 2 / INo. primitives, PQN, No. contracted fhs (O p set) 17.72 0.0433004 -0.0559967 3.857 0.2330835 -0.4246360 1.046 0.5017961 -0.5598365 0.2752 0.4652332 0.9990806 1 3 1 / INo. primitives, PQN, No. contracted fhs (O d set) 1.213 1.000000 I I The atomic set was obtained by optimizing the (s,p)-exponents in I Hartree-Fock calculations on the ground, (3)P, state. The I contraction coefficients were obtained from the natural orbitals of I an HF+1 +2 calculation on this state. The polarization set was I obtained by optimizing the exponent of a single d primitive in HF+1 +2 I calculations on this state. I August 1997

Figure 3.10: O cc-pVDZ Basis Set

73 Step User CPU System CPU Wail Time Iterations

CIDBG, hmatrix > 999999 sec — sec 23.5 days —

CIDBG, diagon. 114864 10981 7-0 8 6

GUGA, diagon. 110692 5821 2 . 2 276

Table 3.1: SOMRCISD/cc-pVDZ Program Timings for 29 Ag Roots

necessary to obtain all the levels; one each for Ag, Big, B2g, and B^g, the four £>2 A

irreps which the spherical symmetry decomposes into. However, the Big irreps are

degenerate. All but the ^Fi level can be obtained from the Ag calculation. The GUGA

programs were used for all but one calculation, performed with CIDBG. Table 3.1 compares the performance of the respective programs on a Sun Ultra 1 200 MHz,

256 Mb machine. Unfortunately the calculations displayed are atypical because 29 roots were converged — the 5/^ space spans only 28 Ag roots. The extra root takes an inordinate number of iterations to converge for both programs. Furthermore, the comparison is unfair because for the 29th root the GUGA convergence criterion was

10“^ hartree, but for CIDBG the convergence criterion was 10~^ hartree.

The ^G^ level is out of sequence for all the REPs. The absolute errors are as large as 7500 cm"*^. Table 3.3 shows the relative percent errors of the data in Table 3.2 for easy comparison. Considering all REPs the relative errors range from —15.7 % to +23.0 %. E78 and C78 relative errors differ by from 1.0 % to 4.2 %, the C78 values always shifting to lower excitations energies, which means worse agreement with experiment. The C 6 8 energy levels are all shifted up by from 5.5 % to 15.3 % in relation to the C78 values. This results in good agreement with experiment. Were it not for the ’■Jg and ^So levels the agreement would be excellent. The C62 excitations

74 E78 C78 C6 8 062 060 Exp

0 cm“^ 0 0 0 0 0 "Fz 3966 3923 4153 5116 5019 4161 5174 4917 5854 6948 6734 6137 7850 7581 8605 10407 10132 8984 'F4 8336 8037 9372 10748 10483 9434 % 9862 9401 11150 13101 12717 11514 'Ü2 15477 15162 16579 19756 19356 16465 'G4 14515 13922 16417 18862 18363 16656 "Po 16498 16313 17478 20697 20301 17128 'Pi 18727 18385 19978 23838 23308 19819 'Is 22354 21837 24509 27017 26509 22276 'P2 22771 22162 24619 29197 28520 24653 'So 43928 42698 47848 51186 50436 43614

Table 3.2: 0^^+ SOMRCISD/cc-pVDZ Energy Levels

energies are again all shifted up, relative to C 6 8 , by from 7.7 % to 23.2 %. This results in the worst agreement with experiment. The C60 levels are shifted down, relative to C62, by a small amount, from 1.8 % to 3.5 %. The consistent direction of shift in moving between REPS suggests one underlying cause; the variation in amount of shift indicates additional, perturbative, forces. The disagreement between E78 and

C78 is noteworthy and obviously points to the REPs as a determining factor.

Basis set quality is important in these types of calculations. Table 3.4 contains

SOMRCISD calculations with E78 using two larger basis sets. cc-pVDZatom mcludes

75 E78 C78 C6 8 C62 C60 Exp

'H4 0 % 0 0 0 0 0

-4.7 -5.7 -0 . 2 23.0 2 0 . 6 4161 -15.7 -19.9 -4.6 13.2 9.7 6137

"F3 - 1 2 . 6 -15.6 -4.2 15.8 1 2 . 8 8984

"F4 - 1 1 . 6 -14.8 -0.7 13.9 1 1 . 1 9434 -14.3 -18.4 -3.2 13.8 10.4 11514

'D 2 -6 . 0 -7.9 0.7 2 0 . 0 17.6 16465

*^G4 -12.9 -16.4 -1.4 13.2 1 0 . 2 16656

"Po -3.7 -4.8 2 . 0 2 0 . 8 18.5 17128

"Pi -5.5 -7.2 0 . 8 20.3 17.6 19819

'le 0.4 -2 . 0 1 0 . 0 21.3 19.0 22276

"Pz -7.6 - 1 0 . 1 -0 . 1 18.4 15.7 24653

'So 0.7 -2 . 1 9.7 17.4 15.6 43614

Table 3.3: SOMRCTSD/cc-pVDZ Energy Level Relative Errors

76 the 7s and 7p subshells in the valence space of the basis set; thus, they are each rep­ resented by two contractions. cc-pVTZ has the usual smaller valence space. The development of this triple zeta basis set is described in Chapter 4. The cc-pVDZatom energy levels are lower than the cc-pVDZ levels merely by from 10 cm“ ^ to 50 cm~^

This is expected because the 7s and 7p subshells are vacant and too diffuse to signifi­ cantly correlate occupied orbitals. The triple zeta basis is a substantial improvement as is easily seen in Table 3.5. The relative errors are now all under —10 %. Because of the difficulties of developing basis sets, Chapter 4, the expense of cc-pVTZ calcula­ tions, and the belief that all of the error in the SONIRCISD/cc-pVDZ treatment can not be attributed to the basis sets, the study proceeded, under the assumption that the cc-pVDZ basis sets for the five REPs are of equal quality.

Correlation treatment is important in these types of calculations. Table 3.6 con­ tains SOMRCIS calculations with all the REPs using their respective cc-pVDZ basis sets. The ^Sq level is particularly sensitive to dynamical correlation. Table 3.7 displays the relative errors. The trends are almost identical to those for the SOMRCISD/cc- pVDZ calculations. Table 3.8 shows the effect of increasing the reference space and the basis set for the SOMRCIS treatment. Including the 35 5f^6d^ references, in addition to the 5/^, has a significant effect on the ^So level due to dynamical cor­ relation; 5f^6d^ states are of the wrong parity to mix with 5/^. Including the 231

6p^5p references, in addition to the 5/^, affects all levels similarly and ameliorates the agreement with experiment. The triple zeta basis set, with the 5/^ reference space, significantly improves the ^Hs, ^Fa, and energy levels, but worsens the ^Sq somewhat, as is shown in Table 3.9.

77 cc-p\T)Z cc-pVDZatom cc-pVTZ Exp

" H 4 0 cm“^ 0 0 0

" F 2 3966 3954 3975 4161 5174 5169 5576 6137

" F s 7850 7833 8243 8984

" F 4 8336 8314 8569 9434

" H e 9862 9850 10538 11514

' Ü 2 15477 15435 15616 16465

' G 4 14515 14477 15095 16656 "Po 16498 16471 16741 17128 "Pi 18727 18695 19143 19819 'le 22354 22340 22076 22276

"P2 22771 22725 23343 24653 'So 43928 43893 43488 43614

Table 3.4: U'*’^ E78 SOMRCISD Energy Levels

78 cc-pVDZ cc-pVDZatom cc-pVTZ Exp 'H, 0% 0 0 0 -4-7 -5.0 -4.5 4161 -15.7 -15.8 -9.1 6137 "Fa -12.6 -12.8 -8.2 8984

"F4 -11.6 -11.9 -9.2 9434 "He -14.3 -14.5 -8.5 11514

'D2 -6.0 -6.3 -5.2 16465

'G4 -12.9 -13.1 -9.4 16656 "Po -3.7 -3.8 -2.3 17128 "Pi -5.5 -5.7 -3.4 19819 % 0.4 0.3 -0.9 22276

"P2 -7.6 -7.8 -5.3 24653 'So 0.7 0.6 -0.3 43614

Table 3.5: E78 SOMRCISD Energy Level Relative Errors

79 E78 C78 C 6 8 C62 C60 Exp

'H 4 0 0 0 0 0 0

"F2 4124 4081 4602 5416 5322 4161

"H5 5018 4762 5160 6056 5801 6137

"F3 7836 7571 8434 9979 9679 8984

"F4 8890 8604 9518 10674 10398 9434 "He 9709 9245 9997 11636 11179 11514 16012 15727 17413 20387 19981 16465

'G 4 15355 14797 16206 18074 17531 16656 "Po 17190 16989 18785 22148 21783 17128 "Pi 19010 18682 20672 24385 23901 19819 % 23250 22749 23288 25552 25038 22276

"P2 23107 22522 24907 29124 28433 24653 'So 54910 53959 58616 60213 59485 43614

Table 3.6: SOMRCIS/cc-pVDZ Energy Levels

80 E78 C78 C 68 C62 C60 Exp 'Ht 0 % 0 0 0 0 0 -0.9 -1.9 10.6 30.2 27.9 4161 'Hs -18.2 -22.4 -15.9 -1.3 -5.5 6137 'F3 -12.8 -15.7 -6.1 11.1 7.7 8984 'F4 -5.8 - 8.8 0.9 13.1 10.2 9434 'He -15.7 -19.7 -13.2 1.1 -2.9 11514 -2.8 -4.5 5.8 23.8 21.4 16465 'G4 -7.8 - 11.2 -2.7 8.5 5.3 16656 'Po 0.4 -0.8 9.7 29.3 27.2 17128 'Pi -4.1 -5.7 4.3 23.0 20.6 19819 % 4.4 2.1 4.5 14.7 12.4 22276 'P 2 -6.3 -8.6 1.0 18.1 15.3 24653 'So 25.9 23-7 34.4 38.1 36.4 43614

Table 3.7: SOMRCIS/cc-pVDZ Energy Level Relative Errors

81 5/2 5/2, 5f^6d^ 5/2, 6p"5/3 pVTZ Exp

"H4 0 0 0 0 0 "Fg 4115 4075 4192 4337 4161 % 5013 5164 5064 5585 6137 "Fa 7822 7916 7916 8532 8984

"F4 8867 8653 8842 9414 9434 "He 9697 9927 9780 10730 11514

'D 2 15971 15967 16158 16944 16465

'G4 15307 14952 15263 16269 16656 "Po 17178 17055 17225 18539 17128 "Pi 18994 19081 19101 19819 % 23243 23377 23612 23941 22276

"P2 23069 23201 23252 24708 24653 'So 54876 48408 52739 56610 43614

Table 3.8: SOMRCIS/cc-pVDZatom Energy Levels

82 5 f 5 / 2 , 5 f W 5/2, 6p^5/^ pVTZ Exp

"H 4 0 cm"^ 0 0 0 0 -1.1 -2.1 0.7 4.2 4161

'H s -18.3 -15.9 -17.5 -9.0 6137

"F 3 -12.9 -11.9 -11.9 -5.0 8984 -6.0 -8.3 -6.3 -0.2 9434

"He -15.8 -13.8 -15.1 -6.8 11514 -3.0 -3.0 -1.9 2.9 16465 *^G4 -8.1 -10.2 -8.4 -2.3 16656 "Po 0.3 -0.4 0.6 8.2 17128 "Pi -4.2 -3.7 -3.6 19819

He 4.3 4.9 6.0 7.5 22276

"P2 -6.4 -5.9 -5.7 0.2 24653

'S o 25.8 11.0 20.9 29.8 43614

Table 3.9: SOMRCIS/cc-pVDZatom Energy Level Relative Errors

83 To examine only the AREPs non-SO calculations were performed. Tables 3.10 and 3.11 show MRCISD and MRCIS terms for E78, C78, C68 with the electrons frozen and the 9s virtual orbital frozen (the latter to maintain consistent calculation sizes; the basis sets have a Cartesian sd space), and C68 with excitations from the

subshell. The MRCISD information indicates that (1) the E78 and C78 AREPs are similar, (2) the subshell is represented differently by the REPs and a single zeta basis set at the SCF level, (3) core—valence correlation is not large. The latter may be substantially because the cc-pVDZ basis set is only single zeta for the

subsheU and thus inadequate for correlating the subsheU. Regarding point

(2) the two representations of the 5d^° subshell are not so different. The MRCIS results are similar, but the '^So level is askew.

Many calculations were performed where SO operator(s) from REP X were used with REP Y. The validity of mixing and matching AREPs and SO operators has been questioned. Swapping SO operators between REPs with the same core size would seem to be the least suspect. However, all calculations lead to the same con­ clusion: all SO operators, for a particular I, are different. Because these calcula­ tions lend themselves to confusing presentations, only one is given. Table 3.12 shows

SOMRCIS/cc-pVDZ with the E78 AREP energy levels for all E78 SO operators, all

C68 SO operators, E78 p SO operator only (d, / , g SO operators from C68), E78 d

SO operator only (p, / , g SO operators from C68), E78 / SO operator only (p, d, g SO operators from C68), and E78 g SO operator only (p, d, f SO operators from

C68), respectively.

To study the substantial change of the REPs on removing the 5p® subshell from the core, as opposed to the smaller change in the 78 electron cores when the

84 E78 078 068 5d Froze 068 G cm"’^ 0 0 0 3F 2679 2669 2758 3072 4589 4494 5452 5508 10406 10323 10888 11735 3p 13597 13518 14192 15110 T 15990 15811 17312 17171 36233 35479 39309 40585 Relative Energies 0 cm ^ 0 0 0 3p -79 -89 0 314 :G -863 -958 0 56 -482 -565 0 847 3p -595 -674 0 918 T -1322 -1501 0 -141 -3076 -3830 0 1276

Table 3.10: U'^'^ ^IRCISD/cc-pVDZ Energy Levels

85 E78 078 068 5rf Froze 068 3H 0 0 0 0 3F 2743 2760 2903 3187 6398 6325 6516 7004 :D 11126 11143 11435 12330 3p 13694 13743 14529 15127 T 16945 16824 17170 16734 47915 47397 47990 51487 Relative Energies 0 cm ^ 0 0 0 3F -160 -143 0 284 -118 -191 0 488 -309 -292 0 895 3p -835 -786 0 598 T -225 -346 0 -436 -75 -593 0 3497

Table 3.11: MRCIS/cc-pVDZ Energy Levels

86 AUE78 A1IC68 E78 p E78 d E78 / E78p Exp

% 0 cm~^ 0 0 0 0 0 0 "Fz 4124 4256 4269 4257 4110 4256 4161 5018 6076 6093 6073 5003 6076 6137 "Fa 7836 8882 8911 8879 7809 8881 8984

"F4 8890 9950 9974 9946 8869 9950 9434 % 9709 11574 11606 11570 9680 11573 11514 'Da 16012 16937 16998 16931 15958 16937 16465

'G4 15355 17460 17496 17456 15324 17460 16656 'Po 17190 17661 17733 17660 17118 17661 17128

"Pi 19010 20013 20090 2 0 0 1 0 18937 20013 19819 'le 23250 24600 24648 24596 23206 24600 22276

"P2 23107 25121 25206 25119 23024 25121 24653 'So 54910 56480 56526 56474 54869 56480 43614

Table 3.12: U'*"*" SOMRCIS/cc-pVDZ E78 AREP Energy Levels with SO Operators from C6 8

87 subshell is removed, an attempt was made to reach the basis set limit. Energies with the C62 REP from SCF, MRCIS, and SOMRCIS calculations on the ground state of are shown in Table 3.13. Most of the basis sets are modifications in the p set of the C62 cc-pVDZ basis. Figure 3.5. [4sd 4p 2f Ig] has as its fourth p contraction the third p primitive, 1.530. [4sd 6p 2f Igj has as its sixth p contraction an additional primitive of 2.300. [4sd 7p 2f Ig} has as its sixth p contraction an additional primitive of 3.971. [4sd lip 2f Ig] uses the Stuttgart p set. The maximum variation in the energies is: -0.029, -0.030, and -0.408, respectively. The huge energy decrease in the

SO calculations is caused by the SO splitting of the 5p subshell. Tables 3.14 and 3.15 display the MRCIS terms using the various basis sets. The improvement in the p set does not change the term energies; however, the (12s lip lOd 8f Ig)/[12s lip lOd 8f Ig] basis set yields some significant changes because of the representation of the other symmetries. Table 3.16 shows the SOMRCIS levels using the various basis sets. As expected the representation of the 5p® subshell affects the excitation energies.

Table 3.17 shows that the relative errors only change by approximately four percent, usually resulting in better agreement with experiment.

3.4.2 Uranyl Cation

Chapter 5 presents background material on UOg Table 3.18 shows uranyl bond lengths using the five uranium REPs. For the 78 and 68 REPs the basis set is converged. For the 62 and 60 REPs the calculations are not at the basis set limit. As is well known the SO interaction does not contribute to the bond length, as comparison of the SCF and SOCIS results indicates, de Jong bond lengths are from triple zeta quality basis sets. No doubt these are not at the basis set limit. The C68

88 Basis Set Rel. En. Energy 5/^ Average of States SCF

4sd 3p 2f Ig 0 -392-638943 4sd 4p 2f Ig -0.000174 -392.639117

9sd 7p 8 f 2g Ih 4-0.011951 -392.626992 4sd 5p 2f Ig -0.000938 -392.639881

4sd 6 p 2f Ig -0.002431 -392.641374 4sd 7p 2f Ig -0.009095 -392.648038 4sd lip 2f Ig -0-029380 -392.668323

12s lip lOd 8 f Ig -0.021655 -392.660598 5 / ^ CIS

4sd 3p 2f Ig 0 -392.746022 4sd 4p 2f Ig -0.000219 -392.746241

9sd 7p 8 f 2g Ih -0.008637 -392.754659 4sd 5p 2f Ig -0.001129 -392.747151

4sd 6 p 2f Ig 4sd 7p 2f Ig -0.009322 -392.755344 4sd lip 2f Ig -0.029578 -392.775600

12s lip lOd 8 f Ig -0.024910 -392.770932

H4 SOCIS

4sd 3p 2f Ig 0 -392.951027 4sd 4p 2f Ig -0.000162 -392.951189

9sd 7p 8 f 2g Ih -0.223996 -393.175023 4sd 5p 2f Ig -0.272930 -393.223957

4sd 6 p 2f Ig -0.333059 -393.284086 4sd 7p 2f Ig -0.384078 -393.335105 4sd lip 2f Ig -0.393956 -393.344983

12s lip lOd 8 f Ig -0.407803 -393.358830

Table 3.13: U^'*' Energies from SOMRCIS with 062 AREP and SO Operators.

89 Term Calc^ Error Exp“ (6sd 5p 4f lg)/[4sd 3p 2f Ig] 0 cm ^ 0% 0 3p 3751 19.5 3139 6733 34.9 4993 13980 27.0 11018 17661 26.8 13948 3? 17501 19.4 14651 ^S 51710 44.1 35903 (6sd 5p 4f lg)/[4sd 4p 2f Ig] 2R 0 cm ^ 0 % 0 3p 3748 19.5 3139 6725 34.9 4993 13965 27.0 11018 n 17660 26.8 13948 3? 17490 19.4 14651 'S 51696 44.1 35903 (6sd 5p 4f lg)/[4sd 5p 2f Ig] 3R 0 cm ^ 0 % 0 3p 3750 19.5 3139 6736 34.9 4993 13988 27.0 11018 H 17686 26.8 13948 3? 17498 19.4 14651 ^S 51737 44.1 35903

Table 3.14: Terms from MRCIS with. 062 AREP and SO Operators and Small Basis Sets.

90 Term Calc^ Error Exp“ (6sd 7p 4f lg)/[4sd 7p 2f Ig] 0 cm ^ 0 % 0 3p 3750 19.5 3139 "G 6738 34.9 4993 13990 27.0 11018 17688 26.8 13948 3p 17496 19.4 14651 :S 51740 44.1 35903 (6sd lip 4f lg)/[4sd lip 2f Ig] 0 cm ^ 0 % 0 3p 3750 19.5 3139 'G 6736 34-9 4993 13989 27.0 11018 H 17686 26.8 13948 3p 17498 19.4 14651 ^S 51734 44-1 35903 (12s lip lOd 8f lg)/[12s lip lOd 8f Ig] 0 cm ^ 0 % 0 3p 3580 14-0 3139 :G 6787 35.9 4993 13504 22.6 11018 H 17203 23.3 13948 3p 16693 13.9 14651 :S 51008 42-1 35903

Table 3.15: Terms from MRCIS with 062 AREP and SO Operators and Large Basis Sets.

91 pVDZ pVDZ 4p pVDZ 5p pVDZ 7p llp Stuttgart Exp 0 0 0 0 0 0 0 "Fa 5416 5412 5532 5574 5426 5388 4161 "% 6056 6055 5812 5778 5591 5679 6137 "Fa 9979 9975 9870 9876 9583 9627 8984

"F4 10673 10669 10480 10459 10098 10303 9434 "Hs 11636 11634 11209 11149 10786 10965 11514 'Da 20387 20377 20643 20747 20159 20172 16465

'G 4 18074 18068 17605 17542 16935 17304 16656

"Po 22148 22138 22655 22812 22244 22056 17128

"Pi 24384 ----- 24129 19819

'le 25551 25550 25522 25560 24846 25017 22276 "Pa 29124 29113 29245 29346 28533 28532 24653

'So 60213 60202 60007 59993 57847 59184 43614

Table 3.16: SOMRCIS/cc-pVDZ Energy Levels with. C62 AREP and SO Oper­ ators.

92 pVDZ pVDZ 4p pVDZ 5p pVDZ 7p llp Stuttgart Exp "H4 0 cm ^ 0 0 0 0 0 0 "Fz 30.2 30.1 32.9 34.0 30.4 29-5 4161 -1.3 -1.3 -5.3 -5.8 -8.9 -7.5 6137 'Fa 11.1 11.0 9.9 9.9 6.7 7.2 8984 'F, 13.1 13.1 11.1 10.9 7.0 9.2 9434 'He 1.1 1.0 -2.6 -3.2 -6.3 -4.8 11514 'Da 23.8 23.8 25.4 26.0 22.4 22.5 16465 '04 8.5 8.5 5.7 5.3 1.7 3.9 16656 'Po 29.3 29.3 32.3 33.2 29.9 28.8 17128 'P i 23.0 21.7 19819 'le 14.7 14.7 14.6 14.7 11.5 12.3 22276 'P2 18.1 18.1 18.6 19.0 15.7 15.7 24653 'So 38.1 38.0 37.6 37.6 32.6 35.7 43614

Table 3.17: SOMRCIS/cc-pVDZ Energy Level Relative Errors with C62 AREP and SO Operators.

93 REP SCF SOCIS Active Space 6s6p5f 5d6s6p5f 5p5d6s6p5f E78 3.084 3.084 C78 3.091 3.091 C68 3.125 3.125 3.127 C62 3.142 3.144 3.114 3.110 C60 3.141 3.144 3.110 3.106 DHF [128, page 112] 3.120 CCSD(T) [128, page 112] 3.241

Table 3.18: UO 2 Bond Lengths in Bohr

REP gives the best bond length in comparison. The spread between the REPs values

is approximately 0.03 Â.

3.5 Conclusions and Future Work

The subshell of U is best included in the valence space. In lieu of larger

basis sets, better treatment of electron correlation, and more accurate DF uranyl

bond lengths, the present results support this conclusion. excitation energies are

unlikely to shift radically with an improved ab initio treatment. It is reasonable to

expect that excitation energies using other core sizes will shift comparably; the 68

REP energy levels will remain sandwiched between those of the 78 and 62 REPs, and

will probably remain in best agreement with experiment. Furthermore, substantial other REP and pseudopotential work indicates that for most accurate results the subshell should be in the valence space [3.188—190,192]. The uranyl results are not so comforting. A straightforward path to further study of the uranyl bond lengths covers

94 three points: (1) SCF basis set convergence, that is, for both DF and HF/AREP; this

should be easily attainable using numerical SCF programs; systematic basis set im­

provement for actinides is problematic. Chapter 4, and overdue; hopefiiUy, this can be

remedied. (2) A systematic analysis of core—valence correlation to elucidate the role,

if any, of the the 5p®, and the 5s^ subshells in the uranyl bond length, as in [192];

of course, this requires basis sets; otherwise however, because uranyl is a closed shell

system, the present programs, in particular SOGUGA, should be adequate. (3) Four-

and two-component calculations with and without frozen core orbitals to differentiate

relativistic and frozen core effects and to identify core-valence partitioning problems,

if any; the concern here is to pinpoint which REP approximations dominate as was

done in some of the early REP and shape-consistent work [181,196]. Other actinyls should be studied similarly; curium seems a logical choice because it is clearly more

like a lanthanide than an early actinide.

The 5p® subshell of U is probably best included in the REP. Here the tradeoff

between incorporation of relativistic effects, which stems from being in the REP, and inclusion m the valence space, which allows for polarization and treatment of correlation effects, falls on the side of relativity. The SO splitting of the subshell is substantial, yet it has little effect on excitation energies. Removal of the 5p® electrons from the REP, however, greatly shifts all the energy levels. The role of the

5p® subsheU in uranyl is unlikely to be significant.

95 C H A PT E R 4

Basis Set Development

4.1 Introduction

In this chapter relevant background material on correlation consistent basis sets is presented in Section 4.2. Feller and Davidson [222] have pragmatically considered

Gaussian basis sets for molecular calculations circa 1990. Shavitt [223] has reviewed the history of Gaussian basis sets. Helgaker and Taylor [224] have reviewed Gaussian basis sets and the evaluation of their integrals. Methodology is summarized in Sec­ tion 4.3. The development of basis sets for use with effective core potentials (ECPs) is described in Section 4.4. The results of a study of correlation energy and spin-orbit

(SO) energy on the ground state of are presented in Section 4.5 and applied to the construction of a uranium cc-pVTZ basis set. Conclusions and suggestions for future work are stated in Section 4.6.

4.2 Background

In electronic structure calculations the wavefunction is expanded in a set of one- electron functions. For both atomic and molecular calculations the set is devised so that the functions accurately represent the atomic orbitals of the constituent atoms.

96 The resulting basis sets are small, usually from one to a dozen functions per atom, because the computational effort is large. The number of integrals scales formally as the fourth power of the number of basis functions. In the last few decades Cartesian

Gaussian functions,

x“y*z‘^exp(—ar^), (4.1) where a, 6, and c are nonnegative integers and a is a positive real number, have been used almost exclusively because integrals over Gaussians can be rapidly evalu­ ated [223]. Gaussian functions are not ideal for the representation of atomic orbitals.

Consequently, many Gaussians are required to model orbitals. Because the compu­ tational effort of correlation methods is at best proportional to the fifth power of the number of basis functions, fixed linear combinations (contractions) of Cartesian

Gaussian functions (primitives) are the most commonly used type of basis set. Con­ tractions enable many primitives to represent one orbital while keeping the number of basis functions relatively small. A contraction is casually referred to as a basis function.

A substantial body of work exists on contracted Gaussian basis sets. Much of that work is aimed at first-row atoms. Modest numbers of primitives and contractions are requir ed to approach the Hartree-Fock (HF) limit [225], for example, (10s 6p) prim­ itives contracted to [5s 4p] and augmented with functions whose angular momentum quantum number, I, is greater than the maximum I of any occupied orbital of the ground state of the atom (polarization functions), which are 3d functions for first- row atoms [226]. Feller and Davidson [222] have discussed the sources of Gaussian primitives and contraction coefficients. Several methods are popular for determin­ ing the exponents in the primitives. Notable are even-tempered and well-tempered

97 approaches where all K exponents in a basis set fit a simple formula [227], such as,

Offc = st*. A: = 1,2,..., ÜT. (4.2)

The principal advantages are that only a small number of parameters, two in the example above, need to be optimized and that a systematic increétse in the basis set size is straightforward. Another approach is energy optimization where each exponent

is variationally optimized in an atomic HF calculation [228]. Table 3.13 shows that this method yields excellent energies. This is the approach used herein and is discussed below. Several methods are popular for determining the contraction coefficients of the primitives in a fixed linear combination. In the energy optimization method the coefficients of some contractions are determined simultaneously with the exponents.

Those are the contractions that represent occupied atomic orbitals. They are referred to as SCF contractions, and the collection of such contractions form a single-zeta basis set. Increasing the number of contractions which models a single atomic orbital increases the zeta number of that orbital so that a double-zeta (DZ) basis set has two contractions for each occupied atomic orbital. In the general contraction approach, where each primitive may contribute to all contractions, those additional contractions are frequently created by freeing the most diffuse primitives [229].

4.2.1 Correlation Consistent Basis Sets

The treatment of electron correlation is demanding on basis sets; in particular, unlike for the HF limit, large numbers of basis functions are required to approach the electron correlation limit [230]. Two methods are popular for determining basis sets for use in correlated calculations — both use the general contraction scheme.

98 Almlof and Taylor [231], noting that most exponent and contraction coefficient de­ terminations are based on HF optimization, studied contraction errors, that is, for a particular type of calculation the energy difference between the uncontracted result and the contracted result. For first- and second-row atoms using existing large primi­ tive sets for occupied HF orbitals and large even-tempered sequences for polarization functions, they showed that natural orbitals, eigenfunctions of the first-order den­ sity matrix [12, pages 252-257], of the configuration interaction (Cl) (usually singles and doubles, multireference where appropriate) atomic wavefunction (ANOs) define contraction coefficients which have minimal contraction errors and provide an excel­ lent basis for molecular correlated calculations. Surprisingly they also found that the number of contractions to reach the limit of zero contraction error is almost indepen­ dent of the number of primitives. Other workers have applied the AND method to systems involving excited states and ions [232].

The second popular method for determining basis sets for use in correlated cal­ culations is due to Dunning [1,233,234]. Studying valence correlation effects in the oxygen atom [1], he showed that (1) exponents from even-tempered sequences opti­ mized in atomic correlated calculations effectively and efficiently describe correlation effects using two-thirds and one-half the number of primitives as ANO basis sets for occupied and polarization functions respectively, (2) usually the most diflfuse primi­ tives of HF optimized basis sets adequately describe correlation effects, and (3) energy lowerings resulting from successive addition of functions of a given angular momentum separate into well-defined groups, in agreement with earlier work [230] on polariza­ tion functions, and decrease approximately geometrically. Table 4.1 displays some of Dunning’s results to clarify the last point. Starting from a large primitive set.

99 Set ACorr. Is 26.231 2s 4.311 Ip 57.874 2p 12.067 3p 3.582 Id 62.239 2d 14.717 3d 2.641 If 15.849 2f 3.443 Ig 3.658

Table 4.1: Oxygen 2s^2p* Correlation Energies in Millihartrees [1]

(16s 7p), contracted to minimal-zeta in the angular momentum under study, with the

Is^ core frozen, in the case of s functions the first s correlating contraction, i.e., DZ

total for the 2s orbital, lowers the correlation energy by 26.231 mE^. The first p and

first d correlating contractions lower the correlation energy similarly. The s space is anomalous. The second p, second d, and first / correlating functions each lower

the correlation energy approximately equally. Thus Dunning developed correlation consistent (cc) basis sets where primitives are used economically and functions are added by correlating group, e.g., cc-pVDZ=(9s 4p Id)/[3s 2p Id], cc-pVTZ=(10s 5p

2d If)/[4s 3p 2d If], cc-pVQZ=(12s 6p 3d 2f lg)/[5s 4p 3d 2f Ig], etc. The basis set space is partitioned into core and valence (V) regions. For oxygen the core is Is^ and is single-zeta (SZ); so that in the cc-pVDZ basis set the valence space of 2s^2p^ is DZ; p indicates that polarization functions are included. These basis sets display

100 systematic convergence which facilitates extrapolation to the basis set limit and are

quite popular [235].

4.2.2 Cl Basis Sets Requirements

Bartlett and Stanton [10] have discussed some basis set requirements of methods

for electron correlation; Shavitt [236] has too, focusing on Cl. Clearly the basis set

must be adequate for the HF calculation which produces the orbitals to be used in

the Cl expansion. Because integral evaluation dominates the HF computational effort

but is negligible in comparison to the Cl effort, large primitive sets are not prohibitive

in Cl calculations. Effective correlating orbitals have spatial extent similar to that

of reference orbitals, provide spatial flexibility, and have additional nodal surfaces.

In particular, polarization functions, being of high angular momenta, are important.

Thus basis sets for correlation must be of at least pVDZ quality. In opposition to

this, however, the number of virtual orbitals should not be too large if the calculation

is to be manageable; freezing virtuals can be a significant complication and potential

source of error.

Related to basis set selection are orbital transformation techniques [236]. A full

Cl wavefunction is invariant under a unitary transformation of the one-electron basis.

In truncated Cl the quality of the molecular orbitals (MOs) is a factor in the quality of the Cl wavefunction. Several methods are available for adjustment of the MOs.

Improved virtual orbitals (IVOs) are obtained using a modified Hamiltonian in a self- consistent field (SCF) procedure so that the unoccupied orbitals describe the same system as the occupied orbitals and not one containing an additional electron [11, pages 56-58]. The resulting virtual orbitals approximate excited state orbitals [237].

101 Note that a Hartree-Fock wavefunction is invariant under a transformation that does not mix the occupied and unoccupied orbitals. The multiconfiguration self-consistent field (MCSCF) method is a generalization of the HF method in which the wavefunc­ tion is a linear combination of Slater determinants, that is, the wavefunction is a, usually short, Cl expansion [238—240]. Both the expansion coeflficients and the or­ bitals are variationally optimized. Usually a small number of occupied and correlating orbitals of known importance in the final wavefunction, perhaps based on preliminary or cheap calculations, such as, all singles Cl (CIS), participate in the MCSCF calcu­ lation. This technique was used by Zhang and Pitzer [241] on the uranyl ion, UOg^, to rotate some occupied (Tu orbitals and to produce better tt^ orbitals for spin-orbit

Cl singles and doubles (SOCISD) calculations. A related operation is the generation of MOs suitable for representing many states. If several states are to be treated in an electron correlation calculation then an averaged energy expression for all those states can be used in SCF or MCSCF calculations to produce MOs that fairly represent each state. This type of generation of MOs in addition to a balanced reference space was applied in multireference SOCI (SOMRCI) calculations on the ground and excited states of actinyl ions (Chapter 5). Finally, natural orbitals give fast Cl convergence in some cases [242,243].

4.2.3 Effective Core Potential Basis Sets

Effective core potentials (ECPs) remove electrons from explicit consideration (Chap­ ter 3). Thus the number of atomic orbitals to be modeled and, consequently, the sizes of atomic basis sets are reduced. Indeed, the most grandiose view is for relativistic

102 effective potentials (REPs) that replace four-component basis sets with reduced one- component basis sets. In addition to this reduction in the number of basis functions, the atomic orbitals change form. In particular, valence orbitals are replaced by pseu­ doorbitals. Christiansen type pseudoorbitals are smooth and as small as possible in the core region [181]. When there is only one pseudoorbital for a particular sym­ metry then that pseudoorbital is radially nodeless. Additional pseudoorbitals within a particular symmetry have a radial node for each pseudoorbital of lower principal quantum number. (Note that, when more than one pseudoorbital exists for a particu­ lar symmetry, basis set nomenclature can become confusing because the valence space of the EOF can be partioned into core and valence parts vis-a-vis the basis set. For example, in the 78 electron REP of U, the valence space is 6s^ 6p® 5/^ 6d^ 7s^. In a cc-pVDZ basis set the 7s subshell may be in the valence space (V) but the 6s subsheU may not; the latter is then in the core space of the basis set.) These three properties, smoothness, small core magnitude, and decreased number of nodes, imply a reduction in the number of primitives necessary to create an adequate contraction. However, for s orbitals the smaU core magnitude of pseudoorbitals is unnatural (Section 4.4.1).

In general, basis sets provided with ECPs are usuaUy of at least valence DZ quality, foUow the general contraction scheme, and lack polarization functions [187,

244-246]. Wallace, Blaudeau, and Pitzer [247] using the energy optimization method produced generally contracted cc-pVDZ quality basis sets for first- and second-row atoms modeled with the ECPs of Christiansen, Ermler, and coworkers [187]. The major conclusion was that for each atomic symmetry with an occupied orbital in the ground state two primitives are suflScient to model the near core region.

103 4.3 Methods

The atomic SCF program ATMSCF [248] has been used to perform energy based exponent optimization. Documentation [249] and a detailed description [248] are

available. The program has an error-prone formatted input interface. Lack of atten­ tion to detail may be punished, perhaps severely with hours of wasted effort. Total energy, orbital energies, and the number of radial nodes of each subsheU should be noted as weU as the exponents and the coeflScients themselves. Several program op­ tions are helpful, especially lAMP, which prints the radial functions, and EPNCH, which writes the formatted orbital exponents and coefficients. Various utilities, such as make [88], or scripts, written for example in peri [250], can automate the many iterations necessary to systematically increase the number of primitives in a basis set or assist in solving convergence problems when more than one subshell of a particular symmetry is present. Tables 4.2 and 4.3 Ust open-shell energy coefficients for the most common cases where all states from a given configuration are averaged.

4.4 ECP Basis Sets

Section 4.4.1 summarizes our groups’ study of methods for the determination of contraction coefficients for fight elements. It is complementary to the work of

Wallace, Blaudeau, and Pitzer on the determination of primitives. Parallel efforts on the determination of both primitives and contraction coefficients for actinides and lanthanides are presented in Section 4.4.2. The work in this section has been submitted for publication [219] and some of the material below is taken from that manuscript. In general, basis sets have been included as figures in the chapter which describes their application.

104 Configuration Kaao KaA2 KàA4 KaA6 A = G s^ -1 S2 G A = 1 5 2 P 3 15 2 4 P 3 75 r»3 1 2 P 3 75 1 1 P 6 75 r>5 L 2 P 15 375 p ' 0 G A = 2 9 2 2 5 35 35 4 8 8 d2 5 315 315 7 2 2 d^ 15 135 135 3 1 1 d^ 10 105 105 1 2 2 d= 5 315 315 2 4 4 d« 15 945 945 3 2 2 d^ 35 735 735 1 1 1 d® 20 630 630 1 2 2 d" 45 2835 2835 d '° G G G A = 3 fi _ Iâ 4 2 100 7 105 77 3003 £2 6 8 _12_ 200 7 455 1001 13013 11 44 2 100 £3 21 4095 273 10647 f4 5 2 5 250 1 14 273 1001 39039 {5 9 12 18 60 35 2275 5005 13013 4 16 8 400 L 21 4095 3003 117117 1 4 2 100 1f 7 1365 1001 39039

Table 4.2: K-type Open-Shell Energy Coefficients for Average of AU States of s, p, rf, and / Configurations Where Jaao = 0

105 Configuration Kaao KaA2 KaA4 KaA6 K aA8 A == 3 3 1 3 25 ? 28 455 2002 13013 5 4 10 500 f" 63 2457 9009 351351 £lO 2 8 4 40 35 6825 5005 39039 fU 3 4 6 100 77 5005 11011 143143 fl2 1 2 1 50 42 4095 3003 117117 fl3 1 4 2 100 91 17745 13013 507507 fl4 0 G G G À == 4 17 20 18 20 490 9 693 1001 1287 21879 8 160 144 160 3920 f 9 11781 17017 21879 371943 5 100 90 100 2450 g" 9 11781 17017 21879 371943 7 10 9 70 1715 g ' 18 1683 2431 21879 371943 13 52 18 4 98 g"' 45 11781 6545 1683 28611 2 40 36 40 980 g ' 9 11781 17017 21879 371943 11 20 18 20 70 g’’ 63 7497 10829 13923 33813 5 25 45 25 1225 g" 36 11781 34034 21879 743886 1 20 18 20 490 g" 9 11781 17017 21879 371943 4 16 72 16 392 g'° 45 11781 85085 21879 371943 7 20 18 140 3430 g^^ 99 18513 26741 240669 4091373 1 10 9 10 245 g^^ 18 11781 17017 21879 371943 5 100 90 100 2450 g '' 117 153153 221221 284427 4835259 2 40 36 40 140 63 82467 119119 153153 371943 1 4 18 4 98 g '' 45 11781 85085 21879 371943 1 5 9 5 245 g '' 72 23562 68068 43758 1487772 1 20 18 20 490 g '' 153 200277 289289 371943 6323031 g '' 0 G G GG

Table 4.3: K-type Open-Shell Energy Coefl&cients for Average of All States of / and g Confîgnrations Where J a a o = 0

106 4.4.1 First- and Second-Row Atoms

Several methods are available for the detennination. of contraction coefficients so

that the resulting basis function wül be small in the core region: (1) ANO contrac­

tions, (2) SCF virtual orbital contractions, and (3) IVO contractions. These and

the free-the-most-diffiise-primitive procedure suggested by the the general contrac­

tion scheme were tested on the 2s and 2p contractions of the oxygen atom with a Is^

REP [187] and primitives from the earlier cc-pVDZ basis set [247]. The total energies

were computed using MRSOCISD with a reference space. For the IVOs the hole

was created in the 2s subshell. The results are:

s contraction: ANO < IVO < virtual « < diffiise

p contraction: ANO < diffiise < < IVO < virtual

where « < means 13 or more mE^, « means 3 to 5 mE^, and < means 2 or less

mEjj^. For s orbitals the diffiise type of contraction is poor because any Is Gaussian

function, no matter how small its exponent, has its maximum at the nucleus. Or-

thogonalizing to an occupied SCF orbital does not help because the latter is small in

the core region. The ANO contraction worked best for both s and p pseudoorbitals.

That method was used to generate new contractions from the earlier primitives [247].

Figure 3.10 contains the new contraction coefficients for oxygen.

4.4.2 Actinides and Lanthanides

Basis set development for the actinides poses particular problems: (1) a large core region, so that pseudoorbitals have a small magnitude on a wide radial interval starting at the nucleus, (2) several valence subshells of widely varying radii, e.g., 5 /,

6d, and 7s, and (3) a few nearly degenerate electron configurations, especially for

107 the early actinides. During energy-based exponent optimization, the first and second

problems lead to primitive collapse, i.e, the exponents of two primitives approach each

other closely and their coeflScients become huge, equal in magnitude, but opposite in

sign. Carrying out this limiting procedure analytically, for two Is primitives, results

in one Is function and one 3s function, both with the same exponent. While primitive

collapse can and does occur for light elements, it is acute and chronic for actinides

and lanthanides. Problem (3) makes it difficult to create one general purpose basis

set for all atomic and molecular environments and all levels of theory, that is, SCF

through large CL

One approach to solve the primitive collapse problem is to replace the collapsed

pair with two functions of the same exponent where one of the functions has increased

powers of r, e.g., use one Is function and one 3s function as above. While analytically

sound, this is unsystematic and tedious. Furthermore, many integral programs cannot

handle such mixed radial power basis sets. An alternative is to constrain the expo­

nents to stay some distance apart. An energy price must be paid, but this solution is

more general at the outset. For the chlorine cc-pVTZ basis set collapse occurred in

the Is primitives [219]. The former solution was used, and fortuitously, the optimum

3s exponent was close to that expected for a 3d polarization function. Thus an sd or

a Cartesian d set was obtained. This is tidy when it occurs, but there is no a priori

reason to expect it for 3s, 4p, 5s, etc. primitive exponents. With combined sets, such as pf or Cartesian f, all the contractions should be useful for all the symmetries, p and f in this case, to avoid inefficient and expensive correlation calculations. For the actinides and the lanthanides Is primitives led to multiple exponent collapses, and

108 2p primitives occasionally led to collapse. Combined sets were used — Cartesian d and occasionally pf and sdg.

Several cc-pVDZ basis sets for each element were created to solve the nearly de­ generate configurations problem. For the most common uranium compounds, which have IV and VT oxidation states, the actual U orbital population is approximately two 5 / electrons and one or two Qd electrons [27,241]. Since outer orbitals are more profoundly affected by inner orbitals than vice versa, the 5/^6d^ average of aU states energy expression was used to optimize the exponents. The resulting basis set has a valence space of 6p, 5 /, and 6d; these are represented with two contractions.

Because ANO calculations could not be performed for the actinides and the lan­ thanides, non-SCF contractions were formed by freeing the most difihise primitives.

The 6s orbital is in the core part of the basis set and is SZ. These basis sets are sometimes referred to as ion sets because they lack a description of the 7s and 7p orbitals. Atom basis sets were generated by adding an additional 3sd primitive and two 2p primitives and optimizing them for the 5f^6(f7s^ and 5f^6cfi7s^7p^ average of all states energy expressions, respectively, while freezing the previously determined orbital exponents. The 7s and 7p contractions contribute very little for U(IV) and

U(VT) complexes, in either ground or excited states. Consequently, for efficiency and economy, the ion basis sets were usually used in large Cl calculations. For AmCl^ and

EuCl""" and for ionization potentials of Am and Eu, 7s and 7p orbitals are important and the atom basis sets were used [251,252].

The lanthanides are fundamentally different from the actinides because there is no / shell in a lanthanide core. Consequently, many more primitives are required for the lanthanides so that the 4 / shell is adequately described near the nucleus. This

109 is an example of the reduction in number of primitives due to the smoothness and

small core magnitude of the ECPs. Otherwise, however, basis set development for

lanthanides is similar to that for actinides.

4.5 Uranium V Correlation and Spin—Orbit Energies

In analogy with Dunning’s study of oxygen [1], the incremental correlation energy

lowerings and the incremental SO energy lowerings of were studied using the

Ermler 78 electron core REP [2]. The electron configuration of is 6s^6p®5/~.

The cc-pVDZ basis set primitives and SCF contraction coefficients. Figure 3.1,

were used to form a SZ set for the 6s, 6p, and 5 / orbitals by adding a fifth / primitive

and reoptimizing to obtain (4sd 4p 5f)/[Is Ip If]; the fifth / primitive lowered the

5/^6d^ average of all states energy by 0.54 ruE^. The (4sd 4p 5f) primitives are

displayed in the first three blocks of Figure 4.1. The exponents of additional functions

of a given angular momentum were optimized in both MRCISD and SOMRCISD

calculations, where all 5/^ configurations, 28 total, composed the reference space;

excitations were from the 5/^ space into the orbitals of only the additional functions.

MOs for the ground term were obtained from MCSCF calculations. This procedure

is somewhat different than Dunning’s because a small basis set was the starting point

and because additional functions of a particular spherical symmetry were not retained

in the study of higher angular momentum functions. That is, in Dunning’s study of g polarization functions the d and / polarization functions were retained, so that his basis set was (16s 7p 3d 2f Ig)/[6s 4p 3d 2f Igj; whereas in this study of h polarization functions, the basis set was (4sd 4p 5f lh /[ls Ip If Ihj. Starting with a large basis set and including all higher angular momentum functions was computationally too

110 expensive. Analogous procedures were followed for studying the and the 6s^ subsheUs.

Tables 4.4, 4.5, and 4.6 contain the total energies and optimized exponents. Ex­ ponents optimized for both correlation and SO are nearly identical to those optimized for correlation only. The largest differences are for functions of the same symmetry as the Cl active orbitals. In particular, excitations to one additional p function, where the difference is 0.06, and excitations to one additional / function, where the difference is 0.04. The only case with two optimized additional functions was of

/ symmetry, and similar differences are found for both exponents. Correlation only values are smaller. The I orbital has a larger occupation than the I — ^ orbital, so that with SO coupling included the relativistic expansion of the former outweighs the relativistic contraction of the latter.

Tables 4.7, 4.8, and 4.9 list the correlation energies, the SO energies, the correla­ tion energy lowerings, and the SO energy lowerings. Because exponent optimization is not sensitive to the inclusion of SO coupling, the relative energies above are sim­ ilar with or without the inclusion of SO coupling. Incremental correlation energj- decreases as the number of correlation functions in a particular symmetry increases.

For / electrons, incremental correlation energies decrease with increasing angular mo­ mentum. For / electrons, the correlation energy lowerings display a pattern sim ilar to that found in Dunning's work — a cc-pVDZ set should have 2f and Ig contractions, cc-pVTZ should have 3f, 2g, and Ih contractions. For p electrons, the trends are as expected through If polarization function. However, the Ig polarization function energy is significantly larger than that of the If, and the Ih is significantly larger than that of the Ig I A possible explanation is that this is an artifact of starting

III Set % MCSCF CISD ^H4 SOCISD Exponent Correlation and Spin-Orbit

SZ -48.22981723 -48-22981723 -48.25367248 — If -48.22973421 -48.23080995 -48.25495680 0.5701 2f -48.22968444 -48.23108151 -48.25547573 0.4087,1.178 Id -48.22981723 -48.22981723 -48.25411925 0.5624 Ig -48.22981723 -48.23122202 -48.25517734 1.736 2g -48.22981723 -48.23178698 -48.25577783 1.06,3.35 Ib -48.22981723 -48.23055300 -48.25445590 1.726 2h -48.22981723 -48.23070901 -48.25462026 1.00,2.93 Correlation Only If -48.22972089 -48.23081073 -48.25492866 0.5290 2f -48.22967834 -48.23109325 -48.25544143 0.3605,0.9845 Ig -48.22981723 -48.23122202 -48.25517734 1.735 2g -48.22981723 -48.23178703 -48.25577784 1.054,3.339 Ih -48.22981723 -48.23055302 -48.25445587 1.716

Table 4.4: U'*'^ 6s^6p^5f^ / Optimized Exponents and Total Energies in Hartrees

112 Set % MCSCF % CISD SOCISD Exponent Correlation and Spin-Orbit

SZ -48.23293657 -48.23293657 -48.25681452 — Ip -48.23293657 -48.24397744 -48.36489044 1.174 2p -48.23293660 -48.24419048 -48.36904096 0.483,1-131 Id -48.23293657 -48.32455109 -48.34880896 0.501 2d -48.23293657 -48.32663654 -48.35092880 0.538,0.834 If -48.23286741 -48.23555604 -48.25916405 0.614 Ig -48.23293657 -48.23939497 -48-26328160 0.912 Ih -48.23293657 -48.24161800 -48.26545020 1.264 Correlation Only

SZ -48.23293657 -48.23293657 -48.25681452 — Ip -48.23293657 -48.24408705 -48.36377086 1.117 Id -48.23293657 -48.32455167 -48.34880811 0.502 If -48.23286365 -48.23555451 -48.25915790 0.602

Table 4.5: 6s^6p®5/^ p Optimized Exponents and Total Energies in Hartrees

113 Set % MCSCF ^HCISD SOCISD Exponent Correlation and Spin-Orbit

SZ -48.23293657 -48.23293657 -48.25681452 — Is -48-23293677 -48.23365212 -48.25751332 1-079 2s -48.23293769 -48.23377224 -48.25763060 1.27,2.04 Ip -48.23293658 -48.23393081 -48.25781431 1.293 2p -48.23293853 -48.23400224 -48.25788577 1.405,1.44 Id -48.23293657 -48.24133860 -48.26525024 0.6756 2d -48.23293657 -48.24161538 -48.26552735 0.76,1.7 If -48.23290654 -48.23298970 -48.25698707 1.408 Ig -48.23293657 -48.23502881 -48.25886914 1.358 Ih -48.23293657 -48.23325439 -48.25713234 1.454 Correlation Only

SZ -48.23293657 -48.23293657 -48.25681452 — Is -48.23293676 -48.23365213 -48.25751330 1.076 Ip -48.23293658 -48.23393082 -48.25781431 1.295 Id -48.23293657 -48.24133861 -48.26525023 0.6750

Table 4.6: U'*"'" 6s^6p®5/^ s Optimized Exponents and Total Energies in Hartrees

114 Set Corr. Spin-Orbit Corr. 4- SO ACorr. ASO Correlation and Spin-Orbit

SZ 0 23.85525 23.85525 — — If 1.07574 24.14685 25.22259 1.07574 0.29160 2f 1.39707 24.39422 25.79129 0.32133 0.24737 Id 0 24.30202 24.30202 0 0.44677

Ig 1.40479 23.95532 25.36011 1.40479 0.10007 2g 1.96975 23.99085 25.96060 0.56496 0.03553 Ih 0.73577 23.90290 24.63867 0.73577 0.04765 2h 0.89178 23.91125 24.80303 0.15601 0.00835 Correlation Only If 1.08984 24.11793 25.20777 1.08984 0.26268 2f 1.41491 24.34818 25.76309 0.32507 0.23025

Ig 1.40479 23.95532 25.36011 1.40479 0.10007 2g 1.96980 23.99081 25.96061 0.56501 0.03549 Ih 0.73579 23.90285 24.63864 0.73579 0.04760

Table 4.7: U^'*" 6s^6p®5/^ / Correlation and Spin-Orbit Energy Lowerings in Millihartrees

115 Set Corr. Spin-Orbit Corr. -h SO ACorr. ASO Correlation and Spin-Orbit

SZ 0 23.87795 23.87795 — — Ip 11.04087 120.91300 131.95387 11.04087 97.03505 2p 11.25388 124.85048 136.10436 0.21301 3.93748 Id 91.61452 24.25787 115.87239 91.61452 0.37992 2d 93-69997 24.29226 117.99223 2.08545 0.03439 If 2.68863 23.60801 26.29664 2.68863 -0.26994 Ig 6.45840 23.88663 30.34503 6.45840 0.00868 Ih 8.68143 23.83220 32.51363 8.68143 -0.04575 Correlation Only

SZ 0 23.87795 23.87795 — — Ip 11.15048 119.68381 130.83429 11.15048 95.80586 Id 91.61510 24.25644 115.87154 91.61510 0.37849 If 2.69086 23.60339 26.29425 2.69086 -0.27456

Table 4.8: 6s^6p®5/^ p Correlation and Spin-Orbit Energy Lowerings in Millihartrees

116 Set Corr. Spin-Orbit Corr. -f- SO ACorr. ASO Correlation and Spin-Orbit

SZ 0 23.87795 23.87795 —— Is 0-71535 23-86120 24-57655 0.71535 -0-01675 2s 0.83455 23.85836 24-69291 0.11920 -0-00284 Ip 0.99423 23.88350 24-87773 0.99423 0.00555 2p 1.06371 23.88353 24-94724 0.06948 0.00003 Id 8.40203 23.91164 32-31367 8.40203 0-03369 2d 8.67881 23-91197 32.59078 0.27678 0-00033 If 0.08316 23-99737 24.08053 0.08316 0-11942

Ig 2.09224 23-84033 25.93257 2.09224 -0.03762 Ih 0.31782 23-87795 24.19577 0.31782 0.00000 Correlation Only

SZ 0 23.87795 23.87795 — — Is 0.71537 23.86117 24.57654 0.71537 -0.01678 Ip 0.99424 23-88349 24.87773 0.99424 0.00554 Id 8.40204 23-91162 32.31366 8-40204 0.03367

Table 4.9: 6s^6p®5/^ s Correlation and Spin-Orbit Energy Lowerings in Millihartrees

117 with a small basis set and of not retaining the higher angular momentum functions.

Likewise the Id, Ig, and Ih anomalies for the s electrons are probably due to the same shortcomings. Incremental SO energies follow similar trends as for correlation.

For / electrons more study is needed in the f and d symmetries to confirm that the

SO energy lowerings converge.

4.5.1 Uranium REP cc-pVTZ Basis Sets

Three triple zeta (TZ) basis sets were created using the correlating exponents determined firom the study above. Figure 4.1 shows the smallest set which was ob­ tained by adding the two f, two 2g, and one h / correlating functions to the (4sd

4p 5f) primitives and using the cc-pVDZ contraction coefficients. This (4sd 4p 7f 2g lh)/[3sd 2p 3f 2g Ih] basis set is designated cc-pVTfZ because it is only TZ in the f space. Figure 3.7 shows the medium set which was created by converting the four 2p primitives, from cc-pVDZ, to five pf primitives and adding the two p, two d, and one f p correlating functions. Adding a fifth 2p primitive led to exponent collapse. The change firom (4sd 4p 5f) to (4sd 4pf 5f) lowered the energy by 1.35 mE^, and firom

(4sd 4pf 5f) to (4sd 5pf 5f) lowered the energy by 0.03 mEj^. This (4sd 7pf 2d 8f

2g lh)/[2sd 3p 2d 4f 2g Ih] basis set is designated as both cc-pVTpfZ and cc-pVTZ because it is TZ in the p and / spaces and of sufficient quality to be considered va­ lence triple zeta in general. Figure 4.2 shows the largest set which was obtained by converting the four 3s primitives, firom cc-pVDZ, to five sdg primitives and adding the two s, one p, one d, one f, and one g s correlating functions. Adding a fifth 3s primitive led to exponent collapse. The change firom (4sd 5pf 5f) to (5sdg 5pf 5f) lowered the energy by 4.19 mEj^. Some p, d, and f s correlating functions were close

118 Basis Set Energy Size cc-pVDZ -48.55048537 971733 cc-pVDZatom -48.55300515 1670745 cc-pVTfZ -48.59106624 2796276 cc-pVTpfZ -48.66545212 4131045 cc-pVTspfZ-h -48.67054190 5138304 cc-pVTspfZ -48.68441084 6493515

Table 4.10: ^îlt Total Energies in Hartrees from N'IRSOCISD with Ermler [2] REP

enough to those already included in the cc-pVTpfZ basis set that they were omitted.

This (7sdg 8pf 4d 9f 3g lh)/[3s 4p 4d 5f 3g Ih] basis set is designated cc-pVTspfZ because it is TZ for the s, p, and / spaces.

Table 4.10 shows the total energies and Cl sizes for the ion and atom cc-pVDZ, three triple zeta, and the cc-pVTspfZ minus its Ih polarization function basis sets.

The cc-pVTpfZ basis set yields a SOMRCISD total energy 115 mE^ lower than the cc-pVDZ. Because the larger cc-pVTspfZ basis set is only 20 mE^ lower and includes the 6s orbital in the valence part of the basis set, the cc-pVTpfZ basis set was chosen to be the uranium triple zeta basis set.

Table 4.11 lists the SOMRCISD energy levels using the various triple zeta basis sets. The experimental values are from Wyart, Kaufrnan, and Sugar [220] and

Van Deurzen, Rajnak, and Conway [221] (see Section 3.4.1). Differences are more easily seen using the relative errors displayed in Table 4.12. The cc-pVTfZ basis set yields consistently worse errors as compared to the cc-pVDZ basis set. Figure 3.1.

119 Uranium Ion E78 cc-pVTfZ

3 3 /Usd(4sd4p7f2g1h)/[3sd2p3f2g1hl 2.168000 -0.1289505 -0.0195499 0.0 1.009000 0.7955080 -0.0090364 0.0 0.402500 0.3649706 0.5279641 0.0 0.139800 0.0020985 0.5899125 1.0 2 2 /U p (4sd4p7f2g1h)/[3sd2p3fôg1h] 6.728000 -0.0033035 0.0 1.419000 -0.3142991 0.0 0.619900 0.7755420 0.0 0.244500 0.4902717 1.0 4 1 /Uf(4sd4p7f2g1h)/[3sd2p3l2g1hl 4.528000 0.1823236 1.983000 0.4125621 0.891100 0.3902867 0.397800 02300015 0.169600 0.0671855 4 1 / U f (4sd4p7Kg1 h)/[3sd2p3«g1 h] 0.367000 1.0 4 1 /Uf(4sd4p7f2g1h)/[3sd2p3eg1h] 1.190000 1.0 5 1 /Ug(4sd4p7f2g1hV[3sd2p3f2g1h] 1.050000 1.0 5 1 /Ug(4sd4p7f2g1h)/[3sd2p3fêg1h] 3.330000 1.0 6 1 /Uh(4sd4p7f2g1h)/[3sd2p3f2g1h] 1.728000 1.0

Figure 4.1: E78 cc-pVTfZ Basis Set

120 C-S-gg galcES §211 § - § - S 8 If 8. I s | s | s | % b|bl§S § g g gk %% 2% t o 3

§ 1 t o I g g 1 g 1 1 1 a 3 a 3 3 3

Ia I (/} DZ TfZ TpfZ TspfZ TspfZ-h Exp. [220] "H4 0 cm“^ 0 0 0 0 0 3966 3976 3975 3990 3895 4161 5174 5079 5576 5558 5574 6137 "Fa 7850 7781 8243 8246 8177 8984 "F4 8336 8187 8569 8547 8533 9434 "He 9862 9670 10538 10503 10543 11514 'D2 15477 15204 15616 15623 15486 16465 'G4 14515 14236 15095 15053 15060 16656 "Po 16498 16379 16741 16812 16497 17128 "Pi 18727 18545 19143 19221 19819 % 22354 21352 22076 22042 22346 22276 "Pz 22771 22396 23343 23372 23185 24653 'So 43928 42609 43488 43280 42944 43614 [221]

Table 4.11: Energy Levels from MRSOCISD with Ermler [2] REP

The cc-pVTpfZ basis set results in significantly improved energy levels. The cc- pVTspfZ basis set has errors very sim ilar to that of cc-pVTpfZ. Since cc-pVTfZ is only DZ in the p space, these calculations suggest that a triple zeta treatment of the 6p orbital is essential, and that a SZ treatment of the 6s orbital is adequate.

The penultimate column, labeled TspfZ-h, uses the cc-pVTspfZ basis set with its Ih polarization function removed. This polarization frmction has a non-negligible effect on the energy levels. Since the effect differs by as much as 2 % between various energy levels, the Ih function is required in a cc-pVTZ basis set.

122 DZ TfZ TpfZ TspfZ TspfZ-h Exp. [2201

"H4 0 % 0 0 0 0 0 -4.7 -4.4 -4.5 -4.1 -6.4 4161 "Hs -15.7 -17.2 -9.1 -9.4 -9.2 6137

- 1 2 . 6 -13.4 -8 . 2 -8 . 2 -9.0 8984

"F4 - 1 1 . 6 -13.2 -9.2 -9.4 -9.6 9434

% -14.3 -16.0 -8.5 - 8 . 8 -8.4 11514

'D2 -6 . 0 -7.7 -5.2 -5.1 -5.9 16465

'G4 -12.9 -14.5 -9.4 -9.6 -9.6 16656

'Po -3.7 -4.4 -2.3 - 1 . 8 -3.7 17128 'Pi -5.5 -6.4 -3.4 -3.0 19819

'le 0.4 -4.1 -0.9 - 1 . 1 0.3 22276

'P 2 -7.6 -9.2 -5.3 -5.2 -6 . 0 24653

'So 0.7 -2.3 -0.3 -0 . 8 -1.5 43614 [221]

Table 4.12: Energy Level Relative Errors from MRSOCISD with Ermler [2] REP

123 4.6 Conclusions and Future Work

Contractioa coefficients for Gaussian basis sets for use with ECPs are best deter­ mined from ANOs, as indicated by a study on the oxygen atom of ANO, SCF virtual orbital, IVO, and free-the-most-diffuse-primitive methods for obtaining contraction coefficients [219]. cc-pVDZ and cc-pVTZ basis sets for light elements using ANO contractions and energy optimized primitives [247] tave been developed. Further study is necessary to verify that conclusion for actinides with REPs when software is available.

The energy optimization method for simultaneously determining exponents and contraction coefficients, while tractable for hght elemeats, has serious difficulties when applied to the development of series of correlation consistent basis sets for actinide and lanthanide atoms: exponent collapse necessitates the use of Gaussians with high powers of r. The resulting sd or Cartesian d, pf or Cartesian f, etc. combined basis sets are not acceptable for most quantum chemistry packages and pose a threat to basis function economy, which is necessary in Cl calculations. The quality of cc-pVDZ energy optimized sets is excellent, and they are ideal for use in Cl calculations. Such basis sets for the first half of the actinides and a few lanthanides have been developed by our group. Study is needed to examine the quality of actinide and lanthanide basis sets whose exponents are determined by other means, such as even-tempered sequences; if they are suitable in comparison with energy optimized sets and amenable to systematic generation of series of correlation consistent sets then such basis sets should be produced for all the actinides and lanthanides.

Exponents of Gaussian primitive functions optimized for both correlation and SO energy are nearly identical to those optimized for correlation only. In a study on the

124 uranium (rV) cation, trends in correlation and SO energies are similar to those found in Dunning's study of correlation energy in the 03Qrgen atom. Several anomalies, such as p correlation energy which increases with the increasing angular momentum of polarization functions, may be due to the small basis sets employed. Further study is necessary to confirm that hypothesis and to establish more clearly the convergence of the SO energy. Recent advances of the COLUMBUS SOGUGA programs should enable such calculations to be performed. A uranium cc-pVTZ basis set has been developed and applied to the calculation of the energy levels. Agreement with experiment is significantly improved as compared to the same calculations with cc- pVDZ basis sets. Further work is needed to study the representation of the 6d orbital.

125 C H A P T E R 5

Actinyl Ions

5.1 Introduction

Actinide chemistry is of prime importance. Locally, plans are underway to seek proposals for conversion of approximately 250 000 tons of depleted uranium hexafluoride, from isotope separation for nuclear fuel production, stored in southern

Ohio to uranium oxide or metal, which could have industrial value [253,254]. Na­ tionally, 215 million curies of high-level radioactive waste, from plutonium generation for nuclear weapons production, is stored in underground tanks at Pacific Northwest

National Laboratory, Richland, WA [255,256]; 34 million gallons of mixed radioactive waste is stored at the Savannah River Site near Aiken, SC [257]; approximately 704

000 metric tons of depleted UFg, a low-level radioactive waste, is contained in 14-ton steel cylinders in outdoor storage yards, such as the one in Piketon Ohio [256,258].

Almost half of the underground tanks and seven of the UFg containers are leaking.

The search continues for technologies to separate radioactive elements from stable ones in saltcakes formed inside tanks, such as at the Savannah River Site, when wa­ ter evaporates [257]. Some of the highly radioactive sludge at the bottom of such tanks is already being vitrified [257,259]. Globally, substantially more radioactive waste is

126 stored or dumped iu the former Soviet Union and its former satellite states [260].

This waste disposal problem is immense. Furthermore, the economic depression in

Eastern Europe seriously aggravates the national security threat.

When UFg is exposed to air it reacts with water vapor to form uranyl fluoride. This is luminescent, 270 fj.s lifetime in aqueous hydrofluoric acid [261], as are many uranyl,

UO2 compounds. Uranyl species have been well studied spectroscopically because of their characteristic green fluorescence [71,241]. However, only comparatively re­ cently have the detailed electronic and vibrational transitions been identified [262].

The understanding of the chemistry and the physics of the transuranium actinides is less developed than that of uranium. This situation and the urgency of the nuclear waste disposal problem prompted the Department of Energy Computational Grand

Challenge Application: Relativistic Quantum Chemistry of Actinides [25] initiative.

Because of the difficulties and hazards of working with highly radioactive materials, actinide chemistry is an ideal candidate for the use of theoretical methods. Indeed, two experimental actinide chemists, J. V. Beitz and D. L. Clark, organized a sym­ posium at a recent ACS national meeting to encourage more collaboration between theoreticians and experimentalists [256,263].

This work is part of our group’s ab initio study of the electronic states of the dioxoactinide(VI) or actinyl, AnOg and dioxoactinide(V), AnOz"*", cations: uranyl

[71,241], neptunyl and dioxoneptunium(V) [264], and plutonyl and dioxoplutonium(V)

[265]. Electronic states and vibrational properties of the ground states of the americyl, dioxoamericium(V), and curyl cations are presented in Section 5.4 after a description of our methodology in Section 5.3. In the next section some relevant background is illuminated. Conclusions and suggestions for future work are stated in Section 5.5.

127 5.2 Background

The early actinides exist in a wide range of oxidation states, firom EE to VII [266,

267]. The most stable oxidation states are: III for actinium, IV for thorium, V for protactinium, VI for uranium, V for neptunium, IV for plutonium, and III for the rest of the series with the possible exception of II for nobelium. In aqueous solutions without complexing agents the penta- and hexavalent cations are too acidic to exist as hydrated ions. They hydrolyze to form weakly acidic dioxoactinide(V) and moderately acidic actinyl cations, respectively. The dioxoactinide(V) cations are known firom Pa through Am. The actinyl cations are known fi’om U through Am.

However, PaOg is perhaps better formulated as PaO(OH ) 2 [268, page 1482]; it is a base [268, page 1489] and is not considered further. The aqueous dioxoactinide(V) cations, mainly the non-neptunium species, tend to disproportionate into the actinyl and either the tetra- or the trivaient cations [269,270]. The aqueous actinyl cations, the non-uranium species, are easily reduced; oxidizing strength increases markedly in the series: U < Np < Pu < Am. Autoreduction via radiation-induced formation of H2 O2 , HO2 , and OH is especially common for Am [269]. Dioxoactinide(V) and actinyl cations form aqueous complexes. The perchlorate anion affinity is especially weak; aqueous perchloric acid is a popular medium for the study of uncomplexed, i.e., hydrated penta- and hexavalent actinides. The carbonate anion affinity is especially strong. Crystalline alkali carbonates of dioxoactinide(V) and actinyl cations tiave been characterized [271,272]. Phosphates and arsenates of actinyl cations have also been crystallized and studied with X-ray diflhaction [273—276]. Of course, uranyl and dioxoneptunium(V) compounds abound. Much of the spectroscopy of those cations

128 has been measured in the solid state [262], in particular, dicesium tetrachlorodioxo-

uranium(VI) [277]. Some crystalline non-dioxo penta- and hexavalent compounds

exist, for example, the hexafluorides. The bare uranyl ion has been isolated and

thermodynamically characterized [278], as have several isoelectronic moieties, such as

NUO+ [279].

A simple Lewis structure [280] for uranyl indicates that it is a linear closed shell

cation with U -0 double bonds. In this model the formal charge of plus two on U

accounts for the actual charge of the cation. The dioxoactinide(V) and the actinyl

cations are, in fact, linear, as established most clearly by IR and Raman spectroscopy.

In Dooh the 5 / orbitals split into cr^, tFu, S^, and (j)^ orbitals, and the 6d orbitals split

into ag, TTg, and Sg orbitals. A simple molecular orbital bonding scheme for uranyl

indicates that it is a closed shell cation with two ir bonds and one a bond between

U and each O, that is, with U -0 triple bonds. A typical empirical bond length is

1.77 Â [241]. Based on atomic orbital energies, electrons beyond the uranyl closed shell occupy the 5 / 5^ and orbitals. Neptunyl has an 12 = 5/2u ground state stemming from spin-orbit (SO) splitting and mixing of the 6^ and 4>u orbitals, as established by single crystal spectroscopy [281,282]. The bonding in thorium dioxide, which is bent, and in the dioxoactinide (V) and the actinyl ions is controversial and has been the subject of much speculation and study [283-288]. The 6p hole, a 0.1 to 0.5 electron deficit in this expected closed shell, is noteworthy and may indicate 6p involvement in bonding.

The electronic absorption spectrum of Am02 has been observed in several acidic aqueous solutions: nitric acid [289], perchloric acid [289-292], sulfuric acid [269,289], and other aqueous media [293, page 75]. These spectra, as well as those of all other

129 AnOg and AnOg cations, contain two characteristic categories of transitions: nar­

row, usually weak peaks and broad, usually strong peaks. The narrow peaks are the

actinide fingerprint / <— / transitions; the broad and strong peaks are ligand-to-

metal charge transfer excitations [294]. The transitions are either gerade to gerade

or ungerade to ungerade for AnOg and AnOg in general. They are thus electric

dipole forbidden and are vibronic, magnetic dipole, or electric quadrupole in origin.

The selection rules for the latter are Afi = 0, ± I and AQ = 0, ±1, ±2, respectively.

For AmOg the different acidic aqueous media do not significantly alter the tran­ sitions. The following peaks have been observed: 10100 cm"^, strong and narrow;

15080 c m -\ narrow; 13200,13660,16160, 18250 cm "\ weak and broad; 22310, 22790,

23350, 24480 cm~^, a strong and broad band. In not strongly complexing, acidic aqueous solution AmOg is vibrant yellow; Am^'*' is pink. The absorption spectra of

AmOg complexed with carbonate is significantly different firom that of the acidic environments; the complex is red-brown [295]. However, the complex with phosphate yields a spectrum that is similar to those in acid [295]. The electronic absorption spectrum of AmOg has been observed in several aqueous solutions: potassium car­ bonate, nitric acid, and sulfuric acid [296], perchloric acid [291], and other aqueous media [293, page 72]. For AmOg'*' the different aqueous media do not significantly alter the transitions. The following peaks have been observed: 13980, 19490 cm~^, strong and narrow; 24080, 27170, 28900 cm“^, weak and broad; 30600, 32100 cm~^, a broad band. No assignments have been offered; even speculation is sparse [283,297].

Searches for oxidation states higher than III for Cm have been unsuccessful [298].

The actinyl ions have 3N — 5 vibrational degrees of freedom. The four normal vibrations are, where the spectroscopic notation corresponding to the T>oc/i irreducible

130 representations are noted in parentheses, the symmetric stretch which is in­ frared (IR) inactive and Raman active, 1^2 (!!«}, the bend which is IR active and

Raman inactive, and 1 /3 (S„), the asymmetric stretch which is IR active and Raman inactive [299, page 6 6 ]. The two components of the doubly degenerate bend will have equal frequencies for a free actinyl ion. Note that if the actinyl ions were nonlinear then aU three normal vibrations would be both IR and Raman active. The IR and Ra­ man spectra in acidic and basic aqueous solutions confirm that the dioxoactinide(V) and actinyl cations are linear [270,300,301]. The symmetric stretch frequencies are listed in Table 5.3 on page 142. IR spectroscopy of crystalline sodium actinyl acetates and carbonates has also been performed [295,302].

5.3 Methods

The Am relativistic effective potential (REP) employed has a core size of 78 elec­ trons, leaving 17 valence space electrons [194]; the same core size for Cm leaves 18 valence electrons. The Is^ core of the O REP requires explicit treatment of six elec­ trons [187]. Thus, Am 0 2 ^"^ has 27 electrons; Am0 2 and Cm0 2 have 28 electrons.

This is three and four electrons in addition to the uranyl closed shell. Correlation consistent double zeta plus polarization basis sets were developed [219], using the methods described in Chapter 4, Figures 5.1, 5.2, 5.3, 3.10 and 5.4. The Am atom and O augmented sets were used in initial calculations that indicated that the Am

7s and 7p and O diffuse functions played little role in the bonding description. Con­ sequently, later investigations applied the smaller Am^'*' and unaugmented basis sets.

The Cm atom basis set was used by omitting the 7s and 7p contractions.

131 The Americium +3 Ion Core pVDZ Correlation Consistent Set

Basis S et (4sd4p4ftg)/[3sd2p2f1g]

Core Potential: C. S. Nash. B. E. Burster), and W.C. Ermler, J. Chem. Phys. 106,5133-5142 (1997).

State: 6s(2)6p(6)5f(6), av. of config.

Hartree-Fdck Cakulahons Total HF Energy -80.50306745

(HF+1+2) Calculations Total Cl Energy HF Coeffioent Spin-orbit Splitting J -80.83399434 0.81233801 0.00000000 ev 0 -80.82963057 0.8620 0.11874427 1 -80.82347352 0.9131 0.28628620 2 -80.81670367 0.94271747 0.47050327 3 -80.80984383 0.9469 0.65716909 4 -80.80317785 0.9387 0.83855972 5 -80.79694467 0.91455994 1.00817325 6

4 3 3 / INo. primitives, PQN, No. contracted fhs (Am sd set) 3.424000 -0.0751760 -0.0119296 0.0 1.103000 0.7164989 -0.0105725 0.0 0.490400 0.3695951 0.4601084 0.0 0.203700 0.0201664 0.6239167 1.0 4 2 2 /INo.. primitives. PQN, No. contracted 15.160000 -0.0021822 0.0 1.327000 -0.5619931 0.0 0.824100 0.9307736 0.0 0.296100 0.5850749 1.0 4 4 2 / INo. primitives, PQN, No. contracted 5.230000 0.1966952 0.0 2.219000 0.4567554 0.0 0.902600 0.4213684 0.0 0.335300 0.1988090 1.0 1 5 1 / INo. primitives, PQN, No. contracted (hs (Am g set) 2.075000 1.0 I I T he alom ic s e t was obtained by optimizing the (sd,p,f)-exponents in I Hartree-Fdck calculations on the 5f(6) av. of config., with the idea ! of obtaining an f basis dose to optimum (dr an f(6) Am(lll) or I higher oxidation-state idn. The second sd contractidn was obtained I from a single Hartree-Fock calculatidn on the 5f(5)6d(1) av. of I config. The polarization set was obtained by optimizing the exponent I of a single g primitive in HF+1+2 calculatidns on the lowest state of 15f(6), allowing for correlation of the 5f shell only. I I December 1997

Figure 5.1: cc-pVDZ Basis Set

132 The Americtuin Atom Core pVDZ Correlation Coneiatent Set Ain(lll)

Basis Set: (5sd6p4f1g)/[4sd4p2f1g]

Core Potential: C. S. Nash, B. E. Bursten, and W.C. Ermler, J. Chem. Phys. 106.5133-5142 (1997).

State: 6s(2)6p(6)5f(6), av. of config.

Hartree-Pbck Calculations T otal HF Energy -80.50353075 (contracted)

5 3 4 / INo. . primitives. PQN. No. contracted fhs (Am sd set) 3.451000 -0.0738775 0.0262142 -0.0090596 0.0 1.096000 0.7256891 -0.3661812 -0.0308136 0.0 0.479400 0.3652257 -0.0080944 0.5594950 0.0 0.171700 0.0149600 0.6364434 0.5531929 1.0 0.068880 -0.0010024 0.4589944 0.0184150 0.0 6 2 4 /INo. . primitives. PQN. No. contracted fhs (Am p set) 15.160000 -0.0021458 0.0010249 0.0 0.0 1.327000 -0.5646449 0.1960859 0.0 0.0 0.824100 0.9365265 -0.3511988 0.0 0.0 0.296100 0.5768442 -0.2882332 1.0 0.0 0.077530 0.0097267 0.6539052 0.0 0.0 0.035170 0.0002168 0.4900237 0.0 1.0 4 4 2 / INo. primitives. PQN. No. contractedfhs (Am f set) 5.227000 0.1970987 0.0 2.216000 0.4573135 0.0 0.901000 0.4211283 0.0 0.334700 0.1980760 1.0 1 5 1 / !No. primitives. PQN, No. contracted fhs (Am g set) 2.075000 1.0 ! 1 The atomic set was obtained by optimizing the (sd,p,f)-exponents in I Hartree-Pbck calculations on the 5f(6) av. of config.. with the I idea of obtaining an f basis dose to optimum for an f(6) Am(lll) or I higher oxidation-state ion. I The fourth and fifth sd primitives were chosen to optimize the energy I of the 5f(6)7s(2) av. of config. Trying to optimize four sd I primitives for the 5f(6) av. of config. led to intractable exponent- I collapse problems. The third sd contraction was obtained from a I single Hartree-Fock calculation on the 5f(5)6d(1) av. of config. The I outer two p primitives were chosen to optimize the energy of 15f(6)7s(1 )7p(1 ) av. of config. The polarization set was obtained by ! optimizing the exponent of a single g primitive in HF+1+2 I calculations on the lowest state of 5f(6), allowing for correlation I of the 5f shell only. ! I ...... November 1997

Figure 5.2: Am cc-pVDZ Atom Basis Set

133 Tiw Curium Atom Com pVOZ Corrélation Con#l*t#nt Sut

Basis Set; (5sd6p4f1g)/[4sd4p2f1g]

Core Potential; C. S. Nash. B. E. Bursten. and W.C. Ermler. J. Chem. Phys. 106.5133-5142 (1997).

State; 6s(2)6p(6)5f(7]6d(1). av. of config.

Hartree-Pbck Calculations Total HF Energy i -93.93345506 -93.27541468 6s(2)6p(6)5f(7). av. of conRg.

(HF+1+2) Calculations 6s(2)6p(6)Sf(7) J =7/2 Total Cl Energy HF CoefWent Spin-orbit Splitting -93.70300347 0.92321663 0.00000000

5 3 4 / INo. primitives. PQN, No. contracted fhs (Cm sd set) 3.935000 41.0703875 0.0219458 -0.0068131 0.0 1.109000 0.7602028 -0.3204306 41.0213829 0.0 0.454700 0.3355374 0.0340575 0.5145380 0.0 0.146600 0.0053512 0.6300239 0.5916286 1.0 0.050300 0.0007933 0.4703855 0.0522092 0.0 4 4 / INo. primitives. PQN, No. contracted fhs (Cm 4229000 0.0189904 -0.0058627 0.0 0.0 1.157000 0.5337681 -0.1313266 0.0 0.0 0.535000 0.4626998 -0.1008228 1.0 0.0 0243900 0.1222378 0.1319570 0.0 0.0 0.089950 0.0011091 05616499 0.0 0.0 0.033020 0.0003439 0.5011633 0.0 1.0 4 2 / INo. primitives. PQN. No. contracted Ins (Cm 5225000 0.2189629 0.0 2.217000 0.4656283 0.0 0.893300 0.4077205 0.0 0.320600 0.1881485 1.0 1 5 1 / INo. primitives. PQN, No. contracted fhs (Cm g set) 2.221000 1.0000000 I I The atomic set was obtained by optimiang the first four sd. the I first four p. and the four f exponents in Hartree-Fbck calculations I on the 5f(7)6d(1 ) av. of config.. with the idea of obtaining an f I basis dose to optimum for an f(7) Cm(lll) or higher oxidation-state I ion. The use of n = 2 p primitives led to unacceptable exponent and I coefRdent patterns and near exponent-collapse. The fifth sd I primitive was chosen to optimize the energy of 5f(7)6d(1)7s(2) av. of I config. The outer two p primitives were chosen to optimize the 1 energy of 5f{7) 6d(1 )7s(1 )7p(1 ) av. of config. The polarization set I was obtained by optimizing the exponent of a single g primitive in I HF+1 +2 calculations on the lowest state of 5f(7), allowing for ! correlation of the 5f shell only. I I ...... February 1998

Figure 5.3: Cm cc-pVDZ Atom Basis Set

134 The Oxygen Atom Cora Augnwntad pVDZ Comlation Consistent Set

Basis Set: (5s5p2d)/[3s3p2d]

C ore Potential: 1_F. P ad o s & PVL C hristiansen. J . Chem. P hys. 8 2 .2 6 6 4 (1985).

State: 2s(2)2p(4),(3)P 2s(2)2p(5), (2)P

Hartree-Fbck Calculatibns Total HF Energy -15.65804902 (uncontracted) -15.64007369 (uncontracted) -15.65793794 (contracted) -15.63084766 (contracted)

(HF+1+2) Calculations (J=2,0^3/2,1/2) Total Cl Energy Lead Coefficient Spin-orbit Splitting -15.80183327 0.97901040 0.00093813 -15.82029373 0.97254010 0.00082861

5 1 3 /INo. primitives. PQN, No. contracted fins (O s set) 41.04 -0.0097069 0.0209595 -0.0853925 7.161 -0.1326923 0.1236435 0.2298016 0.9074 0.6012559 -1.5370912 0.8281264 0.2807 0.4896966 1.3610831 -1.9242331 0.07725 0.0261184 0.1600015 1.6160078 5 2 3 / INo. primitives, PQN. No. contracted (ns (O p set) 17.72 0.0433381 -0.0516010 0.0772819 3.857 0.2316810 -0.3876031 0.7163096 1.046 0.5068728 -0.5352938 -0.5138613 0.2752 0.4349053 0.8038801 -0.4469688 0.06853 0.0553951 0.2622864 0.9096179 1 3 1 / INo. primitives. PQN. No. contracted fris (O d set) 1.213 1.000000 1 3 1 / INo. primitives. PQN. No. contracted fins (O d set) 0.3596 1.000000 1 I The cc-pVDZ set was augmented by optimizing one addibbnai s and p I exponent each in Hartree-Fock calculations on the ground (2)P state I of the negative ion. The contraction coefficients were obtained from I the natural orbitals of an HF+1+2 calculation on the ground state of I th e neutral atom . The additional d exponent w as optim ized in HF+1 +2 1 calculations on the ground state of the negative ion. I September 1997

Figure 5.4: O Augmented cc-pVDZ Basis Set

135 For AmOa small multireference SO configuration interaction (MRSOCI) cal­ culations using molecular orbitals (MOs) from the state and from the

state yielded significantly difierent results. This underscored the need for a balanced treatment of the Su, 4>u reference space. State-averaged multiconfiguration self-consistent field (MCSCF) calculations were employed to average the above sets of MOs. For A m 02‘*' and CmOg the SCF MOs were used in the SOCI calculations.

Both the non-direct CIDBG and the direct GUGA program s were applied. Due to restrictions in both programs, the most sensible yet economical calculations could not be performed — for AmOg all singles and doubles from the reference space containing three electrons in the 5^ and orbitals plus four electrons in the and and one in the MO, 3a^, from which chaxge-transfer excitations occur. This would give a fair treatment of the low lying states arising from the three 5 / electrons and the charge transfer states arising from an excitation of one orbital to the

5 / orbitals. Various compromise calculations were attempted. For consideration of charge transfer states SOCI singles calculations were done with a large reference space of three electrons in the 6^ and (pu orbitals plus four electrons in the Sy, and

(pu and one in Za^ plus these same configurations with the and the 2x„3x„ orbital populations. This provides a fair treatment of the states discussed above and includes the most important double excitations, which involves the one, two, and three

TTy orbitals. For the ground state potential energy curve calculations, the reference space was and the lcr^lcr^2(7|2<7^1x^3

136 For AmOg '*■ and CmOg analogous calculations with, four electrons in the and

orbitals were performed.

5.4 Results and Discussion

The common uranyl closed shell core of Am02 AmÜ 2 and Cm02 is l(T„l(T^l7r„2cr^2cr^3o^27r^l7r^3(7^. This is consistent with our group’s other work. The ordering is that of ascending orbital energy. Various arguments from empirical data and ab initio studies have established that 3<%g is the orbital from which the low- lying excitations occur for UO 2 Orbital energies are not quantum mechanical observables; electron excitations are quantum mechanical observables. In the SOCI calculations on uranyl, the lowest transition is, in fact, from 3cr„ [71,241].

The population analyses for Am02^'*', Am02^, and Cm02 are displayed in

Table 5.1. Substantial 6p holes are found: 0.7, 0.5, and 0.3 electrons, respectively.

In Am02^'^ and Cm02 the metals have a partial charge of 4-1.66 and 4-1.87, respectively, significantly lower than in UO 2 [241] and Np02 [264], so that the oxygens are slightly positive: -b0.16 and 4-0.06. In Am02 the charges are 4-1.29 for

Am and —0.14 for O.

5.4.1 A m 02

Table 5.2 shows the electronic spectrum at the SCF/ (cc-pVDZ Atom) level of theory for the ground state SCF equilibrium bond length. It indicates that the low- lying states come from the and electron configurations as expected.

is the ground state. (5^0^ ‘^Au is the first excited state. The charge transfer states could not be investigated using SCF theory because of open shell and symmetry restrictions in SCFPQ. The MRSOCIS/cc-pVDZ electronic spectrum at the ground

137 Atom s P d f 9 Total AmOg^^ Am 1.88 5.32 1.70 6.42 0.005 15.33

0 + 0 3.75 7.82 0.095 — — 11.67 A m 02‘‘‘ Am 1.90 5.45 1.61 6.74 0.004 15.71

0 + 0 3.77 8.45 0.071 — — 12.29 CmOî Cm 2.08 5.68 1.47 6.89 0.003 16.13

0 + 0 3.74 8.06 0.070 — — 11.87

Table 5.1: Ground State SCF/(cc-pVDZ Atom) Population Analyses

Occupancy State Energy ISl 10i 0 cm"*- 9799 is i "A^ 20790 47260 %. 99930 168700 ^A„ 181100

3 < 4

Table 5.2: AmOg^^ SCF/ (cc-pVDZ Atom) Electronic Spectrum

138 State MRSOCISD/cc-pVDZ equilibrium bond length is listed in Figure 5.5.

‘^*^3 /2 u is the ground state. This state should have a negligible magnetic moment due to cancellation of the spin and orbital contributions. The multiplet is regular and has no interpenetrating states. However, the SO splittings are in the ratio 1:4:4, not

1:1:1 as expected [303, page 215], suggesting some intermingling of states. The rest of the spectrum is complicated. In particular, the multiplet is convoluted. The first charge transfer states are the six 3cr„^^„ states. This multiplet is inverted, and its components are not evenly spaced. The parenthesized values on the left are assignments of the observed AmOg transitions. The ungerade to gerade transitions begin at 36,000 cm~*^ based on MRSOCIS/cc-pVDZ calculations.

At the SCF/cc-pVDZ level the equilibrium bond distance. Re, is 1.53 Â.

Re for 4 ^ is 1.52 Â. At the MRSOCISD/cc-pVDZ level the Re is

1.57 Â. The potential energy curve at this level of theory is graphed in Figure 5.6. A symmetric stretch frequency of 1240 cm~^ is obtained from a quadratic fit. Table 5.3.

The 56% relative error is large. The most important sources of error are: basis set quality, electron correlation treatment, fitting method, and comparison to empirical data from solution. The experimental frequency from crystalline sodium americyl acetate is 749 cm“^ [302].

5.4.2 Am02 and CmOg

SCF calculations yield ground states. The MRSOCIS/cc-pVDZ elec­ tronic spectra at the MRSOCISD/ cc-pVDZ ground state equilibrium bond lengths are listed in Figures 5.7 and 5.8. is the ground state. The multi­ plet is regular, has no interpenetrating states, and its components are evenly spaced.

139 ( 2 3 3 5 0 ) 2 5 3 0 0 c m ”

( 2 2 7 9 0 ) 2 4 2 0 0 «S3/2U ( 2 2 3 1 0 ) 2 3 5 0 0 3cr^ 4>^ , ®Ss/2,x

22200 T /2 .U

2 0 9 0 0 «^2 4 - . ^ A 3 /2u

5 / 2 u 1 8 1 0 0

1 7 6 0 0 1 6 9 0 0 <52

1 5 9 0 0 <5^2 -f- <^2

<52 0 2 . ‘^Ai /2« 1 2 9 0 0 1 2 4 0 0 <52 02 , ^ATi3/2« 12100 ^2 + <52 02 4- 62 02, ^A s/2« ( 10100) 1 1 8 0 0 <52 02, “A3/2«

7 2 7 0 <52 02 ,

3 9 5 0 <52 02 I **^7/2 ti

7 9 5

(0 ) 0

Figure 5.5: AmOg^^ MRSOCIS/cc-pVDZ Electronic Spectrum

140 -1 0 9 .6 7 2

-1 0 9 .6 7 4

-1 0 9 .6 7 6

-1 0 9 .6 7 8

-1 0 9 .6 8

-1 0 9 .6 8 2 lU -1 0 9 .6 8 4 -1 0 9 .6 8 6

-1 0 9 .6 8 8

-1 0 9 .6 9

-1 0 9 .6 9 2

-1 0 9 .6 9 4 2.8 2 .8 5 2 .9 2 .9 5 3 3 .0 5 3-1 Am-O Displacement, a.u.

Figure 5.6: MRSOCISD/cc-pVDZ Ground State Potential Energy

141 2+ An02 An02^"^ An Exp. [30Ij Calc. Exp. [270,301] Calc.

U 872 cm~^ 1010 — — Np 863 1059 767 913

Pu 835 996 748 — Am 796 1240 730 932

Cm — 1103 ——

Table 5.3: Symmetric Stretch. Frequencies

The rest of the spectrum is complicated. The first charge transfer states are the

states. That state for CmOg lies 16,000 cm“^ higher than for

AmOg The parenthesized values on the left are assignments of the observed AmOo + transitions.

At the SCF/cc-pVDZ level the equilibrium bond distances are 1.55 Â and 1.58 Â for AmOa'*' and CmOg At the MRSOCISD/cc-pVDZ level the

ground state R e’s are 1.59 and 1.62 Â for Am02 and Cm02 Sym m etric stretch frequencies, Table 5.3, are 932 and 1103 cm“^, respectively. The 28% relative error for Am02 is large. The potential energy curve for Cm02 at this level of theory is graphed in Figure 5.9.

5.5 Conclusions and Future Work

The ground state of the americyl cation is *^ 3/2u~ The first charge transfer state is 3a^5^4>u ®^5 / 2 u- The ground states of the isoelectronic dioxoactinide(V) and curyl cations are ^2+o+g- The first charge transfer states are 3al$l(f>lZTr^ ^Ilo+g.

142 (28900) 26600 ______

8310 %o_,

7270 +

2860 ------

1390 ------

0 cm-: ------

Figure 5.7: AmOg^ MRSOCIS/cc-pVDZ Electronic Spectrum

143 42400 ’'rio+3

11800

10500

3550

1660

-1 0 c m

Figure 5.8: CmOg MRSOCIS/cc-pVDZ Electronic Spectrum

144 -1 2 5 .5 2 8

-1 2 5 .5 3

-1 2 5 .5 3 2

-1 2 5 .5 3 4

-1 2 5 .5 3 6 <à3 i -12 5 .5 3 8 (D m -1 2 5 .5 4

-1 2 5 .5 4 2

-1 2 5 .5 4 4

-1 2 5 .5 4 6

-1 2 5 .5 4 8 2 .9 2 .9 5 33 .0 5 3.1 3 -1 5 3 .2 Cm-O Displacement, a.u.

Figure 5.9: Cm02 MRSOCISD/cc-pVDZ Ground State Potential Energy

145 Agreement with likely experimental charge transfer excitation energies is good for those that have been observed experimentally, namely, AmOg and AmOg Other electronic / <— / transitions have been tentatively assigned. Software development on computing transition moments, in progress in this group, will help in further study of these important spectral features. Symmetric stretch frequencies. Table 5.3, have been calculated for the ground states. Obvious avenues of improvement of this work have been outlined in Section 5.3. Further study might include the dioxocurium(V) cation. The Am02 """ and CmOa^’'" charge transfer states, 3cr„5^0„3x„ ’rio+^, suggest that CmO^^ may have a 5^(^^37t„ ®II_3 / 2 ÿ ground state. If that is the case then curiousity demands a study of einsteinyl, ESO 2 to determine whether its ground state is an octet.

146 C H A PTE R 6

Lanthanide and Actinide Ions: Free and Doped into Crystalline Calcium Fluoride

6.1 Introduction

Most elements are metals. These can be partitioned into four classes: (1) alkali

and alkaline earth, (2) p-block, (3) transition metals, and (4) lanthanides and ac­

tinides. The chemistry of the metals, indeed all elements, is largely determined by

the nature of their valence electrons. The lanthanides axe unique in that their filling

4 / subshell is chemically inert. Owing to this inertness and their open 4 / subshell,

most lanthanides are paramagnetic. The complicated electronic spectroscopy of the

lanthanides, mainly due to / i— / transitions, as well as their electron paramag­

netic resonance spectroscopy, have been intensely studied since World War II. Pure

rare earths have been available in large quantities only in the postwar period [304].

The second optical laser discovered was the samarium dication doped into calcium

fluoride. This further fueled the effort to measure experimentally and understand

theoretically the lanthanide energy levels, which are most easily obtained when the

lanthanides are impurity ions in various host crystals. The second half of the actinides are very similar to the lanthanides. The first half of the actinides are more involved

147 in bonding leading to a broad range of oxidation states as in the transition metals.

Their chemistry, a mix of lanthanide and transition metal behaviors, is quite inter­ esting and has been a subject of study because of their importance in nuclem fuels, nuclear weapons, and nuclear waste disposal. Enough data has been accumulated to make comparisons with the lanthanides firuitful. Ab initio calculations of /-element energy levels and magnetic moments are relatively rare. In this chapter the free ion energy levels of various actinide and lanthanide cations are calculated. Section 6.4, as are the CaF2 crystal field split energy levels of the same cations. Section 6.5. First background material on a few specific lanthanide and actinide ions is presented in Sec­ tion 6.2. Methodology including a description of the calcium fl.uoride crystal model is summarized in Section 6.3.

6.2 Background

Classic treatments of the theory of rare earth spectroscopy and energy levels are the textbooks of Dieke [304], Wyboume [305], and Judd [306], in ascending order of theoretical depth. Martin, Zalubas, and Hagan [307] is the reference for free atom and free ion energy levels. Solution spectra of rare earth ions have been reviewed by

Camall [308], circa 1979. Morrison and Leavitt [309] is a compendium of interpreted experimental data on impurity lanthanide trications for 26 host crystals, circa 1982.

Boatner and Abraham [310] is a review of electron paramagnetic resonance of actinide ions doped into various crystals. Edelstein and Goffart [311] introduce the theory of and summarize the data on the magnetic properties of the actinides. Camall [312] has reviewed the optical absorption spectroscopy of the actinides. Low [313] is an early presentation of electron paramagnetic resonance in solids. Abragam and Bleaney [314]

148 is the reference on the electron paramagnetic resonance of transition metals, including

the lanthanides and actinides, circa 1970. Griffith [315] is a treatment of the theory of

transition metal spectroscopy and energy levels. Pulde [316] has reviewed the theory

of crystal fields.

Calcium fiuoride occurs naturally as the mineral fluorite. The fluorescence of this

mineral is caused by traces of rare-earth metal ions [317, page 281]. The crystal structure of calcium fluoride is simple, consisting of a cubic array of fluoride anions

with calcium cations occupying alternate body centers [267, page 5]. It is known as

the fluorite structure and is common when the coordination number of the cation is eight. The space group is O^.

In 1956 Bleaney, Llewellyn, and Jones [318] observed the electron paramagnetic resonance (EPR) spectra of in single crystals of both CaFg and SrF 2 at 20 K.

Three types of magnetic ions each with four-fold symmetry about one of the cube axes produced = 3.501 ± 0.008 and g±_ = 1.866 ± 0.002 for CaFa and pn = 3.433 ± 0.008 and g±_ = 1.971 ± 0 . 0 0 2 for SrFg, implying that the ground state was probably a

Kramers doublet. The spectral details strongly indicated that a trivalent uranium ion had replaced a calcium ion and that a fluoride ion occupied an adjacent interstitial position reducing the site symmetry from cubic to tetragonal. The crystals were grown by the Stockbarger method and contained approximately 0.1 and 1 % uranium fluoride by weight.

In 1959 Conway [319] observed the absorption spectrum from 200 nm to 8 gxa. of

UF4 in CaF2 crystals at 77 K. Transitions from the ground state to all other levels in the 5/^ configuration were found and assigned. Using an intermediate coupling model the F2 radial integral was 206 cm~^, the Cs/ spin-orbit coupling constant was

149 1870 and the ground state wave function was composed of 87.7 % 11.3 %

and 1 % Despite sharp lines, Zeeman sphttings could not be observed. The site symmetry was not known; crystal field splittings were not assigned. The crystals were described as optically clear containing approximately 0.1, 1, and 10 % UF^ by weight.

Several papers on the electron paramagnetic resonance (EPR) of uranium ions in CaF2 crystals were published in 1962. Title et al. [320] conclusively determined that uranium centers having trigonal symmetry about the < 111 > direction were tetravalent not trivalent. A correlation was established between the intensity of the

EPR absorption of trigonal centers and the intensity of characteristic lines in that crystal’s optical absorption spectrum which matched Conway’s [319] UF4 in CaF 2 spectrum. Furthermore, they demonstrated that a 5/^ configuration could not pro­ duce the asymmetric EPR line shapes and zero g±_ value obtained firom the trigonal centers. A charge compensation model was proposed in which two oxide anions re­ place two fluoride anions at opposite body comers of the nearest neighbor cube of eight F“ ions. They noted that UO 2 has the fluorite crystal structure with a lattice constant of 545 pm compared to 546 pm for CaF 2 . The following EPR transitions were observed in CaF2 : some undescribed low symmetry resonances, in tetrago­ nal site symmetry as previously found by Bleaney et al. [318], in cubic symmetry which accounted for about 10 % of all observed centers with a graph of the angular variation in the (110) plane of the g value for transitions within the lowest lying quar­ tet, and in trigonal symmetry with asymmetric line shapes presumably caused by non-Kramers doublets spht by random fields of low symmetry or by the Jahn-Teller

150 effect, where = 3.27 ± 0.03 and = 0 ± 0 .2 , In SrF% the in tetragonal sym­

metry resonance of Bleaney et al. [318] and the in trigonal symmetry transition were observed. For the latter = 2.87 ifc 0.03 and = 0 ± 0.2. Crystals containing

almost exclusively tetravalent uranium ions were pale green and were produced by holding the crystal at the melt for only a short time to prevent reduction. A deep red color in CaFg or a deep orange-red color in SrFg were indicative of

Yariv [321] also observed the EPR of in sites of trigonal symmetry about the

< 1 1 1 > direction, in addition to that of U^"*" in tetragonal symmetry as previously found by Bleaney et al. [318], and lines in sites lacking axial symmetry hypothesized to be caused by Yariv concluded that the EPR of uranium ions in trigonal sites was caused by and not by as he and coworkers had previously reported [322,323].

The crystals were grown in a graphite crucible and contained approximately 0.1 % uranium.

Uranium ion spectra have been extensively studied experimentally. Additional references on paramagnetic resonance of tetravalent uranium doped into Cap2 are

[324-329]. EPR spectra of pentavalent uranium has been observed in CaFg [330—

332]. Hexavalent uranium as a dopant in crystalline materials can exist as octahedral

UOe®“ [333-336]. This uranate ion, like the uranyl ion, produces green luminescence.

The uranate emission is coupled to more than one vibrational mode, however, unlike the uranyl emission which is strongly coupled to only the symmetric stretching mode.

The high symmetry of octahedral hexacoordinate complexes of / elements sim­ plifies both the experimental and theoretical study of their spectra. The / <— / electric dipole transitions are parity forbidden; the fingerprint region thus consists of weak vibronic transitions [337].

151 Thirty-five of the forty energy levels of doped into hexacoordinate octahedral sites of CsgZrBrg have been observed experimentally [338]. All forty energy levels have been calculated by a non-linear least squares fit of the experimental data to the variable parameters from several crystal field models [338,339]. The best fit includes configuration interaction between 5/^ and 5/^7p^.

Most of the forty hexacoordinate octahedral crystal field spht energy levels of

Np^"*" in CsNpFs have been observed in absorption spectra of mulls [340]. At room temperature strong transitions from the (free ion designation) ground state to all magnetic dipole and electric quadrupole free ion allowed states were identified.

Weak forbidden transitions to ^Pq and ^Fi states were discernible. At Uquid nitrogen temperature increased intensity and resolution enabled the observation of crystal field split levels. With the exception of the ^Ai level from the ^Sq level, which was not observed, the and levels, which gave incorrect numbers of spht levels, and the level, which was not in the range of the experiment, the sphtting agreed with that expected from a cubic field. The 5/^ levels of Np^"*" are sandwiched between those of and Pu®'*' with regard to intermediate coupling.

6.3 Methods

Lanthanide and actinide ions are represented with relativistic effective potentials

(REPs) both to reduce the number of electrons exphcitly treated and to include the major relativistic effects, notably spin-orbit (SO) which is energetically on a par with interelectron repulsion for these / elements. The Ermler [2] uranium REP, henceforth denoted E78, has a 78 electron core of [Kr]4d^°4/*^'^5s^5p®5d^° leaving 14 valence elec­ trons in the neutral: 6s^6p®5/^6d^7s^. The plutonium [2], the americium [194], and

152 the curium [194] REPs also have 78 core electrons. Their valence configurations for the

neutrals are: 6s^6p®5/^6d°7s^, and 6s^6p^5f^6d}-7s^, respectively.

The Stuttgart Am pseudopotential [3] was also used in initial calculations on Am^"^

doped into CaFg. It was the only core potential available at the time for Am, and

has a small 60 electron core and a large valence space of 5s^5p®5d^°6s^6p®5/^6d®7s^.

W. E. Ermler developed a 46 electron REP for europium. P. A. Christiansen created a

46 electron REP for praseodymium. Their valence configurations are 5s^5p®4/^5d°6s^

for Eu and 5s^5p®4/^5d^6s^ for Pr.

Existing cc-pVDZ basis sets for the uranium and plutonium REPs were produced

by Z. Zhang, J.-P. Blaudeau, and R. M. Pitzer [219], Figures 3.1 and 6 .1 . cc-pVDZ

basis sets for the Am pseudopotential and the Am, Cm, Eu, and Pr REPs were developed. They are displayed in Figures 6.2, 5.3, 6.3, and 6.4. AH basis sets were created using the methods described in Chapter 4.

All SOCI calculations were performed using the procedure described in Chapters

1 and 2 . Open shell / electrons necessitate multireference treatments. The 5/^

free ion calculations are described in Chapter 3 and include the complete 5/^ reference space. The 4/^ Pr^"^ calculations are performed identically. The free ion computations used a complete active space of 5 /, 6 d, 7s, and 7p containing one electron — 16 references total. is closed shell. This is not an interesting cation experimentally, so even though open-shell excited states were computed, the reference space was a single closed shell. and Pr^"*" are cations. A balanced treatment requires 77 references. Including all singles and doubles is beyond the capabilities of both SOCI programs, CIDBG and SOGUGA, even with small cc-pVDZ basis sets consisting of 47 functions. For simplicity, all singles calculations were performed.

153 Tiw Plutonium +2 Ion Cors pVOZ Corralatibn Consistont Sot

Basis S et (4sd4p4flg)/[3sd2p2f1g]

Core Potential: W.C. Ermler, R.B. Ross, & P,A. Christiansen, Int J. Quantum Chem. 40,829-846 (1991).

State: 6s(2)6p(6)6d(2)5f(4), av. of confîg.

Hartree-Fdck Calculations Total HF Energy -69.36137932 I (HF+1+2) Calculations (J=5/2) Total Cl Energy HF Coeflîoent Spin-orbit Splitting -69JOOOOOOOC Ojo o o o o o o c x jo o o o o g o c

4 3 3 / !No. primitives, PQN. No. contracted fns (Pu sd set) 2.798000 -0.1268045 -0.0115915 0.0 1.152000 0.7206011 -0.0242601 0.0 0.490700 0.4148308 0.4327487 0.0 0.178600 0.0183674 0.6769954 1.0 4 2 2 / INo. primitives, PQN, No. contracted fns (Pu p set) 9.374000 0.0000422 0.0 1.710000 -0.2668691 0.0 0.624200 0.8012919 0.0 0.237200 0.4175420 1.0 4 4 2 / INo. primitives, PQN, No. contracted Iris (Pu f set) 4.514000 0.2432891 0.0 1.913000 0.4720941 0.0 0.782800 0.3958826 0.0 0Z84400 0.1581343 1.0 1 5 1 / INo. primitives, PQN. No. contracted fns (Pu g set) 2.025 1.0000000

The atomic set was obtained by optimizing the (sd,p,f)-exponents in Hartree-Fock calculations on the 6d(2)5f(4) av. of conlig., with the idea of obtaining an f basis dose to optimum for an f(4) Pu(IV) or higher oxidation-state ion. The polarization set was obtained by optimizing the exponent of a single g primitive in HF+1 +2 calculations on the lowest state of 5f(4)6d(2), allowing for correlation of the 5f shell only.

Ju n e 1998

Figure 6.1: cc-pVDZ Basis Set

154 AiiMricium km 60 Etectron Psmidopotmtfal cc-pVDZ

10 1 3 / Am s (10s8p8d4f)/[3s2p2d2f] based on Kuechle. JCP 100 p7535 (1994). 86.646959 0.0110589 0.0042201 0.0010856 36.331211 -0.2532557 -0.1221381 -0.0326395 23.357740 0.8421846 0.4465796 0.1210118 11.845143 -1.3726238 -0.8266780 -0.2282752 3.284375 1.0787505 1.0547855 0.3127283 I.763231 0.4286281 0.3961198 0.1270377 0.704153 0.0196827 -0.9150166 -0.3722432 0.313841 -0.0020922 -0.5491748 -0.3760532 0.061853 0.0003496 -0.0133401 0.6771835 0.024942 -0.0001472 0.0039388 0.5199691 8 2 2 / Am p (10s8p8d4f)/[3s2p2d2f] based on Kuechle. JCP 100 p7535 (1994). 480.581630 0.0035860 0.0025442 112.414600 0.0181758 0.0143203 28.693325 0.0513171 0.0317175 7.288971 -0.4298899 -0J2503916 4.333191 0.6626961 0.5333252 1.970423 0.6522171 0.3110023 0.677628 0.0666096 -0.6425102 0.250915 -0.0064009 -0.5608309 8 3 2 / Am d (10s8p8d4f)/[3s2p2d2fl based on Kuechle. JCP 100 p7535 (1994). 97.484377 0.0024677 0.0008938 24.684333 0.0188059 0.0043432 II.055688 -0.0598731 -0.0128656 3.972069 0.4767953 0.1537652 1.928029 0.5025445 0.1517753 0.858770 0.1426233 -0.1664624 0.279300 0.0048859 -0.5653769 0.081115 -0.0001076 -0.5051935 5 4 2 / Am f (10s8p8d4f)/[3s2p2d2fl based on Kuechle. JCP 100 p7535 (1994). 11.645485 0.0538509 0.0 3.638305 0.3749720 0.0 1.329806 0.5265939 0.0 0.409359 0.3494673 0.0 0.510000 0 .0 1.0 1 5 1 /Am g (10s8p8d5f1g)/[3s2p2d2f1g] ccoptSg 2.070000 1.0 4 3 /Am ECP60MWB Kuechle. JCP 100 p7535 (1994). QR 35-valence-e- pseudopotential

Figure 6.2: Stuttgart Pseudopotential [3] cc-pVDZ Basis Set

155 I Th« Europium Atom Coro pVDZCurfolollon Com itli nt Sot ! ! Basis S e t (5sd6p7ftg)/[4sd4p2flgl ! ! Core Potential: ! 46 electron core. ! W.C. Ermler ! Unpublisned. Sun. 14 Jun 1998 2122:11. ! lEuon I State: 5s(2)5p(6)4fC7). av. of config. ! ! Hartree-Fock Caiculatians 1 Total HFEriergy State I -93.03589397 5sP)5p{6)4fC7). av. of config. ! -92.98165976 5sP)5p(6)4f(8)5d(1).av. of config. -9339007210 5s(2)5p(6)4t(7)6s(2).av. of config. -9333731274 5s(2)5p(6)4f(7)6s(1)6p(1),av. of config.

(HF+1+2) Calculations (J = 7 /2 ground state) Total Cl Biergy HF Coefficient Descriptian ! -9331083207 0386912 4fC7)CASto5g ! -9332675326 0384481 41(7) S = 7/2 tOSf ! 5 3 4 /INo.. primitives. PQN. No. contracted fhs (Eu sd set) 2.798000 -0.0762040 0.0190910 -0.0242985 0.0 1307000 0.8788533 -03311595 0.0721271 0.0 0.467200 0.4514028 •0.0649069 03489359 0.0 0.138800 0.0150650 03520045 03503919 1.0 0.046280 -0.0004574 03814318 0.0055554 0.0 6 2 4 /INo.. primitives. PQN. No. contracted fhs (Eu p set) 22.040000 -0.0022989 0.0008192 0.0 0.0 1.774000 -03484627 0.0548326 0.0 0.0 0.781400 0.8853267 •0.1867537 0.0 0.0 0380600 03432780 •0.1779390 1.0 0.0 0.057400 0.0155093 0.4845824 0.0 0.0 0.022080 -0.0032987 0.8228101 0.0 1.0 7 4 2 /INo.. primitives. PQN. No. contracted kis (Eu f set) 117300000 0.0079117 0.0 38.710000 0.0530313 0.0 15.090000 0.1765327 0.0 6.341000 0.3318602 0.0 2.836000 0.3949121 0.0 1.029000 0.3188224 1.0 0.350700 0.1578245 0.0 1 5 1 /INo. primitives. RON. No. contracted Ins (Eu g set) 3.889000 1.0 ! ! The atomic se t was otitaned by optimizing the (3sd.4p.7f)-exponents ! in Hartree-Fock calculations on the 4f(7) av. of config.. wilti ttie I ideaof obtaining an flMSis dose to optimum for an f(^ Eu(ll) or ! higtier oxidation-slate ion. An additional f. p. or sd primitive I lowered the energy by 0.0049.0.0038. or0.0023. respecfiveiy. ! resulting in exponent collapse forp orsd. The outer two sd I primitives were cfiosen to optimize 8ie energy of tlie 4f(7)6s(2) av. I of config. The outer two p primitives were cfiosen to optimize the ! energy of the 4f(7)6s(l )8p(l ) av. of config. The second sd ! contraction was obtained tram a single Hartree-Fock calculation on I the 4f(7)6s(2) av. of config. The Itiird sd contraction was obtained ! from a single Hartree-Fock calculation on the 4t(8)5d(l) av. of ! config. The second p contraction was obtained from a single ! Hartree-Fock I calculation on the 4f(7)6s(l)6p(1) av. of config. The I second f contraction was based on HF+1+2 caicuiations on tfie lowest I state of 4t(7) afiowing forconelatian of the 4f stiell only. Tfie I optimal single f exponent tor correlation was 0.88 and w as 0.0019 ! lower in energy than tfie general contraction choiceO f 1.029. The ! polarization se t was ofWahed by optsnizing tfie exprxient of a single I g primitive in HF+1+2 calculations on tfie lowest state of 4f(7). allowing fOr correlation of Ifie 4f sfiel only.

Octofier1998

Figure 6.3: Eu cc-pVDZ Atom Basis Set

156 PrasMdymium Ion 46 Electron REP cc-pVOZ

4 3 3 /Prsd(4sd4p7flg)/[3sd2p2f1g] 3.451000 -0.0734221 -0.0215908 0.0 1.013000 0.6811809 0.0536632 0.0 0.423100 0.4187985 0.5555578 0.0 0.143234 0.0094944 0.5281131 1.0 4 2 2 /Prp(4sd4p7f1g)/[3sd2p2f1gl 1.895000 -0.2277142 0.0 0.731100 0.4870169 0.0 0.364700 0.5492288 1.0 0.183600 0.1502833 0.0 7 4 2 /Prf(4sd4p7f1g)/[3sd2p2f1gl 108.400000 0.0063669 0.0 34.940000 0.0448790 0.0 13.650000 0.1522553 0.0 5.851000 0.3083567 0.0 2.514000 0.3989144 0.0 1.035000 0.3352577 1.0 0.387000 0.1515556 0.0 1 5 1 /Prg (4sd4p7f1gV[3sd2p2f1g] cc optimized 5g from Eu 3.889000 1.0000000

Figure 6.4: cc-pVDZ Basis Set

157 For computing many energy levels the direct SOGUGA diagonalization program can

require thousands of iterations. Even for SOMRCIS wall times can be prohibitive.

The Pu^'^free ion was also treated with all singles from the 266 5/® references. Am^"*"

and Eu^"^ have six f electrons. A balanced reference space of 357 / configurations can

be treated with SOMRCIS and with all singles and doubles if the and p® electrons

are frozen. Am^""", Eu^'*', and Cm^"^ are 5 p cations. 393 references makes even all singles calculations large, beyond the capabilities of the serial CEDBG and the early

versions of SOGUGA. However, SOMRCIS and larger calculations on 5f^ systems are tractable with the parallel CIDBG [81] and the current version of SOGUGA.

Because of the problems of computing many energy levels, only the ground state and the lowest excited states were obtained. Unfortunately, the parallel CIDBG could not be applied to the systems under study here. COLUMBUS 5.5 SOGUGA will enable various calculations which include many double excitations.

6.3.1 Crystal Model

The crystal is represented as a three-layer cluster. The first or innermost layer consists of a central metal cation and its nearest-neighbor anions. These are treated with REPs and basis sets. This enables the study of ligand field effects as well as the crystal field. The next-nearest neighbors compose the second layer, and these cations are modeled with only REPs. The REPs insure that the nearest-neighbor anions feel the repulsive cores of the surrounding cations. The last layer consists of many point charges to reproduce the Madelung potential of the infinite crystal lattice. Figure 6.5 shows the two inner layers. The central Ca^'"' cation is octahedrally coordinated by the cube of eight F~ anions. The ten core electrons of the central Ca^'*" are replaced

158 by the Stuttgart quasi-relativistic pseudopotential [341]. The basis set provided with that pseudopotential was energy optimized and reduced to single zeta, Figure 6 .6 , for the 3s^3p®4s^ valence space of the neutral. Because a Stuttgart SO operator was not available, the one from the Christiansen, Ermler and coworkers REPs was used [188]. The eight nearest-neighbor F~ anions have valence spaces of 2s^2p® using the Stuttgart two electron pseudopotential [246]. The basis set provided with that pseudopotential was energy optimized and augmented with a diffuse p function to produce an aug-VDZ set. Figure 6.7. Polarization functions were not added because they might lead to artificial attractive interactions between the fluorides and the point charges. Because a Stuttgart SO operator was not available, the one from the Christiansen, Ermler and coworkers REPs was used [187]. In CaFg each F“ is tetrahedraUy coordinated by Ca^^ cations. The 12 next-nearest-neighbor Ca^"*" ions were represented by 18 electron pseudopotentials from the Stuttgart group [342]. The

4s^ valence space is ionized in the crystal and is not represented by basis functions.

The cluster model of the crystal was constructed using the experimental positions of the Ca^""" and F~ ions. CaFg has a lattice constant of 10.32 bohr [317]. Several criteria were used to determine the number of point charges in the cluster’s third layer. ( 1 ) The overall octahedral symmetry must be preserved. (2) The total cluster charge should be close to zero. (3) The Madelung potential of the crystal lattice must be adequately reproduced at the central metal ion. (4) The equilibrium C a-

F distance should be converged with respect to the number of point charges and reasonably close to the experimental distance. Additional point charges were added in spherical shells constrained by conditions ( 1 ) and (2 ). The Madelung potential of a few clusters was calculated and compared to that of the infinite lattice. It was found

159 Figure 6.5: Crystalline CaF 2 Cluster Model: Inner Two Layers [4]

160 Calcium Ion 10 Electron Pseudopotentlal pVSZ

6 1 2 /Cas(6s4p5d)/[2s1p1d]basedonKaupp,JCPv94n2p1360. 12.082714 0.0599808 -0.0137427 4.444413 -0.3900563 0.0963118 0.843042 0.7191131 -0.2051825 0.360341 0.4635712 -0.3060531 0.066337 0.0181540 0.4870933 0.026363 -0.0052836 0.6638342 4 2 1 /Cap(6s4p5d)/[2s1p1d]basedonKaupp, JCPv94n2p1360. 5.843366 -0.0846689 1.625570 0.3221734 0.674737 0.5614247 0261291 02580727 5 3 1 / Ca d (6s4p5d)/[2s1p1d] based on Kaupp, JCP v94n2p1360. 9.197134 0.0383692 2.422329 0.1594789 0.721177 0.3293365 0.200440 0.4538111

Figure 6.6: Ca^'*' Stuttgart 10 Electron Pseudopotential pVSZ Basis Set

161 Ruorfdclon 2 Electron Pseudopotentlal aug>VDZ

4 1 2 / F s (4s5p)/t2s3pl based on Bergner, Mdec. Rhys. v80n6p1431. 51.6435280 0.0086554 0.0 9.4143390 -0.1530064 0.0 1.2140520 0.5898864 0.0 0.3700530 0.5247526 1.0 5 2 3 / Fp(4s5p)/[2s3p] based on Bergner, Mdec. Phys.v80n6p1431. 222945200 0.0518722 0.0 0.0 4.9530120 02373568 0.0 0.0 1.3419120 0.5076902 0.0 0.0 A * » A C A J t O n A 4C4A40-1 4 A A A

Figure 6.7: F Stuttgart 2 Electron Pseudopotential aug-VDZ Basis Set

162 Cluster Ions Ca—F (bohr) SCF SOCIS

Cai3 F3 2 ® 45 4.59

Ca?9 F i6o^~ 239 4-477 Cai4i Faso^^ 421 4-497

CaiTT F36o®~ 537 4.490 4.490

Cas3i F io64^~ 1595 4.494 4.494 Cagos F1208^" 1811 4.494

CaFa 0 0 4.4701 UCai77 Faeo^^ 537 4.272 4.269 UCas3o F1064° 1595 4.273 4.270

UCagoa Fi208° 1811 4.273 4.270

Table 6 .1 : CaFg Cluster Size

that a relatively small cluster, 239 total ions, was sufficient to reproduce the potential at the central ion. The equilibrium Ca—F distance was computed for various clusters by symmetrically stretching the eight Ca—F bonds. Table 6.1 shows the results for

SCF and SOCIS calculations on CaFg and for SOMRCIS calculations on CaFa with

doped into the central ion position. Both systems reach convergence at 1595 total ions. This cluster was used for all succeeding calculations. As expected for

Ca^""" SO effects are not significant and the SCF and SOCIS distances are identical.

Even for SO effects are small. The equilibrium distance is 0.22 bohr shorter for

because of its greater charge. Figure 6 . 8 shows the cluster of 239 ions.

163 a

Figure 6.8: Crystalline CaF 2 Cluster Model: CaygF^go [4]

164 6.4 Free Ion Energy Levels

Tables 6.2, 6.3, 6.4, and 6.5 show the calculated and experimental energy levels

for uranium cations +6 through +3, respectively. No experimental data is available

for The excited levels are probably significantly too high because the ground

state configuration is the only reference. The term SO splitting has a large relative

error. A higher quality basis set might improve the agreement with experiment. The

7p^ states were not easily identifiable because of many 6p®5/^ states. The cc-pVTZ

basis set is necessary to bring the agreement within ten percent of experiment for

The experimental values for are from spectra of the cation doped into

CaF2 [343]. A parametric fit was employed to obtain some of the levels. The free ion

spectrum of in the visible and near ultraviolet regions has been published but no

assignments were made [344]. Although better calculations could be performed for

U®'*' and with current software, the three / electrons of prohibit calculations

beyond SOMRCIS. Double excitations are necessary to reduce the large errors seen

in the ground '*‘1 term. Tables 6.6, 6.7, and 6.8 show the corresponding results for

and Pr^^, respectively. The agreement with experiment is similar to that found for uranium. The tables confirm that L—S coupling is adequate for the low lying levels of Pr cations, but for U cations the coupling is intermediate. Table 6.9 lists some low lying levels of Eu^^ with similar results. Table 6.10 display these same low lying levels for Am^'*'. The actinide energy levels are in error by as much as —50 %; however, the experimental values are from solution spectra [345].

165 Level Calc Error Exp 6s^ 6p® 'So 0 — — 6s^ 6p“ 5/^ 100975 — — 6s^ 6p^ Qd} "Po 224190 — — 6s^ 6p^ 7s^ 304382 — —

Table 6.2: U®"*" Energy Levels from SOCISD/cc-pVDZ with. E78 REP

Level Calc Error Exp [346]

5 /' ^Fs/2 0 cm“^ 0 % 0 5 /' "FT/2 6268 -17.6 7608.6

6d^ "D3 / 2 103916 14.2 90999.6

6d^ "D5 / 2 114654 14.1 100510.5 7s^ "Si/2 155011 9.6 141447.5 Tp' "Pl/2 — — 193340.2

7p' "Ps/ 2 — — 215885.8

Table 6.3: U®'*' Energy Levels from SOMRCISD/cc-pVDZ with E78 REP and a (5/, 6d, 7s, 7p)^ Reference Space

166 Level Calc Error Exp [220] 0 cm^^ 0 % 0 3990 -4.1 4161 5558 -9.4 6137 "F, 8246 -8.2 8984 'F4 8547 -9.4 9434 10503 -8.8 11514 'Dz 15623 -5-1 16465 'G4 15053 -9.6 16656 "Po 16812 -1.8 17128 "Pi 19221 -3.0 19819 % 22042 -1.1 22276 "P2 23372 -5.2 24653 'So 43280 -0.77 43614 [221]

Table 6.4: Energy Levels from 5/^ Reference Space SOMRCISD/cc-pVDZ with E78 REP

Level Calc Error Exp [347]

■‘I9 / 2 0 cm“^ 0 % 0

‘'I1 1 /2 3615 -19.7 4500

‘'I1 3 /2 7091 -24.8 9433 ^Fs/2 8338 --- —

‘'I1 5 /2 10402 -31.3 15133

^Fs/2 10556 — —

"H9 / 2 10769 — — --- "F7 / 2 12105 —

Table 6.5: Low Lying Energy Levels from 5/^ Reference Space SOMRCIS/cc- pVDZ with E78 REP

167 Level Calc Error Exp [307]

4/1 "F5 / 2 0 cm“i 0 % 0

4/1 'F 7 / 2 2811 -7.1 3027 5rfl %3/2 158126 37.4 115052

5di 2 Ds/ 2 160814 35.7 118514

6 si "Si/ 2 194919 8.9 178971

6 pi 1 / 2 — — 223478

6 pi ^Pa/ 2 — — 230039

Table 6 .6 : Energy Levels from SOMRCISD/cc-pVDZ with. Ermler 46 Electron REP and a (4 /, 5d, 6 s, 6 p)^ Reference Space

Level Calc Enror Exp [307]

0 cm“i 0 % 0 % 1904 -11.5 2152

3923 - 1 0 . 6 4389

"F2 5024 0.54 4997 6244 -2.67 6415

"F4 6685 -2.48 6855

'G4 9415 -5.10 9921 "Da 18041 4.08 17334 "Po 22366 4.56 21390 'Pi 22887 4.00 22007

"le 24133 8.65 2 2 2 1 2

'P 2 23931 3.32 23161 "So 52977 5.76 50090*

Table 6.7: Pr^"*" Energy Levels from 4/^ Reference Space SOMRCISD/cc-pVDZ with Ermler 46 Electron REP

168 Level Calc Error Exp [307] 0 cm~^ 0 % 0 ‘^In/2 1150 -17-7 1398 2429 -16-0 2893 '^Il5/2 3811 -14-4 4454 11133 18-8 9371 %9/2 12579 25-4 10033

"Fs/2 11716 15-6 10138 "FT/2 12371 13-9 10859 % /2 13419 22-5 10950 "Fg/2 13365 13-6 11762

"Hu /2 14808 18-5 12495

Table 6.8: Low Lying Energy Levels from 4/^ Reference Space SOMRCIS/cc- pVDZ with Ermler 46 Electron REP

169 Level Calc Error Exp [307] 'Fo 0 cm“^ 0% 0 309 -16.5 370 "F2 873 -16.1 1040 ^F3 1625 -14.0 1890 2512 -12.2 2860 'Fs 3491 -10.7 3910 4531 -8.3 4940 "Do 18078 4.7 17270 "Dr 19434 2.1 19030 "Dz 21550 0.19 21510 "Da 24163 -0.93 24390 "Le 27070 — 25182 [304] "D4 27159 -1.7 27640

Table 6.9: Eu^'*' Low Lying Energy Levels from, 4/® Reference Space, Frozen Core, SOMRCISD with. cc-pVDZ Basis Set and Ermler 46 Electron REP.

170 Level Calc Error Exp [345] 'Fo 0 cm“^ 0 % 0 1227 -50.9 2500 "F2 2775 -45-6 5100 "Fa 4364 -40.2 7300 5900 -36.6 9300 "Fa 7330 -33.4 11000 "Fa 8596 -28.4 12000 'Do 11276 -9.8 12500 14674 -13.7 17000 'Da 18320 -15.2 21600 'La 20907 — — 'Ga 21975 — — 'Da 22014 — — 'Ga 22919 — —

Table 6.10: Low Lying Energy Levels from, 5/® Reference Space, 6s^6p® Frozen Core, SOMRCISD with cc-pVDZ Basis Set and Nash 78 Electron REP.

171 6.5 Crystal Doped Ion Energy Levels

In the octahedral site symmetry of CaF 2 an '‘^Ig/ 2 level spUts into two quartets,

Gzjiu-, and one doublet, j&i/2u [348]. Interstitial doping of oue F“ or 0^“ into an

empty cube of fluoride anions lowers the symmetry to C 4 „. Interstitial doping of two

F“ or anions into empty opposite cubes of fluoride anions lowers the symmetry to

In either case the quartets are split into two doublets. Such interstitial dopings

are one of many forms of charge compensation when a cation with a higher charge

than 4-2 replaces a Ca^'*' cation in CaF2 [349—353]. Table 6.11 shows the calculated

and experimental levels split by these three crystal fields for doped into CaF 2 -

Table 6.12 lists the energy levels of 4/^ Pr^"^ split by a C^t, crystal field. The

agreement with experiment is similar to that found for the free ions. The calculated

values suggest that some lines were absent or mis-assigned in the experimental spectra.

In the octahedral site symmetry of CaF 2 an * 8 7 / 2 level splits into one quartet,

G3/2UJ two doublets, Ei/2u and £'5/214 [348,356]. The splittings of this ground state can be determined from electron spin resonance measurements of the magnetic moment [357-359] and from optical spectroscopy [360,361]. Table 6.13 shows the calculated and experimental levels split by the octahedral crystal field for Am^"*",

Cm^'*', and Eu^"^ as dopants in CaF 2 . The agreement with experiment is poor. Initial studies on Am^^ did not significantly correlate the / electrons. However, similar calculations, with the SOGUGA programs, did correlate the seven / electrons with singles and doubles. Surprisingly, the ground state splitting energies did not change.

The reasons for this large discrepancy are not apparent at this time.

The uranate ion, UOg was studied by replacing the eight fluoride anions with six octahedrally coordinated oxide anions. The green luminescence of UOs is an

172 0/. Dih Q v Exp. [343,354] %/2 0 0 0 0 0 456 91 57 419 712 353 410 419 791 607 497 2003 2241 1969 602 3601 3976 3673 4433 3601 3986 3797 4452 3752 4057 3855 4470 3837 4292 3980 4496 4952 5061 4862 4511 4952 5480 5122 4638 “^113/2 6719 6996 6792 9433

Table 6.11: in CaF2 Low Lying Energy Levels

173 4/2 Cale Exp. [355] Exp. Fit "H4 Bi 150 0 0 B2 179 35 36 A-i 591 83 58 E 0 99 61 E 524 134 141 A-i 641 154 259

A2 188 208 262 "H5 Bi 1956 2179 2179

B2 1950 2233 2185 E 2072 2297 2206 E 2177 2315 2229

A2 2445 — 2257 Al 2156 — 2380

A2 2489 — 2506 E 2448 2381 2711 "He E 3819 4268 4268

Table 6.12: in CaFg Low Lying Energy Levels

174 g Factor (cm"^) (cm~^)

Calc. 1.26 3.32 1.982 Exp. [357] 18.6 49.6 1.9258 Cm^+ Calc. 0.975 2.79 Exp. [357] 13.4 35.7 1.9261 Eu^ • Calc. 0.0038 0.0096 — Exp. [362] 0.071 0.19 1.9926

Table 6.13: Ground State Splitting Energies of Ions in CaFg.

E — > A transition of 19180 cm“^ in CaFg [336]. The ground and excited state symmetric stretch frequencies are approximately 770 cm"^ and 670 cm~^, respectively, in various perovskite crystals [335]. At the undoped CaFa geometry, the calculated excitation energy is 18910 cm"^. The calculated ground and excited state symmetric stretch frequencies are 1107 cm~^ and 1052 cm~^, respectively.

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