loos/BSE-PES
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

first draft of theory

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-01-25 21:58:47 +01:00
parent 872904b5ca
commit fa7f0cb97a
2 changed files with 102 additions and 110 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

212
BSE-PES.tex
View File

@ -63,7 +63,7 @@
\newcommand{\Enuc}{E^\text{nuc}} \newcommand{\Enuc}{E^\text{nuc}}
\newcommand{\Ec}{E_\text{c}} \newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E^\text{HF}} \newcommand{\EHF}{E^\text{HF}}
\newcommand{\EBSE}[1]{E_{#1}^\text{BSE}} \newcommand{\EBSE}{E^\text{BSE}}
\newcommand{\EcRPA}{E_\text{c}^\text{dRPA}} \newcommand{\EcRPA}{E_\text{c}^\text{dRPA}}
\newcommand{\EcBSE}{E_\text{c}^\text{BSE}} \newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
\newcommand{\EcsBSE}{{}^1\EcBSE} \newcommand{\EcsBSE}{{}^1\EcBSE}
@ -80,51 +80,41 @@
\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}} \newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
\newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}} \newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}}
\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} \newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
\newcommand{\Om}[1]{\Omega_{#1}} \newcommand{\Om}[2]{\Omega_{#1}^{#2}}
% Matrix elements % Matrix elements
\newcommand{\A}[1]{A_{#1}} \newcommand{\A}[2]{A_{#1}^{#2}}
\newcommand{\B}[1]{B_{#1}} \newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
\newcommand{\B}[2]{B_{#1}^{#2}}
\renewcommand{\S}[1]{S_{#1}} \renewcommand{\S}[1]{S_{#1}}
\newcommand{\ABSE}[1]{A^\text{BSE}_{#1}} \newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}}
\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}} \newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}}
\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}} \newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}}
\newcommand{\BRPA}[1]{B^\text{RPA}_{#1}} \newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}}
\newcommand{\G}[1]{G_{#1}} \newcommand{\G}[1]{G_{#1}}
\newcommand{\LBSE}[1]{L_{#1}} \newcommand{\LBSE}[1]{L_{#1}}
\newcommand{\XiBSE}[1]{\Xi_{#1}} \newcommand{\XiBSE}[1]{\Xi_{#1}}
\newcommand{\Po}[1]{P_{#1}} \newcommand{\Po}[1]{P_{#1}}
\newcommand{\W}[1]{W_{#1}} \newcommand{\W}[2]{W_{#1}^{#2}}
\newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}} \newcommand{\vc}[1]{v_{#1}}
\newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}}
\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} \newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
\newcommand{\Z}[1]{Z_{#1}} \newcommand{\Z}[1]{Z_{#1}}
\newcommand{\MO}[1]{\phi_{#1}} \newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\sERI}[2]{[#1|#2]}
% excitation energies % excitation energies
\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}} \newcommand{\OmRPA}[2]{\Omega_{#1}^{#2,\text{RPA}}}
\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}} \newcommand{\OmRPAx}[2]{\Omega_{#1}^{#2,\text{RPAx}}}
\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}} \newcommand{\OmBSE}[2]{\Omega_{#1}^{#2,\text{BSE}}}
\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}}
\newcommand{\spinup}{\downarrow} \newcommand{\spinup}{\downarrow}
\newcommand{\spindw}{\uparrow} \newcommand{\spindw}{\uparrow}
\newcommand{\singlet}{\uparrow\downarrow} \newcommand{\singlet}{\uparrow\downarrow}
\newcommand{\triplet}{\uparrow\uparrow} \newcommand{\triplet}{\uparrow\uparrow}
\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}}
\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}}
\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}}
\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}}
\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}}
\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}}
\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}}
\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}}
\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}}
\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}}
% Matrices % Matrices
\newcommand{\bO}{\mathbf{0}} \newcommand{\bO}{\mathbf{0}}
\newcommand{\bI}{\mathbf{1}} \newcommand{\bI}{\mathbf{1}}
@ -139,16 +129,15 @@
\newcommand{\bde}{\mathbf{\Delta\epsilon}} \newcommand{\bde}{\mathbf{\Delta\epsilon}}
\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}} \newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
\newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}} \newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}}
\newcommand{\bOm}{\mathbf{\Omega}} \newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}}
\newcommand{\bA}{\mathbf{A}} \newcommand{\bA}[1]{\mathbf{A}^{#1}}
\newcommand{\bAs}{{}^1\bA} \newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}}
\newcommand{\bAt}{{}^3\bA} \newcommand{\bB}[1]{\mathbf{B}^{#1}}
\newcommand{\bB}{\mathbf{B}} \newcommand{\bX}[1]{\mathbf{X}^{#1}}
\newcommand{\bX}{\mathbf{X}} \newcommand{\bY}[1]{\mathbf{Y}^{#1}}
\newcommand{\bY}{\mathbf{Y}} \newcommand{\bZ}[1]{\mathbf{Z}^{#1}}
\newcommand{\bZ}{\mathbf{Z}}
\newcommand{\bK}{\mathbf{K}} \newcommand{\bK}{\mathbf{K}}
\newcommand{\bP}{\mathbf{P}} \newcommand{\bP}[1]{\mathbf{P}^{#1}}
% units % units
\newcommand{\IneV}[1]{#1 eV} \newcommand{\IneV}[1]{#1 eV}
@ -190,8 +179,8 @@ However, we also observe, in some cases, unphysical irregularities on the ground
\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction} %\section{Introduction}
\label{sec:intro} %\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
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} 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}
@ -236,22 +225,22 @@ Embracing this definition, the purpose of the present study is to investigate th
The location of the minima on the ground- and (singlet and triplet) excited-state PES is of particular interest. The location of the minima on the ground- and (singlet and triplet) excited-state PES is of particular interest.
This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.} This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@{\GW} formalism.}
The paper is organized as follows. %The paper is organized as follows.
In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism. %In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
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}. %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}.
Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study. %Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}. %Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
Section \ref{sec:PES} reports ground-state PES for various diatomic molecules. %Section \ref{sec:PES} reports ground-state PES for various diatomic molecules.
Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}. %Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory} %\section{Theory}
\label{sec:theo} %\label{sec:theo}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Bethe-Salpeter equation} %\subsection{The Bethe-Salpeter equation}
\label{sec:BSE} %\label{sec:BSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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} In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016}
\begin{multline} \begin{multline}
@ -272,120 +261,120 @@ which takes into account the self-consistent variation of the Hartree potential
In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables. In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2') = - i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables.
In the {\GW} approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have In the {\GW} approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
\begin{equation} \begin{equation}
\SigGW{\xc}(1,2) = i \G{}(1,2) \W{}(1^+,2), \SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
\end{equation} \end{equation}
where $\W{}$ is the screened Coulomb operator, and hence the BSE reduces to where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to
\begin{equation} \begin{equation}
\XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}(3,4), \XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4),
\end{equation} \end{equation}
where, as commonly done, we have neglected the term $\delta \W{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982} where, as commonly done, we have neglected the term $\delta \W{}{}/\delta \G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982}
Finally, the static approximation is enforced, \ie, $\W{}(1,2) = \W{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}$ to its static limit, \ie, $\W{}(1,2) = \W{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{BSE in a finite basis} %\subsection{BSE in a finite basis}
\label{sec:BSE_basis} %\label{sec:BSE_basis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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} 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}
\begin{equation} \begin{equation}
\label{eq:LR} \label{eq:LR}
\begin{pmatrix} \begin{pmatrix}
\bA & \bB \\ \bA{\IS} & \bB{\IS} \\
-\bB & -\bA \\ -\bB{\IS} & -\bA{\IS} \\
\end{pmatrix} \end{pmatrix}
\begin{pmatrix} \begin{pmatrix}
\bX_m \\ \bX{\IS}_m \\
\bY_m \\ \bY{\IS}_m \\
\end{pmatrix} \end{pmatrix}
= =
\Om{m} \Om{m}{\IS}
\begin{pmatrix} \begin{pmatrix}
\bX_m \\ \bX{\IS}_m \\
\bY_m \\ \bY{\IS}_m \\
\end{pmatrix}, \end{pmatrix},
\end{equation} \end{equation}
where $\Om{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. 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 $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
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. 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.
In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
In the absence of instabilities (\ie, $\bA - \bB$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension 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
\begin{equation} \begin{equation}
\label{eq:small-LR} \label{eq:small-LR}
(\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ, (\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS} = (\bOm{\IS})^2 \bZ{\IS},
\end{equation} \end{equation}
where the excitation amplitudes are where the excitation amplitudes are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ, \bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS},
\\ \\
\bX - \bY = \bOm^{1/2} (\bA - \bB)^{-1/2} \bZ. \bX{\IS} - \bY{\IS} = (\bOm{\IS})^{1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
\end{align} \end{align}
\end{subequations} \end{subequations}
In the case of BSE, the specific expression of the matrix elements are In the case of BSE, the specific expression of the matrix elements are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE} \label{eq:LR_BSE}
\ABSE{ia,jb} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \W{ia,bj}(\omega = 0) - (ia|jb), \ABSE{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \IS \W{ia,bj}{\IS}(\omega = 0) - \lambda \ERI{ia}{jb},
\\ \\
\BBSE{ia,jb} & = \W{ia,jb}(\omega = 0) - (ib|ja) , \BBSE{ia,jb}{\IS} & = \W{ia,jb}{\IS}(\omega = 0) - \lambda \ERI{ib}{ja} ,
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\eGW{p}$ are the {\GW} quasiparticle energies, where $\eGW{p}$ are the {\GW} quasiparticle energies,
\begin{multline} \begin{multline}
\label{eq:W} \label{eq:W}
\W{ia,jb}(\omega) = 2 (ia|jb) \W{ia,jb}{\IS}(\omega) = 2 \ERI{ia}{jb}
\\ \\
+ \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta}) + \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})
\end{multline} \end{multline}
are the elements of the screened Coulomb operator, are the elements of the screened Coulomb operator $\W{}{\IS}$,
\begin{equation} \begin{equation}
[pq|m] = \sum_i^{\Nocc} \sum_a^{\Nvir} (pq|ia) (\bX_m+\bY_m)_{ia} \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
\end{equation} \end{equation}
are the screened two-electron integrals, are the screened two-electron integrals,
\begin{equation} \begin{equation}
(pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
\end{equation} \end{equation}
are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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 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
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE} \label{eq:LR_BSE}
\ARPA{ia,jb} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + (ia|bj), \ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \ERI{ia}{bj},
\\ \\
\BRPA{ia,jb} & = (ia|jb), \BRPA{ia,jb}{\IS} & = \IS \ERI{ia}{jb},
\end{align} \end{align}
\end{subequations} \end{subequations}
and $\eHF{p}$ are the HF orbital energies. where $\eHF{p}$ are the HF orbital energies.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ground-state BSE energy} %\subsection{Ground-state BSE energy}
\label{sec:BSE_energy} %\label{sec:BSE_energy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The key quantity to define in the present context is the total ground-state BSE energy. The key quantity to define in the present context is the total ground-state BSE energy $\EBSE$.
Although this choice is not unique, \cite{Holzer_2018} we propose to define it as Although this choice is not unique, \cite{Holzer_2018} we propose to define it as
\begin{equation} \begin{equation}
\label{eq:EtotBSE} \label{eq:EtotBSE}
\EBSE{m} = \Enuc + \EHF + \EcBSE \EBSE = \Enuc + \EHF + \EcBSE
\end{equation} \end{equation}
where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and
\begin{equation} \begin{equation}
\label{eq:EcBSE} \label{eq:EcBSE}
\EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP_\IS) d\IS \EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP{\IS}) d\IS
\end{equation} \end{equation}
is the ground-state BSE correlation energy computed in the adiabatic connection framework, where is the ground-state BSE correlation energy computed in the adiabatic connection framework, where
\begin{equation} \begin{equation}
\bK = \bK =
\begin{pmatrix} \begin{pmatrix}
\bA' & \bB \\ \btA{\IS=1} & \bB{\IS=1} \\
\bB & \bA' \\ \bB{\IS=1} & \btA{\IS=1} \\
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (ia|bj)$], is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS} = \IS \ERI{ia}{bj}$],
\begin{equation} \begin{equation}
\bP_\lambda = \bP{\IS} =
\begin{pmatrix} \begin{pmatrix}
\bY_\IS \T{\bY}_\IS & \bY_\IS \T{\bX}_\IS \\ \bY{\IS} \T{(\bY{\IS})} & \bY{\IS} \T{(\bX{\IS})} \\
\bX_\IS \T{\bY}_\IS & \bX_\IS \T{\bX}_\IS \\ \bX{\IS} \T{(\bY{\IS})} & \bX{\IS} \T{(\bX{\IS})} \\
\end{pmatrix} \end{pmatrix}
- -
\begin{pmatrix} \begin{pmatrix}
@ -393,18 +382,21 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (i
\bO & \bI \\ \bO & \bI \\
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace. 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, it is unnecessary to compute the triplet contribution as it is strictly zero. 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}
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} Several important comments are in order here.
However, this is not the case at the BSE level. 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} %\section{Computational details}
\label{sec:comp_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. 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. 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. This step is, by far, the computational bottleneck in our current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Potential energy surfaces} %\section{Potential energy surfaces}
\label{sec:PES} %\label{sec:PES}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%% %%% TABLE I %%%
@ -512,23 +504,23 @@ Additional graphs for other basis sets and within the frozen-core approximation
\end{figure*} \end{figure*}
%%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hydrogen molecule} %\subsection{Hydrogen molecule}
\label{sec:H2} %\label{sec:H2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lithium hydride and lithium fluoride} %\subsection{Lithium hydride and lithium fluoride}
\label{sec:LiX} %\label{sec:LiX}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}} %\subsection{The isoelectronic sequence: \ce{N2}, \ce{CO}, and \ce{BF}}
\label{sec:isoN2} %\label{sec:isoN2}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} %\section{Conclusion}
\label{sec:conclusion} %\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\titou{A nice conclusion here saying that what we have done is pretty awesome.} \titou{A nice conclusion here saying that what we have done is pretty awesome.}

BIN
Data/N2_GS_VDZ.pdf
View File

Binary file not shown.