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Spin-Orbit Graphical Unitary Group Approach Configuration Interaction and Applications to Uranium Compounds

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

Zhiyong Zhang, M. S.

*****

The Ohio State University

1998

Dissertation Committee: Approved by Professor Russell M. Pitzer, Adviser / ; Professor Anil K. Pradhan Adviser Professor Emeritus Isaiah Shavitt G raduate Program in Chemical Phvsics UMI Number: 9911295

UMI Microform 9911295 Copyright 1999, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103 ABSTRACT

In this work we implemented the Graphical Unitary Group Approach (GL’GA)

based spin-orbit configuration interaction (Cl) programs in a new release of the

COLUMBUS programs. .Ab initio spin-orbit Cl calculations on several uranium-

containing compounds. U0|'^, UOgOH^ and U(BH 4 )4 , were carried out using rela-

tivistic effective core potentials and correlation consistent basis sets.

Potential curves of the ground state, Ig 3crj), and several low-lying excited-

states, Ig (^Ag. 3(ril(5i), 2g and 3g (^Ag, 3

3cr^lcj|j), of UQo'*' were obtained at the spin-orbit Cl level. .All 24 valence electrons

were correlated by single and double excitations to all the virtual orbitals. Symmetric- stretch frequencies of the ground and low-lying excited states were obtained. The

transition energies to the low-lying excited states were calculated and compared with experimental single-crystal CsgUOgCU values. It is found that, due to the strong spin- orbit effects, multireference (3cr„l(J^, 3cr^l(^l) Cl calculations are needed for most of the excited states. The calculated transition energies from the multireference cal­ culations agree very well with single-crystal experiments. It is found that the first excited state is linear and the Renner-Teller splitting is very small. The first ungerade state, accessible by electric-dipole-allowed transitions from the first excited state, was found to be from the 3crjl(i^ configuration and the calculated transition energy from

ii the first excited state to the first ungerade state agrees reasonably with experiment.

UOgOH^. a very simple model of complexion in solutions, was studied also.

The ground- and the first excited-state geometries were optimized at the spin-orbit Cl

level. The calculated vertical transition energ}' at the ground-state geometry agrees

well with the experimental values but the calculated adiabatic transition energy is

only about half of the experimental value. In the ground state, the interaction be­

tween and 0H“ is strong and the optimized geometry is bent by nearly

10°. In the excited state at the ground-state geometry, one electron is transferred

from an orbital with both L'Oo^ and OH" character to a U /-like orbital. In the ex­

cited state at the optimized excited-state geometry, one electron is transferred from

an OH~-like orbital to an orbital on UO^^. and there is little additional interaction.

We also carried out calculations on which has been studied extensively

by spectroscopy and which is one of the few uranium compounds to form molecular cr\'stals. .A.11 electrons with substantial uranium 5/ character are correlated. The

Cl reference space included all the p configurations. The ground state was found

to be E in Tj symmetry and all electronic-dipole-allowed /- / transitions from the ground state were obtained. The calculated energies agree reasonably well with the experimental values and crystal-field assignments.

Ill Dedicated to Xiaojing and Ningv'uan

IV ACKNOWLEDGMENTS

There are many people I would like to express my heart-felt thanks, but among them I can name only a few here. I thank Professor Zhenyi Wen of Northwest Univer­ sity in China for initiating me into the field of Quantum Chemistry and for preparing for the unitary group approach related work. I thank Dr. Satoshi Vabushita of Japan, on whose work the current implementation of spin-orbit GUG.A. Cl is largely based. I thank Dr. Larry Curtiss of Argonne National Lab for providing me with the funding for my GRA appointment during the last year of my graduate study. I thank Dr.

Isaiah Shavitt of Ohio State for many stimulating discusions. Most of all I would like to thank Dr. Russell M. Pitzer. my research advisor, for his inspiring guidance and incredible patience with me, and for his assistance with my personal matters. What 1 have learned from his character and rigorous scientific style will certainly benefit me for many years to come.

1 also want to take this opportunity to thank the faculty for the stimulating re­ search environment. 1 thank many colleagues for their friendship and help, especially

Jun Li, Troy Wu and former and current Pitzer group students: Ke Zhao, Jean

Blaudeau, Scott Brozell and Spiridoula Matsika. I owe everything to my family. My most valuable asset is the everlasting love and support of my family. Particularly I thank my wife, Xiaojing Shi, for her love, for taking most of the responsibility of taking care of our son. for her unrelenting support for and trust in me even during her struggle with her pain.

VI VITA

March 10, 1966 ...... Born - Hebei. China

1985 ...... B. S. - Hebei University, Baoding, C hina 1988 ...... M. S. - Northwest University. Xian. C hina 1995 ...... M. S. (Chemical Physics) - Ohio State U niversity 1998 ...... M. S. - (Computer and Information Sci­ ence) Ohio State University 1992-1998 ...... Graduate School Fellowship, Teach­ ing Associate and Research .Associate. Ohio State Universitv

PUBLICATIONS

FIELDS OF STUDY

Major Field: Chemical Physics

Studies in Theoretical Chemistrv: Professor Russell M. Pitzer

Vll TABLE OF CONTENTS

Page

A b s tra c t ...... ü

D edication ...... iv

Acknowledgments ...... v

\ ' i t a ...... vii

List of Tables ...... x

List of Figures ...... xii

Chapters:

1. Introduction ...... 1

2. Spin-O rbit GUGA CI P ro g ra m s ...... 11

2.1 Review of Graphical Unitary Group Approach ...... 11 2.2 The Spin-Orbit Interaction in the Graphical Unitary Group Approach 17 2.3 Spin-Orbit GUGA Cl: Implementation and Performance ...... 29

3. Theoretical Calculation on Uranyl Ion U O ^^ ...... 43

3.1 Background ...... 43 3.2 Computational Details ...... 54 3.3 Results and Discussion ...... 57

via 4. Calculation on Uranyl (mono)hydroxide: U02(0H)'*' 78

4.1 Background ...... 78 4.2 Results and Discussion ...... 80

5. Calculation on Uranium Borohydride U(BH.,)., ...... 90

5.1 Background ...... 90 5.2 Computational Description and Discussion of R esults ...... 92

6. Conclusion ...... 99

Bibliography ...... 102

IX LIST OF TABLES

Table Page

2.1 Symmetry Properties of "Real Spherical" Spin Functions ...... 24

2.2 Timing Data for the CI-Diagonalization S tep ...... 36

3.1 Properties of the Electronic States of CsoUOoCh ...... 50

3.2 U cc-pV'DZ Basis Set (4srf4p4/l^) —> [3.s

3.3 O cc-pV'DZ Basis Set; (4s4pld) — [2s2pld] ...... 55

3.4 O cc-pVTZ Basis Set {ôsôp2d) —> [3s3p2dlf] ...... 56

3.5 G round-state SCF O rbitals: P rincipal MO C o e ffic ie n ts ...... 58

3.6 Ground-state Uranyl SCF Population Analysis ...... 59

3.7 Low Gerade Excited-State M anifold ...... 60

3.8 Summary of Calculated Excitation Energies and Symmetric-Stretch Vibrational Frequencies from Single-reference Spin-orbit CISD Calcu­ lations, Double-^ B asis ...... 62

3.9 Calculations with Triple-^ B asis ...... 63

3.10 Cl Wave Function Properties ...... 63

3.11 Comparison of Single-Reference and Multireference Calculations at 3.39 U q ...... 65

X 3.12 Comparison of Single-Reference and Multireference Calculations at 3.27 o o ...... 66

3.13 Multireference Calculations with D ouble-B asis ...... 6

3.14 Energ}' Differences between Multireference Calculations and Single­ reference Calculations ...... 68

3.15 Higher Excited-States M anifold ...... 69

3.16 Low Ungerade Excited-State Manifold ...... 70

4.1 Ground-State Occupied SCF Orbitals: Principal MO Coefficients . . 83

4.2 Ground-State Unoccupied SCF Orbitals: Principal MO Coefficients . 84

4.3 Optimized Ground- and Excited-State Geometries of UOoOH^ .... 85

4.4 Excited-State SCF Orbitals of UOoOH"*": Principal MO Coefficients . 86

4.5 Comparison of Spin-Orbit and Xon-Spin-Orbit Calculations With Dif­ ferent References ...... 87

5.1 U(BH^); Geometry. Tj sym m etry ...... 93

5.2 U Population Analysis of U(BH. 4)4 MOs ...... 94

5.3 Calculated f — f Excitations from the Ground S ta te ...... 96

XI LIST OF FIGURES

Figure Page

2.1 DRT Graph With n = 6. N = 5. S = ^ ...... 37

2.2 Graphical Representation of Spin-orbit Coupling ...... 38

2.3 8(b) Type Loop from [30,48] ...... 39

2.4 Internal Part of the DRT G raph ...... 40

2.5 Spin-Orbit DRT G raph ...... 41

2.6 Graphical Representation of Spin-Orbit Coupling for a System with an Odd Number of Electrons ...... 42

3.1 SCF Ground-State Orbitals of ÜO 2 ''’ ...... 71

3.2 Ground- and Excited-State Potential Curves. Single-Reference Spin- Orbit CISD Calculation, Double-C Basis ...... 72

3.3 Low-Lying Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Double-^ B asis ...... 73

3.4 Ground- and Excited-State Potential Curves. Single-Reference Spin- Orbit CISD Calculation, Oxygen Triple-(^ Basis ...... 74

3.5 Low-Lying Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Oxygen Triple-^ B asis ...... 75

3.6 Renner-Teller Effect in the First Excited State of U ran y l ...... 76

3.7 Low-Lying Excited-State Curves From Multireference Calculations. . 77

x i i 4.1 Schematic Drawing of UOgOH'"' ...... 88

4.2 Orbital Correlation Diagrams of and UOgOH^ ...... 89

5.1 Structure of U(BH 4 ) 4 ...... 97

5.2 SCF Ground-State Orbitals of U{BH 4 ) 4 ...... 98

xm CHAPTER 1

Introduction

The discovery of relativity by Einstein in 1905 [1] immediately captured the imag­

ination of the scientific world and revolutionalized our view of fundamental physics.

But its relevance to chemical systems was not fully recognized until many years later.

In fact, shortly after formulating the famous Dirac equation [2], Dirac stated [3] that relativistic effects would be "of no importance in the consideration of atomic and molecular structures and ordinary chemical reactions'' since the valence-electron speed is small compared with the speed of light. .A.s is widely recognized now and discussed later in this chapter, relativistic effects have a fundamental influence even on valence electrons, particularly for systems containing heavy elements.

The importance of relativistic effects is readily recognized in very accurate cal­ culations even on the lightest elements. For example, to explain the very accurate measurements of the ionization energy of the helium atom, the ground state energy of He must be calculated to the order of [4], where a = is the fine-structure constant. The intrinsic spin of the electron, most simply demonstrated in the Stern-Gerlach experiment [5], is a direct consequence of relativistic quantum mechanics. Conspicuous daily-life phenomena that can be explained only in a rela­ tivistic framework include the color of gold (silver without relativistic effects) and the liquid state of mercury (solid state without relativistic effects). [6]

Intuitive insight into relativistic effects in chemistry can readily be obtained by considering the consequences of the finite speed of light on atomic shell structure qualitatively. In classical relativity theor\', the mass of a particle increases with its speed v:

(1,1) - ? The influence of this mass-velocity effect on atomic orbitals can be seen approximately by examining the Bohr radius:

a o = 4 7 r e o - ^ . (1-2) me^

Xs a result, the increased mass will shrink the atomic orbitals.

In general there are three major effects on the atomic orbitals due to relativity [7|.

First, the inner-shell s and p orbitals are contracted due to the large speed of electrons close to the nuclei. The higher s and p shells are similarly contracted since they must be orthogonal to the inner shells. Even though the average speeds of valence-shell electrons may be small, the inner parts of valence s and p orbitals still come close to the nuclei so these valence electrons also have high speeds in the core region. In fact, the gold 6s orbital is contracted more than the gold inner-shell s orbitals on a percentage basis [7]. Second, shells with / > 0 (p, d, /, • ■ •) will be split by the spin-orbit interaction into two subshells

j = (1-3) with the I — \ subshell relatively contracted and stabilized and the / + ^ subshell rel­ atively expanded and destabilized. Finally, due to the centrifugal potential . d and / electrons are less influenced by the mass-velocity contraction. The contraction of the s and p orbitals will nonetheless shield the d and / electrons more effectively from the nuclei. Consequently the d and / orbitals will be destabilized in energy and further expanded radially. The interplay of these three major effects largely deter­ mines the chemical properties due to relativity.

Following two pioneering articles, one by K. S. Pitzer [8] and the other by P.

P\ykkd and J. P. Desclaux [9], explicitly recognizing the importance of relativity in determining the chemical properties of molecules containing heavy elements, numer­ ous studies have been carried out to clarify the effect of relativity on various aspects of chemistry. Some of the recent reviews are listed in the bibliography [7.10-14].

The many-electron relativistic atomic system can be most properly described by the Dirac-Breit Hamiltonian [15],

H = yhu[i) + y { — + Bij] (1.4) i i

hu{i) = cQi ■ Pi + (A - l)c^ - ^ (1.5)

B., = + (1.6) where h^ii) is the Dirac Hamiltonian for electron i and a and 5 are 4 x 4 matrices:

j - ( i ’,)■ " « and â are the Pauli matrices and / is the identity matrix:

■"* = ( 10)

, 0 )

= (Ô - 1 )

/ = ( J ° ) (1.12)

The Bij of eq 1.6 is the magnetic counterpart of the Coulomb interaction Its effect on chemical properties is rather small, e.g., only ca. 3% of the Dirac relativity change in the bond length of PbH^ [16]. Thus for chemical applications the problem reduces to the Dirac-Coulomb Hamiltonian

H = Y ^ h o { i) + Y ,— (1.13) t i < J for molecules.

Pvykko reviewed up to thirteen methods for relativistic quantum chemical cal­ culations in 1988 [7j. Noticeable recent developments are the all-electron relativistic programs of K. G. Dyall [17] and W. C. Nieuwpoort et al. [18]. The most success­ ful method though has proven to be the relativistic effective core potential (RECP) method so far. The reason for the success of RECPs is twofold. First, the effects of the inner-shell electrons and the major relativistic effects are both folded into the core

4 potentials, so only the valence electrons need to be treated explicitly. Second, many existing non-relativistic algorithms can be easily adapted to relativistic calculations using RECPs. In the following we briefly describe this method.

The RECP method is closely analogous to the non-relativistic effective core po­ tential approach as developed by Kahn. Bay butt and Truhlar [19] and reviewed by

Krauss and Stevens [20]. The following presentation will mostly follow the paper by

Christiansen, Ermler and Pitzer [13].

Neglecting the Breit interaction term, the atomic relativistic Hamiltonian is:

H = J2hu{i)+Y,— (1.14) I i

PnkXkm{9, O) t^nkmir.9.0) = - (1.15) r ^Qnk^—kmifi 1 ®)

The angular and spin functions are given by

.Vkm(e.C))= ^ (1.16) where m — a.a) are Clebsch-Gordan coefficients, are spherical har­ monics. and 0? are Pauli spinors (i.e. electron spin wave functions): 2

and is eigenfunction of the operators p , and j., as well as the operator K =

I + a-f:

KXkm{0,(f>) = {I + â-r)Xkm{e,) = -kXkm{d:Ci>) (1-18) The relativistic quantum number k is related to the total angular momentum j and orbital angular momentum I by;

( U 9 ,

As in the Hartree-Fock equations, the relativistic X-electron atomic wave functions

can be constructed from antisymmetrized products of one-electron functions, in this

case spinors similar to eq 1.15. The one-electron spinors can be partitioned into core-

and valence-spinors. and li-'J. Then we have

- - - <)j (1.20)

The total energ}' then is separated into core, valence and core-valence interaction

energies:

E^ = E^ + E^, + E^ (1.21)

The last two terms can be combined using a modified valence-space Hamiltonian:

/fr' = + E — + (1-22) V ''w ' VC + = (1.23) where Jc(v) and A'c(v) are coulomb and exchange operators. The effect of the core electrons can be combined into an effective potential For one valence electron we have:

(/zD + (/"")x^ = 6vX^ (1.24) where Xv is a pseudospinor. The pseudospinors are defined to reproduce exactly the valence spinors in the valence region and to be nodeless in the core region [21].

6 Expanding in terms of the angular functions (1.16) yields the coupled radial equations for the large components and small components Q^:

pps d ■ pp* ■ I + Q[ev - U^] ^ V (1.25) dr —a[ev — Q r where and are large- and small-component pseudopotentials and the super­ script ps stands for pseudo.

Since the effects of the small components are negligible in the valence region, there is no need to include direct relativistic effects, which are important only in the core region, in the valence-electron Hamiltonian. Thus the Hamiltonian for the valence electrons is composed of the nonrelativistic Hamiltonian for the valence electrons plus the relativistic effective core potentials (RECPs), which include the effects of the core electrons as well as the relativistic effects on the valence electrons in the core region

[22]. The corresponding radial equation is then:

( 1.2 6 )

For more than one valence electron the corresponding core potentials are given by

[13]

riU 2 ^ ^ - I V , . 4- rrREP + 7 ~ + ^u)Xlj (1.27) - 7, where \ij is the large component of the pseudospinor and Wij represents the two- electron integrals of xij with all the rest of the pseudospinors. The pseudospinors are obtained using the shape-consistent algorithm of Christiansen et al. [21]. The potentials obtained from eq 1.26 have the form

^—0 j= (/-il

Oij = 51 \ljm){ljm\ (1-28) m = —j

A This form of core potential, with the spin-dependent projection operator Oij- is suit­ able to be used in j — j coupling calculations. However, this can also be expressed in

A terms of the spin-independent projection operator Oi

Of = 5Z = 5Z Oij m = - l J

1=0 1 = 0 = + (1.29) as the sum of core potentials and spin-orbit operators [23]. It is found [22] that, the

^'AREP^^^ are approximately independent of I when / > L + 1, where L is the largest angular momentum of the core electrons. Then and Hso can be reduced to

fjAREP _ (1.30) 1=0

H s o = X;^,(r)r-sO, (1.31) f=I

With RECPs and spin-orbit operators given in the forms of (1.30) and (1.31), exist­ ing programs for nonrelativistic calculations can be adapted to include the relativistic effects. The spin-orbit interaction in the form of (1.31) can be included in the corre­ lation step. Such methodology is adopted in the pioneering work of R. M. Pitzer and

N. W. W inter's spin-orbit C l program , CIDBG [24]. For molecules containing heavy elements, the presence of d and / valence shells with possibly large numbers of open-shell electrons will give rise to a large number of closely spaced configurations, so multireference Cl calculations are expected in gen­ eral. In addition, the coupling of electrons in the cases of heavy elements are likely to be neither \-S nor J-j coupling. A reeisonable way of treating intermediate cou­ pling cases would be to include the spin-orbit interaction in Cl calculations. States that are of different symmetries in spin-orbit-free calculations will be mixed by the strong spin-orbit interaction in spin-orbit Cl. As a result, the reference space should in general also include, besides the near-degenerate configurations in a spin-orbit-free description, all the configurations that strongly interact through spin-orbit effects.

For such multireference calculations. Cl is the best correlation method. As a conse­ quence of the large number of closely spaced configurations and the strong spin-orbit interaction for systems containing heavy elements, the spin-orbit Cl spaces may be an order of magnitude larger than those for systems containing only lighter elements.

One limitation to CIDBG mentioned before is that it calculates and stores the whole Hamiltonian matrix, so the calculations are limited to matrices of order 500.000 on today's mainstream workstations. It has been shown by Yabushita [25, 26] that the Columbus Cl programs based on the Graphical Unitary Group Approach can be adapted to include spin-orbit effects and thus make larger calculations possible.

In this work we describe the spin-orbit Cl based on the Graphical Unitary Group

.Approach and its implementation in a recent release of the COLUMBUS program package. Using these programs we carried out calculations on several uranium compounds.

UOo'*'. UOgOH^ and UC^^ fluorescence was studied in 1849 [27], even be­ fore the word ‘‘fluorescence" was coined. Most uranium compounds contain UO^'*' and most uranyl compounds in solids or solutions exhibit characteristic fluorescence pat­ terns due to the UO^"^ moiety. UO^OH^ is the simplest model of a complexion

in solutions. U(BH 4 ) 4 is one of the few molecular (as opposed to ionic) solids among uranium compounds and its spectra have been extensively studied [28,29]. Following the discussion of the spin-orbit GUGA Cl, our calculations on these systems will be presented.

1 0 CHAPTER 2

Spin-Orbit GUGA Cl Programs

In this chapter we first review the Graphical Unitary Group Approach (GUGA) as applied to the correlation problem in electronic systems [30-35], followed by the discussion of the spin-orbit interaction in the Unitary Group Approach formalism.

Finally we will discuss the implementation of the spin-orbit interaction in the cur­ rent version of the COLUMBUS Graphical Unitary Group Approach Configuration

Interaction (Cl) (singles and doubles) programs.

2.1 Review of Graphical Unitary Group Approach

The spin-independent Hamiltonian of an .\-electron system.

+ 4 . ( 2 1 ) can be expressed in second-quantized form as

ffo = E E + i E Ek; ki]x*x^,x,rX,„ {2.2) i ,j

[ij: kl] are the usual one-electron and two-electron integrals over the spatial orbitals:

11 The one-body and two-body operators and eij.^ki are defined by [30]:

E A%A;,(z.j = L2.. .,n). (2.3) cr=Q.^ and

ei],ki = EijEki - SkjEti (2 .6 ) and satisfy the commutation relation

Eki] = EijEki - EkiEtj

= SkjEii - diiEkj, (2.7)

The Hamiltonian can be written in terms of these unitary-group generators and generator products e,

H q = Y 1 ^ E (‘-- 8 ) ij ~ tj,k,i

The evaluation of the matrix elements of H q over a chosen Hilbert space is then re­ duced to the calculation of the matrix elements of the unitary-group generators Etj and generator products Cjj^ki in that same space.

The iV-electron Hilbert space is usually constructed from the antisymmetric .V-th rank tensor product of the one-electron Hilbert space defined by an orthonormal set of n spatial orbitals, and its basis can be labeled by the subgroup chain [31]:

Uin)DUin-l)D---DUi2)DU{l). (2.9)

1 2 Such a basis is known as the Gel'fand-Tsetlin basis and the individual basis functions

are referred to as Gel’fand states. The GelTand state can be represented by a Gel’fand

tableau: rn.\n ^ 3n rUr.

[m] = ( 2 . 10)

TTI\2 TÎI22 m il where the row vector

^nn ) ( 2. 11)

is the highest weight vector uniquely labeling the irreducible representation r(m „) of

lj{n). For electronic systems. m,„ can only be 0. 1 . or 2. so the Gel'fand tableau can

be simplified to the Paid us ABC tableau [32]:

Qn—1 1 Cji — I

[a 6 c ] = (2 . 12)

Oi 61 Cl 0 0 0

where a, b. and c denote the number of 2’s, Ts, and O’s of the highest weight vectors

respectively and satisfy the following conditions:

0,1 + bi + Ci — i,

2a„ + bn — N, (2.13)

bn = 25, (2.14) where N is the total number of electrons and 5 is the total spin quantum number.

Noticing that a given subtableau may appear in many different Gel’fand tableaux

13 or Paid us ABC tableaux representing the GelTand-Tsetlin basis. Shavitt proposed and developed [33J an elegant scheme of representing the basis graphically. VVe next describe the Shavitt distinct row graph.

For the electronic problem under consideration, the irreps of the unitary group

U{n) are uniquely determined by the highest-weight vectors and can be represented by the distinct rows. {an. bn)- The unitary-group irreps can be reduced to irreps of

its subgroup U{n — 1 ) with the property that each irrep of the subgroup, which is

in turn specified by the distinct row (a„_i, 6 n_i), only appears at most once in the reduction. For electronic problems, there are only four possible ways of reducing an

irrep of U{n). F „ ((a „ , 6„)), to the irreps of U{n — 1 ) [34]:

rn((Qn-^n)) — ^ ^ ® fri—I ( (®n—11 I ) ) d fi — j ,b n — I

= r„_i((a„, 6 „)) © r„_i((an, 6 „ — l))© F „ _ i((a „ — 1 , 6 „ + 1 ))

®rn-l((^n —

(2.15)

For each of the four cases in reduction (2.15), a step number can be defined:

dj = 3A aj + Abj = 0,1, 2, 3 (2.16)

A oj = Qj — Oj-I,

Abj = bj — bj-i

The step numbers dj represent the spin-coupling scheme of orbital level j: dj = 0

corresponds to ASj = 0 and ANj = 0, dj = 1 to ASj = ^ and ANj = 1 , d_, = 2 to

ASj = — 5 and ANj = 1 , and dj = 3 to ASj = 0 and ANj = 2 , where ASj = 5j —5j_i

14 and ANj = Nj — The step vector formed from the step numbers.

d = (dn dn-i di) (2.17) then provides a unique labeling of the Gel'fand states corresponding to the subgroup chain (2.9) and its relation to the representation in terms of Slater determinants can be found in [30].

The step vector form of the Gel'fand basis can be represented compactly as a graph

and such a graph for N = 5 electrons and n = 6 orbitals with spin S = ^ is shown in

Fig. 2.1. The vertices at a level j of the graph are the distinct rows representing irreps of the unitar}' group U{J) and the arcs connecting the vertices at successive levels j

and j — I correspond to the reduction of an irrep of U{j) to irreps of U{j — 1 ) as given by (2.15). Thus there are only four possible arcs with increasing slopes connecting a vertex to the lower level, corresponding to the four step numbers. Each level of the graph, comprising the vertices at that level and the arcs connecting them with the vertices of the lower level, represents one of the orbitals of the one-electron basis. A path from the tail of the graph to the head of the graph then represents a member of the Gel’fand basis. The Gel’fand basis represented by the graph can be enumerated conveniently by assigning weights Xk and y^k to a vertex k and to the arcs. Assigning

the weight x = 1 to the vertex at the bottom of the graph, the weights can be defined recursively in order of increasing level index j [34]:

yok = 0 ,

ydk = yd-i,k + ^k^^u^ (d = 1 , 2 ,3,4) (2.18)

^k — y4ki

15 where k is the running index of a vertex at the level j, Xk is the weight of that vertex.

Tjdk is the weight of the arc with step number d and is the weight of the vertex at the lower level connected by that same arc. In this scheme the weight, ij^k- of an arc that connects a vertex k at level j with a vertex at level j — \ represents the number of walks that pass to the left of that arc at level j and the sum of y^k over the step numbers is just the weight of vertex k (i.e. the number of walks leading from the tail to that vertex), spanning a basis for the irrep of U{j) denoted by the corresponding distinct row. Based on the arc weights, a lexical index for each walk can be defined:

m{d) = \ + '^yd,kj- (2.19)

Once the graphical representation is defined it can be shown [30,35] that the ma­ trix elements of the operators (2.5) and (2.6), can be derived entirely graphically.

Knowing the one- and two-electron integrals and the values of the matrix elements of the generators (2.5) and (2.6), the non-zero matrix elements of the Hamiltonian can easily be identified and calculated. With this compact representation of the Hilbert space as a graph, the structure of the calculations can easily be visualized and very efficient algorithms based on the Shavitt graph can be designed and implemented

[36-43]

16 2.2 The Spin-Orbit Interaction in the Graphical Unitary Group Approach

Considering the spin-orbit interaction explicitly, the total Hamiltonian is:

Hiotax = Ho + Hso (2 .2 0 )

where the spin-free Hamiltonian H q is given by (2.1) and (2.2) and Hso is given by

V

Hso = ^ hso(lJ-) (2 .2 1 ) /I=l with the one-particle spin-orbit interaction hso for electron fj, given by:

hsoifi) = ?(//) • s{n)

= ( 2 .2 2 ) 7 ==0 ,± 1

where the 7 summation here and in succeeding equations is over the component indices of a vector in spherical tensor form. In the simplest case, the H atom. q =^^ - For the general form of spin-orbit operator actually used in the ECP formalism.

.4 U = l

A where .4 denote the atomic centers and Ou ^re projection operators defined earlier.

In all cases, is a pseudovector (even to inversion) like /, and is pure imaginary.

In second-quantized form, the spin-orbit part of the Hamiltonian can be written in terms of the U{2n) generators as

Hso = (2.23) i j 7

17 where i.j = 1 . ■ • •. n and cr. r = q. j are the orbital and spin indices respectively and

V ^ iicr(/x)>OV(//)| /l=l = (2.24)

.A.ccording to the Pauli exclusion principle, the eigenfunctions of the spin-orbit

Hamiltonian for an .V-electron system with n orthonormal spatial functions span the irrep defined by the totally antisymmetric component of the N-th rank tensor product of the one-electron Hilbert space which exhibits the unitary symmetry U{2n) and the wave function can be expanded in terms of a suitable basis for this irrep. We can simply choose the basis to be the Slater determinants, which are not spin-adapted in general. On the other hand we can choose a spin-adapted basis by considering the subgroup chain:

D'(2 M) D &'(n) X &;(2 ). (2.25)

In this representation, a basis function of the U{'2n) irrep is simply the direct product

of a basis function of an irrep r(a„, 6 n) of U{n), with specific total spin quantum

number S, and a basis function of the irrep of U{2) conjugate to r(a„, 6 „), which is an eigenfunction of the total spin operator:

|(d)5A/) = |d)|5A/> (2.26)

The advantage of using such a basis is that the solution of the spin-orbit problem can be carried out in the same framework as the spin-orbit-free problem, a point that will be taken up in more detail later. Next we discuss the calculation of matrix elements

18 over the basis (2.26).

.A.S has been shown by a number of authors [25,44-47], matrix elements of spin-

dependent operators can be expressed in terms of those of spin-independent operators.

In particular, the matrix elements of the spin-orbit interaction (2.23) are given by;

{{d')S'M'\H,o\{d)SM)

I.J O’-'T 7

= ÊZ(-WX(l9-7l;)(K)-^'Ar|z^(i.;)l(d)SA/), ( 2 .2 7 ) I . j 7

where

Z^(z\j) = (2.28) a . T

behaves like a rank-one tensor operator when applied to the spin space. Then by the

Wigner-Eckart theorem.

((d')S'A/'|Z,(i,j)|(rf)S.U) = ( - l f - " ' ( _ J: .) J ^ (K)S'||Z(:J)||(d)S)

(2.29)

For the matrix element {{d')S'.\r\Hso\{d)SM) to be nonzero, d' and d must differ

by exactly one orbital, i.e. d' must be obtainable from d by substituting orbital i

for orbital j in state d, and 7 must satisfy 7 = M' — M. Thus, only one term in

the summation in (2.27), of specific i, j and 7 , contributes to the spin-orbit matrix element between given states \{d')S'M') and \{d)SM). So eq 2.27 can be further simplified to

{{d')S'M'\Hso\{d)SM)

= (-l)Xz|9_^|j>

19 = f _ J ! I J j ((/)5'||Z(.,;)||(j)g) (2.30)

The reduced matrix element. ((d') 5 '||Z ( i , 7 )l|(d ) 5 ), can be evaluated in terms of

the U{n + 1) group generators (see. for example. [45]):

((d').9l|Z(f.;)||(d)S) L }" {{d').\+iSx+iM:^+i\En-i-i,jEt,n+i + -£ ’ijl(d),v+i5»iA/;v+i). (2.31)

where dx+i denotes the step vector in an (N + l)-electron system with ri + I orbitals

and 5 .V+ 1 and d/;v+i are the corresponding total spin and spin projection quantum

num bers in th e .V + 1 electron system. For the reduced matrix elements to be nonzero.

S' and 5 in (2.31) must satisfy

S' — 5 = 0. ± 1 {but not S' = S = 0) (2.32)

in accordance with the selection rule. Correspondingly, the quantum number S\+i for the (-V + l)-electron system must satisfy: r S + 4, S' = S, (a)

_ I S - S' = S, (6 ) . 1 S + I, S' = S + 1. (c) . s - | , S ' = S-L (d)

Case (d) can be obtained from case (c) by interchanging S and S', so we will not carry' it further. The corresponding step numbers, at the (n + l)th level, are =

dn+i = 1 , d'„+i = dn+i = 2 , and = 2 , d„+i = 1 respectively, for cases (a), (b) and (c), and are illustrated graphically in Fig. 2.2. Expressing the 6-j symbols in eq

2.31 explicitly in terms of S gives:

(а)

(((f)S'||Z(i,;)||(d)S> = (fu)(Ar+i) X (б ) (2.34) x/S+T (c)

2 0 where we have defined

{p'ij)(\ + l) = {{d')x^iSy+iMt\+i\En+i,jEt,n+l + 2 K^)‘V+l‘?iV+l A/^,V + l)- (2.35)

Using the commutation relation (2.7) and the definition (2.6), this can be re-expressed

as

(•f'ij)(.v+D = ((d').v+i5v+i-V/,v+i-f -£',j|(d);V+i5;V+iA/:Vj-i)- (2.36)

The first part of the above matrix elements for is represented by the 8 (b)

type of loop defined by Shavitt [30,48], as shown in Fig. 2.3. The corresponding

value is given by

{{d' ) . \ - + 1 1 M x + 1 |ei,„+ 1;„+ i,j 1(d) ;v+iS ^ + 1 ATv+ 1 )

=u-„(i) n n "'«6 ( ' - ) » # ( " +1 ) + r=i+l r=j+l

n + (2.37) r=j + I

where the lU’s are the one- and two-body segment values and the superscripts ( 0 ) and

( 1 ) denote the direct and exchange contributions of the loop respectively [30, 48]. The

direct contribution is identically zero unless the bra and ket coincide above the level j and so is the contribution from ^Eij. So we need consider the direct contribution

only when the bra and ket are the same above the level j,

w S ( j ) n + = --L ' ' Ù 1 ' A ? r= i+ l r=j + l V Z

= -\w-ifU)- (2.38)

Then the first term of eq 2.37 becomes

-hW ') n lf'B('-)VC(j). (2.39) ^ r=»+l

21 exactly cancelling the contribution from Thus (Fy)(jv+i) in the reduced matrix element (2.34) is entirely given by the exchange contribution of

= »£(<) n «'«(OWstO) n (2.40) r = t + l r~ j+ l

For cases (a-c) in (2.33). the segment values fF ^(n + 1) are given by [30.48|:

- ~\/2(fiT) case{a)

*. 4 (3 . 1) = = v w

1 case(c)

Combining eqs 2.34. 2.40 and 2.41, we get for the reduced matrix elements

((d-lS'II^C,,j)ll(d)S) = { ^ (2.42)

The value (F,_,)iv denotes the product of exchange segment values of the 8 (b) type loop for the two-body operator e^,n+i;n+i.j up to the nth level:

( F I = / ■ ncL (W s ( r ) . » l2 ( i ) ■ ^v^dr) ‘ < j j , , ' ' 1 W£(i)-n:;i+,14L(r).4Ç(i).nP=,+,44«l(r) i > j

With the reduced matrix elements given by eq 2.42, the spin-orbit matrix element

(2.30) becomes

{{d')SM'\Hso\{d)SM)

= { - i f ^ ) \J 2

(2.44)

{{d'){S+l)M'\Hso\{d)SM)

= ,vF - M M ) '/5+l(Fÿ),v

(2.45)

2 2 To further simplify the expression for the spin-orbit matrix elements, we need to discuss the choice of the spin functions \SM). The spherical form of the spin functions

\SM) is not symmetry adapted to the point group irreps and the spin-orbit matrix elements are complex in general. To have an efficient algorithm for solving the many- electroii problem, it is essential to choose a symmetrv-adapted many-electron basis and to use a real Hamiltonian matrix. For this purpose, the following Teal spherical" form of spin functions can be chosen:

ISA/-) = ,^-^[|S, -A/) - (-1)'"^|S. M)l M = l to S (2.46) v 2 ISA/+) = -, — [|S. -A/) + (-1)''^|S, A/)], A/ = 0 to S (2.47) / 2 4- 26AA0

-\s will be discussed later, the odd number of electrons case can be adapted to the formalism for the even number of electrons case, so the following discussion will be concentrated on systems with even numbers of electrons. The point groups used

in the COLUMBUS programs are the D 2/1 group and its subgroups, so our discussion will be confined to the D-2 h group also, even though some conclusions are not specific to this requirement.

Under the (double) group, the Cartesian components of the angular mo­ m entum {Rx,Ry and R^) transform as Bzg,B 2 g and Big respectively and the many- electron spin functions of a system with an even number of electrons transform ac­ cording to Ag + Big + B 2 g + Bsg [24] (The 5 = 0 spin function transforms as Ag).

For groups with lower than D 2 or C2 V symmetry (in our case, the C 2 , C 5 , C, and

Cl groups), as well as for C 2/1 symmetry, more than one component of the angular

23 momentum transforms according to the same irrep, so we classify the irreps by the

transformation property of the components of the angular momentum. Thus for a

system with an even number of electrons, the many-electron spin functions transform

as

.4. -j- "b Aÿ 4- R~ (2.48)

For integral values of total spin 5, it can be shown, by examining the effect of the

D '2 group operators C2 i,C 2 y and C2 :, that the spin functions (2.46) and (2.47) are

already symmetry-adapted to the irreps, as listed in Table 2.1.

S=even S = o d d M spin function sym m etry spin function symmetry M = 0 150) -Aj ISO) R. |S M -> R. ISA/-) AI1 f — 1 i.V.U 1 = . . . Ry \SM+) Ry ISA /+) R, ISA/-) R. ISA/-) Ag \ / — 9 4 6 • ■ • IS.1/+) Ag ISA /+) Rz

Table 2.1: Symmetry Properties of "Real Spherical” Spin Functions

Using the Cartesian form of the spin-orbit integrals over real orbitals. (a|gl 6 ), we can easily show that the spin-orbit matrix elements are purely real and that only one of the Cartesian components, q^Sx-, qySy or ç-S; will contribute to a given matrix element if the spin functions (2.46) and (2.47) are used. First, notice that, by the selection rule, we expect that q.s~ can couple states with the same M values and

or qySy can only couple states with M values that differ exactly by 1 . Substituting the 3-j symbols, the matrix elements (2.44) and (2.45) become:

24 (^ ' = S) 1 {{d')S..\r-l\Hso\{d)S.M) = ■‘2 \| .1/ {{d')S.M\Hso\{d)S.M) = V « S ( S + 1 )

1 {S 4- XI 4- l) ( 5 — -V/) y ( 1). . (2.49) 2 \l 5(5 4-1)

{S' = 5 + 1) (5 4- XI 4-1)(5 -h XI + 2) , . (I), ,. „ . {S, MlHsolS + L M + I) = - 2(25 + l)(25-f 3) (I*'

(5 + XI + l) ( 5 — XI + 1 ) y (1). . {S.M\H,o\S + 1.M) = - (25 + l)(25 + 3) (5--V/ + l)(5-A / + 2) (5. A/1 /fso 15 4-1, XI — 1) = — 2(25 + l)(25 + 3)

Here we have dropped the indices, d' and d, for the spatial part of the wave functions for simplicity. Transforming to "real spherical” spin functions gives

(5,A/.+|//so|5.A/-l,±)

= +-[(5, -.V/)| + (-1)‘^^5, XI)\]Hso[\S.. -{XI - 1)) ± (-1) "-M5. XI - 1)]

= -A/|//,o|5, -XI + 1) + (5, A/|//so|5, XI - 1)]

z , 1, (5 - XI + 1)(5 + XI) - - ( - g ) 5(5 + 1) (5 - XI + 1)(5 + XI) 25(5 + 1)

(2.51)

25 where the purely real spin-orbit integrals

i) - \= -iq (2.52) are used instead. Similarly

{S.M.t \H,o\S.M - l .f )

= \[{S. -A/| ? (-1)‘''(5. A/l]//so[|5. -A / + I) T M - 1)]

-[{S.-M -hi) - (S, M\Hso\S. M - I)]

I |(S-A/-M)(5-h. iV I (4 2^\) 5(5-hi)

{ S - M+l)(S + M) 2 5 (5 + 1)

1 (2.53) 2 \|

{S..\L+\Hso\S,.\L~)

^[(5. -A/| + (-I)"'(5. A/|]//so[15. -A/) - (-1)"'|5. A/)]

^[(5.-M\Hso\S. -M ) - (S. M\Hso\S. A/)]

\/25(5 + l) M (2.54) \/25(5 + l)

(5,A/,=f1//so15 + 1 , - V / - L + )

= ^[(5, -A /| f (-1)'^(5, iV/|]//3o[|5 + 1. -(M - 1)) f (-1)'""-'|5 + 1. (A/ - 1))]

= ^[(5,-A/|^3o|5 + 1, -(M - 1)) - (5, M|/fso|5 + L {M - 1))]

2 \

26 2(25+ l)(25 + 3)

(5 - M + 1)(5 - M + 2) 2(25 + l)(25 + 3) ^

(5.A/.+|i/so|5 + l.A/- l.±)

1 = ±:j[(5. -A/| + (-1) "(5. M\]H^[\S + 1, -{M - 1 )) ± {-iy^‘~^\S + 1, (.U - l))]

= ± ^ [(5 . -M\Hso\S + 1.-{M - 1)) + (5. M\Hso\S + 1. {M - 1))]

1

^ 2 \| 2(25+ l)(25 + 3)

( 5 - A/ + l)(5-A/ + 2) iV 2(25 + l)(25 + 3)

1 ( 5 - A/ + l)(5-A/ + 2) = + (2.56) 2(25+ l)(25 + 3) V 2 \

(5. A/ + |i5so|5+ LAF+)

| [ ( S . -M\ T (-1)” (5, .V/|l//„l|5 + 1 . -M) T ( - 1 ) " |S + 1 . -U)|

;([(5. -A /|H „|S + 1, -V/) + (S, + 1, .V/)]

(5 + A/ + 1)(5 — A/ + 1) N N (25 + l)(25 + 3) (5 + A/+ 1)(5- A/+ 1) (25+ l)(25 + 3)

(2.57) (25 + l)(25 + 3)

In sum m ary

(5, \L t \Hso\S.m - 1, ±) = (5, M - 1, +i/7so|5, A/, ±)

27 (5, M. 1=|/f„|S, M - I. T) = -(S , ,V/ - 1. T|//„1S. -Ï/. T) 1 2 \

(5.-U.±|/fso|5,.U. f)

= ±^=ü=(z|A,U)(F.,)x (2-60) /2 5 (S + 1 )

( 5 ..U . t |F so 1 S + L A / - L t ) 1 (2.61) / 2 \

(5.A/.=F|-f^so|5 + L-U + l.T ) 1 (5 4- M + 1)(5 4- M + 2), ^ i.wr \

2(25+l)(25 + 3 ) ^ I- 4 2 )( v)'V V ^ \

(5.A/.+|Fso|5 + l.A/-L±)

1

(5,A/,+|//sol5+l,iV/ + l,±) 1 (£±AL±y|l^(=|,X„|,-)(F,), (2.64,

(5,M ,+|Hsol5 + l,iV/, +)

28 Thus these matrix elements are shown to be purely real and have contributions from

only one component qxSx, Qy^y or qzS^. Simple symmetry arguments can be used to

get the same conclusion in cases where Rx, Ry and i?- transform as different species

[24]. Notice that the above arguments are independent of the symmetry group used

for the system, so implementations based on this formalism can handle cases with

symmetry groups lower than D-i or C^v

2.3 Spin-Orbit GUGA Cl: Implementation and Performance

The work on GUGA-based Cl programs started in the early 1980s [42]. Satoshi

Yabushita [26] added the spin-orbit interaction to an early version of the COLUMBUS

GUGA programs. Over the past two decades, the COLUMBUS system of programs

has evolved into one of several popular and highly efficient ab initio quantum chemical computational packages available and is portable to most major computers. Many enhancements have been added to the original version [49]. In this section we de­ scribe the implementation of the spin-orbit GUGA Cl in the impending release of the

COLUMBUS programs (version 5.4. 0 .2 )

The post-Hartree-Fock Cl part of the COLUMBUS programs consists of several separate program units, namely the CIDRT, CISRT, CIUFT and CIUDG programs.

The programs use the integral-driven scheme [36] in which the contributions of each of the one- and two-electron integrals are considered in turn in canonical order. The

CIDRT program generates and stores the information about the Shavitt graph for

29 later uses. The CIUFT program calculates the so called loop values (the matrix el­

ements of the unitary-group generators and generator products), i.e.. the coupling

coefficients for the matrix elements. The CISRT program sorts the integrals into the

order in which they are to be used. The CIUDG program diagonalizes the Hamilto­

nian matrix.

Before we discuss the implementation of these program units as relevant to the spin-orbit formalism, a brief review of the algorithms of the non-relativistic GUG.A.

Cl is in order. This discussion will be mostly based on reference [34]. The multiref­ erence Cl space is generated by exciting one or two electrons from a set of reference configurations. The orbital space is then divided into internal and external spaces where the internal space consists of all the singly- and doubly-occupied orbitals in the set of reference configurations and the rest of the orbitals constitute the external space and are placed at the bottom of the graph. The internal space can be further divided into inactive and active orbital spaces. The inactive orbitals are all doubly occupied and are usually placed at the top of the graph. The active orbitals have variable occupation numbers from zero to two (in the reference configurations), and can be placed either below or above the inactive orbitals. This partition results in a simple structure of the distinct row graph as illustrated in Fig. 2.4. for the internal part only. There are at most four vertices at each level in the external part of the graph with singles and doubles excitations. The four boundary vertices between the

internal part and external part of the graph are named Z vertex (a = 6 = 0 ), U vertex

(a = 0 , 6 = I), X vertex {a — 0,b = 2), and W vertex {a = 1 , 6 = 0) going from right to left, corresponding to no electron, a single electron, two electrons with triplet

30 coupling, and two electrons with singlet coupling in the external space respectively.

A walk representing a Gel'fand state then is divided into an internal path and an external path lying within the internal and external parts of the graph respectively.

The ordinal index of a CSF is then first determined by the ordering of the internal path and then by the ordering of its external path among all the external paths shar­ ing that same internal path. The internal paths are grouped by the boundary vertices from right to left and each group is ordered by reverse lexical ordering. Two vectors, the index vector and the symmetry' vector, are used to represent the internal paths and their combined symmetr)% There is one entry for each of the internal paths in these two vectors. An entry in the index vector represents the ordinal number of the first Gel'fand state that shares that internal path and any internal path that is to be excluded is represented by the value -1. The symmetry of a walk is obtained as the direct product of the internal walk symmetry, given in the symmetry vector, and the external walk symmetry, calculated on the fly.

With the partitioning of the orbital space into internal and external spaces, the calculation of the loop values required for the evaluation of the matrix elements can be confined to the internal space only and the contribution of the part of the loop in the external space can be combined easily due to the simple structure of the ex­ ternal part of the distinct row graph [42]. As the one- or two-body operators in the

Hamiltonian can couple configuration state functions (CSFs) that differ by at most two orbital indices, the corresponding loops may have zero, one, two, three, or all four orbital indices in the external space and the calculation is structured accordingly. The integrals are sorted into groups with zero external, one external, two external, three

31 external and four external orbital indices and so are the corresponding loop values.

We now proceed to discuss the modification required for the spin-orbit interaction.

Without the spin-orbit interaction, the wave function is an eigenfunction of the 5“ operator with fixed value of total spin quantum number S and the Cl space spans

the irrep specified by the distinct row (a„, 6 „) so the resulting distinct row graph has

a single head (a„. 6 „). The spin-orbit interaction, however, will couple CSF’s with different total spin values 5. In general, the appropriate Cl space spans the totally antisymmetric irrep of U{2n). A suitable basis then is that adapted to the group chain

(2.25). So we need to consider all the irrep spaces of the U{n) with different total spin values S = 26„ in the reduction of the totally antisymmetric irrep r(A(l®^’*)) of

U{2n):

r(.4(l®''')) = ^er((a„,o„)). (2.66)

With the restriction to single- and double-excitations from the reference space, the allowed spin quantum values in the above reduction are determined by the num­

ber of singly occupied orbitals 6 „, i.e. the possible number of unpaired electrons.

The CIDRT program is modified to reflect the choice of the basis of (2.25) instead.

The generation of the DRT table is no longer restricted to a single top row of spe­

cific (a n , 6 „) and there will be multiple such top rows consistent with the number of unpaired electrons. Such a graph for four electrons and three internal orbitals is schematically illustrated in Fig. 2.5.

Consistent with the use of U{n) ® U{2), each walk in the distinct row graph, representing the spatial part of the CSF, must be combined with the spin functions

32 as defined in eqs 2.46-2.47. So the internal path of a CSF will be defined to be the internal part of the walk representing the spatial part of the CSF and the spin func­ tion associated with it. Accordingly, instead of a vertex weight of one, each top level vertex is assigned a weight equal to the spin multiplicity of that particular vertex and the vertex weights and arc weights ydk in eq 2.18 are calculated similarly.

The index vector then will have one entry for each internal path of a CSF and the symmetry' vector contains the internal path symmetry, which is the direct product of the symmetry of the internal part of the walk and that of the spin function as given in Table 2.1.

The CISRT program is modified to sort the the spin-orbit integrals in addition to the ordinary integrals. In the case of the one-body spin-orbit interaction, exactly one orbital index must be different in any two given CSFs for the spin-orbit matrix element between them to be nonzero, so we need to consider only the zero-external, one-external and two-external cases in which zero, one, and two of the differing or­ bitals are in the external space, respectively. Each set of % and q, type of spin-orbit integrals is sorted into groups of zero-external, one-external and two-external inte­ grals similar to the ordinary one-electron integrals in CISRT.

Three subroutines are added to the CIUFT program to calculate the spin-orbit loop values i.j = l---n in eqs 2.58-2.65. These are directly adapted from the zero-external, one-external and two-external subroutines for the spin-orbit-free

case. Only the exchange type of contributions of the 8 (6 ) type of loops need to be considered in the spin-orbit case and the loops for are only evaluated

33 up to level n since the contribution of the (n + l)th level is explicitly folded into eq

2.41 and the loop may be open-ended at the nth level. The quantities stored in the formula tape file for each spin-orbit loop are the weights of the internal part of the ket and bra walks and the internal loop values, as in the spin-orbit-free case. .4.11 the

CSFs with the same spatial ket and bra walks and different spin functions share the same loop value. Instead of the weight of the loop head in the spin-orbit-free case, the spin values corresponding to the ket and bra walks will be saved.

Similarly three subroutines adapted from zero-external, one-external and two- external subroutines for the spin-orbit-free case are added to the diagonalization program CIUDG. In this program, the contributions of each integral to the matrix elements of all the CSFs that share the same loop are considered during each itera­ tion. For each pair of ket and bra walks specified for a given loop, the possible spin functions need to be considered also. The contribution of the spin functions to the spin-orbit matrix elements, i.e. the factors other than the spin-orbit integrals and the spin-orbit loop values in eqs 2.58-2.65, are stored in an array, and array look-up is used to speed up the calculation.

Finally we discuss the implementation of the case of systems with an odd number of electrons. For such systems, a given eigenfunction (pk and its time reversal partner

K(pk, where K is the time reversal operator, are degenerate but independent. The

irreps D{R) and D{R)’ spanned by { 0 } and {Kcp} respectively belong to one of the following three categories, according to Wigner [50]:

(a) D{R) and D{R)' are equivalent to the same real irrep.

34 (b) D{R) and D[RY are inequivalent (2.67)

(c) D{R) and D{RY are equivalent but cannot be made real

The double point groups Ci, C 2 or D2 belongs to case (a), (b) or (c) respectively.

In cases (6 ) and (c), the two components of the Kramers pair belong to different symmetry species, so only one component will be needed in the eigenvalue problem.

In case (a), the Kramers pair belong to the same irrep and the matrix elements

between them are nonzero in general so both components are required. .A.Iso only in case (c) can the Hamiltonian matrix be made real. A simple procedure to overcome these difficulties is to add an extra non-interacting electron to the system to make the number of electrons even, as proposed by Yabushita [26], so that the Hamiltonian matrix can be made real and the calculation can be done using the single point group formalism in the COLUMBUS system. With one extra electron added, we are considering an (.V -t- l)-electron. (n + l)-orbitaI problem instead. The extra electron always occupies the {n + l)th orbital, either with spin up or with spin down, as schematically illustrated in Fig. 2.6. The Hamiltonian is then:

+ (2.68 )

I.J o-.T 7 where the summation over the orbital indices i.j goes to n instead of n 4- 1. Equiv­ alently only the loops and integrals with orbital indices less than or equal to n are needed in the implementation. The dimension of the composite system, without con­ sidering the symmetry, is twice that of the original odd-number of electron system and the Hamiltonian matrix is real. In the case of symmetry, the matrix elements can be made real and only one component of the Kramer’s pair need be considered

35 while in the actual implementation we can carry out the calculation by chosing any of the four symmetry species so the number of real multiplications is still the same.

Similar considerations show that in all symmetry groups under consideration, the number of real multiplications in the actual implementation is the same as that of the original system.

Finally in Table 2.2 we list the timing data for several calculations. All these calculations were done on a Sun Ultra 1 Model 200E Workstation. In all the cases only one root is converged and the convergence criterion is approximately

size of Cl(million) Wall-clock time (hours) 9.16 5.74

4.16 2 . 0 0 1.77 0.76 0.25 0.083

Table 2.2: Timing Data for the CI- Diagonalization Step

36 a = 2 2 111 0 0 0 0

b = 1 0 2 1 0 3 2 1 0

j =6

11X12X13.

14X15. 16X17X18. 19X20X21X22

24X25, 26X27X28.

30X31

Figure 2 . 1 : DRT Graph With n = 6 , iV = 5, 5 = ^. The a and b values of the vertices are shown at the top, and the level indices j are shown on the left. Vertex labels k are circled.

37 level n + 1 'n + l =25+1

" — r1 , — • n + l ~ ~ “ n + l ------n

case (a) case (b)

n + l

■n+l ■n+l = 2

n bn = 2 = 25 + 2

case (c)

Figure 2.2: Graphical Representation of Spin-orbit Coupling

38 Figure 2.3: 8 (b) Type Loop from [30,48]

39 a = 2 2 2 1 1 1 1 0 0 0 0 0

b=210 3 2 104 3 210

J = 0

28X27 .29X30,

22)023, 26j^4)C23

1 # # 5 X 2 # # # 7

Figure 2.4: Internal Part of the DRT Graph

40 a = 2 1 1 1 0 0 0 0 0 b = 0 2 1 0 4 3 2 1 0

Figure 2.5: Spin-Orbit DRT Graph.

41 level

n + l S -

n S + 1

Figure 2.6: Graphical Representation of Spin-orbit Coupling for a System With an Odd Number of Electrons: for formal purposes only, an additional electron is added to make the svstem have an even number of electrons

42 CHAPTER 3

Theoretical Calculation on Uranyl Ion UO^'*’

3.1 Background

The uranyl ion (UO^*^), with a formal oxidation state of VI for U. is very prominent in the chemistr>' of uranium compounds. In compounds, the uranium atom may have different oxidation states. U(VI), U (\’). U(IV) and U(III). The most common oxidation state of uranium is VI and the number of U(VI) compounds is much larger than the number of compounds with other uranium oxidation states. More strikingly,

with a few exceptions (notably UFg, UClg and UOF 5 ), most solid U(VI) compounds are salts derived from the divalent ion. .\s a survey of 180 structures shows [51], actinyl ions are remarkably close to linear, rarely deviating from linearity by more than 5° even in low-symmetry environments. In solids, ions have rather short bond lengths ranging from 1.7 .4 to 1.8 .4 and are coordinated with various ligands at much larger distances in the equatorial plane. The most common coordination number in the equatorial plane is six. Free Ü'*’® ions do not exist in solution. .\ compound derived from this ion, such as UFg, is immediately hydrolysed and leads to the divalent uranyl ion:

+ 2H 2O -4. U0^+ + 4H+ (3.1)

or + 20^- -)■ U0^+ (3.2)

43 For transition metals, the most closely analogous very tight binding between a metal

atom and oxygen occurs in the vanadyl ion [52].

The uranyl ion forms complexes with the anions of water and of various acids. Free

(hydrated) uranyl ions exist at pH below 2 in noncomplexed solutions and further

hydrolysis occurs above pH 2. Near pH 3.5. first dimers then trimers with bridging

hydroxide groups are formed [53] but Raman spectroscopy shows [54] that the dioxo

group is little changed. ,\t pH 5 uranyl hydroxide is precipitated and the dioxo group

is intact in the solid state structure [55]. Certain anions, such as citrate ion. can

prevent precipitation up to pH 10 by forming soluble complexes with uranyl ions. To obtain free (hydrated) uranyl ion one can add an excess amount of perchloric acid.

The presence of sufficient ClOj prevents association of UOo'*’ with anions of water

without forming CIO 4 complexes since ClOj has the least tendency for complexing.

Even though U(\T) in UCle is strongly oxidizing, this property is much attenu­ ated in uranyl compounds with a UO|'*'/U(IV) redox couple of only about 4-0.32 V

[56]. In aqueous solutions, the uranyl ion is strongly acidic (forming hydroxo com­ plexes). comparable to cadmium or lead. .A.n excited species with lifetime longer than

10“' s is expected to thermalize to the Boltzmann distribution and can have its own chemistly. .A. consequence of this is that the excited-state species is more oxidizing as an oxidizing agent and more reducing as a reducing agent than the corresponding ground-state species. The uranyl ion in the first excited state, ‘UOo"^, has a long

life greater th an 1 ms. has an estimated standard oxidizing potential around

4-2.6 V [52] and is as oxidizing as free fluorine [57-59], Notably, it extracts hydrogen from organic molecules [60,61] and forms various exciplexes [52]. 44 The history of human use of uranium compounds is long. Evidence of the earliest

known such use was found at an imperial villa on the Bay of Naples, dated 79 AD.

where the pale green glass from a mosaic was measured [62] to contain approximately

1.5% UOo. According to E. R. Caley of the Ohio State University [63], "... probable

that the U was separately added in the form of a mineral ... and cannot be said that

the coloration was an entirely accidental procedure.’’ "... taken as fixing the approx­

imate time of the first use o f... any kind of material containing U.’’

The first recorded study of uranyl salts came in 1789 when Klaproth character­

ized uranyl salts such as chloride, nitrate, sulphate. ... and obtained brown UO 2 by

strong reduction of pitchblende, an ore containing UsOg (mixed UO 2 and UO3 ), by

hydrogen at red heat [64]. He thought th at this UO 2 was the element and that the

salts contained a metal ion such as U-"*". This was not corrected until many years

later when Péligot [65] reducued UO 2 to metallic uranium using molten potassium.

D. Brewster [27] first studied the optical properties of uranium compounds. His study of a yellow Bohemian glass (canary glass, known to contain uranium) shows

that it absorbs light in the blue and transmits light in the yellow and that the lumi­

nescence ( “internal dispersion” or “dispersive reflexion” ) is in the green range.

In 1852, G. G. Stokes [ 66] studied the optical properties of many solid and solu­

tion samples using a candle, sunlight, and prisms. The best data came from samples

containing uranium: canary glass, several U minerals, and several UOl’*’ salts. These

showed several absorption bands in the blue and several emission bands in the yellow

45 with the green band in common. Stokes coined the term ‘‘fluorescence" for this phe­ nomenon and formulated the following law to describe the shift between absorption and emission: ‘‘The refrangibility of the incident light is a superior limit to the re- frangibility of the components of the dispersed light.” This came to be known as the famous "Stokes law” [67]: ‘The emitted radiation is displaced to longer wavelengths compared to the absorbed radiation.”, which we now know is a result of transitions to different excited vibrational levels.

In 1872 E. Becquerel (père) published fluorescence spectra of several uranyl com­

pounds [ 68]. More studies of uranyl fluorescence by H. Becquerel (fils) appeared in

1885 [69]. A consequence of these fluorescence studies was the discovery of radioactiv­

ity when a sample of K 2U 02(S 0 4 ) 2 was placed next to a covered photographic plate and the image of the sample was observed on the developed plate [70].

Over the years, numerous studies have been undertaken to understand the spec­ troscopic properties of uranyl compounds. The work prior to 1919 was summarized in the monograph by Nichols and Howes [71]. During World War II. as part of the

Manhattan project, pioneering work on single crystals was initiated to study the spectroscopy of several uranyl compounds at liquid hydrogen temperatures in order to develop an optical method of isotope separation. The results were detailed in the monograph by Dieke and Duncan [72] which included a bibliography for 1919-1948.

A later book covered both the spectroscopy and photochemistry of uranyl compounds

[61]. Improved measurements came in the late 1970’s from the low-temperature laser spectroscopic studies by Denning and coworkers [73-77]. In the following we briefly

46 discuss the spectroscopic aspects of uranyl compounds.

.A.I1 uranyl salts absorb light of wavelength shorter than 4800 A and fluoresce brightly when illuminated. The absorption in the blue region gives the salts a yellow­ ish color. Both the absorption and fluorescence show regular band structures even at room temperature. The absorption extends from the blue to the ultraviolet and the fluorescence goes beyond the green toward the red. The spacing is ca. 830 cm“^ in absorption and ca. 710 cm~^ in fluorescence and the exact position varies from compound to compound. These transitions are attributed to the uranyl ion since all the uranyl salts show the same essential features and these features persist even in solutions and the differences are attributed to the interaction of uranyl ions with ligands and crystal fields. The fluorescence is an essential feature (independent of impurities present) and is due to the first excited state since it is independent of illuminating light.

There are three vibrational modes for the linear uranyl ion, the symmetric-stretch mode (z/s mode, infrared inactive, Raman active), the asymmetric-stretch mode (i/a mode, infrared active, Raman inactive), and the bending mode (z/y mode, doubly degenerate, infrared active, Raman inactive). The harmonic frequencies of these are:

z/s = \ --- — (3.3) V ttio

47 where mo and mu are the masses of the oxygen and uranium atoms, / is the force

constant of the Ü - 0 bond, f' is the interaction constant between the two U - 0 bonds

and d is the force constant for the bending of O-U-O. The constant / is often neglected compared to /. giving the simplified relationships

Us = J — (3.6) V mo

+ -—-) (3.7) V mu

n, = (3.8)

Raman frequencies cluster near 860 cm~^ and 210 cm“ ‘ in solutions of uranyl chloride, sulfate, and nitrate [78] and infrared frequencies cluster near 930 cm~^ (strong) and

860 cm~^ (weak) in solid uranyl chloride and acetate [78] and some other compounds

[79]. These are assigned as z/g % 860 cm~^ % 930 cm '\ and f/y % 210 cm"^

Since both the ground and excited states have Dock symmetry, it follows from the

Franck-Condon principle that only totally symmetric vibrational quanta can be ex­ pected to accompany the electronic transitions. Indeed, the spectra show simple band

progressions mostly from a single (symmetric) vibrational mode (up to 8 quanta) in­ stead of the complicated structure expected if the two closely spaced symmetric and antisymmetric frequencies were both excited. Only one or two quanta of the antisym­ metric vibration are observed and this must be due to the anharmonicity. The fact that two quanta of antisymmetric vibration are usually observed is due to the more pronounced anharmonic character of excited electronic states.

Denning [77] has done a series of measurements on crystalline CS 2 UO 2 CI4 and clarified much of the electronic structure of the uranyl ion. The crystal structure is

48 monoclinic with space group Co/m and there is only one molecule in the unit ceil. The

site sym m etry a t the U atom is C 2 /, with four chloride ions in the equatorial plane of the uranyl ion. At low temperature (4.2 K), absorption spectra of low intensity with long progressions are recorded. The excited symmetric-stretch frequency of 710 cm“^ is lower than the ground-state value of 830 cm 'k When '*’0 is replaced by the uranyl symmetric-stretch frequency drops from 710 cm“^ to 675 cm“^ and the 0-0

transition shifts 1 0 cm"^ to higher frequencies, so the vibronic states from different origins are easily distinguishable and the the origins are easily identified. Twelve electronic excited states are thus observed between 20,000 cm“^ and 29,000 cm"'.

With the strong spin-orbit interaction in the uranium atom, there is no effective spin selection rule so the weak transitions are most likely due to the parity selection rule, indicating that the transitions are gerade to gerade with an electron excited from an ungerade occupied orbital to an ungerade unoccupied orbital. With a ger­ ade ground state, the gerade excited states can only be magnetic-dipole or electric- quadrupole allowed. Polarization experiments confirm that all the observed transi­ tions are gerade to gerade. The uranyl ion with D^h symmetry is slightly perturbed by the almost tetragonal field of the chloride ions and the electronic states can be approximately descibed by Doh symmetry^

Denning’s observed electronic states [77], along with the transition mechanism for one-photon and two-photon absorption, are listed in Table 3.1. Because the first two origins are allowed by the x and y components of the magnetic dipole moment, they

49 O rigin Sym m etry(D 2 /i) Energy/ cm ^ Mechanism OPA“ TPA* ground state Ag 0 I B2g 20095.7 hy xz II 20097.3 .y- III B\g 20406.5 Qxy xy IV Ag 21316.0 XX. y y \- B2g 22026.1 xz \T B^g 22076.0 \TI Ag 22406.0 XX. y y M il Bxg 22750.0 xy IX Bzg 26197.3 xz X Bzg 26247.6 fJ'jj y- XI Big 27719.6 yz XII Ag 27757.0 XX, yy XX XIII -■^9 29277.0 h i XI\' Big 29546.0 yy

“One-Photon Absorption *’Two-Photon Absorption

Table 3.1: Properties of the Electronic States of CsgUO^CU, reproduced from [

are expected to be a Ig state in cylindrical symmetry. In the lowered crystal symme- tr\', Ig states can be mixed with 3g states. The shifts of the individual components are too small to be measured in a magnetic field of 5 T, but the mixing of these two components is large enough for the magnetic moment to be determined to be 0.16

/ZB and it thus can be concluded that the first excited state is predominantly Ig. To have a near-zero magnetic moment with two singly occupied orbitals, the orbital and spin magnetic quantum numbers along the uranyl axis must be A = ±2 and E = =Fl, giving a ^Aig state in \-S coupling.

50 The uranium /<^ and orbitals interact with oxygen orbitals, leaving the non­ bonding fs and orbitals as the lowest unoccupied orbitals. So only two excitations can give rise to an excited state with = 1 and near-zero magnetic moments; either

one electron is excited from cTu to d'„ or from tTu to 0 u (tTu^J^). .A. notable difference between these two electron configurations is that there are twice as many states in the latter configuration within approximately the same energy range. The observed state density between 20,000 cm"^ and 29.000 cm~‘ is consistent with the

configuration. The observed large equatorial field splitting (states III and IV in Table 3.1) also confirms that these states are from this configuration. This can be seen from the following arguments [76]. A tetragonal equatorial field Vêq can be expanded in terms of V)*"* with m = 0.4, 8 ... Then the diagonal matrix elements in state Q are constant and the degeneracy can only be resolved if (— / 0. In the A — ü coupling scheme, the wave function can be specified by | ±Q) = |A. T. ±Q).

Since Véq is spin-independent. (Qlléq] — fi) ^ 0 if and only if S = 0. States with T = 0

are given by [A) = i ( |A i A 2 l ± IA1 A2 I), so we need to consider matrix elements of the

type (AiA 2 |V ^| — Ai — A2 ). Since Véq is a one-electron operator, the two wave functions in the above matrix element must be single replacements of each other, i.e. either

Ai = 0 or A2 = 0. This can be true only for the configuration Taken together, it can be concluded that the observed excited states must be from the configuration

instead of

Higher energy excited states have also been studied experimentally by absorption from excited states. Strong absorptions around 17,500 cm”^ are observed both in

solid state CsU 0 2 (N 0 3 ) 3 [80] and in aqueous solutions [81]. The essential features

51 of these absorptions are independent of the ligand environment and the absorption occurs after the rapid relaxation to the first excited state so the absorption must be intrinsically from the first excited state of the uranyl ion. The absorption shows clear progressions in 570 cm“^ and therefore a much-weakened bond in the upper state.

The intensity is very strong so the absorption must be from the gerade first excited state to a (parity-allowed) ungerade state. This can happen only if an electron is either excited from a gerade orbital to an ungerade orbital or vice versa. In the first case one electron may be excited from the fs orbital to the nonbonding ds orbital.

Since ds is nonbonding, the change in the bond length cannot be as large as suggested by the large decrease of the vibrational frequency. In the second case, one electron in the filled ag or -g shells may be promoted to occupy the hole left in the a^ shell and a large increase in bond length may be expected. Preliminary experiments indicate that this transition is polarized along the ion axis, so the excited electron configura­ tion may be expected to be a^Sl [77].

Given the importance of the uranyl ion in uranium chemistry and the wealth of experimental data it is not surprising to see a large number of theoretical calcula­ tions on the uranyl ion to understand various aspects of it. The uranyl ion is also an important test case for benchmarking theoretical methods for heavy elements that re­ quire relativistic treatments. Both nonrelativistic and relativistic methods have been

used, including MS-Xq [82-85], Discrete Variational Method [ 8 6 - 8 8 ], Effective Core

Potential Method [89-91], Extended Hiickel Method [92-94], local approximation for relativistic scalar operators [95], density functional method [96], and four-component

52 spinor calculations [97]. Most of these calculations are devoted to the understand­

ing of the bonding and orbital orders in the ground state. The role of / orbitals in

the bonding of uranyl was first investegated by Belford and Belford [98]. Without

relativity, the overlap of / orbitals with the oxygens is much smaller compared with

the d orbitals. If relativistic orbitals are used instead [99], the contraction of .s and p

orbitals will shield the nucleus from the valence orbitals, producing a large expansion

of the / orbitals and. to a lesser extent, the d orbitals. The overlap of the / orbitals

becomes comparable to that of the d orbitals. The strong bonding of oxygen p orbitals

with the uranium / orbitals is seen to be responsible for the strong bond in uranyl

ion. The linear geometry of uranyl ion is generally attributed to the interaction of the

Ü 6d with the O 2 p orbitals, which “pushes from below” [52] the O 2 p orbitals so that

the overlap between the / orbitals and O 2 p orbitals become more effective, as origi­ nally suggested by Tatsumi and Hoffmann [92] and supported by various calculations

[88.93.100]. Just recently, Ken Dyall [101] carried out all-electron DHF calculations comparing iso-electronic ThOo, PaOj and In the linear configurations, strong

.A.n6d-02p bonds and strong .\n5/-02p bonds with antibonding .A.n6p-02p character were found. The efficient 5f-6d hybridization in ThOg upon bending due to the sim­ ilarity in energv' and radial extent of Th 5 / and 6d orbitals, contrary to the case of

PaO j and UOg"*", where 5/ is lower in energy than 6 d, is found to be responsible for the bent structure of ThOg.

As far as we know, no systematic calculations on the excited-state properties of the uranyl ion are available so far. In the rest of this chapter, we discuss our results for

53 such calculations. Specifically we are interested in the bond lengths and symmetric- stretch frequencies of the low-lying excited states correlating with the observed crystal transitions and the lowest electric-dipole allowed transition from the first excited state.

We are also interested in the type of spin coupling found in these states.

3.2 Computational Details

In all the calculations on the uranyl ion, the core electrons of uranium and oxygen are replaced by relativistically derived core potentials. Only the uranium

6s^6p^5p6d''7s^ and oxygen 2.s^2p^ valence electrons are explicitly treated. Uranium and oxygen core potentials and spin-orbit operators for the valence electrons are those developed by Christiansen and coworkers [102,103]. The basis sets used are the cc- p\'DZ basis set developed recently in our group. Our cc-pVTZ basis set for oxygen is also used. These are listed in tables 3.2. 3.3 and 3.4. The uranium basis set was opti-

orbital primitives contraction contraction contraction

sd 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 P 6.728000 -0.0033035 0 . 0 1.419000 -0.3142991 0 . 0

0.619900 0.7755420 0 . 0

0.244500 0.4902717 1 . 0

f 4.436000 0.1957684 0 . 0

1.860000 0.4559656 0 . 0

0.755200 0.4265113 0 . 0

0.277000 0.1970811 1 . 0 g 1.690000 1 . 0 0 0 0 0 0 0

Table 3.2: U cc-pVDZ Basis Set (4sd4p4flg) -> [3sd2p2flg]

54 mized for the U^'*’ ion. The sd. p. and / primitives were optimized in an atomic SCF

calculation on the 6cPôf^ averaged configurations. The polarization g function was

obtained by optimizing the correlation energy of the / orbitals. In the case of effective

orbital primitives contraction contraction s 41.04 -0.0097241 0.0222003 7.161 -0.1318703 0.1265661 0.9074 0.5903463 -1.6261307 0.2807 0.5169632 1.5531546 P 17.72 0.0433004 -0.0559967 3.857 0.2330835 -0.4246360 1.046 0.5017961 -0.5598365 0.2752 0.4652332 0.9990806

d 1.213 1 . 0 0 0 0 0 0

Table 3.3: O cc-p\'DZ Basis Set: (4s4pld) —>■ [2s2pld]

core potential calculations, the pseudo-orbitals are required to be small in the core region. The correlating orbitals obtained by freeing the most diffusive primitives, as

is usually done in the case of all-electron calculations, are not necessarily small in the core region. This is especially pronounced in the case of s orbitals. This implies that the contractions should be obtained by energy-related methods. In our investigation

[104] it is found that the best results are obtained if natural orbitals from correlated atomic calculations are used. The oxygen double-^ s, p primitives were optimized in atomic Hartree-Fock calculations on the ground state of the oxygen atom. The contractions were obtained from the natural orbitals. The polarization d functions are optimized for the correlation energy. The triple-oxygen basis was optimized in the same way.

00 orbital primitives contraction contraction contraction s 37.70 -0.0111059 0.0214722 -0.0641494 6.840 -0.1384833 0.1635464 0.0384204 1.053 0.4495813 -1.5180990 2.0857688 0.4163 0.4865708 0.6751041 -3.9609883 0.1706 0.1874286 0.7603202 2.4202440 P 34.57 0.0160866 -0.0223468 0.0307516 7.760 0.1007112 -0.1344920 0.3181273 2.282 0.3153183 -0.5815336 0.8048192 0.7160 0.4776778 -0.1619810 -1.5818126 0.2140 0.3477725 0.9092533 0.9599946

d 2.025 1 . 0 0 . 0

0.5425 0 . 0 1 . 0

f 1.260 1 . 0

Table 3.4: O cc-pVTZ Basis Set (5s5p2rf) [3.s3p2rfl/]

Ali calculations are done using the COLUMBUS programs. The integrals over atomic orbitals are generated by ARGOS [105]. These integrals are then converted to the supermatrix form to be used in the SCF calculation by the CXVRT program.

Molecular orbital coefficients are generated from SCFPQ in ground-state SCF cal­ culations. These MO coefficients are used as the initial guess in averaged MCSCF calculations. MO coefficients either from SCFPQ or from MCSCF can be used to transform the atomic integrals to MOs to be used in correlated calculations. The transformation is done by the program TRAN. Spin-orbit CI singles as well as sin­ gles and doubles calculations are carried out. The CIS calculations are done using program CIDBG and the CISD calculations are done using the spin-orbit GUGA CI method described in the previous chapter. The MOs used in the CISD calculations are the natural orbitals from state-averaged MCSCF calculations.

56 3.3 Results and Discussion

The ground-state SCF orbital energ\' levels at the optimized equilibrium bond

length are shown in Fig. 3.1. .\ group of closely spaced higher occupied MOs. span­

ning an energ}' range of about 0.95 eV'. and a group of closely spaced lower unoccupied

MOs. spanning an energy range of about 1.64 eV . are well separated from the rest of

the occupied and virtual orbitals. The strong bonding in the ground-state uranyl ion can be seen from the large LUMO-HOMO gap of about 17.84 eV . Detailed analysis of the orbitals is given in Table 3.5.

The HOMO. Zog, is bonding mainly between uranium d and oxygen p orbitals.

Slightly (0.007 eV ) below the HOMO is the 2wu orbital, bonding between uranium

/ and oxygen p orbitals but with significantly larger oxygen composition. .\lmost degenerate with the orbital is the orbital, bonding between uranium / and oxygen p, but contrary to the 2iTu orbital, with larger uranium character ( 54% vs.

35%). The strongest bonding effect is between uranium dn and oxygen pir orbitals, as seen from the orbital energ}' of —1.097 eV of Itt^, but it is only 0.035 eV below the

HOMO. The energ}- levels obtained are in qualitative agreements with previous ECP calculations [89], even though the ordering and the energy span are slightly different.

These differences may be attributed to the different core potentials and basis set used. The four-component spinor calculation of De Jong [97] at a different geometr}'

(3.25 Qq instead of 3.16 gq) placed the ungerade orbitals above the gerade orbitals, with the HOMO being 3ui with dominant Ou character and some character. .A, closer look at the SCF results show that as the bond distance decreases, the 30u orbital is raised toward the 30g orbital. From the triple-^ results it is found that,

57 higher occupied MOs lower unoccupied MOs

l7Tg 3

Table 3.5: Ground-state SCF Orbitals: Principal MO Coefficients

58 gross atomic populations atom s P d / 9 total U 2.034 5.546 1.469 2.524 0.005 11.577

0 3.863 8.499 0.062 0 . 0 0 0 0 . 0 0 0 12.423

Table 3.6: Ground-state üranyl SCF Population Analysis

for example, at 3.10 oq, 3

basis set used will play a significant role in the ordering of the orbitals in the case of

uranyl. The nonbonding UfS^ and U / 0 % orbitals are separated by 0.12 eV with the

nonbonding Udôg and the antibonding and partially 7s-like 4cTg orbitals in between.

The antibonding 3TTu orbital is slightly above the Ufcpu orbital. The gross population analysis is also shown in Table 3.6. The charge on the oxygen is seen to be only

about - 0 .2 1 . The U6 p hole is about 0.45 electrons in close agreement with the four-

component spinor results. The mixing of U 6 p, U 5/ and 02p orbitals results in transfer

of electrons from U 6 p to Ü5/, a result of the "pushing from below” effect.

For excited-state calculations we first carried out a CI singles calculation to locate the lowest excited states. The results are shown in Table 3.7. From Table 3.7 we see that the lowest excited states are from the 3<7„15„ and 3cr„l

CS2 ÜO 2 CI4 single-crystal data [77]. We also found that these excited-state wave func­ tions had contributions from nonreference configurations with coefficients as large as

0.39. This is mainly due to the ground-state orbitals being used for the excited-state calculations. Using orbitals optimized for the excited states reduced the magnitude

59 Energy S tate (0) A 5 term Configuration

-81.5374796J 0 3CTÜ

-81.4071685 1 -81.4024586 2 'A, ScrAWA -81.3950088 3 -81.3864499 2 -81.3770738 3 3^,1 WA -81.3676414 4 -81.3433264 3 3(rAW^ -81.3283020 2 'A, 3o-ii5^

Table 3.7: Low Gerade Excited-State Manifold: results from CIS cal­ culation. calculated at 3.21 qq

of the nonreference Cl coefficients to the range below 0.05. .A.s seen from the ground-

state SCF population analysis, exciting one electron from the 3 cr„ MO (0.348 electron

from the O atom and 0.644 electron from the Ü atom) to the ld„ or 1 0 u MO trans­ fers 0.348 electron from the O atoms to the U atom. This process has been referred to as "charge-transfer” [106,107], but a quantitative value of the amount of charge transferred is not clearly defined. Mixing of the 3cTu and 2cTu MOs in the excited states will partially compensate the amount of charge transferred as determined from the ground-state MOs. Excited-state SCF calculation on the state shows that the charge on ü is about the same as in the ground state, so the principal electron change is from fcr (bonding) to fS (nonbonding).

CISD calculations for the ground state also showed that the largest single-excitation

Cl coefficient was for 27Tu —> 3 n*. If the orbitals are optimized in multiconfiguration

SCF calculations with both 27Tu and included in the active space then the largest

60 singles contribution is reduced from about 0.047 to below 0.03. Since we are mainly interested in the lowest excited states from the two configurations 3o-„ldy and 3a„10„, especially the triplets, we optimized the orbitals to be used in Cl calculations with averaged MCSCF calculations. The active space included 2%^, 3

SCF calculation, we expect that the excited states and the ground state are treated equally and the excitation energies will be from a balanced treatment of all the states.

Single-reference spin-orbit CISD calculations were used to calculate the ground- and excited-state potential energy curves in the symmetric-stretch coordinate with a step size of 0.01 Oq . The curves are shown in Fig. 3.2. The enlarged low-lying excited-state energ\" curves are plotted in Fig. 3.3. The results are also summarized in Table 3.8. We also repeated the calculation using the oxygen triple-^ basis set.

The potentials curves are plotted in Figs. 3.4 and 3.5 and the results are summa­ rized in Table 3.9 together with the double-^ results. The analysis of the typical Cl wave functions in double-C single-reference spin-orbit CISD calculations is presented in Table 3.10. The composition of the Cl wave function is roughly the same across the whole curve. The contribution from the references is around 83%, about the same in all states. There is little mixing of 2g(^Ag) and 2g(^Ag), only about 1.3% of ^A,

61 S tate % (A ) Frequency ( cm ^) Te(adiab.) CS2 UO 2 CI4 " Vertical ground state (closed shell)

1 . 6 6 8 0 . 1103

1.735 923 20895 20096 22694 22296 20861 24079 2 , 1.735 922 3, 1.735 923 24075 22051 25835

26666 2 , 1.749 984 23590 22578 3, 1.755 940 24997 26222 28138 ■Iff 1.753 941 28223 27738 31262

“From reference [77] '’Note: The order between 3, from zmd 2g from of the adiabatic transition is reversed compared with the vertical transition and experimental assignment.

Table 3.8: Summary of Calculated Excitation Energies and Symmetric-Stretch Vibrational Frequencies from Single-reference Spin-orbit CISD Calculations, Double-^ Basis^

compared with 82.3% of ^Ag. The mixing of 3,(^0,) with 3g(^0g) is considerably

larger, about 1 1 .8 % ^0 ^, and is due to the smaller spacing between these two states

compared with 2g{^Ag) and 2 g(^Ag), as can be seen from the Cl singles results shown

in Table 3.7. Presumably the exchange integral between 3 <7 u and l

Since there is no gas-phase data on the transition energies and vibrational frequen­

cies for the uranyl ion, we compare the calculation with the single-crystal CS 2 UO 2 CI4 experimental values of Denning [77] (averaged for Dooh symmetry). The calculated ground-state vibrational frequency of 1103 cm“^ is about 200 cm"^ higher (~ 25%) than the observed crystal value of 830 cm“K The equatorial ligands are expected

62 S tate &(A) Frequency( cm ^) Te(adiab.) ground state (closed shell) 0., 1.645(1.668) 1204(1103) 3(7^ 1<^^ 1.691(1.735) 966(923) 23155(20895) 29 1.691(1.735) 972(922) 24560(22296) 39 1.691(1.734) 966(923) 26316(24075)

29 1.709(1.749) 978(984) 26074(23590) 39 1.710(1.755) 976(940) 27523(24997) ■^9 1.708(1.753) 977(941) 30683(28223)

“Note: The order between 3g from and 2, from of the adiabatic transition is reversed compared with the vertical transition and experimental assignment. ‘’Numbers in parentheses are double-results

Table 3.9: Calculations with Triple-C Basis^’^

S tate i?e(A) Frequency( cm *) Te(adiab.) ground state (closed shell) ^^9 total reference 83.3% O9 83.3%

^Ag ' ^ 9 total reference 83.4% I 9 83.4%

2 9 82.3% 1.3% 83.6%

3 9 83.4% 83.4%

^ $9 , 3 ail(^ i

' $ 9 total reference 83.4% 2 9 83.4%

3 9 83.4% 83.4%

4 9 71.6% 11.8% 83.4%

Table 3.10: Cl Wave Function Properties

63 to elongate the U-0 bond and decrease the vibrational frequencies accordingly. We

see roughly the same difference in the excited-state vibrational frequencies (around

920 cm~‘ calculated compared with about 830 cm“‘ observed). The vibrational fre­

quencies and equilibrium bond lengths of the three states from the state are

about the same as is also true for the three states. The differences between the

vibrational frequencies and bond lengths of the ^Ag states and those of the are

rather small, as may be expected from the fact that these two states both result

from exciting one electron to a nonbonding U/ orbital. The changes in vibrational

frequencies and thus the changes in bond lengths in going from the ground state to

the excited states are consistent with the experimental vibrational frequencies.

We see the crossing of the 3, (^A) and 2g (^$) curves near the optimized equilib­

rium position both in the oxygen double-and triple-calculations. The calculated transition energies with the double-(^ basis set are in reasonable agreement with the the observed transitions, and are in general only 1.7% to 6.9% higher than the ob­ served values. The notable difference is that the calculated 3g (^$) transition is lower than the observed transition. W ith the oxygen triple-<^ basis, the calculated transi­ tion energies are shifted uniformly higher by about 2300 c m '\ This difference may be attributed to the larger basis improving the ground state more than the excited states. The effect of the larger basis on the vibrational frequencies and bond lengths is rather small, decreasing the bond length by several pm and increasing the vibrational frequency by about 5%.

64 State Energy '^ 9 % total reference MR -82.104168 0.835 0.835 ^9 SR -82.103395 0.836 0.836 2.9 MR -82.102467 0.464 0.371 0.835 29 SR -82.097068 0.823 0.013 0.836 3, MR -82.097335 0.424 0.331 0.080 0.835 39 SR -82.088852 0.837 0.837 29 MR -82.092150 0.356 0.445 0.034 0.835 ^9 SR -82.094720 0.835 0.835 39 MR -82.081786 0.402 0.404 0.030 0.836 3g SR -82.088457 0.718 0.117 0.835 ■^9 MR -82.074573 0.835 0.835 ■^9 SR -82.073547 0.836 0.836

Table 3.11: Comparison of Single-Reference and Multireference Calculations at 3.39 Oo

The potential curves of the two 3g states cross at the bond distance of 3.40 G q

(these two curves are calculated in different Cl spaces). To account for this avoided curve crossing, multireference calculations are needed. The calculations were re­ peated with reference configurations 3cr„l^i and 3(7^10^. The results at 3.39 gq are summarized in Table 3.11. Overall the total contributions from all the reference con­ figurations remained the same as in the single-reference calculation. The Ig and 4g states are the least affected, remaining single-reference, and the energies are lowered by only 170 cm~^ and 225 cm“^ respectively. The 2g and 3g are most affected due to the strong interaction of these states from the two different reference configurations.

Even though the total contribution from the two reference configurations remained the same, these two references are strongly mixed together, with roughly the same contribution from each configuration. The interaction pushed the 3g(^A) down by

1862 cm“^ and pushed the 3g(^0) up by 1464 cm "\ Similarly the 2g(^A) state was

65 State Energy 'A, '^ 9 '* 9 total reference MR -82.110094 0.839 0.839 1. SR -82.109322 0.840 0.840 2, MR -82.106810 0.617 0.221 0.839 2p SR -82.102936 0.829 0.011 0.840 3g MR -82.101182 0.551 0.233 0.055 0.839 3. SR -82.094833 0.840 0.840 2, MR -82.095062 0.209 0.599 0.030 0.838 2. SR -82.096087 0.839 0.839 3g MR -82.085007 0.283 0.513 0.043 0.839 3g SR -82.089579 0.735 0.104 0.839 ■^9 MR -82.076055 0.838 0.838 ■^9 SR -82.075022 0.839 0.839

Table 3.12: Comparison of Single-Reference and Multireference Calcula­ tions at 3.27 ao

pushed down by 1185 cm ' and the 2,(^0) state was pushed up by 564 cm -1

The results at 3.27 oq, the optimized first excited-state geometry, are shown in

Table 3.12. Essentially the same effects of multireference calculations are seen both at 3.39 oq, where the curves cross in single-reference calculations, and 3.27 ao, around the optimized excited-state geometries. The potential curves from the multireference calculations are plotted in Fig. 3.7. The curves no longer cross in the multireference calculations. The transition energies, vibrational frequencies, and optimized excited- state geometries are summarized in Table 3.13, along with the experimental crystal results. The energy changes in going from single-reference to multireference calcu­ lations are compared in Table 3.14 both at 3.27 Qq and 3.39 oq . With the reference space including the 3a„l5„ and 3cr„10„ configurations, the calculated transition en­ ergies compare rather favorably with the experimental crystal results. The largest

66 S tate &(A) Frequency ( cm * ) 7^(adiab.) CS2 UO 2 CI4 ^ Ag, 3cr* 1.733(1.735) 867(923) 20719(20895) 20096 1.739(1.735) 845(922) 21421(22296) 20861 ^9 1.742(1.734) 847(923) 22628(24075) 22051 3<7^10i 29 1.749(1.749) 900(984) 23902(23590) 22578 39 1.747(1.755) 898(940) 26118(24997) 26222 4 1.755(1.753) 880(941) 27983(28223) 27738 31710 29411 3 9 “

“The crystal-held assignment is a2g state ^Numbers in parentheses are single-reference results

Table 3.13: M ultireference C alculations w ith Double-<^ Basis*’

difference is seen in the 2g state. The calculated transition energ\' to this state is 1324 cm“ * higher, while all the other calculated transition energies agree very well with the experimental values. The inclusion of the 3cr„37r* configuration is expected to push down the 2g (^$g) state more than other triplet states and further improve the calculated transition energy to this particular state. The next higher excited state, a 3g state in our calculation but determined to be a 2g state from crystal-held assignment [77], is also included in Table 3.13. The multireference geometries differ from the single-reference values by less than 0.01 A. The multireference frequencies are substantially better than the single-reference values. The energy differences be­ tween single-reference and multireference calculations are listed in Table 3.14.

We also investigated the linearity of uranyl in the hrst excited state. The ion was bent up to an angle of 160° and the potential curve is shown in Fig. 3.6. The

67 'A, 3 $ , 2, 3g 2, 3, 4 3.27 A +169 +850 +1393 -225 -1003 +227 3.39 A +170 +1185 + 1862 -564 -1464 +225

multireference energy* is lowered (increased) with respect to single reference calculation ‘energj- in cm“ ^

Table 3.14: Energ>' Differences between Multireference Calculations and Single-reference Calculations^'^

results clearly show that uranyl in the first excited state has a linear structure and that the Renner-Teller splitting of the two components of the first excited state is almost indiscernable.

Finally we discuss the calculation of the gerade to ungerade transition to high- lying excited states from the lowest excited state lg{^Ag). To find the lowest ungerade excited state, we carried out MCSCF calculations. The active space included all the occupied MOs from iTr^ to 3

3.1). Only single excitations from the occupied MOs to the unoccupied MOs were included and all the roots in the (small) Cl space are obtained in a single MCSCF calculation. The orbitals are optimized for the lowest state of the Cl space. The calculations are done at the optimized first excited-state geometry of 3.28 oq. The results are summarized in Table 3.15.

The lowest ungerade excited state was found to be the 3

68 Energy State Configuration -81.323828 3(7^ i^i -81.320856 3cr^lô'i -81.305712 3(T^1< -81.297843 -81.254752 -81.254036 -81.253476 -81.247415 -81.238482 3o-]37ri -81.236761 -81.236566 iT rh ^ i -81.233949 T , -81.232886 ^A„ -81.213546 SalZTvl -81.192236 l7rj37T^ -81.182682 ^A„ ln23%^ -81.173260 l<37r^ -81.171596 l<37Ti -81.170691 ^A„ l7r^37Ti -81.089494 ' s : l7 rj3 < -80.940942 3(^u4oy -80.934061 ^A« -80.893137 -80.876132 ^A„ -80.765183 -80.758155 -80.159544 2<4a^ -81.048633 2 < H ^ -81.039345 2%^4o4 -81.028210 2Tr'tlSl

Table 3.15: Higher Excited-States Manifold

69 Energy S tate (fi) Configuration -82.046902 lu -82.046836 2u -82.032869 3u -82.032779 2u

“Results from CISD calculations, calculated at 3.28 uo, the optimized first excited-state geometry.

Table 3.16: Low Ungerade Excited-State Manifold^

state. The next higher configuration is 3 cTj 10u , with a larger triplet-singlet splitting of

1727 cm“U The state is almost 4000 cm“^ higher than the state. Spin-Orbit single-reference CISD calculations of the 3cTgl5u states at 3.28 Qq , the optimized bond length of the first excited state, are shown in Table 3.16. The separations between lu and 2u and between 3„ and 2„ are only 14 cm“^ and 20 cm“L much smaller than the lu and 3u separation, a manifestation of the spin-orbit interaction for Id'u being larger than the exchange integral between Id'u and 3crg, resulting in w-w coupling.

The transition from the first excited state to the lowest ungerade state, lu <— Ig, polarized along the uranyl axis, is at 13744 cm"^. The experimental absorption is strongly peaked around 17500 cm“^ but usually with a wide range of several thousand wave numbers. The large discrepancy between the observed value and the calculated value may also be explained by the basis set used. .A.s observed earlier, the oxygen triple-»^ basis lowers the 3cTg level with respect to the 3cTu level. .A. lowering of the 3

70 - 0 . 2 - I

3n..

0.4 - I4>u IS, 4*, 15..

(U I - 0.6 -

- 1.0 - 3 a , 27T„

- 1.2

Figure 3.1: SCF Ground-State Orbitals of

71 -8 2 .0 4 -

- 8 2 .0 6

- 8 2 .0 8

eu 2 -8 2 .1 0 - c ce JO - 8 2 .1 2 - o> CD C -8 2 .1 4 - LU

-8 2 .1 6

- 8 2 .1 8 -

- 8 2 .2 0 -

3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 U-O Distance (Bohr)

Figure 3.2: Ground- and Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Double-^ Basis

72 - 82.04

r"*.) \ \

-82.06 \ \

CD 2 c -82.08

ca> LU -82.10

-82.12 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 U-O Distance (Bohr)

Figure 3.3: Low-Lying Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Double-Basis

73 -8 2 .2 2

4, ('*.) - 8 2 .2 4 3. r’f.)

- 8 2 .2 6 3 , ("a .) 2,r’A,> 1, fa,, - 8 2 .2 8 s - 8 2 .3 0 - (O

g ) - 8 2 .3 2 CD

- 8 2 .3 4

- 8 2 .3 6 0,- es.-) - 8 2 .3 8

- 8 2 .4 0 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 U-O Distance (Bohr)

Figure 3.4: Ground- and Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Oxygen T rip le-Basis

74 ® - 82.25

I 2, ('-».> 2>

2,r"V

1, (*a.)

* 82.28 3.06 3 06 3.10 3.12 3.14 3 16 3.18 3.20 3.22 3.24 3 26 3 28 3 30 3 32 U-O Distance (Bohr)

Figure 3.5: Low-Lying Excited-State Potential Curves. Single-Reference Spin-Orbit CISD Calculation, Oxygen Triple-C Basis

75 - 82.100

- 82.102

- 82.104

- 82.106

- 82.108

- 82.110

180 175 170 165 160 155 O-U-O angle (degree)

Figure 3.6: Renner-Teller Effect in the First Excited State of Uranyl

76 Excited State Curves from Multi-Reference Calculations

\

- 82.08

*c \ CO 3, fa.) \

- 82.10 - I 2, fa.) i.fa.)

- 82.12 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 U-O Distance (Bohr)

Figure 3.7: Low-Lying Excited-State Curves From Multireference Calculations.

77 CHAPTER 4

Calculation on Uranyl (mono)hydroxide: U02(0H)+

4.1 Background

One interesting and complex aspect of uranium chemistry is uranyl fluorescence in various media. The uranyl ion will associate with the anions of the solution to form various complexes, strongly influenced by the type of solution, pH value, concentration and ionic strength. The most common form of complexing is hydrolysis. The uranyl ion is much more readily hydrolyzed than most divalent ions due to the cylindrical distribution of charge and the high positive charge on the uranium. In most cases the hydrolysis is extensive even at low pH values. Even in solutions with relatively low uranyl concentration, hydrolysis results in polynuclear species [108]:

mUO^ + nHgO = (U02)m(OH)g^-")+ + nH+ (4.1)

The existence of mononuclear U02(0H)+ has long been postulated (see, e.g., [108] and references therein). To limit the presence of polynuclear species and to avoid precipitation, low concentrations (100 //g/L, 4 x 10~^ M) of uranyl in noncomplexing media (0.1 M NaClO^ and pco2=10“^-^ atm) have been used to study uranyl fluores­ cence as a function of pH [109]. Nine species, primarily hydroxides and carbonates,

78 have been identified. At pH 1. only is present, at pH 4 UOgOH^ appears, and at pH 7 several uranyl hydroxides (UOgOH^. U02(OH)2, U02(0H).7, U02(0H)i^".

(U02)20H^'*‘) and uranyl carbonates (ÜO 2 CO 3 , U 0 2 (C 0 3 )2 ~) appear. Presumably,

these species have various numbers of H 2 O molecules associated with them. .A.s the pH increases, red shifts and longer lifetimes are observed, due to the stablization of the excited state by complexation [110]. Another study [111] observed bi-exponential behavior in the excited-state fluorescence and attributed it to the acid-base process:

U0^+ + H 2 O = UO 2 OH+ -k H+ (4.2)

possibly with a change of the number of coordination H 2 O (from U 0 2 (H 2 0 )^'*" to

U02(H20)„0H+)

To assess the safety of nuclear waste disposal, it is important to understand the migration of the radionuclides, mainly by complexing with various reagents in the aquifer system, in the natural environment. Knowledge of the species formed can be used to model the transportation of radionuclides [112-114] and spectroscopic information is widely used to identify trace amount of various radionuclide species.

Specifically, the free uranyl ion and the mono hydroxide can be characterized by their spectral shifts and fluorescence properties [110]. At pH 7 and above, the most common range for natural waters, the spectrum consists of fluorescence from a large number of species. The spectral information of and U020H'^ can be used to deconvolute the complex spectra at higher pH.

79 We are interested in the calculation of the ground- and first excited-state (the fluorescent state) geometries and symmetric-stretch frequencies as well as the tran­ sition energies. The primary purpose of this study is to understand the fluorescence properties of UO^OH""" as a first step to understanding the complexation of uranyl ion with ligands computationally. The level of treatment for U020H^ was the same as was used for uranyl ion and most of the computational details have already been described in the previous chapter. The geometry optimization was confined to a pla­ n ar C 2 V symmetry with the z axis along the OH bond. A schematic drawing of planar

UOgOH^ is shown in Fig. 4.1.

4.2 Results and Discussion

To understand the interaction between and OH'*’, the properties of the higher occupied MOs and the lower unoccupied MOs are listed in Table 4.1 and

4.2 and the orbital level correlation diagram between ÜO^'*’ and U020H^ at their equilibrium geometries is shown in Fig. 4.2. The irreps of D^h. (for UO^"*") reduce to

the irreps of C^v (for UO 2 OH’*’) according to:

(jg —>■ (4.3)

^ ("1-4)

TTg, • • • and 6u, 7 m • • • —> 0 2 , 6 2 (4.5)

^Uî 0ui ' * * and Sg^ 7g? * ' * ' ^ (^ b)

Going from D^oh symmetry to C 2 V symmetry, cr,, Sg, , tt^, • • • orbitals will be mixed together as will be cr„, • • -, tt^, 0g, • • • orbitals. The strongest interactions

80 between and 0H~ are seen in the four highest occupied MOs, 27 Tu and Z(7g, of UOl’*’ discussed in the previous chapter and the three oxygen p orbitals

of OH". The two highest occupied MOs, Zb\ and 0 6 2 , are mainly oxygen p orbitals on OH", with OH" oxygen p populations of 1.74 and 1.53 respectively. Mixed into

the 36i orbital are mainly the nonbonding UO^"*" Sg (6 d) and àu {of) orbitals while

some mixing of the bonding orbital of UOg""" is found in the 0 6 2 HOMO. The most strongly bonding OH" orbital is oxygen contributing the most to 5oi, the lowest in energy among the highest occupied MOs. This orbital has a population of

1.26 from OH", ca. 1.02 from OH" oxygen and ca. 0.24 from OH" hydrogen, and is strongly bonding with the mixed UOf'*’ 3ag and 27Tu orbitals. Uranium d and /

orbitals, non-interacting under D^oh symmetry, are strongly hybridized in U 0 2 0 H"*".

The Goi and Toi orbitals are a result of the strong mixing of the 3og and 27Tu or­

bitals in ai symmetry. The 6 % component of UO^'*’ 27Tu remains as the 26i orbital in

U 0 2 0 H""", mostly nonbonding between UO^"*" and OH". The Ixg orbital splits into

luo and 3 6 2 , w ith 36 2 ca. 0.28 eV lower than lu 2 , mainly as a first-order ligand field

splitting since the contribution to 3 6 2 from orbitals other than liVg is rather small.

The 3 (Tu orbital of largely remains as the 4 6 2 orbital, with some mixing with iTTg and OH" oxygen Py.

Compared to the group of five closely spaced lowest virtual MOs in UOg" with a total of eight components, we see only five closely spaced lowest unoccupied MOs

well separated from the rest of the virtual orbitals in UO 2 OH'''. As a result of the interaction of 15^ with the highest occupied MOs of UO^"^ and the oxygen p orbitals

from OH" and the interaction between 4 (7 , and iSg, the lowest unoccupied MO of

81 UOaOH"*" becomes 8 ai, a mixture of mainly 4 (7 , and 1

the 46i orbital, principally composed of W,, 1 0 u and 3 7 t„. Next come the 2 o 2 and 6 6 2 orbitals, mainly and separated only by 0.033 eV.

Now we turn to the discussion of the first excited-state SCF results. The re­ sults are obtained at the optimized geometry, as shown, together with the optimized

ground-state geometry, in Table 4.3. The optimized excited state shows little interac­

tion between the two moieties as can be seen in the highest occupied MOs shown in

Table 4.4. An electron has been transferred so the system is essentially UO 2 and OH and there is little bonding between them. The three OH orbitals lie right below the four highest occupied MOs discussed in the previous chapter. The highest occupied

MOs mainly of UOJ character are in the order tt,, ct,, tTu, cTu and with increasing energy. The HOMO, (Ju: is a nonbonding U / orbital and the excited-state configura­ tions are and Zb^la^., corresponding to 30^16% for UO^"^.

The ground-state geometry of U 0 2 0 H‘*' has a bent UO 2 ''’ structure with the 0-Ü-

O angle at 171.4° while UO^^ is linear in the excited state, as a result of the strong

interaction between UO^"*" and 0H “ in the ground state and almost non-existent in­ teraction in the excited state. Compared with the optimized UO^'*’ geometries, there is little change in the excited-state UO bond length (1.735 A in UO^^ and 1.741 A in

U 0 2 0 H"^) while the ground-state UO^""" bond distance is increased from 1 . 6 6 8 A to

1.704 A, a change of 0.036 A. The UO^'*’ bond length difference between the excited state and the ground state is less in UO 2 OH'*' (0.037 A) than in (0.067 A). The

U-O bond between UO^''’ and 0H~ is much longer than the uranyl U-O bond and longer still in the excited state. 82 higher occupied MOs

5a 1 302 lU2 6 oi 4&2 2b, 7oi 36i 5 6 2

Energ}' - 0 . 8 6 8 -0.825 -0.815 j -0.786 -0.785 -0.779 -0.743 -0.681 -0.651 lUStTg 0.113 -0.205 -0.256 2\Jsag 0.189 0.223 3US(Tg -0.191 -0.142 2U dag 0.236 0.565 0.359 2Ud~g 0.446 0.452

2L^d5g 0 . 2 0 1 -0.142 0 . 2 0 0 3\Jdag -0.126 -0.367 -0.2.30 SUd-Kg -0.156 -0.154 3\JdSg lUpTTu 0.180 0.138 -0.266 -0.376 -0.156 lÜpCTu -0.232 -0.207 2Üp7Tu 2\Jpau -0.173 lUfcTu 0.673 0.255 lU/TTu 0.136 0.246 0.327 0.177 lU/<5„ 0.128 IL'fOu 0.108 2Ufa^

2 U/ 7r„ 2 U/ÔU

2 U/ 0 „ 1 0 SCTg -0.225 -0.131 lOsCTu lOpag 0.180 0.694 0.340 lOpVTu 0.335 -0.370 0.813 0.672 0.131

lOpTTg 0.784 0.829 0 . 1 0 2 -0.293 lOpCTu 0.574 0.235 10*S -0.138 0.172 lO *poi 0.655 -0.379

1 0 *p 6 i - 0 . 1 1 0 0.904

I 0 'p & 2 0.223 -0.292 0.880 IH stti 0.364 -0.209 2H soi -0.129

Table 4.1: Ground-State Occupied SCF Orbitals: Principal MO Coefficients

83 LUMOs

8a 1 46i 2o.2 662 9a 1 Energ}' -0.148J -0.121 -0.115 -0.114 -0.112 lUSCTg -0.433 -0.173 2USCTg 0.265 SUSCTg 0.611 0.298 2U da g 2\Jd~g 2Ud5g 0.418 -0.499 -0.238 3\J da g -0.244 SUdiTg -0.121 3Udôg 0.169 -0.207 -0.133 lUpTTu 0.118 lUpCTu 2Up7Tu 2\Jpau lU/cTu

IU/ tTu -0.167 0.271 1U/(J„ 0.915 0.913 lU /0 „ 0.630 -0.753 2 U /a „

2 U /7 T u 2U /f u 0.143 0.136 2 U /0 „ 0.11^ -0.116 1 0 SCTg lOsCTu lOpCTg 0.150 lOpVTu 0.123 -0.141 lOpiTg 0.131 lO p a u 10*S lO *poi I0*p6i lO*pÔ 2 -0.198 iH so i 2H sai

Table 4.2; Ground-State Unoccupied SCF Orbitals: Principal MO Coefficients

84 U - 0 (A ) U - 0 *“(A) ZÜ - U - 0 0 ‘H'’

ground state 1.704 2 . 0 2 1 171.4 0.958 excited state 1.741 2.519 179.92 0.958

“O* is the oxygen from OH ^0*H distance is optimized in the ground-state calculation and kept unchanged in the excited-state calculation

Table 4.3: Optimized Ground- and Excited-State Geometries of UOoOH^

The excited-state wave function is strongly mixed with 6 3 6 2 and 6 2 ^ 2 configura­ tions, e.g., at the optimized excited-state geometry, the wave function is 40.6%

and 42.7% 6 3 ni- This mixing is attributed to the spin-orbit effect. To see this quan­ titatively, we compare the spin-orbit calculation with the spin-free calculation using

the same reference space of 6 3 6 3 and b^a^ as well as calculations with and without

spin-orbit effects using only 6 3 6 3 as a reference at the same optimized excited-state geometry. The results are summarized in Table 4.5.

The adiabatic transition energy from the ground state to the excited state is calculated to be only about 7963.5 cm“^ The TRLIF [110] experiments identified fluorescence with vibrational bands at 20,120 cm "\ 19,231 cm "\ 18,382 cm~^ and

17,544 cm "\ which are attributed to UOgOH^. The calculated adiabatic transition energy thus does not fall in the range of the experimental values. The calculated vertical transition energy from the ground state geometry is 19874 cm "\ in good agreement with experimental values. Examination of the ground- and excited-state orbitals shows that at the equilibrium position of the excited state, the electron transferred to the U /-like orbital is from an almost purely 0H “ orbital while at the

85 higher occupied MOs

5a 1 3 6 2 26i 402 l ü 2 6 a 1 36i 7oi 5 6 2 6 6 2

Energy -0.930 - 0 . 8 8 8 -0.785 -0.735 -0.734 -0.707 - 0 . 6 8 8 -0.687 -0.655 0.627 lUsCTj -0.253 2U5CTp 0.199 SÜSCTÿ -0.232 0.684

2 \ 2 d % g 0.411 0.414

2 \ j d 6 g

Z U d C g -0.409

Z \ j d T \ g -0 . 1 0 0 -0 . 1 0 1

Z \ 5 d ô g lUpTTu 0.183 -0.274 -0.281 lUpCTu -0.320 2Up7Tu

2 Upcr„ -0.183 lU/cTu 0.638 lU /V u 0.263 0.259 l U / J . 0.984 lU /O u

2 U /a „ 2 U /7 t„ 0.104 0.107 2 U /6 *

2 Ü/ 0 U l O S C T g -0.243 lOsCTu lOpCTg 0.817 lOpTTu 0.854 0.851 lOpTTg 0.826 0.826 lOpCTu 0.684 lO 'g -0.277 lO 'p o i 0.812

1 0 "p 6 i 0.990

I 0 *p 6 2 0.998 iH sa i 0.423

2 H sai -0.190

Table 4.4: Excited-State SCF Orbitals of UOgOH^: Principal MO Coefficients

86 5 6 2 (6 6 2 2 0 2 ) 5 6 5 6 6 2 spin-orbit no spin-orbit spin-orbit no spin-orbit Energy -99.142254 -99.104331 -99.134886 -99.103703

Leading Terms 6 ^6 | (0.406), 6 ^0 ^ (0.427) 6 'ai (0.842) 6 ^6 ‘ (0.834) 6 ^6 ^ (0.843)

Table 4.5: Comparison of Spin-Orbit and Non-Spin-Orbit Calculations With Different References

equilibrium position of the ground state, the electron transferred is from an orbital

with substantial UO 2 '*' character. The most likely reason for the poor agreement between the calculated adiabatic transition energy and the experimental value is that

the H 2 O ligands typically found around the equatorial plane of UO^'*' will increase the separation between the ground and excited states. The water ligands are expected to influence the ground and excited state in such a way that the ground state would have a nearly linear structure and longer UO distances and that in the excited state the excited electron would be in a UO^'^'-like orbital. Consequently, the transition energy of the complexed uranyl would closely resemble that of the bare

87 Figure 4.1: Schematic Drawing of UOjOH^

88 9^1 0 1 6b,

2 a,

4b,

8a,

371u

15 9 4a 9 15.

ë c3 X 5b, BP 3b c

-1 - 3a

l7 t 9

Figure 4.2: Orbital Correlation Diagrams of UOg and UO 2 OH+

89 CHAPTER 5

Calculation on Uranium Borohydride U(BH4)4

5.1 Background

Metal tetrakis(tetrahydroborates), [115], exhibit physical and chemical properties typical of covalent compounds, such as high volatility and solubility in nonpolar solvents. The first five members of the actinide series, Th-Pu, have been prepared [116-118]. Single crystals of these actinide borohydrides are easily obtained since they are volatile near room temperature. The other two of the seven known

metal borohydrides. Zr(BH 4 ) 4 and Hf(BH 4 )4 , are diamagnetic high-symmetry species and can be used as diluents [119,120] for the study of the actinide borohydrides.

All the actinide borohydrides are monomeric in the gas phase and have tetrahe­ dral {Td) symmetry with tridendate hydrogen atoms bridging the central metal atom and each of the four boron atoms [121,122] (For a schematic drawing of the struc­ ture, see Fig. 5.1). In the solid phase, two structural types are found in metal borohydrides. All seven compounds contain tridendate hydrogen bridge bonds con­

necting the boron atoms with the central metal atom. Np(BH 4 ) 4 and Pu(BH 4 ) 4 form monomeric crystals that are isomorphic to the transition-metal counterparts,

Hf(BH 4 ) 4 and Zr(BH 4 ) 4 . The 12-coordinated central metal atoms are surrounded by four BHj groups, forming a high-symmetry Td structure [115,122-124]. On the

90 other hand. Th(BH 4 )4 , Pa(B H 4 ) 4 and U(BH 4 ) 4 form polymeric and low-symmetry- structure crystals with additional double-bridged borohydride groups linking pairs of

metal atoms together [122,125]. Four compounds, M(BH 3 CH 3 ) 4 (M = T h, U, Np and Zr). with four methylborohydride groups tetrahedrally coordinated to the metal atom , have also been synthesized [126].

There have been a number of experimental studies of the optical and magnetic properties of metal borohydrides and methylborohydrides [28,127-132]. Particularly attractive are the uranium compounds. As occurs frequently with inorganic molecular

crystals, U(BH 4 ) 4 has a large number of low-lying electronic states that can be stud­ ied optically and its high-symmetry structure can be preserved in host crystals. The

diamagnetic crystals, Hf(BH 4 )4 , Zr(BH 4 ) 4 or T h(B H 4 )4 , can be used as host crystals.

These cry stals all have high symmetry at low temperatures and form substitutional

structures with U(BH 4 )4 . Having no nonbonded electrons, the host typically has its lowest electronic transitions above 40,000 cm“^ so the d-d or /- / electronic transi­ tions of the guest molecules, typically below 25,000 cm "\ can be easily isolated and studied.

U(BH 4 ) 4 and T h(B H 4 ) 4 were first synthesized during the Manhattan project

[116,117]. The first electronic structure study of U(BH 4 ) 4 was the pioneering work

of Bernstein and Keiderling [29]. They obtained the absorption spectra of U(BH 4 ) 4

doped into Hf(BH 4 ) 4 at several temperatures. Over 200 lines were found between

5000 cm~^ and 20,000 cm~^ Their EPR and Zeeman effect measurements showed

91 no positive results. The ground state was therefore assigned as an E state in Tj sym-

metrv' and transitions to T\ and Tg states are then electric-dipole allowed from the

ground state. Of the 19 possible /- / transitions from the E ground state, the origin

at 5932 cm~^ is separated from states below by more than 1500 cm“^ and the origin at 24. 795 cm“‘ is separated from the higher energy states by at least 8000 cm“*.

Based on their crystal field model, they could assign 11 origins with an rms deviation of 62 cm“‘ but their fit for 18 energies increased the rms deviation to 158 cm“^ In contradiction to their optical and magnetic data, their calculated ground state was 7^ w ith the E level only 14 cm“^ higher. Based on the normal mode analysis of Banks and Edelstein [128], Rajnak, Gamp, Shinomoto and Edelstein [28] later reanalyzed the data of Bernstein and Keiderling.

There have been a number of theoretical studies of metal borohydrides mostly considering the bonding properties [133-135]. In our study, we will concentrate on

the calculation of electronic transitions of U(BH 4 ) 4 as observed in the spectroscopic studies.

5.2 Computational Description and Discussion of Results

The calculations were done at the experimental geometry shown in Table 5.1

[125,133]. The U-B bond distance is taken from the known structure [125] and the borohydride geometries are idealized averaged values [133]. The same geometry is used for all the excited states. In general, equilibrium geometries are different for different electronic states. The choice of the same geometry can be justified to some extent by the simple Franck-Condon intensity pattern of progressions expected for

92 d(U-B) d(B-Hb) = d(B-Ht) ZHbBHb ZHbBHt 2.520 1.250 107 113

“Bond length inA. angle in degrees ‘Terminal hydrogen denoted Ht. bridging hydrogen Hy

Table 5.1: Geometry, Tj symmetry^’^

small displacements of equilibrium geometry [29].

U(BH 4 ) 4 can be formally described as a system with closed-shell BH^ ligands. To obtain all the / - / electronic transitions, averaged SCF calculation of all the p states were undertaken and the same molecular orbitals were used throughout to have a balanced treatment of all the excited states. Under Td symmetry, the uranium, boron and hydrogen atomic orbitals transform according to:

Us —y

Up —> ( 2

Ud —y i

U f —y + il + ^ 2

Bs —y Ui + ^ 2

Bp —y CLi + e -f* il + 2 i 2

HtS —y oi + i2

Hys —y ill + e + il + 2 i 2 (5.1)

93 2 ai 2 ( 2 le 4(2 4a 1 5(2 l(i 2 (i orbital energy -0.780 -0.670 -0.645 -0.466 -0.408 -0.393 -0.379 -0.309 U sai 0.492

Up < 2 0 . 1 2 1 Ude 0.515

\Jdt2 0.313 U fa i 0.272

U / ( 2 0.260 U /ti 0.265 0.197

“Only the U population is listed, the corresponding population of can be deduced from the total population of each orbital *’The 3ai and 4<2 orbitals, mostly BHJ and which lie between le and 4<>, are omitted.

Table 5.2: U Population Analysis of U(BH 4 ) 4 MOs^’^

The configuration will give rise to 40 energy levels in Td symmetry. 7.4 1 + 3 . 4 2 +

9E + 9Ti + I 2 T2 , with a total degeneracy of 91.

The doubly occupied orbitals are loi, 1 ^2 , 2oi, 2 ^2 , le, 3ai, 3 ^ 2 and 2ti. The low­

est occupied orbitals, loi and 1 ^ 2 with orbital energies of -2.222 Eh and —1.192 Eh,

are mostly U 6 s and B H J ( 2 with little bonding between the metal and BHJ. The occupied orbitals, with energies between —0.780 Eh and —0.309 Eh are shown in Fig.

5.2 and the population analysis is summarized in Table 5.2. The closely spaced 3oi

and 3 ^ 2 at —0.551 Eh and —0.534 Eh are virtually nonbonding BH 4 orbitals. The

strongest bonding with uranium 6 d is in the le and 4 ^ 2 molecular orbitals. Moderate

bonding is seen with uranium 6 p orbitals in 1 ^ 2 and 2 ( 2 - The uranium 5/ orbitals

mostly participate in the bonding through the 2 ti molecular orbital. The 4oi, ôto and Iti orbitals are almost purely nonbonding uranium / orbitals in the averaged P

94 configuration SCF description. The total uranium / population is 2.855.

In the excited state calculations, all the p configurations, with two electrons

distributed in 4oi, ot-i and Iti in all possible ways, are used as references since the

spin-orbit interaction is expected to strongly couple the states. Altogether ten

electrons. are correlated. This includes all orbitals with significant

uranium / character; 4oi, ôt,2 , I ti and 2ti. The ground state is found to be the doubly degenerate E state. All 21 electric-dipole-allowed excited states, nine Ty and

twelve T2 , from the ground E state are calculated. The calculated transition energies,

together with the experimental transitions assigned with the crystal field model, are summarized in Table 5.3. The first three calculated excited states have not been

found experimentally. Only three of the calculated values differ more than 1096 from

the experimental values.

95 S tate T( cm Experiment ( cm

E 0 0 IT, 131 —

1 ^ 2 2497 —

2 T 2 2817 — 2T, 6086 5932

3 T 2 7310 6557 3Ti 7320 7809 4T2 8673 8725 4Ti 9137 8968

ÔT2 10106 9589

6 T 2 10835 10416 5T, 10843 11164

6 T, 12142 12628

IT 2 12911 13839

8 T 2 14752 14288

9 T 2 15713 16057 TT, 16288 17622 ST, 20563 18280

IQT2 21596 20541

IIT 2 21780 20854

9T, 22174 2 1 2 2 1

I 2 T 2 27743 24795

“The assignments from the crystal field calculations [28] are reversed for the ptiirs 4ro, 4Ti and GTo, oTi

Table 5.3: Calculated f — f Excitations from the Ground State^

96 Ht

© Ht

Figure 5.1: Structure of U(BH 4 )4 : Only the central U atom and one of the four T^- symmetry BH 4 moieties tetrahedrally coordinated to the U atom are shown. Each BH4 is coordinated to the Ü atom through three (tridendate) hydrogens (denoted Hb), resulting in twelve hydrogens in the first coordination sphere. The terminal hydrogens are denoted Ht

97 0.0 n

2e 6 t2

- 0.2 -

- 0 .4 - & cd PC ^ * 2 3t2 3a. I - 0.6 - 1e 2 t„

2a. - 0.8 -

- 1.0

Figure 5.2: SCF Ground-State Orbitals of U(BH 4 ) 4

98 CHAPTER 6

Conclusion

The spin-orbit interaction has been implemented in the most recent version of the

COLUMBUS Graphical Unitary Group Approach Singles and Doubles Configuration

Interaction programs. Currently the spin-orbit operators are derived from the rela- tivistic effective core potential formalism. Calculations with Cl dimensions up to ten million (several million spatial configurations) are routinely performed on a worksta­ tion. The program can handle D^h symmetry and all its subgroups. Calculations on several uranium-containing compounds were carried out using these spin-orbit GUGA

Cl programs.

( 1 )U 0 2 '*': We have optimized the geometries of the ground and several excited states of UOg"*" at the spin-orbit CISD level and calculated the symmetric-stretch fre­ quencies and transition energies. The spin-orbit interaction is seen to strongly couple different spatial configurations. Systems with strong spin-orbit interaction, even if single reference in character otherwise, must be properly described by multireference spin-orbit CL The calculated transition energies with relativistic effective core poten­ tials in a double-C basis and MRSD spin-orbit Cl agree well with the experimental

CS2 UO 2 CI4 crystal transition energies. Triple-C oxygen basis set calculations shift

99 the transition energies to the blue rather uniformly by about 2000 cm"^ The larger basis set seems to describe the ground state better than the excited states. Precise experimental comparisons will need gas-phase experiments or solid-state calculations.

The Renner-Teller effect in the first excited (fluorescent) state was also investigated.

The structure of UO|'^ in its first excited state was found to be linear with a very small Renner-Teller splitting.

(2 )U 0 2 0 H'*’: To understand the interaction of with ligands, we carried out calculations on UO^OH'*'. The ground- and first excited-state geometries were opti­ mized at the spin-orbit CISD level. The interaction between and 0H~ in the ground state is strong and the is bent with an angle of ca. 170° and the UO distance is increased by 0.036 A. The excited state at its minimum geometry has one electron transferred from 0H “ to UO^^. It has a linear QUO linkage and the UO distance in UOH is quite long so there is little interaction between UO? and OH. The adiabatic excitation energy is only about half of the value of observed in fluorescence experiments for the first hydroxide while the vertical transition energ}' at the opti­ mized ground-state geometry is very close to the experimental value. It was found that at the equilibrium position of the first excited state, one electron is excited from a OH“-like orbital to the /-like orbital on UOj'*’. In comparison, a vertical transition from the ground state involves excitation from an orbital with both UOg^ and 0H ~ character.

(3 )U(BH 4 )4 : All the low-energy electronic transitions were calculated with spin- orbit CISD. Due to the limitation of the current programs, only ten electrons are

100 correlated. References used in the calculation included all the configurations.

.A.11 occupied orbitals with substantial U 5/ character are included. The calculated energies are compared with crystal-held assignments of the experimental values. The calculated ground state is an E state, in agreement with experimental assignments.

The calculated transition energies are in reasonable agreement with experimental values.

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