Stefan Rohra ——————————————————————————————————— Exact-Exchange Kohn-Sham Spin-Current Density-Functional Theory ——————————————————————————————————— Exact-Exchange Kohn-Sham Spin-Current Density-Functional Theory Den Naturwissenschaftlichen Fakult¨aten der Friedrich–Alexander-Universit¨at Erlangen–N¨urnberg zur Erlangung des Doktorgrades vorgelegt von Stefan Bruno Rohra aus M¨unchen Als Dissertation genehmigt von den Naturwissenschaftlichen Fakult¨aten der Universit¨at Erlangen–N¨urnberg Tag der m¨undlichen Pr¨ufung:20. Juni 2006 Vorsitzender der Promotionskommission: Prof. Dr. D.-P. H¨ader Erstberichterstatter: Prof. Dr. A. G¨orling Zweitberichterstatter: Prof. Dr. P. Otto F¨urmeine lieben Eltern Acknowledgements First of all I want to express my gratitude to Prof. Dr. Andreas G¨orling for letting me join his group and supervising this thesis. His guidance and support were essential for writing this work. I am grateful to Dr. Eberhard Engel for providing his pseudopotential program and patiently answering upcoming questions with respect to its use. Furthermore, I want to thank Dr. Pierre Carrier for reading the first drafts of this work and contributing with useful improvement suggestions. I generally want to thank all colleagues of the groups in Munich, Bonn and Erlangen for the friendly working atmosphere and useful scientific discussions. Within the framework of this thesis the following articles have been written (status of each article at the time of the printing of this thesis is explicitly indicated): [1] S. Rohra, E. Engel and A. G¨orling. Exact-Exchange Kohn-Sham formal- ism applied to one-dimensional periodic electronic systems, Phys. Rev. B (accepted), cond-mat/0512299. [2] S. Rohra and A. G¨orling. Exact-Exchange Spin-Current Density-Functional Theory, Phys. Rev. Lett. (accepted), cond-mat/0511156. [3] S. Rohra, E. Engel and A. G¨orling. Treatment of spin-orbit interactions within a non-collinear exact-exchange spin-current Kohn-Sham method, in preparation (to be submitted to Phys. Rev. Lett.). [4] P. Carrier, S. Rohra and A. G¨orling. A simple, efficient, and general treat- ment of the singularities in Hartree-Fock and exact-exchange Kohn-Sham methods for solids, submitted to Phys. Rev. B, cond-mat/0603632. Contents List of Abbreviations v 1 Introduction 1 I Formalism 7 2 Foundations of Density-Functional Theory 9 2.1 Hohenberg-Kohn Theorems . 10 2.2 Constrained-Search Formulation . 12 2.3 Kohn-Sham Scheme . 13 2.4 Approximate Exchange-Correlation Functionals . 20 3 Density-Functional Theory for Periodic Systems 27 3.1 Pseudopotential Approach . 27 3.2 Plane-Wave Formalism . 32 4 Exact Exchange Kohn-Sham Formalism 37 5 Spin-Current Density-Functional Theory 47 5.1 N-electron system in a magnetic field . 48 5.2 HK-Theorem and Constrained-Search Formulation for SCDFT . 52 5.3 Kohn-Sham Scheme for SCDFT . 54 6 Exact Exchange Kohn-Sham Formalism for SCDFT 59 i ii Contents II Implementation 65 7 The Program 67 7.1 Overview . 67 7.2 Technical details . 68 8 The SCF Cycle 73 8.1 Hamiltonian Operator . 73 8.2 Spin-Current Densities . 76 8.3 Response Function . 77 8.4 Hartree Potential . 78 8.5 LDA- and GGA Exchange-Correlation Potentials . 79 8.5.1 LDA Exchange- and Correlation Potentials . 79 8.5.2 GGA Exchange Potentials . 81 8.5.3 GGA Correlation Potentials . 81 8.6 Exact Exchange Potential . 84 III Applications 87 9 One-dimensional Periodic Systems 89 9.1 T rans-polyacetylene as a Test System . 89 9.2 Computational Details . 90 9.3 Results . 91 9.3.1 Isolated, Infinite Chain of T rans-polyacetylene . 91 9.3.2 Bulk T rans-polyacetylene . 94 9.3.3 Response Functions . 96 9.4 Discussion . 98 10 Atom in an External Magnetic Field 101 11 Spin-orbit Splittings in Diamond-like Semiconductors 111 12 Summary 121 13 Outlook 123 Contents iii 14 Zusammenfassung 127 A Geometric Data for T rans-polyacetylene 129 List of Tables 131 List of Figures 133 Bibliography 135 iv Contents List of Abbreviations au hartree atomic units BZ Brillouin zone CSDFT current-spin density-functional theory DFT density-functional theory EXX exact exchange FT Fourier transformation GGA generalized gradient approximation HF Hartree-Fock HK Hohenberg-Kohn KS Kohn-Sham KB Kleinman-Bylander LDA local density approximation LSDA local spin-density approximation LYP Lee-Yang-Parr OEP optimized effective potential OPM optimized potential method PBE Perdew-Burke-Ernzerhof PW Perdew-Wang rhs right hand side (of EXX equation) SCDFT spin-current density-functional theory SDFT spin density-functional theory SCF self consistent field SO spin-orbit TDDFT time-dependent density-functional theory VWN Vosko-Wilk-Nusair v vi List of Abbreviations Chapter 1 Introduction Properties of ground and excited states of many-electron systems like atoms, molecules and solids are one of the major interests in theoretical physics and chemistry. The physics of stationary state properties, within nonrelativistic quan- tum mechanics, is determined by the time-independent many-body Schr¨odinger equation that describes the many-body wave functions of the underlying physical system. Within this description, the behavior of an individual electron of the many-electron system is coupled to all other electrons. Therefore, an exact ana- lytical solution to the many-body Schr¨odinger equation becomes impracticle for systems consisting of two or more electrons. Alternative approaches are therefore needed for the description of many-electron systems. One of the most attractive and successful approaches is the Kohn-Sham (KS) density-functional theory (DFT). Within this scheme, the system of real, inter- acting electrons is mapped onto a model system of noninteracting electrons that has the same ground state density as the real system. This artificial system is mathematically described by single-particle equations, since the behavior of all particles becomes decoupled. The first crucial step in the development of DFT was the work of Hohen- berg and Kohn [1], who proved that the ground state electron density, a quantity depending only on three spatial coordinates, can be used as basic variable that describes all properties of a stationary electron system rather than the many- body wave function that depends on many parameters. With their work they legitimized the use of the ground state density as the basic variable and showed 1 2 Chapter 1. Introduction that all other ground state properties of the system, like lattice constants, band structures or cohesive energies are functionals of the density. However, the work did not yet provide a practically realizable scheme. This problem was solved with the work of Kohn and Sham [2], who introduced the above mentioned mapping to a system of independent particles. The associated KS equations have a similar form as the stationary Schr¨odinger equation, but are based on a modified op- erator, the KS Hamiltonian operator. Therein, the electron-electron interaction operator is replaced by the usual Hartree potential, the Coulomb potential of the electron density, and the exchange-correlation potential, whose form is unknown. All unknown energy contributions are collected within this exchange-correlation contribution and the quest for it became the actual challenge of DFT. Within their work, Kohn and Sham also provided the first approximation to the exchange-correlation contribution. Based on the results of exchange- and correlation energies of the homogeneous electron gas, this “local density approxi- mation” (LDA) is not very accurate for the description of highly inhomogeneous systems. The LDA is mostly employed in solid state physics and in this field is still in use and of importance. The search for an alternative to the rather crude LDA, in order to treat inhomogeneous density distributions, lead to the family of “generalized gradient approximations” (GGA). Therein, additional cor- rection terms depending on the gradient and higher derivatives of the density are considered for the construction of the exchange-correlation functional. Being the key quantity for the study of the electronic structure of solids, the band structure is of major interest. Nowadays, KS band structures are mostly calculated within the framework of the LDA [3,4] or some GGA [3,4,5] scheme. The currently most popular approach for the calculation of quasi-particle band structures and band gaps, the GW method [6, 7, 8, 9, 10] that approximates the electron self-energy within many-body perturbation theory is usually based on LDA or GGA band structures as well. Although the possibility of self-consistent GW calculations exists, such computations are usually not performed in practice, because they are computationally more demanding than non self-consistent GW calculations based on one-particle states and eigenvalues gained via the LDA or GGA scheme. Moreover, self consistent GW calculations often lead to worse results in comparison to experimental values. The good results gained via non 3 self-consistent GW calculations emphasize the importance and central role of the KS band structure. Schemes like the Bethe-Salpeter method [10,11,12,13,14] or time-dependent density-functional theory (TDDFT) [15,16,17,18,19,20] used for investigating optical properties usually start from the KS band structure as well. Although LDA and GGA band structures are crucial for the investigation of electronic systems, they contain severe shortcomings like the underestimation of the band gap for semiconductors and insulators [21,22,23,24,25]. A common ex- planation of this underestimation was the intrinsic property of the KS formalism to yield a band gap not identical to the physical band gap, even in the case of an exact exchange-correlation potential [21,22]. This difference is commonly known as derivative discontinuity. The KS orbitals, together with their associated eigenvalues for a long time were considered to have no or only little physical meaning. However, recent pub- lications [26,27,28,29,30,31,32] showed that KS eigenvalues and their differences can be viewed as well-defined zeroth order approximations for ionization and excitation energies and therefore represent meaningful physical quantities.