diff --git a/Manuscript/MRGW.tex b/Manuscript/MRGW.tex index c25f50d..46bc607 100644 --- a/Manuscript/MRGW.tex +++ b/Manuscript/MRGW.tex @@ -1,4 +1,4 @@ -\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress,onecolumn]{revtex4-1} +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx} \usepackage[version=4]{mhchem} @@ -148,26 +148,26 @@ Here comes the introduction. In the case of {\GOWO}, the quasiparticle equation reads \begin{equation} \label{eq:qp_eq} - \eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0 + \eps{p}{} + \SigC{p}(\omega) - \omega = 0 \end{equation} -where $\eps{p}{\HF}$ is the HF one-electron energy of the spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is +where $\eps{p}{}$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is \begin{equation} \label{eq:SigC} \SigC{p}(\omega) - = \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}} - + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}} + = \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{} + \Om{m}{}} + + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{} - \Om{m}{}} \end{equation} where \begin{equation} - \ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA + \ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m} \end{equation} -are the screened two-electron repulsion integrals where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system: +are the screened two-electron repulsion integrals where $\Om{m}{}$ and $\bX{m}{}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation: \begin{equation} - \bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA} + \bA{}{\RPA} \cdot \bX{m}{} = \Om{m}{\RPA} \bX{m}{} \end{equation} with \begin{equation} - A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj} + A_{ia,jb}^{} = (\eps{a}{} - \eps{i}{}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj} \end{equation} and \begin{equation} @@ -183,7 +183,7 @@ with the following sum rules: \begin{align} \sum_{s} Z_{p,s} & = 1 & - \sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF} + \sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{} \end{align} As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution: @@ -195,7 +195,7 @@ with \label{eq:Hp} \bH^{(p)} = \begin{pmatrix} - \eps{p}{\HF} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} + \eps{p}{} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} \\ \T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\ @@ -203,27 +203,38 @@ with \end{pmatrix} \end{equation} and where the expressions of the 2h1p and 2p1h blocks reads +\begin{subequations} \begin{align} - C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{\HF} + \eps{J}{\HF} - \eps{A}{\HF}) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK} + \label{eq:C2h1p} + C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{} + \eps{J}{} - \eps{A}{}) \delta_{JL} \delta_{AC} - 2 \ERI{JA}{CL} ] \delta_{IK} \\ - C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{\HF} + \eps{B}{\HF} - \eps{I}{\HF}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD} + \label{eq:C2p1h} + C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{} + \eps{B}{} - \eps{I}{}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD} \end{align} +\end{subequations} with the following expressions for the coupling blocks: +\begin{subequations} \begin{align} + \label{eq:V2h1p} V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL} - & + \\ + \label{eq:V2p1h} V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC} \end{align} +\end{subequations} +Here, we use lower case letters for the electronic configurations belonging to the reference model state and upper case letters for the external determinants (\ie, the perturbers). + By solving the secular equation \begin{equation} \det[ \bH^{(p)} - \omega \bI ] = 0 \end{equation} we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, -\begin{equation} +\begin{multline} \SigC{p}(\omega) = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + \\ + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} -\end{equation} +\end{multline} with \begin{equation} \label{eq:Z_proj} @@ -236,9 +247,9 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $ \label{eq:Hp_qia} \bH^{(p,qia)} = \begin{pmatrix} - \eps{p}{\HF} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} + \eps{p}{} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}} \\ - V_{qia,p} & C_{qia,qia} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}} + V_{qia,p} & \eps{qia}{} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}} \\ \T{(\bV{p}{\text{2h1p}})} & \T{(\bC{qia}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\ @@ -247,51 +258,68 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $ \end{equation} with new blocks defined as \begin{gather} - V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia} + V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia} \\ - C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ] + \eps{qia}{} \equiv C_{qia,qia} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ] \\ C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK} \\ C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD} \end{gather} +The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has remove the contribution from the 2h1p or 2p1h configuration $qia$. +While $\eps{p}{\HF}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration. +Therefore, when $\eps{p}{\HF}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems. -Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy +Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix \begin{equation} \label{eq:Hp} \bSigC{p,qia}(\omega) = \begin{pmatrix} - \eps{p}{\HF} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega) + \eps{p}{} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega) \\ - V_{qia,p} + \SigC{qia,p}(\omega) & C_{qia,qia} + \SigC{qia}(\omega) + V_{qia,p} + \SigC{qia,p}(\omega) & \eps{qia}{} + \SigC{qia}(\omega) \\ \end{pmatrix} \end{equation} with +\begin{subequations} \begin{gather} - \SigC{p}(\omega) - = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} - + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} + \begin{split} + \SigC{p}(\omega) + & = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + \\ + & + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} + \end{split} \\ + \begin{split} \SigC{qia}(\omega) - = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} - + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} + & = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} + \\ + & + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} + \end{split} \\ + \begin{split} \SigC{p,qia}(\omega) - = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} - + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} + & = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} + \\ + & + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} + \end{split} \\ + \begin{split} \SigC{qia,p}(\omega) - = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} - + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} + & = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} + \\ + & + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} + \end{split} \end{gather} +\end{subequations} Of course, the present procedure can be generalized to any number of states. %%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} %%%%%%%%%%%%%%%%%%%%%% Here comes the conclusion. -d + %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}