652 lines
39 KiB
TeX
652 lines
39 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,bbold}
|
|
\usepackage[version=4]{mhchem}
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage{txfonts}
|
|
|
|
\usepackage[
|
|
colorlinks=true,
|
|
citecolor=blue,
|
|
breaklinks=true
|
|
]{hyperref}
|
|
\urlstyle{same}
|
|
|
|
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
|
\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
|
|
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
|
\newcommand{\ant}[1]{\textcolor{green}{#1}}
|
|
|
|
% addresses
|
|
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
|
|
|
\begin{document}
|
|
|
|
\title{A Similarity Renormalization Group Approach To Many-Body Perturbation Theory}
|
|
|
|
\author{Antoine \surname{Marie}}
|
|
\email{amarie@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
|
|
\author{Pierre-Fran\c{c}ois \surname{Loos}}
|
|
\email{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
|
|
\begin{abstract}
|
|
Here comes the abstract.
|
|
%\bigskip
|
|
%\begin{center}
|
|
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
|
|
%\end{center}
|
|
%\bigskip
|
|
\end{abstract}
|
|
|
|
\maketitle
|
|
|
|
%=================================================================%
|
|
\section{Introduction}
|
|
\label{sec:intro}
|
|
%=================================================================%
|
|
|
|
One-body Green's functions provide a natural and elegant way to access the charged excitations energies of a physical system. \cite{Martin_2016,Golze_2019}
|
|
The one-body non-linear Hedin's equations give a recipe to obtain the exact interacting one-body Green's function and therefore the exact ionization potentials and electron affinities. \cite{Hedin_1965}
|
|
Unfortunately, fully solving Hedin's equations is out of reach and one must resort to approximations.
|
|
In particular, the $GW$ approximation, \cite{Hedin_1965} which has first been mainly used in the context of solids \cite{Strinati_1980,Strinati_1982,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely used for molecules as well \ant{ref?}, provides fairly accurate results for weakly correlated systems\cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2021}
|
|
|
|
The $GW$ approximation is an approximation for the self-energy $\Sigma$ which role is to relate the exact interacting Green's function $G$ to a non-interacting reference one $G_0$ through the Dyson equation
|
|
\begin{equation}
|
|
\label{eq:dyson}
|
|
G = G_0 + G_0\Sigma G.
|
|
\end{equation}
|
|
The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken in account in the reference system.
|
|
%Throughout this manuscript the references are chosen to be the Hartree-Fock (HF) ones so that the self-energy only account for the missing correlation.
|
|
Approximating $\Sigma$ as the first order term of its perturbation expansion with respect to the screened interaction $W$ gives the so-called $GW$ approximation. \cite{Hedin_1965, Martin_2016}
|
|
Alternatively one could choose to define $\Sigma$ as the $n$-th order expansion in terms of the bare Coulomb interaction leading to the GF($n$) class of approximations. \cite{Hirata_2015,Hirata_2017}
|
|
The GF(2) approximation is also known as the second Born approximation. \ant{ref ?}
|
|
|
|
Despite a wide range of successes, many-body perturbation theory is not flawless.
|
|
It has been shown that a variety of physical quantities such as charged and neutral excitations energies or correlation and total energies computed within many-body perturbation theory exhibits some discontinuities. \cite{Veril_2018,Loos_2018b}
|
|
Even more worrying these discontinuities can happen in the weakly correlated regime where $GW$ is thought to be valid.
|
|
These discontinuities are due to a transfer of spectral weight between two solutions of the quasi-particle equation. \cite{Monino_2022}
|
|
This is another occurrence of the infamous intruder-state problem. \cite{Roos_1995,Olsen_2000,Choe_2001} \ant{more ref}
|
|
In addition, systems for which two quasi-particle solutions have a similar spectral weight are known to be particularly difficult to converge for partially self-consistent $GW$. \cite{Forster_2021}
|
|
|
|
In a recent study, Monino and Loos showed that these discontinuities could be removed by introduction of a regularizer inspired by the similarity renormalisation group (SRG) in the quasi-particle equation. \cite{Monino_2022}
|
|
Encouraged by this result, this work will investigate the application of the SRG formalism to many-body perturbation theory in its $GW$ and GF(2) variants.
|
|
The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
|
|
This formalism has been been introduced in quantum chemistry by White \cite{White_2002} before being explored in more details by Evangelista and his co-workers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,Li_2015, Li_2016, Li_2017, Li_2018, Li_2019a}
|
|
The SRG has also been successful in the context of nuclear theory, \cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016} see Ref.\onlinecite{Hergert_2016a} for a recent review in this field. \ant{Maybe search for recent papers of T. Duguet as well.}
|
|
|
|
The SRG transformation aims at decoupling a reference space from an external space while folding information about the coupling in the reference space.
|
|
This is often during such decoupling that intruder states appear. \ant{ref}
|
|
However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space.
|
|
Because intruder states have energies really close to the reference energies they will be the last ones decoupled.
|
|
Therefore the SRG continuous transformation can be stopped once every external configurations except the intruder ones have been decoupled.
|
|
Doing so, it gives a way to fold in information about the coupling in the reference space while avoiding intruder states.
|
|
|
|
The aim of this manuscript is to investigate whether SRG can treat the intruder-state problem in many-body perturbation theory as successfully as it has been in other fields.
|
|
We begin by reviewing the $GW$ formalism in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
|
|
Section~\ref{sec:theoretical} is concluded by a perturbative analysis of the SRG formalism applied to $GW$ (see Sec.~\ref{sec:srggw}).
|
|
The computational details of our implementation are provided in Sec.~\ref{sec:comp_det} before turning to the results section.
|
|
This section starts by
|
|
|
|
%=================================================================%
|
|
\section{Theoretical background}
|
|
\label{sec:theoretical}
|
|
%=================================================================%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Many-body perturbation theory in the GW approximation}
|
|
\label{sec:gw}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasi-particle equation
|
|
\begin{equation}
|
|
\label{eq:quasipart_eq}
|
|
\left[ \bF + \bSig(\omega = \epsilon_p) \right] \psi_p = \epsilon_p \psi_p,
|
|
\end{equation}
|
|
where $\bF$ is the Fock matrix, \cite{SzaboBook} and $\bSig(\omega)$ is the self-energy, both are $K \times K$ matrices with $K$ the number of one-body basis functions.
|
|
The self-energy can be physically understood as a dynamical screening correction to the Hartree-Fock (HF) problem represented by $\bF$.
|
|
Similarly to the HF case, this equation needs to be solved self-consistently.
|
|
Note that $\bSig$ is dynamical, \ie it depends on both the eigenvalues $\epsilon_p$ and eigenvectors $\psi_p$ while $\bF$ depends only on the eigenvectors.
|
|
|
|
Because of this $\omega$ dependence, fully solving this equation is a rather complicated task, hence several approximate solving schemes has been developed.
|
|
The most popular one is probably the one-shot scheme, known as $G_0W_0$ if the self-energy is the $GW$ one, in which the off-diagonal elements of Eq.~(\ref{eq:quasipart_eq}) are neglected and the self-consistency is abandoned.
|
|
In this case, there are $K$ quasi-particle equations which read as
|
|
tr\begin{equation}
|
|
\label{eq:G0W0}
|
|
\epsilon_p^{\HF} + \Sigma_{p}(\omega) - \omega = 0,
|
|
\end{equation}
|
|
where $\Sigma_{p}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
|
|
The previous equation is non-linear with respect to $\omega$ and therefore can have multiple solutions $\epsilon_{p,s}$ for a given $p$.
|
|
These solutions can be characterised by their spectral weight defined as the renormalisation factor $Z_{p,s}$
|
|
\begin{equation}
|
|
\label{eq:renorm_factor}
|
|
0 \leq Z_{p,s} = \left[ 1 - \pdv{\Sigma_{p}(\omega)}{\omega}\bigg|_{\omega=\epsilon_{p,s}} \right]^{-1} \leq 1.
|
|
\end{equation}
|
|
The solution with the largest weight is referred to as the quasi-particle solution while the others are known as satellites or shake-up solutions.
|
|
However, in some cases Eq.~(\ref{eq:G0W0}) can have two (or more) solutions with similar weights and the quasi-particle solution is not well-defined.
|
|
In fact, these cases are related to the discontinuities and convergence problems discussed earlier because the additional solutions with large weights are the previously mentioned intruder states.
|
|
|
|
One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
|
|
Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
|
|
Therefore, one could try to optimise the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
|
|
Alternatively, one could solve this equation self-consistently leading to the eigenvalue only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
|
|
To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached.
|
|
However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
|
|
Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
|
|
|
|
To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~(\ref{eq:quasipart_eq}).
|
|
To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
|
|
The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
|
|
Various choice for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
|
|
\begin{equation}
|
|
\label{eq:sym_qsgw}
|
|
\Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right).
|
|
\end{equation}
|
|
This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
|
|
One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form, this will be done in the next section.
|
|
|
|
In this case as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
|
|
Multiple solutions arise due to the $\omega$ dependence of the self-energy.
|
|
Therefore, by suppressing this dependence the static qs approximation relies on the fact that there is one well-defined quasi-particle solution.
|
|
If it is not the case, the qs scheme will oscillates between the solutions with large weights. \cite{Forster_2021}
|
|
|
|
Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
|
|
The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
|
|
The $\ii eta$ term that is usually added in the denominator of the self-energy (see below) is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
|
|
Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages, in the $GW$ case, over the imaginary shift one. \cite{Monino_2022}
|
|
But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
|
|
This is the aim of this work.
|
|
|
|
Therefore if we apply it, the SRG would gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
|
|
However, to do so one needs to identify the coupling terms in Eq.~(\ref{eq:quasipart_eq}), which is not straightforward.
|
|
The way around this problem is to transform Eq.~(\ref{eq:quasipart_eq}) to its upfolded version and the coupling terms will elegantly appear in the process.
|
|
From now on, we will restrict ourselves to the $GW$ in the Tamm-Dancoff approximation (TDA) case but the same derivation could be done for the non-TDA $GW$ and GF(2) self-energies.
|
|
The corresponding formula are given in Appendix~\ref{sec:nonTDA} and \ref{sec:GF2}, respectively.
|
|
The upfolded $GW$ quasi-particle equation is the following
|
|
\begin{equation}
|
|
\label{eq:GWlin}
|
|
\begin{pmatrix}
|
|
\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
|
|
(\bV^{\text{2h1p}})^{\mathrm{T}} & \bC^{\text{2h1p}} & \bO \\
|
|
(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bC^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY^{\text{2h1p}} \\
|
|
\bY^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY^{\text{2h1p}} \\
|
|
\bY^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\boldsymbol{\epsilon},
|
|
\end{equation}
|
|
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
|
|
\begin{subequations}
|
|
\begin{align}
|
|
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_{i}^{\GW} + \epsilon_{j}^{\GW} - \epsilon_{a}^{\GW}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
|
|
\\
|
|
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_{a}^{\GW} + \epsilon_{b}^{\GW} - \epsilon_{i}^{\GW}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
|
|
\end{align}
|
|
\end{subequations}
|
|
and the corresponding coupling blocks read
|
|
\begin{align}
|
|
V^\text{2h1p}_{p,klc} & = \eri{pc}{kl},
|
|
&
|
|
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
|
|
\end{align}
|
|
Throughout the manuscript $p,q,r,s$ indices are used for general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
|
|
The indices $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
|
|
The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
|
|
\begin{equation}
|
|
\label{eq:GWnonlin}
|
|
\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
|
|
\end{equation}
|
|
with
|
|
\begin{align}
|
|
\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
|
|
&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
|
|
\end{align}
|
|
which can be further developed to give
|
|
\begin{equation}
|
|
\label{eq:GW_selfenergy}
|
|
\Sigma_{pq}(\omega)
|
|
= \sum_{iv} \frac{W_{p,(i,v)} W_{q,(i,v)}}{\omega - \epsilon_i + \Omega_{v} - \ii \eta}
|
|
+ \sum_{av} \frac{W_{p,(a,v)}W_{q,(a,v)}}{\omega - \epsilon_a - \Omega_{v} + \ii \eta},
|
|
\end{equation}
|
|
with the screened integrals defined as
|
|
\begin{equation}
|
|
\label{eq:GW_sERI}
|
|
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia},
|
|
\end{equation}
|
|
where $\bX$ is the matrix of eigenvectors of the particle-hole direct RPA (dRPA) problem in the TDA defined as
|
|
\begin{equation}
|
|
\bA \bX = \boldsymbol{\Omega} \bX,
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
A^\dRPA_{ij,ab} = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}.
|
|
\end{equation}
|
|
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues and its elements $\Omega_v$ appear in Eq.~(\ref{eq:GW_selfenergy}).
|
|
|
|
Equations~(\ref{eq:GWlin}) and~(\ref{eq:GWnonlin}) have exactly the same solutions but one is linear and the other not.
|
|
The price to pay for this linearity is that the size of the matrix in the former equation is $\mathcal{O}(K^3)$ while it is $\mathcal{O}(K)$ in the latter one.
|
|
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Chapter 8 of Ref.~\onlinecite{Schirmer_2018} for the GF(2) case).
|
|
|
|
As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $ V^\text{2p1h}$ are coupling the 1h and 1p configuration to the dressed 2h1p and 2p1h configurations.
|
|
Therefore, these blocks will be the target of our SRG transformation but before going in more details we will review the SRG formalism.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{The similarity renormalization group}
|
|
\label{sec:srg}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The similarity renormalization group method aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form.
|
|
Therefore, the transformed Hamiltonian
|
|
\begin{equation}
|
|
\label{eq:SRG_Ham}
|
|
\bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
|
|
\end{equation}
|
|
depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block)-diagonal Hamiltonian.
|
|
An evolution equation for $\bH(s)$ can be easily obtained by deriving Eq~(\ref{eq:SRG_Ham}) with respect to $s$.
|
|
This gives the flow equation
|
|
\begin{equation}
|
|
\label{eq:flowEquation}
|
|
\dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)},
|
|
\end{equation}
|
|
where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
|
|
\begin{equation}
|
|
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
|
|
\end{equation}
|
|
To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$.
|
|
Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian.
|
|
The Hamiltonian is separated in two parts as
|
|
\begin{equation}
|
|
\bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}},
|
|
\end{equation}
|
|
and by definition we have the following condition on $\bH^\text{od}$
|
|
\begin{equation}
|
|
\bH^\text{od}(s=\infty) = \boldsymbol{0}.
|
|
\end{equation}
|
|
|
|
In this work, we will use Wegner's canonical generator which is defined as \cite{Wegner_1994}
|
|
\begin{equation}
|
|
\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}.
|
|
\end{equation}
|
|
If this generator is used, the following condition is verified \cite{Kehrein_2006}
|
|
\begin{equation}
|
|
\label{eq:derivative_trace}
|
|
\dv{s}\text{Tr}\left[ \bH^\text{od}(s)^2 \right] \leq 0
|
|
\end{equation}
|
|
which implies that the matrix elements of the off-diagonal part will decrease in a monotonic way.
|
|
Even more, the coupling coefficients associated with the highest energy determinants are removed first as will be evidenced by the perturbative analysis after.
|
|
The main flaw of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \ant{ref}
|
|
However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014, Hergert_2016}
|
|
|
|
Let us now perform the perturbative analysis of the SRG equations.
|
|
For $s=0$, the initial problem is
|
|
\begin{equation}
|
|
\bH(0) = \bH^\text{d}(0) + \lambda ~ \bH^\text{od}(0)
|
|
\end{equation}
|
|
where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the perturber.
|
|
For finite values of $s$, we have the following perturbation expansion of the Hamiltonian
|
|
\begin{equation}
|
|
\label{eq:perturbation_expansionH}
|
|
\bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \dots
|
|
\end{equation}
|
|
Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as well.
|
|
Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) and solve analytically the low-order differential equations.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Renormalised GW}
|
|
\label{sec:srggw}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Finally, the SRG formalism exposed above will be applied to $GW$.
|
|
First, one needs to defined the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
|
|
As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as
|
|
\begin{align}
|
|
\label{eq:diag_and_offdiag}
|
|
\bH^\text{d}(s) &=
|
|
\begin{pmatrix}
|
|
\bF & \bO & \bO \\
|
|
\bO & \bC^{\text{2h1p}} & \bO \\
|
|
\bO & \bO & \bC^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
&
|
|
\bH^\text{od}(s) &=
|
|
\begin{pmatrix}
|
|
\bO & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
|
|
(\bV^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
|
|
(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
|
|
\end{pmatrix}
|
|
\end{align}
|
|
where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
|
|
Then, the aim of this section is to solve order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are
|
|
\begin{align}
|
|
\bHd{0}(0) &= \begin{pmatrix}
|
|
\bF{}{} & \bO \\
|
|
\bO & \bC{}{}
|
|
\end{pmatrix} &
|
|
\bHod{0}(0) &=\begin{pmatrix}
|
|
\bO & \bO \\
|
|
\bO & \bO
|
|
\end{pmatrix} \notag \\
|
|
\bHd{1}(0) &=\begin{pmatrix}
|
|
\bO & \bO \\
|
|
\bO & \bO
|
|
\end{pmatrix} &
|
|
\bHod{1}(0) &= \begin{pmatrix}
|
|
\bO & \bV{}{} \\
|
|
\bV{}{\dagger} & \bO \notag
|
|
\end{pmatrix} \notag
|
|
\end{align}
|
|
where we have defined the matrix $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
|
|
Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalised quasi-particle equation.
|
|
In particular, in this manuscript the focus will be on the second-order renormalized quasi-particle equation.
|
|
|
|
%///////////////////////////%
|
|
\subsubsection{Zero-th order matrix elements}
|
|
%///////////////////////////%
|
|
|
|
There is only one zero-th order term in the right hand side of the flow equation
|
|
\begin{equation}
|
|
\dv{\bH^{(0)}}{s} = \comm{\comm{\bHd{0}}{\bHod{0}}}{\bH^{(0)}},
|
|
\end{equation}
|
|
and performing the block matrix products gives the following system of equations
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\dv{\bF^{(0)}}{s} &= \bO \\
|
|
\dv{\bC^{(0)}}{s} &= \bO \\
|
|
\dv{\bV^{(0),\dagger}}{s} &= 2 \bC^{(0)}\bV^{(0),\dagger}\bF^{(0)} - \bV^{(0),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bV^{(0),\dagger} \\
|
|
\dv{\bV^{(0)}}{s} &= 2 \bF^{(0)}\bV^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bV^{(0)} - \bV^{(0)}(\bC^{(0)})^2
|
|
\end{align}
|
|
\end{subequations}
|
|
where the $s$ dependence of $\bV^{(0)}$ and $\bV^{(0),\dagger}$ has been dropped in the last two equations.
|
|
$\bF^{(0)}$ and $\bC^{(0)}$ do not depend on $s$ as a consequence of the first two equations.
|
|
The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)} = \bU \bD^{(0)} \bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
|
|
\begin{equation}
|
|
\dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
|
|
\end{equation}
|
|
where $\bW^{(0)}= \bV^{(0)} \bU$.
|
|
Note that the matrix $\bU$ is also used in the downfolding process of Eq.~(\ref{eq:GWlin}). \cite{Bintrim_2021}
|
|
Due to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, these equations can be easily solved and give
|
|
\begin{equation}
|
|
W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
|
|
\end{equation}
|
|
Due to the initial conditions $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
|
|
The two first equations of the system are trivial and finally we have
|
|
\begin{equation}
|
|
\bH^{(0)}(s) = \bH^{(0)}(0)
|
|
\end{equation}
|
|
which shows that the zero-th order matrix elements are independent of $s$.
|
|
The matrix elements of $\bU$ and $\bD$ are
|
|
\begin{align}
|
|
U_{(p,v),(q,w)}^{(0)} &= \delta_{pq} \bX_{v,w} \\
|
|
D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw}
|
|
\end{align}
|
|
where $\epsilon_F$ is the Fermi level.
|
|
|
|
%///////////////////////////%
|
|
\subsubsection{First order matrix elements}
|
|
%///////////////////////////%
|
|
|
|
Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
|
|
\begin{equation}
|
|
\dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}}
|
|
\end{equation}
|
|
which gives the same system of equations as in the previous subsection except that $\bV^{(0)}$ and $\bV^{(0),\dagger}$ should be replaced by $\bV^{(1)}$ and $\bV^{(1),\dagger}$.
|
|
|
|
Once again the two first equations are easily solved
|
|
\begin{align}
|
|
\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO.
|
|
\end{align}
|
|
and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
|
|
\begin{align}
|
|
W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\
|
|
W_{p,(q,v)}^{(1)}(s) &= \left( \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia} \right) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s}
|
|
\end{align}
|
|
Note that at $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) and that for $s\to\infty$ they go to zero.
|
|
Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals.
|
|
Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
|
|
|
|
%///////////////////////////%
|
|
\subsubsection{Second-order matrix elements}
|
|
% ///////////////////////////%
|
|
|
|
The second-order renormalised quasi-particle equation is given by
|
|
\begin{equation}
|
|
\label{eq:GW_renorm}
|
|
\left( \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) \right) \bX = \omega \bX
|
|
\end{equation}
|
|
with
|
|
\begin{align}
|
|
\widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s)\\
|
|
\label{eq:srg_sigma}
|
|
\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \mathbb{1} - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}
|
|
\end{align}
|
|
|
|
As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasi-particle equation.
|
|
Collecting every second-order terms and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
|
|
\begin{equation}
|
|
\label{eq:diffeqF2}
|
|
\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} .
|
|
\end{equation}
|
|
This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
|
|
\begin{equation}
|
|
F_{pq}^{(2)}(s) = \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right).
|
|
\end{equation}
|
|
with $\Delta_{pqv} = \epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v$.
|
|
|
|
At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
|
|
\begin{equation}
|
|
\label{eq:static_F2}
|
|
F_{pq}^{(2)}(\infty) = \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
|
|
\end{equation}
|
|
Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
|
|
Therefore, the SRG flow gradually transforms the dynamic degrees of freedom of $\bSig(\omega)$ in static ones, starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
|
|
Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
|
|
Yet, both are closely related as they share the same diagonal terms.
|
|
Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.
|
|
|
|
However, as will be discussed in more details in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor.
|
|
This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
|
|
Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable. \ant{ref fabien ?}
|
|
Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
|
|
\begin{align}
|
|
\label{eq:SRG_qsGW}
|
|
\Sigma_{pq}^{\text{SRG}}(s) &= \epsilon_p \delta_{pq} + \sum_{r,v} \frac{\left(\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)\right) W_{p,(r,v)} W_{q,(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} \notag \\
|
|
&\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
|
|
\end{align}
|
|
which depends on one regularising parameter $s$ analogously to $eta$ in the usual case.
|
|
The fact that the $s\to\infty$ static limit does not well converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
|
|
Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
|
|
|
|
To conclude this section, we will discuss the case of discontinuities.
|
|
Indeed, we have previously said that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
|
|
So is it possible to remove discontinuities by using the SRG machinery developed above?
|
|
In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
|
|
However, doing a change of variable such that
|
|
\begin{align}
|
|
e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t}
|
|
\end{align}
|
|
hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
|
|
In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
|
|
|
|
%=================================================================%
|
|
\section{Computational details}
|
|
\label{sec:comp_det}
|
|
% =================================================================%
|
|
|
|
The two qs$GW$ variants considered in this work has been implemented in a in-house program.
|
|
The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
|
|
The geometry have been optimized at the CC3 level in the aug-cc-pvtz basis set without frozen core using the CFOUR program.
|
|
The reference CCSD(T) IP energies have obtained using default parameters of Gaussian 16.
|
|
This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism.
|
|
|
|
%=================================================================%
|
|
\section{Results}
|
|
\label{sec:results}
|
|
%=================================================================%
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Conclusion}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
Here comes the conclusion.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\acknowledgements{This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Data availability statement}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The data that supports the findings of this study are available within the article.% and its supplementary material.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\bibliography{SRGGW}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\appendix
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Non-TDA $GW$ equations}
|
|
\label{sec:nonTDA}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
The $GW$ self-energy without TDA is the same as in Eq.~(\ref{eq:GW_selfenergy}) but the screened integrals are now defined as
|
|
\begin{equation}
|
|
\label{eq:GWnonTDA_sERI}
|
|
W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
|
|
\end{equation}
|
|
where $\bX$ and $\bY$ are the matrix of eigenvectors of the full particle-hole dRPA problem defined as
|
|
\begin{equation}
|
|
\label{eq:full_dRPA}
|
|
\begin{pmatrix}
|
|
\bA & \bB \\
|
|
- \bB & \bA \\
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix} = \boldsymbol{\Omega}
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY \\
|
|
\end{pmatrix},
|
|
\end{equation}
|
|
with
|
|
\begin{align}
|
|
A_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\
|
|
B_{ij,ab} &= \eri{ij}{ab}.
|
|
\end{align}
|
|
$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ in this case has the same size as in the TDA because we consider only the positive excitations of the full dRPA problem.
|
|
|
|
Defining an unfold version of this equation which does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
|
|
However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
|
|
\begin{equation}
|
|
\label{eq:nonTDA_upfold}
|
|
\begin{pmatrix}
|
|
\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
|
|
(\bW^{\text{2h1p}})^{\mathrm{T}} & \bD^{\text{2h1p}} & \bO \\
|
|
(\bW^{\text{2p1h}})^{\mathrm{T}} & \bO & \bD^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY^{\text{2h1p}} \\
|
|
\bY^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
=
|
|
\begin{pmatrix}
|
|
\bX \\
|
|
\bY^{\text{2h1p}} \\
|
|
\bY^{\text{2p1h}} \\
|
|
\end{pmatrix}
|
|
\cdot
|
|
\boldsymbol{\epsilon},
|
|
\end{equation}
|
|
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~(\ref{eq:full_dRPA}).
|
|
|
|
|
|
where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasi-particle and satellite energies, the 2h1p and 2p1h matrix elements are
|
|
\begin{subequations}
|
|
\begin{align}
|
|
D^\text{2h1p}_{ija,klc} & = \delta_{ik}\delta_{jl} \delta_{ac} \qty[\epsilon_i - \Omega_{ja}] ,
|
|
\\
|
|
D^\text{2p1h}_{iab,kcd} & = \delta_{ik}\delta_{ac} \delta_{bd} \qty[\epsilon_a + \Omega_{ib}] ,
|
|
\end{align}
|
|
\end{subequations}
|
|
and the corresponding coupling blocks read
|
|
\begin{align}
|
|
W^\text{2h1p}_{p,klc} & = \sum_{ia}\eri{pi}{ka} \qty( \bX_{lc} + \bY_{lc})_{ia} \\
|
|
W^\text{2p1h}_{p,kcd} & = \sum_{ia}\eri{pi}{ca} \qty( \bX_{kd} + \bY_{kd})_{ia}
|
|
\end{align}
|
|
|
|
Using the SRG on this matrix instead of Eq.~(\ref{eq:GWlin}) gives the same expression for $\bW^{(1)}$, $\bF^{(2)}$ and $\bSig^{\text{SRG}}$ but now the screened integrals are the one of Eq.~\eqref{eq:GWnonTDA_sERI} and the eigenvalues $\Omega$ and eigenvectors $\bX$ and $\bY$ are the ones of the full RPA problem defined in Eq.~\eqref{eq:full_dRPA}.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{GF(2) equations}
|
|
\label{sec:GF2}
|
|
%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
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.
|
|
The matrix elements of its second-order version read as
|
|
\begin{align}
|
|
\label{eq:GF2_selfenergy}
|
|
\Sigma_{pq}^{\text{GF(2)}}(\omega)
|
|
&= \frac{1}{\sqrt{2}} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \\
|
|
&+ \frac{1}{\sqrt{2}} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta}
|
|
\end{align}
|
|
This self-energy can be upfolded similarly to the $GW$ case and one obtain the following ``super-matrix''
|
|
\begin{equation}
|
|
\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 $\bC{}{}$ is given below.
|
|
\begin{align}
|
|
\label{eq:GF2_unfolded}
|
|
V^\text{2h1p}_{p,ija} & = \frac{1}{\sqrt{2}}\aeri{pa}{ij}
|
|
\\
|
|
V^\text{2p1h}_{p,iab} & = \frac{1}{\sqrt{2}}\aeri{pi}{ab}
|
|
\\
|
|
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}
|
|
\end{align}
|
|
Note that this matrix is exactly the ADC(2) matrix for charged excitations.
|
|
The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.
|
|
|
|
Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
|
|
We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasiparticle equations.
|
|
|
|
\begin{equation}
|
|
(V^\text{2h1p}_{p,ija})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pa}{ij} e^{- (\epsilon_p + \epsilon_a - \epsilon_i - \epsilon_j)^2 s}
|
|
\end{equation}
|
|
\begin{equation}
|
|
(V^\text{2h1p}_{p,iab})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pi}{ab} e^{- (\epsilon_p + \epsilon_i - \epsilon_a - \epsilon_b)^2 s}
|
|
\end{equation}
|
|
|
|
We define $ \Delta_{pq,rs} = \epsilon_p + \epsilon_q - \epsilon_r - \epsilon_s $
|
|
|
|
\begin{align}
|
|
F_{pq}^{(2)}(s) &= \sum_{ria} \frac{\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W_{p,(r,v)} \notag \\
|
|
&\times W^{\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).
|
|
\end{align}
|
|
|
|
|
|
\end{document}
|