Based on an exact functional form derived for the three-point vertex function $Gamma$, we propose a self-consistent calculation scheme for the electron self-energy with $Gamma$ always satisfying the Ward identity. This scheme is basically equivalent
to the one proposed in 2001, but it is improved in the aspects of computational costs and its applicability range; it can treat a low-density electron system with a dielectric catastrophe. If it is applied to semiconductors and insulators, we find that the obtained quasiparticle dispersion is virtually the same as that in the one-shot $GW$ approximation (or $G_0W_0$A), indicating that the $G_0W_0$A actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner.
By treating the electron-ion interaction as perturbation in the first-principles Hamiltonian, we have calculated the density response functions of a fluid alkali metal to find an interesting charge instability due to anomalous electronic density fluc
tuations occurring at some finite wave vector ${bi Q}$ in a dilute fluid phase above the liquid-gas critical point. Since $|{bi Q}|$ is smaller than the diameter of the Fermi surface, this instability necessarily impedes the electric conduction, implying its close relevance to the metal-insulator transition in fluid alkali metals.
