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Pierre-Francois Loos 2020-07-22 15:50:05 +02:00
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BSEdyn.tex
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@ -524,7 +524,6 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\\ \\
\times \qty[ \frac{1}{\Om{ib}{S} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{S} - \Om{m}{\RPA} + i\eta} ]. \times \qty[ \frac{1}{\Om{ib}{S} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{S} - \Om{m}{\RPA} + i\eta} ].
\end{multline} \end{multline}
One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to their static expression One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to their static expression
\begin{equation} \begin{equation}
\label{eq:Wstat} \label{eq:Wstat}
@ -535,7 +534,7 @@ One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the ma
evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit. evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit.
Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{S} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{S}{})$ has no resonances. Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{S} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{S}{})$ has no resonances.
This property holds for a few low lying $\Om{s}{}$ excitations but special care must be taken for higher ones. This property holds for low-lying excitations but special care must be taken for higher ones.
Furthermore, $\Om{ib}{S}$ and $\Om{ja}{S}$ are necessarily negative quantities for in-gap low-lying BSE excitations. Furthermore, $\Om{ib}{S}$ and $\Om{ja}{S}$ are necessarily negative quantities for in-gap low-lying BSE excitations.
Thus, we have $\abs*{\Om{ib}{S} - \Om{m}{\RPA}} > \Om{m}{\RPA}$. Thus, we have $\abs*{\Om{ib}{S} - \Om{m}{\RPA}} > \Om{m}{\RPA}$.
As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies. As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies.
@ -743,7 +742,7 @@ The $GW$ calculations performed to obtain the screened Coulomb operator and the
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations. Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected. These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}. Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
Note that, for the present (small) molecular systems, {\GOWO}@HF and ev$GW$@HF yield similar quasiparticle energies. Note that, for the present (small) molecular systems, {\GOWO}@HF and ev$GW$@HF yield similar quasiparticle energies and fundamental gap.
Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$. Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions. As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations. Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.