Exact-Exchange Kohn-Sham Spin-Current Density Functional

Stefan Rohra ——————————————————————————————————— Exact-Exchange Kohn-Sham Spin-Current Density-Functional Theory

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Exact-Exchange Kohn-Sham Spin-Current Density-Functional Theory

Den Naturwissenschaftlichen Fakultäten der Friedrich–Alexander-Universität Erlangen–Nürnberg zur Erlangung des Doktorgrades

vorgelegt von Stefan Bruno Rohra aus München Als Dissertation genehmigt von den Naturwissenschaftlichen Fakultäten der Universität Erlangen–Nürnberg

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 Görling 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 formalism 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 treatment 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 quantum mechanics, is determined by the time-independent many-body Schrödinger 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 analytical solution to the many-body Schrödinger 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, interacting 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 Schrödinger equation, but are based on a modified operator, 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 approximation” (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 correction 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 explanation 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. The Coulomb energy and potential contains an unphysical self-interaction of each electron with itself. Within the KS framework, this self-interaction is exactly cancelled by the exchange potential. In case of the LDA and GGAs, this cancellation is incomplete, because the exchange potential is approximated and as a result Coulomb self-interactions raise the KS eigenvalues. Since the valence bands are more localized than the conduction bands, their eigenvalues increase more strongly, which then leads to an artificially decreased band gap for the LDA- and GGA case. An approach that treats the exchange energy and potential exactly for the case of solids was introduced in [33, 34, 35, 36, 37]. This formalism is usually referred to as exact-exchange (EXX) and yields band gaps that are larger and closer to experimental data than their LDA and GGA counterparts. Obviously, the reason for this improvement lies in the cancellation of all self-interactions. The obtained results for standard semiconductors showed almost perfect agreement with experimental data [36, 37, 38, 39, 40], constituting the EXX approach to be a good improvement to the LDA and GGAs. Moreover, the calculated band structures were also used as a starting point for GW calculations [41,42], as well 4 Chapter 1. Introduction as Bethe-Salpeter and TDDFT methods [20]. The EXX approach has been applied to atoms [43], molecules [44, 45], and three-dimensional periodic systems [36, 37]. For the case of three-dimensional periodic systems, existing implementations always exploited additional internal symmetries of the investigated systems beyond periodicity, in particular the cubic symmetry of simple semiconductors like silicon or germanium, in order to reduce computational costs. Within the framework of this thesis, a program that does not rely on any symmetries except translational symmetry was developed. With this software the EXX scheme can be applied to any type of periodic structure in one- to three dimensions. The standard DFT approach described above is based on the ground state density as the only basic variable. While it is possible to calculate various material properties like band structures, cohesive energies or lattice constants, this scheme is not sufficient to describe magnetic properties of many-electron systems. Applications like magneto-optics [46], spintronics [47] or nuclear magnetic resonance (NMR) spectroscopy [48], however, make an appropriate description of magnetic effects within the framework of DFT highly desirable. An extension of the standard DFT scheme to spin polarized systems was firstly introduced by von Barth and Hedin [49]. This spin density-functional theory (SDFT) additionally incorporates collinear spin polarization, i.e. the difference between the densities of electrons with α- and β-spin as basic variable and describes spin polarized systems and the case of the presence of magnetic fields that are not strong enough to induce orbital currents. For the general case of non-collinear SDFT a local spin density approximation (LSDA) [50] and GGAs [51] are at hand. If the magnetic field becomes so strong that an induction of orbital currents occurs, an explicit inclusion of paramagnetic currents as basic variables becomes necessary in order to describe the coupling of the magnetic field to the induced orbital currents. A current spin density-functional theory (CSDFT) depending on seven variables (electron density, x, y, z-components of the spin-magnetization and x, y, z-components of the paramagnetic current density) has been introduced by Vignale and Rasolt [52, 53]. So far, the CSDFT approach did not play a significant role for practical calculations, because no sufficiently accurate ap- 5 proximations for the exchange-correlation energy and potential are available. An application of the LDA scheme to CSDFT, for instance, leads to a functional with the associated potential showing a discontinuous behavior [54]. In this thesis it will be shown that besides currents of the electron density also currents of the spin-magnetization can occur in the presence of magnetic fields. Therefore, an inclusion of these nine “spin-currents” (i.e. the x, y, z- components of the currents of the x, y, z-components of the spin magnetization) as additional basic variables seems to be desirable for a more accurate and more complete treatment of magnetic effects. This new scheme is firstly introduced within the framework of this thesis and shall be termed spin-current density- functional theory (SCDFT) [55]. It represents a new DFT formalism that in total depends on sixteen basic variables and includes the SDFT, as well as the CSDFT formalism as special cases. In addition, the new SCDFT scheme will turn out to be capable of describing spin-orbit effects [56]. The success of the EXX scheme within standard DFT furtheron suggests an introduction of EXX into the newly developed SCDFT formalism. This is one of the major objectives of this work, since this introduction for the first time gives current-density functionals at hand that produce reasonable results [55,56]. Previously available functionals, due to severe shortcomings, were not suitable for practical purposes. For instance, an introduction of the LDA into CSDFT [54] lead to a pathological nature of the associated functional. Applications presented in later chapters will underpin the correctness of the EXX scheme within SCDFT and it will be shown that the new approach produces results that are in good agreement with experimental data. This thesis is divided into three main parts. Within the first part the formalism and methods are developed. For this purpose, chapter 2 quickly reviews the foundations of standard DFT. A specific treatment of periodic structures via the pseudopotential approach and the plane-wave scheme is given in chapter 3. Chapter 4 introduces the EXX formalism for the case of standard DFT. The work then switches to the treatment of magnetic effects via the newly introduced SCDFT in chapter 5. The first part of the thesis concludes with an introduction of the EXX scheme into SCDFT in chapter 6. The second part deals with the implementation of the formalism, i.e. with 6 Chapter 1. Introduction the software that was developed in the framework of this thesis. It starts with chapter 7 that provides a general overview over the program, as well as some technical details. The calculations of all relevant physical quantities are explicitly described in chapter 8. Therein, all required formulas are described and the program sequences for the implementation are given in diagrammatic form. The implementation part is mainly written for persons that want to apply the program or want to extend its present version in the future. The third part summarizes the results for different applications that were performed using the developed software. In chapter 9 the EXX scheme of standard DFT is applied to a one-dimensional periodic system for the first time [57]. As a test system trans-polyacetylene is chosen. Chapter 10 underpins the correctness of the new SCDFT formalism applying it to an isolated oxygen atom in an external magnetic field [55]. The last application is dedicated to the investigation of spin-orbit splittings in diamond-like semiconductors [56]. Results for the cases of silicon and germanium are given in chapter 11. The work concludes with a short summary on the achieved results and a short outlook in view of possible extensions and improvements of the developed software. Part I

Formalism

Chapter 2

Foundations of Density-Functional Theory

One of the central problems in theoretical solid state physics is the prediction of ground state properties for a system of N interacting electrons exposed to an external local potential V (r). This requires the solution of the Schr¨odinger equation for the ground state wave function |Ψ(r1, ..., rN)i of the electronic system

H|Ψi = [T + Vee + V ]|Ψi = N N N h X 1 X 1 X i = − ∇2 + + V (r ) |Ψi = E|Ψi, (2.1) 2 i |r − r | i i=1 i

Coulomb interaction Vee and an external potential V . The energy of the system is denoted by E and the index i labels the individual electrons. Equation (2.1) is a diﬀerential equation of second order with 3N independent variables and it is quite obvious that the solution of this problem is an extremely diﬃcult task. As a matter of fact, the exact analytical solution of this equation is only possible for N =1. An alternative method to determine the ground state wave function |Ψi is the Ritz variational principle, in which the ground state wave function is obtained by minimizing the expectation value of the Hamiltonian operator, i.e. the energy of the ground state

E0 = MinhΨ|H|Ψi = MinhΨ|T + Vee + V |Ψi. (2.2) Ψ→N Ψ→N

9 10 Chapter 2. Foundations of Density-Functional Theory

The notation “Ψ → N” hereby indicates a search over all allowed, normalizable and antisymmetric N-electron wave functions. In practice, an application of this variational principle leads to no simplification of the problem itself. A possibility to circumvent these difficulties is the strategy to focus on the electronic ground state density Z 2 ρ(r) = N dσdx2...dxN |Ψ(r, σ, x2, ..., xN )| (2.3) as the central parameter of the many-particle problem. The quantity xi = (ri, σi) now holds an additional spin variable σ. Utilizing this approach, one can leave behind redundant information stored in the N-particle wave function and utilize the fact that ρ(r) solely depends on the three spatial variables r = (x, y, z), which is obviously a remarkable simplification of this problem. The first physical model following this idea was originally introduced within the framework of the Thomas-Fermi model, in which the energy of an atom is completely determined by the electron density [58,59]. However, this theory was not formulated on an exact physical basis and expressing the energy as a functional of ρ(r) was at that time formally not justified. This changed with two fundamental theorems by Hohenberg and Kohn [1], which represent the foundations of Density-Functional Theory (DFT) as an exact theory.

2.1 Hohenberg-Kohn Theorems

The essential insight of Hohenberg and Kohn was the fact that the electronic density of a physical system suffices to determine all of its properties. This fact is formulated in the first Hohenberg-Kohn (HK) theorem that legitimizes the use of the electron density as the basic variable. HK-Theorem 1 The external potential V (r) is, up to an additive constant, determined by the electronic ground state density ρ(r). The density ρ(r) thus determines uniquely the corresponding Hamiltonian operator, therefore the ground state wave function |Ψ[ρ]i (as well as all other stationary states) and all electronic properties of the underlying system. As a consequence, the kinetic energy and the electron-electron interaction energy can be expressed via the density, and for a system of electrons subjected 2.1. Hohenberg-Kohn Theorems 11 to an external potential V (r) the functional for the total energy results in the expression Z Z E[ρ] = T [ρ] + Vee[ρ] + drV (r)ρ(r) ≡ F [ρ] + drV (r)ρ(r) (2.4) with the HK-functional F [ρ] defined to

F [ρ] = T [ρ] + Vee[ρ] = hΨ[ρ]|T + Vee|Ψ[ρ]i. (2.5)

By deﬁnition the electron-electron term can be separated in two terms

Vee[ρ] ≡ U[ρ] + Exc[ρ] (2.6) containing the classical Coulomb energy 1 ZZ ρ(r)ρ(r0) U[ρ] = drdr0 (2.7) 2 |r − r0| and the functional Exc[ρ] representing the nonclassical contribution to the electron-electron interaction. Exc[ρ] contains a correction for the self-interaction, as well as quantum-mechanical exchange and all correlation eﬀects. It is important to realize that all intrinsic properties of the electronic system are completely absorbed in the HK-functional F [ρ], whose knowledge is therefore equivalent to the exact solution of the Schr¨odinger equation (2.1).

HK-Theorem 2 The functional E[ρ] of the total energy satisﬁes a variational principle with respect to the density, i.e. the total energy E[ρ] reaches its minimal value E0 for the correct ground state density ρ0 :

E0 = MinE[ρ] = E[ρ0]. (2.8) ρ→N The notation “ρ→N” hereby indicates a variation over all ground state densities of arbitrary N-electron systems. The ﬁrst HK-Theorem implies the existence of an external potential V (r) for any arbitrary density ρ(r), which is then called v-representable. Since it is possible to construct densities that are not v-representable [60], i.e. that cannot be related to a corresponding external potential V (r), this implication clearly leads to formal problems, which can however be circumvented by a more general formulation of the HK-Theorem called “constrained”-search formulation [60], which will be described in the next section. 12 Chapter 2. Foundations of Density-Functional Theory

2.2 Constrained-Search Formulation

In analogy to the deﬁnition of v-representability, a density is called to be N- representable if it can be constructed from an antisymmetric N-particle wave function via equation (2.3). A modiﬁed minimization scheme for the total energy functional E[ρ] based on N-representable densities, instead of v-representable densities then clearly shows no explicit connection to an external potential V (r) and the now more general energy functional becomes Z Z E[ρ] = Min hΨ|T + Vee|Ψi + V (r)ρ(r)dr = F [ρ] + V (r)ρ(r)dr. (2.9) Ψ→ρ(r)

The notation “Ψ→ρ(r)” now indicates a minimization over all wave functions |Ψi leading to the density ρ(r) via (2.3). Compared to the Ritz variational principle (2.2), which comprises all allowed wave functions within its search, the variation (2.9) is “constrained” to wave funtions that have to generate the density ρ(r). The minimization now runs over wave functions without requiring v-representability. A connection of the Ritz variational principle (2.2) with the variational principle of the second HK-Theorem (2.8) can be established by simply separating the search over all wave functions “Ψ→N”, as it is performed in the Ritz variation, into two separate searches that lead to a variation performed using the second HK-Theorem

E0 = MinhΨ|T + Vee + V |Ψi (2.10) Ψ→N h i = Min Min hΨ|T + Vee + V |Ψi (2.11) ρ(r)→N Ψ→ρ(r) h Z i = Min Min F [ρ] + V (r)ρ(r)dr (2.12) ρ(r)→N Ψ→ρ(r) = Min E[ρ]. (2.13) ρ(r)→N

The ﬁrst (inner) minimum search in (2.11) goes over all wave functions yielding the density ρ(r). The second search then lifts the constraint to a particular density and extends the new search “ρ(r)→N” over all densities. The Hohenberg- Kohn functional F now is deﬁned via the constrained search

F [ρ] = MinhΨ|T + Vee|Ψi (2.14) Ψ→ρ 2.3. Kohn-Sham Scheme 13 with the minimizing wave function Ψ[ρ] being a functional of the density ρ(r).

The minimizing wave function Ψ[ρ0] for the ground state density ρ0 represents the ground state wave function Ψ0 of the system, that is Ψ0 = Ψ[ρ0]. Though the constrained-search formulation and the two HK-Theorems es- tablish a strict mathematical framework and the existence of the total energy functional is now ensured, one still has no practically realizable scheme to treat the problem of N interacting electrons at hand at this point. This shortcome was solved with the work of Kohn and Sham [2], whose ideas are explained in the next section.

2.3 Kohn-Sham Scheme

The main idea of the Kohn-Sham (KS) method is to map the real interacting system of electrons to an artificial system of noninteracting electrons that yields exactly the same ground state density and ground state energy as the interacting one. In this noninteracting KS system, the electrons then behave like uncharged particles that by definition do not interact with each other through the Coulomb repulsion. Instead, each of the electrons is subjected to a field averaged over all other electrons. In other words, the KS method represents a mean-field theory. The KS wave function Φ[ρ] of the KS system is defined as the wave function that minimizes the constrained search minimization

Ts[ρ] = MinhΨ|T |Ψi. (2.15) Ψ→ρ

This yields the functional Ts[ρ] of the kinetic energy of the KS system. By deﬁning the exchange energy

Ex[ρ] = hΦ[ρ]|Vee|Φ[ρ]i − U[ρ0] (2.16) and the correlation energy

Ec[ρ] = hΨ[ρ]|T + Vee|Ψ[ρ]i − hΦ[ρ]|T + Vee|Φ[ρ]i, (2.17) the total energy can then be written as Z E[ρ] = T [ρ] + Vee[ρ] + drV (r)ρ(r) ≡ (2.18) Z ≡ Ts[ρ] + U[ρ] + Ex[ρ] + Ec[ρ] + drV (r)ρ(r). (2.19) 14 Chapter 2. Foundations of Density-Functional Theory

The exchange-correlation functional

Exc[ρ] = Ex[ρ] + Ec[ρ] (2.20) contains the diﬀerence of the kinetic energies and the diﬀerence of the expectation values of the electron-electron interaction of the two systems and reads

Exc[ρ] ≡ T [ρ] − Ts[ρ] + Vee[ρ] − U[ρ]. (2.21)

At this point one already succeeded in reformulating the problem in a way that the main part of the energy is treated in an exact manner and all other unknown contributions are absorbed in the exchange-correlation functional Exc. This con- ceptual feature can be seen as one of the main merits of KS-DFT. At the same time, one has to be aware of the fact that up to now, the original problem of solving the many-body Schr¨odinger equation (2.1) has only been transformed onto the problem of ﬁnding the exact expression for the exchange-correlation functional Exc[ρ], which is unknown. The variation of the total energy reads h Z i δ E[ρ] − µ drρ(r) − N = 0 (2.22) with the Lagrange parameter µ coming from the constraint of constant particle number N. Varying the expression of the total energy (2.19) with respect to the density gives the Euler-Lagrange equation of KS theory δE[ρ] δT [ρ] δU[ρ] δE [ρ] µ = = s + + xc + V (r). (2.23) δρ(r) δρ(r) δρ(r) δρ(r)

(Note that the density ρ in the denominators of equation (2.23) represents the physical observable and not the associated operator. Throughout this work no symbolic distinction between observables and operators is performed.) Deﬁning the KS potential as δU[ρ] δE [ρ] V (r) = V (r) + + xc = V (r) + V (r) + V (r) (2.24) s δρ(r) δρ(r) H xc the Lagrange multiplier becomes δT [ρ] µ = s + V (r) (2.25) δρ(r) s 2.3. Kohn-Sham Scheme 15 with the conventional Hartree potential Z ρ(r0) V (r) = dr0 (2.26) H |r − r0| and the exchange-correlation potential δE [ρ] V (r) = xc . (2.27) xc δρ(r)

Since the explicit functional Ts[ρ] is not known, the direct solution of the Euler- Lagrange equation is not pursued in practice. Kohn and Sham originally introduced orbitals for solving the problem and the corresponding variation then reads

N Z δ h X 0 † 0 0 i † E[ρ] − jk dr φj(r )φk(r ) = 0 (2.28) δφi (r) j,k=1 with the Lagrange multipliers jk ensuring a constraint of orthonormality of the orbitals. The variation ﬁnally results in the equations

N h 1 i X H (r)φ (r) = − ∇2 + V (r) φ (r) = φ (r) (2.29) s i 2 s i ij j j=1 with Vs representing the potential in equation (2.24). Naturally, the Hamiltonian

Hs is hermitian, which then likewise holds for the matrix ij. It can therefore be brought to diagonal form by an unitary transformation, which leaves the associated physical observables invariant. In this manner one gets the KS equations in their canonical form h 1 i H (r)φ (r) = − ∇2 + V (r) + V (r) + V (r) φ (r) = φ (r). (2.30) s i 2 H xc i i i

This is a set of decoupled equations for the KS orbitals φi with the Hamiltonian having the typical form of a Hamiltonian characterizing a system of noninteracting electrons. For such a noninteracting system the (nondegenerate) ground state is expressed by a Slater determinant 1 |Φi = √ det[φ1, φ2, ..., φn], (2.31) N! which is in this case built from the KS orbitals φi determined by the one-particle equations (2.30). With the ground state expressed as a Slater determinant, the 16 Chapter 2. Foundations of Density-Functional Theory density ρ(r) is simply built out of the KS orbitals via

N X 2 ρ(r) = |φi(r)| (2.32) i=1 and is per construction equivalent to the density of the interacting system ρ0. In terms of the variational principle scheme, the KS wave function of equation (2.31) can equivalently be deﬁned as the minimizing wave function in the constrained search minimization

Ts[ρ0] = Min hΨ|T |Ψi (2.33) Ψ→ρ0(r) yielding the kinetic energy functional of the noninteracting system having the real ground state density ρ0. All potentials in (2.30), and especially the exchange-correlation potential, are local multiplicative, i.e. they depend only on the spatial variable r. This fact results in a quite simple structure of the KS equation themselves, as opposed to the equations underlying e.g. the Hartree-Fock (HF) approximation with their nonlocal exchange term. The total energy can explicitly be written as

N Z X 1 1 ZZ ρ(r)ρ(r0) Z E[ρ] = − dr φ†(r) ∇2φ (r)+ drdr0 +E [ρ]+ drV (r)ρ(r) i 2 i 2 |r − r0| xc i=1 (2.34) with the orbitals depending implicitly on the density. Thus, the energy within the KS formalism depends on both, density and orbitals. The KS Hamiltonian operator within the KS equation depends on the density, which in turn results from the orbitals. Thus, the system of KS equations (2.30) has to be solved in a selfconsistent way. A natural question that arises concerns the physical meaning of the KS orbitals gained by this selfconsistent scheme. It is a fact that they yield the exact ground state density, which is, for instance, not the case in the HF formalism. On the other hand it is clear that the Slater determinant built out of the KS orbitals does not represent the true many-electron wave function. The related KS eigenvalues originally were considered to have no (or little) physical meaning. More recent works, however, showed that the KS eigenvalues are 2.3. Kohn-Sham Scheme 17 well deﬁned zeroth order approximations for the ionization and excitation energies [26, 27, 28, 29, 30, 31, 32] and therefore can be seen as physically meaningful quantities. The associated fundamental (quasiparticle) band gap can be expressed in terms of the KS-eigenvalues as

Egap = N+1(N + 1) − N (N) (2.35) with the quantity N+1(N+1) denoting the eigenvalue of the (N+1)-th orbital for the (N+1)-electron system. In practical calculations the KS-equations are solved for the N-particle system yielding the corresponding eigenvalue set {i(N)}. The band gap within the KS-formalism, contrary to equation (2.35), then reads

S Egap = N+1(N) − N (N). (2.36)

This band gap obviously diﬀers from the quasiparticle band gap by a diﬀerence of

∆xc = N+1(N + 1) − N+1(N) (2.37) called the discontinuity term [21, 22]. As one will see later on, the intrinsic neglectance of this term within the KS-formalism is said to be one of the reasons for band gap underestimations using certain approximations for the exchange- correlation functional. One has to keep in mind that the knowledge of the exchange-correlation functional Exc[ρ] (2.21) is equivalent to solving exactly the many-particle Schr¨odinger equation. Therefore the complexity of the underlying problem has been absorbed in a very complex, nonlocal dependence of the KS potential on the density. In practical calculations the exchange-correlation functional always is approximated. There are no simple systematic methods for constructing such energy functionals, but it is possible to separate the exchange from the correlation part

Exc[ρ] = Ex[ρ] + Ec[ρ]. (2.38)

The same then holds for the exchange and correlation potentials as functional derivatives of the energy functionals with respect to the density

δE [ρ] δE [ρ] V (r) = x ; V (r) = c . (2.39) x δρ(r) c δρ(r) 18 Chapter 2. Foundations of Density-Functional Theory

In DFT the exchange energy functional is deﬁned by the relation [4]

Here, the term hΦ[ρ]|Vee|Φ[ρ]i is the expectation value of the electron-electron interation of the KS state, not the true electron-electron interation energy. Com- parison of equation (2.40) with equation (2.21) shows that the diﬀerence between the true electron-electron interaction energy and the expectation value hΦ[ρ]|Vee|Φ[ρ]i is absorbed in the correlation energy, which also contains the kinetic energy diﬀerences of the real and the KS-noninteracting electronic systems. The correlation energy can then be written as

Ec[ρ] = T − Ts + Vee − hΦ[ρ]|Vee|Φ[ρ]i. (2.41)

For the sake of completeness and to avoid any confusion of the terms “exchange” and “correlation” as they are used within the DFT frame with the same terms used in HF, their usage within HF is brieﬂy reviewed. The selfconsistent one-particle HF equations read [3]

h 1 i Z − ∇2 + V (r) + V (r) φHF (r) + dr0V NL(r, r0)φHF (r0) = HF φHF (r) (2.42) 2 H i x i i i with the nonlocal exchange operator Z Z 0 NL 0 0 HF 0 NL 0 HF 0 dr hr|Vx |r ihr |φi i = dr Vx (r, r )φi (r ) ≡ (2.43)

N Z HF † 0 HF X φj (r )φj (r) ≡ − dr0 φHF (r0). (2.44) |r − r0| i j=1

HF The Slater determinant for the ground state |Φ0 i is is constructed from the set HF HF of N HF orbitals {φ1 , ..., φN } yielding the lowest value for the total energy.

The corresponding HF density ρHF (r) is constructed in the usual way

N X HF 2 ρHF (r) = |φi (r)| (2.45) i=1 2.3. Kohn-Sham Scheme 19

and is, in general, not equivalent to the exact ground state density ρ0(r) of the electronic system. In HF theory the exchange energy is deﬁned by

HF HF HF Ex = hΦ0 |Vee|Φ0 i − U[ρHF ], (2.46) which has obviously the analog structure as equation (2.40) of the KS framework, but now contains the HF orbitals in the Slater determinant and the density. HF The Slater determinant |Φ0 i also yields the total HF energy of the system

EHF , which is, according to the variational principle always larger than the exact ground state energy E0. The correlation energy in HF is then expressed as [61]

HF Ec = E0 − EHF . (2.47)

Since the terms “exchange” and “correlation” have different meanings in HF and DFT, they always depend on the context in which they are used. Beside the ground state properties of the investigated electronic systems, it is desirable to gain further knowledge about the behavior of the system when exposed to an external perturbation. The dielectric function (r, r0) describes the infinitesimal change of the total potential δVtot(r) of the electronic system at point r in case of an infinitesimal change of the external potential δV (r0) at another point r0 δV (r) −1(r, r0) = tot . (2.48) δV (r0)

Within the framework of KS-DFT one identiﬁes δVtot = δVs and the associated density change is described via the independent particle response function

0 δρ(r) χ0(r, r ) = 0 . (2.49) δVs(r ) First order perturbation theory applied to the KS equations (2.30) yields [62]

N ∞ † † 0 0 0 X X φi (r)φj(r)φj(r )φi(r ) χ0(r, r ) = + c.c. (2.50) i − j i=1 j=N+1 with the the index i running over all occupied orbitals and the index j running over all unoccupied orbitals. The density change due to a change in the external potential is given analogously to (2.49) as δρ(r) χ(r, r0) = (2.51) δV (r0) 20 Chapter 2. Foundations of Density-Functional Theory with χ(r, r0) standing for the response function of the real electron system. Vary- ing the KS potential with respect to the density gives its inﬁnitesimal change δV (r0) δV (r0) δV (r) = δV (r) + H δρ(r) + xc δρ(r) = s δρ(r) δρ(r) 0 0 = δV (r) + KH (r, r )δρ(r) + Kxc(r, r )δρ(r). (2.52)

In the last line the Hartree kernel is given by 1 K (r, r0) = . (2.53) H |r − r0| The functional derivative of the exchange-correlation functional with respect to the density, termed exchange-correlation kernel, is deﬁned by δV (r0) K (r, r0) = xc . (2.54) xc δρ(r)

Analogous to its energy and potential counterparts, the kernel Kxc can be separated in an exchange and a correlation part

0 0 0 Kxc(r, r ) = Kx(r, r ) + Kc(r, r ). (2.55)

Combining equations (2.49), (2.51) and (2.52) gives

−1 χ = (1 − χ0KH − χ0Kxc) χ0 (2.56) for the response function and together with (2.48) we further get

−1 = 1 − (KH + Kxc)χ0 = [1 + (KH + Kxc)χ)] (2.57) for the dielectric function. Finding good approximations for the exchange- correlation kernel Kxc is obviously a challenging task. A nice review article on kernels proposed in the literature can be found in [10]. Furthermore it has to be mentioned that for the exchange part Kx an exact, explicit expression based on the corresponding exact-exchange potential, which will be introduced in chapter 4 has been derived [63].

2.4 Approximate Exchange-Correlation Func- tionals

The KS formalism itself makes no assertion on the form of the exchange- correlation functional and therefore gives no practical method at hand. In order 2.4. Approximate Exchange-Correlation Functionals 21 to take advantage of the simplifications and the framework gained by introducing the KS formalism and to be able to solve the corresponding KS equations, the exchange-correlation functional has to be approximated. The oldest and most spread approximation is the local density approximation (LDA), which is at the roots of DFT and the KS-formalism and has been proposed by Kohn and Sham themselves. The underlying physical system for the LDA is the homogeneous electron gas, i.e. a system of electrons moving on a uniform positive background charge such that the whole system becomes electrically neutral. The number of electrons N and the volume of the gas V approach infinity, while the electron density ρ = N/V reaches a constant value, which is the same for each spatial point r. Within the LDA an inhomogeneous system like an atom, molecule or solid is then treated as a homogeneous system at each spatial point r with the associated LDA exchange and correlation energy per electron xc (ρ(r)) at each point r given by the corresponding value of a homogeneous electron gas with the density at the point r. Thus, the LDA approximation should be well applicable to systems with a slowly varying density, but formally not justified for highly inhomogeneous systems. Indeed, the LDA is mostly employed in solid state physics and had no comparable impact in computational chemistry that treats atoms and molecules characterized by a more rapidly varying density. Since the homogeneous electron gas is completely determined by the value of its density, the exchange correlation energy within the LDA approximation is gained by integrating the parameter LDA xc (ρ(r)) weighted with the local density ρ(r) at each point in space Z LDA LDA Exc [ρ] = drρ(r)xc (ρ(r)). (2.58)

This energy is again separated between an exchange and correlation contribution

Z Z LDA LDA LDA Exc [ρ] = drρ(r)x (ρ(r)) + drρ(r)c (ρ(r)) (2.59) and the LDA exchange-correlation potential is obtained by the functional derivative of (2.59)

δELDA[ρ] δLDA(ρ(r)) V LDA(r) = xc = LDA(ρ(r)) + ρ(r) xc . (2.60) xc δρ(r) xc δρ(r) 22 Chapter 2. Foundations of Density-Functional Theory

LDA The exchange part of the energy per particle x (ρ(r)) is known and was originally derived by Dirac [64]

1/3 1/3 LDA 3 3 1/3 3 9 1 x (ρ) = − ρ(r) = − 2 (2.61) 4 π 4 4π rs with the quantity rs determining the radius of a sphere with the eﬀective volume of an electron. The exact density dependence of the correlation energy is not known, but LDA proper values for c (ρ) are available, the most widely used are those of Vosko, Wilk and Nusair (VWN) [65]. Ceperley and Alder calculated the total energy for a uniform electron gas using the quantum Monte Carlo method and yielded the correlation energy by subtracting the corresponding kinetic and exchange energies [66]. This data was then interpolated by VWN and the ﬁnal form for the LDA correlation reads An x 2b Q LDA(r ) = ln + arctan − c s 2 X(x) Q 2x + b bx h (x − x )2 2(b + 2x ) Q io − 0 ln 0 + 0 arctan , (2.62) X(x0) X(x) Q 2x + b √ √ 2 2 where x = rs, X(x) = x + bx + c, Q = 4c − b and for zero spin polarization

A = 0.0621814, x0 = −0.409286, b = 13.0720 and c = 42.7198. The next step in treating inhomogeneous systems in a more accurate way is to include the gradient of the density in the exchange-correlation functional. By doing so, one hopes to account for the nonhomogeneity of the true electron density. The general approach for these “generalized gradient approximations” (GGA) is given by a functional of the form Z GGA GGA Exc [ρ] = drfxc (ρ(r), ∇ρ(r)), (2.63) which explicitly contains the density gradient ∇ρ(r). Again, the exchange and correlation term can be treated separately

GGA GGA GGA Exc [ρ] = Ex [ρ] + Ec [ρ]. (2.64)

In most cases the GGA functionals for the exchange part can be written in the form Z GGA LDA Ex [ρ] = drρ(r)x (ρ(r))F (s) (2.65) 2.4. Approximate Exchange-Correlation Functionals 23 with the dimensionless parameter

|∇ρ(r)| s = (2.66) 2kf (r)ρ(r) and a scaling function F that can have a pretty complicated form in practice. 2 1/3 The quantity kf (r) denotes the Fermi wave vector given by kf (r) = (3π ρ(r)) . GGA functionals often yield results that are distinctively better than those obtained by the LDA. However, one has to be aware of the fact that some of these functionals are not necessarily based on new physical ideas. Some are of semiempirical character, contain ﬁtting parameters and are sometimes exclusively constructed in a way to satisfy required boundary conditions and gain good results in an acceptable computing time. Mostly, they are optimized to gain good results for the energy value and do not produce bearable results for the corresponding potentials. In the following, the most important and most widespread GGAs are listed. For further information on the concrete form of the relevant scaling functions F (s), the original works can be consulted:

Becke’s exchange functional (B88) [67].

Perdew, Burke and Ernzerhof’s exchange-correlation functional (PBE) [68].

Lee, Yang and Parr’s correlation functional (LYP) [69].

Perdew’s 1986 correlation functional (P86) [70,71].

Perdew and Wang’s correlation functional (PW91) [72].

The functional derivative of the general GGA functional (2.63) is given by [73]

GGA hδfxc δfxc i Vxc (r) = − ∇ = δρ δ∇ρ ρ(r),∇ρ(r) 1/3 3 3 1 h4 −1 dF 4 3 d −1 dF i = − ρ 3 F − ts − (u − s ) s (2.67) 4 π 3 ds 3 ds ds with the two dimensionless parameters

∇2ρ(r) t = t(r) = 2 (2.68) 4kf (r) ρ(r) 24 Chapter 2. Foundations of Density-Functional Theory and ∇ρ(r)∇|∇ρ(r)| u = u(r) = 3 2 . (2.69) 8kf (r) ρ(r) The GGA potentials therefore contain even higher derivatives of the density than the associated energy functionals and along with the already complicated forms of some energy functionals, the analytical expressions for the potentials can become highly complicated. The approximate exchange-correlation functionals presented within this chapter usually yield band structures that underestimate the associated band gap [21, 22, 23, 24, 25]. In case of LDA calculations, this underestimation is typically of order 30 – 100%. One possible explanation is related to the treatment of the discontinuity ∆xc (2.37). Assuming a continuous particle number N, the discontinuity can be expressed as the limit [4] δExc[ρ] δExc[ρ] ∆xc = lim − = (2.70) →0+ δρ(r) δρ(r) N+ N− ρ=ρ0 = Vxc([ρ], r)N+ − Vxc([ρ], r)N− . (2.71) ρ=ρ0

The proposed LDA- and GGA exchange-correlation functionals explicitly depend on the density (LDA) or the density plus its derivatives (GGA) per definition. Adding an electron to an N-particle system, the change in the density is of order 1/N and due to formula (2.70) the discontinuity obviously vanishes for these functionals, i.e. the LDA and GGA approximations simply neglect it. Another reason for the mentioned underestimation of the band gap is the fact that the LDA and GGA contain merely approximations of the KS exchange potential, whose exact form cancels self-interactions contained in the Coulomb potential (see chapter 4). This cancellation is incomplete in the LDA and GGA case and therefore the associated KS eigenvalues are energetically raised. Due to their stronger localization, the valence bands are raised more strongly in energy compared to the conduction bands, which then results in a decreased band gap. A complete exact expression for the exchange potential will be presented in chapter 4. Within this approach Coulomb self-interactions are cancelled out and only correlation effects have to be approximated. The underlying functional is constructed out of KS-orbitals rather than the density itself and gains strongly improved band structures. For the treatment of the exact exchange case in a 2.4. Approximate Exchange-Correlation Functionals 25 manner appropriate for the needs of this work, the next chapter first provides some information on the application of DFT to periodic systems. 26 Chapter 2. Foundations of Density-Functional Theory Chapter 3

Density-Functional Theory for Periodic Systems

A widespread technique for performing DFT calculations for periodic systems has turned out to be the pseudopotential approach in combination with a plane wave basis set. The pseudopotential method exploits the fact that the core electrons are more or less chemically inert, i.e. the chemical properties of atoms and the solid states composed out of them are mostly determined by their valence electrons. Within the pseudopotential framework the physical eﬀect of the core electrons is therefore summarized in an eﬀective pseudopotential for the core so that in practice the one particle equations (2.30) have to be explicitly solved only for the valence states. The decision to choose a plane wave basis set for the treatment of periodic systems is quite natural, because the corresponding representation of orbitals is then done by usual Fourier series that are the common mathematical objects for describing periodic functions.

3.1 Pseudopotential Approach

The valence and core orbitals as solutions of the Kohn-Sham equation, a Schr¨odinger equation with a hermitian Hamiltonian operator, are orthogonal to each other. To satisfy this orthogonality requirement the corresponding valence and core orbitals have to diﬀer in their number of nodal surfaces. With the energy

27 28 Chapter 3. Density-Functional Theory for Periodic Systems of the eigenvalues rising from core to valence orbitals, the valence electrons then naturally own a high number of nodes and therefore show a strong oscillatory behavior in the core region. In order to represent this behavior in an appropriate way, a high number of plane waves has to be incorporated in the underlying Fourier series. This results in an increased dimension and subsequently increased computational eﬀort of the self consistent calculations for the periodic structure. A strategy to get rid of the strong oscillatory behavior of the valence orbitals, i.e. gain smoother orbitals with no nodes was proposed by Philips and Kleinman [74]. The idea is to project out the overlap between core and valence electrons by the use of an appropriate projection operator and thus extract a smooth part

φv from the valence orbital ψv [75] X |φvi = |ψvi + hψc|φvi|ψci. (3.1) c

The smooth part |φvi of the valence state has no overlap with the core states

|ψci by construction, therefore satisﬁes the orthogonality condition and is free of disturbing oscillations. Both valence and core electrons satisfy the atomic Schr¨odinger equation

core H|ψc,vi = [T + Vee + V ]|ψc,vi = c,v|ψc,vi (3.2) with the atomic Hamiltonian H and the core potential V core. An application of the atomic Hamiltonian on the new, smoothed valence states φv results in a modiﬁed Schr¨odinger equation with a new “Pseudo”hamiltonian

h X i H + (v − c)|ψcihψc| |φvi = v|φvi. (3.3) c

The core potential V core is thus modiﬁed by an additional repulsive potential term of projector form resulting in

ps core X V = V + (v − c)|ψcihψc|. (3.4) c

For the construction of the pseudopotential V ps and the associated pseudoorbitals φps(r) one ﬁrst considers the single atom. As the pseudopotential V ps contains a projection operator, it has a nonlocal character. The real space representation 3.1. Pseudopotential Approach 29 of this nonlocal pseudopotential can be expressed via a hermitian kernel V (r, r0) acting on a pseudoorbital φps(r) Z hr|V ps|φpsi = dr0V ps(r, r0)φps(r0). (3.5)

Assuming spherical symmetry of the investigated system, the kernel can be expanded in spherical harmonics

ps 0 X ∗ ps 0 0 0 V (r, r ) = Ylm(θ, ϕ)Vl (r, r )Ylm(θ , ϕ ). (3.6) l,m Accordingly, the corresponding pseudoorbitals can be written as φps(r) φps (r) = l Y (θ, ϕ). (3.7) lm r lm The pseudoorbitals then obey a radial Schr¨odinger equation for the pseudoatom h 1 d2 l(l + 1) i − + + V˜ ps φps(r) = psφps(r) (3.8) 2 dr2 2r2 l l l l ˜ ps with Vl denoting an integral operator deﬁned by equations (3.5) and (3.6). ps Equations (3.6) and (3.7) have only one kernel Vl and only one pseudoorbital ps φl for each angular momentum l. For high l-values l > lmax the centrifugal barrier l(l + 1)/2r2 in equation (3.8) dominates the energy of the system and a local part can be separated from the nonlocal potential V (r, r0)  V ps(r)δ(r − r0) + ∆V (r, r0) for l ≤ l ps 0  loc l max Vl (r, r ) = (3.9) ps 0 Vloc(r)δ(r − r ) for l > lmax.

For the few l-values with l ≤ lmax, the potential is corrected by short-ranged 0 nonlocal corrections ∆Vl(r, r ). P Introducing operators m |YlmihYlm| that project on the individual angular momentums, equation (3.9) can be rewritten in a form that explicitly emphasizes ps that each l-component of the orbital φl is under the inﬂuence of an orbital ps dependent pseudopotential Vl

ps ps X X V = Vloc + ∆Vl |YlmihYlm|. (3.10) l m Often the pseudopotential is chosen to be of semilocal form (see e.g. [76])

0 0 ∆Vl(r, r ) = ∆Vl(r)δ(r − r ), (3.11) 30 Chapter 3. Density-Functional Theory for Periodic Systems i.e. radially local, but angularly nonlocal. For the case of “fully separable” pseudopotentials the r and r0 dependencies of the nonlocal potential in (3.9) are disconnected 0 † 0 ∆Vl(r, r ) = Γl (r)Γl(r ). (3.12) Kleinman and Bylander (KB) [75] proposed an operator representing such fully ps separable pseudopotentials, i.e. the correction to the local part Vloc. It has a projector form as proposed in (3.10) and sums over all values for l and m

The real space representation of (3.13) can be computed to

Z −1 KB 0 X h 2 ps† −1 ps ps i ∆V (r − R, r − R) = drr φl (r)) Vl (r)φl (r) × l,m ps 0 ps 0 0 × Vl (|r − R|)φl (|r − R|)Ylm(|r − R|) × ps ps × Vl (|r − R|)φl (|r − R|)Ylm(|r − R|) (3.14) with the spatial coordinate r shifted by the additionally introduced core coordinate R representing an arbitrary atom in the unit cell of the periodic structure. With a reduced effort to calculate its contributions to the matrix elements of the Hamiltonian, separable pseudopotentials emerged to be very useful in large-scale electron-structure calculations. However, they hold a danger by sometimes induc- ing additional, unphysical “ghost states”, which destroy the eigenvalue spectrum and therefore the usage of the pseudopotentials. Shortly spoken, the equation to be solved to obtain separable pseudopotentials is of integrodifferential type and does not satisfy the Wronskian theorem of differential equation theory. As a consequence, it becomes possible that additional, unphysical states at lower energies than the zero-node reference atomic level appear. Usually one can get rid of these states by choosing a different local component or varying the cutoff radius during the construction of the pseudopotentials. A more thorough analysis of this problem has been carried out in [77,78]. ps Obviously one first has to construct the pseudoorbitals φl together with ps their pseudopotentials Vl to really calculate the separable pseudopotential of (3.14). The starting point in practice is an all-electron calculation within the 3.1. Pseudopotential Approach 31

KS framework for the corresponding atom to obtain the atomic KS orbitals.

Assuming again spherical symmetry, the atomic orbitals φnlm(r) can be written as φ (r) φ (r) = nl Y (θ, ϕ) (3.15) nlm r lm with the radial part φnl(r) obeying the one-dimensional KS equation h 1 d2 i − + V at (r) φ (r) = φ (r). (3.16) 2 dr2 eff nl nl nl at The eﬀective, atomic potential Veff is given by Z l(l + 1) V at (r) = − + + V (r) + V (r) (3.17) eff r 2r2 H xc ps and the pseudoorbitals φl , satisfying the corresponding radial pseudoequation

(3.8), are then constructed out of the all-electron orbitals φnl of the highest given value of n. The pseudoorbitals have by construction no nodes and coincide with the all-electron atomic orbital outside a characteristic cutoﬀ radius rcut,l that determines the size of the core region. In practice, this radius is chosen diﬀerently for each l-value. Within this radius, the atomic- and pseudoorbital have to enclose the same charge via the normconservation condition [76]

rcut rcut Z φps 2 Z φ 2 dr l r2 = dr l r2. (3.18) r r 0 0 ps Furthermore the two orbitals shall have the same eigenvalue l = l and coincide in their logarithmic derivatives at the cutoﬀ radius rcut,l, which guarantees identical scattering properties of the atomic- and pseudoorbital

0 φps (r) φ0(r) l l ps = . (3.19) φl (r) φl(r) r=rcut,l r=rcut,l The constraints (3.18) and (3.19) do not uniquely determine the pseudoorbitals and therefore ab-initio pseudopotentials can be constructed in various ways. Worth mentioning are the works of Bachelet, Hamann, Schl¨uter[79], Vander- bilt [80], Troullier, Martins [81,82] and Bl¨ochl [83]. Using one of these procedures ps ps to obtain the pseudoorbital φl , the appropriate pseudopotential Vl can then be constructed via inversion of the KS equation (3.8) 2 ps ps l(l + 1) 1 d ps Vl (r) = l − 2 + ps 2 φl (r). (3.20) 2r 2φl (r) dr 32 Chapter 3. Density-Functional Theory for Periodic Systems

In order to gain an optimal transferability of the pseudopotentials among various chemical environments, it is desirable to ﬁnally yield ionic pseudopotentials. Thus the valence electron contributions, which determine the actual physical behavior of the electronic system have to be eliminated. They are introduced back in the selfconsistent calculation of the periodic structure in which the generated pseudopotentials are ﬁnally employed. For simplicity one considers the unscreening procedure for a conventional density-functional of LDA or GGA type. Denoting the valence density as ρval, the ionic potential is constructed by the unscreening procedure ps ps ps ps Vl,ion(r) = Vl (r) − VH (ρval, r) − Vxc(ρval, r). (3.21)

Equation (3.21) implies the additive decompositions

ps ps ps VH (ρ , r) = VH (ρval, r) + VH (ρc , r) (3.22)

ps ps ps Vxc(ρ , r) = Vxc(ρval, r) + Vxc(ρc , r) (3.23) for the potentials VH and Vxc into a valence and a core part, which is fulﬁlled in the case of the Hartree potential VH , but not necessarily for the exchange- correlation part Vxc, which is in general a nonlinear functional of the density. This nonlinearity is often taken into account by the nonlinear core-correction [84].

3.2 Plane-Wave Formalism

In order to solve the KS equations (2.30) numerically, the one particle equations have to be discretized by introducing a set of basis functions. For an investigation of periodic systems the obvious approach is the introduction of a plane-wave basis, i.e. to represent the KS orbitals, the electron density and other quantities by their Fourier expansions. According to Bloch’s theorem, the eigenstates of a system described by a Hamiltonian including a periodic external potential can be expressed as a plane ikr wave e times a function unk(r) with the periodicity of the underlying lattice [85].

An expansion of the unk(r) in a plane-wave basis leads to the following expression 3.2. Plane-Wave Formalism 33

for the KS orbitals φnk(r) of the periodic system

1 ikr 1 X i(k+G)r φnk(r) = √ e unk(r) = √ Cnk(G)e . (3.24) Ω Ω G The quantum numbers labelling the eigenstates are the wavevector k belonging to the first Brillouin zone (BZ) and the band index n. Ω represents the crystal volume. In practice the sum running over all reciprocal lattice vectors G is only considered up to a fixed energy cutoff Ecut and all expansion coefficients Cnk(G) 1 2 related to a plane wave with a kinetic energy 2 (k + G) > Ecut are neglected. Applying this transformation to the real space KS equations h 1 i − ∇2 + V (r) + V (r) + V ps(r, r0) φ (r) = (r)φ (r) (3.25) 2 H xc nk nk nk leads to a discretized formulation and the corresponding equation system for the Fourier coefficients becomes

X h1 2 0 0 (k + G) δ 0 + V (G − G ) + V (G − G ) + 2 G,G H xc G0 ps 0 i 0 + V (k + G, k + G ) Cnk(G ) = nkCnk(G). (3.26)

This equation system is built out of matrices with finite dimensions (due to the finite energy cutoff) and represents an eigenvalue problem that can be treated by standard techniques. Since the KS equations describe a noninteracting system, the kinetic energy matrix is diagonal. The number of quantum numbers k, at least for a regular grid of k-points, can be associated with the number of unit Q3 cells composing the periodic system. Assuming i=1 Ni unit cells, the allowed values for k are [85] 3 X mi k = b (3.27) N i i=1 i with the bi denoting the reciprocal lattice vectors. The conditions for the integers mi are given by N N − i ≤ m ≤ i (3.28) 2 i 2 in case of an even Ni and (N − 1) (N − 1) − i ≤ m ≤ i (3.29) 2 i 2 34 Chapter 3. Density-Functional Theory for Periodic Systems

in case of an odd Ni. For an ideal infinitely extended system this gives an infinite set of k-points and the charge density ρ(r) is obtained by an integration over the first BZ [85] Z Ω0 X ρ(r) = dk|φ (r)|2 (3.30) (2π)3 vk v BZ with Ω0 denoting the volume of the Wigner-Seitz cell of the system. For the discrete case this integral turns into a sum over k-points

X 2 ρ(r) = |φvk(r)| . (3.31) vk One can interpret the sum (3.31) as a numerical integration formula of the integral (3.30). In this case, depending on the choice of k-points, one has to assign to each k-point a lattice dependent weight wvk. In practice, the k-points with their corresponding weights are then chosen to reproduce the integral (3.30) as accurately as possible. For a regular grid the weights all equal one. If the investigated system owns internal symmetries in reciprocal space, it is possible to apply the concept of special k-points [86, 87]. In this scheme, the required quantities are exclusively calculated for certain high symmetry k-points. The corresponding values for further k-points, which are related to these special k-points via internal symmetries, can then be calculated in a quite simple way by applying the associated symmetry transformations. If no internal symmetries shall be exploited, a uniform mesh of k-points covering the ﬁrst BZ can be used. Without loss of generality, i.e. any inﬂuence on the further development of the formalism, this latter case is considered and the weights are set to a value of one. Inserting the Fourier expansion of the KS orbitals (3.24) in (3.31), the density can be evaluated by

1 X X 0 ρ(r) = C (G)ei(k+G)rC∗ (G0)e−i(k+G )r (3.32) Ω vk vk vk GG0 1 X X 0 = C (G)C∗ (G0)ei(G−G )r (3.33) Ω vk vk vk GG0 1 X X h X i 00 = C (G00 + G0)C∗ (G0) eiG r (3.34) Ω vk vk vk G00 G0 with the substitution G00 =G−G0 applied for the last step. Therefore, the plane wave representation ρ(G) of the density can be calculated by summing up the 3.2. Plane-Wave Formalism 35 expansion coeﬃcients according to 1 X X ρ(G) = C (G + G0)C∗ (G0). (3.35) Ω vk vk vk G0

The Fourier transform of the Hartree potential VH (r) reads 1 Z V (G) = dre−iGrV (r) (3.36) H Ω H and is related to the electron density through the momentum space version of electrostatic’s Poisson equation 4πρ(G) V (G) = (3.37) H G2 that describes the electrostatic field induced by the density. The Hartree potential can thus be calculated in a straightforward way via direct usage of the density coefficients. A direct insertion of G=0 into (3.36) shows that the value VH (G=0) can be identified as the spatial mean value of the Hartree potential. For an infinitely extended, periodic system this value is ill defined and is set to zero like the corresponding term in the potential of the nuclei, respectively the local pseudopotential, which has opposite sign. For a calculation of traditional (i.e. LDA or GGA) exchange correlation potentials Vxc(G) in momentum space, the density is first transformed to real space. Then, the potentials are computed via the formulas described in section 2.4 and afterwards transformed back to momentum space. The last quantity in (3.26) that needs to be calculated is the contribution of the pseudopotential V ps(G + k, G0 + k) in momentum space. For separable KB pseudopotentials the real space representation was already described in section 3.1 by formula (3.14). In the required matrix elements for momentum space, the core coordinate can be split off and resolves in an additional phase factor e−i(G−G0)R

KB 0 i(G+k)r KB 0 i(G0+k)r0 Vl (G + k, G + k) = he |Vl (r − R, r − R)|e i = −i(G−G0)R i(G+k)r KB 0 i(G0+k)r0 = e he |Vl (r, r )|e i. (3.38) Employing an expansion of an ordinary plane wave in terms of Bessel functions jl and spherical harmonics Ylm ∞ l iGr X X l ∗ e = 4π i jl(Gr)Ylm(G)Ylm(G), (3.39) l=0 m=−l 36 Chapter 3. Density-Functional Theory for Periodic Systems in (3.14), the FT of the pseudopotential (3.38) is given by

2 (4π) 0 h Z † i−1 ∆V KB(G + k, G0 + k) = e−i(G−G )R drr2φps (r)V ps(r)φps(r) × l Ω l l l Z Z h 2 ps ps ih 2 0 ps ps i × drr jl(|G + k|r)Vl (r)φl (r) drr jl(|G + k|r)Vl (r)φl (r) ×

l X ∗ 0 × Ylm(|G + k|)Ylm(|G + k|). (3.40) m=−l

The FT of the local component of the pseudopotential can be carried out by direct integration using spherical coordinates and results in 1 Z 4π Z V ps(G) = drV ps(|r − R|)e−iGr = e−iGr drrV (r) sin(Gr). (3.41) loc Ω loc ΩG loc An additional momentum space representation of the independent particle response function χ0 will be useful later on. Its real space representation was given by formula (2.50) and can now be transformed to momentum space as well. The new indices vk and ck for valence and conduction bands are identiﬁed with the former indices i and j used in (2.50) and one gets

−i(q+G)r i(q+G0)r 0 2 X hvk|e |ck + qihck + q|e |vki χ0(q, G, G ) = (3.42) Ω vk − ck+q vck with the matrix elements in the plane wave basis given by

−i(q+G)r X ∗ 0 0 hvk|e |ck + qi = Cvk(G )Cck+q(G + G ). (3.43) G0 At this point, all quantities forming the equation system (3.26) are described in an appropriate way. Together with the momentum space representation of the response function, one is now able to formulate the exact exchange KS formalism, which will be presented in the following chapter. Chapter 4

Exact Exchange Kohn-Sham Formalism

This chapter shall describe the exact treatment of the local exchange potential

Vx for periodic systems as it has been introduced in section 2.3. The corresponding scheme is commonly denoted as “EXX”(exact exchange) scheme and was ﬁrstly introduced in [34, 35, 88]. A ﬁrst implementation and application to three-dimensional semiconductors [36,37] yielded improved results for band gaps compared to calculations done within the LDA- and GGA-framework.

As a starting point, the exchange energy Ex is considered. An expression for it was already introduced in formula (2.40). The sums now run over the valence states v and v0 ZZ † † 0 0 1 X X φ (r)φv0k0 (r)φ 0 0 (r )φvk(r ) E [ρ] = − drdr0 vk v k . (4.1) x 2 |r − r0| vk v0k0 Constructing the exchange energy in this way, the self-interaction contributions in the Hartree energy are cancelled out and one can already predict at this point that an exchange potential directly derived from the exchange energy (4.1), will be free of self-interactions, in contrast to the usual LDA and GGA potentials. By deﬁnition the exchange potential can be obtained by calculating the functional derivative of the exchange energy with respect to the electron density. Unfortunately, expression (4.1) denotes the energy in terms of the orbitals rather than the electron density itself. The explicit dependence of the orbitals on the density, however, is unknown and a trick has to be applied to really get a usable

37 38 Chapter 4. Exact Exchange Kohn-Sham Formalism expression for the exchange potential. A double application of the functional chain rule leads to an exact expression containing new functional derivatives that can be calculated exactly ZZ 0 00 δEx[ρ] X 0 00h δEx[ρ] δφvk(r ) iδVs(r ) Vx(r) = = dr dr 0 00 + c.c. . (4.2) δρ(r) δφvk(r ) δVs(r ) δρ(r) vk 0 The first derivative δEx[ρ]/δφvk(r ) can be obtained by straightforward differen- tiation of the exchange energy (4.1) with respect to the orbital φvk(r) Z † 0 † δEx[ρ] X φv0k0 (r )φvk(r1)φv0k0 (r1) 0 = − dr1 0 . (4.3) δφvk(r ) |r − r1| v0k0 0 00 The second derivative δφvk(r )/δVs(r ) represents the first order change of the one particle wave function φvk(r) by an infinitesimal change of the local KS-potential

Vs(r). This physical situation can be treated by applying first order perturbation theory to the underlying KS-equations. Note that the derivative, a first order quantity, is given exactly by first-order perturbation theory

0 † 00 00 δφvk(r ) X 0 φn00k0 (r )φvk(r ) 00 0 00 = φn k (r ) . (4.4) δVs(r ) vk − n00k0 n00k0(6=vk) The sum runs over all, i.e. valence and conduction, bands n00, except n00k0 = vk. Inserting equations (4.3) and (4.4) into (4.2) leads to a ﬁrst interim result for the exchange potential ZZZ 00 0 00 δVs(r ) X X X 1 Vx(r) = dr dr dr1 × δρ(r) vk − n00k0 vk v0k0 n00k00 † † 0 0 † 00 00 φvk(r1)φv0k0 (r1)φv0k0 (r )φn00k00 (r )φn00k00 (r )φvk(r ) × 0 + c.c. . (4.5) |r − r1| In analogy to the nonlocal exchange operator of the HF-equations (2.42), a NL similar operator Vx can be introduced at this point, which provides further simpliﬁcation. The real space kernel of this integral operator reads

† 0 X φv0k0 (r)φ 0 0 (r ) hr|V NL|r0i = V NL(r, r0) = − v k (4.6) x x |r − r0| v0k0 and has the same form as its HF counterpart, but is constructed out of KS- orbitals. Inserting (4.6) into equation (4.5) then yields Z † 00 00 00 00 X X h NL 00 00 φn00k00 (r )φvk(r ) iδVs(r ) Vx(r) = dr hvk|Vx |n k i + c.c. . (4.7) vk − n00k00 δρ(r) vk n00k00 39

The second sum arose from first order perturbation theory and runs over all bands n00, including valence as well as conduction bands. Since the terms in brackets cancel out pairwise in case of n00 representing a valence band v, the second sum in (4.7) can be reduced to a summation over conduction bands. Furthermore, the NL matrix element of the nonlocal exchange operator Vx can be modified by taking into account its invariance under translations by a lattice vector. A translation by a lattice vector R leads to an additional phase factor eiR(k−k00) in the matrix NL 00 00 NL element hvk|Vx |n k i of equation (4.7). (Note that the operator Vx (4.6) is invariant with respect to translations by a lattice vector R.) On the other hand, the matrix element has to be invariant under discrete translations, i.e. the phase factor has to reduce to one, which is fulfilled for k = k00. With these additional informations, the potential can be rewritten as

Z † 0 0 0 0 X h NL φck(r )φvk(r ) iδVs(r ) Vx(r) = dr hvk|Vx |cki + c.c. . (4.8) vk − ck δρ(r) vck

This quite compact formula for Vx explicitly reveals the nonlocal internal structure of the (in itself) local exchange potential. To receive the value of the potential at a specific point r in space, the wave functions need to be known in the whole unit cell. This is a remarkable difference to the LDA or GGA methods, in which only the local density or its local derivatives are implied in the construction of the particular exchange- and correlation functionals. It has also to be noted that the exchange potential now explicitly depends on the orbitals and their eigenvalues, rather than the density itself. Linear response theory for independent particles provides a relation between the change of the electron density under influence of an infinitesimal change in the local KS potential Z 0 0 0 δρ(r) = dr χ0(r, r )δVs(r ), (4.9) i.e. the independent particle response function equals the inverse of the remaining 0 functional derivative δVs(r )/δρ(r) that needs to be evaluated

0 δρ(r) χ0(r, r ) = 0 . (4.10) δVs(r ) 40 Chapter 4. Exact Exchange Kohn-Sham Formalism

The response function has been expressed in equation (2.50) and using the notation containing valence (v) and conduction (c) bands it becomes

† † 0 0 0 X φvk(r)φck(r)φck(r )φvk(r ) χ0(r, r ) = + c.c. (4.11) vk − ck cvk

Using (4.10) and multiplying (4.8) by χ0 one gets

Z Z † 0 0 0 0 0 0 X h NL φck(r )φvk(r ) i dr χ0(r, r )Vx(r ) = dr hvk|Vx |cki + c.c. , (4.12) vk − ck vck which is called EXX-equation. Deﬁning the quantity

Z † 0 0 0 X h NL φck(r )φvk(r ) i t(r) = dr hvk|Vx |cki + c.c. (4.13) vk − ck vck as the “right hand side”, equation (4.12) takes the form Z 0 0 0 dr χ0(r, r )Vx(r ) = t(r). (4.14)

Since the inverse of the response function is used, its invertibility has to be guaranteed. Changing the potential Vs by a constant leads to a shift of the energy, but leaves the wave functions and the density identical. Therefore equation (4.9) becomes Z 0 0 dr χ0(r, r ) = 0 (4.15) and χ0 cannot be inverted. Hence it is necessary to consider a restricted function space excluding constant inﬁnitesimal variations δρ(r) = δρ0 and δVs(r) = δVs,0. This can be best put into practice by switching to momentum space and neglecting the (G = 0) component of the involved quantities. The analogue of (4.9) in Fourier space reads X 0 0 δρ(G) = χ0(G, G )δVs(G ) (4.16) G0 with the Fourier transformation of χ0 being deﬁned as

1 X 0 0 χ (r, r0) = eiGrχ (G, G0)e−iG r . (4.17) 0 Ω 0 G,G0 The transformation of (4.17) to momentum space reads [62]

−iGr iG0r 0 1 X hvk|e |ckihck|e |vki χ0(G, G ) = . (4.18) Ω vk − ck vck 41

0 Neglecting the G = 0 and G = 0 row and column of the singular matrix χ0, one continues working with a reduced submatrixχ ˜0 and together with a direct transformation of equation (4.8) to reciprocal space the exchange potential can now be written in a compact form

X ˜ 0 ˜† 0 −1 0 Vx(G) = (t(G ) + t (−G ))˜χ0 (G, G ) (4.19) G06=0 with the quantity t˜(G) deﬁned as

1 X hvk|V NL|ckihck|e−iGr|vki t˜(G) = x . (4.20) Ω vk − ck vck

The (G = 0)-component of the exchange potential, which is not taken into account in (4.19), only causes a uniform shift of the exchange potential and its value can therefore be chosen arbitrarily. In this work the value Vx(G = 0) is chosen to be zero. Such a shift merely shifts the zero level of energies, which is arbitrary anyway. In particular, the value of Vx(G=0) does not alter important quantities like eigenvalue differences, i.e. band structures and band gaps, effective masses, etc. By construction, formula (4.19) together with (4.20) provides an exact expression for the exchange potential Vx(G), but one has also to be aware of the fact that for a truly exact construction of the exchange potential via (4.19), the exact values for the orbitals and eigenvalues of the KS-system need to be provided. As an exact expression for the correlation potential is still unknown, correlation effects need to be approximated and in practical calculations one will not be able to get exact values for the orbitals, resp. exact eigenvalues. Therefore Vx(G) can strictly speaking not be known exactly at the moment, but since correlation effects are significantly smaller than classical coulomb and exchange effects (at least for the systems investigated in the framework of this thesis), the orbitals and eigenvalues used for the construction of the exchange potential via (4.19) probably cause only small deviations from the exact exchange potential. The contribution of exchange effects to the total energy was already given in (4.1) for the real space case. The approach to calculate the exchange energy, as 42 Chapter 4. Exact Exchange Kohn-Sham Formalism it has been followed in the works [36,37], is based on the expression 1 X E [ρ] = hvk|V NL|vki = x 2 x vk 1 X = C∗ (G)hk + G|V NL|k + G0iC (G0) (4.21) 2 vk x vk vk,GG0 with the included matrix element given by [89]

∗ 0 NL 0 4π X Cvq(G + G1)Cvq(G + G1) hk + G|Vx |k + G i = − 2 . (4.22) Ω |k − q + G1| vq,G1

The expansion coeﬃcients Cvq(G) of the underlying Fourier transformation have been introduced in section 3.2. Matrix elements of the same type, but combining valence with conduction bands are needed for the construction of the exchange potential (4.20) and are calculated in a similar way to

NL X ∗ NL 0 0 hvk|Vx |cki = Cvk(G)hk + G|Vx |k + G iCck(G ). (4.23) GG0

For the condition q−k = G1, the matrix element (4.22) holds an integrable singularity, which has been treated correctly for the case of a face-centered-cubic (fcc) lattice system in [89]. The calculation of the matrix elements (4.22) has been the most time consuming step for calculations performed in the framework of [36,37] and similar HF-calculations. The required matrix elements can, however, be calculated in a more eﬃcient way. Starting directly from equation (4.1) and separating the k-dependent phase ikr ˜ ˜ ikr e from the periodic part φvk of the Bloch function φvk(r) = φvk(r)e leads to ZZ ˜† ˜ −i(k−k0)r ˜† 0 ˜ 0 i(k−k0)r0 1 X X φ (r)φv0k0 (r)e φ 0 0 (r )φvk(r )e E = − drdr0 vk v k . x 2 |r − r0| vk v0k0 (4.24)

A new quantity Ank,nk0 is introduced

˜† ˜ Ank,n0k0 (r) = φnk(r)φn0k0 (r) (4.25) with the associated Fourier transformations reading

1 X iGr A 0 0 (r) = A 0 0 (G)e (4.26) nk,n k Ω nk,n k G 43

Z 1 −iGr A 0 0 (G) = drA 0 0 (r)e . (4.27) nk,n k Ω nk,n k Inserting (4.25) and (4.26) into equation (4.24) yields

ZZ i(G+k0−k)r † 0 −i(G0+k0−k)r0 1 X X X Avk,v0k0 (G)e Avk,v0k0 (G )e E = − drdr0 . x 2Ω |r − r0| vk v0k0 GG0 (4.28) With the use of the identity

ZZ iGr iG0r0 0 e e 4π drdr = δ 0 , (4.29) |r − r0| |G|2 GG which takes into account translational symmetry and periodic boundary conditions, and together with further simpliﬁcations, a compact form of the exchange energy is obtained

† 2π X X X Avk,v0k0 (G)Avk,v0k0 (G) E = − . (4.30) x Ω |G + k0 − k|2 vk v0k0 G The matrix elements of the nonlocal exchange operator can be calculated in a similar way

ZZ † † 0 0 X φ (r)φv0k0 (r)φ 0 0 (r )φck(r ) hvk|V NL|cki = drdr0 vk v k = (4.31) x |r − r0| v0k0 † 4π X X Avk,v0k0 (G)Ack,v0k0 (G) = − . (4.32) Ω |G + k0 − k|2 v0k0 G

The new expression for the exchange energy (4.30) is computationally more eﬃ- cient than (4.21), but still holds an integrable singularity. A scheme for treating this singularity for any underlying crystal symmetries was presented in [90]. In contrast, the treatment of the matrix elements for the nonlocal exchange operator (4.32) between valence and conduction bands does not contain singularities. The denominator in (4.32) becomes zero for the case k=k0 and G=0, but at † the same time the quantity Ack,v0k0 (G) in the numerator is zero, i.e. those terms are actually not present in (4.32). The reason is that for G=0 the Fourier trans- † formation (4.27) of Ack,v0k0 (G) simply reduces to an integration of the orbitals ˜† ˜ φnk(r)φn0k(r) over space, which is zero due to their orthogonality. Similar to the 44 Chapter 4. Exact Exchange Kohn-Sham Formalism

Table 4.1: Comparison of LDA and EXX eigenvalue gaps (in eV) with experiment. Values are taken from reference [37].

LDA EXX Exp.

L Γ XL Γ XL Γ X Si 1.43 2.56 0.64 2.36 3.46 1.43 2.4 3.34 1.25 Ge 0.13 -0.09 0.75 1.01 1.28 1.34 0.84 1.00 1.3 GaAs 0.89 0.32 1.41 1.93 1.82 2.15 1.85 1.63 2.18

exchange energy, the computational effort for calculating the nonlocal exchange operator matrix elements is significanctly reduced by this new scheme. The EXX scheme introduced has first been applied to three-dimensional semiconductors with an extensive investigation of fundamental physical quantities like band gaps, effective masses and dielectric functions [36, 37, 88]. Here, strongly improved band structures and band gaps quite close to experimental values were obtained. To become a first glimpse of the quality of EXX band gaps in contrast to band gaps gained via LDA calculations, values for the three semiconductors Si, Ge and GaAs are given in Table 4.1. As a further orientation, an EXX band structure of silicon for the LDA and the EXX case (calculated with the software developed in the framework of this thesis) is provided in Fig. 4.1. As usual, the LDA band gaps clearly underestimate the experimental results for reasons already explained in section 2.4. The EXX results for these sytems were quite close to experiment and an additional consideration of correlation effects via LDA or GGA almost had no effect on the band structure. For the investigated systems the neglect of the derivative discontinuity together with an approximate treatment of correlation effects therefore had little effect. A different picture showed up in an application of the EXX scheme to solid noble gases [91]. A comparison to LDA showed improved band gaps, which, however, were still too small compared to experimental data. The increased value for the EXX band gap can be explained by the fact that the exact exchange KS potential is free of self-interactions, which is not the case 45

-5 Energy [eV]

-10

L Γ X K,U Γ

Figure 4.1: Band structure of silicon for the EXX (solid lines) and the LDA (dashed lines) case. in the LDA or GGA. Within the EXX scheme the total energy holds no self- interactions, as can be seen by the expression for the exchange energy

ZZ † † 0 0 1 X X φ (r)φv0k0 (r)φ 0 0 (r )φvk(r ) E [ρ] = − drdr0 vk v k . (4.33) x 2 |r − r0| vk v0k0

The contributions to the exchange energy for v = v0, i.e. the self-interaction contributions, equal the corresponding contributions of the Coulomb energy, but have opposite sign. Since the exchange potential is defined by the functional derivative of the energy with respect to the density, the self-interactions cancel out for the potential as well. As a consequence, the bands are not self-repulsive anymore and become more localized, which causes a lowered energetic value in comparison to the LDA case. Comparing valence with conduction bands, the lowering of the energy is stronger for the valence states, which then leads to an increased value for the band gap. Because the exact exchange KS potential is local multiplicative, this cancellation of self-interaction holds for all bands. This is different for the HF case with its nonlocal exchange potential. All 46 Chapter 4. Exact Exchange Kohn-Sham Formalism electrons are under the influence of the local Hartree potential

N Z † 0 0 X φj(r )φj(r )φi(r) V (r)φ (r) = dr0 . (4.34) H i |r − r0| j=1

The action of the nonlocal HF-exchange potential (2.44) on an orbital φi reads

N Z † 0 0 X φj(r )φj(r)φi(r ) V NL,HF φ (r) = − dr0 . (4.35) x i |r − r0| j=1

For the valence states, the (i=j)-terms in (4.34) and (4.35) cancel out, whereas this is not the case for the conduction bands. The HF potential therefore is self-interaction free for the occupied states, but not for the unoccupied states, which are then, as a consequence, energetically raised. This leads to the usual overestimation of band gaps in HF theory. The implementation for the three-dimensional semiconductors used in [36,37] explicitly exploited internal symmetries of the investigated periodic systems by using the concept of special k-points. This strategy obviously leads to a reduced computation time, but restricts the developed software to a limited set of systems that can be investigated. Besides the mentioned systems of three-dimensional semiconductors and solid noble gases, it is desirable to investigate other types of systems. The EXX scheme has not been applied to one- or two-dimensional electronic systems so far and in the framework of this thesis an extended program that can also handle systems without inhering symmetry has been developed. As a test system for organic one-dimensional periodic systems, trans-polyacetylene has been investigated [57]. The results of this investigation are shown in chapter 9. Chapter 5

Spin-Current Density-Functional Theory

The methods presented in the previous chapters are based on the electronic ground state density as the only basic variable and turned out to be a pow- erful tool for investigating electronic ground state properties. For the treatment of magnetic properties and properties related to internally induced currents of an electronic system, the standard DFT approach is not sufficient, because these properties are related to currents and to the electron spin, rather than the electron density alone. If the magnetic field is not strong enough to induce orbital currents, the magnetism can directly be included in the Hamiltonian via an effective magnetic field. This “spin-only” theory is usually referred to as spin density-functional theory (SDFT) [49]. If the magnetic field becomes sufficiently strong to induce orbital currents, an extended approach has to be applied. A first attempt for this was introduced about 20 years ago by Vignale and Rasolt [52,53]. This theory is called current spin density-functional theory (CSDFT). However, in practice the CSDFT approach did not play an important role, since good approximations for required exchange-correlation functionals are not at hand. For instance, the LDA within CSDFT leads to an associated functional unveiling a pathological nature [54]. In this chapter, the CSDFT scheme will be extended to a new scheme, denoted as spin-current density-functional theory (SCDFT) [55], in which, besides the usual spin-density and the paramagnetic density current, additional spin-currents

47 48 Chapter 5. Spin-Current Density-Functional Theory are included. In chapter 6, as a second step, the exact-exchange scheme of chapter 4 will be employed within the new SCDFT formalism.

5.1 N-electron system in a magnetic ﬁeld

The Hamiltonian operator for a system of N electrons subjected to a magnetic ﬁeld B is given by [92]

N X 1 2 1 1 H = p + A(r ) + v (r ) + σ · B(r ) + σ · [(∇v )(r ) × p ] + 2 i i ext i 2 i 4c2 ext i i i=1 N X 1 + . (5.1) |r − r | i

The vector potential A enters the Hamiltonian via the kinetic momentum p+

A(ri) [93], while the magnetic ﬁeld B couples to the spin and contributes to the 1 Hamiltonian by a term of form 2 σ · B. The vector σ is built out of the Pauli spin matrices deﬁned by       0 1 0 −i 1 0 σx =   , σy =   , σz =   . (5.3) 1 0 i 0 0 −1

The last term of the ﬁrst summation in (5.1) 1 HSO = σ · [(∇v ) × p] (5.4) 4c2 ext denotes the general, i.e. independent of any system symmetries, spin-orbit (SO) coupling term that in case of a spherically symmetric potential vext reduces to 1 1 dv HSO = ext L · S ≡ ξ(r)L · S. (5.5) 4c2 r dr

The remaining terms in (5.1) mark the external potential vext and the Coulomb interaction N X 1 V = , (5.6) ee |r − r | i

N X 1 H = (p2 + p · A(r ) + A(r ) · p + A(r ) · A(r )) + v (r ) + 2 i i i i i i i ext i i=1 N 1 1 X 1 + σ · B(r ) + σ · [(∇v )(r ) × p ] + . (5.7) 2 i 4c2 ext i i |r − r | i

N X ρ(r) ≡ J0(r) = δ(r − ri), (5.8) i=1 the operator of the paramagnetic current density

N 1 X j(r) = p δ(r − r ) + δ(r − r )p , (5.9) 2 ν,i i i ν,i i=1 as well as the spin-magnetization operator

N 1 X m(r) = − σδ(r − r ) (5.10) 2 i i=1 are introduced. Note that the paramagnetic current density operator j(r) is not invariant under gauge transformations and does not represent the real, physical current density operator jph. The physical current is actually given by

jph(r) = j(r) + ρ(r)A(r) (5.11) and satisfies the associated continuity equation ∂ρ(r) ∇j (r) + = 0. (5.12) ph ∂t The reason for defining the paramagnetic current j is that it can be calculated directly from the wave function without any knowledge of the magnetic field or its vector potential, while jph cannot. Next, a new operator smn is introduced via the relation N 1 X s (r) = − σ [p δ(r − r ) + δ(r − r )p ] (5.13) mn 4c2 m n,i i i n,i i=1 50 Chapter 5. Spin-Current Density-Functional Theory with the subscripts m and n taking the values 1, 2, 3. Representing a combination of the operators (5.9) and (5.10), the operator smn holds nine parameters named spin-current densities, i.e. paramagnetic currents of the spin-density or the magnetization. Introducing the definitions (5.8), (5.9), (5.10) and (5.13) into the Hamiltonian (5.7), one obtains Z Z Z H = T + Vee + dr ρ(r)vext(r) − dr m(r)B(r) + dr j(r)A(r) + 1 Z Z X + dr ρ(r)A2(r) + dr s (r)V . (5.14) 2 mn SO,mn m,n The newly introduced potential   vext,z(r) vext,y(r) 0 − 4c2 4c2   V (r) = V (r) =  vext,z(r) vext,x(r)  (5.15) SO SO,mn  4c2 0 − 4c2    vext,y(r) vext,x(r) − 4c2 4c2 0 determines the spin-orbit term. Formally, it is also possible to treat the quantities smn and VSO,mn as vectors with superindex mn. For the sake of a consistent notation with all other integrals in (5.14), the integrand of the last integral is then written as a scalar product Z X Z dr smn(r)VSO,mn ≡ dr s(r)VSO(r). (5.16) m,n It is now possible to switch to a compact form of the Hamiltonian (5.14). For this purpose the two vectors Σ = (1, σx, σy, σz) and J = (J0, j), each holding four entries, together with the 4 × 4-matrix V(r) defined by

 A2(r)  vext(r)+ 2 Ax(r) Ay(r) Az(r)    Bx(r) vext,z(r) vext,y(r)   2 0 − 4c2 4c2  V(r) = Vµν(r) =   (5.17)  By(r) vext,z(r) vext,x(r)   2 4c2 0 − 4c2    Bz(r) vext,y(r) vext,x(r) 2 − 4c2 4c2 0 are introduced. The index µ labels the rows, while the index ν indicates the columns of V. The Hamiltonian (5.14) then takes the form Z T H = T + Vee + dr Σ V(r) J(r). (5.18) 5.1. N-electron system in a magnetic ﬁeld 51

By explicitly writing out expression (5.18) via matrix multiplication and carrying out the spatial integration that eliminates the delta functions contained in the operators (5.8), (5.9), (5.10) and (5.13), all terms in (5.7) are reproduced.

Multiplication of the submatrix Vµν (µ, ν = 1, 2, 3), i.e. VSO,mn from the left with the vector Σ˜ = (σx, σy, σz) and from the right with the vector j reproduces the spin-orbit contribution. The first wing V0n (n=1, 2, 3) of the matrix V yields the mixed terms p·A(r) and A(r)·p, while the second wing Vm0 (m = 1, 2, 3) 1 reproduces the coupling of the magnetic field to the spin, i.e. 2 σ·B. The V00- component yields the external potential vext together with the quadratic term of the vector potential A. The representation (5.17) holds the advantage that parts of the potential V can easily be switched on and off for a selfconsistent calculation. That way special cases like SDFT or CSDFT are contained within the new formalism and can easily be reproduced. With the definitions (5.9), (5.10) and (5.13) the set of basic variables has been extended to a system of 16 quantities that can be captured in a 4 × 4- matrix ρµν(r). With the subscripts m and n taking the values 1, 2, 3, one can identify the physical variables

ρ00(r) = ρ(r) (5.19)

ρm0(r) = 2m(r) (5.20)

ρ0n(r) = j(r) (5.21)

ρmn(r) = smn(r). (5.22)

The associated expectation values with respect to the many-body ground state wave function |Ψi are given by

ρµν(r) = hΨ|ΣµJν(r)|Ψi. (5.23)

In accordance to the Hamiltonian (5.14) the total energy of the system can now be expressed as X Z E = T + Vee + drVµν(r)ρµν(r), (5.24) µν which has a similar form to the energy expression of standard DFT. With a treatment of the matrix V as vector with superindex µν, the energy can also be 52 Chapter 5. Spin-Current Density-Functional Theory written as Z T E = T + Vee + drV (r)ρ(r). (5.25)

Similar to the case of a system of N interacting electrons not subjected to an external magnetic ﬁeld (see chapter 2), the physical problem described by (5.1) is not exactly solvable, because of the electron-electron interaction coupling all electrons. It is therefore desirable to adopt and extend the strategy of conventional DFT to tackle this problem. This means that one needs to focus on the

16 components of the spin-current density ρµν as central parameters, i.e. formulate an analogon of the HK-Theorem and the Constrained-Search formulation for the SCDFT case. Next, these parameters shall be installed into a modiﬁed KS-scheme. An extension of the conventional DFT-scheme, depending on the seven variables of electron density ρ00, the x-, y-, z-components of the spin-density ρm0 and the x-, y-, z-components of the paramagnetic density current ρ0n has already been introduced by Vignale and Rasolt [52,53]. In the framework of these works, an extended HK-Theorem has been formulated. The formalism on which this thesis is based, however, holds nine additional spin-current parameters ρmn and therefore the HK-Theorem and the Constrained-Search formulation have to be further extended.

5.2 HK-Theorem and Constrained-Search For- mulation for SCDFT

With the basic variables collected in ρµν=(b ρ, j, m, s), the ﬁrst theorem can be stated in analogy to section 2.1 as

Theorem 1 The ground state wave function Ψ(r1, ..., rN ) of an N-particle system subjected to an external potential vext(r), including the associated spin-orbit contribution, an external magnetic ﬁeld B(r) and its vector potential A(r) is a uniquely determined function Ψ[ρµν] of the 16 spin-current densities ρµν(r).

For the CSDFT-case, i.e. the case of seven basic variables, this theorem has been proven by Vignale in [53]. Note that the paramagnetic current is not invariant 5.2. HK-Theorem and Constrained-Search Formulation for SCDFT 53 under gauge transformations, i.e. the addition of the gradient of an arbitrary function to the vector ﬁeld A(r). However, none of the physical observables are changed due to such a gauge transformation. Here, the gauge is determined by the choice of the external vector potential A(r). For a strict gauge invariance of the presented approach a gauge invariant basis set would be required. Plane waves, which are employed as basis functions here, are not gauge invariant. How- ever, the number of plane waves used for calculations carried out within the framework of this thesis was always chosen large enough, such that the resulting plane wave basis can be considered complete enough to guarantee almost perfect gauge invariance of all observables. A version of the Hohenberg-Kohn theorem holding for the physical current density has been proven in [94]. Similar to (2.5) a new Hohenberg-Kohn functional can be deﬁned, now additionally depending on j(r), m(r) and s(r) rather than the density ρ(r) alone

F [ρµν] = T [ρµν] + Vee[ρµν] = hΨ[ρµν]|T + Vee|Ψ[ρµν]i. (5.26)

Hereby, the wave function Ψ[ρµν] is deﬁned by the minimization

F [ρµν] = Min hΨ[ρµν]|T + Vee|Ψ[ρµν]i (5.27) Ψ→ρµν (r) and again all intrinsic properties of the electronic system are completely absorbed in the functional (5.26). The total energy functional becomes

Z 1 Z E[ρ ] = F [ρ ] + dr ρ(r)v (r) + A2(r) − dr m(r)B(r) + µν µν ext 2 Z Z + dr j(r)A(r) + dr s(r)VSO(r) = (5.28) X Z = F [ρµν] + drVµν(r)ρµν(r). (5.29) µν

The second theorem handles the variational principle and can be formulated as

Theorem 2 The functional E[ρµν] of the total energy satisﬁes a variational principle with respect to the 16 quantities ρµν, i.e. the total energy reaches its minimal value E0 for the correct ground state values of the spin-current density ρµν,0 :

E0 = Min E[ρµν] = E[ρµν,0]. (5.30) ρµν →N 54 Chapter 5. Spin-Current Density-Functional Theory

The notation “ρµν → N” indicates a variation over all ground state spin-current densities ρµν of arbitrary N-electron systems. Similar to the generalization of the Hohenberg-Kohn Theorems, the constrained- search formulation introduced in section 2.2 can be extended to the spin-current case. The Hohenberg-Kohn functional F [ρµν] is then deﬁned identically to (2.9) via the variation

F [ρµν] = Min hΨ[ρµν]|T + Vee|Ψ[ρµν]i. (5.31) Ψ→ρµν (r) In contrast to standard DFT, the minimization (5.31) now runs over all wave functions yielding the spin-current densities ρµν, i.e., the wave functions are constrained not only to yield a given reference electron density ρ00, but to yield all

16 components of the spin-current density ρµν. The minimizing wave function

Ψ[ρµν] in (5.31) is a functional of the spin-current density ρµν. The minimizing wave function Ψ[ρµν,0] for the ground state spin-current density ρµν,0 is the ground state wave function Ψ0 of the system, Ψ0 = Ψ[ρµν,0], i.e., in case of the presence of magnetic ﬁelds, the ground state spin-current density determines the ground state wave function via the constrained-search (5.31) and thus all ground state properties of the system. The latter statement represents the extended Hohenberg-Kohn theorem formulated above. A Kohn-Sham wave function

Φ[ρµν] corresponding to a wave function Ψ[ρµν] is deﬁned as the minimizing wave function in the constrained search minimization

Ts[ρµν] = Min hΨ[ρµν]|T |Ψ[ρµν]i (5.32) Ψ→ρµν (r) yielding the functional Ts[ρµν] of the noninteracting kinetic energy. The given extension to the HK-Theorem and the constrained-search formulation now legitimizes the use of the spin-current density ρµν as basic variable, but again one needs a practically realizable scheme. Therefore, the Kohn-Sham scheme, which showed to be very successful within conventional DFT, shall be expanded to systems under the inﬂuence of a magnetic ﬁeld.

5.3 Kohn-Sham Scheme for SCDFT

The real, interacting electronic system shall again be mapped on an artiﬁcial KS system consisting of noninteracting electrons, with both systems now gaining the 5.3. Kohn-Sham Scheme for SCDFT 55

same spin-current densities ρµν for the ground state. As a result, one desires to gain decoupled single-particle equations (for the single-particle states building the many-electron KS wave function in form of a single Slater determinant) that can be handled in a simple way. The total energy can be written in the modiﬁed way

Z 1 E[ρ ] = T [ρ ] + U[ρ ] + E [ρ ] + dr ρ(r)v (r) + A2(r) − µν s µν µν xc µν ext 2 Z Z Z − dr m(r)B(r) + dr j(r)A(r) + dr s(r)VSO(r) (5.33)

with the exchange-correlation functional Exc deﬁned as

Exc[ρµν] = T [ρµν] − Ts[ρµν] + Vee[ρµν] − U[ρµν]. (5.34)

The quantities T and Ts denote the kinetic energies of the interacting and noninteracting system, respectively. Vee is the electron-electron interaction energy with its classical Coulomb contribution U. Again, the main part of the kinetic energy, i.e. Ts, is expressed in a way that can be calculated exactly. All remaining unknown energy contributions are absorbed in the exchange-correlation functional

Exc. Variation of the total energy with respect to the orbitals φi yields a new set of single particle Kohn-Sham equations

1 1 1 p+A (r)2+v (r)+ σ·B (r)+ σ·[(∇v )(r) × p] φ (r) = φ (r) (5.35) 2 s s 2 s 4c2 ext i i with the associated Kohn-Sham potentials reading

1 v (r) = v (r) + V (r) + V (r) + A2(r) − A2(r) (5.36) s ext,0 H xc 2 0 s As(r) = A0(r) + Axc(r) (5.37)

Bs(r) = B0(r) + Bxc(r) (5.38)

VSO,s(r) = VSO,0(r) + VSO,xc(r). (5.39)

The variation for the seven basic variables of CSDFT has explicitly been carried out in [53]. A generalization to the new SCDFT scheme including all 16 quantities

ρµν is straightforward and can be performed in a similar way. The 16 associated exchange-correlation potentials are then given by the functional derivatives of the 56 Chapter 5. Spin-Current Density-Functional Theory exchange-correlation energy

δE V (r) = xc (5.40) xc δn(r) δE A (r) = xc (5.41) xc δj(r) δE B (r) = − xc (5.42) xc δm(r) δE V (r) = − xc . (5.43) SO,xc δs(r)

Following the same procedure as it has been applied to the Hamiltonian (5.1), the presented Kohn-Sham formalism can as well be expressed in a more compact way. The KS Hamiltonian hereby reads Z T Hs = Ts + dr Σ Vs(r)J(r) (5.44)

with the associated KS potential Vs given by

Vs(r) = V(r) + VH (r) + Vxc(r). (5.45)

The single particle KS equations (5.35) can therefore be written as Z T Ts + dr Σ Vs(r)J(r) φi(r) = φi(r). (5.46)

The potential V(r) has already been deﬁned in (5.17) and the Coulomb potential

U, together with the exchange-correlation potential Vxc, can be summarized in a 4 × 4-matrix as well   VH (r) + Vxc,00(r) Vxc,01(r) Vxc,02(r) Vxc,03(r)      Vxc,10(r) Vxc,11(r) Vxc,12(r) Vxc,13(r) VH (r) + Vxc(r) =  . (5.47)    Vxc,20(r) Vxc,21(r) Vxc,22(r) Vxc,23(r)   Vxc,30(r) Vxc,31(r) Vxc,32(r) Vxc,33(r)

The set of equations (5.40)-(5.43) can be compactiﬁed to

δExc Vxc,µν(r) = . (5.48) δρµν(r) 5.3. Kohn-Sham Scheme for SCDFT 57

The functional derivatives (5.48) uniquely deﬁne the exchange-correlation potentials Vxc,µν and like in the conventional Kohn-Sham formalism, it is assumed that they exist. The exchange part can be separated from the correlation part

δEx δEc Vx,µν(r) = ; Vc,µν(r) = . (5.49) δρµν(r) δρµν(r)

The ground state wave function |Ψ0i of the noninteracting system is expressed by a Slater determinant built out of the KS orbitals φi gained by a self-consistent solution of the KS-equations (5.35). The expectation values for the spin-current densities (5.23) then become a sum over all occupied states with the form occ X ρµν(r) = hφa|ΣµJν|φai. (5.50) a At this point, a reasonable and practically realizable scheme for treating magnetic and spin effects in a many-particle system is at hand and similar to standard DFT the quest for an accurate approximation of the exchange-correlation functionals Vxc,µν has become the actual problem. Consequently, the first attempt to tackle this problem should be an application of the LDA scheme to the new SCDFT-KS formalism developed above. For the CSDFT case with its seven basic variables, an attempt has been performed in [54]. Therein, the authors state that in the case of a uniform electron liquid, the exchange-correlation part of the energy functional only depends on the paramagnetic current density via the vorticity, which is defined by j(r) ν(r) = ∇ × . (5.51) ρ(r) The resulting energy functional can thus be written in the general form Z CSLDA CSLDA Exc [ρ, j] = drρ(r)xc (ρ(r), ν(r)). (5.52) The resulting potential for the state of a uniform spin density and vorticity was computed and plotted against the magnetic field. The result showed a discontinuous behavior of the potential at Landau subband edges, which naturally gives rise to problems in the usage of the underlying energy functional. A practical use of functionals depending explicitly only on the density and the paramagnetic current therefore seem to be problematic, but one can hope that a use of orbital-dependent functionals can solve this problem. Therefore, an approach for an exact treatment of the exchange part will be given in the next chapter. 58 Chapter 5. Spin-Current Density-Functional Theory Chapter 6

Exact Exchange Kohn-Sham Formalism for SCDFT

In chapter 4 the exact exchange Kohn-Sham formalism within the framework of conventional DFT was introduced. As already mentioned, the application of this scheme to various physical systems gained good results for band structures and other physical quantities. Therefore, the transfer of exact exchange methods to the new SCDFT formalism presented in chapter 5 seems desirable and a promising approach. This transfer can be put into practice in a straightforward way.

The exact exchange energy Ex retains the form of (4.1), but now depends on the 16 component vector ρµν, rather than the electronic ground state density ρ alone

ZZ † † 0 0 1 X X φ (r)φv0k0 (r)φ 0 0 (r )φvk(r ) E [ρ ] = − drdr0 vk v k . (6.1) x µν 2 |r − r0| vk v0k0

The orbitals φvk(r) are solutions of the SCDFT-KS equations (5.35) and represent two-component spinors. The integral equation for a component of the 16 component vector potential Vx,µν(r) is obtained by taking the functional derivative of the exchange energy (6.1) with respect to the associated ρµν via double application of the functional chain rule

ZZ 0 00 δEx[ρµν] X X 0 00hδEx[ρµν] δφvk(r ) iδVs,κλ(r ) Vx,µν(r) = = dr dr 0 00 + c.c. . δρµν(r) δφvk(r ) δVs,κλ(r ) δρµν(r) κλ vk (6.2)

59 60 Chapter 6. Exact Exchange Kohn-Sham Formalism for SCDFT

The functional derivatives contained in (6.2) can in principle be obtained similarly to chapter 4. The ﬁrst derivative δEx[ρµν]/δφvk is again obtained by a straightforward diﬀerentiation of the exchange energy (6.1) with respect to the orbital φvk

Z † 0 † δEx[ρµν] X φv0k0 (r )φvk(r1)φv0k0 (r1) 0 = − dr1 0 . (6.3) δφvk(r ) |r − r1| v0k0

The derivative of the state φvk with respect to a component of the 16-component

KS-potential Vs,µν is obtained by ﬁrst order perturbation theory. This yields the 16 equations

0 00 δφvk(r ) X 0 hφn00k0 |ΣµJν(r )|φvki 00 00 00 = φn k (r ) . (6.4) δVs,µν(r ) vk − n00k0 n00k0(6=vk)

NL The nonlocal exchange operator Vx remains identical as given in (4.6)

† 0 X φv0k0 (r)φ 0 0 (r ) hr|V NL|r0i = V NL(r, r0) = − v k . (6.5) x x |r − r0| v0k0

Inserting identities (6.3), (6.4), and (6.5) into (6.2) and proceeding with the calculation exactly the same way as in chapter 4 one obtains a real space representation of the 16 exact exchange potentials

Z 0 0 X 0 X h NL hφck|ΣκJλ(r )|φvki iδVs,κλ(r ) Vx,µν(r) = dr hvk|Vx |cki +c.c. . (6.6) vk − ck δρµν(r) κλ vck

The sum over the superindex κλ in (6.6) contains 16 terms. The change of the component ρµν under the influence of an infinitesimal change in a component of the local KS potential Vs,κλ can be expressed via linear response theory. The µν,κλ 16×16 component response function χ0 is defined by Z X 0 µν,κλ 0 0 δρµν(r) = dr χ0 (r, r )δVs,κλ(r ). (6.7) κλ

00,00 The component χ0 hereby represents the conventional KS-response function µν,κλ deﬁned by (4.9). The newly deﬁned response function χ0 can be written as

µν,κλ 0 δρµν(r) χ0 (r, r ) = 0 (6.8) δVs,κλ(r ) 61 with its inverse equal to the last functional derivative in (6.2). An application of ﬁrst order perturbation theory to the SCDFT-KS equations (5.46) yields a real space representation of the response function

0 µν,κλ 0 X hφvk|ΣµJν(r)|φckihφck|ΣκJλ(r )|φvki χ0 (r, r ) = + c.c. (6.9) vk − ck cvk

Note, that for µν = κλ = 0, equation (6.9) reproduces the conventional KS response function (4.11), because in that case the matrix elements of form † hφvk|ΣµJν(r)|φcki reduce to the orbital product φvk(r)φck(r). Using the inverse µν,κλ of the response function χ0 given in (6.8), equation (6.6) can be reduced to

Z Z 0 X 0 µν,κλ 0 0 0 X h NL hφck|ΣµJν(r )|φvki i dr χ0 (r, r )Vx,κλ(r ) = dr hvk|Vx |cki +c.c. , vk − ck κλ vck (6.10) which represents the EXX-equation of SCDFT. Following the approach of chapter

4, the 16-component vector tµν is deﬁned as

Z 0 0 X h NL hφck|ΣµJν(r )|φvki i tµν(r) = dr hvk|Vx |cki + c.c. (6.11) vk − ck vck and shall again be called “right hand side”. The integral equation (6.10) can therefore be written as Z X 0 µν,κλ 0 0 dr χ0 (r, r )Vx,κλ(r ) = tµν(r) (6.12) κλ with the sum over κλ consisting of 16 terms. The 16 terms of the exchange potential can be calculated via Z X 0 −1 µν,κλ 0 0 Vx,µν(r) = dr (χ0 ) (r, r )tκλ(r ). (6.13) κλ

To provide a complete picture of this chapter’s formalism, the considered quantities shall also be treated in momentum space. The change of a component ρµν in momentum space, i.e. the Fourier transformation of equation (6.8) is given by

X µν,κλ 0 0 δρµν(G) = χ0 (G, G )δVs,κλ(G ). (6.14) G0 62 Chapter 6. Exact Exchange Kohn-Sham Formalism for SCDFT

µν,κλ A connection between the real space representation of the response function χ0 and its counterpart in momentum space is given by

1 X 0 0 χµν,κλ(r, r0) = eiGrχµν,κλ(G, G0)e−iG r . (6.15) 0 Ω 0 G,G0

The Fourier transformation of (6.9) then yields

0 µν,κλ 0 1 X hvk|ΣµJν(G)|ckihck|ΣκJλ(G )|vki χ0 (G, G ) = , (6.16) Ω vk − ck vck which reduces to the KS response function of conventional DFT (4.18), when µν = κλ = 0. The prefactors in equations (4.18) and (6.16) diﬀer by a factor of 2, because of an explicit inclusion of the spin. A straightforward Fourier transformation of the exchange potentials Vx,µν (6.13) to momentum space yields

X X ˜ 0 ˜† 0 −1 µν,κλ 0 Vx,µν(G) = (tκλ(G ) + tκλ(−G ))(χ0 ) (G, G ) (6.17) κλ G06=0 with the quantity t˜µν deﬁned by

NL 1 X hvk|Vx |ckihck|ΣµJν(G)|vki t˜µν(G) = . (6.18) Ω vk − ck vck

In order to construct the 16 exchange potentials given by (6.17), the inverse of µν,κλ the response functions χ0 are required, so that their invertibility has to be guaranteed. Like in the conventional EXX case of chapter 4 singularities, i.e. zero eigenvalues can occur. For the conventional EXX case, this problem came up for the (G=0)-component and was resolved by simply neglecting these terms, i.e. working with a reduced response function. Similarly, one gets rid of the singularity problem for the SCDFT case by neglecting all G-components with a zero eigenvalue. It has, however, to be guaranteed that the same G-components of the associated right hand side have zero values as well. For further comments and details on the technical realization of this treatment, see section 8.6 of the implementation part of this thesis. For the calculation of the response function (6.16) and the 16 components of the exchange potential (6.17), several matrix elements have to be calculated. A scheme for the calculation of the nonlocal exchange operator matrix element 63

NL hvk|Vx |cki in (6.18) has already been deduced in chapter 4. The only new quantity, which is required for the calculation of the exchange potentials Vx,µν µν,κλ and the response functions χ0 is the matrix element of form hck|ΣµJν(G)|vki.

The action of the operator Σµ = (1, σx, σy, σz) on a spinor |nki resolves in a projection of its x-, y- or z-component (µ = 1, 2, 3) or simply leaves the spinor untouched (µ=0). With the spinor |nki consisting of two components

↑ ↓ 2 2 |nki = α+|nk i + α−|nk i (|α+| + |α−| = 1), (6.19)

↑↓ ↑↓ the four types of matrix elements hck |ΣµJν(G)|vk i can therefore be con- ↑↓ ↑↓ structed out of the four types of matrix elements hck |Jν(G)|vk i. The Fourier transformation for the 4 components Jµ reads 1 Z J (G) = drJ (r)e−iGr. (6.20) µ Ω µ

Inserting the particle density operator J0(r) given by equation (5.8) into equation (6.20) and carrying out the spatial integration yields occ X −iGrv J0(G) = e . (6.21) v For this case, the matrix element for one of the summands in (6.21) has already been given by equation (3.43) as

−iGr X ∗ 0 0 hck|e |vki = Cck(G )Cvk(G + G ). (6.22) G0 The Fourier transformation (6.20) of the paramagnetic current operator (5.9), i.e. Jµ with µ=1, 2, 3, is calculated to occ 1 X ∂ ∂ j(G) = − e−iGrv + e−iGrv . (6.23) 2 ∂r ∂r v v v The related matrix element is provided by X 1 hck|j(G)|vki = ( G0 + G + k)C∗ (G0)C (G + G0). (6.24) 2 ck vk G0

The last quantity to be treated is the exchange energy Ex. Its calculation remains similar to the conventional EXX case. It is carried out via a relation similar to equation (4.30) † 2π X X X Avk,v0k0 (G)Avk,v0k0 (G) E = − . (6.25) x Ω |G + k0 − k|2 vk v0k0 G 64 Chapter 6. Exact Exchange Kohn-Sham Formalism for SCDFT

The quantity Avk,v0k0 (G) hereby represents the product of the periodic part of the Bloch function. Note again that the orbitals now are two-component spinors and depend on the spin. Part II

Implementation

Chapter 7

The Program

7.1 Overview

In order to apply the formalism presented in Part I, a new software has been developed within the framework of this thesis, which shall be referred to as program SCEXX. This software has more or less been implemented from scratch and is capable to calculate various ground state properties of periodic systems in one- to three dimensions as they have been described in Part I. Some applications of the developed methods will be discussed in Part III. The present chapter shall give a brief overview over the main structure of the developed program, which solves the KS equations in a self consistent manner. Further details on the implementation can be found in chapter 8. A top-level description of the program, i.e. the structure of the main routine is shown in Fig. 7.1. As usual, the required input data is read from a regular textfile. The most important parameters to specify the calculation to be performed are the coordinates of the unit cell vectors, the coordinates of the atoms constituting the investigated system and the two energy cutoffs for the wave functions and the response function. Further input specifies, e.g. the type of applied exchange- correlation potentials, magnetic field strength or the fineness of the k-point mesh covering the first Brillouin zone. In a second step, quantities that do not depend on the coordinates of a specific k-point are calculated. These parameters are calculated only once before the self consistent field (SCF) loop and are stored in associated arrays throughout the

67 68 Chapter 7. The Program execution of the program. Examples for k-point independent quantities are the local part of the pseudopotential (3.41) or the matrix elements in the denominator of formula (3.40) for the separable, nonlocal pseudopotential. The following SCF loop shall then solve the KS equations in a self-consistent way. It starts with a loop over the set of k-points. In each loop the Hamiltonian is built and diagonalized as a first step. A detailed description of the associated procedure is given in section 8.1. With the orbitals and their associated eigenvalues at hand, the spin-current densities and the response function are updated for each k-point (see sections 8.2 and 8.3 for further details). The spin-current densities are then transformed to momentum space and using its 00-component, i.e. the regular electron density, the Hartree potential can be calculated quite easily (see section 8.4). Furthermore, exchange- and correlation potentials only depending on the electronic density, resp. derivatives of it, can be calculated via the procedures described in section 8.5. The next step in the program sequence is the calculation of the right hand side of the (SC)EXX-equation. This is put into practice by a double loop over the considered k-point set, in which the required quantities are first calculated in their real space representation (see section 8.6 and Fig. 8.8). After being constructed in real space within the double loop, the right hand side is transformed to momentum space and the exact exchange potential can be calculated by a matrix multiplication with the already computed response function. After the SCF loop, various output files that contain converged quantities of interest are written. Additionally, the program calculates plotting data, e.g. to display the behavior of potentials throughout the unit cell. With the converged KS potential at hand, it is also possible to calculate and plot band structures on certain paths through the first Brillouin zone.

7.2 Technical details

In order to exploit features like dynamical memory management, the program was exclusively implemented in the programming language FORTRAN90. In case of linear algebra operations, routines of the BLAS- [95, 96] and LAPACK [97] libraries were used. All Fourier transformations were performed using the C 7.2. Technical details 69

SCEXX — Program sequence

• Readin input data

• Build k-point independent quantities

SCF loop

loop over k-points

• Build and diagonalize Hamiltonian matrix H(G+k, G0 +k)

• Update spin-current densities ρµν (r) in real space

µν,κλ 0 • Update response functions χ0 (G, G ) in momentum space

• Transform spin-current densities to momentum space → ρµν (G)

• Calculate Hartree potential VH (G) in momentum space

• Calculate LDA- and GGA potentials Vxc(G) in momentum space

double loop over k-points

• Calculate rhs tµν (r) of (SC)EXX equation in real space

• Transform rhs to momentum space → tµν (G)

• Calculate (SC)EXX potential Vexx,µν (G) in momentum space

• Write outputﬁles

• Generate plotting data

• Calculate band structures

Figure 7.1: Top-level description of program SCEXX. subroutine library FFTW [98]. The serial version of the software was parallelized using the Message Passing Interface (MPI) communications protocol [99]. The parallelization exclusively takes place on the top-level of the program, i.e. during the main routine sequence. For this purpose the set of k-points is divided between the processors and the k-point loop, as well as the double loop over k-points indicated in Fig 7.1 are processed in a parallel manner. Each processor calculates the required quantities for its assigned set of k-points. After all processors have passed their individual 70 Chapter 7. The Program loop, the slave processors send their interim results to the master processor that sums up everything and sends the final result back to the slaves. Since only the main routine is comprised in the parallelization and no further subroutines are touched, the described parallelization procedure holds the advantage of being quite clear and simple. Furthermore, the chosen parallelization scheme turned out to be very effective, because almost the whole computational workload is located in the mentioned k-point loops. A diagnosis via profiling tools showed that steps between the k-point loops only require very little computation time. The sequence of the parallelization scheme is explicitly shown in Fig. 7.2. 7.2. Technical details 71

SCEXX — Parallelization

. . .

SCF loop

• divide k-points between processors

loop over k-points Z Z each processor calculates quantities for its assigned k-point set Z . . • Diagonalize H ......

• Update ρµν

µν,κλ • Update χ0

µν,κλ • Slaves send their interim results for ρµν and χ0 to master, master sums up and sends ﬁnal result back to slaves

. . .

• divide k-points between processors

loop over k-points Z Z each processor loops over all k0-points Z

. . • Calculate tµν ......

• Slaves send their interim results for tµν to master, master sums up and sends ﬁnal result back to slaves

. . .

Figure 7.2: Parallelization of program SCEXX. 72 Chapter 7. The Program Chapter 8

The SCF Cycle

The following sections shall give a detailed description of individual parts of the implemented program. For this purpose, the relevant formulas are recapitulated. A detailed development of the formalism, i.e. a derivation of the used formulas can be found in Part I.

8.1 Hamiltonian Operator

The KS Hamiltonian has explicitly been treated in chapter 5 and in its compact real space form reads Z T Hs = Ts + dr Σ Vs(r)J(r). (8.1)

During the execution of the program the Hamiltonian has to be built and diagonalized in its momentum space representation for each individual k-point

0 0 KB 0 Hs(G+k, G +k) = Ts(G+k)+Vs(G+k, G +k)+∆V (G+k, G +k). (8.2)

The kinetic energy Ts exclusively contributes to the diagonal parts of the Hamiltonian matrix (8.2) and reads

1 2 T (G + k) = (G + k) δ 0 . (8.3) s 2 G,G The second contribution to the Hamiltonian operator are the matrix elements of the local KS potential Vs in the chosen plane wave basis

0 T 0 Vs(G + k, G + k) = hG + k|Σ VsJ|G + ki. (8.4)

73 74 Chapter 8. The SCF Cycle

The matrix elements (8.4) can be constructed from the matrix element containing the regular density ρ

0 0 hG + k|VsJ0|G + ki = Vs(G − G ) (8.5) and the matrix element containing the paramagnetic current density operator j 1 hG + k|V j|G0 + ki = (G + G0) + kV (G − G0). (8.6) s 2 s

If the Fourier coeﬃcients of the local KS potential Vs are known, the Hamiltonian (8.2) can be set up via the previous equations. The complete local KS potential

Vs comprises the contribution of the external magnetic ﬁeld B, the spin-orbit ps contribution, the local part of the pseudopotential Vloc, the Hartree potential

U and the exchange-correlation potential Vxc. The contribution of the local pseudopotential, as well as the contribution of the optional magnetic ﬁeld are k- point independent and calculated before the SCF loop. The local pseudopotential part is calculated via 4π Z V ps(G) = e−iGr drrV (r) sin(Gr) (8.7) loc ΩG loc and can be merged with the magnetic ﬁeld contribution to

 ps A2(G)  Vloc(G)+ 2 Ax(G) Ay(G) Az(G)    Bx(G) 0 − vext,z(G) vext,y(G)  ˜  2 4c2 4c2  Vloc(G) =   . (8.8)  By(G) vext,z(r) vext,x(r)   2 4c2 0 − 4c2    Bz(G) vext,y(G) vext,x(G) 2 − 4c2 4c2 0 The local KS potential can therefore be written as

Vs(G) = V˜ loc(G) + VH (G) + Vxc(G). (8.9) The last contribution to the Hamiltonian (8.2) is the k-point dependent contribution of the nonlocal pseudopotential. The calculation of this contribution follows formula (3.40), which in the case of lj-dependent pseudopotentials has to be modiﬁed to 2 (4π) 0 h Z † i−1 ∆V KB(G + k, G0 + k) = e−i(G−G )R drr2φps (r)V ps(r)φps(r) × lj Ω lj lj lj Z Z h 2 ps ps ih 2 0 ps ps i × drr jl(|G + k|r)Vlj (r)φlj (r) drr jl(|G + k|r)Vlj (r)φlj (r) ×

j X † 0 × Ωljm(|G + k|)Ωljm(|G + k|). (8.10) m=−j 8.1. Hamiltonian Operator 75

ps ps In equation (8.10) the pseudopotentials Vlj and the pseudoorbitals φlj now are lj-dependent and the Ωljm denote the angular momentum eigenfunctions that are given by spherical spinors [100].

In case of a calculation starting from scratch, the potential (8.8) can be used as an initial guess for the local KS potential. A detailed sequence for building and diagonalizing the Hamiltonian is given in Fig. 8.1.

SCEXX — Building the Hamiltonian

ps • Build local pseudopotential Vloc(G) via equation (8.7)

• Include contributions of magnetic ﬁeld B, its vector potential A and spin-orbit

coupling in V˜ loc(G) via equation (8.8)

• Build local KS potential Vs(G) via equation (8.9)

loop over k-points

loop over G-vectors

• Calculate kinetic energy contribution Ts via equation (8.3)

double loop over G-vectors

• Calculate contribution of nonlocal separable pseudopotential KB 0 ∆Vl (G+k, G +k) via equation (8.10)

double loop over G-vectors

0 • Calculate matrix elements of local KS potential Vs(G+k, G +k) (8.4) using equations (8.5) and (8.6)

• Update Hamiltonian matrix H(G+k, G0 +k) via equation (8.2)

• Obtain orbitals and eigenvalues via diagonalization of k-dependent Hamiltonian H(G+k, G0 +k)

Figure 8.1: Generation and diagonalization of the k-point dependent Hamilto- nian. 76 Chapter 8. The SCF Cycle

8.2 Spin-Current Densities

The real space representation of the 16 spin-current densities ρµν is updated in each cycle of the ﬁrst k-point loop indicated in Fig. 7.1. For each k-point the contribution to the spin-current densities ρµν is calculated via

occ X ρµν,k(r) = hφvk|ΣµJν|φvki (8.11) v with the sum running over the valence states. The action of the operator Σµ merely resolves in projecting out speciﬁc components of a spinor φnk. With the spinor consisting of two components

↑ ↓ 2 2 φnk = α+φnk + α−φnk (|α+| + |α−| = 1), (8.12)

↑↓ ↑↓ the four types of matrix elements hφvk|ΣµJν|φvki can be constructed via the four ↑↓ ↑↓ types of matrix elements hφvk|Jν|φvki. For ν = 0, i.e. the case of the particle density operator (5.8), the matrix element is calculated to

occ occ X X † hφvk|ρ(r)|φvki = hφvk|δ(r − rvk)|φvki = φvk(r)φvk(r). (8.13) v v

For the case of the paramagnetic current operator (5.9), i.e. ν =1, 2, 3, the matrix element becomes

To calculate the matrix element (8.14), the derivative of the orbital φvk with respect to the spatial coordinate r has to be calculated. The state φvk can be expanded in plane waves

1 X i(G+k)r φvk(r) = √ Cvk(G)e ≡ F(Cvk(G)). (8.15) Ω G

Hereby, the symbol F(f) denotes the Fourier transformation of a function f from momentum to real space or vice versa. Note that an expression of the 8.3. Response Function 77 form F(f(G)) represents a function in real space and an expression of the form F(f(r)) represents a function in momentum space. A straightforward diﬀerentiation of expression (8.15) yields

∂ F(C (G)) = F(GC (G)) + kF(C (G)). (8.16) ∂r vk vk vk

After passing the k-point loop, the spin-current densities are transformed to momentum space. The explicit calculation sequence for the spin-current densities is shown in Fig. 8.2.

SCEXX — Calculation of ρµν

loop over valence states vk

loop over points of real space mesh

∂ • Calculate ∂r F(Cvk(G)) via equation (8.16)

∂ • Calculate matrix element hφvk| ∂r |φvki

∂ • Obtain complex conjugation of hφvk| ∂r |φvki

• Calculate contribution to spin-current densities

ρµν,k(r) (8.11) using equations (8.13) and (8.14)

Figure 8.2: Calculation of the k-point dependent contribution to the spin- current densities.

8.3 Response Function

For the calculation of the exact exchange potential, the KS response function µν,κλ χ0 is required in its momentum space representation. During each cycle of the ﬁrst k-point loop depicted in Fig. 7.1, the response function is updated. Hereby, the construction follows the formula

0 µν,κλ 0 1 X hvk|ΣµJν(G)|ckihck|ΣκJλ(G )|vki χ0 (G, G ) = . (8.17) Ω vk − ck vck 78 Chapter 8. The SCF Cycle

The required matrix elements of type hck|ΣµJν(G)|vki can be constructed using the matrix elements of the particle density operator

−iGr X ∗ 0 0 hck|e |vki = Cck(G )Cvk(G + G ) (8.18) G0 and the paramagnetic current density operator X 1 hck|j(G)|vki = ( G0 + G + k)C∗ (G0)C (G + G0), (8.19) 2 ck vk G0 which were already treated in chapter 6. The calculation sequence for the response function is given by Fig. 8.3.

µν,κλ SCEXX — Calculation of χ0

loop over valence bands vk

double loop over G-vectors

0 • Calculate Fourier coeﬃcient Cvk(G+G )

loop over conduction bands ck

loop over G-vectors

• Calculate matrix element hck|ΣκJλ(G)|vki using equation (8.18) and (8.19)

• Obtain matrix element hvk|ΣµJν (G)|cki via complex conjugation

• Divide product of matrix elements by eigenvalue diﬀerence vk −ck

double loop over G-vectors

• Update sum (8.17)

Figure 8.3: Calculation of the k-point dependent contribution to the response function.

8.4 Hartree Potential

After the regular electron density ρ has been obtained, the Hartree potential VH is calculated in its momentum space representation. With the electron density 8.5. LDA- and GGA Exchange-Correlation Potentials 79 at hand, the calculation follows the formula 4πρ(G) V (G) = . (8.20) H G2 The associated calculation sequence is shown in Fig. 8.4.

SCEXX — Calculation of VH

loop over G-vectors

• Calculate VH (G) via equation (8.20)

Figure 8.4: Calculation of the Hartree potential.

8.5 LDA- and GGA Exchange-Correlation Po- tentials

The program oﬀers the possibility to include various LDA- and GGA exchange- correlation potentials within the SCF loop. The associated functionals only depend on the regular electron density and derivatives of it. Therefore, these potentials should only be used within pure DFT calculations, i.e. calculations without inclusion of currents and spin-magnetization. Three types of functionals were implemented and are described in the next subsections. The exchange-correlation potentials shall be calculated in their momentum space representation Vxc(G). The calculation scheme is of the same type for all provided kinds of functionals. The regular electron density ρ has already been obtained in its real space representation and additionally its derivative ∇ρ can be calculated. With this information at hand, the exchange-correlation potentials can be constructed via the formulas given in the following subsections.

8.5.1 LDA Exchange- and Correlation Potentials

The following exchange and correlation potentials of LDA type are available:

1. Dirac-Slater exchange [101], 80 Chapter 8. The SCF Cycle

2. Vosko-Wilk-Nusair (VWN) correlation [65],

3. Perdew-Wang (PW) parametrization of the homogenous electron gas correlation energy [102],

4. Vosko-Wilk-Nusair (VWN) correlation within the random phase approximation (RPA) [65].

Since LDA functionals only depend on the local value of the density, the exchange or correlation energy can be expressed as Z Exc,LDA = drf(ρ). (8.21)

The functional derivative of the energy (8.21) with respect to the density yields the corresponding exchange or correlation potential in real space, which in the LDA case simply reads

δE ∂f V (r) = xc,LDA = . (8.22) xc,LDA δρ ∂ρ

The Fourier transformation of equation (8.22) to momentum space is given by

Z 1 ∂f −iGr ∂f Vxc,LDA(G) = √ dr e = F . (8.23) Ω ∂ρ ∂ρ

The calculation sequence for the LDA potentials is shown in Fig. 8.5.

SCEXX — Calculation of Vxc,LDA

loop over points of real space mesh

∂f • Calculate derivative ∂ρ in real space

` ∂f ´ • Calculate F ∂ρ , i.e. potential Vxc,LDA(G) via equation (8.23)

Figure 8.5: Calculation of LDA exchange- and correlation potentials. 8.5. LDA- and GGA Exchange-Correlation Potentials 81

8.5.2 GGA Exchange Potentials

The following exchange potentials of GGA type are available:

1. Becke88 exchange [67],

2. Perdew-Burke-Ernzerhof (PBE) exchange [68].

GGA exchange functionals of the above type depend on the density ρ and the quantity |∇ρ|2. The functional for the exchange energy can therefore be expressed as Z 2 Ex,GGA = drf(ρ, |∇ρ| ). (8.24)

Calculating the functional derivative of the energy (8.24) with respect to the density yields the exchange potential in its real space representation δE ∂f ∂f V (r) = x,GGA = − 2∇· ∇ρ . (8.25) x,GGA δρ ∂ρ ∂|∇ρ|2

The Fourier transformation of equation (8.25) to momentum space reads Z 1 ∂f ∂f −iGr Vx,GGA(G) = √ dr − 2iG· ∇ρ e = Ω ∂ρ ∂|∇ρ|2 ∂f ∂f = F − 2iG·F ∇ρ . (8.26) ∂ρ ∂|∇ρ|2 The sequence for the calculation of the GGA exchange potentials can be seen in Fig. 8.6.

8.5.3 GGA Correlation Potentials

The following correlation potentials of GGA type are available:

1. Lee-Yang-Parr (LYP) correlation [69],

2. Perdew-Burke-Ernzerhof (PBE) correlation [68,102],

3. Perdew86 correlation gradient correction.

The correlation energy can be expressed via Z 2 2 Ec,GGA = drf(ρα, ρβ, |∇ρα| , |∇ρβ| , ∇ρα∇ρβ) (8.27) 82 Chapter 8. The SCF Cycle

SCEXX — Calculation of Vx,GGA

loop over points of real space mesh

∂f • Calculate derivative ∂ρ in real space

• Calculate derivative ∂f in real space and build ∂|∇ρ|2 ∂f quantity 2 ∇ρ ∂|∇ρα|

` ∂f ´ ` ∂f ´ • Calculate F and F 2 ∇ρ ∂ρ ∂|∇ρα|

loop over G-vectors

“ ” • Calculate quantity 2iG·F ∂f ∇ρ ∂|∇ρ|2

• Calculate potential Vx,GGA(G) via equation (8.26)

Figure 8.6: Calculation of GGA exchange potentials.

with the quantities ρα and ρβ denoting spin-polarized densities. The used routines for obtaining derivatives of the function f in case of the above given GGA correlation functionals are based on spin-polarized densities. It is, however, straightforward to obtain the correlation potential for the non-spin-polarized case by simply using 1 ρ = ρ = ρ. (8.28) α β 2

In the non-spin-polarized case the potentials Vα, Vβ and the value for the non- spin-polarized potential V turn out to have the same value

∂Ec,GGA ∂ρα ∂Ec,GGA ∂ρβ Vc,GGA(ρα, ρβ) = + = ∂ρα ∂ρ ∂ρβ ∂ρ 1 = (V + V ) = V = V . (8.29) 2 α β α β

Therefore, the functional derivative of (8.27) with respect to the density ρα yields the correlation potential in real space

δEc,GGA ∂f ∂f 1 ∂f Vc,GGA(r) = = −∇· 2 ∇ρ − ∇· ∇ρ . (8.30) δρα ∂ρα ∂|∇ρα| 2 ∂(∇ρα∇ρβ) 8.5. LDA- and GGA Exchange-Correlation Potentials 83

The Fourier transformation of (8.30) reads

1 Z ∂f ∂f √ Vc,GGA(G) = dr − iG· 2 ∇ρ − Ω ∂ρα ∂|∇ρα| 1 ∂f − iG· ∇ρ e−iGr. (8.31) 2 ∂(∇ρα∇ρβ)

This can also be written as

∂f ∂f 1 ∂f Vc,GGA(G) = F − iG·F 2 ∇ρ − iG·F ∇ρ . (8.32) ∂ρα ∂|∇ρα| 2 ∂(∇ρα∇ρβ)

The sequence for the calculation of the GGA correlation potentials is summarized in Fig. 8.7.

SCEXX — Calculation of Vc,GGA

loop over points of real space mesh

• Calculate derivative ∂f in real space ∂ρα

∂f • Calculate derivative 2 in real space and build ∂|∇ρα| ∂f quantity 2 ∇ρ ∂|∇ρα|

• Calculate derivative ∂f in real space and ∂(∇ρα∇ρβ ) build quantity ∂f ∇ρ ∂(∇ρα∇ρβ )

` ∂f ´ ` ∂f ´ • Calculate F , F 2 ∇ρ and ∂ρα ∂|∇ρα| F` ∂f ∇ρ´ ∂(∇ρα∇ρβ )

loop over G-vectors

“ ∂f ” • Calculate iG·F 2 ∇ρ ∂|∇ρα|

“ ” • Calculate 1 iG·F ∂f ∇ρ 2 ∂(∇ρα∇ρβ )

• Calculate potential Vc,GGA(G) via equation (8.32)

Figure 8.7: Calculation of GGA correlation potentials. 84 Chapter 8. The SCF Cycle

8.6 Exact Exchange Potential

In order to update the local KS potential in each SCF cycle, the exact exchange potential Vx,µν is required in its momentum space representation. For its calcula- µν,κλ tion the KS response functions χ0 and the right hand side tµν of the (SC)EXX equation are required. The response function has already been obtained following the procedure described in section 8.3. The values for tµν are updated in each cycle of the double k-point loop indicated in Fig. 8.1. The right hand side is constructed via the relation Z 0 0 X h NL hφck|ΣµJν(r )|φvki i tµν(r) = dr hvk|Vx |cki + c.c. . (8.33) vk − ck vck

NL The matrix elements of the nonlocal exchange operator Vx in (8.33) are given by † 4π X X Avk,v0k0 (G)Ack,v0k0 (G) hvk|V NL|cki = − , (8.34) x Ω |G + k0 − k|2 v0k0 G with Avk,v0k0 denoting products of the periodic part of the Bloch functions. Sim- ilar to the calculation of the spin-current densities, the matrix elements of type 0 hφck|ΣµJν(r )|φvki in (8.33) can be constructed using the relation

N i X ∂ hφ |j(r)|φ i = − hφ | |φ i + c.c. (8.35) ck vk 2 ck ∂r vk i=1

The spatial derivative of the orbital φvk can be obtained via the relation ∂ F(C (G)) = F(GC (G)) + kF(C (G)). (8.36) ∂r vk vk vk The calculation sequence for obtaining the right hand side due to formula (8.33) is given by Fig. 8.8.

After having obtained tµν, the exact exchange potential Vx,µν can be calculated using the relation Z X 0 µν,κλ −1 0 0 Vx,µν(r) = dr (χ0 ) (r, r )tκλ(r ). (8.37) κλ Treating µν as a superindex, equation (8.37) can be rewritten as Z 0 −1 0 0 Vx(r) = dr χ0 (r, r )t(r ) (8.38) 8.6. Exact Exchange Potential 85

SCEXX — Calculation of tµν

loop over k-points k

loop over k-points k0

double loop over valence bands v and v’

• Calculate product of valence bands Avk,v0k0 (r) in real space

• FT of Avk,v0k0 (r) to momentum space gives Avk,v0k0 (G)

0 2 • multiply Avk,v0k0 (G) by Coulomb term 4π/|G+k −k|

0 2 • FT of Avk,v0k0 (G)/|G+k −k| to real space

• Multiply by valence band φvk(r) and update quantity 0 2 φvk(r)F(Avk,v0k0 (G)/|G+k −k| )

loop over conduction bands c

∂ • Calculate ∂r F(Cck(G)) via equation (8.36)

loop over valence bands v

• Multiply by conduction band φck(r) and update quantity † 0 2 F(Avk,v0k0 (G)Ack,v0k0 (G)/|G+k −k| )

• Divide by eigenvalue diﬀerence vk −ck

∂ • Multiply by ∂r F(Cck(G)) and φvk(r) and update ﬁrst part of right hand side, i.e. summand in equation (8.33) in real space F(t˜µν (G))

• Complete right hand side by adding complex conjugate of F(t˜µν (G)): † F(tµν (G)) = F(t˜µν (G)) + F (t˜µν (G))

Figure 8.8: Calculation of the right hand side of the (SC)EXX equation. and with the use of the plane wave basis set, the associated algebraization leads to the matrix equation −1 Vx = χ0 t (8.39) that actually has to be solved in momentum space. Solving this linear equation

system, one encounters the problem of χ0 holding singularities, i.e. zero eigenvalues. This issue has already been discussed in chapter 6 and can be circumvented by projecting out the G-components that are associated to zero eigenvalues. For 86 Chapter 8. The SCF Cycle

this purpose, the response function χ0 is diagonalized, yielding a diagonal matrix d that contains the eigenvalues λi of χ0. Additionally, one gets a matrix S holding the associated orthonormalized eigenvectors. The inverse of the response function χ0 can then be expressed via its spectral representation

−1 −1 −1 χ0 = S d S . (8.40)

The diagonal matrix d can be inverted via

−1 −1 (d )ij = δijλi . (8.41)

For the case of a zero eigenvalue λi, or in practice an eigenvalue being close to −1 zero, its inverse is set to zero, i.e. λi = 0. The modiﬁed spectral representation then reads −1 ˜−1 −1 χ˜0 = S d S . (8.42) The calculation of the exact exchange potential (8.39) becomes

˜−1 −1 Vx = (S d S ) t. (8.43)

In order to guarantee that no information is lost by projecting out the components with zero eigenvalues, it has to be ensured that the same components do not contribute to the right hand side. For this purpose, an additional test has been implemented. First, a unity matrix 1˜ with zero entries at the components associated to zero eigenvalues in d˜ is built. Afterwards, the same spectral analysis, as it has been applied for the inverse of the diagonal matrix d˜ is performed, i.e. S 1˜−1 S−1. The obtained matrix S 1˜−1 S−1 multiplied with the right hand side then projects out the same components as it has already been done in the above description for the inverse of the response function. This means that if the condition (S 1˜−1 S−1)t = t (8.44) is satisﬁed, the performed projection did not cause any information loss. The validity of equation (8.44) is explicitly checked during each cycle of the SCF loop. If the associated test is not passed, the execution of the program stops. Part III

Applications

Chapter 9

One-dimensional Periodic Systems

9.1 T rans-polyacetylene as a Test System

The EXX formalism has, so far, only been applied to atoms [43], molecules [44,45] and simple three-dimensional periodic systems with small unit cells [36, 37]. In this chapter the EXX approach is applied to the electronic structure of trans- polyacetylene as a typical representative of a one-dimensional periodic organic system [57]. One-dimensional organic oligomeres or polymeres are of great interest due to their high potential as active materials in new optoelectronic devices. T rans-polyacetylene is the standard example for an organic one-dimensional periodic system. It is therefore not surprising that a number of theoretical investigations considering the optical properties of this system have already been carried out [103, 104, 105, 106, 107, 108, 109]. In Ref. [107] the band gap and the optical spectrum of an isolated trans-polyacetylene chain was calculated with the GW and the Bethe-Salpeter method, respectively. The results, e.g., the calculated band gap of 2.1 eV and the absortion spectrum with a single peak at 1.7 eV, agree well with experimental values [110, 111], which, however, refer to bulk trans-polyacetylene. In Ref. [109], on the other hand, a much larger band gap of 4.1 eV for an isolated trans-polyacetylene chain was obtained by a multireference conﬁguration interaction approach. Finally, in Refs. [108] and [105], band gaps of 3.68 and 3.96 eV, respectively, were determined with Møller-Plesset perturbation

89 90 Chapter 9. One-dimensional Periodic Systems theory for an isolated trans-polyacetylene chain. Because most the above mentioned calculations, which yield quite diﬀerent results, consider isolated chains but are compared to experimental data from bulk trans-polyacetylene, it seems desirable to investigate the inﬂuence of the geometry and the arrangement of the chains on the band structure. Beside an evaluation of the performance of the EXX approach for one-dimensional periodic organic systems, such an investigation is the second goal of this chapter.

9.2 Computational Details

Originally the EXX formalism was implemented in a plane-wave code optimized for three-dimensional periodic cubic systems [37] that exploited the cubic symmetry and used the concept of special k-points. For the investigation of non-cubic systems, including, via the super cell ansatz, one- or two-dimensional periodic systems, a new plane-wave code, which does not rely on symmetry or special k-points has been implemented as described in Part II. For the calculations carried out, the plane-wave cutoffs for the orbitals and the KS response function χs in all cases were chosen to be 32 Ry and 12 Ry, respectively. While these cutoffs are not sufficient for calculations of the total energy, they turned out to be sufficient for the calculation of the band structure. Errors of the obtained band gaps can be estimated to be below the order of about 0.05 eV, which is more than accurate enough for the purpose of the performed investigations. Instead of special k-points, uniform grids of k-points covering the first Bril- louin zone were used. For isolated chains of trans-polyacetylene 24 k-points in direction of the chain were used and a very good convergence with the number of k-points was reached. For bulk trans-polyacetylene additional k-points were placed in the two directions along the reciprocal unit cell vectors not associated with the chain direction. In this case, a uniform 2×32×2 mesh (32 k-points along the chain direction, 2 k-points along the two other directions) was applied. In all calculations all unoccupied states were taken into account for both the construction of the response function (4.18) and the right hand side of the EXX equation (4.20). Typical values for the number of conduction bands were 3600 9.3. Results 91

(1500) for the isolated chain (bulk) case. For all calculations normconserving separable pseudopotentials on the basis of the optimized potential method (OPM) were employed. The density-functionals (LDA, exact exchange-only, exact exchange plus approximate correlation) used in the generation of the pseudopotentials and in the self-consistent ground state calculations of the regarded periodic system were always chosen to be identical. The needed pseudopotentials were generated with a program package based on a formalism presented in [112]. For hydrogen the pseudopotentials contained exclusively an angular momentum component with l = 0, i.e., an s-component. For carbon pseudopotentials with angular momentum components l = 0, 1, i.e., s- and p-components, were employed. The s-component of the pseudopotential was chosen as local component for all calculations. Additional test calculations choosing the p-component as local component for the carbon atom were carried out, but did not lead to H significant changes in the results. For the cutoff radii, values of rc,l=0 = 0.9, C C rc,l=0 = 1.2 and rc,l=1 = 1.1 in units of Bohr radii were used. The criterion in optimizing the cutoff radii was the agreement, i.e. the graphical fit between the resulting pseudoorbitals and the corresponding all-electron orbitals.

9.3 Results

9.3.1 Isolated, Inﬁnite Chain of T rans-polyacetylene

As a first system within a super cell ansatz, an isolated, infinite chain of trans- polyacetylene was considered. The polymere is planar and exhibits alternating single and double bonds between the carbon atoms, which are accompanied by different bond lengths. The unit cell contains a C2H2 unit. The bond distances of the carbon backbone were taken from experimental X-ray scattering data of crystalline trans-polyacetylene [113]. This geometry, which shall be named G1, has also been used in an investigation of electronic correlation effects in trans- polyacetylene [109]. The carbon-carbon bond lengths in geometry G1 are given by d(C−C) = 1.45A˚ and d(C=C) = 1.36A.˚ The lattice constant along the chain direction equals 2.455A˚ [114]. A (C−H)–bond length of 1.087A˚ has been found 92 Chapter 9. One-dimensional Periodic Systems

-10 Energy [eV]

-20 Γ k X

Figure 9.1: Band structure of an isolated, infinite chain of trans-polyacetylene for the LDA (dashed lines) and the EXX (solid lines) case. by optimization on the HF and MP2 level [104]. The angle 6 (C−C, C=C) (with respect to the short (C=C)– double bond) has been determined with 120.057° via HF optimization [109]. The system was placed in the the xy-plane with the chain oriented along the y-direction. The unit cell is defined by the vectors a1 =(a, 0, 0), a2 =(0, c, 0) and a3 =(a/2, 0, b) with the parameter c=2.455A˚ defining the lattice vector along the chain. The parameter a was set to a=8A.˚ The third lattice vector was chosen in a way that the p-orbitals placed at the carbon atoms of two adjacent chains do not lie on top of each other, but at maximal distance. The parameter b was set to b = 9A.˚ The values of a = 8A˚ and b = 9A˚ turned out to be large enough that the chains can be considered as isolated. Effects on the band gap due to coupling of the chains can be estimated to be below 0.05 eV from varying the size of the the supercell. For further details of the chain geometry see Appendix A. The calculated LDA band gap equals 0.78 eV, whereas exact exchange-only calculations yield a band gap of 1.63 eV. The band structure for these two cases is displayed in Fig. 9.1. The LDA and EXX band structures are qualitatively similar, however, the energetical difference between valence and conduction bands 9.3. Results 93

Table 9.1: Band gaps (in eV) for an isolated, infinite chain of trans- polyacetylene calculated with various density functionals, the EXX case has been calculated for the two different geometries G1 and G2. For comparison, values of Hartree-Fock (HF), Møller-Plesset theory (MP2), multireference configuration interaction (MRCI) and quasiparticle (GW) calculations are given as well (Labeled values are taken from references: aRef. [108], bRef. [105], cRef. [109], d Ref. [107]).

X HF 6.06a MP2 3.68a, 3.96b MRCI 4.11c GW 2.1d LDA(G1) 0.78 EXX(G1) 1.62 EXX(G2) 1.56 EXX-VWN(G1) 1.65 EXX-PBE(G1) 1.61

is significantly smaller in the LDA case leading to the observed smaller band gap. This is in line with the finding for three-dimensional semiconductors that the EXX approach increases the commonly too small LDA band gap. Addition of an LDA correlation potential to the EXX potential leads to a band gap of 1.65 eV, i.e. increases the gap marginally by 0.01 eV, whereas addition of a GGA-correlation potential, here the PBE correlation potential, to the EXX potential marginally decreases the gap to 1.61 eV. This shows, again in line with the observation in three-dimensional semiconductors, that the treatment of the correlation potential has no significant influence on the band gap. Within the EXX scheme, the band gap comes close to the value of 2.1 eV resulting from GW calculations [107], but still stays somewhat smaller. The calculated band gaps as well as those from several other methods are collected in Table 9.1. To investigate 94 Chapter 9. One-dimensional Periodic Systems the sensitivity of the band structure of an isolated trans-polyacetylene chain on the geometry, we calculated the EXX band structure also in a slightly modified geometry, denoted G2, which has also been used in Ref. [107]. The geometry G2 is characterized by the carbon-carbon bond lengths d(C−C) = 1.44A˚ and d(C=C) = 1.36A˚ [115]. Optimization yielded [107] a lattice constant along the chain direction of 2.473A.˚ A carbon hydrogen bond length d(C=H)=1.1A˚ [107] and an angle 6 (C−C, C=C) of 118° have been used (for geometric data see Ap- pendix A). In geometry G2 an EXX band gap of 1.56 eV is obtained, which differs only slightly from the result observed for the geometry G1.

9.3.2 Bulk T rans-polyacetylene

The geometries of the chains building the considered bulk trans-polyacetylene are those of the above discussed isolated chains in geometry G1, i.e., the geometry based on the X-ray structure [114]. Within the crystalline trans-polyacetylene, each unit cell contains two chains. The unit cell according to Ref. [113] corresponds to a simple monoclinic Bravais lattice with the values of a = 4.24A,˚ b = 7.32A˚ and c = 2.46A˚ for the lattice constants (for geometric data see Ap- pendix A). Again LDA and EXX calculations were carried out, which yielded band gaps at the X-point of 0.98 eV and 1.64 eV. The corresponding band structures are displayed in Fig. 9.2. However in both cases the band gap at the X-point is not the fundamental one. Along the line from the Γ- to the X-point, one observes band gaps (0.81 eV for LDA and 1.55 eV for EXX), which are reduced in comparison to gaps at the X-point. However, this band gap also is not the fundamental one. For LDA as well as EXX calculations, the smallest gaps we found are located at the point (0.5,0.5,0.0), in multiples of the reciprocal lattice vectors, i.e., at the edge of the Brillouin zone. At this point band gaps of 0.43 eV for LDA and 1.18 eV for EXX were found, which represent, compared to the gap at the X- point, a remarkable dispersion of about 0.55 eV for LDA, resp. 0.46 eV for EXX. This ﬁnding is in agreement with results from HF calculations, which estimated a reduction of the band gap by 0.6 eV due to interband interactions [109]. The LDA and EXX band structures along the line from point (0.5,0.5,0.0) to point 9.3. Results 95

0 10 0

0 -10

-10

Energy [eV] -20 Energy [eV]

-20 Γ k X Γ k X

Figure 9.2: Band structure of bulk trans-polyacetylene for the LDA (left) and the EXX (right) case.

(0.5,0.5,0.0), in multiples of the reciprocal lattice vectors, are shown in Fig. 9.3. Furthermore, the inﬂuence of the addition of an LDA or PBE correlation potential to the EXX potential was checked. Like in the case of an the isolated chain, the correlation potential turned out to have an only negligible eﬀect on the band structure.

Optical absorption experiments for trans-polyacetylene in the considered geometry [110] give an absorption coeﬃcient rising sharply at 1.4 eV and having a peak at about 1.9 eV. A second similar measurement [111] gives an identical value for the peak and a value of 1.5 eV for the onset of the absorption. The lattice constants mentioned within the framework of this second measurement [114], diﬀer only slightly from the used geometry of Ref. [110] underlying the present calculations (a = 4.18A,˚ b = 7.34A,˚ c = 2.455A).˚ Nevertheless, the experimental results for the two distinct measurements agree more or less and are in good agreement with the values obtained via EXX calculations. The data of the band gaps for all discussed cases have been compiled in Table 9.2. 96 Chapter 9. One-dimensional Periodic Systems

0 0 10

0 5

0 -10 -5 Energy [eV] Energy [eV] -10 -20

0 0 -15 k k

Figure 9.3: Dispersion of the band gap for bulk trans-polyacetylene for the LDA (left) and the EXX (right) case. The band structure goes along the line from point (0.5,0.5,0.0) to point (0.5,0.5,0.5) in multiples of the reciprocal lattice vectors.

9.3.3 Response Functions

The optical absorption strength obtained by the independent particle, i.e., Kohn- Sham, response function for an isolated trans-polyacetylene chain in geometry G1 is shown in Fig. 9.4 for the EXX case. The onset of the peak coincides with the calculated fundamental band gap, the absorption maximum lies about 0.1 eV higher than the band gap. A similar behavior was obtained for the Kohn- Sham response function for the LDA case with the corresponding peak shifted to a lower energy. The absorption spectrum resulting from the EXX response function is quite similar to the one obtained via the Bethe-Salpeter equation [107], which includes electron-hole interactions missing in an independent particle spectrum. This could be due to a cancellation of two eﬀects. For an isolated trans-polyacetylene chain the neglect of electron-hole interactions seems to shift [107] the peak of the absorption spectrum to higher energy values. This could cancel the eﬀect that the EXX band gap is smaller than the one obtained within 9.3. Results 97

Table 9.2: Band gaps (in eV) for bulk trans-polyacetylene. Beside the value for the gap at the X-point of the Brioullin zone, a smaller gap between the Γ- and the X-point (X → Γ) has been observed. The smallest gap was found at the point (0.5, 0.5, 0.0) (in units of the reciprocal lattice vectors) lying on the edge of the Brillouin zone. The experimental values of the band gap correspond to the energy, where a sharp rise in the absorption spectrum is observed (Labeled values are taken from references: a Ref. [110], b Ref. [111]).

∆X ∆X→Γ ∆min LDA 0.98 0.81 0.43 EXX 1.64 1.55 1.18 EXX-VWN 1.65 1.57 1.21 EXX-PBE 1.61 1.52 1.17 Experiment — — 1.4a, 1.5b

the GW scheme [107]. The absorption spectrum of the isolated chain obtained via the Bethe-Salpeter equation agrees quite well with the experimental spectrum observed for bulk trans-polyacetylene. However, in the previous subsection it was shown that packing eﬀects in the bulk reduce the fundamental band gap by about 0.5 eV. Therefore it remains to be seen, whether Bethe-Salpeter calculation for bulk trans-polyacetylene would yield an absorption spectrum in agreement with experiment or a spectrum shifted to lower energy. The optical absorption spectrum resulting from the EXX independent particle response function for bulk trans-polyacetylene is displayed in Fig. 9.5. With a value of about 1.6 eV, the maximum of the response function lies close to the calculated values of the two band gaps at the X-point and on the path between the X- and the Γ-point (see Table 9.2). Compared to the Kohn-Sham response function for the case of an isolated, inﬁnite chain of trans-polyacetylene, the displayed result for the bulk shows a broader peak. This widened shape, as well as the maximum position at about 1.6 eV is in agreement with the experimental data for bulk trans-polyacetylene [110,111]. 98 Chapter 9. One-dimensional Periodic Systems

0 0 0 0

0 Absorption strength

0 0 1 2 3 Energy [eV]

Figure 9.4: Kohn-Sham response function of an isolated, inﬁnite chain of trans- polyacetylene for the EXX case.

9.4 Discussion

The exact exchange KS approach, implemented within a plane-wave pseudopotential framework, was shown to yield band gaps for a typical one-dimensional periodic organic polymere, which agree quite well with experimental data in the case of bulk trans-polyacetylene. Like in the case of three-dimensional periodic semiconductors EXX band gaps are distinctively increased compared to LDA or GGA band gaps and thus agree distinctively better with experimental data. The EXX approach therefore promises to be a viable tool for the investigation of the electronic structure of organic polymeres, a class of materials with high potential as active compound in new optoelectronic devices. The results of this chapter sug- gest that also for an isolated chain of trans-polyacetylene EXX band gaps seem to be close to experimental ones, although in this case no direct measurements of the latter could be found. On the other hand, EXX band gaps differ from experimental ones only by the derivative discontinuity of the exchange-correlation potential and the effects due to the approximation of the correlation potential. Like for semiconductors the latter approximation seems to have little influence on 9.4. Discussion 99 Absorption strength

1 1.5 2 2.5 Energy [eV]

Figure 9.5: Kohn-Sham response function of bulk trans-polyacetylene for the EXX case. the band structure and thus could be considered to be negligible. Then agreement of EXX- and experimental band gaps would mean that also the derivative discontinuity of the exchange-correlation potential is small. However, this point needs further investigation, because it could be that all presently available approximate correlation potentials are missing features affecting band structures. Another important outcome of this chapter’s investigation is that packing effects in bulk trans-polyacetylene have a substantial quantitative effect on the band structure of trans-polyacetylene, the band gap in bulk trans-polyacetylene is by about 0.5 eV smaller than in isolated chains. In future investigations of organic polymeres such packing effects therefore should be taken into account. 100 Chapter 9. One-dimensional Periodic Systems Chapter 10

Atom in an External Magnetic Field

As a first system to which the new exact-exchange spin-current formalism presented in chapters 5 and 6 shall be applied, an isolated atom subjected to an external magnetic field is chosen [55]. This example shall firstly show that the new SCDFT approach works correctly in practice and secondly demonstrate the role of spin-currents in calculations of magnetic properties. For light atoms, the Hartree energy is much greater than the spin-orbit contribution and therefore the explicit spin-orbit term can be neglected in the Hamiltonian (5.1) N N X 1 2 1 X 1 H = p + A(r ) + v (r ) + σ · B(r ) + . (10.1) 2 i i ext i 2 i |r − r | i=1 i

 A2(r)  vext(r)+ 2 Ax(r) Ay(r)Az(r)    Bx(r)   2 0 0 0  V(r) =   . (10.3)  By(r)   2 0 0 0    Bz(r) 2 0 0 0

Note that as a consequence of the lack of spin-orbit interactions the terms Vµν with µ, ν = 1, 2, 3, in contrast to the originally introduced matrix (5.17), now

101 102 Chapter 10. Atom in an External Magnetic Field equal zero. The usual mapping of the real, interacting electronic system to the noninterating KS system gives the associated KS-Hamiltonian Z T Hs = Ts + drΣ Vs(r)J(r) (10.4) with the corresponding KS-potential

Vs(r) = V(r) + VH (r) + Vxc(r). (10.5)

According to (5.47) Vs constitutes a 4 × 4-matrix with components Vs,µν. The contributions Vs,µν with µ, ν =1, 2, 3 are the contributions coupling to the actual spin-currents. Although the fact that the corresponding components in the real external potential V (10.3) are zero, the example presented in this chapter will show that these components will differ from zero in Vs and therefore spin-currents can have an effect in calculations of magnetic properties. From atomic physics [92] it is known that the discussion of magnetic field effects on atoms and molecules has to be divided into two parts, describing “weak” and “strong” fields separately. The scale for this distinction is set by the magni- tude of the spin-orbit energy. If the resulting energy change due to the application of a magnetic field is small compared to the spin-orbit energy, the total values for spatial angular momentum L and total spin angular momentum S can be coupled to the total angular momentum J, which then precesses around the axis of the applied magnetic field. In case of strong magnetic fields, the individual coupling of L and S to the magnetic field is stronger than the spin-orbit coupling itself. Therefore, L and S cannot be coupled to J and each of them precesses around the magnetic field axis on its own. The above described distinction for magnetic fields depends on the nuclear charge of the regarded atom or molecule. The weak field case is described by the Zeeman formula

∆E = βgJ BmJ (10.6) with β denoting the Bohr magneton and mJ denoting the quantum number for the total angular momentum. The Land´efactor gJ can be calculated via [92]

j(j + 1) + s(s + 1) − l(l + 1) g = 1 + . (10.7) J 2j(j + 1) 103

For the weak field case, mJ is the “good” quantum number, in contrast to the strong field case, where the magnetic quantum numbers m` and ms, for spatial and spin angular momentum respectively, become the good quantum numbers. For the case of strong fields, the Paschen-Back formula has to be applied

∆E = βB(m` + 2ms). (10.8)

Since spin-orbit effects are neglected in the Hamiltonian (10.1), a light atom is chosen for the considerations of this chapter, i.e. an atom of small nuclear charge. Furthermore, both the total spin S and total orbital angular momentum L shall differ from zero, so that all contributions given in (10.1) are considered in the calculations. A good choice to satisfy these requirements is oxygen with its (2s2, 2p4)- valence configuration. In the absence of spin-orbit coupling, the total angular momentum J = L + S can be calculated via LS(Russel-Saunders)-coupling, since the total spin and total orbital angular momentum are conserved separately. Within the LS-scheme, the spin and spatial angular momentum are regarded separately and afterwards their total values L and S are coupled to the total angular momentum J. Closed shells and subshells do not contribute to spin or orbital angular momentum and therefore only the 4p-electrons of the regarded (2s2, 2p4)-configuration must be considered to find the actual state one is dealing with. The orbital angular momentum of the 4p-electrons can couple to the values of L = 0, L = 1 or L = 2, i.e. to an S, P , or D-state, with P denoting the energetically lowest. The spins of the electrons couple to S = 0, S = 1 or S = 2 giving a singlet, triplet and quintet (the spin quantum number running from −S to S for each case). It follows that in the absence of magnetic fields and spin- orbit interactions the electronic ground state of oxygen is a nine-fold degenerate 3P -state with the possibility of spin and spatial angular momenta to be coupled 3 3 3 to P0, P1 or P2 LS-coupled states. In the presence of a constant magnetic field directed along the z-axis, the axis of quantization, the state with magnetic angular momentum number ML=–1 and with magnetic spin quantum number 3 MS=–1 becomes the ground state. Within LS-coupling this state is the P2- state with total magnetic quantum number MJ =–2, i.e. with the total angular momentum aligned antiparallel to the magnetic field direction. 104 Chapter 10. Atom in an External Magnetic Field

In order to treat the oxygen atom as an isolated system, it was centered in simple cubic supercells of 9, 10, and 11 au side length. Plane-wave cutoffs of 7, 10, and 12 au for the orbitals and of 2, 3, and 3.6 au for the response function, the exchange potential, and the r.h.s of the underlying EXX-equation Z 0 0 0 dr χs(r, r )Vx(r ) = t(r) (10.9) were chosen. All results shown in this chapter are converged with respect to supercell size and plane-wave cutoffs. In analogy to the investigations of trans- polyacetylene presented in chapter 9, the 1s-electrons of the oxygen atom were taken into account via separable EXX pseudopotentials. The s-component was O O chosen as local and for the cutoff radii values of rc,l=0=0.9 and rc,l=1=0.7 in units of Bohr radii were used. For spin-polarized calculations the j-values take the two possible values j = l−1/2 and j = l+1/2. If such relativistic spin-orbit dependent pseudopotentials are not averaged over the two j-values corresponding to one l-quantum number, the spin-orbit contribution would be considered in the SCDFT calculation via the pseudopotentials. If one wishes to omit this spin-orbit contribution in spin-polarized calculations, one simply can use for the two values j =l−1/2 and j =l+1/2 the same pseudopotential obtained by averaging over the original two pseudopotentials. Within this chapter the spin-orbit contribution of the pseudopotentials was left out except for one test that will be explained later. The oxygen atom was subjected to a constant magnetic field B directed along the z-axis, which can be generated by a vector potential A of the form Ax(r) =

−y/2, Ay(r) = x/2, and Az(r) = 0. Such a vector potential is incompatible with periodic boundary conditions and therefore was replaced by the corresponding zig-zag potential, which is discontinuous at the cell boundaries, but inside the cell provides the correct value. It is however necessary to ensure that the wave function and density of the oxygen atom are decayed at the supercell boundaries, in which the atom is put in practice, to make sure that no errors in the performed calculations are introduced. Formulas (10.6) and (10.8) provided a good test of the proper implementation of the magnetic ﬁeld and a correct reproduction of the spin and angular momenta of the orbitals obtained from the pseudopotentials. Only for this test spin-orbit dependent pseudopotentials were required and the eigenvalues of the orbitals ob- 105

-2.2053 ml=1, ms=1/2

m =0, m =1/2 -2.2056 l s p3/2

ml=-1, ms=1/2 m =1, m =-1/2 -2.2059 l s p

Energy [eV] 1/2

ml=0, ms=-1/2

-2.2062 ml=-1, ms=-1/2 0 20 40 60 80 100 Magnetic Field [Tesla]

Figure 10.1: Eigenvalues of oxygen orbitals p1/2 and p3/2 extended to the bare pseudopotential as functions of magnetic field strength. tained with the bare pseudopotential plus the magnetic field were investigated. These values then represent energies of free electrons without any additional interactions like Coulomb repulsion or exchange between them.1 The corresponding values of E/(β B) for the p1/2 and p3/2 orbitals can be calculated by (10.6) and (10.7) and were exactly reproduced by the calculations (see Fig. 10.1). More- over the transition from the Zeeman to the Paschen-Back regime is indicated in the mentioned figure (see approach of the two lines for m` = 1, ms = −1/2 and m` = −1, ms = 1/2). 3 Applying usual angular momentum coupling [92], it turns out that the P2- state itself is a one-determinantal state with occupied p-orbitals p−1,−1/2, p0,−1/2, p+1,−1/2, and p−1,+1/2. This one-determinantal character has to be ensured for a KS treatment of the problem. A KS treatment of this state in the limit of zero magnetic field leads to a symmetry breaking KS wave function because of the partially filled p-shell accompanied by a nonspherical, but cylindrical elec-

1Technically, this is done after the ﬁrst iteration step of the SCF cycle, where interactions between the electrons are not yet comprised. 106 Chapter 10. Atom in an External Magnetic Field

p0,+1/2 0.00 p+1,+1/2 -0.05 p-1,+1/2 -0.10

-0.15 p+1,-1/2 p -0.20 -1,-1/2 p0,-1/2 s -0.95 0,+1/2 Energy [au] -1.00 -1.05 -1.10 s0,-1/2 0 2000 4000 6000 8000 10000 Magnetic Field [T]

Figure 10.2: Eigenvalues of oxygen KS orbitals sm`,ms and pm`,ms as functions of magnetic field strength. tron density. Correspondingly, the symmetry of the KS Hamiltonian operator is reduced from spherical to cylindrical symmetry around the z-axis. In the presence of the magnetic field the latter is the actual symmetry of the system. The energetic degeneracy of the p-orbitals is lifted even in the absence of a magnetic field, because of this symmetry breaking. This can be seen in Fig. 10.2, which displays the KS orbital eigenvalues as a function of the magnetic field strength. In the absence of spin-orbit interactions, the change ∆E of orbital energies and total energy due to the magnetic field of strength B is described according to the Paschen-Back formula. The values of ∆E/(βB) for the curves of the

KS orbital eigenvalues in Fig. 10.2 therefore have to equal (m` + 2ms). Indeed the calculations yielded these values in all cases. For these calculations high magnetic fields were used, because the energy differences due to magnetic fields are extremely small compared to the total electronic energy. The high fields are still in the linear regime as can be seen in Fig. 10.2 and therefore guarantee 107

0.6 ρ00 ρ01 0.4 [au]

01 0.2 / ρ 00

ρ 0

-0.2 -4 -2 0 2 4 y[au]

Figure 10.3: Electron density ρ00 of oxygen and its current ρ01 in x-direction displayed along the y-axis for a magnetic field strength of 104T. numerically more stable calculations. After consideration of the KS orbital energies, one now considers the total 3 energy. For the special case of the P2-state with MJ =–2 the Paschen-Back and the Zeemann formula yield the same energy splitting with respect to the magnetic field strength. It is thus possible to obtain the Landéfactor of oxygen from the performed calculations, despite the neglect of spin-orbit interactions. 3 The Landéfactor for the P2-state of the oxygen was calculated to a value of 1.5 that coincides with the experimental one [116]. These results clearly demonstrate that the new SCDFT approach works correctly in practice. It has to be emphasized that the orbitals in the used plane-wave supercell method are completely arbitrary two-dimensional spinors and therefore the correct spatial and spin angular momenta including their correct alignment along the magnetic field are an outcome of the SCDFT treatment and not enforced by the ansatz for the orbitals.

In Fig. 10.3 the electron density ρ00 and its current ρ01 in x-direction are displayed along the y-axis. As usual in pseudopotential calculations, ρ00 exhibits a minimum at the nucleus. The displayed current ρ01 corresponds to a current 108 Chapter 10. Atom in an External Magnetic Field

0.1

ρ30 ρ 0.05 31 [au]

31 0 / ρ 30 ρ -0.05

-0.1 -4 -2 0 2 4 y [au]

Figure 10.4: Spin density ρ30 of oxygen and its current ρ31 in x-direction displayed along the y-axis for a magnetic ﬁeld strength of 104T.

around the z-axis. The current ρ02 is zero in the yz-plane (but not elsewhere), the current ρ03 is zero everywhere. Thus, as expected, the magnetic ﬁeld aligns the spatial angular momentum along the z-axis. In Fig. 10.4 the spin-density ρ30, i.e. the magnetization, along the z-direction and its current ρ31 in x-direction are displayed along the y-axis. The spin densities ρ10 and ρ20 are zero, thus also the magnetization is correctly aligned along the z-axis by the magnetic ﬁeld. The most important fact that Fig. 10.4 clearly shows is that the spin-current

ρ31 differs from zero. This demonstrates that spin-currents, which can only be correctly described by the presented SCDFT scheme, are far from negligible in a treatment of magnetic systems. They influence the exchange potential and therefore the eigenvalue spectrum of the regarded system. Within this chapter it was shown that spin-currents occur, even if the corresponding components in the real external potential (10.3) vanish. With the inclusion of spin-orbit interactions, these components directly couple to the Hamilto- nian operator as has been shown in chapter 5. In the next chapter spin-orbit splittings in three-dimensional semiconductors will be investigated using the SCDFT 109 approach. Since this formalism explicitly includes the required spin-currents, it offers the possibility to treat spin-orbit effects in a thorough and accurate manner. 110 Chapter 10. Atom in an External Magnetic Field Chapter 11

Spin-orbit Splittings in Diamond-like Semiconductors

Spin-orbit effects clearly affect band structures of semiconductors, especially those that contain heavier elements. Even for a semiconductor like germanium that consists of “light” atoms spin-orbit energy splittings of about 0.30 eV appear [117]. Therefore, an accurate treatment of band structures and related properties within electronic structure calculations should explicitly include spin-orbit effects. Furthermore, these splitting energies are important for various applications, like e.g. the determination of optical transitions [118, 119]. The SCDFT approach presented in chapter 5 offers the possibility to include spin-orbit effects and as a further illustration of this formalism the band structures of silicon and germanium, as well as spin currents induced by spin-orbit effects will be investigated within this chapter [56]. The Hamiltonian of a many electron system in a magnetic field including spin-orbit interactions was already introduced in chapter 5 and reads N X 1 2 1 1 H = p + A(r ) + v (r ) + σ · B(r ) + σ · [(∇v )(r ) × p ] + 2 i i ext i 2 i 4c2 ext i i i=1 N X 1 + . (11.1) |r − r | i

111 112 Chapter 11. Spin-orbit Splittings in Diamond-like Semiconductors

By introducing the 4 × 4-matrix

 A2(r)  vext(r)+ 2 Ax(r) Ay(r) Az(r)    Bx(r) vext,z(r) vext,y(r)   2 0 − 4c2 4c2  V(r) =   (11.3)  By(r) vext,z(r) vext,x(r)   2 4c2 0 − 4c2    Bz(r) vext,y(r) vext,x(r) 2 − 4c2 4c2 0 the Hamiltonian (11.1) can again be expressed in the compact form Z T H = T + Vee + drΣ V(r)J(r). (11.4)

The mapping of the real, interacting electronic system to the noninterating KS system yields the associated KS-Hamiltonian Z T H = Ts + drΣ Vs(r)J(r), (11.5) together with the KS-potential

Vs(r) = V(r) + VH (r) + Vxc(r). (11.6)

The spin-orbit interaction HSO can be obtained from a relativistic many- electron operator by applying a Foldy-Wouthuysen transformation [120]. In the framework of self-consistent all-electron KS calculations this would however lead to difficulties, because of singularity problems of the Foldy-Wouthuysen transformation [121]. For the treatment of relativistic effects within all-electron calculations one therefore would have to combine the newly presented SCDFT approach with procedures of Douglas-Kroll [122,123] or zeroth-order regular approximation (ZORA) [124] type. These transformations are free of singularities and provide a decoupling of the small and big component of the Dirac equation, i.e. transform the four-component Dirac Hamiltonian to effective regular two-component Hamiltonians. For the calculations performed within the framework of this thesis such a combination is however not necessary, because the SCDFT method is implemented within a pseudopotential framework. Spin-orbit effects mostly affect core electrons, but valence electrons are af- fected as well in the regions close to the nuclei. In selfconsistent solid state calculations that employ pseudopotentials, the core electrons are not treated explicitly but taken into account via the employed pseudopotentials. Furthermore, 113 the properties of the valence orbitals in the regions close to the nuclei are predom- inantely determined by the pseudopotential. Thus spin-orbit interactions have to be introduced through the pseudopotentials that are used in the solid state calculations. Therefore, the employed pseudopotentials should be generated in a fully relativistic framework taking into account spin-orbit interactions. At first glance the use of spin-orbit pseudopotentials seems to be sufficient to treat spin-orbit effects within performed solid state calculations. As a consequence, the SCDFT approach then would be superfluous for such investigations. It will be shown that a neglect of the spin-orbit contribution HSO in the Hamilto- nian operator for the valence electrons (11.1) has only little effect on the spin-orbit splitting energies, if the valence electrons are treated within SCDFT. However, the changes of the core orbitals and of the valence orbitals in the vicinity of the nuclei due to spin-orbit interactions also change the valence orbitals in the bond- ing region between the nuclei. As a consequence, this leads to indirect spin-orbit effects in the valence electronic structure and the valence band structure. Effects in the valence band structure correspond to the spin-orbit interaction HSO in the all electron Hamiltonian operator, but are actually induced by the pseudopotentials. In other words, the used pseudopotentials, rather than the operator HSO, give rise to spin-orbit effects and currents of the spin-density. However, an accurate and correct description of these spin-orbit effects, especially a description of currents of the spin-magnetization, and their influence on the band structure, can best be provided by the SCDFT formalism. The solid state calculations presented in this chapter all employed normconserving separable pseudopotentials that were constructed via the relation

j Z −1 sep 0 X h 2 † i Vlj (r, r ) = drr φlj(r)Vlj(r)φlj(r) × m=−j 0 0 † 0 × Vlj(r)φlj(r)Ωljm(ˆr)Vlj(r )φlj(r )Ωljm(ˆr ). (11.7)

In Eq. (11.7) Ωljm denotes 2-dimensional spinors of well deﬁned j-quantum number with j = l ± 1/2 that are obtained by angular momentum coupling from spherical harmonics with quantum numbers l and ml and spineigenfunctions with quantum numbers s = 1/2 and ms = ±1/2. For the performed calculations relativistic atomic lj-dependent pseudopotentials Vlj and pseudoorbitals φlj were 114 Chapter 11. Spin-orbit Splittings in Diamond-like Semiconductors

0 Energy [eV]

-10

L Γ X K,U Γ

Figure 11.1: EXX band structure of germanium without spin-orbit splitting. employed that are based on the relativistic atomic optimized eﬀective potential (OPM) method of Ref. [125].

The pseudoorbitals φlj as solutions of the radial KS equations of relativistic DFT consist of a large and a small component. For the construction of the separable pseudopotentials (11.7), the small component was neglected and only the renormalized large component was taken into account. The pseudopotential generation for the investigated cases of silicon and germanium employed angular momentum components of l = 0, 1, 2 with the j-quantum number taking the Si Si values j = l +1/2 and j = l −1/2. Cutoff radii of rc,l=0 = 1.8, rc,l=1 = 2.0 and Si Ge Ge Ge rc,l=2 = 2.0 for silicon, as well as rc,l=0 = 1.8 rc,l=1 = 2.0 and rc,l=2 = 0.6 for germanium in au, i.e. units of Bohr radii were used. For both cases, the valence space contained no d-electrons, i.e. included the 3s and 3p-states in the case of silicon and the 4s and 4p-states in the case of germanium. For the self consistent solid state calculations plane wave energy cutoffs of 12.5 au for the orbitals and 7 au for the KS response function were chosen. The set of used k-points was chosen as a uniform 4 × 4 × 4 mesh covering the first Brillouin zone. For the lattice constants of silicon and germanium with their diamondlike 115

5 Γ8c

Γ Γ8v 6c 0

Γ7v -5 Energy [eV]

-10

L Γ X K,U Γ

Figure 11.2: EXX-SCDFT band structure of germanium including spin-orbit effects. Energy values for the indicated spin-orbit splittings are given in Table 11.1 crystal structure, experimental values of 5.43A˚ and 5.66A˚ were used respectively. Throughout the calculations, all conduction states were taken into account for the construction of the response function and the right hand side of the SCEXX equation. The calculated EXX band structure for germanium including spin-orbit effects is shown in Fig. 11.2 with the explicit energy values for the indicated spin-orbit splittings listed in Table 11.1. For a comparison, an EXX band structure for germanium without spin-orbit effects is shown in Fig. 11.1. The deviation of a SCDFT calculation that takes into account all 16 spin-currents, as well as the spin-orbit contribution HSO from the experimental value is at about 13.5%. Neglecting the spin-orbit term HSO in the Hamiltonian operator (11.1) leads to no significant changes in the resulting splitting energies as can be seen in Table 11.1. A noncollinear “spin-only” calculation (SDFT) yielded almost similar results as the corresponding SCDFT calculations. In future investigations, however, it has to be seen, if such a similarity of the results still will be obtained in a treatment 116 Chapter 11. Spin-orbit Splittings in Diamond-like Semiconductors

Table 11.1: Spin-orbit splitting energies (in meV) for germanium at the Γ-point, lowered indices indicate the symmetry of the degenerate bands. Experimental data measured via electroreﬂectance (Labeled values are taken from reference [117]).

∆(Γ7v − Γ8v) ∆(Γ6c − Γ8c) SDFT 256.233 185.382 SCDFT 258.097 173.255 SCDFT + SO 258.084 173.221 Experiment 297a 200a

of heavier elements. Besides that, the occurence of spin-currents itself represents a physical phenomenon of high interest, which naturally cannot be treated by “spin-only” calculations. The mentioned deviation from the experimental values might be caused by the fact that for the germanium calculations the 3d-states were not included in the chosen valence space. Calculations within the framework of SCDFT based on such an extended valence space are technically possible, but result in a computational overload, since the representation of the wave-, as well as the response function then clearly requires higher energy cutoffs. A future investigation of the effect of additional d-states within the valence space is highly desirable. Further improvement of the program efficiency by, e.g., ultra-soft pseudopotentials to reduce the required energy cutoffs could solve these problems. An indication of the neglected d-states to be the actual reason for the deviation of the splitting energies from their experimental values in the germanium case are the corresponding results obtained for silicon. In silicon no d-states are present and in this case the calculations achieve almost perfect agreement with experiment, i.e. a deviation of about only 4% (see Table 11.2 for numerical values).

Figures 11.3 and 11.4 show the electron density ρ00 and its current ρ12 along the bond axis, i.e. the unit cell’s diagonal. The plot for ρ00 shows, as usual for pseudopotential calculations, a minimum of the electron density at the positions 117

0.08

0.06 = ρ

00 0.04 ρ

0.02

0 Ge Ge [111] direction

Figure 11.3: Electron density ρ00 (in au) of germanium displayed along the bond axis ([111] direction).

of the germanium nuclei. The spin-currents ρ0µ, ρµ0 and ρµµ (for µ = 1, 2, 3) turned out to be zero. The displayed spin-current ρ12 clearly diﬀers from zero and is identical to the components ρ23 and ρ31. This shows that the used spin- orbit pseudopotentials induce currents of the spin-density that on their part can inﬂuence the resulting electron structures and therefore should not be neglected

Table 11.2: Spin-orbit splitting energies (in meV) for silicon at the Γ-point, lowered indices indicate the symmetry of the degenerate bands. Experimental data measured via wavelength modulated absorption (Labeled value is taken from reference [117]).

∆(Γ25v) SDFT 42.988 SCDFT 42.481 SCDFT + SO 42.587 Experiment 44.1a 118 Chapter 11. Spin-orbit Splittings in Diamond-like Semiconductors

-4 10

12 0 ρ

-4 -10

Ge Ge [111] direction

Figure 11.4: Spin-current ρ12 (in au) of germanium displayed along the bond axis ([111] direction). in a rigorous treatment of spin-orbit eﬀects. At the same time, it is clear that associated components of the exchange potential are induced as well, which change the KS Hamiltonian and to some extent its eigenvalue spectrum.

The regular exchange potential Vx,00, as well as the potential Vx,12 are displayed in Fig. 11.5 and Fig. 11.6. Similar to the spin-currents, the components

Vx,0µ,Vx,µ0 and Vx,µµ (for µ = 1, 2, 3) of the exchange potential showed zero values. Furthermore, the symmetry of the investigated system lead to the relation

ρ12 = ρ23 = −ρ13 = −ρ21 = −ρ32 = ρ31 (11.8) between the spin-currents with nonzero value. The same symmetry relation holds for the nonzero components of the exchange potential, i.e.

Vx,12 = Vx,23 = −Vx,13 = −Vx,21 = −Vx,32 = Vx,31. (11.9)

In summary, it turned out that using the SCDFT approach, together with spin-orbit pseudopotentials, spin-orbit eﬀects can be treated in a proper way. The results for silicon are in excellent agreement with experiment, while the deviation from experiment in case of germanium most likely is due to the restriction of the considered valence space. Although an inclusion of the spin-orbit term in 119

0.3

0.2

0.1

x,00 0 V -0.1

-0.2

Ge Ge [111] direction

Figure 11.5: Exact exchange potential Vx,00 (in au) of germanium displayed along the bond axis ([111] direction). the Hamiltonian operator showed only negligible eﬀect on the spin-orbit splitting energy, a treatment of spin-orbit eﬀects within the SCDFT approach seems to be necessary, because it was shown that currents of the spin density are induced. The

-3 10

x,12 0 V

-3 -10

Ge Ge[111] direction

Figure 11.6: Exact exchange potential Vx,12 (in au) of germanium coupling to a spin-current displayed along the bond axis ([111] direction). 120 Chapter 11. Spin-orbit Splittings in Diamond-like Semiconductors

SCDFT scheme oﬀers an explicit treatment of these spin-currents and therefore represents a thorough formalism to describe spin-orbit eﬀects in many-electron systems. Chapter 12

Summary

This work can thematically be divided into two parts. The first major objec- tive was to provide the option to apply the exact exchange (EXX) Kohn-Sham (KS) formalism within the framework of conventional density-functional theory (DFT) to any kind of periodic system in one- to three dimensions. So far, the EXX scheme has exclusively been applied to atoms, molecules and highly symmetric three-dimensional periodic structures. To provide the possibility for an application of the EXX scheme in practice, a new software was developed that does not rely on any specific symmetries of the investigated system. For a one-dimensional periodic test-system to be investigated, trans- polyacetylene in the form of an isolated chain and in its bulk geometry was chosen. The analysis of this system contained band structures and independent particle response functions that were compared to experimental data and results obtained by various other methods. Similar as in prior calculations on three-dimensional semiconductors, comparison with results obtained via the local density approximation (LDA) showed an increase of the band gap. The EXX results showed good agreement with experiment. The second goal of this work was to provide a sophisticated DFT framework to treat magnetic effects. For this purpose, a new spin-current density-functional theory (SCDFT) scheme was introduced for the first time. Besides the electronic ground state density, the non-collinear spin-density and the paramagnetic current density, which are considered in standard current spin-density-functional theory (CSDFT), the new SCDFT formalism additionally takes into account currents

121 122 Chapter 12. Summary of the spin-density and thus currents of the magnetization. Depending on this set of sixteen basic variables, this new theory shall provide a rigorous description of any magnetic system. Furthermore, the successful EXX scheme of standard DFT was incorporated into the new SCDFT formalism, providing a possibility to apply the new theory in practice. The above mentioned software was extended accordingly. As a first application of the new EXX-SCDFT approach, an isolated oxygen atom subjected to an external magnetic field was chosen. The correct behavior of the eigenvalues of the oxygen KS orbitals with the magnetic field strength, as well as the correct calculated value of the Landéfactor of the oxygen ground state showed that the new SCDFT formalism works correctly in practice. The EXX treatment within SCDFT gained correct spatial and spin angular momenta that were not enforced by the ansatz for the orbitals. A further important outcome of this first investigation was the fact that the newly introduced spin-currents differed from zero and therefore should not be neglected in a proper treatment of magnetic systems. As a second application spin-orbit splittings in the semiconductors silicon and germanium were calculated using the EXX-SCDFT scheme. Here, spin-orbit pseudopotentials were employed. The calculated results for silicon and germanium showed good agreement compared to experimental data. An explicit inclusion of the spin-orbit contribution in the KS Hamiltonian showed only negligible effect on the spin-orbit splitting energies. It was, however, shown that currents of the spin density were induced. The new SCDFT approach offers the opportunity to explicitly take into account spin-currents as fundamental variables. It therefore represents an accurate formalism to describe spin-orbit effects in many-electron systems. Chapter 13

Outlook

The software developed within the framework of this thesis was, except for some few routines, written from scratch. Naturally, a variety of improvements and extensions of the present program version are possible. For an application of the EXX scheme to more complicated systems like inter- or surfaces, a performance improvement of the program is highly desirable. One possibility could be a replacement of the presently used pseudopotentials by ultrasoft- [80] or PAW- [83] pseudopotentials. This type of pseudopotential would provide a more rapid convergence of the wave functions in the used plane-wave representation leading to smaller energy cutoﬀs. So far, a simple linear mixing scheme was implemented. A displacement of this scheme by a more sophisticated mixing scheme like those of Broyden [126,127] or Pulay [128] would deﬁnitely resolve in a reduced number of required iteration steps for the convergence of a performed self consistent procedure. For the diagonalization of the Hamiltonian matrix, i.e. the calculation of the orbitals and their energy eigenvalues a routine of the LAPACK library was used. The scaling behavior of this traditional method is only favorable for a small number of plane waves. While the required memory increases quadratically with the number of plane waves, the computation time scales with the cube of the plane wave number. Especially in view of the application to more complex systems, this direct matrix diagonalization scheme should be replaced by an iterative method like the Davidson algorithm [129, 130]. Using this scheme, it is possible to re- strict the calculation to a reduced set of eigenvectors and energy eigenvalues.

123 124 Chapter 13. Outlook

Furtheron, the accuracy of the eigenvalue calculation can be adapted to the accuracy of the local potential of the respective iteration step, which would reduce the computational costs as well. The parallel version of the program was written using the Message Passing In- terface (MPI) communications protocol and in its current version is not capable of exploiting shared-memory computer architectures. Especially for investigations using a supercell ansatz huge memory amounts are required. A further parallelization using the OpenMP application programming interface (API) [131] that supports multi-platform shared memory multiprocessing programming would provide more available memory for the mentioned architecture type. Besides these technical improvements and extensions, the program could also be extended to calculate additional physical quantities. An already completed project was the development of a generalized singularity correction to the total energy within the EXX approach [90]. This extension makes it possible to calculate the correct total energy for systems with general lattice parameters and provides the possibility of volume optimization. With the total energy at hand, structural and cohesive properties like the lattice constant, bulk modulus and cohesive energies can be calculated. An investigation of these quantities within the EXX scheme for three-dimensional properties has already been performed in [88]. An implementation of these properties into the new program could provide these features for a bigger variety of systems. In the context of the total energy, it would also be worthwhile to provide a calculation of Hellman-Feynman forces [132,133,134,135] for the determination of the equilibrium positions of the nucleii within any unit cell. Another current project is the introduction of fractional occupation numbers to provide the possibility of investigating metals. Here, the Fermi broadening is provided via the Fermi-Dirac function [136] and a replacement of the total energy by a generalized free energy would also make ﬁnite temperature calculations possible. Several other properties, like for instance the density of states and the partial density of states, remain to be implemented in the future. One simply has to look at the capabilities of sophisticated software packages, like ABINIT [137], VASP [138] or WIEN2K [139] to get an impression of possible extensions. 125

For the case of three-dimensional semiconductors, EXX band structures have been used as a starting point for GW [41,42] and TDDFT [20] calculations. Since the new program is not restricted to any kind of symmetry, EXX band structures can now be calculated for a large set of new systems and could also be used as a starting point for GW- or TDDFT calculations. The EXX scheme has been extended to the TDDFT case in [20] yielding an exact expression for the exchange kernel. In the cited work, excitonic peaks in the experimental optical absorption spectrum were reproduced. This was not possible using independent particle response functions. As a next step, one could apply the new SCDFT scheme to the TDDFT case and perform investigations of optical properties using a generalized EXX kernel. 126 Chapter 13. Outlook Chapter 14

Zusammenfassung

Diese Arbeit lässt sich thematisch in zwei Teile gliedern. Das erste Hauptan- liegen war es, die Möglichkeit zu bieten, den Kohn-Sham (KS) Formalismus mit exaktem Austausch (EXX) im Rahmen der konventionellen Dichtefunktio- naltheorie (DFT) auf beliebige periodische Systeme in ein- bis drei Dimensio- nen anwenden zu können. Bis jetzt wurde das EXX-Schema ausschliesslich auf Atome, Moleküleund hochsymmetrische, dreidimensionale periodische Struk- turen angewendet. Um den EXX-Formalismus praktisch anwenden zu können, wurde eine neue Software entwickelt, welche unabhängig von spezifischen Sym- metrien des zu untersuchenden Systems benutzt werden kann. Als eindimensionales, periodisches Testsystem wurde T rans-polyacetylen in Form einer isolierten Kette und in seiner kristallinen Geometrie gewählt. Die Analyse dieses Systems beeinhaltete Bandstrukturen und Kohn-Sham- Responsefunktionen, die mit experimentellen Daten und Ergebnissen anderer theoretischer Modelle, verglichen wurden. Ahnlich¨ zu früherenBerechnungen fürdreidimensionale Halbleiter, zeigten Vergleiche mit Resultaten der lokalen Dichtenäherung (LDA) eine Vergrößerung der Bandlücke. Die Ergebnisse mit exaktem Austausch zeigten eine gute Ubereinstimmung¨ mit dem Experiment. Das zweite Ziel dieser Arbeit war die Formulierung eines ausgereiften DFT-Ansatzes zur Beschreibung magnetischer Effekte. Zu diesem Zweck wurde erstmalig eine neues Spinstromdichtefunktionaltheorie(SCDFT)-Schema vorgestellt. Neben der elektronischen Grundzustandsdichte, der nichtkollinearen Spindichte und der paramagnetischen Stromdichte, welche in der Standard-

127 128 Chapter 14. Zusammenfassung

Stromspindichtefunktionaltheorie (CSDFT) berücksichtigt werden, betrachtet der neue SCDFT-Formalismus darüberhinaus Strömeder Spindichte und damit Ströme der Magnetisierung. Abhängig von diesen sechzehn Basisvariablen soll diese neue Theorie eine rigorose Beschreibung beliebiger magnetischer Systeme bieten. Darüberhinaus wurde das erfolgreiche EXX-Schema der Standarddichte- funktionaltheorie in den neuen SCDFT-Formalismus eingearbeitet, wodurch die Möglichkeit einer praktischen Anwendung der neuen Theorie realisiert wurde. Die oben erwähnte Software wurde entsprechend erweitert. Fürdie erste Anwendung des neuen EXX-SCDFT-Verfahrens wurde ein isoliertes Sauerstoffatom in einem externen Magnetfeld gewählt. Das korrekte Verhalten der Eigenwerte der KS-Orbitale des Sauerstoffs in Abhängigkeit von der magnetischen Feldstärke, sowie der richtig berechnete Wert des Landéfaktors fürden Sauerstoffgrundzustand zeigten, dass der neue SCDFT-Formalismus in der Praxis fehlerfrei arbeitet. Das EXX-Verfahren im Rahmen der SCDFT lieferte korrekte Bahn- und Spindrehimpulse, die nicht durch einen Ansatz fürdie Or- bitale erzwungen wurden. Ein weiteres, wichtiges Ergebnis dieser ersten Unter- suchung war die Tatsache, dass die neu eingeführtenSpinströmevom Nullwert abwichen und aus diesem Grunde bei einer angemessenen Beschreibung magnetischer Systeme nicht vernachlässigtwerden sollten. Als zweite Anwendung wurden Spin-Orbit-Aufspaltungen in den Halbleitern Silizium und Germanium unter Verwendung des EXX-SCDFT-Schemas berech- net. Hierbei wurden Spin-Orbit-Pseudopotentiale verwendet. Die berechneten Werte fürSilizium und Germanium zeigten eine gute Ubereinstimmung¨ mit experimentellen Daten. Eine explizite Berücksichtigung des Spin-Orbit-Beitrags im KS-Hamiltonoperator resultierte lediglich in einem vernachlässigbaren Effekt auf die Spin-Orbit-Aufspaltungsenergien. Es wurde jedoch gezeigt, dass Strömeder Spindichte induziert wurden. Das neue SCDFT-Verfahren bietet die Möglichkeit Spinströme als fundamentale Basisvariable explizit zu berücksichtigen. Aus diesem Grunde stellt das Verfahren einen präzisenund vollständigen Formalismus zur Beschreibung von Spin-Orbit-Effekten in Vielelektronsystemen dar. Appendix A

Geometric Data for T rans-polyacetylene

Table A.1: Geometric data for bulk trans-polyacetylene given in atomic units.

Real space unit cell vectors are denoted as ai, coordinates of hydrogen (carbon) atoms are denoted as Hi (Ci).

x y z

a1 8.01244060 0.0 0.0

a2 -0.12144217 4.63768905 0.0

a3 0.0 0.0 13.83279947

H1 -1.76389103 -1.153571655 -5.367986055

H2 -2.18160819 -1.165272865 1.548413675

H3 2.18160821 1.165272865 -1.548413665

H4 1.88533319 -3.484117395 5.367986055

C1 -3.44604796 -1.194232545 -6.546186695

C2 -0.49945126 -1.124611975 0.370213055

C3 0.49945126 1.124611975 -0.370213055

C4 3.56749012 -3.443456505 6.546186695

129 130 Appendix A. Geometric Data for T rans-polyacetylene

Table A.2: Geometric data for isolated chain of trans-polyacetylene (geometry G1) given in atomic units. Real space unit cell vectors are denoted as ai, coordinates of hydrogen (carbon) atoms are denoted as Hi (Ci).

x y z

a1 15.11781328 0.0 0.0

a2 0.0 4.63927896 0.0

a3 7.55890664 0.0 17.00753994

H1 -2.69959124 1.10770723 0.0

H2 2.69959124 -1.10770723 0.0

C1 -0.64546122 1.11114406 0.0

C2 0.64546122 -1.11114406 0.0

Table A.3: Geometric data for isolated chain of trans-polyacetylene (geometry G2) given in atomic units. Real space unit cell vectors are denoted as ai, coordinates of hydrogen (carbon) atoms are denoted as Hi (Ci).

x y z

a1 15.11781328 0.0 0.0

a2 0.0 4.67329404 0.0

a3 7.55890664 0.0 17.00753994

H1 -2.69868711 1.24389067 0.0

H2 2.69850018 -1.24389067 0.0

C1 -0.61981899 1.21122435 0.0

C2 0.61981899 -1.21122435 0.0 List of Tables

4.1 Comparison of EXX and LDA eigenvalues ...... 44

9.1 Band gaps for trans-polyacetylene chain caluclated with various methods ...... 93 9.2 Band gaps for bulk trans-polyacetylene ...... 97

11.1 Spin-orbit splitting energies for Ge ...... 116 11.2 Spin-orbit splitting energies for Si ...... 117

A.1 Geometric data for bulk trans-polyacetylene ...... 129 A.2 Geometric data for trans-polyacetylene chain (geometry G1) . . . 130 A.3 Geometric data for trans-polyacetylene chain (geometry G2) . . . 130

131 132 List of Tables List of Figures

4.1 LDA- and EXX band structures of Si ...... 45

7.1 Program sequence ...... 69 7.2 Parallelization ...... 71

8.1 Building the Hamiltonian ...... 75

8.2 Calculation of ρµν ...... 77 µν,κλ 8.3 Calculation of χ0 ...... 78

8.4 Calculation of VH ...... 79

8.5 Calculation of Vxc,LDA ...... 80

8.6 Calculation of Vx,GGA ...... 82

8.7 Calculation of Vc,GGA ...... 83

8.8 Calculation of tµν ...... 85

9.1 Band structure of trans-polyacetylene chain ...... 92 9.2 Band structure of bulk trans-polyacetylene ...... 95 9.3 Band gap dispersion of bulk trans-polyacetylene ...... 96 9.4 Kohn-Sham response function of trans-polyacetylene chain . . . . 98 9.5 Kohn-Sham response function of bulk trans-polyacetylene . . . . 99

10.1 Eigenvalues of oxygen orbitals (bare pseudopotential case) . . . . 105 10.2 Eigenvalues of oxygen orbitals ...... 106 3 10.3 Electron density ρ00 and its current ρ01 of oxygen P2-state . . . . 107 3 10.4 Spin density ρ30 and its current ρ31 of oxygen P2-state ...... 108

11.1 EXX band structure of Ge without spin-orbit splitting ...... 114 11.2 EXX-SCDFT band structure of Ge with spin-orbit splitting . . . 115

133 134 List of Figures

11.3 Electron density ρ00 of Ge ...... 117

11.4 Spin-current ρ12 of Ge ...... 118

11.5 EXX potential Vx,00 of Ge ...... 119

11.6 EXX potential Vx,12 of Ge ...... 119 Bibliography

[1] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev. 136, 864 (1964).

[2] W. Kohn and L. Sham. Self-consistent equations including exchange and correlation eﬀects. Phys. Rev. 140, 1133 (1965).

[3] R. G. Parr and W. Yang. Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York (1989).

[4] R. M. Dreizler and E. K. U. Gross. Density-Functional Theory. Springer, Berlin (1990).

[5] W. Koch and M. C. Holthausen. A Chemist’s Guide to Density Functional Theory. Wiley, New York (2000).

[6] L. Hedin. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys. Rev. 139, A796 (1965).

[7] M. S. Hybertsen and S. G. Louie. First-principles theory of quasiparticles: calculation of band gaps in semiconductors and insulators. Phys. Rev. Lett. 55, 1418 (1985).

[8] F. Aryasetiawan and O. Gunnarsson. The GW method. Rep. Prog. Phys. 61, 237 (1998).

[9] W. G. Aulbur, L. J¨onssonand J. W. Wilkins. Quasiparticle calculations in solids. Solid State Phys.: Adv. Res. Appl. 54, 1 (2000).

135 136 Bibliography

[10] G. Onida, L. Reining and A. Rubio. Electronic excitations: density- functional versus many-body Green’s-function approaches. Rev. Mod. Phys. 74, 601 (2002).

[11] W. Hanke and L. J. Sham. Many-particle eﬀects in the optical excitations of a semiconductor. Phys. Rev. Lett. 43, 387 (1979).

[12] S. Albrecht, L. Reining, R. DelSole and G. Onida. Ab initio calculation of excitonic eﬀects in the optical spectra of semiconductors. Phys. Rev. Lett. 80, 4510 (1998).

[13] L. X. Benedict, E. L. Shirley and R. B. Bohn. Optical absorption of insulators and the electron-hole interaction: An ab initio calculation. Phys. Rev. Lett. 80, 4514 (1998).

[14] M. Rohlﬁng and S. G. Louie. Electron-hole excitations in semiconductors and insulators. Phys. Rev. Lett. 81, 2312 (1998).

[15] E. Runge and E. K. U. Gross. Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52, 997 (1984).

[16] E. K. U. Gross, J. F. Dobson and M. Petersilka. in Density Functional Theory II, Springer Series in Topics in Current Chemistry, Band 181, ed. R.F. Nalewajski. Springer, Heidelberg (1996).

[17] M. E. Casida. in Recent Advances in Density Functional Methods, Band I, eds. D. P. Chong. World Scientiﬁc, Singapore (1995).

[18] A. G¨orling. Exact exchange kernel for time-dependent density-functional theory. Int. J. Quantum Chem. 69, 265 (1998).

[19] A. G¨orling, H. H. Heinze, S. P. Ruzankin, M. Staufer and N. R¨osch. Density- and density-matrix-based coupled Kohn-Sham methods for dynamic polar- izabilities and excitation energies of molecules. J. Chem. Phys. 110, 2785 (1999).

[20] Y. H. Kim and A. G¨orling. Excitonic optical spectrum of semiconductors obtained by time-dependent density-functional theory with the exact- exchange kernel. Phys. Rev. Lett. 89, 096402 (2002). Bibliography 137

[21] J. P. Perdew and M. Levy. Physical content of the exact Kohn-Sham orbital energies: Band gaps and derivative discontinuities. Phys. Rev. Lett. 51, 1884 (1983).

[22] L. J. Sham and M. Schl¨uter. Density-functional theory of the energy gap. Phys. Rev. Lett. 51, 1888 (1983).

[23] C. Filippi, D. J. Singh and C. J. Umrigar. All-electron local-density and generalized-gradient calculations of the structural properties of semiconductors. Phys. Rev. B 50, 14947 (1994).

[24] G. Ortiz. Gradient-corrected pseudopotential calculations in semiconductors. Phys. Rev. B 45, 11328 (1992).

[25] L. J. Sham and M. Schl¨uter. Density-functional theory of the band gap. Phys. Rev. B 32, 3883 (1985).

[26] J. P. Perdew, R. G. Parr, M. Levy and J. L. Balduz. Density-functional theory for fractional particle number: Derivative discontinuities of the energy. Phys. Rev. Lett. 49, 1691 (1982).

[27] C. O. Almbladh and U. von Barth. Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues. Phys. Rev. B 31, 3231 (1985).

[28] D. P. Chong, O. V. Gritsenko and E. J. Baerends. Interpretation of the Kohn-Sham orbital energies as approximate vertical ionization potentials. J. Chem. Phys. 116, 1760 (2002).

[29] O. V. Gritsenko and E. J. Baerends. The analog of Koopmans’ theorem in spin-density functional theory. J. Chem. Phys. 117, 9154 (2002).

[30] A. G¨orling. Density-functional theory for excited states. Phys. Rev. A 54, 3912 (1996).

[31] C. Filippi, C. J. Umrigar and X. Gonze. Excitation energies from density functional perturbation theory. J. Chem. Phys. 107, 9994 (1997). 138 Bibliography

[32] A. G¨orling and M. Levy. Hardness of molecules and the band gap of solids within the Kohn-Sham formalism: A perturbation-scaling approach. Phys. Rev. A 52, 4493 (1995).

[33] T. Kotani. Exact exchange potential band-structure calculations by the linear muﬃn-tin orbital-atomic-sphere approximation method for Si, Ge, C, and MnO. Phys. Rev. Lett. 74, 2989 (1995).

[34] A. G¨orling. Exact treatment of exchange in Kohn-Sham band-structure schemes. Phys. Rev. B 53, 7024 (1995).

[35] A. G¨orling. Erratum: Exact treatment of exchange in Kohn-Sham band- structure schemes. Phys. Rev. B 59, 10370 (1999).

[36] M. St¨adele, J. A. Majewski, P. Vogl and A. G¨orling. Exact Kohn-Sham exchange potential in semiconductors. Phys. Rev. Lett. 79, 2089 (1997).

[37] M. St¨adele,M. Moukara, J. A. Majewski, P. Vogl and A. G¨orling. Exact exchange Kohn-Sham formalism applied to semiconductors. Phys. Rev. B 59, 10031 (1999).

[38] A. Qteish, A. I. Al-Sharif, M. Fuchs, M. Scheﬄer, S. Boeck and J. Neuge- bauer. Exact-exchange calculations of the electronic structure of AlN, GaN and InN. Comp. Phys. Comm. 169, 28 (2005).

[39] P. Rinke, A. Qteish, J. Neugebauer, C. Freysoldt and M. Scheﬄer. Com- bining GW calculations with exact-exchange density-functional theory: an analysis of valence-band photoemission for compound semiconductors. New J. Phys. 7, 126 (2005).

[40] S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl. All-electron exact exchange treatment of semiconductors: Eﬀect of core-valence interaction on band-gap and d-band position. Phys. Rev. Lett. 95, 136402 (2005).

[41] W. G. Aulbur, M. St¨adeleand A. G¨orling. Exact-exchange-based quasiparticle calculations. Phys. Rev. B 62, 7121 (2000). Bibliography 139

[42] A. Fleszar. LDA, GW, and exact-exchange Kohn-Sham scheme calculations of the electronic structure of sp semiconductors. Phys. Rev. B 64, 245204 (2001).

[43] J. D. Talman and W. F. Shadwick. Optimized eﬀective atomic central potential. Phys. Rev. A 14, 36 (1976).

[44] A. G¨orling. New KS method for molecules based on an exchange charge density generating the exact local KS exchange potential. Phys. Rev. Lett. 83, 5459 (1999).

[45] S. Ivanov, S. Hirata and R. J. Bartlett. Exact exchange treatment for molecules in ﬁnite-basis-set Kohn-Sham theory. Phys. Rev. Lett. 83, 5455 (1999).

[46] S. Sugano and N. Kojima. Magneto-optics. Springer, Berlin (2000).

[47] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova and D. M. Treger. Spintronics: A spin-based electronics vision for the future. Science 294, 1488 (2001).

[48] R. R. Ernst, G. Bodenhausen and A. Wokaun. Principles of nuclear magnetic resonance in one and two dimensions. Oxford University Press, New York (1990).

[49] U. von Barth and L. Hedin. A local exchange-correlation potential for the spin-polarized case. J. Phys. C: Solid State Phys. 5, 1629 (1972).

[50] J. K¨ubler,K. H. H¨ock, J. Sticht and A. R. Williams. Density functional theory of non-collinear magnetism. J. Phys. F 18, 469 (1988).

[51] K. Capelle and E. K. U. Gross. Spin-density functionals from current- density functional theory and vice versa: A road towards new approximations. Phys. Rev. Lett. 78, 1872 (1997).

[52] G. Vignale and M. Rasolt. Density-functional theory in strong magnetic ﬁelds. Phys. Rev. Lett. 59, 2360 (1987). 140 Bibliography

[53] G. Vignale and M. Rasolt. Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic ﬁelds. Phys. Rev. B 37, 10685 (1988).

[54] P. Skudlarski and G. Vignale. Exchange-correlation energy of a three- dimensional electron gas in a magnetic ﬁeld. Phys. Rev. B 48, 8547 (1993).

[55] S. Rohra and A. G¨orling. Exact-Exchange Spin-Current Density-Functional Theory. Phys. Rev. Lett. (accepted), cond-mat/0511156 (2005).

[56] 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.) (2006).

[57] S. Rohra, E. Engel and A. G¨orling. Exact-Exchange Kohn-Sham formalism applied to one-dimensional periodic electronic systems. Phys. Rev. B (accepted), cond-mat/0512299 (2005).

[58] L. Thomas. The calculation of atomic ﬁelds. Proc. Cambridge Phil. Soc. 23, 542 (1927).

[59] E. Fermi. Un metodo statistice per la determinazione di alcune proprieta dell’atomo. Rend. Accad. Lincei 6, 602 (1927).

[60] M. Levy. Electron densities in search of Hamiltonians. Phys. Rev. A 26, 1200 (1982).

[61] P. O. L¨owdin. Correlation problem in many-electron quantum mechanics. Adv. Chem. Phys. 2, 207 (1959).

[62] M. S. Hybertsen and S. G. Louie. Ab initio static dielectric matrices from the density-functional approach. I. Formulation and application to semiconductors and insulators. Phys. Rev. B 35, 5585 (1987).

[63] Y. H. Kim and A. G¨orling. Exact Kohn-Sham exchange kernel for insulators and its long-wavelength behavior. Phys. Rev. B 66, 35114 (2002).

[64] P. A. M. Dirac. Note on exchange phenomena in the Thomas atom. Proc. Camb. Phil. Soc. 26, 376 (1930). Bibliography 141

[65] S. J. Vosko, L. Wilk and M. Nusair. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis. Can. J. Phys. 58, 1200 (1980).

[66] D. M. Ceperly and B. J. Alder. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566 (1980).

[67] A. D. Becke. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 38, 3098 (1988).

[68] J. P. Perdew, K. Burke and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).

[69] C. Lee, W. Yang and R. G. Parr. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37, 785 (1988).

[70] J. P. Perdew. Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys. Rev. B 33, 8822 (1986).

[71] J. P. Perdew. Erratum: Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys. Rev. B 34, 7406 (1986).

[72] J. P. Perdew. in Electronic Structure of Solids. Akademie Verlag, Berlin (1991).

[73] J. P. Perdew and W. Yue. Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. Phys. Rev. B 33, 8800 (1986).

[74] J. C. Phillips and L. Kleinman. New method for calculating wave functions in crystals and molecules. Phys. Rev 116, 287 (1959).

[75] L. Kleinman and D. M. Bylander. Eﬃcacious form for model pseudopotentials. Phys. Rev. Lett. 48, 1425 (1982).

[76] D. R. Hamann, M. Schl¨uter and C. Chiang. Norm-conserving pseudopotentials. Phys. Rev. Lett. 43, 1494 (1979). 142 Bibliography

[77] X. Gonze, P. K¨ackell and M. Scheﬄer. Ghost states for separable, normconserving, ab initio pseudopotentials. Phys. Rev. B 41, 12264 (1990).

[78] X. Gonze, R. Stumpf and M. Scheﬄer. Analysis of separable potentials. Phys. Rev. B 44, 8503 (1991).

[79] G. B. Bachelet, D. R. Hamann and M. Schl¨uter. Pseudopotentials that work: From H to Pu. Phys. Rev. B 26, 4199 (1982).

[80] D. Vanderbilt. Optimally smooth norm-conserving pseudopotentials. Phys. Rev. B 32, 8412 (1985).

[81] N. Troullier and J. L. Martins. Eﬃcient pseudopotentials for plane-wave calculations. Phys. Rev. B 43, 1993 (1991).

[82] N. Troullier and J. L. Martins. Eﬃcient pseudopotentials for plane-wave calculations. II. Operators for fast iterative diagonalization. Phys. Rev. B 43, 8861 (1991).

[83] P. E. Bl¨ochl. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).

[84] S. G. Louie, S. Froyen and M. L. Cohen. Nonlinear ionic pseudopotentials in spin-density-functional calculations. Phys. Rev. B 26, 1738 (1982).

[85] N. W. Ashcroft and N. D. Mermin. Solid State Physics. Thomson Learning, USA (2001).

[86] A. Baldereschi. Mean-value point in the Brillouin zone. Phys. Rev. B 7, 5212 (1973).

[87] H. J. Monkhorst and J. D. Pack. Special points for Brillouin-zone integra- tions. Phys. Rev. B 13, 5188 (1976).

[88] M. Städele. Dichtfunktionaltheorie mit exaktem Austausch fürHalbleiter. Ph.D. thesis, Technische UniversitätMünchen (1998).

[89] F. Gygi and A. Baldereschi. Self-consistent Hartree-Fock and screened- exchange calculations in solids: Application to silicon. Phys. Rev. B 34, 4405 (1986). Bibliography 143

[90] P. Carrier, S. Rohra and A. G¨orling. Generalised singularity correction to the total energy within exact exchange. submitted to Phys. Rev. B, cond- mat/0603632 (2006).

[91] R. J. Magyar, A. Fleszar and E. K. U. Gross. Exact-exchange density- functional calculations for noble-gas solids. Phys. Rev. B 69, 045111 (2004).

[92] M. Weissbluth. Atoms and Molecules. Academic Press, New York (1978).

[93] J. D. Jackson. Classical Electrodynamics. Wiley, New York (1999).

[94] G. Diener. Current-density-functional theory for a nonrelativistic electron gas in a strong magnetic ﬁeld. J. Phys.: Condens. Matter 3, 9417 (1991).

[95] C. L. Lawson, R. J. Hanson, D. Kincaid and F. T. Krogh. Basic linear algebra subprograms for FORTRAN usage. ACM Trans. Math. Soft. 5, 308 (1979).

[96] L. S. Blackford, J. Demmel, J. Dongarra, I. Duﬀ, S. Hammarling, G. Henry, M. Heroux, L. Kaufman, A. Lumsdaine, A. Petitet, R. Pozo, K. Remington and R. C. Whaley. An updated set of basic linear algebra subprograms (BLAS). ACM Trans. Math. Soft. 28, 135 (2002).

[97] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. D. Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA (1999).

[98] M. Frigo and S. G. Johnson. The design and implementation of FFTW3. Proceedings of the IEEE 93, 216 (2005).

[99] M. P. I. Forum. MPI: A message-passing interface standard. The In- ternational Journal of Supercomputer Applications and High Performance Computing 8, 159 (1994).

[100] M. E. Rose. Elementary theory of angular momentum. Dover, New York (1995). 144 Bibliography

[101] P. A. M. Dirac. Quantum Mechanics of Many-Electron Systems. Proc. Royal Soc.(London) A 123, 714 (1929).

[102] J. P. Perdew and Y. Wang. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244 (1992).

[103] C. M. Liegener. Ab initio calculations of correlation eﬀects in trans- polyacetylene. J. Chem. Phys. 88, 6999 (1988).

[104] S. Suhai. Structural and electronic properties of inﬁnite cis- and trans- polyenes: Perturbation theory of electron correlation eﬀects. Int. J. Quant. Chem. 42, 193 (1992).

[105] J. Q. Sun and R. J. Bartlett. Second-order many-body perturbation-theory calculations in extended systems. J. Chem. Phys. 104, 8553 (1996).

[106] W. F¨orner,R. Knab, J. Cizek and J. Ladik. Numerical application of the coupled cluster theory with localized orbitals to polymers. IV. Band structure corrections in model systems and polyacetylene. J. Chem. Phys. 106, 10248 (1997).

[107] M. Rohlﬁng and S. G. Louie. Optical excitations in conjugated polymers. Phys. Rev. Lett. 82, 1959 (1999).

[108] P. Y. Ayala, K. N. Kudin and G. E. Scuseria. Atomic orbital Laplace- transformed second-order Møller-Plesset theory for periodic systems. J. Chem. Phys. 115, 9698 (2001).

[109] V. Bezugly and U. Birkenheuer. Multireference conﬁguration interaction treatment of excited-state electron correlation in periodic systems: the band structure of trans-polyacetylene. Chem. Phys. Lett. 399, 57 (2004).

[110] C. R. Fincher, M. Ozaki, M. Tanaka, D. Peebles, L. Lauchlan, A. J. Heeger and A. G. MacDiarmid. Electronic structure of polyacetylene: Optical

and infrared studies of undoped semiconducting (CH)x and heavily doped

metallic (CH)x. Phys. Rev. B 20, 1589 (1979). Bibliography 145

[111] G. Leising. Anisotropy of the optical constants of pure and metallic polyacetylene. Phys. Rev. B 38, 10313 (1988).

[112] E. Engel, A. H¨ock, R. N. Schmid and R. M. Dreizler. in Reviews of Modern Quantum Chemistry. World Scientiﬁc, Singapore (2002).

[113] C. R. Fincher, C. E. Chen, A. J. Heeger, A. G. MacDiarmid and J. B. Hastings. Structural determination of the symmetry-breaking parameter

in trans-(CH)x. Phys. Rev. Lett. 48, 100 (1982).

[114] H. Kahlert, O. Leitner and G. Leising. Structural properties of trans- and

cis-(CH)x. Synth. Met. 17, 467 (1987).

[115] C. S. Yannoni and T. C. Clarke. Molecular geometry of cis- and trans- polyacetylene by nutation NMR spectroscopy. Phys. Rev. Lett. 51, 1191 (1983).

[116] H. E. Radford and V. W. Hughes. Microwave Zeeman spectrum of atomic oxygen. Phys. Rev 114, 1274 (1959).

[117] O. Madelung. Semiconductors: Data Handbook. Springer, Berlin (2004).

[118] S. H. Wei and A. Zunger. Optical anisotropy and spin polarization in ordered GaInP. Appl. Phys. Lett. 64, 1676 (1994).

[119] A. Janotti and S. H. Wei. Computational design of a material for high- eﬃciency spin-polarized electron source. Appl. Phys. Lett. 81, 3957 (2002).

[120] L. L. Foldy and S. A. Wouthuysen. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Phys. Rev. 78, 29 (1950).

[121] M. Reiher and A. Wolf. Exact decoupling of the Dirac Hamiltonian. I. General theory. J. Chem. Phys. 121, 2037 (2004).

[122] M. Douglas and N. M. Kroll. Quantum electrodynamical corrections to the ﬁne structure of helium. Ann. Phys. 82, 89 (1974).

[123] B. A. Hess. Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-ﬁeld projection operators. Phys. Rev. A 33, 3742 (1986). 146 Bibliography

[124] E. van Lenthe, E. J. Baerends and J. G. Snijders. Relativistic regular two-component Hamiltonians. J. Chem. Phys. 99, 4597 (1993).

[125] E. Engel, A. H¨ock and S. Varga. Relativistic extension of the Troullier- Martins scheme: Accurate pseudopotentials for transition-metal elements. Phys. Rev. B 63, 125121 (2001).

[126] C. G. Broyden. A class of methods for solving nonlinear simultaneous equations. Math. Comp. 19, 577 (1965).

[127] D. D. Johnson. Modiﬁed Broyden’s method for accelerating convergence in self-consistent calculations. Phys. Rev. B 38, 12807 (1988).

[128] P. Pulay. Convergence acceleration of iterative sequences. The case of scf iteration. Chem. Phys. Lett. 73, 393 (1980).

[129] E. R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 87 (1975).

[130] N. Kosugi. Modiﬁcation of the Liu-Davidson method for obtaning one or simultaneously several eigensolutions of a large real symmetric matrix. J. Comput. Phys. 55, 426 (1984).

[131] R. Chandra, L. Dagum, D. Kohr, D. Maydan, J. McDonald and R. Menon. Parallel Programming in OpenMP. Academic Press, San Diego, USA (2000).

[132] R. P. Feynman. Forces in Molecules. Phys. Rev. 56, 340 (1939).

[133] H. Hellmann. Einf¨uhrungin die Quantenchemie. Deuieke, Leipzig (1937).

[134] R. Yu, D. Singh and H. Krakauer. All-electron and pseudopotential force calculations using the linearized-augmented-plane-wave method. Phys. Rev. B 43, 6411 (1991).

[135] P. Bendt and A. Zunger. Simultaneous relaxation of nuclear geometries and electric charge densities in electronic structure theories. Phys. Rev. Lett. 50, 1684 (1983). Bibliography 147

[136] N. D. Mermin. Thermal properties of the inhomogeneous electron gas. Phys. Rev. 137, A1441 (1965).

[137] http://www.abinit.org/.

[138] http://cms.mpi.univie.ac.at/vasp/.

[139] http://www.wien2k.at/. 148 Bibliography Lebenslauf Stefan Bruno Rohra

Personliche¨ Daten Geburtstag 4. April, 1975 Geburtsort M¨unchen Nationalit¨at deutsch Familienstand ledig

Ausbildung seit 07/2002 Doktorarbeit an der Friedrich–Alexander-UniversitätErlangen–Nürnberg, Rheinischen Friedrich–Wilhelms-Universität Bonn, Technischen UniversitätMünchen 11/1995–12/2000 Studium der Physik an der Ludwig–Maximilians-Universität München 09/1985–06/1994 Asam-Gymnasium in München 09/1981–07/1985 Grundschule in München

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