Correlation effects in valence electron spectroscopy of transition metal oxides: many-body perturbation theory and alternative approaches Matteo Gatti La science n’est pas que scientifique (Edgar Morin, Science avec conscience) Da chimico un giorno avevo il potere di sposare gli elementi e di farli reagire, ma gli uomini mai mi riusc`ıdi capire perch´esi combinassero attraverso l’amore. Affidando ad un gioco la gioia e il dolore. Guardate il sorriso guardate il colore come giocan sul viso di chi cerca l’amore. (Fabrizio de Andr´e,Un chimico, Non al denaro non all’amore n´eal cielo) Moi, se dit le petit prince, si j’avais cinquante-trois minutes `ad´epenser, je marcherais tout doucement vers une fontaine... (Antoine de Saint-Exup´ery, Le petit prince) Contents Preface ix 1 Introduction 1 1.1 The many-body problem ..................... 1 1.2 Spectroscopies .......................... 7 1.2.1 Photoemission ...................... 8 1.2.2 Absorption and electron energy loss .......... 13 1.3 Beyond the independent-particle picture ............ 16 1.3.1 From Hartree-Fock to dynamical quasiparticles .... 16 1.3.2 From Wigner and Mott to strong correlations ..... 20 2 Ground-state properties 27 2.1 The variational principle ..................... 27 2.2 Density-functional theory .................... 30 2.2.1 The Kohn-Sham system ................. 32 2.2.2 Approximations of the exchange-correlation potential 34 2.2.3 Optimized effective potential method .......... 36 2.2.4 DFT in practice ..................... 38 3 One-particle excitations 41 3.1 The Green’s function ....................... 41 3.2 Hedin’s equations and the GW approximation ......... 45 3.3 GW in practice .......................... 51 3.3.1 Calculations in the quasiparticle framework ...... 51 3.3.2 Dynamical screening and the frequency integration .. 54 3.3.3 Core-valence interaction ................. 56 3.4 Self-consistent quasiparticle calculations ............ 57 3.5 The COHSEX approximation .................. 59 3.5.1 Self-consistent COHSEX in practice .......... 62 4 Models for strongly correlated systems 67 4.1 LDA+U .............................. 67 4.1.1 What is U? A GW perspective ............. 70 4.2 Dynamical mean-field theory .................. 72 v 4.2.1 DMFT for the Hubbard model ............. 72 4.2.2 DMFT for electronic structure calculations (LDA + DMFT) .......................... 76 4.3 Spectral density-functional theory ............... 78 5 Neutral excitations 83 5.1 Linear response .......................... 84 5.2 Microscopic-macroscopic connection .............. 86 5.3 The Bethe-Salpeter equation .................. 90 5.3.1 BSE in practice ...................... 92 5.4 Time-dependent density-functional theory ........... 95 5.4.1 A kernel from many-body perturbation theory .... 98 5.5 Time-dependent current-density-functional theory ...... 102 6 Vanadium dioxide 105 6.1 The phase transition ....................... 106 6.2 Peierls or Mott-Hubbard? .................... 110 6.3 Measurements of the electronic properties ........... 112 6.3.1 Valence states ....................... 112 6.3.2 Conduction states .................... 115 6.4 LDA ground-state calculations ................. 116 6.4.1 Pseudopotential generation ............... 116 6.4.2 Ground-state atomic structures ............. 119 6.4.3 Kohn-Sham eigenvalues ................. 121 6.4.4 LDA validation of the molecular orbital picture .... 124 6.5 Standard GW calculations .................... 125 6.6 Self-consistent quasiparticle calculations ............ 133 6.7 Comparison with experiments and previous results ...... 141 6.7.1 X-ray absorption spectra ................ 146 6.8 Calculation of electron energy-loss spectra ........... 148 6.8.1 Methodological issues .................. 148 6.8.2 The satellite in the photoemission spectrum of the metal152 7 Effective potentials and kernels for spectroscopy 159 7.1 Generalization of the Sham-Schluter¨ equation ......... 160 7.2 The photoemission potential ................... 164 7.3 Transforming nonlocality into frequency dependence ..... 166 7.3.1 A scissor operator in a two-level system ........ 167 7.3.2 Screened exchange in homogeneous electron gas .... 173 7.4 Spectral density-functional theory ............... 178 7.5 Density-matrix functional theory ................ 179 7.6 Time-dependent density-functional theory ........... 180 7.6.1 The absorption kernel .................. 182 7.7 Time-dependent current-density-functional theory ...... 184 8 Conclusions 189 A Derivation of some equations in MBPT 193 A.1 Equation of motion of the Green’s function .......... 193 A.2 Schwinger’s functional derivative ................ 195 A.3 The Dyson equation ....................... 196 A.4 Derivative of the identity .................... 197 A.5 Hedin’s equations ......................... 197 A.6 How to get a two-point Bethe-Salpeter equation? ....... 199 B Density functional in the Dyson equation 201 C Integral equations 203 C.1 Linearization of the generalized Sham-Schluter¨ equation ... 204 C.2 Screened-exchange self-energy in HEG ............. 206 List of Publications 209 Bibliography 211 List of Figures 231 Notations and units 235 vii Preface Theoretical spectroscopy [1] has become one of the most active research field in condensed-matter physics. In spectroscopic measurements an electronic system is excited by means of an external perturbation and its subsequent response is measured. These measurements give access to a great deal of information about the properties of the system. The calculation of the elec- tronic excitations [2] is hence a very powerful tool to study the properties of the materials. On the one hand, it permits to provide an interpretation of the experimental spectra and organize the information one can extract from the experimental measurements. On the other hand, when experimental results are still missing, reliable theoretical simulations allow one to formu- late predictions about the electronic and optical properties of new materials. For this reason it is essential to adopt parameter-free approaches that can be used as a “theoretical microscope” to understand the physical properties of the materials. In particular, in this thesis I will deal mostly with photoemission spec- troscopy. In a photoemission experiment a photon is absorbed by a sample and an electron is emitted. By measuring the kinetic energy of the electron, one has access to the electronic properties of the system. If the electrons in the sample were independent particles, the emitted electron would give rise just to a delta peak in the experimental spectrum, in correspondence to the one-particle energy level it was occupying in the system. Instead, structures displayed by photoemission spectra are much more complex and hence richer of information. Therefore, their interpretation allows one to gain physical insights on the electronic interactions in the system. In a photoemission experiment the emitted electron leaves a hole (i.e. a positive charge in correspondence to a depletion of negative charge) in the system. The presence of the hole induces a relaxation of the other electrons that screen this new positive charge. So the measured kinetic energy of the emitted electron is different with respect to the independent- particle situation. This in turn leads to a shift and a renormalization of the independent-particle peak, which broadens and looses part of its weight. When in the spectrum a main structure is still identifiable as deriving from the independent-particle peak, it can be still associated to a one-particle-like excitation, a quasiparticle. In this case, one can describe these excitations by ix means of a one-particle Sch¨odinger equation with a complicated effective po- tential, the self-energy, which is a nonlocal, complex and energy-dependent operator. Moreover, the photon energy, besides creating and screening a hole, can be used also to simultaneously induce other excitations in the sys- tem. In this case, other structures, the satellites, appear in the spectrum at lower energies (in this case in fact the electron is emitted with a smaller kinetic energy). When, on the contrary, the effect of the Coulomb interaction is very strong, the independent-particle peak is completely smeared out. The elec- trons are so correlated each other that a picture based on one-particle ex- citations is no more a convenient starting point in the description of the electronic properties of the system. In this case, one generally resorts to models, like the Hubbard model, that have been specifically devised to deal with strongly correlated materials. First-principles methods currently employed in theoretical spectroscopy are the result of a long history. Essentially, on one side they are based on the density-functional theory (DFT), and its extension to time-dependent potentials (TDDFT); on the other side, they make use of the techniques developed in the context of the many-body perturbation theory (MBPT) in the 1960s. In this thesis I have tried to explore new applications of these first- principles theoretical spectroscopy methods to systems where electronic cor- relation plays an important role. In particular, I have considered the elec- tronic properties of vanadium dioxide (VO2), a paradigm for materials where one expects to find a strong influence of the electronic correlations on the properties of the system. The description of the complex phase transition that characterizes vanadium dioxide represents a longstanding problem, in which the importance of electronic correlation