865 lines
45 KiB
TeX
865 lines
45 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,mleftright,bbold}
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]{hyperref}
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%============================================================%
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%%% NEWCOMMANDS %%%
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% ============================================================%
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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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\newcommand{\ant}[1]{\textcolor{green}{#1}}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
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\title{Notes on the project: Similarity Renormalization Group formalism applied to Green's function theory}
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\author{Antoine \surname{Marie}}
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\email{amarie@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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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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%\end{center}
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%\bigskip
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%\end{abstract}
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\maketitle
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%=================================================================%
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\section{Introduction}
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\label{sec:intro}
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%=================================================================%
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This document is a compilation of various notes related to the similarity renormalisation group (SRG) and its application to many-body perturbation theory (MBPT).
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Before tackling its application to MBPT, we give the main SRG equations in Sec.~\ref{sec:intro}.
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Then, we derive in details the perturbative expressions obtained by Evangelista by applying the SRG to the electronic Hamiltonian (see Sec.~\ref{sec:srg}).
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Now turning to MBPT, its various flavors are presented in Sec.~\ref{sec:folded} and the corresponding unfolded equations are given in Sec.~\ref{sec:unfolded}.
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Then, a SRG perturbative analysis is performed on general matrices in Sec.~\ref{sec:matrix_srg}.
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Finally, in Sec.~\ref{sec:second_quant_mbpt} we investigate the possibility to find a second quantized effective Hamiltonian corresponding to the unfolded MBPT equations.
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%=================================================================%
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\section{The similarity renormalisation group}
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\label{sec:srg}
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%=================================================================%
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The similarity renormalization group aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form.
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Therefore, the transformed Hamiltonian
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\begin{equation}
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\bH(s) = \bU(s) \, \bH \, \bU^\dag(s)
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\end{equation}
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depends on a flow parameter $s$.
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The resulting Hamiltonian possess up to $N$-body operators with $N$ the number of particle.
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\begin{equation}
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\bH(s) = E_0(s) + \bF{}{}(s) + \bV{}{}(s) + \bW(s) + \dots
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\end{equation}
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In the following, we will truncate every contribution superior to two-body operators.
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We can easily derive an evolution equation for this Hamiltonian by taking the derivative of $\bH(s)$. This gives
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\begin{equation}
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\label{eq:flowEquation}
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\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)}
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\end{equation}
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where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
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\begin{equation}
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\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s) .
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\end{equation}
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To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$.
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Before doing so, we need to define what is the blocks to suppress in order to obtain a block-diagonal Hamiltonian.
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Therefore, the Hamiltonian is separated in two parts as
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\begin{equation}
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\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}.
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\end{equation}
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By definition, we have the following condition on $\bH^\text{od}$
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\begin{equation}
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\bH^\text{od}(\infty) = \boldsymbol{0}.
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\end{equation}
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In this work, we will use Wegner's canonical generator which is defined as
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\begin{equation}
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\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}.
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\end{equation}
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This generator has the advantage of defining a true renormalisation scheme, \ie the coupling coefficients with the highest energy determinants are removed first.
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One of the flaws of this generator is that it generates a \titou{stiff} set of ODE which is difficult to solve numerically.
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However, here we consider analytical perturbative expressions so we will not be affected by this problem.
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%=================================================================%
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\section{The SRG electronic Hamiltonian}
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\label{sec:electronic_ham}
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%=================================================================%
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In this part, we derive the perturbative expression for the SRG applied to the non-relativistic electronic Hamiltonian
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\begin{equation}
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\label{eq:hamiltonianSecondQuant}
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\hH = \sum_{pq} f_{pq} \cre{p}\ani{q} + \frac{1}{4} \sum_{pqrs} \aeri{pq}{rs} \cre{p}\cre{q}\ani{r}\ani{s}
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\end{equation}
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which can also be written in normal order wrt a reference determinant as
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\begin{equation}
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\label{eq:hamiltonianNormalOrder}
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\hH = E_0 + \sum_{pq} + f_p^q\no{q}{p} + \frac{1}{4} \sum_{pqrs}v_{pq}^{rs}\no{rs}{pq}.
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\end{equation}
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In this case, we want to decouple the reference determinant from every singly and doubly excited determinants.
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Hence, we define the off-diagonal Hamiltonian as
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\begin{equation}
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\label{eq:hamiltonianOffDiagonal}
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\hH^{\text{od}}(s) = \sum_{ia} f_i^a(s)\no{a}{i} + \frac{1}{4} \sum_{ijab}v_{ij}^{ab}(s)\no{ab}{ij}.
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\end{equation}
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Note that each coefficients depend on $s$.
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The perturbative parameter $\lambda$ is such that
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\begin{equation}
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\bH(0) = E_0(0) + \bF{}{}(0) + \lambda \bV{}{}(0)
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\end{equation}
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In addition, we know the following initial conditions.
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We use the HF basis set of the reference such that $\bF{}{\text{od}}(0) = 0$ and $\bF{}{\mathrm{d}}(0)_{pq}=\delta_{pq}\epsilon_p$.
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Therefore, we have
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\begin{align}
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\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)
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\end{align}
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Now, we want to compute the terms at each order of the following development
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\begin{equation}
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\label{eq:hamiltonianPTExpansion}
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\bH(s) = \bH^{(0)}(s) + \bH^{(1)}(s) + \bH^{(2)}(s) + \bH^{(3)}(s) + \dots
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\end{equation}
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by integrating Eq.~\eqref{eq:flowEquation}.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Zeroth order Hamiltonian}
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%%%%%%%%%%%%%%%%%%%%%%
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First, we start by showing that the zeroth order Hamiltonian is independant of $s$ and therefore equal to $\bH^{(0)}(0)$
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\begin{align}
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\bH(\delta s) &= \bH(0) + \delta s \dv{\bH(s)}{s}\bigg|_{s=0} + O(\delta s^2) \\
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\dv{\bH(s)}{s}\bigg|_{s=0} &= [ \boldsymbol{\eta}(0), \bH(0)] \\
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\end{align}
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However, we have seen that $\bH^\text{od}(0)$ is of order 1.
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Hence, $\boldsymbol{\eta}(0)$ does not have a zero order contribution.
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Which gives us the following equality
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\begin{equation}
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\bH^{(0)}(\delta s) = \bH^{(0)}(0)
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\end{equation}
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meaning that the zeroth order Hamiltonian is independant of $s$.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{First order Hamiltonian}
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%%%%%%%%%%%%%%%%%%%%%%
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The right-hand side of \eqref{eq:flowEquation} has no zeroth order, so we want to compute its first order term.
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To do so, we first need to compute the first order term of $\boldsymbol{\eta}(s)$.
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We have seen that $\bH^\text{od}(0)$ has no zeroth order contribution so we have
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\begin{equation}
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\boldsymbol{\eta}^{(1)}(s) = \comm{\bH^{\text{d},(0)}(0)}{\bH^{\text{od},(1)}(s)}
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\end{equation}
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According to the appendix the one-body part of $\boldsymbol{\eta}^{(1)}(s) $ has four contributions. However, two of them involve the two-body part of $\bH^{\text{d},(0)}(0)$ which is equal to zero.
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In addition, the term $\left[A_{1}, B_{2}\right]_{p}^{q}$ involves the coefficients $A_i^a = f_i^a(0) = 0$.
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So finally we only have one term
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\begin{align}
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\eta_a^{i,(1)}(s) &= \comm{\bH_1^{\text{d},(0)}(0)}{\bH_1^{\text{od},(1)}(s)}_a^i \\
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&= \sum_r \left( f_a^r(0) f_r^{i,(1)}(s) - f_r^i(0) f_a^{r,(1)}(s) \right) \\
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&= (\epsilon_a - \epsilon_i)f_a^{i,(1)}(s)
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\end{align}
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Now turning to the two-body part of $\boldsymbol{\eta}^{(1)}(s) $ and once again two terms are zero because the two-body part of $\bH^{\text{d},(0)}(0)$ is equal to zero.
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So we have
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\begin{align}
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\eta_{ab}^{ij,(1)}(s) &= \comm{\bH_1^{\text{d},(0)}(0)}{\bH_2^{\text{od},(1)}(s)}_{ab}^{ij} \\
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&= \sum_t [P(ab) f_a^t(0) v_{tb}^{ij,(1)}(s) - P(ij) f_t^i(0) v_{ab}^{tj,(1)}(s) ] \notag \\
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&= \sum_t [ P(ab) \epsilon_a \delta_{at}v_{tb}^{ij,(1)}(s) - P(ij) \epsilon_i \delta_{it} v_{ab}^{tj,(1)}(s) ] \notag \\
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&= P(ab) \epsilon_a v_{ab}^{ij,(1)}(s) - P(ij) \epsilon_i v_{ab}^{ij,(1)}(s) \notag\\
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&= \left( \epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j \right) v_{ab}^{ij,(1)} \notag
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\end{align}
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We can now compute the first order contribution to Eq.~\eqref{eq:flowEquation}. We have seen that $\eta$ has no zeroth order contribution so
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\begin{equation}
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\dv{\bH^{(1)}(s)}{s} = \comm{\boldsymbol{\eta}^{(1)}(s)}{\bH^{(0)}(s)}
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\end{equation}
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We start with the scalar contribution, \ie the PT1 energy
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\begin{equation}
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\dv{E_0^{(1)}(s)}{s} = \mel{\phi}{\comm{\boldsymbol{\eta}_1^{(1)}(s)}{\bH_1^{(0)}(s)}}{\phi} + \mel{\phi}{\comm{\boldsymbol{\eta}_2^{(1)}(s)}{\bH_2^{(0)}(s)}}{\phi}
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\end{equation}
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where the second term is equal to zero.
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Thus we have
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\begin{align}
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\label{eq:diffEqScalPT1}
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\dv{E_0^{(1)}(s)}{s} &= \sum_{ip}\eta_i^{p,(1)}f_p^i(0) - \eta_p^{i,(1)}f_i^p(0) \\
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&= \sum_i \epsilon_i(\eta_i^{i,(1)} - \eta_i^{i,(1)}) \notag \\
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&= 0 \notag
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\end{align}
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Using the exact same reasoning as above we can show that there is only of the four terms of the one-body part of the commutator that is non-zero.
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\begin{align}
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\dv{f_a^{i,(1)}(s)}{s} &= \comm{\boldsymbol{\eta}_1^{(1)}(s)}{\bH_1^{(0)}(s)}_a^i \\
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&= \sum_r \eta_a^{r,(1)}(s)f_r^i(0) - \eta_r^{i,(1)}f_a^r(0) \notag \\
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&= (\epsilon_i - \epsilon_a) \eta_a^{i,(1)}(s) \notag \\
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&= -(\epsilon_i - \epsilon_a) ^2f_a^{i,(1)}(s) \notag
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\end{align}
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The derivation for the two-body part to first order is once again similar
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\begin{align}
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\dv{v_{ab}^{ij,(1)}(s)}{s} &= \comm{\boldsymbol{\eta}_2^{(1)}(s)}{\bH_1^{(0)}(s)}_{ab}^{ij} \\
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&= -(\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b) ^2v_{ab}^{ij,(1)}(s) \notag
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\end{align}
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These three differential equations can be integrated to obtain the analytical form of the Hamiltonian coefficients up to first order of perturbation theory.
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\begin{align}
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E_0^{(1)}(s) &= E_0^{(1)}(0) = 0 \\
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f_a^{i,(1)}(s) &= f_a^{i,(1)}(0) e^{-s (\Delta_a^i )^2} = 0 \\
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v_{ab}^{ij,(1)}(s) &= v_{ab}^{ij,(1)}(0) e^{-s (\Delta_{ab}^{ij})^2} = \aeri{ij}{ab} e^{-s (\Delta_{ab}^{ij})^2}
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\end{align}
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Second order Hamiltonian}
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%%%%%%%%%%%%%%%%%%%%%%
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Now turning to the differential equations, we start by computing the scalar part of Eq.~\eqref{eq:flowEquation}, \ie the differential equation for the second order energy.
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\begin{align}
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\dv{E_0^{(2)}(s)}{s} & = \mel{\phi}{\comm{\boldsymbol{\eta}_1^{(2)}(s)}{\bH_1^{(0)}(s)}}{\phi} + \mel{\phi}{\comm{\boldsymbol{\eta}_2^{(2)}(s)}{\bH_2^{(0)}(s)}}{\phi} \\
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&+ \mel{\phi}{\comm{\boldsymbol{\eta}_1^{(1)}(s)}{\bH_1^{(1)}(s)}}{\phi} + \mel{\phi}{\comm{\boldsymbol{\eta}_2^{(1)}(s)}{\bH_2^{(1)}(s)}}{\phi}
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\end{align}
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The two first terms are equal to zero for the same reason as the PT1 scalar differential equation (see Eq.~\eqref{eq:diffEqScalPT1}).
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In addition, the one-body hamiltonian has no first order contribution so
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\begin{align}
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\dv{E_0^{(2)}(s)}{s} &= \mel{\phi}{\comm{\boldsymbol{\eta}_2^{(1)}(s)}{\bH_2^{(1)}(s)}}{\phi} \\
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&= \frac{1}{4} \sum_{i j} \sum_{a b}\left(\eta_{i j}^{ab,(1)} H_{ab}^{i j,(1)} - H_{i j}^{ab,(1)} \eta_{ab}^{i j,(1)}\right) \\
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&=\frac{1}{4} \sum_{i j} \sum_{a b}\left(\eta_{i j}^{ab,(1)} - \eta_{ab}^{i j,(1)}\right) v_{ab}^{ij,(1)} \\
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&= \frac{1}{4} \sum_{i j} \sum_{a b}\left(\Delta_{ab}^{ij}v_{ij}^{ab,(1)} - (-\Delta_{ab}^{ij} v_{ab}^{ij,(1)}) \right) \aeri{ij}{ab} e^{-s (\Delta_{ab}^{ij})^2} \\
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&= \frac{1}{2} \sum_{i j} \sum_{a b} \Delta_{ab}^{ij} (v_{ab}^{ij,(1)})^2 \\
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&= \frac{1}{2} \sum_{i j} \sum_{a b} \Delta_{ab}^{ij} \aeri{ij}{ab}^2 e^{-2s (\Delta_{ab}^{ij})^2}
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\end{align}
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After integration, using the initial condition $E_0^{(2)}(0)=0$, we obtain
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\begin{equation}
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E_0^{(2)}(s) = \frac{1}{4} \sum_{i j} \sum_{a b} \frac{\aeri{ij}{ab}^2}{\Delta_{ab}^{ij}}\left(1-e^{-2s (\Delta_{ab}^{ij})^2}\right)
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\end{equation}
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We can continue the derivation further than Evangelista's paper.
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To compute the second order contribution to the Hamiltonian coefficients, we first need to compute the second order contribution to $\boldsymbol{\eta}(s)$.
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\begin{equation}
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\boldsymbol{\eta}^{(2)}(s) = \comm{\bH^{\text{d},(0)}(0)}{\bH^{\text{od},(2)}(s)} + \comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}
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\end{equation}
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The expressions for the first commutator are computed analogously to the one of the previous subsection.
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We focus on deriving expressions for the second term.
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The one-body part of $\bH^{\text{od},(1)}(s)$ is equal to zero so two of the four terms contributing to the one-body part of $\comm{\bH^{\text{d},(1)}(s)}{\bH^{\text{od},(1)}(s)}$ are zero.
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In addition, the term $\comm{A_1}{B_2}$ is equal to zero as well because the coefficients $(A_1)_{i}^a$ are zero (see expression in Appendix).
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So we have
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\begin{align}
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\eta_a^{i,(2)}(s) &= \comm{\bH_1^{\text{d},(0)}(0)}{\bH_1^{\text{od},(2)}(s)}_a^i + \comm{\bH_2^{\text{d},(1)}(s)}{\bH_2^{\text{od},(1)}(s)}_a^i \\
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&= (\epsilon_a - \epsilon_i)f_a^{i,(2)}(s) + \dots
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\end{align}
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Need to continue this derivation but this not needed for EPT2.
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%=================================================================%
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\section{The various flavors of MBPT}
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\label{sec:mbpt}
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% =================================================================%
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The central equation of MBPT in practice is the following
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\begin{equation}
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\label{eq:quasipart_eq}
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\bF{}{} + \bSig(\omega) = \omega \mathbb{1}.
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\end{equation}
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However, in order to use it we need to rely on approximations of the self-energy $\bSig(\omega)$.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Self-energies and quasiparticle equations}
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\label{sec:folded}
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%%%%%%%%%%%%%%%%%%%%%%
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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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\end{align}
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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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\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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\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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\begin{equation}
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\label{eq:GT_selfenergy}
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\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
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\end{equation}
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with
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\begin{align}
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\label{eq:GT_sERI}
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\eri{pq}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\
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\eri{pq}{\chi^{N-2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag
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\end{align}
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The two RPA problems giving the eigenvectors needed to build the GW and GT self-energies are given in Appendix~\ref{sec:rpa}.
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The unfolded equations}
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\label{sec:unfolded}
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%%%%%%%%%%%%%%%%%%%%%%
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Following Schirmer for the GF(2) case or Bintrim \etal, the non-linear quasi-particle equations for each approximations can be unfolded in larger linear problems
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\begin{equation}
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\label{eq:unfolded_equation}
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\bH \bc_{(s)} = \epsilon_s \bc_{(s)}
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\end{equation}
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where $\bH$ depends on the approximation chosen for the self-energy.
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For the three approximations considered here, the three matrices $\bH$ share the general form
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\begin{equation}
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\label{eq:unfolded_matrice}
|
|
\bH =
|
|
\begin{pmatrix}
|
|
\bF{}{} & \bV{}{\text{2h1p}} & \bV{}{\text{2p1h}} \\
|
|
(\bV{}{\text{2h1p}})^{\mathsf{T}} & \bC{}{\text{2h1p}} & \bO \\
|
|
(\bV{}{\text{2p1h}})^{\mathsf{T}} & \bO & \bC{}{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ in the different cases is given below.
|
|
\begin{itemize}
|
|
\item \textbf{GF(2)}
|
|
\begin{align}
|
|
\label{eq:GF2_unfolded}
|
|
V^\text{2h1p}_{p,klc} & = \frac{1}{\sqrt{2}}\aeri{pc}{kl}
|
|
&
|
|
V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc} \\
|
|
C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
|
|
&
|
|
C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd} \notag
|
|
\end{align}
|
|
\item \textbf{GW}
|
|
\begin{align}
|
|
\label{eq:GW_unfolded}
|
|
V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
|
|
&
|
|
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc} \notag \\
|
|
C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} & &
|
|
\\
|
|
C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd} \notag & &
|
|
\end{align}
|
|
\item \textbf{GT}
|
|
\begin{align}
|
|
\label{eq:GT_unfolded}
|
|
V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
|
|
&
|
|
V^\text{2p1h}_{p,kcd}&= \aeri{pk}{cd} \notag \\
|
|
C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac} & & \\
|
|
C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik} & & \notag
|
|
\end{align}
|
|
\end{itemize}
|
|
The downfolding procedure to obtain the GW self-energy is derived in details in Appendix~\ref{sec:downfolding}.
|
|
|
|
\textbf{\textcolor{red}{That would be nice to add electron-hole T matrix to see if it also correspond to one term that can be found in the CI below.}}
|
|
|
|
\textbf{\textcolor{red}{That would be nice to add the self-energies of the various post-RPA correction.}}
|
|
|
|
% =================================================================%
|
|
\section{A first quantization approach to SRG for MBPT}
|
|
\label{sec:matrix_srg}
|
|
%=================================================================%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Initial conditions}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Finding a second quantized effective Hamiltonian for MBPT is far from being trivial so we start the project with matrix perturbation theory.
|
|
A general upfolded MBPT matrix can be written as
|
|
\begin{equation}
|
|
\label{eq:H_MBPT}
|
|
H =
|
|
\begin{pmatrix}
|
|
\bF{}{} & \bV{}{} \\
|
|
\bV{}{\dagger} & \bC{}{}
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
Using SRG language, we define the diagonal and off-diagonal parts as
|
|
\begin{equation}
|
|
\label{eq:H_MBPT_partitioning}
|
|
H(0) =
|
|
\begin{pmatrix}
|
|
\bF{}{} & \bO \\
|
|
\bO & \bC{\text{d}}{}
|
|
\end{pmatrix}
|
|
+ \lambda
|
|
\begin{pmatrix}
|
|
\bO & \bV{}{} \\
|
|
\bV{}{\dagger} & \bC{\text{od}}{}
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
which gives the following conditions
|
|
\begin{align}
|
|
\bHd{0}(0) &= \begin{pmatrix}
|
|
\bF{}{} & \bO \\
|
|
\bO & \bC{\text{d}}{}
|
|
\end{pmatrix} & \bHod{0}(0) &= \bO \notag \\
|
|
\bHd{1}(0) &= \bO & \bHod{1}(0) &= \begin{pmatrix}
|
|
\bO & \bV{}{} \\
|
|
\bV{}{\dagger} & \bC{\text{od}}{} \notag
|
|
\end{pmatrix}
|
|
\end{align}
|
|
At this point, we aren't sure if the off-diagonal part of $\bC{}{}$ should be included or not in the off-diagonal part of the Hamiltonian $\bH_\text{od}$.
|
|
In the following derivation, we choose to do so because the other case can be obtained simply by taking $\bC{\text{od}}{} = \boldsymbol{0}$ and $\bC{\text{d}}{} = \bC{}{}$.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Zeroth order Hamiltonian}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The zeroth-order commutator of the Wegner generator therefore gives
|
|
\begin{equation}
|
|
\bEta{0} = \comm{\bHd{0}}{\bHod{0}} = \bO
|
|
\end{equation}
|
|
and similarly
|
|
\begin{equation}
|
|
\dv{\bH^{(0)}}{s} = \comm{\bEta{0}}{\bH^{(0)}} = \bO
|
|
\end{equation}
|
|
Finally, we have
|
|
\begin{equation}
|
|
\color{red}{\boxed{
|
|
\color{black}{\bH^{(0)}(s) = \bH^{(0)}(0)}
|
|
}}
|
|
\end{equation}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{First order Hamiltonian}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Now turning to the first-order contribution to the MBPT matrix, we start by computing the first order part of the Wegner generator.
|
|
|
|
\begin{align}
|
|
&\bEta{1} = \comm{\bHd{0}}{\bHod{1}} \\
|
|
&= \begin{pmatrix}
|
|
\bO & \bF{}{(0)}\bV{}{(1)} - \bV{}{(1)}\bC{\text{d}}{(0)}\\
|
|
\bC{\text{d}}{(0)}\bV{}{(1),\dagger} - \bV{}{(1),\dagger} \bF{}{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} - \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag
|
|
\end{pmatrix}
|
|
\end{align}
|
|
|
|
\begin{align}
|
|
\dv{\bH^{(1)}}{s} &= \comm{\bEta{1}}{\bHd{0}} = \begin{pmatrix}
|
|
\dv{\bF{}{(1)}}{s} & \dv{\bV{}{(1)}}{s} \\
|
|
\dv{\bV{}{(1),\dagger}}{s} & \dv{\bC{}{(1)}}{s}
|
|
\end{pmatrix} \\
|
|
\dv{\bF{}{(1)}}{s} &= \bO \\
|
|
\dv{\bV{}{(1)}}{s} &= 2 \bF{}{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 \\
|
|
\dv{\bV{}{(1),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(1),\dagger}\bF{}{(0)} - \bV{}{(1),\dagger}(\bF{}{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(1),\dagger} \\
|
|
\dv{\bC{}{(1)}}{s} &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(1)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(1)} - \bC{\text{od}}{(1)}(\bC{\text{d}}{(0)})^2
|
|
\end{align}
|
|
The last two equations can be solved differently depending on the form of $\bF{}{}$ and $\bC{}{}$.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Second order Hamiltonian}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
|
|
\begin{align}
|
|
&\bEta{2} = \comm{\bHd{0}}{\bHod{2}} + \comm{\bHd{1}}{\bHod{1}} \\
|
|
&= \comm{\bHd{0}}{\bHod{2}} \notag \\
|
|
&= \begin{pmatrix}
|
|
\bO & \bF{}{(0)}\bV{}{(2)} - \bV{}{(2)}\bC{\text{d}}{(0)}\\
|
|
\bC{\text{d}}{(0)}\bV{}{(2),\dagger} - \bV{}{(2),\dagger}\bF{}{(0)} & \bC{\text{d}}{(0)} \bC{\text{od}}{(2)} - \bC{\text{od}}{(2)} \bC{\text{d}}{(0)} \notag
|
|
\end{pmatrix}
|
|
\end{align}
|
|
|
|
\begin{align}
|
|
\dv{\bH^{(2)}}{s} &= \comm{\bEta{2}}{\bHd{0}} + \comm{\bEta{1}}{\bHd{1}} \\
|
|
&= \begin{pmatrix}
|
|
\dv{\bF{}{(2)}}{s} & \dv{\bV{}{(2)}}{s} \\
|
|
\dv{\bV{}{(2),\dagger}}{s} & \dv{\bC{}{(2)}}{s}
|
|
\end{pmatrix} \notag \\
|
|
\dv{\bF{}{(2)}}{s} &= \bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger}\\
|
|
\dv{\bC{}{(2)}}{s} &= 2 \bC{\text{d}}{(0)}\bC{\text{od}}{(2)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(2)} - \bC{\text{od}}{(2)}(\bC{\text{d}}{(0)})^2 \\
|
|
&-2 \bC{\text{d}}{(1)}\bC{\text{od}}{(0)}\bC{\text{d}}{(1)} + (\bC{\text{d}}{(1)})^2\bC{\text{od}}{(0)} + \bC{\text{od}}{(0)}(\bC{\text{d}}{(1)})^2 \notag \\
|
|
&+ \bC{\text{d}}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bC{\text{d}}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \notag \\
|
|
\dv{\bV{}{(2)}}{s} &= 2 \bF{}{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
|
|
&- 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} + \bF{}{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \notag \\
|
|
\dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{\text{d}}{(0)}\bV{}{(2),\dagger}\bF{}{(0)} - \bV{}{(2),\dagger}(\bF{}{(0)})^2 - (\bC{\text{d}}{(0)})^2\bV{}{(2),\dagger} \\
|
|
&- 2 \bC{\text{od}}{(1)} \bC{\text{d}}{(0)} \bV{}{(1),\dagger} + \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \bF{}{(0)} + \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} \bV{}{(1),\dagger} \notag
|
|
\end{align}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Downfolding the SRG-transformed matrix}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
In order to choose what to do with $\bC{\text{od}}{}$ we look at the downfolded SRG quasiparticle equation.
|
|
|
|
\begin{equation}
|
|
\label{eq:H_SRGMBPT}
|
|
H(s) =
|
|
\begin{pmatrix}
|
|
\bF{}{(0)}(0) + \bF{}{(2)}(s) & \bV{}{(1)}(s) + \bV{}{(2)}(s) \\
|
|
\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s) & \bC{}{(0)}(0) +\bC{}{(2)}(s)
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
|
|
\begin{widetext}
|
|
\begin{equation}
|
|
\left\{
|
|
\begin{aligned}
|
|
(\bF{}{(0)}(0) + \bF{}{(2)}(s)) \bR^{1h/1p} + (\bV{}{(1)}(s) + \bV{}{(2)}(s)) \bR^{2h1p/2p1h} &= \omega \bR^{1h/1p} \\
|
|
(\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s)) \bR^{1h/1p} + (\bC{}{(0)}(0) +\bC{}{(2)}(s) ) \bR^{2h1p/2p1h}&= \omega \bR^{2h1p/2p1h}
|
|
\end{aligned}
|
|
\right.
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
(\bF{}{(0)}(0) + \bF{}{(2)}(s)) + (\bV{}{(1)}(s) + \bV{}{(2)}(s)) (\omega \mathbb{1} - \bC{\text{d}}{(0)}(0) - \bC{\text{od}}{(1)}(s) -\bC{}{(2)}(s) )^{-1} (\bV{}{(1),\dagger}(s) + \bV{}{(2),\dagger}(s)) \bR^{1h/1p} = \omega \bR^{1h/1p}
|
|
\end{equation}
|
|
|
|
If we want to truncate the quasiparticle equation to the second order we obtain
|
|
|
|
\begin{equation}
|
|
(\bF{}{(0)}(0) + \bF{}{(2)}(s)) + \bV{}{(1)}(s)(\omega \mathbb{1} - \bC{\text{d}}{(0)}(0))^{-1} \bV{}{(1),\dagger}(s) \bR^{1h/1p} = \omega \bR^{1h/1p}
|
|
\end{equation}
|
|
\end{widetext}
|
|
So if we choose to put the off-diagonal part of $\bC{}{}$ in the off-diagonal $\bH{}{}$ we see that the off diagonal part of $\bC{}{}$ is not present in the second order quasi-particle equation.
|
|
We believe that this is not desirable.
|
|
In the following, we will integrate order by order the differential equations obtained above in the case $\bC{\text{od}}{} = \boldsymbol{0}$ and $\bC{\text{d}}{} = \bC{}{}$.
|
|
The expression in the other case are given in Appendix~\ref{sec:diagC}.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Integrating order by order}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
In the following, upper case indices correspond to the 2h1p and 2p1h sectors while lower case indices correspond to the 1h and 1p sectors. Also the $\Delta\epsilon_R$ corresponds to the diagonal elements of the 2h1p and 2p1h sectors.
|
|
|
|
\subsubsection{First order}
|
|
|
|
\begin{align}
|
|
\dv{\bF{}{(1)}}{s} &= \bO \Longleftrightarrow \bF{}{(1)}(s) = \bF{}{(1)}(0) \Longleftrightarrow \color{red}{\boxed{\color{black}{\bF{}{(1)}(s)= \bO}}} \\
|
|
\dv{\bV{}{(1)}}{s} &= 2 \bF{}{(0)}\bV{}{(1)}\bC{}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{}{(0)})^2 \\
|
|
\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} \\
|
|
\dv{\bC{}{(1)}}{s} &= \bO \Longleftrightarrow \bC{}{(1)}(s) = \bC{}{(1)}(0) \Longleftrightarrow \color{red}{\boxed{\color{black}{\bC{}{(1)}(s)= \bO}}}
|
|
\end{align}
|
|
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}).
|
|
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 $\bF{}{} = \epsilon_p$.
|
|
\begin{align}
|
|
\label{eq:matrixdiffeq}
|
|
\dv{\bV{}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \\
|
|
\dv{\bV{}{(1),\dagger}}{s} &= (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
|
|
\end{align}
|
|
|
|
\begin{itemize}
|
|
\item \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}
|
|
|
|
\item \textbf{GW}
|
|
In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ 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}
|
|
Therefore the eigenvalues of $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_p \Omega_{(q,v)} - \epsilon_p^2 - \Omega_{q,v}^2 = -(\epsilon_p - \Omega_{(q,v)})^2$.
|
|
And finally the analytical expressions for the GW coupling blocks at first order are
|
|
\begin{align}
|
|
\bV{}{\hhp,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s})\bU^{-1}\bV{}{\hhp,(1)}(0) \\
|
|
\bV{}{\pph,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(a,v)})^2s})\bU^{-1}\bV{}{\pph,(1)}(0)
|
|
\end{align}
|
|
|
|
Therefore the downfolded SRG self-energy is
|
|
\begin{align}
|
|
&\bSig(\omega)^{\hhp} = \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) \\
|
|
&\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
|
|
&= \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_p - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag
|
|
\end{align}
|
|
Renaming the index $p$ as $r$ and using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
|
|
\begin{align}
|
|
\label{eq:SRGGW_selfenergy}
|
|
\Sigma(\omega)_{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 \\
|
|
&+ \sum_{(a,v)} \frac{M_{ap}^{(v)}M_{aq}^{(v)}}{\omega - \epsilon_a - \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}
|
|
\end{align}
|
|
where the $\pph$ part has been obtained by an analog derivation.
|
|
|
|
\item \textbf{GF(2)}
|
|
|
|
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.
|
|
A compeltely analog derivation gives
|
|
\begin{align}
|
|
\label{eq:SRGGF2_selfenergy}
|
|
\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} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
|
|
&+ \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
|
|
\end{align}
|
|
|
|
\item \textbf{GT}
|
|
|
|
\textcolor{red}{\textbf{TODO Give analytical expression for the GT case.}}
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|
|
|
\end{itemize}
|
|
|
|
\subsubsection{Second order}
|
|
|
|
\begin{align}
|
|
\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} \\
|
|
\dv{\bC{}{(2)}}{s} &= \bC{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bC{}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \\
|
|
\dv{\bV{}{(2)}}{s} &= 2 \bF{}{(0)}\bV{}{(2)}\bC{}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{}{(0)})^2 \\
|
|
\dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(2),\dagger}\bF{}{(0)} - \bV{}{(2),\dagger}(\bF{}{(0)})^2 - (\bC{}{(0)})^2\bV{}{(2),\dagger}
|
|
\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.
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|
|
|
% =================================================================%
|
|
\section{Towards second quantized effective Hamiltonians for MBPT?}
|
|
\label{sec:second_quant_mbpt}
|
|
%=================================================================%
|
|
|
|
The many-body perturbation theory formalism and its various approximations are naturally derived using time-dependent Feynman diagrams.
|
|
These derivation are quite different from wave function methods like configuration interaction (CI) and coupled-cluster (CC) which are naturally expressed in second quantization.
|
|
One can study the link between these formalisms by expanding the MBPT Feynman diagrams into time-independent Goldstone diagrams and then compare them to the ones that appear in WFT.
|
|
However, that would be valuable to extend this connection by expressing the MBPT approximations in the second quantization formalism.
|
|
This is the aim of this section.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{The IP/EA CI}
|
|
\label{sec:ip_ea_ci}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
We start by expressing the electronic Hamiltonian in the IP/EA basis to compare its expressions to the matrices of Sec.~\ref{sec:unfolded}.
|
|
|
|
\appendix
|
|
|
|
%=================================================================%
|
|
\section{Matrix elements of $C=[A, B]_{1,2}$}
|
|
%=================================================================%
|
|
|
|
An operator $A$ containing at most two-body terms may be written in normal ordered form with respect to the reference $\Phi$ as
|
|
|
|
$$
|
|
A=A_{0}+A_{1}+A_{2},
|
|
$$
|
|
|
|
where $A_{0}$ is a scalar, and
|
|
|
|
$$
|
|
\begin{gathered}
|
|
A_{1}=\sum_{p q} A_{p}^{q}\left\{\hat{a}_{q}^{p}\right\}, \\
|
|
A_{2}=\frac{1}{4} \sum_{p q r s} A_{p q}^{r s}\left\{\hat{a}_{r s}^{p q}\right\},
|
|
\end{gathered}
|
|
$$
|
|
|
|
with the second quantization operator written compactly as $\hat{a}_{q}^{p}=\hat{a}_{p}^{\dagger} \hat{a}_{q}$ and $\hat{a}_{r s}^{p q}=\hat{a}_{p}^{\dagger} \hat{a}_{q}^{\dagger} \hat{a}_{s} \hat{a}_{r}$. The commutator $C=[A, B]_{1,2}$ contains contributions from the following terms:
|
|
|
|
$$
|
|
\begin{gathered}
|
|
C_{0}=\left\langle\Phi\left|\left[A_{1}, B_{1}\right]\right| \Phi\right\rangle+\left\langle\Phi\left|\left[A_{2}, B_{2}\right]\right| \Phi\right\rangle, \\
|
|
C_{p}^{q}=\left[A_{1}, B_{1}\right]_{p}^{q}+\left[A_{1}, B_{2}\right]_{p}^{q}-\left[B_{1}, A_{2}\right]_{p}^{q}+\left[A_{2}, B_{2}\right]_{p}^{q}, \\
|
|
C_{p q}^{r s}=\left[A_{1}, B_{2}\right]_{p q}^{r s}-\left[B_{1}, A_{2}\right]_{p q}^{r s}+\left[A_{2}, B_{2}\right]_{p q}^{r s},
|
|
\end{gathered}
|
|
$$
|
|
|
|
where the unique contributions to the matrix elements are
|
|
|
|
$$
|
|
\begin{gathered}
|
|
\left\langle\Phi\left|\left[A_{1}, B_{1}\right]\right| \Phi\right\rangle=\sum_{p} \sum_{i}\left(A_{i}^{p} B_{p}^{i}-B_{i}^{p} A_{p}^{i}\right), \\
|
|
\left\langle\Phi\left|\left[A_{2}, B_{2}\right]\right| \Phi\right\rangle=\frac{1}{4} \sum_{i j} \sum_{a b}\left(A_{i j}^{a b} B_{a b}^{i j}-B_{i j}^{a b} A_{a b}^{i j}\right), \\
|
|
{\left[A_{1}, B_{1}\right]_{p}^{q}=\sum_{r}\left(A_{p}^{r} B_{r}^{q}-B_{p}^{r} A_{r}^{q}\right),} \\
|
|
{\left[A_{1}, B_{2}\right]_{p}^{q}=\sum_{i} \sum_{a} A_{i}^{a} B_{p a}^{q i}-A_{a}^{i} B_{p i}^{q a},} \\
|
|
{\left[A_{2}, B_{2}\right]_{p}^{q}=\frac{1}{2} \sum_{i j} \sum_{a}\left(A_{a p}^{i j} B_{i j}^{a q}-A_{i j}^{a q} B_{a p}^{i j}\right)} \\
|
|
+\frac{1}{2} \sum_{i} \sum_{a b}\left(A_{i p}^{a b} B_{a b}^{i q}-A_{a b}^{i q} B_{i p}^{a b}\right), \\
|
|
{\left[A_{1}, B_{2}\right]_{p q}^{r s}=\sum_{t}\left[P(p q) A_{p}^{t} B_{t q}^{r s}-P(r s) A_{t}^{r} B_{p q}^{t s}\right],} \\
|
|
{\left[A_{2}, B_{2}\right]_{p q}^{r s}=\frac{1}{2} \sum_{a b}\left(A_{p q}^{a b} B_{a b}^{r s}-A_{a b}^{r s} B_{p q}^{a b}\right)} \\
|
|
-\frac{1}{2} \sum_{i j}\left(A_{p q}^{i j} B_{i j}^{r s}-A_{i j}^{r s} B_{p q}^{i j}\right) \\
|
|
+\sum_{i} \sum_{a} P(p q) P(r s)\left[A_{p i}^{r a} B_{q a}^{s i}-A_{p a}^{r i} B_{q i}^{s a}\right] .
|
|
\end{gathered}
|
|
$$
|
|
|
|
In these equations $P(r s)$ is the antisymmetric permutation operator.
|
|
|
|
%=================================================================%
|
|
\section{The ph- and pp-RPA problems}
|
|
\label{sec:rpa}
|
|
%=================================================================%
|
|
|
|
The particle-hole RPA is defined as
|
|
\begin{equation}
|
|
\begin{pmatrix}
|
|
\bA^\phRPA & \bB^\phRPA \\
|
|
- \bB^\phRPA & \bA^\phRPA \\
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix} = \boldsymbol{\Omega}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
|
|
with
|
|
|
|
\begin{align}
|
|
A^\phRPA_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \aeri{ib}{aj} \\
|
|
B^\phRPA_{ij,ab} &= \aeri{ij}{ab}
|
|
\end{align}
|
|
|
|
The equation above correspond to the RPAx version while the direct RPA version used in GW has the same element but with non-antisymmetrized Coulomb integrals.
|
|
|
|
The particle-particle RPA reads as
|
|
\begin{equation}
|
|
\begin{pmatrix}
|
|
\bA^\ppRPA & \bB^\ppRPA \\
|
|
- (\bB^\ppRPA)^\mathsf{T} & - \bC{}{\ppRPA} \\
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix} = \boldsymbol{\Omega}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
|
|
with
|
|
|
|
\begin{align}
|
|
A_{ab,cd}^\ppRPA &= \delta_{ab}\delta_{cd}(\epsilon_a + \epsilon_b) + \aeri{ab}{cd} \\
|
|
B_{ab,ij}^\ppRPA &= \aeri{ab}{ij} \\
|
|
C_{ij,kl}^\ppRPA &= -\delta_{ik}\delta_{jl}(\epsilon_i + \epsilon_j) + \aeri{ij}{kl}
|
|
\end{align}
|
|
|
|
%=================================================================%
|
|
\section{Downfolding procedure of the linear GW matrix}
|
|
\label{sec:downfolding}
|
|
%=================================================================%
|
|
|
|
\begin{equation}
|
|
\left\{ \begin{aligned}
|
|
\bF{}{}\bR^{\hp} + \bV{}{\hhp} \bR^{\hhp} + \bV{}{\pph} \bR^{\pph} &= E \bR^{\hp} \\
|
|
(\bV{}{\hhp})^{\mathsf{T}}\bR^{\hp} + \bC{}{\hhp} \bR^{\hhp} &= E \bR^{\hhp} \\
|
|
(\bV{}{\pph})^{\mathsf{T}}\bR^{\hp} + \bC{}{\pph} \bR^{\pph} &= E \bR^{\pph}
|
|
\end{aligned} \right.
|
|
\end{equation}
|
|
|
|
\begin{equation}
|
|
\left( \bF{}{} + \bSig(E) \right) \bR^{\hp} = E \bR^{\hp}
|
|
\end{equation}
|
|
|
|
\begin{align}
|
|
\bSig(E) &= \bV{}{\hhp} (E \mathbb{1} - \bC{}{\hhp})^{-1} (\bV{}{\hhp})^{\mathsf{T}} \\
|
|
&+ \bV{}{\pph} (E \mathbb{1} - \bC{}{\pph})^{-1} (\bV{}{\pph})^{\mathsf{T}} \notag
|
|
\end{align}
|
|
|
|
We introduce the matrices of eigenvectors $\bU^{\hhp}$ and $\bU^{\pph}$ such that
|
|
\begin{align}
|
|
\bC{}{\hhp} &= \bU^{\hhp} \text{diag}(\epsilon_i - \Omega^{(v)}) (\bU^{\hhp})^{-1} \\
|
|
\bC{}{\pph} &= \bU^{\pph} \text{diag}(\epsilon_a + \Omega^{(v)}) (\bU^{\pph})^{-1}
|
|
\end{align}
|
|
with
|
|
\begin{align}
|
|
U_{i[ja],(k,v)}^{\hhp} &= \bX_{ja}^{(v)}\delta_{pi} & U_{[ia]b,(c,v)}^{\pph} &= \bX_{ia}^{(v)}\delta_{bp}
|
|
\end{align}
|
|
|
|
Therefore we have
|
|
\begin{align}
|
|
(E \mathbb{1} - \bC{}{\hhp})^{-1} &= [\bU^{\hhp} (E \mathbb{1} - \text{diag}(\epsilon_i - \Omega^{(v)}))^{-1} (\bU^{\hhp})^{-1}]^{-1} \notag \\
|
|
&= \bU^{\hhp} (\text{diag}(E - \epsilon_i + \Omega^{(v)}))^{-1}(\bU^{\hhp})^{-1} \notag \\
|
|
&= \bU^{\hhp} \text{diag}(\frac{1}{E - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1}
|
|
\end{align}
|
|
|
|
\begin{equation}
|
|
\bSig(E)^{\hhp} = \bV{}{\hhp} \bU^{\hhp} \text{diag}(\frac{1}{E - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp})^{\mathsf{T}}
|
|
\end{equation}
|
|
|
|
We have
|
|
\begin{align}
|
|
&(\bV{}{\hhp} \bU^{\hhp})_{p,(k,v)} = \sum_{i[ja]} v_{p,i[ja]} U_{i[ja],(k,v)} \\
|
|
&= \sum_{i[ja]} \eri{pa}{ij} \bX_{ja}^{(v)}\delta_{ik} \notag \\
|
|
&= \sum_{[ja]} \eri{pa}{kj} \bX_{ja}^{(v)} \notag \\
|
|
&(\bV{}{\hhp} \bU^{\hhp} \text{diag}(\frac{1}{E - \epsilon_i + \Omega^{(v)}}) )_{p,(k,v)} \\
|
|
&=\sum_{l,(w)} \left(\sum_{[ja]} \eri{pa}{lj} \bX_{ja}^{(w)}\right) \text{diag}(\frac{1}{E - \epsilon_i + \Omega^{(v)}})_{(l,w),(k,v)} \delta_{lk}\delta_{(w),(v)} \notag \\
|
|
&= \left(\sum_{[ja]} \eri{pa}{kj} \bX_{ja}^{(v)}\right) \frac{1}{E - \epsilon_k + \Omega^{(v)}} \notag \\
|
|
&(\bSig(E)^{\hhp})_{p,q} \\
|
|
&= \sum_{(k,v)} \left(\sum_{[ja]} \eri{pa}{kj} \bX_{ja}^{(v)}\right) \frac{1}{E - \epsilon_k + \Omega^{(v)}} \left(\sum_{[ja]} \eri{qa}{kj} \bX_{ja}^{(v)}\right) \notag \\
|
|
&= \sum_{(k,v)} \frac{M_{kp}^{(v)}M_{kq}^{(v)}}{E - \epsilon_k + \Omega^{(v)}} \notag
|
|
\end{align}
|
|
|
|
%=================================================================%
|
|
\section{Perturbative matrix coefficients for $C^{(0)}$ diagonal}
|
|
\label{sec:diagC}
|
|
%=================================================================%
|
|
|
|
\begin{align}
|
|
(\dv{\bV{}{(1)}}{s})_{pQ} &= (2 \bF{}{(0)}\bV{}{(1)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(1)} - \bV{}{(1)}(\bC{\text{d}}{(0)})^2 )_{pQ}\\
|
|
&= \sum_{rS} 2 f^{(0)}_{pr} v^{(1)}_{rS}c^{(0)}_{SQ} - \sum_{rs} f^{(0)}_{pr} f^{(0)}_{rs} v^{(1)}_{sQ} - \sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS}c^{(0)}_{SQ} \\
|
|
&= \sum_{rS} 2 \epsilon^{(0)}_p\delta_{pr} v^{(1)}_{rS}\Delta\epsilon^{(0)}_Q\delta_{SQ} \\
|
|
&- \sum_{rs} \epsilon^{(0)}_p\delta_{pr} \epsilon^{(0)}_r\delta_{rs} v^{(1)}_{sQ} \\
|
|
&- \sum_{RS} v^{(1)}_{pR} \Delta\epsilon^{(0)}_R\delta_{RS} \Delta\epsilon^{(0)}_Q\delta_{SQ} \\
|
|
&= (2 \epsilon^{(0)}_p\Delta\epsilon^{(0)}_Q - (\epsilon^{(0)}_p)^2 - (\Delta\epsilon^{(0)}_Q )^2) v^{(1)}_{pQ} \\
|
|
\dv{v^{(1)}_{pQ}}{s} &= - (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2 v^{(1)}_{pQ} \\
|
|
&\color{red}{\boxed{\color{black}{v^{(1)}_{pQ}(s) = v^{(1)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} }}}
|
|
\end{align}
|
|
Note the close similarity with Evangelista's expressions for the off-diagonal part at first order!
|
|
\begin{align}
|
|
(\dv{\bC{}{(1)}}{s})_{PQ} &= (2 \bC{\text{d}}{(0)}\bC{\text{od}}{(1)}\bC{\text{d}}{(0)}- (\bC{\text{d}}{(0)})^2\bC{\text{od}}{(1)} - \bC{\text{od}}{(1)}(\bC{\text{d}}{(0)})^2)_{PQ} \\
|
|
&= \sum_{RS} 2 c^{(0)}_{PR} c^{(1)}_{RS} c^{(0)}_{SQ} - c^{(0)}_{PR} c^{(0)}_{RS} c^{(1)}_{SQ} - c^{(1)}_{PR} c^{(0)}_{RS} c^{(0)}_{SQ} \\
|
|
&= 2 \Delta\epsilon^{(0)}_Pc^{(1)}_{PQ}\Delta\epsilon^{(0)}_Q - (\Delta\epsilon^{(0)}_P)^2 c^{(1)}_{PQ} - c^{(1)}_{PQ} (\Delta\epsilon^{(0)}_Q)^2 \\
|
|
&= - (\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2 c^{(1)}_{PQ} \\
|
|
&\color{red}{\boxed{\color{black}{c^{(1)}_{PQ}(s) = c^{(1)}_{PQ}(0) e^{-s(\Delta\epsilon^{(0)}_P - \Delta\epsilon^{(0)}_Q )^2} }}}
|
|
\end{align}
|
|
|
|
\begin{align}
|
|
&(\dv{\bF{}{(2)}}{s})_{pq} = (\bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{\text{d}}{(0)}\bV{}{(1),\dagger})_{pq} \notag \\
|
|
&= \sum_{rS} f^{(0)}_{pr} v^{(1)}_{rS} v^{(1),\dagger}_{Sq} + \sum_{Rs} v^{(1)}_{pR} v^{(1),\dagger}_{Rs} f^{(0)}_{sq} - 2\sum_{RS} v^{(1)}_{pR} c^{(0)}_{RS} v^{(1),\dagger}_{Sq} \notag \\
|
|
&= \sum_{S} \epsilon^{(0)}_{p} v^{(1)}_{pS} v^{(1)}_{qS} + \sum_{R} \epsilon^{(0)}_{q} v^{(1)}_{pR} v^{(1)}_{qR} - 2\sum_{R} \Delta\epsilon^{(0)}_R v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
|
|
&= \sum_R (\epsilon^{(0)}_{p} + \epsilon^{(0)}_{q} - 2 \Delta\epsilon^{(0)}_R) v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
|
|
&= \sum_R (\epsilon^{(0)}_{p} + \epsilon^{(0)}_{q} - 2 \Delta\epsilon^{(0)}_R) v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) e^{-s [ (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_R)^2+ (\epsilon^{(0)}_q - \Delta\epsilon^{(0)}_R)^2]} \notag \\
|
|
&f^{(2)}_{pq}(s) = \notag \\
|
|
&\color{red}{\boxed{\color{black}{- \sum_R\frac{ v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) (\epsilon^{(0)}_{p} + \epsilon^{(0)}_{q} - 2 \Delta\epsilon^{(0)}_R)}{(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_R)^2+ (\epsilon^{(0)}_q - \Delta\epsilon^{(0)}_R)^2}(1 - e^{-s [ (\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_R)^2+ (\epsilon^{(0)}_q - \Delta\epsilon^{(0)}_R)^2]})}}} \notag
|
|
\end{align}
|
|
|
|
\begin{align}
|
|
(\dv{\bV{}{(2)}}{s})_{pQ} &= (2 \bF{}{(0)}\bV{}{(2)}\bC{\text{d}}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{\text{d}}{(0)})^2 \\
|
|
& - 2 \bV{}{(1)} \bC{\text{d}}{(0)} \bC{\text{od}}{(1)} + \bF{}{(0)} \bV{}{(1)} \bC{\text{od}}{(1)} + \bV{}{(1)} \bC{\text{od}}{(1)} \bC{\text{d}}{(0)})_{pQ} \notag \\
|
|
v^{(2)}_{pQ}(s) &= v^{(2)}_{pQ}(0) e^{-s(\epsilon^{(0)}_p - \Delta\epsilon^{(0)}_Q )^2} + \text{Non-homogeneous solution} \notag \\
|
|
v^{(2)}_{pQ}(s) &= \text{Non-homogeneous solution}
|
|
\end{align}
|
|
|
|
\end{document} |