AIR FORCE INSTITUTE OF TECHNOLOGY

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1 THEORETICAL COMPARISON OF THE EXCITED ELECTRONIC STATES OF THE URANYL (UO + ) AND URANATE (UO 4 - ) IONS USING RELATIVISTIC COMPUTATIONAL METHODS THESIS Eric V. Beck, Captain, USAF AFIT/GNE/ENP/3-1 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wriht-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

2 The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government.

3 AFIT/GNE/ENP/3-1 THEORETICAL COMPARISON OF THE EXCITED ELECTRONIC STATES OF THE LINEAR URANYL (UO + ) AND TETRAHEDRAL URANATE (UO 4 - ) IONS USING RELATIVISTIC COMPUTATIONAL METHODS THESIS Presented to the Faculty Department of Enineerin Physics Graduate School of Enineerin and Manaement Air Force Institute of Technoloy Air University Air Education and Trainin Command In Partial Fulfillment of the Requirements for the Deree of Master of Science (Nuclear Science) Eric V. Beck, BS Captain, USAF March 3 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

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5 Acknowledments There are many people I d like to thank for their help and contributions to this research. First and foremost, I d like to thank my advisors, Dr. Larry Burraf, Dr. Jean- Phillipe Blaudeau, and Dr. David Weeks. Dr. Burraf, for havin the faith in my ability to undertake such a dauntin project despite my doubts, and for his entle uidance and support for this work. Dr. Blaudeau for the countless hours he spent teachin me literally everythin I know about relativistic quantum mechanics. Without him, this project would have surely failed. And last, but not least, Dr. Weeks, for his probin comments and suestions that forced me to be precise in my lanuae and develop a true understandin of many of the theoretical concepts. In addition to my advisors, I d like to thank Dave Doaks and Jim Gray. They have assembled a powerful, and unfortunately, under-utilized 96 processor parallel computin cluster at AFIT, where I performed most of my calculations. I hope that the abuse I put the cluster throuh will lead to a more capable system for future students. Dave and Jim have done a tremendous job. I can t ive them enouh praise. I also have to thank Dr. Frank Duan at the MSRC for compilin and installin NWChem on their Compaq ES 4/45s for my use, as well as providin several useful suestions durin my eometry optimizations. Last, but definitely not least, I have to thank my wife and my two children,. They raciously accepted the demands AFIT placed on my time, and their support and love is unconditional. I am truly blessed to have them. Eric V. Beck iv

6 Table of Contents Acknowledments... iv List of Fiures... vii List of Tables...viii Abstract... x I. Introduction... 1 II. Theory... 4 Relativistic Effects in Chemistry... 5 The Dirac Equation... 1 Relativistic Many-Electron Hamiltonians... Dirac Hartree-Fock Theory... 4 Relativistic Effective Core Potentials... 3 Electron Correlation Models Summary III. Resources and Methodoloy Hardware Resources Software Resources Geometry Optimizations Overview of COLUMBUS Calculations... 5 COLUMBUS Calculations on Uranyl (UO + ) COLUMBUS Calculations on Uranate (UO 4 - ) IV. Results and Discussion Uranyl Geometry Optimization Results Lare-core Uranyl (UO + ) Results Small-core Uranyl (UO + ) Results... 7 Small-core Uranate (UO 4 - ) Results... 8 V. Conclusions and Recommendations Appendix A: Symmetry Considerations for the Linear Uranyl Ion Appendix B: Symmetry Considerations of the Tetrahedral Uranate Ion v

7 Appendix C: State Assinments Usin MR-CISD Wave Functions Uranyl MR-CISD State Assinment Uranate MR-CISD State Assinment List of Abbreviations Biblioraphy Vita vi

8 List of Fiures Fiure 1. Schematic Representation of the Assumptions in Russell-Sanders and j- j Spin-Orbit Couplin Schemes (Gerloch, 1986: 61)...9 Fiure. Correlation Diaram of the Various States Arisin From a d Electronic Confiuration Usin Both Russell-Sanders and j-j Spin-Orbit Couplin (Gerloch, 1986: 78)...1 Fiure 3. Electronic Ground State for Uranyl Usin Both Non-relativistic HF and Four-component, Fully Relativistic DHF Methods (de Jon, 1999: 45)...31 Fiure 4. Lare-core Uranyl SCF Potential Enery Surface...64 Fiure 5. Lare-core Uranyl MCSCF Potential Enery Surface...65 Fiure 6. Lare-core MR-CISD Uranyl Electronic States from Molecular Orbitals...67 Fiure 7. Lare-core MR-CISD Uranyl Electronic States from Natural Orbitals...68 Fiure 8. Small-core Uranyl SCF Potential Enery Surface...7 Fiure 9. Small-core Uranyl MCSCF Potential Enery Surface...73 Fiure 1. Small-core Uranyl MR-CISD Electronic States From Molecular Orbitals...74 Fiure 11. Small-core Uranyl MR-CISD Electronic States From Natural Orbitals...75 Fiure 1. Uranate SCF Potential Enery Surface...81 vii

9 List of Tables Table 1. Uranyl Geometry Optimization Results...61 Table. COLUMBUS Lare-Core Uranyl Ground-State Calculation Results...6 Table 3. COLUMBUS Small-Core Uranyl Ground-State Calculation Results...6 Table 4. Uranate NWChem 4..1 Geometry Optimization Results...63 Table 5. Lare-core Uranyl SCF Symmetric Stretch Vibrational Frequencies...64 Table 6. Lare-core Uranyl MCSCF Symmetric Stretch Vibrational Frequencies...66 Table 7. Lare-core Uranyl MR-CISD Symmetric Stretch Vibrational Frequencies...69 Table 8. Lare-core Uranyl Wave Function Compositions...7 Table 9. Electronic Transition Eneries from Lare-core MR-CISD Results...71 Table 1. Lare-core Uranyl Results from Zhan et al (Zhan, 1999: 6884)...71 Table 11. Small-core Uranyl SCF Equilibrium Bond Lenth and Symmetric Stretch Vibrational Frequency...73 Table 1. Small-core Uranyl MCSCF Symmetric Stretch Vibrational Frequencies...74 Table 13. Small-core Uranyl MR-CISD Symmetric Stretch Vibrational Frequencies...76 Table 14. Small- and Lare-core Uranyl Wave Function Compositions...77 Table 15. Electronic Transition Eneries from Small-core MR-CISD Results...78 Table 16. Comparison Between Theoretical and Experimental Uranyl Fluorescent Electronic Spectra and Symmetric Stretch Vibrational Frequencies (Rabinowitch et al, 1964: 48)...79 Table 17. Small-core Uranate SCF Symmetric Stretch Vibrational Frequencies...8 viii

10 Table 18. Small-core Uranate MR-CISD Double Group Terms Of A Symmetry in D and Their ΓS Compositions...84 Table 19. Small-core Uranate MR-CISD Double Group Terms Of B 1 Symmetry in D and Their ΓS Compositions...84 Table. Small-core Uranate MR-CISD Low-Lyin Vertical Electronic Transition Eneries at Å...85 Table 1. Correlation Between D h and D h Symmetry Point-Groups (Cotton, 1971: )...89 Table. Uranium Atomic Orbitals in D h and D h Symmetry Point Groups...9 Table 3. Two Oxyen Atomic Orbitals Alon z-axis in D h and D h...91 Table 4. Uranyl Possible States from CISD References in both ΛS and ωω Couplin Schemes...9 Table 5. Total Wave Function Symmetry in D h From Reference Electronic Confiurations...93 Table 6. Correlation Table Between T d and D Symmetry Point-Groups (Cotton, 1971: )...95 Table 7. Linear Combination of Uranium Atomic Orbitals in T d and D Symmetry Point Groups...96 Table 8. Combination of Four Tetrahedral Oxyen Atomic s-orbitals in T d and D...97 Table 9. Uranate Total Wave Function Symmetry in D From Reference Electronic Confiurations...98 ix

11 AFIT/GNE/ENP/3-1 Abstract This thesis examines the round and excited electronic states of the uranyl (UO + ) and uranate (UO - 4 ) ions usin Hartree-Fock self-consistent field (HF SCF), multi-confiuration self-consistent field (MCSCF), and multi-reference sinle and double excitation confiuration interaction (MR-CISD) methods. The MR-CISD calculation included spin-orbit operators. Molecular eometries were obtained from self-consistent field (SCF), second-order perturbation theory (MP), and density functional theory (DFT) eometry optimizations usin the NWChem 4.1 massively parallel ab initio software packae. COLUMBUS version was used to perform in-depth analysis on the HF SCF, MCSCF, and MR-CISD potential enery surfaces. Excited state calculations for the uranyl ion were performed usin both a lareand small core relativistic effective core potential (RECP) in order to calibrate the method. This calibration included comparison to previous theoretical and experimental work on the uranyl ion. Uranate excited states were performed usin the small-core RECP as well as the methodoloy developed usin the uranyl ion. x

12 THEORETICAL COMPARISON OF THE EXCITED ELECTRONIC STATES OF THE LINEAR URANYL (UO + ) AND TETRAHEDRAL URANATE (UO 4 - ) IONS USING RELATIVISTIC COMPUTATIONAL METHODS I. Introduction The chemical properties of uranium and plutonium oxides are critically important to nuclear applications. Of particular importance in the chemistry of these actinide compounds is the oxidation state. Uranium, like all the actinides, can possess a wide rane of oxidation states, ranin from +3 to +6, due to chemical activation of the uranium 5f orbitals via relativistic effects. As a result, the oxidation state of uranium can be influenced by its local chemical environment, which in turn influences the eometry of the uranium oxide compounds. Uranium oxidation state plays a very important role throuhout the nuclear fuel cycle, and it plays a critical role in the mobility of uranium in the environment. This oxidation state can be inferred throuh spectroscopic measurements, providin a simple and inexpensive tool for use in such areas as nuclear forensics and environmental monitorin. Additionally, the stockpile stewardship 1 proram demands a thorouh understandin of the processes by which uranium and plutonium components ae, as well as the effect this ain has on the reliability and performance of nuclear weapons. A cornerstone of the stockpile stewardship proram is theoretical modelin and simulation 1 Stockpile stewardship refers to the substantial effort undertaken by the U.S. Department of Enery to maintain and certify the U.S. nuclear weapon arsenal without resortin to underround nuclear testin-- 1

13 of the basic physics and chemistry involved in the desin, manufacture, maintenance, and operation of a nuclear weapon. Aain, non-invasive electronic spectroscopic methods can be used to dianose the extent of nuclear weapon component ain, based on the relationship between uranium oxidation state and its local chemical environment. Uranium oxidation is of particular interest. Oxyen and uranium readily react, formin a wide rane of complex oxides over a rane of temperatures and pressures (Wanner, 199). The uranyl ion, UO +, is an unusually stable oxide of uranium, and it is present in a majority of complex uranium(vi) oxides (Pyykkö, 1998: ; Zhan 1999:688). While there is a lare amount of experimental data on the various properties of uranium and plutonium (Katz et al, 1986; Wanner, 199), theoretical understandin of the spectra of these elements has proressed slowly. Ab initio quantum mechanical theoretical techniques have made reat strides in understandin of molecules consistin of lihter elements, and computational methods have been quite successful in predictin thermodynamic and spectroscopic properties of these compounds. Unfortunately, such proress in the actinide compounds has proressed more slowly, for two main reasons. The first difficulty is the sheer number of electrons to deal with in actinide compounds. Common uranium oxide compounds such as UO have 18 electrons, while more complex oxides such as U 3 O 8 have over 3 electrons. Accurately treatin such lare numbers of electrons becomes computationally intensive, and it has only been in the last decade that such molecules can be treated with the accuracy necessary to compare theoretical and experimental electronic spectra. A second difficulty is the fact that relativistic effects must be accounted, not as perturbations to, but on an equal footin with electron correlation in these heavy molecular systems for even moderate accuracy. This is in stark contrast to lihter molecules where relativistic effects can be nelected in Latin for "from the beinnin", Merriam-Webster's Colleiate Dictionary,,

14 all but hih-precision theoretical calculations (Pyykkö, 1998: ; Balasubramanian, 1997: 1-7). With the development of faster computers, especially massively parallel computer systems, as well as quantum chemistry software codes desined to take advantae of these computer architectures, there has been an increasin number of theoretical investiations of actinide compounds. However, the theoretical calculation of excited electronic states for actinide compounds is a difficult problem, and there are relatively few in-depth studies of the excited states of uranium oxides necessary for understandin the electronic spectra. This research focuses on two uranium oxide ions in particular: uranyl (UO + ) and uranate (UO - 4 ) ions. Startin from optimized, theoretical, as-phase molecular eometries, electronic spectra calculations from sinle and double excitations with spinorbit couplin included were computed and compared with experimental and other theoretical results. The calculations involvin uranyl were used to calibrate and validate the method, while those involvin UO - 4 were an attempt to bein understandin the influence of the local oxyen coordination on the electronic spectra of uranium oxides. Particular interest was paid to the first excited states of both. The theory relevant to calculations of the electronic spectra of uranium oxides is laid out in chapter two. Next, the hardware and software resources used in this research, as well as the methodoloy behind the study of uranyl and uranate electronic spectra is described in chapter three. Results and discussion of the results is included in chapter four, followed by conclusions drawn from this research and recommendations for further research in chapter five. 3

15 II. Theory Ab initio quantum mechanical theoretical techniques have been applied to molecules containin liht elements with increasin success in the past several decades. Application of these theoretical techniques to liht-atom molecules, especially oranic molecules, has yielded vast insiht into the properties of these molecules such as molecular eometries for round and transition states, electron affinities, ionization potentials and more. Advances in computin power, coupled with quantum chemistry software desined specifically to take maximum advantae of these computers has increased the applicability to larer molecules. Such calculations have become an indispensable tool to theoreticians and experimentalists alike. Complications arise when applyin theory to molecules containin heavy elements, especially actinide molecules. The two most difficult complications to the theoretical treatment of heavy-element molecules are increased electron correlation and relativistic effects. First, these heavy-element molecules contain a lare number of electrons whose motions are coupled throuh electrostatic and quantum mechanical interactions. Electron correlation effects can contribute rouhly 1 ev (3 kcal/mol) to the total electronic enery per electron pair (Rahavachari, 1996: 196). Usin this rule of thumb, electron correlation accounts for 46 ev of the total electronic enery in the uranium 4

16 atom. An accurate treatment of electronic correlation is critical in order to perform meaninful comparisons between theoretical and experimental spectra. A second complicatin factor in the theoretical treatment of heavy-element molecules is the increasin importance relativistic effects play in the accurate description of round and excited electronic states with increasin atomic number, Z. Several reviews examine relevant chemical effects due to relativistic quantum mechanical treatments (Pyykkö, 1988: ; Pepper et al, 1991: ; Kaltsoyannis, 1997: 1-11). Relativistic Effects in Chemistry There are three main relativistic effects in atomic and molecular chemistry, all of which are rouhly the same manitude, and they approximately scale as Z 4 (Pyykkö, 1988, 564). The first main relativistic effect is considered a direct relativistic effect, and it consists of a radial contraction of atomic orbitals, alon with a lowerin of the enery level of the electronic state. This effect is due primarily to the relativistic mass increase as electron velocities become appreciable fractions of the speed of liht. Simple replacement of the relativistic mass expression for the electron in the Bohr radius formula yields a = 4πε h m e 1 v ( ) c. (1) Here, h is Planck s constant divided by π, ε is the permittivity of free space, e is the electron chare, and m is the electron mass. As electron speeds, v, approach the 5

17 speed of liht, c, the Bohr radius, a, shrinks. Electron orbitals with hih densities near the nucleus experience the larest contractions, where electron speeds are larest. For electrons in the hydroenic 1s shell, the averae fraction of the speed of liht is iven by Z 137 (Pyykkö, 1988: 563). For uranium, this is.67c, yieldin a 1s orbital radial contraction of rouhly 6%. All atomic orbitals have some density near the nucleus, therefore, all atomic orbitals experience some contraction. However, the inner s- and p- orbitals nearest the nucleus experience the most contraction (Pyykkö, 1988: 563). In liht-element molecules, this orbital contraction is small and neliible in all but the hihest precision calculations, but the effect becomes dramatic in actinide elements such as uranium. The second relativistic effect is considered to be an indirect effect, and it consists of a radial expansion and increase in the electronic enery levels of outer atomic orbitals. This is due to more effective nuclear chare screenin by the inner, contracted electrons, reducin the effective nuclear chare experienced by the outer electrons. Additionally, relativistic contraction of the inner s- and p- electron shells increase the electron density near the nucleus, crowdin out the outer d- and f- electron shells. This is due to the fact that there is a decrease in electron density near the nucleus for orbitals with increasin orbital anular momentum. Thus, the direct orbital contraction competes with the indirect orbital expansion. In eneral, the result of this interplay between relativistic effects is to contract and stabilize s- and p- atomic orbitals, while d- and f- orbitals expand and destabilize in enery. The orbital expansion and contraction can affect bond lenths (Pyykkö, 1988: 571) and force constants (Pyykkö, 1988: 58), which in turn 6

18 affect molecular vibrational frequencies. These relativistic effects obviously affect the observed spectra of heavy-element molecules, but not as much as the splittin of states caused by the third major relativistic effect, spin-orbit couplin. Intrinsic electron spin is a natural result of a Lorentz-covariant description of the quantum mechanically wave equation (Balasubramanian, 1997: 76-78). This spin anular momentum couples with the electron orbital anular momentum, liftin deeneracy in atomic orbitals with anular momentum. Thus, the three deenerate p- orbitals in non-relativistic theory split into one p 1 orbital and two deenerate p 3 orbitals. Of the three effects, spin-orbit couplin has the larest impact in atomic and molecular spectra, even for low-z atoms and molecules. For liht atoms, a perturbative treatment of spin-orbit couplin known as Russell-Sanders couplin or L-S couplin often yields sufficient accuracy for electronic transition eneries. This couplin scheme treats manetic spin-orbit couplin as a small perturbation to the electron-electron electrostatic interaction. Orbital anular momentum and spin anular momentum are still nearly ood quantum numbers in this couplin scheme, and both L and S commute with the Hamiltonian in Russell-Sanders couplin scheme. Atomic states are described by term symbols S + 1 L J S =,1,, K L = S, P, D, F, K J = L + S with S equal to the total spin multiplicity, and L is the total orbital anular momentum (, 1,,..). Traditional spectroscopic notation is used for the total orbital anular 7

19 momentum, with S representin zero total orbital anular momentum, P representin one unit of orbital anular momentum and so on. J is the total anular momentum of the electron, iven by the sum of orbital and spin anular momenta. Examples of Russell- Sanders term symbols include 3 P, 1 S, and 3 D 4 (Gerloch, 1986: 69-74). On the other end of the perturbation spectrum, more appropriate for very heavy atoms, the electron-electron electrostatic interaction is treated as a perturbation to the manetic spin-orbit couplin. This couplin scheme is known as j-j couplin. In this couplin scheme, neither L nor S commute with the Hamiltonian. However, the total anular momentum, J, still commutes with the atomic Hamiltonian, and hence, is a ood quantum number. The term symbol for j-j couplin is iven by the J value for the state (Gerloch, 1986: 74-76). Fiure 1 contains a schematic representation of the two spin-orbit couplin extremes (Gerloch, 1986: 61). The horizontal sprins represent electrostatic couplin between the electrons, while the vertical sprins represent manetic couplin between the electron intrinsic manetic moments. In Russell-Sanders couplin, the electron orbital anular momenta couple stronly, as do each electron s spin anular momenta. These total orbital and spin anular momenta then couple weakly. The opposite is true of j-j couplin. In j-j couplin, each electron s orbital and spin anular momenta couple stronly, and this individual total anular momentum couples weakly with the other electrons total anular momenta. 8

20 Fiure 1. Schematic Representation of the Assumptions in Russell-Sanders and j-j Spin- Orbit Couplin Schemes (Gerloch, 1986: 61) Most elements on the periodic table fall between these two perturbation extremes, and so intermediate couplin is more appropriate than either perturbative treatment. Intermediate couplin is not a separate couplin scheme, but occurs as deviations from the separate perturbative treatments iven by L-S and j-j couplin (Gerloch, 1986: 77). Fiure illustrates the effect of Russell-Sanders, intermediate, and j-j spin-orbit couplin on a d electronic confiuration. Fiure shows the effect of spin-orbit couplin on an atomic electronic state with two electrons in the d-shell. The left-hand side shows the term symbols that arise due to Russell-Sanders couplin, while the riht-hand side shows 9

21 the effect of j-j couplin on the same electronic confiuration. Here, the importance of spin-orbit couplin to electronic spectroscopy is evident. Without takin into account spin-orbit couplin, both the number of states and their relative orderin will be incorrect. Fiure. Correlation Diaram of the Various States Arisin From a d Electronic Confiuration Usin Both Russell-Sanders and j-j Spin-Orbit Couplin (Gerloch, 1986: 78) Relativity also affects the symmetry of molecules, because of electron spin. Under the assumption that the total electronic wave function can separated (the product of a spatial and spin wave functions), each wave function may possess separate symmetry, and the total, observable state symmetry is iven by the product of the spatial 1

22 and spin symmetries. For example, sinlet spin states are completely symmetric, while triplet spin states transform like the components of the anular momentum operator. Thus, a completely symmetric spatial wave function multiplied by a triplet spin wave function will not be totally symmetric. For systems with a spin-orbit Hamiltonian, the symmetry point roups can have twice the number of symmetry operations, and are called double point roups. This doublin of the order of the symmetry point roups is due to the introduction of 1 -interal anular momentum values. Systems possessin an even number of electrons obey Bose-Einstein (bosons) statistics, and the total wave function of bosonic systems is symmetric with respect to rotations by π. Systems possessin an odd number of electrons obey Fermi-Dirac statistics (fermions), and fermionic wave functions chane sin up the exchane of two particles. This exchane is equivalent to a rotation by π, and so a rotation of 4π returns a fermionic system to its oriinal state. While bosonic systems transform accordin to the irreducible representations of the sinle point roups, the rotation by π is a new symmetry operation for fermionic systems, doublin the order of the symmetry point roup. For example, rotations of a closed-shell molecule, such as uranyl (UO + ) transforms accordin to the normal irreducible representations of the D h point roup. Rotatin the molecule by π leaves the molecule (wave function) unchaned. However, for an open-shell molecule, such as UO, such is not the case. Such molecules transform accordin to the extra irreducible representations enerated by a rotation of π. Rotatin the UO molecule by π introduces a phase factor into the total electronic wave function. A rotation by 4π in this case returns the molecule (wave function) to its oriinal confiuration. 11

23 These three main effects, orbital contraction and enery stabilization, orbital expansion and enery destabilization, and spin-orbit couplin, alon with the consequential double roup symmetry constitute the chemically relevant relativistic effects in atoms and molecules. The most important, from a spectroscopic standpoint is spin-orbit couplin, even in the spectra of the lihtest elements. A quantum mechanical treatment of the electron must account for the intrinsic manetic moment of the electron, and the Dirac equation accomplishes this quite eleantly. The Dirac Equation Relativity has played a role in quantum mechanical systems since the inception of the theory. Attempts at findin a Lorentz invariant form for Schrödiner s equation led to two Lorentz-covariant equations: the Klein-Gordon equation, and the Dirac equation. Schrödiner s equation, a non-relativistic quantum mechanical wave equation, is iven by H Ψ = EΨ, () where the Hamiltonian, H, and enery, E, operators are iven by r r p (, t) r H (, t) = + V (, t), (3) m and E( t) = h. (4) i t The momentum operator, p, is defined by r h p (, t ) =. (5) i 1

24 Here, m is the electron mass, i is the imainary number, h is Planck s constant divided by π, V ( r, t) is the potential enery operator, and p(r,t r ) is the electron momentum. Ψ is the electronic wave function. This equation, because of the non-equivalent treatment of the spatial and temporal variables, is not Lorentz invariant, and therefore is limited to non-relativistic phenomena. Early attempts at makin a Lorentz-covariant equation bean by quantizin the Lorentzcovariant relativistic enery expression E = p c + m c 4. (6) Aain, p is the electron momentum, c is the speed of liht, and m is the electron rest-mass, and E is the electron enery. Replacin the enery and momentum expressions with their quantized counterparts leads to the Klein-Gordon wave equation for a free particle (Balasubramanian, 1997: 99-11; Messiah, 1999: ; Bjorken and Drell, 1964: 4-6,198-6) Ψ 4 h = h c Ψ + m c Ψ. (7) t While this scalar wave function is Lorentz-covariant, it has several undesirable properties, makin it unacceptable as a wave function for the electron. First, the probability density associated with it is not positive definite, resultin in possible neative probability densities. Additionally, both positive and neative enery solutions to this equation exist, complicatin early interpretation of the wave function. The fact that the probability density is not positive definite makes this equation a poor choice for an electronic wave function; however, the Klein-Gordon turns out to be a valid 13

25 relativistic wave equation for spin-free fields, such as pi mesons (Messiah, 1999: 888; Balasubramanian, 1997: 18). Dirac took a different approach in formulatin a Lorentz-covariant equation for a free electron (Dirac, 198: 61-64; Dirac, 198: ; Balasubramanian, 1997: ). He bean with the same Lorentz-covariant expression for the enery of a free particle as the Klein-Gordon equation, E = p c + Takin the square root yields the Dirac Hamiltonian, m c 4. (8) H Dirac + 4 = ± p c mc. (9) Quantizin this expression by the usual substitutions for the enery and momentum operators yields a Hamilton that involves a first-order time derivative. However, the square root in the operator makes application problematic and hopelessly complicated. Dirac circumvented this problem by introducin a new deree of freedom into the relativistic Hamiltonian, effectively completin the square. This yielded a more tractable Hamiltonian operator The Dirac equation, t r t r t H Dirac = c( α1 p1 + α p + α 3 p3) + βmc. (1) ih c Ψ t t r t r t r t = c( α p + α p + α p Ψ + βm c ) results from this Hamiltonian (Kello, 1997: 4-6). r t Ψ, (11) Requirin solutions to this equation to simultaneously satisfy the Klein-Gordon equation places restrictions on the components of the t α i and β matrices: 14

26 ij i j j i δ α α α α = +, (1) =1 β, (13) and = + k k βα β α. (14) In order to satisfy these restrictions both i α t and β must be at least four-by-four matrices, which operate on a four-component vector wave function. The i α t and β matrices are defined as, (15) = α t 1, (16) = i i i i α t, (17) = α t 3 and. (18) = β t 15

27 The Dirac equation is then a set of four, coupled, first order partial differential equations in space and time. The four-component wave function solution to this equation corresponds to two positive enery components and two neative enery components, each with a spin-up and spin-down component (Messiah, 1999: , 9-94). The t α i operators are velocity operators, while β is an parity operator. Observable quantities associated with this new internal deree of freedom are t he enery, relativistic mass, current density, total anular momentum, spin, and parity. The operators associated with these observables are (Messiah, 1999: 91) enery: (19) r t r ea H = eφ + α ( p ) + βmc, c where H is the Dirac Hamiltonian, φ and A r are external electric and maentic potentials, respectively. The relativistic mass is iven by M = H eφ. () Current density is found usin r t r r j( ) = α δ, (1) ( r ) where ( r r δ ) is the Dirac delta function. Electron spin is iven by Si t 1 = σ i, () where t σ i is the i th Pauli spin matrix. Finally, the parity of the wave function is iven by P = βp, (3) 16

28 where P is the initial parity. These Pauli spin matrices can be expressed in terms of the t α i matrices (Messiah, 1999: 891): t σ t σ t t =, (4) z iα xα y t t =, (5) x iα yα z and t σ t t =. (6) y iα zα x In the presence of an external field, the Dirac Hamiltonian, H D, becomes t c r r ea t ( p m c c H D = eφ + α ) + β. (7) For the hydroen atom, in the absence of an external manetic field, this equation reduces to t r t EΨ = [ c( α p) + βm c + e ] Ψ. (8) φ While it is possible to construct an exact solution to this equation in terms of spherical harmonics for the anular coordinates and hypereometric functions for the radial coordinate, such a construction does not shed much liht on the nature of the bound enery states. The details of the solution can be found in various sources (Messiah, 1999: ; Balasubramanian, 1997: ; Bethe et al, 1957: 63-71). The electronic enery levels for the Dirac hydroen atom are iven by (Bethe et al, 1956: 67-68) mc E nj =. (9) + Zα 1 1 n j + + ( j + 1 ) Z α 17

29 Here, α is the fine structure constant, defined as e α =, (3) 4πε hc and the total anular momentum quantum number, j, takes on the values j = l + 1, l. (31) 1 The bindin enery of the hydroen atom is iven by E nj E, where E = m. c Expandin E nj in powers of ( Z α ), where α is the fine structure constant iven E above, and assumin Z α << 1, yields (Bethe et al, 1956: 84) E nj ( Zα ) ( j 8n)( Zα ) + n 8( 1+ j) n = 4 4 K. (3) The first term is the non-relativistic enery for the bound electronic states of the hydroen atom. Hiher order corrections involve both the principle quantum number n, as well as the total anular momentum quantum number j. This illustrates the importance of a relativistic picture of the atom. Corrections to the non-relativistic enery increase rouhly as 4 Z. Note that this Taylor series expansion in powers of ( α ) Z is appropriate for Z α << 1. This expansion leads to the Russell-Sanders spin-orbit couplin scheme. Such an approximation is not valid for uranium, where Z α =. 6. In this case, 1 Z << ( Zα ), and the electrostatic electron-electron interaction can be treated as a perturbation to the manetic interaction between the electron and the field of the nucleus. This approximation leads to the j-j spin-orbit couplin scheme, which is more appropriate for very heavy elements. 18

30 Detailed examination of the neative enery component solutions to the Dirac equation for the free electron shows in the non-relativistic limit where E m c << m, c are much larer than the neative enery components, especially in the valence reion (Balasubramanian, 1997: ). Thus, the four-component Dirac wave function naturally separates into two lare and two small components. Rewritin the Dirac equation in terms of two, coupled differential equations with two, two-component wave functions yields the Pauli approximation to the Dirac Hamiltonian in the absence of an external manetic field (Balasubramanian, 1997: ) H Pauli 1 = E + eϕ + m µ m c 1 + m c ( E + eϕ ) r r r r r [ σ ( E p) µ ( σ H )] µ v r + i E p m c (33) where H r is the manetic field, E r is the electric field, and µ is the Bohr maneton, defined by eh µ =. (34) m c While somewhat cumbersome, the separate terms have simple interpretations. The first three terms are the non-relativistic Schrödiner Hamiltonian. The next term is the mass-velocity correction that accounts for the variation in electron mass with speed. The fifth term is known as the Darwin term, and is a result of zitterbeweun, or tremblin motion. It is a result of the Heisenber uncertainty principle. Nonrelativistically, the uncertainty in the location of an electron can be measured to any accuracy usin hiher and hiher enery photons. Relativistically, there is a limit to this 19

31 photon enery used to locate the electron, because at photon eneries above m c, pair production can occur. This results in an effective smearin of the chare of the electron (Balasubramanian, 1997: 186). The final two terms account for the spin-orbit couplin between the intrinsic electron manetic moment and the orbital anular momentum. The successes of the Dirac equation is the prediction of electron spin as an observable property in the non-relativistic limit, as well as accountin for the correct value for the electron manetic moment. Thus, the inclusion of electronic spin in the non-relativistic theory as an additional assumption is validated and explained in the non-relativistic limit of the Dirac equation. Relativistic Many-Electron Hamiltonians Now that a Lorentz-covariant electronic wave function is available for the hydroen atom, the next loical step is to try to extend this approach to larer atoms and molecules. A relativistic wave function for a many-electron atom can be constructed as the sum of the one-electron Dirac Hamiltonians alon with an electron-electron interaction term where the i th one-electron Dirac Hamiltonian, i H = h d + B, (35) i i h d i< j ij is iven by h i d t r r ea c t = eφ + cα ( pi ) + βmc, (36) while B ij represents a eneral electron-electron interaction term.

32 This approach is analoous with the non-relativistic approximation to the many-electron Hamiltonian based upon the one-electron Schrödiner equation for the hydroen atom. Next, a Lorentz-covariant description of electron-electron interactions, B ij, is required. Unfortunately, the electron-electron interaction requires the more detailed treatment afforded by quantum electrodynamics, where vacuum interactions, virtual photon exchanes, and electron self-interactions are treated perturbatively. Even then, there is no closed form for a Lorentz-covariant electron-electron interaction. Such interactions must be treated approximately (Balasubramanian, 1997: 18; Messiah, 1999: ; Bethe et al, 1956: 17). The first such approximation, widely used, is the approximation that relativistic corrections to the electron-electron interaction are small and neliible, and that the Coulomb interaction is an appropriate description, correct to zeroth order. This leads to the Dirac-Coulomb Hamiltonian (Kello, 1997: 15) H DC = i h i d + i< j e 4πε r ij. (37) Here, r ij is the interelectron distance. This Hamiltonian is not Lorentz-covariant; however, corrections to the Columbic interaction are small for lare electron separation, and the Dirac-Coulomb Hamiltonian is quite successful. Another approach to determinin the electron-electron interaction is to perturbatively expand the quantum electrodynamics interaction term in powers of the fine structure constant, and retain those α terms of order. This yields the Dirac-Coulomb-Breit Hamiltonian (Bethe et al, 1956: 17; Balasubramanian, 1997: 18; Jackson, 1975: ; Breit, 193: ) 1

33 H DCB i = hd + i e 1 πε rij r r r r r r α ( )( ) i α j α i ij α j ij. (38) 3 r ij r i< j 4 ij This approximate Hamiltonian is still not Lorentz-covariant; however, it accounts for most of the chemically relevant electron-electron interaction effects. The first term is the electrostatic Coulomb interaction between two electrons, the second term accounts for first order manetic interactions between the intrinsic manetic moments of the electron. The last term accounts for the retardation of the propaation of the electromanetic field of the electron due to the finite speed of liht. Another method, based on perturbative expansion of the Dirac-Coulomb-Breit Hamiltonian in powers of α yields the Breit-Pauli Hamiltonian, iven by (Balasubramanian, 1997: ) H = H + H + H + H + H + H + H, (39) BP nr mv retardation Darwin SO SS external where H nr = i pi Ze e + m 4πε r 4πε r i i< j ij, (4) 4 i H mv 3 = i p 8m c, (41) r r H Ze pi p j ( rij pi )( rij p j ) = retardation + 8πε m c i< j rij r 3 ij r r r r, (4) H Darwin H SO ieh = (m c ) i r r p E, (43) i i µ r = ], (44) m c r [ r e r r s + i E i pi ij p 3 j i i< j rij

34 r r r r r r eh s < i s j ( si ij )( s j ij ) 8π r r 3 r H = SS 3 ( si s j ) δ ( ij ), (45) 3 5 mc i j rij rij 3 and H external eh = m c i r r H s + i i e m c i r r A p. (46) i i Here, µ is the electron manetic moment, and s r i is the spin anular momentum of the i th electron. The first term, H nr, is the non-relativistic many-electron Hamiltonian. The second term, H mv, is the mass-velocity term, which corrects for the relativistic variation in electron mass near the speed of liht. The third term, H retardation, corrects for the finite propaation speed of the electromanetic field of the electron. The fourth term, H Darwin, is the Darwin correction, described earlier. These four terms comprise scalar relativistic effects, and do not require a two-component wave function to implement. The fifth term, H SO, is the spin-orbit couplin between the intrinsic spin anular momentum of the electron and the orbital anular momenta of all the electrons. The next term, H SS, is the spin-spin couplin between the intrinsic spin-anular momenta of multiple electrons. In order to incorporate these terms, a two-component wave function is required. The last term, H external, involves the interaction with an external electric and manetic field. While only a perturbative treatment, valid for liht atoms and molecules, the Breit-Pauli Hamiltonian sheds some liht on the expected effects present in relativistic many-electron Hamiltonians. With a well defined, albeit approximate, many-electron Hamiltonian, the next step in constructin relativistic wave functions is based on the Hartree-Fock mean field 3

35 theory. This provides the first theoretical method to many-electron systems; however, electron correlation is not explicitly included. Dirac Hartree-Fock Theory The Hartree-Fock (HF) self-consistent field (SCF) method provides the theoretical framework to determine, non-relativistically, the properly antisymmetrized many-electron sinle-determinant wave function for atoms and molecules (Szabo et al, 1989: 18-15; Levine, : 35-31). It also provides the basis for correlation calculations throuh multi-confiuration and perturbation methods. The non-relativistic many-electron Hamiltonian is iven by (Levine, : 35) H h = m e i i Ze 4πε r i + n 1 n e 4πε r i= 1 j= i+ 1 ij. (47) The oal of the HF approximation is to find a set of spin-orbitals, which minimize the round state electronic enery. It is a variational theory, in that the exact round state enery of an atom or molecule is a lower bound to the HF enery. Additionally, HF theory is a sinle determinant theory. This means that the round state wave function obtained from the variationally optimized set of spin orbitals contains only a sinle electron confiuration. The optimized set of spin-orbitals satisfy the equations (Szabo et al, 1989: ) E h k = Ψ H Ψ = χa χa χa χa + m e a k 4πε a rak 4πε a< b Z e 1 e 1 χaχb r ab χ χ, (48) a b where χ a is the spatial wave function of the a th electron. The total wave function is the product of individual electron spatial functions, 4

36 Ψ χ χ Kχ χ Kχ. (49) = 1 a b N The first two terms are one-electron operators, and correspond to the expectation values of the kinetic enery and potential enery of the a th electron in the field of the k th nucleus. The last term is the expectation value of the Coulomb interaction between the two electrons. It can be expanded into two terms r * e dxa χa (1) 4πε a< b b b e 4πε r dx χ () a< b 1 χaχb r ab χ χ 1 e * χ a (1) χa r1 4πε a b = () 1 a< b r * 1 dx bχb () χa() χb(1) r 1 (5) Writin equation 5 in operator form yields d r r * e xadxbχ a (1) Jb(1) Kb(1) χa (1) 4πε. (51) a b This final term results in two, one-electron operators, J and K. The first operator is the Coulomb operator and represents the averae electric field due to electron two experienced by electron one. This is the oriin of the mean-field concept. Each electron experiences an averae potential due to all the other electrons. The second term is an exchane potential arisin from the Pauli Exclusion Principle. Because the total wave function must be antisymmetric with respect to the exchane of two electrons, the motion between electrons with parallel spins is correlated in the HF theory. As a result, electrons experience an exchane potential, quantum mechanical in nature, which repels electrons with parallel spins and prevents them from occupyin the same orbital (Szabo et al, 1989: ). In its eienvalue form, the HF equations are 5

37 h + J b K a b a b b χ a = ε a χ a, (5) where ε a is the enery of the a th electron. The one-electron operators are defined by h = h m k χ a χ a e a k 4 Z e πε a χ a 1 r ak χ a, (53) J χ (1) = b a e 4πε a< b r dx b χ () b 1 χ a (1) r, (54) 1 and e K bχ a (1) = 4πε a< b r * 1 dxbχ b () χ a () χ a (1) r. (55) 1 The term in parentheses of the first equation is the Fock operator, operator form, this equation is f r f ( 1 ). In r r r ) ψ ( ) = ε ψ ( ). (56) ( 1 i 1 a i r1 r Expandin the unknown molecular orbitals, ψ (r ), in terms of a finite set of known basis functions, φ k, yields i K ψ ( r 1) = φk. (57) i c ki k = 1 ki Here, the c are molecular orbital coefficients. This reduces the HF interodifferential equation to a set of alebraic equations: the Roothan equations (Szabo et al, 1989: ) ν F c = S c, (58) µν νi ε i µν νi ν where F µν is the Fock matrix, defined by 6

38 * Fµν dr r = φ 1 µ (1) f (1) φν (1), (59) and S µν is the overlap matrix, defined by r * S µν = d 1φ µ (1) φν (1). (6) Thus, findin the optimal orbitals that minimize the Hartree-Fock enery consists of solvin the Roothan equations in a self-consistent manner. The non-relativistic HF theory and relativistic Dirac-Hartree-Fock (DHF) theory are analoous. In DHF theory (Saue et al, 1997: ; Ore, 1975: ; Aoyama et al, 198: ; Matsuoka et al, 198: ; Kim, 1967: ; Lee et al: 1977: ; Dyall et al, 1991: ), the non-relativistic Hamiltonian is replaced with the Dirac-Coulomb Hamiltonian, and the spin-orbitals have four components instead of one. These spinors can be complex, unlike the nonrelativistic case, where the spin-orbitals were real. This four-component wave function is also expanded in a real basis, as was done in the non-relativistic case, where (Saue, 1997: 939) ψ k L χ = S χ L χ c c c S χ c α k β k α k β k. (61) Here, L and S represent the lare and small components, respectively. There are two c k α β sets of expansion coefficients now, and for the spin-up and spin-down components, respectively. In a manner analoous to the non-relativistic theory, these four-spinors are varied until the total electronic enery is minimized. This leads to a c k 7

39 matrix equation similar to the Hartree-Fock theory, except that the Fock matrix and expansion coefficients are now complex. The Fock matrix splits into two parts, a oneelectron matrix (Saue, 1997: 94) F 1 LL V icd = icd SL z SL + icd W icd LS z SS SL + icd V icd SL LL SL z icd icd W LS LS z SS, (63) where XY X X V µν = χ µ V χ ν, (64) XY X X d z, µν = χ µ χν, (65) z XY X X d±, µν = χ µ ± i χν, (66) x y and W. (67) XY X X µν = χ µ V c χν As in the non-relativistic case, V represents the potential enery of the electron. Atomic units are used here, where h = e = m 1, in order to simplify the expressions. The two-electron Fock matrix, is iven by (Saue, 1997: 94): = F Lα J K S K = L K S K LαLα αlα βlα βlα J K Sα K K K LαSα LβSα SβSα SαSα J K K Lβ K K LαLβ SαLβ SβLβ LβLβ J K K K Sβ LαSβ SαSβ LαLβ K SβSβ. (68) Lα Lβ Here, J, J represents the Coulomb operator of the spin-up ( α ) and spin-down ( β ) electrons respectively, while Sα Sβ J, J represents the mean-field due to positrons with 8

40 spin-up and spin-down, respectively. The exchane operator, K, becomes more complicated, reflectin the possibility of exchane between electronic ( L ) components and positronic ( S ) components with various spins. One important difference exists between the non-relativistic Hartree-Fock and its relativistic counterpart. In the non-relativistic theory, the enery eienvalue correspondin to the Fock operator was uaranteed to be reater than or equal to the exact enery by the variational principle. In the relativistic case, the existence of positronic neative enery solutions means that the DHF enery is not bounded from below. In the non-relativistic case, there were no neative enery solutions. In the relativistic case, electronic solutions look like excited positronic states, and unless care is taken durin the solution of the relativistic Roothan equations, variational collapse can occur. This occurs because of the fact that a bound electronic-positronic state is deenerate with an unbound electronic-postronic state in the relativistic theory. Thus, instead of variationally optimizin the orbitals as in the non-relativistic case, the orbitals are minimized with respect to electronic states and simultaneously maximized with respect to positronic states. Another complication with the relativistic DHF theory is that the basis sets for the lare and small components are related via a kinetic balance requirement (Dyall, 1991: 585) S χ L = t σ χ. (69) Here, the small component (positronic) wave functions are related to the x Pauli spin-matrix, σ t, operatin on the radient of the lare component (electronic) wave function. 9

41 DHF methods suffer several computational difficulties (Saue, 1997: ). First, spin and spatial wave functions are coupled, resultin in complex total wave functions in eneral. Thus, spatial and spin symmetry can not be handled separately, which has resulted in substantial computational savins for non-relativistic computations. Another computational difficulty lies in the fact the DHF basis sizes are enerally much larer than their non-relativistic counterparts. Both the lare and small components are expanded in separate, but coupled basis sets. The small basis set size can be enerally twice the size of the lare component basis. Computationally, the Hartree-Fock method scales minimally as N 4, where N is the number of basis functions. Thus, Dirac-Hartree- Fock calculations typically involve an order of manitude or more increase in computational complexity over non-relativistic Hartree-Fock computations. These difficulties currently limit DHF methods to atoms and some small molecules. A recent + DHF calculation (de Jon, et al, 1999: 45) for UO compares the non-relativistic and relativistic results for the round electronic states. Electron correlation was included in this calculation via coupled cluster sinles and doubles with some triples (CCSD(T)) (Rahavachari, 1996: ), and this calculation represents perhaps the allelectron computational state-of-the-art on the round state of uranyl (de Jon, 1999: 41-5). Fiure 3 reproduces the electronic states from the paper. 3

42 Fiure 3. Electronic Ground State for Uranyl Usin Both Non-relativistic HF and Fourcomponent, Fully Relativistic DHF Methods (de Jon, 1999: 45) Dirac-Hartree-Fock theory can result in fairly accurate, fully-relativistic calculations of molecular electronic round states. Like the non-relativistic counterpart, DHF provides the best, sinle-determinant wave function, inorin electronic correlation effects. As such, HF and DHF can describe only the electronic round state. In order to describe the excited electronic states, electronic correlation needs to be incorporated, and the wave function must be expanded in a series of determinants. As was evident in the DHF equations, the fully relativistic treatment can become quite complicated and computationally intensive. What is needed is a method that is a compromise between the non-relativistic and fully-relativistic methods that also provides computational savins 31