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Pierre-Francois Loos 2020-01-20 23:37:18 +01:00
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BSE-PES.tex
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@ -386,14 +386,17 @@ As a final remark, we point out that Eq.~\eqref{eq:EtotBSE} can be easily genera
\section{Computational details} \section{Computational details}
\label{sec:comp_details} \label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we consider here. All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we consider here.
Dunning's basis sets are defined in cartesian gaussians. Dunning's basis sets are defined in cartesian gaussians.
Both perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent {\evGW} \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} calculations are employed as starting point to compute the BSE neutral excitations. Both perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent {\evGW} \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} calculations are employed as starting point to compute the BSE neutral excitations.
These will be labeled as BSE@{\GOWO} and BSE@{\evGW}, respectively. These will be labeled as BSE@{\GOWO} and BSE@{\evGW}, respectively.
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 non-linear, frequency-dependent quasiparticle equation.
For {\evGW}, the quasiparticle energies are obtained self-consistently and we have used the DIIS convergence accelerator technique proposed by Pulay \cite{Pulay_1980,Pulay_1982} to avoid convergence issues. For {\evGW}, the quasiparticle energies are obtained self-consistently and we have used the DIIS convergence accelerator technique proposed by Pulay \cite{Pulay_1980,Pulay_1982} to avoid convergence issues.
Further details about our implementation of {\GOWO} and {\evGW} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}. Further details about our implementation of {\GOWO} and {\evGW} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}.
Finally, the infinitesimal $\eta$ has been set to $10^{-3}$ for all calculations. Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
\titou{For sake of comparison, no frozen core approximation.
The numerical integration required to compute the correlation energy along the adiabatic path has been computed with a 21-point Gauss-Legendre quadrature.
This number of points is probably too big...}
Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3}$ computational cost. Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3}$ computational cost.
This step is, by far, the computational bottleneck in our current implementation. This step is, by far, the computational bottleneck in our current implementation.
@ -406,24 +409,39 @@ This step is, by far, the computational bottleneck in our current implementation
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\caption{ \caption{
Equilibrium distances of ground and excited states of diatomic molecules obtained at various levels of theory.} Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.}
\label{tab:Req} \label{tab:Req}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llcccccccc} \begin{tabular}{llcccccccc}
& & \mc{2}{c}{FCI} & \mc{2}{c}{CC3} & \mc{2}{c}{BSE@{\GOWO}} & \mc{2}{c}{BSE@{\evGW}} \\ & & \mc{8}{c}{Molecules} \\
\cline{3-4} \cline{5-6} \cline{7-8} \cline{9-10} \cline{3-10}
Molecule & State & cc-pVDZ & cc-pVTZ & cc-pVDZ & cc-pVTZ & cc-pVDZ & cc-pVTZ & cc-pVDZ & cc-pVTZ \\ Method & Basis & \ce{H2} & \ce{LiH} & \ce{LiF}& \ce{N2} & \ce{CO} & \ce{BF} & \ce{F2} & \ce{HCl} \\
\hline \hline
\ce{H2} & $S_0$ & 1.438 & 1.403 & & & 1.440 & & 1.432 & \\ CC3 & cc-pVDZ & 1.438 & 3.043 & 3.012 & 2.114 & 2.166 & 2.444 & 2.740 & 2.435 \\
& $S_2$ & & & & & 1.451 & & 1.442 & \\ & cc-pVTZ & 1.403 & 3.011 & 2.961 & 2.079 & 2.143 & 2.392 & 2.669 & 2.413 \\
& $S_5$ & & & & & 1.781 & & 1.778 & \\ & cc-pVQZ & 1.402 & 3.019 & & & & & & \\
\ce{LiH} & & & & \\ CCSD & cc-pVDZ & & & & & & & & \\
\ce{LiF} & & & & \\ & cc-pVTZ & & & & & & & & \\
\ce{HCl} & & & & \\ & cc-pVQZ & & & & & & & & \\
\ce{N2} & & & & \\ CC2 & cc-pVDZ & & & & & & & & \\
\ce{CO} & & & & \\ & cc-pVTZ & & & & & & & & \\
\ce{BF} & & & & \\ & cc-pVQZ & & & & & & & & \\
MP2 & cc-pVDZ & & & & & & & & \\
& cc-pVTZ & & & & & & & & \\
& cc-pVQZ & & & & & & & & \\
BSE@{\GOWO}@HF & cc-pVDZ & & & & & & & & \\
& cc-pVTZ & & & & & & & & \\
& cc-pVQZ & & & & & & & & \\
RPA@{\GOWO}@HF & cc-pVDZ & & & & & & & & \\
& cc-pVTZ & & & & & & & & \\
& cc-pVQZ & & & & & & & & \\
RPAx@HF & cc-pVDZ & & & & & & & & \\
& cc-pVTZ & & & & & & & & \\
& cc-pVQZ & & & & & & & & \\
RPA@HF & cc-pVDZ & & & & & & & & \\
& cc-pVTZ & & & & & & & & \\
& cc-pVQZ & & & & & & & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}