Helium Atom Excitations by the $GW$ and Bethe-Salpeter Many-Body Formalism

The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body $GW$ approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the $GW$ and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and $GW$ approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.

Figure 1

He atom excitation spectrum. The 0 of the energy is set to the ground state $1{}^{1}S$. (Inset) $2{}^{1}P$ electron-averaged hole (blue) and hole-averaged electron (red) distributions, isosurfaces taken at $5\times {10}^{-4}\mathrm{a}.\mathrm{u}.$.