Current-Density Functionals in Extended Systems Arjan Berger Het promotieonderzoek beschreven in dit proefschrift werd uitgevoerd in de theoretische chemie groep van het Materials Science Centre (MSC) aan de Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen. Het onderzoek werd financieel mogelijk gemaakt door de Stichting voor Fundamenteel Onderzoek der Materie (FOM). Arjan Berger, Current-Density Functionals in Extended Systems, Proefschrift Rijksuniversiteit Groningen. c J. A. Berger, 2006. RIJKSUNIVERSITEIT GRONINGEN Current-Density Functionals in Extended Systems Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op maandag 2 oktober 2006 om 16.15 uur door Jan Adriaan Berger geboren op 24 februari 1978 te Steenwijk Promotor: prof.dr.R.Broer Copromotores: dr. R. van Leeuwen dr. ir. P. L. de Boeij Beoordelingscommissie: prof. dr. A. G¨orling prof. dr. J. Knoester prof. dr. G. Vignale ISBN: 90-367-2758-8 voor mijn ouders vi Contents General Introduction xi 0.1 Introduction................................. xi 0.2 OverviewoftheThesis.. .. .. .. .. .. .. .. .. .. .. .. xiv 1 Density-Functional Theory 1 1.1 Introduction................................. 1 1.2 Preliminaries ................................ 1 1.3 Hohenberg-Kohn Theorem for Nondegenerate Ground States ..... 3 1.4 TheFunctionalDerivative . 6 1.5 Kohn-ShamTheory............................. 7 1.6 Hohenberg-Kohn Theorem for Degenerate Ground States . ..... 10 1.7 The Lieb Functional and Noninteracting v-Representability . 13 1.8 Exchange-CorrelationFunctionals. .... 17 2 Time-Dependent (Current-)Density-Functional Theory 21 2.1 Introduction................................. 21 2.2 Preliminaries ................................ 21 2.3 TheRunge-GrossTheorem . 23 2.4 Time-DependentKohn-ShamTheory. 26 2.5 The Adiabatic Local Density Approximation . ... 27 2.6 Time-Dependent Current-Density-Functional Theory . ........ 28 2.7 TheKeldyshActionFunctional . 34 2.8 ExactConstraints ............................. 40 3 Linear Response within TD(C)DFT 43 3.1 Introduction................................. 43 3.2 LinearResponseTheory . .. .. .. .. .. .. .. .. .. .. .. 43 3.3 The LinearResponseKohn-ShamEquations. .. 46 viii CONTENTS 3.4 The Exchange-CorrelationKernelin TDDFT . .. 49 3.5 Exact Constraints within the Linear Response Formulationof TDDFT 50 3.6 Examples of Approximate Exchange-Correlation Kernels ........ 51 3.7 TheUltra-NonlocalityProblem . 52 3.8 The Exchange-CorrelationKernel in TDCDFT . .. 53 3.9 Exact Constraints within the Linear Response Formulation of TDCDFT 56 4 The Vignale-Kohn Functional 59 4.1 Introduction................................. 59 4.2 SummaryoftheMainResults. 59 4.2.1 The Exchange-Correlation Kernel of the Homogeneous Electron Gas ................................. 59 4.2.2 TheWeaklyInhomogeneousElectronGas . 60 4.2.3 The Exhange-Correlation Kernel to First Order in γ ...... 61 4.2.4 The Vignale-Kohn Functional in Real Space . 62 4.3 TheHomogeneousElectrongas . 63 4.4 TDCDFTfortheHomogeneousElectronGas . 66 4.5 TheWeaklyInhomogeneousElectronGas . 68 4.5.1 The Expansion of the Exchange-Correlation Kernel . .... 68 4.5.2 First Order in the Inhomogeneity and the Ward Identity .... 69 4.5.3 The Onsager Relation and the Conservation Laws . 72 4.5.4 Explicit Form of the Exchange-Correlation Kernel to First Or- der in γ ............................... 75 4.5.5 Construction of the Vignale-Kohn Functional in Real Space . 77 h 4.6 The Response Coefficients fxcL,T (ρ,ω).................. 84 5 Analysis of the Viscoelastic Coefficients in the Vignale-Kohn Func- tional: The Cases of One- and Three-Dimensional Polyacetylene 93 5.1 Introduction................................. 94 5.2 Theory.................................... 98 5.2.1 TDCDFT.............................. 98 5.2.2 LinearResponse .......................... 99 5.2.3 TheVignale-KohnFunctional . 102 5.2.4 TheResponseCoefficients . 104 5.3 Implementation............................... 108 5.4 ComputationalDetails . 111 5.5 Results.................................... 112 5.6 Conclusions ................................. 119 CONTENTS ix 6 A Physical Model for the Longitudinal Polarizabilities of Polymer Chains 123 6.1 Introduction................................. 124 6.2 Theory.................................... 127 6.2.1 ExtrapolationMethods . 127 6.2.2 Infinite Quasi-one-dimensional Dielectric Media . ...... 128 6.2.3 TheDielectricNeedle . 130 6.2.4 Approximate Solution of the Fredholm Equation . 134 6.3 ComputationalDetails . 137 6.3.1 PeriodicBoundaryCalculations. 137 6.3.2 Extrapolation. .. .. .. .. .. .. .. .. .. .. .. .. 138 6.4 Results.................................... 138 6.5 Conclusions ................................. 142 7 Analysis of the Vignale-Kohn Current Functional in the Calculation of the Optical Spectra of Semiconductors 147 7.1 Introduction................................. 147 7.2 Theory.................................... 149 7.2.1 TDCDFTLinearResponseEquations . 149 7.2.2 TheVignale-KohnFunctional . 152 h 7.2.3 Limiting behavior of fxcL,T .................... 155 h 7.2.4 Parametrizations for fxcL,T .................... 157 7.3 ComputationalDetails . 158 7.4 Results.................................... 159 7.5 Conclusions ................................. 162 8 Performance of the Vignale-Kohn Functional in the Linear Response of Metals 163 8.1 Introduction................................. 164 8.2 Theory.................................... 166 8.2.1 Time-Dependent Current-Density-Functional Theory ...... 166 8.2.2 LinearResponse . .. .. .. .. .. .. .. .. .. .. .. 167 8.2.3 TheVignale-KohnFunctional . 171 8.2.4 RelativisticCorrections . 174 8.3 Implementation............................... 176 8.4 Results.................................... 178 8.5 Conclusions ................................. 183 x CONTENTS 9 A Nonlocal Current-Density Functional 191 9.1 Introduction................................. 191 9.2 Theory.................................... 193 9.2.1 TDCDFT .............................. 193 9.2.2 Zero-ForceandZero-TorqueTheorems . 194 9.2.3 The Exchange-Correlation Vector Potential of the Homogeneous ElectronGas ............................ 197 9.2.4 A Nonlocal Current-Denstity Functional . 199 9.2.5 The Exchange-Correlation Kernel of the Homogeneous Electron Gas ................................. 201 9.3 Implementation............................... 203 9.3.1 Evaluation of the Exchange-Correlation Vector Potential. 203 9.3.2 Fitfunctions............................. 204 9.4 ComputationalDetails . 207 9.5 Results.................................... 207 9.6 Conclusions ................................. 211 List of Acronyms 213 Samenvatting 215 List of Publications 223 Dankwoord 225 Bibliography 227 General Introduction 0.1 Introduction In quantum chemistry we are interested in the physical properties of many-particle systems like atoms, molecules and solids. These systems consist of electrons, protons and neutrons which are all spin-1/2 fermions. The protons and neutrons together form the nuclei of these systems. A complete (nonrelativistic) description of the physics of these many-particle systems is governed by the quantum-mechanical wave function Ψ(x , x , , x ,t), where n is the number of particles in the system and 1 2 · · · n the coordinates xi consist of the space-coordinate ri and the spin-coordinate σi. This n-particle wave function can be obtained by solving the time-dependent Schr¨odinger equation, ∂ Hˆ (t)Ψ(x , x , , x ,t)= i Ψ(x , x , , x ,t), (1) 1 2 · · · n ∂t 1 2 · · · n where Hˆ (t) is the Hamiltonian which consists of a kinetic energy operator and a potential operator. Its expectation value describes the total energy of the system. The exact form of the potential operator depends on the physical problem at hand. If the potential is time-independent then the Schr¨odinger equation can be solved by the method of separation of variables for separable initial states. The solution is then a simple product of a purely time-dependent function and a function that only depends on the positions of the particles. The solution for the latter function can be obtained from the time-independent Schr¨odinger equation, Hˆ Ψ(x , x , , x )= EΨ(x , x , , x ), (2) 1 2 · · · n 1 2 · · · n where E is an eigenenergy of the n-particle system. The solution for the time- dependent function is simply a phase factor. The expectation values of any oper- ator can therefore be obtained without knowledge of this phase factor since it can- cels out. To obtain these expectation values one can therefore solve Eq. (2) for Ψ(x , x , , x ). However, for many-particle systems this equation is still much too 1 2 · · · n xii General Introduction difficult to solve. A widely used and accepted approximation to simplify this equation is the so-called Born-Oppenheimer approximation. This approximation is based on the large difference in mass of the electrons and nuclei. Since the nuclei are much heavier than the electrons their movement is much slower than that of the electrons. This means that for each instantaneous configuration of the nuclei the electrons are approximately in a stationary state. We can therefore consider the nuclei to provide a fixed frame for the electrons. The Schr¨odinger equation for the electronic system in principle has to be solved for all possible configurations of the nuclei. However, since the nuclei are very localized around