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Pierre-Francois Loos 2021-01-14 15:15:01 +01:00
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Manuscript/sfBSE.tex
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\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
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\begin{abstract} \begin{abstract}
Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe-Salpeter equation (BSE) formalism in its static approximation is ``blind'' to double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes. Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe-Salpeter equation (BSE) formalism in its static approximation is ``blind'' to double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes.
Here, we apply the spin-flip technique (which consists in considering the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism of many-body perturbation theory in order to access double excitations. Here, we apply the spin-flip technique (which consists in considering the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism of many-body perturbation theory in order to access double excitations.
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\subsection{Hydrogen molecule} \subsection{Hydrogen molecule}
\label{sec:H2} \label{sec:H2}
%=============================== %===============================
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The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule. The second system of interest is the \ce{H2} molecule where we stretch the bond. The ground state of the \ce{H2} molecule is a singlet with $(1\sigma_g)^2$ configuration. Three excited states are investigated during the stretching: the singly excited state B${}^1 \Sigma_u^+$ with $(1\sigma_g )~ (1\sigma_u)$ configuration, the singly excited state E${}^1 \Sigma_g^+$ with $(1\sigma_g )~ (2\sigma_g)$ configuration and the doubly excited state F${}^1 \Sigma_g^+$ with $(1\sigma_u )~ (1\sigma_u)$ configuration. Three methods with and without spin-flip are used to study these states. These methods are CIS, TD-BHHLYP and BSE and are compared to the reference, here the EOM-CCSD method. %that is equivalent to the FCI for the \ce{H2} molecule.
Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BHHLYP calculation with and without spin-flip. TD-BHHLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip. Left panel of Fig ~\ref{fig:H2} shows results of the CIS calculation with and without spin-flip. We can observe that both SF-CIS and CIS poorly describe the B${}^1 \Sigma_u^+$ state, especially at the dissociation limit with an error of more than 1 eV. The same analysis can be done for the F${}^1 \Sigma_g^+$ state at the dissociation limit. EOM-CSSD curves show us an avoided crossing between the E${}^1 \Sigma_g^+$ and F${}^1 \Sigma_g^+$ states due to their same symmetry. SF-CIS does not represent well the E${}^1 \Sigma_g^+$ state before the avoided crossing. But the E${}^1 \Sigma_g^+$ state is well describe after this avoided crossing. SF-CIS describes better the F${}^1 \Sigma_g^+$ state before the avoided crossing than at the dissociation limit. In general, SF-CIS does not give a good description of the double excitation. As expected CIS does not find the double excitation to the F${}^1 \Sigma_g^+$ state. The rigth panel gives results of the TD-BHHLYP calculation with and without spin-flip. TD-BHHLYP shows bad results for all the states of interest with and without spin-flip. Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip.