From 9676c5ac60a1d84d3ef7cd10f32c2e413cae284c Mon Sep 17 00:00:00 2001 From: Antoine MARIE Date: Tue, 8 Nov 2022 14:18:17 +0100 Subject: [PATCH] saving work, GW and GF(2) up to first order, working on second order --- Notes/Notes.tex | 362 ++++++++++++++++++++++++++++-------------------- 1 file changed, 215 insertions(+), 147 deletions(-) diff --git a/Notes/Notes.tex b/Notes/Notes.tex index 6aa4a6b..2a42051 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -13,126 +13,15 @@ ]{hyperref} \urlstyle{same} -\newcommand{\ie}{\textit{i.e.}} -\newcommand{\eg}{\textit{e.g.}} -\newcommand{\etal}{\textit{et al.}} -\newcommand{\alert}[1]{\textcolor{red}{#1}} +%============================================================% +%%% NEWCOMMANDS %%% +% ============================================================% + +\input{Commands} + \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\ant}[1]{\textcolor{green}{#1}} -\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}} -\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} - -\newcommand{\mc}{\multicolumn} -\newcommand{\fnm}{\footnotemark} -\newcommand{\fnt}{\footnotetext} -\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} -\newcommand{\QP}{\textsc{quantum package}} -\newcommand{\T}[1]{#1^{\intercal}} -\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} -\newcommand{\dRPA}{\text{dRPA}} - -% coordinates -\newcommand{\br}{\boldsymbol{r}} -\newcommand{\bx}{\boldsymbol{x}} -\newcommand{\dbr}{d\br} -\newcommand{\dbx}{d\bx} - -% methods -\newcommand{\GW}{\text{$GW$}} -\newcommand{\GT}{\text{$GT$}} -\newcommand{\evGW}{ev$GW$} -\newcommand{\qsGW}{qs$GW$} -\newcommand{\GOWO}{$G_0W_0$} -\newcommand{\Hxc}{\text{Hxc}} -\newcommand{\xc}{\text{xc}} -\newcommand{\Ha}{\text{H}} -\newcommand{\co}{\text{c}} -\newcommand{\x}{\text{x}} -\newcommand{\KS}{\text{KS}} -\newcommand{\HF}{\text{HF}} -\newcommand{\RPA}{\text{RPA}} -\newcommand{\Om}[2]{\Omega_{#1}^{#2}} -\newcommand{\sERI}[2]{(#1|#2)} -\newcommand{\e}[2]{\epsilon_{#1}^{#2}} - -% -\newcommand{\Ne}{N} -\newcommand{\Norb}{K} -\newcommand{\Nocc}{O} -\newcommand{\Nvir}{V} - -% operators -\newcommand{\hH}{\Hat{H}} -\newcommand{\hS}{\Hat{S}} -\newcommand{\ani}[1]{\hat{a}_{#1}} -\newcommand{\cre}[1]{\hat{a}_{#1}^\dagger} -\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} } - -% energies -\newcommand{\Enuc}{E^\text{nuc}} -\newcommand{\Ec}[1]{E_\text{c}^{#1}} -\newcommand{\EHF}{E^\text{HF}} - -% orbital energies -\newcommand{\eps}{\epsilon} -\newcommand{\reps}{\Tilde{\epsilon}} - -% Matrix elements -\newcommand{\SigC}{\Sigma^\text{c}} -\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}} -\newcommand{\MO}[1]{\phi_{#1}} -\newcommand{\SO}[1]{\psi_{#1}} -\newcommand{\eri}[2]{\braket{#1}{#2}} -\newcommand{\aeri}[2]{\mel{#1}{}{#2}} -\newcommand{\ERI}[2]{(#1|#2)} -\newcommand{\rbra}[1]{(#1|} -\newcommand{\rket}[1]{|#1)} - - -% Matrices -\newcommand{\bO}{\boldsymbol{0}} -\newcommand{\bI}{\boldsymbol{1}} -\newcommand{\bH}{\boldsymbol{H}} -\newcommand{\bSig}{\boldsymbol{\Sigma}} -\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}} -\newcommand{\be}{\boldsymbol{\epsilon}} -\newcommand{\bOm}{\boldsymbol{\Omega}} -\newcommand{\bA}{\boldsymbol{A}} -\newcommand{\bB}{\boldsymbol{B}} -\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}} -\newcommand{\bD}{\boldsymbol{D}} -\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}} -\newcommand{\bR}{\boldsymbol{R}} -\newcommand{\bU}{\boldsymbol{U}} -\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}} -\newcommand{\bW}{\boldsymbol{W}} -\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}} -\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}} -\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}} -\newcommand{\bc}{\boldsymbol{c}} - -% orbitals, gaps, etc -\newcommand{\IP}{I} -\newcommand{\EA}{A} -\newcommand{\HOMO}{\text{HOMO}} -\newcommand{\LUMO}{\text{LUMO}} -\newcommand{\Eg}{E_\text{g}} -\newcommand{\EgFun}{\Eg^\text{fund}} -\newcommand{\EgOpt}{\Eg^\text{opt}} -\newcommand{\EB}{E_B} - -% shortcuts for greek letters -\newcommand{\si}{\sigma} -\newcommand{\la}{\lambda} - - -\newcommand{\RHH}{R_{\ce{H-H}}} -\newcommand{\ii}{\mathrm{i}} - -\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)} -\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}} -\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}} % addresses \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} @@ -185,7 +74,7 @@ Therefore, the transformed Hamiltonian depends on a flow parameter $s$. The resulting Hamiltonian possess up to $N$-body operators with $N$ the number of particle. \begin{equation} - \bH(s) = E_0(s) + \bF(s) + \bV{}{}(s) + \bW(s) + \dots + \bH(s) = E_0(s) + \bF{}{}(s) + \bV{}{}(s) + \bW(s) + \dots \end{equation} In the following, we will truncate every contribution superior to two-body operators. We can easily derive an evolution equation for this Hamiltonian by taking the derivative of $\bH(s)$. This gives @@ -240,15 +129,15 @@ Hence, we define the off-diagonal Hamiltonian as \end{equation} Note that each coefficients depend on $s$. -The perturbative parameter $\la$ is such that +The perturbative parameter $\lambda$ is such that \begin{equation} - \bH(0) = E_0(0) + \bF{}{}(0) + \la \bV{}{}(0) + \bH(0) = E_0(0) + \bF{}{}(0) + \lambda \bV{}{}(0) \end{equation} In addition, we know the following initial conditions. We use the HF basis set of the reference such that $\bF{}{\text{od}}(0) = 0$ and $\bF{}{\mathrm{d}}(0)_{pq}=\delta_{pq}\epsilon_p$. Therefore, we have \begin{align} - \bH^\text{d}(0)&=E_0(0) + \bF{}{\mathrm{d}}(0) + \la \bV{}{\mathrm{d}}(0) & \bH^\text{od}(0)&= \la V^{\mathrm{od}}(0) + \bH^\text{d}(0)&=E_0(0) + \bF{}{\mathrm{d}}(0) + \lambda \bV{}{\mathrm{d}}(0) & \bH^\text{od}(0)&= \lambda \bV{}{\mathrm{od}}(0) \end{align} Now, we want to compute the terms at each order of the following development \begin{equation} @@ -398,30 +287,30 @@ In the following, we will focus on the GF(2), GW and GT approximations. 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. \begin{align} \label{eq:GF2_selfenergy} - \Sig{pq}{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \eps _a -\eps_i -\eps_j - \ii \eta} \notag \\ - &+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \eps _i -\eps_a -\eps_b + \ii \eta} \notag + \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 \\ + &+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag \end{align} 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. \begin{equation} \label{eq:GW_selfenergy} - \Sig{pq}{\GW}(\omega) = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta} + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta} \notag + \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 \end{equation} with \begin{equation} \label{eq:GW_sERI} - \sERI{pq}{m} = \sum_{ia} \eri{pq}{ia} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \notag + \ceri{pq}{m} = \sum_{ia} \eri{pi}{qa} \qty( \bX_{m}^{\dRPA} + \bY_{m}^{\dRPA} )_{ia} \notag \end{equation} Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA The corresponding self-energies read as \begin{equation} \label{eq:GT_selfenergy} - \Sig{pq}{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \e{i}{} - \Om{m}{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \e{a}{} - \Om{m}{N-2} - \ii \eta} \notag + \Sigma_{pq}^{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \epsilon_i - \Omega_{m}^{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \epsilon_a - \Omega_{m}^{N-2} - \ii \eta} \notag \end{equation} with \begin{align} \label{eq:GT_sERI} - \eri{pq}{\chi^{N+2}_m} &= \sum_{c