New derivation working in the general case for F
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Notes/Notes.rty
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Notes/Notes.rty
Executable file
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%============================================================%
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%%% NEWCOMMANDS %%%
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% ============================================================%
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%%% This latex gather all the latex commands that I use to facilitate the writing of electronic structure manuscript, report...
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%%% You can import these commands by adding the following line in you .text
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%%% \input{commands}
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%%% Latin %%%
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\newcommand{\ie}{\textit{i.e.}~}
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\newcommand{\eg}{\textit{e.g.}~}
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\newcommand{\etal}{\textit{et al.}~}
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%%% Operators %%%
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\newcommand{\hH}{\Hat{H}} % Hamiltonian operator
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\newcommand{\HN}{\Hat{\mathnormal{H}}_{\text{N}}} % Normal ordered Hamiltonian
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\newcommand{\Hsim}{\hat{\bar{H}}} % Similarity transformed Hamiltonian
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\newcommand{\hC}{\Hat{C}} % CI operator
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\newcommand{\hT}{\Hat{T}} % Cluster operator
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\newcommand{\T}[1]{\Hat{\mathnormal{T}}_{#1}} % Cluster operator of a given excitation number
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\newcommand{\hsig}{\Hat{\sigma}} % Unitary cluster operator
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\newcommand{\hK}{\Hat{K}} % Anti-hermitian orbital rotation operator
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\newcommand{\hS}{\Hat{S}} % Anti-hermitian CI coefficients rotation operator
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\newcommand{\hP}[1]{\Hat{\mathnormal{P}}_{#1}} % Permutation operators
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\newcommand{\hE}{\Hat{E}} % Spin averaged single excitation operator
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\newcommand{\cre}[1]{a_{#1}^\dagger} % Creation operator
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\newcommand{\ani}[1]{a_{#1}} % Annihilation operator
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\newcommand{\bcre}[1]{b_{#1}^\dagger} % Boson creation operator
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\newcommand{\bani}[1]{b_{#1}} % Boson annihilation operator
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\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} }
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%%% Matrices %%%
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
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\newcommand{\bD}{\boldsymbol{D}}
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\newcommand{\bE}{\boldsymbol{E}}
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\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bJ}{\boldsymbol{J}}
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\newcommand{\bK}{\boldsymbol{K}}
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\newcommand{\bL}{\boldsymbol{L}}
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\newcommand{\bM}{\boldsymbol{M}}
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\newcommand{\bN}{\boldsymbol{N}}
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\newcommand{\bP}{\boldsymbol{P}}
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\newcommand{\bQ}{\boldsymbol{Q}}
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\newcommand{\bR}{\boldsymbol{R}}
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\newcommand{\bS}{\boldsymbol{S}}
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\newcommand{\bT}{\boldsymbol{T}}
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\newcommand{\bU}{\boldsymbol{U}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bc}{\boldsymbol{c}}
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bx}{\boldsymbol{x}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)}
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\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}}
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\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}}
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\newcommand{\FC}[1]{F_{#1}^{\text{C}}} % Core Fock matrix
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\newcommand{\FA}[1]{F_{#1}^{\text{A}}} % Active Fock matrix
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%%% Wave functions %%%
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\SO}[1]{\psi_{#1}}
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%%% Matrix and tensor elements %%%
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\newcommand{\oa}{O_{\alpha}}
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\newcommand{\ob}{O_{\beta}}
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\newcommand{\eri}[2]{\braket{#1}{#2}} % Electron repulsion integral physician notation
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\newcommand{\ceri}[2]{\mleft(#1|#2\mright)} % Electron repulsion integral chemist notation
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\newcommand{\aeri}[2]{\mel{#1}{}{#2}} % Double bar integral
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\newcommand{\kron}[1]{\delta_{#1}} % Kronecker delta
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\newcommand{\cbra}[1]{(#1|} % Chemist bra
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\newcommand{\cket}[1]{|#1)} % Chemist ket
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%%% Mathematics %%%
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\newcommand{\ii}{\mathrm{i}}
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%%% Text acronyms and abbreviations %%%
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\MCSCF}{\text{MCSCF}}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\nuc}{\text{nuc}}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\phRPA}{\text{ph-RPA}}
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\newcommand{\ppRPA}{\text{pp-RPA}}
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\newcommand{\h}{\text{1h}}
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\newcommand{\p}{\text{1p}}
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\newcommand{\hp}{\text{1h+1p}}
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\newcommand{\hhp}{\text{2h1p}}
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\newcommand{\pph}{\text{2p1h}}
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\newcommand{\dRPA}{\text{dRPA}}
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\newcommand{\RPAx}{\text{RPAx}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\x}{\text{x}}
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\newcommand{\GW}{\text{GW}}
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\newcommand{\GF}{\text{GF(2)}}
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\newcommand{\GT}{\text{$GT$}}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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%%% Notations %%%
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\newcommand{\Ne}{N}
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\newcommand{\Norb}{K}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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241
Notes/Notes.tex
241
Notes/Notes.tex
@ -17,7 +17,7 @@
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%%% NEWCOMMANDS %%%
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% ============================================================%
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\input{Commands}
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%\input{Commands}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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@ -287,18 +287,23 @@ In the following, we will focus on the GF(2), GW and GT approximations.
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The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory.
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\begin{align}
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\label{eq:GF2_selfenergy}
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\Sigma_{pq}^{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\
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&+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag
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\Sigma_{pq}^{GF(2)}(\omega) &= \sum_{ija} \frac{W_{pa,ij}W_{qa,ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\
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&+ \sum_{iab} \frac{W_{pi,ab}W_{qi,ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag
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\end{align}
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with
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\begin{equation}
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\label{eq:GF2_sERI}
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W^{\GF}_{pq,rs}= \frac{1}{\sqrt{2}}\aeri{pq}{rs}
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\end{equation}
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On the other hand, the GW self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy.
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\begin{equation}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{\ceri{pi}{v} \ceri{qi}{v}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{am} \frac{\ceri{pa}{v} \ceri{qa}{v}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag
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\Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v} W_{qi,v}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}W_{qa,v}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag
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\end{equation}
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with
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\begin{equation}
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\label{eq:GW_sERI}
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\ceri{pq}{m} = \sum_{ia} \eri{pi}{qa} \qty( \bX_{m}^{\dRPA} + \bY_{m}^{\dRPA} )_{ia} \notag
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W_{pq,v}^\GW = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v}^{\dRPA} + \bY_{v}^{\dRPA} )_{ia}
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\end{equation}
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Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA
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The corresponding self-energies read as
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@ -549,81 +554,59 @@ In the following, upper case indices correspond to the 2h1p and 2p1h sectors whi
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\dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(1),\dagger}\bF{}{(0)} - \bV{}{(1),\dagger}(\bF{}{(0)})^2 - (\bC{}{(0)})^2\bV{}{(1),\dagger} \\
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\dv{\bC{}{(1)}}{s} &= \bO \Longleftrightarrow \bC{}{(1)}(s) = \bC{}{(1)}(0) \Longleftrightarrow \color{red}{\boxed{\color{black}{\bC{}{(1)}(s)= \bO}}}
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\end{align}
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The differential equation for the coupling blocks can be solved in the GF(2) case because in this case $\bC{}{}$ is diagonal (see Appendix~\ref{sec:diagC}).
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However, in the general case this matrix differential equation is not trivial to solve.
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In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_r$.
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The differential equation for the coupling blocks can be solved in the GF(2) case because in this case $\bC{}{}(0)$ is diagonal (see Appendix~\ref{sec:diagC}).
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Inspired by this remark we transform $\bC{}{(0)}$ to its diagonal representation using
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$\bC{}{(0)} = \bU \bD^{(0)} \bU^{-1}$ and insert it in the differential equation for $\bV{}{(1)}$
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\begin{align}
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\label{eq:matrixdiffeq}
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\dv{\bV{r}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \\
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\dv{\bV{r}{(1),\dagger}}{s} &= (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
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\dv{\bV{}{(1)}}{s} &= 2 \bF{}{(0)}\bV{}{(1)}\bU \bD^{(0)} \bU^{-1}- (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}\bU (\bD^{(0)})^2 \bU^{-1}\\
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\dv{\bV{}{(1)}}{s}\bU &= 2 \bF{}{(0)}\bV{}{(1)}\bU \bD^{(0)} - (\bF{}{(0)})^2\bV{}{(1)} \bU - \bV{}{(1)}\bU (\bD^{(0)})^2 \\
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\dv{\bW^{(1)}}{s} &= 2 \bF{}{(0)}\bW^{(1)} \bD^{(0)} - (\bF{}{(0)})^2\bW^{(1)} - \bW^{(1)} (\bD^{(0)})^2
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\end{align}
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where in the last line we have defined the matrix of screened integral.
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Note that in the GF(2) case $\bU = \mathbb{1}$ and thus $\bW = \bV{}{}$.
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\begin{align}
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&(\dv{\bW^{(1)}}{s})_{p,(q,v)} = (2 \bF{}{(0)}\bW^{(1)} \bD^{(0)} - (\bF{}{(0)})^2\bW^{(1)} - \bW^{(1)} (\bD^{(0)})^2)_{p,(q,v)} \notag \\
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&= \sum_{r,(s,x)} 2 F_{pr}^{(0)}W_{r,(s,x)}^{(1)}D_{(s,x),(q,v)}^{(0)} - \sum_{r,s} F_{pr}^{(0)} F_{rs}^{(0)}W_{s,(q,v)}^{(1)} \notag \\
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&- \sum_{(r,x),(s,y)} W_{p,(r,x)}^{(1)}D_{(r,x),(s,y)}^{(0) } D_{(s,y),(q,v)}^{(0)} \notag \\
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&= 2 F_{pp}^{(0)}W_{p,(q,v)}^{(1)}D_{(q,v),(q,v)}^{(0)} - (F_{pp}^{(0)})^2 W_{p,(q,v)}^{(1)} \notag - W_{p,(q,v)}^{(1)} (D_{(q,v),(q,v)}^{(0)})^2 \\
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&(\dv{\bW^{(1)}}{s})_{p,(q,v)} = - (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 W_{p,(q,v)}^{(1)}
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\end{align}
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This equation can be integrated to give
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\begin{equation}
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W_{p,(q,v)}^{(1)}(s) = W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
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\end{equation}
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\begin{itemize}
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\item \textbf{Matrix differential equation}
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A general differential matrix equation of the form
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\begin{equation}
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\dv{\bX}{s} = \bA \bX
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\end{equation}
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admits a solution
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\begin{align}
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\bX(s) &= c_1e^{\lambda_1 s}\bU_1 + c_2e^{\lambda_2 s}\bU_2 + \dots + c_ne^{\lambda_n s}\bU_n \\
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&= \bU \text{diag}(e^{\lambda_i s}) \bc \notag
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\end{align}
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where $c_i$ are coefficients to determine, $\lambda_i$ are the eigenvalues of $\bA$ and $\bU_i$ the corresponding eigenvectors.
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The $c_i$ are determined by the initial condition which gives
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\begin{equation}
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\label{eq:solution_eqdiff}
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\bX(s) = \bU \text{diag}(e^{\lambda_i s}) \bU^{-1}\bX(0)
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\end{equation}
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\item \textbf{GW}
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In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
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The elements of the matrix $\bC{}{(0)}$ in the GW case are given in Eq.~(\ref{eq:GW_unfolded}) and can be written equivalently as
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\begin{align}
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\label{eq:GW_unfolded}
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C^\text{2h1p}_{i[ja],k[lc]} &= \epsilon_i \delta_{jl} \delta_{ac} \delta_{ik} - A_{ja,lc}^{\phRPA}\delta_{ik} \notag \\
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C^\text{2p1h}_{[ia]b,[kc]d} &= \epsilon_i \delta_{ik} \delta_{ac} \delta_{bd} + A_{ia,kc}^{\phRPA}\delta_{bd} \notag
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\end{align}
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So the matrix $\bC{}{(0)}$ is a diagonal block matrix with each block corresponding to a shifted RPA problem.
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Because we know the eigenvectors of the RPA problem we can buil the eigenvectors of $\bC{}{(0)}$ as
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\begin{align}
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U_{i[ja],(p,v)} &= \bX_{ja}^{(v)}\delta_{pi} & U_{[ia]b,(p,v)} &= \bX_{ia}^{(v)}\delta_{bp}
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\end{align}
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where the eigenvectors are indexed by a collective index $(n,v)$ where $n$ refers to the block number and $v$ refers to the eigenvector inside the block.
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The corresponding eigenvalues are
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\begin{align}
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\Omega_{(i,v)} &= \epsilon_i - \Omega_v & \Omega_{(a,v)} &= \epsilon_a + \Omega_v
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\end{align}
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Therefore the eigenvalues of $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_r \Omega_{(q,v)} - \epsilon_r^2 - \Omega_{q,v}^2 = -(\epsilon_r - \Omega_{(q,v)})^2$.
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And finally the analytical expressions for the GW coupling blocks at first order are
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\begin{align}
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\bV{r}{\hhp,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s})\bU^{-1}\bV{r}{\hhp,(1),\dagger}(0) \\
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\bV{r}{\pph,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(a,v)})^2s})\bU^{-1}\bV{r}{\pph,(1),\dagger}(0)
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\end{align}
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The downfolded correlation part of the self-energy is
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\begin{align}
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\bSig^{\hhp} (\omega) &= \bV{}{\hhp,(1)} \bU^{\hhp} \text{diag}(\frac{1}{\omega - D_{(i,v),(i,v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)})^{\mathsf{T}} \notag \\
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&+ \bV{}{\pph,(1)} \bU^{\pph} \text{diag}(\frac{1}{\omega - D_{(a,v),(a,v)}}) (\bU^{\pph})^{-1} (\bV{}{\pph,(1)})^{\mathsf{T}} \notag \\
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&= \bW^{\hhp,(1)} \text{diag}(\frac{1}{\omega - D_{(i,v),(i,v)}}) (\bW^{\hhp,(1)})^{\mathsf{T}} \notag \\
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&+ \bW^{\pph,(1)} \text{diag}(\frac{1}{\omega - D_{(a,v),(a,v)}}) (\bW^{\pph,(1)})^{\mathsf{T}} \notag
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\end{align}
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Therefore the downfolded SRG self-energy is
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\begin{align}
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&\bSig(\omega)^{\hhp}_r = \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) \\
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&\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
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&= \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag
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\end{align}
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Using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
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\begin{align}
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\label{eq:SRGGW_selfenergy}
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(\Sigma(\omega)_r^{\GW})_{pq} &= \sum_{(i,v)} \frac{M_{ip}^{(v)}M_{iq}^{(v)}}{\omega - \epsilon_i + \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(i,v)})^2s} \notag \\
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&+ \sum_{(a,v)} \frac{M_{ap}^{(v)}M_{aq}^{(v)}}{\omega - \epsilon_a - \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}
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\end{align}
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where the $\pph$ part has been obtained by an analog derivation.
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\begin{itemize}
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\item \textbf{GF(2)}
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The GF(2) case is much easier because $\bC{}{(0)}$ is already diagonal so the matrices $\bU$ and $\bU^{-1}$ are equal to the identity.
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A compeltely analog derivation gives
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In the GF(2) case we have $D_{(i,v),(i,v)} = D_{ija,ija} = \epsilon_i + \epsilon_j - \epsilon_a$ and $D_{(a,v),(a,v)} = D_{iab,iab} = \epsilon_a + \epsilon_b - \epsilon_i $ and the $W$ matrix elements have been defined in Eq.~(\ref{eq:GF2_sERI}).
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\begin{align}
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\label{eq:SRGGF2_selfenergy}
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(\Sigma_r^{\GF2})_{pq}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
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&+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} e^{-2(\epsilon_r - \Delta_{i}^{ab})^2s} \notag
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\end{align}
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(\bSig^{(2)} (\omega,s))_{pq} &= \sum_{ija} W_{p,i[ja]}^{(1)} \frac{1}{\omega - D_{ija,ija}}(W^{\mathsf{T}})_{i[ja],q}^{(1)} \notag \\
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&+ \sum_{iab} W_{p,[ia]b}^{(1)} \frac{1}{\omega - D_{iab,iab}}(W^{\mathsf{T}})_{[ia]b,q}^{(1)} \notag \\
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&= \sum_{ija} \frac{W_{pa,ij}^{(1)} W_{qa,ij}^{(1)} }{\omega - (\epsilon_i + \epsilon_j - \epsilon_a)} \notag \\
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&+ \sum_{iab} \frac{W_{pi,ab}^{(1)} W_{qi,ab}^{(1)}}{\omega - (\epsilon_a + \epsilon_b - \epsilon_i)} \notag \\
|
||||
&= \sum_{ija} \frac{W_{pa,ij}^{(0)} W_{qa,ij}^{(0)} }{\omega - (\epsilon_i + \epsilon_j - \epsilon_a)} e^{-(\epsilon_p - \Delta_{ij}^a)^2s}e^{-(\epsilon_q - \Delta_{ij}^a)^2s} \notag \\
|
||||
&+ \sum_{iab} \frac{W_{pi,ab}^{(0)} W_{qi,ab}^{(0)} }{\omega - (\epsilon_a + \epsilon_b - \epsilon_i)} e^{-(\epsilon_p - \Delta_{i}^{ab})^2s}e^{-(\epsilon_q - \Delta_{i}^{ab})^2s} \notag
|
||||
\end{align}
|
||||
|
||||
\item \textbf{GW}
|
||||
|
||||
A similar derivation gives
|
||||
|
||||
\begin{align}
|
||||
\label{eq:SRGGW_selfenergy}
|
||||
\Sigma_{pq}^{\GW}(\omega) &= \sum_{iv} \frac{W_{pi,v}^{(0)} W_{qi,v}^{(0)}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta}e^{-(\epsilon_p - \epsilon_i + \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_i + \Omega_v)^2s} \notag \\
|
||||
&+ \sum_{av} \frac{W_{pa,v}^{(0)} W_{qa,v}^{(0)} }{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} e^{-(\epsilon_p - \epsilon_a - \Omega_v)^2s}e^{-(\epsilon_q - \epsilon_a - \Omega_v)^2s} \notag
|
||||
\end{align}
|
||||
|
||||
\item \textbf{GT}
|
||||
|
||||
@ -641,55 +624,64 @@ Using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally
|
||||
\end{align}
|
||||
|
||||
The two first equations can be solved by simple integrations.
|
||||
The two last equations admit the same solutions as the first order coupling blocks differential equations with different initial conditions.
|
||||
|
||||
We focus on $\bF{}{(2)}$ because it is the only second order block contributing to the second order quasiparticle equation.
|
||||
We are still in the case where $\bF{}{(2)} = \epsilon_r^{(2)}$
|
||||
The two last equations admit the same solutions as the first order coupling blocks differential equations with different initial conditions.
|
||||
|
||||
\begin{align}
|
||||
\dv{\epsilon_r^{(2)}}{s} &= \epsilon_r^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\epsilon_r^{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger} \\
|
||||
&= 2\bV{}{(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{(0)} \right) \bV{}{(1),\dagger} \notag \\
|
||||
&= 2\bV{}{\hhp,(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\hhp,(0)} \right) \bV{}{\hhp,(1),\dagger} \notag \\
|
||||
&+ 2\bV{}{\pph,(1)} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\pph,(0)} \right) \bV{}{\pph,(1),\dagger} \notag
|
||||
\dv{\bF{}{(2)}}{s} &= \bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger} \\
|
||||
&= \bF{}{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF{}{(0)} - 2 \bW^{(1)}\bD^{(0)}\bW^{(1),\dagger} \notag
|
||||
\end{align}
|
||||
We focus on the $\hhp$ part
|
||||
|
||||
\begin{align}
|
||||
&= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1} \left(\epsilon_r^{(0)} \mathbb{1} - \bC{}{\hhp,(0)} \right) \bU \notag \\
|
||||
&\times e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
|
||||
&= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} (\epsilon_r^{(0)} \mathbb{1} - \boldsymbol{\Omega}) e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
|
||||
\dv{F_{pq}^{(2)}}{s} &= F_{pp}^{(0)}(\bW^{(1)}\bW^{(1),\dagger} )_{pq} + (\bW^{(1)}\bW^{(1),\dagger})_{pq}F_{qq}^{(0)}\\
|
||||
&- 2 (\bW^{(1)}\bC{}{(0)}\bW^{(1),\dagger})_{pq}
|
||||
\end{align}
|
||||
|
||||
\begin{itemize}
|
||||
\item \textbf{GW}
|
||||
|
||||
In the GW case this evaluates to
|
||||
\begin{align}
|
||||
&= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag
|
||||
\end{align}
|
||||
The other part is analog and we obtain
|
||||
\begin{align}
|
||||
\dv{\epsilon_r^{(2)}}{s} &= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
|
||||
&+ 2 \sum_{(a,v)} M_{ar}^{(v)}M_{ar}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(a,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} \notag
|
||||
\begin{align}
|
||||
&\dv{F_{pq}^{(2)}}{s} = \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)})W^{(1)}_{p,(r,v)}W^{(1),\dagger}_{(r,v),q} - 2 W^{(1)}_{p,(r,v)}C^{(0)}_{(r,v),(r,v)}W^{(1),\dagger}_{(r,v),q} \notag \\
|
||||
&= \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)})W^{(1)}_{p,(r,v)}W^{(1),\dagger}_{(r,v),q} \notag \\
|
||||
&= \sum_{r,v}(\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)})W^{(0)}_{p,(r,v)} \notag \\
|
||||
&\times W^{(0),\dagger}_{(r,v),q} e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s} \notag
|
||||
\end{align}
|
||||
|
||||
which can be integrated as
|
||||
\begin{align}
|
||||
\epsilon_r^{(2)}(s) &= - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
|
||||
&- \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} + \text{Cte} \notag
|
||||
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
|
||||
&\times W^{(0),\dagger}_{(r,v),q} e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s} + \text{Cte} \notag
|
||||
\end{align}
|
||||
|
||||
The constant is determined as
|
||||
\begin{equation}
|
||||
\epsilon_r^{(2)}(0) = 0 = - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} - \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} + \text{Cte} \notag
|
||||
\end{equation}
|
||||
\begin{align}
|
||||
&F_{pq}^{(2)}(0) = 0 \notag \\
|
||||
&= - \sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} W^{(0),\dagger}_{(r,v),q} + \text{Cte} \notag
|
||||
\end{align}
|
||||
Which finally gives
|
||||
\begin{align}
|
||||
\epsilon_r^{(2)}(s) &= \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r - \Omega_{(i,v)}} \left(1 - e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}\right) \notag \\
|
||||
&+ \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r - \Omega_{(a,v)}} \left(1 - e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}\right) \notag
|
||||
F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 C^{(0)}_{(r,v),(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
|
||||
&\times W^{(0),\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right) \notag
|
||||
\end{align}
|
||||
|
||||
In the previous formula we can see that the diagonal elements at $s \to \infty$ correspond to the same values as in the usual diagonal static approximation.
|
||||
Therefore, the SRG as a renormalization group method removes the coupling $\bV{}{}$ while incorporating some of its physics in the non-coupled problem $\bF{}{}$.
|
||||
This formalism gives us a rationalization of the diagonal static approximation from a RG perspective.
|
||||
In addition, this gives us a way to define a non-diagonal static approximation which is not straightforward to define by simply looking at Eq.~(\ref{eq:GW_selfenergy}).
|
||||
Even more, the SRG formalism defines a hierarchy of static approximation by considering higher and higher perturbation order for $\bSig$.
|
||||
|
||||
One of the con of the static approximation is that we loose information about the satellites and this is true for the SRG also when the coupling has been totally removed.
|
||||
However, SRG allows us to stop at a finite value $s$ corresponding to a renormalized coupling but the coupling is still present.
|
||||
Therefore the satellites can still be observed due to the non-linearity induced by the renormalized coupling.
|
||||
|
||||
\item \textbf{GF(2)}
|
||||
|
||||
The expression for the GF(2) case is
|
||||
\begin{align}
|
||||
\epsilon_r^{(2)}(s) &= \frac{1}{2} \sum _{ija} \frac{\aeri{ra}{ij}^2}{\epsilon_r ^{(0)}- \Delta_{ij}^a} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{ij}^a)^2s}\right) \notag \\
|
||||
&+ \frac{1}{2} \sum _{iab} \frac{\aeri{ri}{ab}^2}{\epsilon_r^{(0)} - \Delta_{i}^{ab}} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{i}^{ab})^2s}\right) \notag
|
||||
\begin{align}
|
||||
F_{pq}^{(2)}(s) &= - \frac{1}{2} \sum_{ija} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - \Delta_{ij}^a}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{ij}^a)^2} \aeri{pa}{ij}\notag \\
|
||||
&\times\aeri{qa}{ij} \left(1 - e^{-(\epsilon_p - \Delta_{ij}^a)^2s} e^{-(\epsilon_q - \Delta_{ij}^a)^2s}\right) \notag \\
|
||||
& - \frac{1}{2} \sum_{iab} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - \Delta_{i}^{ab}}{(\epsilon_p^{(0)} - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)} - \Delta_{i}^{ab})^2} \aeri{pi}{ab}\notag \\
|
||||
&\times\aeri{qi}{ab} \left(1 - e^{-(\epsilon_p - \Delta_{i}^{ab})^2s} e^{-(\epsilon_q - \Delta_{i}^{ab})^2s}\right) \notag
|
||||
\end{align}
|
||||
|
||||
\end{itemize}
|
||||
@ -913,4 +905,51 @@ Note the close similarity with Evangelista's expressions for the off-diagonal pa
|
||||
v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
|
||||
\end{align}
|
||||
|
||||
% \section{Old stuff}
|
||||
|
||||
% However, in the general case this matrix differential equation is not trivial to solve.
|
||||
% In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_r$.
|
||||
% \begin{align}
|
||||
% \label{eq:matrixdiffeq}
|
||||
% \dv{\bV{r}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \\
|
||||
% \dv{\bV{r}{(1),\dagger}}{s} &= (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
|
||||
% \end{align}
|
||||
% These equations can be solved if
|
||||
|
||||
% \textbf{Matrix differential equation}
|
||||
% A general differential matrix equation of the form
|
||||
% \begin{equation}
|
||||
% \dv{\bX}{s} = \bA \bX
|
||||
% \end{equation}
|
||||
% admits a solution
|
||||
% \begin{align}
|
||||
% \bX(s) &= c_1e^{\lambda_1 s}\bU_1 + c_2e^{\lambda_2 s}\bU_2 + \dots + c_ne^{\lambda_n s}\bU_n \\
|
||||
% &= \bU \text{diag}(e^{\lambda_i s}) \bc \notag
|
||||
% \end{align}
|
||||
% where $c_i$ are coefficients to determine, $\lambda_i$ are the eigenvalues of $\bA$ and $\bU_i$ the corresponding eigenvectors.
|
||||
% The $c_i$ are determined by the initial condition which gives
|
||||
% \begin{equation}
|
||||
% \label{eq:solution_eqdiff}
|
||||
% \bX(s) = \bU \text{diag}(e^{\lambda_i s}) \bU^{-1}\bX(0)
|
||||
% \end{equation}
|
||||
|
||||
% In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
|
||||
% The elements of the matrix $\bC{}{(0)}$ in the GW case are given in Eq.~(\ref{eq:GW_unfolded}) and can be written equivalently as
|
||||
% \begin{align}
|
||||
% \label{eq:GW_unfolded}
|
||||
% C^\text{2h1p}_{i[ja],k[lc]} &= \epsilon_i \delta_{jl} \delta_{ac} \delta_{ik} - A_{ja,lc}^{\phRPA}\delta_{ik} \notag \\
|
||||
% C^\text{2p1h}_{[ia]b,[kc]d} &= \epsilon_i \delta_{ik} \delta_{ac} \delta_{bd} + A_{ia,kc}^{\phRPA}\delta_{bd} \notag
|
||||
% \end{align}
|
||||
% So the matrix $\bC{}{(0)}$ is a diagonal block matrix with each block corresponding to a shifted RPA problem.
|
||||
% Because we know the eigenvectors of the RPA problem we can buil the eigenvectors of $\bC{}{(0)}$ as
|
||||
% \begin{align}
|
||||
% U_{i[ja],(p,v)} &= \bX_{ja}^{(v)}\delta_{pi} & U_{[ia]b,(p,v)} &= \bX_{ia}^{(v)}\delta_{bp}
|
||||
% \end{align}
|
||||
% where the eigenvectors are indexed by a collective index $(n,v)$ where $n$ refers to the block number and $v$ refers to the eigenvector inside the block.
|
||||
% The corresponding eigenvalues are
|
||||
% \begin{align}
|
||||
% \Omega_{(i,v)} &= \epsilon_i - \Omega_v & \Omega_{(a,v)} &= \epsilon_a + \Omega_v
|
||||
% \end{align}
|
||||
|
||||
|
||||
\end{document}
|
||||
Loading…
Reference in New Issue
Block a user