From 573330e2b150ad83bbf1b5514eab20f640d5e365 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 18 May 2020 12:21:08 +0200 Subject: [PATCH] saving work --- BSEdyn.tex | 309 +++++++++++++++++++++++++++++++++++------------------ 1 file changed, 203 insertions(+), 106 deletions(-) diff --git a/BSEdyn.tex b/BSEdyn.tex index d39535d..bb61ee7 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -24,13 +24,10 @@ \definecolor{darkgreen}{HTML}{009900} \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} -\newcommand{\denis}[1]{\textcolor{purple}{#1}} \newcommand{\xavier}[1]{\textcolor{darkgreen}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} -\newcommand{\trashDJ}[1]{\textcolor{purple}{\sout{#1}}} \newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} -\renewcommand{\DJ}[1]{\denis{(\underline{\bf DJ}: #1)}} \newcommand{\XB}[1]{\xavier{(\underline{\bf XB}: #1)}} \newcommand{\mc}{\multicolumn} @@ -63,6 +60,10 @@ % operators \newcommand{\hH}{\Hat{H}} +% methods +\newcommand{\RPA}{\text{RPA}} +\newcommand{\BSE}{\text{BSE}} + % energies \newcommand{\Enuc}{E^\text{nuc}} \newcommand{\Ec}{E_\text{c}} @@ -109,7 +110,7 @@ \newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} -\newcommand{\sERI}[3]{[#1|#2]^{#3}} +\newcommand{\sERI}[2]{[#1|#2]} %% bold in Table \newcommand{\bb}[1]{\textbf{#1}} @@ -117,9 +118,9 @@ \newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} % excitation energies -\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}} -\newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}} -\newcommand{\OmBSE}[2]{\Omega_{#1}^{#2,\text{BSE}}} +\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}} +\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}} +\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}} \newcommand{\spinup}{\downarrow} \newcommand{\spindw}{\uparrow} @@ -144,8 +145,8 @@ \newcommand{\bA}[1]{\mathbf{A}^{#1}} \newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}} \newcommand{\bB}[1]{\mathbf{B}^{#1}} -\newcommand{\bX}[1]{\mathbf{X}^{#1}} -\newcommand{\bY}[1]{\mathbf{Y}^{#1}} +\newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}} \newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}} \newcommand{\bK}{\mathbf{K}} \newcommand{\bP}[1]{\mathbf{P}^{#1}} @@ -170,9 +171,7 @@ \newcommand\hOms{\frac{{\Omega}_s}{2}} \newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France} -\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} -\newcommand{\CEA}{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France} \begin{document} @@ -206,6 +205,9 @@ This is the abstract \section{Theory} \label{sec:theory} %%%%%%%%%%%%%%%%%%%%%%%% +%================================ +\subsection{Theory for physics} +%================================= The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables: \begin{align*} @@ -273,138 +275,233 @@ $$ in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of $[ij|m] [ab|m]$ ?? } -\titou{This is the theory section from the previous paper.} +%In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016} +%\begin{multline} +%\label{eq:BSE} +% \LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2') +% \\ +% + \int d3 d4 d5 d6 \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2') +%\end{multline} +%as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential +%\begin{equation} +% \XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)}, +%\end{equation} +%which takes into account the self-consistent variation of the Hartree potential +%\begin{equation} +% \vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+), +%\end{equation} +%(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$. +%In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables. +%In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have +%\begin{equation} +% \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2), +%\end{equation} +%where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to +%\begin{equation} +% \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4), +%\end{equation} +%where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} +%Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. -In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016} -\begin{multline} -\label{eq:BSE} - \LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2') - \\ - + \int d3 d4 d5 d6 \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2') -\end{multline} -as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential -\begin{equation} - \XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)}, -\end{equation} -which takes into account the self-consistent variation of the Hartree potential -\begin{equation} - \vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+), -\end{equation} -(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$. -In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables. -In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have -\begin{equation} - \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2), -\end{equation} -where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to -\begin{equation} - \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4), -\end{equation} -where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} -Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. +%================================ +\subsection{Theory for chemists} +%================================= -For a closed-shell system in a finite basis, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$), one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} +For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \begin{equation} -\label{eq:LR} +\label{eq:LR-dyn} \begin{pmatrix} - \bA{\IS} & \bB{\IS} \\ - -\bB{\IS} & -\bA{\IS} \\ + \bA{}(\omega) & \bB{}(\omega) \\ + -\bB{}(\omega) & -\bA{}(\omega) \\ \end{pmatrix} + \cdot \begin{pmatrix} - \bX{\IS}_m \\ - \bY{\IS}_m \\ + \bX{m}{}(\omega) \\ + \bY{m}{}(\omega) \\ \end{pmatrix} = - \Om{m}{\IS} + \omega \begin{pmatrix} - \bX{\IS}_m \\ - \bY{\IS}_m \\ + \bX{m}{}(\omega) \\ + \bY{m}{}(\omega) \\ \end{pmatrix}, \end{equation} -where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interaction strength $\IS$, $\T{}$ is the matrix transpose, and we assume real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. -The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively. +where the dynamical matrices $\bA{}(\omega)$, $\bB{}(\omega)$, $\bX{}{}(\omega)$, and $\bY{}{}(\omega)$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. -In the absence of instabilities (\ie, when $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension -\begin{equation} -\label{eq:small-LR} - (\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{m}{\IS} = (\Om{m}{\IS})^2 \bZ{m}{\IS}, -\end{equation} -where the excitation amplitudes are +The BSE matrix elements read \begin{subequations} \begin{align} - (\bX{\IS} + \bY{\IS})_m = (\Om{m}{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{m}{\IS}, - \\ - (\bX{\IS} - \bY{\IS})_m = (\Om{m}{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{m}{\IS}. -\end{align} -\end{subequations} -Introducing the so-called Mulliken notation for the bare two-electron integrals -\begin{equation} - \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', -\end{equation} -and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$ -\begin{equation} - \W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}', -\end{equation} -the BSE matrix elements read -\begin{subequations} -\begin{align} - \label{eq:LR_BSE-A} - \ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \W{ij,ab}{\IS} ], + \label{eq:BSE-Adyn} + \A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{}(\omega), \\ - \label{eq:LR_BSE-B} - \BBSE{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ], + \label{eq:BSE-Bdyn} + \B{ia,jb}{}(\omega) & = 2 \ERI{ia}{bj} - \W{ib,aj}{}(\omega), \end{align} \end{subequations} -where $\eGW{p}$ are the $GW$ quasiparticle energies. -In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie, -\begin{subequations} -\label{eq:wrpa} -\begin{align} - \W{}{\IS}(\br{},\br{}') - & = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , - \\ - \epsilon_{\IS}(\br{},\br{}'; \omega) - & = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' , -\end{align} -\end{subequations} -with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals +where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, +\begin{equation} + \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}' +\end{equation} +are the bare two-electron integrals in the molecular orbital basis $\lbrace \MO{p}(\br{}) \rbrace_{1 \le p \le \Norb}$, and the dynamically-screened Coulomb potential reads \begin{multline} \label{eq:W} - \W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m}{\IS} \sERI{ab}{m}{\IS} + \W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m} \\ - \times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}), + \times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}), \end{multline} -where the spectral weights at coupling strength $\IS$ read +where $\eta$ is a positive infinitesimal, and \begin{equation} - \sERI{pq}{m}{\IS} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}. +\label{eq:sERI} + \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia} \end{equation} -In the case of complex orbitals, we refer the reader to Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$. -Note that, in the case of {\GOWO}, the RPA neutral excitations in Eq.~\eqref{eq:W} are computed using the HF orbital energies. - -In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements +are the spectral weights. +In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (linear) static response problem +\begin{equation} +\label{eq:LR-stat} + \begin{pmatrix} + \bA{\RPA} & \bB{\RPA} \\ + -\bB{\RPA} & -\bA{\RPA} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{\RPA} \\ + \bY{m}{\RPA} \\ + \end{pmatrix} + = + \OmRPA{m} + \begin{pmatrix} + \bX{m}{\RPA} \\ + \bY{m}{\RPA} \\ + \end{pmatrix}, +\end{equation} +with \begin{subequations} \begin{align} \label{eq:LR_RPA-A} - \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb}, + \A{ia,jb}{\RPA} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 \ERI{ia}{jb}, \\ \label{eq:LR_RPA-B} - \BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj}, + \B{ia,jb}{\RPA} & = 2 \ERI{ia}{bj}, \end{align} \end{subequations} -where $\eHF{p}$ are the Hartree-Fock (HF) orbital energies. +where the $\e{p}$'s are taken as the Hartree-Fock (HF) orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent scheme such as ev$GW$. -The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$. -In this limit, the $GW$ quasiparticle energies reduce to the HF eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations: +Now, let us decompose, using basis perturbation theory, the eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static part and a first-order dynamic part, such that +\begin{equation} +\label{eq:LR-dyn} + \begin{pmatrix} + \bA{}(\omega) & \bB{}(\omega) \\ + -\bB{}(\omega) & -\bA{}(\omega) \\ + \end{pmatrix} + = + \begin{pmatrix} + \bA{(0)} & \bB{(0)} \\ + -\bB{(0)} & -\bA{(0)} \\ + \end{pmatrix} + + + \begin{pmatrix} + \bA{(1)}(\omega) & \bB{(1)}(\omega) \\ + -\bB{(1)}(\omega) & -\bA{(1)}(\omega) \\ + \end{pmatrix} +\end{equation} +where \begin{subequations} \begin{align} - \label{eq:LR_RPAx-A} - \ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \ERI{ij}{ab} ], + \label{eq:BSE-0} + \A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}}, \\ - \label{eq:LR_RPAx-B} - \BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ]. + \label{eq:BSE-0} + \B{ia,jb}{(0)} & = 2 \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}, \end{align} \end{subequations} +and +\begin{subequations} +\begin{align} + \label{eq:BSE-1} + \A{ia,jb}{(1)}(\omega) & = - \W{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}}, + \\ + \label{eq:BSE-1} + \B{ia,jb}{(1)}(\omega) & = - \W{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}}, +\end{align} +\end{subequations} +The static version of the screened Coulomb potential reads +\begin{equation} +\label{eq:Wstat} + \W{ij,ab}{\text{stat}} = \ERI{ij}{ab} - 4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}. +\end{equation} +The $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as +\begin{subequations} +\begin{gather} + \Om{m}{} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots + \\ + \begin{pmatrix} + \bX{m}{} \\ + \bY{m}{} \\ + \end{pmatrix} + = + \begin{pmatrix} + \bX{m}{(0)} \\ + \bY{m}{(0)} \\ + \end{pmatrix} + + + \begin{pmatrix} + \bX{m}{(1)} \\ + \bY{m}{(1)} \\ + \end{pmatrix} + + \ldots +\end{gather} +\end{subequations} +Solving the zeroth-order static problem yields +\begin{equation} + \begin{pmatrix} + \bA{(0)} & \bB{(0)} \\ + -\bB{(0)} & -\bA{(0)} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{(0)} \\ + \bY{m}{(0)} \\ + \end{pmatrix} + = + \Om{m}{(0)} + \begin{pmatrix} + \bX{m}{(0)} \\ + \bY{m}{(0)} \\ + \end{pmatrix}, +\end{equation} +Thanks to first-order perturbation theory, the first-order correction to the $m$th excitation energy is +\begin{equation} + \Om{m}{(1)} = + \T{\begin{pmatrix} + \bX{m}{(0)} \\ + \bY{m}{(0)} \\ + \end{pmatrix}} + \cdot + \begin{pmatrix} + \bA{(1)}(\Om{m}{(0)}) & \bB{(1)}(\Om{m}{(0)}) \\ + -\bB{(1)}(\Om{m}{(0)}) & -\bA{(1)}(\Om{m}{(0)}) \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{(0)} \\ + \bY{m}{(0)} \\ + \end{pmatrix}. +\end{equation} +From a practical point of view, if one enforces the Tamm-Dancoff approximation (TDA), we obtain the very simple expression +\begin{equation} + \Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}. +\end{equation} + +This correction can be renormalized by computing, at basically no extra cost, the renormalization factor +\begin{equation} + Z_{m} = \qty[ \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}. +\end{equation} +which finally yields +\begin{equation} + \Om{m}{} \approx \Om{m}{(0)} + Z_{m} \Om{m}{(1)}. +\end{equation} +This is our final expression. %%% FIG 1 %%% %\begin{figure}