Orbital-dependent exchange-correlation functionals in density-functional theory realized by the FLAPW method

Von der Fakultät für Mathematik, Informatik und NaturwissenschaŸen der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der NaturwissenschaŸen genehmigte Dissertation

vorgelegt von

Dipl.-Phys. Markus Betzinger

aus Fröndenberg

Berichter: Prof. Dr. rer. nat. Stefan Blügel Prof. Dr. rer. nat. Andreas Görling Prof. Dr. rer. nat. Carsten Honerkamp Tag der mündlichen Prüfung: Ô¥.Ôò.òýÔÔ

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. is document was typeset with LATEX. Figures were created using Gnuplot, InkScape, and VESTA. Das Unverständlichste am Universum ist im Grunde, dass wir es verstehen können. Albert Einstein

Abstract

In this thesis, we extended the applicability of the full-potential linearized augmented-plane- wave (FLAPW) method, one of the most precise, versatile and generally applicable elec- tronic structure methods for solids working within the framework of density-functional the- ory (DFT), to orbital-dependent functionals for the exchange-correlation (xc) energy. In con- trast to the commonly applied local-density approximation (LDA) and generalized gradient approximation (GGA) for the xc energy, orbital-dependent functionals depend directly on the Kohn-Sham (KS) orbitals and only indirectly on the density. Two dišerent schemes that deal with orbital-dependent functionals, the KS and the gen- eralized Kohn-Sham (gKS) formalism, have been realized. While the KS scheme requires a local multiplicative xc potential, the gKS scheme allows for a non-local potential in the one- particle Schrödinger equations. Hybrid functionals, combining some amount of the orbital-dependent exact exchange en- ergy with local or semi-local functionals of the density, are implemented within the gKS scheme. We work in particular with the PBEý hybrid of Perdew, Burke, and Ernzerhof. Our implementation relies on a representation of the non-local exact exchange potential – its cal- culation constitutes the most time consuming step in a practical calculation – by an auxiliary mixed product basis (MPB). In this way, the matrix elements of the Hamiltonian correspond- ing to the non-local potential become a Brillouin-zone (BZ) sum over vector-matrix-vector products. Several techniques are developed and explored to further accelerate our numerical scheme. We show PBEý results for a variety of semiconductors and insulators. In compar- ison with experiment, the PBEý functional leads to improved band gaps and an improved description of localized states. Even for the ferromagnetic semiconductor EuO with local- ized ¥ f electrons, the electronic and magnetic properties are correctly described by the PBEý functional. Subsequently, we discuss the construction of the local, multiplicative exact exchange (EXX) potential from the non-local, orbital-dependent exact exchange energy. For this purpose we employ the optimized ešective potential (OEP) method. Central ingredients of the OEP equation are the KS wave-function response and the single-particle density response func- tion. A formulation in terms of a slightly modied MPB enables to solve the OEP integral

i equation for the local potential without any shape approximations for the potential. We show that a balance between the LAPW and mixed product basis is mandatory for a smooth and physical local EXX potential. e LAPW basis must be converged to an accuracy which is far beyond that for LDA or GGA calculations. We demonstrate that this is necessary to lend the LAPW basis and thus the KS wave functions and density su›cient žexibility to react ad- equately to the changes of the ešective potential, which are described in our formalism by the MPB. If both basis sets are properly balanced, our results for C, Si, SiC, Ge, GaAs as well as solid Ne and Ar are in favorable agreement with plane-wave pseudopotential results. Be- cause of the exceedingly large LAPW basis sets the EXX-OEP approach is computationally expensive. We propose a correction, the nite basis-set correction (FBC), for the density and wave-function response, which explicitly considers the dependence of the LAPW basis on the ešective potential and which vanishes in the limit of an innite, complete basis. For the example of ScN, we demonstrate that the FBC leads to converged potentials at much smaller LAPW basis sets and thus turns the EXX-OEP approach into a practical method. Finally, we discuss a generalization of the formalism to metals and report results for the cubic perovskites

CaTiOç, SrTiOç, and BaTiOç, the transition-metal oxides MnO, FeO, and CoO as well as the metals Al, Na, and Cu.

ii Zusammenfassung

Im Rahmen dieser Arbeit wurde die full-potential linearized augmented-plane-wave (FLAPW) Methode, eine der präzisesten, vielseitigsten und allgemein anwendbarsten elektronischen Strukturmethoden der Dichtefunktionaltheorie (DFT) für Festkörper, um orbitalabhängige Funktionale für die Austausch-Korrelations (xc) Energie erweitert. Im Gegensatz zu den geläugen Funktionalen, wie der lokalen Dichtenährung (LDA) und der verallgemeinerten Gradientennährung (GGA), hängen orbitalabhängige Funktionale direkt von den Kohn-Sham (KS) Orbitalen ab und sind damit nur indirekte Funktionale der Dichte. Für die numerische Umsetzung orbitalabhängiger Funktionale haben sich zwei unter- schiedliche Ansätze etabliert, der KS und der verallgemeinerte KS (gKS) Formalismus. Sie unterscheiden sich in der Behandlung des xc Potentials. Während der KS Formalismus ein lokales, multiplikatives xc Potential erfordert, lässt das gKS Schema auch ein nicht-lokales Potential zu. Im Rahmen dieser Arbeit werden beiden Ansätze diskutiert und untersucht. Hybridfunktionale, die typischerweise im gKS Schema behandelt werden, kombinieren orbitalabhängigen exakten Austausch mit dichteabhängigen LDA oder GGA Funktionalen. Im Speziellen konzentrieren wir uns in dieser Arbeit auf das Hybridfunktional von Perdew, Burke und Ernzerhof (PBEý). Die entwickelte Implementierung des PBEý Funktionales beruht auf einer Darstellung des nicht-lokalen Austauschpotentials durch eine Hilfsbasis, die gemischte Produktbasis (MPB). Mittels dieser werden die Matrixelemente des nicht- lokalen Austauschpotentials zum Hamiltonian eine Brillouin-Zone (BZ) Summe über Vektor-Matrix-Vektor Produkte. Zur beschleunigten Berechnung dieser Matrixelemente werden weitere Techniken entwickelt und erforscht. Im Vergleich zu LDA und GGA führt das PBEý Funktional zu einer verbesserten Beschreibung der experimentellen Bandlücke und zu einer verbesserten Beschreibung von lokalisierten Zuständen, wie d- und f -Elektron Zuständen. Zum Beispiel beschreibt das PBEý Funktional die elektronischen und mag- netischen EigenschaŸen des ferromagnetischen Halbleiters EuO mit seinen lokalisierten ¥ f Zuständen im Einklang mit dem Experiment. Die Konstruktion eines lokalen, multiplikativen Austauschpotentials im Rahmen des KS Formalismus ausgehend von der nicht-lokalen, orbitalabhängigen, exakten Austauschen- ergie (EXX) erfordert hingegen die optimierte-ešektive-Potential (OEP) Methode. Sie führt

iii zu einer Integralgleichung für das lokale Austauschpotential, deren zentrale Bestandteile die KS Wellenfunktionsresponse und die KS Dichteresponse sind. Eine Darstellung des lokalen Potentials durch eine leicht modizierte MPB gestattet die Lösung der Integralgleichung. Wir zeigen, dass ein glattes und physikalisches Austauschpotential eine Balance zwischen LAPW und gemischter Produkt Basis erfordert, d.h. die LAPW Basis muss zu einer Genauigkeit konvergiert werden, die weit jenseits derer für LDA oder GGA Rechnungen liegt. Diese Genauigkeit der LAPW Basis ist notwendig, damit die KS Wellenfunktion und damit die Dichte adäquat auf die Potentialänderungen, welche in unserem Formalismus durch die MPB beschrieben werden, antworten können. Sofern beide Basen balanciert sind, nden wir eine gute Übereinstimmung zwischen unseren All-Elektronen FLAPW Resultaten für C, Si, SiC, Ge, GaAs, festes Ne und Ar und Pseudopotentialergebnissen der Literatur. Aufgrund der nötigen Balance beider Basen ist der EXX-OEP Ansatz numerisch sehr aufwendig. Wir haben eine nite Basissatzkorrektur (FBC) entwickelt, welche die Abhängigkeit der LAPW Basis vom Potential explizit berücksichtigt und im Grenzfall einer vollständigen LAPW Basis verschwindet. Anhand von ScN demonstrieren wir, dass die FBC zu konvergierten Austauchpotentialen bei kleinerer LAPW Basis führt, was eine EXX-OEP Rechnung auf der Basis der FLAPW-Methode erst praktisch ermöglicht. Desweiteren diskutieren wir eine Verallgemeinerung des Formalismus für Metalle und berichten Ergebnisse für die kubischen

Perovskite CaTiOç, SrTiOç und BaTiOç, die Übergangsmetalloxide MnO, FeO und CoO, wie auch für die Metalle Al, Na und Cu.

iv Contents

1. Introduction 1

2. Basics of density-functional theory 7 ò.Ô. Hohenberg-Kohntheorem ...... Ôý ò.ò. omas-Fermiapproach ...... ÔÔ ò.ç. Kohn-Shamsystem ...... Ôò ò.¥. SpinDFT ...... Ô¥

3. The exchange-correlation energy functional of KS DFT 17 ç.Ô. Adiabatic-connectionmethod ...... ԗ ç.ò. LDAandGGA...... òý ç.ç. Uniform coordinate scaling and the adiabatic connection ...... òÔ ç.¥. Hybridfunctionals...... ò ç. . Görling-Levyperturbationtheory...... ò— ç.â. Sham-Schlüterequation...... çý ç.Þ. Summary ...... çÔ

4. Electronic structure methods: the FLAPW approach 33 ¥.Ô. APWmethod...... ç¥ ¥.ò. LAPWmethod...... çâ ¥.ò.Ô. Local-orbitalextension ...... ç— ¥.ò.ò. Mixedproductbasis ...... ¥Ô

5. Hybrid functionals within the generalized Kohn-Sham scheme 45 .Ô. GeneralizedKSsystem ...... ¥â .ò. Implementation of hybrid functionals ...... ¥— .ò.Ô. SparsityoftheCoulombmatrix ...... ý .ò.ò. Symmetry...... ò .ò.ç. Singularity of the Coulomb matrix ...... ç. Self-consistenteldcycle ...... Þ

v Contents

.¥. Convergencetests ...... À . . Results for prototype semiconductors and insulators ...... âÔ .â. Calculation of band structures and density of states ...... âç .Þ. EuO:acasestudy ...... ◠.—. Summary ...... Þâ

6. Exact exchange within the optimized effective potential method 79 â.Ô. Optimizedešectivepotentialmethod...... —ý â.ò. ImplementationoftheEXXfunctional ...... —ç â.ç. BalanceofMPBandFLAPWbasis ...... —À â.¥. Comparison of EXX and LDA exchange potential ...... À â. . EXX KS transition energies of prototype semiconductors and insulators . . . À— â.â. EXX approach and the band gap problem of DFT: the III-V nitrides ...... ÔýÔ â.Þ. Finitebasiscorrection(FBC)...... Ôý¥ â.Þ.Ô. DerivationoftheFBC ...... Ôý â.Þ.ò. EšectoftheFBCfortheexampleofScN ...... ÔÔý

â.—. Perovskites: CaTiOç, SrTiOç, BaTiOç ...... ÔÔâ â.À. Transition metal oxides: MnO, FeO, and CoO ...... ÔòÔ â.Ôý.Metals...... Ôò— â.Ôý.Ô. EXX-OEPformalismformetals ...... ÔòÀ â.Ôý.ò. Metals:Na,Al,andCu...... Ôçò â.ÔÔ. Summary ...... ÔçÞ

7. Summary & Conclusion 141

Appendix

A. Incorporation of constraints in the mixed product basis 147

B. Scalar-relativistic approximation 151

List of Abbrevations 155

Bibliography 157

Acknowledgement 171

Publications 173

vi 1. Introduction

A description of the electronic structure of a material from rst-principles requires to solve the quantum-mechanical Schrödinger equation for the electrons and atomic nuclei consti- tuting the material. e large mass dišerence between the electrons and the atomic nuclei permits to separate the electronic motion from that of the nuclei [Ô], which nally results in a Schrödinger equation solely for the electrons. e electrons move in an external potential created by the atomic nuclei and interact among themselves via the electrostatic Coulomb interaction. As a consequence the motion of the electrons is correlated, i.e., the motion of an individual electron is inžuenced by all other electrons and, conversely, it ašects the mo- tion of all others. e Coulomb interaction impedes a direct solution of the many-electron Schrödinger equation, a partial dišerential equation for the many-electron wave function, except for a very limited number of (small) systems. Density-functional theory (DFT) provides an ingenious reformulation of the many- electron problem and thus avoids the direct solution of the many-electron Schrödinger equation. e central quantity of DFT is the electron ground-state density, a function of the three spatial coordinates, which is much less complex than the many-electron wave function, which depends on çN coordinates, where N denotes the number of electrons of the system. Despite this reduction in complexity, the electron ground-state density incorporates all information about the system, in principle. Hohenberg and Kohn [ò] showed that the ground-state density uniquely determines the external potential (up to a constant) of the many-electron system so that the many-electron Hamiltonian, its ground state, and excited states can be regarded as functionals of the ground-state density. Ultimately, the expectation value of any observable becomes a functional of the density. However, the ground-state density of a material is unknown a priori and must be calculated. It can be determined by applying the variational principle for the total energy, which constitutes the second part of the Hohenberg and Kohn theorem [ò] and states that the total energy becomes minimal for the ground-state density. Yet, the explicit functional dependence of the total energy on the density is unknown. erefore, Kohn and Sham [ç] proposed an indirect procedure to minimize the total energy and determine the ground-state density. Kohn and Sham introduced an auxiliary system of non-interacting electrons, the Kohn-

Ô Ô. Introduction

Sham (KS) system, whose ešective potential is adjusted such that its ground-state density coincides with that of the real interacting system. In order to determine the ešective po- tential, they partitioned the total energy of the interacting system into the kinetic energy of non-interacting electrons, the classical Hartree energy, the energy arising from the external potential and an everything-else term, the so-called exchange-correlation (xc) energy. e latter comprises exchange and correlation ešects of the many-electron system as well as a correction for the kinetic energy. Application of the variational principle then leads to the identication of the ešective, local potential as the sum of three terms: external, Hartree, and xc potential, where the latter is dened as the functional derivative of the xc energy with re- spect to the density. In this way, Kohn and Sham mapped the complicated interacting electron problem onto an auxiliary system of non-interacting electrons with the same ground-state density. is mapping is, in principle, exact. All complicated many-electron ešects are hidden in the xc energy functional. However, its mathematical form is unknown, and we must resort to approximations for practical calculations. e local-density approximation (LDA) [¥–â] and the generalized gradient approximation (GGA) [Þ, —] are nowadays routinely employed ap- proximations. While in the LDA the xc energy is approximated by a (direct) functional of the density, the GGA additionally incorporates the density gradient. Despite their success, they sušer from several shortcomings: (a) the Hartree potential contains an unphysical interaction of the electron with itself, which should be compensated exactly by the exchange potential. However, the only approximate treatment of exchange in the LDA and the GGA leads only to a partial compensation and a spurious self-interaction remains [À]. As a consequence, lo- calized states are typically bound too loosely and appear too high in energy. (b) LDA and GGA do not exhibit a derivative discontinuity of the xc energy functional with respect to the particle number and, hence, underestimate the experimental fundamental band gap by ¥ý% or even more [Ôý–Ôò]. (c) e inherent locality of LDA and GGA prohibits to describe van-der-Waals bonded systems [Ôç]. Orbital-dependent xc functionals constitute a new and promising class of xc function- als [Ô¥, Ô ]. ey are functionals of the KS orbitals and are only indirect functionals of the density. In contrast to the LDA and the GGA, orbital-dependent functionals can be improved systematically. ey naturally arise from an expansion of the exact xc energy in powers of the electron-electron interaction [Ôâ]. In rst order, this expansion gives the orbital-dependent exact exchange energy, which is formally identical to the Hartree-Fock (HF) exchange en- ergy, but evaluated with the KS orbitals. e higher-order terms dene an orbital-dependent correlation functional. Already, the rst-order term,the exact exchange energy,cures the self- interaction problem of LDA and GGA and gives rise to a derivative discontinuity, whereas

ò the second-order term of the expansion enables a description of the van-der-Waals interac- tion [ÔÞ]. A combination of density- and orbital-dependent functionals into one functional yields the hybrid functionals. ey usually mix orbital-dependent exact exchange with LDA and GGA functionals and are motivated by the adiabatic-connection method [ԗ–òý]. e rst hybrid functional has been proposed by Becke in the year ÔÀÀç [òÔ] and consists of ý% ex- act exchange and ý% LDA exchange and correlation. Since then, many hybrid functionals have been developed. For example, Perdew, Burke, and Ernzerhof [òò] proposed a hybrid, named PBEý, which combines ò % of exact exchange with Þ % PBE exchange and Ôýý% PBE correlation, where PBE [Þ] denotes one specic GGA functional. e amount of orbital- dependent exact exchange has been inferred from theoretical arguments. Typically, hybrid functionals are employed within the generalized Kohn-Sham (gKS) system [òç], in which the orbital-dependent exact exchange energy is treated as a non-local HF-like potential in the one-particle Schrödinger equations. In this thesis, we have developed an implementation of the PBEý hybrid functional in the gKS system and of the exact exchange energy functional in the KS formalism, which in contrast to the gKS formalism demands a local instead of a non-local potential. Both ap- proaches and functionals are realized within the all-electron full-potential linearized aug- mented-plane-wave (FLAPW) method [ò¥–òÞ], which treats core and valence electrons on an equal footing. e FLAPW method does not employ any shape approximations for the density and potential and allows a treatment of open as well as close-packed multicompo- nent systems including d- and f -electron states. Frequently, FLAPW results are considered as the “gold standard” for the problem at hand. e rst hybrid functional implementations and calculations for solids within the gKS system were reported in the late ÔÀÀýs and the early òýýýs [ò—–çý]. ey employed Gaussian-type functions or plane-waves in the pseudopotential or the projector augmented- wave (PAW) approach. In the all-electron FLAPW method, so far only an approximate scheme [çÔ] has been proposed due to the methodological complexity involved in the im- plementation as well as the large computational demand required for the calculation of the non-local potential. In this approximation, the non-local exchange term is restricted to cer- tain atoms and selected angular momenta l. We have developed an implementation of hybrid functionals, in particular of the PBEý hybrid, without the limitations of Ref. çÔ. It relies on an additional auxiliary basis, the mixed product basis (MPB), which is constructed from products of LAPW basis functions. e MPB is suited for the representation of products of LAPW wave functions. It retains the all-electron character of the wave function product and can be converged in a systematic manner. e usage of the MPB turns the matrix elements

ç Ô. Introduction of the Hamiltonian corresponding to the non-local exchange potential into vector-matrix- vector products, where the matrix can be calculated once at the beginning of a calculation. Only the vectors are computed in each iteration. Due to the large computational demand for the calculation of the non-local exchange potential, we develop several techniques to accelerate our numerical scheme. In order to verify our implementation, we compare our results of PBEý hybrid-functional calculations for prototype semiconductors and insulators with theoretical results from the literature. As an application we discuss and compare the structural, electronic, and magnetic properties of rock-salt EuO obtained with the PBE and PBEý functional. While the PBE functional predicts EuO to be a metal in contradiction to experiment, the PBEý hybrid opens a band gap and leads to a consistent description of the structural and magnetic properties of EuO.

A treatment of hybrid functionals and orbital-dependent functionals within the KS for- malism, in contrast to the gKS formalism, requires a local, multiplicative xc potential. e only indirect dependence of orbital functionals on the density, however, does not allow for a direct dišerentiation of the xc energy with respect to the density. Instead one has to apply the optimized ešective potential (OEP) method [çò–ç¥], which yields an integral equation for the local xc potential. For periodic systems and the exact exchange (EXX) functional, this integral equation was solved rst by Kotani in ÔÀÀ¥ [ç ]. He employed the linearized mu›n- tin orbital (LMTO) method with the additional approximation that the potential is purely spherical around each atom (atomic sphere approximation). Implementations of the EXX- OEP approach within the pseudopotential plane-wave method followed [çâ–ç—]. e rst all-electron full-potential implementation, realized within the FLAPW method, goes back to Sharma et al. [çÀ] in òýý . However, their results deviated substantially from the pseu- dopotential values reported before, which initiated a controversy about the adequacy of the pseudopotential approximation in the context of the EXX-OEP approach. Engel [¥ý] demon- strated that all-electron and pseudopotential EXX-OEP results for lithium and diamond dif- fer only slightly. He mimicked the all-electron results by pushing the pseudopotential and plane-wave cutošs to their all-electron limit. Reasonable agreement between all-electron and pseudopotential EXX-OEP calculations has also been observed by Makmal et al. [¥Ô] for the diatomic molecules BeO and CO employing a real-space grid approach. We have developed an alternative implementation of the EXX-OEP approach within the FLAPW method – dif- ferent and independent from that in Ref. çÀ. It employs a slightly modied version of the MPB for the representation of the local exact exchange potential. So, the integral equation becomes an algebraic equation, which can be solved for the local potential by standard numer- ical techniques. We discuss and demonstrate the requirement of a balance between MPB and LAPW basis to obtain physical and stable EXX potentials. Only if both basis sets are properly

¥ balanced, a favorable agreement between our all-electron full-potential and pseudopotential plane-wave EXX KS transition energies reported in literature is observed. is balance de- mands a very žexible LAPW basis set and, thus, leads to a high computational cost of EXX calculations in practice. In order to reduce the demand on the LAPW basis we develop and explore a nite basis correction (FBC) for the KS response function, which arises from the de- pendence of the LAPW basis functions on the spherical ešective potential in the MT spheres. It leads to stable exchange potentials at much smaller LAPW basis sets. is FBC turns out to be a very general concept that paves the road to response quantities and density functional perturbation theory applicable to electronic structure methods with potential-dependent ba- sis sets. We also report results for the cubic transition-metal perovskites CaTiOç, SrTiOç, and

BaTiOç as well as the transition-metal monoxides MnO, FeO, and CoO. We will see that the EXX functional leads to an opening of the KS band gap. Most notably, antiferromagnetic FeO and CoO are correctly predicted to be insulating. Finally, we present a generalization of the EXX-OEP formalism for metals and show results for Na, Al, and Cu. e thesis is organized as follows. e second chapter provides a short introduction to KS DFT, where the basic equations and quantities are dened. e xc energy functional is thoroughly discussed in the third chapter. Orbital-dependent as well as hybrid functionals are motivated and the basic concepts for the development of functionals are introduced. e FLAPW method is explained in chapter ¥, where emphasis lies on the local-orbital extension of the LAPW approach and the construction of the MPB. Chapter introduces the gKS system and discusses the implementation of the hybrid functionals within the gKS system. e OEP method and its numerical realization within the FLAPW approach for the EXX functional is described in chapter â. e thesis is concluded in chapter Þ. Parts and results of chapter and â have recently been published in Physical Review B [¥ò, ¥ç].

2. Basics of density-functional theory

e physical properties of a material, be they of mechanical, electrical or magnetic character, rest on the motion of the electrons under the inžuence of the ions constituting the material. is motion is governed by the time-dependent Schrödinger equation

∂ H Ψ i Ψ (ò.Ô) S ⟩= ∂t S ⟩ with the many-body wave function Ψ and the Hamiltonian S ⟩

Ô ò Ô ò Ô Ô Zα Ô Zα Zβ H i α , (ò.ò) =−ò Qi ∇ − Qα òMα ∇ + ò Qi, j ri rj − Qi,α ri Rα + ò Qα,β Rα Rβ i≠ j S − S S − S α≠β S − S where Zα and Mα denote the atomic number and mass of the α-th atomic ion, Rα its space Ô coordinates and ri the coordinates of the i-th electron. In the time-independent case Eq. (ò.Ô) becomes H Ψ E Ψ (ò.ç) S ⟩= S ⟩ with the total energy E. In ÔÀòÀ Dirac [¥¥] already noted that with Eqs. (ò.Ô)–(ò.ç) “the laws necessary for [...] a large part of physics and the whole of chemistry are thus completely known, and the di›culty is only that exact application of these laws lead to equations which are to complicated too be soluble”. Indeed, even —ò years aŸer Dirac the exact solution of the Schrödinger equation for the enormous number of interacting electrons and atomic ions present in real materials is impossible. It is therefore necessary to develop models or approx- imations that are able to describe the main aspects of the physical problem under consider- ation. e density-functional theory (DFT) is such an approximation and has been proven to be very successful over the last decades. But before the DFT is introduced, the Born- Oppenheimer approximation, which allows to separate the electronic from the ionic motion, is discussed. It is essential for DFT, as well. In order to contrast DFT with wave function based approaches, we will also briežy discuss the Hartree and Hartree-Fock (HF) approach. e enormous mass dišerence between the electrons and ions – the mass of the latter is at

ÔWe use atomic units except where noted otherwise: ħ = m = e = Ô.

Þ ò. Basics of density-functional theory leasta factor of Ôýýýlarger thanthat of theelectrons – causes the motion of the electrons and ions to take place in dišerent time scales. On the time scale of the electron motion the ions appear xed, which justies to decouple the electronic from the ionic motion and to dene an electronic Hamiltonian

Ô ò H i V ri Vee , (ò.¥) = Qi −ò∇ + ( ) +

Zα Ô Zα Zβ where we have introduced the external potential V ri α r R ò α,β R R and ( )=− ∑ S i− α S + ∑ α≠β S α− β S Ô Ô the abbreviation Vee ò i, j r r for the electron-electron interaction. e ion positions = ∑ i≠ j S i − jS enter in (ò.¥) solely as xed parameters. is is the so-called Born-Oppenheimer approxi- mation [Ô].

However, solving Eq. (ò.ç) with the Born-Oppenheimer Hamiltonian (ò.¥) is still too com- plex for systems with many electrons. Actually, a simple brute-force solution, which samples each coordinate direction of the wave function with only Ôý grid points, requires for an atom with N electrons the storage of ÔýçN numbers. For an iron atom N òâ this amounts to ( = ) Ôýޗ numbers. is ’gedankenexperiment’ already indicates that from a practical point of view the many-body wave function Ψ might not be the optimal quantity to describe the many S ⟩ electron system with.

In the Hartree theory [¥ ] the many-electron wave function Ψ is approximated by a prod- S ⟩ uct of one-particle wave functions

Ψ r , r ,..., r ϕ r ϕ r ... ϕ r (ò. ) H( Ô ò N )= nÔ ( Ô) nò ( ò) nN ( N ) where ni denotes a set of quantum numbers. Due to the separation of the variables, the demand of storing ÔýçN numbers in our ’gedankenexperiment’ for the true many-electron wave functions is reduced to N ⋅ Ôýç numbers. Using ansatz (ò. ) in combination with the Hamiltonian (ò.¥) yields the total energy

N ò ò Ô Ô ϕn r ϕn r′ ò S i ( )S S j ( )S ç ç EH ΨH H ΨH ϕni − ∇ +V r ϕni + d r d r′ . (ò.â) = ⟨ S S ⟩= Qi Ô⟨ S ò ( )S ⟩ ò Qi, j ∫∫ r − r′ = i≠ j S S

Minimization of E with respect to the one-particle wave functions ϕ r under the con- H ni ( i ) straint that the latter are normalized leads to a Schrödinger equation for the one-particle wave functions Ô − ∇ò + V r + V r ϕ r є ϕ r (ò.Þ) ‹ ò ( ) i( ) ni ( )= i ni ( )

— ϕ r′ ò S n j ( )S ç with the electrostatic potential Vi r j j i ∫ r r′ d r′. e set of Eqs. (ò.Þ) must be ( ) = ∑ ( ≠ ) S − S solved self-consistently, as the electrostatic potential already depends on the one particle wave functions and thus on the solution of Eq. (ò.Þ). Starting with a guess of wave functions the electrostatic potential is calculated and a new set of orbitals is gained by solving Eqs. (ò.Þ), which is then used to calculate a new electrostatic potential. is scheme is iterated until the orbitals do not alter anymore. According to the Ritz variational principle the total energy EH constitutes an upper bound for the exact one. In the Hartree approach the Pauli principle is only considered insofar that two one-particle wave functions are not allowed to agree in all quantum numbers. A rigorous consideration of the Pauli principle however requires that the electron wave function Ψ is antisymmetric S ⟩ under the exchange of two particles. A corresponding renement of the Hartree-ansatz (ò. ) leads to a single Slater determinant as an approximation for the many-electron wave function

Ô ΨHF rÔ, rò,..., rN det ϕn rj . (ò.—) ( )= √N! S i ( )S is is the HF approach [¥â, ¥Þ]. e explicit anti-symmetrization of the wave function en- tails an additional energy contribution to the total energy, the so-called exchange energy or

HF exchange energy Ex E E + E (ò.À) HF = H x with ϕ∗ r ϕ r ϕ∗ r′ ϕ r′ Ô ni n j n j ni ç ç Ex − ( ) ( ) ( ) ( )d r d r′ . (ò.Ôý) = ò Qi, j ∫∫ r − r′ i≠ j S S

In analogy to the Hartree approach minimization of EHF with respect to the one-particle wave functions yields Schrödinger-type equations for the orbitals ϕ r ni ( )

ϕ∗ r′ ϕ r′ Ô ò n j ni ç − ∇ + V r + V r ϕ r − ( ) ( )d r′ ϕ r є ϕ r , (ò.ÔÔ) i ni ∫ − n j i ni ‹ ò ( ) ( ) ( ) j Qj i r r′ ( )= ( ) ( ≠ ) S S where the HF exchange energy results in an additional non-local exchange potential.ò e HF approach considers exchange exactly, but completely neglects correlation ešects. A linear combination of Slater determinants [conguration interaction (CI)] is able to give the exact many-electron wave function in the limit of innite many determinants. However, the com- putational demand scales exponentially with the system size. As a consequence, only small

ò In Eq. (ò.ÔÔ) the diagonal term j = i can be added to the local potential Vi (r) and the non-local exchange potential as they cancel out mutually.

À ò. Basics of density-functional theory systems (atoms, molecules) can be treated. In contrast to the Hartree, Hartree-Fock or CI approach, the DFT pursues a dišerent ap- proach. e electron density instead of the complex many-electron wave function is the cen- tral quantity. e density depends only on three spatial coordinates in contrast to the çN coordinates of the many-electron wave function, where N denotes the number of electrons in the system. is is a tremendous simplication and DFT still represents an exact theory of the many-electron system, in principle. e central theorem of DFT is the Hohenberg and Kohn theorem and will be discussed in the next section.

2.1. Hohenberg-Kohn theorem

e Hohenberg and Kohn theorem [ò] forms the foundation of DFT. It comprises the follow- ing two statements:

• e ground-state density of any electronic system n r uniquely determines the ex- ý( ) ternal potential V r (up to a constant) and, thus, with Eqs. (ò.ç) and (ò.¥) the ground- ( ) state wave function Ψ r . Consequently, the expectation value of any observable of a ( ) many-electron system can be regarded as a functional of its ground-state density, in particular the total energy.

• e total energy functional

E n F n + V r n r dçr (ò.Ôò) = ∫ [ ] [ ] ( ) ( ) with the universal functional

F n min Ψ T + Vee Ψ , (ò.Ôç) = Ψ→n

where the minimization is over[ all] N-electron⟨ S waveS functions⟩ yielding the density n, becomes equal to the exact ground-state energy E for n r n r . For all other ý = ý densities with n r n r ≠ ý ( ) ( ) E n E n (ò.Ô¥) ( ) ( ) > ý holds. is is the variational principle[ of] DFT[ leading] to the Euler-Lagrange equation

δ E n − µ n r dçr − N ý , (ò.Ô ) δn r ∫ =  [ ] ‹ ( )  where the Lagrange( ) multiplier µ ensures particle conservation.

Ôý ò.ò. omas-Fermi approach

F n is universal in the sense that it is independent from the actual external po- tential. It is the same for all electronic systems. [ ] ese theorems alone are, however, of little practical use, as for a practical calculation of the ground-state density and energy the universal functional F n has to be evaluated. So far, besides its formal denition in terms of the many-electron wave function no method has [ ] been given to nd or approximate F n . A simple but crude approximation for the universal functional is given within the omas-Fermi approach. [ ]

2.2. Thomas-Fermi approach

e omas-Fermi approach [¥—, ¥À] goes back to the year ÔÀòÞ. is is çÞ years before the rigorous fundament of DFT was laid by Hohenberg and Kohn. In retrospect, however, it can be understood as the rst realization of DFT. In the following, the main ideas of the omas- Fermi method are discussed in the terminology of DFT. e omas-Fermi method approximates the universal functional by an explicit functional of the density

ç ò ò ç Ô n r n r′ ç ç FTF n çπ ç n r ç d r + d r d r′ . (ò.Ôâ) = Ôý ∫ ò ∫∫ r − r′ ( ) ( ) [ ] ( ) ( ) e rst term is an approximation to the kinetic energy assuminS S g that it can be described locally by that of a homogeneous electron gas. e second term is the classical electrostatic Coulomb energy, which is in the context of DFT usually denoted by Hartree energy

Ô n r n r′ ç ç UH n d r d r′ . (ò.ÔÞ) = ò ∫∫ r − r′ ( ) ( ) [ ] Application of the variational principle (ò.Ô )S leadsS to the omas-Fermi equation for the density n r Ô ò ò ò çπ ç n r ç + V r + V r − µ ý , (ò.ԗ) ( ) ò H = where we additionally introduced( ) the( Hartree) ( potential) ( )

δUH n n r′ ç VH r d r′ . (ò.ÔÀ) = δn r = ∫ r − r′ [ ] ( ) ( ) Solution of the integral equation (ò.ԗ) results( ) in the grouS nd-stateS density and with Eqs. (ò.Ôâ) and (ò.Ôò) the ground-state total energy can be calculated. But it turns out that the omas- Fermi approach is not able to describe the bonding of molecules or solids, because the ki-

ÔÔ ò. Basics of density-functional theory netic energy functional is only a poor approximation for the true kinetic energy and essential physics like electron exchange and correlation are neglected. e decisive step for an accurate, practical realization of DFT is the Kohn-Sham approach [ç].

2.3. Kohn-Sham system

e brilliant idea of Kohn and Sham [ç] was to map the interacting many-electron system onto an auxiliary system of non-interacting electrons, the Kohn-Sham (KS) system. e in- dependent electrons of the KS system move in an ešective, local potential Veš r , which is adjusted such that the ground-state density of the KS system equals the ground-state density ( ) of the interacting system. e Hohenberg and Kohn theorem is independent of the particular form of the electron- electron interaction. Especially, it holds for the non-interacting KS system

Ô ò Hs − ∇i + Veš ri . (ò.òý) = Qi ò ‹ ( ) Consequently, Veš r is uniquely determined by the ground-state density, and the KS single Slater determinant Φ ( ) H Φ r ,..., r E Φ r ,..., r (ò.òÔ) s Ô N = KS Ô N as well as its single-particle wave( functions )ϕi r ( )

Ô ( ) − ∇ò + V r ϕ r є ϕ r (ò.òò) ò eš i = i i [ ( )] ( ) ( ) can be regarded as functionals of the density. e total energy of the KS system is given by

E n T n + V r n r dçr (ò.òç) KS = s ∫ eš [ ] [ ] ( ) ( ) with the kinetic energy functional

Ô ò ç Ts n ϕ∗i r − ∇ ϕi r d r . (ò.ò¥) = Qi ∫ ò [ ] ( ) ‹  ( ) In order to guarantee that the KS system has the same ground-state density as the interacting one, Kohn and Sham rewrite the universal functional F n of the interacting, many electron system as a sum of the kinetic energy T n of the non-interacting system, the Hartree energy s [ ] [ ]

Ôò ò.ç. Kohn-Sham system

UH n , and an exchange-correlation (xc) energy

[ ] F n T n + U n + E n , (ò.ò ) = s H xc [ ] [ ] [ ] [ ] where Exc subsumes all many-electron exchange and correlation ešects and the dišerence between the kinetic energy of the interacting and the non-interacting system. en, the stationary equation (ò.Ô ) becomes

δTs δExc + VH r + + V r − µ ý . (ò.òâ) δn r δn r = ( ) ( ) is Euler-Lagrange equation( is) formally equivalent( ) to that obtained for the non-interacting KS system, if the ešective potential fullls

V r V r + V r + V r (ò.òÞ) eš = H xc ( ) ( ) ( ) ( ) with the xc potential given by δExc n Vxc r . (ò.ò—) = δn r [ ] ( ) Consequently, Kohn and Sham recast the problem( such) that the ground-state density ný r of the interacting system can be found by solving the one-particle KS equations (ò.òò) with ( ) the potential (ò.òÞ) and subsequently occupying the electron states according to the Pauli principle. e density is then given by a sum over the occupied KS orbitals

occ. ò ný r ϕi r . (ò.òÀ) = Qi ( ) S ( )S As Veš r itself depends on the density, the set of equations (ò.òò) must be solved self- consistently. e ground-state energy E is nally obtained from ( ) ý

E E n T n + U n + E n + V r n r dçr . (ò.çý) ý = ý = s ý H ý xc ý ∫ ý [ ] [ ] [ ] [ ] ( ) ( ) Formally, the mapping of the interacting system onto the KS system is exact. However, except for the formal denition of the xc energy functional

E n F n − T n − U n (ò.çÔ) xc = s H [ ] [ ] [ ] [ ] its exact form is unknown. Exc n is usually split into an exchange and a correlation part

[ ] E n E n + E n (ò.çò) xc = x c [ ] [ ] [ ]

Ôç ò. Basics of density-functional theory which are dened by E n Φ n V Φ n − U n (ò.çç) x = ee H E n Ψ n[ ]T +⟨V[ Ψ]S n S −[ Φ]⟩ n T +[ V] Φ n . (ò.ç¥) c = ee ee Here, Φ n denotes the[ ] KS⟨ single[ ]S Slater determinantS [ ]⟩ ⟨ that[ ]S yields theS density[ ]⟩ n r and Ψ n is the many-electron ground-state wave function of density n. [ ] ( ) [ ] In practice, the xc energy functional must be approximated. Chapter ç gives some phys- ical insight into the xc functional, discusses the nowadays routinely used local-density ap- proximation (LDA) and generalized gradient approximation (GGA), and nally introduces orbital-dependent functionals. In contrast to the omas-Fermi approach, the KS method approximates the kineticenergy by the exact orbital-dependent, non-interacting kinetic energy term instead of an explicit density functional. In this way, the dominant contribution to the total energy is described well and only the smaller dišerence to the exact kinetic energy must be approximated by Exc. is explains, at least partially, the success of KS DFT, as already relatively simple approximations for Exc lead to reliable results for a wide range of materials.

2.4. Spin DFT

e DFT formalism, developed so far, does not incorporate the spin oftheelectronasadegree of freedom. However, the spin is essential to describe, for example, the magnetism in the çd metals Fe, Co, Ni, and Cr. Here, we sketch the extension of the Hohenberg and Kohn theorem as well as the KS formalism to spin DFT. Instead of the spin-independent interacting Hamiltonian (ò.¥), the interacting Pauli-type Hamiltonian

Ô ò σ Ô Ô H − ∇i + V ri + µB ⋅ B ri + (ò.ç ) = Qi ò ò Qi Qj i ri − rj ≠  ( ) ( ) S S forms the starting point of spin DFT, where µB is the Bohr magneton, σ is the vector of the Pauli spin matrices, and B r denotes a magnetic eld. Von Barth and Hedin [ ý] showed that the ground-state wave function Ψ of the Hamiltonian (ò.ç ) and hence all ground-state ( ) properties are functionals of the density matrix, which is dened by S ⟩

σσ′ n r Ψ ψ+ r ψ ′ r Ψ . (ò.çâ) = σ σ ( ) ⟨ S ( ) ( )S ⟩ Here, ψ r denotes the eld operator for an electron of spin σ (σ ↑, ↓). is statement σ = ( )

Ô¥ ò.¥. Spin DFT corresponds to the rst part of the Hohenberg and Kohn theorem for non-spin-polarized systems. Accordingly, the second part – the variational principle – can be generalized: the ′ ground-state energy E is stationary with respect to variations of the density matrix nσσ r . e concept of the KS system as an auxiliary system of non-interacting electrons, which ( ) yield the same ground-state density (matrix) can be carried over, too. e KS equations for ϕ↑ r the KS orbitals, which are then two-component Pauli spinors i , thus become ϕ↓ r ⎛ i ( ) ⎞ ⎝ ( ) ⎠ Ô n r ′ ′ δE ′ ′ ′ − ∇ò + ′ dçr + V r σσ + µ σ ⋅ B σσ + xc σ r єσ σ r . ′ ext δ B σσ′ ϕi i ϕi ò ∫ r − r′ δn r = ( ) œ ( ) ( ) ¡ ( ) ((ò.çÞ)) e density matrixS inS terms of the occupied KS spinors is given by( )

occ. σσ′ σ σ′ n r ϕi ∗ r ϕi r . (ò.ç—) = Qi ( ) ( ) ( ) For collinear structures, on which we will focus in this thesis, i.e., ferromagnetic or antifer- romagnetic materials, the KS Hamiltonian becomes diagonal and Eq. (ò.çÞ) decouples into two equations

Ô n r − ∇ò + ′ ç + + + ↑ ↑ ↑ ↑ d r′ Vext r µBB r Vxc r ϕi r єi ϕi r (ò.çÀ) ò ∫ r − r′ = ( ) œ Ô n r ( ) ( ) ( )¡ ( ) ( ) − ∇ò + ′ ç + − + ↓ ↓ ↓ ↓ S Sd r′ Vext r µBB r Vxc r ϕi r єi ϕi r , (ò.¥ý) ò ∫ r − r′ = ( ) œ ( ) ( ) ( )¡ ( ) ( ) σ where Vxc r is the spin-dependentS S generalization of the xc potential dened as

( ) δE V σ r xc (ò.¥Ô) xc = δnσ r ( ) and the N lowest states are occupied without regard( to) the spin such that the particle number is conserved σ ç σ σ ç N n r d r ϕi ∗ r ϕi r d r . (ò.¥ò) = Qσ ∫ = Qσ Qi ∫ єσ ≤E ( ) i F ( ) ( ) e energy of the highest occupied state corresponds to the Fermi energy EF of the system.

Ô

3. The exchange-correlation energy functional of KS DFT

Kohn-Sham (KS) density-functional theory (DFT) relies on an exact mapping of the interact- ing, many-electron system onto an auxiliary one of non-interacting electrons such that the ground-state densities of the auxiliary and true systems coincide. All complicated many-body ešects are taken into account in the KS formalism, in principle, via the exchange-correlation (xc) energy functional. e exact form of the latter, however, is unknown, so that approxi- mations for Exc are mandatory in practice. Fortunately, already simple approximations give reliable results for a wide range of materials. In this chapter we give some physical insight into the xc functional of DFT. We begin with the adiabatic-connection method, which connects the KS with the true, fully interacting sys- tem by gradually switching on the electron-electron interaction. It provides a denition of the xc energy in terms of the xc hole and constitutes the basis for the rest of this chapter. en, we introduce the nowadays standard functionals the local-density approximation (LDA) and the generalized gradient approximation (GGA) and shortly discuss their success and failures. For the construction of approximate functionals, exact constraints for the xc functional are of utmost importance. Such constraints can, for example, be developed by the technique of uniform coordinate scaling. e latter will be explained and a link between uniform coordi- nate scaling and the adiabatic-connection method will be established. AŸerwards, based on the adiabatic-connection method, we introduce the hybrid functionals that combine orbital- dependent exact exchange with (semi-)local xc functionals. While the hybrids exactly de- scribe the coupling constant integrand in the non-interacting limit λ ý , a systematic = expansion of the xc functional in the electron-electron interaction around λ ý is provided ( =) by Görling-Levy perturbation theory. e perturbation theory yields, in principle, an exact expansion of the xc functional. Finally, we point out a dišerent route for the construction of approximate functionals by combining many-body perturbation theory (MBPT) and KS DFT. e Sham-Schlüter equation connects the self-energy of MBPT with the xc potential. In principle, a diagrammatic expansion of the dynamical and non-local self-energy in higher and higher orders of the electron-electron interaction leads to xc potentials of ever increasing

ÔÞ ç. e exchange-correlation energy functional of KS DFT accuracy. We close this chapter by a short summary. For the sake of simplicity, we have suppressed the spin index throughout this chapter.

3.1. Adiabatic-connection method

In the adiabatic connection [ԗ–òý], which is sometimes also called coupling-constant inte- gration method, the KS system is connected with the fully interacting many-electron system by adiabatically switching on the electron-electron interaction Vee

Hλ T + λV + V λ (ç.Ô) = ee and adjusting the external potential V λ such that the ground-state density equals the real physical density of the interacting system for any λ ý,Ô . For λ ý the external potential ∈ = V λ then corresponds to the ešective KS potential, whereas for λ Ô it is the true external [ ] = potential. According to the coupling-constant Hamiltonian Hλ, the universal functional F n of Eq. (ò.Ôç) may be generalized to [ ]

λ F n min Ψ T + λVee Ψ . (ç.ò) = Ψ→n [ ] ⟨ S S ⟩ In analogy to Eq. (ò.çÔ), the xc energy functional that corresponds to Hλ is dened by

Eλ n Fλ n − T n − λU n . (ç.ç) xc = s H

λ [ ]λ ý [ ] [ ] [ ] In the limit λ ý, E is zero, as F = n T n holds, which režects that the KS system of = xc = s a non-interacting system is the system itself. Consequently, the xc energy functional can be [ ] [ ] written as Ô λ Ô dExc λ λ Exc n dλ Ψ Vee Ψ − UH n dλ , (ç.¥) = ∫ý dλ = ∫ý where Ψλ denotes the[ many-electron] ground-state
⟨ S waveS function⟩ [ of]H λ.

At rst glance, this exact equation for Exc seems to be of little practical use, as the many- electron wave function Ψλ enters for each λ. But we will see in the following that it is a central equation for developing and understanding functionals. We dene the λ dependent pair-density

λ λ λ nò r, r′ Ψ δ r − ri δ r′ − rj Ψ (ç. ) = Qi, j (i≠ j) ( ) ⟨ S ( ) ( )S ⟩

ԗ ç.Ô. Adiabatic-connection method and the λ dependent xc hole

λ λ nò r, r′ − n r n r′ n r, r′ . (ç.â) xc = n r ( ) ( ) ( ) ( ) e latter describes the reduced probability of nding( ) an electron at r′ if a given electron sits at r. is hole is, on the one hand, created by the Coulomb repulsion and, on the other hand, by the Pauli principle separating two electrons of equal spin in space. Combining Eq. (ç.¥) with Eqs. (ç. ) and (ç.â) we obtain for the xc energy

Ô Ô λ ç ç Exc n nò r, r′ − n r n r′ dλ d r d r′ (ç.Þ) = ò ∫∫ r − r′ ∫ [ ] Ô n r nxc r, r′ ( ç )ç ( ) ( )  S S d r d r′ , (ç.—) = ò ∫∫ r − r′ ( ) ( ) where we additionally introduced theS λ-averagedS xc hole

Ô λ nxc r, r′ nxc r, r′ dλ . (ç.À) = ∫ý

us, the xc energy is the Coulomb( energy) of the( electron) density with the surrounding xc hole. e xc hole around an electron at r does not depend on the absolute position r′. us, we can represent the xc hole nxc r, r′ by

( ) l ∞ nxc r, r′ nxc,lm r, r − r′ Ylm r − r′ . (ç.Ôý) = Ql ý mQl = =− ( ) ( S S) (Å) Using this expansion in Eq. (ç.—) it becomes evident that only the spherical average of the xc hole nxc,ýý r, r′ determines the xc energy

( ) Ô ç E n ¥π n r n r, r′ r′ dr′ d r . (ç.ÔÔ) xc = ò ∫ ∫ xc,ýý √ [ ] ( ) ( ) Approximate xc functionals therefore must not mimic the exact xc hole, it is su›cient if they are able to reproduce its spherical average. Moreover, the exact xc hole obeys the sum rule

ç n r, r′ d r′ −Ô . (ç.Ôò) ∫ xc = ( )

In the next section the nowadays routinely used approximations for the xc functional, the LDA and the GGA, are discussed.

ÔÀ ç. e exchange-correlation energy functional of KS DFT

3.2. LDA and GGA

e LDA assumes that the interacting electron system locally acts as a homogeneous electron gas. Correspondingly, the xc energy is written as an integral over space, where at each point LDA in space the xc energy is approximated by the xc energy density єxc of the homogeneous electron gas ELDA n n r єLDA n r dçr . (ç.Ôç) xc = ∫ xc LDA e exchange part of єxc can be[ derived] exactly( ) within( ( the)) Hartree-Fock (HF) method

LDA ç ò Ô є n − çπ n ç , (ç.Ô¥) x = ¥π ( ) ( ) while the remaining correlation energy is parametrized by a t to quantum Monte-Carlo cal- culations [¥]. e dišerent available LDA xc functionals, e.g. Vosko-Wilk-Nusair (VWN) [ ], Perdew-Zunger (PZ) [À], and Perdew-Wang (PW) [â] dišer in details of the parametrization LDA for єc . Despite its simplicity, the LDA gives reliable results for a wide range of materials and prop- LDA erties. Its success goes back to the fact that Exc provides (nearly) an exact description of the xc energy for the model system of the homogeneous electron gas. erefore, the LDA xc hole fullls the sum rule (ç.Ôò) and its spherical average is usually a reasonable approximation to the exact average, whereas the full non-spherical hole is usually in poor agreement with the exact one [ Ô]. A simple expansion around the uniform limit by taking into account the inhomogeneity of the system via its density gradient leads in practice to less accurate results as the LDA [ ò, ç]. Explicit incorporation of the sum rule (ç.Ôò) and additional exact constraints nally lead to the generalized-gradient approximation (GGA) with an improved accuracy in general. e GGA is oŸen called a semi-local approximation and formally given by

EGGA n n r єGGA n r , ∇n r dçr. (ç.Ô ) xc = ∫ xc [ ] ( ) ( ( ) ( )) A large variety of GGA functionals exists nowadays: Perdew-Wang-ÀÔ (PWÀÔ) [—], Perdew- Burke-Ernzerhof (PBE) [Þ], Becke-Lee-Yang-Parr (BLYP) [ ¥, ] and many others. For a comparison of the performance of LDA and GGA we cite some well known trends [ â– —]:

• LDA as well as GGA give ground-state structural properties with a deviation of a few percent to the experimental value. e LDA usually underestimates the lattice param- eter. It over-binds. On the contrary, GGA has a tendency to under-bind, where the

òý ç.ç. Uniform coordinate scaling and the adiabatic connection

magnitude of the error is typically not as large as the LDA one.

• With respect to the experimental values cohesive energies are in general signicantly overestimated in LDA. e GGA gives usually better energies.

• KS band gaps of LDA and GGA are far too small in comparison with experimental results. is is oŸen called the band gap problem of LDA and GGA.

• Due to the self-interaction error inherent in LDA and GGA, localized d- and f -electron states are oŸen bound to loosely and their binding energies appear too high in energy. In the worst case, this can lead to a metallic instead of an insulating ground state.

e self-interaction error arises from the separation of the Hartree energy in Eq. (ò.ò ) and the only approximate treatment of exchange in LDA and GGA. Hence the total energy E n contains an unphysical self-interaction, i.e., the electrons move in an ešective potential which [ ] is partially created by its own charge. e error is more severe for localized states, like d- and f -electrons of transition metal oxides, and tends to delocalize these electrons.

3.3. Uniform coordinate scaling and the adiabatic connection

As already pointed out, exact constraints for the xc functional are essential for the construc- tion of approximate xc functionals Exc. Each constraint limits the ’phase space’ of possible forms for Exc. In particular, the technique of uniform coordinate scaling has been proven as a useful tool to nd such constraints [ À, âý].

e uniformly scaled density nγ r arises from n r by scaling all length scales by γ and ensuring that n r is still normalized to N electrons γ ( ) ( )

( ) ç n r γ n γr γ ý, ∞ . (ç.Ôâ) γ = ∈ ( ) ( ) [ [ For γ ý,Ô the scaled density n r is stretched in comparison with n r , whereas γ Ô, ∞ ∈ γ ∈ corresponds to a compression of the density. Accordingly, the uniformly scaled KS single [ [ ( ) ( ) ] [ Slater determinant Φγ and the many-electron wave function Ψγ are dened by

çN ò Φ r ,..., r γ ~ Φ γr ,..., γr (ç.ÔÞ) γ Ô N = Ô N çN ò Ψ r ,..., r γ ~ Ψ γr ,..., γr . (ç.ԗ) γ( Ô N ) = ( Ô N ) ( ) ( )

òÔ ç. e exchange-correlation energy functional of KS DFT

e kinetic energy of the scaled wave function depends quadratically on the scaling factor γ

T Ψ Ψ T Ψ γò Ψ T Ψ γòT Ψ . (ç.ÔÀ) γ = γ γ = = [ ] ⟨ S S ⟩ ⟨ S S ⟩ [ ] On the contrary, the expectation value of the electron-electron interaction is linear in γ

V Ψ Ψ V Ψ γ Ψ V Ψ γV Ψ . (ç.òý) ee γ = γ ee γ = ee = ee [ ] ⟨ S S ⟩ ⟨ S S ⟩ [ ] e scaling relations of T and Vee are here given for the physical many-electron wave func- tion, but they hold as well for the KS wave function. e question arises which is the KS or physical ground-state wave function belonging to the scaled density? We rst answer the question for the KS wave function. e KS wave function of density n is the wave function that minimizes Φ n T Φ n . As the scaled KS wave function Φ also minimizes the kinetic energy γ ⟨ [ ]S S [ ]⟩

Φ n T Φ n γò Φ n T Φ n , (ç.òÔ) γ γ = ⟨ [ ]S S [ ]⟩ ⟨ [ ]S S [ ]⟩ the KS ground-state wave function for the scaled density is the scaled KS wave function of density n Φ n Φ n . (ç.òò) γ = γ For the physical interacting problem this[ is not] the case[ ] anymore,i.e., the scaled wave function

Ψγ is not the ground-state wave function that gives the scaled density nγ. e many-electron wave function Ψ of density n minimizes Ψ n T + Vee Ψ n . Due to the electron-electron interaction Ψ does not minimize Ψ n T + V Ψ n . γ γ ⟨ [ ]See γ S [ ]⟩ However, the scaled ground-state wave function Ψλ of the coupling-constant Hamiltonian ⟨ [ ]S S γ[ ]⟩ (ç.Ô) fullls

Ô Ψλ n T + V Ψλ n Ψλ n γòT + γV Ψλ n γò Ψλ n T + V Ψλ n . (ç.òç) γ ee γ = ee = γ ee ⟨ [ ]S S [ ]⟩ ⟨ [ ]S S [ ]⟩ ⟨ [ ]S S [ ]⟩ Here, we used the scaling relations Eqs. (ç.ÔÀ) and (ç.òý). We note that the ground-state wave λ λ λ function Ψ of the coupling constant Hamiltonian minimizes Ψ T + λVee Ψ . So, if we choose the coupling constant λ as the reciprocal value of the scaling factor ⟨ S S ⟩ Ô λ , (ç.ò¥) = γ

Ô γ Ô ~ + Ô γ + the wave function Ψγ minimizes T Vee , since Ψ ~ minimizes T γ Vee. Consequently, the many-electron ground-state wave function of density nγ is the scaled ground-state wave

òò ç.ç. Uniform coordinate scaling and the adiabatic connection function of the coupling constant Hamiltonian with λ Ô γ = Ô γ ~ Ψ n Ψ ~ n . (ç.ò ) γ = γ

Conversely, the many-electron ground-state[ ] wave[ function] Ψλ of the coupling-constant

Hamiltonian (ç.Ô) is the uniformly scaled ground-state wave function of density nÔ λ of the ~ fully interacting system λ Ψ n Ψλ nÔ λ . (ç.òâ) = ~ is manifests an important relation between[ ] coordinate[ ] scaling and the adiabatic connection approach. From Eq. (ç.òò) the scaling relations for the non-interacting kinetic energy (ò.ò¥) and the exchange energy (ò.çç) follow immediately

T n T Φ n T Φ n γòT n (ç.òÞ) s γ = s γ = s γ = s E n E Φ n E Φ n γE n . (ç.ò—) x[ γ] = x[ [ γ]] = x[ γ[ ]] = x [ ]

Combination of Eq. (ç.òâ)[ with] Eqs.[ (ç.ÔÀ)[ ]] and (ç.òý)[ [ yields]] the[relation] for the universal functional (ç.ò) of the coupling-constant Hamiltonian

λ λ λ ò F n Ψ n T + λVee Ψ n λ F nÔ λ . (ç.òÀ) = = ~ [ ] ⟨ [ ]S S [ ]⟩ [ ] e coupling-constant dependent xc energy functional (ç.ç) fullls an equivalent relation

λ ò ò ò ò Exc n λ F nÔ λ − λ Ts nÔ λ − λ UH nÔ λ λ Exc nÔ λ . (ç.çý) = ~ ~ ~ = ~ [ ] [ ] [ ] [ ] [ ] e xc energy of the Hamiltonian Hλ with coupling constant λ is identical to λò times the xc energy at λ Ô but evaluated at the scaled density nÔ λ. is last relation can be used to = ~ λ calculate the coupling constant integrand dExc dλ (ç.¥)

λ ~ dExc d ò λ Exc nÔ λ (ç.çÔ) dλ = dλ ~
 for any approximate xc functional. [ ] For small systems it is possible to calculate the coupling constant integrand

λ dExc λ λ Ψ V Ψ − UH n (ç.çò) dλ = ee ⟨ S S ⟩ [ ] exactly. A comparison of the exact integrand with its approximate counterpart gives insight

òç ç. e exchange-correlation energy functional of KS DFT

-0.85 He -0.9

-0.95

(htr) -1 λ /d λ

xc -1.05 dE -1.1 exact -1.15 LDA

-1.2 0 0.2 0.4 0.6 0.8 1 λ

λ Figure ç.Ô.: Exact and LDA coupling-constant integrand Exc of the He atom. (Data of graph taken from Ref. âý.)

into the power and shortcomings of the approximate xc functional. e calculation of the exact integrand requires to solve the interacting electron system Hλ many times. In a rst step for each coupling constant λ the external potential, that keeps the density xed at the ground-state density of the fully interacting system, must be found. en the many electron ground-state wave function Ψλ can be determined. As an example Fig. ç.Ô shows the exact and λ LDA integrand dExc dλ for the He atom. e LDA integrand has been evaluated by means of Eq. (ç.çÔ). For λ ý the LDA overestimates the exact value quite notable. Here, the curve is = ~ solely determined by exchange, as dEλ dλ at λ ý is identical to the exact exchange energy xc = (ò.çç). With increasing λ the distance between the LDA and the exact integrand decreases. ~ e LDA can, thus, be considered as more accurate in the limit λ → Ô.

Moreover, Fig. ç.Ô demonstrates the oŸen cited error cancellation between the exchange and correlation part of the LDA. According to Eqs. (ç.ò—) and (ç.çÔ) the exchange contribution

ò¥ ç.¥. Hybrid functionals dEλ dλ to the coupling-constant integrand is a constant and given by the integrand at λ ý x = ~ dEλ dEλ x xc . (ç.çç) dλ = dλ λ ý = W Hence, the area between the constant dEλ dλ and the curve y ý corresponds to the ex- x = change energy, whereas the correlation energy can be identied with the area between the λ ~ λ coupling-constant integrand dExc dλ and the constant dEx dλ. A simple comparison of both areas for the LDA and the exact integrand shows that the LDA overestimates the ex- ~ ~ change energy for He, but underestimates the correlation energy. eir sum, however, is a reasonable approximation for the exact xc energy, since the correlation energy counterbal- ances the overestimation of the exchange energy. is error cancellation between exchange and correlation energy, here observed for He, is a quite general statement. It holds for larger systems as well as for the GGA. As there is a quite substantial deviation between the exact integrand and the LDA in the limit λ → ý, one might think of using the exact orbital-dependent expression for the exchange energy (ò.çç) instead of an explicit density-dependent approximation as in LDA or GGA. In this way, the λ ý limit would be described exactly. As in the opposite limit λ → Ô the LDA = performs quite well, a suitable combination of exact exchange with LDA or GGA might be a good approximation for Exc. is is the underlying idea of hybrid functionals and will be discussed in more detail in the next section.

3.4. Hybrid functionals

Hybrid functionals combine local or semi-local functionals with the exact orbital-dependent exchange term Ex (ò.çç). is class of functionals relies on the observation that LDA as well as GGA perform better in the λ → Ô limit of the coupling-constant integration method than in the limit λ → ý, where the exact integrand is known and is given by the exact exchange energy (ò.çç). Nowadays, a whole zoo of hybrid functionals is available. We concentrate our discussion here on three prototype hybrid functionals. λ e simplest way to approximate the coupling constant integrand dExc dλ is a linear in- terpolation between its limits λ ý and λ Ô = = ~

dEλ dEλ dEλ dEλ xc xc + λ xc − xc . (ç.ç¥) dλ = dλ λ ý dλ λ Ô dλ λ ý = = = W Œ W W ‘

ò ç. e exchange-correlation energy functional of KS DFT

AŸer integration we obtain for the xc energy functional

λ λ λ dExc Ô dExc dExc Exc ∫ dλ + . (ç.ç ) = dλ = ò dλ λ ý dλ λ Ô = = Œ W W ‘ For the ý end the orbital dependent exact exchange energy is used dEλ d E , λ xc λ λ ý x = = = whereas in the opposite limit dEλ d is usually approximated by a local or semi-local xc λ λ Ô = ~ T density functional (DF) dEλ d EDF. We note, however, that the latter is an additional xc λ λ Ô xc =~ = T approximation beyond the linearization of the integrand Eq. (ç.ç¥). Consequently, one nally ~ T obtains the functional Ô E E + EDF , (ç.çâ) xc = ò x xc which is known as half and half (HaH) mixing‰ and whichŽ has been proposed by Becke [òÔ].

Perdew et al. [òò] proposed to interpolate the λ dependence of the integrand by

λ DF,λ dExc dExc DF n Ô + Ex − E Ô − λ − (ç.çÞ) dλ = dλ x

DF,λ ( )( ) where dExc dλ denotes the λ-dependent integrand of any local or semi-local functional, which is given by (ç.çÔ), and EDF is the exchange part of the DF functional. e integer n ~ x DF,λ → determines how rapid the integrand tends to dExc dλ for λ Ô. It is argued [òò], that the integer n should be the lowest order of Møller-Plesset (MP) perturbation theory, which ~ provides a realistic description of the many-electron system. As for a large class of materials MP perturbation theory of fourth order (MP¥) gives accurate results, it follows

Ô E EDF + E − EDF . (ç.ç—) xc = xc ¥ x x ( ) With the density functional of Perdew,Burke, and Ernzerhof EDF EPBE [Þ] Eq. (ç.ç—) denes xc = xc the PBEý functional.

In Fig. ç.ò the exact coupling-constant integrand for the He atom is compared with the integrand of the LDA and the integrand of the hybrids introduced so far. e HaH-I curve corresponds to the half and half mixing of Eq. (ç.ç ), where dEλ d dELDA,λ d xc λ λ Ô xc λ λ Ô = = = has been used. e curve denoted with HaH-II is the hybrid of Eq. (ç.çâ) with EDF ELDA, ~ T xc =~ xcT and the one labelled with Perdew corresponds to Eq. (ç.çÞ). In the last case we have employed the LDA for EDF and dEDF,λ dλ. By construction the hybrids match the exact curve at λ ý. x xc = is graphical analysis reveals that the HaH-I hybrid should be the most accurate approxi- ~ mation for the xc functional for the He atom. It is closely followed by the functional labelled with Perdew. Furthermore, Fig. ç.ò clearly demonstrates that the additional approximation

òâ ç.¥. Hybrid functionals

-0.85 He -0.9

-0.95

(htr) -1 λ /d λ

xc -1.05

dE exact -1.1 LDA HaH I -1.15 HaH II Perdew -1.2 0 0.2 0.4 0.6 0.8 1 λ

λ Figure ç.ò.: e exact coupling-constant integrand Exc for the He atom is compared with the integrand in the LDA, the half-and-half hybrid according to Eq. (ç.ç ) denoted by HaH-I, the half-and-half hybrid of Eq. (ç.çâ) denoted by HaH-II, and nally with the hybrid proposed by Perdew et al. with n ¥ [Eq. (ç.çÞ)]. For all three = hybrids EDF ELDA is used. xc = xc made in the HaH-II of replacing the integrand at dEλ d by the full functional worsens xc λ λ Ô = the description of the integrand. ~ T e hybrid functionals introduced so far are derived from theoretical considerations. us, they are oŸen considered as ab-initio functionals. However, there exist a few hybrid function- als, whose parameters are tted to experimental data sets. e most famous one is the BçLYP hybrid [âÔ, âò], which has the generic form

E ELDA + a E − ELDA + a EB—— − ELDA + a ELYP − ELDA . (ç.çÀ) xc = xc ý x x x x x c c c ( ) ( ) ( ) e B stands for its inventor Becke, ç denotes the three parameters aý, ax, and ac, which are determined by a t to â atomization energies, ¥ò ionization energies, — proton a›nities and the Ôý rst-row total atomic energies of the molecules of the GÔ database [âç, â¥], and LYP species the correlation functional of Lee, Yang, and Parr [ ].

òÞ ç. e exchange-correlation energy functional of KS DFT

In conclusion, hybrid functionals interpolate the coupling constant integrand between its exact exchange limit for λ ý and the fully interacting limit λ Ô, where local or semi- = = local functionals are supposed to work well. For the hybrid functionals introduced so far correlation is treated on the level of LDA and GGA. Alternatively, one can search for orbital- dependent expressions for the correlation energy. Görling-Levy perturbation theory for ex- ample ošers a systematic expansion of the xc energy functional in terms of orbital-dependent functionals and thus provides an orbital denition of correlation.

3.5. Görling-Levy perturbation theory

Görling-Levy perturbation theory [Ôâ] expresses the ground-state energy of the coupling- constant Hamiltonian Hλ [Eq. (ç.Ô)] by a perturbation expansion in the scaling parameter λ of the electron-electron interaction around λ ý. e coupling constant λ appears in the in- = teractiontermandintheexternalpotential V λ r , which keeps the density independent from λ. With the help of uniform coordinate scaling Görling and Levy have been able to specify ( ) the λ-dependent external potential V λ r in the coupling-constant integration Hamiltonian (ç.Ô) and carry out the perturbation theory, which nally provides a systematic expansion of ( ) the xc energy functional in terms of KS wave functions and eigenvalues. In section ç.ç all necessary equations are already developed in order to derive Görling-Levy perturbation theory. Combination of Eqs. (ç.çý) and (ç.ò—) links the xc energy of the scaled system Eλ with those at λ Ô xc =

λ ò ò Exc n λ Ex nÔ λ + Ec nÔ λ λEx n + λ Ec nÔ λ , (ç.¥ý) = ~ ~ = ~ where we split up the[ xc] energy( into[ exchange] [ and]) correlation[ ] contributions.[ ] e ešective po- λ λ tential Veš of the KS system, that corresponds to the scaled interacting system H , is therefore given by λ λ ò δEc nÔ λ V r V r + λVH r + λVx r + λ ~ . (ç.¥Ô) eš = δn r [ ] ( λ) ( ) ( ) ( ) is ešective potential Veš ensures that the ground-state density( of) the KS system is identi- cal to the ground-state density of the interacting scaled system Hλ. Per construction of the adiabatic connection, however, the density of the latter is identical to the true ground-state density for λ Ô and hence independent of λ. According to the Hohenberg and Kohn theo- = λ rem Veš must, thus, be the same for each λ up to a constant. Consequently, the λ-dependent external potential can be written as

ò— ç. . Görling-Levy perturbation theory

δEc n Ô λ λ ý λ V r V = r − λ VH r + Vx r + λ . (ç.¥ò) = δn r ⎛ [ ]⎞ λ ý ( ) ( ) ( ) ( ) e potential V = r is identical to the ešective⎝ potential of the KS( system,) ⎠ which mimics the fully interacting many-electron system. According to Eq. (ò.òÞ) it is partitioned into external, ( ) Hartree, and xc part

λ ý λ Ô V = r V = r + V r + V r V r + V r + V r . (ç.¥ç) = H xc = H xc ( ) ( ) ( ) ( ) ( ) ( ) ( ) us, the coupling constant Hamiltonian Hλ can be written as

δEc n Ô λ λ H Hý − λ VH ri + Vx ri + λ + λVee , (ç.¥¥) = Q δn ri i ⎛ [ ]⎞ ( ) ( ) ⎝ ( ) ⎠ where Hý is independent of λ

Ô ò Hý − ∇ + V ri + VH ri + Vxc ri . (ç.¥ ) = Qi ò ‹ ( ) ( ) ( ) By means of standard perturbation theory the ground-state energy of Hλ can be expressed in terms of the scaling parameter λ of the electron-electron interaction. At full coupling strength λ Ô it nally becomes =

( ) ∞ i ç E n E n Ts n + n r V r d r + UH n + Ex n + Ec n . = Qi ý = ∫ = [ ] [ ] [ ] ( ) ( ) [ ] [ ] [ ] While the zero-order i ý terms of the expansion are given by the kinetic energy T n = s and the energy contribution arising from the external potential V r , the rst-order terms ( ) [ ] i Ô are the Hartree energy U n and the exact exchange energy = H ( ) ( ) occ.[ ] ϕ r ϕ r ϕ r ϕ r Ô ∗i j ∗j ′ i ′ ç ç Ex n − d r d r′ . (ç.¥â) = ò Q ∫∫ r − r i,j ( ) ( ) ′( ) ( ) [ ] All higher-order terms i ò,..., ∞ dene theS correlationS energy in terms of the KS wave = functions and eigenvalues ( ) ∞ j Ec Ec n . (ç.¥Þ) = Qj ò = ò [ ] e leading term of the correlation energy Ec is given by

òÀ ç. e exchange-correlation energy functional of KS DFT

Φ V − V − V Φi ò Eò − ∞ ee H x , (ç.¥—) c i − = Qi Ô EKS EKS = S⟨ S S ⟩S where Φ is the ground-state KS single Slater determinant and Φi denotes the i-th excited KS i Slater determinant. EKS and EKS are the corresponding total energies. e correlation energy ò Ec is very similar to the energy expression of second-order MP perturbation theory. In fact, Eq. (ç.¥—) can be decomposed into a contribution, which is formally identical to the energy in second-order MP perturbation theory, but evaluated with the KS orbitals and KS eigenvalues, and a second term, which arises from the dišerence between non-local and local treatment of exchange in MP perturbation theory and KS DFT, respectively [â ]. In contrast to Eq. (ç.¥â), ò Ec and all higher-order correlation terms depend on the unoccupied states. e complexity of the mathematical expressions increases with the order of the expansion. However, the consid- ò eration of Ec already enables a description of dispersion forces [ÔÞ, ââ]. Unfortunately, there is no guarantee that the higher-order terms are negligible. In fact, a partial re-summation of the perturbation series is required for a variationally stable functional [âÞ]. Taking everything into consideration, Görling-Levy perturbation theory provides a sys- tematic and exact expansion of the xc energy in terms of orbital-dependent functionals. One order aŸer the other can be used to dene xc functionals of increasing accuracy, in principle. e leading term in this expansion is the exact exchange energy. Furthermore, Görling-Levy perturbation theory can be combined with the idea of hybrid functionals, as the Görling- Levy expansion of the xc energy simultaneously denes a series for the coupling constant integrand

λ dExc ò ò Ex + òE λ + O λ . (ç.¥À) dλ = c While E is the exact value of the coupling constant integrand( ) at λ ý, the slope of the inte- x = grand at λ ý is given by twice the correlation energy Eò. In elaborate hybrid functionals [â—] = c both informations, the exact value together with the exact slope at λ ý, are incorporated. =

3.6. Sham-Schlüter equation

An alternative approach for a systematic improvement of the xc potential of DFT is provided by the Sham-Schlüter equation [âÀ, Þý], which connects the xc potential of DFT with MBPT. By construction of the KS system its ground-state density equals the true ground-state density of the many-electron system. e KS density as well as the density of the many-

çý ç.Þ. Summary electron system can be expressed by the respective Green function

Ô Ô ý n r − nKS r I dωG r, r, ω − I dωGKS r, r, ω . (ç. ý) = = π ∫ π ∫ With the Dyson equation( ) that( ) connects many-electron( ) and KS Green function( )

G r, r′, ω G r, r′, ω (ç. Ô) = KS ç ç ( ) − ∫∫ (GKS r, r)′′, ω Σxc r′′, r′′′, ω − Vxc r′′ δ r′′ − r′′′ G r′′′, r′, ω d r′′d r′′′ ( )
( ) ( ) ( ) ( ) the density condition (ç. ý) becomes

ç ç G r, r′, ω Σ r′, r′′, ω − V r′ δ r′ − r′′ G r′′, r, ω d r′ d r′′ dω ý. (ç. ò) ∫∫∫ KS xc xc = ( )
( ) ( ) ( ) ( ) is is the so-called Sham-Schlüter equation. It establishes a direct link between the self- energy Σxc of MBPT and the xc potential of DFT. e xc potential corresponding to a given self-energy Σxc can be understood as the best local approximation to the non-local dynamic self-energy. As the self-energy can be improved diagrammatically, Eq. (ç. ò) ošers a way for a systematic improvement of the xc potential. However, the solution of Eq. (ç. ò) is by far not trivial. e linear-response Sham-Schlüter equation, which is obtained by combining

Eqs. (ç. Ô) and (ç. ò) and neglecting all non-linear terms in Σxc r, r′, ω − Vxc r δ r − r′ , has been solved to our knowledge so far only for bulk silicon [ÔÔ, ÞÔ] and for a jellium sur- [ ( ) ( ) ( )] face [Þò].

3.7. Summary

Orbital-dependent xc functionals of DFT have been introduced and motivated in this chap- ter. It has been shown that theorbital-dependentexact exchange energy Ex, which is formally identical to the HF exchange energy, but evaluated with the KS orbitals, improves the descrip- tion of the coupling-constant integrand in the λ → ý limit. So, it is reasonable to combine Ex with pure density functionals like LDA or GGA, which usually perform well in the opposite limit λ → Ô, into one functional, a hybrid functional. We have demonstrated that Ex is, fur- thermore, the leading term in a systematic expansion of the xc functional in powers of the electron-electron interaction. e higher order terms in this expansion dene an orbital and eigenvalue dependent correlation functional. e introduction of orbital dependent xc functionals is conceptionally comparable with the transition from the omas-Fermi method to the KS approach, where the density func- tional for the kinetic energy has been replaced by the orbital dependent kinetic energy of

çÔ ç. e exchange-correlation energy functional of KS DFT non-interacting electrons. Any orbital dependent xc functional in analogy to the KS kinetic energy is still an implicit functional of the density, because the KS orbitals are functionals of the ešective potential through the KS single-particle Schrödinger equation and the ešective potential is a functional of the density through the Hohenberg and Kohn theorem. e same argumentation holds for functionals, which are additionally eigenvalue-dependent. e construction of the xc potential from an orbital-dependent functional has not been discussed so far. Two substantially dišerent approaches are available: on the one hand, a strict treatment of orbital-dependent functionals within the KS framework, which requires a local multiplicative xc potential, per denition, for the KS system as a non-interacting electron system. However, a straight-forward analytical functional derivative of the xc energy with respect to the density is not feasible in contrast to LDA and GGA. is problem is resolved in the optimized ešective potential (OEP) method, which yields the local xc potential as the solution of an integral equation. e OEP approach as well as its numerical implementation within the full-potential linearized augmented-plane-wave (FLAPW) method is discussed in detail in chapter â. On the other hand, instead of constructing a local xc potential in agreement with the KS formalism one can dene a dišerent reference system, the so-called generalized Kohn- Sham (gKS) system. Hybrid functionals are typically treated within the latter. e orbital- dependent exchange energy, which is characteristic for hybrid functionals, leads to a non- local HF like exchange term in the one-particle equations of the gKS system. e gKS scheme, its implementation as well as its application are discussed in chapter . Before, we will intro- duce the FLAPW method in the next chapter.

çò 4. Electronic structure methods: the FLAPW approach

A wide spectrum of electronic structure methods exists for the numerical solution of the Kohn-Sham (KS) equation. ese methods can be divided into all-electron and pseudopo- tential approaches. In the latter only the valence electrons are described explicitly, while the core electrons are frozen in their atomic conguration, and the Coulomb potential of the nuclei is replaced by a smooth ešective one, the pseudopotential. e smoothness of the po- tential, which in turn leads to smooth pseudo wave functions, enables to use plane waves for the representation of the wave functions. Plane waves have several advantages: each power of the momentum operator is diagonal, in particular the kinetic energy, the charge density can be calculated e›ciently via a fast Fourier transform (FFT), and the Hartree potential can be obtained by a simple algebraic expression in reciprocal space. e plane-wave pseudopo- tential approach, however, faces its limits in the precise treatment of localized electron-states, e.g. çd or ¥ f electron-states. e all-electron approaches, on the contrary, treat core and valence electrons on an equal footing without any pseudization of the potential. e presenceof theÔ r-like behavior of the true potential in the vicinity of the atomic nuclei leads to rapidly varying wave functions there. ~ us, an enormous number of plane waves would be required for the accurate description of the all-electron wave function. is shows that plane waves do not constitute the optimal basis set in an all-electron method. Typically, in all-electron methods space is partitioned into atom-centered spheres, the so-called mu›n-tin (MT) spheres, and the remaining inter- stitial region (IR). In order to cope with the Ô r-like potential in the MT spheres, one usually uses numerical functions, which explicitly incorporate the rapid oscillations caused by the ~ Ô r singularity of the potential. ese numerical functions are then matched to an envelope function in theIR. e mu›n-tin orbital (MTO) and augmented plane-wave (APW) method ~ and its linearized variants LMTO and LAPW belong to this kind of approach. In the following, we will introduce the LAPW method step by step. We will begin with the APW approach, the precursor of the LAPW method. en, we discuss the LAPW method, where we concentrate in particular on the formulation for bulk solids, and its local-orbital

çç ¥. Electronic structure methods: the FLAPW approach

Ra'

Ra IR

MT

Figure ¥.Ô.: Division of space into mu›n-tin (MT) spheres and the interstitial region (IR). Each atomic nucleus lies in a center Ra of a MT sphere of radius Sa. e radii of the spheres are chosen such that the spheres do not overlap. extension. Finally, we introduce the mixed product basis (MPB), which is an auxiliary basis for the representation of LAPW wave-function products. e MPB fully preserves the all- electron character of the product and can be converged in a systematic manner.

4.1. APW method

As already pointed out in the introduction of the chapter, the APW approach [Þç] partitions space into atom-centered MT spheres and the IR (s. Fig. ¥.Ô). e potential in the IR is rela- tively smooth, which allows to use plane waves as basis functions there. In the MT spheres, solutions of the scalar-relativistic, spin-dependent, spherical Schrödinger equation are em- ployed and matched to the interstitial plane wave. us, for a given unit cell the APW basis function of Bloch vector k, reciprocal lattice vector G and spin index σ is given by

Ô exp i k + G ⋅ r r IR φσ r, E √Ω ∈ , (¥.Ô) kG aσ aσ = ⎧ α k, G, E u ra , E Ylm ra r MT a ⎪ ∑lm lm [ (l ) ] ∈ ( ) ⎨ ⎪ ̂ where the vector ra has its⎩⎪ origin at the( center) of the(S sphereS ) R(a, Ω) is the unit-cell( ) volume, and aσ ul r, E is the radial solution of the Schrödinger equation

( ) Ô ∂ò l l + Ô − + + V aσ r − E ruaσ r, E ý (¥.ò) ò ∂rò òrò eš,ý l = ( )  ( ) ( )

ç¥ ¥.Ô. APW method

aσ with the spin-dependent, spherical ešective potential Veš,ý r at atom a and the energy pa- rameter E.Ô e coe›cient αaσ k, G, E is solely determined from the requirement of conti- lm ( ) nuity at the sphere boundary ( ) ¥π j k + G S aσ + ⋅ l + l a αlm k, G, E exp i k G Ra i Ylm∗ k G aσ , (¥.ç) = Ω u Sa, E (Sl S ) ( ) √ [ ( ) ] ( Å ) where Sa denotes the radius of the MT sphere of atom a and jl r are( the spherical) Bessel functions. e latter arise from the Rayleigh expansion of the plane wave exp i k + G ⋅ r . ( ) e KS wave function of Bloch vector k and band index n is then expressed in the APW basis [ ( ) ]

σ σ σ ϕnk r zG n, k φkG r, E . (¥.¥) = QG ( ) ( ) ( ) In this way, the dišerential KS equation [Eq. (ò.òò)] is cast into an algebraic form

σ σ σ σ HGG′ k, E − єnk SGG′ k, E zG′ n, k ý (¥. ) QG′ = [ ( ) ( )] ( ) with the Hamiltonian

σ σ Ô ò σ σ ç H ′ k, E φ ∗ r, E − ∇ + V r φ ′ r, E d r (¥.â) GG = ∫ kG ò eš kG ( ) ( )  ( ) ( ) and the overlap matrix

σ σ σ ç S ′ k, E φ ∗ r, E φ ′ r, E d r , (¥.Þ) GG = ∫ kG kG ( ) ( ) ( ) respectively. It turns out, that the APW basis only leads to an accurate description of the wave σ σ function ϕnk if the energy parameter E is set to the eigenvalue єnk. As the eigenvalue is not known a priori, Eq. (¥. ) constitutes a non-linear problem. Starting with a initial guess for σ the eigenvalue єnk the APW basis (¥.Ô), the Hamiltonian (¥.â), and the overlap matrix (¥.Þ) are constructed with E єσ . e root of the determinant = nk

σ σ σ det H ′ k, E − є S ′ k, E ý (¥.—) GG nk GG =

S ( ) σ ( )S then yields a rened guess for the eigenvalue єnk. is scheme is iterated until input and σ output guess coincide. In this way, the true eigenvalue єnk is found. AŸerwards the eigen- σ vector zG n, k can be calculated by solving Eq. (¥. ). is procedure must be repeated for

( ) ÔFor simplicity, we give the non-relativistic equations here. A discussion of the scalar-relativistic equations is deferred to Appendix B.

ç ¥. Electronic structure methods: the FLAPW approach each band n, Bloch vector k, and spin σ, which makes the APW method computationally demanding.

Additionally, the APW approach sušers from the so-called asymptote problem, i.e., the aσ decoupling of the radial function and the plane wave at the MT sphere boundary, if ul r, E vanishes for r S . A more detailed discussion of the APW method can be found in Ref. Þ¥. = a ( ) e non-linearity and the asymptote problem of the APW approach are overcome in the LAPW method by using an additional radial function in the MT spheres as described in the next section.

4.2. LAPW method

e LAPW method [ò¥–òÞ] is the linearized version of the APW approach. It uses, similar to the APW method, piecewise dened functions to deal simultaneously with the atomic-like potential close to the nuclei and the smooth potential in the IR. However, the žexibility in the MT spheres in contrast to the APW basis is enhanced by adding for each angular momentum aσ l and each MT sphere a second radial function u˙l r , which is the energy derivative of uaσ r . Additionally, a separate energy parameter Eaσ is introduced for each atom a, angular l l ( ) momentum l, and spin σ. In this way, for a given unit cell the LAPW basis function of Bloch ( ) vector k and reciprocal lattice vector G is given by

Ô exp i k + G ⋅ r r IR φσ r √Ω ∈ . (¥.À) kG aσ aσ aσ aσ = ⎧ α k, G u ra + β k, G u˙ ra Ylm ra r MT a ⎪ ∑lm lm l [ ( lm ) ]l ∈ ( ) ⎨ ⎪
( ) (S S) ( ) (S S) (̂) aσ( ) aσ For the sake⎩ of clarity, we suppressed the dependence of the radial functions ul r , u˙l r as well as of the coe›cients αaσ k, G , βaσ k, G on the energy parameter Eaσ . lm lm l ( ) ( ) While the radial function uaσ r is the solution of Eq. (¥.ò) for E Eaσ , the energyderiva- l ( ) ( ) = l tive u˙aσ r obeys the inhomogeneous Schrödinger equation l ( ) ( ) Ô ∂ò l l + Ô − + + V aσ r − Eaσ ru˙aσ r ruaσ r . (¥.Ôý) ò ∂rò òrò eš,ý l l = l ( )  ( ) ( ) aσ( ) Its solution becomes unique by the additional requirement that u˙l r is orthogonal to uaσ r . e coe›cients αaσ k, G and βaσ k, G ensure continuity in value and rst radial l lm lm ( ) ( ) ( ) ( )

çâ ¥.ò. LAPW method derivative at the MT sphere boundary

aσ αlm k, G (¥.ÔÔ) ¥π Ô + ⋅ l + aσ + ( ) exp i k G Ra i Ylm∗ k G aσ aσ u˙l Sa , jl k G Sa = Ω u˙l Sa , ul Sa βaσ k√, G [ ( ) ] ( Å ) [ ( ) (S S (¥.Ôò))] lm [ ( ) ( )] ¥π −Ô + ⋅ l + aσ + ( ) exp i k G Ra i Ylm∗ k G aσ aσ ul Sa , jl k G Sa . = Ω u˙l Sa , ul Sa √ [ ( ) ] ( Å ) [ ( ) (S S )] e square brackets denote the Wronskian [ ( ) ( )]

dg r df r f r , g r f r − g r . (¥.Ôç) = dr dr ( ) ( ) [ ( ) ( )] ( ) ( ) e addition of the energy derivative in the MT spheres enables to represent a solution aσ aσ ul r, E of the radial Schrödinger equation (¥.ò) of energy E by the radial functions ul r and u˙aσ r ( l ) ( ) ( ) uaσ r, E uaσ r, Eaσ + u˙aσ r, Eaσ E − Eaσ + O E − Eaσ ò (¥.Ô¥) l = l l l l l l − aσ up to an error which( ) is quadratic( in) the energy( dišerence) ‰ E Ž El .‰( With this) increasedŽ žex- aσ ibility the energy parameters El need not equal the band energies as in the APW method, so that the algebraic equation (¥. ) becomes linear in the LAPW basis. All eigenvalues and eigenvectors for a given Bloch vector and spin are then obtained by a single diagonalization. Furthermore, there is no asymptote problem in the LAPW approach. By using Eqs. (¥.ò) and (¥.Ôý) it can be shown that the denominator in Eqs. (¥.ÔÔ) and (¥.Ôò) does not vanish. Instead it fullls the relation

ò ˙aσ aσ ul Sa , ul Sa ò . (¥.Ô ) = Sa [ ( ) ( )] σ e LAPW basis functions φkG r are usually employed to represent the valence-electron aσ states accurately. erefore, the energy parameters El are placed in the valence band region. aσ ( ) One possible choice is to set El in the center of gravity of the l-like band. However, the LAPW method is an all-electron approach comprising valence and core states. e orthogonality of the tightly bound core states with respect to the LAPW basis [òÞ, Þ ] allows to separate the core from the valence states. Due to the spatial connement of the core electrons it is an excellent approximation to describe them as atomic states by solving a fully relativistic Dirac equation with the spherically symmetric part of the ešective potential.

çÞ ¥. Electronic structure methods: the FLAPW approach

Although the LAPW basis functions are constructed with respect to the spherical average of the ešective potential, the full potential enters in Eq. (¥.â) without shape approximations. It is commonly accepted that the LAPW basis possesses su›cient žexibility to represent the full- potential KS wave function within the MT spheres. A comparison between the full-potential LAPW (FLAPW) method and the full-potential Korringa-Kohn-Rostocker (KKR) method, where the basis functions are constructed with respect to the full non-spherical potential, yields a perfect agreement [Þâ] and is thus a practical justication of this procedure. For the representation of the all-electron potential and density, similar to the LAPW ba- sis, a dual representation is employed, i.e., plane waves in the IR and radial functions times spherical harmonics in the MT spheres. Density and potential are invariant under symmetry operations of the system, which enables to decrease the basis set size considerably. is is achieved by constructing symmetry-adapted linear combinations of plane waves, so-called stars, in the IR and symmetry-adapted spherical harmonics, so-called lattice harmonics, in the MT spheres for the representation of the density and potential [òÞ]. e number of stars and lattice harmonics is usually much smaller than the number of plane waves and spherical harmonics, respectively. Furthermore, for a practical calculation cutoš values for the reciprocal lattice vectors k + G G and the angular momentum l l are introduced. ≤ max ≤ max S S 4.2.1. Local-orbital extension

aσ States that lie in the vicinity of the energy parameters El are accurately described by the LAPW basis. However, the basis becomes less adequate for states that are energetically far aσ away from El . ese are for example semi-core states, i.e., high-lying core states, whose wave functions extend considerably over the MT sphere. Due to their spatial extent a treatment as core states is not justied. Nevertheless, the linearization error of the standard LAPW basis prohibits an accurate description of such states, as well. High-lying unoccupied states are another example, where the linearization error matters. In order to improve the representation of such states, the basis may be extended with local orbitals (LOs) [ÞÞ–ÞÀ]. Let us assume that we want to improve the LAPW basis for states aσ with an angular momentum l in the MT sphere of atom a around an energy ELO. en, an aσ aσ additional radial function ul r, ELO is constructed from Eq. (¥.ò) for the energy parameter Eaσ . is radial function is combined with uaσ r and u˙aσ r to LO ( ) l l ( ) ( ) uaσ r αaσ uaσ r + βaσ u˙aσ r + γaσ uaσ r, Eaσ , (¥.Ôâ) l,LO = l,LO l l,LO l l,LO l LO

aσ ( ) ( ) ( ) ( ) such that ul,LO r is normalized and its value and radial derivative vanish at the MT sphere ( )

ç— ¥.ò. LAPW method boundary. In this way, the local orbital is completely conned to theMTsphereand does not aσ need to be matched to a plane wave. Multiplication of the radial function ul,LO r with the spherical harmonic Y rˆ then leads to òl +Ô additional LAPW basis functions in the unit lm ( ) cell ( ) ( )ý r IR aσ,LO r aσ ∈ φk,lm , ELO aσ . (¥.ÔÞ) = ⎧ u ra Ylm ra r MT a ⎪ l,LO ∈ ( ) ⎨ ⎪ For the improved description of a semi-core⎩⎪ (S stateS) of( atom̂) a,which( is) nearly dispersion-less, aσ the energy parameter ELO is xed at the semi-core energy level. In order to improve the linearization error for the unoccupied states, two dišerent schemes aσ aσ are possible. On the one hand, local orbitals can be added, where ul r, ELO is replaced by a higher energy derivative evaluated at Eaσ Eaσ [Þ ]. erewith the order of the Taylor LO = l ( ) series expansion (¥.Ô¥) is increased and the LAPW basis becomes more and more žexible in aσ a window around El . is ansatz does not require the choice of an additional energy pa- aσ rameter, but it has the disadvantage that due to the improved variational freedom around El with respect to higher as well as lower energies, core states can appear in the valence energy window. Alternatively, the LAPW basis can be extended by local orbitals, where the energy aσ aσ aσ parameter ELO is determined such that the logarithmic derivative of the solution ul r, ELO of Eq. (¥.ò) fullls ( ) d aσ aσ ln ul r, ELO − l + Ô (¥.ԗ) dr r S = = a at the MT sphere boundary r
S‰ [—ý].( is) conditionŽV yields( for) each l quantum number = a [cf. Fig. ¥.ò] a series of orthogonal solutions of increasing energies and number of nodes. Consecutively, local orbitals constructed from these series of radial functions can be added for a systematic convergence of the LAPW basis with respect to the unoccupied states. e advantage of the latter approach is that the LAPW basis is only improved for the unoccupied states. Core states do not occur in the valence window. In order to demonstrate that the addition of local orbitals to the LAPW basis makes the MT aσ LAPW basis more and more complete, we calculate the radial function ul r, E that solves Eq. (¥.ò) at energy mesh points E over an energy range from −ò to ¥ýhtr. At each energythe aσ ( ) exact solution ul r, E is represented by the radial MT functions of the LAPW basis set. We dene the error ∆ in the representation of the exact radial function uaσ r, E at the energy (l ) l E by the integral S ( ) a aσ aσ ò ò ∆l E ul r, E − cpul p r r dr , (¥.ÔÀ) = ∫ý Qp

( ) S ( ) ( )S aσ where cp are the expansion coe›cients of the exact solution ul r, E in the radial LAPW functions uaσ r . For a unied notation, we have subsumed the dišerent radial functions l p ( ) ( )

çÀ ¥. Electronic structure methods: the FLAPW approach

òl çl ¥l l a S = r ))S E , r ( σ a l − l + Ô u (

ln ( ) d dr

aσ aσ aσ aσ ELO,ò ELO,ç ELO,¥ ELO,

aσ Figure ¥.ò.: Logarithmic derivative of the radial function ul r as a function of the energy parameter E at the MT sphere boundary Sa. e logarithmic derivative splits up aσ ( ) into branches, in which the radial functions ul r have a constant number of nodes for r S . A branch of type nl has n − l + Ô nodes. e intersection < a points of the red line, which corresponds to − l +(Ô ), with the curve representing the logarithmic derivative dene a series of energies,( which) are used to converge the LAPW basis with respect to the unoccupied( states.)

¥ý ¥.ò. LAPW method into the composite index p

uaσ r p Ô l = uaσ r ⎧u˙aσ r p ò (¥.òý) l p ⎪ l ( ) = ⎪ = ⎪ aσ aσ ( ) ⎨ul (r), ELO p ç. ⎪ ≥ ⎪ ⎪ ( ) Figure ¥.ç shows the representation error⎩ ∆l E for the case of diamond for dišerent LAPW basis-set congurations. As an example, we have chosen the angular momentum l ý. A ( ) = similar behavior is observed for the higher angular momenta, as well. e red curve corre- sponds to the conventional LAPW basis without any local orbitals. For the green, blue, and magenta curves one, two, and three local orbitals have been added according to the prescrip- tion (¥.ԗ), respectively. e energy parameters of the conventional LAPW basis and local orbitals correspond to the spikes in the curves. e conventional LAPW basis set provides an accurate description around the energy parameter of the conventional radial LAPW func- aσ − tions El ý ý.òçhtr . However, with increasing energy the representation error increases = = and approaches Ô at about Ôý htr, which means that the radial functions uaσ r (p Ô, ò) ( ) l p = are (nearly) orthogonal to the exact solution u r, E . By adding one local orbital (green l ( ) curve) the representation error can be reduced to Ôh in an energy range of about Ôý htr. Yet, ( ) above an energy of òýhtr the exact radial function cannot be represented. e addition of more and more local orbitals reduces the linearization error ∆l E over an increasing energy range. ree local orbitals are necessary to restrict the error ∆ E below h over an energy l( ) range of ¥ý htr. In conclusion, Fig. ¥.ç demonstrates that the LAPW basis becomes more and ( ) more complete over a wide energy range by adding local orbitals constructed according to Eq. (¥.ԗ).

4.2.2. Mixed product basis

e mixed product basis (MPB) is specically designed for the representation of products of LAPW wave functions. Such products naturally occur in orbital-dependent functionals. e MPB retains the all-electron description of the product and can be converged in a systematic manner. We will describe the construction of the MPB in the following. In order to accurately describe the products of two LAPW wave functions in the MT spheres the MPB must be capable of describing the products

aσ aσ aσ aσ ul p r Ylm∗ r ul′ p′ r Yl′m′ r ul p r ul′ p′ r GLM,l′ m′,lmYLM r , (¥.òÔ) = L∈{Sl−lQ′S,...,l+l′ } M=−L,...,L ( ) ( ) ( ) ( ) ( ) ( ) ( )

¥Ô ¥. Electronic structure methods: the FLAPW approach

100 10-1 10-2 10-3

) 10-4 E ( l -5 ∆ 10 10-6 10-7 10-8 10-9 0 5 10 15 20 25 30 35 40 E (htr)

Figure ¥.ç.: Representation error ∆ E for the case of diamond and for l ý as a function l = of the energyand the number of local orbitals added to the LAPW basis set. e red curve corresponds to( an) LAPW basis without any local orbitals, whereas for the green, blue, and magenta curves we consecutively add one local orbital more with energy parameters according to Eq. (¥.ԗ).

where we exploit the addition theorem of spherical harmonics and GLM,l′m′,lm are the Gaunt coe›cients

G ′ ′ Y ∗ rˆ Y ′ ′ rˆ Y ∗ rˆ dΩ . (¥.òò) LM,l m ,lm = ∫ LM l m lm For a given angular momentum L we dene the( ) set of( radial) ( functions) by

aσ aσ aσ U r u r u ′ ′ r , (¥.òç) LP = l p l p ( ) ( ) ( ) where P counts all combination of p, p′, l, and l′ that contribute to L. is set of radial func- tions exhibits usually a high degree of linear dependence. In order to remove the linear de- pendence, we diagonalize the overlap matrix

aσσ′ aσ aσ′ ò O ′ U r U ′ r r dr (¥.ò¥) L,PP = ∫ LP LP ( ) ( ) for each a and L, and retain only those eigenfunctions whose eigenvalues exceed a given threshold value (typically ý.ýýýÔ). us, the (nearly) linearly dependent combinations are removed and a smaller, but still žexible basis is obtained. In the case of spin-polarized cal- culations the resulting radial functions become spin-independent as products of spin-up and

¥ò ¥.ò. LAPW method spin-down radial functions enter in the overlap matrix. Additionally, we add a constant func- tion to the optimized set of spherical radial product functions. From the resulting radial MT a functions MLP r we then construct Bloch functions

a ( ) Ô a Mk,LMP r MLP r − Ra − T YLM r − Ra − T exp ik T + Ra , (¥.ò ) = N QT ( ) √ (S S) ( Æ ) [ ( )] where the sum runs over the lattice vectors T and N denotes the number of unit cells. In the IR, the product of two LAPW wave functions gives rise to products of interstitial plane waves (IPWs), which yield another IPW. erefore, the MPB in the IR consists of IPWs

Ô Mk,G r exp i k + G r Θ r (¥.òâ) = NΩ ( ) √ [ ( ) ] ( ) with the step function Ô r IR Θ r ∈ . (¥.òÞ) = ⎪⎧ý r MT ⎪ ∈ ( ) ⎨ From here we can proceed in two ways: we⎪ can either combine the MT and IR functions to a ⎩⎪ full MPB, that consists of two spatially separate basis sets, which are not matched at the MT sphere boundary, or we can alternatively construct continuous functions with a continuous radial derivative over all space.

e former concept is used for the implementation of hybrid functionals (cf. Chap. ), where the MPB is employed to represent products of wave functions. e partition of the MPB in MT and IR functions, which are either zero in the IR or in the MT spheres, enables to accelerate the most time consuming part of a hybrid functional calculation, the compu- tation of the non-local exchange potential (s. Sec. .ò.Ô). On the contrary, in the optimized ešective potential (OEP) approach (cf. Chap. â), a continuous MPB is essential in order to avoid discontinuities in the resulting local potential. erefore, we form linear combinations of the MT functions and IPWs that are continuous in value and rst radial derivative at the MT sphere boundary. We can use the same construction as for the LAPW basis, i.e., two radial functions per lm channel are used to augment the IPWs in the MT spheres. As there are usually far more than two radial functions per lm channel in the MPB, local orbitals are constructed from the remaining MT functions. A more general approach, which allows to elegantly incorporate further constraints, is deferred to Appendix A.

In both cases, the nal MPB functions are not orthogonal and hence give rise to a non- diagonal overlap matrix ç O k M∗ r M r d r , (¥.ò—) IJ = ∫ k,I k,J ( ) ( ) ( )

¥ç ¥. Electronic structure methods: the FLAPW approach where I and J are used to index the MPB functions. Due to the non-orthogonality the com- pleteness relation of the MPB, which is only valid in the subspace spanned by the MPB, be- comes Ô Ô Mk,I OIJ− k Mk,J . (¥.òÀ) = QI,J With the denition of the biorthogonalS set ⟩ ( )⟨ S

˜ Ô Mk,I r OJI− k Mk,J r , (¥.çý) = QJ ( ) ( ) ( ) the completeness relation can be written in a more compact form

Ô M˜ k,J Mk,J Mk,J M˜ k,J . (¥.çÔ) = QJ = QJ Biorthogonal and normal MPB areS pairwise⟩⟨ orthogonalS S ⟩⟨ S

M M˜ δ . (¥.çò) k,I k,J = IJ ⟨ S ⟩ In chapter and â, the MPB with its completeness relation (¥.çÔ) is employed to implement hybrid functionals as well as the OEP method in the FLAPW method.

¥¥ 5. Hybrid functionals within the generalized Kohn-Sham scheme

In chapter ç, we already introduced the hybrid functionals, which approximate the coupling- constant integrand by combining exact exchange with (semi-)local approximations for the exchange-correlation (xc) functional of density-functional theory (DFT). Typically, hybrid functionals are treated within the generalized Kohn-Sham (gKS) system instead of the Kohn- Sham (KS) system. While the latter requires a local, multiplicative potential by construction (cf. Chap. ò), the gKS system allows a non-local Hartree-Fock(HF)-like exchange potential to be used in the one-particle Schrödinger equations. AŸer a short introduction to the gKS system the focus of this chapter is on the implementation of the non-local exchange potential within the full-potential linearized augmented-plane-wave (FLAPW) method. It employs the mixed product basis (MPB) (s. Sec. ¥.ò.ò) with which the matrix elements of the non- local potential, which correspond to state-dependent six-dimensional integrals, decompose into two three-dimensional and one state-independent six-dimensional integral. While the three-dimensional integrals must be calculated for each self-consistent eld (SCF) cycle, the state-independent six-dimensional integral is evaluated only once at the beginning of the cal- culation. By subsuming the three- and six-dimensional integrals into state-dependent vectors and a matrix, the Coulomb matrix, the matrix elements of the non-local exchange potential can be formulated as a Brillouin-zone (BZ) sum over vector-matrix-vector products. e calculation of the non-local potential constitutes the most-time consuming step in a practical hybrid functional calculation. We present several techniques to accelerate the computation. For example, the Coulomb matrix is made sparse by a unitary transformation of the mu›n-tin (MT) MPB functions. Its sparsity is then exploited to reduce the computational demand of the vector-matrix-vector product considerably. Furthermore, spatial and time- reversal symmetries are used to decide in advance which matrix elements are non-zero, and only these are calculated. Symmetry arguments also allow to restrict the BZ sum in the non- local exchange potential to a smaller, irreducible set of k points. e convergence of the BZ sum with respect to the k-point sampling, however, is hampered by a divergence of the Coulomb matrix at the Γ point. For a favorable convergence of the BZ sum with respect to the

¥ . Hybrid functionals within the generalized Kohn-Sham scheme k-point sampling, the divergence of the Coulomb matrix at the Γ point must be properly taken into account. For this purpose, the BZ sum is separated into a divergent and a non-divergent contribution. e divergent part, whose leading term is proportional to Ô qò, is integrated analytically, while the non-divergent contribution is treated numerically by a weighted sum. ~ Furthermore, we report the convergence behavior of the gKS transition energies employing the PBEý hybrid functional for Si and SrTiOç as well as of the total energy for Si. en, we apply the PBEý hybrid functional to a set of prototype semiconductors and insulators and nd a reasonable agreement with recent results from the literature. Moreover, we show band structure and density-of-states calculations with the PBEý hybrid functional exploiting the Wannier interpolation technique [—Ô, —ò]. Finally, we apply the PBEý hybrid functional to EuO and discuss its structural, electronic, and magnetic properties.

5.1. Generalized KS system

e central idea of the KS formalism is the introduction of an auxiliary system of non- interacting electrons, whose single Slater determinant yields the ground-state density of the real interacting system. So instead of the complicated many-electron system, a set of one-particle equations must be solved. e complexity of the interacting electron system, however, is hidden in the xc energy functional of the KS system (s. Chap. ç). Yet, the KS system is only one specic, arbitrary reference system. Dišerent choices of reference systems are possible [òç]. For example, we can dene a reference system, whose many-body wave function is still a single Slater determinant of one-particle orbitals, but a fraction of exchange is treated by a orbital-dependent, non-local HF-like exchange potential. is corresponds to an approximation of the universal spin-dependent functionalÔ F n↑, n↓ as the minimum of the sum of kinetic and scaled electron-electron interaction over all Slater determinants Φ [ ] yielding the spin densities n↑ r and n↓ r

gKS ↑ ↓ ( ) ( ) ↑ ↓ F n , n min ↑ ↓ Φ T + aV Φ + Ô − a U n + n . ( .Ô) = Φ→n ,n ee H

e parameter a [a ý,Ô] scales the⟨ electron-electronS S ⟩ ( interaction) [ V and] will be xed ∈ ee later. By adding Ô − a times the Hartree energy U n↑ + n↓ in Eq. ( .Ô), we pick up in com- ( [ ]) H bination with the Hartree term contained in the expectation value of aV the Hartree energy ( ) [ ] ee completely. e dišerence between the exact universal functional F n↑, n↓ and FgKS n↑, n↓

[ ] [ ] ÔFor the spin-independent case the denition of the universal functional is given in Eq. (ò.Ôç).

¥â .Ô. Generalized KS system denes in analogy to the KS system the xc energy functional of the gKS system

EgKS n↑, n↓ F n↑, n↓ − FgKS n↑, n↓ . ( .ò) xc =

If Ψ n↑, n↓ denotes the true[ many-electron] [ ground-state] [ wave] function of spin-densities n↑ r and n↓ r and if ΦgKS n↑, n↓ is accordingly the ground-state single Slater determinant [ ] of the gKS system, the xc energy ( .ò) can be written as ( ) ( ) [ ]

EgKS n↑, n↓ Ψ n↑, n↓ T + V Ψ n↑, n↓ − ΦgKS n↑, n↓ T ΦgKS n↑, n↓ xc = ee − gKS ↑ ↓ − ↑ ↓ [ ] ⟨ aE[ x n]S, n USH [n , n ]⟩.⟨ [ ]S S [ ]⟩ ( .ç)

gKS ↑ ↓ [ ] [ ] Here, Ex n , n is the exact exchange energy (ò.çç) evaluated with the gKS Slater determi- nant ΦgKS n↑, n↓ . Using the denition of the xc functional of the KS system [cf. Eq. (ò.ò )] [ ] the xc energy of the gKS system turns into a sum of the xc energy of the KS system, the exact [ ] gKS exchange energy Ex , and a correction for the kinetic energy, which arises from the dišer- ence between KS and gKS single Slater determinant

EgKS n↑, n↓ EKS n↑, n↓ − aEgKS n↑, n↓ xc = xc x + KS ↑ ↓ KS ↑ ↓ − gKS ↑ ↓ gKS ↑ ↓ [ ] Φ[ n ,]n T Φ [ n , n] Φ n , n T Φ n , n .( .¥) ⟨ [ ]S S [ ]⟩ ⟨ [ ]S S [ ]⟩ Usually, this dišerence between the KS and the gKS single Slater determinant is neglected and Eq. ( .¥), thus, becomes

EgKS n↑, n↓ Ô − a EKS n↑, n↓ + EKS n↑, n↓ . ( . ) xc = x c [ ] ( ) [ ] [ ] e one-particle orbitals of band index n, Bloch vector k, and spin index σ that correspond to the minimizing Slater determinant ΦgKS of Eq. ( .Ô) are solutions of

Ô ò σ σ NL,σ σ ç σ σ − ∇ + V r ϕ r + a V r, r′ ϕ r′ d r′ є ϕ r ( .â) ò eš nk ∫ x nk = nk nk  ( ) ( ) ( ) ( ) ( ) with the non-local exchange potential

occ. BZ σ σ NL,σ ϕnk r ϕnk∗ r′ V r, r′ − ( .Þ) x = Q Q r − r n k ( ) ′ ( ) ( ) and the local ešective potential S S

V σ r V r + V r + V gKS,σ r . ( .—) eš = H xc ( ) ( ) ( ) ( )

¥Þ . Hybrid functionals within the generalized Kohn-Sham scheme

gKS,σ e xc potential of the gKS system Vxc r is dened in analogy to the KS system by

( ) δEgKS V gKS,σ r xc . ( .À) xc = δnσ r ( ) We note that Eq. ( .Ô) together with ( .ò) denes a whole( ) series of reference systems. Each choice of a corresponds to a dišerent reference system with a dišerent xc energy functional EgKS. In thecase a ý, the gKS system is identical to the KS system. e opposite limit a Ô xc = = corresponds to a HF scheme with a DFT correlation functional. In principle, however, the gKS formalism is exact for each a. e gKS system is usually employed for the hybrid functionals (cf. Sec. ç.¥). For exam- ple, for the PBEý hybrid functional [òò] the amount of orbital-dependent non-local HF-type exchange is set to a ý.ò and the xc energy functional EgKS is approximated by EgKS = xc xc = PBE PBE ý.Þ Ex + Ec .

5.2. Implementation of hybrid functionals

In the LAPW basis [Eq. (¥.À)] the dišerential equation ( .â) becomes a generalized eigenvalue problem

σ + NL,σ σ σ σ σ HGG′ k aVx,GG′ k zG′ n, k єnk SGG′ k zG′ n, k , ( .Ôý) QG′ = QG′
( ) ( ) ( ) ( ) ( ) σ − Ô ∇ò + σ NL,σ where HGG′ k is the representation of the local part ò Veš r of ( .â), Vx,GG′ k de- σ notes the matrix of the non-local exchange potential Eq. ( .Þ), and SGG′ k is the overlap ma- ( ) σ σ ( ) ( ) trix of the LAPW basis functions. H ′ k and S ′ k are nowadays calculated routinely GG GG ( ) in FLAPW DFT codes. erefore, we will concentrate in the following on the computation of ( ) ( ) the non-local exchange matrix elements, whose evaluation is by far the most time-consuming step in DFT calculations employing hybrid xc functionals. e reason is the non-locality of the operator in Eq. ( .Þ), which gives rise to six-dimensional integrals

σ σ σ σ occ. BZ φ ∗ r ϕ ′′ r ϕ ′′∗ r′ φ ′ r′ NL,σ − kG n q n q kG ç ç Vx,GG′ k d r d r′ . ( .ÔÔ) = Q′′ Q ∫∫ r − r n q ( ) ( ) ′ ( ) ( ) ( ) In contrast, for the local operators in standard DFTS calculationS s only three-dimensional in- tegrals must be evaluated. A representation of the exchange operator in terms of the wave functions instead of the

¥— .ò. Implementation of hybrid functionals

LAPW basis

σ σ σ σ occ. BZ ϕ ∗ r ϕ ′′ r ϕ ′′∗ r′ ϕ ′ r′ NL,σ − nk n q n q n k ç ç Vx,nn′ k d r d r′ ( .Ôò) = Q′′ Q ∫∫ r − r n q ( ) ( ) ′ ( ) ( ) ( ) S S is advantageous for two reasons: (a) If the states n and n′ fall into dišerent irreducible sym- metry representations, the corresponding matrix element is zero and need not be calculated at all. (b) Although very important, the exchange energy is a relatively small energy contri- bution compared to kinetic and potential energies. erefore, we can ašord to describe the non-local exchange potential in a subspace of wave functions up to a band cutoš nmax. e matrix ( .Ôò) is then only constructed for the elements with n, n′ n and the rest is set to ≤ max zero. nmax is a convergence parameter and we will show in section .¥ that the results con- NL,σ verge reasonably fast with respect to this parameter. e matrix Vx,GG′ k is obtained from NL,σ V ′ k by multiplying with the inverse matrix of eigenvectors and its adjoint from the right x,nn ( ) and leŸ, respectively ( )

NL,σ σ σ NL,σ σ σ Vx,GG′ k SGG∗ ′′ k zG′′ n, k Vx,nn′ k zG∗′′ n′, k SG′′G′ k . ( .Ôç) = nQ,n′ QG′′ QG′′ ( )  ( ) ( ) ( )  ( ) ( ) e inverse matrix of eigenvectors results from the normalization condition

σ σ σ zG∗ n, k SGG′ k zG′ n, k Ô . ( .Ô¥) GQ,G′ = ( ) ( ) ( )

e sum over the occupied states in Eq. ( .Ôò) involves core and valence states. e core states in the FLAPW method are the solutions of the fully relativistic Dirac equation with the spherically averaged ešective potential. ey are, thus, classied by the principal quantum number n, the angular momentum quantum number l, the total angular momentum quan- tum number j l ± Ô ò, and the magnetic quantum number m m −j,..., j . Duetothe = j j = sphericity of the potential the core states are degenerate with respect to the magnetic quan- ~ ( ) tum number. For each n and l we average over the two dišerent values of j weighted by its degeneracy of òj + Ô. With this, we go over to a non-relativistic description of the core states and can apply the formula by Dagens and Perot [—ç] to compute the core contribution to the matrix elements of Eq. ( .Ôò). Its evaluation is simple and computationally cheap, in contrast to the valence electron contribution.

For the latter, we employ the MPB (s. Sec. ¥.ò.ò). e products of valence-electron wave functions are represented by the MPB such that the six-dimensional integral, Eq. ( .Ôò), de-

¥À . Hybrid functionals within the generalized Kohn-Sham scheme composes into a vector-matrix-vector product

occ. BZ NL,σ − σ σ σ σ Vx,nn′ k ϕnk ϕn′′k qMq,I vIJ q Mq,J ϕn′′k q ϕn′k . ( .Ô ) = Qn′′ Qq QIJ − − ( ) ⟨ S ⟩ ( )⟨ S ⟩ σ σ ç e vectors Mq,J ϕn′′k q ϕn′k are denedby f g ∫ f ∗ r g r d r and represent the wave − = σ σ − function product ϕn′′∗k q r ϕn′k r in the MPB. In order to ensure that k q is again a mem- ⟨ − S ⟩ ⟨ S ⟩ ( ) ( ) ber of the k-point set, we choose an equidistant k-point mesh. Additionally, we require that ( ) ( ) the k-point set contains the Γ point. is is necessary for an accurate treatment of the diver- gence of the Coulomb matrix

M˜ r M˜ r q∗,I q,J ′ ç ç v q d r d r′ ( .Ôâ) IJ = ∫∫ r − r ( ) ′ ( ) ( ) around q ý (s. Sec. .ò.ç). e Coulomb matrixS doesS not dependent on the wave functions. = As the MPB does not change during the SCF, the Coulomb matrix is constructed once at the beginning of the SCF cycle, while the vectors are calculated in each iteration. Due to the partition of the MPB in MT functions and interstitial plane waves (IPWs) the Coulomb matrix consists of four distinct blocks: the diagonal parts MT-MT and IPW-IPW as well as the two oš-diagonal part MT-IPW and IPW-MT, which are the complex conjungates of each other. e evaluation of the dišerent blocks is discussed in detail in Ref. —¥.

5.2.1. Sparsity of the Coulomb matrix

e vector-matrix-vector products of Eq. ( .Ô ) must be evaluated in every iteration of the self-consistent eld cycle for each combination of band indices n, n′, and n′′ as well as Bloch vectors k and q. is easily amounts to billion matrix-vector operations or more. We will show that the matrix-vector product becomes considerably faster, if the Coulomb matrix is made sparse. is is achieved by a unitary transformation of the MT functions that acts in the subspace of each atom and LM channel.

a a Let us consider the two radial MT functions MLÔ r and MLò r , whose electrostatic mul- tipole moments are given by ( ) ( ) S a a a L ò µLP MLP r r + dr ( .ÔÞ) = ∫ý ( )

ý .ò. Implementation of hybrid functionals for P Ô, ò. If we apply the unitary transformation =

′ Ô Ma r µa Ma r + µa Ma r ( .ԗ) LÔ = a ò + a ò LÔ LÔ òL Lò µLÔ µLò a′ ( ) » Ô [ a a ( ) a a ( )] M ò r ( ) ( ) µ òM Ô r − µ Ô M ò r ( .ÔÀ) L = a ò + a ò L L L L µLÔ µLò ( ) » [ ( ) ( )] ( ) ( ) a′ the multipole moment of the second function MLò r will vanish. With this procedure we can generally transform a set of MT functions so that the resulting functions are multipole- ( ) free for all except one function. For example, out of ten functions we obtain nine multipole- free functions and one with a non-vanishing multipole moment. We denote the sets of these transformed functions by MT µ ý and MT µ ý , respectively. By construction, the for- = ≠ mer does not generate a potential outside the MT spheres. So, the Coulomb matrix elements ( ) ( ) involving such a function can only be non-zero, if the other function is a MT function resid- ing in the same MT sphere. Matrix elements with a MT function in a dišerent MT sphere or an interstitial plane wave vanish. is leads to a very sparse, nearly block-diagonal form of the Coulomb matrix as illustrated in Fig. .Ô. We have ordered the MPB according to: MT µ ý , MT µ ý , and IPWs. ere are on-site blocks (one for each LM channel) for = ≠ the MT µ ý part and one big block for the combined set of the MT µ ý and the IPWs. ( )= ( ) ≠ Only a few oš-diagonal elements exists between MT µ ý and MT µ ý functions at ( ) = ( ≠ ) the same atom. Exploiting this sparsity in the matrix-vector products of Eq. ( .Ô ) drastically ( ) ( ) reduces the number of žoating point operations and, thus, the computational demand. In order to demonstrate the reduction in the computational demand and, thus, the speed- up we report timings for Ôýý ýýý matrix-vector products with and without the sparse matrix technique. e conventional matrix-vector product scales quadratically with the size N of the MPB N N + N , whereas the timing of the sparse-matrix vector product is usually = MT IPW dominated by the large block of IPWs and MT µ ý functions. e size of this block is given ( ) ò ≠ by NIPW MT µ ý NIPW + Natom ⋅ Lmax + Ô . Hence, we can ideally obtain a speed-up of + ( ≠ ) =ò ( ) N NIPW MT µ ý . For the simple case of diamond with ò atoms in the unit cell and a MPB + ( ≠ ) ( ) Ô that consists of À interstitial plane waves (G′ ç.òa− ) and òÔý MT function (Lmax ¥), ( ~ ) max = ý = the sparse matrix approach is a factor ò.— faster than the conventional matrix-vector product.ò is factor is much smaller than the ideal speed-up of â.Ô, which, however, does not consider the time for the multiplication of the block-diagonal matrix MT µ ý with the vector. = Due to the relatively small number of interstitial plane waves in this example the timing for [ ( )] the block-diagonal part MT µ ý is not negligible. For the perovskite SrTiO with ç—À = ç Ô interstitial plane waves (G′ ç.—a− ) and â—ç MT functions (L ¥) the conventional [ max(= )]ý max = òe calculations have been performed on a single core of a Intel Core I processor with ò.âÞ GHz.

Ô . Hybrid functionals within the generalized Kohn-Sham scheme

ΜΤ ΜΤ (µ=0) (µ=0) IPWs

ΜΤ (µ=0)

ΜΤ (µ=0)

IPWs

Figure .Ô.: Illustration of the Coulomb matrix aŸer transforming the MPB according to Eqs. ( .ԗ) and ( .ÔÀ). e elements that are in general non-zero are marked. e matrix is predominantly block-diagonal. approach requires about ¥ý.Þ s to execute the matrix product Ôýýýýý times, while the sparse matrix approach needs only ÔÔ.¥ s. Consequently, the sparse-matrix approach is a factor of ç.â faster, which is quite close to the ideal factor of ¥.¥. We note that the number of matrix vector products easily amounts to Ôýý ýýý: a combination of merely Ôý k points, Ôý q points, òý occupied bands, and ý unoccupied bands already yields in total Ôýý ýýý products.

5.2.2. Symmetry

NL,σ In addition to the sparse-matrix technique, we accelerate the calculation of Vx,nn′ k by ex- ploiting spatial and time reversal symmetries. ese symmetries can be used to facilitate the ( ) computation in three ways: (a) Inversion symmetry leads to real-valued quantities. (b) If the σ σ wave functions ϕnk and ϕn′k in Eq. ( .Ô ) fall into dišerent irreducible representations, the corresponding exchange matrix element vanishes. is can be used as a criterion whetheran element must be calculated explicitly or not. And (c), for each k chosen from the irreducible wedge of the BZ the q summation in Eq. ( .Ô ) is restricted to a smaller set of Bloch vectors giving rise to an extended irreducible BZ. We will discuss each item in the following in detail.

ò .ò. Implementation of hybrid functionals

(a) Inversion symmetry

In general, the Coulomb matrix ( .Ôâ) is Hermitian. However, if the system under consider- ation exhibits inversion symmetry and the MPB functions fulll the condition

f −r f ∗ r , ( .òý) = ( ) ( ) it becomes real-symmetric. Similarly, the vectors in ( .Ô ) are then real instead of complex. is reduces the computational demand in terms of both CPU time and memory consider- ably. However, presently the condition only holds for the IPWs, but not for the MT functions. We, therefore, combine the MT functions of each pair of atoms a and −a, which are related via inversion symmetry,

a Ô a L M a Mk,′LMP r Mk,LMP r + −Ô + Mk−,L M P r ( .òÔa) = ò (− ) a ( ) √i  a ( ) ( )L M a ( ) Mk,′L M P r Mk,LMP r − −Ô + Mk−,L M P r . ( .òÔb) (− ) = ò (− ) ( ) √  ( ) ( ) ( ) If the atom is placed in the origin, i.e., the atom indices a and −a correspond to the same atom, the transformations ( .òÔa) and ( .òÔb) only hold for the integer index M ý, and we < dene

a a MkLýP r , if L even Mk,′LýP r ( .òò) = ⎧ i ⋅ Ma r , if L odd ⎪ kLýP( ) ( ) ⎨ ⎪ for M ý. It is then easy to show that⎩⎪ the transformed( ) functions will fulll the condition = ( .òý). We note that this symmetrization does not destroy the form of the Coulomb matrix shown in Fig. .Ô.

(b) Irreducible representations

e great orthogonality theorem of group theory [— ] demands that the matrix elements σ σ ϕnk A ϕn′k of any operator A, which commutes with the symmetry operations of the system, are zero, if the wave functions fall into dišerent irreducible representations. In particular, this ⟨ S S ⟩ holds for the exchange operator ( .Þ). Instead of evaluating the irreducible representations explicitly, we exploit the fact that the great orthogonality theorem applies to any operator that has the full symmetry of the system. A suitable operator is given by the MT step function

ΘMT r Ô − Θ r , ( .òç) = ( ) ( )

ç . Hybrid functionals within the generalized Kohn-Sham scheme where Θ r is dened according to Eq. (¥.òÞ). e calculation of its matrix elements σ MT σ ϕ Θ ϕ ′ is elementary and takes negligible CPU time. If the matrix elements between nk ( )n k two groups of degenerate wave functions are numerically zero, we conclude that these ⟨ S S ⟩ two groups belong to dišerent irreducible representations. en the corresponding matrix elements of the non-local exchange potential ( .Ô ) must be zero, too. e question remains whether, conversely, the matrix elements of ( .òç) are always non-zero, if the two irreducible representations are identical. is is not fullled in only two cases. First, either of the two wave functions is completely conned to the MT sphere. is can be ruled out, since we deal with valence or conduction states. Second, the matrix elements are zero by accident: the overlaps in the interstitial and the MT spheres exactly cancel. is is extremely unlikely, verging on the impossible. We nd that the procedure provides a fast and reliable criterion to decide in advance, which exchange matrix elements are non-zero and must be calculated explicitly.

(c) Extended irreducible BZ

In general, if a symmetry operation, which leaves the Hamiltonian invariant, acts on a wave function, it generates another wave function with the same energy. In other words, the so- lutions of the one-particle equations at two dišerent k points are equivalent, if the k vectors are related by a symmetry operation. is can be used to restrict theset of k points, at which the Hamiltonian must be diagonalized, to a smaller set, whose members are not pair-wise re- lated. is denes the so-called irreducible Brillouin zone (IBZ), which is routinely employed in calculations with periodic boundary conditions. In a similar way, the summation over q points in the non-local exchange term can be conned, too. However, due to the additional k dependence on k and k−q, we can only employ those symmetry operations Pi that leave the k k k given k vector invariant, i.e., Pi k k + Gi , where Gi is a reciprocal lattice vector. is subset k = of operations Pi is commonly called little group LG k . In the same way as for the IBZ the little group gives rise to a minimal set of inequivalent q points, which we denote by the { } ( ) extended IBZ [EIBZ k ]. e exchange potential in the LAPW basis can then be written as

NL,σ ( ) Vx,GG′ k ( .ò¥) Ô Ô LG k EIBZ k σ k σ k σ σ occ. φ r P − ϕ r P − ϕ r φ ′ r (( ) ( ) Ô kG∗ i nk q i nk∗ q ′ kG ′ ç ç − − − d r d r′ = Q Q N Q ∫∫ r − r i q k,q n ( )[ ( )][ ′ ( )] ( ) σ σ σ σ LG k EIBZ k occ. φ ∗ r ϕ r ϕ ∗ r′ φ r′ ( ) ( ) Ô k PkG Gk nk q S nkS q k PkG′ Gk − ( i + i ) − − ( i + i ) dçr dçr , ∫∫ − ′ = Qi Qq Nk,qQn ( ) ( r) r′ ( ) ( ) S S

¥ .ò. Implementation of hybrid functionals

where Nk,q is the number of symmetry operations that are members of both LG k and LG q . As a result, we can restrict the q summation to the EIBZ k and add the contri- ( ) bution of all other q points by transforming the nal matrix with the operations LG k and ( ) ( ) summing up the transformed matrices. is takes very little computation time, because a ( ) symmetry operation acts as a one-to-one mapping in the space of the augmented plane waves as indicated in Eq. ( .ò¥). Local orbitals transform in a similar way. is is why we apply the NL,σ NL,σ symmetrization to Vx,GG′ k instead of Vx,nn′ k . In the latter case we would need the irre- ducible representations again. We note that the whole formalism can be easily extended to ( ) ( ) the case of non-symmorphic and time-reversal symmetry operations. NL,σ In conclusion, we compute the non-local exchange potential Vx,nn′ k in the space of the wave functions, where we restrict the q summation to the EIBZ k and evaluate only those ( ) band combinations n and n′, which can be expected to be non-zero. We then apply the trans- ( ) formation ( .Ôç) and sum up the dišerent matrix elements according to ( .ò¥).

5.2.3. Singularity of the Coulomb matrix

Due to the long-range nature of the Coulomb interaction the matrix vIJ q , Eq. ( .Ôâ), is singular at q ý, which leads to a divergent integrand in ( .Ô ). As the divergence is propor- = ( ) tional to Ô qò, a three-dimensional integration over the BZ yields a nite value. However, in a practical calculation the q summation in ( .Ô ) is not an integral, but a weighted sum over ~ the discrete BZ mesh. A simple way to avoid the divergence is to exclude the point q ý = from the k-point set. en all terms in ( .Ô ) are nite, and the q sum can be evaluated easily. is, however, leads to very poor convergence with respect to the BZ sampling, because the quantitatively important region around q ý is not properly taken into account. Hence, it is = advantageous to explicitly treat the Γ point and the singularity of the Coulomb matrix at Γ.

is is possible by a decomposition of vIJ q into a divergent and a non-divergent part [—¥]

¥π Ô ( )iq⋅r iq⋅r ′ v q M˜ q e e M˜ q + v q , ( .ò ) IJ = V qò ,I ,J IJ ( ) ⟨ S ⟩⟨ S ⟩ ( ) where the second term v′ q is nite for all q and is identical to v q for q ý. e IJ IJ ≠ divergent rst term is exact in the limit q → ý, as the MPB contains the constant basis function ( ) ( ) explicitly (s. Sec. ¥.ò.ò). Correspondingly, the combination of Eq. ( .ò ) and ( .Ô ) yields two NL,σ contributions, a divergent and a non-divergent one, to the matrix element Vx,nn′ k

NL,σ NL,σ NL,σ ( ) V ′ k V ′ k + V ′ k , ( .òâ) x,nn = x,nn non−div x,nn div ( ) ( )T ( )T

. Hybrid functionals within the generalized Kohn-Sham scheme where the non-divergent part is given by

occ. BZ NL,σ σ σ ′ σ σ V ′ k − ϕ ϕ ′′ M v q M ϕ ′′ ϕ ′ . ( .òÞ) x,nn non−div nk n k−q q,I IJ q,J n k−q n k = Qn′′ Qq QIJ ( )T ⟨ S ⟩ ( )⟨ S ⟩ NL,σ Its q summation can be performed numerically. For the divergent part V ′ k we switch x,nn div ⋅ from a representation with the MPB to a representation with plane waves eiq r. For the im- NL,σ ( )T portant region close to q ý this is exact. us, V ′ k is given by = x,nn div

occ. ( )T NL,σ Ô σ σ iq⋅r Ô iq⋅r σ σ ç V ′ k − ϕ ϕ ′′ e e ϕ ′′ ϕ ′ d q − d.c. , ( .ò—) x,nn div ò nk n k−q ò n k−q n k = òπ Qn′′ ∫BZ q ( )T Œ ⟨ S ⟩ ⟨ S ⟩ ‘ where d.c. denotes a double-counting correction for the nite q points (see below). In the limit q → ý we can replace ⋅ ⋅ in ( .ò—) by δnn′′ and δn′n′′ , respectively. For an expansion of ⋅ ⋅ beyond the leading term in q we refer to Refs. ¥ò and —â. e integration in Eq. ( .ò—) ⟨ S ⟩ is restricted to the rst part of the BZ. An extension over the whole reciprocal space requires ⟨ S ⟩ to take the periodicity of the integrand into account explicitly. erefore, we replace Ô qò by the function ~ − + ò e β q G q S S F ò , ( .òÀ) = QG q + G ( ) which was proposed by Massidda et al. in Ref. —Þ.S In contrastS to Ref. —Þ we choose the param- eter β as small as possible such that ( .òÀ) is su›ciently close to Ô qò. AŸer inserting ( .òÀ) in ( .ò—) we obtain ~ −β q ò −β q ò NL,σ σ Ô e S S ç Ô e S S V ′ k −δ ′ f d q − , ( .çý) x,nn div nn nk ò ò ò = òπ ∫ q NkΩ qQý q ⎛ ≠ ⎞ ( )T where the summation over q ý avoids⎝ double counting, N denotes the number⎠ of k points ≠ k σ and fnk is the occupation number. Eq. ( .çý) shows that the divergent contribution is only non-zero for diagonal, occupied matrix elements. In order to evaluate the integral and the sum in Eq. ( .çý) we introduce a reciprocal cutoš radius qý and nally obtain

−β q ò NL,σ σ Ô Ô e S S V ′ k −δ ′ f erf βq − . ( .çÔ) x,nn div nn nk ý ò = πβ NkΩ ý Qq q q ⎛ » < ≤ ý ⎞ ( )T » Š  ⎝ − ò ⎠ We get rid oš the convergence parameter q by relating β and q by e βqý β. is relation ý ý = ensures that a small parameter β, which corresponds to a slow exponential decay, entails a

â .ç. Self-consistent eld cycle large q . We nd that β ý.ýý is a good choice in practice. ý = In Fig. .ò the convergence of the exchange energy ENL ò occ. V NL k with respect to x = ∑nk x,nn the k-point sampling for NaCl is demonstrated. While the separate contributions from the ( ) divergent term, Eq. ( .çÔ), and the remainder converge poorly,their sum nearly looks constant on the energy scale of Fig. .ò(a). As shown in Fig. .ò(b), the k-point convergence can be improved further by taking corrections at q ý into account that arise from multiplying Ô qò = with second-order terms of ⋅ ⋅ ⋅ ⋅ derived by k ⋅ p perturbation theory [¥ò, —â]. ~ ⟨ S ⟩⟨ S ⟩ 5.3. Self-consistent field cycle

e one-particle equation ( .â) must be solved self-consistently, as the ešective potential σ σ NL,σ Veš r , which enters in HGG′ k , and the non-local HF potential Vx,GG′ k depend on the density and density matrix, respectively. In standard DFT calculations employing a local or ( ) ( ) ( ) semi-local xc functional, the one-particle KS equation solely depends on the density. In or- der to achieve self-consistency in this case an elaborate mixing scheme, for example Broyden mixing [——, —À], is applied, where a new density out of the densities of the previous iterations is constructed as the input density for the next iteration. Hybrid functionals, in principle, re- quire a mixing of the density matrix, as well. Indeed, we nd that a standard Broyden mixing of the density and simply using the output density matrix as the input for the next iteration leads to a poor convergence. òÞ iterations for Si and more than òýý iterations for SrTiOç are required to obtain converged PBEý results (s. Fig. .ç), where we consider a calculation as converged if the root-mean square of the density dišerence ∆n between input and output − ç densities fall below Ôý me aý (e denotes the elementary charge and aý is the Bohr radius). e denition of a mixing scheme for the density matrix in the FLAPW method, however, ~ is di›cult, maybe impossible, as the LAPW basis for the wave functions changes in each it- eration. We propose here an alternative approach, which leads to a fast density convergence. It consists of an outer self-consistency cycle for the density-matrix and an inner one for the density (cf. Fig. .¥). AŸer the construction of the non-local exchange potential we keep its NL,σ matrix representation Vx,GG′ xed and iterate Eq. ( .â), until self-consistency in the density is NL,σ reached; only then the exchange potential Vx,GG′ is updated from the current wave functions, which starts a new set of inner self-consistency iterations. With this nested iterative proce- dure the outer loop converges aŸer eight steps for a PBEý calculation of Si (s. Fig. .ç) and aŸer only twelve steps for SrTiOç. One iteration of the inner loop lasts only Ô.ý s for Si and

—.ç s for SrTiOç on a single Intel Xeon X ç at ò.ââ GHz using a ¥×¥×¥ k-point set. is is negligible compared with the cost for the construction of the non-local potential in the outer loop, which amounts to ÔÔ.À s for Si and Þç.Ô s for SrTiOç. In summary, the nested iteration

Þ . Hybrid functionals within the generalized Kohn-Sham scheme

(a) 0 NaCl -10 -20 -30 10 2 3 4 5 6 7 8 × × × × × × × ×

-40 2 3 4 5 6 7 8 10 × × × × × × × × (eV) NL 2 3 4 5 6 7 8 x -50 10 E -60 -70 -80 -90 1 1.5 2 2.5 3 Number of k points (log. scale) (b) NaCl -87.6

-87.8 10 2 3 4 5 6 7 8

-88 × × × × × × × × 2 3 4 5 6 7 8 (eV) 10 NL x × × × × × × × × 2 3 4 5 6 7 8 E

-88.2 10

-88.4

-88.6 1 1.5 2 2.5 3 Number of k points (log. scale)

Figure .ò.: (a) Exchange energy as a function of the k-point mesh for NaCl. e green dashed and blue dotted curves correspond to the divergent contribution Eq. ( .çÔ) and the remaining numerical sum, respectively. e sum of both is shown by the red solid curve. (b) Convergence of the exchange energy with (blue dashed curve) and without (red solid curve) higher-order corrections at q ý. = Please note the dišerent scale of the exchange energy in gures (a) and (b).

— .¥. Convergence tests

101 Si 100

10-1 ) 3 0 10-2

-3 n (me/a 10 ∆

10-4

10-5

10-6 5 10 15 20 25 30 Number of iterations

Figure .ç.: Convergence behavior of the electron density for Si: the change in the density ∆n is shown as a function of the self-consistent-eld cycles. e red solid and blue dashed curves correspond to calculations with and without the nested density convergence scheme (see text). scheme constitutes a practical SCF scheme which leads to a considerably faster convergence.

5.4. Convergence tests

In order to assess the e›ciency of our numerical scheme, we analyze the convergence of the gKS transition energies and of the total energy with respect to the numerical cutoš parameters ′ Lmax and Gmax of the MPB as well as the number of bands nmax, which are used to represent the non-local exchange potential. In particular, we show the convergence of the Γò ′v → ΓÔ c and Γò ′v → XÔc transitions for Si and the ΓÔ c → Γò ′c and RÔ ′v → Γò ′c transition energies for SrTiOç as functions of the convergence parameters for the PBEý functional. Figure . (a) and (b) demonstrate that the convergence of these transition energies, obtained from a self- ′ −Ô ′ consistent solution of Eq. ( .â), is achieved to within ý.ýÔ eV for Gmax ò.ýaý and Gmax −Ô = = ò.Þaý for Si and SrTiOç, respectively. It is remarkable that these cutoš values are far below ′ −Ô the exact limit of Gmax òGmax for the wave function products (Gmax ç.âaý for Si and − = = G ¥.ça Ô for SrTiO ) and even below the reciprocal cutoš radius G for the wave max = ý ç max functions themselves. A similar observation can be made for the cutoš parameter for the angular momentum L . For both materials L ¥ [s. Fig. ( . )] is su›cient, while an max max =

À . Hybrid functionals within the generalized Kohn-Sham scheme

Outer SCF loop(i)

use converged DFT i ni(r),φi calculation as input if = 1 nk

i Construct Vx,NL Inner SCF loop(j)

Construct local potential Veff

j j Solve gKS equation [{ǫnk, φnk}] Mix density j r occ. j r 2 n ( )= Pn,k |φnk( )|

knj(r) − nj−1(r)k < δ No

Yes

kni(r) − ni−1(r)k <ǫ No

Yes

Self-consistency

Figure .¥.: Schematic illustration of the nested SCF cycle for hybrid functional calculations.

âý . . Results for prototype semiconductors and insulators exact representation of the wave function products would require L òl ( l — for max = max max = Si and l ÔÔ for SrTiO ). e number of bands n that denes the Hilbert space, in max = ç max which the exchange potential is represented, can be restricted to only ý bands per atom. ∼ is amounts to Ôýý and ò ý bands for Si and SrTiOç, respectively. Furthermore, we examine the convergence of the total energy dišerence between the di- amond and wurtzite crystal structure of Si. e total energy dišerence is converged to an ′ − accuracy of Ô meV with a reciprocal cutoš radius G ò.ò a Ô, an angular-momentum max = ý cutoš L ¥, and n òý. We note that the accuracy of Ô meV is one order of magnitude max = max = smaller than the tolerance for the transition energies and well below the error resulting from the BZ discretization of the ¥×¥×¥ k-point sampling. With an —×—×— k-point mesh the cal- culations are converged to within ò meV. e diamond structure is ÔÔò meV lower in energy than the wurtzite structure. For the PBE functional the energy dišerence amounts to Àò meV. ′ In conclusion, as demonstrated for the example of Si and SrTiOç we found that Gmax can ′ be chosen universally smaller than G . Typically G ý.Þ G is a reasonable choice. max max = max e angular-momentum cutoš Lmax and the number of bands nmax is more material specic. orough convergence tests are necessary for these parameters.

5.5. Results for prototype semiconductors and insulators

For the semiconductors and insulators Si, C, GaAs, MgO, NaCl, and crystalline Ar we report in Table .Ô gKS transition energies obtained with the PBEý hybrid functional.ç All calcu- lations are performed at the experimental lattice constant. e transition energies are con- ′ verged to within ý.ýÔ eV with respect to the MPB parameters Gmax, Lmax and the band cutoš nmax.AÔò×Ôò×Ôò k-point mesh ensures converged results with respect to the sampling of the BZ. While the PBE functional (s. Table .Ô) underestimates the Γ → Γ, Γ → L, and Γ → X transitions for this set of materials substantially, the admixture of ò % HF exchange in the PBEý functional leads to a consistent increase of the energies. Consequently, the PBEý values come closer to the experimental results. A slight overestimation of the energies is observed for the semiconductors Si and GaAs. In contrast, the transition energies for the insulators are still underestimated by PBEý. e mean absolute error of the PBE transition energies with respect to the available experimental values of ÔÀÞ% is reduced to —â% by the PBEý functional. We, furthermore, compare our results with recent projector augmented-wave (PAW) cal- culations [çý]. For Si and GaAs the FLAPW PBE and PBEý transition energies deviate by

çFor crystalline Ar we augmented the LAPW basis set in the MT spheres by ¥ local orbitals. Two of them are added to the angular momentum l = ý and the other two to l = Ô. e energy parameters of these local orbitals have been determined according to Eq. (¥.ԗ).

âÔ . Hybrid functionals within the generalized Kohn-Sham scheme

(a) 4.1 Si 2.08 4.095 2.07 4.09 (eV) (eV)

2.06 1c 15c

4.085 X →Γ → 2.05

25'v 4.08 25'v Γ Γ 4.075 2.04

4.07 2.03 1.5 2.5 3.5 4.5 3 4 5 20 40 60 80 -1 L Bands per atom G©max (a0 ) max

(b) 4.5 4.10 SrTiO3 4.05

4.45 (eV) (eV) 25'c 25c' 4.00 →Γ →Γ

15'v

15v 4.4 Γ 3.95 R

4.35 3.90 1.5 2.5 3.5 4.5 3 4 5 30 50 70 90 -1 L Bands per atom G©max (a0 ) max

(c) 78 Si

77 (meV) dia

- E 76 wz

E = 75 ∆

74 2.5 3 3.5 4 4 5 6 20 40 60 80 100 120 -1 L Bands per atom G©max (a0 ) max

Figure . .: Convergence of (a) the Γò ′v → ΓÔ c (red solid line, leŸ scale) and Γò ′v → XÔc (blue dashed line, right scale) transitions for Si, (b) the direct (red solid line, leŸ scale) and indirect band gaps (blue dashed line, right scale) of SrTiOç, and (c) the total energy dišerence ∆E between the wurtzite (wz) and diamond (dia) ′ phases of Si with respect to the reciprocal cutoš values Gmax and the angular momentum cutoš Lmax for the MPB, as well as the number of bands per atom used to construct the exchange potential in the space of the wave functions. For these convergence tests we have employed a ¥×¥×¥ k-point set.

âò .â. Calculation of band structures and density of states maximally ý.ýò eV from the corresponding PAW values. Larger discrepancies are observed for the residual systems with larger band gaps. However, these dišerences are already present on the PBE level. So, we assume that they result from the dišerent basis sets used in the FLAPW and PAW approach. Overall, we nd a good agreement between the PBEý results in both methods.

5.6. Calculation of band structures and density of states

In a DFT calculation with a purely local ešective potential the strategy for constructing a band structure is a follows: in a rst step the calculation is converged with a moderate k- point mesh. en, the converged density and thus the converged ešective potential is used to setup and diagonalize the Hamiltonian along a k-point path through the BZ, which nally results in the band structure. In contrast, for hybrid functionals the diagonalization of the Hamiltonian at an arbitrary k-point in the BZ would require the knowledge of all occupied states at the points k − q. ese wave functions are, however, unknown in general. erefore, we employ the Wannier interpolation technique as realized in the WannierÀý code [À ] to interpolate the band energies between the k-points of the nite mesh. We give a short introduction to the construction of Wannier functions and then discuss the Wannier interpolation technique. As examples, we nally show the PBE and PBEý inter- polated band structure of Si and the Wannier interpolated PBEý density of states (DOS) for ZnO. σ e Wannier function wnR r is dened as the Fourier transform of the KS wave function ϕσ nk ( ) Ω wσ r exp −ikR ϕσ r dçk , ( .çò) nR = ¾—πç ∫ nk ( ) ( ) ( ) σ where R denotes a lattice vector and Ω is the unit cell volume. While the wave functions ϕnk σ are Bloch functions and represented in reciprocal space, the Wannier functions wnR r are characterized by the real space lattice vector R. Similar to the KS wave functions the Wannier ( ) functions form an orthonormal, complete basis

σ σ w w ′ ′ δ ′ δ ′ ( .çç) nR n R = n,n R,R σ σ∗ ′ − ′ wnR⟨ r wSnR r ⟩ δ r r . ( .ç¥) QnR = ( ) ( ) ( ) However, the Wannier functions as dened in Eq. ( .çò) are not unique, because the wave σ σ functions ϕnk r are only dened up to an arbitrary phase factor. If ϕnk r is an eigenfunction of the Hamiltonian, exp iθ ϕσ r is also a solution of the Hamiltonian corresponding to the ( ) nk ( ) [ ] ( )

âç . Hybrid functionals within the generalized Kohn-Sham scheme

is work PAWa PBE PBEý PBE PBEý Expt. Si Γ → Γ ò. â ç.Àâ ò. Þ ç.ÀÞ ç.¥b Γ → X ý.ÞÔ Ô.Àç ý.ÞÔ Ô.Àç — Γ → L Ô. ¥ ò.—Þ Ô. ¥ ò.—— ò.¥b

C Γ → Γ .⥠Þ.Þ¥ . À Þ.âÀ Þ.çb Γ → X ¥.ÞÀ â.âÀ ¥.Þâ â.ââ — Γ → L —. — Ôý.—— —.¥â Ôý.ÞÞ —

GaAs Γ → Γ ý. ò.ýò ý. â ò.ýÔ Ô.âçb Γ → X Ô.¥Þ ò.âÀ Ô.¥â ò.âÞ ò.ԗb,ò.ýÔb Γ → L Ô.ýò ò.ç— Ô.ýò ò.çÞ Ô.—¥b,Ô.— b

MgO Γ → Γ ¥.—¥ Þ.çÔ ¥.Þ Þ.ò¥ Þ.Þc Γ → X À.Ô ÔÔ.âç À.Ô ÔÔ.âÞ — Γ → L —.ýÔ Ôý. Ô Þ.ÀÔ Ôý.ç— —

NaCl Γ → Γ .ý— Þ.Ôç .òý Þ.ò⠗. d Γ → X Þ.çÀ À. À Þ.âý À.ââ — Γ → L Þ.òÀ À.çç Þ.çò À.¥Ô —

Ar Γ → Γ —.ÞÔ ÔÔ.Ô —.◠ÔÔ.ýÀ Ô¥.Ô e

aReference Àý bReference ÀÔ cReference Àò dReference Àç eReference À¥

Table .Ô.: PBE and PBEý transition energies in eV for Si, C, GaAs, MgO, NaCl, and Ar com- pared with theoretical and experimental values from the literature. All results are obtained with a Ôò×Ôò×Ôò k-point set.

⥠.â. Calculation of band structures and density of states same eigenvalue. Furthermore, for degenerate wave functions any unitary transformation within the degenerate subspace leads to another valid set of wave functions. While physical observables do not depend on the choice of the phase, the spatial localization of the Wannier functions substantially varies with the phase. e freedom in the phase of the wave function σ can be used to localize the Wannier functions wnR r around the lattice vector R. erefore, denition ( .çò) may be generalized to ( )

Ω σ r − kR σ k σ r ç wnR ç exp i Umn ϕmk d k , ( .ç ) = ¾—π ∫ Qm which explicitly considers( ) the freedom in( the phase)  by the( unita) ry( matrix) U σ k . e latter has the dimension of the eigenspace of ϕσ . Following an approach proposed by N. Marzari nk ( ) and D. Vanderbilt [—Ô], the unitary transformations U σ k are determined such that the spread σ ò σ σ ( σ) ò S wný r wný − wný r wný ( .çâ) = Qn of the Wannier functions is minimized.
⟨ SSe sum⟩ in⟨ Eq.S S ( .çâ)⟩  runs over all Wannier func- tions n. In this way, maximally localized Wannier functions (MLWFs) are obtained. e algo- rithm for minimizing the spread S by Marzari and Vanderbilt [—Ô] is only applicable to isolated groups of bands, but it has been extended to entangled energy bands by Souza et al. [—ò]. Due to their spatial localization, the MLWFs are an e›cient basis for an interpolation of the band structure. In a rst step, the Hamiltonian H is expressed in the MLWFs

Hσ R wσ H wσ . ( .çÞ) nm = ný mR

We note that one Wannier function can( be) chosen⟨ S toS reside⟩ at ý without loss of generality. In order to interpolate onto an arbitrary k-point, we perform the inverse Fourier transform

σ σ Hnm k exp ikR Hnm R . ( .ç—) = QR ( ) ( ) ( ) σ Diagonalization of Hnm k thenleadsto theband energiesat k. Due to theexponentialdecay of the MLWFs, which has been conrmed numerically for several materials [Àâ], the matrix σ ( ) elements Hnm R exhibit this exponential decay with R , as well. is permits to truncate the sum over R in Eq. ( .ç—) without loosing (much) accuracy. ( ) S S As an example, we show in Fig. .â the PBE and PBEý interpolated band structure for Si. We have generated — spç-like MLWFs, four for the valence and four for the lowest conduction bands. To demonstrate the accuracy of the Wannier interpolation, we compare in Fig. .â(a) the PBE band structure of Si obtained either by explicitly diagonalizing the PBE Hamilto-

â . Hybrid functionals within the generalized Kohn-Sham scheme

4 Si 8 Si 6 2 4 0 2 -2 0

-4 -2 (eV) (eV) F F -6 -4 E-E E-E -6 -8 -8 -10 -10 -12 -12

-14 -14 W L Γ X W K W L Γ X W K

(a) (b) Figure .â.: (a) Comparison of routinely generated PBE (orange) and Wannier interpolated (blue) band structure for Si. (b) Comparison of PBE (blue) and the Wannier- interpolated PBEý band structure (red) for Si. nian along the k-point path in the BZ or by employing the Wannier interpolation technique. Only minor dišerences between both approaches are observable. Hence, we expect a similar accuracy for the interpolated PBEý band structure. A comparison of PBE and PBEý band structure [s. Fig. .â(b)] reveals that the PBEý hybrid functional opens the gap from ý.¥ÞeV in PBE to Ô.Þ¥ eV in PBEý. In comparison, the experimental value amounts to Ô.ÔÞ eV [ÀÔ]. e conduction band minimum is for both functionals close to the X point. Moreover, the PBEý hybrid increases the band width of the occupied states from ÔÔ.À— eV for the PBE func- tional to Ôç.çÀ eV. e experimentally measured band width of Si is Ôò.¥ ± ý.â eV [ÀÞ]. Apart from the upward shiŸ, the dispersion of the lowest conduction bands in PBE and PBEý are very similar (nearly indistinguishable). e Wannier interpolation technique can be used to calculate the DOS, as well. e DOS at energy E measures the number of states per energy

Ô σ ç D E δ E − єnk d k . ( .çÀ) = Qσ Qn VBZ ∫BZ ( ) ( )

ââ .â. Calculation of band structures and density of states

15 ZnO Zn 3d 10 PBE0

5 O 2p 0

DOS (states/eV) 5

10 PBE

15 -7 -6 -5 -4 -3 -2 -1 0

E-EF (eV)

Figure .Þ.: Comparison of PBE (blue) and PBEý (red) density of states (DOS) for ZnO.

e δ-function is usually smeared out by a Gaussian distribution function, and the BZ integral becomes a sum over a nite number of k points. For a highly resolved DOS a large number of k-points is necessary, which in turn leads to a large computational cost in the case of the σ hybrid functionals. However, the MLWFs permit to interpolate the eigenvalues єnk accurately from a coarse to a ne k-point mesh. Hence, we can calculate a highly resolved DOS for hybrid functionals at a moderate computational expense. We use this approach to calculate the DOS of the II-VI semiconductor ZnO in its wurtzite structure. Using ve MLWFs for the Zn çd and three MLWFs for O òp states the eigenvalues σ × × × × єnk are interpolated from a ¥ ¥ ¥ to a dense Ôò Ôò Ôò k-point mesh. e eigenvalues on the dense mesh are then used to evaluate Eq. ( .çÀ). e corresponding DOS is shown in Fig. .Þ. In the case of the PBE functional the Zn çd states hybridize strongly with the O òp states. e PBEý functional, on the contrary, binds the Zn çd more strongly. e center of gravity of the d bands changes from .Ô eV in PBE to â.ç eV in PBEý, whereas the experimen- tal value is Þ.— eV [À—]. Due to the stronger binding of the Zn çd states, the d-p hybridization becomes less pronounced and the O p valence band width is increased from ¥.ò eV in PBE to .ò eV in PBEý, which is in excellent agreement with the experimental value [À—]. Con- comitantly, the band gap opens from ý.À¥ eV to ç.çò eV. e wrong d band position relative to the p states is commonly attributed to the unphysical self-interaction error inherent in

âÞ . Hybrid functionals within the generalized Kohn-Sham scheme the local-density approximation (LDA) and the generalized gradient approximation (GGA) (cf. Sec. ç.ò), which is larger for localized than delocalized electrons. e admixture of HF exchange into the hybrid functionals partly cancels this error and leads for ZnO to a lower- ing of the relative d band position. We note that we also observe a stronger binding of the d electrons in Ge and GaAs.

5.7. EuO: a case study

EuO is a ferromagnetic semiconductor with a band gap of ý.À eV in the limit of ý K [ÀÀ], which crystallizes in the rock-salt structure. It is of major interest in the eld of spintronics due to its ability to generate a highly spin-polarized current by combining non-magnetic elec- trodes with EuO in-between as a magnetic tunnel barrier. e spin-dependent tunnel barrier of EuO acts as a spin-lter, since the tunnel current depends exponentially on the respective barrier height [Ôýý, ÔýÔ]. EuO exhibits a remarkable spectrum of magnetic properties which are induced by carrier doping, including colossal magnetoresistance ∆ρ ρ Ôýâ [Ôýò], a ≈ metal-insulator transition ∆ρ ρ ÔýÔç [Ôýç, Ôý¥], spin polarized carriers Àý% [Ôý , Ôýâ], ≈ ( ~> ) and an enhancement of the Curie temperature T [ÔýÞ]. ( ~ ) c ( ) From a theoretical point of view, however, the description of EuO from rst-principles ( ) calculations constitutes a challenge. LDA and GGA functionals predict ferromagnetic EuO to be metallic. is is commonly attributed to the self-interaction error of LDA and GGA, which becomes severe for the strongly localized f electrons of Eu. e LDA+U approach represents a practical solution by adding on-site Coulomb repulsion through the parameter U. If the parameter U is chosen properly, the correct electronic ground state of EuO is ob- tained [Ôý—, ÔýÀ]. Actually, Ingle et al. [ÔýÀ] applied two dišerent U, one on theO òp and one on the Eu ¥ f state to describe EuO reasonably well. However, the LDA+U approach is unsatisfactory, because the parameter U is not universal. It strongly depends on the material and the material property under consideration [ÔÔý].¥ In the following, we will demonstrate that the PBEý hybrid functional, which does not contain an adjustable parameter, leads to a consistent description of EuO with respect to its structural, electronic, and magnetic proper- ties. We have determined the numerical parameters of the LAPW basis for EuO such that the dišerence in the total energy calculated for the experimental and a Ô% larger lattice constant − is converged up to Ô meV. is requires a reciprocal cutoš of G ¥.ça Ô, an angular max = ý momentum of l —, and the addition of one local orbital for each atom and each angular max = momentum l from l ý,...,ç. e energy parameters of these local orbitals have been = ¥We note that methods have been developed to calculate U from rst principles [ÔÔÔ, ÔÔò].

◠.Þ. EuO: a case study

atom lmax Lmax Sa Eu — â ò.âý O — â ò.Ôâ −Ô Gmax ¥.çaý ′ −Ô Gmax ç. aý

Table .ò.: Numerical parameters of the LAPW and mixed product basis for EuO. determined according to Eq. (¥.ԗ). While the Eu ¥ f states are treated explicitly by the LAPW basis, the semi-core s and p states of Eu are described by local orbitals. In this case, the energy parameters of the local orbital lie at the corresponding energy of the semi-core states. e MPB has been converged in a similar manner. e resulting parameters are summarized in Table .ò. In order to nd the optimal lattice constant of ferromagnetic rock-salt EuO, we calculate the total energy for À dišerent lattice constants in the vicinity of the experimental lattice constant. e parabolic shape of the total energy as function of the lattice parameter a is shown in Fig. .—. By a t to a Murnaghan equation of state [ÔÔç]

B′ BýV ′ Vý Vý ý E V ′ ′ Bý Ô − + − Ô + E Vý ( .¥ý) = Bý Bý − Ô V V ( )  ‹  ‹  ( ) we determine the equilibrium( unit-cell) volume Vý, which is related to the equilibrium lattice ç ò ò constant aý by aý ¥Vý, the equilibrium bulk modulus Bý Bý Vý∂ E ∂V V Vý and its ′ = = = derivative B . While√ the PBE functional predicts an equilibrium lattice constant of À. ÞÔ a , ý ( ~ S ) ý the PBEý hybrid gives a larger lattice constant of À.âޗ aý. In comparison with the experi- mental lattice constant at room temperature of À.ÞòÔ aý [ÔÔ¥], PBE and PBEý functional tend to over-bind. However, the thermal expansion, which leads to an increaseof theexperimental lattice constant at room temperature, is not taken into account in the calculations. A lattice constant of À.â—À a is observed at T ¥.ò K [ÔÔ¥]. In comparison with this value, the PBE still ý = underestimates the experimental value by Ô.ò%. e optimized PBEý lattice constant deviates from the experimental value at T ¥.òK by only Ô.Ôh. It is an order of magnitude more ac- = curate than the PBE result. e bulk modulus Bý, which corresponds to the curvature of the energy-versus-volume curve at the relaxed volume Vý, amounts to À¥.—À GPa for the PBE and Àâ.¥â GPa for the PBEý functional. Both values lie within the error bar of the experimentally determined value of ÀÔ ± — GPa [ÔÔ ]. We note that the experiment was performed at ÞÞK. In Fig. .À the spin-resolved DOS for ferromagnetic EuO is shown for the PBE and PBEý functionals. While EuO is metallic in the PBE, the PBEý functional opens a gap between the

âÀ . Hybrid functionals within the generalized Kohn-Sham scheme

100 EuO 90 PBE 80 PBE0 70 9.571 9.678 60 50 (meV)

E 40 30 20 10 0 9.5 9.55 9.6 9.65 9.7 9.75 9.8 9.85 9.9 9.95

Lattice constant a (a0)

Figure .—.: PBE and PBEý total energy E as a function of the lattice constant a for ferro- magnetic EuO. For both curves the energy zero has been chosen to lie at the minimum of the curve. spin-up Eu ¥ f states at the Fermi energy and the spin-up states of Eu d and âs character. e gap amounts to ý.— eV. Experimentally, a band gap of Ô.Ôò eV at room temperature [ÀÀ] and of ý.À eV in the limit T ýK [ÔÔâ] is observed. In analogy to the lattice constant, the theo- = retical gap is in much better agreement with the experimental result in the zero Kelvin limit. Apart from the opening of the band gap, the dišerent energetic position of the unoccupied spin-down Eu ¥ f states becomes evident. While spin-up and spin-down Eu ¥ f states are split by about ¥ eV in the PBE, they are separated by ÔÔ eV in the PBEý approach. Moreover, the bottom of the conduction bands is split by Ô.Ôý eV. It is this quantity which determines the e›ciency of EuO as a spin lter. Experimentally, values of ý. ¥ eV [Ôýý] and ý.âý eV [Ôýâ] are reported, respectively. e opening of the band gap and, thus, the total lling of the spin-up

Eu ¥ f states gives rise to an increase of the magnetic moment from â.ÞÞ µB for the PBE to

â.Àý µB for the PBEý functional. In order to describe the magnetic order of EuO, we apply a classical Heisenberg Hamilto- nian Ô H − Jijeiej , ( .¥Ô) = ò Qi j ≠ where Jij describes the exchange coupling between two Eu atoms at the sites i and j and ei is a unit vector pointing in the direction of the magnetic moment at site i. Due to thedenition of the Heisenberg Hamiltonian with a minus sign in Eq. ( .¥Ô) a positive exchange constant

Þý .Þ. EuO: a case study

EuO PBE Eu-4f 4 total O-2p Eu O 2 Eu-5d6s

0

DOS (states/eV) -2

-4 Eu-4f

-6 -4 -2 0 2 4 6 8 10 12

E-EF (eV) (a)

EuO PBE0 Eu-4f 4 total O-2p Eu O 2 Eu-5d6s

0

DOS (states/eV) -2

-4 Eu-4f

-6 -4 -2 0 2 4 6 8 10 12

E-EF (eV) (b)

Figure .À.: Density of states (DOS) of ferromagnetic EuO for (a) the PBE and (b) the PBEý functional. While the total DOS is shown in black, the partial DOSof EuandO is drawn in red and blue, respectively.

ÞÔ . Hybrid functionals within the generalized Kohn-Sham scheme

(a) (b) (c)

Figure .Ôý.: Nearest and next-nearest neighbors of a representative Eu atom (colored in red) for (a) the FM, (b) the AFM-I, and (c) the AFM-II conguration. e nearest and next-nearest neighbors are shown in pink and blue, respectively. All re- maining Eu atoms are indicated by the small circles. For simplicity, the oxygen atoms are not drawn.

corresponds to ferromagnetic coupling. We restrict the coupling to nearest neighbor (nn) and next-nearest neighbor (nnn) interactions

Ô H − ei JÔ ej + Jò ej . ( .¥ò) = ò Qi ⎡ jQnn j Qnnn ⎤ ⎢ ∈ ∈ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ In order to extract the coupling constants⎣JÔ and Jò from a DFT⎦ calculation we calculate EuO in its dišerent magnetic congurations. e rock-salt structure of EuO allows the Eu atoms to couple in four dišerent collinear magnetic congurations [ÔÔÞ]: simple ferromagnetic (FM) ordering, antiferromagnetic (AFM) ordering of type AFM-I, AFM-II, and AFM-III. In the AFM-I conguration the Eu atoms along the ýýÔ direction are coupled antiferromagneti- cally. e AFM-II type structure corresponds to an antiferromagnetic coupling in the ÔÔÔ [ ] direction, and the AFM-III exhibits two layers of Eu atoms with alternating spins in the ýýÔ [ ] direction. We restrict ourselves here to the AFM-I, AFM-II, and FM conguration, as they [ ] are su›cient for a determination of the exchange coupling constants JÔ and Jò. According to the Heisenberg model [s. Eq. ( .¥ò)], the energy per Eu atom for the FM spin conguration amounts to Ô E − ÔòJ + âJ −âJ − çJ , FM = ò Ô ò = Ô ò because each Eu atom is surrounded by[ Ôò nn and] â nnn with parallel spin conguration [cf. Fig. .Ôý]. In the case of the AFM-I structure each Eu atom has ¥ nn which are coupled ferromagnetically, while — nn exhibit a antiferromagnetic spin alignment. e â nnn are all

Þò .Þ. EuO: a case study

EuO PBE PBEý Expt. JÔ meV ¥.À¥ ò.âÔ Jò meV ¥. ¥ ý.⥠MFA( ) Tc K çç Ôçâ a RPA( ) âÀ Tc K òÞò Ôý ( ) aReference( ) Ôԗ

Table .ç.: Nearest and next-nearest neighbor exchange coupling constants JÔ and Jò calcu- lated with the PBE and PBEý functional as well as the resulting Curie temperature MFA in the mean-eld approximation (MFA) Tc and the random-phase approxima- RPA tion (RPA) Tc . coupled ferromagnetically, so that the energy per Eu atom for the AFM-I is given by

Ô E − − −¥J + âJ òJ − çJ . ( .¥ç) AFM I = ò Ô ò = Ô ò ( ) In the AFM-II phase the contribution of the nn mutually cancels out as each Eu atom is surrounded by â Eu atoms with parallel and â atoms with antiparallel spin. As the nnn are all aligned antiferromagnetically, we obtain

Ô E − − −âJ çJ ( .¥¥) AFM II = ò ò = ò ( ) for theenergyperEu. e energy dišerence between the AFM-I and FM conguration allows to calculate the nn coupling constant

J E − − E — . ( .¥ ) Ô = AFM I FM ( )~ Combination of Eq. ( .¥ ) with the energy dišerence between AFM-II and FM results in Jò

J E − − E â − E − − E — . ( .¥â) ò = AFM II FM AFM I FM ( )~ ( )~ In order to avoid numerical inaccuracies, the energy dišerencebetweenAFMandFMcong- uration is computed in the corresponding AFM unit cell. Moreover, we apply the previously optimized PBE and PBEý lattice constants. e exchange coupling constants extracted in this way are shown in Table .ç. As the FM conguration turns out to be energetically the most favorable conguration, the nn as well as nnn coupling constants are positive. is is in accordance with the ferromagnetic order- ing observed in experiment below a Curie temperature of âÀ K [Ôԗ]. At least for the PBEý

Þç . Hybrid functionals within the generalized Kohn-Sham scheme

functional the nnn exchange coupling constant Jò is signicantly smaller than JÔ, which jus- ties to disregard contributions from neighbors further afar. e mean-eld approximation (MFA) [ÔÔÀ, Ôòý], which neglects the žuctuation of the spins around its thermal average, al- lows to estimate the Curie temperature from the exchange constants JÔ and Jò by

MFA Ô Tc ÔòJÔ + âJò , ( .¥Þ) = çkB ( ) where kB denotes the Boltzmann constant. It is known that the MFA tends to overestimate the experimental Curie temperature [ÔòÔ]. Indeed, the PBE and PBEý exchange constants

JÔ and Jò yield a too large Tc in comparison with experiment (s. Table .ç). Nevertheless, the deviation between MFA and experimental Tc is drastically reduced by the PBEý functional. A further improvement can be achieved by employing the random-phase approximation (RPA) instead of the MFA. e RPA was introduced by Tyablikov [Ôòò] for spin Ô ò and generalized by Callen [Ôòç] to Bravais lattices with general spin quantum number. e Curie temperature ~ in the RPA is nally given by

−Ô RPA Ô Ô Tc , ( .¥—) = çkB Q J ý − J q ⎛ q ⎞ where J q is the Fourier transform of the⎝ exchange( ) coupling( )⎠ constants

( ) J q Jýj exp iqRj JÔ exp iqRj + Jò exp iqRj . ( .¥À) = Qj i = jQnn j Qnnn ≠ ∈ ∈ ( )
 [ ] [ ] For the fcc lattice of Eu atoms, Eq. ( .¥À) becomes

a a a a a J q ¥J cos q cos q + cos q + cos q cos q = Ô x ò z ò y ò z ò y ò ( ) +òJ›ò cos( qx a) + cos( qy)a + cos( qz a) .( ) ( ) ( . ý)

™ ž As shown in Table .ç, the RPA( leads) to( a reduction) ( of the) Curie temperature in compari- son with the MFA. For the PBEý functional we obtain a Tc of Ôý K in the RPA, which still overestimates the experimental value of âÀ K.

e exchange coupling constants, moreover, allow to calculate the energy dispersion of spin waves. A spin wave is a propagating collective excitation of spins, which can be characterized by its frequency ω and wave vector q. In a classical Heisenberg model both quantities are connected by the dispersion relation

Þ¥ .Þ. EuO: a case study

6 EuO

4.5

3

Spin-wave energy (meV) 1.5 Expt. PBE PBE0 [111] [100] 0 L Γ X

Figure .ÔÔ.: PBE and PBEý spin-wave energies are compared with experiment in the ÔÔÔ and Ôýý direction. e experimental data points are taken from Ref. Ôԗ. [ ] [ ]

Ô ω q J ý − J q , ( . Ô) = M where M corresponds to the magnetization( ) of[ ( the) Eu( atoms)] M Þ and J q is the Fourier = transform of the coupling constants as dened above. Insertion of Eq. ( . ý) into ( . Ô) en- ( ) ( ) ables to calculate the spin-wave dispersion for any q. In Fig. .ÔÔ we compare the PBE and PBEý spin-wave dispersions with experimental data along the directions Ôýý and ÔÔÔ . e experimental values are measured by inelastic neutron scattering [Ôԗ]. For small q around [ ] [ ] Γ the spin-wave dispersion is proportional to qò. e proportionality factor is the spin-wave stišness D, which is given by D J + J aò M. As already observable from Fig. .ÔÔ the PBE = Ô ò functional substantially overestimates the spin-wave stišness DPBE Ôò¥.Ô meV aò with re- ( ) ~ = ý spect to the experiment Dexp ç—.âmeVaò [Ôԗ]. e PBEý functional slightly overestimates = ý the experimental stišness DPBEý ¥ç. meV aò, but reproduces the spin-wave dispersion in = ý the [Ôýý] and [ÔÔÔ] direction quite accurately. We note the small energy scale from ý up to â meV.

In conclusion, the PBEý functional constitutes a denite improvement over the PBE func- tional for the rst-principles description of the rare-earth oxide EuO. e structural, elec- tronic, and magnetic properties are predicted in good agreement with experiment.

Þ . Hybrid functionals within the generalized Kohn-Sham scheme

5.8. Summary

A numerical scheme for the calculation of the non-local exchange potential, which is the central ingredient of hybrid functionals, within the all-electron FLAPW method has been introduced. Our algorithm relies on the auxiliary mixed product basis (MPB) for the repre- sentation of wave-function products. In this way, the six-dimensional integrals of the non- local potential transform into vector-matrix-vector products. e matrix that corresponds to the representation of the Coulomb interaction in the MPB can be made sparse by a unitary transformation of the basis leading to a reduction of the computational cost for the matrix- vector product. e algorithm is accelerated additionally (a) by a redenition of the MPB in the case of inversion symmetry such that the Coulomb matrix and the vectors become real-valued quantities, (b) by using spatial and time-reversal symmetries. e latter allows to identify those exchange matrix elements that are non-zero and must be calculated explicitly and, furthermore, to restrict the q-point summation arising from the sum over all occupied states to an irreducible wedge of the BZ. e aforementioned techniques are used to accelerate the calculation of the non-local ex- change potential. Self-consistency of a hybrid functional calculation, though, requires to cal- culate the potential in each iteration of the self-consistent eld cycle until self-consistency is achieved. We have shown that, while a direct iteration scheme requires many steps to con- verge, a nested self-consistent eld cycle consisting of an outer density-matrix and an inner density-only iteration loop accelerates the convergence of the self-consistent eld cycle con- siderably.

For the PBEý hybrid functional and the two materials Si and SrTiOç we have demonstrated that PBEý transition and total energies are easily converged with respect to the numerical pa- rameters of the MPB. Hence, the MPB provides a small but accurate all-electron basis for the construction of the exchange potential. Moreover, we have reported PBEý transition energies for a set of prototype semiconductors and insulators. In comparison with PBE they lie much closer to the experimental values. Furthermore, they are in favorable agreement with recent PBEý calculations within the PAW method [çý]. e non-locality of the exchange potential hampers the calculation of a band structure or a density of states (DOS). In order to circumvent this, we have employed the Wannier interpo- lation technique. It allows to construct an accurate band structure and DOS at a reasonable numerical cost. We have demonstrated this for the PBEý band structure of Si and the DOS of ZnO. e band structure of Si shows that the PBEý functional opens the band gap, but leaves the dispersion of the bands nearly unchanged apart from an increase of the valence band width. For ZnO the reduced self-interaction of the PBEý functional gives rise to a stronger

Þâ .—. Summary binding of the Zn d states. As a consequence, their position comes closer to the experimental value. e stronger binding, in turn, entails a reduced hybridization of theZn çd states with theOòp states, such that the band width of the O p states is increased. Moreover, we have applied the PBEý functional to EuO. In comparison with the PBE func- tional, the latter provides a signicant improvement with respect to the structural, electronic, and magnetic properties of EuO. e PBEý equilibrium lattice constant deviates by only Ôh from the experimental one at T ¥.ò K. Concerning the electronic properties, EuO is cor- = rectly predicted to be a ferromagnetic semiconductor with an indirect band gap of ý.— eV, while the experimental gap amounts to ý.À eV in the limit of ý K. Application of a classi- cal Heisenberg model with nearest and next-nearest-neighbors coupling gives access to the magnetic properties. In contrast to the PBE functional, the PBEý spin-wave dispersion agrees qualitatively and even quantitatively with experiment. e Curie temperature of ferromag- netic EuO is overestimated by both functionals in the mean-eld as well as the random-phase approximation. Yet, the PBEý Curie temperature is in signicant better agreement with ex- periment. We note that the numerical scheme is not restricted to the PBEý hybrid functional. It can be used to implement any other hybrid functional with an unscreened non-local exchange potential, for example the BçLYP functional [âò]. Moreover, the numerical procedure can be generalized to also enable the calculation of a screened non-local exchange potential, that is, for example, employed in the HSE functional [Ôò¥, Ôò ]. e Coulomb matrix must be simply replaced by the matrix of the screened Coulomb interaction [Ôòâ]. Furthermore, the matrix elements of the non-local exchange potential are an essential ingredient in the EXX- OEP formalism, in which a local potential is constructed from the non-local exact exchange energy. e EXX-OEP method will be discussed in the next chapter.

ÞÞ

6. Exact exchange within the optimized effective potential method

In the previous chapter, hybrid functionals combining a fraction of non-local exact exchange with local approximations for the remaining exchange-correlation (xc) energy functional were discussed. ey are usually applied in the generalized Kohn-Sham (gKS) scheme, where theelectrons experiencea local as well as a non-local potential. e Kohn-Sham (KS) system, however, requires by construction a purely local potential. erefore, the treatment of hybrid functionals as described in the previous chapter is outside the realm of KS density-functional theory (DFT). e construction of a local xc potential from any orbital-dependent and maybe eigenvalue- dependent approximation to the xc energy will be described in the present chapter for the example of the orbital-dependent exact exchange energy. e optimized ešective potential (OEP) method, which is sometimes called optimized potential method (OPM) in the litera- ture, forms the general theoretical framework for the construction of a local potential from an orbital- and (or) eigenvalue-dependent approximation to the xc energy. e chapter is organized as follows. AŸer a theoretical introduction into the OEP method we discuss the numerical implementation of the exact exchange OEP (EXX-OEP) approach within the full-potential linearized augmented-plane-wave (FLAPW) method. It employs a specically adjusted mixed product basis (MPB) for the representation of the local exchange potential. For the case of diamond we carefully analyze the conditions and requirements on the LAPW and mixed product basis to obtain stable and physical local EXX potentials. e LAPW basis must be converged with respect to the MPB. We compare our all-electron full potential KS transition energies for C, Si, SiC, Ge, GaAs, solid Ne and Ar with pseudopoten- tial plane-wave and experimental results from the literature and nd a favorable agreement. For the III-V nitrides and rock-salt ScN we analyze the ešect of the derivative discontinu- ity of the EXX functional on the theoretical fundamental band gap. In order to improve the convergence of the EXX potential with respect to the LAPW basis set size, we propose and explore a nite basis correction (FBC) for the response functions of density and KS wave-

ÞÀ â. Exact exchange within the optimized ešective potential method function, which are substantial ingredients of the EXX-OEP equation. We then apply the

EXX functional to the cubic perovskites CaTiOç, SrTiOç, and BaTiOç as well as the antiferro- magnetic transition metal oxides MnO, FeO, and CoO. Finally, we present a generalization of the EXX-OEP approach to metals and report on calculations for Na, Al, and Cu.

6.1. Optimized effective potential method

e optimized ešective potential (OEP) method is a quite general approach, which allows to construct a local, multiplicative xc potential from any orbital- and (or) eigenvalue-dependent xc functional. In order to derive the basic equations for the OEP approach, we go back to thedenition of the xc potential of KS DFT as the functional derivative of the xc energy with respect to the density [cf. Eq. (ò.ò—) for the paramagnetic and Eq. (ò.¥Ô) for the spin-polarized case] δE V σ r xc . (â.Ô) xc = δnσ r ( ) In the case of the local-density approximation (LDA)( ) or the generalized gradient approxi- mation (GGA), where the xc energy depends locally on the spin densities and for the GGA also on their gradients, the functional derivative translates to a derivative of a function and is evaluated in a straightforward way. However, an orbital- and (or) eigenvalue-dependent func- σ tional is only an indirect functional of the spin densities, as the KS orbitals ϕnk r and the KS eigenvalues єσ are functionals of the ešective potential V σ r through the KS equation and nk eš ( ) the ešective potential V σ r is a functional of the spin density nσ r through the Hohenberg eš ( ) and Kohn theorem. is interdependence of the KS orbitals and eigenvalues on the density ( ) ( ) via the ešective potential is exploited to evaluate the functional derivative by means of the chain rule for functional derivatives [çç, ç¥]

σ ′ σ σ ′′ δE δϕ r ′ δE δє δV r ′′ V σ r xc nk dçr + c.c. + xc nk eš dçr . xc = Q ∫ ∫ δϕσ r′ δV σ r′′ δєσ δV σ r′′ δnσ r nk nk eš ( ) nk eš ( ) ( ) Œ ‘ (â.ò) σ ( ) ( ) ( ) ( ) As the KS orbitals ϕnk r are in general complex, we can either dišerentiate with respect to σ σ the real and imaginary part of ϕnk r) or with respect to ϕnk r) and its complex conjungate ∗ ( ) (c.c.) ϕσ r . Here, we performed the dišerentiation with respect to ϕσ r) and its complex nk ( ( nk conjungate. e sum in Eq. (â.ò) runs over all KS states present in a given approximate func- ( ) ( σ′ ′ σ ′′ tional Exc. Moreover, we have exploited that the wave-function response δϕnk r δVeš r , σ′ σ ′′ the eigenvalue response δєnk δVeš r , and the inverse single-particle response function ′ ′′ ( )~ ( ) δV σ r δnσ r are diagonal in spin space for semiconductors and insulators. An exten- eš ~ ( ) sion of the formalism to metals is discussed in section â.Ôý. ( )~ ( )

—ý â.Ô. Optimized ešective potential method

σ ′ e linear response of the wave function ϕnk r due to a change in the ešective potential at ′′ r is exactly accessible in rst-order perturbation theory and given by ( ) σ ′ σ∗ ′′ σ ′′ δϕnk r ϕn′k r ϕnk r σ ′ ϕ ′ r . (â.ç) σ ′′ σ − σ n k δVeš r = nQ′ n єnk єn′k ( ) ≠ ( ) ( ) ( ) ′ ′ With this, the single-particle( spin-density) response function χσ r, r δnσ r δV σ r s = eš becomes occ. σ∗ σ σ∗ ′ σ ′ ′ ϕ r ϕ ′ r ϕ ′ r ϕ r( ) ( )~ ( ) χσ r, r nk n k n k nk + c.c. . (â.¥) s σ − σ = Qnk nQ′ n єnk єn′k ≠ ( ) ( ) ( ) ( ) ( ) ′ e combinations of occupied (occ.) states n with occupied states n cancel mutually in ′ Eq. (â.¥), so that the sum over n can be restricted to the unoccupied (unocc.) states. Ad- σ∗ σ ditionally, we can exploit time-reversal symmetry ϕn −k r ϕnk r and, thus, Eq. (â.¥) ( ) = turns into occ. unocc. σ∗ σ[ σ(∗ ) ′ σ ( ′)] σ ′ ϕnk r ϕn′k r ϕn′k r ϕnk r χs r, r ò σ σ . (â. ) = Q Q′ є − є ′ nk n ( ) (nk) n k( ) ( ) ( ) σ ′ Multiplication of Eq. (â.ò) with the single-particle spin-density response function χs r, r , ′ ′ integration, and use of the symmetry χσ r, r χσ r , r result in an integral equation for s = s ( ) the xc potential ( ) ( ) σ ′ σ ′ ′ ′ δE δϕ r ′ δE δє χσ r, r V σ r dçr xc nk dçr + c.c. + xc nk . (â.â) ∫ s xc = Q ∫ δϕσ r′ δV σ r δєσ δV σ r nk nk eš( ) nk eš ( ) ( ) Œ ‘ ( ) ( ) ( )

So far, the derivation holds for any orbital- and (or) eigenvalue-dependent xc functional. In the following, we will concentrate on the orbital-dependent exact-exchange (EXX) functional

σ∗ σ σ∗ ′ σ ′ occ. occ. ϕ r ϕ ′ r ϕ ′ r ϕ r Ô nk n q n q nk ç ç ′ Ex − ′ d r d r . (â.Þ) = ò Q Q Q′ ∫∫ r − r σ nk n q ( ) ( ) ( ) ( ) S S It has been shown in section ç. that Ex is theleading term of theexact xc functional in terms of the electron-electron interaction. As Ex is only orbital-dependent, the functional derivative σ of Ex with respect to the KS eigenvalues єnk is zero, whereas the derivative of Ex with respect to the KS wave function is given by

δE ∗ ′′ ′′ ′ ′′ x ϕσ r V NL,σ r , r dçr . (â.—) σ ′ ∫ nk x δϕnk r = ( ) ( ) NL,σ ( ) Vx is the non-local exchange potential as dened in Eq. ( .Þ). Consequently, the integral

—Ô â. Exact exchange within the optimized ešective potential method equation (â.â) for the EXX functional (â.Þ) becomes

′ ′ ′ χσ r, r V σ r dçr tσ r (â.À) ∫ s x = with the right-hand side ( ) ( ) ( )

occ. unocc. σ∗ σ σ σ NL,σ σ ϕn′k r ϕnk r t r ò ϕnk Vx ϕn′k σ σ (â.Ôý) = Q Q′ є − є ′ nk n n(k ) n k( ) ( ) ⟨ S S ⟩ and σ NL,σ σ σ∗ NL,σ ′ σ ′ ç ç ′ ϕ V ϕ ′ ϕ r V r, r ϕ ′ r d r d r . (â.ÔÔ) nk x n k = ∫∫ nk x n k We have exploited⟨ time-reversalS S ⟩ symmetry( for) the right-hand( ) side,( ) as well.

e integral equation for the local EXX potential in the form of Eq. (â.À) is oŸen called OEP equation. So far, we have derived the latter within the framework of KS DFT. However, originally this equation goes back to Sharp and Horton and the year ÔÀ ç [çò]. Sharp and Horton searched for an approximation to the Hartree-Fock (HF) approach, where the orbitals forming the single Slater determinant, are the solution of a Schrödinger equation with a local instead of the non-local HF potential. erefore, they minimized the HF total energy with respect to the orbitals under the constraint that the orbitals move in a local, ešective potential. As the orbitals are functionals of the local, ešective potential, the constrained minimization is equivalent to an unconstrained minimization of the total HF energy with respect to the local potential. With a partition of the local potential into external, Hartree, and exchange parts, in analogy to the KS potential, the ansatz of Sharp and Horton nally leads to Eq. (â.À), as well. e equivalence of the variational principle of Hohenberg and Kohn and the condition of Sharp and Horton can be easily demonstrated. By applying the chain rule the variational principle of Hohenberg and Kohn δE ý (â.Ôò) δn r = can be written as ( ) ′ δE δE δVeš r ç ′ ý ′ d r . (â.Ôç) = δn r = ∫ δVeš r δn r ( ) As Eq. (â.Ôç) holds for an arbitrary( ) density variation,( ) which( ) does not change the particle number (∫ δn r dçr ý), it follows that = ( ) δE ′ ý (â.Ô¥) δVeš r = must be fullled. is is the condition of Sharp( ) and Horton. Sahni et al. [ÔòÞ] realized in

—ò â.ò. Implementation of the EXX functional

ÔÀ—ò that the approach of Sharp and Horton is equivalent to the EXX formalism of KS DFT. According to Eq. (â.Ô¥), the local exact exchange potential can be understood as the best local potential which minimizes the HF total energy. Moreover, the construction of Sharp and Horton shows that HF total energies must always be lower than EXX total energies,because in the HF approach the total energy is minimized over all possible orbitals without an additional constraint. Before the implementation of the EXX-OEP integral equation is discussed in the next sec- tion, we note that Eq. (â.À) can also be obtained from the linearized Sham-Schlüter equation

[cf. Eq. (ç. ò)] if one approximates the self-energy Σxc by the HF energy.

6.2. Implementation of the EXX functional

In order to solve the integral equation for the local EXX potential numerically, we introduce σ σ σ an auxiliary basis set MI r for the representation of V r [V r V MJ r ]. e x x = ∑J x,J EXX-OEP equation (â.À) then becomes a linear algebraic equation for the vector Vσ repre- { ( )} ( ) ( ) (x) senting the local EXX potential σ σ σ χs,IJ Vx,J tI (â.Ô ) QJ = with the spin-density response matrix

occ. unocc. σ σ σ σ σ MI ϕnk ϕn′k ϕn′k ϕnk MJ χs,IJ ò σ σ (â.Ôâ) = Q Q′ є − є ′ nk n ⟨ S nk ⟩⟨ n k S ⟩ and the vector of the right-hand side

occ. unocc. σ σ σ σ NL,σ σ MI ϕn′k ϕnk tI ò ϕnk Vx ϕn′k σ σ . (â.ÔÞ) = Q Q′ є − є ′ nk n ⟨ nk Sn k ⟩ ⟨ S S ⟩ σ Eq. (â.Ô ) can be solved for the exchange potential by a matrix inversion of χs .

So far, the auxiliary basis set MI r hasnot been specied. e form of the response ma- trix (â.Ôâ) and the vector (â.ÔÞ) shows that the auxiliary basis should be primarily constructed { ( )} from products of KS wave functions. In section ¥.ò.ò we have already introduced such a ba- sis, the mixed product basis (MPB), which was constructed from products of LAPW basis functions. e MPB has been employed in chapter to calculate the matrix elements of the non-local exchange potential. For the solution of the EXX-OEP equation we will slightly modify the MPB. First, the MPB can be restricted to k ý, as the EXX potential is strictly = periodic. Second, atomic EXX calculations show that the local exchange potential has pro- nounced humps, which režect the atomic shell structure. Due to the spatial contraction of

—ç â. Exact exchange within the optimized ešective potential method

0

-5

-10

-15

-20 (htr) x

V -25

-30 C Si -35 Ge Sn -40

-45 0.001 0.01 0.1 1 10 r (a0)

Figure â.Ô.: Atomic EXX potentials obtained with the RELKS code for the C, Si, Ge, and Sn atom. With increasing atomic number from C (Z=â) to Sn (Z= ý) the intershell humps move closer and closer to the atomic nuclei (at r ý). Simultaneously, = more and more intershell humps in the EXX potential appear, one for each atomic shell. the electron orbitals these humps move closer to the atomic nucleus with increasing atomic number [s. Fig. (â.Ô)]. An accurate description of these humps would require a very žexible basis. However, the crystal EXX potential should resemble the atomic one in the vicinity of the nuclei. erefore, we add the atomic EXX potential to the spherical mu›n-tin (MT) MPB functions. e other MT basis functions then only have to describe the dišerence between the spherical atomic and the crystal EXX potential. e atomic EXX potential is obtained from the relativistic atomic-structure program RELKS [ââ, Ôò—, ÔòÀ]. Moreover, we form lin- ear combinations of the MT and IR MPB functions that are continuous in value and derivative over the whole space to avoid discontinuities in the resulting EXX potential at the MT sphere boundaries. is can be achieved by a construction similar to that of the LAPW basis, i.e., two radial functions of the MT MPB in each lm channel are used to augment an interstitial plane wave in the MT spheres. As the MT MPB usually consists of more than two radial functions per lm channel, the remaining radial functions are used to form local orbitals. In a last step, the matched MPB functions are orthogonalized with respect to a constant function in order

—¥ â.ò. Implementation of the EXX functional

σ to guarantee that the response matrix χs,IJ [Eq. (â.Ôâ)] is invertible. is is necessary, because a change of the ešective potential by a constant does not change the electron density. So any constant function is an eigenfunction of the response matrix with eigenvalue zero. erefore, we must go over to a restricted function space that excludes constant potential variations. As a consequence, the solution of the integral equation (â.Ô ) is not unique. If we have found σ σ one solution Vx r , also Vx r + c solves Eq. (â.Ô ), where c denotes a real number. In Appendix A we present a dišerent approach to construct the subspace of the MPB, ( ) ( ) which fullls the constraints of continuity in value and rst radial derivative at the MT sphere boundaries and is orthogonal with respect to a constant function. is approach has the benet that it is more general in the sense that further constraints to the MPB can easily be incorporated as long as the constraint can be formulated as an orthogonality condition to the MPB. We note that the construction of the modied MPB in analogy to the LAPW basis – as discussed above – and that described in Appendix A lead to a MPB spanning the same Hilbert space. e response matrix elements (â.Ôâ) and the right-hand side (â.ÔÞ) of the EXX-OEP equa- tion contain a sum over all occupied states. Here, both core and valence states are taken into account.Ô Consequently, matrix elements of the non-local exchange potential between oc- cupied (core and valence) and unoccupied valence states enter on the right-hand side. In the previous chapter, we already introduced a numerical scheme to compute the matrix ele- ments of the non-local exchange potential between valence states. is scheme is extended to also calculate the matrix elements between core and valence states as well as core and core states. e matrix elements of the latter are, in fact, not needed for the calculation of the local exchange potential but required to compute the all-electron EXX energy. As the non- NL,σ ′ local exchange potential Vx r, r [Eq. .Þ] contains a sum over all occupied states, which comprise core and valence electron states, we distinguish between two cases: the two wave ( )NL,σ functions arising from the sum in Vx correspond to (a) valence or (b) core states. In case (a) we employ a MPB to represent wave function products at the same spatial coordinate. We note that we employ two dišerent MPB sets. One for the representation of the local ex- σ change potential Vx r , as explained above, and the other to calculate the matrix elements of ′ the non-local exchange operator V NL,σ r, r as accurately as possible. e latter is typically ( ) x much larger than the former. With the MPB the six-dimensional integral that corresponds NL,σ ′ ( ) to the matrix element of Vx r, r turns into a vector-matrix-vector product, where the matrix represents the Coulomb kernel Ô r − r′ in the MPB. e vectors correspond to the ( ) representation of the product of two valence states or the product of valence times core state. ~S S

ÔFor the core states we switch to a non-relativistic description by averaging for each principal quantum number n and angular momentum l the radial solutions of the Dirac equation with j = l + Ô~ò and j = l − Ô~ò.

— â. Exact exchange within the optimized ešective potential method

For an accurate representation of valence times core state we generalize the MT MPB for the NL,σ calculation of Vx such that its radial functions are constructed from products of LAPW radial functions as well as products of LAPW and core radial functions. e techniques de- NL,σ veloped in chapter to accelerate the calculation of the matrix elements of Vx can be applied in this more general case, as well. e computation of the contribution type (b) to the matrix elements does not require the MPB. Instead, these matrix elements can be calculated analytically, as the core states are dispersionless and conned to the MT spheres [—ç, —â]. Furthermore, we exploit spatial and time-reversal symmetries to restrict the k-point sums in Eq. (â.Ôâ) and (â.ÔÞ) to the irreducible wedge of the Brillouin zone (BZ) in a similar man- ner as for the matrix elements of the non-local exchange potential (s. Sec. .ò.ò). For the response matrix we will discuss the usage of symmetry in detail. A restriction of the BZ sum in Eq. (â.Ôâ) to the irreducible BZ (IBZ) is possible without changing the value of the matrix element χσ , if we simultaneously run over all symmetry operations P , i Ô,..., N of the s,IJ i = crystal { } N IBZ occ. unocc. σ σ σ σ Ô MI Pi ϕ Pi ϕ ′ Pi ϕ ′ Pi ϕ MJ χσ ò nk n k n k nk . (â.ԗ) s,IJ σ − σ = Qi Ô Qk NP k Qn Qn′ єnk єn′k = ⟨ [ ]S[ ]⟩⟨[ ]S[ ] ⟩  e factor Ô NP k avoids( ) a multi-counting of certain Bloch vectors k. For example, without this factor the Γ point is contained in Eq. (â.ԗ) N-times. Consequently, N k is the number ~ ( ) P of symmetry operations, which leave k invariant, or, to be more precise, N k is the number P( ) of members in the little group of k. Instead of operating with the symmetry operation Pi on σ σ −Ô ( ) ϕnk and ϕn′k, the inverse symmetry operation Pi can be applied to MI and MJ

N IBZ occ. unocc. −Ô σ σ σ σ −Ô Ô P MI ϕ ϕ ′ ϕ ′ ϕ P MJ χσ ò i nk n k n k nk i . (â.ÔÀ) s,IJ σ − σ = Qi Ô Qk NP k Qn Qn′ єnk єn′k = ⟨[ ] S ⟩⟨ S [ ]⟩  Hence, we can restrict the( BZ) sum in the response matrix to the IBZ, but have to rotate the MPB functions aŸerwards to add the contribution of all k points. e rotation of the MPB functions is computationally very cheap, as it simply translates to a summation over dišerent matrix elements. Since an analogous argumentation holds for the vector tσ of the right-hand side we only give the nal result

N IBZ occ. unocc. −Ô σ σ σ Ô σ NL,σ σ Pi MI ϕn′k ϕnk t ò ϕ V ϕ ′ . (â.òý) I nk x n k σ − σ = Qi Ô Qk NP k Qn Qn′ єnk єn′k = ⟨[ ] S ⟩  ⟨ S S ⟩ ( ) e local exact exchange potential is then obtained by a multiplication of the inverse re-

—â â.ò. Implementation of the EXX functional sponse matrix with the vector tσ . Finally, the local, multiplicative exchange potential is trans- formed from a representation in the MPB to a representation in terms of lattice harmonics and stars (cf. Sec. ¥.ò) and added to the external and Hartree potential.

Numerical tests of the implementation

σ σ e spin-density response function χs , the right-hand side t , and the resulting exchange po- σ ′ ′ tential Vx are functional derivatives of the form δA r δB r or δA δB r . ey describe the linear change of the quantity A due to a change in B. To test our implementation, we ( )~ ( ) ~ ( ) calculate the change of A due to a perturbation in B explicitly and compare the full response of A with its linear counterpart arising from δA δB. e dišerence between exact and linear response should then vanish quadratically with the perturbation strength. ~ For the test of the spin-density response function and the right-hand side we perturb the σ σ σ Hamiltonian with the potential V r α V MI r , where V are random num- per = ∑I per,I per,I bers and α controls the strength of the perturbation. e perturbed Hamiltonian is exactly ( ) ( ) diagonalized. From the perturbed wave functions the change in the density ∆nσ r and the σ exact exchange energy ∆Ex is calculated. On the other hand, we compute the linear change lin σ lin σ σ ( ) ∆ n and ∆ Ex arising from the perturbation Vper by

′ ′ ′ ∆linnσ r χσ r, r V σ r dçr (â.òÔ) = ∫ s per ( ) ( ) ( ) and ′ ′ ′ ∆linEσ tσ r V σ r dçr . (â.òò) x = ∫ per Table â.Ô(a) and (b) demonstrate for the case of( diamond) ( ) that the dišerences ∆nσ −∆linnσ and σ lin σ ∆Ex −∆ Ex indeed depend quadratically on the perturbation strength α, if the perturbation is small enough, so that we are in the linear regime.

σ For the check of the exchange potential Vx itself we compare the exact change in the ex- σ lin σ change energy ∆Ex with its linear counterpart ∆ Ex , which is calculated from

lin σ σ ′ σ ′ ç ′ σ ′ σ ′ ç ′ ∆ Ex Vx r ∆n r d r Vx,I MI r ∆n r d r . (â.òç) = ∫ = QI ∫ ( ) ( ) ( ) ( ) We note that ∆nσ in Eq. (â.òç) is the exact change in the density due to the perturbing po- σ σ lin σ tential Vper. e dišerence ∆Ex − ∆ Ex [cf. Table â.Ô(c)] shows a quadratic dependence on the perturbation strength α, as well.

ese three tests conrm the validity and correctness of our implementation.

—Þ â. Exact exchange within the optimized ešective potential method

(a)

α ý.ýÔ ý.ýýÔ ý.ýýýÔ − − − ∆linn-∆n ò.ÔâÞÔý × Ôý ¥ ò.ÔÞýԗ × Ôý â ò.ÔÞýç × Ôý —

(b) S S α ý.ýÔ ý.ýýÔ ý.ýýýÔ −Ô −ò ∆Ex Ô.âÀ¥ Ô.Þ¥âçÀ × Ôý Ô.Þ ÔÞÔ × Ôý lin −Ô −ò ∆ Ex Ô.Þ òçý Ô.Þ òçý × Ôý Ô.Þ òçý × Ôý lin −ò −¥ −â ∆ Ex-∆Ex .ÞÞ¥ÞÀ × Ôý .Àý——ò × Ôý .ÀòòÔÔ × Ôý

(c) S S α ý.ýÔ ý.ýýÔ ý.ýýýÔ −Ô −ò ∆Ex Ô.âÀ¥ Ô.Þ¥âçÀ × Ôý Ô.Þ ÔÞÔ × Ôý lin −Ô −ò ∆ Ex Ô.âÞÀò¥ Ô.Þ¥¥ÞÀ × Ôý Ô.Þ Ô × Ôý lin −ò −¥ −â ∆ Ex-∆Ex Ô. çýÀç × Ôý Ô. ÀÀÔÀ × Ôý Ô.âýâò × Ôý

S S ′ ′ Table â.Ô.: Numerical test of (a) the response function χ r, r δn r δV r , (b) the s = s right-hand side t r δE δV r , and (c) the potential V r δE δn r for = x s x = x the case of diamond. In (a) the exact response of( the) density( ∆)~n r (is compared) ( ) ~ lin( ) lin (lin) ~ ( ) with its linear approximation ∆ n r I nI MI r and nI α J χs,IJ Vper,J lin ò=ç∑Ô ò =( )∑ by the Lò norm ∫ ∆n r − ∆n r d r ~ . In cases (b) and (c) the exact re- ( ) ( ) sponse of the exchange energy ∆Ex is opposed to its linear approximation, which is lin[ S ( )∗ ( )S ] lin ç calculated by ∆ Ex α t Vper,I in (b) or by ∆ Ex Vx,I ∫ MI r ∆n r d r = I I = I in (c). e dišerence between∑ exact and linear response∑ shows in all three cases a quadratic dependence on the perturbation strength α. ( ) ( )

—— â.ç. Balance of MPB and FLAPW basis

6.3. Balance of MPB and FLAPW basis

AŸer the discussion of the implementation of the EXX-OEP approach within the FLAPW method, we demonstrate in this section that LAPW and mixed product basis are not indepen- dent from each other. In fact, both basis sets are intertwined through the response functions Eqs. (â.ç) and (â.¥). We show in particular that a smooth and physical local EXX poten- tial requires a balance of LAPW and mixed product basis, i.e., the LAPW basis for the KS wave functions must be converged with respect to a given MPB until the EXX potential does not change anymore. A similar behavior has been observed in EXX-OEP implementations employing plane-wave and Gaussian basis sets [Ôçý, ÔçÔ]. In Fig. â.ò the local EXX potential for diamond is shown between two neighboring car- bon atoms along (a) the [ÔÔÔ] and (b) the [Ôýý] direction. e position of the corresponding atoms and directions in the cubic unit cell of diamond are indicated in Fig. â.ç. For these calculations we have employed a MPB for the EXX potential with a reciprocal cutoš radius ′ − of G ç.¥a Ô and L ¥, which amounts to ve s−, four p−, four d−, three f −, and max = ý max = two g−type radial functions per carbon atom. Moreover, the LAPW basis-set parameters are − G ¥.òa Ô and l â, and a ¥×¥×¥ k-point sampling has been employed. If we use max = ý max = the conventional LAPW basis set (green dashed curve), the local EXX potential tends to a positive value at the atomic nuclei. is behavior is unphysical, as the exchange interaction is attractive. We say that both basis sets are unbalanced in this case. However, if we add local orbitals to the LAPW basis set, the potential becomes smooth and physical (red solid curve). e EXX potential then starts at the atomic nuclei with a value of about −¥ htr. At roughly

ý.âaý it shows the typical intershell hump and goes smoothly over to the interstitial region. At the MT sphere boundary S Ô.¥òa a tiny discontinuity in the potential is observable, a = ý which results from the nite angular momentum cutoš in the MT spheres. We nd that for ( ) a converged EXX potential six local orbitals per lm channel from l ý, . . . , and m l, = ≤ placed at higher energies according to the criterion (¥.ԗ), are needed. e requirement of σ σ S S local orbitals from l ý, . . . , is reasonable, as in the projections M ϕ ′ ϕ the occupied = I n k nk òs and òp states of diamond and the MT MPB functions with L ¥ can maximally cou- max⟨ = S ⟩ ple to unoccupied states with angular character of l . e need of so many local orbitals = increases the number of basis functions by approximately a factor of . We note that we take all resulting KS bands, about çý, in the sums over the unoccupied states in Eqs. (â.Ôâ) and (â.ÔÞ) into account. For comparison, we also show the atomic EXX potential (blue dotted curve) in Fig. â.ò. As expected, in the direct vicinity of the carbon nucleus the atomic and the (balanced) crys- tal EXX potentials are indistinguishable. However, towards the MT sphere boundary atomic

—À â. Exact exchange within the optimized ešective potential method

0.5 [111] C 0 -0.5 -1 -1.5 (htr) x

V -2 -2.5 -3 balanced -3.5 unbalanced atomic -4 0 0.5 1 1.5 2 2.5

r (a0)

(a)

1 [100] C 0.5 0 -0.5 -1

(htr) -1.5 x V -2 -2.5 -3 balanced -3.5 unbalanced atomic -4 0 1 2 3 4 5 6

r (a0)

(b) Figure â.ò.: Local EXX potential of diamond between two C atoms (a) along the [ÔÔÔ] and (b) along the [Ôýý] directions for the cases where the MPB and LAPW basis set are unbalanced (green dashed line) and balanced (red solid line). Moreover, the atomic EXX potential of C is shown as blue dotted lines. It is shiŸed to align with the crystal potential at the atomic nucleus at r ý. =

Àý â.ç. Balance of MPB and FLAPW basis

Figure â.ç.: Cubic unit cell of diamond with the ýÔÔ plane. e lines along the ÔÔÔ and Ôýý directions corresponding to Fig. â.ò are indicated. ( ) [ ] [ ] and crystal EXX potential start to deviate. ere, the atomic EXX potential already shows the typical Ô r behavior, from which the crystal potential deviates due to the demand of periodic- ity. Hence, it is not surprising, that the MPB function which corresponds to the atomic EXX ~ potential, gets the largest weight in the MT spheres and, thus, contributes most to the MT part of the potential. e constant function in the MT spheres, which enables an alignment of the MT to the interstitial potential, gets the second largest weight. So far, we pointed out that a physical and stable EXX potential in the MT spheres requires a balance of LAPW and mixed product basis, which is achieved by adding local orbitals to the

LAPW basis set. In analogy, in the interstitial region the reciprocal cutoš radius Gmax of the ′ LAPW basis must be converged with respect to the reciprocal cutoš of the MPB Gmax. In or- der to demonstrate this behavior more clearly, we have chosen a rather large reciprocal cutoš ′ − ′ value for the MPB G .—a Ô (s. Fig. â.¥). If G is smaller than G , the EXX potential max = ý max max shows spurious oscillations in the interstitial region. In fact, the potential oscillates around −Ô the converged curve obtained here with Gmax â.ýýaý . Fortunately, the converged EXX = ′ potential is a smooth function, and moderate reciprocal cutoš radii Gmax for the MPB are ′ su›cient. We found that Gmax ý.Þ Gmax is a reasonable choice. For diamond for example ′ =− − the combination of G ç.¥a Ô and G ¥.òa Ô leads to stable results. max = ý max = ý We go back to the EXX-OEP equation (â.À) to understand the requirement of the basis set balance in more detail. is equation contains two response functions: on the leŸ-hand

ÀÔ â. Exact exchange within the optimized ešective potential method

1 0.5 [100] C 0 -0.5 0.4 -1

(htr) -1.5

x 0.2

V -2 -2.5 0 -3

-3.5 2 3 4 5 -4 0 1 2 3 4 5 6

r (a0) G = 4.0 a -1 G = 4.5 a -1 max 0-1 max 0-1 Gmax= 5.0 a0 Gmax= 6.0 a0

Figure â.¥.: e convergence of the interstitial EXX potential is demonstrated for dišerent reciprocal cutoš radii Gmax of the LAPW basis on a line connecting two carbon −Ô atoms along the [Ôýý] direction. e reciprocal MPB cutoš is chosen as .— aý .

side the density response function χs, which must be inverted to obtain Vx, and on the right- hand side the KS wave-function response. ey describe the change of the density and KS wave function, respectively, due to a change in the ešective potential. As we represent the EXX-OEP equation in terms of the MPB, the changes in the ešective potential are given by the MPB. On the contrary, the change in the density and KS wave function are described by rst-order perturbation theory giving rise to a sum over all unoccupied states. In practice, of course, only a nite number N of unoccupied states is available and the sum over the unoccupied states is truncated. us, the response is determined by the quality of the LAPW basis. e latter must provide enough žexibility to enable the density or KS wave function to respond adequately to changes in the ešective potential. is explains the behavior observed above, and it is conrmed by the convergence behavior of the density response function with respect to the LAPW basis as demonstrated in Fig. â. . e relative change in the eigenvalues of the density response function as a function of the number of local orbitals nLO added to the LAPW basis is shown. In each step, òl +Ô local orbitals per l channel from l ý,..., are = added to the LAPW basis resulting in çâ additional functions per atom. While we observe a change in the eigenvalues in the range of Ôýý% between zero and one set of local orbitals, the relative changes are of the order of ý.Ô% to Ô.ý% between n and n â. Fortunately, LO = LO =

Àò â.ç. Balance of MPB and FLAPW basis

10 ) C LO n (

I 1 s, χ )]/

LO 0.1 n ( I s, χ 0.01 +1)- LO n

( 0.001 I s, χ [ 0.0001 0 20 40 60 80 100 I n = 0 n = 2 n = 4 nLO nLO nLO LO= 1 LO= 3 LO= 5

Figure â. .: Convergence of the eigenvalues of the density response function χs for diamond with respect to the number of local orbitals added to the LAPW basis. nLO de- notes the number of local orbitals added in each lm channel from l ý,..., = to the conventional LAPW basis. e eigenvalues χs,I are ordered according to their absolute values along the x axis. e y axis shows the relative change of the eigenvalue.

in particular the small eigenvalues of the response function, which become important in the inverse response function, converge well.

e observation that the orbital LAPW basis and auxiliary MPB are not independent and must be balanced for a physical density-response function and consequently physical EXX potential, can be understood by analyzing a model system of independent electrons in a con- stant potential [Ôçý]. For this system the eigenfunctions are known analytically and given by simple plane waves. But let us assume that we perform an electronic-structure calculation for such a system, where the orbital basis is given by plane waves, too. en, each basis function is also an eigenfunction of the Hamiltonian. In order to be able to describe all occupied states, orb the reciprocal cutoš value Gmax for the orbital basis set is chosen to be larger than the vector GF corresponding to the Fermi energy. Moreover, let us assume that we represent the density aux response function χs by an auxiliary basis of plane waves with the cutoš Gmax. e density

Àç â. Exact exchange within the optimized ešective potential method

G G Gorb orb orb

G G G G F G F G F

orb Gorb -G Gorb -G Gmax -G F max F max F

Gorb +G Gorb +G Gorb +G (a) max F (b) max F (c) max F

Figure â.â.: Visualization of the three limiting cases for the representation of the density re- sponse function χs for the model system of independent electrons in a constant aux orb potential. In case (a) with Gmax Gmax−GF all matrix elements of χs are obtained ≤ ′′ with their correct value as for any G of the auxiliary basis the vector G + G lies orb orb aux within the orbital cutoš radius Gmax. In the case (b) with Gmax − GF Gmax orb orb orb < ≤ Gmax + GF the matrix elements with Gmax − GF G Gmax + GF are incorrect, orb < ≤ while for (c) with G Gmax + GF all matrix elements are zero. > S S S S

response matrix then becomes a diagonal matrix with the entries

Ô χ ′ ¥δ ′ . (â.ò¥) s,GG G,G ′′ ò − ′′ + ò = G′′QG G G G S S≤ F S S S S We now distinguish three cases: in the rst case, Gaux Gorb − G , all matrix elements of max ≤ max F the density response matrix are obtained with their correct analytical value, because all states corresponding to G′′ + G are accessible in the orbital basis set (cf. Fig. â.â). In the second case, Gorb − G Gaux Gorb + G , however all matrix elements with Gorb − G G max F < max ≤ max F max F < ≤ Gorb + G are incorrect, i.e., their entry deviates from their analytical value, as some of the max F S S unoccupied states required for these matrix elements are not contained in the orbital basis set. e third case, Gaux Gorb + G , is the extreme case, where all diagonal elements with max > max F G Gorb +G are zero, as none of the required unoccupied states is accessible by the orbital > max F basis set. In conclusion, this simple model shows that auxiliary and orbital basis cannot be S S chosen arbitrarily for a correct representation of the density response function. It conrms our observation that for a stable and physical EXX potential the orbital LAPW basis set must be converged with respect to the auxiliary basis MPB.

À¥ â.¥. Comparison of EXX and LDA exchange potential

6.4. Comparison of EXX and LDA exchange potential

So far, the local EXX potential has only been shown on a line between two neighboring car- bon atoms. An overall view of the potential on the ýÔÔ plane of diamond is provided in Fig. â.Þ. e ýÔÔ plane contains the ÔÔÔ and Ôýý directions that correspond to Fig. â.ò. ( ) e position of the plane in the cubic unit cell is displayed in Fig. â.ç. ( ) [ ] [ ] Figure â.Þ(a) shows that the EXX potential around the atomic nuclei is predominantly spherical. Towards the MT sphere boundary, however, the potential becomes strongly anisotropic. In particular, the C-C bond axes stand out. Furthermore, Fig. â.Þ(a) demon- strates that the MT potential goes smoothly over to the corrugated interstitial potential at the MT sphere boundary. In comparison with the LDA exchange potential [s. Fig. â.Þ(b)], the EXX potential exhibits much more structure and is considerably more anisotropic in the MT spheres. e LDA exchange potential is exact for the homogeneous electron gas and scales with the electron LDA Ô ç density Vx r ∝ n r ~ , which explains its similarity to the electron density distribution (cf. Fig. â.—). However, diamond is a covalently bonded material and, hence, substantially ( ) ( ) deviates from the homogeneous electron gas. e dišerence between LDA and EXX poten- tial, thus, arises from thefact that in thelatter thefull non-locality of the EXX functional and the full inhomogeneity of the system is taken into account by means of the KS wave func- tions. In particular in the MT spheres, where the KS wave functions are strongly oscillating, the EXX potential features, in contrast to the LDA exchange potential, strong non-sphericity. In the interstitial region, on the other hand, both potentials are similar. is reveals that a full-potential treatment, incorporating the non-spherical contributions of the potential, is considerably more important in the case of the EXX-OEP formalism than in the LDA. Of course, the dišerent shapes of LDA and EXX potentials inžuence the electron density. In Fig. â.—(a) the electron density distribution corresponding to the EXX potential is shown as a color-coded contour plot on the ýÔÔ plane. As naturally expected, the electrons aggregate at the atomic nuclei and along the bond axes. Despite the obvious changes between the LDA ( ) and EXX potential the dišerences between the densities become only clearly visible in a log- arithmic plot of the density dišerence nEXX r − nLDA r [s. Fig. â.—(b)]. e black regions are areas on the plane, where EXX and LDA density coincide (at least up to Ôý−¥ a−ç). e S ( ) ( )S ý regions, where charge has been accumulated in comparison to the LDA density, are marked with a plus sign. e EXX functional tends to accumulate charge at the atomic nuclei and along the bond axes. e LDA, on the contrary, tends to distribute the density more homo- geneously, which we attribute to the self-interaction error, which is inherent in the LDA but completely eliminated in the EXX approach.

À â. Exact exchange within the optimized ešective potential method

(a)

(b)

Figure â.Þ.: Visualization of (a) the EXX potential and (b) the LDA exchange potential on the ç ýÔÔ plane of diamond. Both potentials are gauged such that ∫ Vx r d r ý = holds. e contour lines which lie in the interval between −ý.çý and ý.çýhtr with( ) a spacing of ý.ý htr are shown. e dotted line corresponds to(V) ýhtr. x = Moreover, the MT sphere boundaries are shown.

Àâ â.¥. Comparison of EXX and LDA exchange potential

(a)

(b)

Figure â.—.: (a) EXX total electron density on the ýÔÔ plane of diamond with equidistant −ç contour lines between ý.ýç, ý.ýâ,..., ý.çÀaý . (b) Logarithmic plot of the den- sity dišerence nEXX r −nLDA r . (e area,) where charge is accumulated com- pared with the LDA nEXX r − nLDA r ý , is marked with a plus sign. S ( ) ( )S > [ ( ) ( ) ]

ÀÞ â. Exact exchange within the optimized ešective potential method

6.5. EXX KS transition energies of prototype semiconductors and insulators

Before we will discuss the KS band gap of the prototype semiconductors and insulators C, Si, SiC, Ge, GaAs, crystalline Ne and Ar within the EXX approach and compare our all-electron results with pseudopotential plane-wave results from the literature, we demonstrate the ešect of an unbalanced LAPW and mixed product basis on the KS transition energies for the case of diamond. In Table â.ò the self-consistent KS transition energies from the highest occupied KS state at the Γ point to the lowest conduction band state at the Γ,L, and X points are shown for dišerent

LAPW basis sets. e basis sets have in common the cutoš values lmax â and Gmax −Ô = = ¥.òaý , but dišer in the number of local orbitals nLO. e MPB parameters are Lmax ¥ and ′ − = G ç.¥a Ô. While in the interstitial region balance between both basis sets is given due to max = ý the choice of the reciprocal cutoš radii, balance in the MT spheres is gradually achieved by consecutively increasing the number of local orbitals added for each l from l ý, . . . , and = m l. e addition of more and more local orbitals leads to a decrease in the KS transition ≤ energies. Between an unbalanced and a balanced setup the values change by about ý.ò eV. For S S an accuracy of ý.ýÔeVin the transition energies three local orbitals per lm channel, resulting in òÔâ additional basis functions, are required. We note that the transition energies depend similarly on the balance of the IR basis sets. In the following, we show KS transition energies for the above dened set of prototype semiconductors and insulators employing the EXX and the EXXc functional. For the lat- ter, correlation on the level of the LDA is added to the EXX functional. We here use the parametrization of the LDA correlation functional by Vosko, Wilk, and Nusair [ ]. All calcu- lations are performed at the experimental lattice constants and with an —×—×— k-point sam- pling. In particular, we make sure that the LAPW and mixed product basis sets are balanced. e LDA transition energies (rst column in Table â.ç) systematically underestimate the ex- perimental results. e EXX as well as the EXXc approach lead to KS transition energies that are much closer to experiment. e consideration of correlation on the LDA level changes the transition energies only slightly, typically by ý.Ô − ý.ò eV. For the semiconductors Si, SiC, Ge, and GaAs the results are even quantitatively in reasonable agreement with experiment. For the insulators we observe larger deviations. A comparison of the all-electron FLAPW values with pseudopotential plane-wave results from the literature shows a very good agreement for all materials except Ne. For example, for diamond we obtain a direct band gap (Γ → Γ) of â.òÔeV with the EXX functional. En- gel et al. recently reported a direct band gap of â.ԗ eV using a pseudopotential plane-wave

À— â. . EXX KS transition energies of prototype semiconductors and insulators

nLO Γ → Γ Γ → L Γ → X ý â.ç Ô À.ò¥ç .çýÞ Ô â.ÔÀâ À.ý—â .Ôò ò â.ԗâ À.ýâÀ .Ô¥¥ ç â.ԗý À.ýâç .Ôç— ¥ â.Ôޗ À.ý À .ÔçÀ â.ÔÞÞ À.ý Þ .Ôçâ â â.ÔÞâ À.ý .Ôçâ

Table â.ò.: Convergence of KS EXX transition energies (in eV) for diamond employing LAPW basis sets including zero to six local orbitals per lm channel (l ý,..., , = m l). ≤ S S

approach but pushing the pseudopotential to its all-electron limit [¥ý]. Our result is identi- cal to a standard pseudopotential plane-wave approach, which employs a Troullier-Martins pseudopotential, that has been constructed from atomic EXX calculations [Ôçò]. However, the rst all-electron, full-potential implementation of the EXX functional by Sharma et al. [çÀ] – realized within the FLAPW approach, as well – gives a much larger value of â.âÞ eV for the direct band gap for diamond. In general, their reported EXX KS transition energies deviate substantially from the published pseudopotential values and are further in considerably worse agreement with experiment. From this Sharma et al. draw the conclusion that (a) the success of pseudopotential EXX calculations in predicting accurate transition energies and band gaps is only an artifact of the pseudopotential approximation, which neglects by construction the core-valence exchange interaction; (b) a treatment of core and valence electrons on an equal footing is mandatory for a proper EXX calculation. We, however, obtain EXX KS transition energies, which are in excellent agreement with the pseudopotential and experimental results provided that LAPW and mixed product basis are balanced. As Sharma et al. [çÀ] reported that their calculations are converged up to ý.ýÔ eV with only ò empty states, we speculate that they are far away from a balance between LAPW and auxiliary basis.

In conclusion, we nd a very good agreement between our all-electron FLAPW and pseu- dopotential plane-wave EXX results provided that the basis sets are properly balanced. is conrms that the pseudopotential approximation is adequate for the EXX-OEP approach, at least for the systems examined here.

ÀÀ â. Exact exchange within the optimized ešective potential method

is work Plane-wave PP LDA EXX EXXc EXX EXXc Expt. C Γ → Γ . â â.òÔ â.òâ â.ÔÀa,â.òÔb â.ò—c Þ.çe Γ → L —.¥ç À.ýÀ À.Ôâ À.Ô b À.ԗc – Γ → X ¥.ÞÔ .òý .çç .ç¥b .¥çc –

Si Γ → Γ ò. ç ç.Ôç ç.òÔ ç.Ôòb ç.òâc ç.¥e Γ → L Ô.¥ò ò.òÔ ò.ò— ò.òÔb ò.ç c ò.¥e Γ → X ý.âÔ Ô.çý Ô.¥¥ Ô.ò b Ô. ýc –

SiC Γ → Γ â.òÞ Þ.ԗ Þ.ò¥ – Þ.çÞc – Γ → L .ç— â.Ô¥ â.òÔ – â.çýc – Γ → X Ô.çò ò.òÀ ò.¥¥ – ò. òc ò.¥òe

Ge Γ → Γ −ý.Ô¥ Ô.ò¥ Ô.òÔ – Ô.ò—c Ô.ýe Γ → L ý.ýâ ý.—À ý.À¥ – Ô.ýÔc ý.Þe Γ → X ý.ââ Ô.Ô Ô.ò— – Ô.ç¥c Ô.çe

GaAs Γ → Γ ý.òÀ Ô.Þò Ô.Þ¥ – Ô.—òc Ô.âçe Γ → L ý.— Ô.ÞÀ Ô.—â – Ô.Àçc – Γ → X Ô.ç Ô.À ò.Ôò – ò.Ô c ò.ԗe

Ne Γ → Γ ÔÔ.¥ç Ô¥.ÞÀ Ô .¥â Ô¥.Ô d Ô¥.Þâd òÔ. Ô f Γ → L Ôâ.ÀÞ òý.¥À òÔ.Ôâ – – – Γ → X ԗ.òÞ òÔ.— òò. â – – –

Ar Γ → Γ —.ÔÀ À.â Ôý.ýÀ À.âÔd À.À d Ô¥.Ô f Γ → L ÔÔ.ýâ Ôò.òò Ôò.âý – – – Γ → X Ôý.—â Ôò.ý— Ôò.¥À – – –

aReference ¥ý bReference Ôçò cReference çÞ dReference Ôçç eReference ÀÔ f Reference À¥

Table â.ç.: KS transition energies (in eV) obtained with the local EXX and EXXc potentials and an —×—×— k-point sampling. For comparison, plane-wave PP results and ex- perimental values from the literature are given.

Ôýý â.â. EXX approach and the band gap problem of DFT: the III-V nitrides

6.6. EXX approach and the band gap problem of DFT: the III-V nitrides

In the previous section, we already compared KS eigenvalue dišerences with experimental transition energies. We will focus in the following on a particular transition of the KS system, namely the KS band gap, and its relation to the fundamental band gap of the true interacting system. e fundamentalband gap is denedin terms of thetotal energyof the N +Ô, N, and N − Ô electron systems

E E N + Ô + E N − Ô − òE N , (â.ò ) gap = whereas the KS gap is given by the eigenvalue( ) dišerence( ) of the( lo) west unoccupied and high- est occupied KS state. e denition of the fundamental band gap (â.ò ) in terms of total energy dišerences reveals that it is in principle exactly accessible in ground-state DFT. For an innite periodic system, however, the addition of a single electron to the system – instead of one electron per unit cell – is not practicable. However, Perdew et al. [Ôç¥] showed by a generalization of DFT to non-integer particle numbers, that the exact total energy exhibits a linear behavior as a function of the particle number between two integer particle numbers and a derivative discontinuity at N [s. Fig. (â.À)]. e slope of the line connecting the total energy of the N − Ô and N electron system equals the ionization energy −I

−I E N − E N − Ô , (â.òâ) = ( ) ( ) whereasthe slope between the N and N+Ô electron system corresponds to the electron a›nity −A −A E N + Ô − E N . (â.òÞ) = e fundamental gap in Eq. (â.ò ) can thus( be written) ( as the) dišerence of total-energy deriva- tives from the right and leŸ evaluated at N and at the ground-state density ný

δE δE Egap −A + I − . (â.ò—) = = δN N→N+Ô,ný δN N→N−Ô,ný Œ V V ‘ e additional constraint that the derivatives have to be taken at the ground-state density ný is equivalent to the formulation that the addition or removal of the electron takes place at

xed ešective KS potential and thus xed KS orbitals because the density ný determines the KS potential (up to a constant). Hence, the change in the particle number at xed density ný is only associated with the occupation of a previously unoccupied KS level or with the

ÔýÔ â. Exact exchange within the optimized ešective potential method

E(N-1)

I

E(N) -A E(N+1)

N-1 N N+1

Figure â.À.: e linear behavior of the total energy for fractional particle numbers between two subsequent integers is shown and the derivative discontinuity (kink) at N becomes visible.

emptying of a previously occupied state. With this in mind, the fundamental gap can be KS nally written as the KS band gap Egap plus a derivative discontinuity ∆xc of the xc energy functional [âÀ, Ôç , Ôçâ]

E EKS + ∆ , (â.òÀ) gap = gap xc where the latter is given by

δExc δExc ∆xc − ϕN+Ô Vxc ϕN+Ô − − ϕN−Ô Vxc ϕN−Ô . (â.çý) = δN N→N+Ô,ný δN N→N−Ô,ný Œ V ⟨ S S ⟩‘ Œ V ⟨ S S ⟩‘ Here, ϕN and ϕN+Ô denote the highest occupied and lowest unoccupied KS orbitals and Vxc is the local xc potential. In the case of the LDA or GGA, this discontinuity ∆xc is exactly zero. According to Eq. (â.òÀ), the LDA and GGA KS band gap should then coincide with the true fundamental band gap. Yet, it is well known that both approximations underestimate the experimental gap by about ¥ý% or even more, which is oŸen called the band-gap problem of LDA and GGA [Ôý].

For the EXX-OEP functional the discontinuity ∆xc does not vanish. Instead it is given

Ôýò â.â. EXX approach and the band gap problem of DFT: the III-V nitrides by [òç, Ôçâ]

EXX NL NL ∆ ϕ + V ϕ + − ϕ + V ϕ + − ϕ − V ϕ − − ϕ − V ϕ − . x = N Ô x N Ô N Ô x N Ô N Ô x N Ô N Ô x N Ô (â.çÔ) ‰⟨ S S ⟩ ⟨ S S ⟩Ž ‰⟨ S S ⟩ ⟨ S S ⟩Ž In the following, we demonstrate for the III-V nitrides in their zincblende structure and rock- EXX salt ScN that the discontinuity ∆x results in a substantial, positive contribution to the fun- damental band gap, with the consequence that the sum of the KS band gap and the disconti- nuity [Eq. (â.òÀ)] strongly overestimates the experimental gap. Table â.¥ shows that the LDA seriously underestimates the experimental gap. For InN and ScN it even predicts a metal. e EXX functional, instead, gives KS band gaps, which are closer to the experimental value. In particular, InN and ScN are correctly predicted to be EXX semiconductors. But the discontinuity ∆x amounts to several eV for these materials so that the fundamental gap [Eq. (â.òÀ)] overshoots the experimental value quite drastically. e fundamental gap in the HF and the EXX approach, on the contrary, are very similar. As the discontinuity in Eq. (â.çÔ) is formally equivalent to the linear change of the KS band gap if one replaces the local EXX potential by the non-local HF-like potential, the dišerence between EXX and HF gap merely arises from the additional self-consistency in theHF gap. e simi- larity of EXX and HF band gap can be explained alternatively by the observation that HF and EXX total energies are nearly identical. For example, we found for diamond that the HF total energyismerelyý.ò¥eVsmallerthanthetotalenergyintheEXX approach. Due to the deni- tion of the fundamental gap in terms of total-energy dišerences [Eq. (â.ò )] and the similarity of the HF and EXX total energies, it is obvious that the EXX and HF gaps must be similar, too. Yet, the question arises why does the EXX KS band gap alone give a quantitatively good estimate for the experimental band gap? e EXX method completely ignores correlation. e usage of LDA correlation in combination with EXX leads to a slight increase of the KS band gap but does not contribute to the derivative discontinuity except indirectly through a change in the KS wave function. Consequently, the EXXc approach leads to an overestimation of the experimental fundamental gap, as well. Instead, a correlation functional, that is com- patible with the orbital-dependent EXX functional, is required. Grüning et al. [Ôò] combined the EXX-OEP approach with a correlation functional in the random-phase approximation (RPA) and showed that the ešect of an orbital-dependent correlation functional is twofold: on the one hand, the KS band gap is decreased in comparison to that in the EXX approach. It becomes comparable with the LDA KS band gap again. On the other hand, the discontinu- ity arising from the EXX and the RPA correlation functional shrinks. As a consequence, the combination of both, KS band gap and derivative discontinuity, leads to a fundamental gap that is in excellent agreement with experiment (at least for the systems examined in Ref. Ôò).

Ôýç â. Exact exchange within the optimized ešective potential method

LDA KS,EXX c EXX c EXX c HF Egap Egap ( ) ∆x ( ) Egap ( ) Eg Expt. BN indirect ¥.ç¥ .¥ò . — —.ýò —.ýÀ Ôç.¥¥ Ôç.âÞ Ô¥.ýç â.¥a AlN indirect ç.òò ¥.ÞÞ ¥.Àâ Þ.ý— Þ.ÔÞ ÔÔ.— Ôò.Ôç Ôò.òç .çb GaN direct Ô.Þâ ç.ÔÔ (ç.ò¥ ) â.À— (Þ.ýâ) Ôý.ýÀ (Ôý.çý) Ôý.—Þ ç.òÞc InN direct −ý.¥Ô ý.À—( Ô.Ôò ) .¥¥ ( . ) â.¥ò (â.âÞ ) Þ.¥â ý.âýd ScN indirect −ý.Ô¥ Ô. —( Ô.âÞ) .ÀÀ (â.ýâ) Þ. Þ (Þ.Þç ) —.òÞ Ô.çe ( ) ( ) ( ) ( ) ( ) ( ) aReference ÔçÞ bReference Ôç— cReference ÔçÀ dReference Ô¥ý eReference Ô¥Ô

Table â.¥.: Fundamental band gap for the III-V nitrides in their zincblende structure and rock-salt ScN with the LDA, the EXX, and the EXXc functionals of DFT. Consid- eration of the derivative discontinuity of the EXX(c) functional leads to funda- mental band gaps, which are very close to the corresponding HF values.

So, we conclude that in the EXX approach a fortuitous error cancellation between the neglect EXX of orbital correlation and the neglect of the discontinuity ∆x results in an EXX KS band gap that is quantitatively close to the experimental one.

6.7. Finite basis correction (FBC)

So far, EXX-OEP calculations are computationally demanding because of two reasons: (a) the NL,σ calculation of the matrix elements of the non-local exchange potential Vx , which enter on the right-hand side of Eq. (â.À), is, despite our techniques developed in chapter , an enor- mous computational task for a periodic solid; (b) the demand of a very žexible LAPW basis in order to obtain a physical and stable EXX calculation makes the scheme even more expensive, as a žexible LAPW basis is equivalent to a large number of unoccupied states, which in turn NL,σ means that the matrix elements of Vx have to be evaluated for a large number of unoccu- pied KS states. e high quality of the LAPW basis is required as demonstrated in section â.ç to lend the KS wave functions and thus the density su›cient žexibility to react to changes in the ešective potential. In the following, we derive and examine a numerical correction, the nite basis correction (FBC), for the wave-function and density response function that arises from the observation that the LAPW basis functions themselves depend on the ešective po- tential.

Ôý¥ â.Þ. Finite basis correction (FBC)

6.7.1. Derivation of the FBC

σ e LAPW basis functions φkG r [Eq. (¥.À)] do not depend on the potential in the intersti- tial region. However, the radial functions uaσ r in the MT spheres are the solutions of the ( ) l p spherical Schrödinger equations (¥.ò) and (¥.Ôý) and thus adjusted to the spherical part of the ( ) σ σ σ ešective potential. Hence, the response of the KS wave function ϕnk r G zG n, k φkG r ′ = ∑ to a variation in the ešective potential V σ r consists, in principle, of two contributions eš ( ) ( ) ( ) δϕσ r δzσ n, k( ) δφσ r nk G φσ r + zσ n, k kG , (â.çò) δV σ r′ = Q δV σ r′ kG G δV σ r′ eš ( ) G eš( ) eš ( ) ( ) ( ) ( ) ( ) σ ( ) where the rst describes the response of the eigenvector zG n, k and the second arises from the change in the LAPW basis function. In order to derive a formula for the wave function σ σ ′ ( ) response δϕnk r δVeš r that considers the dependence of the LAPW basis as well as of the eigenvector on the ešective potential, we employ rst-order perturbation theory in the ( )~ ( ) following. σ Let ϕnk r, λ be the solution of the perturbed Hamiltonian H λ

( ) ′ ( ) H λ ϕσ r, λ H + λV σ r ϕσ r, λ єσ λ ϕσ r, λ (â.çç) nk = ý eš nk = nk nk

( ) ( σ′ ) [ ( )]σ ( ) ( ) (σ ) with the perturbation λVeš r and eigenvalue єnk λ . Expansion of ϕnk r, λ and єnk λ in the perturbation strength λ under the consideration that the basis set φσ for the wave ( ) ( ) (kG ) ( ) functions itself depends on the perturbation yields { }

σ σý σÔ σý σÔ ϕnk r, λ zG n, k + λzG n, k + ... φkG r + λφkG r + ... (â.ç¥a) = QG σý
σÔ σý 
σý σÔ ò ( ) ϕnk r (+ λ ) zG n(, k φ)kG r + zG (n,)k φkG r( )+ O λ (â.ç¥b) = QG єσ λ єσý (+ λ)єσÔ + O
λò( . ) ( ) ( ) ( ) ( ) (â.ç¥c) nk = nk nk

We label the unperturbed( ) quantities with( ) an index ý. e higher order terms obtain an index according to their order. e linear change of the wave function consists of two contributions: (a) the part that results from a change in the eigenvector which we abbreviate with ϕσÔ r zσÔ n, k φσý r , nk = ∑G G kG and (b) the change that arises from the linear change of the basis function. e latter is de- ( ) ( ) ( ) noted with ϕ˜σÔ r zσý n, k ϕσÔ r . Let us assume for a while that ϕ˜σÔ r is known. nk = ∑G G kG nk Its calculation will be discussed later. ( ) ( ) ( ) ( ) Insertion of Eqs. (â.ç¥b) and (â.ç¥c) into (â.çç) gives in zeroth order the unperturbed prob-

Ôý â. Exact exchange within the optimized ešective potential method

σÔ lem. e rst order terms in λ yield a equation for ϕnk r

Ô Ô ′ ý ý ( )Ô Ô Ô ý H ϕσ r + ϕ˜σ r + V σ r ϕσ r єσ ϕσ r + ϕ˜σ r + єσ ϕσ r . (â.ç ) ý nk nk eš nk = nk nk nk nk nk

[ ( ) σÔ ( )] ( ) ( ) [ ( ) ( )] ( ) By its denition ϕnk r lies in the Hilbert space spanned by the original LAPW functions φσý r . Hence, a representation of ϕσÔ r in terms of the unperturbed wave functions kG ( ) nk ϕσý r , which span the same Hilbert space as φσý r , is possible without a loss of informa- nk ( ) ( ) kG tion ( ) σÔ σý σ(Ô ) σý ϕnk r ϕn′k ϕnk ϕn′k r . (â.çâ) = Qn′ σý ( )′ ⟨ S ⟩ ( ) By projecting Eq. (â.ç ) onto ϕ ′ n n the expansion coe›cients are obtained n k ≠ ( σ)ý σ′ σý ϕ ′ V ϕ σý σÔ n k eš nk − σý ˜σÔ ϕn′k ϕnk σý σý ϕn′k ϕnk . (â.çÞ) = є − є ′ ⟨ nkS nS k ⟩ ′ ⟨ S ⟩ ⟨ S ⟩ For the case of n n the normalization of ϕσ r, λ requires that = nk

ý Ô ( )ý Ô ϕσ ϕσ − ϕσ ϕ˜σ (â.ç—) nk nk = nk nk holds. In conclusion, the rst-order⟨ changeS ⟩ of the⟨ KSS wave⟩ function becomes

σý σ′ σý σÔ σÔ ϕn′k Veš ϕnk σý ϕ r + ϕ˜ r ϕ ′ r nk nk σý − σý n k = nQ′ n єnk єn′k ≠ ⟨ S S ⟩ ( ) ( ) ( ) ç ′ ′ σý σý∗ ′ ˜σÔ ′ + d r δ r − r − ϕn′k r ϕn′k r ϕnk r . (â.çÀ) ∫ Qn′ Œ ( ) ( ) ( )‘ ( ) e rst term is the usual expression from rst-order perturbation theory, that runs over a - nite number N of available states in a practical calculation. e second term, the FBC, arises from the dependence of the LAPW basis on the ešective potential. It corrects the incom- pleteness of the basis. In the limit of a complete basis corresponding to an innite number of states N → ∞ , the expression in the brackets would vanish and the rst term would give the exact result.ò ( ) σÔ We now turn to the calculation of the linear change of the LAPW basis φkG r ′ due to the perturbing potential V σ , which is required for the calculation of ϕ˜σÔ r eš nk ( =) zσý n, k φσÔ r . According to the denition of the LAPW basis functions, their linear ∑G G kG ( ) ( ) ( ) òBy starting the derivation from the dišerential form of the Schrödinger equation (â.çç) we implicitly assume σý that the zero order wave functions ϕnk(r) are exact pointwise solutions of the Hamiltonian Hý. In practice, this is not the case, as only a nite LAPW basis is employed for representing the wave functions. It will lead to an additional correction, which is however much smaller than the derived FBC.

Ôýâ â.Þ. Finite basis correction (FBC)

aσ response is only non-zero in the MT spheres. While the radial LAPW functions ul p r can in principle respond to spherical and non-spherical variations of the potential, we restrict ( ) ourselves to spherical potential variations. e linear change of the LAPW basis function σ σ′ φkG r due to the spherical perturbation Veš,ý r is given by ( ) ( )

ý if r IR ∈ σÔ aσ aσÔ − φkG r ⎧ lmp Almp k, G ul p r Ra . (â.¥ý) = ⎪ ∑ if r MT a ⎪ + AaσÔ k, G uaσ r − R Y r − R ∈ ( ) ⎨  lmp( ) l p(S aS) lm a ⎪ ( ) ⎪ Æ e index p subsumes⎩⎪ the dišerent( ) radial(S LAPWS) functions( [cf.) Eq. (¥.òý)] and accordingly aσ aσÔ Almp k, G comprises the matching coe›cients Eqs. (¥.ÔÔ) and (¥.Ôò). e quantities ul p r aσÔ and Almp k, G denote the linear change of the radial function and the matching coe›cient, ( ) ′ ( ) respectively, due to the perturbation V σ r . e radial functions uaσÔ r are obtained from ( ) eš,ý l p linearizing Eqs. (¥.ò) and (¥.Ôý). is nally results in the following dišerential equations ( ) ( ) ò Ô ∂ l l + Ô ′ − + + V aσ r − єaσ ruaσÔ r єaσÔ − V aσ r ruaσ r ò ∂rò òrò ý,eš l lý = l eš,ý lý ( ) (â.¥Ô)  ò ( ) ( )
( ) ( ) Ô ∂ l l + Ô ′ − + + V aσ r − єaσ ruaσÔ r єaσÔ − V aσ r ruaσ r + ruaσÔ r . ò ∂rò òrò ý,eš l lÔ = l eš,ý lÔ lý ( )  ( ) ( )
( ) ( ) ( (â.¥ò)) aσÔ aσ′ e variation of the energy parameter єl due to the perturbing potential Veš,ý r is given by the expectation value ′ ( ) єaσÔ uaσ V aσ uaσ . (â.¥ç) l = lý eš,ý lý e scalar-relativistic analogs to Eqs. (â.¥Ô)⟨ andS (â.¥ò)S are⟩ deferred to Appendix B. e radial inhomogeneous dišerential equations can be solved in a similar way as the dišerential equa- tions for uaσ r by integrating from the origin r ý to the MT boundary r Sa. However, the l p = = resulting solutions are not dened uniquely, as we can always add the homogeneous solution ( ) aσ or a multiple of it. But, the normalization of ulý r imposes the additional conditions ( ) ròuaσ r uaσÔ r dr ý (â.¥¥) ∫ lý lý = ( ) ( ) and ròuaσ r uaσÔ r dr − ròuaσ r uaσÔ r dr . (â.¥ ) ∫ lý lÔ = ∫ lÔ lý ese equations nally x the( freedom) ( and) lead to a well dened( ) ( soluti) on. Eq. (â.¥ç) together aσ′ with (â.¥¥) and (â.¥ ) ensure that for a constant potential variation Veš,ý r the linear change ( )

ÔýÞ â. Exact exchange within the optimized ešective potential method

aσÔ ul p r vanishes. e calculation of the linear change in the matching coe›cients AaσÔ k, G is straightfor- ( ) lmp ward aŸer the radial functions uaσÔ r are calculated. Dišerentiation of Eqs. (¥.ÔÔ) and (¥.Ôò) l p ( ) results in ( ) p+Ô ¥π ∗ −Ô aσÔ k, G l k + G exp k + G R Almp i Ylm i a aσ a aσ a = Ω ulÔ S , ulý S + ( ) ( ) √ ( ) [ ( ) ] œ−Ô p Ô × uaσÔ Sa , j k + G Sa − [ ( ) ( )] (â.¥â) l p l uaσ Sa , uaσ Sa ò lÔ ( ) lý [ ( ) (S S )] × aσ a + a aσÔ a aσ a + aσ a aσÔ a ul p S , jl k G S u[lÔ (S ), ulý (S )] ulÔ S , ulý S , [ ( ) (S S )] ‰[ ( ) ( )] [ ( ) ( )]Ž ¡ where j r is the spherical Bessel function, p Ô − p, and the square brackets denote the l = Wronskian as dened in Eq. (¥.Ôç). ( ) σÔ Finally, all constituents for the construction of φkG r [Eq. (â.¥ý)] are at hand. e resulting functions φσÔ r and their radial derivatives go continuously to zero at the MT sphere bound- kG ( ) ary. A convolution of φσÔ r with the wave function coe›cients zσý n, k nally yields the ( ) kG G linear change in the wave function ϕ˜σÔ r that results from the change in the LAPW basis ( ) nk ( ) functions. So, all quantities occurring in Eq. (â.çÀ) are known and the full response, com- ( ) prising the change in the eigenvector and in the LAPW basis function, can be calculated. Our derivation, so far, only concerns the augmented plane waves Eq. (¥.À). But the con- siderations can easily be generalized to include the local orbitals. In contrast to that, the core-state response requires a dišerent treatment. e core states are bound so tightly that they are conned to the MT spheres. is enables to describe them atomic-like, i.e., they are obtained by solving a fully-relativistic Dirac equation with the spher- ical average of the ešective potential under the boundary condition that the radial eigenfunc- tions vanish in the limit r → ∞. Under this condition, the radial Dirac equation only features solutions at certain discrete energies, the core energies. Instead of solving an inhomogeneous Dirac equation for the response of the radial core wave function, in analogy to Eqs. (â.¥Ô) aσ + λ aσ′ and (â.¥ò), we simply use a nite dišerence approach. For the potentials Veš,ý r ò Veš,ý r ′ and V aσ r − λ V aσ r we solve the Dirac equation with the conventional core solver, then eš,ý ò eš,ý ( ) ( ) take the dišerence between the two core wave functions and divide by λ. e nite dišerence ( ) ( ) aσ′ approximates the response of the core wave function due to the perturbing potential Veš,ý r . e accuracy of the nite-dišerence approach can be controlled by the parameter λ. Typi- ( ) cally λ ý.ýýýÔ is a good choice. In a similar way, we can construct the linear response of the = core eigenenergies. A comparison of the change in the core eigenvalue either calculated with the nite dif-

Ôý— â.Þ. Finite basis correction (FBC)

aÔ a a′ a a n l єnl ϕnl Veš,ý ϕnl Sc Ô s -Ô.— Þ â⟨ -Ô.— Þ —S S ⟩ ò s -ý.â ò¥çÀ -ý.â ò¥ç— ò p -ý.âÞý¥Àò -ý.âÞýò—

N Ô s -ý.âÔ¥—ÀÀ -ý.âÔ¥—ÀÀ

a′ Table â. .: Comparison of the core-energy response due to a perturbation Veš,ý for the aÔ case of ScN obtained, on the one hand, with the nite dišerence approach єnl (as described in the text) and, on the other hand, by the analytical formula a a′ a ϕnl Veš,ý ϕnl . ⟨ S S ⟩ aσ aσ′ aσ ference approach or by the analytical formula ϕnl Veš,ý ϕnl reveals the accuracy of this approach [s. Table (â. )]. Here, ϕaσ denotes a core state of atom a, principal quantum num- nl ⟨ S S ⟩ ber n, angular momentum l, and spin σ. It is obtained by averaging over the solutions of the Dirac equation with total angular momentum j l + Ô ò and j l − Ô ò, which corresponds = = to a non-relativistic description of the core states. ~ ~ e consideration of the FBC for the KS wave-function response leads to additional terms in the density response matrix (â.Ôâ)

occ. unocc. σ σ σ σ MI ϕ ϕ ′ ϕ ′ ϕ MJ χσ ò nk n k n k nk s,IJ σ − σ = Qnk nQ′ N єnk єn′k ≤ ⟨ S ⟩⟨ S ⟩ occ. σ ˜σ σ σ σ ˜σ +ò MI ϕnk ϕnkJ − MI ϕnk ϕn′k ϕn′k ϕnkJ (â.¥Þ) Qnk nQ′ N ≤ ⟨ S ⟩ ⟨ S ⟩⟨ S ⟩ and the vector of the right-hand side (â.ÔÞ)

occ. unocc. σ σ σ σ NL,σ σ MI ϕn′k ϕnk t ò ϕ V ϕ ′ I nk x n k σ − σ = Qnk nQ′ N єnk єn′k ≤ ⟨ S ⟩ occ. ⟨ S S ⟩ σ NL,σ ˜σ σ NL,σ σ σ ˜σ +ò ϕnk Vx ϕnkI − ϕnk Vx ϕn′k ϕn′k ϕnkI . (â.¥—) Qnk nQ′ N ≤ ⟨ S S ⟩ ⟨ S S ⟩⟨ S ⟩ σ′ Since in our formalism the linear change of the ešective potential Veš r is represented by the MPB, the additional index J in ϕ˜σ r stands for the MPB function M r . We note that nkJ ( )J the functions ϕ˜σ r are non-zero only for spherical MT functions M r . nkJ ( ) J ( ) In its present form, the expression in Eq. (â.¥Þ) breaks the Hermiticity of σ because ( ) ( ) χs,IJ σ the additional term is slightly asymmetric in the indices I and J. We make χs,IJ explicitly

ÔýÀ â. Exact exchange within the optimized ešective potential method

σ σ∗ Hermitian by averaging over χs,IJ and χs,JI .

6.7.2. Effect of the FBC for the example of ScN

In the following, the ešect of the FBC on the convergence and stability of the EXX potential is demonstrated for the case of ScN. Before, however, we analyze the improved convergence σ of the single-particle response function χs due to the FBC. We will refer to the rst term of Eq. (â.¥Þ) as the sum-over-states (SOS) expression and to the second one as the FBC. For our analysis we split up Eq. (â.¥Þ) into a core and valence electron response part χσ χσv + χσc , (â.¥À) s,IJ = s,IJ s,IJ where the sum over the occupied states in Eq. (â.¥Þ) is either restricted to the valence v or core c states. e contribution of the core electrons to the density response function ( ) will be discussed later on. Figure â.Ôý shows the trace of the matrix χσv as a function of ( ) s,IJ the number of local orbitals added to the LAPW basis. nLO denotes the number of local orbitals per lm channel and atom for l ý, . . . , ¥ and m l. Consequently, nLO ý cor- = ≤ = − responds to the conventional LAPW basis for ScN with a reciprocal cutoš of G ç.—a Ô S S max = ý and l —. e çs and çp semi-core states of Sc are treated explicitly as valence electrons max = and a ¥×¥×¥ k-point sampling has been employed. As the FBC is only non-zero for those σ diagonal elements χs,II , which correspond to spherical MT MPB functions, we restrict the trace tr χs χs,II to these functions. Figure â.Ôý(a) demonstrates the slow convergence = ∑I of the SOS expression (green dashed curve). It režects the importance of adding local orbitals ( ) to describe the valence-density response adequately. e curve corresponding to the SOS ex- pression is monotonically decreasing. In contrast, the trace of the FBC (dotted blue curve) is monotonically increasing. While the LAPW basis becomes more and more complete towards n â the FBC becomes smaller and smaller. e sum of the two curves (red solid line) is LO = nearly constant. e inset of Fig. â.Ôý(a) shows that the sum hardly shows any convergence at all. e convergence, that is achieved by the SOS expression alone with n â, is obtained LO = in the sum of SOS and FBC with n ý. Taking into account a single set of local orbitals LO = n Ô results in an accuracy, which would be di›cult to achieve with the SOS expression LO = alone. ( ) So far, we have demonstrated that the combination of SOS expression and FBC leads to a converged spherical response function at much smaller LAPW basis sets. e question arises if the FBC allows to reduce the number of bands N that enter in the SOS and the FBC term in Eq. (â.¥Þ), as well. Indeed, the number of unoccupied bands can be chosen much smaller as shown in Fig. â.Ôý(b). For these calculations we have employed six local

ÔÔý â.Þ. Finite basis correction (FBC)

0 ScN -0.1 -0.2 -0.9

-0.3 -0.905 -0.4 -0.91 ) s v

χ -0.5

tr( -0.915 -0.6 -0.7 -0.92 0 1 2 3 4 5 6 -0.8 -0.9 -1 0 1 2 3 4 5 6

nLO (a)

0 ScN -0.1 -0.2 -0.88

-0.3 -0.89 -0.4 -0.9 ) s v -0.91 χ -0.5

tr( -0.92

-0.6 -0.93 -0.7 0 50 100 150 200 250 300 350 400 450 -0.8 -0.9 -1 0 50 100 150 200 250 300 350 400 450 Number of unoccupied states (b)

v Figure â.Ôý.: Convergence of tr χs for ScN as a function of (a) the number of local orbitals n per l quantum number ý l ¥ and atom and (b) the number of un- LO ≤ ≤ occupied states n( ) â . We only consider spherical MPB functions, and the LO = contribution of the core states( is neglected.) Dashed (green) and dotted (blue) lines correspond( to the SOS) and the FBC contributions, respectively. e solid (red) lines show their sum, which converges very rapidly in the two cases.

ÔÔÔ â. Exact exchange within the optimized ešective potential method

− orbitals (n â) and increased the reciprocal cutoš of the LAPW basis to G . a Ô. LO = max = ý Similar to Fig. â.Ôý(a) the separate contributions, SOS expression (green dashes curve) and FBC (blue dotted curve), show a slow convergence with respect to the number of unoccupied bands. eir sum, however, converges very rapidly. In fact, already òý unoccupied states are su›cient to reproduce or even improve the accuracy that is achieved with the SOS alone using ¥ ý unoccupied states. e inset of Fig. â.Ôý(b) shows that with about ý states the trace is converged to a precision of Ôh. From this convergence behavior we can conclude ˜σ that the function ϕnk r obtained non-perturbatively contains nearly the whole information about the innite spectrum of unoccupied states. e second term in the FBC merely acts as ( ) a double counting correction for the SOS expression. It removes those parts of the function ˜σ ϕnk r that are already included in the Hilbert space spanned by the KS wave functions up to band N and thus contained in the SOS expression. ( ) Until now, we have restricted ourselves to the valence-electron contribution to the response function. We will discuss the core-electron part in the following. e FBC for the core elec- trons – as already discussed in the previous section – is calculated by a nite-dišerence ap- ˜σ proach. e function ϕnk exhibits in the case of a core state only a formal k dependence and ˜σ n subsumes the principal quantum number n, the angular momentum l etc.. ϕnk already σc contains the exact response of the core state. e sum over states N in χs,IJ can therefore be restricted only to the occupied states so that the SOS expression does not contribute at all and the double counting term in the FBC only projects out the occupied states. is is conrmed in Fig. â.ÔÔ. With zero unoccupied states already the converged full response is obtained. In- creasing the number of unoccupied states transfers a part of the core response to the SOS and accordingly less is accounted by the FBC. Yet, with ¥ ý bands n â the SOS part covers LO = only about òý% of the complete response, which manifests that the LAPW (valence) basis is ( ) incapable of describing the response of the core states. In the inset of Fig. â.ÔÔ we observe variations of the total core response in the order of a few per mill. ese variations might be caused, on the one hand, by the spherical-potential approximation for the core states and by the scalar-relativistic approximation for the valence and conduction band states, on the other hand. In comparison with the valence-electron response, the core part accounts for about —ý% of the response function for the example of ScN. However, the inžuence of the core electrons on the EXX potential is rather small, as the relevant quantity for the EXX potential is not the σ response function, but its inverse in which the large eigenvalues of χs become comparatively unimportant. We will discuss this point later on. In Fig. â.Ôò, the local EXX potential is shown between neighboring Sc and N atoms in the [Ôýý] direction for three dišerent calculations. None of them employs any local orbitals for

ÔÔò â.Þ. Finite basis correction (FBC)

0 ScN -0.5

-1

-4.003 -1.5

) -2 c s -4.004 χ

tr( -2.5 -4.005

-3 0 50 100 150 200 250 300 350 400 450 -3.5

-4

-4.5 0 50 100 150 200 250 300 350 400 450 Number of unoccupied states

c Figure â.ÔÔ.: Convergence of tr χs as a function of the number of unoccupied bands. Dashed green and blue dotted line correspond to the SOS and FBC contribu- tion. e red solid line( ) is their sum. the unoccupied states n ý . For the rst calculation (green dashed line), the FBC is LO = switched oš. e EXX potential then is oscillatory and even tends to an unphysical positive ( ) value at the N atom. is is the situation, where both basis sets are out of balance. As discussed in section â.ç, this can be cured by the addition of local orbitals. But the same ešect can be achieved by employing the FBC (blue dotted curve). We stress again that no local orbitals are used in this calculation. At closer inspection, in the direct vicinity of the N nucleus, one still observes slight anoma- lies of the blue dotted curve. From our experience, the region close to the atomic nuclei is di›cult to converge, which results from the fact that the radial MT potential always enters with an integration weight rò into the EXX-OEP equation. However, the total ešective poten- tial in this region is dominated by the potential of the nucleus −Z r so that slight inaccuracies in the EXX potential are of no or little consequence for the electronic-structure calculation. ~ Yet, we can simply improve the EXX potential and its stability around the atomic nuclei by de- manding that the gradient of the potential vanishes at the position of the atomic nuclei. is is not an ad-hoc postulate. It is motivated by the observation that the local EXX potential (always) shows this behavior (cf. EXX potentials in Refs. Ô¥ò–Ô¥¥ or Fig. â.Ô). e condition of vanishing gradient at the atomic position can easily be incorporated into the MPB by the approach discussed in Appendix A. As only the spherical MT MPB functions are non-zero at the atomic nuclei, it is su›cient to demand a zero gradient only for the spherical MT func-

ÔÔç â. Exact exchange within the optimized ešective potential method

0 ScN

-4 0 0 Sc N

-4 -8 -2 (htr)

x -8 V

-4 -12 -12

-16 -6 -16 0.0001 0.01 1 1 0.01 0.0001

0 0.5 1 1.5 2 2.5 3 3.5 4

r (a0)

Figure â.Ôò.: Local EXX potential for ScN on a line along the [Ôýý] direction connecting neighboring Sc r ý and N r ¥.ò a atoms. We have used a conven- = = ý tional LAPW basis without local orbitals for the unoccupied states. e blue dotted and green( dashed) curves( correspond) to calculations with and without the FBC. For the red solid curve, we have employed an additional constraint for the MPB (see text). Eq. (â.â) denes the potential only up to a constant. Here, ç we use the convention ∫ Vx r d r ý. = ( ) tions at the atomic nuclei. is modication of the MPB nally causes that the irregularities in the EXX potential around the N atom vanish as demonstrated by the red solid curve in Fig. â.Ôò. e FBC only applies for spherical potential variations. So we still need a few local orbitals to fully converge the EXX potential, in particular the non-spherical and interstitial parts of the potential. For the example of ScN, the ešect of these additional local orbitals on the potential is most pronounced around the Sc atom as shown in Fig. â.Ôç(a). e potential with one set of local orbitals (n Ô) is opposed to the potential without any. e slight LO = change in the potential inžuences the KS transition energies (Γ → Γ, Γ → X, Γ → L) in the order of ý.Ôý − ý.Ô eV. e changes in the potential, which emerge from a second set of local orbitals, are not visible on the scale of Fig. â.Ôç(a) and result in a change less than ý.ýç eV in the transitions. We note that the calculations with or without the FBC converge to the same result. Finally, we illustrate the ešect of the exact core response on the local EXX potential. e FBC for the core electrons is switched oš and the resulting potential is compared with the one

ÔÔ¥ â.Þ. Finite basis correction (FBC)

0 Sc 0 Sc -2 -2 -4 -4 -6 -6 -8 -8 (htr) (htr) x x

V -10 V -10 -12 -12 -14 -14 -16 -16 -18 -18

0.0001 0.01 1 0.0001 0.01 1

r (a0) r (a0) (a) (b)

Figure â.Ôç.: (a) Comparison of Vx r around the atomic nucleus of Sc for calculations with (green dashed line, n Ô) and without (red solid line, n ý) local orbitals. LO = LO = (b) Local exact exchange( ) potential Vx r with (red solid line) and without (green dashed line) the explicit core response of the FBC. ( )

ÔÔ â. Exact exchange within the optimized ešective potential method obtained with the full core response [s. Fig. â.Ôç(b)]. Despite the large contribution of the FBC to the core response (cf. Fig. â.ÔÔ), it has a rather small inžuence on the shape of the potential because the inverse response function, instead of the response function, determines Vx r . Interestingly, the changes in the potential are spatially restricted to the region close to the ( ) Sc nuclei, which is not self-evident due to the non-locality of the EXX-OEP equation. ese changes due to the dišerent treatment of the core states ašect the KS transition energies by less than ý.ý¥eV.

6.8. Perovskites: CaTiO3, SrTiO3, BaTiO3

e perovskites form a large class of materials with various physical properties, for example, colossal magneto-resistance, ferroelectricity, charge ordering etc.. eir unifying property is their chemical composition ABOç, where A and B are typically metals and O is oxygen. e metals A and B behave cationic and transfer electrons to the O atoms. In the ideal perovskite crystal, the A cations sit at the corners, while the B cation is located in the center of the cubic unit cell. e oxygen anions are placed at the centers of the cube faces (s. Fig. â.Ô¥). us, the B cation is surrounded by an octahedron of â oxygen atoms, while the A atom has Ôò oxygen neighbors. e perovskite SrTiOç exhibits this ideal cubic structure at room temperature. However, many perovskites dišer from this cubic arrangement, for example by a displacement of the B cation within the oxygen octahedron or by tilting of the octahedron.

e former distortion, lowering the symmetry from cubic to tetragonal, is typical for BaTiOç below Ôòý ○C. Due to the displacement of the Ti atom towards one of the oxygen atoms, an electric dipole moment is formed that makes BaTiOç ferroelectric. A tilting of the complete oxygen octahedron is observable for CaTiOç. In the following, we investigate the electronic structure of the transition metal (TM) per- ovskites ATiOç, where A stands for Ca, Sr, and Ba. To our knowledge, these are the rst rst-principles calculations of these perovskites employing the EXX functional. Deviations from the ideal cubic structure, which are observed experimentally for CaTiOç and BaTiOç, are neglected in the calculations. For each material, we employ its experimental lattice con- stant (CaTiOç: Þ.ç aý; SrTiOç: Þ.¥âaý; BaTiOç: Þ.âýaý). e çs and çp semi-core states of the transition metal Ti as well as the rst subshell of the A cations (Ca: çsçp; Sr: ¥s¥p; Ba: s p) are treated as valence states. e LAPW and MPB cutoš parameters are summarized in Table â.â and a â×â×â k-point mesh has been used for the calculations. LAPW and MPB are carefully balanced to obtain converged and stable EXX results. Figures â.Ô , â.Ôâ, and â.ÔÞ show the KS band structure and density of states (DOS) for all three perovskites obtained with the LDA and the EXX functionals. e contribution of the

ÔÔâ â.—. Perovskites: CaTiOç, SrTiOç, BaTiOç

Figure â.Ô¥.: Crystal structure of the ideal cubic perovskite ABOç.

O òp and Ti çd states to the total DOS (black line) are drawn in red and green, respectively. Moreover, the partial DOS of the A cation nd-states (Ca: çd; Sr: ¥d; Ba: d) is shown as a blue line. e bands forming the highest valence bands are mainly of O òp character and ac- commodate ԗ electrons. e band width of these states reduces over this series of elements. e lowest conduction band states show Ti çd behavior. e crystal eld generated by the oxygen octahedron splits the ve Ti çd states into three states of tòg symmetry (labelled aŸer their irreducible representation) and two states of eg symmetry. e three lowest conduction bands have tòg character and are energetically separated from the two bands with eg charac- ter. e bands corresponding to the A-cation d states lie several eV above the Fermi energy. Consequently, independently from the A cation, the highest occupied valence and lowest un- occupied conduction bands exhibit O òp and Ti çd character. With the consequence that the shape of the band structure around the Fermi energy is nearly identical for all three materi- als. In particular, all three oxides possess an indirect band gap between the highest occupied valence band at R and the lowest conduction band at Γ, and the direct gap lies at the Γ point. e discussion of the band structure above holds for the LDA as well as the EXX func- tional. In comparison to the LDA, the band width of the O òp bands is reduced by the EXX functional. is reduction amounts to ý.òÀeV for CaTiOç, ý.ýÞ eV for SrTiOç, and ý.Ô eV for

BaTiOç. Moreover, the EXX functional leads to a distinct opening of the band gap by about ò eV for each of these three oxides. e exact values of direct and indirect band gap are given

ÔÔÞ â. Exact exchange within the optimized ešective potential method

CaTiOç SrTiOç BaTiOç −Ô −Ô −Ô Gmax ¥.âaý ¥.çaý ¥.çaý lmax — (Ti); Ôò(Ca); â (O) À (Ti); ÔÔ(Sr); Þ (O) — (Ti); Ôò(Ba); — (O) ′ −Ô Gmax ç.ýýaý Lmax â

Table â.â.: Numerical input parameters of LAPW and MPB for the three perovskites CaTiOç, SrTiOç, and BaTiOç.

10

CaTiO3 8

6

4 (eV) F 2 LDA EXX EXX E - LDA 0

-2

-4

DOS ΓΓ X M R X DOS

Figure â.Ô .: LDA and EXX KS band structure and DOS for the perovskite CaTiOç. e total DOS (in units of states/eV) is shown in black, whereas the partialDOSofOòp, Ti çd,andCaçd is drawn in red, green, and blue.

Ôԗ â.—. Perovskites: CaTiOç, SrTiOç, BaTiOç

10

SrTiO3 8

6

4 (eV) F 2 LDA EXX EXX E - LDA 0

-2

-4

DOS Γ X M Γ R X DOS

Figure â.Ôâ.: Same as Fig. â.Ô for SrTiOç. in Table â.Þ. By substituting Ca by Sr and nally by Ba both gaps, direct and indirect, shrink and become more and more similar so that for BaTiOç direct and indirect EXX gap are even identical (at least with an accuracy of ý.ýÔ eV). e addition of LDA correlation to the EXX functional leads to a slightly dišerent direct and indirect gap again. If there is no hybridization of the states of the A cation (Ca, Sr, Ba) with theO òp and Ti çd bands, between which the gap is formed, the question arises what causes the band gap to de- crease if one replaces Ca by Sr and nally by Ba. In order to analyze whether this ešect goes back to the increase of the lattice constant from CaTiOç to BaTiOç or whether it is caused by a change in the potential induced through the replacement of the A cation, we calculate the band gap of CaTiOç at the lattice constant of the two other materials. Additionally, we replace the cation Ca by Sr and Ba at the lattice constant of CaTiOç. e corresponding changes in the gaps employing the LDA are shown in Table â.—. By increasing the lattice constant of CaTiOç the direct and the indirect band gap shrink. e indirect gap of CaTiOç at the lattice constant of SrTiOç or BaTiOç is slightly smaller than those obtained for the real material, while the direct gap exhibits the opposite behavior. e replacement of the A cation at the xed lattice constant of CaTiOç yields an increase in the indirect gap and a decrease in the direct gap. If one combines both ešects, the band gap of the ’true’ material can be constructed (up to small inaccuracies). For example, the change of the lattice constant from CaTiOç to SrTiOç leads to a reduction of the direct gap of ý.ýâ eV, whereas the interchange of Ca by Sr at xed lattice

ÔÔÀ â. Exact exchange within the optimized ešective potential method

10

BaTiO3 8

6

4 (eV) F 2 LDA EXX EXX E - LDA 0

-2

-4

DOS Γ X M Γ R X DOS

Figure â.ÔÞ.: Same as Fig. â.Ô for BaTiOç.

constant decreases the direct gap by ý.Ôý eV. In combination the direct gap of SrTiOç should be ý.ÔâeVsmaller thanthedirect gap of CaTiOç, which deviates from the ’true’ gap of SrTiOç by only ý.ýÔ eV. In conclusion, the reduction of the band gap from CaTiOç to BaTiOç is not merely caused by the increase in the lattice constant. e change in the potential due to the change of the A cation is as important as the increase in the lattice constant. In comparison with experiment (s. Table â.—), the LDA underestimates the direct and in- direct band gap by roughly ý%. e EXX as well as the EXXc functional yield KS gaps that are closer to the experimental values. However, we observe a pronounced overestimation by about ò %. is might have two reasons: (a) the error cancellation between the neglect of cor- relation on the one hand and of the derivative discontinuity on the other hand – as discussed in section â.â – may not work for these transition metal oxides, (b) deviations of the experi- mental crystal structure at room temperature from the ideal cubic structure employed in the calculations may inžuence the gap. At room temperature, CaTiOç is orthorhombic, BaTiOç is tetragonal, and SrTiOç is cubic. In Ref. Ô¥—, PBE DFT calculations are reported for the cubic as well as the room-temperature crystal structure of the three perovskites. For orthorhombic

CaTiOç a band gap, which is about ý. eV smaller than that of cubic CaTiOç, is found. If we assume that we can simply transfer this reduction observed forthePBEfunctional totheEXX band gap, we obtain for CaTiOç a signicant improvement with the experimental result. Yet, the discrepancies for the other two materials, in particular for SrTiOç, can not be explained

Ôòý â.À. Transition metal oxides: MnO, FeO, and CoO

LDA EXX EXXc Expt.

CaTiOç Γ → Γ ò.ò¥ ¥.Þý ¥.Þ¥ – R → Γ Ô.ÞÞ ¥.ò— ¥.çÔ ç. Þa b SrTiOç Γ → Γ ò.ýÀ ¥. Ô ¥. ¥ ç.—— R → Γ Ô.Þç ¥.òý ¥.òò ç.òýb c BaTiOç Γ → Γ Ô.—Ô ¥.ý— ¥.Ôò ç.Àò R → Γ Ô.Þ ¥.ý— ¥.ÔÔ ç.òÞc

aReference Ô¥ bReference Ô¥â cReference Ô¥Þ

Table â.Þ.: Direct and indirect KS band gaps (in eV) for the cubic perovskites CaTiOç, SrTiOç, and BaTiOç and comparison with experiment.

CaTiOç LDA aCaTiOç LDA Γ → Γ R → Γ Γ → Γ R → Γ

aCaTiOç ò.ò¥ Ô.ÞÞ CaTiOç ò.ò¥ Ô.ÞÞ

aSrTiOç ò.ԗ Ô.âÀ SrTiOç ò.Ô¥ Ô.—ý

aBaTiOç ò.ýÀ Ô.âý BaTiOç Ô.—À Ô.—À (a) (b)

Table â.—.: Direct and indirect LDA KS gap (in eV) for (a) CaTiOç at the lattice constant of CaTiOç, SrTiOç and BaTiOç, (b) the three perovskites at the lattice constant of CaTiOç. in this way. In conclusion, we have investigated the electronic structure of the three cubic perovskites

CaTiOç, SrTiOç, and BaTiOç using the EXX(c) functional. In comparison with the LDA the EXX(c) approach leads to an opening of the band gap for all three materials. e dispersion of the bands is nearly unašected except for a slight reduction of the valence band width. While the LDA underestimates the band gaps by about ý%, the EXX(c) approach overcompensates this error and leads to an overestimation of ò %. Possible reasons for this overestimation have been discussed.

6.9. Transition metal oxides: MnO, FeO, and CoO

e transition-metal (TM) oxides MnO, FeO, and CoO are antiferromagnetic insulators, which crystallize in the rock-salt structure with antiferromagnetic ordering along the [ÔÔÔ] di-

ÔòÔ â. Exact exchange within the optimized ešective potential method

Figure â.ԗ.: Cubic unit-cell of the transition metal oxides MnO, FeO, and CoO in their an- tiferromagnetic order (AFM-II). e transition metal atoms with their actual spin orientation are depicted in blue, while the oxygen atoms are red. rection (AFM-II). e TM ions within the (ÔÔÔ) plane are coupled ferromagnetically, whereas TM ions in adjacent (ÔÔÔ) planes are aligned antiferromagnetically (s. Fig. â.ԗ). Experimen- tally, all three TM oxides remain insulating even above the Néel temperature TN [Ô¥À]. In particular, the latter is in contradiction to band theory, that would predict metallic behavior ò+ ò+ â ò+ for the paramagnetic state above TN due to the partially lled d states (Mn d , Fe d , Co dÞ). Peierls [Ô ý] pointed out in ÔÀçÞ that "it is quite possible that the electrostatic interaction between the electrons prevents them from moving at all". Based on this idea Mott developed a model how electron correlation may explain the insulating behavior. Metallic behavior of a material requires mobile electrons, that move through the crystal, hopping from site to site. In the case of the relatively localized d electrons, the hopping is determined by two competing mechanisms: on the one hand, the reduced localization of the electrons due to the hopping leads to a gain in kinetic energy (−t). On theother hand, one has to pay the Coulomb energy U arising from the electrostatic repulsion of the hopping electrons with those residing at the sites. If U ≫ t, the energy loss of this process exceeds the gain so that the electrons avoid to hop and the material is insulating. is is a sketch of the physical picture developed by Mott in order to explain the insulating behavior of the TM oxides even above TN. erefore, this class of materials is nowadays coined Mott insulators. In DFT, which is in principle an exact theory, electron correlation is taken into account

Ôòò â.À. Transition metal oxides: MnO, FeO, and CoO

by the xc energy functional Exc. In the following, we will demonstrate that if the latter is approximated by the EXX functional, DFT predicts an insulating behavior for this set of TM oxides in their antiferromagnetic conguration, whereas in the LDA and the GGA, FeO and CoO are metallic and the gap of MnO is seriously underestimated. For all materials, we treat the complete M shell of the TM comprising the çs, çp, and çd electrons as valence states. e electronic wave functions are described by an LAPW basis −Ô −Ô set with the reciprocal cutoš radius Gmax ç.Àýaý (MnO), Gmax ¥.ýýaý (FeO), and − = = G ¥. ýa Ô (CoO) in the interstitial region. An angular momentum cutoš (l ) of max = ý max — is employed for all materials in the MT spheres. e MPB for the representation of the ′ − local EXX potential is formed by IPWs with G ç.ýýa Ô and numerical functions with max = ý L â in the MT spheres. e parameters of the MPB apply to all three TM oxides and max = the BZ is sampled by a ¥×¥×¥ k-point set. e structure of the compounds is of rock-salt type. eir experimental lattice constants are —.¥ý aý for MnO, —.ÔÀ aý for FeO, and —.ý¥ aý for CoO.[Ô Ô] e antiferromagnetic ordering along the [ÔÔÔ] direction reduces the symmetry to the space group Rçm. is fact will become important later. In experiment, small lattice distortions are observed [Ô ò]. ese distortions, which are rhombohedral for MnO and FeO and tetragonal for CoO, are neglected in the calculations. For the EXXc calculations, LDA correlation is combined with the orbital-dependent EXX functional. In particular, we use the parametrization of the LDA correlation energy of Vosko, Wilk, and Nusair [ ]. In Figs. â.ÔÀ, â.òý, and â.òÔ PBE and EXX band structures and DOS are shown for each of the three TM oxides. e corresponding path in the rhombohedral BZ is shown in Fig. â.òò. Due to the antiferromagnetic order spin-up and spin-down eigenvalue spectra are identical. While FeO and CoO are metallic in the PBE functional, the EXX approach predicts a semi- conducting behavior with a direct gap at the Γ point for all three oxides. e location of the indirect gap between the valence-band maximum and the conduction-band minimum is ma- terial specic. While the conduction-band minimum lies at the Γ point for all three materials, the valence-band maximum is located at T for MnO and at F for CoO. e highest valence band for FeO is extremely narrow and hardly shows any dispersion. For all three oxides, the highest valence band exhibits TM çd character with some admixture of O òp (cf. Figs. â.ÔÀ, â.òý, and â.òÔ). Due to the localization of the çd states, these bands possess little dispersion. On the contrary, the lowest conduction band exhibits a parabolic shape in particular around its minimum at the Γ point režecting the TM ¥s character of this band. Above this ¥s-type state, the bands corresponding to the unoccupied d states follow. In the case of MnO, each of the two Mn atoms in the magnetic unit cell posseses ve d elec- trons. At one of the Mn atoms they occupy ve spin-up d states, whereas at the other atom ve spin-down states are lled. Occupied and unoccupied d states of the same Mn atom are

Ôòç â. Exact exchange within the optimized ešective potential method

15 MnO

10

PBE EXX 5 (eV) F

E - 0

-5

-10 DOS F Γ T K L Γ DOS

Figure â.ÔÀ.: PBE (dashed black lines) and EXX (solid black lines) KS band structure and DOS (in units of states/eV) for the transition-metal oxide MnO. e contribu- tion of the TM d states to the total DOS (black line) is drawn in red. e partial DOSfor theOòp states is shown in blue.

15 FeO

10

PBE EXX 5 (eV) F

E - 0

-5

-10 DOS F Γ T K L Γ DOS

Figure â.òý.: Same as Fig. â.ÔÀ but for FeO.

Ôò¥ â.À. Transition metal oxides: MnO, FeO, and CoO

15 CoO

10

PBE EXX 5 (eV) F

E - 0

-5

-10 DOS F Γ T K L Γ DOS

Figure â.òÔ.: Same as Figs. â.ÔÀ and â.òý but for the CoO.

Figure â.òò.: Path in the rhombohedral BZ of the antiferromagnetic TM oxides for the band structures of Figs. â.ÔÀ, â.òý, and â.òÔ.

Ôò â. Exact exchange within the optimized ešective potential method split by ¥.â eV in the PBE and by Ôý.— eV in the EXX approach, where we measure from the center of the occupied to the center of the unoccupied d states. is exchange splitting of the d states makes MnO an antiferromagnetic semiconductor. As Fe provides an additional d electron in contrast to Mn, one would naively expect that FeO is metallic. However, the antiferromagnetic ordering reduces the symmetry from Fmçm in the chemical unit cell to

Rçm in the magnetic unit cell. While the former splits the ve d bands into three tòg and two eg states at the Γ point, the latter divides the d states into a single degenerate state of symmetry aÔg and two doubly degenerate bands of symmetry eg and eu. erefore, the additional d elec- tron of Fe can occupy the singly degenerate d state of aÔg symmetry instead of occupying a multiply degenerate state fractionally. Moreover, the occupation of this d state in the EXX approach causes the corresponding single-particle energy to fall such that it is energetically separated from the remaining unoccupied d states. In this way, a gap is formed in FeO. e single, nearly dispersionless d band at the Fermi energy is separated from the unoccupied states by about Ô.âÞ eV and from the remaining occupied bands by about ò. eV. We note that in the case of the LDA or GGA one usually observes that a single-particle energy rises if the level is occupied and falls if it is emptied [Ô ç]. is is exactly the opposite behavior as observed here for the EXX functional. Each Co atom in CoO provides yet another d electron. At a given Co atom the seven d electrons occupy ve spin-up and two spin-down d states. e two spin-down states have eg symmetry at the Γ point. In analogy to FeO, the energy levels of the two occupied spin- down d states are lowered with respect to the unoccupied ones so that they are separated from the unoccupied states by ò.—ò eV. In Table â.À, we summarize the direct and indirect gap for the three oxides and compare them with experimental values, which are either measured by optical absorption or photoe- mission experiments. We note again that, in contrast to the PBE functional, the EXX ap- proach predicts a semiconducting behavior for all three oxides. e addition of LDA correla- tion to theEXX functional (EXXc) leadsto a furtheropeningof the gap by about ý.ò−ý.ç eV.

According to Hund’s rst rule, we would expect a magnetic moment of µB for MnO, ¥ µB for FeO, and ç µB for CoO. Within the TM mu›n-tin spheres the calculated moments are smaller (s. Table â.Ôý). e magnetic moments in the EXX as well as the EXXc approach are consistently larger than those in the PBE and closer to Hund’s rst rule, while EXX and EXXc moments are (nearly) identical. e opening of the band gap, in particular in the case of FeO and CoO, and the increased localization of the KS wave functions due to removal of the self- interaction error in the EXX method are responsible for the enhancement of the magnetic moment with respect to the PBE functional. We note, however, that the moment depends

Ôòâ â.À. Transition metal oxides: MnO, FeO, and CoO

PBE EXX EXXc Expt. MnO direct Ô.çâ ¥.òÔ ¥.çÞ ç.â...ç.—a indirect ý.— ç.—ý ç.Àâ ç.Àý ± ý.¥b FeO direct – Ô.—â ò.ÔÞ ò.¥c indirect – Ô.âÞ Ô.ÀÀ – CoO direct – ç.◠ç.ÀÀ .¥çd indirect – ò.—ò ç.Ôç ò. ± ý.çe

aReference Ô ¥ bReference Ô cReference Ô â dReference Ô Þ eReference Ô —

Table â.À.: Direct and indirect band gaps (in eV) for the TM oxides MnO, FeO, and CoO cal- culated with the PBE, EXX, and EXXc functional. For comparison, experimental values from the literature are also given. e indirect (direct) gap is measured by photoemission (optical absorption) measurements.

PBE EXX EXXc Expt. MnO ¥.ԗ ¥.âý ¥.âý ¥. —a FeO ç.ç— ç.Þâ ç.Þ ç.çòb; ¥.òc CoO ò.¥ç ò.— ò.—¥ ç.ç d; ç.—b; ç.À—e

aReference Ô⥠bReference Ôâ cReference Ô À dReference Ôââ eReference ÔâÞ

Table â.Ôý.: Spin magnetic-moments (in µB) of the antiferromagnetic TM oxides for the PBE, EXX, and EXXc functionals and a comparison with experimental values.

on the choice of the MT sphere radius Sa. Moreover, we note that the experimental magnetic moments comprise spin and orbital contributions. For example for FeO and CoO, it is argued in Refs. Ô À–Ôâç that the orbital moment may contribute to the total magnetic moment with a value of about Ô µB. In summary, the EXX approach predicts all three transition-metal monoxides to be insu- lating in their anti-ferromagnetic (AFM-II) conguration. While for MnO the spin splitting of the çd states is su›cient for the formation of the gap, in the case of FeO and CoO it is the interplay of the reduced symmetry of the magnetic unit cell with the property of the EXX functional that the energy of a single particle level falls if it is occupied, which nally makes FeO and CoO insulating as well. Moreover, we have shown that the EXX functional, in com- parison with the PBE, leads to an enhancement of the spin magnetic moment.

ÔòÞ â. Exact exchange within the optimized ešective potential method

6.10. Metals

Until now, we have restricted our discussion to semiconducting or insulating materials. In metals, in contrast to semiconductors and insulators, occupied and unoccupied states are not clearly separated by an energy gap so that an oŸen complex Fermi energy surface is formed. is has the consequence that the convergence of BZ integrals with respect to the k-point sampling is slow. A common approach to improve the convergence is to smear out the sharp, step-like distinction between occupied and unoccupied sates. States lying a little above or be- low theFermi energy EF are fractionally occupied. For example, the step-function is replaced by a Fermi-Dirac distribution function

Ô σ,FD (â. ý) fnk єσ −E = exp nk F + Ô kB T σ σ describing the occupation of a state ϕnk with energy[ ]єnk. Alternatively, a Gaussian distribu- tion function can be used

Ô єσ − E σ,G − nk F fnk Ô erf . (â. Ô) = ò kBT  Œ ‘ In both approaches, the articial temperature T or the articial energy kBT with the Boltz- mann factor kB controls the degree of smearing. In the literature, further broadening schemes are discussed [Ôâ—], but we restrict ourselves here to those mentioned above. In order to retain the variational property of the total energy with respect to density vari- ations an additional term must be added to the total energy that takes into account the frac- tional occupations [ÔâÀ, ÔÞý]. In the case of the Fermi-Dirac distribution, this additional term has the following form

FD FD FD FD −TS kBT fnk ln fnk + Ô − fnk ln Ô − fnk . (â. ò) = Qnk
( ) ( ) ( ) While for a Gaussian distribution it is given by

ò Ô єnk − EF −TS −kBT exp − . (â. ç) = Qnk π kBT √  ‹  For the Fermi-Dirac distribution, the variational total energy E −TS is identical in its form to the total energy in nite-temperature density-functional theory for grand canonical ensem- bles [ÔÞÔ]. In this special case and only in this case, the variational total energy corresponds to the thermodynamic free energy F and the articial temperature T can be associated with

Ôò— â.Ôý. Metals the real temperature of the system. Besides an improved k-point convergence, the introduction of fractional occupation also stabilizes the convergence of the SCF cycle. While in the case of a sharp step-like occupation an interchange of an occupied and an unoccupied state at the Fermi energy in two subse- quent SCF iterations lead to a jump in the density, which oŸen impedes the convergence of the SCF cycle, the usage of a Fermi-Dirac or Gaussian distribution function suppresses the dependence of the density on such interchanges at the Fermi level.

6.10.1. EXX-OEP formalism for metals

In the following, a generalization of the EXX-OEP formalism for metals, which allows for fractional occupation, will be discussed. A formulation for spin-unpolarized systems can be found in Ref. ÔÞò. For its extension to spin-polarized systems, we refer to Ref. ÔÞç. Let us rst concentrate on the spin-density response function. Due to the introduction of σ σ the smearing function fnk (here and in the following fnk stands for Fermi-Dirac or Gaussian distribution) the electron spin density turns into a sum over all states and k points

BZ σ σ σ ò n r fnk ϕnk r , (â. ¥) = Qk Qn ( ) S ( )S where f σ controls if a state is totally f σ Ô or fractionally occupied f σ ý,Ô or un- nk nk = nk ∈ occupied f σ ý . e density has to be varied with respect to the ešective potential for nk = ( ) ( ] [) the calculation of the spin-density response function. In contrast to a semiconductor or in- ( ) sulator, an additional term appears in the spin-density response function that arises from the σ σ′ variation of fnk with respect to the ešective potential Veš

σ ′ ′ δn r χσσ r, r (â. a) s = δV σ′ r′ eš ( ) ( ) BZ σ∗ σ σ∗ ′ σ ′ ϕnk r ϕn′k r ϕn′k r ϕnk r δ ′ ( ) f + c.c. σσ nk σ − σ = Qk Qn nQ′ n єnk єn′k ≠ ( ) ( ) ( ) ( ) BZ δ f σ + nk σ∗ σ σ′ ′ ϕnk r ϕnk r . (â. b) Qk Qn δVeš r ( ) ( ) While the rst term is diagonal in the( spin) σ, the second additional term is not, as will be illustrated later on. In the case of a spin-polarized metal with fractional occupation, the po- ↑ ↓ tentials Vx r and Vx r are, therefore, not obtained by solving two independent EXX-OEP equations for each spin channel. Instead, one equation that determines both V↑ r and V ↓ r ( ) ( ) x x ( ) ( )

ÔòÀ â. Exact exchange within the optimized ešective potential method must be solved

↑↑ ′ ↑↓ ′ ↑ ′ ↑ χs r, r χs r, r Vx r ç ′ t r ′ ′ ′ d r . (â. â) ∫ χ↓↑ r, r χ↓↓ r, r V ↓ r = t↓ r ⎛ s ( ) s ( ) ⎞ ⎛ x ( ) ⎞ ⎛ ( ) ⎞ In a similar way as⎝ for the( single-particle) ( ) response⎠ ⎝ ( function,) ⎠ an additional⎝ ( ) ⎠ term is present in the spinor of the right-hand side

σ δ Ô σ′ σ′ NL,σ′ σ′ t r σ ′ fnk ϕnk Vx ϕnk (â. Þa) = δVeš r ò Qσ′ Qnk ( ) BZ  ⟨ S σS∗ ′⟩ σ σ σ NL,σ σ ϕn′k r ϕnk r (f ) ϕ V ϕ ′ + c.c. nk nk x n k σ − σ = Qk Qn nQ′ n єnk єn′k ≠ ( ) ( ) BZ σ′  δ⟨f S ′ S ⟩′ ′ + nk σ NL,σ σ σ ′ ϕnk Vx ϕnk , (â. Þb) Qσ′ Qk Qn δVeš r ⟨ S S ⟩ where the denition of the non-local exchange( ) potential has been generalized to

σ σ∗ ′ BZ ϕ ′ r ϕ ′ r NL,σ ′ σ n q n q Vx r, r − fn′q ′ . (â. —) = Q Q′ r − r q n ( ) ( ) ( ) ′ In contrast to semiconductors and insulators the sumS overS n in the square brackets in σ Eqs. (â. b) and (â. Þb) can not be restricted solely to the unoccupied states f ′ ý , as n k = σ the occupation number fnk prohibits, in general, that combinations of occupied bands n σ ′ σ σ ( ) ( f ý with occupied bands n f ′ f ý cancel each other. nk ≠ n k ≠ nk ≠ ) ( )

σ e calculation of the variation of fnk due to a change in the ešective potential is discussed σ,FD in the following for the Fermi-Dirac distribution fnk . However, the derivation can be easily σ,G transferred to the Gaussian distribution function fnk . According to the rules of functional derivatives, the occupation number response obeys

σ,FD δ fnk Ô σ,FD σ,FD σ∗ ′ σ ′ δEF − − ′ − σ′ ′ fnk Ô fnk ϕnk r ϕnk r δσσ σ′ ′ . (â. À) δVeš r = kBT δVeš r [ ] œ ( ) ( ) ¡ ( ) ( ) σ e rst term in the curled brackets arises from the variation of the eigenvalue єnk and is, therefore, diagonal in spin space. On the contrary, the second term couples spin-up and ′ spin-down electrons, because a variation of the ešective potential of spin σ can change the σ′ σ eigenvalues єnk with respect to the eigenvalues of the opposite spin direction єnk sothata dif- σ ferent occupation of the energy levels єnk becomes preferable. In order to explicitly calculate

Ôçý â.Ôý. Metals

the variation of the Fermi energy EF, we exploit that the total particle number N is conserved

BZ σ,FD δN δ fnk ý σ′ ′ σ′ ′ , (â.âý) = δVeš r = Qσ Qk Qn δVeš r which together with Eq. (â. À) gives ( ) ( )

BZ δE Ô ′ ′ ′∗ ′ ′ ′ F σ ,FD − σ ,FD σ σ σ′ ′ fnk Ô fnk ϕnk r ϕnk r , (â.âÔ) δVeš r = Z Qk Qn [ ] ( ) ( ) where we have introduced( )Z as an abbreviation for

BZ σ,FD − σ,FD Z fnk Ô fnk . (â.âò) = Qσ Qk Qn [ ] Inserting Eqs. (â. À) and (â.âÔ) into Eq. (â. b), the components of the density response ma- trix nally become

BZ σ∗ σ σ∗ ′ σ ′ σσ′ ′ ϕnk r ϕn′k r ϕn′k r ϕnk r χ r, r δ ′ f + c.c. s σσ nk σ − σ = Qk Qn nQ′ n єnk єn′k ≠ ( ) ( ) ( ) ( ) ( ) BZ  Ô σ,FD σ,FD σ∗ σ σ∗ ′ σ ′ − ′ − δσσ fnk Ô fnk ϕnk r ϕnk r ϕnk r ϕnk r kBT Qk Qn BZ Ô Ô [ ] ∗ ( ) ( ) ( ) ( ) + σ,FD − σ,FD σ σ fnk Ô fnk ϕnk r ϕnk r kBT Z Qk Qn  BZ [ ] ( ) ( ) × σ′,FD − σ′,FD σ′∗ ′ σ′ ′ fn′k′ Ô fn′k′ ϕn′k′ r ϕn′k′ r . (â.âç) Qk′ Qn′  [ ] ( ) ( ) Similarly, the spinor of the right-hand side Eq. (â. Þb) turns into

BZ σ∗ ′ σ σ σ σ NL,σ σ ϕn′k r ϕnk r t r f ϕ V ϕ ′ + c.c. nk nk x n k σ − σ = Qk Qn nQ′ n єnk єn′k ≠ ( ) ( ) BZ ( ) Ô  ⟨ S ′ S ⟩ ∗ − σ NL,σ σ σ,FD − σ,FD σ σ ϕnk Vx ϕnk fnk Ô fnk ϕnk r ϕnk r kBT Qk Qn BZ BZ Ô Ô ⟨ S S ⟩′ [ ′ ′ ] ′ ( ) (′ ) + σ NL,σ σ σ ,FD − σ ,FD ϕn′k′ Vx ϕn′k′ fn′k′ Ô fn′k′ kBT Z Qk Qn Qσ′ Qk′ Qn′ × σ,FD −⟨ σ,FDS σ∗ S σ ⟩ [ ] fnk Ô fnk ϕnk r ϕnk r . (â.â¥)

→ [ ] ( ) σσ( ′ ) ′ σ In the limit of T ý, all temperature-dependent terms in χs r, r and t r vanish, as f σ,FD is either Ô or ý. In particular, the oš-diagonal elements of the density response matrix nk ( ) ( )

ÔçÔ â. Exact exchange within the optimized ešective potential method are zero, and Eq. (â. â) decouples into two separate equations for the spin-up and spin-down potential. Finally, the equations of section â.Ô are recovered. In order to solve Eq. (â. â) numerically for nite T,we expandthespinor of thelocal EXX potential in the MPB, which is by construction spin-independent (cf. Sec. ¥.ò.ò). e basis set is thus given by

MI r ý , i Ô,..., NMPB , , i Ô,..., NMPB . (â.â ) ⎡ ý = ⎤ ⎡ MI r = ⎤ ⎪⎧⎢ ⎥ ⎢ ⎥⎪⎫ ⎪⎢⎛ ( ) ⎞ ⎥ ⎢⎛ ⎞ ⎥⎪ ⎨⎢ ⎥ ⎢ ⎥⎬ ⎪⎢⎝ ⎠ ⎥ ⎢⎝ ( ) ⎠ ⎥⎪ Here, NMPB ⎩denotes⎣ the total number of spin-independent⎦ ⎣ MPB functions. In⎦⎭ analogy to the spin-polarized case with integer occupation, the spinor corresponding to a constant func- Ô tion c is an eigenfunction of the response matrix in Eq. (â. b) with eigenvalue zero Ô ⎛ ⎞ ′ and must be excluded from the Hilbert space in order to make χσσ invertible. is is al- ⎝ ⎠ s ready fullled by the MPB (â.â ) as each function MI r is by construction orthogonal to a constant function (s. Sec. â.ò). But the Hilbert space spanned by the two constant functions ( ) Ô ý Ô c and c additionally contains the function c . e latter is orthogonal to ý Ô −Ô ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Ô the⎝ constant⎠ spinor⎝ ⎠c and enables an alignment of the⎝ spin-up⎠ part of the potential with Ô ⎛ ⎞ Ô respect to the spin-down⎝ part.⎠ erefore, we add the spinor to the MPB [Eq. (â.â )]. −Ô ⎛ ⎞ Consequently, the integral equation that couples the spin-up and spin-down EXX potentials ⎝ ⎠ turns into an algebraic equation of dimension òNMPB + Ô × òNMPB + Ô , that can be solved by numerical inversion of the response matrix. ( ) ( )

6.10.2. Metals: Na, Al, and Cu

Before we show EXX-OEP results for the metals Na, Al, and Cu at the experimental lattice constants (Na —.ÔÔ aý;AlÞ.â aý;Cuâ.—òaý), we discuss the homogeneous electron gas (HEG). e HEG is a model system, in which the positive charges of the atomic nuclei are smeared out uniformly over space. Consequently, the electron density is distributed uniformly in space, too. e HEG provides a simple model for delocalized electrons in a metal. We discuss the HEG in the HF as well as the EXX-OEP approach in order to point out the dišerences between both approaches for metals. It is known that the HF method, which approximates the many-electron wave function by a single Slater determinant instead of the many-electron Hamiltonian, gives for the HEG an

Ôçò â.Ôý. Metals eigenvalue dispersion that consists of two terms. e rst one corresponds to the energy of a free electron, which is corrected by the second term arising from the explicit consideration of the electron-electron interaction

HF Ô ò òkF k єHEG k k − F (â.ââ) = ò π kF ( ) ( ) with the function Ô Ô − xò Ô + x F x + log (â.âÞ) = ò ¥x Ô − x ( ) Œ ‘ and the Fermi wavevector kF. e correction term F isS continuous,S but has a derivative dis- continuity at k k . Because of this the HF method predicts a vanishing density of states = F Ô V kò D є δ є − є k dçk . (â.â—) ∫ ò ′ = VBZ = òπ є k є k є ( )= ( ) ( ( )) W ( ) at the Fermi level EF, which is in contradiction to the conductivity or specic heat of metals.

e EXX-OEP approach, on the contrary, employs formally the same total energy func- tional as the HF approach, but requires a local instead of a non-local exchange potential. As the EXX energy is, in the case of the HEG, identical to the exchange energy of the LDA, the Ô ç local EXX potential is proportional to n ~ and therefore a constant. Moreover, the Hartree potential and the external potential arising from the positive homogeneously smeared out charge background are constant, too. So, the KS electrons move in a constant potential and behave as free electrons with the dispersion

Ô єEXX k kò . (â.âÀ) HEG = ò ( ) us, the density of states for the HEG in the EXX-OEP method is proportional to є and does not vanish at the Fermi energy EF. √

According to this argumentation, we expect that the EXX-OEP approach gives a reasonable description at least for simple metals that behave nearly free-electron-like, in contrast to the HF method. is assumption is indeed conrmed by our calculations for bcc Na and fcc Al. −Ô For the calculation of Na (Al) we use an LAPW basis set with the parameters Gmax ¥.ýaý −Ô = (Gmax ç.Þaý ) and lmax Ôò (lmax Ôý). e corresponding cutoš values of the MPB are ′ = − ′ = − = G ç.ýa Ô (G ò.—a Ô) and L â (L â). In both calculations we have em- max = ý max = ý max = max = ployed an —×—×— k-point mesh for the self-consistency cycle and occupy the states according to the Fermi-Dirac distribution with a temperature of T çÔâ K k T Ô mhtr . Figure â.òç = B = shows LDA and EXX band structures and densities of states for both materials. e occupied ( )

Ôçç â. Exact exchange within the optimized ešective potential method

LDA EXX Expt. d center -ç.Ôý -¥.Þ¥ −ç.â a d width ç.¥â ò.¥À ç.Ô ± ý.òa

aReference ÔÞ¥

Table â.ÔÔ.: Comparison of the center (with respect to EF)andthewidthofthe d states (in eV) of Cu from LDA, EXX, and experimental band structures. We used thedenition of the d-band width as given in Ref. ÔÞ¥ as the dišerence between the highest and lowest occupied state at the X point in the BZ. Accordingly, the center of the d bands is dened as the midpoint of the highest and lowest occupied state at X. spectrum in the EXX approach is nearly indistinguishable from the LDA one. EXX and LDA bands cross the Fermi energy at almost the same points in the BZ. However with increasing energy, the LDA and EXX band structures start to dišer, but the shape of the band dispersions still remains very similar. Due to the similarity in the band structure between the LDA and EXX approaches, the densities of states is nearly identical, too. It shows a free-electron-like square root behavior with only minor variations. e question arises what happens for a metal that substantially deviates from the HEG as for example a transition metal with localized d electrons. For the example of Cu, we demon- strate in Fig. â.ò¥ that LDA and EXX band structures and densities of states dišer signif- icantly. e calculations employ an LAPW basis (MPB) with reciprocal cutoš parameter − ′ − G ¥. a Ô G ç.ýa Ô and angular momentum cutoš l — L â . e BZ max = ý max = ý max = max = zone is sampled by an —×—×— k-point mesh and a Fermi-Dirac distribution with a temper- ( ) ( ) ature of T çÔâ K k T Ô mhtr is used. Figure â.ò¥ shows that the lowest valence state = B = of Cu çs character is bound much stronger in the LDA than with the EXX functional. In ( ) contrast to that the Cu d states are bound more tightly in the EXX approach and their band width is reduced with respect to the LDA (cf. Table â.ÔÔ). Moreover, the dispersion of the bands around the L point close to the Fermi energy is apparently dišerent in the LDA and

EXX method. While the former results in an occupied state at roughly Ô.ý eV below EF, the latter yields an unoccupied state at about ý.çç eV above EF at the L point. A comparison with the experimentally measured band dispersion along the BZ path L → Γ → X is shown in Fig. â.ò . While in the LDA the Cu d states are bound to shallow with respect to experiment, the trend of the EXX functional to bind them more strongly is in ac- cordance with experiment, even though the underbinding of the LDA is overcorrected by the EXX approach. Similarly, the tendency of the EXX method to reduce the valence band width is correct. However, quantitatively the EXX functional underestimates the experimental band

Ôç¥ â.Ôý. Metals

10 Na 8

6

4 (eV) F 2 E - LDA EXX 0

-2 EXX LDA

-4 DOS ΓΓ H N P N DOS (a)

10 Al

5

0 (eV) F E - LDA EXX -5

EXX -10 LDA

DOS X K ΓΓ L W X DOS (b)

Figure â.òç.: Comparison of LDA and EXX band structures and DOS (in units of states/eV) for (a) bcc Na and (b) fcc Al.

Ôç â. Exact exchange within the optimized ešective potential method

10 Cu 7.5

5

2.5 LDA EXX (eV)

F 0 E - -2.5

-5

EXX -7.5 LDA

-10 DOS X K ΓΓ L W X DOS

Figure â.ò¥.: Band structures and DOS (in units of states/eV) for copper calculated with the LDA and EXX-OEP functional.

Ôçâ â.ÔÔ. Summary

10 LDA 10 EXX 8 8 6 6 4 4 2 2 (eV)

F 0 0

E - -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 L Γ X L Γ X

Figure â.ò .: LDA and EXX band structures for Cu and comparison with experimental data points taken from Refs. ÔÞ¥ and ÔÞ . width. e addition of LDA correlation to the orbital-dependent EXX functional does not signi- cantly change the EXX band dispersion of Cu. Kotani [ÔÞâ] showed that an orbital-dependent RPA correlation functional in combination with the EXX functional leads to an eigenvalue spectrum of Cu, which is similar to the LDA spectrum again.

6.11. Summary

We have developed an all-electron full-potential implementation of the EXX-OEP method within the FLAPW approach. It is based on a representation of the local, multiplicative ex- change potential by the auxiliary mixed product basis (MPB), which has been adjusted for its present purpose. In this way, the integral equation for the exchange potential becomes an al- gebraic equation. AŸer exclusion of the constant function from the Hilbert space spanned by the MPB, this algebraic equation can be solved for the potential by a simple matrix inversion. We found that a numerically stable and physical EXX potential requires a balance between LAPW and mixed product basis sets. Otherwise, the potential shows spurious oscillations. is balance can be achieved by converging the LAPW basis with respect to the MPB. In the interstitial region, the reciprocal cutoš radius of the LAPW basis must be converged with

ÔçÞ â. Exact exchange within the optimized ešective potential method respect to that of the MPB. Typically Ô.çç times the reciprocal cutoš of the MPB leads to converged results. In the MT spheres, on the contrary, the LAPW basis must be enhanced by local orbitals. For example for diamond, we must add six local orbitals for each lm channel from l ý to l to obtain a smooth potential in the spheres. is increases the number = = of LAPW basis functions by a factor of ¥. e enhanced žexibility of the LAPW basis, in contrast to LDA or GGA calculations, is necessary to enable the wave functions and, thus, the density to respond adequately to the changes of the ešective potential, which are described in our formalism by the MPB. An analysis of the convergence behavior of the density response function conrms this statement. In comparison with the LDA exchange potential, the (balanced) EXX potential exhibits a much stronger anisotropy in particular within the MT spheres. is indicates that a full- potential treatment in the case of the EXX OEP formalism is even more important than in the LDA or GGA. e smooth or oscillatory shape of the potential naturally ašects the KS eigenvalues and transition energies. For the materials C, Si, SiC, Ge, GaAs, and crystalline Ar we found a very good agreement between our all-electron full potential EXX calculations and pseudopoten- tial plane wave results from the literature provided that both basis sets are properly balanced. is is in contradiction to a previous implementation within the FLAPW method [çÀ], which initiated a controversy about the adequacy of the pseudopotential approximation within the EXX-OEP approach. Our results, instead, conrm the reliability of the pseudopotential ap- proximation (at least for this set of materials) and, thus, contribute to the clarication of the contradiction. Furthermore, we discussed the connection of the KS band gap, which corresponds to the transition of the highest occupied to the lowest unoccupied KS state, with the fundamental gap of the true interacting system. In exact DFT, the sum of KS gap and the derivative discon- tinuity of the xc energyfunctional equals the exact fundamental gap. While the discontinuity vanishes for the LDA and GGA, we demonstrated for the III-V nitrides that the discontinuity of the EXX functional amounts to a signicant contribution of several eV. is has the con- sequence that the sum of EXX KS band gap and derivative discontinuity overestimates the experimental gap. We argued that a fortuitous error cancellation between the neglect of the discontinuity and the neglect of orbital correlation is responsible for the quantitative agree- ment of the EXX KS gap alone with experiment. A further improved description of the band gap can be obtained by combining exact exchange with the random-phase approximation (RPA) for the correlation energy of DFT [Ôò]. In order to reduce the computational demand of EXX-OEP calculations, we have devel- oped a numerical nite-basis correction (FBC) for response quantities within the LAPW ap-

Ôç— â.ÔÔ. Summary proach. It considers the change of the LAPW MT functions due to a change in the ešective potential explicitly. e nal formula for the KS wave-function and density response consists of two terms: the usual sum-over-states expression and a correction term, the FBC, which vanishes in the limit of a complete basis. Moreover, it provides, in principle, an exact treat- ment of the core-state response. For the example of ScN, we demonstrated that the FBC improves the convergence of the EXX-OEP potential. Considerably fewer local orbitals are required to obtain a stable and smooth potential. We note that theFBC is not restrictedto the EXX-OEP method. For example, a slightly dišerent construction of the FBC might improve the slow convergence of GW calculations [ÔÞÞ] with respect to the basis set and the number of unoccupied states.

We applied the EXX-OEP approach to the transition metal perovskites CaTiOç, SrTiOç, and BaTiOç as well as the antiferromagnetic oxides MnO, FeO, and CoO. For all materials, the EXX functional opens the band gap. Most notably, FeO and CoO are predicted to be semiconducting, while they are metallic in the LDA and GGA. e band gap of the three perovskites is quantitatively overestimated by about ò %, which we attribute, on the one hand, to an incomplete error cancellation between the neglect of correlation and the neglect of the xc derivative discontinuity and, on the other hand, to dišerences in the crystalline structure between theoretical and experimental setup. Finally, we have presented a generalization of the EXX-OEP formalism for the calcula- tion of metals. In contrast to semiconductors and insulators, the variation of the occupation number due to a change in the ešective potential has to be taken into account. As a proof of principle, we showed results for Na, Al, and Cu. While for the simple metals bcc Na and fcc Al LDA and EXX band structures (nearly) coincide, at least for the occupied part of the spectrum, both approaches give rise to substantially dišerent eigenvalue spectra for Cu.

ÔçÀ

7. Summary & Conclusion

In this thesis, we have implemented and explored two approaches, the Kohn-Sham (KS) [ç] and the generalized Kohn-Sham (gKS) [òç] formalism, for dealing with orbital-dependent exchange-correlation (xc) functionals of density-functional theory (DFT). We developed an implementation of hybrid functionals, which combine orbital-dependent exact exchange (EXX) with local or semi-local approximations for the xc energy of DFT, in the gKS for- malism. In this formalism, the EXX energy leads to a non-local exchange potential in the one-particle equations. We worked, in particular, with the PBEý hybrid functional of Perdew, Burke, and Ernzerhof [òò]. Furthermore, we presented an implementation of the orbital-dependent EXX functional within the KS formalism of DFT. e KS, in contrast to the gKS formalism, requires a local instead of a non-local potential in the one-particle equations. e local, multiplicative exchange potential is constructed from the non-local exact exchange energy by the optimized ešective potential (OEP) method [çò–ç¥]. Both approaches and functionals, PBEý and EXX-OEP, have been realized within the all- electron full-potential linearized augmented-plane-wave (FLAPW) method [ò¥–òÞ]. ey rely on an additional auxiliary basis, the mixed product basis (MPB). e latter has been built from products of LAPW basis functions and constitutes an e›cient all-electron basis for the representation of wave-function products. For the implementation of the hybrid functionals, the MPB is employed to calculate the matrix elements of the non-local exchange potential. Each element corresponds to a Brillouin zone (BZ) sum over six-dimensional integrals. Exploiting the MPB to represent the products of LAPW wave functions, the integral becomes a vector-matrix-vector product, where the matrix is the representation of the Coulomb interaction in the MPB. As the calculation of the non-local exchange potential is the most time-consuming step in a practical calculation, we have developed several techniques to accelerate our numerical scheme: the Coulomb ma- trix is made sparse by a unitary transformation of the MPB, which is constructed such that nearly all mu›n-tin (MT) MPB functions are multipole-free. e sparsity is then used to reduce the computational demand for the matrix-vector multiplications. Moreover, we ex- ploit spatial and time-reversal symmetries to (a) restrict the BZ sum to an irreducible part of the BZ and (b) to identify those matrix elements in advance that are zero and need not

Ô¥Ô Þ. Summary & Conclusion be calculated. e k-point convergence of the BZ sum occurring in the non-local exchange potential is improved by separating the divergent part, resulting from the long-range nature of the Coulomb interaction, from the non-divergent one. While the latter is integrated nu- merically, the divergent part is treated analytically. So far, the techniques contribute to the acceleration of a single iteration. However, the one-particle equations of the gKS system must be solved self-consistently, as the local and non-local potential are functionals of the density and density matrix, respectively. We found that a nested self-consistent eld (SCF) cycle con- sisting of an outer density-matrix and an inner density-only iteration scheme leads to a much faster convergence of the PBEý calculations than a direct iteration scheme. e developed implementation of the PBEý hybrid functional is applied to Si, C, GaAs, MgO, NaCl, and crystalline Ar. For this set of materials, the PBEý functional gives transition energies and band gaps that lie much closer to the experimental value than the respective PBE [Þ] value. ough, we observe an overestimation of the transitions for the semiconduc- tors Si and GaAs and an underestimation for the large gap insulators. Our results compare well with recent PBEý calculations within the projector augmented-wave (PAW) method [çý]. Moreover, the PBEý functional improves the description of localized states as illustrated for ZnO. We have applied the PBEý functional to EuO, a ferromagnetic semiconductor with local- ized ¥ f -states that could not be described properly by the conventional local and semi-local xc functionals commonly used in DFT calculations. We examined the structural, electronic, and magnetic properties of EuO in the rock-salt structure. In contrast to the semi-local PBE functional, the PBEý hybrid correctly predicts EuO to be a ferromagnetic semiconductor. e PBEý and experimental lattice constant measured at T ¥.ò K deviate by only Ôh, which is = one order of magnitude closer to the experimental value than the PBE result. While an ex- perimental gap of ý.À eV is observed in the limit of T → ý K, the theoretical gap amounts to ý.— eV, whereas the PBE functional predicts a metal. In order to measure the quality of PBEý total-energy dišerences between dišerent magnetic states, we map the total-energy dif- ferences between ferromagnetic and antiferromagnetic congurations of EuO onto a Heisen- berg model with nearest and next-nearest neighbor interactions. is mapping enables to calculate the Curie temperature and magnon dispersion of EuO. e resulting Curie temper- ature of Ôý K evaluated in the random-phase approximation overestimates the experimental value of âÀ K but is in much better agreement than the corresponding PBE value of òÞò K. Particularly impressive is the magnon dispersion calculated with the PBEý functional, where ò the PBEý spin stišness of ¥ç. meV aý compares extremely well with the experimental data ò of ç—.âmeVaý. is gives condence in the energetics obtained by the PBEý functional for dišerent magnetic phases. From this we conclude that the PBEý functional has the potential

Ô¥ò to describe the properties of rare-earth chalcogenides without the need for an adjustable pa- rameter as the Hubbard U in theLDA+U approach. e reduced self-interaction error of the PBEý hybrid in contrast to the PBE functional seems to be pivotal for the improved descrip- tion of EuO. It recties, in particular, the description of the strongly localized ¥ f electrons of Eu. Summarizing, an e›cient implementation of the PBEý hybrid functional within the FLAPW method has been developed. e PBEý functional leads to an improved description of band-gap materials and materials with localized d- and f -electron states in comparison to the PBE. For the implementation of the EXX-OEP approach within the FLAPW method, we have employed a slightly modied MPB. e basis has been specically adjusted for the repre- sentation of the local exact exchange potential. So, the OEP integral equation becomes an algebraic equation, which can be solved for the local exact exchange potential, aŸer the con- stant function has been eliminated from the MPB. We found that the LAPW basis for the representation of the KS wave functions and the MPB for the representation of the local po- tential are not independent. Instead, the LAPW basis must be converged with respect to the MPB to obtain a smooth and stable EXX potential. is interdependence of the basis sets arises from the density response function on the leŸ-hand side and the KS wave function re- sponse on the right-hand side of the OEP equation. ey describe theresponse of thedensity or KS wave function due to a change in the ešective potential, which is given in our formal- ism by the MPB. If the LAPW basis, which parametrizes the KS wave function and thus also the density, is not žexible enough, the KS wave function cannot follow the changes of the ešective potential, and a corrupted response function is obtained. is is the reason why the LAPW basis must be converged to an accuracy which is beyond that for conventional DFT calculations employing the local-density approximation (LDA) or generalized gradient ap- proximation (GGA). In particular, the žexibility of the MT LAPW basis must be enlarged, which is achieved by adding an exceedingly large number of local orbitals to the LAPW basis. e requirement of a žexible LAPW basis to obtain a stable and physical local EXX poten- tial makes the EXX-OEP approach computationally demanding. In this thesis, we explored a novel route circumventing this problem by developing a nite basis correction (FBC), an expression that corrects the conventional formulation of rst-order perturbation theory for the KS wave-function and density response functions and vanishes in the limit of a innite, complete LAPW basis. It arises from the fact that the LAPW basis in the MT spheres it- self depends on the ešective potential as the radial MT functions are the solutions of radial Schrödinger equations taking into account the spherical part of the ešective potential. e change in the radial LAPW functions due to a change in the ešective potential is explicitly calculated by solving radial Sternheimer equations. Finally, the response consists of two con-

Ô¥ç Þ. Summary & Conclusion

tributions: a term which is identical to the conventional perturbation-theory result and the FBC term. e FBC acts only within the MT spheres. As demonstrated for ScN, the FBC leads to numerical stable EXX potentials at much smaller LAPW basis sets. Similarly to the PBEý hybrid functional, the EXX-OEP functional leads to KS transition energies and KS band gaps which are in much better agreement with experiment. More- over, we observe – provided that both basis sets are properly balanced – a very good agree- ment between our all-electron FLAPW and pseudopotential plane-wave transition energies for the semiconductors and insulators C, Si, SiC, Ge, GaAs, and crystalline Ar. is nding is in contradiction to a previous implementation of the EXX functional within the FLAPW method [çÀ], which incited a controversy about the adequacy of the pseudopotential approx- imation in the context of the EXX-OEP method [¥ý, ¥Ô]. Our results, however, conrm the pseudopotential approximation and thus contribute to the clarication of this controversy. In contrast to LDA and GGA, the EXX functional exhibits a derivative discontinuity at integral particle numbers, which contributes to the theoretical fundamental band gap. e latter is given by the sum of the KS band gap and the derivative discontinuity. For the III- V nitrides and ScN, we showed that the discontinuity amounts to several eV so that the sum of EXX KS band gap and EXX discontinuity overestimates the experimental band gap. e theoretical fundamental band gap becomes as large as the Hartree-Fock (HF) gap. is sug- gests that the quantitative agreement of the EXX KS band gap with experiment (at least for the simple semiconductors) is caused by a fortuitous error cancellation between the neglect of the discontinuity and the neglect of orbital correlation.

Furthermore, we have applied the EXX approach to the cubic perovskites CaTiOç, SrTiOç,

and BaTiOç. In comparison with the LDA, we observe an opening of the KS band gap and a slight reduction of the valence band width. Apart from this, the dispersion of the bands is nearly unašected. Quantitatively, the LDA KS gap underestimates the experimental gap for the three perovskites by about ý%. e EXX approach leads to an overestimation of ò %. is pronounced overestimation may result either from an incomplete error cancellation be- tween the neglect of orbital correlation and the neglect of the xc derivative discontinuity or from the dišerence in the crystalline structure assumed in the experimental realization and the theoretical idealization by a cubic lattice. Moreover, we have reported calculations of the antiferromagnetic transition-metal oxides MnO, FeO, and CoO in the rock-salt structure, which are insulating in experiment. LDA as well as GGA erroneously predict CoO and FeO to be metallic. With the EXX functional all three oxides become insulators with an indirect band gap. While for MnO the spin splitting of the Mn çd states is responsible for the formation of the gap, we attribute the gap in FeO and CoO tothefactthatintheEXXcalculations theoccupation ofapreviously unoccupied single

Ô¥¥ particle level causes its energy to fall, whereas LDA and GGA exhibit exactly the opposite behavior. Finally, we have generalized the EXX formalism to metals and performed EXX calculations for the metals Na, Al, and Cu. e calculations for the simple metals Na and Al show that the band structure for these simple metals is equally well described within the EXX and the LDA functional. Small deviations between the EXX and the LDA band structures are seen for the unoccupied spectrum. However, we demonstrate for Cu with its localized d electrons that the LDA and EXX functionals give rise to signicantly dišerent KS band structures. For future work, we propose to combine the orbital-dependent exact exchange energy with an orbital-dependent correlation functional. e latter can be either determined by Görling- Levy perturbation theory [Ôâ], by many-body perturbation theory (MBPT) [âÀ, Þý], or by the adiabatic-connection-žuctuation-dissipation (ACFD) theorem [Ôޗ–ԗý]. From such a functional we might expect (a) a better description of the band gap in the sense that thesum of KS band gap and derivative discontinuity yield a reasonable estimate for the experimental gap, (b) an improved description of complex metals, and (c) a seamless description of the van-der-Waals interaction. Last but not least, the FBC introduced in this thesis is a general concept to evaluate re- sponse functions within electronic structure methods employing potential dependent basis functions. It opens the vista to electric eld response or the calculation of phonon dispersion using incommensurate phonons within the framework of the density functional perturbation theory. Furthermore, the FBC might be generalized to the GW approximation of many body perturbation theory, which involves the calculation of a dynamical density response function and the single-particle Green function. In this way, the slow convergence of GW calculations with respect to the basis set and the number of unoccupied states [ÔÞÞ] might be improved.

Ô¥

A. Incorporation of constraints in the mixed product basis

e mixed product basis (MPB) as introduced in section ¥.ò.ò consists of two sets of func- tions, plane waves in the interstitial region and products of numerical, radial functions times spherical harmonics in the mu›n-tin (MT) spheres, that are simply grouped together. So, the interstitial plane waves abruptly go to zero at the MT sphere boundary and vice versa. In order to guarantee that any quantity represented by the MPB is continuous and contin- uous dišerentiable at the MT sphere boundary, it is desirable to form linear combinations of the originally discontinuous MPB functions. Moreover, the EXX-OEP approach demands that the MPB is orthogonal with respect to a constant function (s. Sec. â.ò). In the following, we will discuss a quite general numerical approach [ԗÔ], which allows to construct the sub- space of the MPB fullling the above mentioned constraints. e approach easily permits to incorporate further constraints in the construction of the MPB.

Our aim is to form linear combinations of the original MPB functions Mk,J r

′ ( ) Mk,I r aJI Mk,J r , (A.Ô) = QJ ( ) ( ) that are continuous and continuous dišerentiable at the MT sphere boundary and orthogonal to a constant function. Each constraint can be understood as an orthogonality demand to the matrix of coe›cient vectors aJI. For example, the demand of orthogonality of the nal MPB ′ Mk,I r with respect to a constant function requires that the integral

( ) ′ M r dçr ý (A.ò) ∫ k,I = ( ) vanishes for all indices I. Combination of Eq. (A.Ô) with (A.ò) results in the following condi- tion for the coe›cient matrix

cJ aJI ý (A.ç) QJ =

ç with cJ ∫ Mk,J r d r. Consequently, each column of the matrix a must be orthogonal to = the constrain vector c to guarantee that Eq. (A.ò) holds. In analogy to that the requirement of ( )

Ô¥Þ A. Incorporation of constraints in the mixed product basis continuity in value and rst radial derivative at the MT sphere boundary leads to additional orthogonality demands on the coe›cient matrix a. To be more precise, for each lm channel and each atom a in the unit cell we obtain two additional orthogonality constraints, one for the requirement of continuity and another for the requirement of continuous dišerentiability at the MT sphere boundary. e coe›cient vector c, which enforces that the interstitial plane wave is continuously matched at the MT sphere boundary of atom a to the MT function with angular momentum l and magnetic quantum number m, is given by

− a MlmP Sa if J almP ∗ = c ¥π il exp iGR j G S Y k + G if J G (A.¥) J ⎪⎧ √Ω a l a lm = ⎪ ( ) = ⎪ ý otherwise, ⎨ ( ) (S S ) ( ) ⎪ ⎪ where Ra denotes the⎩⎪ center of the MT sphere around atom a with radius Sa, Ω is the unit cell volume, and jl x is the spherical Bessel function arising from the Rayleigh decomposition of the interstitial plane wave. A similar constraint vector c arises from the demand of continuity ( ) in the rst radial derivative at the MT boundary of atom a

− d a dr MlmP Sa if J almP ∗ = c ¥π il exp iGR d j G S Y k + G if J G . (A. ) J ⎪⎧ √Ω a dr l a lm = ⎪ ( ) = ⎪ ý otherwise ⎨ ( ) (S S ) ( ) ⎪ ⎪ ò us, we obtain ò ⎩Lmax + Ô constraints for each atom a in the unit cell plus the constraint arising from the removal of a constant function, which amounts in total to N ò L + ( ) c = max Ô òN + Ô constraints. For example, in the case of Si with an angular momentum cutoš of atom ( Lmax ¥ for the MPB and ò atoms in the unit-cell we nally end up with ÔýÔ constraint vectors ) = c i Ô,..., N ÔýÔ . e original MPB functions, represented in terms of a unit matrix i = c = a, are then orthogonalized with respect to the constraints c . In order to simplify the orthog- ( ) i onalization process, we orthogonalize the constraints in a rst step. en, the original MPB functions are orthogonalized with respect to the orthogonalized constraints. AŸerwards, the ′ linear dependentfunctions in the set Mk,I are removed. As there are always as many linear dependent functions as constraints, the nal mixed product basis is of the size N − N , if N is { } c the size of the original basis Mk,I . Consequently, we end up with a rectangular coe›cient matrix a, which is of the dimension N × N − Nc and mediates between the sets Mk,I and ′ { } M . It can be understood as a projector to the subspace of continuous and continuous k,I ( ) { } dišerentiable functions, which are additional orthogonal to a constant function. { } e algorithm easily allows to incorporate any further condition in the MPB at least if it can be formulated as an orthogonality demand on the coe›cient vectors. For example, the

Ô¥— requirement that the slope of the MPB has to vanish at the position of the atomic nuclei Ra.

ԥ
B. Scalar-relativistic approximation

Despite the relatively small (band) energy of the electrons in a solid, relativistic ešects yet can become signicant within the MT spheres close to the atomic nuclei, where the potential is deep and the electrons exhibit a large kinetic energy. Koelling and Harmon [ԗò] devel- oped an approximation, the scalar relativistic approximation (SRA), for the fully-relativistic Dirac equation with a spherical external potential which allows to incorporate relativistic ef- fects in the full-potential linearized augmented-plane-wave (FLAPW) method. In the SRA kinematic relativistic ešects are fully taken into account, whereas the spin-orbit interaction ò ò is neglected. Due to the neglect of the spin-orbit term S , Sz, L , and Lz commute with the scalar relativistic equation and form, in turn, a set of good quantum numbers. e scalar relativistic wave function can be written as

Ô pσ r Y rˆ χσ ϕσ r l lm , (B.Ô) lm = r qσ r Y rˆ χσ ⎛ l ( ) lm( ) ⎞ ( ) σ ⎝ ( ) ( ) ⎠ σ where χ is a ò-component spin-up or spin-down spinor. e radial functions pl r and qσ r are the solution of the coupled radial dišerential equations l ( ) ( ) dpσ r pσ r l òM r, E cqσ r + l (B.ò) dr = l r σ ( ) ( ) σ dq r Ô l l + Ô q r l ( ) ( )+ V σ r − E pσ r − l . (B.ç) dr = c òM r, E rò l r ( ) ( ) ( )  ( ) ( ) with the abbreviation ( ) Ô M r, E Ô + E − V σ r , (B.¥) = òcò the speed of light c, and the spherical,( ) spin-dependent[ ( potential)] V σ r . A derivation of Eqs. (B.ò) and (B.ç) is discussed in detail in Refs. ԗò and ԗç. ( ) aσ In order to incorporate the SRA in the MT LAPW basis the radial function ul p Ô r ( = ) uaσ r is replaced by the ò-component function l ( )

( ) aσ aσ → Ô pl r ul p Ô r aσ (B. ) ( = ) r q r ⎛ l ( ) ⎞ ( ) ⎝ ( ) ⎠

Ô Ô B. Scalar-relativistic approximation which is obtained from Eqs. (B.ò) and (B.ç) by substituting the spherical potential V σ r and aσ the energy E by the spherical, ešective Kohn-Sham (KS) potential Veš,ý r of atom a and the aσ aσ aσ( ) energy parameter El , respectively. In analogy, the energy derivative ul p ò r u˙l r is (()= ) replaced by the two components ( )
( ) ˙aσ aσ → Ô pl r ul p ò r aσ , (B.â) ( = ) r q˙ r ⎛ l ( ) ⎞ ( ) ˙aσ aσ ⎝ ( ) ⎠aσ aσ where pl r q˙l r denotes the energy derivative of pl r ql r , which is given by the set of equations ( ) [ ( )] ( ) [ ( )] d p˙aσ r Ô p˙aσ r l òM r, Eaσ cq˙aσ r + qaσ r + l (B.Þ) dr = l l c l r aσ ( ) ( ) dq˙ r Ô l l + Ô l ( ) ( +) V aσ r( −)Eaσ p˙aσ r dr = c òM r, Eaσ rò eš,ý l l ( ) ( l ) Ô l l + Ô ( ) q˙aσ( r) − Ô +( ) paσ r − l . (B.—) c ¥còM r, Eaσ òrò l r ( l) ( )  ( ) e matching of the scalar-relativistic two-component( ) MT functions to the non-relativistic, one-component interstitial plane wave at the MT sphere boundary is carried out only for aσ ˙aσ the large component, that corresponds to the functions pl r and pl r , while the small component remains unmatched. e notation large and small component goes back to the aσ aσ ( ) ( ) fact that the radial function ql r is connected with pl r according to Eq. (B.ò) by

( ) Ô d Ô ( ) qσ r − pσ r . (B.À) l = òM r, E c dr r l ( ) ‹  ( ) aσ ( ) aσ us, the function ql r is a factor of Ô c smaller than pl r . is justies not to match the small component at the MT sphere boundary (r S ), because it is very small, nearly zero at ( ) ~ = a ( ) Sa. We note that in the non-relativistic limit (c → ∞) the set of Eqs. (B.ò) and (B.ç) reduces aσ to the radial Schrödinger equation (¥.ò) for the large component pl r , while the small component qaσ r is exactly zero. Accordingly p˙aσ r is given in this limit by Eq. (¥.Ôý) and l l ( ) q˙aσ r vanishes. l ( ) ( ) Due( ) to the SRA the construction of the local orbitals changes in a similar manner.

Finite-basis set correction e nite-basis set correction considers the response of the LAPW basis functions due to a change in the spherical ešective potential explicitly. In section â.Þ we have only given the

Ô ò equations to calculate the response of the radial LAPW functions in the non-relativistic limit. Here, we develop the corresponding equations in the scalar-relativistic approximation. Lin- earizing Eqs. (B.ò) and (B.ç) in the potential results in a set of dišerential equations for the aσ′ response of the large and small component due to a given spherical perturbation V eš,ý r ( ) aσ′ aσ′ dp r ′ p r Ô ′ ′ l òM r, Eaσ cqaσ r + l + Eaσ − V aσ r qaσ r (B.Ôý) dr = l l r c l eš,ý l aσ′( ) ( ) aσ′ dq r Ô l l + Ô ′ q r l ( ) (+)V aσ r − Eaσ
paσ r − l( ) ( ) dr = c òM r, Eaσ rò eš,ý l l r ( ) ( l ) ( ) Ô l l + Ô ( ) ′ ′ ( ) − Ô +( ) Eaσ − V aσ r paσ r . (B.ÔÔ) c ¥M r, Eaσ òròcò l eš,ý l ( l ) 
( ) ( ) aσ′ ( ) aσ aσ′ While pl r denotes the response of the large component pl r , ql r describes the aσ aσ′ corresponding change in the small component ql r . El is the change in the parameter ( ) ′ ( ) ( ) Eaσ caused by the perturbation V aσ . It is computed according to Eq. (â.¥ç), where uaσ r l eš,ý ( ) pý has to be replaced by the two component relativistic function. ( ) ˙aσ In a similar way, the dišerential equations for the change in the energy derivatives pl r and q˙aσ r are obtained from Eqs. (B.Þ) and (B.—) l ( ) aσ′ aσ′ d p˙ (r ) ′ Ô ′ ′ ′ p˙ r l òM r, Eaσ cq˙aσ r + Eaσ − V aσ r q˙aσ r + qaσ r + l (B.Ôò) dr = l l c l eš,ý l l r aσ ( ) aσ′ ( ) dq˙ r q˙ r Ô l l + Ô ′ l − l( +) ( ) ™
+ V aσ r( −)Eaσ (p˙)aσ r ( )ž dr = r c òM r, Eaσ rò eš,ý l l ( ) ( ) ( l ) Ô ll + Ô ′ ( ) ′ ′ ( ) − Ô + ( ) paσ r + Eaσ − V aσ r p˙aσ r c ¥còM r, Eaσ òrò l l eš,ý l ( l) Ô  òl l + Ô
′ ( )′
( ) ( ) + ( ) Eaσ − V aσ r paσ r . (B.Ôç) c —c¥M r, Eaσ çrò l eš,ý l ( l ) 
( ) ( ) ( )

Ô ç

List of Abbrevations

AFM antiferromagnetic. APW augmented plane-wave.

BZ Brillouin zone.

CI conguration interaction.

DF density functional. DFT density-functional theory. DOS density of states.

EXX exact exchange.

FBC nite basis correction. FFT fast Fourier transform. FLAPW full-potential linearized augmented-plane-wave. FM ferromagnetic.

GGA generalized gradient approximation. gKS generalized Kohn-Sham.

HEG homogeneous electron gas. HF Hartree-Fock.

IBZ irreducible Brillouin zone. IPW interstitial plane wave. IR interstitial region.

KKR Korringa-Kohn-Rostocker. KS Kohn-Sham.

LAPW linearized augmented-plane-wave. LDA local-density approximation.

Ô List of Abbrevations

LMTO linearized mu›n-tin orbital.

MBPT many-body perturbation theory. MFA mean-eld approximation. MLWF maximally localized Wannier function. MP Møller-Plesset. MPB mixed product basis. MT mu›n-tin. MTO mu›n-tin orbital.

nn nearest neighbor. nnn next-nearest neighbor.

OEP optimized ešective potential. OPM optimized potential method.

PAW projector augmented-wave.

RPA random-phase approximation.

SCF self-consistent eld. SRA scalar relativistic approximation.

TM transition metal.

xc exchange-correlation.

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ÔÞý Acknowledgement

First and foremost, I would like to thank my supervisor Prof. Dr. Stefan Blügel. He has given me the possibility to write this PhD thesis at the Forschungszentrum Jülich. His great support, encouragement, and numerous ideas essentially contribute to the success of this work. I am truly indebted to Prof. Dr. Andreas Görling for reviewing this thesis and explaining me during a four month stay in his working group at the University Erlangen-Nuremberg the numerical nitty gritties of the EXX-OEP formalism. I will not forget the granted hospitality and pleasant inclusion in his working group. Next, I would like to thank Prof. Dr. Carsten Honerkamp for his willingness to be the third referee of this thesis at short notice. Furthermore, this thesis would have been impossible without the many discussions with and the steady support of my direct supervisor Dr. Christoph Friedrich. I thank Dr. Gustav Bihlmayer for his guidance concerning the FLAPW method and the Fu¶§ code. Moreover, I would like to thank all the members of our institute Quantum theory of Materials. ey made my stay in Jülich in academic and personal matters unforgettable. I am obliged to Dr. Christoph Friedrich, Dr. Ersoy Şaşıoğlu and Gregor Michalicek for proof- reading this manuscript with outmost care. Finally, I would like to thank my family, in particular my parents, for their unbreakable faith in me and their steady support. I hope they will accompany me for a long period of time. Dinah, you had to bear numerous weekends, at which I spend more time at the desk than with you. ank you for your inestimable patience and moral support!

Jülich, June òýÔÔ Markus Betzinger

ÔÞÔ

Publications

Parts and results of the thesis have already been published (the status of each article is indi- cated):

• M. Betzinger, C. Friedrich, and S. Blügel, Hybrid functionals within the all-electron FLAPW method: Implementation and applications of PBEý, published in Phys. Rev. B —Ô, ÔÀ ÔÔÞ (òýÔý)

• M. Betzinger, C. Friedrich, S. Blügel, and A. Görling, Local exact exchange potentials within the all-electron FLAPW method and a comparison with pseudopotential results, published in Phys. Rev. B —ç, ý¥ Ôý (òýÔÔ)

• M. Betzinger, C. Friedrich, A. Görling, and S. Blügel, Precise response functions in all- electron methods: Application to the optimized-ešective-potential approach, submitted to PRB

• M. Schlipf, M. Betzinger, C. Friedrich, M. Ležaić, and S. Blügel, HSE hybrid functional within the FLAPW method and its application to GdN, published in Phys. Rev. B —¥, Ôò Ô¥ò (òýÔÔ)

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