537 lines
20 KiB
TeX
537 lines
20 KiB
TeX
\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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% coordinates
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% methods
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{x}}
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%
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\newcommand{\Norb}{N}
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\newcommand{\Nvir}{V}
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% operators
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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{\EBSE}{E^\text{BSE}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcRPAx}{E_\text{c}^\text{RPAx}}
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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% orbital energies
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^{GW}_{#1}}
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\newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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% Matrix elements
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\newcommand{\A}[2]{A_{#1}^{#2}}
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\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
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\newcommand{\B}[2]{B_{#1}^{#2}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}}
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\newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}}
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\newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}}
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\newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}}
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\newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}}
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\newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\LBSE}[1]{L_{#1}}
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\newcommand{\XiBSE}[1]{\Xi_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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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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\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
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\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
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% excitation energies
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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{\bOm}[1]{\mathbf{\Omega}^{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\DeclareMathOperator*{\argmin}{argmin}
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\newcommand\vari{{\varepsilon}_i}
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\newcommand\vara{{\varepsilon}_a}
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\newcommand\varj{{\varepsilon}_j}
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\newcommand\varb{{\varepsilon}_b}
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\newcommand\varn{{\varepsilon}_n}
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\newcommand\varm{{\varepsilon}_m}
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\newcommand\Oms{{\Omega}_s}
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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{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
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\title{Dynamical Correction to the Bethe-Salpeter Equation}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Xavier \surname{Blase}}
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\email{xavier.blase@neel.cnrs.fr }
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\affiliation{\NEEL}
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\begin{abstract}
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This is the abstract
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%\\
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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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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[iL_0]( \omega_1 ) = \frac{ 1 }{ 2\pi } \int d \omega \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
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\end{align*}
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with $\tau_{34} = t_3 - t_4$ and
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$t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g.
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$$
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G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) }
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$$
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and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component
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\begin{align*}
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\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times \\
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& \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta }
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\Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau }
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+ \theta( - \tau ) e^{i (\vara - \hOms \tau } \Big)
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\end{align*}
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with $\tau = \tau_{34}$.
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We further obtain the spectral representation of
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$\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$
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expanding the field operators over a complete orbital basis creation/destruction operators:
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\begin{align*}
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\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) & | N,s \rangle = - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \times \nonumber \\
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\times & \Big( \theta( \tau ) e^{- i ( \varepsilon_m - \hOms ) \tau }
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+ \theta( -\tau ) e^{ - i ( \varepsilon_n + \hOms) \tau } \Big)
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\end{align*}
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with $\tau = \tau_{34}$ and where the $ \lbrace \varepsilon_{n/m} \rbrace$ are proper addition/removal energies such that e.g.
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$$
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e^{ i H \tau } {\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau } {\hat a} _m^{\dagger} | N \rangle
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$$
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Selecting (n,m)=(j,b) yields the largest components
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$A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b | N,s \rangle $, while (n,m)=(b,j) yields much weaker
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$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ leads to the Tamm Dancoff approximation (TDA). Obtaining similarly the spectral representation of $ \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle$ ($t_{1'} = t_1^{+}$) projected onto $\phi_a^*(x_1) \phi_i(x_{1'})$,
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one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (DBSE) :
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\begin{align}
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( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s}
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&+ \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \\
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&+ \sum_{bj} \Big( v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) \Big) B_{jb}^{s}
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= 0
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\end{align}
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with an effective dynamically screened Coulomb potential (see Pina eq. 24):
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\begin{align}
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\widetilde{W}_{ij,ab}(\Oms) &= { i \over 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
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\hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - ( \varb - \vari ) +i \eta } + \frac{1}{ (\Oms + \omega) - ( \vara - \varj ) + i\eta } \right] \nonumber
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\end{align}
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In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
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\begin{align*}
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W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big)
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\end{align*}
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so that
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\begin{align}
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\widetilde{W}_{ij,ab}( \Oms ) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
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\right] \nonumber
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\end{align}
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with e.g. $ \Omega_{ib}^{s} = \Oms - ( \varepsilon_b - \varepsilon_i) $. \textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that
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e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and cannot diverge. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that
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$$
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\left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
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\right]
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<
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\Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) < 0
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$$
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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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%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 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-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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\cdot
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\begin{pmatrix}
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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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\omega
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\begin{pmatrix}
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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 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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The BSE matrix elements read
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\begin{subequations}
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\begin{align}
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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: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 $\eGW{ia} = \eGW{a} - \eGW{i}$ 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}{}(\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}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}),
|
|
\end{multline}
|
|
where $\eta$ is a positive infinitesimal, and
|
|
\begin{equation}
|
|
\label{eq:sERI}
|
|
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
|
|
\end{equation}
|
|
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}
|
|
\A{ia,jb}{\RPA} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2 \ERI{ia}{jb},
|
|
\\
|
|
\label{eq:LR_RPA-B}
|
|
\B{ia,jb}{\RPA} & = 2 \ERI{ia}{bj},
|
|
\end{align}
|
|
\end{subequations}
|
|
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$GW$.
|
|
|
|
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:BSE-0}
|
|
\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
|
|
\\
|
|
\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}
|
|
% \includegraphics[width=\linewidth]{}
|
|
%\caption{
|
|
%\label{fig:}
|
|
%}
|
|
%\end{figure}
|
|
%%% %%% %%%
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Conclusion}
|
|
\label{sec:conclusion}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
This is the conclusion
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\acknowledgements{
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
|
|
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
|
|
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Supporting Information}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
See {\SI} for plenty of stuff
|
|
|
|
\bibliography{BSEdyn}
|
|
|
|
\end{document}
|