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BSEdyn.tex
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BSEdyn.tex
@ -24,13 +24,10 @@
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\denis}[1]{\textcolor{purple}{#1}}
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\newcommand{\xavier}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\trashDJ}[1]{\textcolor{purple}{\sout{#1}}}
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\newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\renewcommand{\DJ}[1]{\denis{(\underline{\bf DJ}: #1)}}
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\newcommand{\XB}[1]{\xavier{(\underline{\bf XB}: #1)}}
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\newcommand{\mc}{\multicolumn}
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@ -63,6 +60,10 @@
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% operators
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\newcommand{\hH}{\Hat{H}}
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% methods
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\BSE}{\text{BSE}}
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% energies
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\newcommand{\Enuc}{E^\text{nuc}}
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\newcommand{\Ec}{E_\text{c}}
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@ -109,7 +110,7 @@
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\newcommand{\Z}[1]{Z_{#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}[3]{[#1|#2]^{#3}}
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\newcommand{\sERI}[2]{[#1|#2]}
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%% bold in Table
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\newcommand{\bb}[1]{\textbf{#1}}
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@ -117,9 +118,9 @@
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\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
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% excitation energies
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\newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
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\newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}}
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\newcommand{\OmBSE}[2]{\Omega_{#1}^{#2,\text{BSE}}}
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\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}}
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\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}}
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\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}}
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\newcommand{\spinup}{\downarrow}
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\newcommand{\spindw}{\uparrow}
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@ -144,8 +145,8 @@
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\newcommand{\bA}[1]{\mathbf{A}^{#1}}
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\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}}
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\newcommand{\bB}[1]{\mathbf{B}^{#1}}
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\newcommand{\bX}[1]{\mathbf{X}^{#1}}
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\newcommand{\bY}[1]{\mathbf{Y}^{#1}}
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\newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}}
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\newcommand{\bK}{\mathbf{K}}
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\newcommand{\bP}[1]{\mathbf{P}^{#1}}
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@ -170,9 +171,7 @@
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\newcommand\hOms{\frac{{\Omega}_s}{2}}
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\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
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\newcommand{\CEISAM}{Laboratoire CEISAM - UMR CNRS 6230, Universit\'e de Nantes, 2 Rue de la Houssini\`ere, BP 92208, 44322 Nantes Cedex 3, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\CEA}{Universit\'e Grenoble Alpes, CEA, IRIG-MEM-L Sim, 38054 Grenoble, France}
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\begin{document}
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@ -206,6 +205,9 @@ This is the abstract
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%================================
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\subsection{Theory for physics}
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%=================================
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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:
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\begin{align*}
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@ -273,138 +275,233 @@ $$
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in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of
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$[ij|m] [ab|m]$ ?? }
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\titou{This is the theory section from the previous paper.}
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%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}
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%\begin{multline}
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%\label{eq:BSE}
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% \LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2')
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% \\
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% + \int d3 d4 d5 d6 \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2')
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%\end{multline}
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%as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential
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%\begin{equation}
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% \XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)},
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%\end{equation}
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%which takes into account the self-consistent variation of the Hartree potential
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%\begin{equation}
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% \vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+),
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%\end{equation}
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%(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$.
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%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.
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%In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
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%\begin{equation}
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% \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
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%\end{equation}
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%where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to
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%\begin{equation}
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% \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4),
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%\end{equation}
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%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}
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%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, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
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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}
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\begin{multline}
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\label{eq:BSE}
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\LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2')
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\\
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+ \int d3 d4 d5 d6 \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2')
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\end{multline}
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as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential
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\begin{equation}
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\XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)},
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\end{equation}
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which takes into account the self-consistent variation of the Hartree potential
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\begin{equation}
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\vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+),
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\end{equation}
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(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$.
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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.
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In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
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\begin{equation}
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\SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
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\end{equation}
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where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to
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\begin{equation}
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\XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4),
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\end{equation}
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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}
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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, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
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%================================
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\subsection{Theory for chemists}
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%=================================
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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}
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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
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\begin{equation}
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\label{eq:LR}
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\label{eq:LR-dyn}
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\begin{pmatrix}
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\bA{\IS} & \bB{\IS} \\
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-\bB{\IS} & -\bA{\IS} \\
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\bA{}(\omega) & \bB{}(\omega) \\
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-\bB{}(\omega) & -\bA{}(\omega) \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{\IS}_m \\
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\bY{\IS}_m \\
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\bX{m}{}(\omega) \\
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\bY{m}{}(\omega) \\
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\end{pmatrix}
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=
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\Om{m}{\IS}
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\omega
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\begin{pmatrix}
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\bX{\IS}_m \\
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\bY{\IS}_m \\
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\bX{m}{}(\omega) \\
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\bY{m}{}(\omega) \\
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\end{pmatrix},
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\end{equation}
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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 $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
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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.
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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.
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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.
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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
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\begin{equation}
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\label{eq:small-LR}
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(\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},
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\end{equation}
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where the excitation amplitudes are
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The BSE matrix elements read
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\begin{subequations}
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\begin{align}
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(\bX{\IS} + \bY{\IS})_m = (\Om{m}{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{m}{\IS},
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\\
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(\bX{\IS} - \bY{\IS})_m = (\Om{m}{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{m}{\IS}.
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\end{align}
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\end{subequations}
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Introducing the so-called Mulliken notation for the bare two-electron integrals
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\begin{equation}
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\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
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\end{equation}
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and the corresponding (static) screened Coulomb potential matrix elements at coupling strength $\IS$
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\begin{equation}
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\W{pq,rs}{\IS} = \iint \MO{p}(\br{}) \MO{q}(\br{}) \W{}{\IS}(\br{},\br{}') \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}',
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\end{equation}
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the BSE matrix elements read
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-A}
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\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \W{ij,ab}{\IS} ],
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\label{eq:BSE-Adyn}
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\A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{}(\omega),
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\\
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\label{eq:LR_BSE-B}
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\BBSE{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \W{ib,aj}{\IS} ],
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\label{eq:BSE-Bdyn}
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\B{ia,jb}{}(\omega) & = 2 \ERI{ia}{bj} - \W{ib,aj}{}(\omega),
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\end{align}
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\end{subequations}
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where $\eGW{p}$ are the $GW$ quasiparticle energies.
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In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie,
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\begin{subequations}
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\label{eq:wrpa}
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\begin{align}
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\W{}{\IS}(\br{},\br{}')
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& = \int \frac{\epsilon_{\IS}^{-1}(\br{},\br{}''; \omega=0)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
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\\
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\epsilon_{\IS}(\br{},\br{}'; \omega)
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& = \delta(\br{}-\br{}') - \IS \int \frac{\chi_{0}(\br{},\br{}''; \omega)}{\abs*{\br{}' - \br{}''}} \dbr{}'' ,
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\end{align}
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\end{subequations}
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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
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where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies,
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\begin{equation}
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\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
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\end{equation}
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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
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\begin{multline}
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\label{eq:W}
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\W{ij,ab}{\IS}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m}{\IS} \sERI{ab}{m}{\IS}
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\W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
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\\
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\times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}),
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\times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}),
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\end{multline}
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where the spectral weights at coupling strength $\IS$ read
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where $\eta$ is a positive infinitesimal, and
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\begin{equation}
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\sERI{pq}{m}{\IS} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
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\label{eq:sERI}
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\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
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\end{equation}
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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 $\W{}{}$.
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Note that, in the case of {\GOWO}, the RPA neutral excitations in Eq.~\eqref{eq:W} are computed using the HF orbital energies.
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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
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are the spectral weights.
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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
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\begin{equation}
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\label{eq:LR-stat}
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\begin{pmatrix}
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\bA{\RPA} & \bB{\RPA} \\
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-\bB{\RPA} & -\bA{\RPA} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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\end{pmatrix}
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=
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\OmRPA{m}
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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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{align}
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\label{eq:LR_RPA-A}
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\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb},
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\A{ia,jb}{\RPA} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 \ERI{ia}{jb},
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\\
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\label{eq:LR_RPA-B}
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\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj},
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\B{ia,jb}{\RPA} & = 2 \ERI{ia}{bj},
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\end{align}
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\end{subequations}
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where $\eHF{p}$ are the Hartree-Fock (HF) orbital energies.
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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$GW$.
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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 $\vc{}$.
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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:
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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
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\begin{equation}
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\label{eq:LR-dyn}
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\begin{pmatrix}
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\bA{}(\omega) & \bB{}(\omega) \\
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-\bB{}(\omega) & -\bA{}(\omega) \\
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\end{pmatrix}
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=
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\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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user