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