clean up last section of theory (more to come)

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Pierre-Francois Loos 2020-01-07 22:42:45 +01:00
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@ -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.
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\section{Computational details}