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Pierre-Francois Loos 2020-05-27 19:51:13 +02:00
parent f2e09d1532
commit 7822bc83cb
1 changed files with 11 additions and 7 deletions
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BSEdyn.tex
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@ -442,7 +442,7 @@ In Eq.~\eqref{eq:BSE-final},
\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'), \ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'),
\end{equation} \end{equation}
are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
\begin{multline} \begin{multline} \label{eq:wtilde}
\widetilde{W}_{ij,ab}(\Om{s}{}) \widetilde{W}_{ij,ab}(\Om{s}{})
= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) = \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)
\\ \\
@ -507,8 +507,8 @@ with
where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$. where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$.
The RPA matrices $\bA{\RPA}$ and $\bB{\RPA}$ in Eq.~\eqref{eq:LR-RPA} are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}$, and $\bY{m}{}$ are (eigen)vectors of length $\Nocc \Nvir$. The RPA matrices $\bA{\RPA}$ and $\bB{\RPA}$ in Eq.~\eqref{eq:LR-RPA} are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}$, and $\bY{m}{}$ are (eigen)vectors of length $\Nocc \Nvir$.
Because $\Om{m}{\RPA} > 0$, we have The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\begin{multline} \begin{multline}
\widetilde{W}_{ij,ab}( \Om{s}{} ) \widetilde{W}_{ij,ab}( \Om{s}{} )
= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m} = \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}
\\ \\
@ -527,8 +527,10 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
\begin{equation} \begin{equation}
\label{eq:LR-dyn} \label{eq:LR-dyn}
\begin{pmatrix} \begin{pmatrix}
\bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\ \bA{}(\Om{s}{}) & \bB{}(\Om{s}{})
-\bB{}(\titou{-}\Om{s}{}) & -\bA{}(\titou{-}\Om{s}{}) \\ \\
-\bB{}(\titou{-}\Om{s}{}) & -\bA{}(\titou{-}\Om{s}{})
\\
\end{pmatrix} \end{pmatrix}
\cdot \cdot
\begin{pmatrix} \begin{pmatrix}
@ -577,8 +579,10 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
\\ \\
= =
\begin{pmatrix} \begin{pmatrix}
\bA{(0)} & \bB{(0)} \\ \bA{(0)} & \bB{(0)}
-\bB{(0)} & -\bA{(0)} \\ \\
-\bB{(0)} & -\bA{(0)}
\\
\end{pmatrix} \end{pmatrix}
+ +
\begin{pmatrix} \begin{pmatrix}