PTEROSOR/sfBSE
10
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

modifs again

Browse Source
This commit is contained in:
Pierre-Francois Loos 2021-01-13 10:15:19 +01:00
parent 4048b358af
commit 9257c70b8e
1 changed files with 33 additions and 29 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

60
Manuscript/sfBSE.tex
View File

@ -565,22 +565,24 @@ For a given single excitation $m$, the explicit expressions of $\Delta \expval{\
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the closed-shell systems under investigation here, we consider a triplet reference state.
We then adopt the unrestricted formalism throughout this work.
The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
Note that, in any case, the entire set of orbitals and energies is corrected.
Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
This is left for future work.
However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
%\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
We then adopt the unrestricted formalism throughout this work.
The {\GOWO} and ev$GW$ calculations performed to obtain the screened Coulomb potential and the quasiparticle energies required to compute the BSE neutral excitations are performed using an unrestricted HF (UHF) starting point, while, by construction, the corresponding qs$GW$ quantities are independent from the starting point.
For {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation [see Eq.~\eqref{eq:G0W0_lin}].
Note that, in any case, the entire set of orbitals and energies is corrected.
Further details about our implementation of {\GOWO}, ev$GW$, and qs$GW$ can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020e,Loos_2020h}.
Here, we do not investigate how the starting orbitals affect the BSE@{\GOWO} and BSE@ev$GW$ excitation energies.
This is left for future work.
However, it is worth mentioning that, for the present (small) molecular systems, HF is usually a good starting point, \cite{Loos_2020a,Loos_2020e,Loos_2020h} although improvements could certainly be obtained with starting orbitals and energies computed with, for example, optimally-tuned range-separated hybrid functionals. \cite{Stein_2009,Stein_2010,Refaely-Abramson_2012,Kronik_2012}
In the following, all linear response calculations are performed within the TDA to ensure consistency between the spin-conserved and spin-flip results.
%\titou{As one-electron basis sets, we employ Pople's 6-31G basis or the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.}
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
The SF-ADC, EOM-SF-CC and SF-TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and the EOM-CCSD calculation with Gaussian 09. \cite{g09}
As a consistency check, we systematically perform the SF-CIS calculations with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
Throughout this work, all spin-flip calculations have been performed with a UHF reference.
All the static and dynamic BSE calculations (labeled in the following as SF-BSE and SF-dBSE respectively) are performed with the software \texttt{QuAcK}, \cite{QuAcK} developed in our group and freely available on \texttt{github}.
The standard and extended spin-flip ADC(2) calculations [SF-ADC(2)-s and SF-ADC(2)-x, respectively] as well as the SF-ADC(3) \cite{Lefrancois_2015} are performed with Q-CHEM 5.2.1. \cite{qchem4}
Spin-flip TD-DFT calculations \cite{Shao_2003} considering the BLYP, \cite{Becke_1988,Lee_1988} B3LYP, \cite{Becke_1988,Lee_1988,Becke_1993a} and BH\&HLYP \cite{Lee_1988,Becke_1993b} functionals with contains $0\%$, $20\%$, and $50\%$ of exact exchange are labeled as SF-TD-BLYP, SF-TD-B3LYP, and SF-TD-BH\&HLYP, respectively, and are also performed with Q-CHEM 5.2.1.
EOM-CCSD excitation energies \cite{Koch_1990,Stanton_1993,Koch_1994} are computed with Gaussian 09. \cite{g09}
As a consistency check, we systematically perform the SF-CIS calculations \cite{Krylov_2001a} with both \texttt{QuAcK} and Q-CHEM, and make sure that they yield identical excitation energies.
Throughout this work, all spin-flip calculations are performed with a UHF reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
@ -625,9 +627,9 @@ The excitation energies corresponding to the first singlet and triplet single ex
FCI\fnm[3] & 2.862 & 6.577 & 7.669 & 8.624 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Excitation energies extracted from Ref.~\onlinecite{Casanova_2020}.}
\fnt[1]{Value from Ref.~\onlinecite{Casanova_2020}.}
\fnt[2]{This work.}
\fnt[3]{Excitation energies taken from Ref.~\onlinecite{Krylov_2001a}.}
\fnt[3]{Value from Ref.~\onlinecite{Krylov_2001a}.}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
@ -637,7 +639,7 @@ The excitation energies corresponding to the first singlet and triplet single ex
\includegraphics[width=\linewidth]{Be}
\caption{
Excitation energies [with respect to the $^1S(1s^2 2s^2)$ singlet ground state] of \ce{Be} obtained with the 6-31G basis for various levels of theory:
SD-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black).
SF-TD-DFT \cite{Casanova_2020} (red), SF-BSE (blue), SF-CIS \cite{Krylov_2001a} and SF-ADC (orange), and FCI \cite{Krylov_2001a} (black).
All the spin-flip calculations have been performed with a UHF reference.
\label{fig:Be}}
\end{figure}
@ -787,7 +789,8 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
%%% TABLE ?? %%%
\begin{table}
\caption{
Vertical excitation energies (with respect to the singlet $X\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $X\,{}^1 A_{g}$ singlet ground state.
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ singlet ground state.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lccc}
@ -802,12 +805,12 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
SF-ADC(2)-s\fnm[2] & 1.572& 3.201& 4.241\\
SF-ADC(2)-x\fnm[2] &1.576 &3.134 &3.792 \\
SF-ADC(3)\fnm[2] & 1.455&3.276 &4.328 \\
SF-BSE@{\GOWO}@UHF\fnm[3] & 1.438 & 2.704 &4.540 \\
SF-dBSE@{\GOWO}@UHF\fnm[3] & 1.403 &2.883 &4.621 \\
SF-BSE@{\GOWO}\fnm[3] & 1.438 & 2.704 &4.540 \\
SF-dBSE@{\GOWO}\fnm[3] & 1.403 &2.883 &4.621 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[3]{This work.}
\end{table}
%%% %%% %%% %%%
@ -815,7 +818,8 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
%%% TABLE ?? %%%
\begin{table}
\caption{
Vertical excitation energies (with respect to the singlet $X\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $X\,{}^1B_{1g}$ singlet ground state.
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state.
All the spin-flip calculations have been performed with a UHF reference.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lccc}
@ -830,12 +834,12 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
SF-ADC(2)-s\fnm[2] & & & \\
SF-ADC(2)-x\fnm[2] & & & \\
SF-ADC(3)\fnm[2] & & & \\
SF-BSE@{\GOWO}@UHF\fnm[3] & -0.049 & 1.189 & 1.480 \\
SF-dBSE@{\GOWO}@UHF\fnm[3] & 0.012 & 1.507 & 1.841 \\
SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\
SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value from Ref.~\onlinecite{Manohar_2008} using a UHF reference.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015} using a UHF reference.}
\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
\fnt[2]{Value from Ref.~\onlinecite{Lefrancois_2015}.}
\fnt[3]{This work.}
\end{table}
%%% %%% %%% %%%