Electron-Hole Problem

Abstract

In order to specify the Bethe-Salpeter equation of the macroscopic polarization function PM to account for the relevant excitonic effects in optical spectra, spin effects are treated within the collinear approximation. The quasiparticle wave functions are used to represent the space dependence of the polarization function. The resulting integral equation can be formally solved by means of the eigenvectors and eigenvalues of a generalized eigenvalue problem for pairs of particles. Together with the matrix elements of the optical transition operator calculated with the single-particle wave functions, the pair eigenvectors determine the optical oscillator strengths whereas the eigenvalues can be interpreted as the oscillator frequencies. The generalized eigenvalue problem is ruled by a matrix in the single-particle pair states, whose diagonal blocks represent the pairs and antipairs while the off-diagonal blocks describe their coupling. The reduction of the complexity within the Tamm-Dancoff approximation leads to a Hermitian electron-hole-pair problem with screened Coulomb attraction and unscreened electron-hole exchange. The resulting Hamiltonian represents a high-rank matrix which asks for efficient strategies to solve the eigenvalue problem or to compute directly the dielectric function.