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done with BSE

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Pierre-Francois Loos 2021-01-19 10:05:09 +01:00
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18
Manuscript/sfBSE.tex
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@ -70,7 +70,7 @@ Here we apply the spin-flip technique to the BSE formalism in order to access, i
The present BSE calculations are based on the spin-unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method.
Moreover, we also go beyond the static approximation by taking into account dynamical effects (Sec.~\ref{sec:dBSE}) via an unrestricted generalization of our recently developed (renormalized) perturbative correction which builds on the seminal work of Strinati, \cite{Strinati_1982,Strinati_1984,Strinati_1988} Romaniello and collaborators, \cite{Romaniello_2009b,Sangalli_2011} and Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Lettmann_2019}
We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).
We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval*{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).
Computational details are reported in Sec.~\ref{sec:compdet} and our results for the beryllium atom \ce{Be} (Subsec.~\ref{sec:Be}), the hydrogen molecule \ce{H2} (Subsec.~\ref{sec:H2}), and cyclobutadiene \ce{C4H4} (Subsec.~\ref{sec:CBD}) are discussed in Sec.~\ref{sec:res}.
Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
Unless otherwise stated, atomic units are used.
@ -540,28 +540,28 @@ For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix e
\label{sec:spin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One of the key issues of linear response formalism based on unrestricted references is spin contamination or the artificial mixing with configurations of different spin multiplicities.
As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for a high-spin triplets, and ii) spin-contamination of the excited states due to spin incompleteness of the CI expansion.
As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval*{\hS^2} > 2$ for high-spin triplets, and ii) spin contamination of the excited states due to spin incompleteness of the CI expansion.
The latter issue is an important source of spin contamination in the present context as BSE is limited to single excitations with respect to the reference configuration.
Specific schemes have been developed to palliate these shortcomings and we refer the interested reader to Ref.~\onlinecite{Casanova_2020} for a detailed discussion on this matter.
In order to monitor closely how contaminated are these states, we compute
\begin{equation}
\expval{\hS^2}_m = \expval{\hS^2}_0 + \Delta \expval{\hS^2}_m
\expval*{\hS^2}_m = \expval*{\hS^2}_0 + \Delta \expval*{\hS^2}_m
\end{equation}
where
\begin{equation}
\expval{\hS^2}_{0}
\expval*{\hS^2}_{0}
= \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 )
+ n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2
\end{equation}
is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval{\hS^2}$, and
is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval*{\hS^2}$, and
\begin{equation}
\label{eq:OV}
(p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br
\end{equation}
are overlap integrals between spin-up and spin-down orbitals.
For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the vectors $\bX{m}{}$ and $\bY{m}{}$ as well as the orbital overlaps defined in Eq.~\eqref{eq:OV}.
For a given single excitation $m$, the explicit expressions of $\Delta \expval*{\hS^2}_m^{\spc}$ and $\Delta \expval*{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the vectors $\bX{m}{}$ and $\bY{m}{}$ as well as the orbital overlaps defined in Eq.~\eqref{eq:OV}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
@ -629,7 +629,7 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
\caption{
Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained for various methods with the 6-31G basis set.
All the spin-flip calculations have been performed with a UHF reference.
The $\expval*{S^2}$ value associated with each state is reported in parenthesis (when available).
The $\expval*{\hS^2}$ value associated with each state is reported in parenthesis (when available).
\label{tab:Be}}
\begin{ruledtabular}
\begin{tabular}{lcccccccccc}
@ -646,9 +646,9 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002) \\
SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006) \\
SF-BSE@{\GOWO} & (0.004) & 2.399(1.999) & 6.191(0.023) & 7.792(1.000) & 9.373(0.013) \\
SF-BSE@{\evGW} & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
% SF-BSE@ev$GW$ & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
SF-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424 \\
SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\
% SF-dBSE@ev$GW$ & & 2.369 & 6.273 & 7.820 & 9.441 \\
SF-ADC(2)-s & & 2.433 & 6.255 & 7.745 & 9.047 \\
SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 \\
SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618 \\