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@ -60,7 +60,8 @@
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\begin{document}
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\title{A Similarity Renormalization Group Approach To Many-Body Perturbation Theory}
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% \title{A Similarity Renormalization Group Approach To Many-Body Perturbation Theory}
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\title{Tackling The Intruder-State Problem In Many-Body Perturbation Theory: A Similarity Renormalization Group Approach/Perspective}
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\author{Antoine \surname{Marie}}
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\email{amarie@irsamc.ups-tlse.fr}
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@ -86,8 +87,6 @@ Here comes the abstract.
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\label{sec:intro}
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% =================================================================%
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\ANT{Should we introduce the acronym MBPT?}
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One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
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The non-linear Hedin equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, to a wealth of properties such as the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965}
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Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations.
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@ -108,9 +107,6 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
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\end{equation}
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Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
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\trashant{Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017}
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The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013} }
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Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
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For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
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Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter equation. \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} However, the accuracy is not yet satisfying for triplet excited states, where instabilities often occur. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b,Holzer_2018a}
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@ -692,33 +688,21 @@ The data that supports the findings of this study are available within the artic
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\label{sec:nonTDA}
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%%%%%%%%%%%%%%%%%%%%%%
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The matrix elements of the $GW$ self-energy without TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
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The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
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\begin{equation}
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\label{eq:GWnonTDA_sERI}
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W_{p,(q,v)} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v} + \bY_{v})_{ia},
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W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
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\end{equation}
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where $\bX$ and $\bY$ are the eigenvector matrices of the full particle-hole dRPA problem defined as
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where $\bX$ ais the eigenvector matrix of the TDA particle-hole dRPA problem defined as
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\begin{equation}
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\label{eq:full_dRPA}
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\begin{pmatrix}
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\bA & \bB \\
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- \bB & \bA \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix} = \boldsymbol{\Omega}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix},
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\bA \bX = \bX \boldsymbol{\Omega}
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\end{equation}
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with
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\begin{align}
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A_{ij,ab} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj}, \\
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B_{ij,ab} &= \eri{ij}{ab}.
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A_{ia,jb} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj} \\
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\end{align}
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$\boldsymbol{\Omega}$ 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.
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$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ has the same size as without the TDA because in RPA we consider only the positive excitations of the full dRPA problem.
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Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
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However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
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@ -763,59 +747,59 @@ and the corresponding coupling blocks read
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Using the SRG on this matrix instead of Eq.~\eqref{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}.
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%%%%%%%%%%%%%%%%%%%%%%
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\section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}
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\label{sec:GF2}
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%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%
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% \section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}
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% \label{sec:GF2}
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% %%%%%%%%%%%%%%%%%%%%%%
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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.
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The matrix elements of its second-order version read as
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\begin{align}
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\label{eq:GF2_selfenergy}
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\Sigma_{pq}^{\text{GF(2)}}(\omega)
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&= \frac{1}{\sqrt{2}} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \\
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&+ \frac{1}{\sqrt{2}} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta}
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\end{align}
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This self-energy can be upfolded similarly to the $GW$ case and one obtain the following ``super-matrix''
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\begin{equation}
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\label{eq:unfolded_matrice}
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\bH =
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\begin{pmatrix}
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\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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(\bV^{\text{2h1p}})^{\mathsf{T}} & \bC^{\text{2h1p}} & \bO \\
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(\bV^{\text{2p1h}})^{\mathsf{T}} & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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\end{equation}
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The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ is given below.
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\begin{align}
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\label{eq:GF2_unfolded}
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V^\text{2h1p}_{p,ija} & = \frac{1}{\sqrt{2}}\aeri{pa}{ij}
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\\
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V^\text{2p1h}_{p,iab} & = \frac{1}{\sqrt{2}}\aeri{pi}{ab}
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\\
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C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
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\\
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C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd}
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\end{align}
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Note that this matrix is exactly the ADC(2) matrix for charged excitations.
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The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.
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% 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.
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% The matrix elements of its second-order version read as
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% \begin{align}
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% \label{eq:GF2_selfenergy}
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% \Sigma_{pq}^{\text{GF(2)}}(\omega)
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% &= \frac{1}{\sqrt{2}} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \\
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% &+ \frac{1}{\sqrt{2}} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta}
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% \end{align}
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% This self-energy can be upfolded similarly to the $GW$ case and one obtain the following ``super-matrix''
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% \begin{equation}
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% \label{eq:unfolded_matrice}
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% \bH =
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% \begin{pmatrix}
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% \bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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% (\bV^{\text{2h1p}})^{\mathsf{T}} & \bC^{\text{2h1p}} & \bO \\
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% (\bV^{\text{2p1h}})^{\mathsf{T}} & \bO & \bC^{\text{2p1h}} \\
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% \end{pmatrix}
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% \end{equation}
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% The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ is given below.
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% \begin{align}
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% \label{eq:GF2_unfolded}
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% V^\text{2h1p}_{p,ija} & = \frac{1}{\sqrt{2}}\aeri{pa}{ij}
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% \\
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% V^\text{2p1h}_{p,iab} & = \frac{1}{\sqrt{2}}\aeri{pi}{ab}
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% \\
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% C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
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% \\
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% C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd}
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% \end{align}
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% Note that this matrix is exactly the ADC(2) matrix for charged excitations.
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% The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.
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Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
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We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasi-particle equations.
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% Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
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% We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasi-particle equations.
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\begin{equation}
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(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}
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\end{equation}
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\begin{equation}
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(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}
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\end{equation}
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% \begin{equation}
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% (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}
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% \end{equation}
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% \begin{equation}
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% (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}
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% \end{equation}
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We define $ \Delta_{pq,rs} = \epsilon_p + \epsilon_q - \epsilon_r - \epsilon_s $
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% We define $ \Delta_{pq,rs} = \epsilon_p + \epsilon_q - \epsilon_r - \epsilon_s $
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\begin{align}
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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 \\
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&\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).
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\end{align}
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% \begin{align}
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% 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 \\
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% &\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).
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% \end{align}
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\end{document}
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