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Numerical and Exact Density Functional Studies of Light Atoms in Strong Magnetic Fields

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PAGE 1

NUMERICAL AND EXACT DENSITY FUNC TIONAL STUDIES OF LIGHT ATOMS IN STRONG MAGNETIC FIELDS By WUMING ZHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Wuming Zhu

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Dedicated to Mom, to Dad, and to my wife.

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iv ACKNOWLEDGMENTS First of all, I would like to thank Professo r Samuel B. Trickey, my research advisor and committee chair, for the guidance he provi ded throughout the course of my graduate study at the University of Florida. His pa tience and constant encouragement are truly appreciated. Besides physics, I have also learned a lot from him about life and language skills which also are indispensable for becoming a successful physicist. I would also like to thank Professor Hai-Ping Cheng, Professor Jeffrey L. Krause, Professor Susan B. Sinnott, and Professor Davi d B. Tanner for serving in my supervisory committee, and for the guidance and advice th ey have given me. Professor David A. Micha is acknowledged for his help when I wa s in his class and for being a substitute committee member in my qualifying exam. My gratitude goes to Dr. John Ashley Alford II for many helpful discussions, and to Dr. Chun Zhang, Dr. Lin-Lin Wang, and Dr Mao-Hua Du for their academic and personal help. Besides them, many other fr iends have also enriched my life in Gainesville. They are Dr. Rongliang Liu, Dr. Linlin Qiu, Dr. Xu Du, Dr. Zhihong Chen, and Dr. Lingyin Zhu, who have moved to other places to advance their academic careers, and Guangyu Sun, Haifeng Pi, Minghan Chen, Yongke Sun, and Hui Xiong, who are continuing to make progress in their Ph.D. research. Many thanks go to the incredible staff at the Department of Physics and at QTP. I especially would like to thank Darlene Latimer, Coralu Clements, and Judy Parker for the

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v assistance they provided duri ng my graduate study. Financ ial support from NSF grants DMR-0218957 and DMR-9980015 is acknowledged. Lastly, I thank my parents, who will never read this dissertation but can feel as much as I do about it, for their boundless love I thank my wife for her special patience and understanding during the days I wrote my dissertation, and fo r all the wonderful things she brings to me.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................xi ABSTRACT.....................................................................................................................xi ii CHAPTER 1 BASICS OF DENSITY FUNCTIONAL THEORY AND CURRENT DENSITY FUNCTIONAL THEORY............................................................................................1 Introduction................................................................................................................... 1 Density Functional Theory...........................................................................................3 Foundations for DFT.............................................................................................4 The Kohn-Sham Scheme.......................................................................................7 Current Density Functional Theory (CDFT)..............................................................10 Basic Formulations..............................................................................................10 Vignale-Rasolt-Geldart (VRG) Functional.........................................................13 Survey on the Applic ations of CDFT..................................................................15 Other Developments in CDFT.............................................................................16 2 ATOMS IN UNIFORM M AGNETIC FIELDS — THEORY...................................18 Single Particle Equations............................................................................................18 Hartree-Fock Approximation..............................................................................18 Simple DFT Approximation................................................................................19 CDFT Approximation.........................................................................................19 Exchange-correlation Potentials.................................................................................20 3 BASIS SET AND BASIS SET OPTIMIZATION.....................................................25 Survey of Basis Sets Used in Other Work..................................................................25 Spherical-GTO and Anisotropic-GTO Representations.............................................27 Spherical GTO Basis Set Expansion...................................................................27 Anisotropic GTO (AGTO) Basis Set Expansion.................................................29 Connection between GTOs and AGTOs.............................................................30

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vii Primary and Secondary Sequences in AGTO.............................................................31 Optimized AGTO Basis Sets......................................................................................33 4 ATOMS IN UNIFORM MAGNETIC FIELDS — NUMERICAL RESULTS.........48 Comparison with Data in Literature...........................................................................48 Magnetic Field Induced Ground State Transitions.....................................................51 Atomic Density Profile as a Function of B .................................................................53 Total Atomic Energies and Their Ex change and Correlation Components...............55 Ionization Energies and Hi ghest Occupied Orbitals for Magnetized Atoms.............61 Current Density Correction and Other Issues.............................................................65 5 HOOKE’S ATOM AS AN INSTRUCTIVE MODEL...............................................71 Hooke’s Atom in Vanishing B Field..........................................................................71 Hooke’s Atom in B Field, Analytical Solution...........................................................75 Hooke’s Atom in B Field, Numerical Solution..........................................................84 Phase Diagram for Hooke’s Atom in B Field.............................................................89 Electron Density and Paramagnetic Current Density.................................................91 Construction of Kohn-Sham Orbitals from Densities................................................95 Exact DFT/CDFT Energy Components a nd Exchange-correlation Potentials...........97 Comparison of Exact and A pproximate Functionals................................................102 6 SUMMARY AND CONCLUSION.........................................................................110 APPENDIX A HAMILTONIAN AND MATRIX ELEM ENTS IN SPHERICAL GAUSSIAN BASIS.......................................................................................................................112 B ATOMIC ENERGIES FOR ATOMS He, Li, Be, B, C AND THEIR POSITIVE IONS Li+, Be+, B+ IN MAGNETIC FIELDS...........................................................115 C EXCHANGE AND CORRELATION ENERGI ES OF ATOMS He, Li, Be, and POSITIVE IONS Li+, Be+ IN MAGNETIC FIELDS..............................................132 D EFFECTIVE POTENTIAL INTEGRALS WITH RESPECT TO LANDAU ORBITALS IN EQUATION (5.30).........................................................................140 E ENERGY VALUES FOR HOOKE’S ATOM IN MAGNETIC FIELDS...............143 LIST OF REFERENCES.................................................................................................154 BIOGRAPHICAL SKETCH...........................................................................................160

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viii LIST OF TABLES Table page 3-1 Basis set effect on the HF ener gies of the H and C atoms with B = 0.........................35 3-2 Basis set errors for the ground state energy of the H atom in B = 10 au.....................36 3-3 Optimized basis set and expansion coe fficients for the wavefunction of the H atom in B = 10 au.....................................................................................................37 3-4 Test of basis sets including 1, 2, and 3 se quences on the energies of the H atom in B fields......................................................................................................................42 3-5 Energies for high angular mome ntum states of the H atom in B fields.......................43 3-6 Basis sets for the H atom in B fields with accuracy of 1 H.......................................44 3-7 Basis set effect on the HF energies of the C atom in B = 10 au..................................45 3-8 Construction of the AGTO ba sis set for the C atom in B = 10 au...............................47 3-9 Overlaps between HF orbitals for the C atom in B = 10 au and hydrogen-like systems in the same field..........................................................................................47 4-1 Atomic ionization ener gies in magnetic fields............................................................62 4-2 Eigenvalues for the highest occupi ed orbitals of magnetized atoms...........................63 4-3 CDFT corrections to LDA re sults within VRG approximation..................................68 4-4 Effect of cutoff function on CDFT corrections for the He atom 1 s 2 p-1 state in magnetic field B = 1 au............................................................................................69 5-1 Confinement frequencies for HA that have analytic al solutions to eqn. (5.5)........74 5-2 Confinement frequencies which have analytical solutions to eqn.(5.12)....................82 5-3 Some solutions to eqn. (5.12)......................................................................................84 5-4 Field strengths for configuration changes for the ground states of HA.......................89 5-5 SCF results for HF and approximate DFT functionals..............................................109

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ix B-1 Atomic energies of the He atom in B fields..............................................................115 B-2 Atomic energies of the Li+ ion in B fields................................................................121 B-3 Atomic energies of the Li atom in B fields...............................................................122 B-4 Atomic energies of the Be+ ion in B fields...............................................................124 B-5 Atomic energies of the Be atom in B fields..............................................................125 B-6 Atomic energies of the B+ ion in B fields.................................................................126 B-7 Atomic energies of the B atom in B fields................................................................128 B-8 Atomic energies of the C atom in B fields................................................................130 C-1 Exchange and correlation energies of the He atom in magnetic fields.....................132 C-2 Exchange and correlation energies of the Li+ ion in magnetic fields.......................135 C-3 Exchange and correlation energies of the Li atom in magnetic fields......................136 C-4 Exchange and correlation energies of the Be+ ion in magnetic fields......................137 C-5 Exchange and correlation energies of the Be atom in magnetic fields.....................138 D-1 Expressions for Vs( z ) with |z| 2 and 0 s 8..............................................................141 E-1 Relative motion and spin energies for the HA in B fields ( = 1/2)........................143 E-2 As in Table E-1, but for = 1/10...........................................................................145 E-3 Contributions to the total energy for the HA in Zero B field ( B = 0, m = 0)...........147 E-4 Contributions to the total energy for the HA in B fields ( = , m = 0, singlet)..147 E-5 Contributions to the total energy for the HA in B fields ( = 1/10, m = 0, singlet)148 E-6 Contributions to the total energy for the HA in B fields ( = , m = -1, triplet)...149 E-7 Contributions to the total energy for the HA in B fields ( = 1/10, m = -1, triplet)150 E-8 Exact and approximate XC energies for the HA in Zero B field ( B = 0, m = 0, singlet)....................................................................................................................151 E-9 Exact and approximate XC energies for the HA in B fields ( =1/2, m = 0, singlet)151 E-10 Exact and approximate XC energies for the HA in B fields ( = 1/10, m = 0, singlet)....................................................................................................................152

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x E-11 Exact and approximate XC energies for the HA in B fields ( = 1/2, m = -1, triplet).....................................................................................................................15 2 E-12 Exact and approximate XC energies for the HA in B fields ( = 1/10, m =-1, triplet).....................................................................................................................15 3

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xi LIST OF FIGURES Figure page 3-1 Exponents of optimized basis sets for the H, He+, Li++, Be+++, C5+, and O7+ in reduced magnetic fields = 0.1, 1, 10, and 100.......................................................40 3-2 Fitting the parameter b ( =1) using the function (3.26)...............................................41 4-1 UHF total energies for different el ectronic states of the He atom in B fields.............52 4-2 Cross-sectional view of the HF tota l electron densities of the He atom 1 s2 and 1s2 p-1 states as a function of magnetic field strength...............................................54 4-3 Differences of the HF and DFT tota l atomic energies of the He atom 1 s2, 1s2 p0, and 1s2 p-1 states with respect to the corres ponding CI energies as functions of B field strength.............................................................................................................56 4-4 Differences of DFT exchange, correlati on, and exchange-correlation energies with HF ones, for the He atom in B fields........................................................................58 4-5 Atomic ground state ionizati on energies with increasing B field................................64 4-6 Various quantities for the helium atom 1 s 2 p-1 state in B = 1 au..................................66 5-1 Confinement strengths subject to analytical solution to eqn. (5.12)...........................83 5-2 Phase diagram for the HA in B fields.........................................................................90 5-3 Cross-sectional view of the electron de nsity and paramagnetic current density for the ground state HA with = 1/10 in B = 0.346 au.................................................94 5-4 Energy components of HA with B = 0........................................................................99 5-5 Comparison of exact and approximate XC functionals for the HA with different confinement frequency in vanishing B field ( B = 0)..........................................103 5-6 Comparison of exact and approximate ex change, correlation, and XC energies of the HA with = 1/2 in B fields..............................................................................105 5-7 Same as Fig. 5-6, except for = 1/10.......................................................................106

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xii 5-8 Cross-sectional views of the exact a nd approximate XC potentials for the ground state HA with = 1/10 in B = 0.346 au.................................................................107

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xiii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NUMERICAL AND EXACT DENSITY FUNC TIONAL STUDIES OF LIGHT ATOMS IN STRONG MAGNETIC FIELDS By Wuming Zhu August 2005 Chair: Samuel B. Trickey Major Department: Physics Although current density functional theory (CDFT) was proposed almost two decades ago, rather little progress has been made in development and application of this theory, in contrast to many successful applicat ions that ordinary density functional theory (DFT) has enjoyed. In parallel with early DFT exploration, we have made extensive studies on atom-like systems in an external magnetic field. The objectives are to advance our comparative understanding of the DFT and CDFT descriptions of such systems. A subsidiary objective is to provide extensiv e data on light atoms in high fields, notably those of astrophysical interest. To address the cylindrical symmetry indu ced by the external field, an efficient, systematic way to construct high quality ba sis sets within anis otropic Gaussians is provided. Using such basis sets, we did exte nsive Hartree-Fock and DFT calculations on helium through carbon atoms in a wide range of B fields. The applicability and

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xiv limitations of modern DFT and CDFT functionals for atomic systems in such fields is analyzed. An exactly soluble two-electron model syst em, Hooke’s atom (HA), is studied in detail. Analogously with known results for ze ro field, we develope d exact analytical solutions for some specific confinement and field strengths. Exact DFT and CDFT quantities for the HA in B fields, specifically exchange and correlation functionals, were obtained and compared with results from approximate f unctionals. Major qualitative differences were identified. A major overall c onclusion of the work is that the vorticity variable, introduced in CDFT to ensure gauge invariance, is rather difficult to handle computationally. The difficulty is severe enough to suggest that it might be profitable to seek an alternative gauge-invariant formul ation of the current-dependence in DFT.

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1 CHAPTER 1 BASICS OF DENSITY FUNCTIONAL THEORY AND CURRENT DENSITY FUNCTIONAL THEORY Introduction Ambient and low-temperature properties of normal bulk materials are largely determined by knowledge of the motion of the nu clei in the field of the electrons. In essence, this is a statement that the Bo rn-Oppenheimer approximation [1] is widely relevant. For materials drawn from the light er elements of the periodic table, the electrons even can be treated non-relativis tically [2]. While doing some electronic structure calculations on -quartz [3], and some classical inter-nuclear potential molecular dynamics (MD) simulations on silica -like nano-rods [4], a feature of modern computational materials physics became obvious Very little is done with external magnetic fields. This scarcity seems like a missed opportunity. Even with no external field, within the Born-Oppenheimer approximation, a nonrelativistic approach to solution of the Nelectron Schrdinger equation is not a trivial task. For simple systems, e.g. the He at om, highly accurate approximate variational wavefunctions exist [5], but these are too comp licated to extend. Much of the work of modern quantum chemistry involves ex tremely sophisticated sequences of approximations to the exact system wave function [6]. The Hartree-Fock (HF) approximation, which uses a single Slater de terminant as the approximation to the manyelectron wavefunction, usually constitutes th e first step toward a more accurate, sophisticated method. Several approaches, such as configuration in teraction (CI), many-

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2 body perturbation theory (MBPT), and coupled cl uster (CC), are widely used in practice to improve HF results. It is worthwhile mentioning that such methods are extremely demanding computationally. Their computational cost scales as some high power of the number of electrons, typically 5-7th power. Thus these methods are only affordable for systems having up to tens of electrons. An external magnetic fiel d which could not be treated perturbatively would make things much worse. The largest system that has been investigated with the full CI method as of today is a four-electron system, beryllium atom [7, 8]. On the other hand, people always ha ve interests in larger systems and more accurate results than those achievable, no matter how fast and how powerful the computers are; thus theorists continue to c onceive all kinds of clever approximations and theories to cope with this problem. Density functional theory (D FT) [9, 10] is an alternat ive approach to the manyelectron problem that avoids explicit contact with the N -electron wavefunction. DFT developed mostly in the materials physic s community until the early 1990s when it reappeared in the quantum chemistry comm unity as a result of the success of new approximate functionals. These aspects will be discussed below. Two other aspects are worth emphasizing. DFT has been remarkably successful in predic ting and interpreting materials properties. Almost none of thos e predictions involve an external magnetic field. Particularly in Florida, with the Na tional High Magnetic Field Laboratory, that is striking. Even for very simple atoms, inclusion of an external B field is not easy. Only recently have the calculations on the helium atom in a high field been pushed beyond the HF approximation [11, 12, 13]. Although a ve rsion of DFT called current density

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3 functional theory (CDFT) [14, 15, 16] exists for external magnetic fields, it has seen very little application or development. As di scussed below, there is a lack of good approximate CDFT functionals and a lack of studies on which to try to build such improved functionals. One of the foundations of the success of ordinary DFT has been the availability of exact an alytical and highly precise numerical data for atoms for comparison of various functionals and unders tanding their behavior. The main purpose of this dissertation is to find how the eff ect of an external magnetic field on electron motion should be incorporated in the DFT f unctional. In particular I obtain numerical results on various atom-like systems in an external field, with and without CDFT approximate functionals. In addition, I gi ve exact solutions for a model two-electron atom in a nonzero external B field, the so-called Hooke's at om (HA), that has provided valuable insight for DFT at B = 0. Density Functional Theory Attempts to avoid calculation of the many-electron wavefunction began almost simultaneously with the emergence of modern quantum mechanics. In 1927, Thomas and Fermi proposed a model in which the electron kinetic energy is expressed as a functional of the electron density, totally neglecting ex change and correlation effects [17]. The kinetic energy density is assumed to be solely determined by the electron density at that point, and approximated by the kinetic ener gy density of a non-interacting uniform electron gas having the same density. Later th is approach was called the "local density approximation" (LDA) in DFT. The Thomas-Fer mi (TF) model was refined subsequently by Dirac to include exchange effects (anti-sy mmetry of identical-p article wavefunction) and by von Weizscker to include spatial grad ient corrections for the kinetic energy. The result is called the TFDW model. Though usef ul, it fails as a candidate for a model of

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4 materials behavior. Teller proved that the model will not provide binding even for a simple diatomic molecule [17]. The modern form of DFT is rooted in the 1964 paper of Hohenberg and Kohn [9] which put forth two basic theorems, and th e subsequent paper by Kohn and Sham [10], which gave an ingenious scheme for the use of those theorems. A difficulty with the KS scheme is that it lumps all of the subtlety of the many-electron problem, exchange and correlation, in one approximation. The popularit y of DFT depends on the availability of reasonably accurate, tractable approximate functionals. To make the point clear and establish notation, next I give the ba re essentials of ordinary DFT. Foundations for DFT The Hamiltonian of an interacting N -electron system is 2 11,1111 ˆ 22NNN ii iiij ij ijHr rr (1.1) where ir labels the space coordinate of the i th electron. Hartree atomic units are used throughout. The Schrdinger equation specifies the map from the external potential ir to the ground state many-body wavefunction, and the electron number density can be obtained by integrating out N -1 space variables. Schematically, 12,,,Nrrrrnr (1.2) Hohenberg and Kohn noticed that the inversio n of the above maps is also true [9], even though it is not as obvious as above. Because of the key importance of this observation, their proof is included here. For simplicity, they considered the spin independent, non-degenerate ground states. Let the Hamiltonianˆ H, ground state

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5 wavefunction, density nr and energy E associated with the specific external potential r, ˆ :,,,. rHnrE (1.3) Similarly define a primed system, ˆ :,,,. rHnrE (1.4) where rrC and hence By the variational principle, ˆˆˆ EHHHnrrr (1.5) Interchanging the primed and unprimed syst ems gives us another inequality. Summation of those two inequalities leads to a contradiction EEEE if we assume nrnr Thus, different potentials must ge nerate different ground state electron densities. Equivalently speaking, th e knowledge of the ground state density nr uniquely determines the external potential r up to a physically irrelevant addictive constant. This asse rtion is referred as Hohenberg-Kohn (HK) theorem I. Now the maps in eqn. (1.2) are both bijective, 12,,,Nrrrrnr (1.6) An immediate consequence of HK theorem I is that the ground state electron density nr can be chosen as the basic vari able to describe the interacting N-electron system, since it is as good as the many-body wa vefunction. Here “as good as” means that the ground state density nr contains no more or less info rmation about the system than the wavefunction does. It does not mean the density is a variable as easy as, or as hard as, the many-body wavefunction to handle. Actually since the density is a 3-dimensional

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6 physically observable variable, whereas th e spatial part of wavefunction is a 3Ndimensional variable, the density is a much simpler variable to ma nipulate and to think about. On the other hand, by switching from the wavefunction to the density, we also lose some tools from quantum mechanics (QM) wh ich we are quite adept at using. For example, in QM, an observable can be calcu lated by evaluating the expectation value of its corresponding operator. This approach ofte n does not work in DFT. The best we can say is that the observable is a functional of the ground state density. In contrast to the explicit dependence on the wavefunction in the QM formulas for the expectation value, the implicit dependence on the ground state densit y in DFT is rarely expressible in a form useful for calculation. Most such functiona ls are not known as of today. Among them, the most exploited and the most successful one is the exchange-correlation energy functional, which is amenable to approximations for larg e varieties of system s. Another one being extensively studied but not so successfully is the kineti c energy functional, already mentioned in the paragraph about TF-type models. While DFT is a whole new theory that does not need to resort to the many-body wavefunction, to make use of the Rayleigh-R itz variational princi ple to find the ground state energy E0, we retain that concept for a while. By the so-called constrained search scheme independently given by Lieb [18] and Levy [19] in 1982, all the trial wavefunctions are sorted into classes according to the densities nr to which they give rise. The minimization is split into two steps, 0ˆ minminLL nr E HrnrdrFnr (1.7) where ˆ minLL nFnTU (1.8)

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7 The Lieb-Levy functional FLL is defined on all the possibl e densities realizable from some anti-symmetric, normalized N -particle functions, or N representable densities. Both the densities of degenerate states and even excited states are included. One good thing about FLL is that we have a simple criterion for N representable densities: all nonnegative, integrable densities are N representable. The Kohn-Sham Scheme The HK theorem showed that the ground st ate energy of a many-electron system can be obtained by minimizing the energy functional Enr TF-type models constitute a direct approach to attack th is problem, in which energy functionals are constructed as explicit approximate forms dependent upon the electron density. However, the accuracy of TF-type models is far from acceptable in most appli cations, and there are seemingly insurmountable difficulties to impr ove those models significantly. The reason is that the kinetic energy functionals in TF-type models bring in a large error. To circumvent this difficulty, an ingenious i ndirect approach to the kinetic energy was invented by Kohn and Sham [10] A fictitious non-interac ting system having the same ground state electron density as the one under study is introduced. Because the kinetic energy of this KS system, Ts can be calculated exactly, and because Ts includes almost all the true kinetic energy T, the dominant part of the error in TF-type models is eliminated. Since then, DFT has become a practical tool for realistic calculations. It is advantageous to decompose th e total energy in the following way, ()()LL E nFnrrnrdr ()()eeTnEnrnrdr

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8 ()()[][]HF sxcsTnJrnrdrEVnTnTn (1.9) where Eee is the total QM electron-electron interaction energy, 1()() 2 nrnr Jdrdr rr is the classical electron-electron repulsion energy, H F x E is the conventional exchange energy, and Vc is the conventional correlation energy. Ts is defined in terms of the noninteracting system as usual, 2 11 2N s ii iT (1.10) Then H F xEis replaced by Ex[n], the exchange energy calculated using the singledeterminant HF formula but w ith the same orbitals in Ts, and Ec is defined as all the remaining energy [][][]HF ccsxx E nVnTnTnEEn (1.11) The total energy is fi nally expressed as []()()sxc E nTnJrnrdrEn (1.12) where [][][]xcxcEnEnEn (1.13) Each of the first three terms in eqn. (1. 12) usually makes a large contribution to the total energy, but they all can be calculated exactly. The remainder, Exc, is normally a small fraction of the total energy and is more amenable to approximation than the kinetic energy. Even though the equations in this se ction are all exact, approximations to the Exc functional ultimately must be introduced. The variational principle leads to th e so-called Kohn-Sham self-consistent equations,

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9 ) ( ) ( ) ( 22r r ri i i s (1.14) where ()()()()dft sHxcrrrr (1.15) r d r r r n rH ) ( ) ( (1.16) [()] () ()dft xc xcEnr r nr (1.17) and 2 1()()N i inrr (1.18) Again, Hartree atomic units are used throughout Equations (1.14) (1 .18) constitute the basic formulas for KS calculations. The deta iled derivation and ela borations can be found in the abundant literatu re on DFT, for example, references 20-22. Since the foundations of DFT were es tablished, there have been many generalizations to this theory. The most impor tant include spin density functional theory (SDFT) [23], DFT for multi-component syst ems [24, 25], thermal DFT for finite temperature ensembles [26], DFT of excited states, superconductors [27], relativistic electrons [28], time-dependent density func tional theory (TDDFT) [29], and current density functional theo ry (CDFT) for systems with external magnetic fields [14-16]. Among them, SDFT is the most well-develope d and successful one. TDDFT has attracted much attention in recent years and shows gr eat promise. Compared to the thousands of papers published on DFT and SDFT, we have fewer than 80 papers on CDFT in any form. Thus CDFT seems to be the leas t developed DFT generalization, perhaps surprisingly since there is a gr eat deal of experimental work on systems in external B fields. That disparity is the unde rlying motivation for this thesis.

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10 Current Density Functional Theory (CDFT) Basic Formulations One of the striking features of the very limited CDFT literature is the extremely restricted choice of functionals. A second striki ng feature is that most of the work using CDFT has been at B = 0, in essence using CDFT e ither to gain access to magnetic susceptibility [30, 31] or to provide a richer parameterization of the B = 0 ground state than that provided by SDFT [32]. In orde r to comprehend the challenge it is first necessary to outline the essentials of CDFT. For an interacting N -electron system under both a scalar potential r and a vector potential A r its Hamiltonian reads, 2111 ˆ (()) 22i totii iij ijHArr i rr (1.19) The paramagnetic current density pjr is the expectation value of the corresponding operator ()op pJr 1212()(,,,)|()|(,,,)op pNpNjrrrrJrrrr (1.20) where 1 ˆˆˆˆ ()()()()() 2op pJrrrrr i (1.21) in terms of the usual fermion field operators. CDFT is an extension of DFT to include the vector potentialAr The original papers [14-16] followed the HK argument by contradiction and purported thereby to prove not only that the ground state is uniquely parameterized by the density nr and paramagnetic current density pjr but also that the r and A r are uniquely

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11 determined. Later it was found th at a subtlety was overlooked. It is obvious that two Hamiltonians with different scalar extern al potentials cannot even have a common eigenstate, e.g., the first map in eqn. (1.6) is bijective, but this is not true when a vector potential is introduced. It is possi ble that two sets of potentials rAr and rAr could have the same ground state wa vefunction. This non-uniqueness was later realized [33]. Fortunat ely, the HK-like variational pr inciple only needs the one-toone map between ground state wavefunction an d densities, without recourse to the system’s external potentials being functionals of densities [34]. To avoid the difficulties of representability problems, we follow th e Lieb-Levy constrained search approach. Sort all the trial wavefuncti ons according to the densities nr and pjr they would generate. The ground state wavefunction, which generates corre ct densities, will give the minimum of the total energy. 0 2 ,ˆ min min, 2ptot pp nrjrEH Ar rnrdrArjrFnrjr (1.22) where ,ˆ ,minpp njFnjTU (1.23) A non-physical non-interacting KS system is now introduced, which generates correct densities nr and pjr The functional F is customarily decomposed as ,,,cdft pspxcpFnjTnjJEnj (1.24) The variational principle gives us the self-consistent equations ) ( ) ( ) ( ) ( 2 12r r r r A ii i i cdft eff eff (1.25)

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12 where ) ( ) ( ) ( r A r A r Axc eff (1.26) 221 ()()()()()() 2cdftcdft effHxceffrrrrArAr (1.27) It is easy to see that ) ( rcdft effreduces to ) ( rs when we set 0 ) ( r Axc Exchangecorrelation potentials are defi ned as functional derivatives, ()(),() () ()cdft xcp xc p nrEnrjr Ar jr (1.28) ()(),() () ()pcdft xcp cdft xc jrEnrjr r nr (1.29) The electron density nr can be calculated just as in eqn. (1.18). The paramagnetic current density is constructed fr om the KS orbitals according to **1 ()()()()() 2piiii i j rrrrr i (1.30) The total energy expression of the system is 2() (),()()()() 2cdftcdft totsxcppAr ETJEnrjrnrrdrjrArdr r d r A r j r d r n r r j r n E Jxc p cdft xc p cdft xc i ) ( ) ( ) ( ) ( ) ( ), ( (1.31) Equation (1.25) can be rewritten as 211 ()()()()()()()() 22cdft HxcxcxciiiArrrrArArrr ii (1.32) which is more suitable for application.

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13 Vignale-Rasolt-Geldart (VRG) Functional On grounds of gauge-invariance, Vignal e and Rasolt argued that the exchangecorrelation functional cdft xcE should be expressed as a functional of nr and the so-called voticity [14-16] pjr r nr (1.33) Following their proposal, if we choose r as the second basic variable in CDFT, [(),()][(),()]cdftcdft xcpxcEnrjrEnrvr (1.34) exchange-correlation potentials can be found from functional derivatives ()[,] 1 ()| ()()cdft xc xcnrEnv Ar nrvr (1.35) ()[,] ()| ()()cdft p cdft xc xcrxc j r Env rAr nrnr (1.36) To make use of the already proven successf ul DFT functionals, it is useful to separate the exchange-correla tion functional into a curren t-independent term and an explicitly current-dependent term, [(),()][()][(),()]xccdftdft xcxcEnrvrEnrEnrvr (1.37) The current-independent term can be any widely used XC functional, such as LDA, or the generalized gradient approxi mation (GGA) functional. The current-dependent term is presumably small, and should vanish for zer o current system, for example, the ground state of the helium atom.

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14 Next we proceed in a slightly differe nt, more general way. By homogeneous scaling of both the nr and pjr Erhard and Gross deduced that the current-dependent exchange functional scales homogeneously as [35] [,][,]cdftcdft xpxp E njEnj (1.38) where is the scaling factor. 3nn and 4pp j j are scaled charge density and paramagnetic current density, respectively. A ssuming that the exchange part dominates the exchange-correlation energy, a local approximation for the cdft xcE takes the form 2[(),()][()]([(),()],)|()|xccdftdft xc E nrvrEnrgnrvrrvrdr (1.39) The foregoing expression is deri ved based on the assumption that r is a basic variable in CDFT. A further (dras tic) approximation is to assume ([(),()],)(()) gnrvrrgnr (1.40) which is done in the VRG approximation. By considering the perturbative energy of a homogeneous electron gas (HEG) in a uniform B field (the question whether the HEG remains uniform after the B field is turned on was not di scussed), Vignale and Rasolt[1416] gave the form for g 2 0(()) ()(())[1] 24(())Fknr grgnr nr (1.41) Here kF is the Fermi momentum, and 0 are the orbital magnetic susceptibilities for the interacting and non-interacting HEG, re spectively. From the tabulated data for 110sr in reference 36, Lee, Colwell, and Handy (LCH) obtained a fitted form [31], 0.042 0/(1.00.028) s r LCHssre (1.42)

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15 where 1/33 4sr n (1.43) accordingly, 2(1) 24F LCHLCHk gs (1.44) Orestes, Marcasso, and Capelle proposed two other fits, bo th polynomial [37] 2 30.99560.012540.0002955OMCsssrr (1.45) 1/32 51.10380.4990.44230.066960.0008432OMCsssssrrrr (1.46) Those fits all give rise to divergence problems in the low density region. A cutoff function needs to be introduced, which will be discussed in the next chapter. Survey on the Applications of CDFT The VRG functional has been applied in th e calculation of magnetizabilities [30, 31], nuclear shielding constant s [38], and frequency-dependent polarizabilities [39, 40] for small molecules, and ionization energies fo r atoms [37, 41]. In those calculations, the vector potential was treated perturbatively. Fully self-consistent calculations are still lacking. None of those studies has a conclu sive result. The first calculation in Handy’s group was plagued with problems arising from an insufficiently large basis set [31]. In their second calculation, they found that the VRG functiona l would cause divergences and set (())0 gnr for rs >10. The small VRG contribution was overwhelmed by the limitations of the local density functional [ 38]. The VRG contribu tion to the frequencydependent polarizability was also found to be negligible, and several other issues emerge as more important than explicitly including the current density functional [40]. Contrary to the properties of small molecule s studied by Handy’s group, Orestes et al. found that the current contribution to th e atomic ionization energy is non-negligible, even though use of VRG did not improve the energy systematically [37].

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16 All those investigations are based on the assumption that the VRG functional at least can give the correct order of magnitude of current contributi ons to the properties under study. Actually this is not guaranteed. The errors from ordinary DFT functionals and from the current part are always intert wined. To see how much the current term contributes, an exact solution is desired. Vignale’s group has never done any actual num erical calculation based on the VRG functional. Either for an electr onic system [42, 43] or for an electron-hole liquid [44, 45], they used Danz and Glasser’s approximation [46] for the exchange, and the random phase approximation for the correlation energy, whic h is known to be problematic in the low density regime. The kinetic energy was appr oximated from a non-in teracting particle model or a TF-type model. Even though corre lation effects were included in their formulas, the numerical errors introduced in each part were uncontrollable, and their calculations could only be t hought as being very crude at best. This is somewhat inconsistent with invoking CDFT to do a better calculation than the ordinary DFT calculation does. While the fully CDFT calculations on three-dimensional (3D) systems are scarce, there are more applications of CDFT to 2D systems. Ex amples include the 2D Wigner crystal transition [47], quantum dots [48], and quantum rings [ 49] in a magnetic field. In these cases, the cdft xcE was interpolated between the zero-field value from the Monte Carlo calculation by Tanatar and Ceperley [ 50], and the strong field limit [51]. Other Developments in CDFT Some formal properties and virial theorems for CDFT have been derived from density scaling arguments [35, 52-54] or density matrix theory [55]. A connection

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17 between CDFT and SDFT functionals is also established [56]. Those formal relations could be used as guidance in the construction of CDFT functionals, but as of today, there is no functional derived from them as far as I know. CDFT is also extended to TDDFT in the linear response regime [57], which is called time-dependent CDFT (TD-CDFT). Th ere is not much connection between TDCDFT [57] and the originally proposed CDFT formulation [14-16]. Notice an important change in reference 57, the basi c variables are electron density nr and physical current densityjr as opposed to the paramagnetic current density pjr which is argued in references 14-16 to be the basic variable In TD-CDFT, the frequency dependent XC kernel functions are approximated from the HE G [58, 59], and the formalism is used in the calculations of polari zabilities of polymers and op tical spectra of group IV semiconductors [60, 61, 62]. TD-CDFT has also been extended to weakly disordered systems [63] and solids [64].

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18 CHAPTER 2 ATOMS IN UNIFORM MAGNETIC FIELDS – THEORY Single Particle Equations When a uniform external magnetic field B which we choose along the z direction, is imposed on the central field atom, its symmetry goes over to cylindrical. The Hamiltonian of the system commutes with a rotation operation about the direction of the B field, so the magnetic quantum number m is still a good quantum number. The natural gauge origin for an atom-like system is its cen ter, e.g. the position of its nucleus. In the coulomb gauge, the external vect or potential is expressed as r B r A 2 1 (2.1) The total many-electron Hamiltonian (in Hartree atomic units) then becomes 2 222 1 ,1111 ˆ 2 2822N N iiiisi i ij i ij ijZBB Hxymm r rr (2.2) where Z is the nuclear charge, ir mi, ms,i are the space coordinate, magnetic quantum number, and spin z component for the i th electron. Hartree-Fock Approximation In the Hartree-Fock (HF) approximation, correlation effects among electrons are totally neglected. The simplest case is restri cted Hartree-Fock (RHF), which corresponds to a single-determinant va riational wavefunction of doubl y occupied orbitals. We useri to denote single-particle orbitals. For spin-unrestricted Hartree-Fock (UHF)

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19 theory, different spatial orbitals are assigned to the spin-up ( ) and spin-down ( ) electrons. The resulting si ngle-particle equation is 22 22 ,2 282HF Hisii HFHF ji HFHFHF jii jZBB rxymmr r rr drrr rr (2.3) Notice that exchange contribute s only for like-spin orbitals. Simple DFT Approximation It seems plausible to assume that the xc A contribution to the total energy is small compared to the ordinary DFT Exc. Then the zeroth-order approximation to the CDFT exchange-correlation functional p cdft xcj n E can be taken to be the same form as the XC functional in ordinary DFT, n Exc, with the current depende nce in the XC functional totally neglected. Notice that in this scheme, the interaction between the B field and the orbitals is still partially included. From eqn. (1.28), we s ee that this approximation amounts to setting the XC vector pot ential identically zero everywhere, 0 r Axc The corresponding single-particle Kohn-Sham equation is ,ˆdftdftdftdft iiihrr (2.4) where 22 22 ,ˆ 2 282dft dft HisixcZBB hrxymmr r (2.5) The scalar XC potential is defined as in eqn. (1.17). CDFT Approximation In this case, both the density dependen ce and the current dependence of the XC energy functional p cdft xcj n E are included. If we knew the exact form of this functional,

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20 this scheme in principal would be an exact theory including all many-body effects. In practice, just as in ordinary DFT, the XC functional must be approximated. Unfortunately very little is known about it. A major theme of this work is to develop systematic knowledge about the exact CDFT functiona l and the one available general-purpose approximation, VRG. The CDFT KS equation reads ,ˆcdftcdftcdftcdft iiihrr (2.6) where 1 ˆˆ 2cdftdftdftcdft xcxcxcxchhrrAA i (2.7) and cdft xcr is defined by eqn. (1.29). Notice the last term means ,i xcA when the operator is applied to a KS orbital. Exchange-correlation Potentials For ordinary DFT, both LDA and GGA approximations were implemented. Specifically, the XC functionals include HL [65], VWN [66, 67], PZ [68], PW92 [69], PBE [70], PW91 [71], and BLYP [72-75]. jPBE is an exte nsion of the PBE functional that includes a current term [32], but does not treat pj as an independent variable, which means 0 r Axc For GGAs, the XC scalar potential is calculated according to r n E r n E r n r n r n E rGGA xc GGA xc GGA xc dft xc (2.8) Before considering any specific approximate XC functional in CDFT, we point out several cases for which CDFT should reduce to ordinary DFT. The errors in those DFT calculations are solely introduced by the approximate DFT functionals, not by neglecting the effects of the current. Such systems can provide estimates of the accuracy of DFT

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21 functionals. Comparing their re sidual errors with the erro rs in corresponding currentcarrying states can give us some clues about the magnitude of current effects. The ground states of several small atom s have zero angular momentum for sufficiently small external fields. These are the hydrogen atom in an arbitrary field, the helium atom in B < 0.711 au. [76] (1 au. of B field =52.350510 Tesla), the lithium atom in B < 0.1929 au. [77], and the beryllium atom in B < 0.0612 au. [8]. Since their paramagnetic current density pj vanishes everywhere, the proper CDFT and DFT descriptions must coincide. Notice (for future reference) that thei r density distributions are not necessarily spherically symmetric. This argument also holds for positive ions with four or fewer electrons a nd any closed shell atom. If we admit the vorticity r to be one basic variable in CDFT, as proposed by Vignale and Rasolt [14-16], there is another kind of system for which the DFT and CDFT descriptions must be identical. As Lee, Handy, and Colwell pointed out [38], for any system that can be described by a single complex wavefunction r r vanishes everywhere. The proof is trivial, 0 ln 2 1 2 1* * i i r n r j rp Cases include any single elec tron system, and the singlet states for two-electron systems in which the two electrons have the same spatial parts, such as H2 and HeH+ molecules. Notice that the system can have non-va nishing paramagnetic current density,0 r jp A puzzling implication would seem to be that the choice of parameterization by r is not adequate to capture all the physics of imposed B fields.

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22 For CDFT calculations, we have mainly investigated the VR G functional already introduced. It is the only e xplicitly parameterized CDFT functional designed for B > 0 and applicable to 3D systems that we have encountered in the literature: r d r r n g r r n EVRG xc 2, (2.9) where gnr and r are defined in (1.41) and (1.33). Substitution of (2.9) into (1.35) gives the expression for the vector XC potential, 2 ()()() ()xcArgnrvr nr (2.10) In actual calculations it generally was neces sary to compute the curl in this equation numerically. In CDFT, the scalar potentia l has two more terms beyond those found in ordinary DFT, namely 2() ()()() ()p cdftdft xcxcxcjr dgn rrrAr dnnr (2.11) There are three fits for gn to the same set of data tabulated in the range of 110sr from random phase approximation (RPA) on the diamagnetic susceptibility of a uniform electron gas [36], namely eqns. (1.42) (1.45) and (1.46). Their derivatives are LCHsLCH sdgdrdg dndndr 1 3 0.042 22219111 0.0420.028 3244sr s sssr e nrrr (2.12) 1 3 3 22190.0044 0.0002955 3244OMCs sdgr dnnr (2.13)

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23 1 3 5 3 2 5 3 2 219 0.10380.33270.22120.0008432 3244OMCs sssdgr rrr dnn (2.14) The three fits are very close in the range of 110sr but differ wildly in other regions due to different c hosen functional forms for g ( n ). They all cause divergence problems in any low density region. Without improving the reliabilit y, precision, and the valid range of the original data set, it s eems impossible to improve the quality of the fitted functions. It is desirable to know its be havior in the low density region, especially for finite system calculations, but unfortunate ly, reference 36 did not give any data for rs >10, nor do we know its asymptotic form. Because dg/dn is required for all r hence for all n yet g ( n ) is undefined for low densities, we must introduce a cutoff function. After some numerical experiment we chose s cutoffr a s F cutoffe r c c k g 2 1 224 (2.15) wherecutoff is the cutoff exponent, which determines how fast the function dies out. The two constants c1 and c2 are determined by the smooth connection between () gn and cutoffg at the designated cutoff densitycutoffn / /,cutoff cutoffcutoff LCHOMC cutoffcutoffLCHOMCcutoff n ndg dg gngn dndn (2.16) In this work, we use 1 0 3 00 2 001 0 a a ncutoff cutoff, unless other values are explicitly specified. There is an identity abou t the vector XC potential xcA derived from the VRG functional,

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24 r r n E r d r n r j r r n g r d r j r r n g r n r j r AVRG xc p p p xc 2 2 2 1 (2.17) Since xcp A rjr and VRG xcE can be computed independently, this equation can provide a useful check in the code for wh ether the mesh is adequate and whether numerical accuracy is acceptable.

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25 CHAPTER 3 BASIS SET AND BASIS SET OPTIMIZATION Survey of Basis Sets Used in Other Work For numerical calculations, the single particle orbitals in eqn. (2.3), or (2.4), or (2.6), can be represented in several ways. One is straightforw ard discretization on a mesh. For compatibility with extended syst em and molecular techniques, however, we here consider basis set expansions. For zero B field, the usual choices are Gaussian-type orbitals (GTO), or, less commonl y, Slater-type orbitals (STO). Plane wave basis sets are more commonly seen in calculati ons on extended systems. Large B fields impose additional demands on the basis set, as di scussed below. Here we summarize various basis sets that have been us ed for direct solution of the few-electron Schrdinger equation and in variational approaches such as the HF approximation, DFT, etc. For the one-electron problem, the hydrogen atom in an arbitrary B field, the typical treatment is a mixture of numerical mesh and basis functions. The wavefunction is expanded in spherical harmonics ,lmY in the low field regime, and in Landau orbitals (,)Lan nm for large B fields. Here r , are spherical coordinates, and z, are cylindrical coordinates. Th e radial part (for low B ) or the z part (for high B ) of the wavefunction is typically represented by nu merical values on a one-dimensional mesh [78]. In Chapter 5 we will also use this te chnique for the relative motion part of the Hooke’s atom in a B field. Of course, the hydrogen atom has also been solved algebraically, an approach in which the wavefunction takes the form of a polynomial

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26 multiplying an exponential. This is by no m eans a trivial task. To get an accurate description for the wavefunction, the polynomial may have to include thousands of terms, and the recursion relation fo r the polynomial coefficients is complicated [79-82]. The multi-channel Landau orbital expansion was also used in DFT calculations on many-electron atoms [83]. Another approach is the two-dimensional finite element method [84]. Dirac exchange-only or simila r functionals were used in those two calculations. In the series of Hartree-Fock calculations on the atoms hydrogen through neon by Ivanov, and by Ivanov and Schmelcher, the wavefunctions were expressed on two-dimensional meshes [85-90, 76]. Slater-typ e orbitals were chos en by Jones, Ortiz, and Ceperley for their HF orbitals to provide the input to quantum Monte Carlo calculations, with the aim to develop XC func tionals in the context of CDFT [91-93]. Later they found that the STO basis was not suff icient and turned to anisotropic Gaussian type orbitals (AGTO) [94]. Apparently th eir interests changed since no subsequent publications along this thread were found in the literatu re. Schmelcher’s group also employed AGTOs in their full CI calculati ons on the helium [11-13], lithium [77], and beryllium [8] atoms. At present, AGTOs seem to be the basis set of choice for atomic calculations which span a wide range of field strengths. This basis has the flexibility of adjusting to different field st rengths, and the usual advant age of converting the one-body differential eigenvalue problem into a matr ix eigenvalue problem. Moreover, the onecenter coulomb integral can be expressed in a closed form in this basis, though the expression is lengthy [11, 12]. The disadvantage of AGTOs is that one has to optimize their exponents nonlinearly for each value of the B field, which is not an easy task, and a

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27 simple, systematic optimization is lacking. We will come back to this issue and prescribe an efficient, systematic procedure. Spherical-GTO and Anisotropic-GTO Representations As with any finite GTO basis, there is also the improper representation of the nuclear cusp. Given the predominance of GTO basis sets in molecular calculations and the local emphasis on their use in periodic system calculations, this limitation does not seem to be a barrier. Spherical GTOs are most widely used in electronic structure calculations on finite systems without external magnetic field. The periodic system code we use and develop, GTOFF [95], also uses a GTO basis. Several small molecules in high B fields were investigated by Runge and Sa bin with relatively small GTO basis sets [96]. To understand the perfor mance of GTOs in nonzero field and make a connection to the code GTOFF, our implementation includes both GTO and AGTO basis sets. The former is, of course, a special case of the la tter, in which the expone nts in the longitudinal and transverse directions are the same. Spherical GTO Basis Set Expansion The form of spherical Gaussian basis we used is 2,lr lmlmlmGrNreY (3.1) where lmN is the normalization factor. The KS or Slater orbitals (DFT or HF) are expanded in the r Glm , ,,iliillmlm llraGRrY (3.2) where 2, ,illr illmRrraNe (3.3)

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28 Notice m is understood as mi, the magnetic quantum number of the ith orbital. For simplicity, the subscript i is omitted when that does not cause confusion. The electron density and its gradient can be evaluated conveniently as il l m l lm l i il il lm il i iY Y r R r R Y r R r r n , ,* 2 2 (3.4) il l m l lm l i il il l m l lm l i il l i ilY Y r r R r R Y Y r R r R r R r R r r n , ˆ , ˆ* (3.5) The paramagnetic current density is ˆ , sin 1 ˆ*r j Y Y r R r R m r r jp il l m l lm l i il i p (3.6) and the curl of r jp is ** 2 *ˆ sinsin 1 ˆ sinilillmlmlmlm pi ill iilililillmlm illRRYYYY jrrm r mRRRRYY r (3.7) The vorticity is evaluated analytically according to ˆ ˆ ,2r r r r n r j r j r n r n r n r j rr p p p (3.8) For the VRG functional, the vector XC potentia l is expressed in spherical coordinates as ˆ , , 1 2 ˆ r A r r n g r r n g r r r r n r Axc r xc (3.9) The last term in eqn. (2.7) becomes

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29 r r A r A r r r r A r r Ai xc xc i i xc i xc , ,, sin 1 (3.10) Appendix A includes the matrix elements in th is basis for each term in the Hamiltonian. Anisotropic GTO (AGTO) Basis Set Expansion An external B field effectively increases the conf inement of the electron motions in the xy plane, and causes an el ongation of the electron dens ity distribution along the z axis. It is advantageous to reflect this effect in the basis set by having different decay rates along directions parallel and perpendicular to the B field. AGTOs are devised precisely in this way: 22(,,)z jjjjjnn zim jjzNzee 1,2,3, j (3.11) where ||2, 2,j jjjj zzjnmk nl with 0,1, 0,1,j jk l ,2,1,0,1,2, 0,1.jj zm and Nj is the normalization factor. If we letjj this basis of course recovers the isotropic Gaussian ba sis, appropriate for B = 0. The basis sets used in reference 94 were limited only to0jjkl which are more restrictive th an those used by Becken and Schmelcher [11-13] and ours. Single-particle wavefuncti ons expanded in AGTOs have the general form ,,iijj jrbz (3.12) Various quantities can be calculated in this basis according to their expressions in cylindrical coordinates,

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30 ˆ ˆ ˆ ˆ ˆ ˆ 22,,jjiii ii z i ijjjj jim rz z nn im bzzz z (3.13) 222 2,z jjjjnn z iijj iijnrrbNzenz (3.14) i i i i iz z z r n z r n r n *ˆ ˆ 2 ˆ ˆ (3.15) 222 211 ˆˆ ˆ ,z jjjjnn z piiiijj iij pjrmrmbNze jz (3.16) r j z z r j z z m r jp p i i i i i i p ˆ ˆ ˆ ˆ ˆ ˆ 2 1* (3.17) 22ˆ ˆ ˆ ˆp p ppn n j j jzj z rz nrnrnrnr (3.18) z z zzˆ ˆ Again, for the VRG functional we have z z n g z z n g z z n r Az xc, , , 2 ˆ (3.19) We follow the scheme in references 11-12 for evaluation of matrix elements, in which all the integrals, including Coulomb in tegrals, are expressed in closed form. Connection between GTOs and AGTOs As pointed out before, a GTO basis is a special case of an AGTO basis. Conversely, a particular AGT O can be expanded in GTOs.

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31 0 2! ,2k k j j im r n n j jk z e e z N zj j j z j 22 2 0sincos !z z jjjj jjk nnk nnk jj rim j kNree k (3.20) It is easy to see this is a linear combination of j jlmG with k n n lj z j2 , 2 1 0 k An ordinary contracted Ga ussian basis is a fixed lin ear combination of several primitive Gaussians having same the l and m but different exponentsj Similarly, an AGTO can also be thought as a contracted GT O that contains infinitely many GTOs (in principle) having the same exponent and m but different l values with increment of 2. This establishes the equivalence of the two ki nds of orbitals. The relative efficiency of the AGTO basis in cylindrically c onfined systems is apparent for B 0. Primary and Secondary Sequences in AGTO While the AGTO basis provides extra flexibility, its optimization is more complicated than for a GTO set of comparable size. Kravchenko and Liberman investigated the performance of AGTO basis sets in one-electron systems, the hydrogen atom and the hydrogen molecular ion, and show ed that they could provide accuracy of 10-6 Hartree or better [97]. Jone s, Ortiz, and Ceperley estimated their basis set truncation error for the helium atom in B < 8au. to be less than one milihartree in the total atomic energy of about 2 Hartrees [94]. Even-tempered Gaussian (ETG) sequences of ten are used in zero-field calculations. For a sequence of primitive spherical Gaussian s having the same quantum numbers, their exponents are given by j j jpq 1,2,.b j N (3.21)

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32 where p and q are determined by lnln(1) ln(ln)lnb p aqa qbNb (3.22) and Nb is the basis size. For the hydrogen atom, Schmidt and Ruedenburg [98] recommended the following parameters: a = 0.3243, a' = -3.6920, b = -0.4250, b' = 0.9280. Since the external magnetic field only increases the confinem ent in the horizontal direction, we may expect eqn. (3.21) to be equally useful for generating longitudinal exponents j for the AGTO basis. The choice of j is more subtle. Jones, Ortiz, and Ceperley [94] used several tempered sequences of the types ,2,4,8,jjjjj (3.23) For convenience, we refer to the first sequence (jj ) as the primary sequence, and the second, the third, the fourth, … seque nces as the secondary sequences in our discussion. The primary JOC sequence in e qn. (3.23) is obviously as same as the spherical GTOs, for which the transverse and longitudinal exponents are the same. However for the second JOC sequen ce, the transverse exponents j ’s are twice the longitudinal exponents j ’s, and for the third sequence, 4jj etc. The basis set is the sum of all those sequences. The total number of basis functions is Nb multiplied by the number of different sequences. Reference 94 used 2-5 sequences in the expansion of HF orbitals. Obviously, when several sequenc es are included, which is necessary for large B fields, very large basis sets can result.

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33 Kravchenko and Liberman [97] chose KL j KL j KL j KL j KL j jB B B B B 6 0 4 1 8 0 2 1 (3.24) whereKL is a value between 0 and 0.3 which mi nimizes the basis se t truncation error compared to more accurate results. Here we st ill refer to the sequences in eqn. (3.24) as primary and secondary sequences. In each KL sequence, the differences between the transverse and longitudinal exponents are th e same for all the basis functions. An improvement of KL basis sets over JOC basi s sets is that the former have shorter secondary sequences, which helps to keep ba sis size within reason. Namely, the second and the third KL sequences have lengths of one-half of KL primary sequence, e.g., Nb /2 is used in eqns. (3.21) and (3.22) to genera te them, and the fourth and fifth KL sequences have lengths of Nb /4. Becken et al. [11] used a seemingly differe nt algorithm to optimize both j and j in the same spirit of minimizing the one-particle HF energy, H atom or He+ in a B field, but they did not give enough details for one to repeat their op timization procedure. Optimized AGTO Basis Sets In this section, I give so me numerical illustrations of the basis set issues. These examples illustrate the importance, difficu lties, and what can be expected from a reasonably well-optimized basis. Our goal is set to reduce basis set error in the total energy of a light atom to below one milihartree. This criterion is based on two considerations. One is the observation by Orestes, Marcasso, and Capelle that the magnit ude of current effects in CDFT is of the same order as the accuracy reached by m odern DFT functionals [37]. They compared atomic ionization energies from experime nt with DFT-based cal culations. A typical

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34 difference is 0.4eV, or 15 m H. To study the current effect in CDFT, we need to reduce the basis set errors to considerably below th is value. Another fact or considered is the well-known standard of chemical accuracy, usually taken to be 1 kcal/mol, or 1.6 m H. It turns out that this goal is much harder to reach for multi-electron atoms in a large B field than for the field-free case. Two systems I choose for comparison are the hydrogen and carbon atoms. There are extens ive tabulations for the magnetized hydrogen atom [78], and even more accurate data from the algebraic method [80] against which to compare. However, the hydrogen atom does not include electron-electron inte raction, which is exactly the subject of our in terest. For the carbon atom, our comparison mainly will be made with numerical Hartree-Fock data [ 90]. Without external field, the correlation energy of the C atom is about 0.15 Hartree [99], two orders of magnitude larg er than our goal. This difference also makes the choice of one m H basis set error plausible. Examine the zero-field case fi rst. It is well known that th e non-relativistic energy of the hydrogen atom is exactly -0.5 Hartree. For the carbon atom, the numerical HF data taken from reference 90 are treated as the ex act reference. Calculated HF energies in various basis sets are listed in Table 3-1, toge ther with basis set errors in parentheses. We first tested the widely used 6-31G basis se ts. Those basis sets are obtained from the GAMESS code [100]. In primitive Gau ssians, they include up through 4 s for the hydrogen, and 10 s 4 p for the carbon atom. As expected, th e accuracies in total energy that they deliver increase only slightly after de -contraction. A sequence of exponents derived from eqn. (3.21) with length Nb = 8 has a comparable size with the 6-31G basis for the carbon atom. It gives rather bad re sults, but recall that a signif icant deficiency of GTOs is that they cannot describe the nuclear cusp condition. By adding five tighter s orbitals

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35 extrapolated from eqn. (3.21) with j = 9,10,…,13, the basis set error is reduced by 99%. To further reduce the remaining 1.6 m H error, higher angular momentum orbitals are required. Addition of four d orbitals and removal of the tightest, unnecessary p orbital gives a 13 s 7 p 4 d basis set, with only 0.8 m H truncation error left. A larger basis set, 20 s 11 p 6 d similarly constructed from the Nb = 16 sequence by adding 4 tighter s orbitals has error only of 0.05 m H. Table 3-1 Basis set effect on the HF energies of the H and C atoms with B = 0 (energies in Hartree) Basis Set a Hydrogen atom Carbon atom 6-31G -0.498233 (0.001767) -37.67784 (0.01312) De-contracted 6-31G -0.498655 (0.001345) -37.67957 (0.01139) Sequence Nb = 8 -0.499974 (0.000026) -37.51166 (0.17930) Nb = 8, plus 5 tighter s -0.499989 (0.000011) -37.68938 (0.00158) 13 s 7 p 4 d -0.499989 (0.000011) -37.69018 (0.00078) Sequence Nb = 16 -0.49999992(0.00000008) -37.68949 (0.00147) 20 s 11 p 6 d -0.49999996(0.00000004) -37.69091 (0.00005) -0.5 -37.69096 b (a) see text for definitions; (b) from reference 90; (c) numbers in parentheses are basis set errors. The situation changes greatly when a substantial external B field is added. Let us first take the example of the H atom ground state in a field B = 10 au. Its energy is known accurately to be -1.747 797 163 714 Hartree [80]. The sequence of Nb = 16 included in Table 3-1 works remarkably well for the fieldfree energy, but gives 24% error in the B = 10 au field. See Table 3-2. Adding a sequence of d orbitals that has same length and same exponents as that for the s orbitals, which doubles the basis size, reduces the error by 80%. Further supplementation by g and i orbitals in the same way decreases the error by another order of magnitude. But this is sti ll far from satisfactory. To reduce the basis set error below 1 H, higher angular momentum basis with l up to 20 must be included. Obviously, this is a very inefficient appro ach. The basis sets used by Jones, Ortiz, and

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36 Ceperley [94] (see eqn. 3.23) converge the to tal energy more rapidly than these spherical bases. The primary sequence in the JOC basis se ts is the same as the spherical basis, but subsequent secondary sequences double the transv erse exponents j’s successively. With four sequences the error is less than 1% of the error in a spheri cal basis set having the same size. Another significant gain can be obtai ned if we move to the KL basis sets [97] (see eqn. 3.24). Here we choose KL = 0.18, which is obtained by searching with a step of 0.01 to minimize the basis set truncation erro r. Including only the primary KL sequence gains about the accuracy of the three-sequence JOC basis set. Recall that the subsequent KL secondary sequences have shorter leng ths than the primary one (refer to the discussion after eqn. 3.24). Specifically, the second and the third sequences have length of Nb /2 = 8, and the fourth and the fifth sequences have length of Nb /4 = 4. Thus, the basis size will be 16 + 8 + 8 + 4 + 4 = 40 if we include five KL sequences, with accuracy of 1 H. Table 3-2 Basis set errors for the gro und state energy of the hydrogen atom in B = 10 au. (energies in Hartree) Basis size Spherical JOC a KL b Optimized Eqn. (3.26) 16 0.419 8 0.419 787 28 0.003 738 20 0.000 000 60 0.001 044 51 32 0.081 5 0.027 124 87 0.000 005 39 0.000 000 36 0.000 000 50 48 0.021 7 0.001 008 57 64 0.008 1 0.000 075 02 40 0.000 001 12 0.000 000 30 0.000 000 28 (a) Jones-Ortiz-Ceperley basis sets, see ref [94] and eqn. (3.23); (b) Kravchenko-Liberman basis sets, see ref [97] and eqn. (3.24). KL is chosen to be 0.18. However, this does not mean there is no oppor tunity left for basis set optimization. Starting from the primary sequence in the KL ba sis set, we then sear ched in the parameter space { j} to minimize the total energy of the H atom. First, the energy gradient in

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37 parameter space is calculated, then a walk is ma de in the steepest de scent direction. These two steps are repeated until 0jE (3.25) The error left in this optimized basis set is only 0.6 H, six orders of magnitude smaller than the error of a s pherical basis set of the same size! The resulting exponents are listed in Table 3-3, together with the co efficients used in the wavefunction expansion. Addition of the same secondary KL sequences can further reduce the remaining error by one half. This improvement is not as spectac ular as that for the KL basis set because those exponents have already been optimized. It is worth mentioning that, while it is easy to optimize the basis set for the H atom fully, it is very hard to do so for multi-electron atoms. We usually only ge t partially optimized results but by including secondary sequences, the basis error can be greatly reduced, as demonstrated here. Table 3-3 Optimized basis set and expansion coefficients for the wavefunction of the hydrogen atom in B = 10 au. j Coefficients j j jj 1 0.000493 1.88860.05731.8313 2 0.011007 2.86400.1247 2.7393 3 0.184818 2.44620.2717 2.1745 4 0.372811 2.55410.5917 1.9624 5 0.277663 2.64421.2890 1.3552 6 0.132857 3.76902.8077 0.9613 7 0.050662 6.78556.1159 0.6696 8 0.019285 13.904813.3221 0.5827 9 0.007139 29.328729.0190 0.3097 10 0.002772 64.470263.21111.2591 11 0.000955 139.3854137.69041.6950 12 0.000418 301.7122299.92601.7862 13 0.000114 655.1166653.31801.7986 14 0.000082 1424.89881423.09881.8000 15 -0.000001 3101.68453099.88451.8000 16 0.000021 6754.16876752.36591.8028

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38 From Table 3-3, we see that the wave function is mainly expanded in basis j = 3, 4, 5, 6, and jj is not a constant as the KL sequences suggest. The smaller the exponent, the larger th e difference between the transverse and the longitu dinal exponents. This is quite understandable. A smaller expone nt means that the electron density extends far from the nucleus, and the magnetic field wi ll overpower the nuclear attraction, thus the distortion from the field-free spherical shap e will be relatively la rger. In the limit of j 0, which can be equivalen tly thought of as the large B limit, or zero nuclear charge, the electron wavefunction is a Landau orbital with an exponential parameter 4jB B a The opposite limit, j corresponds to B = 0, for which j = j. A natural measure of th e orbital exponents is Ba Now we can make an explicit construction (discussed below) which inco rporates all these behaviors, namely 1 2 244 411 20jj jjB bBbB (3.26) where 2 110.16tan0.77tan0.74 b (3.27) j j p q 1,2,.b j N (3.28) and 2B Z is the reduced field strength for an effective nuclear charge Z The parameters p and q are defined in eqn. (3.22). For the innermost electrons, Z is close to the bare nuclear charge; for th e outmost electrons in a neutra l atom, it is close to unity. Nevertheless, accurate Z values do not need to be provided. Nominal values turn out to be good enough for the input to eqn. (3.26). Second ary sequences are defined similarly as in

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39 KL basis set, using a factor of 1.2, 0.8, 1.4, and 0.6 for the second, third, fourth, and fifth sequences, respectively. Next I show how eqn. (3.26) was obtained. Start from the basis set of one sequence Nb = 16. Full basis set optimizations were done for H, He+, Li++, Be+++, C5+ and O7+ in reduced fields = 0.1, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 800, 1000, 2000, and 4000. Results for = 0.1, 1, 10, and 100 are plotted in Fig. 3-1. The first obser vation is that data points for different nuclear charges with the same are on the same curve. One can show that this must be the case. Suppose th e wavefunction for the hydrogen atom in an external B field is ,HrB, 22 221 ,, 28HHHB x yrBErB r (3.29) Scaling rZr leads to 222 22 21 ,, 28HHHZB x yZrBEZrB ZZr 2 2 2 222,, 28HHHZB Z x yZrBZEZrB r The Hamiltonian on the left side is the same as that of a hydrogen-like atom with nuclear charge Z in an external field2 B ZB The scaled hydrogen-atom wavefunction is precisely the eigenfunction of th is Hamiltonian with energy of 2 H Z E Now we expand ,HrB in the optimized basis set, 22,z jjjjnn z im Hjj jrBaNzee (3.30) and the scaled wavefunction is

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40 222,,zz jjjjjjnnnn z im ZHjj jrZBZrBaNZzee (3.31) where 22,jjjjZZ Obviously, jjjjBB and jj B B Another observation is that th e curvatures for different values are slightly different. To describe the ra pid decrease in the small /j B region and the slowly decaying long tail, we used a fixed combination of inverse square and inverse square root terms, which proved to be superior to a si ngle reciprocal function. In Fig. 3-2, the functional forms are compared with data points from optimized basis sets for = 1. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 j/B( j j)/B FIG. 3-1 Exponents of optimized basis sets for the H(+), He+(x), Li++(o), Be+++( ), C5+( ) and O7+( ) in reduced magnetic fields = 0.1 (blue) 1 (black), 10 (green) and 100 (red).

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41 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 j/B( j j)/B optimized for =1 Eqn. (3.26) using b=1.286 Eqn. (3.26) using b=1.246 1/4/(1+4/0.722* /B) FIG. 3-2 Fitting the parameter b ( =1) using the func tion (3.26). Fitted result is 1.286, compare to the calculated value 1.246 fr om eqn. (3.27). Two curves are shown by dotted black line and solid green line. They are almost indistinguishable in the graph. A reciprocal fitting result 14 1 40.722jB B is also shown by a dashed blue line. A calculation using the basis set derived from eqn. (3.26) also is included in Table 3-2. The new primary sequence outperforms the KL primary sequence by a factor of four, but by including only the second and th e third sequences, the basis set is almost saturated, compared to other, more slowly converging basis sets. Another advantage of the present basis set is the explicit expression s eqns. (3.26) and (3.27), whereas searching for the best parameter KL in the KL basis sets is quite time consuming.

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42Table 3-4 Test of basis sets including 1, 2, and 3 sequences on the energies of the hydrogen atom in B fields (energies in Hartree, negative signs are omitted) State B (au.) 1 sequence 2 sequences 3 sequences Reference 80 Reference 78a Reference 76 1 s 0.1 0.5475263 0.5475263 0.547526461 0.547526480401 0.5475265 1 0.83109 0.8311680 0.83116886 0.831168896733 0.831169 0.83116892 10 1.74675 1.747781 1.74779694 1.747797163714 1.747797 1.74779718 100 3.78933 3.789790 3.78980395 3.789804236305 3.78905/90250 3.7898043 1000 7.66224 7.662419 7.66242306 7.662423247755 7.66205/65 7.6624234 4000 11.20372 11.204139 11.20414499 11.204145206603 10000 14.14037 14.140959 14.14096829 14.14097 2 p0 1 0.25991 0.2600055 0.26000652 0.260006615944 0.2600066 0.260007 10 0.38263 0.382641 0.38264977 0.382649848306 0.38264875/5180 0.382650 1000 0.49248 0.4924948 0.49249495 0.4924950 0.492495 2 p-1 1 0.45658 0.456596 0.45659703 0.456597058424 0.45659705 10 1.12521 1.125415 1.12542217 1.125422341840 1.1254225 100 2.63472 2.634756 2.63476052 2.634760665299 2.634740/95 1000 5.63792 5.638413 5.63842079 5.63842108 5.638405/35 3 d-1 10 0.33890 0.3389555 0.33895610 0.3389561898 0.33895610/45 1000 0.48696 0.4869775 0.48697789 0.4869777 0.48697795 3 d-2 10 0.90813 0.908212 0.90821466 0.9082147755 0.9082115/235 1000 4.80432 4.805094 4.80511012 4.80511067 4.8051095/125 4 f-2 100 0.42767 0.427756 0.42775840 0.4277585 4 f-3 10 0.78773 0.787768 0.78776910 0.7877685/705 1000 4.30738 4.308344 4.30836962 4.3083700/05 5 g-4 1 0.26570 0.266184 0.26618782 0.26618770/875 100 1.75448 1.754848 1.75485593 1.754856 1000 3.96338 3.964471 3.96450833 3.9645095 (a) Numbers before/after slashes ar e the upper/lower bounds to the energy.

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43 The basis set constructed from eqn. (3.26) not only works well for 1 s electrons, but also for higher excited states. Table 3-4 in cludes some test results on the hydrogen atom in a wide range of B fields. The primary sequence was de rived from eqns. (3.26) through (3.28) using Nb = 16. Extrapolations to j > Nb or j 0 were made for extremely tight or diffuse orbitals whenever necessary. The averaged basis set error for the primary sequence is 0.3 m H, which is reduced to 7 H if the second sequence is added. With three sequences, the accuracy of our basis sets reaches 1 H level. Notice that energies quoted from reference 76 are slightly lower than th e more accurate algebraic results [80]. This implies that Ivanov and Schmelcher’s data are not necessarily upper bounds to the energy. We need to keep this in mind wh en we compare our results with theirs. Table 3-5 Energies for hi gh angular momentum states of the hydrogen atom in B fields ( B in au; energy in Hartree) State B = 1 B = 10 B = 100 B = 1000 5 g-4 -0.2661880 -0.7080264 -1.7548563 -3.9645100 6 h-5 -0.2421928 -0.6499941 -1.6252244 -3.7061998 7 i-6 -0.2239757 -0.6051943 -1.5238725 -3.5018527 8 j-7 -0.2095131 -0.5691841 -1.4415788 -3.3343126 9 k-8 -0.1976562 -0.5393750 -1.3728860 -3.1933035 10 l-9 -0.1876974 -0.5141408 -1.3143222 -3.0722218 The accuracy of the previous basis sets can be improved further by increasing Nb and including the fourth and the fifth seque nces. Using larger basis sets having five sequences with the primary sequence derived from Nb = 28, we obtained energies of -5.638 421 065, -4.805 110 65, -4.308 370 6, and -3.964 510 0 for the H atom 2 p-1, 3 d-2, 4 f-3, and 5 g-4 states in B = 1000, respectively. Th eir accuracy is at th e same level as the best available data in the literature (see Ta ble 3-4). There are no specific difficulties for the higher angular momentum states in our e xpansion. The energy values for the excited

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44 states with quantum numbers l = m = 4, 5, …, 9 of the hydrogen atom in B fields are listed in Table 3-5. Those values will be used in the next step, cons truction of Table 3-6. Actually we do not need to use the entir e sequences generated from eqns. (3.26) through (3.28). In the expansi on of nonzero angular momentum orbitals, very tight basis functions ( j close to Nb ) are not necessary, but extrapolation to negative j may be required in order to include very diffuse basi s functions. Table 3-6 lists the subsets which have accuracy of 1 H. The one to five segments in each basis set means the ranges of j values selected from the primary and th e subsequent secondary sequences. Numbers underlined identify the negative j values. Nb values used for the five sequences are 16, 8, 8, 4, and 4. Also recall a factor of 1.2, 0.8, 1.4, and 0.6 is used for the second, third, fourth, and fifth sequences, respectively. Nu mbers in parentheses are the sizes of the basis sets. Table 3-6 Basis sets for the hydrogen atom in B fields with accuracy of 1 H ( B in au) State B = 0 B = 1 B = 10 B = 100 B = 1000 1 s 1-14(14) 2-14,1-3,1-2(18) 1-16,2-6,2-6( 26) 4-20,2-8,2-6(29) 4-21,4-10,5-10(31) 2 p0 1 -7(9) 0-8,0-3,3(14) 0-10,1-5(16) 1-12,2-5(16) 1-14,3-5(17) 2 p-1 1 -7(9) 1-9,0-3,3(14) 1-11,5,2-6(17) 3-14,4-8,3-6(21) 4-16,4-10,5-10(26) 3 d-1 3 -4(8) 1 -6,0-3,0-3(16) 0-8,1-5,2-3(16) 0-10,1-6,2(18) 1-12,2-6,3(18) 3 d-2 2 -4(7) 0-7,1-3,0-3(15) 1-8,3-5,2-6(16 ) 3-12,3-7,3-7(20) 4-15,4-10,4-7(23) 4 f-2 5 -2(8) 1 -5,0-3,0-3(15) 0-7,1-4,1-5(17) 0-10,1-6,2-3(19) 1-12,2-6,3(18) 4 f-3 4 -2(7) 0-4,0-3,0-4(14) 1-7,2-5,2-6(16) 3-11,1-7,1-7(23) 3-14,2-8,4-7,3,3(25) 5 g-3 4 -0(5) 1 -5,1 -2,0-1(13) 0-7,1-4,1(13) 0-9,1-5,2-3(17) 1-11,2-6,3-4(18) 5 g-4 4 -0,2 -1 (7) 0-4,0-3,0-4(14) 1-6,2-5,1-3(13) 2-11,1-7,3-6(21) 3-13,3-8,4-7,3,3(23) 6 h-5 5 -2 ,4 -1 (8) 0-4,0-3,0-3(13) 1-6,2-5,1-3(13) 2-10,3-7,2-5(18) 3-13,3-8,4-7,3,3(23) 7 i-6 6 -2 ,5 -3 (8) 1 -2,0-2,0-3(11) 1-5,2-5,1-3(12) 2-10,1-5,3-6(18) 3-13,3-8,4-7,3,3(23) 8 j-7 1 -2,0-2,0-3(11) 1-5,2-5,1-3(12) 2-10,2-6,3,2,2(17) 3-13,3-8,4-7,3,3(23) 9 k-8 1-5,2-5,1-2(11) 2-9,2-6,3,2,2(16) 3-12,3-8,4-6,3,3(21) 10 l-9 1-5,2-5,1-2(11) 2-9,2-6,3,2,2(16) 3-12,3-8,4-6,3,3(21) While the previous optimization scheme is quite impressive for the hydrogen atom in a B field, we want to know whether it also works equivalently well for multi-electron atoms. Thus we do another case study, the carbon atom in the same field B = 10 au. Its

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45 ground state configuration is2 123412345 spdfg and HF energy is -44.3872 Hartree from calculations on a numerical grid [90]. The performance of various basis sets is summarized in Table 3-7. Table 3-7 Basis set effect on the HF energies of the carbon atom in B = 10 au. Basis set Basis size HF energy (Hartree) Error(Hartree) Spherical( spdfghi ) 112 -43.6157 0.7715 2sequences, JOC 160 -44.1572 0.2300 3sequences, JOC 240 -44.3529 0.0343 1sequence, KL 80 -44.1629 0.2243 2sequences, KL 120 -44.3824 0.0048 3sequences, KL 160 -44.3863 0.0009 5sequences, KL 200 -44.3867 0.0005 1sequence, present 50 -44.3859 0.0013 2sequences, present 72 -44.38704 0.0002 3sequences, present 91 -44.38714 <0.0001 The spherical basis set includes 16 s 16 p 16 d 16 f 16 g 16 h 16 i orbitals. Again it gives a large basis set error. For the KL basis se ts, I used the same parameter as before, KL = 0.18. Its accuracy can also be improved grea tly by systematic augmentation of secondary sequences. However, the basis size will be in creased considerably. Based on the previous detailed study on the basis set for the hydroge n atom, here we pres cribe a procedure to construct the basis set for a multi-electron atom in a B field, with the C atom as the example. We first assign the effective nuclear charge Zeff for each electron roughly. The approach is by approximate isoe lectronic sequences. Since the 1 s electrons feel the whole strength of the nuclear attracti on, we use 6 for them. For the 2 p electron, the nucleus is screened by the two 1 s electrons, so we use 4, and so forth. Next basis functions are generated according to eqns. (3.26) through (3.28) using Nb = 16, 8, 8, 4, 4 for the primary, the second, …, the fifth sequences. To use Table 3-6 as guidance in selecting

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46 subsets of basis functions from the previously generated sequences, first recall the scaling argument after eqn. (3.29). The 1 s wavefunction with an e ffective nuclear charge Zeff ,1 s = 6 in a field B = 10 can be scaled from the hydrogen atom 1 s wavefunction in a field2 ,1/0.28effsBBZ The value of B falls in the range of 0 to 1. By inspection of Table 3-6, we find a sufficient choice of basis set includes th e first through the fourteenth elements in the primary, 1-3 elements in the second, and 1-2 in the third sequences. But do not forget the scaling f actor. Since the C atom 1 s wavefunction is tighter than the H atom 1 s wavefunction approximately by ,16effsZ times, the basis function exponents used in the expansion of the C atom 1 s orbital should be larger than those used for the H atom by 2 ,136effsZ times (recall eqns. 3.30 and 3. 31). Remember the exponents j’s consist of a geometrical series (eqn. 3.28) The increase of the exponents amounts to a shift of 2 ,12ln6 log4.6 ln2.18qeffsZ elements in the primary sequence, and a shift of 2ln6 3.4 ln2.84 elements in the second and the third sequences. Hence, we should pick the 5-19, 4-7, and 4-6 elements in the prim ary, the second, and the third sequences, respectively. Basis function selections for ot her electron orbitals ar e similar. The final basis set is 22 s 19 p 16 d 21 f 13 g, which is summarized in Tabl e 3-8. Among the total of 91 gaussians in this basis set, 50 are from the primary, 22 from the second, and 19 from the third sequences. From Table 3-7, we can see th e accuracy of this basis set is remarkably higher than that of the others. By using only the primary sequence, the error left is close to 1 m H. Supplementation with the second sequence reduces th e error to 0.2 m H. We estimate the error of the 3-sequence present basis to be less than 0.1 m H.

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47 Table 3-8 Construction of the AGTO basis set for the carbon atom in B = 10 au. Orbital Zeff 2/eff B BZ H atom basis shifts C atom basis 1 s 6 0.28 1-14, 1-3, 1-2 4.6, 3.4, 3.4 5-19, 4-7, 4-6 2 p-1 4 0.63 1 -9, 0-3, 3 3.6, 2.6, 2.6 2-13, 2-6, 5-6 3 d-2 3 1.1 0-7, 1-3, 0-3 2.8, 2.1, 2.1 2-10, 3-5, 2-5 4 f-3 2 2.5 0-7, 0-5, 0-6 1.8, 1.3, 1.3 2-9, 1-6, 1-7 5 g-4 1 10 1-6, 2-5, 1-3 0, 0, 0 1-6, 2-5, 1-3 One may wonder why this procedure works so well, or even why it works at all. The main reason is that each electron orbital can be approximated by a hydrogen-like problem fairly closely. For exam ple, the overlap between the 1 s HF orbital for the carbon atom in B = 10 and the 1 s orbital for C5+ in the same field is 0.998. See Table 3-9. Actually by adjusting the nuclear charge to 5.494 and 5.572, the overlaps for 1 s spin down and spin up orbitals with their corre sponding hydrogen-like counterparts reach the maxima of 0.9998 and 0.9999, respectively. Other orbitals are similar. Table 3-9 Overlaps between HF or bitals for the carbon atom in B = 10 au and hydrogenlike systems in the same field Orbital eff Z overlap eff Z overlap 1 s 6 0.99773 5.494 0.99984 1 s 6 0.99838 5.572 0.99989 2 p-1 4 0.99967 3.944 0.99968 3 d-2 3 0.99974 3.145 0.99983 4 f-3 2 0.99731 2.644 0.99986 5 g-4 1 0.98224 2.108 0.99981 Now that we have a systematic way to c onstruct reasonably accu rate basis sets for atoms in a B field. In the next section, I will use those basis sets for the DFT and CDFT studies.

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48 CHAPTER 4 ATOMS IN UNIFORM FIELDS – NUMERICAL RESULTS Comparison with Data in Literature I did extensive unrestricted Hartr ee-Fock (UHF) and conventional DFT calculations on the atoms He, Li, Be, B, C, and their positive ions Li+, Be+, B+ in a large range of B fields with basis sets constructed acco rding to the procedure outlined in the previous chapter. Total ener gies are compiled in appendix B. Ground states are indicated in orange. Data available from the liter ature are also included for comparison. The UHF calculations were primarily for purposes of validation. The agreement of our calculations with those from other groups is excellent. For the helium atom, our HF energies are generally slightly lower than those by Jones, Ortiz, and Ceperley [91, 94]. Their earlier calculations used an STO expans ion [91] which is labeled as JOC-HF1 in Table B-1. Later they utilized JOC basis sets within AGTO [94] (also refer eqn. 3.23), which we call JOC-HF2. Presumably the small distinction between their data and ours results from better optimized basis sets I generated, as already illustrated in the previous chapter. This observation is supported by the generally closer agreement of their anisotropic basis set results w ith ours (in contrast to their spherical basis results). One notable exception to the ove rall agreement is the 1 s 4 f-3 state in B = 800 au. Our result is -23.42398 Hartree versus theirs -23.4342. Another is 1 s 3 d-2 at B = 560 au: -21.59002 versus their -21.5954 Hartree. The reason for these discrepancies is unclear. It may be some peculiarity of the basis for a particular field strength. For other atoms and ions, the data for comparison are mainly from the seri es of studies by Ivanov and Schmelcher [86,

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49 88-90]. Our HF energies generally match or are slightly higher than theirs, and the overall agreement is quite satisfactory. Differences are usually less than 0.1 m H, far surpassing our goal of 1 m H accuracy for the basis set. The remaining differences arise, presumably from our basis set truncation e rror and their numerical mesh errors. As for any basis-set based calculations, we can only use a finite number of basis functions, which will cause basis set truncation error. Since this error in our calculation is always positive (by the variational principle), one can use our da ta as an upper bound for the HF energies. However, the numerical error in Ivanov and Schmelcher’s 2D HF mesh method seems to tend to be negative. Recall the comparison made in Table 3-4 for the hydrogen atom. Their energies are always lower than the more accurate algebraic result [80]. Another indication is the zero field atomic energies. For example, the HF energy for the beryllium atom is known accurately to be -14.57302316 Ha rtree [101], which agrees well with our result of -14.57302, but Ivanov and Schmelch er gave a lower value of -14.57336 [88]. They commented that this configuration ha s large correlation ener gy and the contribution from the 1 s22 p2 configuration should be considered but did not specify whether their result was from single determinant or multi-determinant calculation. Since multiconfiguration HF (MCHF) is not our main interest here, our data are solely from single configuration HF calculations. Th ey also noted that the precis ion of their mesh approach decreases for the 1 s22 s2 configuration in a st rong field. This can also contribute to the discrepancy. From previous observations, one ma y speculate that the true HF energies lie between our data and theirs. Furthermore, th e accuracy of our da ta is ready to be improved by invoking a larger basis set, but this is not necessary for the purpose of the present study.

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50 There are fewer DFT calculations for atoms in B fields than HF studies. As far as I know, appendix B is the first extensive compila tion of magnetized atomic energies based on modern DFT functionals. I chose PW 92 [69] for LDA and PBE [70] for GGA calculations. For the field-free case, the pres ent results agree well with published data [102]. Since reference 102 only gave spin non-polarized DFT energies for spherically distributed densities (which is no problem fo r the helium and beryllium atoms), one needs to use fractional occupation numbers in other atoms for comparison. For example, in the carbon atom, two p electrons are placed in six spin orbitals,,, 10, p p and 1 p with equal occupation number of 1/3 for each of th em. Actually there are more accurate data for the VWN functional [67]. My results differ from those from reference 67 by no more than 5 H if I choose the VWN functi onal, either for spin-polarized or spin non-polarized energies, neutral atoms or their positive ions. Comparison of non-vanishing B field DFT calculations is handicapped by the different magnetic field grids on which different authors pres ent their results, and by the different functionals implemented. The func tional due to Jones [103], which is at the level of the local density approximation, was used by Neuhauser, Koonin, and Langanke [104], and by Braun [84]. The simple Di rac exchange-only functional was used by Relovsky and Ruder [83], and by Braun [ 84]. When I choose the Dirac exchange functional, good agreement is found with Br aun’s data. However, no application of density-gradient-dependent fiunctionals on at oms in a strong magnetic field is found in the literature so far as I can tell. For the CDFT study, the present work appare ntly is the first fu lly self-consistent calculation on atoms in large B fields based on the VRG fu nctional. The most closely

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51 related study is the perturbative implemen tation of the VRG functional by Orestes, Marcasso, and Capelle [37] on the atom ic ionization energi es in vanishing B field. For comparison, CI results are available for helium, lithium, and beryllium atoms, and their positive ions in external fields [7 8, 11-13, 77, 105]. Thos e data will be treated as the most reliable ones in my comparison. Magnetic Field Induced Ground State Transitions The most drastic change of the ground stat e atoms caused by an external magnetic field is a series of configur ation transitions from the fiel d-free ground state to the highfield limit ground state. It is well known th at for the field-free case, two competing factors — the spherical nucleus attractive pote ntial and the electron-electron interactions — lead to the shell structure of atoms. This structure is perturbed slightly if the external field is relatively small, but when the fi eld is strong enough that the Lorentz forces exerted on the electrons are comparable to nuclear attraction and electron repulsions, the original shell structure is crushed, and th e electrons make a ne w arrangement. Thus, a series of configuration tr ansitions happens as the B field becomes arbitrarily large. In the large-field limit, the ground state is a fully spin -polarized state in wh ich the electrons take the minimum value of spin Zeeman energy. Ivanov and Schmelcher further analyzed the spatial distribution of electrons and describe d the high-field limit ground state “... with no nodal surfaces crossing the z axis (the field axis), and with nonpositive magnetic quantum numbers decreasing from m = 0 to m = N + 1, where N is the number of electrons” [76]. In Hartree-Fock language, the high field configuration is 1231234 spdf. In this regime, the magnetic field is the dominant fact or, and coulombic forces can be treated as small perturbations. A cylindrical separation of the z part from x and y parts is usually

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52 made for the electron state. Its motion in the xy plane is described by a Landau orbital, and the question is reduced to a quasi-1D probl em. This technique is often referred to as the adiabatic approximation, valid only in the limit of very large field. Many early investigations on matter in a B field concentrated on this regime [103, 104, 106]. FIG. 4-1 UHF total energies for different el ectronic states of the helium atom in B fields. Curves 1 to 9 represent configurations 1 s2, 1s2s, 1s2 p0, 1s2 p-1, 1s3 d-1, 1s3 d-2, 1s4 f-2, 1s4 pf-3, and 1s5 g-3, respectively. The most difficult part is the region of intermediate B field, for which both cylindrical and spherical expansions are inefficient, and where the ground state configurations can only be de termined by explicit, accura te calculations. Figure 4-1 displays the HF energies of various configurations for the helium atom in B fields as listed in Table B-1, which includes one singlet state and eight triplet states. Curves 1 to 9 represent configurations of 1 s2, 1s2s, 1s2 p0, 1s2 p-1, 1s3 d-1, 1s3 d-2, 1s4 f-2, 1s4 pf-3, and 1s5 g-3, respectively. Each configuration belongs to a spin subspace according to its total

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53 spin z component. For convenience, we use th e inexact terminology of “local ground state” and “global ground state”. For a specific field strength, the c onfiguration which has the lowest energy within a spin subspace is called the local ground state for this spinmultiplet. Among them, the one taking the minimu m regardless of its spin is called the global ground state. Thus in Fig. 4-1, the singlet state remains as the global ground state until B reaches 0.71 au., then a conf iguration transition to 1 s 2 p-1 occurs. This triplet state is the global ground state for B fields larger than 0.71 au. Atoms with more electrons can have more complicated series of configura tion transitions. For example, the carbon atom undergoes six transitions with seven electronic configuratio ns involved being the ground states in different regions of B field strengths [90]. This scenario is basically the same if one uses DFT or CI energies instead of HF energies, except the crossing points for different configuratio ns usually change. Atomic Density Profile as a Function of B Within each configuration, the elect ron density is squeezed toward the z axis with increasing B field. This follows from energy mini mization: the electr on density shrinks toward the z axis to alleviate the corres ponding diamagnetic energy increment (expectation of B2( x2+ y2)/8). Figure 4-2 shows the density profiles for the 1 s2 and 1s2 p-1 states of the helium at om at field strengths B = 0, 0.5, 1, and 10 au. The transverse shrinkage is quite evident. However, this shrinkage increases the electron repulsion energy. A configuration transi tion therefore will happen at some field strength (for He, B = 0.71 au.), accompanied by a change of quantum numbers, and eventually a spin flip. Note Figs. 4-2 (a), (b), and (g). The energy increase in the diamagnetic term caused by

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54 FIG. 4-2 Cross-sectional view of the HF to tal electron densities of the helium atom 1 s2 (panels a-d ) and 1s2 p-1 (panels e-h ) states as a function of magnetic field strength. Each large tick mark is 2 bohr radii. The B field orientation is in the plane of the paper from bottom to top. The density at the outermost contour lines is63 010 a, with a factor of 10 increase for each neighboring curves. Panels ( a ), ( b ), ( g ), and ( h ) are global ground states. See text for details.

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55 the electron density expansion in the new sp in-configuration for the global ground state is more than compensated by the accompany en ergy lowering of the Zeeman and electronelectron repulsion terms. In fact, the same net lowering can occur (and sometimes does occur) without change of spin symmetry. Total Atomic Energies and Their Exchange and Correlation Components Atomic total energies of He, Li, Be B, C, and their positive ions Li+, Be+, B+ in a large range of B fields within the HF, DFT-LDA, and DFT-GGA approximations are compiled in Appendix B. The exchange-correlation energy Exc, and its exchange and correlation components Ex, Ec corresponding to the total en ergies in appendix B, are given in appendix C. As is conventional, the HF correlation energy is defined as the difference between the CI total atomic ener gy and HF total energy tabulated in appendix B. DFT exchange and correlation energies ar e defined at the self-consistent electron densities within the corresponding XC energy functionals (LDA or PBE). Keep in mind that exchange and correlation energies are no t defined identically in the wavefunction and DFT approaches; recall the discussion in Chap.1, particularly eqn. (1.11). Because energies of different states in di fferent field strengths vary considerably, we compare their differences from the corr esponding CI total energies. Figure 4-3 shows those differences for the HF and DFT to tal atomic energies of helium atom 1 s2, 1s2 p0, and 1s2 p-1 states as functions of B field strength. Since the HF calculation includes exchange exactly, the difference for the HF energy is the negative of the conventional correlation energy H F cE (the superscript “HF” is for clar ify). First we observe that the conventional correlation ener gies for the states 1s2 p0 and 1s2 p-1 are extremely small.

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56 102 101 100 101 102 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 B (au)EECI (Hartree)a) He 1s2 HF LDA PBE 102 101 100 101 102 0.25 0.2 0.15 0.1 0.05 0 0.05 B (au)EECI (Hartree) b) He 1s2p0 HF LDA GGA 102 101 100 101 102 0.1 0.05 0 0.05 0.1 B (au)EECI (Hartree) c) He 1s2p1 HF LDA GGA FIG. 4-3 Differences of the HF and DFT total atomic energies of the helium atom 1 s2, 1s2 p0, and 1s2 p-1 states with respect to th e corresponding CI energies as functions of B field strength. Blue squares : HF; Black circles o: DFT-LDA; Red triangles : DFT-GGA.

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57 This is because the 1s and 2 p electrons in the atom are we ll separated, unlike the two 1 s electrons in the 1 s2 configuration. The increase of the absolute value of H F cE in the large B field regime for the two states 1 s2 and 1s2 p-1 seems to be the result of the compression of electron densities illustrated in Fig. 4-2. The PBE generalized gradient functional gives extremely good results for both the singlet state 1 s2, and triplet sates 1s2 p0 and 1s2 p-1 when the B field magnitude is less than 1 au. Both LDA and GGA approximations fail in the large field regime, B > 10 au. Notice the similar performance of DFT functionals for the two triplet states 1s2 p0 and 1s2 p-1. The former one does not carry paramagnetic current density, thus there is no CDFT current correction for this configuration, whereas the later one is a cu rrent-carrying state. Th is observation implies that the success or failure of these partic ular LDA and PBE functi onals is not because they omit current terms. The success of DFT calculations mainly depends on accurate approximations for the system exchange and correlation energi es. As given in detail in Chap. 1, DFT exchange and correlation energies differ subtly from conventional exchange and correlation energies. The DFT quantities refe r to the auxiliary KS determinant (and include a kinetic energy cont ribution) whereas the conven tional quantities are defined with respect to the HF determinant. Neve rtheless, conventional exchange-correlation energies often are used as the quantity to approximate in DFT exchange-correlation functionals, mostly for pragmatic reasons. The difficulty is that exact DFT quantities and KS orbitals are only available for a few, very small systems. One of those is discussed in the next chapter. For most systems, the ex act KS orbitals are unknow n. However, there

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58 102 101 100 101 102 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 B (au)Ex DFT Ex HF (Hartree) 1s2, LDA 1s2, PBE 1s2p0, LDA 1s2p0, PBE 1s2p1, LDA 1s2p1, PBE 102 101 100 101 102 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 B (au)Ec DFT Ec HF (Hartree) 1s2, LDA 1s2, PBE 1s2p0, LDA 1s2p0, PBE 1s2p1, LDA 1s2p1, PBE 102 101 100 101 102 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 B (au)Exc DFT Exc HF (Hartree) 1s2, LDA 1s2, PBE 1s2p0, LDA 1s2p0, PBE 1s2p1, LDA 1s2p1, PBE FIG. 4-4 Differences of DFT exchange (t op panel), correlation (middle panel), and exchange-correlation (bottom panel) en ergies with HF ones, for the helium atom in B fields.

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59 are abundant HF and correlated calculati ons for many finite systems, providing good reference densities and energies. To gain a better understanding of the behavior of DFT exchange and correlation approximations, we make a separate comparison of DFT exchange-correlation energies with the HF one s in Fig. 4-4. Note that “DFT exchange” here means the Ex term in a particular functional a nd not exact DFT exchange calculated from KS orbitals. We can see from Fig. 4-4 that th e LDA approximation shows its typical underestimation of exchange and overesti mation of correlation energies. The PBE functional gives good approximations to the exch ange and correlation energies separately when the B field is less than 1 au., but it serious ly overestimates the exchange when B >10 au., while the correlation energy does not depe nd on the field strength very much. Since exchange dominates the XC energies, the er ror in the exchange term overwhelms the correlation term in large B fields. Of course both the L DA and PBE functionals are based on analysis of the field-free electron gas, in which the exchange-correlation hole is centered at the position of th e electron, and only the spherically averaged hole density enters. This picture breaks down for an atom in a large B field. Because of the strong confinement from the B field, there is strong angula r correlation among electrons. The XC hole is not centered at the electron, and is not isotropic. Moreover, the external B field will effectively elongate XC hole as well as the electron density. If one wishes to improve XC functionals for applications in the large B field regime, those factors need to be considered. Another observation from Fi g. 4-4 is that PBE overestimates the correlation energies for the two triplet states, in which the HF correlation is very small. This presumably is due to the imperfect cancellation of self-interactions in the functional.

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60 The lithium positive ion is a two-electron syst em isoelectronic with neutral He. It has approximately the same correlation energy as that of the helium atom in the field-free case. For non-vanishing field, recall the scaling argument for the wavefunction of hydrogen-like atoms in a B field. The deformation of the atomic density induced by the B field is measured by its reduced strength 2/ B Z rather than by its absolute value B where Z is the nuclear charge (refer to eqns. 3. 30 and 3.31 and discussion). Of course, the atomic configuration is anothe r important factor. For the same electronic c onfiguration, the helium atom in B = 5 au. and the lithium positive ion in B = 10 au. have about the same values. Indeed they have about the same HF and PBE correlation energies. On the other hand, an attempt at a similar comp arison between the lithium atom and the beryllium ion is obscured by two factors. On e is the large correlation energy between the two 1 s electrons for the doublet states The effect of the external B field on its correlation energy is hardly discernable in the studied range. For the quadrupl et state, notice the tabulated conventional correlation energy for the beryllium ion is much smaller in magnitude than that of the lithium atom, giving rise to the suspicion of systematic errors existing in those data. Also notice that the conventional correlation energy of the lithium atom 1 s 2 p-13 d-2 state in vanishing B field is even larger than that of its ground state 1 s22 s Since the electrons are well-separated in the 1 s 2 p-13 d-2 state, its correlation energy is expected to be smaller than that of a more compact state. Even for vanishing B field, large discrepancies on the correl ated energies are found in the literature. For example, AlHujaj and Schmelcher gave -14.6405 Hartree for the ground state of the beryllium atom from a full CI calculation [8], versus -14.66287 Hartree from a frozen-core approximation by Guan et al. [7]. The difference is more than 20 m H. This shows it is

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61 difficult to systematically extract atomic correlation energies from the literature, especially for non-vanishing field data. The DFT functionals investigated here fail spectacularly in a large B field, mainly from their exchange part. However, the PBE correlation still gives a large portion of the correlation energy even though its performan ce degrades somewhat with increasing B field. On the other hand, the HF appr oximation is more robust than DFT-based calculations and includes exchange exactly, bu t it totally neglects correlation. From those analyses, it seems a better estimation to the total atomic energies in a large B field may be achieved by combining HF exact exchange a nd PBE correlation energy rather than using solely the HF or DFT approximations. Ionization Energies and Highest Occupi ed Orbitals for Magnetized Atoms Because of the magnetic-field-induced configuration transitions for both magnetized atoms and their positive ions, at omic ionization energi es are not monotonic increasing or decreasing smoot h functions of the applied B field. This is already obvious from Koopmans’ theorem and the UHF total ener gies in Fig. 4-1. Here we use the total energy difference between the neutral atom and its positive ion, ESCF, for estimation of the ionization energy. For each field streng th, the ground state configurations for the atom and its positive ion must be determined first. Table 4-1 and Fig. 4-5 show the change of ionization energies of the atoms He, Li, and Be with increasing B field. Results from different methods are close. For the be ryllium atom, a frozen -core calculation [7] gave a larger ionization energy, by 26 m H, in the near-zero-field region than the one derived from Al-Hujaj and Schmelcher’s data [8]. This difference is mainly caused by the lower ground state atomic energy obtained in the former reference, which has already been mentioned near the end of the previous section.

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62 Table 4-1 Atomic ionization energies in magnetic fields (energy in Hartree) Atom Configurations B (au) HF CI LDA PBE Hea 1 s2 1 s 0 0.8617 0.9034 0.8344 0.8929 0.02 0.8516 0.8933 0.8244 0.8828 0.04 0.8415 0.8831 0.8142 0.8727 0.08 0.8208 0.8625 0.7935 0.8520 0.16 0.7782 0.8199 0.7506 0.8092 0.24 0.7340 0.7756 0.7059 0.7647 0.4 0.6409 0.6824 0.6113 0.6706 0.5 0.5798 0.6212 0.5492 0.6089 1 s 2 p-1 1 s 0.8 0.4687 0.4741 0.4199 0.4734 1 0.5187 0.5245 0.4685 0.5225 1.6 0.6438 0.6504 0.5887 0.6452 2 0.7132 0.7201 0.6549 0.7136 5 1.0734 1.0816 0.9978 1.0739 10 1.4394 1.4493 1.3519 1.4527 20 1.9061 1.9190 1.8161 1.9554 50 2.7182 2.7378 2.6627 2.8829 100 3.5161 3.5442 3.5445 3.8593 Li 1 s22 s 1 s2 0 0.1963 0.2006 0.2011 0.2054 0.01 0.2012 0.2056 0.2059 0.2102 0.02 0.2068 0.2136 0.2135 0.2178 0.05 0.2177 0.2254 0.2226 0.2269 0.1 0.2329 0.2403 0.2380 0.2425 1 s22 p-1 1 s2 0.2 0.2587 0.2614 0.2691 0.2729 0.5 0.3699 0.3750 0.3844 0.3870 1 0.5025 0.5111 0.5216 0.5226 2 0.6995 0.7113 0.7226 0.7229 1s2 p-13 d-2 1 s 2 p-1 3 0.7525 0.7572 0.7635 5 0.9475 0.9555 0.9558 0.9644 10 1.2877 1.2982 1.3074 1.3219 Be 1 s22 s2 1 s22 s 0 0.2956 0.3158 0.3318 0.3306 0.001 0.2951 0.3159 0.3313 0.3302 0.01 0.2905 0.3112 0.3267 0.3255 0.02 0.2852 0.3313 0.3214 0.3203 0.05 0.2683 0.2911 0.3047 0.3035 1 s22 s 2 p-1 1 s22 s 0.1 0.3234 0.3242 0.3304 0.3312 0.2 0.3941 0.3941 0.4010 0.4019 0.3 0.4531 0.4597 0.4603 0.4612 1 s22s2 p-1 1 s22 p-1 0.4 0.4687 0.4758 0.4677 0.4717 0.5 0.4710 0.4749 0.4713 0.4758 0.6 0.4696 0.4767 0.4718 0.4766 0.8 0.4593 0.4650 0.4663 0.4718 1 s22 p-13 d-2 1 s22 p-1 1 0.4559 0.4455 0.4575 0.4636 2 0.6231 0.6217 0.6257 0.6336 1 s 2 p-13 d-24 f-3 1 s 2 p-13 d-2 5 0.8787 0.8772 0.8895 0.9019 10 1.1973 1.1959 1.2223 1.2401 (a) Exact energies are used for the one-electron system He+.

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63 Table 4-2 Eigenvalues for the highest occupied orbitals of magnetized atoms (energy in Hartree) Atom Configuration B (au) HOMO HF LDA PBE He 1 s2 0 1 s -0.91795 -0.5702 -0.5792 0.02 -0.90789 -0.5601 -0.5692 0.04 -0.89771 -0.5499 -0.5589 0.08 -0.87699 -0.5289 -0.5379 0.16 -0.83412 -0.4848 -0.4940 0.24 -0.78942 -0.4383 -0.4476 0.4 -0.69501 -0.3387 -0.3485 0.5 -0.63298 -0.2728 -0.2829 1 s 2 p-1 0.8 2 p-1 -0.47120 -0.3184 -0.3184 1 -0.52183 -0.3529 -0.3532 1.6 -0.64824 -0.4389 -0.4408 2 -0.71820 -0.4867 -0.4900 5 -1.07974 -0.7379 -0.7502 10 -1.44629 -0.9994 -1.0230 20 -1.91388 -1.3424 -1.3824 50 -2.72841 -1.9648 -2.0381 100 -3.52994 -2.6077 -2.7192 Li 1 s22 s 0 2 s -0.19636 -0.1162 -0.1185 0.01 -0.20122 -0.1211 -0.1234 0.02 -0.20668 -0.1282 -0.1306 0.05 -0.21778 -0.1364 -0.1389 0.1 -0.23293 -0.1487 -0.1516 1 s22 p-1 0.2 2 p-1 -0.25885 -0.1728 -0.1751 0.5 -0.37038 -0.2529 -0.2549 1 -0.50398 -0.3472 -0.3489 2 -0.70293 -0.4873 -0.4903 1 s 2 p-13 d-2 5 3 d-2 -0.95259 -0.6626 -0.6763 10 -1.29348 -0.9139 -0.9355 Be 1 s22 s2 0 2 s -0.30927 -0.2058 -0.2061 0.01 -0.30417 -0.2007 -0.2010 0.02 -0.29888 -0.1953 -0.1957 0.05 -0.28186 -0.1779 -0.1783 1 s22 s 2 p-1 0.1 2 p-1 -0.33120 -0.1959 -0.1961 0.2 -0.40159 -0.2560 -0.2566 0.3 -0.46016 -0.3046 -0.3054 0.4 2 s -0.47732 -0.3108 -0.3163 0.5 -0.47908 -0.3099 -0.3161 0.6 -0.47722 -0.3064 -0.3134 0.8 -0.46613 -0.2953 -0.3037 1 s22 p-13 d-2 1 3 d-2 -0.46092 -0.3105 -0.3180 2 -0.62799 -0.4283 -0.4391 1 s 2 p-13 d-24 f-3 5 4 f-3 -0.88345 -0.6259 -0.6421 10 -1.20284 -0.8671 -0.8908

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64 102 101 100 101 0 0.5 1 1.5 B (au)EI (Hartree) He Li Be FIG. 4-5 Atomic ground state ioni zation energies with increasing B field. Data plotted are from CI calculations shown in Table 41. Dotted lines are the guides to the eye. Even though the ionization energies in bot h the low and intermediate field regions are rather complicated as the result of atomic configuration changes, their behaviors are similar for the strong field limit configurations This is an indication that the original atomic shell structure has been e ffectively obliterated by the field. Eigenvalues of the highest o ccupied orbitals are reported in Table 4-2. In all the cases, the HF orbital energies give the closest approximation to the atomic ionization energies, with an average deviation of only 7.6 m H. KS eigenvalues, as usual, significantly underestim ate the ionization energy. This is because both LDA and PBE

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65 functionals give approximate potentials too shallow compared with the exact DFT XC potential. Self-interaction correction (SIC) c ould significantly improve these values, but we will not pursue it further here, because our focus is on Exc functionals that are not explicitly orbitally dependent. Current Density Correction and Other Issues Advancement in CDFT, especially in a pplications, is hindered by the lack of reliable, tractable functionals. In comparison with the vast literature of ordinary DFT approximate XC functionals, the total num ber of papers about CDFT approximate functional is substantia lly less than 50. The earliest propos ed functional, also the most widely investigated one as of today, is th e VRG functional [14-16]. Even for it there are no conclusive results for B 0 in the literature. Here we examine this functional in detail for atoms in a B field, and show the problems inherent in it. The analysis indicates that the VRG functional is not cast in a suitable form, at least for magnetized atoms. The choice of vorticity p j n as the basic variable to ensu re gauge-invariance, which is the central result of references 14-16, needs to be critically re-examined. The challenge to implementing CDFT is, so mewhat paradoxically, that the current effect is presumably small. We do not expect that the current correction within CDFT will drastically change the DFT densities. Ther efore the first question we encountered is which DFT functional should be used as a re ference for the CDFT calculations. If the variation in outcomes that re sults from different choices of DFT functionals is much larger than the CDFT corrections, which seems to be the case in many situations, the predictability of the calculation is jeopardized. Of course, there is no easy answer to this question. Indeed, DFT functionals themse lves are still undergoing improvement.

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66 Here we made the conventional DFT choice of using the LDA as the starting point. Even though not the most accurate one, the LDA is among the best understood DFT functionals. It is also easy to implement. Us ing self-consistent KS orbitals obtained from LDA calculation for the helium atom 1 s 2 p-1 state in a field B = 1 au., I plotted various quantities that are impor tant in CDFT along the z and directions in Fig. 4-6. 0 0.5 1 1.5 2 2.5 3 3.5 4 1010 108 106 104 102 100 102 104 z or (a0)n or |jp| or | | or |g 2| (au) n(0,z) (0,z) gLCH(n) 2(0,z) n( ,0) jp( ,0) ( ,0) gLCH(n) 2( ,0) FIG. 4-6 Various quantiti es (electron density n paramagnetic current density jp, vorticity and the current correction to the exchange-correlation energy density, g2, in the VRG functional) for the helium atom 1 s 2 p-1 state in B = 1 au. All quantities are evaluated from the LD A KS orbitals and plotted along the z and axes (cylindrical coordinates). Exponential decay of electron density was already seen in Fig. 4-2. Because the current density along the z axis is zero, it does not appear in Fig. 4-6. However, is not zero on that axis. On the cont rary, it diverges at the tw o poles of the atom. This

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67 divergence causes large values of 2gn the energy density corre ction within the VRG functional (recall eqns. 1.39 a nd 2.9). If the pre-factor gn does not decay fast enough to cancel this divergence, a convergence problem in the SCF solution of the CDFT KS equation will happen. Also notice that the el ectron density decays very rapidly along the z axis. At z = 3 a0, the density is already smaller than 43 010 a Remember the function gn was fitted to data points with 010sra thus in the low-density region it is not well-defined. Different fits to the same set of original data points vary considerably (refer to eqns. 1.41 through 1.46). Furthermore, even the accuracy of the original data set to be fitted is questionable at 010sra. Even were these problems to be resolved, the underlying behavior shown in Fig. 4-6 would remain. The largest co rrection relative to ordinary DFT given by the VRG functional is at the places where the electron density and the current density are both almost zero, which is obviously pecu liar if not outright unphysical. This peculiar (a nd difficult) behavior is rooted in the choice of as the basic variable in the CDFT functional. To avoid the divergence problem, we intr oduced a rapidly decaying cutoff function. Details were given in chapter 2 (also r ecall eqns. 2.15 and 2.16). Using parameters 331 0010,2cutoffcutoffnaa for the cutoff function, Ta ble 4-3 lists some of the calculated results within the VRG approximation for the fully spin-polarized states of the helium, lithium, and beryllium atoms at several selected field strengths. An estimation of the current effect is to evaluate the VRG functional using the LDA Kohn-Sham orbitals. Results for that estimation are listed in the th ird column of Table 4-3. This scheme can be thought as a non-self-consist ent post-DFT calculation. Fu lly self-consistent CDFT

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68 calculations were also accomplished when the B field is not too larg e, and they verified the LDA-based estimates. When the B field is larger than roughly 5 au., SCF convergence problems return because of the pa thological behavior of VRG functional. Table 4-3 CDFT corrections to LDA resu lts within VRG approximation (parameters 331 0010,2cutoffcutoffnaa are used for the cutoff function, E in Hartree) Atom and State B ( au ) No nSC F VRGE SC F VRGE H OMO He 1 s 2 p-1 0 -0.0022 -0.0021 0.0001 0.24 -0.0031 -0.0031 -0.0013 0.5 -0.0045 -0.0047 -0.0029 1 -0.0077 -0.0081 -0.0071 5 -0.036 10 -0.074 100 -0.81 Li 1 s 2 p-13 d-2 0 -0.0070 -0.0071 0.0002 2 -0.027 -0.029 -0.0077 5 -0.065 10 -0.129 Be 1 s 2 p-13 d-24 f-3 1 -0.025 -0.026 -0.0017 5 -0.085 10 -0.166 Putting these concerns aside, consider Tabl e 4-3. By design, the current correction given by the VRG functional is negative. It strongly depends on the particular atomic configuration. Within each c onfiguration, the VRG contribu tion increases with increasing B field. Besides total atomic energies, the eige nvalues of the highest occupied KS orbitals are also slightly lowered by including the curr ent term, but it helps li ttle in bringing the HOMO energies closer to the ionization energies This error, of course, is the well-known self-interaction problem. Because of the use of a cutoff function, these CDFT calculations can at best be thought of as semi-quantitative. This is b ecause the current corrections strongly depend

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69 on the chosen cutoff function. Use of differe nt cutoff parameters gives quite different results (see Table 4-4), an outcome which is re ally undesirable. Of c ourse, all of this is because the VRG functional does not provide a su itable form for either the low density or the high-density regions, nor do we know its correct asymptotic expression. Table 4-4 Effect of cutoff function on CD FT corrections for the helium atom 1 s 2 p-1 state in magnetic field B = 1 au. (energy in Hartree) 3 0cutoffna 1 0cutoffaNon-SCF VRGE 0.005 2.0 -0.004 0.001 2.0 -0.008 0.001 1.0 -0.010 0.0001 2.0 -0.025 0.00001 2.0 -0.064 It is unsurprising that the VRG functional fails when applied to atomic systems in a strong magnetic field. It was developed from the study of the moderately dense to dense HEG in a weak magnetic field, for which La ndau orbitals were used as approximations. This physical situation is quite different fr om a finite system. First, electron and paramagnetic current densities vary considerably within an atom, and the low density regions (010sra ) are non-negligible. Secondly, ther e is not a direct relationship between pjr and the external B field as there is for the HEG. The question whether the electron gas remains homogeneous after imposing a substantial B field is even unclear. If the field induces some form of crystallizat ion, the basic picture based on which the VRG functional proposed is completely lost. The analysis and numeri cal studies in this chapter suggest the picture of Landau or bitals used for the HEG may not be applicable at all for the atomic-like systems. Unlike the LDA, also based on the HEG, it seems that the VRG functional is too simple to encompass the essential physics of the atomic systems.

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70 A more fundamental question is whether r is a suitable basic variable in gauge-invariant CDFT as Vignale and Rasolt required [14-16]. While it is appealing from a purely theoretical perspective, our nu merical results on atomic systems in B fields suggest it is an inappropriat e choice, or at least an awkw ard one, from the application perspective. Largely due to the choice of r as the basic variable in the VRG functional, it gives unphysical results in our tests. Recently, Becke proposed a currentdependent functional to resolve the discrepa ncy of atomic DFT en ergies of different multiplicity of open-shell atoms [107]. Since this functional is based on the analysis of atomic systems, it may be more suitable for application to magnetized atoms than the VRG functional. There are signif icant technical barriers to its use. Nonetheless, we hope to investigate this functional in the future. Before attempting (sometimes in effect, guessing) better forms for the CDFT functional, we need to know some exact CD FT results to serve as touchstones for any possible proposed functional, This is the major task of the next chapter. Finally, one additional comment remains to be made about the results presented in this chapter. Relativistic effects and the effects due to finite nuclear mass are not considered. Those effects can be importa nt for matter in super-strong fields (410 B au), in which regime the adiabatic approximation will be applicable. But for the field strengths involved in this chapter, both effects should be negligible.

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71 CHAPTER 5 HOOKE’S ATOM AS AN INSTRUCTIVE MODEL In DFT, the need for accurate approxim ations to the electronic exchangecorrelation energy Exc has motivated many studies of a model system often called Hooke’s atom (HA) in the DFT literature [108-121]. The basic HA is two electrons interacting by the Coulomb potential but confin ed by a harmonic potential rather than the nuclear-electron attraction. This system is significant for DFT because, for certain values of the confining constant, exact analytical solutions for various states of the HA are known [108, 111]. For other confin ing strengths, it can be solved numerically also with correlation effects fully included [113]. Since the DFT universal functional is independent of the external potential and th e HA differs from atomic He (and isoelectronic ions) only by that potential, exact solutions of the HA allow construction of the exact Exc functional and comparative tests of approximate functionals for such twoelectron systems. Given that much less is known about the approxi mate functionals in CDFT than ordinary DFT, it would be of considerable value to the advancement of CDFT to have corresponding ex act solutions for the HA in an external magnetic field. Hooke’s Atom in Vanishing B Field There is a long history of i nvestigating this problem. The system Hamiltonian reads 2 2222 1212 1211 ˆ () 22totHrr r (5.1)

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72 where 1,2iri are the spatial coordinate s of the electrons, and the confinement frequency. Hartree atomic units are used th roughout. By introducing ce nter of mass (CM) and relative coordinates, 21 21121 () 2 R rr rrrr (5.2) (when I deal with the relative motion part, r always means 12r ), the Hamiltonian eqn. (5.1) becomes ˆˆˆtotCMrHHH (5.3) where 2221 ˆ 4CM RHR (5.4) 22211 ˆ 4rrHr r (5.5) The solution to the three-dimensional osci llator problem (5.4) can be found in any undergraduate QM textbook. It is the re lative motion Schrdinger problem, defined by eqn. (5.5), that has been tr eated variously by different aut hors. Laufer and Krieger used the numerical solution to the relative motion problem to construct the exact DFT quantities, and found that, although most a pproximate functionals generate rather accurate total energies for this model system, the corresponding approximate XC potentials are significantly in error [113]. In 1989, Kais, Herschbach, and Levine found one analytical solution to the HA relative motion problem by dimensional scaling [108]. Samanta and Ghosh obtained solutions by adding an extra linear term to the Hamiltonian [110]. Later, Taut obtained a sequence of exact solutions for certain specific confinement frequencies [111], and used them in st udies of DFT functionals [114-116].

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73 A basic observation about the HA follows fr om the Pauli principle, which requires the total wavefunction to be antisymmetric. Because the CM part is always symmetric under particle exchange 12rr if the relative motion part is symmetric (e.g. s or d –like orbitals), the spin part must be anti-symmetric, thus a spin singlet state; otherwise a spin triplet state. Thus we can concern oursel ves with the spatial relative motion problem alone. Since the Hamiltonian (5.5) is spherically symmetric, the relative motion wavefunction can be written as the product of a spherical harmonic and a radial part. The radial part is in turn decomposed in to a gaussian decaying part (ground state wavefunction of a harmonic osc illator) and a polynomial part. In some special conditions, the polynomial has only a finite number of term s, and thus the wavefunction is expressed explicitly in a closed form. Here I proceed slightly differently from the approach in reference 111. Insertion of the relative motion wavefunction 2/4 0,lrk rlmk krYrear (5.6) in ˆrrrrHrEr and a little algebra give the recursion relation 213 2230 2kkrkkklaaklEa (5.7) Suppose the polynomial part in eqn. (5.6) terminates at the n th term, e.g. 0na and 0kna The recursion relation (5.7) for k = n immediately gives ,,3 2rnlEln (5.8)

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74 Repeatedly invoking eqn. (5.7) for k = n -1, n -2, …, 0, -1, we get an expression for a-1 which by definition must be zero, in terms of Frequencies which make this expression be zero are the ones that correspond to analyt ical solutions with eigenvalues given by eqn. (5.8). Table 5-1 Confinement frequencies for HA that have analytical solutions to eqn. (5.5) (see eqn. 5.8 for their eigenvalues) n l = 0 l = 1 l = 2 l = 3 1 0.50000000000000 0.25000000000000 0.16666666666667 0.12500000000000 2 0.10000000000000 0.05555555555556 0.03846153846154 0.02941176470588 3 0.03653726559926 0.02211227585113 0.01583274147996 0.01232668925503 0.38012940106740 0.20936920563036 0.14620429555708 0.11267331074497 4 0.01734620322217 0.01122668987403 0.00827862455572 0.00655187269690 0.08096840351940 0.04778618566245 0.03423838224700 0.02675808522737 5 0.00957842801556 0.00653448467629 0.00494304416061 0.00397054409092 0.03085793692937 0.01942406484507 0.01426990657388 0.01130595881607 0.31326733875878 0.18237478381198 0.13126853074700 0.10313090450042 6 0.00584170375528 0.00415579376716 0.00321380796521 0.00261635006133 0.01507863770249 0.01002629075547 0.00753956664388 0.00605214259197 0.06897467166559 0.04231138533718 0.03104720074689 0.02465640755471 7 0.00382334430066 0.00281378975218 0.00221804190308 0.00182765149628 0.00849974006449 0.00591291799966 0.00454144170886 0.00369051896040 0.02696238772621 0.01743843557070 0.01305357779085 0.01048109720372 0.26957696177107 0.16282427466688 0.11975836716865 0.09546288545482 8 0.00263809218012 0.00199650951781 0.00160027228709 0.00133306668779 0.00526419387919 0.00380045768734 0.00297472273003 0.00244506933776 0.01342801519820 0.00910586669888 0.00694983975395 0.00564110607268 0.06058986425144 0.03819659970201 0.02852875778009 0.02293923879074 9 0.00189655882218 0.00146924333165 0.00119499503998 0.00100531643239 0.00348659634110 0.00259554123244 0.00206608391617 0.00171616534853 0.00767969351968 0.00542189229787 0.00421416844040 0.00345660786555 0.02409197815100 0.01589809508448 0.01207319134740 0.00979689471802 0.23835310967398 0.14785508696009 0.11054467575052 0.08912851417778 10 0.00140897933719 0.00111335083551 0.00091728359398 0.00077860389150 0.00242861494144 0.00185491303393 0.00149877344939 0.00125702179525 0.00481042669358 0.00351289521069 0.00277638397339 0.00230004951343 0.01216213038015 0.00837269574743 0.00646564069324 0.00529550074157 0.05433349965263 0.03496458370680 0.02647749580751 0.02150263695791

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75 An example may be helpful. Consider l = 0, and n = 3. According to the previously prescribed procedure, we get 9 2rE 3 2a a 3 23 1 2112 12 22 a aa a 3 12 0 3124 6 36 a aa a 2 3 01 1 413072 2 424 a aa a To ensure that the last expression vanishes requires that the confinement frequency be 517 24 or = 0.3801294, 0.0365373. The solution corresponding to the smaller frequency turns out to be a ground state, wh ile the other one is an excited state. Confinement frequencies correspondi ng to analytical solutions for l = 0, 1, 2, 3 and n 10 are compiled in Table 5-1. This tabu lation includes more angular momentum quantum numbers and more signifi cant figures than that presented by Taut [111]. For n 3, there are several solutions. The sma llest frequency corresponds to a ground state, the others are for excited states. Hooke’s Atom in B Field, Analytical Solution When the HA is placed in an external ma gnetic field, its lateral confinement can exceed its vertical confinement. It is we ll known that the magnetic field can greatly complicate the motion of a columbic system. Ev en for the one-electr on system (H atom), substantial effort is required to get highly accurate results in a B field [79-82]. Only recently have calculations on the He atom in a high field been pushed beyond the HF approximation [11-13]. As far as I know, no exact solutions ar e reported in the literature for the 3D Hooke’s atom in an external ma gnetic field. Taut on ly gave analytical solutions for a 2D HA in a perpendicular B field [117]. Here I present the exact analytical solutions to the magnetized HA [122 ]. When the nuclear attraction in the He

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76 atom is replaced by a harmonic potential, our exact analytical results can serve as a stringent check on the accuracy of the co rrelated calculations just mentioned. With an external magnetic field chosen along the z axis, the system Hamiltonian becomes 2 2222 12 1212 1211 ˆˆ (())(())() 22tot spinHArArrrH iir (5.9) where 12ˆspinzzHssB is the spin part of the Hamiltonian, 1,2izsi are the z components of the spin, and ) ( r A is the external vector potential. A similar separation of the CM and the relative motion part s is done as in the case of the B = 0 HA: ˆˆˆˆtotCMrspinHHHH (5.10) 22222222211 ˆ (2())(4sin2) 44R CM RHARRRBRMB i (5.11) 2222222221111111 ˆ (())sin 244162r rrHArrrBrmB irr 222221 2rzm zB r (5.12) Here 22, 4162zB ; (5.13) 22, x y m and M are magnetic quantum numbers fo r the relative and CM motion parts. In these expressions, th e Coulomb gauge has been chosen. r B r A 2 1 ) ( (5.14) The solution for the CM part is (un-normalized) ) exp( ) 1 ( ) 2 ( ) 2 exp( ) (2 1 1 2 2 iM M N F Z H Z RL N M L CMZ (5.15)

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77 with eigenvalue of 1 1 () 222CMZLM M ENBN (5.16) where 2 24 sin B RL ZNH is ZNth order Hermite polynomial. 1 1F is the confluent hypergeometric function (o r Kummer function) [123]. ,0,1,2,ZNN are the quantum numbers. The relative motion eignvalue problem from eqn. (5.12) generally cannot be solved analytically in either spheri cal or cylindrical coordinates. The difficulty of solving the Schrdinger equation that correspon ds to eqn. (5.12) lies in the different symmetries of the confining potential (cylin drical) and electron-electron interaction term (spherical). Since the effective potential in eqn. (5.12) r z r Vz12 2 2 2 is expressed as a combination of cylindrical coordinate variables z, and the spherical coordinate variable r it proves convenient also to express the relative-mo tion wavefunction in those combined, redundant variables ,,,,rrrzrz In part motivated by the expected asymptotic behavi or, we choose the form 2222,z zz m im rrezurze (5.17) where 0 z for even z parity, 1 for odd z parity. Then ˆ 0rrrHEr yields 222 22 22221 12(,)0z zzzz mzrzEurz rzrzrrrzzr (5.18) where 2121 2rzzm EEBm (5.19)

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78 To avoid messy notation, no quantum numbers are appended to E. This differential equation is not easy to solve either. To pr oceed, we make a direct, double power-series expansion 0 ,) (z r z r z rn n n n n nz r A z r u (5.20) to transform eqn. (5.18) into a recurrence relation, 2,22,1,22232rz rzrzrznnrrzznnnnnAnnmnAA ,,22120rzrzrzznnzzzznnnnEAnnA (5.21) where 0,j iA for 0 i, or 0 j or 1 2 k j We seek values of ,,zE for which the right side of eqn. (5.20) terminates at finite order. Assume the highest power of z that appears is zN, 0, zN j iA where zN is an even number. For 2 zN, generally there is no solution to the set of equations that follow from eqn. (5.21). However, a judicious choice, 2,2zzzEN (5.22) allows us to set 0,j iA for zN j i 2, since there are 1 2zNrecurrence relations of eqn. (5.21) with z z rN n n 2 that then are satisfied automatically. This is the special case, z2, which corresponds to imposition of an external field B upon a HA with magnitude 3 2 B (5.23) Now we find values of z that correspond to analytical solutions. Repeated application of eqn. (5.21) for each combination of 12, 2z rN n

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79 2 1 2 z r zn n N, allows us to express all the coefficients 0,02 2z zN ijNiA in terms of zN jA 0 0. Invoking eqn. (5.21) for 0zn 11 2z rN n gives 1 2zN homogenous linear equations involving 0,0zjNA. To have non-trivial solutions, the determinant of this set of equations must be zero, a requirement which reduces to finding the roots of a polynomial equation in z Energy eigenvalues can be easily found by substituting eqns. (5.22), (5. 23) and (5.13) into (5.19), 5 23 2rzzENmm (5.24) Here I give an explicit example for 0zm ,6zN Its relative motion energy is 5 617 2rzE First we set all the coefficients 0, j iA for 26 ij The four equations derived from eqn. (5.21) are already satisfied for ,rznn = (3,0), (2,2), (1,4), and (0,6). Repeatedly invoking the recursion rela tion (5.21) for 1rn ,6,4,2zn we find A1,4, A1,2, and A1,0, expressed in terms of A0,2, A0,4, and A0,6. 0,6 1,4, 2zA A 0,41,40,40,6 1,2 2105 22z zzAAAA A 2 1,20,20,20,40,6 1,0 36315 22zz zzAAAAA A ; Use eqn. (5.21) again twice for 0rn ,4,2zn

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80 0,41,40,6 2,20,60,4 2430 160 48z z zzAAA AAA 2,21,20,20,40,40,6 2,0 0,2 314812241(42017) 2 416zzzz zzAAAAAA AA ; Employ eqn. (5.21) one more time for 1rn ,2zn to obtain 1,22,21,40,4 3,0 0,6 2412 128 6248z z zzzAAAA AA Now, all the coefficients are expressed in terms of A0,2, A0,4, and A0,6. The next step is to apply eqn. (5.21) for0zn ,1,0,1,2rn We have a set of four homogeneous equations, 0,01,020 AA 1,00,02,00,212620zAAAA 1,03,02,01,281220zAAAA 2,03,02,2420zAAA Substitute the expressions for 1,02,03,01,22,2,,,, A AAAA and rearrange, 32 0,00,20,40,6 31 3150zzz zAAAA 42 0,00,20,40,6 31 96412063411112600 8zzzzzz zAAAA 3 0,20,40,6 31 9621140371160 16zzzz zAAA 422 0,20,40,6 31 38448116443200 48zzzz zAAA To have non-trivial solutions for A0,0, A0,2, A0,4, and A0,6, the determinant for their coefficients must be zero. This requirem ent is equivalent to the polynomial equation

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81 643282061158419465605025642010zzzzz There is a standard procedure for solving the fourth power polynomial equation [123]. Here I give the nonzero solutions to the above equation 1/2 1,2,3 2 1,2,31,2,357433337575 35812423 34506 14345 81234762704184032zx xx where 1/2 12347591563390234759235812423 coscos1 402573120818808133704288112ixi Numerical evaluation gives z 0.0584428577856519844381713514636827195996701651, (third excitation) 0.0230491519033815661266886064985880747559374948, (second excitation) 0.0040457351480954861583832529737697350502295354, (ground state) 0.0089023525372406381159151962974840750107860006. (first excitation) The smallest frequency corresponds to a ground state, others correspond to excited states. Remember those states are not for th e same confinement strength, hence not the same physical system. Table 5-2 lists all the frequencies that correspond to closed-form analytical solutions for m = 0, 1, 2 and 2,4,6,8,10zN including both positive and negative z parities. For each frequency found in the previous step, the corresponding eigenvector 0,0zjNA determines the vector of all the coefficients 0,02 2z zN ijNiA Table 5-3 gives explicitly some of the solutions to eqn. (5.12).

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82 Table 5-2 Confinement frequenciesz which have analytical so lutions to eqn. (5.12) ( /2,23zB see eqn. 5.24 for their eigenvalues) z zN state m = 0 m = 1 m = 2 0 2 g 0.08333333333333(1) 0.05000000000000 0.03571428571429 4 g 0.01337996093554(5) 0.01000000000000 0.00778702514725 e 0.03958614075938 0.02500000000000(4) 0.01938688789623 6 g 0.00404573514810 0.00343014626071 0.00291641684372 e 0.00890235253724 0.00606707008623 0.00468524599259 e 0.02304915190338 0.01711506721549 0.01417651105557 e 0.05844285778565 0.03965895252427 0.03017655861841 8 g 0.00169910717517 0.00151575652301 0.00135563312154 e 0.00326661504755 0.00253598704899 0.00203265499562 e 0.00563875253622 0.00412145134063 0.00339412325617 e 0.01040942255739 0.00824400136608 0.00670121318725 e 0.01653602158660 0.01329159554766 0.01137236016318 e 0.03180329263192 0.02138083910939 0.01706956908545 10 g 0.00086575722262 0.00079244596296 0.00072696957550 e 0.00147907803884 0.00126081972751 0.00107688670338 e 0.00241664809384 0.00181507622152 0.00147682860631 e 0.00334850037594 0.00292405665076 0.00255098333227 e 0.00397594182269 0.00318249275883 0.00273000792956 e 0.00736429574821 0.00526294300755 0.00417905341741 e 0.01303363861242 0.01091689148853 0.00954248742578 e 0.01973176390413 0.01510846119750 0.01275915629739 e 0.04608252623918 0.03333142555505 0.02636159496028 1 2 g 0.03571428571429 0.02777777777778(2) 0.02272727272727(3) 4 g 0.00707894326171 0.00591390023109 0.00504676075803 e 0.02581579358039 0.01964372058675 0.01616733845868 6 g 0.00254580870206 0.00223524705023 0.00197793631977 e 0.00563638108171 0.00451759211533 0.00379720216639 e 0.01883419465413 0.01489720978500 0.01259718274291 e 0.02725999959457 0.02265588831769 0.01934687713121 8 g 0.00119038526800 0.00107801657618 0.00097964051831 e 0.00217041875920 0.00183191437804 0.00158115925765 e 0.00436490208339 0.00352808023992 0.00300865314483 e 0.00573812529618 0.00498357626059 0.00437993535596 e 0.01453566471300 0.01198041559842 0.01037221561899 e 0.02126492440705 0.01697086960871 0.01435335804110 10 g 0.00064917903905 0.00059949900156 0.00055475656842 e 0.00105825552161 0.00093052390433 0.00082578168452 e 0.00178623911003 0.00148750195578 0.00128288767589 e 0.00215173389288 0.00193434188304 0.00174595176146 e 0.00344914023386 0.00286801805941 0.00249502584282 e 0.00477115358052 0.00395655886157 0.00340206703840 e 0.01179464597863 0.01002739489760 0.00884019961249 e 0.01632356905207 0.01328778382152 0.01143529433090 e 0.02247241353174 0.01936642452391 0.01698248977049 (a) g = ground state; e = excited state; (b) Numbers in parentheses are the listing number in Table 5-3.

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83 104 103 102 101 100 101 104 103 102 101 100 101 z 1D 3D HA in B, =2 z 3D HA, B=0, = z 3D QD in B, = z/2 2D FIG. 5-1 Confinement strengths subject to an alytical solution to eqn. (5.12). For 1D, has been shifted to 1 For 2D, z has been shifted to 1z Hexagon, square, up-triangle, diamond, dow n-triangle, circle, left-triangle, plus, right-triangle, and x-mark stand for the highest order of z in 1D, in 2D, r in the 3D spherical HA t = ( r z ) / 2 in QD, in the polynomial part of the relative motion wavefunctions bei ng 1,2,…,10, respectively. For HA in a B field, they stand for zzN Black, blue, and red symbols are for m = 0, 1, 2, respectively. For spherical HA, only 0z is included. Notice its odd parity 1,zmand even parity states 0,1zm are degenerate. For another case of /2z which can be thought as a two-electron quantum dot (QD) in a suitable magnetic field, one can also find analytical solu tions to eqn. (5.12) for some specific confinement frequencies. Together with two lim iting cases of 1D and 2D, they are summarized in reference 122. Fi gure 5-1 shows those frequencies subject to analytical solutions to the electron relative motion part.

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84 Table 5-3 Some solutions to eqn. (5.12) for confinement potential 2,23zB # Relative Motion Wavefunction 1 6 / 1 12 / 2 / 12 24 / 22 2z r ez 2 18 / 1 222/72 21/6/108z iezrze 3 22 / 1 222/88 2221/8/176z iezrze 4 20 / 1 222/80 22241/4/40/80/160/3200z ierzrzze 5 472 17 3 25 4 2 2 2 4 / 211328 314 11 8 18 1 24 2 1 48 22 1 2 12 2z z r rz r ez Hooke’s Atom in B Field, Numerical Solution For arbitrary and B values, eqn. (5.12) does not have an analytical solution. To have a clear picture for the dependence of the HA system behavior upon increasing or B field, more data points are essential. Expansion of the wavefunction in terms of spherical harmonics is satisfactory when the B field is not too large. For large B values, Landau orbitals are used for expansion. Consider the low-field expansion first. Spherical expansion: l lm lm m rY r f r r ) ( ) ( 1 ) ( (5.25) Insertion of the foregoing expansion together with the Hamiltonian (5.12) in the relative motion Schrdinger equation gives a set of coupled differential equations, l m l m l l eff lm lm rr f r V r f dr d r f E 0 ) ( ) ( ) ( ) (2 2 ( l = 0, 2, 4,… or l = 1, 3, 5,…) (5.26) where the effective potential is

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85 m l lm l l l l r B r B r B m r r l l Y r B r B m r r l l Y r Vl l m l lm m l l eff | 20 0 | 20 0 1 2 1 2 24 24 4 2 1 ) 1 ( | sin 16 4 2 1 ) 1 ( | ) (2 2 2 2 2 2 2 2 2 2 2 2 2 (5.27) and m l m l lm | is a Clebsch-Gordon coefficient. This procedure is very similar to Ruder et al .’s method for treating the hydrogen atom in strong magnetic fields[78]. The numerical solver for eqn. (5.26) was obtai ned from reference 124, with appropriate modifications made to adapt it to this problem. Next turn to the strong field case, whic h requires a cylindrical expansion. The expansion used is: Cylindrical expansion: n Lan nm nm m rz g r ) ( ) ( ) ( (5.28) n m n m n n eff nm nm rz g z V z g dz d z g E 0 ) ( ) ( ) ( ) (2 2 ( n = 0, 1, 2, …) (5.29) where the effective potential 1 2 2 2 4 ) ( | 1 | ) ( ) (2 2m n B m z r z VL n n Lan m n Lan nm m n n eff (5.30) and 4 4 42 2BL L Calculating the effective potential ) ( z Vm n n eff is not trivial. I followed Prschel et al. ’s scheme [125]. Details are included in appendix D. By use of a similar argument as in reference 90, we can screen out the configurations pertaining to the HA global ground states in B field, which are ( m = 0, 0z ), ( m = -1, 0z ), and ( m = -3, 0z ). Energies for the relative motion and spin

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86 parts are compiled in Table E-1 and E.2 for two angular frequencies, = 0.5, 0.1, respectively. States are labeled by their conserved quantities as 21 z Svm where (2 S +1) is the spin multiplicity, and is the degree of excitation w ithin a given subspace. Their field-free notations are also included (e.g. 1 s 2 p …). The larger confinement frequency corresponds to the first analytical solution found by Kais, Herschbach, and Levine [108], and is also the most widely studied one. The smaller frequenc y has two analytical solutions, one for the 1 s state in B = 0 and another for the 2 p-1 state in 3/5 B A sixteen spherical function expans ion gives the relative motion energy of the latter state to be 0.4767949192445 Hartree, pleasingly accurate co mpared to the analytical result 55 232023*0.10.47679491924311 22rzzENmm For Tables E.1 and E.2, numbers in pa rentheses denote the number of radial functions used in expansion (5.25); numb ers in brackets are the number of Landau orbitals used in eqn. (5.2 8). It is easy to see that, in the low field regime ( B < 1 au.), the spherical expansion outperforms the cylindrical expansion. However, its quality degrades as the B field increases. As Jones et al. have found and as is phys ically obvious, the high field regime is very demanding for a spheri cal basis [93]. Note in Table E-2, for B = 10 au., the spherical expans ion corresponds to 49 48max l Clearly, it cannot go much further on practical grounds. The analytical solution for the singlet state of 10 1 in vanishing B field is 22 401 ()1 220 10240615r rrr re (5.31)

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87 with eigenvalue 20 7 rE ( 10 1 m = 0, B = 0, singlet) Its density distribution is 2 2423 2 51022190287535 ()4510 51005 10 10240615rrrrrr nrrererfe r (5.32) For very small B field, the diamagnetic term in eqn. (5.12) can be treated as a perturbation. Thus the first order correction 22 2 (1)22242 08 (0)()sin()2.228 16163rrrBB EBrrdrrrdrB Compared with the second column in Table E-2, this diamagnetic correction term is quite accurate up to B = 0.1 au. In super-strong fields ( B >> 1), one expects the adiabatic approximation to be applicable. The electron-electron interaction te rm is treated as a perturbation. By omitting the r 1 term in eqn. (5.12), the resulting Hamilt onian looks very similar to eqn.(5.11), and an analytical unperturbed solution exists 22(0) 2 42 11lim,1, 2L zz m im rnL BeHzFnme (5.33) B m m n n B EL z r B2 4 2 1 2 1 ) ( lim) 0 ( (5.34) For the three configurations included in Table E-2, 0 n nz, the preceding two equations are reduce to 22(0) 42limLz m im r Bee (5.33')

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88 (0)lim()21 22rL Bm EBmB 22221 44 222 m m B BB m (5.34') 21 22 B m B As can be seen in Table E-2, the first term 2 B in eqn. (5.34') dominates all other energy components in the high field limit. Th e first two terms comprise the zero point energy of the magnetized oscillator. One might attempt a perturbative ca lculation for the interacting term r 1 using the wavefunction (5.33'). The result is really disappointing. Choose the singlet state ( m = 0, 10 1 B = 1000), and evaluate th e expectation value of r 1 We get 1.25 Hartree, which is far off from the desired value of ~0.3. The immediate res ponse is to go to the second order, but soon we will see that the result does not get any better. From eqn. (5.34) we can see that the most important states involved are 2,4,6,zn 782 0 487 2 4 4 | 1 | 0 2 2 | 1 | 02 2 ) 2 ( z z z z rn r n n r n E A standard perturbative calc ulation does not perform very well. This is because the most significant contribution of the term r 1 is from the small r region, exactly the place where the B field can least affect the wavefunction. If we “graft” the correlation part in

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89 eqn. (5.31) to eqn. (5.33'), and use this guessed wavefunction 222 421 220Lzrr e to evaluate the expectation value of r 1 term, we get 0.365 Hartree, which is not too bad. This is one demonstration that the B field does not affect the short-distance electron interaction too much. Phase Diagram for Hooke’s Atom in B Field As with what we have seen in the previous chapter for real atoms, application of an external magnetic field also causes a series of configuration changes for HA. For the helium atom, there is only one transition from 1 s2 state to 1 s 2 p-1 state, but there are more configurations involved for the HA. To see a complete picture of the evolution of this model system with increasing B field, the spin energy should also be considered (recall Table 5-4 Field strengths for c onfiguration changes for the gro und states of Hooke’s atom 1c B (au) 2c B ** (au) 0.001896558822 0.0002047 0.00177 0.0038233443 0.000522 0.00445 0.01 0.00187 0.01625 0.02 0.00466 0.04172 0.05 0.01523 0.14596 0.1 0.03642 0.3785 0.2 0.0849 0.987 0.5 0.2493 3.54 1 0.547 9.36 2 1.172 24.91 4 2.47 67.05 10 6.48 247.1 20 13.29 40 27.06 100 68.74 For the transition from singlet state (m = 0) to triplet state (m = -1); ** For the transition within triplet state, from m = -1 configuration to the maximum density droplet (MDD) state, which has m = -3.

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90 eqn. 5.10). However, the CM motion part can be neglected, since it is the same for all the states that could possibly be the ground state. For vanishing B field, the ground state of a HA is a singlet state ( one electron spin up, the other spin down). With increasing B field, sooner or later the spin down electron will be flipped, and a triplet state becomes the ground state. Further increase of the field will cause configuration transitions within triplet states, and the m value becomes more and more negative. Table 5-4 lists the critical field strengths for the first two transitions. The ground state phase diagram in the B plane is shown in Fig. 5-2. For the confinement frequency 0.1 considered in the prev ious section, the first transition occurs at 0.036 B accompanied by a spin flip (singlet to triplet). The m = -1 configuration remains as the ground state until B is increased to 0.38 au., then the m = -3 state takes over. This establishes that the m = -1 configuration is the global ground state at 3 0.34641 5 B au. 103 102 101 100 101 102 104 103 102 101 100 101 102 103 B (au)Singlet state Triplet state m = 0 m = 1 m = 3 FIG. 5-2 Phase diagram for the Hooke’s atom in B fields

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91 Electron Density and Paramagnetic Current Density With the wavefunction of the Hooke’s Atom we can easily get its electron density distribution by integrat ing out one variable. 2()2 2CMrr nrrrdr (5.35) This 3D integral can be reduced to a 2D inte gral. First we express the CM motion part as the product of a Landau orbi tal and a function about Z ,Lan CMNMNMRgZP (5.36) Next notice that 2rr does not depend on Thus (), nrnz 2 1/2 2 222 004/4cos,,, 2Lan NMNMrz dgzzdzd (5.37) where is the azimuthal angle of 2 r r but it is a dummy variable, thus not need to be evaluated. Another dummy variable is in 2rr They facilitate the separation of integrals d and dz Otherwise, the separation in eqn. (5.37) is not possible. Recall from Chap. 1 that the paramagnetic current density pjr is the expectation value of the op pJr operator ** 121212()op pCMrpCMr j rRrJrRrdRdr (5.38) where 1 ˆˆˆˆ ()()()()() 2op pJrrrrr i (5.39)

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92 Here the relative coordinate is explicitly labeled as 1221rrr to avoid confusion, and ˆ () r and ˆ () r are second quantized field creation and annihilation operators. After substitution we reach the following expression ˆ (),prpjrjz 2 1/2 2 222 00ˆ 4/4cos,cos,, 2Lan rNM NMrz mdgzzdzd (5.40) The integrals in eqns (5.37) and (5.40) are evaluated numerically, using a 20-point Gauss-Legendre integral for d and a 40-point Gauss-Lague rre quadrature integral for z d andd [126]. Next I use the ground state of HA with confinement frequency 1 10 in 3 5 B for example, which has an analytical wavefuncti on. Its CM part can be easily obtained from eqn. (5.15) and eqn. (5.16), 22105 3/42 () 5Z CMRe with eigenvalue 4 1 CME (5.41) Its relative motion part is 222 2 40()1 440z i rrz rAee (5.42) where the normalization factor is 2 / 1 18 3 tanh Re 2 55 300 5 114 4 25 A. 63 0075692273 0

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93 with eigenvalue of 20 3 2 13 rE For Hooke’s atom, the total electron number N = 2. The pair density for this example is 2 21 4rrnrrd N 2 22 2/2032/1070 241035562120 32 20rrAir rreirrrerfe r (5.43) where () erfx is the error function. One can integrate this expression to make a check, 2 2 01 4 2 nrrdr, just as expected, since 2111 4 42rrdr NN Its electron density is 2()2 2CMrr nrrrdr 2 2 22 22 2 2 22 2 22 cos 54 2 52010 3/2 0 04 1 440 5z z zz Az eedzeedd (5.44) and paramagnetic current density 2 2 22 22 2 2 22 2 22 cos 54 2 52010 3/2 0 04 ˆ ()1cos 440 5z z z prz Az j reedzeedd (5.45) Their distributions are shown in Fig. 5-3. The difficulty involved in using pj n as a CDFT parameter is evident from studying that figure.

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94 FIG. 5-3 Cross-sectional vi ew of the electron density nz (upper) and paramagnetic current density ,pjz (bottom) for the ground state HA with = 1/10 in 3 5 B The B field orientation is in the plane of the paper from bottom to top.

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95 Their large r limits are (large r means that both z and become arbitrarily large): 22242 1052 lim()5 400z znrAze (5.46) 2224 1052 ˆ lim() 400z pr zjrAze (5.47) Now we are ready to construct exact KS orbitals for this state of the HA. Construction of Kohn-Sham Orbitals from Densities The Hooke’s atom singlet ground state can be inverted easily to obtain KS orbitals [113], since the spatial parts ar e the same for two KS particle s. Each of them contributes one half of the total density, thus the KS orbita l is just the square root of one half of the total density. (for KS orbitals, r always means 1r or 2r ) Singlet state ( m = 0): 12() ()() 2KSKSnr rr (5.48) However, we need to use a current-carrying ground state to study the paramagnetic current effect within the fr amework of CDFT. Those are tr iplet states. According to CDFT, the KS system must ge nerate the exact electron dens ity and paramagnetic current density. A choice of triplet KS orbitals that fulfills this requirement for this two-electron system is as follows: Triplet state ( m = -1): 2 112()()()()KSKSKSrrnrr 22()()(,)KSiKSi prerejz (5.49)

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96 I did this constructi on independently before becoming aware of a similar procedure having been used in reference 127. Their model is a 2D quantum dot, whereas my model is the 3D magnetized HA. An issue not addresse d in reference 127 is that the orbitals so constructed should be properly normalized, othe rwise the construction is invalid (or at least, incomplete). Numerical checking proves that the constructed KS orbitals in eqn. (5.49) indeed are normalized. For the special analytical solution to the ground state of 5 3 10 1 B, one can demonstrate this rigorously as follows. With the expression (5.45) for()pjr let us rearrange the integrals, 2 2,KS prdrjzdr 22 2 10 12 3/2 02 22 5 A I Ied where 2 22 /2 22 2 520 11 440zz zz z I eedzdz 222 /4cos 2 2 5 2 00cos I edd To evaluate I2, consider the following two identities, 2222 cos/10/10 3 0cos210 1cos1 10 Ierfede 2222cos 2 22 51010 4 0cos51 15cos1cos4101 22 10 Ierfedee Thus, 2 22cos 2 22 10 10 2 0510cos cos5cos1cos 816 10 Ieerfed

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97 210 43510 10 816 eII 5 4 The integral dz is easy, 2/2 55zzedz Finally, 222 2 2 22 2010 2 021 440z KSrz rdrAddze 21rrdr By construction, 1()KSr is also properly normalized provided 2()KSr is normalized. The large r limits for the two KS orbitals in the previous example are, 221/4 2 2010 125 lim() 20z KS zrAze (5.50) 221/4 2 2010 22 lim() 20z K Si zrAzee (5.51) Exact DFT/CDFT Energy Components and Exchange-correlation Potentials Each of the energy components can be cal culated straightforwardly according to their definitions. Recall discussion in Chapte r 1, the kinetic energy for the KS system, Ts, is the expectation value of th e kinetic energy operator with re spect to the KS orbitals. The difference between Ttot and Ts, namely Tc, is the kinetic energy contribution to the DFT correlation energy. The exact exchange ener gy is calculated from the single-determinant (i.e. HF) exchange formula but with the KS orbitals. J is the classical el ectrostatic energy, and Ec the DFT correlation energy.

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98 22 12112222 K SKSKSKS sTtt (5.52) 22 24 4ctotsrrsB TTTT (5.53) ** 2 1212 12 ,1 121 2xKSKSKSKS ijji exact ijrrrr Edrdr r (5.54) 1()() 2 nrnr Jdrdr rr (5.55) If we define 2xxr exactexactexact ceer VEJEdrJE r (5.56) then ccexactexact cEVT (5.57) exactexactexact xcxcEEE (5.58) Notice that for a spin singlet state ( m = 0), the exchange ener gy cancels the electron selfinteraction exactly, and 2xexactJ E Various energy components for the HA without B field and in a B field for the two confinement frequencies = 1/2, 1/10 are listed in Tables E.3 through E.7. Other energy components can be obtained from eqns. (5.52) through (5.58). As already noted, only result s for the HA in a vanishing B field are found in the literature. The most widely studied example is = 1/2 [109, 112, 118, 119]. Another frequency = 0.0019, which also has an analytical solution (refer to Table 5-1 for the more accurate value corresponding to l = 0, n = 9), was used as a strong-correlation example [114, 115, 120]. Several other fre quencies which do not have analytical solutions were investigated from numerical so lutions [113]. Cioslows ki and Pernal even

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99 did Pad approximants for various quantities in a large range of frequencies [121]. The calculated data in appendix E agree with those given in the lit erature quite well. 103 102 101 100 101 102 0 2 4 6 8 10 12 Energy components in unit of Er/ Ex/ Eee/ (A) B = 0 103 102 101 100 101 102 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Energy components in percentage KE/EtotVE/EtotEee/Etot (B) B = 0 FIG. 5-4 Energy components of HA with B = 0. Curves are guides to the eye. In the upper panel, the relati ve motion total energy Er ( ), negative of the exchange energy Ex ( o ), and two-electron interaction energy Eee ( ) are in units of In panel B, the total kinetic KE ( ), potential VE ( o ), and two-electron Eee ( ) energies are shown as percentages of the total energy.

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100 Figure 5-4 shows the change in energy cont ributions with respect to confinement frequency without a magnetic field. In the fi rst panel, the energy is divided by The horizontal dashed line is three halves, which is the energy of a 3D harmonic oscillator, and also the relative motion energy for a non-interacting HA. The blue curve is the relative motion total energy. Clearly it goes to the non-interacting limit as goes to infinity. The dash dotted curve is the negati ve of the exchange energy. The dotted curve shows the total electron-electr on interaction energy. The difference between them is the static part of the correlation energy, whose percentage contri bution to the total energy is significant for very small but negligible in the largelimit. Because of this, the smallregion is also referred to as the strong-correlation regime, while the largeregion is the weak-correlation regime. This trend is also obvious in panel (B), which displays the percentages of kinetic, external potential, and electro n-electron interaction energies in the total energy. Again, both kine tic and potential energies tend to their noninteracting limit, 50%, when goes to infinity. There are only a few papers on the nume rical construction of CDFT exchangecorrelation vector potential from a supplied density and paramagnetic current density. Besides the work in reference 127 on a 2D QD, Lee and Handy constructed exchangeonly vector potentials for 3D systems from HF reference densities by introducing Lagrange multipliers [128]. Again they got invo lved in the insufficient basis set problem. In the previous section we have already constructed exact KS orbitals, hence we can invert the CDFT Kohn-Sham eqns. (2.6) an d (2.7) to obtain the exact XC potentials for the HA. Since the spin singlet state ( m = 0) does not carry a current density, there is no CDFT correction to this state. Consider the inversion of the spin triplet state with

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101 m = -1. Notice that 1()KSr is an s -like orbital and ˆ (),xcxcArAz thus the s -like orbital does not “feel” the vect or XC potential, that is 11 ()0KS xcAr i The KS equations for the two KS orbitals therefore are 2 111()()()()() 22cdftKSKS extHxcB rrrrr 2 2221 ()()()()()() 2cdftKSKS extHxcxcrrrBArrr i where 22 222()sin 28extB rrr Hence, 2 1 1 1() 1 ()()() 2()2KS cdft xcextH KSr B rrr r (5.59) 22 21 21 21() ()() 1 ˆ 22()()KSKS xc r KSKSAr rr B rr (5.60) As Laufer and Krieger pointed out, KS eigenvalues can be found by exploring the larger limiting behaviors of the KS orbitals [113]. We again use the example of the triplet ground state m = -1, = 1/10 in a field 3 5 B Use eqns. (5.50) and (5.51) for their larger expressions, 2 1 1 11 ()() 923 22 lim ()20KS ext KS zB rr r 2 2 2 21 ()() 1343 2 lim ()20KS ext KS zrr B r

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102 It is interesting to observe that 2 is just as same as the relative motion energy (excluding the spin part), like Laufer and Krieger’s conclusion for the field-free case [113]. Besides, 1 equals the relative motion energy (e xcluding Zeeman energies) for the non-interacting HA with same parameters 5 3 10 1 B, e.g. by omitting r 1 repulsion term. Comparison of Exact and Approximate Functionals There are two ways to compare the exact KS results obtained in the previous sections with various approximate functionals The first is to evaluate the approximate functionals using exact densitie s. The other is to use self-c onsistent orbitals. Here we choose the former method. Exact and approxima te exchange and correlation energies are compiled in Tables E-8 through E-12. Excha nge-correlation energies evaluated at the SCF densities are found to be clos e to those using exact densities. The chosen functionals include the wide ly used LDA approximation [69], as well as the gradient-dependent PBE [70], and BL YP [72-75]. A recently proposed currentcorrected functional, jPBE [32], is also included. As already emphasized there are far fewer CDFT functionals th an DFT functionals. For practical purposes, the only one availabl e is the VRG local approximation already discussed [14-16]. As shown in previous chapte rs, this function is rather ill-behaved and requires introduction of a cutoff function. Disapp ointingly, the curren t correction term is rather sensitive to the decay ra te of that cutoff function. Ne vertheless, this is the only functional realistically available, so I tested it.

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103 103 102 101 100 101 102 0.8 0.85 0.9 0.95 1 1.05 Ex(app.)/Ex(exact) LDA PBE B88 (A) Ex for HA with B = 0 103 102 101 100 101 102 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Ec (Hartree) Exact LDA PBE LYP (B) Ec for HA with B = 0 103 102 101 100 101 102 0.85 0.9 0.95 1 1.05 1.1 Exc(app.)/Exc(exact) LDA PBE BLYP (C) Exc for HA with B = 0 FIG. 5-5 Comparison of exact and approximate XC functionals for the HA with different confinement frequency in vanishing B field ( B = 0). All energy values are evaluated at the exact densities. Excha nge and XC are expressed as ratios to the exact ones. Correlation uses absolute values.

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104 Look at the field-free case first. Since it is a singlet stat e, there is no CDFT current correction. Exact and approximate exchange, correlation, and XC energies are compared in Fig. 5-5. The exchange and XC panels us e the ratio of approxima te energies to the exact ones. It is well known that LDA unde restimates exchange. In this example LDA misses about 15% of the exchange energy, while GGA omits less than 5%. However, in the strong-correlation regime, the GGA can slightly overestimate Ex. In the correlation panel, an absolute ener gy scale was used. The blue curve presents the exact correlation energy. We can see that PBE slightly overestimates and LYP underestimates correlation energy, but both ar e more accurate than LDA. For the sum of exchange and correlation, both PBE and BLYP give quite accurate results in the weakcorrelation regime, but there is little impr ovement over LDA in the strong-correlation regime. Now turn to the non-vanishing B field cases. Figures 5-6 a nd 5-7 show the effect of the B field on Exc for HA with = 1/2, 1/10, respectively. The left hand panels are for singlet states; right hand ones for triplet st ates. From top to bottom, the panels are exchange, correlation, and XC energies. We see that as the B field increases, GGAs tend to overestimate exchange. For correlati on energy, both LDA and GGA give results almost independent of B field, hence are incapable of including the field effect. Apparently because of error cancellati on, GGAs can still predict an accurate Exc for the singlet state, but overestimate Exc significantly for the triplet state at large B field.

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105 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.9 0.8 0.7 0.6 0.5 0.4 B (au)Ex (Hartree) Exact LDA PBE B88 (A) Ex for HA in B ( =1/2, singlet state) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.9 0.8 0.7 0.6 0.5 B (au)Ex (Hartree) Exact LDA PBE B88 (D) Ex for HA in B ( =1/2, triplet state) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.12 0.1 0.08 0.06 0.04 0.02 0 B (au)Ec (Hartree) Exact LDA PBE LYP (B) Ec for HA in B ( =1/2, singlet state) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.06 0.05 0.04 0.03 0.02 0.01 0 B (au)Ec (Hartree) Exact LDA PBE (E) Ec for HA in B ( =1/2, triplet state) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.9 0.8 0.7 0.6 0.5 B (au)Exc (Hartree) Exact LDA PBE BLYP (C) Exc for HA in B ( =1/2, singlet state) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0.9 0.8 0.7 0.6 0.5 B (au)Exc (Hartree) Exact LDA PBE BLYP (F) Exc for HA in B ( =1/2, Triplet state) FIG. 5-6 Comparison of exact (curve) a nd approximate (symbols) exchange (upper panels), correlation (middle panels), a nd XC (bottom panels) energies of the HA with = 1/2 in B fields. Left panels are for si nglet state, right panels for triplet state. Blue lines are exact values. Black squares ( ) are for LDA, red circles (o) for PBE and green triangles ( ) for BLYP

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106 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.35 0.3 0.25 0.2 0.15 B (au)Ex (Hartree) Exact LDA PBE B88 (A) Ex for HA in B ( = 1/10, singlet state) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.35 0.3 0.25 0.2 B (au)Ex (Hartree) Exact LDA PBE B88 (D) Ex for HA in B ( = 1/10, triplet) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.08 0.06 0.04 0.02 0 B (au)Ec (Hartree) Exact LDA PBE LYP (B) Ec for HA in B ( = 1/10, singlet state) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.04 0.03 0.02 0.01 0 B (au)Ec (Hartree) Exact LDA PBE (E) Ec for HA in B ( = 1/10, triplet state) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.45 0.4 0.35 0.3 0.25 0.2 B (au)Exc (Hartree) Exact LDA PBE BLYP (C) Exc for HA in B ( = 1/10, singlet state) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.35 0.3 0.25 0.2 B (au)Exc (Hartree) Exact LDA PBE BLYP (F) Exc for HA in B ( = 1/10, Triplet state) FIG. 5-7 Same as Fig. 5-6, except for = 1/10.

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107 ,(),cdftexact xcaz ,ˆ (),cdftexact xcxceArAz (),LDA xcbz ˆˆ ()/p f zrzjn (),PBE xccz (),VRG xcgAz (),BLYP xcdz ,()/cdftexactexact xcxchA FIG. 5-8 Cross-sectional views of the ex act and approximate XC potentials for the ground state HA with = 1/10 in 3/5 B The B field orientation is in the plane of the paper from bottom to top. Each large tick mark is 2 bohr radii.

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108 Exact and approximate DFT and CDFT XC potentials for the ground state HA with = 1/10 in 3/5 B are shown in Fig. 5-8. All the quantities are evaluated with exact densities. All the approximate potentials (panels b – d ) are too shallow compared with the exact CDFT scalar XC potential, and none of them shares the sh ape of the exact one. This is similar to the obs ervation for the vanishing B field case [120]. Matters are worse for the vector XC potential. The VRG approximation for the vector potential is very wrong. As with what we have found for real atoms, this functional gives essentially nothing but two singu lar points (actually si ngular cuts if my plot were larger) at two poles. The origin of this pathological beha vior becomes clear if one looks at the distribution of p j n which has only significant values at two poles of either a HA or a real atom. Once again we see that even though is a gauge-invariant combination, an attractive feature from a pur ely theoretical perspective, it seems an awkward choice (at best) as the basic variable in CDFT for practic al calculation. One measure for the paramagnetic current contribution is pxcjrArdr Using exact values for the previous example, I get 0.0282 Hartree, which is clearly non-negligible. However, from the VRG functional, ,0.0024VRGexactexact xcEnrvr Hartree. Previously I mentioned that none of the approximate functionals generates a high quality XC scalar potential. This remark n eeds some modification in the case that both the scalar and vector potent ials are considered simultan eously. Recall that the ground state is a triplet. For the two KS orbitals, the p -like orbital “feels” both the XC scalar and the vector potentials. Their total effect is s hown in the last panel of Fig. 5-8. LDA gives

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109 closest approximation to the total XC pot entials in the outer part, where the p -like orbital has significant distribution, ev en though the LDA potential is still too shallow. For the slike orbital 1 KSr, which “feels” only the scal ar XC potential, the smallr region is more important. Again LDA di d a good job at this region. Table 5-5 SCF results for HF and approxima te DFT functionals (energy in Hartree) system energy HF LDA/VWNGGA/PBEGGA/jPBE GGA/BLYP sT 0.63303 0.62746 0.63280 0.63226 2 / 1 J 1.02983 1.02258 1.02696 as 1.02896 0 B xcE -0.51492 -0.52377 -0.54317 left -0.53642 totE 2.03844 2.02623 2.00904 2.01653 sT 0.30470 0.30763 0.31110 0.31190 0.31017 10 / 1 J 0.49797 0.48241 0.48427 0.48560 0.48285 5 / 3 B xcE -0.29804 -0.29775 -0.31035 -0.31288 -0.29442 totE 0.38969 0.38072 0.36966 0.36775 0.38478 To check the non-SCF effects, Table 5-5 gives energy contributions from the SCF calculation using HF and seve ral approximate DFT functionals on two special cases which have analytical solutions. For HF, xcE means exchange energy only. Those SCF results are consistent with previous direct evaluation of the approxi mate functionals. This suggests our previous observations will remain the same were self-consistent orbitals to be used.

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110 CHAPTER 6 SUMMARY AND CONCLUSION In this dissertation, atom-like systems in strong magnetic fields are investigated by different theoretical methods, with special emphasis on DFT and CDFT descriptions. An atom in strong magnetic fields is an interesting and challe nging problem itself. Most current theoretical studies are still at the level of HF calculations. Besides the inherent complexity of the sy stem, one barrier of the appli cation of well-developed first principle methods on this system is the lack of well-adapted, high quality basis sets. The anisotropic Gaussian basis is su itable for the description of this system, but an efficient, systematic way for basis set optimization is de sired, since the optimizing step needs to be done for each atomic configuration and for each field strength. The procedure presented in Chapter 3, specifically, eqns. (3.26) and (3 .27), give an explicit construction. I have shown that basis sets constructed thus work extremely well when compared to fully numerical calculations. They may facilitate more accurate numerical studies on atoms in strong fields. Comparison HF and DFT calculations are done systematically for helium through carbon atoms in a wide range of magnetic fi elds. Extensive tabulations are given in appendices on the DFT calculations based on modern functionals, LDA and PBE. A CDFT functional, VRG, is also tested. Those investigations contribute to our understanding on the performance and limitatio ns of modern DFT and CDFT functionals, and are suggestive for improving and extendi ng those functionals to describe atomic systems in a strong field.

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111 An exactly soluble model system in DFT and CDFT, Hooke’s atom, is studied in detail. For some specific confinement strengths and B fields, we developed exact analytical solutions that provide benchmarks against which other methods can be compared. Exact KS orbitals are constructed for the HA from its electron density and paramagnetic current density. Exact DFT and CDFT energy components, especially exchange and correlation energies, togeth er with approximate ones from modern functionals, are compiled extensively. Exact and approximate XC scalar and vector potentials are also compared. Those comp arisons provide useful guidance for the advancement of DFT and CDFT functionals. We are aware that this study is onl y at the inception stage of CDFT development. Little is known about the curr ent-correction part. The quantity vr does not seem to be a wholly suitable variable in CDFT. Ne vertheless, some formal properties about [,]cdft xcp E nj are known for years, mostly from scaling arguments. With our exact KS orbitals, detailed examin ation of the effect of B field on the exchange -correlation hole is also possible. The combination of these two a pproaches will be helpful to the progress on better CDFT exchange-correlation functionals, or, perhaps, an equiva lent gauge-invariant formulation of current contri bution to DFT that is better adapted to computational implementation.

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112 APPENDIX A HAMILTONIAN MATRIX ELEMENTS IN SPHERICAL GAUSSIAN BASIS In an ordinary GTO basis, the overlap, ki netic, and nuclear attraction integrals are easily obtained in closed form: 3 23 2 ,,;,,| 2lmlmlmlmllmm ll IlmlmGGNN (A.1) 2 5 25 1 2 ||2 2lmlmlmlmllmm ll GGNN (A.2) 3 21! ||NlmlmNlmlmllmm ll ZGGZNN r (A.3) It is also possible to work out the coul omb integral directly in the spherical Gaussian basis, but that woul d be hard and tedious. To make use of the general formula of the electron repulsion integrals in Hermite Gaussians, we first transform a spherical Gaussian or the product of two spherical Gaus sians to a linear combination of Hermite Gaussians. A Hermite Gaussian centered at the point A is defined as 2; ,A r n Ae r A n f (A.4) where the operator n A is for the abbreviation of y x z y x z n n n n n n n A xz yAA A (A.5)

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113 and z y xn n n n , For example, 25 2 2 4 1111 11322 3rGrGrreY 3 5 2 4 20,,0;200,,0;020,,0; 2 2 4 f rfrfr Thus, we only need to evaluate the coul omb integral of two Hermite Gaussians, r r r C n f r A n f r d r d r C n f r A n f ; , ; , ; , | ; , (A.6) Specifically for0 AC 0,,0;|,,0;0,,;|0,,;nn AC ACfnrfnrfArfCr 2 25 2 1 0 02ACu n nn A ACdue z y x i i i i i n nn n n n n n, 2 2 5! 2 1 2 for i in neven integers ; 0, otherwise. (A.7) where ,z y xn n n n and z y xn n n n The diamagnetic term integral, 22224 0sinr ll lmlmlmlmlmlmGxyGNNYYredr

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114 2 / 52 5 20 0 20 0 1 2 1 2 3 1 l m l lm l l l l N Nl l m m m l lm (A.8) where m l m l lm is a Clebsch-Gordon coefficient. Both vector and scalar XC potential terms were integrated numerically on a mesh, 600 radial points and 180 poi nts in the polar angle. 21* 002,r ll lmxclmmmlmlmlmlmxciGArGmNNreYYArddr (A.9) 2/2*/ 002,sinr dftcdftlldftcdft lmxclmmmlmlmlmlmxcGrGNNreYYrddr (A.10)

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115 APPENDIX B ATOMIC ENERGIES FOR ATOMS HE, LI, BE, B, C AND THEIR POSITIVE IONS LI+, BE+, B+ IN MAGNETIC FIELDS Table B-1 Atomic energies of the Helium atom in B fields. Energies in Hartree, B in au. JOC-HF1 is from reference 91 with spherical STO expansion; JOC-HF2 from reference 94 in which JOC basis sets (s ee eqn. 3.23) within AGTO were used; Correlated data are from CI calculations in references 11 through 13, except those labeled by “d”, which are from fixed-phase quantum Monte Carlo calculations in reference 93. All ot her columns are from present study. Ground states are indicated in orange clolor. See notes following the table. State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s2 0 -2.86168 -2.8617 -2.903351 -2.8344 -2.8929 0.02 -2.86160 -2.903270 -2.8344 -2.8928 0.04 -2.86136 -2.903036 -2.8341 -2.8926 0.08 -2.86042 -2.8604 -2.902083 -2.8331 -2.8916 0.16 -2.85665 -2.898290 -2.8290 -2.8876 0.24 -2.85043 -2.8504 -2.892035 -2.8223 -2.8811 0.4 -2.83101 -2.8310 -2.872501 -2.8014 -2.8607 0.5 -2.81445 -2.855859 -2.7838 -2.8435 0.8 -2.74684 -2.7468 -2.787556 -2.7124 -2.7736 1 -2.68888 -2.729508 -2.6518 -2.7142 1.6 -2.46739 -2.507952 -2.4224 -2.4890 2 -2.28914 -2.329780 -2.2393 -2.3088 4 -1.17629 -1.1762 -1.21901d -1.1068 -1.1908 5 -0.53244 -0.574877 -0.4554 -0.5462 8 1.59127 0.168 1.54628d 1.6860 1.5764 10 3.11063 3.064582 3.2142 3.0932 20 11.31961 11.267051 11.4496 11.2809 40 29.01211 29.1592 28.9175 50 38.14391 38.076320 38.2920 38.0198 80 66.09209 66.2303 65.8794 100 85.00418 84.918313 85.1300 84.7336 160 142.45118 142.5276 142.0151 200 181.10639 181.1452 180.5661 240 219.94017 219.9398 219.2999 400 376.32909 376.1681 375.3224 500 474.58300 474.3228 473.3680 560 533.65483 533.3361 532.3207 800 770.54396 769.9995 768.7679 1000 968.44540 967.7225 966.3336 2000 1961.1249 1959.610 1957.599

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116 Table B-1 ( continued ) State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s2s 0 -2.17425 -2.1742 -2.175220 -2.1203 -2.1768 0.02 -2.19348 -2.194461 -2.1395 -2.1961 0.04 -2.21124 -2.212236 -2.1573 -2.2139 0.08 -2.24291 -2.2429 -2.2425 -2.243958 -2.1891 -2.2459 0.16 -2.29512 -2.296318 -2.2421 -2.2994 0.24 -2.33824 -2.339560 -2.2866 -2.3446 0.4 -2.41124 -2.4112 -2.4111 -2.412723 -2.3640 -2.4239 0.5 -2.45283 -2.454347 -2.4090 -2.4705 0.8 -2.57211 -2.5712 -2.5720 -2.573615 -2.5382 -2.6038 1 -2.64918 -2.650655 -2.6204 -2.6882 1.6 -2.86616 -2.8632 -2.8659 -2.867620 -2.8477 -2.9219 2 -2.99824 -2.999708 -2.9846 -3.0629 4 -3.54362 -3.5114 -3.5435 -3.54353d -3.5466 -3.6437 5 -3.76667 -3.768199 -3.7761 -3.8817 8 -4.32036 -4.2891 -4.3202 -4.31429d -4.3463 -4.4744 10 -4.62594 -4.627450 -4.6616 -4.8030 20 -5.77102 -5.772448 -5.8480 -6.0441 40 -7.25443 -7.2543 -7.3984 -7.6751 50 -7.81400 -7.815256 -7.9874 -8.2972 80 -9.13864 -9.1385 -9.3909 -9.7848 100 -9.84199 -9.843074 -10.1416 -10.5831 160 -11.49473 -11.4945 -11.9202 -12.4820 200 -12.36634 -12.8667 -13.4967 240 -13.12269 -13.1223 -13.6929 -14.3847 400 -15.46711 -15.4668 -16.2827 -17.1819 500 -16.60183 -17.5522 -18.5600 560 -17.20526 -17.2046 -18.2315 -19.2992 800 -19.22890 -19.2286 -20.5313 -21.8113 1000 -20.59486 -22.1027 -23.5360 2000 -25.36396 -27.7150 -29.7493 1s2p0 0 -2.13133 -2.1314 -2.132910 -2.0873 -2.1439 0.02 -2.15087 -2.152378 -2.1071 -2.1639 0.04 -2.16929 -2.170822 -2.1257 -2.1825 0.08 -2.20350 -2.2035 -2.2031 -2.205130 -2.1604 -2.2173 0.16 -2.26467 -2.266575 -2.2230 -2.2801 0.24 -2.31988 -2.322032 -2.2796 -2.3370 0.4 -2.41975 -2.4196 -2.4197 -2.422361 -2.3820 -2.4401 0.5 -2.47732 -2.480172 -2.4408 -2.4995 0.8 -2.63483 -2.6347 -2.6347 -2.638222 -2.6011 -2.6617 1 -2.73016 -2.733813 -2.6977 -2.7597 1.6 -2.98302 -2.9828 -2.9827 -2.987185 -2.9530 -3.0199 2 -3.13076 -3.135142 -3.1018 -3.1723 4 -3.71910 -3.7141 -3.7186 -3.72389d -3.6952 -3.7830 5 -3.95398 -3.959235 -3.9329 -4.0289 8 -4.52888 -4.4861 -4.5282 -4.53277d -4.5175 -4.6357 10 -4.84263 -4.848590 -4.8383 -4.9697 20 -6.00482 -6.011488 -6.0370 -6.2234 40 -7.49328 -7.4932 -7.5940 -7.8623

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117 Table B-1 ( continued ) State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s2p0 50 -8.05222 -8.059466 -8.1843 -8.4864 80 -9.37260 -9.3724 -9.5901 -9.9776 100 -10.07281 -10.079973 -10.3416 -10.7777 160 -11.71716 -11.7170 -12.1223 -12.6810 200 -12.58420 -13.0700 -13.6981 240 -13.33662 -13.3364 -13.8973 -14.5883 400 -15.66948 -15.6691 -16.4911 -17.3924 500 -16.79906 -17.7627 -18.7744 560 -17.39990 -17.3996 -18.4433 -19.5159 800 -19.41540 -19.4150 -20.7480 -22.0364 1000 -20.77640 -22.3233 -23.7673 2000 -25.53090 -27.9510 -30.0030 1s2p-1 0 -2.13133 -2.1314 -2.133149 -2.0821 -2.1370 0.02 -2.16035 -2.162112 -2.1113 -2.1664 0.04 -2.18730 -2.189128 -2.1383 -2.1934 0.08 -2.23646 -2.2365 -2.2353 -2.238504 -2.1877 -2.2427 0.16 -2.32260 -2.325189 -2.2744 -2.3291 0.24 -2.39927 -2.402393 -2.3516 -2.4059 0.4 -2.53674 -2.5336 -2.5366 -2.540763 -2.4894 -2.5431 0.5 -2.61555 -2.620021 -2.5681 -2.6216 0.8 -2.83021 -2.8299 -2.8301 -2.835619 -2.7814 -2.8349 1 -2.95969 -2.965504 -2.9095 -2.9635 1.6 -3.30222 -3.3017 -3.3016 -3.308774 -3.2471 -3.3036 2 -3.50205 -3.508911 -3.4437 -3.5024 4 -4.29844 -4.2950 -4.2980 -4.30587d -4.2273 -4.2977 5 -4.61725 -4.625491 -4.5417 -4.6178 8 -5.40041 -5.3793 -5.4000 -5.40452d -5.3162 -5.4077 10 -5.82951 -5.839475 -5.7420 -5.8428 20 -7.42770 -7.440556 -7.3377 -7.4770 40 -9.48827 -9.4882 -9.4184 -9.6149 50 -10.26449 -10.28410 -10.2090 -10.4292 80 -12.10132 -12.1011 -12.0950 -12.3754 100 -13.07665 -13.10478 -13.1050 -13.4198 160 -15.36930 -15.3690 -15.5026 -15.9049 200 -16.57907 -16.7810 -17.2331 240 -17.62932 -17.6289 -17.8983 -18.3956 400 -20.88777 -20.8876 -21.4084 -22.0585 500 -22.46665 -23.1332 -23.8638 560 -23.30677 -23.3066 -24.0572 -24.8326 800 -26.12655 -26.1264 -27.1911 -28.1255 1000 -28.03209 -29.3371 -30.3870 2000 -34.69865 -37.0248 -38.5309 1s3d-1 0 -2.05557 -2.0556 -2.055629 -2.0035 -2.0592 0.02 -2.08225 -2.082319 -2.0305 -2.0865 0.04 -2.10413 -2.104234 -2.0530 -2.1094 0.08 -2.14084 -2.1408 -2.1403 -2.141017 -2.0910 -2.1481 0.16 -2.20200 -2.202291 -2.1542 -2.2127 0.24 -2.25559 -2.256006 -2.2094 -2.2690

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118 Table B-1 ( continued ) State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s3d-1 0.4 -2.35154 -2.3508 -2.3512 -2.352208 -2.3083 -2.3695 0.5 -2.40669 -2.407521 -2.3651 -2.4270 0.8 -2.55775 -2.5525 -2.5528 -2.559005 -2.5202 -2.5843 1 -2.64947 -2.650973 -2.6141 -2.6796 1.6 -2.89411 -2.8741 -2.8938 -2.896192 -2.8636 -2.9335 2 -3.03792 -3.040304 -3.0099 -3.0827 4 -3.61580 -3.6053 -3.6156 -3.61588d -3.5963 -3.6839 5 -3.84820 -3.851883 -3.8321 -3.9267 8 -4.41969 -4.3470 -4.4193 -4.40580d -4.4126 -4.5267 10 -4.73271 -4.737490 -4.7313 -4.8570 20 -5.89611 -5.902110 -5.9217 -6.0960 40 -7.39013 -7.3900 -7.4646 -7.7118 50 -7.95152 -7.959094 -8.0487 -8.3258 80 -9.27773 -9.2773 -9.4375 -9.7908 100 -9.98090 -9.989376 -10.1791 -10.5757 160 -11.63164 -11.6315 -11.9342 -12.4402 200 -12.50169 -12.8673 -13.4351 240 -13.25654 -13.2562 -13.6812 -14.3053 400 -15.59593 -15.5955 -16.2305 -17.0422 500 -16.72819 -17.4790 -18.3884 560 -17.33035 -17.3302 -18.1468 -19.1112 800 -19.34981 -19.3495 -20.4068 -21.5638 1000 -20.71314 -21.9503 -23.2428 2000 -25.47417 -27.4561 -29.3136 1s3d-2 0 -2.05517 -2.0556 -2.055635 -2.0024 -2.0571 0.02 -2.09057 -2.090760 -2.0382 -2.0933 0.04 -2.11905 -2.119167 -2.0670 -2.1225 0.08 -2.16632 -2.1663 -2.1659 -2.166519 -2.1148 -2.1712 0.16 -2.24437 -2.244776 -2.1938 -2.2513 0.24 -2.31214 -2.312786 -2.2624 -2.3206 0.4 -2.43230 -2.4310 -2.4320 -2.433466 -2.3845 -2.4433 0.5 -2.50087 -2.502362 -2.4542 -2.5132 0.8 -2.68753 -2.6824 -2.6871 -2.689916 -2.6441 -2.7031 1 -2.80039 -2.803296 -2.7589 -2.8178 1.6 -3.10079 -3.0801 -3.1005 -3.104935 -3.0636 -3.1234 2 -3.27738 -3.282141 -3.2423 -3.3031 4 -3.98909 -3.9611 -3.9890 -3.99144d -3.9603 -4.0288 5 -4.27663 -4.284050 -4.2500 -4.3227 8 -4.98705 -4.8724 -4.9866 -4.97560d -4.9660 -5.0509 10 -5.37808 -5.387931 -5.3607 -5.4531 20 -6.84153 -6.854811 -6.8428 -6.9669 40 -8.73864 -8.7385 -8.7792 -8.9512 50 -9.45533 -9.476057 -9.5157 -9.7076 80 -11.15470 -11.1540 -11.2732 -11.5158 100 -12.05871 -12.088566 -12.2148 -12.4864 160 -14.18753 -14.1875 -14.4506 -14.7958 200 -15.31278 -15.6429 -16.0302 240 -16.29062 -16.2905 -16.6851 -17.1106

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119 Table B-1 ( continued ) State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s3d-2 400 -19.32952 -19.3286 -19.9597 -20.5142 500 -20.80454 -21.5688 -22.1915 560 -21.59002 -21.5954 -22.4310 -23.0915 800 -24.22930 -24.2283 -25.3550 -26.1503 1000 -26.01525 -27.3575 -28.2510 2000 -32.27676 -34.5328 -35.8136 1s4f-2 0 -2.03125 -2.0313 -2.031254 -1.9776 -2.0321 0.02 -2.06215 -2.062177 -2.0103 -2.0651 0.04 -2.08431 -2.084414 -2.0336 -2.0891 0.08 -2.12033 -2.1197 -2.1201 -2.120532 -2.0712 -2.1279 0.16 -2.18008 -2.180271 -2.1331 -2.1914 0.24 -2.23257 -2.232769 -2.1870 -2.2466 0.4 -2.32686 -2.3118 -2.3268 -2.327166 -2.2836 -2.3454 0.5 -2.38120 -2.381601 -2.3392 -2.4021 0.8 -2.53039 -2.5253 -2.5299 -2.531094 -2.4914 -2.5571 1 -2.62115 -2.622067 -2.5837 -2.6512 1.6 -2.86378 -2.8503 -2.8637 -2.865215 -2.8297 -2.9022 2 -3.00670 -3.008432 -2.9743 -3.0500 4 -3.58255 -3.5648 -3.5823 -3.58366d -3.5557 -3.6465 5 -3.81470 -3.817900 -3.7900 -3.8877 8 -4.38646 -4.3398 -4.3861 -4.38303d -4.3677 -4.4844 10 -4.70001 -4.704423 -4.6852 -4.8133 20 -5.86663 -5.872256 -5.8714 -6.0469 40 -7.36567 -7.3654 -7.4089 -7.6554 50 -7.92889 -7.935933 -7.9907 -8.2664 80 -9.25902 -9.2549 -9.3736 -9.7237 100 -9.96402 -9.971845 -10.1117 -10.5041 160 -11.61837 -11.6182 -11.8577 -12.3571 200 -12.48998 -12.7855 -13.3455 240 -13.24601 -13.2459 -13.5947 -14.2094 400 -15.58829 -15.5879 -16.1278 -16.9261 500 -16.72161 -17.3679 -18.2634 560 -17.32426 -17.3240 -18.0311 -18.9786 800 -19.34508 -19.3448 -20.2748 -21.4099 1000 -20.70912 -21.8066 -23.0781 2000 -25.47179 -27.2681 -29.0676 1s4f-3 0 -2.03125 -2.0313 -2.031255 -1.9764 -2.0303 0.02 -2.06949 -2.069509 -2.0172 -2.0715 0.04 -2.09691 -2.096967 -2.0460 -2.1010 0.08 -2.14149 -2.1412 -2.1406 -2.141582 -2.0914 -2.1475 0.16 -2.21484 -2.214999 -2.1656 -2.2233 0.24 -2.27873 -2.278994 -2.2301 -2.2889 0.4 -2.39242 -2.3916 -2.3923 -2.392988 -2.3447 -2.4052 0.5 -2.45746 -2.458243 -2.4104 -2.4716 0.8 -2.63477 -2.6344 -2.6344 -2.636282 -2.5893 -2.6520 1 -2.74210 -2.744109 -2.6976 -2.7611 1.6 -3.02805 -3.0270 -3.0275 -3.031517 -2.9859 -3.0513 2 -3.19635 -3.200682 -3.1554 -3.2222

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120 Table B-1 ( continued ) State B Present HF JOC-HF1a JOC-HF2b Correlatedc LDA PBE 1s4f-3 4 -3.87602 -3.8689 -3.8759 -3.88056d -3.8388 -3.9132 5 -4.15121 -4.160555 -4.1153 -4.1936 8 -4.83222 -4.7802 -4.8318 -4.83102d -4.7997 -4.8893 10 -5.20761 -5.222402 -5.1774 -5.2741 20 -6.61486 -6.636961 -6.5973 -6.7242 40 -8.44270 -8.4426 -8.4541 -8.6273 50 -9.13392 -9.169700 -9.1605 -9.3529 80 -10.77406 -10.7734 -10.8463 -11.0881 100 -11.64710 -11.697425 -11.7494 -12.0196 160 -13.70423 -13.7044 -13.8936 -14.2360 200 -14.79217 -15.0370 -15.4207 240 -15.73789 -15.7378 -16.0363 -16.4576 400 -18.67851 -18.6784 -19.1756 -19.7238 500 -20.10658 -20.7178 -21.3334 560 -20.86724 -20.8674 -21.5441 -22.1968 800 -23.42398 -23.4342 -24.3457 -25.1316 1000 -25.15479 -26.2639 -27.1469 2000 -31.22698 -33.1339 -34.3978 1s5g-3 0 -2.01997 -1.9605 -2.0146 0.02 -2.05277 -1.9995 -2.0540 0.04 -2.07425 -2.0239 -2.0791 0.08 -2.10912 -2.1089 -2.10950d -2.0607 -2.1171 0.16 -2.16749 -2.1215 -2.1796 0.24 -2.21905 -2.21919d -2.1745 -2.2340 0.4 -2.31208 -2.3120 -2.31133d -2.2698 -2.3316 0.5 -2.36585 -2.3247 -2.3877 0.8 -2.51380 -2.5133 -2.51328d -2.4751 -2.5415 1 -2.60397 -2.5665 -2.6349 1.6 -2.84546 -2.8451 -2.84601d -2.8105 -2.8843 2 -2.98790 -2.9541 -3.0313 4 -3.56275 -3.5607 -3.74971d -3.5322 -3.6251 5 -3.79480 -3.7656 -3.8656 8 -4.36680 -4.3664 -4.36666d -4.3413 -4.4605 10 -4.68070 -4.6579 -4.7885 20 -5.84934 -5.8413 -6.0193 40 -7.35154 -7.3514 -7.3756 -7.6240 50 -7.91592 -7.9560 -8.2336 80 -9.24860 -9.2476 -9.3356 -9.6868 100 -9.95479 -10.0717 -10.4649 160 -11.61154 -11.6106 -11.8125 -12.3118 200 -12.48420 -12.7374 -13.2968 240 -13.24104 -13.2408 -13.5439 -14.1576 400 -15.58525 -15.5847 -16.0679 -16.8640 500 -16.71928 -17.3031 -18.1935 560 -17.32225 -17.3216 -17.9637 -18.9068 800 -19.34395 -19.3436 -20.1980 -21.3262 1000 -20.70844 -21.7231 -22.9845 2000 -25.47211 -27.1590 -28.9289

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121 Notes for Table B-1: (a) Spherical STO basis expansion. Data from M. D. Jones, G. Ortiz, and D. M. Ceperley, Phys. Rev. A 54, 219 (1996); (b) Anisotropic GTO basis, from M.D. Jone s, G. Ortiz, and D.M. Ceperley, Phys. Rev. A 59, 2875 (1999); (c) Anisotropic GTO basis CI calculations excep t data labled by (d). For magnetic quantum number M = 0, M = -1, and M = -2,-3, data are from W. Becken, P. Schmelcher, and F.K. Diakonos, J. Phys. B 32, 1557 (1999) W. Becken and P. Schmelcher, J. Phys. B 33, 545 (2000), and Phys. Rev. A 63, 053412 (2001), respectively; (d) Fixed-Phase Quantum Monte Carlo calcula tions, from M.D. Jones, G. Ortiz, and D.M. Ceperley, Int. J. Quant. Chem. 64, 523 (1997). Table B-2 Atomic energies of the Lithium singly positive ion, Li+, in B fields. Energies in Hartree, B in au. Numerical HF is from refe rence 86; CI from reference 77. All others from present study. Ground states are indicated in orange. State B Present HF Numerical HF CI LDA PBE 1s2 0 -7.23641 -7.23642 -7.277191 -7.1422 -7.2567 0.01 -7.23643 -7.23641 -7.277327 -7.1422 -7.2568 0.02 -7.23645 -7.23639 -7.277376 -7.1423 -7.2569 0.05 -7.23623 -7.23623 -7.277336 -7.1420 -7.2566 0.1 -7.23567 -7.23567 -7.276897 -7.1414 -7.2560 0.2 -7.23345 -7.23345 -7.274673 -7.1391 -7.2537 0.5 -7.21798 -7.21798 -7.259522 -7.1228 -7.2377 1 -7.16401 -7.16401 -7.205547 -7.0663 -7.1822 2 -6.96299 -6.96300 -7.004453 -6.8577 -6.9769 3 -6.66237 -6.66237 -6.5482 -6.6718 4 -6.28590 -6.1630 -6.2913 5 -5.85050 -5.85051 -5.891947 -5.7192 -5.8525 5.4 -5.66264 -5.704147 -5.5281 -5.6635 7 -4.84724 -4.84725 -4.7005 -4.8441 10 -3.11091 -3.11092 -3.153453 -2.9446 -3.1029 20 3.74896 3.74896 3.9618 3.7599 50 27.96465 27.96465 28.2462 27.9452 100 72.09338 72.09337 72.4153 71.9934 200 164.66868 164.66867 164.9937 164.3926 500 452.00319 452.0032 452.1999 451.2283 1000 939.87972 939.87976 939.7871 938.3857 2000 1925.2140 1924.543 1922.523 1s2p-1 0 -5.02468 -5.02469 -5.026321 -4.9386 -5.0301 0.01 -5.03961 -5.03963 -5.041247 -4.9535 -5.0450 0.02 -5.05440 -5.05442 -5.056040 -4.9683 -5.0598 0.05 -5.09795 -5.09797 -5.099595 -5.0119 -5.1033 0.1 -5.16787 -5.16789 -5.169539 -5.0818 -5.1732 0.2 -5.29872 -5.29873 -5.300455 -5.2125 -5.3038 0.5 -5.64005 -5.64006 -5.643726 -5.5534 -5.6444 1 -6.11462 -6.11462 -6.119216 -6.0264 -6.1171

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122 Table B-2 ( continued ) State B Present HF Numerical HF CI LDA PBE 1s2p-1 2 -6.89407 -6.89408 -6.899768 -6.8001 -6.8922 3 -7.54672 -7.54672 -7.4458 -7.5409 4 -8.11772 -8.0101 -8.1089 5 -8.62942 -8.62943 -8.636273 -8.5155 -8.6183 5.4 -8.82072 -8.827671 -8.7044 -8.8089 7 -9.52491 -9.52492 -9.4000 -9.5112 10 -10.65131 -10.65131 -10.659060 -10.5135 -10.6369 20 -13.42974 -13.42974 -13.2670 -13.4258 50 -18.52547 -18.52548 -18.3463 -18.5834 100 -23.69994 -23.69994 -23.5452 -23.8753 200 -30.26077 -30.26077 -30.1973 -30.6627 500 -41.50392 -41.50393 -41.7541 -42.4946 1000 -52.32301 -52.3230 -53.0628 -54.1191 2000 -65.47655 -67.0601 -68.5692 Table B-3 Atomic energies of the Lithium atom in B fields. Energies in Hartree, B in au. Numerical HF is from reference 86; CI from reference 77, except those labeled by (c), which are from reference 105. All others from present study. Ground states are indi cated in orange. State B Present HF Numerical HFa CI LDA PBE 1s22s 0 -7.43274 -7.43275 -7.477766b -7.3433 -7.4621 0.001 -7.43325 -7.43326 -7.478032b -7.3438 -7.4627 0.002 -7.43364 -7.43375 -7.3442 -7.4631 0.009 -7.43712 -7.43713 -7.4821719c -7.3477 -7.4666 0.01 -7.43759 -7.43760 -7.482888b -7.3481 -7.4670 0.018 -7.44125 -7.44125 -7.4863018c -7.3518 -7.4707 0.02 -7.44328 -7.44214 -7.490983b -7.3558 -7.4747 0.05 -7.45397 -7.45398 -7.502724b -7.3646 -7.4835 0.054 -7.45536 -7.45537 -7.5004678c -7.3660 -7.4849 0.1 -7.46856 -7.46857 -7.517154b -7.3794 -7.4985 0.126 -7.47407 -7.47408 -7.5193718c -7.3851 -7.5043 0.17633 -7.48162 -7.48162 -7.3931 -7.5126 0.18 -7.48203 -7.48204 -7.5275049c -7.3935 -7.5131 0.2 -7.48400 -7.48400 -7.533495b -7.3957 -7.5154 0.5 -7.47740 -7.47741 -7.528055b -7.3948 -7.5170 0.54 -7.47350 -7.47351 -7.5197262c -7.3919 -7.5146 0.900 -7.42504 -7.42504 -7.4710527c -7.3530 -7.4808 1 -7.40878 -7.40879 -7.458550b -7.3392 -7.4683 1.260 -7.36225 -7.36226 -7.2982 -7.4305 5 -6.08810 -6.08811 -6.136918b -6.0481 -6.2195 10 -3.35784 -3.35777 -3.406556b -3.3276 -3.5438 1s22p-1 0 -7.36494 -7.36509 -7.407126b -7.2788 -7.3970 0.009 -7.37380 -7.37387 -7.4185656c -7.2878 -7.4063 0.01 -7.37475 -7.37481 -7.416994b -7.2888 -7.4072 0.018 -7.38213 -7.38218 -7.4268977c -7.2962 -7.4146 0.02 -7.38391 -7.38397 -7.2980 -7.4164

PAGE 137

123 Table B-3 ( continued ) State B Present HF Numerical HFa CI LDA PBE 1s22p-1 0.05 -7.40841 -7.40844 -7.451086b -7.3226 -7.4411 0.054 -7.41138 -7.41141 -7.4562839c -7.3256 -7.4441 0.1 -7.44173 -7.44176 -7.484773b -7.3565 -7.4750 0.126 -7.45648 -7.45650 -7.5018829c -7.3715 -7.4901 0.17633 -7.48160 -7.48162 -7.3973 -7.5158 0.18 -7.48328 -7.48330 -7.5291175c -7.3990 -7.5175 0.2 -7.49218 -7.49220 -7.536032b -7.4082 -7.5266 0.5 -7.58787 -7.58790 -7.634547b -7.5072 -7.6247 0.54 -7.59707 -7.59709 -7.645730c -7.5167 -7.6341 0.9 -7.65627 -7.65628 -7.707054c -7.5775 -7.6944 1 -7.66652 -7.66653 -7.716679b -7.5879 -7.7048 1.26 -7.68285 -7.68288 -7.735013c -7.6040 -7.7211 1.8 -7.67655 -7.67657 -7.729627c -7.5955 -7.7143 2 -7.66245 -7.66246 -7.715709b -7.5803 -7.6998 2.153 -7.64784 -7.64785 -7.5647 -7.6849 2.16 -7.64710 -7.64711 -7.5639 -7.6842 2.5 -7.60350 -7.60351 -7.5179 -7.6399 3 -7.51515 -7.51516 -7.4258 -7.5506 3.6 -7.37637 -7.37638 -7.425857c -7.2824 -7.4108 4 -7.26672 -7.1697 -7.3006 5 -6.94229 -6.94230 -7.002346b -6.8378 -6.9752 5.4 -6.79515 -6.79517 -6.8361629c -6.6879 -6.8279 10 -4.61775 -4.61777 -4.684076b -4.4845 -4.6542 1s2p-13d-2 0 -5.08355 -5.08379 -5.142319b -5.0057 -5.0975 0.1 -5.32136 -5.32140 -5.341030b -5.2440 -5.3357 0.5 -5.97050 -5.97052 -5.982253b -5.8880 -5.9820 1 -6.57079 -6.57081 -6.582361b -6.4853 -6.5805 1.8 -7.34721 -7.34723 -7.2576 -7.3545 2 -7.52002 -7.52003 -7.530125b -7.4293 -7.5268 2.153 -7.64784 -7.64785 -7.5563 -7.6543 2.16 -7.65360 -7.65361 -7.5620 -7.6600 2.5 -7.92531 -7.92532 -7.8318 -7.9311 3 -8.29920 -8.29920 -8.2030 -8.3044 4 -8.97475 -8.8735 -8.9797 5 -9.57693 -9.57694 -9.591769b -9.4713 -9.5827 5.4 -9.80146 -9.80147 -9.6943 -9.8079 10 -11.93900 -11.93902 -11.957294b -11.8209 -11.9588 20 -15.16261 -15.16260 -15.0438 -15.2274 50 -21.05051 -21.0505 -20.9843 -21.2681 100 -27.01926 -27.0192 -27.0769 -27.4789 200 -34.58498 -34.5850 -34.8991 -35.4725 500 -47.55828 -47.5583 -48.5639 -49.4845 1000 -60.05888 -60.0589 -62.0240 -63.3419 2000 -75.28238 -78.7973 -80.6829 (a) From M. V. Ivanov and P. Schmelcher, Phys. Rev. A 57, 3793 (1998); (b) From O.-A. Al-Hujaj and P. Schmelcher, Phys. Rev. A 70, 033411 (2004); (c) From X. Guan and B. Li, PRA 63, 043413 (2001).

PAGE 138

124 Table B-4 Atomic energies of the Beryllium singly positive ion, Be+, in B fields. Energies in Hartree, B in au. Numerical HF is from reference 88; CI from reference 8, except those labeled by (c), which are from frozen core approximation in reference 7. All others from presen t study. Ground states are indicated in orange. State B Present HF Numerical HFa CI LDA PBE 1s22s 0 -14.27746 -14.27747 -14.3247b -14.1147 -14.2993 0.001 -14.27796 -14.27797 -14.3251b -14.1152 -14.2998 0.002 -14.27846 -14.27846 -14.32226c -14.1157 -14.3003 0.004 -14.27953 -14.32326c -14.1169 -14.3015 0.01 -14.28241 -14.28241 -14.3296b -14.1196 -14.3043 0.02 -14.28724 -14.28725 -14.33105c -14.1245 -14.3091 0.05 -14.30110 -14.30111 -14.3482b -14.1383 -14.3230 0.1 -14.32206 -14.32207 -14.3694b -14.1593 -14.3440 0.2 -14.35648 -14.35648 -14.4038b -14.1938 -14.3787 0.3 -14.38211 -14.42588c -14.2197 -14.4048 0.4 -14.40046 -14.40046 -14.44420c -14.2383 -14.4237 0.5 -14.41281 -14.41282 -14.4606b -14.2511 -14.4368 0.6 -14.42021 -14.46390c -14.2590 -14.4452 0.7 -14.42350 -14.46700c -14.2629 -14.4495 0.8 -14.42334 -14.46697c -14.2634 -14.4506 1 -14.41477 -14.41478 -14.4630b -14.2567 -14.4451 2 -14.28223 -14.28225 -14.3300b -14.1376 -14.3353 5 -13.55017 -13.55019 -13.5971b -13.4306 -13.6537 10 -11.57651 -11.57652 -11.6231b -11.4662 -11.7269 1s22p-1 0 -14.13090 -14.13093 -14.1741b -13.9771 -14.1593 0.01 -14.14085 -14.14087 -14.1841b -13.9871 -14.1696 0.05 -14.17913 -14.17916 -14.2216b -14.0254 -14.2079 0.1 -14.22387 -14.22390 -14.2672b -14.0702 -14.2526 0.2 -14.30404 -14.30406 -14.3476b -14.1506 -14.3329 0.3 -14.37399 -14.41738c -14.2208 -14.4030 0.4 -14.43597 -14.43599 -14.47926c -14.2830 -14.4652 0.5 -14.49161 -14.49163 -14.5358b -14.3390 -14.5211 0.6 -14.54207 -14.58512c -14.3897 -14.5717 0.7 -14.58819 -14.63109c -14.4361 -14.6179 0.8 -14.63057 -14.67332c -14.4787 -14.6604 1 -14.70590 -14.70591 -14.7520b -14.5543 -14.7358 2 -14.95180 -14.95181 -15.0000b -14.7988 -14.9807 5 -14.96818 -14.96820 -15.0184b -14.7975 -14.9890 10 -13.75772 -13.75733 -13.8087b -13.5532 -13.7690 20 -9.21789 -9.217910 -8.9640 -9.2287 50 10.42839 10.42836 10.7504 10.3674 1s2p-13d-2 0 -9.41049 -9.41056 -9.4156b -9.2941 -9.4253 0.1 -9.68350 -9.68356 -9.6888b -9.5670 -9.6984 0.2 -9.91871 -9.91878 -9.9243b -9.8010 -9.9327 0.5 -10.51254 -10.51259 -10.5188b -10.3904 -10.5235 1 -11.31310 -11.31312 -11.3203b -11.1854 -11.3205 2 -12.59205 -12.59206 -12.6002b -12.4567 -12.5942 5 -15.42816 -15.42817 -15.4367b -15.2744 -15.4206

PAGE 139

125 Table B-4 ( continued ) State B Present HF Numerical HFa CI LDA PBE 1s2p-13d-2 10 -18.82017 -18.820184 -18.8283b -18.6433 -18.8096 20 -23.61200 -23.612005 -23.4113 -23.6181 50 -32.61958 -32.61959 -32.4163 -32.7191 100 -41.93409 -41.93414 -41.7890 -42.2079 200 -53.90637 -53.90638 -53.9257 -54.5138 500 -74.73620 -74.73619 -75.2754 -76.2067 1000 -95.07510 -95.07513 -96.4008 -97.7251 2000 -120.11944 -120.11947 -122.7839 -124.6704 (a) From M. V. Ivanov and P. Schmelcher, Eur. Phys. J. D 14, 279 (2001); (b) From O.-A. Al-Hujaj and P. Schmelcher, Phys. Rev. A 70, 023411 (2004); (c) FromX. Guan, B. Li, and K. T. Taylor, J. Phys. B 36, 2465 (2003). Table B-5 Atomic energies of the Beryllium atom in B fields. Energies in Hartree, B in au. Numerical HF is from reference 88; CI from reference 8, except those labeled by (c), which are from frozen core approximation in reference 7. All others from present study. Ground states are indicated in orange. State B Present HF Numerical HFa CI LDA PBE 1s22s2 0 -14.57302 -14.57336 -14.6405b -14.4465 -14.6299 0.001 -14.57304 -14.57336 -14.6410b -14.4465 -14.6300 0.01 -14.57288 -14.57322 -14.6408b -14.4463 -14.6298 0.02 -14.57244 -14.57279 -14.66238c -14.4459 -14.6294 0.03 -14.57173 -14.4452 -14.6287 0.04 -14.57072 -14.4442 -14.6277 0.05 -14.56944 -14.56986 -14.6393b -14.4430 -14.6265 0.07 -14.56605 -14.56657 -14.4396 -14.6232 0.1 -14.55897 -14.6298b -14.4327 -14.6163 0.5 -14.30723 -14.32860 -14.3882b -14.1861 -14.3721 1 -13.79426 -13.89120 -13.9220b -13.6858 -13.8762 1s22s2p-1 0.01 -14.52688 -14.52690 -14.5744b -14.3714 -14.5568 0.02 -14.54136 -14.54138 -14.59170c -14.3859 -14.5713 0.03 -14.55551 -14.4000 -14.5854 0.04 -14.56931 -14.4138 -14.5992 0.05 -14.58279 -14.58281 -14.6142b -14.4272 -14.6126 0.07 -14.60878 -14.60879 -14.4531 -14.6386 0.1 -14.64546 -14.6936b -14.4897 -14.6752 0.15 -14.70106 -14.70108 -14.75127c -14.5452 -14.7308 0.2 -14.75063 -14.7979b -14.5948 -14.7806 0.3 -14.83519 -14.83520 -14.88560c -14.6800 -14.8660 0.3185 -14.84904 -14.84905 -14.6940 -14.8801 0.4 -14.90463 -14.95507c -14.7507 -14.9369 0.5 -14.96262 -14.96264 -15.0107b -14.8103 -14.9969 0.6 -15.01170 -15.06181c -14.8615 -15.0483 0.8 -15.08987 -15.13834c -14.9450 -15.1322 1 -15.14895 -15.14899 -15.1982b -15.0103 -15.1981 2 -15.30813 -15.30815 -15.3551b -15.2027 -15.3960

PAGE 140

126 Table B-5 ( continued ) State B Present HF Numerical HFa CI LDA PBE 1s22s2p-1 5 -15.25182 -15.25183 -15.3002b -15.1799 -15.4007 1s22p-13d-2 0.1 -14.37791 -14.4082b -14.2326 -14.4153 0.2 -14.51754 -14.5492b -14.3705 -14.5540 0.3 -14.63365 -14.63369 -14.4854 -14.6698 0.4 -14.73393 -14.5849 -14.7700 0.5 -14.82270 -14.82272 -14.8530b -14.6732 -14.8590 0.6 -14.90260 -14.7529 -14.9391 0.8 -15.04230 -14.8924 -15.0794 1 -15.16176 -15.16178 -15.1975b -15.0118 -15.1994 2 -15.57494 -15.57496 -15.6217b -15.4245 -15.6143 3 -15.79983 -15.6469 -15.8400 4 -15.90158 -15.7448 -15.9423 4.501 -15.91624 -15.91626 -15.7573 -15.9573 5 -15.91025 -15.91027 -15.9493b -15.7491 -15.9517 10 -15.04641 -15.04644 -15.0875b -14.8638 -15.0963 20 -10.97097 -10.97100 -10.7632 -11.0544 50 7.83402 7.83395 8.0441 7.6128 1s2p-13d-24f-3 1 -11.72872 -11.72880 -11.7358b -11.6049 -11.7440 2 -13.16959 -13.16961 -13.1762b -13.0384 -13.1831 3 -14.35015 -14.2142 -14.3635 4 -15.38048 -15.2407 -15.3945 4.501 -15.85572 -15.7143 -15.8704 5 -16.30689 -16.30690 -16.3139b -16.1639 -16.3225 10 -20.01751 -20.01753 -20.0242b -19.8656 -20.0497 20 -25.23248 -25.23250 -25.0850 -25.3181 30 -29.11101 -28.9821 -29.2579 40 -32.28414 -32.28415 -32.1803 -32.4941 50 -35.00766 -35.00768 -34.9326 -35.2806 100 -45.10513 -45.10519 -45.1927 -45.6795 200 -58.08260 -58.08264 -58.5053 -59.1944 500 -80.67355 -80.67357 -82.0025 -83.1010 1000 -102.75472 -102.7548 -105.3481 -106.9144 2000 -129.97904 -129.9790 -134.6277 -136.8618 (a) From M. V. Ivanov and P. Schmelcher, Eur. Phys. J. D 14, 279 (2001); (b) From O.-A. Al-Hujaj and P. Schmelcher, Phys. Rev. A 70, 023411 (2004); (c) FromX. Guan, B. Li, and K. T. Taylor, J. Phys. B 36, 2465 (2003). Table B-6 Atomic energies of the Boron singly positive ion, B+, in B fields. Energies in Hartree, B in au. Numerical HF is from refe rence 89; All others from present study. Ground states are indicated in orange. State B Present HF Numerical HF LDA PBE 1s22s2 0 -24.23757 -24.23758 -24.0373 -24.2934 0.01 -24.23751 -24.23752 -24.0372 -24.2934 0.05 -24.23592 -24.23593 -24.0357 -24.2918

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127 Table B-6 ( continued ) State B Present HF Numerical HF LDA PBE 1s22s2 0.07 -24.23434 -24.23435 -24.0341 -24.2903 0.07811 -24.23355 -24.23356 -24.0333 -24.2895 0.1 -24.23099 -24.23100 -24.0308 -24.2870 0.2 -24.21160 -24.21161 -24.0115 -24.2679 0.3 -24.18034 -24.18047 -23.9805 -24.2372 0.4 -24.13837 -24.13926 -23.9389 -24.1959 0.5 -24.08691 -24.08953 -23.8879 -24.1454 1 -23.72226 -23.75518 -23.5270 -23.7872 1s22s2p-1 0 -24.12015 -24.12078 -23.8855 -24.1418 0.01 -24.13570 -24.13573 -23.9005 -24.1569 0.05 -24.19396 -24.19399 -23.9588 -24.2152 0.07 -24.22223 -24.22226 -23.9870 -24.2435 0.07811 -24.23353 -24.23356 -23.9983 -24.2548 0.1 -24.26357 -24.26360 -24.0283 -24.2848 0.2 -24.39242 -24.39245 -24.1571 -24.4136 0.3 -24.50850 -24.50852 -24.2732 -24.5298 0.4 -24.61321 -24.61324 -24.3780 -24.6348 0.5 -24.70796 -24.70798 -24.4730 -24.7300 0.6 -24.79399 -24.79401 -24.5595 -24.8166 0.8 -24.94405 -24.94408 -24.7109 -24.9685 1 -25.07023 -25.07026 -24.8391 -25.0970 1.5 -25.31062 -25.31064 -25.0867 -25.3456 1.6761 -25.37642 -25.37644 -25.1556 -25.4149 1.8143 -25.42280 -25.42283 -25.2046 -25.4643 2 -25.47886 -25.47889 -25.2643 -25.5246 2.5 -25.60152 -25.3976 -25.6596 3 -25.69387 -25.69390 -25.5005 -25.7648 4 -25.81860 -25.81863 -25.6430 -25.9132 5 -25.88462 -25.88465 -25.7212 -25.9981 10 -25.54205 -25.54207 -25.3982 -25.7121 1s22p-13d-2 0 -23.39306 -23.39320 -23.1756 -23.4268 0.1 -23.61603 -23.6162 -23.3984 -23.6499 0.5 -24.24283 -24.2429 -24.0200 -24.2733 1 -24.78666 -24.78674 -24.5594 -24.8153 1.5 -25.20281 -25.20288 -24.9730 -25.2304 1.8143 -25.42277 -25.42282 -25.1917 -25.4499 2 -25.54102 -25.54108 -25.3092 -25.5679 2.5 -25.82390 -25.5902 -25.8500 3 -26.06393 -26.06397 -25.8282 -26.0892 4 -26.44417 -26.44420 -26.2042 -26.4680 5 -26.71996 -26.71999 -26.4753 -26.7423 7 -27.03668 -27.03671 -26.7819 -27.0567 8 -27.10172 -27.10174 -26.8418 -27.1209 10 -27.08414 -27.08417 -26.8140 -27.1024 20 -25.06408 -25.06410 -24.7520 -25.0900 50 -10.65832 -10.65835 -10.2914 -10.7602 1s2p-13d-24f-3 0 -15.23729 -15.237424 -15.0882 -15.2612 1 -18.07234 -18.07243 -17.9049 -18.0849

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128 Table B-6 ( continued ) State B Present HF Numerical HF LDA PBE 1s2p-13d-24f-3 5 -24.01516 -24.01520 -23.8148 -24.0139 7 -26.16678 -26.16682 -25.9571 -26.1641 8 -27.14062 -27.14064 -26.9270 -27.1381 10 -28.93502 -28.93504 -28.7145 -28.9341 20 -36.02017 -36.02020 -35.7813 -36.0431 30 -41.38293 -41.38296 -41.1416 -41.4429 40 -45.81146 -45.81149 -45.5765 -45.9137 50 -49.63541 -49.63544 -49.4122 -49.7823 100 -63.94724 -63.947265 -63.8180 -64.3235 200 -82.55704 -82.55711 -82.6656 -83.3703 500 -115.33673 -116.1676 -117.2772 1000 -147.71740 -149.6245 -151.1968 2000 -187.99218 -191.7212 -193.9542 Table B-7 Atomic energies of the Boron atom in B fields. Energies in Hartree, B in au. Numerical HF is from reference 89; Ground states are indi cated in orange. State B Present HF Numerical HF LDA PBE 1s22s22p-1 0.01 -24.54011 -24.54018 -24.3632 -24.6174 0.05 -24.57671 -24.57679 -24.3997 -24.6540 0.07 -24.59334 -24.59340 -24.4163 -24.6706 0.07811 -24.59977 -24.4227 -24.6770 0.1 -24.61625 -24.61631 -24.4391 -24.6934 0.2 -24.67629 -24.67634 -24.4989 -24.7534 0.3 -24.71433 -24.71439 -24.5370 -24.7919 0.4 -24.73414 -24.73424 -24.5573 -24.8126 0.5 -24.73877 -24.73975 -24.5627 -24.8185 1 -24.60461 -24.63172 -24.4367 -24.6953 1s22s2p02p-1 0.01 -24.47095 -24.47112 -24.2536 -24.5083 0.05 -24.54806 -24.54823 -24.3305 -24.5853 0.07 -24.58519 -24.58536 -24.3675 -24.6223 0.07811 -24.59998 -24.60015 -24.3822 -24.6370 0.1 -24.63915 -24.63932 -24.4212 -24.6760 0.2 -24.80494 -24.80511 -24.5856 -24.8406 0.3 -24.95164 -24.95180 -24.7307 -24.9862 0.4 -25.08238 -25.08255 -24.8599 -25.1159 0.5 -25.19978 -25.19996 -24.9759 -25.2325 0.6 -25.30594 -25.30612 -25.0808 -25.3380 0.8 -25.49085 -25.49103 -25.2635 -25.5220 1 -25.64692 -25.64711 -25.4179 -25.6777 1.5 -25.94975 -25.94997 -25.7182 -25.9814 1.6761 -26.03495 -26.03522 -25.8030 -26.0674 1.8143 -26.09592 -25.8639 -26.1293 2 -26.17107 -26.17110 -25.9392 -26.2059 5 -26.79526 -26.79531 -26.5919 -26.8851 1s22s2p-13d-2 1 -25.56896 -25.56899 -25.3483 -25.6074 1.5 -25.93109 -25.93113 -25.7194 -25.9812

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129 Table B-7 ( continued ) State B Present HF Numerical HF LDA PBE 1s22s2p-13d-2 1.6761 -26.03517 -26.03522 -25.8276 -26.0903 1.8143 -26.11044 -26.11049 -25.9064 -26.1697 2 -26.20410 -26.20413 -26.0050 -26.2692 2.23984 -26.31433 -26.31437 -26.1219 -26.3873 2.4779 -26.41397 -26.41401 -26.2282 -26.4947 2.5 -26.42279 -26.42283 -26.2376 -26.5043 5 -27.12857 -27.12860 -26.9889 -27.2731 1s22p02p-13d-2 1 -25.36887 -25.36901 -25.2182 -25.4660 1.5 -25.80947 -25.80961 -25.6638 -25.9134 1.8143 -26.04216 -25.8986 -26.1492 2 -26.16559 -26.16572 -26.0246 -26.2759 2.23984 -26.31425 -26.31437 -26.1753 -26.4274 2.4779 -26.45005 -26.45018 -26.3130 -26.5660 2.5 -26.46211 -26.46224 -26.3252 -26.5784 3 -26.71284 -26.71297 -26.5795 -26.8347 4 -27.10845 -27.10860 -26.9807 -27.2408 5 -27.39426 -27.39442 -27.2706 -27.5363 10 -27.77130 -27.77139 -27.6590 -27.9591 1s22p-13d-24f-3 1 -25.20248 -25.20257 -24.9789 -25.2387 2 -26.11853 -26.11859 -25.8903 -26.1567 2.4779 -26.45012 -26.45018 -26.2208 -26.4896 2.5 -26.46426 -26.46431 -26.2348 -26.5038 3 -26.76014 -26.76019 -26.5297 -26.8011 4 -27.23762 -27.23765 -27.0051 -27.2814 5 -27.59735 -27.59737 -27.3627 -27.6439 7 -28.05666 -28.05669 -27.8174 -28.1092 7.957 -28.17992 -28.17996 -27.9384 -28.2355 8 -28.18421 -28.18424 -27.9426 -28.2400 10 -28.27942 -28.27946 -28.0333 -28.3422 50 -13.06551 -13.06555 -12.8236 -13.3414 1s2p-13d-24f-35g-4 1 -18.45800 -18.45809 -18.2958 -18.4798 5 -24.83952 -24.83956 -24.6556 -24.8668 7 -27.12782 -27.12785 -26.9412 -27.1632 7.957 -28.11827 -27.9309 -28.1581 8 -28.16147 -28.16150 -27.9741 -28.2015 10 -30.06359 -30.06363 -29.8759 -30.1138 20 -37.55464 -37.55469 -37.3796 -37.6684 30 -43.21386 -43.213901 -43.0648 -43.3998 40 -47.88388 -47.883924 -47.7676 -48.1443 50 -51.91495 -51.91499 -51.8351 -52.2500 100 -66.99696 -66.99699 -67.1195 -67.6908 200 -86.60730 -86.60738 -87.1424 -87.9436 500 -121.16486 -121.16488 -122.8141 -124.0823 1000 -155.32948 -155.3296 -158.5373 -160.3385 2000 -197.86533 -197.8655 -203.6173 -206.1782

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130 Table B-8 Atomic energies of the Carbon atom in B fields. Energies in Hartree, B in au. Numerical HF is from reference 90; Ground states are indicated in orange. State B Present HF Numerical HF LDA PBE 1s22s22p02p-1 0 -37.69090 -37.69096 -37.4680 -37.7944 0.01 -37.70580 -37.7059 -37.4829 -37.8093 0.05 -37.76304 -37.7633 -37.5400 -37.8664 0.1 -37.82994 -37.8302 -37.6066 -37.9330 0.2 -37.94835 -37.9486 -37.7241 -38.0506 0.3 -38.04780 -38.0479 -37.8224 -38.1492 0.4 -38.13006 -38.1302 -37.9035 -38.2306 0.5 -38.19664 -38.1973 -37.9689 -38.2964 1s22s2p02p-12p+1 0.1 -37.78785 -37.7882 -37.5080 -37.8355 0.2 -37.95500 -37.9552 -37.6737 -38.0013 0.3 -38.10236 -38.1026 -37.8191 -38.1471 0.4 -38.23206 -38.2323 -37.9468 -38.2753 0.5 -38.34613 -38.3464 -38.0591 -38.3881 0.6 -38.44601 -38.4467 -38.1573 -38.4869 0.8 -38.61100 -38.6116 -38.3196 -38.6506 1 -38.73696 -38.7373 -38.4438 -38.7761 1s22s2p02p-13d-2 0.4 -38.16215 -38.1624 -37.8764 -38.2027 0.5 -38.35375 -38.3541 -38.0673 -38.3935 0.6 -38.53370 -38.5339 -38.2462 -38.5727 0.8 -38.86300 -38.8632 -38.5731 -38.9006 1 -39.15752 -39.1577 -38.8652 -39.1941 2 -40.27685 -40.2769 -39.9767 -40.3135 3 -41.04743 -41.0477 -40.7464 -41.0903 4 -41.63173 -41.6319 -41.3355 -41.6861 5 -42.10143 -42.1016 -41.8142 -42.1717 10 -43.47679 -43.4769 -43.2522 -43.6525 1s22p02p-13d-24f-3 1 -38.40409 -38.4043 -38.1922 -38.5107 2 -39.76198 -39.7621 -39.5550 -39.8769 3 -40.77794 -40.7780 -40.5769 -40.9029 4 -41.58847 -41.5886 -41.3931 -41.7233 5 -42.25476 -42.2549 -42.0645 -42.3992 7 -43.27693 -43.2771 -43.0950 -43.4400 8 -43.66834 -43.6685 -43.4900 -43.8406 10 -44.26570 -44.2659 -44.0938 -44.4562 11 -44.48601 -44.3170 -44.6856 12 -44.66133 -44.6615 -44.4952 -44.8702 1s22p-13d-24f-35g-4 1 -37.81283 -37.8130 -37.5149 -37.8497 5 -42.06075 -42.0608 -41.7472 -42.1113 7 -43.21948 -43.2195 -42.9043 -43.2802 8 -43.67332 -43.6734 -43.3576 -43.7393 10 -44.38714 -44.3872 -44.0704 -44.4637 11 -44.66223 -44.3451 -44.7443 12 -44.89040 -44.8905 -44.5730 -44.9780 15 -45.33461 -45.3348 -45.0168 -45.4394 20 -45.44642 -45.4465 -45.1298 -45.5816 1s2p02p-13d-24f-35g-4 1 -28.01924 -28.0195 -27.9066 -28.1223

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131 Table B-8 ( continued ) State B Present HF Numerical HF LDA PBE 1s2p02p-13d-24f-35g-4 5 -35.96005 -35.9601 -35.8547 -36.0840 10 -42.52763 -42.5277 -42.4404 -42.6918 11 -43.64185 -43.5586 -43.8148 12 -44.70921 -44.7094 -44.6300 -44.8911 15 -47.67672 -47.6770 -47.6103 -47.8869 20 -52.02805 -52.0282 -51.9852 -52.2884 30 -59.27456 -59.2747 -59.2851 -59.6420 50 -70.51859 -70.5187 -70.6485 -71.1076 1s2p-13d-24f-35g-46h-4 1 -26.78405 -26.7843 -26.5823 -26.8134 5 -35.18143 -35.1815 -34.9563 -35.2215 10 -42.07983 -42.0799 -41.8560 -42.1517 15 -47.50012 -45.5002 -47.2857 -47.6090 20 -52.08898 -52.0890 -51.8882 -52.2377 30 -59.74324 -59.7433 -59.5777 -59.9760 40 -66.10717 -66.1073 -65.9830 -66.4261 50 -71.62837 -71.6285 -71.5491 -72.0335 100 -92.45517 -92.4552 -92.6194 -93.2756 200 -119.81264 -119.8127 -120.4671 -121.3785 500 -168.52468 -168.5248 -170.4975 -171.9285 1000 -217.14114 -217.1413 -220.9602 -222.9850 2000 -278.16083 -278.1612 -285.0006 -287.8708

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132 APPENDIX C EXCHANGE AND CORRELATION ENERGI ES OF ATOMS HE, LI, BE, AND POSITIVE IONS LI+, BE+ IN MAGNETIC FIELDS All energies in Hartree; B in au; Negative signs are omitted. H F cE values are derived from the energy difference between the correlat ed calculations and HF energies listed in Appendix B. Notice the correlated data may be from different sources. Refer to Appendix B for the cited references. Table C-1 Exchange and correlation energies of the Helium atom in magnetic fields B H F x E H F c E H F xc E LDA x E LDA c E LDA xc E P BE x E P BE c E P BE xc E 1s2 0 1.0258 0.0417 1.0674 0.8618 0.1111 0.9729 1.0051 0.0411 1.0462 0.02 1.0258 0.0417 1.0675 0.8618 0.1111 0.9729 1.0052 0.0411 1.0462 0.04 1.0259 0.0417 1.0676 0.8619 0.1111 0.9730 1.0053 0.0411 1.0464 0.08 1.0264 0.0417 1.0681 0.8624 0.1111 0.9735 1.0058 0.0411 1.0469 0.16 1.0284 0.0416 1.0701 0.8642 0.1112 0.9755 1.0077 0.0412 1.0489 0.24 1.0316 0.0416 1.0732 0.8671 0.1114 0.9786 1.0108 0.0413 1.0520 0.4 1.0410 0.0415 1.0825 0.8756 0.1120 0.9876 1.0197 0.0416 1.0613 0.5 1.0485 0.0414 1.0899 0.8823 0.1124 0.9947 1.0268 0.0419 1.0687 0.8 1.0759 0.0407 1.1166 0.9063 0.1138 1.0201 1.0523 0.0427 1.0950 1 1.0964 0.0406 1.1371 0.9241 0.1148 1.0389 1.0713 0.0433 1.1147 1.6 1.1616 0.0406 1.2021 0.9799 0.1178 1.0977 1.1316 0.0450 1.1766 2 1.2048 0.0406 1.2454 1.0167 0.1197 1.1364 1.1718 0.0460 1.2178 5 1.4830 0.0424 1.5254 1.2555 0.1301 1.3856 1.4357 0.0508 1.4864 10 1.8186 0.0460 1.8647 1.5485 0.1406 1.6891 1.7644 0.0545 1.8189 20 2.2820 0.0526 2.3346 1.9629 0.1527 2.1155 2.2346 0.0576 2.2922 50 3.1300 0.0676 3.1975 2.7514 0.1703 2.9217 3.1404 0.0604 3.2009 100 3.9856 0.0859 4.0715 3.5876 0.1845 3.7721 4.1134 0.0614 4.1747 1s2s 0 0.7435 0.0010 0.7445 0.6322 0.0431 0.6753 0.7310 0.0123 0.7433 0.02 0.7441 0.0010 0.7450 0.6327 0.0432 0.6758 0.7315 0.0123 0.7438 0.04 0.7455 0.0010 0.7465 0.6340 0.0433 0.6773 0.7329 0.0124 0.7453 0.08 0.7502 0.0010 0.7513 0.6384 0.0436 0.6820 0.7375 0.0125 0.7500 0.16 0.7616 0.0012 0.7628 0.6492 0.0442 0.6935 0.7489 0.0129 0.7618 0.24 0.7722 0.0013 0.7735 0.6596 0.0448 0.7044 0.7603 0.0129 0.7733 0.4 0.7884 0.0015 0.7898 0.6772 0.0458 0.7231 0.7806 0.0128 0.7934 0.5 0.7958 0.0015 0.7973 0.6868 0.0464 0.7332 0.7922 0.0128 0.8050 0.8 0.8136 0.0015 0.8151 0.7127 0.0477 0.7604 0.8232 0.0131 0.8363 1 0.8251 0.0015 0.8266 0.7288 0.0485 0.7773 0.8422 0.0132 0.8554 1.6 0.8625 0.0015 0.8639 0.7747 0.0503 0.8249 0.8953 0.0135 0.9088 2 0.8881 0.0015 0.8896 0.8037 0.0513 0.8549 0.9288 0.0136 0.9424 5 1.0589 0.0015 1.0604 0.9871 0.0563 1.0434 1.1419 0.0141 1.1560

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133 Table C-1 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s2s 10 1.2666 0.0015 1.2681 1.2098 0.0612 1.2709 1.4040 0.0144 1.4185 20 1.5510 0.0014 1.5524 1.5239 0.0667 1.5906 1.7782 0.0147 1.7929 50 2.0645 0.0013 2.0657 2.1222 0.0749 2.1971 2.5010 0.0148 2.5158 100 2.5769 0.0011 2.5780 2.7557 0.0816 2.8372 3.2752 0.0147 3.2898 1s2p0 0 0.7332 0.0016 0.7348 0.6356 0.0438 0.6794 0.7358 0.0111 0.7469 0.02 0.7333 0.0015 0.7348 0.6370 0.0439 0.6809 0.7379 0.0111 0.7490 0.04 0.7349 0.0015 0.7364 0.6385 0.0440 0.6825 0.7394 0.0112 0.7506 0.08 0.7400 0.0016 0.7416 0.6432 0.0443 0.6876 0.7442 0.0114 0.7556 0.16 0.7524 0.0019 0.7543 0.6551 0.0451 0.7003 0.7565 0.0118 0.7683 0.24 0.7644 0.0022 0.7666 0.6665 0.0458 0.7124 0.7682 0.0121 0.7803 0.4 0.7869 0.0026 0.7895 0.6890 0.0471 0.7361 0.7917 0.0127 0.8044 0.5 0.7999 0.0029 0.8028 0.7022 0.0477 0.7499 0.8054 0.0131 0.8185 0.8 0.8364 0.0034 0.8398 0.7383 0.0493 0.7876 0.8436 0.0138 0.8574 1 0.8592 0.0037 0.8628 0.7606 0.0502 0.8108 0.8673 0.0142 0.8815 1.6 0.9226 0.0042 0.9267 0.8215 0.0523 0.8738 0.9331 0.0150 0.9481 2 0.9614 0.0044 0.9658 0.8583 0.0534 0.9118 0.9734 0.0155 0.9889 5 1.1898 0.0053 1.1951 1.0755 0.0590 1.1344 1.2172 0.0169 1.2340 10 1.4425 0.0060 1.4484 1.3233 0.0640 1.3872 1.5028 0.0175 1.5203 20 1.7681 0.0067 1.7748 1.6602 0.0694 1.7296 1.8990 0.0175 1.9165 50 2.3227 0.0072 2.3299 2.2838 0.0774 2.3611 2.6469 0.0171 2.6640 100 2.8526 0.0072 2.8597 2.9360 0.0838 3.0198 3.4425 0.0165 3.4591 1s2p-1 0 0.7276 0.0018 0.7295 0.6239 0.0430 0.6669 0.7210 0.0107 0.7317 0.02 0.7281 0.0018 0.7298 0.6249 0.0431 0.6680 0.7223 0.0107 0.7331 0.04 0.7307 0.0018 0.7325 0.6273 0.0432 0.6706 0.7248 0.0108 0.7356 0.08 0.7385 0.0020 0.7405 0.6347 0.0438 0.6785 0.7323 0.0110 0.7433 0.16 0.7562 0.0026 0.7588 0.6515 0.0449 0.6964 0.7495 0.0114 0.7609 0.24 0.7732 0.0031 0.7763 0.6677 0.0459 0.7136 0.7661 0.0117 0.7777 0.4 0.8046 0.0040 0.8086 0.6976 0.0475 0.7451 0.7963 0.0121 0.8085 0.5 0.8229 0.0045 0.8274 0.7150 0.0483 0.7633 0.8139 0.0124 0.8263 0.8 0.8744 0.0054 0.8798 0.7629 0.0504 0.8132 0.8625 0.0133 0.8758 1 0.9066 0.0058 0.9124 0.7924 0.0515 0.8438 0.8928 0.0138 0.9066 1.6 0.9955 0.0066 1.0021 0.8729 0.0541 0.9270 0.9762 0.0153 0.9915 2 1.0495 0.0069 1.0563 0.9215 0.0555 0.9769 1.0271 0.0162 1.0432 5 1.3654 0.0082 1.3736 1.2070 0.0622 1.2692 1.3316 0.0199 1.3514 10 1.7213 0.0100 1.7313 1.5357 0.0681 1.6039 1.6876 0.0226 1.7102 20 2.1979 0.0129 2.2107 1.9892 0.0746 2.0639 2.1829 0.0251 2.2080 50 3.0525 0.0196 3.0721 2.8410 0.0839 2.9249 3.1211 0.0278 3.1489 100 3.9053 0.0281 3.9334 3.7398 0.0912 3.8310 4.1194 0.0294 4.1489 1s3d-1 0 0.6690 0.0001 0.6691 0.5672 0.0379 0.6051 0.6627 0.0097 0.6724 0.02 0.6718 0.0001 0.6718 0.5695 0.0382 0.6077 0.6651 0.0098 0.6749 0.04 0.6765 0.0001 0.6766 0.5739 0.0387 0.6126 0.6700 0.0100 0.6800 0.08 0.6854 0.0002 0.6856 0.5826 0.0397 0.6223 0.6798 0.0104 0.6902 0.16 0.6993 0.0003 0.6996 0.5968 0.0411 0.6379 0.6965 0.0109 0.7073 0.24 0.7104 0.0004 0.7108 0.6082 0.0420 0.6503 0.7095 0.0112 0.7207

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134 Table C-1 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s3d-1 0.4 0.7287 0.0007 0.7293 0.6276 0.0435 0.6711 0.7314 0.0117 0.7431 0.5 0.7387 0.0008 0.7396 0.6384 0.0442 0.6826 0.7435 0.0120 0.7554 0.8 0.7663 0.0013 0.7675 0.6681 0.0459 0.7139 0.7761 0.0126 0.7886 1 0.7835 0.0015 0.7850 0.6866 0.0468 0.7334 0.7963 0.0129 0.8092 1.6 0.8325 0.0021 0.8345 0.7384 0.0489 0.7873 0.8528 0.0135 0.8664 2 0.8632 0.0024 0.8656 0.7705 0.0501 0.8205 0.8880 0.0139 0.9019 5 1.0543 0.0037 1.0580 0.9655 0.0557 1.0212 1.1061 0.0154 1.1215 10 1.2780 0.0048 1.2828 1.1939 0.0608 1.2547 1.3679 0.0164 1.3843 20 1.5783 0.0060 1.5843 1.5077 0.0664 1.5741 1.7343 0.0171 1.7515 50 2.1099 0.0076 2.1175 2.0906 0.0744 2.1649 2.4281 0.0176 2.4457 100 2.6322 0.0085 2.6407 2.6997 0.0808 2.7804 3.1655 0.0176 3.1831 1s3d-2 0 0.6722 0.0005 0.6726 0.5676 0.0380 0.6056 0.6607 0.0103 0.6710 0.02 0.6744 0.0002 0.6746 0.5713 0.0384 0.6097 0.6657 0.0103 0.6761 0.04 0.6799 0.0001 0.6800 0.5766 0.0390 0.6156 0.6716 0.0106 0.6822 0.08 0.6910 0.0002 0.6912 0.5869 0.0401 0.6270 0.6834 0.0110 0.6943 0.16 0.7086 0.0004 0.7090 0.6035 0.0417 0.6451 0.7020 0.0116 0.7136 0.24 0.7227 0.0006 0.7234 0.6170 0.0428 0.6598 0.7170 0.0120 0.7291 0.4 0.7463 0.0012 0.7475 0.6404 0.0444 0.6848 0.7421 0.0127 0.7548 0.5 0.7594 0.0015 0.7609 0.6537 0.0452 0.6989 0.7560 0.0131 0.7690 0.8 0.7951 0.0024 0.7975 0.6907 0.0472 0.7379 0.7940 0.0137 0.8077 1 0.8174 0.0029 0.8204 0.7141 0.0483 0.7624 0.8177 0.0140 0.8318 1.6 0.8806 0.0041 0.8847 0.7797 0.0508 0.8305 0.8847 0.0148 0.8994 2 0.9201 0.0048 0.9248 0.8203 0.0522 0.8725 0.9266 0.0152 0.9417 5 1.1654 0.0074 1.1729 1.0671 0.0589 1.1260 1.1869 0.0175 1.2044 10 1.4563 0.0099 1.4661 1.3577 0.0648 1.4225 1.4995 0.0196 1.5191 20 1.8551 0.0133 1.8684 1.7614 0.0713 1.8327 1.9383 0.0216 1.9598 50 2.5828 0.0207 2.6035 2.5221 0.0806 2.6026 2.7722 0.0237 2.7959 100 3.3180 0.0299 3.3478 3.3260 0.0879 3.4139 3.6606 0.0248 3.6854 1s4f-2 0 0.6506 0.0000 0.6506 0.5533 0.0360 0.5892 0.6452 0.0090 0.6542 0.02 0.6561 0.0000 0.6561 0.5562 0.0365 0.5927 0.6489 0.0091 0.6580 0.04 0.6622 0.0001 0.6623 0.5617 0.0372 0.5989 0.6554 0.0093 0.6647 0.08 0.6715 0.0002 0.6718 0.5712 0.0384 0.6096 0.6669 0.0097 0.6765 0.16 0.6848 0.0002 0.6850 0.5848 0.0399 0.6246 0.6831 0.0101 0.6932 0.24 0.6949 0.0002 0.6951 0.5952 0.0409 0.6360 0.6956 0.0104 0.7060 0.4 0.7115 0.0003 0.7118 0.6125 0.0423 0.6548 0.7162 0.0108 0.7271 0.5 0.7206 0.0004 0.7210 0.6221 0.0430 0.6650 0.7275 0.0110 0.7385 0.8 0.7458 0.0007 0.7465 0.6485 0.0446 0.6930 0.7578 0.0116 0.7694 1 0.7618 0.0009 0.7627 0.6651 0.0454 0.7105 0.7766 0.0119 0.7885 1.6 0.8072 0.0014 0.8086 0.7120 0.0475 0.7595 0.8293 0.0125 0.8419 2 0.8360 0.0017 0.8377 0.7413 0.0487 0.7900 0.8623 0.0129 0.8752 5 1.0180 0.0032 1.0212 0.9240 0.0542 0.9782 1.0691 0.0143 1.0833 10 1.2352 0.0044 1.2396 1.1419 0.0593 1.2011 1.3201 0.0152 1.3353 20 1.5305 0.0056 1.5361 1.4441 0.0649 1.5090 1.6741 0.0159 1.6900 50 2.0590 0.0070 2.0661 2.0088 0.0728 2.0816 2.3463 0.0165 2.3627 100 2.5815 0.0078 2.5894 2.6002 0.0793 2.6795 3.0648 0.0165 3.0813

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135 Table C-1 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s4f-3 0 0.6518 0.0000 0.6518 0.5561 0.0363 0.5925 0.6472 0.0096 0.6568 0.02 0.6588 0.0000 0.6588 0.5594 0.0369 0.5963 0.6517 0.0097 0.6614 0.04 0.6663 0.0001 0.6664 0.5658 0.0378 0.6036 0.6595 0.0099 0.6694 0.08 0.6776 0.0001 0.6777 0.5766 0.0390 0.6157 0.6720 0.0104 0.6823 0.16 0.6941 0.0002 0.6942 0.5921 0.0406 0.6327 0.6903 0.0108 0.7012 0.24 0.7070 0.0003 0.7073 0.6043 0.0417 0.6460 0.7045 0.0112 0.7157 0.4 0.7283 0.0006 0.7289 0.6246 0.0433 0.6679 0.7277 0.0118 0.7395 0.5 0.7401 0.0008 0.7408 0.6360 0.0440 0.6800 0.7403 0.0121 0.7524 0.8 0.7721 0.0015 0.7736 0.6675 0.0459 0.7133 0.7741 0.0130 0.7872 1 0.7920 0.0020 0.7941 0.6874 0.0468 0.7342 0.7952 0.0135 0.8088 1.6 0.8484 0.0035 0.8519 0.7439 0.0493 0.7931 0.8541 0.0148 0.8689 2 0.8838 0.0043 0.8881 0.7794 0.0506 0.8299 0.8914 0.0153 0.9066 5 1.1059 0.0093 1.1152 1.0000 0.0570 1.0570 1.1251 0.0173 1.1424 10 1.3732 0.0148 1.3880 1.2648 0.0628 1.3275 1.4106 0.0187 1.4293 20 1.7436 0.0221 1.7657 1.6356 0.0692 1.7048 1.8150 0.0201 1.8351 50 2.4251 0.0358 2.4608 2.3378 0.0783 2.4161 2.5880 0.0215 2.6096 100 3.1176 0.0503 3.1679 3.0815 0.0856 3.1670 3.4142 0.0222 3.4365 Table C-2 Exchange and correlation energies of the Li+ ion in magnetic fields B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s2 0 1.6517 0.0408 1.6925 1.3962 0.1336 1.5299 1.6170 0.0441 1.6611 0.01 1.6517 0.0409 1.6925 1.3962 0.1336 1.5299 1.6169 0.0441 1.6611 0.02 1.6516 0.0409 1.6925 1.3962 0.1336 1.5298 1.6169 0.0441 1.6611 0.05 1.6517 0.0411 1.6929 1.3963 0.1336 1.5299 1.6170 0.0441 1.6612 0.1 1.6519 0.0412 1.6931 1.3964 0.1336 1.5301 1.6172 0.0442 1.6613 0.2 1.6526 0.0412 1.6938 1.3971 0.1337 1.5307 1.6178 0.0442 1.6620 0.5 1.6574 0.0415 1.6990 1.4013 0.1339 1.5352 1.6224 0.0443 1.6666 1 1.6734 0.0415 1.7149 1.4153 0.1344 1.5498 1.6373 0.0446 1.6819 2 1.7250 0.0415 1.7664 1.4600 0.1362 1.5962 1.6851 0.0456 1.7307 5 1.9233 0.0414 1.9647 1.6297 0.1422 1.7719 1.8698 0.0487 1.9185 5.4 1.9498 0.0415 1.9913 1.6523 0.1429 1.7953 1.8947 0.0491 1.9437 10 2.2288 0.0425 2.2713 1.8923 0.1501 2.0425 2.1601 0.0522 2.2122 1s2p-1 0 1.1486 0.0016 1.1502 0.9956 0.0535 1.0491 1.1393 0.0114 1.1507 0.01 1.1486 0.0016 1.1503 0.9956 0.0535 1.0491 1.1393 0.0114 1.1507 0.02 1.1487 0.0016 1.1504 0.9957 0.0535 1.0492 1.1394 0.0114 1.1508 0.05 1.1495 0.0016 1.1511 0.9964 0.0536 1.0500 1.1402 0.0114 1.1515 0.1 1.1520 0.0017 1.1537 0.9988 0.0537 1.0525 1.1426 0.0114 1.1540 0.2 1.1605 0.0017 1.1623 1.0074 0.0541 1.0615 1.1517 0.0116 1.1633 0.5 1.1963 0.0037 1.1999 1.0412 0.0555 1.0967 1.1861 0.0121 1.1982 1 1.2579 0.0046 1.2625 1.0991 0.0576 1.1567 1.2450 0.0128 1.2578 2 1.3710 0.0057 1.3766 1.2034 0.0606 1.2640 1.3517 0.0142 1.3659 5 1.6540 0.0069 1.6608 1.4595 0.0662 1.5257 1.6195 0.0173 1.6368

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136 Table C-2 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s2p-1 5.4 1.6869 0.0070 1.6939 1.4893 0.0667 1.5560 1.6510 0.0176 1.6686 10 2.0118 0.0077 2.0195 1.7843 0.0715 1.8558 1.9661 0.0201 1.9863 Table C-3 Exchange and correlation energies of the Lithium atom in magnetic fields B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s22s 0 1.7813 0.0450 1.8263 1.5144 0.1502 1.6645 1.7514 0.0510 1.8024 0.001 1.7812 0.0448 1.8260 1.5144 0.1502 1.6645 1.7514 0.0510 1.8024 0.009 1.7813 0.0451 1.8264 1.5145 0.1502 1.6646 1.7515 0.0510 1.8025 0.01 1.7814 0.0453 1.8266 1.5144 0.1502 1.6646 1.7514 0.0510 1.8024 0.018 1.7816 0.0451 1.8266 1.5147 0.1502 1.6649 1.7517 0.0510 1.8027 0.02 1.7801 0.0477 1.8278 1.5145 0.1502 1.6647 1.7516 0.0510 1.8026 0.05 1.7836 0.0488 1.8324 1.5165 0.1504 1.6669 1.7536 0.0511 1.8047 0.054 1.7840 0.0451 1.8291 1.5169 0.1504 1.6673 1.7539 0.0511 1.8051 0.1 1.7891 0.0486 1.8377 1.5216 0.1508 1.6724 1.7589 0.0514 1.8103 0.126 1.7924 0.0453 1.8377 1.5248 0.1511 1.6758 1.7622 0.0515 1.8137 0.18 1.7995 0.0455 1.8450 1.5313 0.1516 1.6829 1.7692 0.0518 1.8210 0.2 1.8021 0.0495 1.8516 1.5338 0.1518 1.6856 1.7719 0.0519 1.8238 0.5 1.8339 0.0507 1.8846 1.5661 0.1541 1.7202 1.8079 0.0524 1.8603 0.54 1.8372 0.0462 1.8834 1.5698 0.1543 1.7241 1.8123 0.0524 1.8647 0.9 1.8616 0.0460 1.9076 1.6009 0.1558 1.7567 1.8493 0.0527 1.9020 1 1.8678 0.0498 1.9175 1.6091 0.1561 1.7653 1.8592 0.0528 1.9120 5 2.1753 0.0488 2.2241 1.9330 0.1675 2.1005 2.2295 0.0563 2.2858 10 2.5201 0.0487 2.5689 2.2720 0.1777 2.4497 2.6187 0.0593 2.6780 1s22p-1 0 1.7502 0.0422 1.7923 1.4905 0.1485 1.6390 1.7261 0.0495 1.7755 0.009 1.7497 0.0448 1.7945 1.4908 0.1485 1.6393 1.7268 0.0495 1.7762 0.01 1.7497 0.0422 1.7920 1.4908 0.1485 1.6393 1.7268 0.0495 1.7763 0.018 1.7503 0.0448 1.7950 1.4913 0.1486 1.6399 1.7273 0.0495 1.7768 0.05 1.7547 0.0427 1.7974 1.4954 0.1490 1.6444 1.7315 0.0497 1.7812 0.054 1.7554 0.0449 1.8003 1.4961 0.1490 1.6451 1.7322 0.0497 1.7819 0.1 1.7648 0.0430 1.8078 1.5049 0.1499 1.6548 1.7414 0.0501 1.7914 0.126 1.7703 0.0454 1.8157 1.5101 0.1505 1.6605 1.7469 0.0503 1.7971 0.18 1.7814 0.0458 1.8272 1.5207 0.1515 1.6722 1.7579 0.0506 1.8086 0.2 1.7854 0.0439 1.8292 1.5245 0.1519 1.6764 1.7619 0.0508 1.8127 0.5 1.8385 0.0467 1.8852 1.5758 0.1566 1.7324 1.8147 0.0527 1.8674 0.54 1.8450 0.0487 1.8937 1.5821 0.1571 1.7392 1.8211 0.0529 1.8740 0.9 1.9010 0.0508 1.9518 1.6354 0.1615 1.7969 1.8754 0.0552 1.9306 1 1.9161 0.0502 1.9663 1.6496 0.1626 1.8121 1.8899 0.0559 1.9457 1.26 1.9549 0.0522 2.0071 1.6857 0.1652 1.8508 1.9267 0.0575 1.9842 1.8 2.0339 0.0531 2.0869 1.7580 0.1698 1.9279 2.0014 0.0606 2.0620 2 2.0626 0.0533 2.1159 1.7841 0.1714 1.9555 2.0285 0.0617 2.0902 3.6 2.2806 0.0495 2.3301 1.9796 0.1814 2.1610 2.2343 0.0685 2.3028 5 2.4529 0.0601 2.5130 2.1335 0.1879 2.3214 2.3985 0.0728 2.4713

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137 Table C-3 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s22p-1 5.4 2.4992 0.0410 2.5402 2.1749 0.1895 2.3644 2.4429 0.0738 2.5167 10 2.9587 0.0663 3.0250 2.5879 0.2036 2.7916 2.8900 0.0819 2.9719 1s2p-13d-2 0 1.2019 0.0588 1.2607 1.0494 0.0634 1.1129 1.1994 0.0158 1.2152 0.1 1.2391 0.0197 1.2588 1.0835 0.0666 1.1501 1.2357 0.0177 1.2535 0.5 1.3816 0.0118 1.3934 1.2104 0.0739 1.2843 1.3677 0.0226 1.3903 1 1.5128 0.0116 1.5243 1.3316 0.0789 1.4105 1.4925 0.0249 1.5174 2 1.7190 0.0101 1.7291 1.5233 0.0851 1.6084 1.6889 0.0275 1.7164 5 2.1788 0.0148 2.1936 1.9497 0.0953 2.0450 2.1320 0.0323 2.1644 10 2.7217 0.0183 2.7400 2.4590 0.1043 2.5633 2.6709 0.0367 2.7076 Table C-4 Exchange and correlation energies of the Be+ ion in magnetic fields B H F x E H F c E H F xc E LDA x E LDA c E LDA xc E P BE x E P BE c E P BE xc E 1s22s 0 2.5072 0.0472 2.5545 2.1431 0.1723 2.3154 2.4662 0.0539 2.5201 0.001 2.5072 0.0471 2.5544 2.1431 0.1723 2.3154 2.4662 0.0539 2.5201 0.002 2.5072 0.0438 2.5510 2.1431 0.1723 2.3154 2.4662 0.0539 2.5200 0.004 2.5071 0.0437 2.5508 2.1431 0.1723 2.3153 2.4662 0.0539 2.5200 0.01 2.5072 0.0472 2.5544 2.1431 0.1723 2.3154 2.4662 0.0539 2.5201 0.02 2.5073 0.0438 2.5511 2.1432 0.1723 2.3154 2.4663 0.0539 2.5201 0.05 2.5077 0.0471 2.5548 2.1435 0.1723 2.3159 2.4667 0.0539 2.5205 0.1 2.5091 0.0473 2.5564 2.1449 0.1724 2.3173 2.4681 0.0539 2.5220 0.2 2.5140 0.0473 2.5613 2.1496 0.1727 2.3223 2.4730 0.0541 2.5271 0.3 2.5208 0.0438 2.5646 2.1560 0.1731 2.3291 2.4798 0.0543 2.5341 0.4 2.5285 0.0437 2.5722 2.1634 0.1736 2.3370 2.4876 0.0546 2.5421 0.5 2.5364 0.0478 2.5842 2.1711 0.1740 2.3451 2.4959 0.0548 2.5507 0.6 2.5445 0.0437 2.5881 2.1790 0.1745 2.3534 2.5043 0.0550 2.5593 0.7 2.5523 0.0435 2.5958 2.1867 0.1749 2.3616 2.5127 0.0552 2.5679 0.8 2.5599 0.0436 2.6035 2.1944 0.1753 2.3697 2.5211 0.0553 2.5764 1 2.5742 0.0482 2.6224 2.2092 0.1761 2.3853 2.5375 0.0555 2.5930 2 2.6308 0.0478 2.6786 2.2751 0.1786 2.4537 2.6140 0.0562 2.6701 5 2.7931 0.0469 2.8400 2.4604 0.1832 2.6436 2.8266 0.0578 2.8844 10 3.0887 0.0466 3.1353 2.7576 0.1906 2.9482 3.1641 0.0601 3.2242 1s22p-1 0 2.4804 0.0432 2.5236 2.1278 0.1726 2.3004 2.4489 0.0524 2.5013 0.01 2.4805 0.0433 2.5237 2.1279 0.1726 2.3005 2.4490 0.0524 2.5013 0.05 2.4813 0.0425 2.5238 2.1292 0.1726 2.3018 2.4507 0.0524 2.5032 0.1 2.4838 0.0433 2.5271 2.1315 0.1728 2.3043 2.4531 0.0525 2.5056 0.2 2.4921 0.0436 2.5356 2.1393 0.1734 2.3127 2.4612 0.0528 2.5140 0.3 2.5027 0.0434 2.5461 2.1495 0.1741 2.3236 2.4716 0.0531 2.5247 0.4 2.5144 0.0433 2.5577 2.1605 0.1749 2.3354 2.4829 0.0534 2.5364 0.5 2.5263 0.0442 2.5705 2.1719 0.1757 2.3476 2.4946 0.0538 2.5484 0.6 2.5384 0.0430 2.5814 2.1833 0.1765 2.3598 2.5063 0.0542 2.5605 0.7 2.5503 0.0429 2.5932 2.1947 0.1773 2.3720 2.5179 0.0545 2.5725

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138 Table C-4 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s22p-1 0.8 2.5622 0.0427 2.6049 2.2059 0.1781 2.3840 2.5295 0.0549 2.5844 1 2.5856 0.0461 2.6317 2.2281 0.1795 2.4076 2.5521 0.0557 2.6077 2 2.6978 0.0482 2.7460 2.3329 0.1859 2.5188 2.6593 0.0593 2.7186 5 3.0137 0.0502 3.0639 2.6189 0.1991 2.8179 2.9573 0.0680 3.0253 10 3.4749 0.0510 3.5259 3.0310 0.2128 3.2439 3.3966 0.0769 3.4735 1s2p-13d-2 0 1.6837 0.0051 1.6888 1.4780 0.0756 1.5536 1.6788 0.0174 1.6962 0.1 1.6984 0.0053 1.7037 1.4920 0.0766 1.5686 1.6942 0.0181 1.7123 0.2 1.7236 0.0056 1.7292 1.5145 0.0779 1.5925 1.7174 0.0191 1.7364 0.5 1.8016 0.0063 1.8078 1.5841 0.0813 1.6654 1.7893 0.0215 1.8109 1 1.9164 0.0072 1.9236 1.6883 0.0853 1.7736 1.8973 0.0238 1.9211 2 2.1048 0.0081 2.1130 1.8623 0.0906 1.9529 2.0762 0.0263 2.1025 5 2.5353 0.0085 2.5439 2.2615 0.0997 2.3612 2.4875 0.0302 2.5177 10 3.0723 0.0081 3.0804 2.7605 0.1081 2.8685 3.0093 0.0343 3.0436 Table C-5 Exchange and correlation energies of the Beryllium atom in magnetic fields B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s22s2 0 2.6669 0.0675 2.7344 2.2903 0.2237 2.5140 2.6336 0.0854 2.7190 0.001 2.6669 0.0680 2.7348 2.2903 0.2237 2.5140 2.6335 0.0854 2.7190 0.01 2.6670 0.0679 2.7349 2.2904 0.2237 2.5141 2.6336 0.0854 2.7191 0.02 2.6672 0.0899 2.7571 2.2906 0.2237 2.5143 2.6338 0.0854 2.7193 0.05 2.6686 0.0699 2.7384 2.2919 0.2239 2.5157 2.6351 0.0856 2.7207 0.1 2.6731 0.0708 2.7440 2.2960 0.2243 2.5204 2.6395 0.0859 2.7254 0.5 2.7390 0.0810 2.8199 2.3583 0.2303 2.5886 2.7054 0.0899 2.7954 1 2.8081 0.1277 2.9358 2.4295 0.2360 2.6655 2.7842 0.0918 2.8761 1s22s2p-1 0.01 2.7211 0.0475 2.7686 2.3394 0.1930 2.5324 2.6752 0.0644 2.7396 0.02 2.7214 0.0503 2.7718 2.3398 0.1930 2.5328 2.6755 0.0644 2.7399 0.05 2.7238 0.0314 2.7552 2.3421 0.1932 2.5353 2.6780 0.0645 2.7425 0.10 2.7312 0.0481 2.7793 2.3494 0.1937 2.5431 2.6854 0.0649 2.7502 0.15 2.7412 0.0502 2.7915 2.3591 0.1943 2.5534 2.6952 0.0653 2.7605 0.2 2.7527 0.0473 2.8000 2.3699 0.1950 2.5649 2.7063 0.0658 2.7721 0.3 2.7769 0.0504 2.8273 2.3929 0.1964 2.5892 2.7297 0.0668 2.7965 0.4 2.8006 0.0504 2.8511 2.4156 0.1978 2.6134 2.7530 0.0677 2.8207 0.5 2.8231 0.0481 2.8712 2.4373 0.1991 2.6364 2.7753 0.0685 2.8438 0.6 2.8439 0.0501 2.8940 2.4577 0.2003 2.6580 2.7962 0.0693 2.8655 0.8 2.8805 0.0485 2.9289 2.4946 0.2026 2.6972 2.8342 0.0706 2.9048 1 2.9109 0.0493 2.9602 2.5269 0.2045 2.7314 2.8675 0.0716 2.9391 2 3.0088 0.0470 3.0558 2.6491 0.2114 2.8605 2.9972 0.0738 3.0710 5 3.3021 0.0484 3.3505 2.9669 0.2252 3.1921 3.3478 0.0787 3.4265 1s22p-13d-2 0.1 2.5723 0.0303 2.6026 2.2162 0.1855 2.4017 2.5458 0.0587 2.6044 0.2 2.6132 0.0317 2.6449 2.2516 0.1880 2.4397 2.5827 0.0606 2.6433

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139 Table C-5 ( continued ) B H F xE H F cE H F xcE LDA xE LDA cE LDA xcE P BE xE P BE cE P BE xcE 1s22p-13d-2 0.5 2.7135 0.0303 2.7438 2.3414 0.1938 2.5352 2.6771 0.0643 2.7413 1 2.8413 0.0357 2.8771 2.4598 0.2007 2.6605 2.8002 0.0680 2.8682 2 3.0441 0.0468 3.0909 2.6495 0.2107 2.8602 2.9952 0.0738 3.0691 5 3.5314 0.0390 3.5704 3.1011 0.2305 3.3316 3.4635 0.0867 3.5501 10 4.1768 0.0411 4.2179 3.6967 0.2503 3.9470 4.0931 0.0995 4.1926 1s2p-13d-24f-3 1 2.1659 0.0071 2.1730 1.9150 0.1064 2.0215 2.1369 0.0371 2.1740 2 2.4496 0.0066 2.4562 2.1769 0.1150 2.2919 2.4085 0.0413 2.4498 5 3.0516 0.0070 3.0586 2.7421 0.1286 2.8707 2.9930 0.0473 3.0403 10 3.7658 0.0067 3.7725 3.4207 0.1406 3.5614 3.7013 0.0527 3.7540

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140 APPENDIX D EFFECTIVE POTENTIAL INTEGRALS WI TH RESPECT TO LANDAU ORBITALS IN EQUATION (5.30) The explicit expression for the Landau orbitals is 2 21 ,Lanim nmns L LeIns a a (D.1) where 0 snm 221 !!xsn s nx ns sd I xexxe dx ns (D.2) and 2La B is the Larmor radius. Consider the expansion '11 ,;00nnm LanLan nmnm smbssnnsss rr (D.3) Comparison of the powers of 2 on both sides immediately yields 1/2 00! !!!!! ,; !!!!!! !!!nn sm kk kksmsm sssnn bssnns nnknkknk s mkmsk (D.4) Here ns and ns are assumed without loss of generality. Three different methods are used fo r the calculation of the potentials 2 011 00 !sx sxe Vzssdx rs xz (D.5) in different regions:

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141 For 2 z and 08s, closed forms for the integrals are used. See Table D-1, and note that erf ( z ) is the usual error function. Table D-1 Expressions for sVzwith 2 z and 08 s s sVz 0 21zeerfz 1 22112 2zzeerfzz 2 23 243 1344 428zzz eerfzzz 3 235 2465 11518128 83648zzzz eerfzzzz 4 2357 246835255 1105120723216 64964824384zzzzz eerfzzzzz 5 23579 24681063719 194510506002408032 12832240401203840zzzzzze erfzzzzzz 6 2357911 24681012231492137 5122563201601440720 110395113406300240072019264 46080zzzzzzz e erfzzzzzzz 7 235791113 246810121442911731373 1 10246412808402016012605040645120 *135135145530793802940084002016448128zzzzzzzze erfz zzzzzzz 8 23579111315 2468101214166435128720996310719 1 163848192409671680358403225689604032010321920 *2027025216216011642404233601176002688053761024256zzzzzzzzze erfz zzzzzzzz For 2100 z and 08 s ,and for 100 z and 8 s the expression (D.5) is integrated with the Gauss-Laguerre qu adrature formula of order 24 [126]. For 100 z we use its large z asymptotic expansion

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142 356 24 24681012111 11 1 11353563231 1 28161282561024ssss ss s Vz zzzzzzz where 112issssi (D.6)

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143 APPENDIX E ENERGY VALUES FOR HOOKE’S ATOM IN MAGNETIC FIELDS Table E-1 Relative motion and spin energies for the Hooke’s atom in B fields ( = 1/2, energy in Hartree). Numbers in parent heses denote the number of functions used in the spherical expansion; Numb ers in brackets denote the number of functions used in the La ndau orbital expansion. B au 1101 s 3 0102 p 3 1112 p 0 1.250000000 (3) 1.609657060 (3) 1.609657060 (3) 0.0001 1.250000003 (3) 1.609557063 (3) 1.609507066 (3) 0.00015 1.250000008 (3) 1.609507066 (3) 1.609432073 (3) 0.0002 1.250000014 (3) 1.609457071 (3) 1.609357083 (3) 0.0003 1.250000031 (3) 1.609357086 (3) 1.609207112 (3) 0.0005 1.250000086 (3) 1.609157132 (3) 1.608907204 (3) 0.0007 1.250000168 (3) 1.608957201 (3) 1.608607342 (3) 0.001 1.250000342 (3) 1.608657347 (3) 1.608157635 (3) 0.0015 1.250000770 (3) 1.608157707 (3) 1.607408354 (3) 0.002 1.250001368 (3) 1.607658210 (3) 1.606659360 (3) 0.003 1.250003079 (3) 1.606659647 (3) 1.605162235 (3) 0.005 1.250008552 (3) 1.604664247 (3) 1.602171434 (3) 0.007 1.250016762 (3) 1.602671147 (3) 1.599185233 (3) 0.01 1.250034208 (3) 1.599685808 (3) 1.594714556 (3) 0.015 1.250076965 (5) 1.594721740 (5) 1.587286421 (5) 0.02 1.250136820 (5) 1.589772041 (5) 1.579887024 (5) 0.03 1.250307795 (5) 1.579915729 (5) 1.565174407 (5) 0.05 1.250854552 (5) 1.560375249 (5) 1.536093504 (5) 0.07 1.251673646 (5) 1.541063723 (5) 1.507470637 (5) 0.1 1.253410092 (5) 1.512523532 (5) 1.465391042 (5) 0.15 1.257642612 (5) 1.466083307 (5) 1.397514744 (5) 0.2 1.263513190 (5) 1.421024408 (5) 1.332407876 (5) 0.3 1.279947919 (5) 1.334879453 (5) 1.210179577 (5) 0.5 1.329535134 (7) 1.176881544 (7) 0.994625266 (7) 0.7 1.396923698 (7) 1.034429994 (7) 0.81085244 (7) 1.39703 [24] 1.034433 [12] 0.810855 [20] 1 1.520416489 (9) 0.841222544 (9) 0.5777205 (9) 1.520519 [24] 0.8412251 [12] 0.577723 [20] 1.5 1.75777857 (9) 0.550811694 (9) 0.2559303 (9) 1.757872 [24] 0.5508135 [12] 0.255933 [20] 2 2.0112016 (11) 0.279793095 (11) -0.0246336 (11) 2.011286 [24] 0.2797945 [12] -0.024631 [20] 3 2.5296152 (13) -0.239648205 (13) -0.539284 (13) 2.529682 [24] -0.2396471 [12] -0.539282 [20] 5 3.5644399 (17) -1.251756371 (19) -1.5219222 (19)

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144 Table E-1 ( continued ) B au 1101 s 3 0102 p 3 1112 p 5 3.5644866 [24] -1.25175534 [12] -1.521919 [17] 7 4.5886380 (21) -2.255103373 (21) -2.497629 (21) 4.5886734 [24] -2.25510106 [12] -2.497623 [14] 10 6.1124882 (25) -3.756420943 (25) -3.9672702 (25) 6.112518 [24] -3.7564155 [12] -3.967261 [12] 15 8.6359755 (25) -6.2564087 (25) -6.4323365 (25) 8.636000 [18] -6.2563985 [12] -6.432323 [10] 20 11.150104 (25) -8.755915 (25) -8.90935 (25) 11.150138 [15] -8.7558996 [12] -8.909340 [9] 30 16.1668 (25) -13.7548 (25) -13.8805 (25) 16.166748 [11] -13.7549175 [12] -13.880982 [7] 50 26.187 (25) -23.747 (25) -23.837 (25) 26.18302 [8] -23.753597 [12] -23.85238 [5] 70 36.22 (25) -33.72 (25) -33.77 (25) 36.19185 [6] -33.75201 [12] -33.83741 [5] 100 51.30 (25) -48.62 (25) -48.64 (25) 51.2025 [5] -48.7464 [12] -48.8221 [4] 150 76.49 (25) -73.38 (25) -73.37 (25) 76.224 [4] -73.726 [12] -73.7971 [3] 200 101.7 (25) -98.1 (25) -98.09 (25) 101.254 [4] -98.695 [12] -98.765 [3] 300 152.2 (25) -147.5 (25) -147.56 (25) 151.332 [3] -148.618 [12] -148.683 [3] 500 253.0 (25) -246.3 (25) -246.6 (25) 251.538 [2] -248.417 [12] -248.474 [2] 700 353.8 (25) -345.1 (25) -345.7 (25) 351.785 [3] -348.177 [12] -348.224 [3] 1000 504.8 (25) -493.4 (25) -494.6 (25) 502.208 [2] -497.764 [12] -497.798 [2] Table E-1 ( continued ) B au 1 1113 d 1 2123 d 3 2124 f 3 3134 f 0 2.043613898 (3) 2.043613898(3) 2.503840941(3) 2.503840941 (3) 0.0001 2.043563903 (3) 2.043513906(3) 2.503640949(3) 2.503590952 (3) 0.00015 2.043538910 (3) 2.043463916(3) 2.503540959(3) 2.503465965 (3) 0.0002 2.043513920 (3) 2.043413930(3) 2.503440973(3) 2.503340984 (3) 0.0003 2.043463947 (3) 2.043313971(3) 2.503241013(3) 2.503091037 (3) 0.0005 2.043364034 (3) 2.043114101(3) 2.502841140(3) 2.502591206 (3) 0.0007 2.043264164 (3) 2.042914297(3) 2.502441330(3) 2.502091459 (3) 0.001 2.043114441 (3) 2.042614712(3) 2.501841734(3) 2.501341999 (3) 0.0015 2.042865119 (3) 2.042115730(3) 2.500842726(3) 2.500093320 (3) 0.002 2.042616070 (3) 2.041617156(3) 2.499844113(3) 2.498845171 (3) 0.003 2.042118784 (3) 2.040621228(3) 2.497848078(3) 2.496350457 (3) 0.005 2.041127471 (3) 2.038634258(3) 2.493860765(3) 2.491367373 (3) 0.007 2.040140502 (3) 2.036653803(3) 2.489879796(3) 2.486392748 (3) 0.01 2.038668190 (3) 2.033695337(3) 2.483920236(3) 2.478946667 (3) 0.015 2.036236052 (5) 2.028797129(5) 2.474019348(5) 2.466578817 (5)

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145 Table E-1 ( continued ) B au 1 1113 d 1 2123 d 3 2124 f 3 3134 f 0.02 2.033831050 (5) 2.023939627(5) 2.464158094(5) 2.454263813 (5) 0.03 2.029102424 (5) 2.014346693(5) 2.444554442(5) 2.429792279 (5) 0.05 2.019970328 (5) 1.995648582(5) 2.405822055(5) 2.381482453 (5) 0.07 2.011270774 (5) 1.977599362(5) 2.367721481(5) 2.334015098 (5) 0.1 1.999028635 (5) 1.951736628(5) 2.311749856(5) 2.264386593 (5) 0.15 1.980756300 (5) 1.911830624(5) 2.221578322(5) 2.152492944 (5) 0.2 1.965100502 (5) 1.875853518(5) 2.135232658(5) 2.045703296 (5) 0.3 1.941337955 (5) 1.815247057(5) 1.973592906(5) 1.846876266 (5) 0.5 1.921179706 (7) 1.735281511(7) 1.690498779(7) 1.502941229 (7) 0.7 1.931222704 (7) 1.701058458(7) 1.452008304(7) 1.218792604 (7) 1.9312234 [12] 1.70105911 [17] 1.45200871 [12] 1.21879300 [13] 1 1.987024019 (9) 1.71190543 (9) 1.15503647 (9) 0.87438094 (9) 1.9870245 [12] 1.71190608 [16] 1.15503698 [12] 0.87438125 [12] 1.5 2.1451176 (11) 1.83029806 (9) 0.75863481 (11) 0.4338451 (11) 2.14511795 [12] 1.8302987 [15] 0.75863507 [12] 0.4338454 [11] 2 2.344898761 (11) 2.0134298 (11) 0.42660426 (13) 0.08101563 (11) 2.34489904 [12] 2.0134306 [14] 0.42660448 [12] 0.0810159 [11] 3 2.795651846 (15) 2.4588635 (15) -0.15632829 (15) -0.5139253 (15) 2.7956525 [12] 2.4588645 [13] -0.15632796 [12] -0.5139245 [10] 5 3.76164459 (19) 3.4447447 (19) -1.21660648 (19) -1.5623472 (19) 3.76164552 [12] 3.4447463 [11] -1.21660545 [12] -1.5623457 [8] 7 4.75047650 (23) 4.4583592 (23) -2.23786510 (23) -2.5626623 (23) 4.75047867 [12] 4.4583624 [10] -2.23786247 [12] -2.5626603 [8] 10 6.24446123 (23) 5.9843736 (23) -3.7504183 (23) -4.0452405 (23) 6.24446471 [12] 5.9843796 [9] -3.7504143 [12] -4.0452364 [7] 15 8.7419092 [12] 8.520735 [7] -6.2570279 [12] -6.512476 [6] 20 11.24164 [12] 11.04731 [7] -8.7588054 [12] -8.98549 [6] 30 16.242351 [12] 16.08239 [6] -13.759101 [12] -13.947178 [5] 50 26.24398 [12] 26.11964 [5] -23.757718 [12] -23.903968 [4] 70 36.24587 [12] 36.13961 [4] -33.75560 [12] -33.880158 [4] 100 51.2518 [12] 51.1588 [4] -48.7497 [12] -48.8571 [3] 150 76.272 [12] 76.1872 [3] -73.729 [12] -73.8259 [3] 200 101.303 [12] 101.221 [3] -98.699 [12] -98.790 [3] 300 151.381 [12] 151.306 [3] -148.620 [12] -148.703 [3] 500 251.582 [12] 251.519 [2] -248.419 [12] -248.487 [2] 700 351.822 [12] 351.771 [3] -348.179 [12] -348.234 [3] 1000 502.236 [12] 502.197 [2] -497.765 [12] -497.806 [2] Table E-2 As in Table E-1, but for = 1/10. B au 1101 s 3 1112 p 3 3134 f 0 0.35000000 (2) 0.40317279 (2) 0.56141220 (2) 0.0001 0.35000002 (2) 0.40302283 (2) 0.56116226 (2) 0.00015 0.35000005 (2) 0.40294787 (2) 0.56103733 (2) 0.0002 0.35000009 (2) 0.40287293 (2) 0.56091243 (2) 0.0003 0.35000020 (2) 0.40272309 (2) 0.56066271 (2)

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146 Table E-2 ( continued ) B au 1101 s 3 1112 p 3 3134 f 0.0005 0.35000056 (2) 0.40242363 (2) 0.56016361 (2) 0.0007 0.35000109 (2) 0.40212443 (2) 0.55966497 (2) 0.001 0.35000223 (2) 0.40167612 (2) 0.55891784 (2) 0.0015 0.35000501 (3) 0.40093028 (3) 0.55767489 (3) 0.002 0.35000891 (3) 0.40018611 (3) 0.55643476 (3) 0.003 0.35002005 (3) 0.39870275 (3) 0.55396296 (3) 0.005 0.35005568 (3) 0.39575601 (3) 0.54905317 (3) 0.007 0.35010912 (3) 0.39283586 (3) 0.54418844 (3) 0.01 0.35022257 (3) 0.38850546 (3) 0.53697577 (3) 0.015 0.35050018 (3) 0.38142058 (3) 0.52517916 (3) 0.02 0.35088769 (4) 0.37450044 (3) 0.51366193 (3) 0.03 0.35198763 (4) 0.36114883 (4) 0.49145730 (4) 0.05 0.35543780 (4) 0.33634307 (4) 0.45028105 (4) 0.07 0.36042632 (5) 0.31391694 (5) 0.41318824 (5) 0.1 0.37036790 (6) 0.28423501 (5) 0.36443746 (6) 0.37039495 [16] 0.28423695 [16] 0.36443759 [11] 0.15 0.39174385 (7) 0.24317081 (7) 0.29824181 (7) 0.39177144 [16] 0.24317281 [16] 0.29824194 [11] 0.2 0.41667766 (8) 0.20941540 (8) 0.24581813 (8) 0.41670496 [16] 0.20941744 [16] 0.24581826 [11] 0.3 0.47039300 (10) 0.15361448 (10) 0.16546431 (10) 0.47041748 [16] 0.15361649 [16] 0.16546444 [11] 0.34641* 0.49566784 (11) 0.13038476 (11) 0.13465525 (11) 0.49568856 [16] 0.13038631 [16] 0.13465538 [11] 0.5 0.57826539 (13) 0.05756353 (13) 0.04646911 (13) 0.57828216 [16] 0.05756521 [16] 0.04646923 [11] 0.7 0.68315870 (16) -0.03562337 (16) -0.05498833 (16) 0.68317018 [16] -0.03562208 [16] -0.05498817 [10] 1 0.83731609 (19) -0.17768011 (20) -0.19983021 (21) 0.83732330 [16] -0.17767924 [16] -0.19983009 [10] 1.5 1.09080151 (23) -0.41987130 (24) -0.43971736 (24) 1.09080565 [16] -0.41987081 [16] -0.43971723 [9] 2 1.34264091 (25) -0.66554881 (25) -0.68207522 (24) 1.34264369 [16] -0.66554844 [14] -0.68207516 [9] 3 1.84457872 (25) -1.16102111 (25) -1.17273995 (24) 1.84458030 [16] -1.16102138 [12] -1.17274770 [8] 5 2.84626 (25) -2.15712831 (25) -2.16347569 (24) 2.84625158 [15] -2.15725040 [9] -2.16442702 [7] 7 3.8473 (25) -3.15404988 (25) -3.15378124 (24) 3.84702797 [13] -3.15556815 [8] -3.16073739 [6] 10 5.35 (25) -4.64435345 (25) -4.62992050 (24) 5.34765193 [11] -4.65425628 [7] -4.65793294 [6] 15 7.84817795 [10] -7.15318255 [6] -7.15571040 [5] 20 10.3484629 [9] -9.65261533 [6] -9.65457004 [5] 30 15.3487725 [8] -14.6520127 [5] -14.6533896 [4] 50 25.3490515 [6] -24.6514848 [5] -24.6523876 [4] 70 35.3491869 [6] -34.6512350 [4] -34.6519267 [4] 100 50.3493002 [5] -49.6510305 [4] -49.6515572 [3]

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147 Table E-2 ( continued ) B au 1101 s 3 1112 p 3 3134 f 150 75.3494006 [5] -74.6508535 [4] -74.6512450 [3] 200 100.349458 [4] -99.6507548 [4] -99.6510748 [3] 300 150.349524 [4] -149.650644 [4] -149.650888 [4] 500 250.349588 [4] -249.650539 [4] -249.650716 [4] 700 350.349621 [5] -349.650486 [5] -349.650630 [5] 1000 500.349652 [5] -499.650439 [5] -499.650557 [5] *The exact field strength used is 3/50.346410161514. Table E-3 Contributions to total energy for Hooke’s atom in Zero B field ( B = 0, m = 0, singlet state, energy in Hartree) Er KEr Eee J Ec Tc 0.001896558822 0.019914 0.000852 0.012140 0.038988 -0.006460 0.000893 0.0038233443 0.032498 0.001735 0.019352 0.060288 -0.009114 0.001678 0.01 0.064205 0.004629 0.036635 0.108665 -0.013945 0.003753 0.02 0.105775 0.009437 0.057934 0.164935 -0.018220 0.006314 0.05 0.207490 0.024500 0.105661 0.283102 -0.024481 0.011409 0.1 0.350001 0.051036 0.165299 0.421934 -0.029242 0.016427 0.2 0.598799 0.107379 0.256027 0.623288 -0.033640 0.021977 0.5 1.250000 0.289414 0.447447 1.030251 -0.038510 0.029168 1 2.230121 0.611502 0.671410 1.493432 -0.041385 0.033921 2 4.057877 1.283267 0.994225 2.151368 -0.043611 0.037849 4 7.523219 2.669785 1.455759 3.083973 -0.045286 0.040942 10 17.448685 6.941110 2.377629 4.936774 -0.046848 0.043911 20 33.492816 14.181479 3.419880 7.026041 -0.047663 0.045478 40 64.970125 28.813145 4.895849 9.981397 -0.048252 0.046597 100 157.902068 73.081043 7.826636 15.845961 -0.048781 0.047564 Table E-4 Contributions to the to tal energy for the Hooke’s atom in B fields ( = , m =0, singlet state, energy in Hartree) B (au) Er KEr Eee J Ec Tc 0 1.250000 0.289400 0.447461 1.030250 -0.038510 0.029154 0.005 1.250009 0.289402 0.447463 1.030255 -0.038510 0.029154 0.01 1.250034 0.289410 0.447470 1.030269 -0.038510 0.029155 0.02 1.250137 0.289442 0.447497 1.030325 -0.038510 0.029155 0.03 1.250308 0.289494 0.447541 1.030418 -0.038511 0.029156 0.04 1.250547 0.289567 0.447603 1.030547 -0.038512 0.029158 0.05 1.250855 0.289661 0.447683 1.030714 -0.038514 0.029160 0.06 1.251230 0.289775 0.447780 1.030918 -0.038515 0.029163 0.08 1.252185 0.290067 0.448028 1.031435 -0.038520 0.029170 0.1 1.253410 0.290442 0.448345 1.032098 -0.038526 0.029179 0.15 1.257643 0.291740 0.449436 1.034383 -0.038546 0.029210 0.2 1.263513 0.293549 0.450938 1.037534 -0.038576 0.029253

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148 Table E-4 ( continued ) B (au) Er KEr Eee J Ec Tc 0.25 1.270970 0.295860 0.452827 1.041508 -0.038617 0.029309 0.3 1.279948 0.298662 0.455076 1.046251 -0.038671 0.029379 0.35 1.290375 0.301944 0.457652 1.051704 -0.038739 0.029461 0.4 1.302172 0.305691 0.460520 1.057800 -0.038823 0.029557 0.45 1.315255 0.309889 0.463644 1.064472 -0.038924 0.029668 0.5 1.329535 0.314521 0.466988 1.071652 -0.039046 0.029792 0.6 1.361344 0.325020 0.474195 1.087264 -0.039353 0.030084 0.8 1.435650 0.350470 0.489798 1.121801 -0.040259 0.030843 1 1.520416 0.380946 0.505673 1.158171 -0.041581 0.031832 1.5 1.757779 0.473051 0.541104 1.245654 -0.046579 0.035144 2 2.011202 0.579719 0.567825 1.320940 -0.053445 0.039200 2.5 2.269769 0.694704 0.586889 1.383789 -0.061421 0.043585 3 2.529615 0.814589 0.600268 1.436538 -0.069952 0.048049 3.5 2.789340 0.937455 0.609591 1.481470 -0.078930 0.052214 4 3.048455 1.062185 0.616035 1.520370 -0.088319 0.055831 4.5 3.306821 1.188087 0.620421 1.554550 -0.097779 0.059075 5 3.564440 1.314712 0.623321 1.584973 -0.106767 0.062399 Table E-5 Contributions to the tota l energy for the Hooke’s atom in B fields ( = 1/10, m = 0, singlet state, energy in Hartree) B (au) Er KEr Eee J Ec Tc 0 0.350001 0.051035 0.165298 0.421934 -0.029242 0.016427 0.001 0.350003 0.051036 0.165299 0.421936 -0.029242 0.016427 0.005 0.350057 0.051046 0.165320 0.421984 -0.029244 0.016428 0.01 0.350224 0.051081 0.165386 0.422134 -0.029247 0.016433 0.02 0.350889 0.051217 0.165648 0.422730 -0.029264 0.016453 0.03 0.351989 0.051443 0.166078 0.423712 -0.029294 0.016484 0.04 0.353511 0.051764 0.166667 0.425064 -0.029336 0.016529 0.05 0.355439 0.052174 0.167403 0.426764 -0.029395 0.016584 0.06 0.357752 0.052678 0.168271 0.428787 -0.029472 0.016650 0.08 0.363439 0.053971 0.170337 0.433676 -0.029686 0.016815 0.1 0.370368 0.055647 0.172720 0.439473 -0.030002 0.017014 0.15 0.391744 0.061524 0.179133 0.456254 -0.031363 0.017631 0.2 0.416678 0.069685 0.184874 0.473808 -0.033673 0.018356 0.25 0.443245 0.079773 0.189134 0.490349 -0.036916 0.019124 0.3 0.470393 0.091310 0.191852 0.505287 -0.040900 0.019891 0.35 0.497620 0.103826 0.193315 0.518638 -0.045382 0.020622 0.4 0.524712 0.116952 0.193876 0.530627 -0.050134 0.021303 0.45 0.551597 0.130422 0.193840 0.541500 -0.054982 0.021928 0.5 0.578266 0.144061 0.193433 0.551467 -0.059805 0.022496 0.6 0.631021 0.171463 0.192068 0.569294 -0.069104 0.023476 0.7 0.683159 0.198730 0.190471 0.584989 -0.077741 0.024283 0.8 0.734846 0.225742 0.188913 0.599073 -0.085671 0.024953 0.9 0.786204 0.252492 0.187485 0.611875 -0.092936 0.025516 1 0.837317 0.279005 0.186208 0.623623 -0.099608 0.025996

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149 Table E-6 Contributions to the to tal energy for the Hooke’s atom in B fields ( = 1/2 m = -1, triplet state, energy in Hartree) B (au) Etot Er KEr Eee J 0 2.359657 1.609657 0.546492 0.344449 0.923482 0.005 2.352178 1.607171 0.546497 0.344451 0.923486 0.01 2.344740 1.604715 0.546515 0.344457 0.923499 0.05 2.286718 1.586094 0.547070 0.344635 0.923908 0.1 2.217885 1.565391 0.548803 0.345191 0.925181 0.2 2.092310 1.532408 0.555672 0.347376 0.930189 0.3 1.982195 1.510180 0.566921 0.350891 0.938245 0.4 1.886425 1.497909 0.582277 0.355570 0.948964 0.5 1.803642 1.494625 0.601402 0.361214 0.961894 0.6 1.732381 1.499286 0.623930 0.367617 0.976564 0.7 1.671180 1.510852 0.649489 0.374583 0.992528 0.8 1.618662 1.528350 0.677722 0.381937 1.009392 0.9 1.573578 1.550897 0.708301 0.389531 1.026823 1 1.534827 1.577720 0.740928 0.397243 1.044549 1.5 1.407318 1.755930 0.926773 0.434924 1.131722 2 1.343400 1.975366 1.136625 0.468075 1.209795 2.5 1.309802 2.213511 1.359833 0.495899 1.277064 3 1.291855 2.460716 1.591124 0.518997 1.334803 3.5 1.282548 2.712521 1.827601 0.538179 1.384693 4 1.278260 2.966707 2.067649 0.554171 1.428264 4.5 1.276993 3.222107 2.310381 0.567564 1.466746 5 1.277588 3.478078 2.555147 0.578824 1.501046 Table E-6 ( continued ) B (au) Ex Ec Tc t1 Self-interaction 0 -0.566825 -0.006452 0.005755 0.334161 0.473258 0.005 -0.566827 -0.006452 0.005755 0.334164 0.473260 0.01 -0.566834 -0.006453 0.005755 0.334172 0.473266 0.05 -0.567061 -0.006454 0.005757 0.334448 0.473474 0.1 -0.567766 -0.006461 0.005764 0.335309 0.474121 0.2 -0.570535 -0.006486 0.005792 0.338722 0.476666 0.3 -0.574985 -0.006529 0.005839 0.344310 0.480759 0.4 -0.580897 -0.006592 0.005905 0.351937 0.486203 0.5 -0.588012 -0.006675 0.005993 0.361436 0.492768 0.6 -0.596065 -0.006781 0.006101 0.372622 0.500212 0.7 -0.604803 -0.006911 0.006231 0.385310 0.508310 0.8 -0.614007 -0.007066 0.006382 0.399323 0.516860 0.9 -0.623490 -0.007248 0.006554 0.414496 0.525692 1 -0.633103 -0.007455 0.006747 0.430681 0.534669 1.5 -0.679926 -0.008828 0.008044 0.522799 0.578743 2 -0.721218 -0.010719 0.009782 0.626700 0.618121 2.5 -0.756291 -0.013343 0.011531 0.737098 0.651969 3 -0.786016 -0.016213 0.013576 0.851443 0.680956 3.5 -0.811429 -0.018955 0.016131 0.968554 0.705960 4 -0.833425 -0.022620 0.018049 1.087906 0.727781 4.5 -0.852694 -0.028757 0.017731 1.209165 0.747048 5 -0.869732 -0.037819 0.014671 1.331989 0.764215

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150 Table E-7 Contributions to the tota l energy for the Hooke’s atom in B fields ( = 1/10, m = -1, triplet state, energy in Hartree) B (au) Etot Er KEr Eee J 0 0.553173 0.403173 0.096200 0.140515 0.391878 0.005 0.545787 0.400756 0.096227 0.140535 0.391925 0.01 0.538630 0.398505 0.096306 0.140596 0.392065 0.02 0.524999 0.394500 0.096623 0.140837 0.392624 0.03 0.512268 0.391149 0.097148 0.141235 0.393547 0.04 0.500417 0.388436 0.097880 0.141784 0.394820 0.05 0.489421 0.386343 0.098814 0.142477 0.396426 0.06 0.479249 0.384846 0.099946 0.143303 0.398344 0.08 0.461230 0.383527 0.102778 0.145314 0.403016 0.1 0.446038 0.384235 0.106326 0.147722 0.408623 0.15 0.418171 0.393171 0.117951 0.154847 0.425321 0.2 0.400837 0.409415 0.132814 0.162525 0.443604 0.25 0.390249 0.430171 0.150170 0.169888 0.461605 0.3 0.383892 0.453614 0.169442 0.176488 0.478385 0.34641* 0.38038476 0.47679492 0.18866537 0.18177953 0.49255431 0.35 0.380183 0.478627 0.190198 0.182154 0.493592 0.4 0.378139 0.504532 0.212112 0.186872 0.507182 0.5 0.376822 0.557564 0.258474 0.193743 0.530008 0.6 0.377270 0.611042 0.307108 0.197884 0.548195 0.7 0.378382 0.664377 0.357120 0.200091 0.563052 0.8 0.379691 0.717380 0.407937 0.201004 0.575571 0.9 0.380997 0.770020 0.459198 0.201083 0.586443 1 0.382222 0.822320 0.510668 0.200644 0.596130 2 0.389439 1.334451 1.023668 0.191409 0.664385 5 0.394749 2.842749 2.536839 0.179678 0.774053 10 0.396743 5.345744 5.046088 0.174667 0.866618 Table E-7 ( continued ) B (au) Ex Ec Tc t1 Self-interaction 0 -0.241488 -0.005554 0.004321 0.059488 0.200023 0.005 -0.241513 -0.005555 0.004322 0.059501 0.200047 0.01 -0.241589 -0.005557 0.004324 0.059539 0.200119 0.02 -0.241892 -0.005564 0.004331 0.059692 0.200404 0.03 -0.242391 -0.005576 0.004345 0.059945 0.200874 0.04 -0.243079 -0.005594 0.004363 0.060298 0.201523 0.05 -0.243946 -0.005617 0.004386 0.060748 0.202342 0.06 -0.244980 -0.005646 0.004415 0.061293 0.203319 0.08 -0.247493 -0.005721 0.004488 0.062658 0.205699 0.1 -0.250498 -0.005821 0.004582 0.064366 0.208555 0.15 -0.259365 -0.006201 0.004909 0.069954 0.217050 0.2 -0.268929 -0.006787 0.005363 0.077086 0.226336 0.25 -0.278183 -0.007600 0.005935 0.085400 0.235462 0.3 -0.286649 -0.008643 0.006605 0.094622 0.243950 0.34641* -0.29366692 -0.00981101 0.00729685 0.10381856 0.25110225 0.35 -0.294176 -0.009909 0.007353 0.104552 0.251625 0.4 -0.300775 -0.011382 0.008154 0.115039 0.258470

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151 Table E-7 ( continued ) B (au) Ex Ec Tc t1 Self-interaction 0.5 -0.311571 -0.014863 0.009831 0.137261 0.269929 0.6 -0.319917 -0.018898 0.011496 0.160651 0.279027 0.7 -0.326598 -0.023297 0.013065 0.184798 0.286436 0.8 -0.332173 -0.027896 0.014497 0.209434 0.292668 0.9 -0.337014 -0.032567 0.015779 0.234380 0.298075 1 -0.341352 -0.037224 0.016911 0.259517 0.302890 2 -0.373422 -0.076231 0.023323 0.512690 0.336881 5 -0.428044 -0.145886 0.020445 1.269841 0.391705 10 -0.474958 -0.210271 0.006722 2.604651 0.438010 *The exact field strength used is 3/50.346410161514. Table E-8 Exact and approximate XC energies for the Hooke’s atom in Zero B field ( B =0, m = 0, singlet state, energy in Hartree) exact xE LDA xE / P BEjPBE xE88 B xE exact cE LDA cE / P BEjPBE cELYP cE 0.00190 -0.0195 -0.0174 -0.0196 -0.0201 -0.0065 -0.0108 -0.0084 -0.0032 0.00382 -0.0301 -0.0265 -0.0297 -0.0304 -0.0091 -0.0152 -0.0117 -0.0048 0.01 -0.0543 -0.0473 -0.0527 -0.0539 -0.0139 -0.0233 -0.0177 -0.0082 0.02 -0.0825 -0.0713 -0.0795 -0.0812 -0.0182 -0.0311 -0.0231 -0.0118 0.05 -0.1416 -0.1218 -0.1359 -0.1389 -0.0245 -0.0437 -0.0310 -0.0184 0.1 -0.2110 -0.1811 -0.2022 -0.2062 -0.0292 -0.0550 -0.0374 -0.0239 0.2 -0.3116 -0.2671 -0.2984 -0.3040 -0.0336 -0.0677 -0.0436 -0.0293 0.5 -0.5151 -0.4410 -0.4931 -0.5021 -0.0385 -0.0862 -0.0514 -0.0350 1 -0.7467 -0.6389 -0.7148 -0.7276 -0.0414 -0.1013 -0.0566 -0.0376 2 -1.0757 -0.9201 -1.0297 -1.0480 -0.0436 -0.1174 -0.0612 -0.0385 4 -1.5420 -1.3186 -1.4761 -1.5021 -0.0453 -0.1342 -0.0653 -0.0380 10 -2.4684 -2.1104 -2.3630 -2.4044 -0.0468 -0.1575 -0.0696 -0.0357 20 -3.5130 -3.0033 -3.3630 -3.4219 -0.0477 -0.1760 -0.0723 -0.0335 40 -4.9907 -4.2663 -4.7777 -4.8611 -0.0483 -0.1949 -0.0744 -0.0312 100 -7.9230 -6.7727 -7.5849 -7.7171 -0.0488 -0.2208 -0.0766 -0.0287 Table E-9 Exact and approximate XC energies for the Hooke’s atom in B fields ( =1/2, m =0, singlet state, energy in Hartree) B (au) exact xE LDA x E P BE x E 88B x E exact cE LDA c E P BE c E LYP c E 0 -0.5151 -0.4410 -0.4931 -0.5021 -0.0385 -0.0862 -0.0514 -0.0350 0.1 -0.5160 -0.4418 -0.4940 -0.5030 -0.0385 -0.0863 -0.0514 -0.0350 0.2 -0.5188 -0.4441 -0.4966 -0.5056 -0.0386 -0.0865 -0.0515 -0.0351 0.3 -0.5231 -0.4479 -0.5008 -0.5099 -0.0387 -0.0868 -0.0516 -0.0351 0.4 -0.5289 -0.4529 -0.5064 -0.5156 -0.0388 -0.0872 -0.0517 -0.0352 0.5 -0.5358 -0.4589 -0.5133 -0.5226 -0.0390 -0.0878 -0.0519 -0.0353 0.6 -0.5436 -0.4658 -0.5211 -0.5305 -0.0394 -0.0884 -0.0521 -0.0354 0.8 -0.5609 -0.4813 -0.5387 -0.5485 -0.0403 -0.0897 -0.0525 -0.0355 1 -0.5791 -0.4982 -0.5580 -0.5682 -0.0416 -0.0910 -0.0528 -0.0356

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152 Table E-9 ( continued ) B (au) exact xE LDA xE P BE xE 88 B xE exact cE LDA cE P BE cE LYP cE 1.5 -0.6228 -0.5408 -0.6072 -0.6188 -0.0466 -0.0944 -0.0534 -0.0353 2 -0.6605 -0.5807 -0.6541 -0.6670 -0.0534 -0.0973 -0.0536 -0.0343 2.5 -0.6919 -0.6166 -0.6970 -0.7113 -0.0614 -0.0999 -0.0537 -0.0333 3 -0.7183 -0.6491 -0.7358 -0.7518 -0.0700 -0.1021 -0.0536 -0.0319 3.5 -0.7407 -0.6786 -0.7714 -0.7891 -0.0789 -0.1040 -0.0533 -0.0301 4 -0.7602 -0.7057 -0.8048 -0.8236 -0.0883 -0.1057 -0.0530 -0.0281 4.5 -0.7773 -0.7308 -0.8361 -0.8558 -0.0978 -0.1072 -0.0528 -0.0263 5 -0.7925 -0.7541 -0.8655 -0.8861 -0.1068 -0.1086 -0.0526 -0.0248 Table E-10 Exact and approximate XC energies for the Hooke’s atom in B fields ( =1/10, m = 0, singlet state, energy in Hartree) B (au) exact xE LDA xE P BE xE 88 B xE exact cE LDA cE P BE cE LYP cE 0 -0.2110 -0.1811 -0.2022 -0.2061 -0.0292 -0.0550 -0.0374 -0.0239 0.05 -0.2134 -0.1832 -0.2045 -0.2085 -0.0294 -0.0554 -0.0375 -0.0240 0.1 -0.2197 -0.1888 -0.2109 -0.2150 -0.0300 -0.0563 -0.0380 -0.0244 0.15 -0.2281 -0.1967 -0.2199 -0.2242 -0.0314 -0.0576 -0.0385 -0.0249 0.2 -0.2369 -0.2056 -0.2303 -0.2349 -0.0337 -0.0590 -0.0389 -0.0255 0.25 -0.2452 -0.2147 -0.2412 -0.2461 -0.0369 -0.0604 -0.0392 -0.0259 0.3 -0.2526 -0.2237 -0.2521 -0.2573 -0.0409 -0.0617 -0.0393 -0.0262 0.35 -0.2593 -0.2324 -0.2626 -0.2682 -0.0454 -0.0630 -0.0394 -0.0265 0.4 -0.2653 -0.2407 -0.2728 -0.2788 -0.0501 -0.0642 -0.0395 -0.0266 0.45 -0.2707 -0.2486 -0.2826 -0.2889 -0.0550 -0.0653 -0.0395 -0.0267 0.5 -0.2757 -0.2562 -0.2919 -0.2986 -0.0598 -0.0663 -0.0395 -0.0268 0.6 -0.2846 -0.2703 -0.3093 -0.3168 -0.0691 -0.0681 -0.0395 -0.0267 0.7 -0.2925 -0.2832 -0.3254 -0.3335 -0.0777 -0.0698 -0.0395 -0.0266 0.8 -0.2995 -0.2952 -0.3402 -0.3490 -0.0857 -0.0712 -0.0395 -0.0263 0.9 -0.3059 -0.3063 -0.3541 -0.3635 -0.0929 -0.0726 -0.0395 -0.0259 1 -0.3118 -0.3167 -0.3670 -0.3772 -0.0996 -0.0738 -0.0395 -0.0255 Table E-11 Exact and approximate XC energies for the Hooke’s atom in B fields ( =1/2, m =-1, triplet state, energy in Hartree) B (au) exact xE LDA x E P BE x E jPBE xE 88B x E exact cE LDA c E P BE c E jPBE cE 0 -0.5668 -0.5039 -0.5487 -0.5552 -0.5557 -0.0065 -0.0443 -0.0241 -0.0210 0.05 -0.5671 -0.5041 -0.5489 -0.5554 -0.5560 -0.0065 -0.0443 -0.0241 -0.0210 0.1 -0.5678 -0.5048 -0.5496 -0.5561 -0.5567 -0.0065 -0.0443 -0.0241 -0.0210 0.2 -0.5705 -0.5073 -0.5523 -0.5589 -0.5594 -0.0065 -0.0444 -0.0241 -0.0211 0.4 -0.5809 -0.5169 -0.5625 -0.5694 -0.5697 -0.0066 -0.0448 -0.0243 -0.0212 0.6 -0.5961 -0.5311 -0.5777 -0.5850 -0.5850 -0.0068 -0.0453 -0.0246 -0.0213 0.8 -0.6140 -0.5482 -0.5960 -0.6038 -0.6036 -0.0071 -0.0460 -0.0249 -0.0214 1 -0.6331 -0.5668 -0.6160 -0.6244 -0.6239 -0.0075 -0.0466 -0.0253 -0.0215 1.5 -0.6799 -0.6144 -0.6679 -0.6779 -0.6764 -0.0088 -0.0483 -0.0260 -0.0218 2 -0.7212 -0.6592 -0.7167 -0.7284 -0.7263 -0.0107 -0.0497 -0.0266 -0.0218

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153 Table E-11 ( continued ) B (au) exact xE LDA xE P BE xE jPBE xE 88 B xE exact cE LDA cE P BE cE jPBE cE 2.5 -0.7563 -0.6998 -0.7615 -0.7747 -0.7721 -0.0133 -0.0510 -0.0270 -0.0218 3 -0.7860 -0.7365 -0.8029 -0.8177 -0.8139 -0.0162 -0.0521 -0.0274 -0.0218 3.5 -0.8114 -0.7697 -0.8406 -0.8568 -0.8522 -0.0190 -0.0531 -0.0277 -0.0217 4 -0.8334 -0.8002 -0.8746 -0.8921 -0.8875 -0.0226 -0.0539 -0.0277 -0.0215 4.5 -0.8527 -0.8286 -0.9056 -0.9246 -0.9204 -0.0288 -0.0547 -0.0277 -0.0212 5 -0.8697 -0.8550 -0.9350 -0.9553 -0.9511 -0.0378 -0.0554 -0.0277 -0.0209 10 -0.9726 -1.0490 -1.1614 -1.1925 -1.1848 -0.0921 -0.0599 -0.0285 -0.0198 20 -1.0701 -1.2993 -1.4679 -1.5144 -1.4903 -0.3119 -0.0655 -0.0267 -0.0163 Table E-12 Exact and approximate XC energies for the Hooke’s atom in B fields ( =1/10, m = -1, triplet state, energy in Hartree) B (au) exact xE LDA xE P BE xE jPBE xE 88 B xE exact cE LDA cE P BE cE jPBE cE 0 -0.2415 -0.2147 -0.2338 -0.2366 -0.2376 -0.0056 -0.0292 -0.0180 -0.0162 0.02 -0.2419 -0.2150 -0.2342 -0.2370 -0.2380 -0.0056 -0.0293 -0.0180 -0.0162 0.04 -0.2431 -0.2161 -0.2354 -0.2382 -0.2392 -0.0056 -0.0293 -0.0181 -0.0163 0.06 -0.2450 -0.2179 -0.2373 -0.2401 -0.2410 -0.0056 -0.0295 -0.0182 -0.0163 0.08 -0.2475 -0.2202 -0.2397 -0.2427 -0.2435 -0.0057 -0.0296 -0.0183 -0.0164 0.1 -0.2505 -0.2231 -0.2427 -0.2458 -0.2466 -0.0058 -0.0298 -0.0184 -0.0165 0.15 -0.2594 -0.2316 -0.2519 -0.2552 -0.2558 -0.0062 -0.0304 -0.0188 -0.0167 0.2 -0.2689 -0.2413 -0.2623 -0.2660 -0.2662 -0.0068 -0.0311 -0.0192 -0.0169 0.25 -0.2782 -0.2511 -0.2729 -0.2770 -0.2770 -0.0076 -0.0317 -0.0196 -0.0171 0.3 -0.2866 -0.2607 -0.2835 -0.2879 -0.2877 -0.0086 -0.0323 -0.0199 -0.0172 0.346* -0.2937 -0.2693 -0.2929 -0.2977 -0.2968 -0.0098 -0.0329 -0.0202 -0.0173 0.35 -0.2942 -0.2699 -0.2936 -0.2984 -0.2979 -0.0099 -0.0329 -0.0202 -0.0173 0.4 -0.3008 -0.2785 -0.3032 -0.3085 -0.3077 -0.0114 -0.0334 -0.0204 -0.0173 0.5 -0.3116 -0.2944 -0.3210 -0.3271 -0.3259 -0.0149 -0.0344 -0.0207 -0.0173 0.6 -0.3199 -0.3085 -0.3372 -0.3440 -0.3425 -0.0189 -0.0352 -0.0209 -0.0172 0.7 -0.3266 -0.3214 -0.3520 -0.3595 -0.3578 -0.0233 -0.0359 -0.0210 -0.0170 0.8 -0.3322 -0.3332 -0.3658 -0.3740 -0.3720 -0.0279 -0.0366 -0.0211 -0.0169 0.9 -0.3370 -0.3443 -0.3787 -0.3876 -0.3853 -0.0326 -0.0372 -0.0212 -0.0167 1 -0.3414 -0.3547 -0.3909 -0.4004 -0.3979 -0.0372 -0.0377 -0.0212 -0.0165 The exact field strength used is 3/50.346410161514.

PAGE 168

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160 BIOGRAPHICAL SKETCH Wuming Zhu was born on September 11, 1970, in Qixian, Henan province, China. He got a B.S. degree in physics in 1991 from East China Normal University, Shanghai, and a M.S. degree in biophysics in 1994 from the same university under the supervision of Professor Jiasen Chen. He taught for 5 year s as an instructor in physics experiments in the Department of Physics, ECNU, before at tending the University of Florida in 1999. He started working under the supe rvision of Professor Samuel B. Trickey in 2000. He married Ms. Xingxing Liu in 2003. He completed his Ph.D. in 2005.


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Title: Numerical and Exact Density Functional Studies of Light Atoms in Strong Magnetic Fields
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Copyright Date: 2008

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NUMERICAL AND EXACT DENSITY FUNCTIONAL STUDIES OF LIGHT ATOMS
IN STRONG MAGNETIC FIELDS















By

WUMING ZHU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

Wuming Zhu

































Dedicated to Mom, to Dad,
and to my wife.















ACKNOWLEDGMENTS

First of all, I would like to thank Professor Samuel B. Trickey, my research advisor

and committee chair, for the guidance he provided throughout the course of my graduate

study at the University of Florida. His patience and constant encouragement are truly

appreciated. Besides physics, I have also learned a lot from him about life and language

skills which also are indispensable for becoming a successful physicist.

I would also like to thank Professor Hai-Ping Cheng, Professor Jeffrey L. Krause,

Professor Susan B. Sinnott, and Professor David B. Tanner for serving in my supervisory

committee, and for the guidance and advice they have given me. Professor David A.

Micha is acknowledged for his help when I was in his class and for being a substitute

committee member in my qualifying exam.

My gratitude goes to Dr. John Ashley Alford II for many helpful discussions, and

to Dr. Chun Zhang, Dr. Lin-Lin Wang, and Dr. Mao-Hua Du for their academic and

personal help. Besides them, many other friends have also enriched my life in

Gainesville. They are Dr. Rongliang Liu, Dr. Linlin Qiu, Dr. Xu Du, Dr. Zhihong Chen,

and Dr. Lingyin Zhu, who have moved to other places to advance their academic careers,

and Guangyu Sun, Haifeng Pi, Minghan Chen, Yongke Sun, and Hui Xiong, who are

continuing to make progress in their Ph.D. research.

Many thanks go to the incredible staff at the Department of Physics and at QTP. I

especially would like to thank Darlene Latimer, Coralu Clements, and Judy Parker for the









assistance they provided during my graduate study. Financial support from NSF grants

DMR-0218957 and DMR-9980015 is acknowledged.

Lastly, I thank my parents, who will never read this dissertation but can feel as

much as I do about it, for their boundless love. I thank my wife for her special patience

and understanding during the days I wrote my dissertation, and for all the wonderful

things she brings to me.
















TABLE OF CONTENTS

page

A C K N O W L ED G M EN T S ............................... ......... ................................................... iv

L IST O F T A B L E S ................. .................................................................. .. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ xi

AB STRA CT.................................................... ............................. xiii

CHAPTER

1 BASICS OF DENSITY FUNCTIONAL THEORY AND CURRENT DENSITY
FU N C TIO N A L TH EO R Y ........................................ ........ ................. .....................

Intro du action ............................................ ..... ............................
D ensity Functional Theory ........................................ ................................. 3
F foundations for D F T ............................................................. ....................... 4
T he K ohn-Sham Schem e ................................................................. ..... ............ 7
Current Density Functional Theory (CDFT)............................................................10
B asic Form ulations ................... ............. ........................... 10
Vignale-Rasolt-Geldart (VRG) Functional .................................... ............... 13
Survey on the Applications of CDFT....................................... ....... ............ 15
Other D evelopm ents in CD FT....................................... .......................... 16

2 ATOMS IN UNIFORM MAGNETIC FIELDS THEORY .................................18

Single P article E qu ation s ................................................................. ..................... 18
H artree-Fock A pproxim ation ........................................ .........................18
Simple DFT Approximation...................................................... 19
C D F T A pproxim ation ........................................... ........................................ 19
Exchange-correlation Potentials ...........................................................................20

3 BASIS SET AND BASIS SET OPTIMIZATION...............................................25

Survey of Basis Sets Used in Other Work..... ........................ ..............25
Spherical-GTO and Anisotropic-GTO Representations.................. .. ............. 27
Spherical G TO Basis Set Expansion ............................. ............................... 27
Anisotropic GTO (AGTO) Basis Set Expansion.....................................29
Connection between GTOs and AGTOs .................................. ............... 30









Primary and Secondary Sequences in AGTO............... ...........................................31
O ptim ized A G T O B asis Sets ........................................................... .....................33

4 ATOMS IN UNIFORM MAGNETIC FIELDS NUMERICAL RESULTS .........48

C om prison w ith D ata in L literature ................................................ .....................48
Magnetic Field Induced Ground State Transitions.............................................51
Atom ic Density Profile as a Function of B ................... ............... .................. 53
Total Atomic Energies and Their Exchange and Correlation Components ..............55
Ionization Energies and Highest Occupied Orbitals for Magnetized Atoms ............61
Current Density Correction and Other Issues...... ............................65

5 HOOKE'S ATOM AS AN INSTRUCTIVE MODEL............... .............. 71

H ooke's Atom in V anishing B Field ............................................... ............... 71
Hooke's Atom in B Field, Analytical Solution ......................................................75
Hooke's Atom in B Field, Numerical Solution .................................. ............... 84
Phase Diagram for Hooke's Atom in B Field................. ........ ......... .......... 89
Electron Density and Paramagnetic Current Density ...........................................91
Construction of Kohn-Sham Orbitals from Densities ............................................. 95
Exact DFT/CDFT Energy Components and Exchange-correlation Potentials...........97
Comparison of Exact and Approximate Functionals..........................................102

6 SUMM ARY AND CONCLUSION .................................................. .............. 110

APPENDIX

A HAMILTONIAN AND MATRIX ELEMENTS IN SPHERICAL GAUSSIAN
B A S IS .............................................................................................1 12

B ATOMIC ENERGIES FOR ATOMS He, Li, Be, B, C AND THEIR POSITIVE
IONS Li+, Be+, B IN MAGNETIC FIELDS ............................... ....................115

C EXCHANGE AND CORRELATION ENERGIES OF ATOMS He, Li, Be, and
POSITIVE IONS Li+, Be IN MAGNETIC FIELDS ...........................................132

D EFFECTIVE POTENTIAL INTEGRALS WITH RESPECT TO LANDAU
ORBITALS IN EQUATION (5.30) ..................... .................................... 140

E ENERGY VALUES FOR HOOKE'S ATOM IN MAGNETIC FIELDS ...............143

L IST O F R E F E R E N C E S ...................................................................... ..................... 154

BIOGRAPHICAL SKETCH ............................................................. ............... 160
















LIST OF TABLES


Table p

3-1 Basis set effect on the HF energies of the H and C atoms with B = 0.........................35

3-2 Basis set errors for the ground state energy of the H atom in B = 10 au ...................36

3-3 Optimized basis set and expansion coefficients for the wavefunction of the H
atom in B = 10 au. .....................................................................37

3-4 Test of basis sets including 1, 2, and 3 sequences on the energies of the H atom in
B fi e ld s ........................................................................ 4 2

3-5 Energies for high angular momentum states of the H atom in B fields....................43

3-6 Basis sets for the H atom in B fields with accuracy of 1 ,H ................... ..............44

3-7 Basis set effect on the HF energies of the C atom in B = 10 au............. ..................45

3-8 Construction of the AGTO basis set for the C atom in B = 10 au.............................47

3-9 Overlaps between HF orbitals for the C atom in B = 10 au and hydrogen-like
sy stem s in the sam e fi eld ............................................................... .....................47

4-1 Atomic ionization energies in magnetic fields ................................. ..................... 62

4-2 Eigenvalues for the highest occupied orbitals of magnetized atoms...........................63

4-3 CDFT corrections to LDA results within VRG approximation ................................68

4-4 Effect of cutoff function on CDFT corrections for the He atom ls2p.l state in
m magnetic field B = 1 au ................................................. ............................... 69

5-1 Confinement frequencies co for HA that have analytical solutions to eqn. (5.5) ........74

5-2 Confinement frequencies which have analytical solutions to eqn.(5.12) ..................82

5-3 Som e solutions to eqn. (5.12) ...................................................................... 84

5-4 Field strengths for configuration changes for the ground states of HA ....................... 89

5-5 SCF results for HF and approximate DFT functionals.................... ............... 109









B-l Atomic energies of the He atom in B fields............... ........................................... 115

B-2 Atomic energies of the Li ion in B fields .......................................... ............121

B-3 Atomic energies of the Li atom in B fields ............................... .... ...........122

B-4 Atomic energies of the Be+ ion in B fields ............. ............................................. 124

B-5 Atomic energies of the Be atom in B fields ............. ............................................. 125

B-6 Atomic energies of the B+ ion in B fields. ...................................... ............126

B-7 Atomic energies of the B atom in B fields............. ................... .... ...........128

B-8 Atomic energies of the C atom in B fields......... .............................................130

C-1 Exchange and correlation energies of the He atom in magnetic fields.....................132

C-2 Exchange and correlation energies of the Li ion in magnetic fields .....................135

C-3 Exchange and correlation energies of the Li atom in magnetic fields...................... 136

C-4 Exchange and correlation energies of the Be+ ion in magnetic fields ......................137

C-5 Exchange and correlation energies of the Be atom in magnetic fields.....................138

D-1 Expressions for Vs(z) with |z1<2 and 0
E-l Relative motion and spin energies for the HA in B fields (0 = 1/2)........................143

E-2 A s in Table E-l, but for = 1/10. ............................. ..... ............................. 145

E-3 Contributions to the total energy for the HA in Zero B field (B = 0, m = 0) ..........147

E-4 Contributions to the total energy for the HA in B fields (co = 1, m = 0, singlet) ..147

E-5 Contributions to the total energy for the HA in B fields (co = 1/10, m = 0, singlet)148

E-6 Contributions to the total energy for the HA in B fields (c = 1/, m = -1, triplet) ...149

E-7 Contributions to the total energy for the HA in B fields (c = 1/10, m = -1, triplet)150

E-8 Exact and approximate XC energies for the HA in Zero B field (B = 0, m = 0,
singlet) .......................... ....... ........ ................ .............. 151

E-9 Exact and approximate XC energies for the HA in B fields (co =1/2, m = 0, singlet)151

E-10 Exact and approximate XC energies for the HA in B fields (co = 1/10, m = 0,
singlet) ............... ........ ........ ....................................... 152









E-11 Exact and approximate XC energies for the HA in B fields (wc = 1/2, m = -1,
triplet) ............................................................... .... ...... ......... 152

E-12 Exact and approximate XC energies for the HA in B fields (co = 1/10, m=-l,
triplet) ............................................................... .... ...... ......... 153
















LIST OF FIGURES


Figure pge

3-1 Exponents of optimized basis sets for the H, He+, Li Be C5 and 0+ in
reduced magnetic fields y = 0.1, 1, 10, and 100.................................................40

3-2 Fitting the parameter b(y =1) using the function (3.26)....................................41

4-1 UHF total energies for different electronic states of the He atom in B fields ............52

4-2 Cross-sectional view of the HF total electron densities of the He atom ls2 and
ls2p.l states as a function of magnetic field strength.................. ............. 54

4-3 Differences of the HF and DFT total atomic energies of the He atom Is2, ls2po,
and s2p.- states with respect to the corresponding CI energies as functions of B
field strength ......... .............. ...... ....................................... ................... .......... 56

4-4 Differences of DFT exchange, correlation, and exchange-correlation energies with
HF ones, for the H e atom in B fields. .................................................................... 58

4-5 Atomic ground state ionization energies with increasing B field.............................64

4-6 Various quantities for the helium atom ls2p.1 state in B = 1 au........................... 66

5-1 Confinement strengths subject to analytical solution to eqn. (5.12) .........................83

5-2 Phase diagram for the HA in B fields......................................................... .. ........ 90

5-3 Cross-sectional view of the electron density and paramagnetic current density for
the ground state HA with co = 1/10 in B = 0.346 au............................................94

5-4 Energy components of HA with B = 0. ................................... .......... ....... ........ 99

5-5 Comparison of exact and approximate XC functionals for the HA with different
confinement frequency wc in vanishing B field (B = 0) .......................................103

5-6 Comparison of exact and approximate exchange, correlation, and XC energies of
the H A w ith co = 1/2 in B fields...................................... ............................ 105

5-7 Sam e as Fig. 5-6, except for co = 1/10. ................... ............................................. 106









5-8 Cross-sectional views of the exact and approximate XC potentials for the ground
state HA with co = 1/10 in B = 0.346 au............... ....................... ............... 107















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NUMERICAL AND EXACT DENSITY FUNCTIONAL STUDIES OF LIGHT ATOMS
IN STRONG MAGNETIC FIELDS

By

Wuming Zhu

August 2005

Chair: Samuel B. Trickey
Major Department: Physics

Although current density functional theory (CDFT) was proposed almost two

decades ago, rather little progress has been made in development and application of this

theory, in contrast to many successful applications that ordinary density functional theory

(DFT) has enjoyed. In parallel with early DFT exploration, we have made extensive

studies on atom-like systems in an external magnetic field. The objectives are to advance

our comparative understanding of the DFT and CDFT descriptions of such systems. A

subsidiary objective is to provide extensive data on light atoms in high fields, notably

those of astrophysical interest.

To address the cylindrical symmetry induced by the external field, an efficient,

systematic way to construct high quality basis sets within anisotropic Gaussians is

provided. Using such basis sets, we did extensive Hartree-Fock and DFT calculations on

helium through carbon atoms in a wide range of B fields. The applicability and









limitations of modern DFT and CDFT functionals for atomic systems in such fields is

analyzed.

An exactly soluble two-electron model system, Hooke's atom (HA), is studied in

detail. Analogously with known results for zero field, we developed exact analytical

solutions for some specific confinement and field strengths. Exact DFT and CDFT

quantities for the HA in B fields, specifically exchange and correlation functionals, were

obtained and compared with results from approximate functionals. Major qualitative

differences were identified. A major overall conclusion of the work is that the vorticity

variable, introduced in CDFT to ensure gauge invariance, is rather difficult to handle

computationally. The difficulty is severe enough to suggest that it might be profitable to

seek an alternative gauge-invariant formulation of the current-dependence in DFT.














CHAPTER 1
BASICS OF DENSITY FUNCTIONAL THEORY AND CURRENT DENSITY
FUNCTIONAL THEORY

Introduction

Ambient and low-temperature properties of normal bulk materials are largely

determined by knowledge of the motion of the nuclei in the field of the electrons. In

essence, this is a statement that the Bom-Oppenheimer approximation [1] is widely

relevant. For materials drawn from the lighter elements of the periodic table, the

electrons even can be treated non-relativistically [2]. While doing some electronic

structure calculations on a-quartz [3], and some classical inter-nuclear potential

molecular dynamics (MD) simulations on silica-like nano-rods [4], a feature of modern

computational materials physics became obvious. Very little is done with external

magnetic fields. This scarcity seems like a missed opportunity.

Even with no external field, within the Bom-Oppenheimer approximation, a non-

relativistic approach to solution of the N-electron Schrodinger equation is not a trivial

task. For simple systems, e.g. the He atom, highly accurate approximate variational

wavefunctions exist [5], but these are too complicated to extend. Much of the work of

modern quantum chemistry involves extremely sophisticated sequences of

approximations to the exact system wavefunction [6]. The Hartree-Fock (HF)

approximation, which uses a single Slater determinant as the approximation to the many-

electron wavefunction, usually constitutes the first step toward a more accurate,

sophisticated method. Several approaches, such as configuration interaction (CI), many-









body perturbation theory (MBPT), and coupled cluster (CC), are widely used in practice

to improve HF results. It is worthwhile mentioning that such methods are extremely

demanding computationally. Their computational cost scales as some high power of the

number of electrons, typically 5-7th power. Thus these methods are only affordable for

systems having up to tens of electrons. An external magnetic field which could not be

treated perturbatively would make things much worse. The largest system that has been

investigated with the full CI method as of today is a four-electron system, beryllium atom

[7, 8]. On the other hand, people always have interests in larger systems and more

accurate results than those achievable, no matter how fast and how powerful the

computers are; thus theorists continue to conceive all kinds of clever approximations and

theories to cope with this problem.

Density functional theory (DFT) [9, 10] is an alternative approach to the many-

electron problem that avoids explicit contact with the N-electron wavefunction. DFT

developed mostly in the materials physics community until the early 1990s when it

reappeared in the quantum chemistry community as a result of the success of new

approximate functionals. These aspects will be discussed below. Two other aspects are

worth emphasizing. DFT has been remarkably successful in predicting and interpreting

materials properties. Almost none of those predictions involve an external magnetic

field. Particularly in Florida, with the National High Magnetic Field Laboratory, that is

striking.

Even for very simple atoms, inclusion of an external B field is not easy. Only

recently have the calculations on the helium atom in a high field been pushed beyond the

HF approximation [11, 12, 13]. Although a version of DFT called current density









functional theory (CDFT) [14, 15, 16] exists for external magnetic fields, it has seen very

little application or development. As discussed below, there is a lack of good

approximate CDFT functionals and a lack of studies on which to try to build such

improved functionals. One of the foundations of the success of ordinary DFT has been

the availability of exact analytical and highly precise numerical data for atoms for

comparison of various functionals and understanding their behavior. The main purpose

of this dissertation is to find how the effect of an external magnetic field on electron

motion should be incorporated in the DFT functional. In particular, I obtain numerical

results on various atom-like systems in an external field, with and without CDFT

approximate functionals. In addition, I give exact solutions for a model two-electron

atom in a nonzero external B field, the so-called Hooke's atom (HA), that has provided

valuable insight for DFT at B = 0.

Density Functional Theory

Attempts to avoid calculation of the many-electron wavefunction began almost

simultaneously with the emergence of modem quantum mechanics. In 1927, Thomas and

Fermi proposed a model in which the electron kinetic energy is expressed as a functional

of the electron density, totally neglecting exchange and correlation effects [17]. The

kinetic energy density is assumed to be solely determined by the electron density at that

point, and approximated by the kinetic energy density of a non-interacting uniform

electron gas having the same density. Later this approach was called the "local density

approximation" (LDA) in DFT. The Thomas-Fermi (TF) model was refined subsequently

by Dirac to include exchange effects (anti-symmetry of identical-particle wavefunction)

and by von Weizsacker to include spatial gradient corrections for the kinetic energy. The

result is called the TFDW model. Though useful, it fails as a candidate for a model of









materials behavior. Teller proved that the model will not provide binding even for a

simple diatomic molecule [17].

The modern form of DFT is rooted in the 1964 paper of Hohenberg and Kohn [9]

which put forth two basic theorems, and the subsequent paper by Kohn and Sham [10],

which gave an ingenious scheme for the use of those theorems. A difficulty with the KS

scheme is that it lumps all of the subtlety of the many-electron problem, exchange and

correlation, in one approximation. The popularity of DFT depends on the availability of

reasonably accurate, tractable approximate functionals. To make the point clear and

establish notation, next I give the bare essentials of ordinary DFT.

Foundations for DFT

The Hamiltonian of an interacting N-electron system is

1 i 1 ( 1
HV2+Zv(i)+- (1.1)
2 ,1 1=1 2 ',,=1 ~' -i


where i labels the space coordinate of the ith electron. Hartree atomic units are used

throughout. The Schr6dinger equation specifies the map from the external potential v (i)

to the ground state many-body wavefunction, and the electron number density can be

obtained by integrating out N-1 space variables. Schematically,

v( (fl, ,..., N -> n() (1.2)

Hohenberg and Kohn noticed that the inversion of the above maps is also true [9],

even though it is not as obvious as above. Because of the key importance of this

observation, their proof is included here. For simplicity, they considered the spin

independent, non-degenerate ground states. Let the Hamiltonian H, ground state









wavefunction Y, density n(F), and energy E associated with the specific external

potential v(F),

v(F) :- -H, T, n (), E. (1.3)

Similarly define a primed system,

v'(f) : -- ',n'( ), E'. (1.4)

where v(F) v'(F)+ C, and hence Y T Y'. By the variational principle,

E=( H )
Interchanging the primed and unprimed systems gives us another inequality. Summation

of those two inequalities leads to a contradiction E + E' < E + E' if we assume

n'() = n(F) Thus, different potentials must generate different ground state electron

densities. Equivalently speaking, the knowledge of the ground state density

n(f) uniquely determines the external potential v(r) up to a physically irrelevant

addictive constant. This assertion is referred as Hohenberg-Kohn (HK) theorem I. Now

the maps in eqn. (1.2) are both bijective,

v(f) (f, ,..f.,F) n(f) (1.6)

An immediate consequence of HK theorem I is that the ground state electron

density n(f) can be chosen as the basic variable to describe the interacting N-electron

system, since it is as good as the many-body wavefunction. Here "as good as" means that

the ground state density n(r) contains no more or less information about the system than

the wavefunction does. It does not mean the density is a variable as easy as, or as hard as,

the many-body wavefunction to handle. Actually, since the density is a 3-dimensional









physically observable variable, whereas the spatial part of wavefunction is a 3N-

dimensional variable, the density is a much simpler variable to manipulate and to think

about. On the other hand, by switching from the wavefunction to the density, we also lose

some tools from quantum mechanics (QM) which we are quite adept at using. For

example, in QM, an observable can be calculated by evaluating the expectation value of

its corresponding operator. This approach often does not work in DFT. The best we can

say is that the observable is a functional of the ground state density. In contrast to the

explicit dependence on the wavefunction in the QM formulas for the expectation value,

the implicit dependence on the ground state density in DFT is rarely expressible in a form

useful for calculation. Most such functionals are not known as of today. Among them, the

most exploited and the most successful one is the exchange-correlation energy functional,

which is amenable to approximations for large varieties of systems. Another one being

extensively studied but not so successfully is the kinetic energy functional, already

mentioned in the paragraph about TF-type models.

While DFT is a whole new theory that does not need to resort to the many-body

wavefunction, to make use of the Rayleigh-Ritz variational principle to find the ground

state energy Eo, we retain that concept for a while. By the so-called constrained search

scheme independently given by Lieb [18] and Levy [19] in 1982, all the trial

wavefunctions are sorted into classes according to the densities n'(F) to which they give

rise. The minimization is split into two steps,

E, = min Yf' Hi '') = min f v(r)n' (i)dr + FL [n'(r)]} (1.7)


where FLL[n'] -min(Y T+U Y' (1.8)
LL J X'T' + /T) 18









The Lieb-Levy functional FLL is defined on all the possible densities realizable from

some anti-symmetric, normalized N-particle functions, or N- representable densities. Both

the densities of degenerate states and even excited states are included. One good thing

about FLL is that we have a simple criterion for N- representable densities: all non-

negative, integrable densities are N- representable.

The Kohn-Sham Scheme

The HK theorem showed that the ground state energy of a many-electron system

can be obtained by minimizing the energy functional E[n'(f)]. TF-type models

constitute a direct approach to attack this problem, in which energy functionals are

constructed as explicit approximate forms dependent upon the electron density. However,

the accuracy of TF-type models is far from acceptable in most applications, and there are

seemingly insurmountable difficulties to improve those models significantly. The reason

is that the kinetic energy functionals in TF-type models bring in a large error. To

circumvent this difficulty, an ingenious indirect approach to the kinetic energy was

invented by Kohn and Sham [10]. A fictitious non-interacting system having the same

ground state electron density as the one under study is introduced. Because the kinetic

energy of this KS system, Ts, can be calculated exactly, and because T, includes almost all

the true kinetic energy T, the dominant part of the error in TF-type models is eliminated.

Since then, DFT has become a practical tool for realistic calculations.

It is advantageous to decompose the total energy in the following way,

E[n] = F, [n(r) + v(r)n(r)dr

= T[n] + [n] + Ju()n(i)d









= T[n] + J +u(F)n(F)c + + V, [n] + (T[n]- T[n]) (1.9)


where Eee is the total QM electron-electron interaction energy, J j= n(f-n(' dFdF' is
2 i r- r'

the classical electron-electron repulsion energy, E is the conventional exchange

energy, and Vc is the conventional correlation energy. Ts is defined in terms of the non-

interacting system as usual,


T = v22 (1.10)


Then EHF is replaced by Ex[n], the exchange energy calculated using the single-

determinant HF formula but with the same orbitals in Ts, and Ec is defined as all the

remaining energy

Ek [n]= V=[n]+(T[n] T [I])+(E -E E [n]) (1.11)

The total energy is finally expressed as

E[n] = T [n]+ J+ f u()n(F)dF + E [n] (1.12)

where E[n] = E[n] + E[n]. (1.13)

Each of the first three terms in eqn. (1.12) usually makes a large contribution to the

total energy, but they all can be calculated exactly. The remainder, Ec, is normally a

small fraction of the total energy and is more amenable to approximation than the kinetic

energy. Even though the equations in this section are all exact, approximations to the E,

functional ultimately must be introduced.

The variational principle leads to the so-called Kohn-Sham self-consistent

equations,










-v + (r (2 )()) (1.14)


where v (t ) v(0) + vH (0) + v~ (i) (1.15)
where )+ (r) (1.15)



vH ()= n(r-' d (1.16)
S-r'


v(f) En()] (1.17)
dn(r)

N
and n(F) = (F)2. (1.18)
i=1

Again, Hartree atomic units are used throughout. Equations (1.14) (1.18) constitute the

basic formulas for KS calculations. The detailed derivation and elaborations can be found

in the abundant literature on DFT, for example, references 20-22.

Since the foundations of DFT were established, there have been many

generalizations to this theory. The most important include spin density functional theory

(SDFT) [23], DFT for multi-component systems [24, 25], thermal DFT for finite

temperature ensembles [26], DFT of excited states, superconductors [27], relativistic

electrons [28], time-dependent density functional theory (TDDFT) [29], and current

density functional theory (CDFT) for systems with external magnetic fields [14-16].

Among them, SDFT is the most well-developed and successful one. TDDFT has attracted

much attention in recent years and shows great promise. Compared to the thousands of

papers published on DFT and SDFT, we have fewer than 80 papers on CDFT in any

form. Thus CDFT seems to be the least developed DFT generalization, perhaps

surprisingly since there is a great deal of experimental work on systems in external B

fields. That disparity is the underlying motivation for this thesis.








Current Density Functional Theory (CDFT)

Basic Formulations

One of the striking features of the very limited CDFT literature is the extremely

restricted choice of functionals. A second striking feature is that most of the work using

CDFT has been at B= 0, in essence using CDFT either to gain access to magnetic

susceptibility [30, 31] or to provide a richer parameterization of the B = 0 ground state

than that provided by SDFT [32]. In order to comprehend the challenge it is first

necessary to outline the essentials of CDFT.

For an interacting N-electron system under both a scalar potential v(r) and a

vector potential A (f), its Hamiltonian reads,


S ( ++A(I))2+ ) (1.19)
2 2, -,

The paramagnetic current density jP (f) is the expectation value of the corresponding

operator Ji(r),

p V/ ) : (ri, r2, f, ),)l | (v) I | (1,f2,", )) (1.20)
1(1.20)

where JJ(F)= I (i)V(7) q()V+ ()] (1.21)

in terms of the usual fermion field operators.

CDFT is an extension of DFT to include the vector potential A () The original

papers [14-16] followed the HK argument by contradiction and purported thereby to

prove not only that the ground state is uniquely parameterized by the density n (F) and

paramagnetic current density jp (f), but also that the vt(f) and A(F) are uniquely








determined. Later it was found that a subtlety was overlooked. It is obvious that two

Hamiltonians with different scalar external potentials cannot even have a common

eigenstate, e.g., the first map in eqn. (1.6) is bijective, but this is not true when a vector

potential is introduced. It is possible that two sets of potentials u(F), A(F) and

v'(F), A' () could have the same ground state wavefunction. This non-uniqueness was

later realized [33]. Fortunately, the HK-like variational principle only needs the one-to-

one map between ground state wavefunction and densities, without recourse to the

system's external potentials being functionals of densities [34]. To avoid the difficulties

of representability problems, we follow the Lieb-Levy constrained search approach.

Sort all the trial wavefunctions according to the densities n(F) and jp (F) they

would generate. The ground state wavefunction, which generates correct densities, will

give the minimum of the total energy.

E, = min (' to
A 12 (1.22)
= min (f)- n'(f>)df+ J (f).J ;(F) +F[n (),J (f) ]
n'() J(r) (2)f+ i (r),j

where FIn',1-] min jw' TU f 'f) (1.23)

A non-physical non-interacting KS system is now introduced, which generates

correct densities n(r) and j (f). The functional F is customarily decomposed as

F n, = T jp] +J + E j] (1.24)

The variational principle gives us the self-consistent equations


[I +A(ff) +uvdft (i)) (1.25)









where Aef (r) = A(r) + A, (r) (1.26)


ff(r) = v[(r+ ()+ (r)] + [2(f) A-f()] (1.27)

It is easy to see that vucf (r) reduces to v, (r) when we set Axc (r) = 0. Exchange-

correlation potentials are defined as functional derivatives,

SE [n(r, j ()] (.2)
Axc (') = (1.28)
s j (tr)

cft 5E' [""(f),('
n((r)

v ()) (1.29)


The electron density n(F) can be calculated just as in eqn. (1.18). The paramagnetic

current density is constructed from the KS orbitals according to

jp(V) = -'*()V, () 0' ()V #(rt)] (1.30)
-1
S2i (1.30)

The total energy expression of the system is


Ef = T, + + Ec [n(), j,(F)] + n() v(F) + A + f J ) A(F)d


= J + E{f [n(Fr), ()]- (i)] f f (i)n(F)r JP (i). A (F)r (1.31)

Equation (1.25) can be rewritten as

V. 1
2 i 2i

(1.32)


which is more suitable for application.









Vignale-Rasolt-Geldart (VRG) Functional

On grounds of gauge-invariance, Vignale and Rasolt argued that the exchange-

correlation functional Ef should be expressed as a functional of n(f) and the so-called

voticity [14-16]


(f) Vx (1.33)


Following their proposal, if we choose v(f) as the second basic variable in CDFT,

Ef [n(i), Ji (l)] = Ef [n(ri), (i)] (1.34)

exchange-correlation potentials can be found from functional derivatives


Axc (i) L v I| )l (1.35)


oftft = ) -A ()- (1.36)
n(r) n(r)

To make use of the already proven successful DFT functionals, it is useful to

separate the exchange-correlation functional into a current-independent term and an

explicitly current-dependent term,

E j [n(ir), v(r)] = Edf [n()] + AE [n(r), v(r)] (1.37)

The current-independent term can be any widely used XC functional, such as LDA, or the

generalized gradient approximation (GGA) functional. The current-dependent term is

presumably small, and should vanish for zero current system, for example, the ground

state of the helium atom.









Next we proceed in a slightly different, more general way. By homogeneous

scaling of both the n(r) and j (f), Erhard and Gross deduced that the current-dependent

exchange functional scales homogeneously as [35]

E [ftn[1 ]J= AEcdf[n, j] (1.38)
4-.
where h is the scaling factor, nz = An and j% = 24 j are scaled charge density and

paramagnetic current density, respectively. Assuming that the exchange part dominates

the exchange-correlation energy, a local approximation for the Eft takes the form


f [n(i), ()]= Edft[n(j)]+ Jg([n(i), (f)],) i(r) 12 F (1.39)

The foregoing expression is derived based on the assumption that v(r) is a basic

variable in CDFT. A further (drastic) approximation is to assume

g([n(r), (r)], ) = g(n(F)) (1.40)

which is done in the VRG approximation. By considering the perturbative energy of a

homogeneous electron gas (HEG) in a uniform B field (the question whether the HEG

remains uniform after the B field is turned on was not discussed), Vignale and Rasolt[14-

16] gave the form for g,

kF (n (F))
g(F) =g(n(F))= [ 1] (1.41)
247T2 Zo(n(f))

Here kF is the Fermi momentum, and Z0 are the orbital magnetic susceptibilities for

the interacting and non-interacting HEG, respectively. From the tabulated data for

1 < r < 10 in reference 36, Lee, Colwell, and Handy (LCH) obtained a fitted form [31],

SLCH = / 0 = (1.0 + 0.028rs)e ... (1.42)










where r, =\( (1.43)


k
accordingly, g, = (LCH -1) (1.44)
24;r

Orestes, Marcasso, and Capelle proposed two other fits, both polynomial [37]

sOC3 = 0.9956-0.01254r -0.0002955r2 (1.45)

soMc5 = 1.1038 0.4991 /3 + 0.4423 0.06696ur + 0.0008432r2 (1.46)

Those fits all give rise to divergence problems in the low density region. A cutoff

function needs to be introduced, which will be discussed in the next chapter.

Survey on the Applications of CDFT

The VRG functional has been applied in the calculation of magnetizabilities [30,

31], nuclear shielding constants [38], and frequency-dependent polarizabilities [39, 40]

for small molecules, and ionization energies for atoms [37, 41]. In those calculations, the

vector potential was treated perturbatively. Fully self-consistent calculations are still

lacking. None of those studies has a conclusive result. The first calculation in Handy's

group was plagued with problems arising from an insufficiently large basis set [31]. In

their second calculation, they found that the VRG functional would cause divergences

and set g(n(r)) = 0 for rs >10. The small VRG contribution was overwhelmed by the

limitations of the local density functional [38]. The VRG contribution to the frequency-

dependent polarizability was also found to be negligible, and several other issues emerge

as more important than explicitly including the current density functional [40]. Contrary

to the properties of small molecules studied by Handy's group, Orestes et al. found that

the current contribution to the atomic ionization energy is non-negligible, even though

use of VRG did not improve the energy systematically [37].









All those investigations are based on the assumption that the VRG functional at

least can give the correct order of magnitude of current contributions to the properties

under study. Actually this is not guaranteed. The errors from ordinary DFT functionals

and from the current part are always intertwined. To see how much the current term

contributes, an exact solution is desired.

Vignale's group has never done any actual numerical calculation based on the VRG

functional. Either for an electronic system [42, 43] or for an electron-hole liquid [44, 45],

they used Danz and Glasser's approximation [46] for the exchange, and the random phase

approximation for the correlation energy, which is known to be problematic in the low

density regime. The kinetic energy was approximated from a non-interacting particle

model or a TF-type model. Even though correlation effects were included in their

formulas, the numerical errors introduced in each part were uncontrollable, and their

calculations could only be thought as being very crude at best. This is somewhat

inconsistent with invoking CDFT to do a better calculation than the ordinary DFT

calculation does.

While the fully CDFT calculations on three-dimensional (3D) systems are scarce,

there are more applications of CDFT to 2D systems. Examples include the 2D Wigner

crystal transition [47], quantum dots [48], and quantum rings [49] in a magnetic field. In

these cases, the Ecf was interpolated between the zero-field value from the Monte Carlo

calculation by Tanatar and Ceperley [50], and the strong field limit [51].

Other Developments in CDFT

Some formal properties and virial theorems for CDFT have been derived from

density scaling arguments [35, 52-54] or density matrix theory [55]. A connection









between CDFT and SDFT functionals is also established [56]. Those formal relations

could be used as guidance in the construction of CDFT functionals, but as of today, there

is no functional derived from them as far as I know.

CDFT is also extended to TDDFT in the linear response regime [57], which is

called time-dependent CDFT (TD-CDFT). There is not much connection between TD-

CDFT [57] and the originally proposed CDFT formulation [14-16]. Notice an important

change in reference 57, the basic variables are electron density n(F) and physical current

density j(f), as opposed to the paramagnetic current density j (f), which is argued in

references 14-16 to be the basic variable. In TD-CDFT, the frequency dependent XC

kernel functions are approximated from the HEG [58, 59], and the formalism is used in

the calculations of polarizabilities of polymers and optical spectra of group IV

semiconductors [60, 61, 62]. TD-CDFT has also been extended to weakly disordered

systems [63] and solids [64].














CHAPTER 2
ATOMS IN UNIFORM MAGNETIC FIELDS THEORY

Single Particle Equations

When a uniform external magnetic field B, which we choose along the z direction,

is imposed on the central field atom, its symmetry goes over to cylindrical. The

Hamiltonian of the system commutes with a rotation operation about the direction of the

B field, so the magnetic quantum number m is still a good quantum number. The natural

gauge origin for an atom-like system is its center, e.g. the position of its nucleus. In the

coulomb gauge, the external vector potential is expressed as

1
A()= -Bx (2.1)
2

The total many-electron Hamiltonian (in Hartree atomic units) then becomes

"I 1 _Z B2 B 1+2 1
H= IV -+- x + y+ n +2n,) +- 1 (2.2)
1=1 2 8 2 2, ,1 -r


where Z is the nuclear charge, M, m,, mi,, are the space coordinate, magnetic quantum

number, and spin z component for the ith electron.

Hartree-Fock Approximation

In the Hartree-Fock (HF) approximation, correlation effects among electrons are

totally neglected. The simplest case is restricted Hartree-Fock (RHF), which corresponds

to a single-determinant variational wavefunction of doubly occupied orbitals. We

use q, (f)to denote single-particle orbitals. For spin-unrestricted Hartree-Fock (UHF)









theory, different spatial orbitals are assigned to the spin-up (a) and spin-down (fl)

electrons. The resulting single-particle equation is

V2 Z B 2 X 7\ B
+ vH () -+ (x2 +2 )+(m, +2m) F (f )

2 r r
L 2 1(2.3)
(f&07 HF (iC) HF10


Notice that exchange contributes only for like-spin orbitals.

Simple DFT Approximation

It seems plausible to assume that the Ax contribution to the total energy is small

compared to the ordinary DFT Exc. Then the zeroth-order approximation to the CDFT

exchange-correlation functional Efj [n, j ] can be taken to be the same form as the XC

functional in ordinary DFT, Ex [n], with the current dependence in the XC functional

totally neglected. Notice that in this scheme, the interaction between the B field and the

orbitals is still partially included. From eqn. (1.28), we see that this approximation

amounts to setting the XC vector potential identically zero everywhere, Ac(r) = 0. The

corresponding single-particle Kohn-Sham equation is

h, 'q (F) ( = )= (f) (2.4)

V2 Z B2 B
where / =--+vH ( r- + 8 2)+x +(m + 2m,) F) (2.5)
2 r 2

The scalar XC potential is defined as in eqn. (1.17).

CDFT Approximation

In this case, both the density dependence and the current dependence of the XC

energy functional Efj [n, jp are included. If we knew the exact form of this functional,








this scheme in principal would be an exact theory including all many-body effects. In

practice, just as in ordinary DFT, the XC functional must be approximated. Unfortunately

very little is known about it. A major theme of this work is to develop systematic

knowledge about the exact CDFT functional and the one available general-purpose

approximation, VRG. The CDFT KS equation reads

Scf (f) = cd (Fr) (2.6)


where kdf =(c-(F))+ (r)+ (i.V+V.,) (2.7)

and vcu (f) is defined by eqn. (1.29). Notice the last term means V (-A,~ c) when the

operator is applied to a KS orbital.

Exchange-correlation Potentials

For ordinary DFT, both LDA and GGA approximations were implemented.

Specifically, the XC functionals include HL [65], VWN [66, 67], PZ [68], PW92 [69],

PBE [70], PW91 [71], and BLYP [72-75]. jPBE is an extension of the PBE functional

that includes a current term [32], but does not treat j, as an independent variable, which

means Axc ()= 0. For GGAs, the XC scalar potential is calculated according to

df X A[n(F),Vn(F)] OE xGA V V G (2.8)
G4n(f) an(F) SVn(Fr)

Before considering any specific approximate XC functional in CDFT, we point out

several cases for which CDFT should reduce to ordinary DFT. The errors in those DFT

calculations are solely introduced by the approximate DFT functionals, not by neglecting

the effects of the current. Such systems can provide estimates of the accuracy of DFT









functionals. Comparing their residual errors with the errors in corresponding current-

carrying states can give us some clues about the magnitude of current effects.

The ground states of several small atoms have zero angular momentum for

sufficiently small external fields. These are the hydrogen atom in an arbitrary field, the

helium atom inB < 0.711 au. [76] (1 au. of B field =2.3505 x105 Tesla), the lithium atom

inB < 0.1929 au. [77], and the beryllium atom inB < 0.0612 au. [8]. Since their

paramagnetic current density j, vanishes everywhere, the proper CDFT and DFT

descriptions must coincide. Notice (for future reference) that their density distributions

are not necessarily spherically symmetric. This argument also holds for positive ions with

four or fewer electrons and any closed shell atom.

If we admit the vorticity 9(F) to be one basic variable in CDFT, as proposed by

Vignale and Rasolt [14-16], there is another kind of system for which the DFT and CDFT

descriptions must be identical. As Lee, Handy, and Colwell pointed out [38], for any

system that can be described by a single complex wavefunction y/(i), v9() vanishes

everywhere. The proof is trivial,


v() = Vx i i -Vx = VxV In =0
n(F) 2i L// 2i

Cases include any single electron system, and the singlet states for two-electron systems

in which the two electrons have the same spatial parts, such as H2 and HeH+ molecules.

Notice that the system can have non-vanishing paramagnetic current density, jp (t) 0O. A

puzzling implication would seem to be that the choice of parameterization by v9() is not

adequate to capture all the physics of imposed B fields.








For CDFT calculations, we have mainly investigated the VRG functional already

introduced. It is the only explicitly parameterized CDFT functional designed for B > 0

and applicable to 3D systems that we have encountered in the literature:
ERG [n(f), i(f)] Jg(n(r))l(rI2dr (2.9)
x G 2vdr (2.9)

where g(n(F)) and i(f) are defined in (1.41) and (1.33).

Substitution of (2.9) into (1.35) gives the expression for the vector XC potential,

A(P) = 2 Vx[g(n(r))V()] (2.10)
n(7)

In actual calculations it generally was necessary to compute the curl in this equation

numerically. In CDFT, the scalar potential has two more terms beyond those found in

ordinary DFT, namely


f (r)= x (r) +dg(n) 2 () (2.11)
dn n(i)

There are three fits for g (n) to the same set of data tabulated in the range of

1 < r, <10 from random phase approximation (RPA) on the diamagnetic susceptibility of

a uniform electron gas [36], namely eqns. (1.42), (1.45) and (1.46). Their derivatives are

dgLCH dr, dgLCH
dn dn dr,


SY e-0 042rs 0.042 0.028 1+ 1 (2.12)
3n 24;42 4 r 2_


9OMC3 i 1(9 Y3 00 0.0002955 (2.13)
dn 3n 24;r21 4) r,










dgoc r 1 9T 2 53 23
dgn 3n 242 (-0.1038r2 +0.3327r3 -0.2212r2 +0.0008432)
dn 3n 242 4

(2.14)

The three fits are very close in the range of 1< r < 10, but differ wildly in other

regions due to different chosen functional forms for g(n). They all cause divergence

problems in any low density region. Without improving the reliability, precision, and the

valid range of the original data set, it seems impossible to improve the quality of the

fitted functions. It is desirable to know its behavior in the low density region, especially

for finite system calculations, but unfortunately, reference 36 did not give any data for r,

>10, nor do we know its asymptotic form. Because dg/dn is required for all r, hence for

all n, yet g(n) is undefined for low densities, we must introduce a cutoff function. After

some numerical experiment we chose


gtf F (c +c2 [sr ac-' (2.15)
24)2

where a.,cOf is the cutoff exponent, which determines how fast the function dies out. The

two constants cl and c2 are determined by the smooth connection between g(n) and

gctoff at the designated cutoff density n cutoff


toff (ncutoff LCH OMC(ncu dgcutoff dgLCHIOMC (2.16)
cun tonff to


In this work, we use ncutof = 0.001ao3, cutoff = 2.0ao', unless other values are explicitly

specified.

There is an identity about the vector XC potential Axc derived from the VRG


functional,





24


Ji, (')- Jp ()= n V x [2g(n())(ir)]. j (i)dF

= -2g(n(i))i(r). V x ()0 d (2.17)

= -2AER [n(f), G (f)

Since Ax (i)C j (r) and AERG can be computed independently, this equation can

provide a useful check in the code for whether the mesh is adequate and whether
numerical accuracy is acceptable.














CHAPTER 3
BASIS SET AND BASIS SET OPTIMIZATION

Survey of Basis Sets Used in Other Work

For numerical calculations, the single particle orbitals in eqn. (2.3), or (2.4), or

(2.6), can be represented in several ways. One is straightforward discretization on a

mesh. For compatibility with extended system and molecular techniques, however, we

here consider basis set expansions. For zero B field, the usual choices are Gaussian-type

orbitals (GTO), or, less commonly, Slater-type orbitals (STO). Plane wave basis sets are

more commonly seen in calculations on extended systems. Large B fields impose

additional demands on the basis set, as discussed below. Here we summarize various

basis sets that have been used for direct solution of the few-electron Schrodinger equation

and in variational approaches such as the HF approximation, DFT, etc.

For the one-electron problem, the hydrogen atom in an arbitrary B field, the typical

treatment is a mixture of numerical mesh and basis functions. The wavefunction is

expanded in spherical harmonics Ym (0, 9) in the low field regime, and in Landau

orbitals 4L (p, ) for large B fields. Here r, 0, p are spherical coordinates, and z, p, p are

cylindrical coordinates. The radial part (for low B) or the z part (for high B) of the

wavefunction is typically represented by numerical values on a one-dimensional mesh

[78]. In Chapter 5 we will also use this technique for the relative motion part of the

Hooke's atom in a B field. Of course, the hydrogen atom has also been solved

algebraically, an approach in which the wavefunction takes the form of a polynomial









multiplying an exponential. This is by no means a trivial task. To get an accurate

description for the wavefunction, the polynomial may have to include thousands of terms,

and the recursion relation for the polynomial coefficients is complicated [79-82].

The multi-channel Landau orbital expansion was also used in DFT calculations on

many-electron atoms [83]. Another approach is the two-dimensional finite element

method [84]. Dirac exchange-only or similar functionals were used in those two

calculations. In the series of Hartree-Fock calculations on the atoms hydrogen through

neon by Ivanov, and by Ivanov and Schmelcher, the wavefunctions were expressed on

two-dimensional meshes [85-90, 76]. Slater-type orbitals were chosen by Jones, Ortiz,

and Ceperley for their HF orbitals to provide the input to quantum Monte Carlo

calculations, with the aim to develop XC functionals in the context of CDFT [91-93].

Later they found that the STO basis was not sufficient and turned to anisotropic Gaussian

type orbitals (AGTO) [94]. Apparently their interests changed since no subsequent

publications along this thread were found in the literature. Schmelcher's group also

employed AGTOs in their full CI calculations on the helium [11-13], lithium [77], and

beryllium [8] atoms. At present, AGTOs seem to be the basis set of choice for atomic

calculations which span a wide range of field strengths. This basis has the flexibility of

adjusting to different field strengths, and the usual advantage of converting the one-body

differential eigenvalue problem into a matrix eigenvalue problem. Moreover, the one-

center coulomb integral can be expressed in a closed form in this basis, though the

expression is lengthy [11, 12]. The disadvantage of AGTOs is that one has to optimize

their exponents nonlinearly for each value of the B field, which is not an easy task, and a









simple, systematic optimization is lacking. We will come back to this issue and prescribe

an efficient, systematic procedure.

Spherical-GTO and Anisotropic-GTO Representations

As with any finite GTO basis, there is also the improper representation of the

nuclear cusp. Given the predominance of GTO basis sets in molecular calculations and

the local emphasis on their use in periodic system calculations, this limitation does not

seem to be a barrier. Spherical GTOs are most widely used in electronic structure

calculations on finite systems without external magnetic field. The periodic system code

we use and develop, GTOFF [95], also uses a GTO basis. Several small molecules in

high B fields were investigated by Runge and Sabin with relatively small GTO basis sets

[96]. To understand the performance of GTOs in nonzero field and make a connection to

the code GTOFF, our implementation includes both GTO and AGTO basis sets. The

former is, of course, a special case of the latter, in which the exponents in the longitudinal

and transverse directions are the same.

Spherical GTO Basis Set Expansion

The form of spherical Gaussian basis we used is

G1 (f)= Nmrlear2Y (0,9) (3.1)

where N, is the normalization factor. The KS or Slater orbitals (DFT or HF) are

expanded in the G1, (r),


w here mI c)= R (r))Y(0, (o )o) (3.2)
I a I

where R (r) r' a,1 (3.3)
(3.3








Notice m is understood as m,, the magnetic quantum number of the ith orbital. For

simplicity, the subscript i is omitted when that does not cause confusion. The electron

density and its gradient can be evaluated conveniently as

n(i) () 2 R, Z (r)Y,(0,(p
SI I (3.4)

r + (
= 1R,, (r)R,,,(r)Y, (0, (p)r,,(, ()

vn()R = ( r [R, (r)R (r) + R' (r)R,, ( r)Y, (, )Y,,(, 3p)
(3.5)
i r 80

The paramagnetic current density is

( r)= 0 mZZ M (r)R, (rfm (0,f m, )= p(r,) (3.6)
rsinO

and the curl of jp (r) is


Vxj (i) = I n +, im
Vx ; r2 80 sin0 80 sin8

r sin8 1 ;;M

The vorticity is evaluated analytically according to

+) V lJp) Vn(_ ) p Vx ()
v(F)=Vx -x v=-^ (F)+--
n(i) n (i) n(i) (3.8)
= r (r, )ri + v (r, 0)

For the VRG functional, the vector XC potential is expressed in spherical coordinates as

A )= 2 1 [r g(n(r, ))vi (r,)]- [ g(n(r,0))v, (r, o)]
= ,q (2r,0)

The last term in eqn. (2.7) becomes









1 aA (r, )
r sin 9o (3.10)



Appendix A includes the matrix elements in this basis for each term in the Hamiltonian.

Anisotropic GTO (AGTO) Basis Set Expansion

An external B field effectively increases the confinement of the electron motions in

the xy plane, and causes an elongation of the electron density distribution along the z axis.

It is advantageous to reflect this effect in the basis set by having different decay rates

along directions parallel and perpendicular to the B field. AGTOs are devised precisely in

this way:

X(p,z,p) =Np "z 'e p pe j =1,2,3,--- (3.11)

where = m +2k, with k= 0,1,... m ,-2,-1,0,1,2,...
where with
nz = Z, +2/1, = 0,1,.- n, = 0,1.

and N, is the normalization factor. If we leta, = /,, this basis of course recovers the

isotropic Gaussian basis, appropriate for B = 0. The basis sets used in reference 94 were

limited only tok, = = 0, which are more restrictive than those used by Becken and

Schmelcher [11-13] and ours.

Single-particle wavefunctions expanded in AGTOs have the general form

0 (Fr)= C bzX, (p,z,,o) Io) (3.12)


Various quantities can be calculated in this basis according to their expressions in

cylindrical coordinates,








v ,(F) 0 + + im, .
V4 (r)=p '+z '+;p
= b \p -2a P + (3Z13)


2
=IA P ill a ) yz ) -'P (p-XAP]^}


n"(F) = ()2 = bV p "zn'e-,2/zL =n (pz) (3.14)


Vn() an() + an() 0 \ 0, + 8, (3.15)
V =n!-,-ap + -= 2 /3 + z (3.15)
p Oz p 8z


(i) M1 0' I 1[ i) 2 m b Np'z7e JZ-,p2_f z2
p (3.16)
=p jp(p, z)

V x JP (r) = 2.m + z 0
S1 v f 8
P O z Op] (3.17)


On .n
S z v j ())+ v j ()) p
S2 + n i + z .i (3.18)


= vP (p, z)p + v, (p, z)2

Again, for the VRG functional we have


f)= { [g(n(p, z))v(p, z) [g(n(p,z)(p,z) (3.19)
n(p,z) 8z Bp?

We follow the scheme in references 11-12 for evaluation of matrix elements, in

which all the integrals, including Coulomb integrals, are expressed in closed form.

Connection between GTOs and AGTOs

As pointed out before, a GTO basis is a special case of an AGTO basis.

Conversely, a particular AGTO can be expanded in GTOs.










X (p, z, (p)= N p' e _,, e -
k= k!

k
=N ( r e ++2k -jr2em (sin O)nf (COS O)"+2k (3.20)
k=0 k

It is easy to see this is a linear combination of Ga' with = n + n + 2k, k = 0, 1, 2,

An ordinary contracted Gaussian basis is a fixed linear combination of several

primitive Gaussians having same the I and m but different exponents a, Similarly, an

AGTO can also be thought as a contracted GTO that contains infinitely many GTOs (in

principle) having the same exponent and m but different / values with increment of 2.

This establishes the equivalence of the two kinds of orbitals. The relative efficiency of

the AGTO basis in cylindrically confined systems is apparent for B 0.

Primary and Secondary Sequences in AGTO

While the AGTO basis provides extra flexibility, its optimization is more

complicated than for a GTO set of comparable size. Kravchenko and Liberman

investigated the performance of AGTO basis sets in one-electron systems, the hydrogen

atom and the hydrogen molecular ion, and showed that they could provide accuracy of

10-6 Hartree or better [97]. Jones, Ortiz, and Ceperley estimated their basis set truncation

error for the helium atom in B < 8au. to be less than one milihartree in the total atomic

energy of about 2 Hartrees [94].

Even-tempered Gaussian (ETG) sequences often are used in zero-field calculations.

For a sequence of primitive spherical Gaussians having the same quantum numbers, their

exponents are given by

a, = ,j = pqJ, j = 1,2,.Nb. (3.21)









where p and q are determined by

In p = aln(q -1)+ a'
(3.22)
In(ln q) = b In Nb+ b'

and Nb is the basis size. For the hydrogen atom, Schmidt and Ruedenburg [98]

recommended the following parameters: a = 0.3243, a'= -3.6920, b = -0.4250, b'=

0.9280. Since the external magnetic field only increases the confinement in the horizontal

direction, we may expect eqn. (3.21) to be equally useful for generating longitudinal

exponents f/ for the AGTO basis.

The choice of a, is more subtle. Jones, Ortiz, and Ceperley [94] used several

tempered sequences of the types

aj = Pf 2,8, 4,j, 8/,j, ... (3.23)

For convenience, we refer to the first sequence (a, = /,) as the primary sequence, and

the second, the third, the fourth, ... sequences as the secondary sequences in our

discussion. The primary JOC sequence in eqn. (3.23) is obviously as same as the

spherical GTOs, for which the transverse and longitudinal exponents are the same.

However for the second JOC sequence, the transverse exponents a, 's are twice the

longitudinal exponents / 's, and for the third sequence, a, = 4/8 etc. The basis set is

the sum of all those sequences. The total number of basis functions is Nb multiplied by

the number of different sequences. Reference 94 used 2-5 sequences in the expansion of

HF orbitals. Obviously, when several sequences are included, which is necessary for

large B fields, very large basis sets can result.









Kravchenko and Liberman [97] chose

a, =,j +BAKL,f +1.2BAKL,/ +0.8BAKL, /, +1.4BAKL,/ +0.6BAKL (3.24)

where AKL is a value between 0 and 0.3 which minimizes the basis set truncation error

compared to more accurate results. Here we still refer to the sequences in eqn. (3.24) as

primary and secondary sequences. In each KL sequence, the differences between the

transverse and longitudinal exponents are the same for all the basis functions. An

improvement of KL basis sets over JOC basis sets is that the former have shorter

secondary sequences, which helps to keep basis size within reason. Namely, the second

and the third KL sequences have lengths of one-half of KL primary sequence, e.g., Nb /2

is used in eqns. (3.21) and (3.22) to generate them, and the fourth and fifth KL sequences

have lengths of Nb /4.

Becken et al. [11] used a seemingly different algorithm to optimize both a, and 8j

in the same spirit of minimizing the one-particle HF energy, H atom or He in a B field,

but they did not give enough details for one to repeat their optimization procedure.

Optimized AGTO Basis Sets

In this section, I give some numerical illustrations of the basis set issues. These

examples illustrate the importance, difficulties, and what can be expected from a

reasonably well-optimized basis.

Our goal is set to reduce basis set error in the total energy of a light atom to below

one milihartree. This criterion is based on two considerations. One is the observation by

Orestes, Marcasso, and Capelle that the magnitude of current effects in CDFT is of the

same order as the accuracy reached by modern DFT functionals [37]. They compared

atomic ionization energies from experiment with DFT-based calculations. A typical









difference is 0.4eV, or 15 mH. To study the current effect in CDFT, we need to reduce

the basis set errors to considerably below this value. Another factor considered is the

well-known standard of chemical accuracy, usually taken to be 1 kcal/mol, or 1.6 mH. It

turns out that this goal is much harder to reach for multi-electron atoms in a large B field

than for the field-free case. Two systems I choose for comparison are the hydrogen and

carbon atoms. There are extensive tabulations for the magnetized hydrogen atom [78],

and even more accurate data from the algebraic method [80] against which to compare.

However, the hydrogen atom does not include electron-electron interaction, which is

exactly the subject of our interest. For the carbon atom, our comparison mainly will be

made with numerical Hartree-Fock data [90]. Without external field, the correlation

energy of the C atom is about 0.15 Hartree [99], two orders of magnitude larger than our

goal. This difference also makes the choice of one mH basis set error plausible.

Examine the zero-field case first. It is well known that the non-relativistic energy of

the hydrogen atom is exactly -0.5 Hartree. For the carbon atom, the numerical HF data

taken from reference 90 are treated as the exact reference. Calculated HF energies in

various basis sets are listed in Table 3-1, together with basis set errors in parentheses. We

first tested the widely used 6-31G basis sets. Those basis sets are obtained from the

GAMESS code [100]. In primitive Gaussians, they include up through 4s for the

hydrogen, and 10s4p for the carbon atom. As expected, the accuracies in total energy that

they deliver increase only slightly after de-contraction. A sequence of exponents derived

from eqn. (3.21) with length Nb = 8 has a comparable size with the 6-31G basis for the

carbon atom. It gives rather bad results, but recall that a significant deficiency of GTOs is

that they cannot describe the nuclear cusp condition. By adding five tighter s orbitals









extrapolated from eqn. (3.21) with = 9,10,...,13, the basis set error is reduced by 99%.

To further reduce the remaining 1.6 mH error, higher angular momentum orbitals are

required. Addition of four d orbitals and removal of the tightest, unnecessary p orbital

gives a 13s7p4d basis set, with only 0.8 mH truncation error left. A larger basis set,

20s1 lp6d, similarly constructed from the Nb = 16 sequence by adding 4 tighter s orbitals

has error only of 0.05 mH.

Table 3-1 Basis set effect on the HF energies of the H and C atoms with B = 0 (energies
in Hartree)

Basis Set a Hydrogen atom Carbon atom
6-31G -0.498233 (0.001767) -37.67784 (0.01312)
De-contracted 6-31G -0.498655 (0.001345) -37.67957 (0.01139)
Sequence Nb = 8 -0.499974 (0.000026) -37.51166 (0.17930)
Nb = 8, plus 5 tighter s -0.499989 (0.000011) -37.68938 (0.00158)
13s7p4d -0.499989 (0.000011) -37.69018 (0.00078)
Sequence Nb = 16 -0.49999992(0.00000008) -37.68949 (0.00147)
20s1 p6d -0.49999996(0.00000004) -37.69091 (0.00005)
oo -0.5 -37.69096 b
(a) see text for definitions;
(b) from reference 90;
(c) numbers in parentheses are basis set errors.

The situation changes greatly when a substantial external B field is added. Let us

first take the example of the H atom ground state in a field B = 10 au. Its energy is known

accurately to be -1.747 797 163 714 Hartree [80]. The sequence ofNb = 16 included in

Table 3-1 works remarkably well for the field-free energy, but gives 24% error in the B

= 10 au field. See Table 3-2. Adding a sequence of d orbitals that has same length and

same exponents as that for the s orbitals, which doubles the basis size, reduces the error

by 80%. Further supplementation by g and i orbitals in the same way decreases the error

by another order of magnitude. But this is still far from satisfactory. To reduce the basis

set error below 1 /H, higher angular momentum basis with I up to 20 must be included.

Obviously, this is a very inefficient approach. The basis sets used by Jones, Ortiz, and









Ceperley [94] (see eqn. 3.23) converge the total energy more rapidly than these spherical

bases. The primary sequence in the JOC basis sets is the same as the spherical basis, but

subsequent secondary sequences double the transverse exponents aj's successively. With

four sequences the error is less than 1% of the error in a spherical basis set having the

same size. Another significant gain can be obtained if we move to the KL basis sets [97]

(see eqn. 3.24). Here we choose AKL = 0.18, which is obtained by searching with a step of

0.01 to minimize the basis set truncation error. Including only the primary KL sequence

gains about the accuracy of the three-sequence JOC basis set. Recall that the subsequent

KL secondary sequences have shorter lengths than the primary one (refer to the

discussion after eqn. 3.24). Specifically, the second and the third sequences have length

of Nb /2 = 8, and the fourth and the fifth sequences have length of Nb /4 = 4. Thus, the

basis size will be 16 + 8 + 8 + 4 + 4 = 40 if we include five KL sequences, with accuracy

of 1 uH.

Table 3-2 Basis set errors for the ground state energy of the hydrogen atom in B = 10 au.
(energies in Hartree)

Basis size Spherical JOC a KL b Optimized Eqn. (3.26)
16 0.4198 0.41978728 0.00373820 0.00000060 0.00104451
32 0.081 5 0.027 124 87 0.000 005 39 0.000 00036 0.000 000 50
48 0.021 7 0.001 008 57
64 0.008 1 0.000 075 02
40 0.000 001 12 0.000 000 30 0.000 000 28
(a) Jones-Ortiz-Ceperley basis sets, see ref [94] and eqn. (3.23);
(b) Kravchenko-Liberman basis sets, see ref [97] and eqn. (3.24). AK is chosen to be 0.18.

However, this does not mean there is no opportunity left for basis set optimization.

Starting from the primary sequence in the KL basis set, we then searched in the parameter

space {a,} to minimize the total energy of the H atom. First, the energy gradient in









parameter space is calculated, then a walk is made in the steepest descent direction. These

two steps are repeated until

OE
= 0 (3.25)
ca]

The error left in this optimized basis set is only 0.6 pH, six orders of magnitude

smaller than the error of a spherical basis set of the same size! The resulting exponents

are listed in Table 3-3, together with the coefficients used in the wavefunction expansion.

Addition of the same secondary KL sequences can further reduce the remaining error by

one half. This improvement is not as spectacular as that for the KL basis set because

those exponents have already been optimized. It is worth mentioning that, while it is easy

to optimize the basis set for the H atom fully, it is very hard to do so for multi-electron

atoms. We usually only get partially optimized results, but by including secondary

sequences, the basis error can be greatly reduced, as demonstrated here.

Table 3-3 Optimized basis set and expansion coefficients for the wavefunction of the
hydrogen atom in B = 10 au.

j Coefficients a aj j
1 0.000493 1.8886 0.0573 1.8313
2 0.011007 2.8640 0.1247 2.7393
3 0.184818 2.4462 0.2717 2.1745
4 0.372811 2.5541 0.5917 1.9624
5 0.277663 2.6442 1.2890 1.3552
6 0.132857 3.7690 2.8077 0.9613
7 0.050662 6.7855 6.1159 0.6696
8 0.019285 13.9048 13.3221 0.5827
9 0.007139 29.3287 29.0190 0.3097
10 0.002772 64.4702 63.2111 1.2591
11 0.000955 139.3854 137.6904 1.6950
12 0.000418 301.7122 299.9260 1.7862
13 0.000114 655.1166 653.3180 1.7986
14 0.000082 1424.8988 1423.0988 1.8000
15 -0.000001 3101.6845 3099.8845 1.8000
16 0.000021 6754.1687 6752.3659 1.8028









From Table 3-3, we see that the wavefunction is mainly expanded in basis = 3, 4,

5, 6, and a, p, is not a constant as the KL sequences suggest. The smaller the

exponent, the larger the difference between the transverse and the longitudinal exponents.

This is quite understandable. A smaller exponent means that the electron density extends

far from the nucleus, and the magnetic field will overpower the nuclear attraction, thus

the distortion from the field-free spherical shape will be relatively larger. In the limit of

fj- 0, which can be equivalently thought of as the large B limit, or zero nuclear charge,

the electron wavefunction is a Landau orbital with an exponential parameter

B
a = aB = The opposite limit, /,-* oo, corresponds to B = 0, for which a,= /,. A
4

natural measure of the orbital exponents is aB Now we can make an explicit

construction (discussed below) which incorporates all these behaviors, namely


B 4 f, 4 ,
a= +- 4 1+4 j-2 + 1+[ ] (3.26)
S 20 bG () B b (y) B


where b(y) =-0.16[tan 1()]2 + 0.77 tan 1(7)+ 0.74 (3.27)


= pq' j = 1,2,-- Nb. (3.28)

and y = B/Z is the reduced field strength for an effective nuclear charge Z. The

parametersp and q are defined in eqn. (3.22). For the innermost electrons, Z is close to

the bare nuclear charge; for the outmost electrons in a neutral atom, it is close to unity.

Nevertheless, accurate Z values do not need to be provided. Nominal values turn out to be

good enough for the input to eqn. (3.26). Secondary sequences are defined similarly as in









KL basis set, using a factor of 1.2, 0.8, 1.4, and 0.6 for the second, third, fourth, and fifth

sequences, respectively.

Next I show how eqn. (3.26) was obtained. Start from the basis set of one sequence

Nb = 16. Full basis set optimizations were done for H, He+, Li+, Be+++, C+ and 07+ in

reduced fields y = 0.1, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, 500, 800, 1000, 2000, and 4000.

Results for y = 0.1, 1, 10, and 100 are plotted in Fig. 3-1. The first observation is that data

points for different nuclear charges with the same y are on the same curve. One can show

that this must be the case. Suppose the wavefunction for the hydrogen atom in an

external B field is WH (, B),


-+ 2 +y2 H (r, B)= EHr H (, B) (3.29)


Scaling r Z r leads to

V2 1 ZZ2B2 2)
2_ 2 H (Z i, B)= EH, H (Z i, B)
2Z2 Zr 8


V2 Z (Z2B)2
[ 2 + z(X)2( +Y H (Z ,B) =(z2EH (Z i,B)
2 r 8

The Hamiltonian on the left side is the same as that of a hydrogen-like atom with nuclear

charge Z in an external fieldB'= Z2B. The scaled hydrogen-atom wavefunction is

precisely the eigenfunction of this Hamiltonian with energy of Z2EH. Now we expand

WH (, B) in the optimized basis set,

/IH (, B) = aNj p" zP e'e-, p2 #e (3.30)


and the scaled wavefunction is







40



VZ (F, ZB)= /H (ZF, B) = aJN. Z" '"%" p z 'e e", (3.31)

a, =2 a' 8j aj --8j 8j' 8

where a' = Zaj, f/ = Z2fj. Obviously, a' and
B' B B' B

Another observation is that the curvatures for different y values are slightly

different. To describe the rapid decrease in the small /8 / B region and the slowly


decaying long tail, we used a fixed combination of inverse square and inverse square root

terms, which proved to be superior to a single reciprocal function. In Fig. 3-2, the

functional forms are compared with data points from optimized basis sets for y= 1.

0.25#-




0.2 -



A
0.15
m
CC-
0


0.1 E
x A
+ x


0.05- +
[] 0

+ 0 O

+ 0 0
0
0 0.5 1 1.5 2 2.5 3 3.5 4



FIG. 3-1 Exponents of optimized basis sets for the H(+), He+(x), Li++(o), Be +(A), C5+(0)
and 07+() in reduced magnetic fields y = 0.1 (blue), 1 (black),
and 100 (red).







41



0 optimized for7=l
0.2 Eqn. (3.26) using b=1.286
2 Eqn. (3.26) using b=1.246
C- 1/4/(1+4/0.722*l/B)
0.18 -
i
0.16 -

0.14 -

m 0.12 -

S0.1 o-
'.\
0.08 0 .\
0o.\

0.06

0.04 .Q
->.. .,
0.02 0.....
@ "- ... -o0-


0 0.5 1 1.5 2 2.5 3 3.5 4
Pj/B


FIG. 3-2 Fitting the parameter b(y =1) using the function (3.26). Fitted result is 1.286,
compare to the calculated value 1.246 from eqn. (3.27). Two curves are shown
by dotted black line and solid green line. They are almost indistinguishable in

the graph. A reciprocal fitting result B + 4 6, is also shown by a
4 0.722 B
dashed blue line.

A calculation using the basis set derived from eqn. (3.26) also is included in Table

3-2. The new primary sequence outperforms the KL primary sequence by a factor of

four, but by including only the second and the third sequences, the basis set is almost

saturated, compared to other, more slowly converging basis sets. Another advantage of

the present basis set is the explicit expressions eqns. (3.26) and (3.27), whereas searching

for the best parameter AKL in the KL basis sets is quite time consuming.













Table 3-4 Test of basis sets including 1, 2, and 3
negative signs are omitted)


sequences on the energies of the hydrogen atom in B fields (energies in Hartree,


State
Is







2po


2p_-



3dl

3d2

4f 2
43

5g4


B (au.)
0.1
1
10
100
1000
4000
10000
1
10
1000
1
10
100
1000
10
1000
10
1000
100
10
1000
1
100
1000


Reference 80
0.547526480401
0.831168896733
1.747797163714
3.789804236305
7.662423247755
11.204145206603

0.260006615944
0.382649848306
0.4924950
0.456597058424
1.125422341840
2.634760665299
5.63842108
0.3389561898
0.4869777
0.9082147755
4.80511067


1 sequence
0.5475263
0.83109
1.74675
3.78933
7.66224
11.20372
14.14037
0.25991
0.38263
0.49248
0.45658
1.12521
2.63472
5.63792
0.33890
0.48696
0.90813
4.80432
0.42767
0.78773
4.30738
0.26570
1.75448
3.96338


Reference 76

0.83116892
1.74779718
3.7898043
7.6624234

14.14097
0.260007
0.382650


(a) Numbers before/after slashes are the upper/lower bounds to the energy.


2 sequences
0.5475263
0.8311680
1.747781
3.789790
7.662419
11.204139
14.140959
0.2600055
0.382641
0.4924948
0.456596
1.125415
2.634756
5.638413
0.3389555
0.4869775
0.908212
4.805094
0.427756
0.787768
4.308344
0.266184
1.754848
3.964471


3 sequences
0.547526461
0.83116886
1.74779694
3.78980395
7.66242306
11.20414499
14.14096829
0.26000652
0.38264977
0.49249495
0.45659703
1.12542217
2.63476052
5.63842079
0.33895610
0.48697789
0.90821466
4.80511012
0.42775840
0.78776910
4.30836962
0.26618782
1.75485593
3.96450833


Reference 78a
0.5475265
0.831169
1.747797
3.78905/90250
7.66205/65


0.2600066
0.38264875/5180
0.492495
0.45659705
1.1254225
2.634740/95
5.638405/35
0.33895610/45
0.48697795
0.9082115/235
4.8051095/125
0.4277585
0.7877685/705
4.3083700/05
0.26618770/875
1.754856
3.9645095









The basis set constructed from eqn. (3.26) not only works well for Is electrons, but

also for higher excited states. Table 3-4 includes some test results on the hydrogen atom

in a wide range of B fields. The primary sequence was derived from eqns. (3.26) through

(3.28) using Nb = 16. Extrapolations toj > Nb orj < 0 were made for extremely tight or

diffuse orbitals whenever necessary. The averaged basis set error for the primary

sequence is 0.3 mH, which is reduced to 7 /H if the second sequence is added. With three

sequences, the accuracy of our basis sets reaches 1 /H level. Notice that energies quoted

from reference 76 are slightly lower than the more accurate algebraic results [80]. This

implies that Ivanov and Schmelcher's data are not necessarily upper bounds to the

energy. We need to keep this in mind when we compare our results with theirs.

Table 3-5 Energies for high angular momentum states of the hydrogen atom in B fields
(B in au; energy in Hartree)

State B = 1 B= 10 B= 100 B = 1000
5g-4 -0.2661880 -0.7080264 -1.7548563 -3.9645100
6h_5 -0.2421928 -0.6499941 -1.6252244 -3.7061998
7i-6 -0.2239757 -0.6051943 -1.5238725 -3.5018527
8j-7 -0.2095131 -0.5691841 -1.4415788 -3.3343126
9k_ -0.1976562 -0.5393750 -1.3728860 -3.1933035
10/19 -0.1876974 -0.5141408 -1.3143222 -3.0722218

The accuracy of the previous basis sets can be improved further by increasing Nb

and including the fourth and the fifth sequences. Using larger basis sets having five

sequences with the primary sequence derived from Nb = 28, we obtained energies of

-5.638 421 065, -4.805 110 65, -4.308 370 6, and -3.964 510 0 for the H atom 2p.i, 3d-2,

4f3, and 5g-4 states in B = 1000, respectively. Their accuracy is at the same level as the

best available data in the literature (see Table 3-4). There are no specific difficulties for

the higher angular momentum states in our expansion. The energy values for the excited









states with quantum numbers I = m = 4, 5, ..., 9 of the hydrogen atom in B fields are

listed in Table 3-5. Those values will be used in the next step, construction of Table 3-6.

Actually we do not need to use the entire sequences generated from eqns. (3.26)

through (3.28). In the expansion of nonzero angular momentum orbitals, very tight basis

functions (j close to Nb ) are not necessary, but extrapolation to negative may be

required in order to include very diffuse basis functions. Table 3-6 lists the subsets which

have accuracy of 1 uH. The one to five segments in each basis set means the ranges ofj

values selected from the primary and the subsequent secondary sequences. Numbers

underlined identify the negative values. Nb values used for the five sequences are 16, 8,

8, 4, and 4. Also recall a factor of 1.2, 0.8, 1.4, and 0.6 is used for the second, third,

fourth, and fifth sequences, respectively. Numbers in parentheses are the sizes of the

basis sets.

Table 3-6 Basis sets for the hydrogen atom in B fields with accuracy of 1 /H (B in au)

State B=0 B= B=10 B = 100 B=1000
Is 1-14(14) 2-14,1-3,1-2(18) 1-16,2-6,2-6(26) 4-20,2-8,2-6(29) 4-21,4-10,5-10(31)
2po 1-7(9) 0-8,0-3,3(14) 0-10,1-5(16) 1-12,2-5(16) 1-14,3-5(17)
2p-1 1-7(9) 1-9,0-3,3(14) 1-11,5,2-6(17) 3-14,4-8,3-6(21) 4-16,4-10,5-10(26)
3di 3-4(8) 1-6,0-3,0-3(16) 0-8,1-5,2-3(16) 0-10,1-6,2(18) 1-12,2-6,3(18)
3d_2 2-4(7) 0-7,1-3,0-3(15) 1-8,3-5,2-6(16) 3-12,3-7,3-7(20) 4-15,4-10,4-7(23)
4f2 5-2(8) 1-5,0-3,0-3(15) 0-7,1-4,1-5(17) 0-10,1-6,2-3(19) 1-12,2-6,3(18)
4f3 4-2(7) 0-4,0-3,0-4(14) 1-7,2-5,2-6(16) 3-11,1-7,1-7(23) 3-14,2-8,4-7,3,3(25)
5g_3 4-0(5) 1-5,1-2,0-1(13) 0-7,1-4,1(13) 0-9,1-5,2-3(17) 1-11,2-6,3-4(18)
5g-4 4-0,2-1(7) 0-4,0-3,0-4(14) 1-6,2-5,1-3(13) 2-11,1-7,3-6(21) 3-13,3-8,4-7,3,3(23)
6h_5 5-2,4-1(8) 0-4,0-3,0-3(13) 1-6,2-5,1-3(13) 2-10,3-7,2-5(18) 3-13,3-8,4-7,3,3(23)
7i-6 6-2,5-3(8) 1-2,0-2,0-3(11) 1-5,2-5,1-3(12) 2-10,1-5,3-6(18) 3-13,3-8,4-7,3,3(23)
8j_7 1-2,0-2,0-3(11) 1-5,2-5,1-3(12) 2-10,2-6,3,2,2(17) 3-13,3-8,4-7,3,3(23)
9k8 1-5,2-5,1-2(11) 2-9,2-6,3,2,2(16) 3-12,3-8,4-6,3,3(21)
1019 1-5,2-5,1-2(11) 2-9,2-6,3,2,2(16) 3-12,3-8,4-6,3,3(21)

While the previous optimization scheme is quite impressive for the hydrogen atom

in a B field, we want to know whether it also works equivalently well for multi-electron

atoms. Thus we do another case study, the carbon atom in the same field B = 10 au. Its









ground state configuration is s22p 13d 24f_ 5g 4, and HF energy is -44.3872 Hartree

from calculations on a numerical grid [90]. The performance of various basis sets is

summarized in Table 3-7.

Table 3-7 Basis set effect on the HF energies of the carbon atom in B = 10 au.

Basis set Basis size HF energy (Hartree) Error(Hartree)
Spherical(spdfghi) 112 -43.6157 0.7715
2sequences, JOC 160 -44.1572 0.2300
3 sequences, JOC 240 -44.3529 0.0343
sequence, KL 80 -44.1629 0.2243
2sequences, KL 120 -44.3824 0.0048
3 sequences, KL 160 -44.3863 0.0009
sequences, KL 200 -44.3867 0.0005
1 sequence, present 50 -44.3859 0.0013
2sequences, present 72 -44.38704 0.0002
sequences, present 91 -44.38714 <0.0001

The spherical basis set includes 16s16p16d16f16g16h16i orbitals. Again it gives a

large basis set error. For the KL basis sets, I used the same parameter as before, AKL =

0.18. Its accuracy can also be improved greatly by systematic augmentation of secondary

sequences. However, the basis size will be increased considerably. Based on the previous

detailed study on the basis set for the hydrogen atom, here we prescribe a procedure to

construct the basis set for a multi-electron atom in a B field, with the C atom as the

example.

We first assign the effective nuclear charge Zf for each electron roughly. The

approach is by approximate isoelectronic sequences. Since the Is electrons feel the whole

strength of the nuclear attraction, we use 6 for them. For the 2p electron, the nucleus is

screened by the two Is electrons, so we use 4, and so forth. Next basis functions are

generated according to eqns. (3.26) through (3.28) using Nb= 16, 8, 8, 4, 4 for the

primary, the second, ..., the fifth sequences. To use Table 3-6 as guidance in selecting









subsets of basis functions from the previously generated sequences, first recall the scaling

argument after eqn. (3.29). The Is wavefunction with an effective nuclear charge Zeffis =

6 in a field B = 10 can be scaled from the hydrogen atom is wavefunction in a

field B' = B/ZbX, = 0.28. The value ofB' falls in the range of 0 to 1. By inspection of

Table 3-6, we find a sufficient choice of basis set includes the first through the fourteenth

elements in the primary, 1-3 elements in the second, and 1-2 in the third sequences. But

do not forget the scaling factor. Since the C atom Is wavefunction is tighter than the H

atom Is wavefunction approximately by Zeffs, = 6 times, the basis function exponents

used in the expansion of the C atom Is orbital should be larger than those used for the H

atom by Z2 = 36 times (recall eqns. 3.30 and 3.31). Remember the exponents fl,'s

consist of a geometrical series (eqn. 3.28). The increase of the exponents amounts to a

21n6
shift of logq Z2 21n, = 4.6 elements in the primary sequence, and a shift of
In 2.18

21n 6
-- = 3.4 elements in the second and the third sequences. Hence, we should pick the
In 2.84

5-19, 4-7, and 4-6 elements in the primary, the second, and the third sequences,

respectively. Basis function selections for other electron orbitals are similar. The final

basis set is 22s19pI6d21f3g, which is summarized in Table 3-8. Among the total of 91

gaussians in this basis set, 50 are from the primary, 22 from the second, and 19 from the

third sequences. From Table 3-7, we can see the accuracy of this basis set is remarkably

higher than that of the others. By using only the primary sequence, the error left is close

to 1 mH. Supplementation with the second sequence reduces the error to 0.2 mH. We

estimate the error of the 3-sequence present basis to be less than 0.1 mH.












Table 3-8 Construction of the AGTO basis set for the carbon atom in B = 10 au.

Orbital Zeff B'= B/Z2 H atom basis shifts C atom basis
eft
is 6 0.28 1-14, 1-3, 1-2 4.6, 3.4, 3.4 5-19, 4-7, 4-6
2p-1 4 0.63 1-9, 0-3, 3 3.6, 2.6, 2.6 2-13, 2-6, 5-6
3d-2 3 1.1 0-7, 1-3, 0-3 2.8, 2.1, 2.1 2-10, 3-5, 2-5
4f3 2 2.5 0-7, 0-5, 0-6 1.8, 1.3, 1.3 2-9, 1-6, 1-7
5g-4 1 10 1-6, 2-5, 1-3 0, 0, 0 1-6, 2-5, 1-3

One may wonder why this procedure works so well, or even why it works at all.

The main reason is that each electron orbital can be approximated by a hydrogen-like

problem fairly closely. For example, the overlap between the Is HF orbital for the carbon

atom in B = 10 and the Is orbital for C5 in the same field is 0.998. See Table 3-9.

Actually by adjusting the nuclear charge to 5.494 and 5.572, the overlaps for Is spin

down and spin up orbitals with their corresponding hydrogen-like counterparts reach the

maxima of 0.9998 and 0.9999, respectively. Other orbitals are similar.

Table 3-9 Overlaps between HF orbitals for the carbon atom in B = 10 au and hydrogen-
like systems in the same field

Orbital Zf overlap Z' overlap
Is, [ 6 0.99773 5.494 0.99984
Is, 6 0.99838 5.572 0.99989
2p-1 4 0.99967 3.944 0.99968
3d-2 3 0.99974 3.145 0.99983
4f3 2 0.99731 2.644 0.99986
5g-4 1 0.98224 2.108 0.99981

Now that we have a systematic way to construct reasonably accurate basis sets for

atoms in a B field. In the next section, I will use those basis sets for the DFT and CDFT

studies.














CHAPTER 4
ATOMS IN UNIFORM FIELDS NUMERICAL RESULTS

Comparison with Data in Literature

I did extensive unrestricted Hartree-Fock (UHF) and conventional DFT

calculations on the atoms He, Li, Be, B, C, and their positive ions Li Be B+ in a large

range of B fields with basis sets constructed according to the procedure outlined in the

previous chapter. Total energies are compiled in appendix B. Ground states are indicated

in orange. Data available from the literature are also included for comparison.

The UHF calculations were primarily for purposes of validation. The agreement of

our calculations with those from other groups is excellent. For the helium atom, our HF

energies are generally slightly lower than those by Jones, Ortiz, and Ceperley [91, 94].

Their earlier calculations used an STO expansion [91] which is labeled as JOC-HF1 in

Table B-1. Later they utilized JOC basis sets within AGTO [94] (also refer eqn. 3.23),

which we call JOC-HF2. Presumably the small distinction between their data and ours

results from better optimized basis sets I generated, as already illustrated in the previous

chapter. This observation is supported by the generally closer agreement of their

anisotropic basis set results with ours (in contrast to their spherical basis results). One

notable exception to the overall agreement is the 1s4f3 state in B = 800 au. Our result is

-23.42398 Hartree versus theirs -23.4342. Another is ls3d-2 at B = 560 au: -21.59002

versus their -21.5954 Hartree. The reason for these discrepancies is unclear. It may be

some peculiarity of the basis for a particular field strength. For other atoms and ions, the

data for comparison are mainly from the series of studies by Ivanov and Schmelcher [86,









88-90]. Our HF energies generally match or are slightly higher than theirs, and the overall

agreement is quite satisfactory. Differences are usually less than 0.1 mH, far surpassing

our goal of 1 mH accuracy for the basis set. The remaining differences arise, presumably

from our basis set truncation error and their numerical mesh errors. As for any basis-set

based calculations, we can only use a finite number of basis functions, which will cause

basis set truncation error. Since this error in our calculation is always positive (by the

variational principle), one can use our data as an upper bound for the HF energies.

However, the numerical error in Ivanov and Schmelcher's 2D HF mesh method seems to

tend to be negative. Recall the comparison made in Table 3-4 for the hydrogen atom.

Their energies are always lower than the more accurate algebraic result [80]. Another

indication is the zero field atomic energies. For example, the HF energy for the beryllium

atom is known accurately to be -14.57302316 Hartree [101], which agrees well with our

result of -14.57302, but Ivanov and Schmelcher gave a lower value of -14.57336 [88].

They commented that this configuration has large correlation energy and the contribution

from the ls22p2 configuration should be considered, but did not specify whether their

result was from single determinant or multi-determinant calculation. Since multi-

configuration HF (MCHF) is not our main interest here, our data are solely from single

configuration HF calculations. They also noted that the precision of their mesh approach

decreases for the is22s2 configuration in a strong field. This can also contribute to the

discrepancy. From previous observations, one may speculate that the true HF energies lie

between our data and theirs. Furthermore, the accuracy of our data is ready to be

improved by invoking a larger basis set, but this is not necessary for the purpose of the

present study.









There are fewer DFT calculations for atoms in B fields than HF studies. As far as I

know, appendix B is the first extensive compilation of magnetized atomic energies based

on modern DFT functionals. I chose PW92 [69] for LDA and PBE [70] for GGA

calculations. For the field-free case, the present results agree well with published data

[102]. Since reference 102 only gave spin non-polarized DFT energies for spherically

distributed densities (which is no problem for the helium and beryllium atoms), one needs

to use fractional occupation numbers in other atoms for comparison. For example, in the

carbon atom, twop electrons are placed in six spin orbitals, p ,p' and p+1 with

equal occupation number of 1/3 for each of them. Actually there are more accurate data

for the VWN functional [67]. My results differ from those from reference 67 by no more

than 5 /H if I choose the VWN functional, either for spin-polarized or spin non-polarized

energies, neutral atoms or their positive ions.

Comparison of non-vanishing B field DFT calculations is handicapped by the

different magnetic field grids on which different authors present their results, and by the

different functionals implemented. The functional due to Jones [103], which is at the

level of the local density approximation, was used by Neuhauser, Koonin, and Langanke

[104], and by Braun [84]. The simple Dirac exchange-only functional was used by

Relovsky and Ruder [83], and by Braun [84]. When I choose the Dirac exchange

functional, good agreement is found with Braun's data. However, no application of

density-gradient-dependent fiunctionals on atoms in a strong magnetic field is found in

the literature so far as I can tell.

For the CDFT study, the present work apparently is the first fully self-consistent

calculation on atoms in large B fields based on the VRG functional. The most closely









related study is the perturbative implementation of the VRG functional by Orestes,

Marcasso, and Capelle [37] on the atomic ionization energies in vanishing B field.

For comparison, CI results are available for helium, lithium, and beryllium atoms,

and their positive ions in external fields [7, 8, 11-13, 77, 105]. Those data will be treated

as the most reliable ones in my comparison.

Magnetic Field Induced Ground State Transitions

The most drastic change of the ground state atoms caused by an external magnetic

field is a series of configuration transitions from the field-free ground state to the high-

field limit ground state. It is well known that for the field-free case, two competing

factors the spherical nucleus attractive potential and the electron-electron interactions

- lead to the shell structure of atoms. This structure is perturbed slightly if the external

field is relatively small, but when the field is strong enough that the Lorentz forces

exerted on the electrons are comparable to nuclear attraction and electron repulsions, the

original shell structure is crushed, and the electrons make a new arrangement. Thus, a

series of configuration transitions happens as the B field becomes arbitrarily large. In the

large-field limit, the ground state is a fully spin-polarized state in which the electrons take

the minimum value of spin Zeeman energy. Ivanov and Schmelcher further analyzed the

spatial distribution of electrons and described the high-field limit ground state "... with no

nodal surfaces crossing the z axis (the field axis), and with nonpositive magnetic quantum

numbers decreasing from m = 0 to m = -N + 1, where Nis the number of electrons" [76].

In Hartree-Fock language, the high field configuration is ls2p _3d 24f3 **. In this

regime, the magnetic field is the dominant factor, and coulombic forces can be treated as

small perturbations. A cylindrical separation of the z part from x and y parts is usually







52


made for the electron state. Its motion in the xy plane is described by a Landau orbital,

and the question is reduced to a quasi-ID problem. This technique is often referred to as

the adiabatic approximation, valid only in the limit of very large field. Many early

investigations on matter in aB field concentrated on this regime [103, 104, 106].








C .24 '1'., s -

-2

m ,f
-22 4

-2.4 -- s. .. .













-32
0 0,5 1 1.5 2
B(au)




FIG. 4-1 UHF total energies for different electronic states of the helium atom in B fields.
Curves 1 to 9 represent configurations Is2, ls2s, 1s2po, ls2p-i, Is3Lj, s3d-2,
ls4f-2, ls4pfJ3, and ls5g-3, respectively.

The most difficult part is the region of intermediate B field, for which both

cylindrical and spherical expansions are inefficient, and where the ground state

configurations can only be determined by explicit, accurate calculations. Figure 4-1

displays the HF energies of various configurations for the helium atom in B fields as

listed in Table B-l, which includes one singlet state and eight triplet states. Curves 1 to 9

represent configurations of Is2, ls2s, ls2po, ls2p-i, ls3Lj, ls3-2, ls4f2, ls4pf3, and

Ss5g-3, respectively. Each configuration belongs to a spin subspace according to its total









spin z component. For convenience, we use the inexact terminology of "local ground

state" and "global ground state". For a specific field strength, the configuration which has

the lowest energy within a spin subspace is called the local ground state for this spin-

multiplet. Among them, the one taking the minimum regardless of its spin is called the

global ground state. Thus in Fig. 4-1, the singlet state remains as the global ground state

until B reaches 0.71 au., then a configuration transition to ls2pil occurs. This triplet state

is the global ground state for B fields larger than 0.71 au. Atoms with more electrons can

have more complicated series of configuration transitions. For example, the carbon atom

undergoes six transitions with seven electronic configurations involved being the ground

states in different regions of B field strengths [90]. This scenario is basically the same if

one uses DFT or CI energies instead of HF energies, except the crossing points for

different configurations usually change.

Atomic Density Profile as a Function of B

Within each configuration, the electron density is squeezed toward the z axis with

increasing B field. This follows from energy minimization: the electron density shrinks

toward the z axis to alleviate the corresponding diamagnetic energy increment

(expectation of B2(x2+y2)/8). Figure 4-2 shows the density profiles for the ls2 and ls2p-l

states of the helium atom at field strengths B = 0, 0.5, 1, and 10 au. The transverse

shrinkage is quite evident. However, this shrinkage increases the electron repulsion

energy. A configuration transition therefore will happen at some field strength (for He, B

= 0.71 au.), accompanied by a change of quantum numbers, and eventually a spin flip.

Note Figs. 4-2 (a), (b), and (g). The energy increase in the diamagnetic term caused by


















































FIG. 4-2 Cross-sectional view of the HF total electron densities of the helium atom Is2
(panels a-d) and s2p-l (panels e-h) states as a function of magnetic field
strength. Each large tick mark is 2 bohr radii. The B field orientation is in the
plane of the paper from bottom to top. The density at the outermost contour
lines isl0 6ao3, with a factor of 10 increase for each neighboring curves.
Panels (a), (b), (g), and (h) are global ground states. See text for details.









the electron density expansion in the new spin-configuration for the global ground state is

more than compensated by the accompany energy lowering of the Zeeman and electron-

electron repulsion terms. In fact, the same net lowering can occur (and sometimes does

occur) without change of spin symmetry.

Total Atomic Energies and Their Exchange and Correlation Components

Atomic total energies of He, Li, Be, B, C, and their positive ions Li Be B+ in a

large range of B fields within the HF, DFT-LDA, and DFT-GGA approximations are

compiled in Appendix B. The exchange-correlation energy Exc, and its exchange and

correlation components Ex, Ec corresponding to the total energies in appendix B, are

given in appendix C. As is conventional, the HF correlation energy is defined as the

difference between the CI total atomic energy and HF total energy tabulated in appendix

B. DFT exchange and correlation energies are defined at the self-consistent electron

densities within the corresponding XC energy functionals (LDA or PBE). Keep in mind

that exchange and correlation energies are not defined identically in the wavefunction and

DFT approaches; recall the discussion in Chap. 1, particularly eqn. (1.11).

Because energies of different states in different field strengths vary considerably,

we compare their differences from the corresponding CI total energies. Figure 4-3 shows

those differences for the HF and DFT total atomic energies of helium atom ls2, 1 2po,

and s2p-l states as functions of B field strength. Since the HF calculation includes

exchange exactly, the difference for the HF energy is the negative of the conventional

correlation energy EcH (the superscript "HF" is for clarify). First we observe that the

conventional correlation energies for the states ls2po and Is2p., are extremely small.










56




0 25



02
0

015- a)He l2 0

0
S 01
o-

0 05 -
005-
A A


0-


0 HF
-0 05-
PBE


-01 .-.--..-.--------
102 10 100 10' 102
B (au)






005- O O o oo oo oo0 o



0a 9o
A A 0
0-A


S-005-



b) He 1s2po

-015-

F HF
O LDA
-0 2 GGA



-0 25'
10 2 10 1 10 101 102
B (au)







01 0 0 -
c) He 1s2p _
0
0o
005- o o o o o oo o

F D
























functions of B field strength. Blue squares o: HF; Black circles o: DFT-LDA;
Red triangles A: DFT-GGA.
-0 1 O LDA
A GGA



10 2 101 100 101 102
B(au)



FIG. 4-3 Differences of the HF and DFT total atomic energies of the helium atom 1s2,

ls2po, and ls2p.i states with respect to the corresponding CI energies as

functions ofB field strength. Blue squares D: HF; Black circles o: DFT-LDA;

Red triangles A: DFT-GGA.









This is because the Is and 2p electrons in the atom are well separated, unlike the two Is

electrons in the ls2 configuration. The increase of the absolute value of EHF in the large

B field regime for the two states Is2 and ls2p-l seems to be the result of the compression

of electron densities illustrated in Fig. 4-2. The PBE generalized gradient functional

gives extremely good results for both the singlet state Is2, and triplet sates 1s2po and

Ss2p.i when the B field magnitude is less than 1 au. Both LDA and GGA approximations

fail in the large field regime, B > 10 au. Notice the similar performance of DFT

functionals for the two triplet states s2po and Is2p.,. The former one does not carry

paramagnetic current density, thus there is no CDFT current correction for this

configuration, whereas the later one is a current-carrying state. This observation implies

that the success or failure of these particular LDA and PBE functionals is not because

they omit current terms.

The success of DFT calculations mainly depends on accurate approximations for

the system exchange and correlation energies. As given in detail in Chap. 1, DFT

exchange and correlation energies differ subtly from conventional exchange and

correlation energies. The DFT quantities refer to the auxiliary KS determinant (and

include a kinetic energy contribution) whereas the conventional quantities are defined

with respect to the HF determinant. Nevertheless, conventional exchange-correlation

energies often are used as the quantity to approximate in DFT exchange-correlation

functionals, mostly for pragmatic reasons. The difficulty is that exact DFT quantities and

KS orbitals are only available for a few, very small systems. One of those is discussed in

the next chapter. For most systems, the exact KS orbitals are unknown. However, there




































1s2, LDA
1s2, PBE


1s2p LDA
1s2p PBE


002



0



-0 02



-004



-0 06

0 0 0

-0 08- ls2, LDA
ls2, PBE
S1s2po, LDA
-01 A s2p, PBE
S s2p I, LDA
S1s2p PBE


02


0 15





005o 8 8 8
00







-005


0 1s2, LDA
1s2, PBE
-0 15

0 1s2p-1' LDA
-02 A 1s2p, PBE


-0 25
102 101


FIG. 4-4 Differences of DFT exchange (top panel), correlation (middle panel), and

exchange-correlation (bottom panel) energies with HF ones, for the helium

atom in B fields.









are abundant HF and correlated calculations for many finite systems, providing good

reference densities and energies. To gain a better understanding of the behavior of DFT

exchange and correlation approximations, we make a separate comparison of DFT

exchange-correlation energies with the HF ones in Fig. 4-4. Note that "DFT exchange"

here means the Ex term in a particular functional and not exact DFT exchange calculated

from KS orbitals.

We can see from Fig. 4-4 that the LDA approximation shows its typical

underestimation of exchange and overestimation of correlation energies. The PBE

functional gives good approximations to the exchange and correlation energies separately

when the B field is less than 1 au., but it seriously overestimates the exchange when B>10

au., while the correlation energy does not depend on the field strength very much. Since

exchange dominates the XC energies, the error in the exchange term overwhelms the

correlation term in large B fields. Of course both the LDA and PBE functionals are based

on analysis of the field-free electron gas, in which the exchange-correlation hole is

centered at the position of the electron, and only the spherically averaged hole density

enters. This picture breaks down for an atom in a large B field. Because of the strong

confinement from the B field, there is strong angular correlation among electrons. The

XC hole is not centered at the electron, and is not isotropic. Moreover, the external B

field will effectively elongate XC hole as well as the electron density. If one wishes to

improve XC functionals for applications in the large B field regime, those factors need to

be considered. Another observation from Fig. 4-4 is that PBE overestimates the

correlation energies for the two triplet states, in which the HF correlation is very small.

This presumably is due to the imperfect cancellation of self-interactions in the functional.









The lithium positive ion is a two-electron system isoelectronic with neutral He. It

has approximately the same correlation energy as that of the helium atom in the field-free

case. For non-vanishing field, recall the scaling argument for the wavefunction of

hydrogen-like atoms in a B field. The deformation of the atomic density induced by the B

field is measured by its reduced strength = B / Z2, rather than by its absolute value B,

where Z is the nuclear charge (refer to eqns. 3.30 and 3.31 and discussion). Of course, the

atomic configuration is another important factor. For the same electronic configuration,

the helium atom in B = 5 au. and the lithium positive ion in B = 10 au. have about the

same y values. Indeed they have about the same HF and PBE correlation energies. On the

other hand, an attempt at a similar comparison between the lithium atom and the

beryllium ion is obscured by two factors. One is the large correlation energy between the

two Is electrons for the doublet states. The effect of the external B field on its correlation

energy is hardly discernable in the studied range. For the quadruplet state, notice the

tabulated conventional correlation energy for the beryllium ion is much smaller in

magnitude than that of the lithium atom, giving rise to the suspicion of systematic errors

existing in those data. Also notice that the conventional correlation energy of the lithium

atom ls2p.13d2 state in vanishing B field is even larger than that of its ground state ls22s.

Since the electrons are well-separated in the ls2p.13d2 state, its correlation energy is

expected to be smaller than that of a more compact state. Even for vanishing B field,

large discrepancies on the correlated energies are found in the literature. For example, Al-

Hujaj and Schmelcher gave -14.6405 Hartree for the ground state of the beryllium atom

from a full CI calculation [8], versus -14.66287 Hartree from a frozen-core

approximation by Guan et al. [7]. The difference is more than 20 mH. This shows it is









difficult to systematically extract atomic correlation energies from the literature,

especially for non-vanishing field data.

The DFT functionals investigated here fail spectacularly in a large B field, mainly

from their exchange part. However, the PBE correlation still gives a large portion of the

correlation energy even though its performance degrades somewhat with increasing B

field. On the other hand, the HF approximation is more robust than DFT-based

calculations and includes exchange exactly, but it totally neglects correlation. From those

analyses, it seems a better estimation to the total atomic energies in a large B field may be

achieved by combining HF exact exchange and PBE correlation energy rather than using

solely the HF or DFT approximations.

Ionization Energies and Highest Occupied Orbitals for Magnetized Atoms

Because of the magnetic-field-induced configuration transitions for both

magnetized atoms and their positive ions, atomic ionization energies are not monotonic

increasing or decreasing smooth functions of the applied B field. This is already obvious

from Koopmans' theorem and the UHF total energies in Fig. 4-1. Here we use the total

energy difference between the neutral atom and its positive ion, AEsCF, for estimation of

the ionization energy. For each field strength, the ground state configurations for the

atom and its positive ion must be determined first. Table 4-1 and Fig. 4-5 show the

change of ionization energies of the atoms He, Li, and Be with increasing B field. Results

from different methods are close. For the beryllium atom, a frozen-core calculation [7]

gave a larger ionization energy, by 26 mH, in the near-zero-field region than the one

derived from Al-Hujaj and Schmelcher's data [8]. This difference is mainly caused by the

lower ground state atomic energy obtained in the former reference, which has already

been mentioned near the end of the previous section.










Table 4-1 Atomic ionization energies in magnetic fields (energy in Hartree)

Atom Configurations B(au) HF CI LDA PBE
Hea Is2 1s 0 0.8617 0.9034 0.8344 0.8929
0.02 0.8516 0.8933 0.8244 0.8828
0.04 0.8415 0.8831 0.8142 0.8727
0.08 0.8208 0.8625 0.7935 0.8520
0.16 0.7782 0.8199 0.7506 0.8092
0.24 0.7340 0.7756 0.7059 0.7647
0.4 0.6409 0.6824 0.6113 0.6706
0.5 0.5798 0.6212 0.5492 0.6089
ls2p_1 -- Is 0.8 0.4687 0.4741 0.4199 0.4734
1 0.5187 0.5245 0.4685 0.5225
1.6 0.6438 0.6504 0.5887 0.6452
2 0.7132 0.7201 0.6549 0.7136
5 1.0734 1.0816 0.9978 1.0739
10 1.4394 1.4493 1.3519 1.4527
20 1.9061 1.9190 1.8161 1.9554
50 2.7182 2.7378 2.6627 2.8829
100 3.5161 3.5442 3.5445 3.8593
Li 1s22s 1s2 0 0.1963 0.2006 0.2011 0.2054
0.01 0.2012 0.2056 0.2059 0.2102
0.02 0.2068 0.2136 0.2135 0.2178
0.05 0.2177 0.2254 0.2226 0.2269
0.1 0.2329 0.2403 0.2380 0.2425
1s22p 1 -- 1s2 0.2 0.2587 0.2614 0.2691 0.2729
0.5 0.3699 0.3750 0.3844 0.3870
1 0.5025 0.5111 0.5216 0.5226
2 0.6995 0.7113 0.7226 0.7229
ls2p_13d2 -I 1s2p_1 3 0.7525 0.7572 0.7635
5 0.9475 0.9555 0.9558 0.9644
10 1.2877 1.2982 1.3074 1.3219
Be 1s22S2 1S22s 0 0.2956 0.3158 0.3318 0.3306
0.001 0.2951 0.3159 0.3313 0.3302
0.01 0.2905 0.3112 0.3267 0.3255
0.02 0.2852 0.3313 0.3214 0.3203
0.05 0.2683 0.2911 0.3047 0.3035
ls22s2pi_ ls22s 0.1 0.3234 0.3242 0.3304 0.3312
0.2 0.3941 0.3941 0.4010 0.4019
0.3 0.4531 0.4597 0.4603 0.4612
1s22s2p1 -* 1s22p 0.4 0.4687 0.4758 0.4677 0.4717
0.5 0.4710 0.4749 0.4713 0.4758
0.6 0.4696 0.4767 0.4718 0.4766
0.8 0.4593 0.4650 0.4663 0.4718
1s22p_13d2 Is22pl 1 0.4559 0.4455 0.4575 0.4636
2 0.6231 0.6217 0.6257 0.6336
ls2p_13d24f3- ls2p_13d2 5 0.8787 0.8772 0.8895 0.9019
10 1.1973 1.1959 1.2223 1.2401
(a) Exact energies are used for the one-electron system He+.










Table 4-2 Eigenvalues for the highest occupied orbitals of magnetized atoms (energy in
Hartree)

Atom Configuration B (au) HOMO HF LDA PBE
He 1s2 0 Is -0.91795 -0.5702 -0.5792
0.02 -0.90789 -0.5601 -0.5692
0.04 -0.89771 -0.5499 -0.5589
0.08 -0.87699 -0.5289 -0.5379
0.16 -0.83412 -0.4848 -0.4940
0.24 -0.78942 -0.4383 -0.4476
0.4 -0.69501 -0.3387 -0.3485
0.5 -0.63298 -0.2728 -0.2829
ls2p_1 0.8 2p_1 -0.47120 -0.3184 -0.3184
1 -0.52183 -0.3529 -0.3532
1.6 -0.64824 -0.4389 -0.4408
2 -0.71820 -0.4867 -0.4900
5 -1.07974 -0.7379 -0.7502
10 -1.44629 -0.9994 -1.0230
20 -1.91388 -1.3424 -1.3824
50 -2.72841 -1.9648 -2.0381
100 -3.52994 -2.6077 -2.7192
Li 1s22s 0 2s -0.19636 -0.1162 -0.1185
0.01 -0.20122 -0.1211 -0.1234
0.02 -0.20668 -0.1282 -0.1306
0.05 -0.21778 -0.1364 -0.1389
0.1 -0.23293 -0.1487 -0.1516
1s22_1 0.2 2p_1 -0.25885 -0.1728 -0.1751
0.5 -0.37038 -0.2529 -0.2549
1 -0.50398 -0.3472 -0.3489
2 -0.70293 -0.4873 -0.4903
1s2p_13d-2 5 3d-2 -0.95259 -0.6626 -0.6763
10 -1.29348 -0.9139 -0.9355
Be 1s22s2 0 2s -0.30927 -0.2058 -0.2061
0.01 -0.30417 -0.2007 -0.2010
0.02 -0.29888 -0.1953 -0.1957
0.05 -0.28186 -0.1779 -0.1783
ls22s2pi_ 0.1 2p_ -0.33120 -0.1959 -0.1961
0.2 -0.40159 -0.2560 -0.2566
0.3 -0.46016 -0.3046 -0.3054
0.4 2s -0.47732 -0.3108 -0.3163
0.5 -0.47908 -0.3099 -0.3161
0.6 -0.47722 -0.3064 -0.3134
0.8 -0.46613 -0.2953 -0.3037
1s22p_13d2 1 3d-2 -0.46092 -0.3105 -0.3180
2 -0.62799 -0.4283 -0.4391
ls2p_13d-24f3 5 4f3 -0.88345 -0.6259 -0.6421
10 -1.20284 -0.8671 -0.8908







64


1.5
O
O He
E Li
O Be '
0 0











0.5 -
O'

















FIG. 4-5 Atomic ground state ionization energies with increasing B field. Data plotted are
from CI calculations shown in Table 4-1. Dotted lines are the guides to the
eye.









Even though the ionization energies in both the low and intermediate field regions

are rather complicated as the result of atomic configuration changes, their behaviors are

similar for the strong field limit configurations. This is an indication that the original

atomic shell structure has been effectively obliterated by the field.
Eigenvalues of the highest occupied orbital are reported in Table 4-2. In all the

cases, the F orbital energies give the closest approximation to the atomic ionization















energies, with an average deviation of only 7.6 mH. KS eigenvalues, as usual,
significantly underestimate the ionization energy. This is because both LDA and P 101
B(au)

FIG. 4-5 Atomic ground state ionization energies with increasing B field. Data plotted are
from CI calculations shown in Table 4-1. Dotted lines are the guides to the
eye.

Even though the ionization energies in both the low and intermediate field regions

are rather complicated as the result of atomic configuration changes, their behaviors are

similar for the strong field limit configurations. This is an indication that the original

atomic shell structure has been effectively obliterated by the field.

Eigenvalues of the highest occupied orbitals are reported in Table 4-2. In all the

cases, the HF orbital energies give the closest approximation to the atomic ionization

energies, with an average deviation of only 7.6 mH. KS eigenvalues, as usual,

significantly underestimate the ionization energy. This is because both LDA and PBE









functionals give approximate potentials too shallow compared with the exact DFT XC

potential. Self-interaction correction (SIC) could significantly improve these values, but

we will not pursue it further here, because our focus is on Exc functionals that are not

explicitly orbitally dependent.

Current Density Correction and Other Issues

Advancement in CDFT, especially in applications, is hindered by the lack of

reliable, tractable functionals. In comparison with the vast literature of ordinary DFT

approximate XC functionals, the total number of papers about CDFT approximate

functional is substantially less than 50. The earliest proposed functional, also the most

widely investigated one as of today, is the VRG functional [14-16]. Even for it there are

no conclusive results for B 0 in the literature. Here we examine this functional in detail

for atoms in a B field, and show the problems inherent in it. The analysis indicates that

the VRG functional is not cast in a suitable form, at least for magnetized atoms. The


choice of vorticity 9 = V x JP as the basic variable to ensure gauge-invariance, which is
n

the central result of references 14-16, needs to be critically re-examined.

The challenge to implementing CDFT is, somewhat paradoxically, that the current

effect is presumably small. We do not expect that the current correction within CDFT

will drastically change the DFT densities. Therefore the first question we encountered is

which DFT functional should be used as a reference for the CDFT calculations. If the

variation in outcomes that results from different choices of DFT functionals is much

larger than the CDFT corrections, which seems to be the case in many situations, the

predictability of the calculation is jeopardized. Of course, there is no easy answer to this

question. Indeed, DFT functionals themselves are still undergoing improvement.










Here we made the conventional DFT choice of using the LDA as the starting point.

Even though not the most accurate one, the LDA is among the best understood DFT

functionals. It is also easy to implement. Using self-consistent KS orbitals obtained from

LDA calculation for the helium atom ls2pl state in a field B = 1 au., I plotted various

quantities that are important in CDFT along the z and p directions in Fig. 4-6.


104



102



100






o
4 10-2
0

10 -4

10

10-6



10-8


10-10
0


0.5 1 1.5 2 2.5 3 3.5 4
zor p (ao)


FIG. 4-6 Various quantities (electron density n, paramagnetic current densityj, vorticity
v, and the current correction to the exchange-correlation energy density, gv2,
in the VRG functional) for the helium atom ls2pl state in B = 1 au. All
quantities are evaluated from the LDA KS orbitals and plotted along the z and
p axes (cylindrical coordinates).

Exponential decay of electron density was already seen in Fig. 4-2. Because the

current density along the z axis is zero, it does not appear in Fig. 4-6. However, i is not

zero on that axis. On the contrary, it diverges at the two poles of the atom. This









divergence causes large values of g(n)v2, the energy density correction within the VRG

functional (recall eqns. 1.39 and 2.9). If the pre-factor g(n) does not decay fast enough

to cancel this divergence, a convergence problem in the SCF solution of the CDFT KS

equation will happen. Also notice that the electron density decays very rapidly along the z

axis. At z = 3ao, the density is already smaller than 10 4a3. Remember the function

g(n) was fitted to data points with rs < 10a thus in the low-density region it is not

well-defined. Different fits to the same set of original data points vary considerably (refer

to eqns. 1.41 through 1.46). Furthermore, even the accuracy of the original data set to be

fitted is questionable at r~ 10a0. Even were these problems to be resolved, the

underlying behavior shown in Fig. 4-6 would remain. The largest correction relative to

ordinary DFT given by the VRG functional is at the places where the electron density and

the current density are both almost zero, which is obviously peculiar if not outright

unphysical. This peculiar (and difficult) behavior is rooted in the choice of v as the basic

variable in the CDFT functional.

To avoid the divergence problem, we introduced a rapidly decaying cutoff function.

Details were given in chapter 2 (also recall eqns. 2.15 and 2.16). Using parameters

ncutoff = 10 3a03, aCtff = 2ao' for the cutoff function, Table 4-3 lists some of the

calculated results within the VRG approximation for the fully spin-polarized states of the

helium, lithium, and beryllium atoms at several selected field strengths. An estimation of

the current effect is to evaluate the VRG functional using the LDA Kohn-Sham orbitals.

Results for that estimation are listed in the third column of Table 4-3. This scheme can be

thought as a non-self-consistent post-DFT calculation. Fully self-consistent CDFT









calculations were also accomplished when the B field is not too large, and they verified

the LDA-based estimates. When the B field is larger than roughly 5 au., SCF

convergence problems return because of the pathological behavior of VRG functional.


Table 4-3 CDFT corrections to LDA results within VRG approximation (parameters
c,,toff, = 10 a3, cutoff = 2ao0 are used for the cutoff function, AE in Hartree)

Atom and State B (au) Non-SCF AERG SCF AERG JHOMO
He ls2p.1 0 -0.0022 -0.0021 0.0001
0.24 -0.0031 -0.0031 -0.0013
0.5 -0.0045 -0.0047 -0.0029
1 -0.0077 -0.0081 -0.0071
5 -0.036
10 -0.074
100 -0.81
Li 1s2p_13d2 0 -0.0070 -0.0071 0.0002
2 -0.027 -0.029 -0.0077
5 -0.065
10 -0.129
Be ls2p_13d24f3 1 -0.025 -0.026 -0.0017
5 -0.085
10 -0.166


Putting these concerns aside, consider Table 4-3. By design, the current correction

given by the VRG functional is negative. It strongly depends on the particular atomic

configuration. Within each configuration, the VRG contribution increases with increasing

B field. Besides total atomic energies, the eigenvalues of the highest occupied KS orbitals

are also slightly lowered by including the current term, but it helps little in bringing the

HOMO energies closer to the ionization energies. This error, of course, is the well-known

self-interaction problem.

Because of the use of a cutoff function, these CDFT calculations can at best be

thought of as semi-quantitative. This is because the current corrections strongly depend









on the chosen cutoff function. Use of different cutoff parameters gives quite different

results (see Table 4-4), an outcome which is really undesirable. Of course, all of this is

because the VRG functional does not provide a suitable form for either the low density or

the high-density regions, nor do we know its correct asymptotic expression.


Table 4-4 Effect of cutoff function on CDFT corrections for the helium atom ls2p._ state
in magnetic field B = 1 au. (energy in Hartree)


ncuoff (ao a.CU (aO) Non-SCF AERG
0.005 2.0 -0.004
0.001 2.0 -0.008
0.001 1.0 -0.010
0.0001 2.0 -0.025
0.00001 2.0 -0.064


It is unsurprising that the VRG functional fails when applied to atomic systems in a

strong magnetic field. It was developed from the study of the moderately dense to dense

HEG in a weak magnetic field, for which Landau orbitals were used as approximations.

This physical situation is quite different from a finite system. First, electron and

paramagnetic current densities vary considerably within an atom, and the low density

regions (r, > 10a0) are non-negligible. Secondly, there is not a direct relationship

between j (r) and the external B field as there is for the HEG. The question whether the

electron gas remains homogeneous after imposing a substantial B field is even unclear. If

the field induces some form of crystallization, the basic picture based on which the VRG

functional proposed is completely lost. The analysis and numerical studies in this chapter

suggest the picture of Landau orbitals used for the HEG may not be applicable at all for

the atomic-like systems. Unlike the LDA, also based on the HEG, it seems that the VRG

functional is too simple to encompass the essential physics of the atomic systems.









A more fundamental question is whether i (r) is a suitable basic variable in

gauge-invariant CDFT as Vignale and Rasolt required [14-16]. While it is appealing from

a purely theoretical perspective, our numerical results on atomic systems in B fields

suggest it is an inappropriate choice, or at least an awkward one, from the application

perspective. Largely due to the choice of v (r) as the basic variable in the VRG

functional, it gives unphysical results in our tests. Recently, Becke proposed a current-

dependent functional to resolve the discrepancy of atomic DFT energies of different

multiplicity of open-shell atoms [107]. Since this functional is based on the analysis of

atomic systems, it may be more suitable for application to magnetized atoms than the

VRG functional. There are significant technical barriers to its use. Nonetheless, we hope

to investigate this functional in the future.

Before attempting (sometimes in effect, guessing) better forms for the CDFT

functional, we need to know some exact CDFT results to serve as touchstones for any

possible proposed functional, This is the major task of the next chapter.

Finally, one additional comment remains to be made about the results presented in

this chapter. Relativistic effects and the effects due to finite nuclear mass are not

considered. Those effects can be important for matter in super-strong fields (B > 104 au),

in which regime the adiabatic approximation will be applicable. But for the field

strengths involved in this chapter, both effects should be negligible.














CHAPTER 5
HOOKE'S ATOM AS AN INSTRUCTIVE MODEL

In DFT, the need for accurate approximations to the electronic exchange-

correlation energy Ex has motivated many studies of a model system often called

Hooke's atom (HA) in the DFT literature [108-121]. The basic HA is two electrons

interacting by the Coulomb potential but confined by a harmonic potential rather than the

nuclear-electron attraction. This system is significant for DFT because, for certain values

of the confining constant, exact analytical solutions for various states of the HA are

known [108, 111]. For other confining strengths, it can be solved numerically also with

correlation effects fully included [113]. Since the DFT universal functional is

independent of the external potential and the HA differs from atomic He (and

isoelectronic ions) only by that potential, exact solutions of the HA allow construction of

the exact Ex functional and comparative tests of approximate functionals for such two-

electron systems. Given that much less is known about the approximate functionals in

CDFT than ordinary DFT, it would be of considerable value to the advancement of

CDFT to have corresponding exact solutions for the HA in an external magnetic field.

Hooke's Atom in Vanishing B Field

There is a long history of investigating this problem. The system Hamiltonian reads


to= lv1 2+v +-(r+r2)+- (5.1)
2 2 r2









where ( (i = 1, 2) are the spatial coordinates of the electrons, and co the confinement

frequency. Hartree atomic units are used throughout. By introducing center of mass (CM)

and relative coordinates,

1
2 (5.2)
r=r2 '- 1=2

(when I deal with the relative motion part, r always means r12), the Hamiltonian eqn.

(5.1) becomes

Hot = HCM + H (5.3)


where H = -V- +co2R2 (5.4)
4

Hr + -V+ 12r2 + (5.5)
4 r

The solution to the three-dimensional oscillator problem (5.4) can be found in any

undergraduate QM textbook. It is the relative motion Schrodinger problem, defined by

eqn. (5.5), that has been treated variously by different authors. Laufer and Krieger used

the numerical solution to the relative motion problem to construct the exact DFT

quantities, and found that, although most approximate functionals generate rather

accurate total energies for this model system, the corresponding approximate XC

potentials are significantly in error [113]. In 1989, Kais, Herschbach, and Levine found

one analytical solution to the HA relative motion problem by dimensional scaling [108].

Samanta and Ghosh obtained solutions by adding an extra linear term to the Hamiltonian

[110]. Later, Taut obtained a sequence of exact solutions for certain specific confinement

frequencies [111], and used them in studies of DFT functionals [114-116].









A basic observation about the HA follows from the Pauli principle, which requires

the total wavefunction to be antisymmetric. Because the CM part is always symmetric

under particle exchange i ++ r2 if the relative motion part is symmetric (e.g. s or d-like

orbitals), the spin part must be anti-symmetric, thus a spin singlet state; otherwise a spin

triplet state. Thus we can concern ourselves with the spatial relative motion problem

alone.

Since the Hamiltonian (5.5) is spherically symmetric, the relative motion wave-

function can be written as the product of a spherical harmonic and a radial part. The

radial part is in turn decomposed into a gaussian decaying part (ground state

wavefunction of a harmonic oscillator) and a polynomial part. In some special conditions,

the polynomial has only a finite number of terms, and thus the wavefunction is expressed

explicitly in a closed form. Here I proceed slightly differently from the approach in

reference 111. Insertion of the relative motion wavefunction


lvr ()= Ylm (O,(p)e r2/4 akk (5.6)
k=0

in Hir,/r (f) = EYf (F) and a little algebra give the recursion relation


-(k+2)(k+21+3)ak 2+ak++ k+L + j -E ak= 0 (5.7)


Suppose the polynomial part in eqn. (5.6) terminates at the nth term, e.g. a, 0 and

ak>n = 0. The recursion relation (5.7) for k = n immediately gives


E,, = 1+n+ (5.8)










Repeatedly invoking eqn. (5.7) for k= n-1, n-2, ..., 0, -1, we get an expression for a.1

which by definition must be zero, in terms of co. Frequencies which make this expression

be zero are the ones that correspond to analytical solutions with eigenvalues given by

eqn. (5.8).


Table 5-1 Confinement frequencies co for HA that have analytical solutions to eqn. (5.5)
(see eqn. 5.8 for their eigenvalues)


n /=0
1 0.50000000000000
2 0.10000000000000
3 0.03653726559926
0.38012940106740
4 0.01734620322217
0.08096840351940
5 0.00957842801556
0.03085793692937
0.31326733875878
6 0.00584170375528
0.01507863770249
0.06897467166559
7 0.00382334430066
0.00849974006449
0.02696238772621
0.26957696177107
8 0.00263809218012
0.00526419387919
0.01342801519820
0.06058986425144
9 0.00189655882218
0.00348659634110
0.00767969351968
0.02409197815100
0.23835310967398
10 0.00140897933719
0.00242861494144
0.00481042669358
0.01216213038015
0.05433349965263


= 1
0.25000000000000
0.05555555555556
0.02211227585113
0.20936920563036
0.01122668987403
0.04778618566245
0.00653448467629
0.01942406484507
0.18237478381198
0.00415579376716
0.01002629075547
0.04231138533718
0.00281378975218
0.00591291799966
0.01743843557070
0.16282427466688
0.00199650951781
0.00380045768734
0.00910586669888
0.03819659970201
0.00146924333165
0.00259554123244
0.00542189229787
0.01589809508448
0.14785508696009
0.00111335083551
0.00185491303393
0.00351289521069
0.00837269574743
0.03496458370680


/=2
0.16666666666667
0.03846153846154
0.01583274147996
0.14620429555708
0.00827862455572
0.03423838224700
0.00494304416061
0.01426990657388
0.13126853074700
0.00321380796521
0.00753956664388
0.03104720074689
0.00221804190308
0.00454144170886
0.01305357779085
0.11975836716865
0.00160027228709
0.00297472273003
0.00694983975395
0.02852875778009
0.00119499503998
0.00206608391617
0.00421416844040
0.01207319134740
0.11054467575052
0.00091728359398
0.00149877344939
0.00277638397339
0.00646564069324
0.02647749580751


/=3
0.12500000000000
0.02941176470588
0.01232668925503
0.11267331074497
0.00655187269690
0.02675808522737
0.00397054409092
0.01130595881607
0.10313090450042
0.00261635006133
0.00605214259197
0.02465640755471
0.00182765149628
0.00369051896040
0.01048109720372
0.09546288545482
0.00133306668779
0.00244506933776
0.00564110607268
0.02293923879074
0.00100531643239
0.00171616534853
0.00345660786555
0.00979689471802
0.08912851417778
0.00077860389150
0.00125702179525
0.00230004951343
0.00529550074157
0.02150263695791









An example may be helpful. Consider = 0, and n = 3. According to the previously

9 a a2 -12a3 (1120)a3
prescribed procedure, we get E, = a a1 =
2 co 2 2co2

a,-6a2 (1-24w)a3 a,-2a, (1-30o +72C2)a3
a0 = 3 a I =
3a 603 4 a 2404

To ensure that the last expression vanishes requires that the confinement frequency be


) = [-- or co = 0.3801294, 0.0365373. The solution corresponding to the smaller
24

frequency turns out to be a ground state, while the other one is an excited state.

Confinement frequencies corresponding to analytical solutions for / = 0, 1, 2, 3 and

n < 10 are compiled in Table 5-1. This tabulation includes more angular momentum

quantum numbers and more significant figures than that presented by Taut [111]. For

n > 3, there are several solutions. The smallest frequency corresponds to a ground state,

the others are for excited states.

Hooke's Atom in B Field, Analytical Solution

When the HA is placed in an external magnetic field, its lateral confinement can

exceed its vertical confinement. It is well known that the magnetic field can greatly

complicate the motion of a columbic system. Even for the one-electron system (H atom),

substantial effort is required to get highly accurate results in aB field [79-82]. Only

recently have calculations on the He atom in a high field been pushed beyond the HF

approximation [11-13]. As far as I know, no exact solutions are reported in the literature

for the 3D Hooke's atom in an external magnetic field. Taut only gave analytical

solutions for a 2D HA in a perpendicular B field [117]. Here I present the exact

analytical solutions to the magnetized HA [122]. When the nuclear attraction in the He








atom is replaced by a harmonic potential, our exact analytical results can serve as a

stringent check on the accuracy of the correlated calculations just mentioned.

With an external magnetic field chosen along the z axis, the system Hamiltonian

becomes

H rot= (Vl +(A())2+ V2 + )2 (2 +r2) +1 + (5.9)
2L i 2 rl

where Hs,,, = -(s, + s )B is the spin part of the Hamiltonian, s, (i = 1,2) are the z

components of the spin, and A(f) is the external vector potential. A similar separation of

the CM and the relative motion parts is done as in the case of the B = 0 HA:

Hto, = H + H, + H (5.10)

1 1
H cM=( ( + 2A(R)) + 2R2 = -(-V +40 2R2 +B2R2 sin20 + 2MB) (5.11)

V. 11 21 1 2 1 1 1
H =( 1 +-A(r))2 +-0r +-= -V 0 r +-B2r2 sin2 0+mB+-
i 2 4 r 4 16 2 r

=-V + p 2 2z2 + B+ (5.12)
2 r
a0)2 B2 0)
Here o = o) (5.13)
4 16 2

p = x + y2, m and M are magnetic quantum numbers for the relative and CM motion

parts. In these expressions, the Coulomb gauge has been chosen.

A(r) = Bx F (5.14)
2

The solution for the CM part is (un-normalized)

Mc (R) =exp(-mZ2 ) PM H ( 2oZ)-, F,(-NP,, +1,LP2) exp(iMD) (5.15)









with eigenvalue of


EM=(N z+ )o+ B p+n +1 (5.16)
2 2 2

where P = R sin 0, Q = 42 + B2 H N is Nz th order Hermite polynomial., F, is the

confluent hypergeometric function (or Kummer function) [123]. Nz,Np = 0, 1, 2,... are

the quantum numbers.

The relative motion eignvalue problem from eqn. (5.12) generally cannot be solved

analytically in either spherical or cylindrical coordinates. The difficulty of solving the

Schrodinger equation that corresponds to eqn. (5.12) lies in the different symmetries of

the confining potential (cylindrical) and electron-electron interaction term (spherical).

Since the effective potential in eqn. (5.12) V()= 22 + 2 expressed as a
r

combination of cylindrical coordinate variables (p, z) and the spherical coordinate

variable r, it proves convenient also to express the relative-motion wavefunction in those

combined, redundant variables y, (r) = v (p, z, r (p, z), p) In part motivated by the

expected asymptotic behavior, we choose the form

_Oz 2 p 2
V r()=e 2 2 pmzf u(r,Zz) e"m (5.17)

where z = 0 for even z parity, 1 for odd z parity. Then (H E) V (r) = 0 yields

{ a2 82 2z 82 2r, j / \2 27| 8 ( n -2 1 ~ o^
-- 2z 0- 1++l +(a o -)z2 J --+2 +2 z- +- -u(r,z)=0 (5.18)
&hr2 E2 r r-r r2z I -+- (519r z


where E =E- mB-(27, +l)c, -2(m +l)op. (5.19)
2









To avoid messy notation, no quantum numbers are appended to E. This differential

equation is not easy to solve either. To proceed, we make a direct, double power-series

expansion


u(r,z)= A,,rzr" z"= (5.20)
n,,nz=0

to transform eqn. (5.18) into a recurrence relation,

-2(n, + 2)(cO cz ) A+2,, 2-(n, + 2)[n, + 3 + 2( m + Tz + n )AA,~ +,, + A4 +,n

+[2(nrw ,+pnW -) Ac]4 (n +r+l1)(n, + +2),, 2= 0 (5.21)

where A, = 0 for i < 0, or j < 0 or j = 2k +1. We seek values of E, o), a) for

which the right side of eqn. (5.20) terminates at finite order. Assume the highest power of

z that appears is N (A,,> = 0), where N, is an even number. For N, > 2, generally

there is no solution to the set of equations that follow from eqn. (5.21). However, a

judicious choice,

op = 2ow, = 2Nw, (5.22)

N
allows us to set A, = 0 for 2i + j > N, since there are recurrencee relations of
2

eqn. (5.21) with 2n, + n, = N, that then are satisfied automatically. This is the special

case, co, = 2o) = o which corresponds to imposition of an external field B upon a HA

with magnitude

B = 2,l (5.23)

Now we find values of c, that correspond to analytical solutions. Repeated

N
application of eqn. (5.21) for each combination of -1< _< 2,
2









N 2(ln +1) nz > 2, allows us to express all the coefficients A O

m i Nz N
terms of tA ,0 2 2

homogenous linear equations involving {Ao,0 oN }. To have non-trivial solutions, the

determinant of this set of equations must be zero, a requirement which reduces to finding

the roots of a polynomial equation in co,. Energy eigenvalues can be easily found by

substituting eqns. (5.22), (5.23) and (5.13) into (5.19),


E, = (N, +-,+2m)+5+3m C (5.24)


Here I give an explicit example for m = rZ = 0, N, = 6. Its relative motion energy


is E, = 6+ = 17o,, First we set all the coefficients A,,j =0 for 2i+ j > 6. The


four equations derived from eqn. (5.21) are already satisfied for (n,nz) = (3,0), (2,2),

(1,4), and (0,6). Repeatedly invoking the recursion relation (5.21) for n, = -1, n = 6,4,2,

we find A1,4, A1,2, and A1,0, expressed in terms of Ao,2, A0,4, and A0,6.

AO,6
A1, 4
2a

2A,4 -1 A1,4 zw-AO4 5A-O,6
A1,2 2 22



20 202 '

Use eqn. (5.21) again twice for n, = 0,n = 4,2,









A2 4 O4 A14 + 30A,6 1-6040
,= A,6 A,4,

1442- A,2 +80A)z,2+12A,4 2 0z(4(0 +1)AO,4+(4200 -17)Ao,6
A,0 = 16o0 2A',2 ;
4w 16w3

Employ eqn. (5.21) one more time for n, = 1, n = 2 to obtain

S40A1, A2,2 +12A1,4 A4 1- 280,
A3, 0 + 80)2
S6w, 2w, 482

Now, all the coefficients are expressed in terms ofAo,2, A0,4, and A0,6. The next step is to

apply eqn. (5.21) forn, = 0, n = -1,0,1,2. We have a set of four homogeneous equations,

4A, 2Al, = 0,

Ao -12 Ao,o 6A2, -2A, = 0,

-80, A, 12A3, + A2 2A,2 = 0,

-4 A2,o + A3, 242, = 0.

Substitute the expressions for A1,0, A2, 0, A 2,, A A2, and rearrange,


[ O- 0z,2 +30 ,4 -15A,61=0,


-[ 9644A, +402(1+20w,)A4,-60(3+40 )A44+(11112600w)A,6]=0,


1[-96 ,34 A+20w (1+1400,)4 3(7+116 )4A,6=0,
16w t

1[384044A, -4802A,4 +(1+164 -432002)A4,61=0.
480)3 Z Z z )AO,6

To have non-trivial solutions for Ao,o, Ao,2, A0,4, and Ao,6, the determinant for their

coefficients must be zero. This requirement is equivalent to the polynomial equation









8m6 (206115840)4 -1946560o3 +50256m2 -420mo +) = 0

There is a standard procedure for solving the fourth power polynomial equation [123].

Here I give the nonzero solutions to the above equation


1 34506x,2,3 2 4345
m/)z = x ,, --' x +-
S8 2,3 1234762704 2,3 184032'



1/2
24759 1 5633902 34759 2( )r 358124231
where x { 425 cos -cos +( -1) +
40257 3 1208188081 3 3704288112

Numerical evaluation gives

),Z = 0.0584428577856519844381713514636827195996701651, (third excitation)

0.0230491519033815661266886064985880747559374948, (second excitation)

0.0040457351480954861583832529737697350502295354, (ground state)

0.0089023525372406381159151962974840750107860006. (first excitation)

The smallest frequency corresponds to a ground state, others correspond to excited

states. Remember those states are not for the same confinement strength, hence not the

same physical system.

Table 5-2 lists all the frequencies that correspond to closed-form analytical

solutions for m = 0, 1, 2 and N. = 2, 4, 6, 8, 10, including both positive and negative z

parties.

For each frequency found in the previous step, the corresponding eigenvector


{A0,0 N } determines the vector of all the coefficients A .N Table 5-3


gives explicitly some of the solutions to eqn. (5.12).










Table 5-2 Confinement frequencies o, which have analytical solutions to eqn. (5.12)

(o = co /2,B = 2,3o, see eqn. 5.24 for their eigenvalues)


T- N state m = 0


0


2 g
4 g
e
6 g
e
e
e
8 g
e
e
e
e
e
10 g
e
e
e
e
e
e
e
e
2 g
4 g
e
6 g
e
e
e
8 g
e
e
e
e
e
10 g
e
e
e
e
e
e
e
e


0.08333333333333(1)
0.01337996093554(5)
0.03958614075938
0.00404573514810
0.00890235253724
0.02304915190338
0.05844285778565
0.00169910717517
0.00326661504755
0.00563875253622
0.01040942255739
0.01653602158660
0.03180329263192
0.00086575722262
0.00147907803884
0.00241664809384
0.00334850037594
0.00397594182269
0.00736429574821
0.01303363861242
0.01973176390413
0.04608252623918
0.03571428571429
0.00707894326171
0.02581579358039
0.00254580870206
0.00563638108171
0.01883419465413
0.02725999959457
0.00119038526800
0.00217041875920
0.00436490208339
0.00573812529618
0.01453566471300
0.02126492440705
0.00064917903905
0.00105825552161
0.00178623911003
0.00215173389288
0.00344914023386
0.00477115358052
0.01179464597863
0.01632356905207
0.02247241353174


(a) g = ground state; e = excited state;
(b) Numbers in parentheses are the listing number in Table 5-3.


m=1

0.05000000000000
0.01000000000000
0.02500000000000(4)
0.00343014626071
0.00606707008623
0.01711506721549
0.03965895252427
0.00151575652301
0.00253598704899
0.00412145134063
0.00824400136608
0.01329159554766
0.02138083910939
0.00079244596296
0.00126081972751
0.00181507622152
0.00292405665076
0.00318249275883
0.00526294300755
0.01091689148853
0.01510846119750
0.03333142555505
0.02777777777778(2)
0.00591390023109
0.01964372058675
0.00223524705023
0.00451759211533
0.01489720978500
0.02265588831769
0.00107801657618
0.00183191437804
0.00352808023992
0.00498357626059
0.01198041559842
0.01697086960871
0.00059949900156
0.00093052390433
0.00148750195578
0.00193434188304
0.00286801805941
0.00395655886157
0.01002739489760
0.01328778382152
0.01936642452391


1


m=2

0.03571428571429
0.00778702514725
0.01938688789623
0.00291641684372
0.00468524599259
0.01417651105557
0.03017655861841
0.00135563312154
0.00203265499562
0.00339412325617
0.00670121318725
0.01137236016318
0.01706956908545
0.00072696957550
0.00107688670338
0.00147682860631
0.00255098333227
0.00273000792956
0.00417905341741
0.00954248742578
0.01275915629739
0.02636159496028
0.02272727272727(3)
0.00504676075803
0.01616733845868
0.00197793631977
0.00379720216639
0.01259718274291
0.01934687713121
0.00097964051831
0.00158115925765
0.00300865314483
0.00437993535596
0.01037221561899
0.01435335804110
0.00055475656842
0.00082578168452
0.00128288767589
0.00174595176146
0.00249502584282
0.00340206703840
0.00884019961249
0.01143529433090
0.01698248977049







83


10-4


x
3D HA in B, c =2co
10 -3



1-2: 1=z1D/
10 -
S3D QD in B, o) = 12
p z
N P


10 p

a 3D HA, B=0, co =co


100 2D -- Av<: ]Av^ XCHd-l>I: X




101
101 100 10-1 10-2 10-3 10-4
p

FIG. 5-1 Confinement strengths subject to analytical solution to eqn. (5.12). For ID,
C, = oc has been shifted to o, = 1. For 2D, o)Z = oc has been shifted to o) = 1.
Hexagon, square, up-triangle, diamond, down-triangle, circle, left-triangle,
plus, right-triangle, and x-mark stand for the highest order ofz in ID, p in
2D, r in the 3D spherical HA, t = ( r z) / 2 in QD, in the polynomial part of
the relative motion wavefunctions being 1,2,...,10, respectively. For HA in a
B field, they stand for N, + ~. Black, blue, and red symbols are for m = 0, 1,
2, respectively. For spherical HA, only r = 0 is included. Notice its odd
parity (, = 1,m) and even parity states (; = 0,m +1) are degenerate.


For another case of cap = o, /2, which can be thought as a two-electron quantum


dot (QD) in a suitable magnetic field, one can also find analytical solutions to eqn. (5.12)

for some specific confinement frequencies. Together with two limiting cases of ID and

2D, they are summarized in reference 122. Figure 5-1 shows those frequencies subject to

analytical solutions to the electron relative motion part.









Table 5-3 Some

# 6)
1 1/6

2 1/18

3 1/22

4 1/20

25 -3 17
5 472


solutions to eqn. (5.12) for confinement potential o) = 2o3,, B = 2,f30

Relative Motion Wavefunction
e (222)/24(l+r/2+2 /12)

e (+2 ):72pz(1+r/6+z2/108)e'

e (2 2P2)j88p2z(l+r/8+z2/176)e2z

e (Z2 +2p2)8p(l+r/4/4-z 2/40+p2/80-rz2/160-z4/3200)e

2 2p24 r 1-22C2 1+2c 2 1-18o2 11-314cz4
e +-+2 4 24 8 112
2 48 24 8 11328 )


Hooke's Atom in B Field, Numerical Solution

For arbitrary o) and B values, eqn. (5.12) does not have an analytical solution. To

have a clear picture for the dependence of the HA system behavior upon increasing cv or

B field, more data points are essential.

Expansion of the wavefunction in terms of spherical harmonics is satisfactory when

the B field is not too large. For large B values, Landau orbitals are used for expansion.

Consider the low-field expansion first.

Spherical expansion:


1
S(r) = (r)(0, )
-7r


(5.25)


Insertion of the foregoing expansion together with the Hamiltonian (5.12) in the relative

motion Schrodinger equation gives a set of coupled differential equations,


-Erfmfi (r)r)-Vf(()fir()) 0
dr2


(= 0, 2, 4,... or = 1, 3, 5,...)

where the effective potential is


(5.26)









"ht ( 1(1 +21) 1 2 12 B2
Vj im(r)= ) m +12 B+ r r2sin2 'm(
r2 r 2 4 16
SI (5.27)
(l+1)m 2 2 B2 2 21+1
= s,1 +-+-B+-r2 2 --r + +(10,201 1'O)(m,20 I'm)
r2 r 2 4 24 24 21'+ 1

and (lm, "m" l'm') is a Clebsch-Gordon coefficient. This procedure is very similar to

Ruder et al.'s method for treating the hydrogen atom in strong magnetic fields[78]. The

numerical solver for eqn. (5.26) was obtained from reference 124, with appropriate

modifications made to adapt it to this problem.

Next turn to the strong field case, which requires a cylindrical expansion. The

expansion used is:

Cylindrical expansion.

,M (r) = I g, (z)O~,L" (p, A ) (5.28)


-Eg(z) 2 g, ()+ 'V,(z)gn(z)= 0 (n 0, 1,2, ...) (5.29)
dz '

where the effective potential


;m (Z)=y m ) t 1 "(p,(0) + z2 + B+2 2 +1I



4 4

Calculating the effective potential eV"m (z) is not trivial. I followed Proschel et

al.'s scheme [125]. Details are included in appendix D.

By use of a similar argument as in reference 90, we can screen out the

configurations pertaining to the HA global ground states in B field, which are (m = 0,

r, = 0), (m = -1, rZ = 0), and (m = -3, zz = 0). Energies for the relative motion and spin









parts are compiled in Table E-1 and E.2 for two angular frequencies, co = 0.5, 0.1,

respectively. States are labeled by their conserved quantities as v 2S+lm where (2S+1)

is the spin multiplicity, and v is the degree of excitation within a given subspace. Their

field-free notations are also included (e.g. Is, 2p, ...). The larger confinement frequency

corresponds to the first analytical solution found by Kais, Herschbach, and Levine [108],

and is also the most widely studied one. The smaller frequency has two analytical

solutions, one for the Is state in B = 0 and another for the 2p-1 state in B = 3/ 5 A

sixteen spherical function expansion gives the relative motion energy of the latter state to

be 0.4767949192445 Hartree, pleasingly accurate compared to the analytical result


E,= L(N +7,z +2m)+5+3m + C = (2+0+2)+5 -, *0.1= 0.47679491924311


For Tables E. 1 and E.2, numbers in parentheses denote the number of radial

functions used in expansion (5.25); numbers in brackets are the number of Landau

orbitals used in eqn. (5.28). It is easy to see that, in the low field regime (B < 1 au.), the

spherical expansion outperforms the cylindrical expansion. However, its quality degrades

as the B field increases. As Jones et al. have found and as is physically obvious, the high

field regime is very demanding for a spherical basis [93]. Note in Table E-2, for B = 10

au., the spherical expansion corresponds to /max = 48,49. Clearly, it cannot go much

further on practical grounds.

1
The analytical solution for the singlet state of o = in vanishing B field is
10

1 I+ r(2 r2 2 (r
lr5(r)= 1+-+- e 40 (5.31)
10 ;(240+61 ,) 2 20