minor corrections up to the end
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dynker.tex
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dynker.tex
@ -186,14 +186,14 @@ In the next section, we illustrate these concepts and the various tricks that ca
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\label{sec:exact}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us consider a two-level quantum system made of two orbitals in its singlet ground state (\ie, the lowest orbital is doubly occupied). \cite{Romaniello_2009b}
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Let us consider a two-level quantum system made of two electrons in its singlet ground state (\ie, the lowest orbital is doubly occupied). \cite{Romaniello_2009b}
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We will label these two orbitals, $\MO{v}$ and $\MO{c}$, as valence ($v$) and conduction ($c$) orbitals with respective one-electron Hartree-Fock (HF) energies $\e{v}$ and $\e{c}$.
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In a more quantum chemical language, these correspond to the HOMO and LUMO orbitals (respectively).
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The ground state $\ket{0}$ has a one-electron configuration $\ket{v\bar{v}}$, while the doubly-excited state $\ket{D}$ has a configuration $\ket{c\bar{c}}$.
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There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations.
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As usual, this can produce a singlet singly-excited state $\ket{S} = (\ket{v\bar{c}} + \ket{c\bar{v}})/\sqrt{2}$, and a triplet singly-excited state $\ket{T} = (\ket{v\bar{c}} - \ket{c\bar{v}})/\sqrt{2}$. \cite{SzaboBook}
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As usual, this produces a singlet singly-excited state $\ket{S} = (\ket{v\bar{c}} + \ket{c\bar{v}})/\sqrt{2}$, and a triplet singly-excited state $\ket{T} = (\ket{v\bar{c}} - \ket{c\bar{v}})/\sqrt{2}$. \cite{SzaboBook}
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For the singlet manifold, the exact Hamiltonian in the basis of the (spin-adapted) configuration state functions reads
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For the singlet manifold, the exact Hamiltonian in the basis of these (spin-adapted) configuration state functions reads
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\begin{equation} \label{eq:H-exact}
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\bH^{\updw} =
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\begin{pmatrix}
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@ -227,7 +227,7 @@ and $\Delta\e{} = \e{c} - \e{v}$.
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The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
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For the sake of illustration, we will use the same numerical example throughout this study, and consider the singlet ground state of the \ce{He} atom in Pople's 6-31G basis set.
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This system contains two orbitals and the numerical values of the various quantities defined above are
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The numerical values of the various quantities defined above are
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\begin{subequations}
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\begin{align}
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\e{v} & = -0.914\,127
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@ -351,7 +351,7 @@ However, following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strin
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Based on the very same two-level model that we employ here, Romaniello and coworkers \cite{Romaniello_2009b} clearly evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent BSE eigenvalue problem.
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For this particular system, they showed that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations, \cite{Romaniello_2009b} attributed to the self-screening problem. \cite{Romaniello_2009a}
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This issue was resolved in the subsequent work of Sangalli \textit{et al.} \cite{Sangalli_2011} via the design of a diagrammatic number-conserving approach based on the folding of the second-RPA Hamiltonian. \cite{Wambach_1988}
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Thanks to a careful diagrammatic analysis of the dynamic kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
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Thanks to a careful diagrammatic analysis of the dynamical kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
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Within the so-called $GW$ approximation of MBPT, \cite{Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
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Assuming that $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is
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@ -400,7 +400,7 @@ and
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\begin{gather}
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W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
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\\
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W^{\co}_C(\omega) = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
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W^{\co}_C(\omega) = \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
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\end{gather}
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\end{subequations}
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are the elements of the correlation part of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian.
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@ -435,7 +435,7 @@ As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears d
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Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}).
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Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^{\dBSE,\updw}$) and two spurious solutions for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$ and $\omega_{3}^{\dBSE,\upup}$).
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It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
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\titou{T2: add comment on how one can detect fake solutions?}
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This makes these two latter solutions quite hard to locate with the Newton-Raphson method.
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%%% TABLE I %%%
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\begin{table*}
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@ -473,6 +473,7 @@ It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slop
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Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the dBSE Hamiltonian \eqref{eq:HBSE}, allows to remove some of these spurious excitations.
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There is thus only one spurious excitation in the triplet manifold ($\omega_{3}^{\BSE,\upup}$), the two solutions of the singlet manifold corresponding now to the single and double excitations.
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The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
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Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
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%%% FIGURE 2 %%%
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@ -487,7 +488,7 @@ Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in th
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%%% %%% %%% %%%
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In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
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Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation, and it shows that the physical single excitation stemming from the dBSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
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Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
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Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} a scheme we label as perturbative BSE (pBSE).
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To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
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@ -617,17 +618,18 @@ and
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f_{\dBSE2}^{\co,\upup} = - \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
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\end{gather}
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\end{subequations}
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Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin-aware with distinct expressions for singlets and triplets, and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
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Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets, and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
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This latter point has an important consequence as this lack of frequency dependence removes one of the spurious pole (see Fig.~\ref{fig:BSE2}).
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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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It is mentioned in Ref.~\onlinecite{Rebolini_2016} that the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
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Numerical results for the two-level model are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
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In the case of BSE2, the perturbative partitioning is simply
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\begin{equation}
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\bH_{\dBSE2}^{\sigma}(\omega)
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= \underbrace{\bH_{\BSE2}^{\sigma}}_{\bH_{\pBSE2}^{(0)}}
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+ \underbrace{\qty[ \bH_{\dBSE2}^{\sigma}(\omega) - \bH_{\BSE2}^{\sigma} ]}_{\bH_{\pBSE2}^{(1)}}
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+ \underbrace{\qty[ \bH_{\dBSE2}^{\sigma}(\omega) - \bH_{\BSE2}^{\sigma} ]}_{\bH_{\pBSE2}^{(1)}(\omega)}
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\end{equation}
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%%% TABLE II %%%
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@ -669,53 +671,53 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The forgotten kernel: Sangalli's kernel}
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\label{sec:Sangalli}
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%\subsection{The forgotten kernel: Sangalli's kernel}
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%\label{sec:Sangalli}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\titou{This section is experimental...}
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In Ref.~\onlinecite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
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We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
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The Hamiltonian with Sangalli's kernel is (I think)
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\begin{equation}
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\bH_\text{S}^{\sigma}(\omega) =
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\begin{pmatrix}
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\bR_\text{S}^{\sigma}(\omega) & \bC_\text{S}^{\sigma}(\omega)
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\\
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-\bC_\text{S}^{\sigma}(-\omega) & -\bR_\text{S}^{\sigma}(-\omega)
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\end{pmatrix}
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\end{equation}
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with
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\begin{subequations}
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\begin{gather}
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R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
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\\
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C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
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\end{gather}
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\end{subequations}
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and
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\begin{subequations}
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\begin{gather}
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f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
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\\
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c_{ia,mn}^{\sigma} = \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
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+ R_{m,kb} R_{n,jc} ] }
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\end{gather}
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\end{subequations}
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where $R_{m,ia}$ are the elements of the RPA eigenvectors.
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For the two-level model, Sangalli's kernel reads
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\begin{align}
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R(\omega) & = \Delta\eGW{} + f_R (\omega)
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\\
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C(\omega) & = f_C (\omega)
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\end{align}
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\begin{gather}
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f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
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\\
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f_C (\omega) = 0
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\end{gather}
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%\titou{This section is experimental...}
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%In Ref.~\onlinecite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
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%We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
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%
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%The Hamiltonian with Sangalli's kernel is (I think)
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%\begin{equation}
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% \bH_\text{S}^{\sigma}(\omega) =
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% \begin{pmatrix}
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% \bR_\text{S}^{\sigma}(\omega) & \bC_\text{S}^{\sigma}(\omega)
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% \\
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% -\bC_\text{S}^{\sigma}(-\omega) & -\bR_\text{S}^{\sigma}(-\omega)
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% \end{pmatrix}
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%\end{equation}
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%with
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%\begin{subequations}
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%\begin{gather}
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% R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
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% \\
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% C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
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%\end{gather}
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%\end{subequations}
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%and
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%\begin{subequations}
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%\begin{gather}
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% f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
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% \\
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% c_{ia,mn}^{\sigma} = \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
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% + R_{m,kb} R_{n,jc} ] }
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%\end{gather}
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%\end{subequations}
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%where $R_{m,ia}$ are the elements of the RPA eigenvectors.
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%
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%For the two-level model, Sangalli's kernel reads
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%\begin{align}
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% R(\omega) & = \Delta\eGW{} + f_R (\omega)
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% \\
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% C(\omega) & = f_C (\omega)
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%\end{align}
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%
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%\begin{gather}
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% f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
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% \\
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% f_C (\omega) = 0
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%\end{gather}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Take-home messages}
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