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tables for Enzo

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Pierre-Francois Loos 2021-01-10 16:30:46 +01:00
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Manuscript/sfBSE.tex
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@ -595,7 +595,7 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
\begin{table*}
\caption{
Spin-flip excitations (in eV) of \ce{Be} obtained for various methods with the 6-31G basis.
The $GW$ calculations are performed with a HF starting point.
The $GW$ calculations are performed with an UHF starting point.
\label{tab:Be}}
\begin{ruledtabular}
\begin{tabular}{lcccccccccccc}
@ -729,13 +729,73 @@ The TD-DFT calculations have been performed with Q-CHEM 5.2.1 \cite{qchem4} and
\label{sec:CBD}
%===============================
Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Manohar_2008,Lefrancois_2015}
Cyclobutadiene (CBD) is an interesting example as its electronic character of its ground state can be tune via geometrical deformation. \cite{Balkova_1994,Manohar_2008,Lefrancois_2015,Casanova_2020}
%with potential large spin contamination.
In its $D_{2h}$ rectangular $^1 A_g$ ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals
However, in its $D_{4h}$ square-planar $^3 A_{2g}$ ground-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals.
In its $D_{2h}$ rectangular $^1 A_g$ singlet ground-state equilibrium geometry, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are non-degenerate, and the singlet ground state can be safely labeled as single-reference with well-defined doubly-occupied orbitals
However, in its $D_{4h}$ square-planar $^3 A_{2g}$ triplet round-state equilibrium geometry, the HOMO and LUMO are strictly degenerate, and the electronic ground state (which is still of singlet nature with $B_{1g}$ spatial symmetry, hence violating Hund's rule) is strongly multi-reference with singly occupied orbitals.
In this case, single-reference methods notoriously fail.
Nonetheless, the lowest triplet state of symmetry $^3 A_{2g}$ remains of single-reference character and is then a perfect starting point for spin-flip calculations.
The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ states of CBD have been extracted from Ref.~\onlinecite{Manohar_2008} and have been obtained at the CCSD(T)/cc-pVTZ level.
EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}.
All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
%%% 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.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lccc}
& \mc{3}{c}{Excitation energies (eV)} \\
\cline{2-4}
Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\
\hline
EOM-SF-CIS\fnm[1] & & & \\
EOM-SF-CCSD\fnm[1] & & & \\
EOM-SF-CCSD(fT)\fnm[1] & & & \\
EOM-SF-CCSD(dT)\fnm[1] & & & \\
SF-ADC(2)-s\fnm[2] & & & \\
SF-ADC(2)-x\fnm[2] & & & \\
SF-ADC(3)\fnm[2] & & & \\
SF-BSE@{\GOWO}@UHF\fnm[3] & & & \\
SF-dBSE@{\GOWO}@UHF\fnm[3] & & & \\
\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[3]{This work.}
\end{table}
%%% %%% %%% %%%
%%% 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.
\label{tab:CBD_D2h}}
\begin{ruledtabular}
\begin{tabular}{lccc}
& \mc{3}{c}{Excitation energies (eV)} \\
\cline{2-4}
Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
\hline
EOM-SF-CIS\fnm[1] & & & \\
EOM-SF-CCSD\fnm[1] & & & \\
EOM-SF-CCSD(fT)\fnm[1] & & & \\
EOM-SF-CCSD(dT)\fnm[1] & & & \\
SF-ADC(2)-s\fnm[2] & & & \\
SF-ADC(2)-x\fnm[2] & & & \\
SF-ADC(3)\fnm[2] & & & \\
SF-BSE@{\GOWO}@UHF\fnm[3] & & & \\
SF-dBSE@{\GOWO}@UHF\fnm[3] & & & \\
\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[3]{This work.}
\end{table}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}