From ecc74c59372528c343fe085a48c764b9426f6f86 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 7 Jan 2020 22:42:45 +0100 Subject: [PATCH] clean up last section of theory (more to come) --- BSE-PES.tex | 14 +++++++++----- 1 file changed, 9 insertions(+), 5 deletions(-) diff --git a/BSE-PES.tex b/BSE-PES.tex index bfb667f..ad2c372 100644 --- a/BSE-PES.tex +++ b/BSE-PES.tex @@ -352,7 +352,7 @@ In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral (direct) RPA excitation energie \subsection{Ground- and excited-state BSE energy} \label{sec:BSE_energy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\titou{The key quantity to define here is the total BSE energy as we are going to compute PES following this definition.} +The key quantity to define in the present context is the total BSE energy. Although not unique, we propose to define the BSE total energy of the $m$th state as \begin{equation} \label{eq:EtotBSE} @@ -364,11 +364,15 @@ where $\Enuc$ and $\EHF$ are the state-independent nuclear repulsion energy and \EcBSE = \frac{1}{2} \qty[ {\sum_m}' \OmBSE{m} - \Tr(\bA) ] \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} +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. -Note that the prime in the summation of Eq.~\eqref{eq:EcBSE} means that we only consider the \textit{positive} excitation energies from the singlet and triplet manifolds. +It has also the indisputable advantage of treating on equal footing (\ie, at the same level of theory) the ground state and the excited states. +Note that the prime in the sum of Eq.~\eqref{eq:EcBSE} means that we only consider in the summation the \textit{positive} excitation energies from the singlet and triplet manifolds. Indeed, in case of 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. -Note also that an alternative formulation does exist which consists in integrating along the adiabatic connection path. \cite{Gell-Mann_1957} -These two formulation 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} \xavier{WELL, THIS IS UNCLEAR TO ME : this is true for direct RPA, but is this true for BSE ... I am really not sure ...} From a practical point of view, it is also convenient to split $\EcBSE = \EcsBSE + \EctBSE$ in its singlet ($\sigma \neq \sigma^{\prime}$) and triplet ($\sigma = \sigma^{\prime}$) contributions, \ie, \begin{equation} @@ -376,7 +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 different level of theory. +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details}