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Pierre-Francois Loos 2020-02-06 18:11:56 +01:00
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BSE-PES.tex
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@ -328,7 +328,6 @@ the BSE matrix elements read
\end{align}
\end{subequations}
where $\eGW{p}$ are the $GW$ quasiparticle energies.
%In the standard BSE implementation, the screened Coulomb potential $\W{}{\IS}$ is taken to be static $(\omega \rightarrow 0)$.
In the standard BSE approach, $\W{}{\IS}$ is built within the direct RPA scheme, \ie,
\begin{subequations}
\label{eq:wrpa}
@ -353,7 +352,7 @@ where the spectral weights at coupling strength $\IS$ read
\end{equation}
In the case of complex molecular orbitals, see Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
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
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{align}
\label{eq:LR_RPA-A}
@ -365,10 +364,8 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ a
\end{subequations}
where $\eHF{p}$ are the HF orbital energies.
The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening
%namely setting $\epsilon_{\IS}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock (HF) eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock (HF) eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
\begin{subequations}
\begin{align}
\label{eq:LR_RPAx-A}
@ -378,7 +375,6 @@ In this limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock (HF)
\BRPAx{ia,jb}{\IS} & = \IS \qty[ 2 \ERI{ia}{bj} - \ERI{ib}{aj} ].
\end{align}
\end{subequations}
%This allows to understand that the strength parameter $\IS$ enters twice in the $\IS W^{\IS}$ contribution, one time to renormalize the screening efficiency, and a second time to renormalize the direct electron-hole interaction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Ground-state BSE energy}
@ -434,11 +430,6 @@ For spin-restricted closed-shell molecular systems around their equilibrium geom
However, singlet instabilities may appear in the presence of strong correlation, \eg, when the bond is stretched, hampering in particular the calculation of atomization energies. \cite{Holzer_2018}
Even for weakly correlated systems, triplet instabilities are much more common, but triplet excitations do not contribute to the correlation energy in the ACFDT formulation. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011}
%\xavier{ IS THIS NEEDED NOW ? However, contrary to the plasmon formulation (an alternative expression of the BSE correlation energy), \cite{Schuck_Book, Sawada_1957b, Rowe_1968} the triplet excitations do not contribute in the ACFDT formulation, which is an indisputable advantage of this approach.
%Indeed, although the plasmon and adiabatic connection formulations are equivalent for RPA, \cite{Sawada_1957b, Gell-Mann_1957, Fukuta_1964, Furche_2008} this is not the case at the BSE and RPAx levels. \cite{Toulouse_2009, Toulouse_2010, Angyan_2011, Li_2020}
%For RPAx, an alternative plasmon expression (equivalent to its ACFDT analog) can be found if exchange is also added to the interaction kernel [see Eq.~\eqref{eq:K}]. \cite{Angyan_2011}
%However, to the best of our knowledge, such alternative plasmon expression does not exist for BSE. }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Computational details}
%\label{sec:comp_details}
@ -516,7 +507,7 @@ Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and \ce{F2}.
The PES of these molecules for various methods are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances for various basis sets are gathered in Table \ref{tab:Req}.
Additional graphs for other basis sets can be found in the {\SI}.
Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
%%% TABLE I %%%
\begin{table*}
@ -563,36 +554,37 @@ When irregularities appear in the PES, the values are reported in parenthesis an
Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and under-shoot the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and under-shoot the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
For example, BSE@{\GOWO}@HF/cc-pVQZ is only off by $0.003$ bohr as compared to FCI/cc-pVQZ, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
This is a general trend that, as one can see, is magnified in larger systems.
This is a general trend that is magnified in larger systems discussed below.
Albeit the shallow nature of the \ce{LiH} PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, theses curves running almost perfectly parallel to one another.
Here again, the BSE@{\GOWO}@HF equilibrium bond length (obtained with cc-pVQZ) is extremely accurate ($3.017$ bohr) as compared to FCI ($3.019$ bohr).
Despite the shallow nature of its PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, theses surfaces running almost perfectly parallel to one another.
Here again, the BSE@{\GOWO}@HF/cc-pVQZ equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to CC3/cc-pVQZ ($3.019$ bohr).
The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemically interesting as they correspond to strongly polarized bonds towards the halogen atoms which are much more electronegative than the first row elements.
For these ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin for these two molecules exhibiting charge transfer, the latter fact being also observed for the other diatomics discussed below.
Interestingly, while CCSD and CC2 systematically underestimates the total energy, the BSE@{\GOWO}@HF energy is always lower than the reference CC3 energy.
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that we have studied here.
This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that we have studied.
For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond lengths are again in very good agreement with the CC3 reference values.
Compared to CCSD which is known to provide slightly too short bond lengths, ACFDT@BSE usually underestimates the bond lengths by a few hundredths of bohr.
Contrary to CCSD which is known to provide slightly too short bond lengths, ACFDT@BSE underestimates the bond lengths by a few hundredths of bohr.
However, in the case of \ce{LiF}, the attentive reader would have observed a small ``glitch'' in the $GW$-based curves very close to their minimum.
As observed in Refs.~\onlinecite{vanSetten_2015,Maggio_2017,Loos_2018} and explained in details in Refs.~\onlinecite{Veril_2018,Duchemin_2020}, these irregularities, which makes particularly tricky the location of the minima, are due to ``jumps'' between distinct solutions of the $GW$ quasiparticle equation.
Including a broadening via the increasing the value of $\eta$ in the $GW$ self-energy and the screened Coulomb operator soften the problem, but does not remove it completely.
Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
In the case of irregularities in the PES, in order to provide an estimate of the equilibrium bond length, we have fitted a Morse potential to the PES.
When irregularities are present in the PES, we have fitted a Morse potential of the form $M(r) = D_e\{1-\exp[-\alpha(r-r_0)]\}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
These value are reported in parenthesis in Table \ref{tab:Req}.
For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.
Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple bond to single bond).
Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple to single bond).
The conclusions drawn for the previous systems also apply to these diatomic molecules.
In particular, the performance of BSE@{\GOWO}@HF are outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
In particular, the performance of BSE@{\GOWO}@HF is outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see {\SI}).
The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain \gb{$2.070$}, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively, for the same set of molecules.
The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain \gb{$2.070$}, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively.
As a final example, we consider the \ce{F2} molecule, a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
Similarly to what we have observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.