Self-interaction in Green’s-function theory of the hydrogen atom

Atomic hydrogen provides a unique test case for computational electronic structure methods, since its electronic excitation energies are known analytically. With only one electron, hydrogen contains no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors introduced by certain computational methods. In this paper we focus on many-body perturbation theory (MBPT) in Hedin’s $GW$ approximation. While the Hartree-Fock and the exact MBPT self-energy are free of self-interaction, the correlation part of the $GW$ self-energy does not have this property. Here we use atomic hydrogen as a benchmark system for $GW$ and show that the self-interaction part of the $GW$ self-energy, while nonzero, is small. The effect of calculating the $GW$ self-energy from exact wave functions and eigenvalues, as distinct from those from the local-density approximation, is also illuminating.

Figure 1

(Color online) Radial probability distributions of the hydrogen $1s$ state: the quasiparticle wave function deviates only slightly from the exact wave function, when the latter is used as a starting point $(\text{exact}+GW)$. The LDA wave function, on the other hand, is more delocalized as a result of the inherent self-interaction. Adding quasiparticle corrections $(\mathrm{LDA}+GW)$ brings the resulting quasiparticle wave function slightly closer to the exact one again. The inset shows the difference to the exact wave function.