From f0525b248310c45f2e0c88fc655305fe8c073956 Mon Sep 17 00:00:00 2001 From: pfloos Date: Wed, 15 Feb 2023 17:32:17 -0500 Subject: [PATCH] change of notations --- Manuscript/SRGGW.tex | 47 ++++++++++++++++++++++++++------------------ 1 file changed, 28 insertions(+), 19 deletions(-) diff --git a/Manuscript/SRGGW.tex b/Manuscript/SRGGW.tex index 4e44540..6c925d2 100644 --- a/Manuscript/SRGGW.tex +++ b/Manuscript/SRGGW.tex @@ -184,13 +184,13 @@ The matrix elements of $\bSig(\omega)$ have the following closed-form expression \begin{equation} \label{eq:GW_selfenergy} \Sigma_{pq}(\omega) - = \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} - + \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}, + = \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} + + \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta}, \end{equation} where $\eta$ is a positive infinitesimal and the screened two-electron integrals are \begin{equation} \label{eq:GW_sERI} - W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia,\nu}, + W_{pq}^{\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia}^{\nu}, \end{equation} with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as \begin{equation} @@ -253,9 +253,11 @@ which was first introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfg The corresponding matrix elements are \begin{equation} \label{eq:sym_qsGW} - \Sigma_{pq}^{\qsGW} = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} ) W_ {p,r\nu} W_{q,r\nu}. + \Sigma_{pq}^{\qsGW} + = \frac{1}{2} \sum_{r\nu} \qty[ \frac{\Delta_{pr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + + \eta^2} +\frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2} ] W_ {pr}^{\nu} W_{qr}^{\nu}. \end{equation} -with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level). +with $\Delta_{pr}^{\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level). One of the main results of the present manuscript is the derivation, from first principles, of an alternative static hermitian form for the $GW$ self-energy. Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level. @@ -371,9 +373,9 @@ where the 2h1p and 2p1h matrix elements are \end{subequations} and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})] \begin{align} - W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu}, + W^\text{2h1p}_{p,i\nu} & = W_{pi}^{\nu}, & - W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}. + W^\text{2p1h}_{p,a\nu} & = W_{pa}^{\nu}. \end{align} The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021} \begin{equation} @@ -492,10 +494,13 @@ Equation \eqref{eq:F0_C0} implies \end{align} and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields \begin{equation} - W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) \titou{e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}} + W_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s} \end{equation} -At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero. -Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals. +At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while, +\begin{equation} + \lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0. +\end{equation} +Therefore, $W_{pq}^{\nu(1)}(s)$ are genuine renormalized two-electron screened integrals. It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}). %///////////////////////////% @@ -522,8 +527,8 @@ with elements \label{eq:SRG-GW_selfenergy} \begin{split} \widetilde{\bSig}_{pq}(\omega; s) - &= \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-(\Delta_{pi\nu}^2 + \Delta_{qi\nu}^2) s} \\ - &+ \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-(\Delta_{pa\nu}^2 + \Delta_{qa\nu}^2) s}. + &= \sum_{i\nu} \frac{W_{pi}^{\nu} W_{qi}^{\nu}}{\omega - \epsilon_i + \Omega_{\nu}} e^{-\qty[(\Delta_{pi}^{\nu})^2 + (\Delta_{qi}^{\nu})^2 ] s} \\ + &+ \sum_{a\nu} \frac{W_{pa}^{\nu} W_{qa}^{\nu}}{\omega - \epsilon_a - \Omega_{\nu}}e^{-\qty[(\Delta_{pa}^{\nu})^2 + (\Delta_{qa}^{\nu})^2 ] s}. \end{split} \end{equation} @@ -535,8 +540,10 @@ Collecting every second-order term in the flow equation and performing the block \end{multline} which can be solved by simple integration along with the initial condition $\bF^{(2)}(0)=\bO$ to yield \begin{multline} - F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \\ - \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}]. + F_{pq}^{(2)}(s) + = \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu} + \\ + \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s}]. \end{multline} %%% FIG 1 %%% @@ -556,7 +563,9 @@ At $s=0$, the second-order correction vanishes, hence giving For $s\to\infty$, it tends towards the following static limit \begin{equation} \label{eq:static_F2} - \lim_{s\to\infty} \widetilde{\bF}(s) = \epsilon_p \delta_{pq} + \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}. + \lim_{s\to\infty} \widetilde{\bF}(s) + = \epsilon_p \delta_{pq} + + \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu}. \end{equation} while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero, \ie, \begin{equation} @@ -565,7 +574,7 @@ while the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends t Therefore, the SRG flow continuously transforms the dynamical self-energy $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}^{(2)}(s)$. As illustrated in Fig.~\ref{fig:flow}, this transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}. -For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{p,r\nu} e^{-(\Delta_{pr\nu})^2 s} \approx 0$, meaning that the state is decoupled, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{p,r\nu}(s) \approx W_{p,r\nu}$, that is, the state remains coupled. +For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \gg \Lambda$, then $W_{pr}^{\nu} e^{-(\Delta_{pr}^{\nu})^2 s} \approx 0$, meaning that the state is decoupled from the 1h and 1p configurations, while, for $\abs*{\Delta_{pr}^{\nu}} \ll \Lambda$, we have $W_{pr}^{\nu}(s) \approx W_{pr}^{\nu}$, that is, the state remains coupled. %%% FIG 2 %%% \begin{figure*} @@ -587,9 +596,9 @@ This yields a $s$-dependent static self-energy which matrix elements read \begin{multline} \label{eq:SRG_qsGW} \Sigma_{pq}^{\SRGqsGW}(s) - = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} + = \sum_{r\nu} \frac{\Delta_{pr}^{\nu}+ \Delta_{qr}^{\nu}}{(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2} W_{pr}^{\nu} W_{qr}^{\nu} \\ - \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ]. + \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ]. \end{multline} Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization. Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator. @@ -691,7 +700,7 @@ This behavior as a function of $s$ can be understood by applying matrix perturba Through second order in the coupling block, the principal IP is \begin{equation} \label{eq:2nd_order_IP} - \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{W_{\text{h},i\nu}^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{\text{h},a\nu}^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} + \text{IP} \approx - \epsilon_\text{h} - \sum_{i\nu} \frac{(W_{\text{h}}^{i\nu})^2}{\epsilon_\text{h} - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{(W_{\text{h}}^{a\nu})^2}{\epsilon_\text{h} - \epsilon_a - \Omega_\nu} \end{equation} where $\text{h}$ is the index of the highest occupied molecular orbital (HOMO). The first term of the right-hand side of Eq.~\eqref{eq:2nd_order_IP} is the zeroth-order IP and the two following terms originate from the 2h1p and 2p1h coupling, respectively.