Diatomic molecule as a testbed for combining DMFT with electronic structure methods such as $GW$ and DFT

We implemented a combination of DMFT and $GW$ in its fully self-consistent way, one shot $GW$ approximation, and quasiparticle self-consistent scheme, and studied how well these combined methods perform on a ${\mathrm{H}}_2$ molecule as compared to more established methods such as LDA+DMFT. We found that most flavors of $GW+\mathrm{DMFT}$ break down in a strongly correlated regime due to causality violation. Among $GW+\mathrm{DMFT}$ methods, only the self-consistent quasiparticle $GW+\mathrm{DMFT}$ with static double-counting, and a method with causal double-counting, correctly recover the atomic limit at large H-atom separation. While some flavors of $GW+\mathrm{DMFT}$ improve the single-electron spectra as compared to LDA+DMFT, the total energy is best predicted by LDA+DMFT, for which the exact double-counting is known, and is static.

Figure 1

Lowest two orbital energies of ${\mathrm{H}}_2^{+}$ cation as a function of $R$, which are taken to define the DMFT projector [Eq. (40)].

Figure 2

Ground state energy of $GW+\mathrm{DMFT}$ vs $R$ presented with the HF, $GW$, LDA+DMFT, and exact result for comparison. (Inset) We also added the total energy results of $G_0{W}_0$+DMFT without charge self-consistency (dashed lines).

Figure 3

Imaginary part of (a) the hybridization function $\mathrm{\Delta }\left(i\omega \right)$, and its component (b) ${\mathrm{\Delta }}_R\left(i\omega \right)$ and (c) $\mathrm{\Delta }{\mathrm{\Sigma }}_{DC}^{GW}\left(i\omega \right)$ of Eq. (52). For the two cases ($R=4.0$ and 5.0) in which the causality break down, we took the result of the first iteration.

Figure 4

Total energy result of $GW+\mathrm{DMFT}$ with causal double-counting (CDC) scheme.

Figure 5

Spectral function for $R=1.4$ where the exact IE is $-1.206\text{a.u}$. Each error bar is presented in the legend. (Upper panel) $GW$, LDA, HF, and LDA+DMFT for comparison. (Lower panel) $GW+\mathrm{DMFT}$ schemes with different self-consistent conditions. The dashed lines indicate the result without charge self-consistency.

Figure 6

Spectral function of $GW+\mathrm{DMFT}$ (CDC) scheme at $R=5.0$ compared to $GW$, HF, LDA, LDA+DMFT, and the exact result.

Figure 7

Ground state energy of the $\mathrm{QS}GW$+DMFT results with two different schemes of double-counting. The inset shows the magnification around the equilibrium distance.

Figure 8

$\mathrm{QS}GW$+DMFT spectral results with two different double-counting schemes at $R=1.4$, compared with $GW$, LDA, HF, and LDA+DMFT.

Figure 9

Spectral function of HF, LDA, $GW$, LDA+DMFT, and $\mathrm{QS}GW$+DMFT for $R=5.0$.