minor corrections
This commit is contained in:
parent
5c074768d2
commit
4003e6ef39
25
BSE-PES.tex
25
BSE-PES.tex
@ -351,7 +351,7 @@ where the spectral weights at coupling strength $\IS$ read
|
||||
\begin{equation}
|
||||
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
|
||||
\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 the case of complex 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
|
||||
\begin{subequations}
|
||||
@ -363,10 +363,10 @@ In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ ar
|
||||
\BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj},
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
where $\eHF{p}$ are the HF orbital energies.
|
||||
where $\eHF{p}$ are the Hartree-Fock (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 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:
|
||||
In this limit, the $GW$ quasiparticle energies reduce to the 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}
|
||||
@ -426,7 +426,7 @@ For RPA, these expressions have been provided in Eqs.~\eqref{eq:LR_RPA-A} and \e
|
||||
In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively.
|
||||
Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA-A} by the $GW$ quasiparticles energies.
|
||||
|
||||
Several important comments are in order here.
|
||||
Before going any further, several important comments are in order.
|
||||
For spin-restricted closed-shell molecular systems around their equilibrium geometry (such as the ones studied here), it is rare to encounter singlet instabilities as these systems can be classified as weakly correlated.
|
||||
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}
|
||||
@ -436,8 +436,8 @@ Even for weakly correlated systems, triplet instabilities are much more common,
|
||||
%\label{sec:comp_details}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) 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.
|
||||
In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
|
||||
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting points to compute the BSE neutral excitations.
|
||||
In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the frequency-dependent quasiparticle equation.
|
||||
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
|
||||
Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
|
||||
The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature.
|
||||
@ -447,7 +447,7 @@ For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Chr
|
||||
The computational cost of these methods, in their usual implementation, scale as $\order*{N^5}$, $\order*{N^5}$, $\order*{N^6}$, and $\order*{N^7}$, respectively.
|
||||
All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
|
||||
As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
|
||||
Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fair comparison between methods.
|
||||
Unless otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fair comparison between methods.
|
||||
However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI} for additional information).
|
||||
|
||||
Because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum for each quadrature point, we perform several complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
|
||||
@ -507,15 +507,16 @@ Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various
|
||||
%\label{sec:PES}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
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 are gathered in Table \ref{tab:Req}.
|
||||
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 and correlation energies are gathered in Table \ref{tab:Req}.
|
||||
Both of these properties have been computed with Dunning's cc-pVQZ basis set.
|
||||
Additional graphs and tables for other basis sets can be found in the {\SI}.
|
||||
Graphs and tables for additional basis sets can be found in the {\SI}.
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{
|
||||
Equilibrium bond length $\Req$ (in bohr) and correlation energy $\Ec$ (in millihartree) for the ground state of diatomic molecules obtained with the cc-pVQZ basis set at various levels of theory.
|
||||
For each system, the correlation energy is computed at its respective equilibrium bond length (\ie, $R = \Req$).
|
||||
When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
|
||||
The error (in \%) compared to the reference CC3 values are reported in square brackets.
|
||||
}
|
||||
@ -554,7 +555,7 @@ RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly unde
|
||||
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, 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 is magnified in larger systems discussed below.
|
||||
This is a general trend that is magnified in larger systems as the ones discussed below.
|
||||
|
||||
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.
|
||||
@ -585,6 +586,8 @@ The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium
|
||||
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.
|
||||
However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
|
||||
Note that, for this system, triplet (and then singlet) instabilities appear for quite short bond lengths.
|
||||
However, around the equilibrium structure, we have not encountered any instabilities.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section{Conclusion}
|
||||
@ -592,7 +595,7 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an estimated bond
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In this Letter, we hope to have illustrated that the ACFDT@BSE formalism is a promising methodology for the computation of accurate ground-state PES and their corresponding equilibrium structures.
|
||||
To do so, we have shown that calculating the BSE correlation energy computed within the ACFDT framework yield extremely accurate PES around equilibrium.
|
||||
(Their accuracy near the dissociation limit remains an open question.)
|
||||
%(Their accuracy near the dissociation limit remains an open question.)
|
||||
We have illustrated this for 8 diatomic molecules for which we have also computed reference ground-state energies using coupled cluster methods (CC2, CCSD, and CC3).
|
||||
However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
|
||||
This shortcoming, which is entirely due to the quasiparticle nature of the underlying $GW$ calculation, has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
|
||||
|
Loading…
Reference in New Issue
Block a user