Ab initioGreen’s function formalism for band structures

Abstract

Using the Green's function formalism, an ab initio theory for band structures of crystals is derived starting from the Hartree-Fock approximation. It is based on the algebraic diagrammatic construction scheme for the self-energy which is formulated for crystal orbitals (CO-ADC). In this approach, the poles of the Green's function are determined by solving a suitable Hermitian eigenvalue problem. The method is not only applicable to the outer valence and conduction bands, it is also stable for inner valence bands where strong electron correlations are effective. The key to the proposed scheme is to evaluate the self-energy in terms of Wannier orbitals before transforming it to a crystal momentum representation. Exploiting the fact that electron correlations are mainly local, one can truncate the lattice summations by an appropriate configuration selection scheme. This yields a flat configuration space; i.e., its size scales only linearly with the number of atoms per unit cell for large systems and, under certain conditions, the computational effort to determine band structures also scales linearly. As a first application of the new formalism, a lithium fluoride crystal has been chosen. A minimal basis set description is studied, and a satisfactory agreement with previous theoretical and experimental results for the fundamental band gap and the width of the F 2p valence band complex is obtained.

Quasiparticle band structure of infinite hydrogen fluoride and hydrogen chloride chains

“…The strict second order approximation of the off-diagonal Fock matrix elements and the twoelectron contributions is applied, which is dubbed CO-ADC(2,2) method. 20,26 Setting out from these equations, we make the transition to the corresponding method for molecules, termed ADC(2,2), by eliminating the lattice summations. The configuration space of the correlation calculations is found to converge rapidly; only nearest-neighbor cells need to be regarded to determine the band gap of the chains within 0.008 eV of the value obtained in the largest calculations performed.…”

“…37,38,42,45 Suitable schemes to express the crystal momentum dependent ADC form (7) using Wannier orbitals are discussed in Refs. 20,26. In this work, we exclusively use the fully translational symmetry adapted form in Eq.…”

“…Consequently, all states of the crystal are N 0 -fold degenerate and the band structure matrix has only to be diagonalized at the Γ point to obtain all distinct energy levels of the crystal. 20,26 These levels can be identified with the energy levels of the isolated molecule. By this line of argument, it is justified to drop the lattice sums and lattice vectors totally in the CO-ADC equations to obtain analytical expressions for the local molecular orbital ADC scheme.…”

“…20,23,24,25 To account for electron correlations, an advanced Green's function method, the crystal orbital algebraic diagrammatic construction (CO-ADC) scheme for Wannier orbitals 20,26 is used. It has been devised recently on the basis of the well-established ADC scheme for atoms, molecules, and clusters.…”

A prioriWannier functions from modified Hartree-Fock and Kohn-Sham equations

“…9,10 Moreover, as soon as electron correlations in the ground state and in excited states are regarded, a local representation offers big advantages. 5,8,11,12,13,14,15,16,17,18 Conventionally, Wannier functions are determined a posteriori from Bloch functions utilizing the Wannier transformation. 1,2,3,4,6,7 It is unique up to a unitary matrix, which can be chosen freely to obtain a certain set of Wannier functions, e.g., those which are maximally localized with respect to a given criterion.…”