Linearized self-consistent GW approach satisfying the Ward identity

We propose a linearized self-consistent GW approach satisfying the Ward identity. The vertex function derived from the Ward-Takahashi identity in the limit of $\mathit{q}=0$ and $\omega -{\omega }^{\prime }=0$ is included in the self-energy and the polarization function as a consequence of the linearization of the quasiparticle equation. Due to the energy dependence of the self-energy, the Hamiltonian is a non-Hermitian operator and quasiparticle states are nonorthonormal and linearly dependent. However, the linearized quasiparticle states recover orthonormality and fulfill the completeness condition. This approach is very efficient, and the resulting quasiparticle energies are greatly improved compared to the nonlinearized self-consistent GW approach, although its computational cost is not much increased. We show the results for atoms and dimers of Li and Na compared with other approaches. We also propose convenient ways to calculate the Luttinger-Ward functional $\Phi$ based on a plasmon-pole model and calculate the total energy for the ground state. As a result, we conclude that the linearization improves overall behaviors in the self-consistent GW approach.

Figure 1

Schematic flow of the linearized self-consistent GW approach.

Figure 2

Feynman diagram of the Luttinger-Ward functional within the LGW approximation. The solid line and dashed line are the renormalized Green's function and the bare Coulomb interaction, respectively.

Figure 3

Total energies measured from the dissociation limit of ${\mathrm{Li}}_2$ (in eV) as a function of the interatomic distance.