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Pierre-Francois Loos 2020-02-07 18:20:39 +01:00
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BSE-PES.tex
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@ -351,7 +351,7 @@ where the spectral weights at coupling strength $\IS$ read
\begin{equation} \begin{equation}
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}. \sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
\end{equation} \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 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{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}, \BRPA{ia,jb}{\IS} & = 2 \IS \ERI{ia}{bj},
\end{align} \end{align}
\end{subequations} \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 $\vc{}$. 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{subequations}
\begin{align} \begin{align}
\label{eq:LR_RPAx-A} \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. 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. 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. 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} 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} 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} %\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. 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. 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 non-linear, frequency-dependent quasiparticle equation. 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}. 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. 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. 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. 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} 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. 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). 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. 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} %\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}. 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. 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 %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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} %\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. 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. 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). 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. 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} 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}