saving work, GW and GF(2) up to first order, working on second order
This commit is contained in:
parent
4fd96bc8d4
commit
9676c5ac60
362
Notes/Notes.tex
362
Notes/Notes.tex
@ -13,126 +13,15 @@
|
||||
]{hyperref}
|
||||
\urlstyle{same}
|
||||
|
||||
\newcommand{\ie}{\textit{i.e.}}
|
||||
\newcommand{\eg}{\textit{e.g.}}
|
||||
\newcommand{\etal}{\textit{et al.}}
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
%============================================================%
|
||||
%%% NEWCOMMANDS %%%
|
||||
% ============================================================%
|
||||
|
||||
\input{Commands}
|
||||
|
||||
\usepackage[normalem]{ulem}
|
||||
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\ant}[1]{\textcolor{green}{#1}}
|
||||
\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}}
|
||||
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
||||
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\fnm}{\footnotemark}
|
||||
\newcommand{\fnt}{\footnotetext}
|
||||
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
||||
\newcommand{\QP}{\textsc{quantum package}}
|
||||
\newcommand{\T}[1]{#1^{\intercal}}
|
||||
\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
|
||||
\newcommand{\dRPA}{\text{dRPA}}
|
||||
|
||||
% coordinates
|
||||
\newcommand{\br}{\boldsymbol{r}}
|
||||
\newcommand{\bx}{\boldsymbol{x}}
|
||||
\newcommand{\dbr}{d\br}
|
||||
\newcommand{\dbx}{d\bx}
|
||||
|
||||
% methods
|
||||
\newcommand{\GW}{\text{$GW$}}
|
||||
\newcommand{\GT}{\text{$GT$}}
|
||||
\newcommand{\evGW}{ev$GW$}
|
||||
\newcommand{\qsGW}{qs$GW$}
|
||||
\newcommand{\GOWO}{$G_0W_0$}
|
||||
\newcommand{\Hxc}{\text{Hxc}}
|
||||
\newcommand{\xc}{\text{xc}}
|
||||
\newcommand{\Ha}{\text{H}}
|
||||
\newcommand{\co}{\text{c}}
|
||||
\newcommand{\x}{\text{x}}
|
||||
\newcommand{\KS}{\text{KS}}
|
||||
\newcommand{\HF}{\text{HF}}
|
||||
\newcommand{\RPA}{\text{RPA}}
|
||||
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
|
||||
\newcommand{\sERI}[2]{(#1|#2)}
|
||||
\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
|
||||
|
||||
%
|
||||
\newcommand{\Ne}{N}
|
||||
\newcommand{\Norb}{K}
|
||||
\newcommand{\Nocc}{O}
|
||||
\newcommand{\Nvir}{V}
|
||||
|
||||
% operators
|
||||
\newcommand{\hH}{\Hat{H}}
|
||||
\newcommand{\hS}{\Hat{S}}
|
||||
\newcommand{\ani}[1]{\hat{a}_{#1}}
|
||||
\newcommand{\cre}[1]{\hat{a}_{#1}^\dagger}
|
||||
\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} }
|
||||
|
||||
% energies
|
||||
\newcommand{\Enuc}{E^\text{nuc}}
|
||||
\newcommand{\Ec}[1]{E_\text{c}^{#1}}
|
||||
\newcommand{\EHF}{E^\text{HF}}
|
||||
|
||||
% orbital energies
|
||||
\newcommand{\eps}{\epsilon}
|
||||
\newcommand{\reps}{\Tilde{\epsilon}}
|
||||
|
||||
% Matrix elements
|
||||
\newcommand{\SigC}{\Sigma^\text{c}}
|
||||
\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}}
|
||||
\newcommand{\MO}[1]{\phi_{#1}}
|
||||
\newcommand{\SO}[1]{\psi_{#1}}
|
||||
\newcommand{\eri}[2]{\braket{#1}{#2}}
|
||||
\newcommand{\aeri}[2]{\mel{#1}{}{#2}}
|
||||
\newcommand{\ERI}[2]{(#1|#2)}
|
||||
\newcommand{\rbra}[1]{(#1|}
|
||||
\newcommand{\rket}[1]{|#1)}
|
||||
|
||||
|
||||
% Matrices
|
||||
\newcommand{\bO}{\boldsymbol{0}}
|
||||
\newcommand{\bI}{\boldsymbol{1}}
|
||||
\newcommand{\bH}{\boldsymbol{H}}
|
||||
\newcommand{\bSig}{\boldsymbol{\Sigma}}
|
||||
\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
|
||||
\newcommand{\be}{\boldsymbol{\epsilon}}
|
||||
\newcommand{\bOm}{\boldsymbol{\Omega}}
|
||||
\newcommand{\bA}{\boldsymbol{A}}
|
||||
\newcommand{\bB}{\boldsymbol{B}}
|
||||
\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
|
||||
\newcommand{\bD}{\boldsymbol{D}}
|
||||
\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
|
||||
\newcommand{\bR}{\boldsymbol{R}}
|
||||
\newcommand{\bU}{\boldsymbol{U}}
|
||||
\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
|
||||
\newcommand{\bW}{\boldsymbol{W}}
|
||||
\newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
|
||||
\newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
|
||||
\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
|
||||
\newcommand{\bc}{\boldsymbol{c}}
|
||||
|
||||
% orbitals, gaps, etc
|
||||
\newcommand{\IP}{I}
|
||||
\newcommand{\EA}{A}
|
||||
\newcommand{\HOMO}{\text{HOMO}}
|
||||
\newcommand{\LUMO}{\text{LUMO}}
|
||||
\newcommand{\Eg}{E_\text{g}}
|
||||
\newcommand{\EgFun}{\Eg^\text{fund}}
|
||||
\newcommand{\EgOpt}{\Eg^\text{opt}}
|
||||
\newcommand{\EB}{E_B}
|
||||
|
||||
% shortcuts for greek letters
|
||||
\newcommand{\si}{\sigma}
|
||||
\newcommand{\la}{\lambda}
|
||||
|
||||
|
||||
\newcommand{\RHH}{R_{\ce{H-H}}}
|
||||
\newcommand{\ii}{\mathrm{i}}
|
||||
|
||||
\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)}
|
||||
\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}}
|
||||
\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}}
|
||||
|
||||
% addresses
|
||||
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
|
||||
@ -185,7 +74,7 @@ Therefore, the transformed Hamiltonian
|
||||
depends on a flow parameter $s$.
|
||||
The resulting Hamiltonian possess up to $N$-body operators with $N$ the number of particle.
|
||||
\begin{equation}
|
||||
\bH(s) = E_0(s) + \bF(s) + \bV{}{}(s) + \bW(s) + \dots
|
||||
\bH(s) = E_0(s) + \bF{}{}(s) + \bV{}{}(s) + \bW(s) + \dots
|
||||
\end{equation}
|
||||
In the following, we will truncate every contribution superior to two-body operators.
|
||||
We can easily derive an evolution equation for this Hamiltonian by taking the derivative of $\bH(s)$. This gives
|
||||
@ -240,15 +129,15 @@ Hence, we define the off-diagonal Hamiltonian as
|
||||
\end{equation}
|
||||
Note that each coefficients depend on $s$.
|
||||
|
||||
The perturbative parameter $\la$ is such that
|
||||
The perturbative parameter $\lambda$ is such that
|
||||
\begin{equation}
|
||||
\bH(0) = E_0(0) + \bF{}{}(0) + \la \bV{}{}(0)
|
||||
\bH(0) = E_0(0) + \bF{}{}(0) + \lambda \bV{}{}(0)
|
||||
\end{equation}
|
||||
In addition, we know the following initial conditions.
|
||||
We use the HF basis set of the reference such that $\bF{}{\text{od}}(0) = 0$ and $\bF{}{\mathrm{d}}(0)_{pq}=\delta_{pq}\epsilon_p$.
|
||||
Therefore, we have
|
||||
\begin{align}
|
||||
\bH^\text{d}(0)&=E_0(0) + \bF{}{\mathrm{d}}(0) + \la \bV{}{\mathrm{d}}(0) & \bH^\text{od}(0)&= \la V^{\mathrm{od}}(0)
|
||||
\bH^\text{d}(0)&=E_0(0) + \bF{}{\mathrm{d}}(0) + \lambda \bV{}{\mathrm{d}}(0) & \bH^\text{od}(0)&= \lambda \bV{}{\mathrm{od}}(0)
|
||||
\end{align}
|
||||
Now, we want to compute the terms at each order of the following development
|
||||
\begin{equation}
|
||||
@ -398,30 +287,30 @@ In the following, we will focus on the GF(2), GW and GT approximations.
|
||||
The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory.
|
||||
\begin{align}
|
||||
\label{eq:GF2_selfenergy}
|
||||
\Sig{pq}{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \eps _a -\eps_i -\eps_j - \ii \eta} \notag \\
|
||||
&+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \eps _i -\eps_a -\eps_b + \ii \eta} \notag
|
||||
\Sigma_{pq}^{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\
|
||||
&+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag
|
||||
\end{align}
|
||||
On the other hand, the GW self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy.
|
||||
\begin{equation}
|
||||
\label{eq:GW_selfenergy}
|
||||
\Sig{pq}{\GW}(\omega) = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta} + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta} \notag
|
||||
\Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{\ceri{pi}{v} \ceri{qi}{v}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{am} \frac{\ceri{pa}{v} \ceri{qa}{v}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
\label{eq:GW_sERI}
|
||||
\sERI{pq}{m} = \sum_{ia} \eri{pq}{ia} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia} \notag
|
||||
\ceri{pq}{m} = \sum_{ia} \eri{pi}{qa} \qty( \bX_{m}^{\dRPA} + \bY_{m}^{\dRPA} )_{ia} \notag
|
||||
\end{equation}
|
||||
Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA
|
||||
The corresponding self-energies read as
|
||||
\begin{equation}
|
||||
\label{eq:GT_selfenergy}
|
||||
\Sig{pq}{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \e{i}{} - \Om{m}{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \e{a}{} - \Om{m}{N-2} - \ii \eta} \notag
|
||||
\Sigma_{pq}^{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \epsilon_i - \Omega_{m}^{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \epsilon_a - \Omega_{m}^{N-2} - \ii \eta} \notag
|
||||
\end{equation}
|
||||
with
|
||||
\begin{align}
|
||||
\label{eq:GT_sERI}
|
||||
\eri{pq}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX{cd,m}{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY{kl,m}{N+2} \notag \\
|
||||
\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
|
||||
\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 \\
|
||||
\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
|
||||
\end{align}
|
||||
The two RPA problems giving the eigenvectors needed to build the GW and GT self-energies are given in Appendix~\ref{sec:rpa}.
|
||||
|
||||
@ -441,40 +330,44 @@ For the three approximations considered here, the three matrices $\bH$ share the
|
||||
\label{eq:unfolded_matrice}
|
||||
\bH =
|
||||
\begin{pmatrix}
|
||||
\bF{}{} & \bV{}{\text{2h1p}} & \bV{}{\text{2p1h}} \\
|
||||
\T{(\bV{}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO \\
|
||||
\T{(\bV{}{\text{2p1h}})} & \bO & \bC{}{\text{2p1h}} \\
|
||||
\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( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} \delta_{ik}
|
||||
C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
|
||||
&
|
||||
C^\text{2p1h}_{iab,kcd} & = \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} \delta_{bd} \notag
|
||||
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( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} & &
|
||||
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( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd} \notag & &
|
||||
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( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac} & & \\
|
||||
C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik} & & \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.}}
|
||||
|
||||
@ -646,7 +539,7 @@ 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 $\Delta\eps_R$ corresponds to the diagonal elements of the 2h1p and 2p1h sectors.
|
||||
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}
|
||||
|
||||
@ -660,13 +553,83 @@ The differential equation for the coupling blocks can be solved in the GF(2) cas
|
||||
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} \\
|
||||
\dv{\bV{}{(1),\dagger}}{s} &= (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger}
|
||||
\end{align}
|
||||
These matrix differential equations can be solved if we know how to diagonalize $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2- (\bC{}{(0)})^2$.
|
||||
We know how to diagonalize $\bC{}{(0)}$ so we know how to diagonalize polynomial of $\bC{}{(0)}$.
|
||||
|
||||
\textcolor{red}{\textbf{TODO Give analytical expression for the different cases.}}
|
||||
\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.}}
|
||||
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{Second order}
|
||||
|
||||
@ -753,6 +716,111 @@ In these equations $P(r s)$ is the antisymmetric permutation operator.
|
||||
\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}
|
||||
@ -779,12 +847,12 @@ Note the close similarity with Evangelista's expressions for the off-diagonal pa
|
||||
|
||||
\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} \eps^{(0)}_{p} v^{(1)}_{pS} v^{(1)}_{qS} + \sum_{R} \eps^{(0)}_{q} v^{(1)}_{pR} v^{(1)}_{qR} - 2\sum_{R} \Delta\eps^{(0)}_R v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
|
||||
&= \sum_R (\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR} v^{(1)}_{qR} \notag \\
|
||||
&= \sum_R (\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R) v^{(1)}_{pR}(0) v^{(1)}_{qR}(0) e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]} \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) (\eps^{(0)}_{p} + \eps^{(0)}_{q} - 2 \Delta\eps^{(0)}_R)}{(\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2}(1 - e^{-s [ (\eps^{(0)}_p - \Delta\eps^{(0)}_R)^2+ (\eps^{(0)}_q - \Delta\eps^{(0)}_R)^2]})}}} \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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user