This commit is contained in:
Pierre-Francois Loos 2020-01-12 22:59:06 +01:00
parent 4157c018a7
commit c45ffd4324

View File

@ -63,7 +63,7 @@
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E^\text{HF}}
\newcommand{\EBSE}[1]{E_{#1}^\text{BSE}}
\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
\newcommand{\EcRPA}{E_\text{c}^\text{dRPA}}
\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
\newcommand{\EcsBSE}{{}^1\EcBSE}
\newcommand{\EctBSE}{{}^3\EcBSE}
@ -236,7 +236,7 @@ The paper is organized as follows.
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}.
Subsection \ref{sec:BSE_energy} defines the BSE total energy, and various other quantities of interest for this study.
In Sec.~\ref{sec:comp_details}, computational details are reported.
Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
Section \ref{sec:PES} reports PES of the ground- and excited-states for various diatomic molecules.
Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
@ -346,7 +346,7 @@ are the bare two-electron integrals, $\delta_{pq}$ is the Kronecker delta, and
1, & \sigma = \sigma^{\prime} \text{ (triplet manifold)}.
\end{cases}
\end{equation}
In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral (direct) RPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal.
In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral direct (\ie, without exchange) dRPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ground- and excited-state BSE energy}
@ -365,8 +365,8 @@ where $\Enuc$ and $\EHF$ are the state-independent nuclear repulsion energy and
\end{equation}
is the ground-state BSE correlation energy computed with the so-called trace formula, \cite{Schuck_Book, Rowe_1968, Sawada_1957b} and $\OmBSE{m}$ is the $m$th BSE excitation energy with the convention that, for the ground state ($m=0$), $\OmBSE{0} = 0$.
An elegant derivation of Eq.~\eqref{eq:EcBSE} has been recently proposed within the BSE formalism by Olevano and coworkers. \cite{Li_2020}
Note that, at the RPA level, an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
These two RPA formulations have been found to be equivalent in practice for both the uniform electron gas \cite{Sawada_1957b, Fukuta_1964, Furche_2008} and in molecules. \cite{Li_2020}
Note that, at the dRPA level, an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957}
These two dRPA formulations have been found to be equivalent in practice for both the uniform electron gas \cite{Sawada_1957b, Fukuta_1964, Furche_2008} and in molecules. \cite{Li_2020}
However, in the case of BSE, there is no guarantee that the two formalisms (trace \textit{vs} adiabatic connection) yields the same values.
Equation \eqref{eq:EtotBSE} defines unambiguously the total BSE energy of the system for both ground and (singlet and triplet) excited states.
@ -380,7 +380,7 @@ From a practical point of view, it is also convenient to split $\EcBSE = \EcsBSE
= \underbrace{\frac{1}{2} \qty[ {\sum_m} \OmsBSE{m} - \Tr(\bAs) ]}_{\EcsBSE}
+ \underbrace{\frac{1}{2} \qty[ {\sum_m}' \OmtBSE{m} - \Tr(\bAt) ]}_{\EctBSE}.
\end{equation}
As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, RPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the these levels of theory.
As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily generalized to other theories (such as CIS, dRPA, or TDHF) by computing $\Ec$ and $\Om{m}$ at the these levels of theory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}