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Pierre-Francois Loos 2022-05-02 16:15:33 +02:00
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Manuscript/MRGW.tex
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@ -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{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
@ -148,26 +148,26 @@ Here comes the introduction.
In the case of {\GOWO}, the quasiparticle equation reads In the case of {\GOWO}, the quasiparticle equation reads
\begin{equation} \begin{equation}
\label{eq:qp_eq} \label{eq:qp_eq}
\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0 \eps{p}{} + \SigC{p}(\omega) - \omega = 0
\end{equation} \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} \begin{equation}
\label{eq:SigC} \label{eq:SigC}
\SigC{p}(\omega) \SigC{p}(\omega)
= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\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}{\HF} - \Om{m}{\RPA}} + \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{} - \Om{m}{}}
\end{equation} \end{equation}
where where
\begin{equation} \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} \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} \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} \end{equation}
with with
\begin{equation} \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} \end{equation}
and and
\begin{equation} \begin{equation}
@ -183,7 +183,7 @@ with the following sum rules:
\begin{align} \begin{align}
\sum_{s} Z_{p,s} & = 1 \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} \end{align}
As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution: 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} \label{eq:Hp}
\bH^{(p)} = \bH^{(p)} =
\begin{pmatrix} \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 \T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
\\ \\
@ -203,27 +203,38 @@ with
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
and where the expressions of the 2h1p and 2p1h blocks reads and where the expressions of the 2h1p and 2p1h blocks reads
\begin{subequations}
\begin{align} \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{align}
\end{subequations}
with the following expressions for the coupling blocks: with the following expressions for the coupling blocks:
\begin{subequations}
\begin{align} \begin{align}
\label{eq:V2h1p}
V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL} V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
& \\
\label{eq:V2p1h}
V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC} V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
\end{align} \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 By solving the secular equation
\begin{equation} \begin{equation}
\det[ \bH^{(p)} - \omega \bI ] = 0 \det[ \bH^{(p)} - \omega \bI ] = 0
\end{equation} \end{equation}
we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
\begin{equation} \begin{multline}
\SigC{p}(\omega) \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{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}})} + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
\end{equation} \end{multline}
with with
\begin{equation} \begin{equation}
\label{eq:Z_proj} \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} \label{eq:Hp_qia}
\bH^{(p,qia)} = \bH^{(p,qia)} =
\begin{pmatrix} \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 \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} \end{equation}
with new blocks defined as with new blocks defined as
\begin{gather} \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,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
\\ \\
C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD} C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
\end{gather} \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} \begin{equation}
\label{eq:Hp} \label{eq:Hp}
\bSigC{p,qia}(\omega) = \bSigC{p,qia}(\omega) =
\begin{pmatrix} \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{pmatrix}
\end{equation} \end{equation}
with with
\begin{subequations}
\begin{gather} \begin{gather}
\begin{split}
\SigC{p}(\omega) \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{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}})}
\\ \\
& + \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) \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{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{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
\end{split}
\\
\begin{split}
\SigC{p,qia}(\omega) \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{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{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
\end{split}
\\
\begin{split}
\SigC{qia,p}(\omega) \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{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{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
\end{split}
\end{gather} \end{gather}
\end{subequations}
Of course, the present procedure can be generalized to any number of states. Of course, the present procedure can be generalized to any number of states.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion. Here comes the conclusion.
d
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{ \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).} 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).}