$GW$ quasiparticle corrections to the $\mathrm{LDA}+U∕\mathrm{GGA}+U$ electronic structure of bcc hydrogen

In this paper, we study the quasiparticle electronic structure of atomic hydrogen in the body-centered cubic structure for various densities. We employ the $GW$ approach to compute the electron self energy. For this model system, we use the local density approximation $\left(\mathrm{LDA}\right)+U$/generalized gradient approximation $\left(\mathrm{GGA}\right)+U$ method as the mean-field solution starting point, which is known to work better than LDA/GGA for systems with strongly correlated electrons. In the low-density insulating phase, we find that the calculated $GW$ quasiparticle gap is quite insensitive to the value of the on-site repulsive $U$ employed over a wide range of physically reasonable values. Moreover, our result for the electronic gap agrees with the measured difference between ionization energy and electron affinity in the atomic limit.

Figure 1

${\mid {\psi }_{\uparrow }\mid }^2$ for the valence band at $\Gamma$ using $\mathrm{LDA}+U$ for $r_s=2.8a_B$ and different values of $U$, plotted along the [111] direction. Solid curve: LDA result $(U=0.0)$. Dashed curves: $\mathrm{LDA}+U$ for different values of $U$ (in eV).

Figure 2

The valence and conduction bands of solid bcc hydrogen calculated within $\mathrm{LDA}+U$ (solid line) and $GW∕\mathrm{LDA}+U$ (dashed line) for $r_s=3.3a_B$ and $U=4.0\mathrm{eV}$.

Figure 3

The Kohn–Sham and quasiparticle gap versus the parameter $U$ using mean field $\mathrm{LDA}+U$ (left) and $\mathrm{GGA}+U$ (right) for a density corresponding to $r_s=3.3a_B$.

Figure 4

The quasiparticle band gap versus the electron density parameter $r_s$.

Figure 5

The antiferromagnetic moment per site versus the electron density parameter $r_s$.

Figure 6

(a) The dependence of the dielectric constant ${ϵ}_0=1∕{ϵ}_{\mathit{G}=0,{\mathit{G}}^{\prime }=0}^{-1}(q=0,\omega =0)$ on the $U$ parameter for $r_s=2.8a_B$. (b) Plot of the expression $({ϵ}_0-1)∕{\omega }_p^2$ versus the inverse of the average gap squared. The dashed line is the expression $y=x$.