first draft of theory
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BSE-PES.tex
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BSE-PES.tex
@ -63,7 +63,7 @@
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\newcommand{\Enuc}{E^\text{nuc}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E^\text{HF}}
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\newcommand{\EBSE}[1]{E_{#1}^\text{BSE}}
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\newcommand{\EBSE}{E^\text{BSE}}
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\newcommand{\EcRPA}{E_\text{c}^\text{dRPA}}
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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\newcommand{\EcsBSE}{{}^1\EcBSE}
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@ -80,51 +80,41 @@
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\newcommand{\eGW}[1]{\epsilon^\text{\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}[1]{\Omega_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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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}[1]{A^\text{BSE}_{#1}}
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\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}}
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\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}}
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\newcommand{\BRPA}[1]{B^\text{RPA}_{#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{\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}[1]{W_{#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{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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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}[2]{[#1|#2]}
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% excitation energies
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\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}}
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\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}}
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\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}}
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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{\spinup}{\downarrow}
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\newcommand{\spindw}{\uparrow}
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\newcommand{\singlet}{\uparrow\downarrow}
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\newcommand{\triplet}{\uparrow\uparrow}
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\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}}
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\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}}
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\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}}
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\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}}
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\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}}
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\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}}
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\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}}
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\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}}
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% Matrices
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\newcommand{\bO}{\mathbf{0}}
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\newcommand{\bI}{\mathbf{1}}
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@ -139,16 +129,15 @@
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\newcommand{\bde}{\mathbf{\Delta\epsilon}}
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\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}}
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\newcommand{\bOm}{\mathbf{\Omega}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bAs}{{}^1\bA}
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\newcommand{\bAt}{{}^3\bA}
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\newcommand{\bB}{\mathbf{B}}
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\newcommand{\bX}{\mathbf{X}}
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\newcommand{\bY}{\mathbf{Y}}
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\newcommand{\bZ}{\mathbf{Z}}
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\newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}}
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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{\bZ}[1]{\mathbf{Z}^{#1}}
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\newcommand{\bK}{\mathbf{K}}
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\newcommand{\bP}{\mathbf{P}}
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\newcommand{\bP}[1]{\mathbf{P}^{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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@ -190,8 +179,8 @@ However, we also observe, in some cases, unphysical irregularities on the ground
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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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%\section{Introduction}
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%\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
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@ -236,22 +225,22 @@ Embracing this definition, the purpose of the present study is to investigate th
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The location of the minima on the ground- and (singlet and triplet) excited-state PES is of particular interest.
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This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.}
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The paper is organized as follows.
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In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
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In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}.
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Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
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Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
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Section \ref{sec:PES} reports ground-state PES for various diatomic molecules.
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Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
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%The paper is organized as follows.
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%In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
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%In particular, the general equations are reported in Subsec.~\ref{sec:BSE}, and the corresponding equations obtained in a finite basis in Subsec.~\ref{sec:BSE_basis}.
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%Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
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%Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
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%Section \ref{sec:PES} reports ground-state PES for various diatomic molecules.
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%Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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\label{sec:theo}
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%\section{Theory}
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%\label{sec:theo}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{The Bethe-Salpeter equation}
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\label{sec:BSE}
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%\subsection{The Bethe-Salpeter equation}
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%\label{sec:BSE}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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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@ -272,120 +261,120 @@ which takes into account the self-consistent variation of the Hartree potential
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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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\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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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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\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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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{BSE in a finite basis}
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\label{sec:BSE_basis}
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%\subsection{BSE in a finite basis}
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%\label{sec:BSE_basis}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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For a closed-shell system, in order to compute the singlet BSE excitation energies within the static approximation in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016}
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For a closed-shell system, in order to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$) in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016}
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\begin{equation}
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\label{eq:LR}
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\begin{pmatrix}
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\bA & \bB \\
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-\bB & -\bA \\
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\bA{\IS} & \bB{\IS} \\
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-\bB{\IS} & -\bA{\IS} \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX_m \\
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\bY_m \\
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\bX{\IS}_m \\
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\bY{\IS}_m \\
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\end{pmatrix}
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=
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\Om{m}
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\Om{m}{\IS}
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\begin{pmatrix}
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\bX_m \\
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\bY_m \\
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\bX{\IS}_m \\
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\bY{\IS}_m \\
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\end{pmatrix},
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\end{equation}
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where $\Om{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
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The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively.
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where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \, \bY{\IS}_m)}$ at interpolation strength $\IS$, and we have assumed 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, $\Norb = \Nocc + \Nvir$), 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, $\bA - \bB$ 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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In the absence of instabilities (\ie, $\bA{\IS} - \bB{\IS}$ 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 - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ,
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(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS} = (\bOm{\IS})^2 \bZ{\IS},
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\end{equation}
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where the excitation amplitudes are
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\begin{subequations}
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\begin{align}
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\bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ,
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\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS},
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\\
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\bX - \bY = \bOm^{1/2} (\bA - \bB)^{-1/2} \bZ.
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\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
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\end{align}
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\end{subequations}
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In the case of BSE, the specific expression of the matrix elements are
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE}
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\ABSE{ia,jb} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \W{ia,bj}(\omega = 0) - (ia|jb),
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\ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \W{ia,bj}{\IS}(\omega = 0) - \lambda \ERI{ia}{jb},
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\\
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\BBSE{ia,jb} & = \W{ia,jb}(\omega = 0) - (ib|ja) ,
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\BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ib}{ja} ,
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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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\begin{multline}
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\label{eq:W}
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\W{ia,jb}(\omega) = 2 (ia|jb)
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\W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb}
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\\
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+ \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta})
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+ \sum_m^{\Nocc \Nvir} \sERI{ia}{m} \sERI{jb}{m} \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} + \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta})
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\end{multline}
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are the elements of the screened Coulomb operator,
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are the elements of the screened Coulomb operator $\W{}{\IS}$,
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\begin{equation}
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[pq|m] = \sum_i^{\Nocc} \sum_a^{\Nvir} (pq|ia) (\bX_m+\bY_m)_{ia}
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\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
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\end{equation}
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are the screened two-electron integrals,
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\begin{equation}
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(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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\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, and $\delta_{pq}$ is the Kronecker delta.
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In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed during the {\GW} calculation by solving the linear eigenvalue problem \eqref{eq:LR} with matrix elements
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are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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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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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE}
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\ARPA{ia,jb} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + (ia|bj),
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\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{bj},
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\\
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\BRPA{ia,jb} & = (ia|jb),
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\BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{jb},
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\end{align}
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\end{subequations}
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and $\eHF{p}$ are the HF orbital energies.
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where $\eHF{p}$ are the HF orbital energies.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Ground-state BSE energy}
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\label{sec:BSE_energy}
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%\subsection{Ground-state BSE energy}
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%\label{sec:BSE_energy}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The key quantity to define in the present context is the total ground-state BSE energy.
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The key quantity to define in the present context is the total ground-state BSE energy $\EBSE$.
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Although this choice is not unique, \cite{Holzer_2018} we propose to define it as
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\begin{equation}
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\label{eq:EtotBSE}
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\EBSE{m} = \Enuc + \EHF + \EcBSE
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\EBSE = \Enuc + \EHF + \EcBSE
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\end{equation}
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where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and
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\begin{equation}
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\label{eq:EcBSE}
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\EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP_\IS) d\IS
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\EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP{\IS}) d\IS
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\end{equation}
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is the ground-state BSE correlation energy computed in the adiabatic connection framework, where
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\begin{equation}
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\bK =
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\begin{pmatrix}
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\bA' & \bB \\
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\bB & \bA' \\
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\btA{\IS=1} & \bB{\IS=1} \\
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\bB{\IS=1} & \btA{\IS=1} \\
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\end{pmatrix}
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\end{equation}
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is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (ia|bj)$],
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is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} = \IS \ERI{ia}{bj}$],
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\begin{equation}
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\bP_\lambda =
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\bP{\IS} =
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\begin{pmatrix}
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\bY_\IS \T{\bY}_\IS & \bY_\IS \T{\bX}_\IS \\
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\bX_\IS \T{\bY}_\IS & \bX_\IS \T{\bX}_\IS \\
|
||||
\bY{\IS} \T{(\bY{\IS})} & \bY{\IS} \T{(\bX{\IS})} \\
|
||||
\bX{\IS} \T{(\bY{\IS})} & \bX{\IS} \T{(\bX{\IS})} \\
|
||||
\end{pmatrix}
|
||||
-
|
||||
\begin{pmatrix}
|
||||
@ -393,18 +382,21 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (i
|
||||
\bO & \bI \\
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace.
|
||||
Note that, it is unnecessary to compute the triplet contribution as it is strictly zero.
|
||||
Note that the present formulation is different from the plasmon formulation. \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020}
|
||||
Note that, at the dRPA level, the plasmon and adiabatic connection formulations are equivalent. \cite{Sawada_1957b, Fukuta_1964, Furche_2008}
|
||||
However, this is not the case at the BSE level.
|
||||
is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace and $\T{}$ the matrix transpose.
|
||||
Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}] has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer et al. \cite{Holzer_2018}
|
||||
|
||||
Several important comments are in order here.
|
||||
For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities.
|
||||
However, they may appear in the presence of strong correlation (\eg, when the bond is stretch).
|
||||
In such a case, this hampers the use of Eq.~\eqref{eq:EcBSE}.
|
||||
Triplet instabilities are much more common.
|
||||
However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} the triplet excitations do not contribute in the adiabatic connection formulation, which is an indisputable advantage of the ACFDT appraoch.
|
||||
Although at the RPA level, the plasmon and adiabatic connection formulations are equivalent, \cite{Sawada_1957b, Fukuta_1964, Furche_2008} this is not the case at the BSE level.
|
||||
|
||||
One of the indisputable advantage of the adiabatic connection formulation is that the triplet does not contribute.
|
||||
Therefore, the triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Computational details}
|
||||
\label{sec:comp_details}
|
||||
%\section{Computational details}
|
||||
%\label{sec:comp_details}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
|
||||
Perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations.
|
||||
@ -421,8 +413,8 @@ Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both s
|
||||
This step is, by far, the computational bottleneck in our current implementation.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Potential energy surfaces}
|
||||
\label{sec:PES}
|
||||
%\section{Potential energy surfaces}
|
||||
%\label{sec:PES}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%% TABLE I %%%
|
||||
@ -512,23 +504,23 @@ Additional graphs for other basis sets and within the frozen-core approximation
|
||||
\end{figure*}
|
||||
%%% %%% %%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Hydrogen molecule}
|
||||
\label{sec:H2}
|
||||
%\subsection{Hydrogen molecule}
|
||||
%\label{sec:H2}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{Lithium hydride and lithium fluoride}
|
||||
\label{sec:LiX}
|
||||
%\subsection{Lithium hydride and lithium fluoride}
|
||||
%\label{sec:LiX}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}}
|
||||
\label{sec:isoN2}
|
||||
%\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}}
|
||||
%\label{sec:isoN2}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Conclusion}
|
||||
\label{sec:conclusion}
|
||||
%\section{Conclusion}
|
||||
%\label{sec:conclusion}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\titou{A nice conclusion here saying that what we have done is pretty awesome.}
|
||||
|
||||
|
||||
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