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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. @@ -720,28 +720,28 @@ 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 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} @@ -804,7 +804,7 @@ 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 $\expval*{\hS^2}$ 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} @@ -872,9 +872,10 @@ Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it At the SF-CIS level, the avoided crossing between the $\text{E}$ and $\text{F}$ states is qualitatively reproduced and placed at a slightly larger bond length than at the EOM-CCSD level. In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results. -Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}. SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD-BH\&HLYP excitation energies of the $\text{B}$ and $\text{E}$ states are only trustworthy around equilibrium but inaccurate at dissociation. Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Vuckovic_2017,Cohen_2008a,Cohen_2008c,Cohen_2012} +Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions. +Notably, one can see that the avoided crossing is not modeled at the SF-TD-BLYP level due to the lack of Hartree-Fock exchange. In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented. SF-BSE provides surprisingly accurate excitation energies for the $\text{B}\,{}^1\Sigma_u^+$ state with errors between $0.05$ and $0.3$ eV, outperforming in the process the standard BSE formalism. @@ -884,27 +885,27 @@ Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\ A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions. The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom). -Overall, one can see that $\expval*{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level. -For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval*{\hS^2}$ values associated with the $\text{F}$ state +Overall, one can see that $\expval{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level. +For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval{\hS^2}$ values associated with the $\text{F}$ state only deviate significantly from zero for short bond length and around the avoided crossing where it strongly couples with the spin contaminated $\text{E}$ state. %%% FIG 2 %%% \begin{figure*} - \includegraphics[width=0.4\linewidth]{H2_CIS} + \includegraphics[width=0.45\linewidth]{H2_CIS} \hspace{0.05\linewidth} - \includegraphics[width=0.4\linewidth]{H2_CIS_S2} + \includegraphics[width=0.45\linewidth]{H2_CIS_S2} \vspace{0.025\linewidth} \\ - \includegraphics[width=0.4\linewidth]{H2_BHHLYP} + \includegraphics[width=0.45\linewidth]{H2_BHHLYP} \hspace{0.05\linewidth} - \includegraphics[width=0.4\linewidth]{H2_BHHLYP_S2} + \includegraphics[width=0.45\linewidth]{H2_BHHLYP_S2} \vspace{0.025\linewidth} \\ - \includegraphics[width=0.4\linewidth]{H2_BSE} + \includegraphics[width=0.45\linewidth]{H2_BSE} \hspace{0.05\linewidth} - \includegraphics[width=0.4\linewidth]{H2_BSE_S2} + \includegraphics[width=0.45\linewidth]{H2_BSE_S2} \caption{ - Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and expectation value of the spin operator $\expval*{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory. + Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and expectation value of the spin operator $\expval{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory. The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip are represented as dashed and dotted lines, respectively. All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference. The raw data are reported in the {\SI}. @@ -949,9 +950,9 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels. %%% FIG 3 %%% \begin{figure*} - \includegraphics[width=0.4\linewidth]{CBD_D2h} + \includegraphics[width=0.45\linewidth]{CBD_D2h} \hspace{0.05\linewidth} - \includegraphics[width=0.4\linewidth]{CBD_D4h} + \includegraphics[width=0.45\linewidth]{CBD_D4h} \caption{ Vertical excitation energies of CBD. Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data). @@ -973,22 +974,23 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels. \cline{2-4} Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\ \hline - SF-TD-B3LYP\fnm[3] & $1.750$ & $2.260$ & $4.094$ \\ - SF-TD-BH\&HLYP\fnm[3] & $1.583$ & $2.813$ & $4.528$ \\ - SF-CIS\fnm[1] & $1.521$ & $3.836$ & $5.499$ \\ - EOM-SF-CCSD\fnm[1] & $1.654$ & $3.416$ & $4.360$ \\ - EOM-SF-CCSD(fT)\fnm[1] & $1.516$ & $3.260$ & $4.205$ \\ - EOM-SF-CCSD(dT)\fnm[1] & $1.475$ & $3.215$ & $4.176$ \\ - SF-ADC(2)-s\fnm[2] & $1.573$ & $3.208$ & $4.247$ \\ - SF-ADC(2)-x\fnm[2] & $1.576$ & $3.141$ & $3.796$ \\ + SF-TD-B3LYP\fnm[1] & $1.750$ & $2.260$ & $4.094$ \\ + SF-TD-BH\&HLYP\fnm[1] & $1.583$ & $2.813$ & $4.528$ \\ + SF-CIS\fnm[2] & $1.521$ & $3.836$ & $5.499$ \\ + EOM-SF-CCSD\fnm[3] & $1.654$ & $3.416$ & $4.360$ \\ + EOM-SF-CCSD(fT)\fnm[3] & $1.516$ & $3.260$ & $4.205$ \\ + EOM-SF-CCSD(dT)\fnm[3] & $1.475$ & $3.215$ & $4.176$ \\ + SF-ADC(2)-s\fnm[4] & $1.573$ & $3.208$ & $4.247$ \\ + SF-ADC(2)-x\fnm[4] & $1.576$ & $3.141$ & $3.796$ \\ SF-ADC(3)\fnm[2] & $1.456$ & $3.285$ & $4.334$ \\ - SF-BSE@{\GOWO}\fnm[3] & $1.438$ & $2.704$ & $4.540$ \\ - SF-dBSE@{\GOWO}\fnm[3] & $1.403$ & $2.883$ & $4.621$ \\ + SF-BSE@{\GOWO}\fnm[1] & $1.438$ & $2.704$ & $4.540$ \\ + SF-dBSE@{\GOWO}\fnm[1] & $1.403$ & $2.883$ & $4.621$ \\ \end{tabular} \end{ruledtabular} - \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} - \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} - \fnt[3]{This work.} + \fnt[1]{This work.} + \fnt[2]{Values from Ref.~\onlinecite{Casanova_2020}.} + \fnt[3]{Values from Ref.~\onlinecite{Manohar_2008}.} + \fnt[4]{Values from Ref.~\onlinecite{Lefrancois_2015}.} \end{table} %%% %%% %%% %%% @@ -1004,22 +1006,23 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels. \cline{2-4} Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\ \hline - SF-TD-B3LYP\fnm[3] & $-0.020$ & $0.547$ & $0.486$ \\ - SF-TD-BH\&HLYP\fnm[3] & $0.048$ & $1.465$ & $1.282$ \\ - SF-CIS\fnm[1] & $0.317$ & $3.125$ & $2.650$ \\ - EOM-SF-CCSD\fnm[1] & $0.369$ & $1.824$ & $2.143$ \\ - EOM-SF-CCSD(fT)\fnm[1] & $0.163$ & $1.530$ & $1.921$ \\ - EOM-SF-CCSD(dT)\fnm[1] & $0.098$ & $1.456$ & $1.853$ \\ - SF-ADC(2)-s\fnm[2] & $0.266$ & $1.664$ & $1.910$ \\ - SF-ADC(2)-x\fnm[2] & $0.217$ & $1.123$ & $1.799$ \\ - SF-ADC(3)\fnm[2] & $0.083$ & $1.621$ & $1.930$ \\ - SF-BSE@{\GOWO}\fnm[3] & $-0.049$ & $1.189$ & $1.480$ \\ - SF-dBSE@{\GOWO}\fnm[3] & $0.012$ & $1.507$ & $1.841$ \\ + SF-TD-B3LYP\fnm[1] & $-0.020$ & $0.547$ & $0.486$ \\ + SF-TD-BH\&HLYP\fnm[1] & $0.048$ & $1.465$ & $1.282$ \\ + SF-CIS\fnm[2] & $0.317$ & $3.125$ & $2.650$ \\ + EOM-SF-CCSD\fnm[3] & $0.369$ & $1.824$ & $2.143$ \\ + EOM-SF-CCSD(fT)\fnm[3] & $0.163$ & $1.530$ & $1.921$ \\ + EOM-SF-CCSD(dT)\fnm[3] & $0.098$ & $1.456$ & $1.853$ \\ + SF-ADC(2)-s\fnm[4] & $0.266$ & $1.664$ & $1.910$ \\ + SF-ADC(2)-x\fnm[4] & $0.217$ & $1.123$ & $1.799$ \\ + SF-ADC(3)\fnm[4] & $0.083$ & $1.621$ & $1.930$ \\ + SF-BSE@{\GOWO}\fnm[1] & $-0.049$ & $1.189$ & $1.480$ \\ + SF-dBSE@{\GOWO}\fnm[1] & $0.012$ & $1.507$ & $1.841$ \\ \end{tabular} \end{ruledtabular} - \fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.} - \fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.} - \fnt[3]{This work.} + \fnt[1]{This work.} + \fnt[2]{Values from Ref.~\onlinecite{Casanova_2020}.} + \fnt[3]{Values from Ref.~\onlinecite{Manohar_2008}.} + \fnt[4]{Values from Ref.~\onlinecite{Lefrancois_2015}.} \end{table} %%% %%% %%% %%% diff --git a/Notebooks/sf-BSE.nb b/Notebooks/sf-BSE.nb index ebf6871..0cbf8df 100644 --- a/Notebooks/sf-BSE.nb +++ b/Notebooks/sf-BSE.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] 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