wiring notes
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@ -86,7 +86,8 @@
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bvc}{\boldsymbol{v}}
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\newcommand{\bSig}[1]{\boldsymbol{\Sigma}^{#1}}
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\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
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\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}_{#1}^{\text{c}}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}}
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\newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}}
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@ -125,6 +126,7 @@
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\affiliation{\LCPQ}
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\begin{abstract}
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Here comes the abstract.
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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@ -134,44 +136,61 @@
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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%%%%%%%%%%%%%%%%%%%%%%
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Here comes the introduction.
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%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%
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In the case of {\GOWO}, the quasiparticle equation reads
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\begin{equation}
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\label{eq:qp_eq}
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\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
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\end{equation}
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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
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega)
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
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\end{equation}
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where
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\begin{equation}
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\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
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\end{equation}
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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:
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\begin{equation}
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\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
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\end{equation}
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with
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\begin{equation}
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A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
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\end{equation}
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and
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\begin{equation}
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\ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2
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\end{equation}
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The spectral weight of the solution $\eps{p,s}{\GW}$ is given by
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\begin{equation}
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\label{eq:Z}
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0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
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\end{equation}
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with the following sum rules:
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\begin{align}
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\sum_{s} Z_{p,s} & = 1
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&
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\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
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\end{align}
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As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
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\begin{equation}
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\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
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\end{equation}
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with
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\begin{equation}
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\label{eq:Hp}
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\bH^{(p)} =
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@ -183,83 +202,96 @@
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\T{(\bV{p}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}}
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\end{pmatrix}
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\end{equation}
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and where the expressions of the 2h1p and 2p1h blocks reads
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\begin{align}
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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}
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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}
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\\
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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}
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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}
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\end{align}
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with the following expressions for the coupling blocks:
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\begin{align}
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V^\text{2h1p}_{p,klc} & = \sqrt{2} \ERI{pk}{cl}
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V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
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&
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V^\text{2p1h}_{p,kcd} & = \sqrt{2} \ERI{pd}{kc}
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V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
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\end{align}
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By solving the secular equation
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\begin{equation}
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\det[ \bH^{(p)} - \omega \bI ] = 0
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\end{equation}
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we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
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\begin{equation}
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\SigC{p}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{equation}
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with
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\begin{equation}
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\label{eq:Z_proj}
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Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
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\end{equation}
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In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
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Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$)
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\begin{equation}
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\label{eq:Hp_qia}
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\bH^{(p,qia)} =
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\begin{pmatrix}
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\eps{p}{\HF} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\\
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V_{qia,p} & C_{qia,qia} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}}
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\\
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\T{(\bV{p}{\text{2h1p}})} & \T{(\bC{qia}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
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\\
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\T{(\bV{p}{\text{2p1h}})} & \T{(\bC{qia}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}}
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\end{pmatrix}
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\end{equation}
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with new blocks defined as
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\begin{gather}
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V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia}
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\\
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C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ]
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\\
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C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
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\\
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C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
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\end{gather}
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Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy
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\begin{equation}
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\label{eq:Hp}
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\bH^{(P,QIA)} =
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\bSigC{p,qia}(\omega) =
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\begin{pmatrix}
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\eps{P}{\HF} & W_{P,QIA} & \bW{P}{\text{2h1p}} & \bW{P}{\text{2p1h}}
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\eps{p}{\HF} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega)
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\\
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W_{QIA,P} & D_{QIA,QIA} & \bD{QIA}{\text{2h1p}} & \bD{QIA}{\text{2p1h}}
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V_{qia,p} + \SigC{qia,p}(\omega) & C_{qia,qia} + \SigC{qia}(\omega)
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\\
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\T{(\bW{P}{\text{2h1p}})} & \T{(\bD{QIA}{\text{2h1p}})} & \bD{}{\text{2h1p}} & \bO
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\\
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\T{(\bW{P}{\text{2p1h}})} & \T{(\bD{QIA}{\text{2p1h}})} & \bO & \bD{}{\text{2p1h}}
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\end{pmatrix}
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\end{equation}
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with
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\begin{gather}
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W_{P,QIA} = \sqrt{2} \ERI{PQ}{IA}
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\SigC{p}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\\
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D_{QIA,QIA} = \text{sgn}(\eps{Q}{\HF} - \mu) \qty[ \qty(\eps{Q}{\HF} + \eps{A}{\HF} - \eps{I}{\HF} ) + 2 \ERI{IA}{IA} ]
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\SigC{qia}(\omega)
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= \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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+ \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\\
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D_{QIA,klc}^\text{2h1p} = - 2 \ERI{IA}{cl} \delta_{Qk}
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\SigC{p,qia}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\\
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D_{QIA,klc}^\text{2p1h} = + 2 \ERI{IA}{kc} \delta_{Qd}
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\SigC{qia,p}(\omega)
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= \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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+ \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{gather}
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Of course, the present procedure can be generalized to any number of states.
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\begin{equation}
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\label{eq:Hp}
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\bH^{(P,QIA)} =
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\begin{pmatrix}
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\eps{P}{\HF} + \SigC{P}(\omega) & W_{P,QIA} + \SigC{P,QIA}(\omega)
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\\
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W_{QIA,P} + \SigC{QIA,P}(\omega) & D_{QIA,QIA} + \SigC{QIA}(\omega)
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\\
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\end{pmatrix}
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\end{equation}
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with
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\begin{align}
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\SigC{P}(\omega)
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& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})}
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+ \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})}
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\\
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\SigC{QIA}(\omega)
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& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})}
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+ \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})}
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\\
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\SigC{P,QIA}(\omega)
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& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})}
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+ \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})}
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\\
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\SigC{QIA,P}(\omega)
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& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})}
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+ \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})}
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\end{align}
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%%
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Here comes the conclusion.
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d
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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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).}
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