Working on notes BSE2

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Pierre-Francois Loos 2020-06-16 23:40:51 +02:00
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@ -56,11 +56,14 @@
\newcommand{\KS}{\text{KS}} \newcommand{\KS}{\text{KS}}
\newcommand{\HF}{\text{HF}} \newcommand{\HF}{\text{HF}}
\newcommand{\RPA}{\text{RPA}} \newcommand{\RPA}{\text{RPA}}
\newcommand{\RPAx}{\text{RPAx}}
\newcommand{\dRPAx}{\text{dRPAx}}
\newcommand{\BSE}{\text{BSE}} \newcommand{\BSE}{\text{BSE}}
\newcommand{\TDABSE}{\text{BSE(TDA)}} \newcommand{\TDABSE}{\text{BSE(TDA)}}
\newcommand{\dBSE}{\text{dBSE}} \newcommand{\dBSE}{\text{dBSE}}
\newcommand{\TDAdBSE}{\text{dBSE(TDA)}} \newcommand{\TDAdBSE}{\text{dBSE(TDA)}}
\newcommand{\GW}{GW} \newcommand{\GW}{GW}
\newcommand{\GF}{\text{GF2}}
\newcommand{\stat}{\text{stat}} \newcommand{\stat}{\text{stat}}
\newcommand{\dyn}{\text{dyn}} \newcommand{\dyn}{\text{dyn}}
\newcommand{\TDA}{\text{TDA}} \newcommand{\TDA}{\text{TDA}}
@ -79,10 +82,13 @@
\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}} \newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}} \newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
\newcommand{\eGW}[1]{\eps^{GW}_{#1}} \newcommand{\eGW}[1]{\eps^{GW}_{#1}}
\newcommand{\eGF}[1]{\eps^{\text{GF2}}_{#1}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}} \newcommand{\Om}[2]{\Omega_{#1}^{#2}}
% Matrix elements % Matrix elements
\newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}}
\newcommand{\SigGW}[1]{\Sigma^{\GW}_{#1}}
\newcommand{\SigGF}[1]{\Sigma^{\GF}_{#1}}
\newcommand{\MO}[1]{\phi_{#1}} \newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)} \newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\sERI}[2]{[#1|#2]} \newcommand{\sERI}[2]{[#1|#2]}
@ -220,31 +226,30 @@ The ground state has a one-electron configuration $v\bar{v}$, while the doubly-e
There is then only one single excitation which corresponds to the transition $v \to c$. There is then only one single excitation which corresponds to the transition $v \to c$.
As usual, this can produce a singlet singly-excited state of configuration $(v\bar{c} + c\bar{v})/\sqrt{2}$, and a triplet singly-excited state of configuration $(v\bar{c} - c\bar{v})/\sqrt{2}$ \cite{SzaboBook}. As usual, this can produce a singlet singly-excited state of configuration $(v\bar{c} + c\bar{v})/\sqrt{2}$, and a triplet singly-excited state of configuration $(v\bar{c} - c\bar{v})/\sqrt{2}$ \cite{SzaboBook}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamical BSE kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Within many-body perturbation theory (MBPT), one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. Within many-body perturbation theory (MBPT), one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is
\begin{equation} \begin{equation}
\e{p}^{\GW} = \e{p} + Z_{p} \Sig{p}(\e{p}) \e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
\end{equation} \end{equation}
where $p = v$ or $c$, where $p = v$ or $c$,
\begin{subequations} \begin{equation}
\begin{align} \label{eq:SigC}
\label{eq:Sigv} \SigGW{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - \Omega}
\Sig{v}(\omega) & = \frac{2 \ERI{vv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cv}^2}{\omega - \e{c} + \Omega} \end{equation}
\\
\label{eq:Sigc}
\Sig{c}(\omega) & = \frac{2 \ERI{vc}{cv}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cc}^2}{\omega - \e{c} + \Omega}
\end{align}
\end{subequations}
are the correlation parts of the self-energy associated with the valence of conduction orbitals, are the correlation parts of the self-energy associated with the valence of conduction orbitals,
\begin{equation} \begin{equation}
Z_{p} = \qty( 1 - \left. \pdv{\Sig{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1} Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation} \end{equation}
is the renormalization factor, and is the renormalization factor, and
\begin{equation} \begin{equation}
\ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br' \ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br'
\end{equation} \end{equation}
are the usual (bare) two-electron integrals. are the usual (bare) two-electron integrals.
In Eqs.~\eqref{eq:Sigv} and \eqref{eq:Sigc}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\e{} = \eGW{c} - \eGW{v}$. In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
\begin{equation} \label{eq:HBSE} \begin{equation} \label{eq:HBSE}
@ -258,7 +263,7 @@ One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, whic
with with
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
R(\omega) & = \Delta\e{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega) R(\omega) & = \Delta\eGW{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega)
\\ \\
C(\omega) & = 2 \sigma \ERI{vc}{cv} - W_C(\omega) C(\omega) & = 2 \sigma \ERI{vc}{cv} - W_C(\omega)
\end{align} \end{align}
@ -266,7 +271,7 @@ with
($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and ($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
W_R(\omega) & = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\e{}} W_R(\omega) & = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
\\ \\
W_C(\omega) & = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega} W_C(\omega) & = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
\end{align} \end{align}
@ -292,7 +297,7 @@ Within the static approximation, the BSE Hamiltonian is
\end{equation} \end{equation}
with with
\begin{align} \begin{align}
R^{\stat} & = R(\omega = \Delta\e{}) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\e{}) R^{\stat} & = R(\omega = \Delta\eGW{}) = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\eGW{})
\\ \\
C^{\stat} & = C(\omega = 0) = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0) C^{\stat} & = C(\omega = 0) = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
\end{align} \end{align}
@ -333,13 +338,17 @@ This system contains two orbitals and the numerical values of the various quanti
& &
\e{c} & = + 1.399\,859 \e{c} & = + 1.399\,859
\\ \\
\ERI{vv}{vv} & = 1.026\,907
&
\ERI{cc}{cc} & = 0.766\,363
\\
\ERI{vv}{cc} & = 0.858\,133 \ERI{vv}{cc} & = 0.858\,133
& &
\ERI{vc}{cv} & = 0.227\,670 \ERI{vc}{cv} & = 0.227\,670
\\ \\
\ERI{vv}{vc} & = 0.255\,554 \ERI{vv}{vc} & = 0.316\,490
& &
\ERI{vc}{cc} & = 0.316\,490 \ERI{vc}{cc} & = 0.255\,554
\end{align} \end{align}
which yields which yields
\begin{align} \begin{align}
@ -429,18 +438,18 @@ A quick configuration interaction with singles and doubles (CISD) calculation pr
\omega_{3}^{\updw} & = 3.47880 \omega_{3}^{\updw} & = 3.47880
\end{align} \end{align}
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree. This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
All these numerical results are gathered in Table \ref{tab:Ex}. All these numerical results are gathered in Table \ref{tab:BSE}.
The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value. The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value.
Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy. Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
%%% TABLE I %%% %%% TABLE I %%%
\begin{table} \begin{table}
\caption{Singlet and triplet excitation energies at various levels of theory. \caption{BSE singlet and triplet excitation energies at various levels of theory.
\label{tab:Ex} \label{tab:BSE}
} }
\begin{center} \begin{center}
\small \footnotesize
\begin{tabular}{|c|ccccccc|c|} \begin{tabular}{|c|ccccccc|c|}
\hline \hline
Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\ Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
@ -460,6 +469,106 @@ Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Second-order BSE kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here, we follow a different strategy and compute the dynamical second-order BSE kernel as illustrated by Yang and collaborators \cite{Zhang_2013}, and Rebolini and Toulouse \cite{Rebolini_2016}.
First, let us compute the second-order quasiparticle energies, which reads
\begin{equation}
\eGF{p} = \e{p} + Z_{p}^{\GF} \SigGF{p}(\e{p})
\end{equation}
where the second-order self-energy is
\begin{equation}
\label{eq:SigGF}
\SigGF{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \e{c} - \e{v}} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - (\e{c} - \e{v})}
\end{equation}
and
\begin{equation}
Z_{p}^{\GF} = \qty( 1 - \left. \pdv{\SigGF{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation}
This expression can be easily obtained in the present case by the substitution $\Omega \to \e{c} - \e{v}$ which transforms the $GW$ self-energy into its GF2 analog.
The static Hamiltonian for this theory is just the usual RPAx (or TDHF) Hamiltonian, \ie,
\begin{equation}
\bH^{\RPAx} =
\begin{pmatrix}
A^{\stat} & B^{\stat}
\\
-B^{\stat} & -A^{\stat}
\end{pmatrix}
\end{equation}
with
\begin{align}
A^{\stat} & = \Delta\eGF{} + 2 \sigma \ERI{vc}{vc} - \ERI{vv}{cc}
\\
B^{\stat} & = 2 \sigma \ERI{vc}{vc} - \ERI{vc}{cv}
\end{align}
The dynamical part of the kernel for BSE2 (that we will call dRPAx for notational consistency) is a bit ugly but it simplifies greatly in the case of the present model to yield
\begin{equation}
\bH^{\dRPAx} = \bH^{\RPAx} +
\begin{pmatrix}
A(\omega) & B
\\
-B & -A(-\omega)
\end{pmatrix}
\end{equation}
with
\begin{align}
A^{\updw}(\omega) & = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\
B^{\updw} & = - \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{align}
and
\begin{align}
A^{\upup}(\omega) & = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\
B^{\upup} & = - \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{align}
Note that the coupling blocks $B$ are frequency independent, as they should.
This has an important consequence as this lack of frequency dependence removes one of the spurious pole.
The singlet manifold has then the right number of excitations.
However, one spurious triplet excitation remains.
Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatmenets.
%%% TABLE II %%%
\begin{table}
\caption{RPAx singlet and triplet excitation energies at various levels of theory.
\label{tab:RPAx}
}
\begin{center}
\footnotesize
\begin{tabular}{|c|ccccccc|c|}
\hline
Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
\hline
$\omega_1$ & 1.84903 & 1.90927 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_2$ & & & & & & & & \\
$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
\hline
Triplets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
\hline
$\omega_1$ & 1.38912 & 1.44267 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
$\omega_2$ & & & & & & & & \\
$\omega_3$ & & & & 4.47797 & & & 4.47767 & \\
\hline
\end{tabular}
\end{center}
\end{table}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sangalli's kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In Ref.~\cite{Sangalli_2011}, Sangalli proposed a norm-conserving kernel without (he claims) spurious excitations.
For the two-level model, this kernel (based on the second RPA) reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Take-home messages}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
What have we learnt here?
% BIBLIOGRAPHY % BIBLIOGRAPHY
\bibliographystyle{unsrt} \bibliographystyle{unsrt}
\bibliography{../BSEdyn} \bibliography{../BSEdyn}