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@ -1,13 +1,47 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-28 14:15:42 +0200 %% Created for Pierre-Francois Loos at 2020-08-28 14:26:14 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Gunnarsson_1976,
Author = {Gunnarsson, O. and Lundqvist, B. I.},
Date-Added = {2020-08-28 14:22:38 +0200},
Date-Modified = {2020-08-28 14:22:38 +0200},
Doi = {10.1103/PhysRevB.13.4274},
Issue = {10},
Journal = {Phys. Rev. B},
Month = {May},
Numpages = {0},
Pages = {4274--4298},
Publisher = {American Physical Society},
Title = {Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism},
Url = {https://link.aps.org/doi/10.1103/PhysRevB.13.4274},
Volume = {13},
Year = {1976},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.13.4274},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.13.4274}}
@article{Langreth_1975,
Author = {D.C. Langreth and J.P. Perdew},
Date-Added = {2020-08-28 14:22:28 +0200},
Date-Modified = {2020-08-28 14:26:09 +0200},
Doi = {https://doi.org/10.1016/0038-1098(79)90254-0},
Issn = {0038-1098},
Journal = {Solid State Commun.},
Number = {8},
Pages = {567 - 571},
Title = {The gradient approximation to the exchange-correlation energy functional: A generalization that works},
Url = {http://www.sciencedirect.com/science/article/pii/0038109879902540},
Volume = {31},
Year = {1979},
Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0038109879902540},
Bdsk-Url-2 = {https://doi.org/10.1016/0038-1098(79)90254-0}}
@article{Teh_2019, @article{Teh_2019,
Author = {H.-H. Teh and J. E. Subotnik}, Author = {H.-H. Teh and J. E. Subotnik},
Date-Added = {2020-08-27 17:31:44 +0200}, Date-Added = {2020-08-27 17:31:44 +0200},

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dynker.nb
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dynker.tex
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@ -44,7 +44,7 @@ Prototypical examples of valence, charge-transfer, and Rydberg excited states ar
\section{Linear response theory} \section{Linear response theory}
\label{sec:LR} \label{sec:LR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_S$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_S$ [extracted from their eigenvectors $\T{(\bX_S \bY_S)}$] via the response of the system to a weak electromagnetic field. \cite{Oddershede_1977,Casida_1995,Petersilka_1996} Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_S$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths [extracted from their eigenvectors $\T{(\bX_S \bY_S)}$] via the response of the system to a weak electromagnetic field. \cite{Oddershede_1977,Casida_1995,Petersilka_1996}
From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995} From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
\begin{equation} \label{eq:LR} \begin{equation} \label{eq:LR}
\begin{pmatrix} \begin{pmatrix}
@ -350,12 +350,14 @@ Although not particularly accurate for the single excitations, Maitra's dynamica
Note that this correlation kernel is known to work best in the weak correlation regime (which is the case here) where the true excitations have a clear single and double excitation character, \cite{Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018} Note that this correlation kernel is known to work best in the weak correlation regime (which is the case here) where the true excitations have a clear single and double excitation character, \cite{Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018}
Its accuracy for the single excitations could be certainly improved in a density-functional theory context. Its accuracy for the single excitations could be certainly improved in a density-functional theory context.
However, this is not the point of the present investigation. However, this is not the point of the present investigation.
In the case of \ce{H2} in a minimal basis, because $\mel{S}{\hH}{D} = 0$, there is no dynamical correction for both singlets and triplets, and one cannot access the double excitation with Maitra's kernel.
It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
Table \ref{tab:Maitra} also reports the slightly improved (thanks to error compensation) CIS and D-CIS excitation energies. Table \ref{tab:Maitra} also reports the slightly improved (thanks to error compensation) CIS and D-CIS excitation energies.
In particular, single excitations are greatly improved without altering the accuracy of the double excitation. In particular, single excitations are greatly improved without altering the accuracy of the double excitation.
Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra}. Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra}.
In the case of \ce{H2} in a minimal basis, because $\mel{S}{\hH}{D} = 0$, there is no dynamical correction for both singlets and triplets, and one cannot access the double excitation with Maitra's kernel.
It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
%%% FIGURE 1 %%% %%% FIGURE 1 %%%
\begin{figure} \begin{figure}
\includegraphics[width=\linewidth]{Maitra} \includegraphics[width=\linewidth]{Maitra}
@ -396,7 +398,7 @@ is the correlation part of the self-energy $\Sig{}$, and
\begin{equation} \begin{equation}
Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1} Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation} \end{equation}
is the renormalization factor. is the renormalization factor (or spectral weight).
In Eq.~\eqref{eq:SigGW}, $\Omega = \Delta\e{} + 2 \ERI{vc}{cv}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$. In Eq.~\eqref{eq:SigGW}, $\Omega = \Delta\e{} + 2 \ERI{vc}{cv}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988,Romaniello_2009b} One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988,Romaniello_2009b}
@ -426,7 +428,7 @@ and where
W^{\co}_C(\omega) = \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega} W^{\co}_C(\omega) = \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
are the elements of the correlation part of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian. are the elements of the correlation part of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian, respectively.
Note that, in this case, the correlation kernel is spin blind. Note that, in this case, the correlation kernel is spin blind.
Within the usual static approximation, the BSE Hamiltonian is simply Within the usual static approximation, the BSE Hamiltonian is simply
@ -443,15 +445,15 @@ with
\begin{gather} \begin{gather}
R_{\BSE}^{\sigma} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - \ERI{vv}{cc} - W_R(\omega = \Delta\eGW{}) R_{\BSE}^{\sigma} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - \ERI{vv}{cc} - W_R(\omega = \Delta\eGW{})
\\ \\
C_{\BSE}^{\sigma} = C_{\dBSE}^{\sigma} = 2 \sigma \ERI{vc}{cv} - \ERI{vc}{cv} - W^{\co}_C(\omega = 0) C_{\BSE}^{\sigma} = C_{\dBSE}^{\sigma}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
It can be easily shown that solving the equation It can be easily shown that solving the secular equation
\begin{equation} \begin{equation}
\det[\bH_{\dBSE}^{\sigma}(\omega) - \omega \bI] = 0 \det[\bH_{\dBSE}^{\sigma}(\omega) - \omega \bI] = 0
\end{equation} \end{equation}
yields 2 solutions per spin manifold (except for \ce{H2} where only one root is observed), as shown in Fig.~\ref{fig:dBSE} for the case of \ce{HeH+}. yields 2 solutions per spin manifold (except for \ce{H2} where only one root is observed, see below), as shown in Fig.~\ref{fig:dBSE} for the case of \ce{HeH+}.
Their numerical values are reported in Table \ref{tab:BSE} alongside other variants discussed below. Their numerical values are reported in Table \ref{tab:BSE} alongside other variants discussed below.
These numbers evidence that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV. These numbers evidence that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV.
As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appear due to the approximate nature of the dBSE kernel. As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appear due to the approximate nature of the dBSE kernel.
@ -459,7 +461,7 @@ Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two
Therefore, there is the right number of singlet solutions but there is one spurious solution for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$). Therefore, there is the right number of singlet solutions but there is one spurious solution for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$).
It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the other solution, $\omega_2^{\dBSE,\sigma}$, stems from a pole and consequently the slope is very large around this frequency value. It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the other solution, $\omega_2^{\dBSE,\sigma}$, stems from a pole and consequently the slope is very large around this frequency value.
This makes this latter solution quite hard to locate with a method like Newton-Raphson (for example). This makes this latter solution quite hard to locate with a method like Newton-Raphson (for example).
Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dynamical effects produce a systematic red-shifting of the static excitations, here we observe both blue- and red-shifted transitions. Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dynamical effects have been shown to produce a systematic red-shifting of the static excitations, here we observe both blue- and red-shifted transitions (see values in square brackets in Table \ref{tab:BSE}).
%%% TABLE III %%% %%% TABLE III %%%
\begin{table*} \begin{table*}
@ -503,9 +505,10 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}). In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is indeed the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$. Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is indeed the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
This can be further verify by switching off gradually the electron-electron interaction as one would do in the adiabatic connection formalism. \cite{Langreth_1975,Gunnarsson_1976}
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the dBSE Hamiltonian \eqref{eq:HBSE} does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations. Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE} does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
However, it does increase significantly the static excitations while the dynamical correction are not altered by the TDA. However, it does increase significantly the static excitations while the magnitude of the dynamical corrections is not altered by the TDA.
%Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA. %Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
%The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation. %The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
@ -561,11 +564,13 @@ and the renormalization factor is
\end{equation} \end{equation}
This corresponds to a dynamical perturbative correction to the static excitations. This corresponds to a dynamical perturbative correction to the static excitations.
The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very effective at reproducing the dynamical value for the single excitations. The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very effective at reproducing the dynamical values for the single excitations.
Because the value of $Z_{1}$ is always quite close to unity in the present systems (evidencing that the perturbative expansion behaves nicely), one could have anticipated the fact that the first-order correction is a good estimate of the non-perturbative result.
However, because the perturbative treatment is ultimately static, one cannot access double excitations with such a scheme. However, because the perturbative treatment is ultimately static, one cannot access double excitations with such a scheme.
For \ce{H2}, there is no dynamical corrections at the BSE, pBSE or dBSE levels. For \ce{H2}, there is no dynamical corrections at the BSE, pBSE or dBSE levels.
Indeed, as $\ERI{vv}{vc} = \ERI{vc}{cc} = 0$ (see Table \ref{tab:params}), we have $W^{\co}_R(\omega) = 0$ [see Eq.~\eqref{eq:WR}]. Indeed, as $\ERI{vv}{vc} = \ERI{vc}{cc} = 0$ (see Table \ref{tab:params}), we have $W^{\co}_R(\omega) = 0$ [see Eq.~\eqref{eq:WR}].
The lack of frequency dependence of the kernel means that one cannot estimate the energy of the doubly-excited state of \ce{H2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Second-order BSE kernel} \subsection{Second-order BSE kernel}
@ -616,34 +621,34 @@ The correlation part of the dynamical kernel for BSE2 is a bit cumbersome \cite{
\begin{equation} \begin{equation}
\bH_{\dBSE2}^{\sigma} = \bH_{\BSE2}^{\sigma} + \bH_{\dBSE2}^{\sigma} = \bH_{\BSE2}^{\sigma} +
\begin{pmatrix} \begin{pmatrix}
+f_{\dBSE2}^{\co,\sigma}(+\omega) & +f_{\dBSE2}^{\co,\sigma} +R_{\dBSE2}^{\co,\sigma}(+\omega) & +R_{\dBSE2}^{\co,\sigma}
\\ \\
-f_{\dBSE2}^{\co,\sigma} & -f_{\BSE2}^{\co,\sigma}(-\omega) -C_{\dBSE2}^{\co,\sigma} & -C_{\BSE2}^{\co,\sigma}(-\omega)
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
with with
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
f_{\dBSE2}^{\co,\updw}(\omega) = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}} R_{\dBSE2}^{\co,\updw}(\omega) = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\ \\
f_{\dBSE2}^{\co,\updw} = \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}} C_{\dBSE2}^{\co,\updw} = \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and and
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
f_{\dBSE2}^{\co,\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}} R_{\dBSE2}^{\co,\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\ \\
f_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}} C_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets. \cite{Rebolini_PhD} As mentioned in Ref.~\onlinecite{Rebolini_2016}, the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
Like in dBSE, dBSE2 generates the right number of excitations for the singlet manifold (see Fig.~\ref{fig:BSE2}). Unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets. \cite{Rebolini_PhD}
However, one spurious triplet excitation remains.
It is mentioned in Ref.~\onlinecite{Rebolini_2016} that the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
Like in dBSE, dBSE2 generates the right number of excitations for the singlet manifold (see Fig.~\ref{fig:BSE2}).
However, one spurious triplet excitation clearly remains.
Numerical results for the two-level models are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments. Numerical results for the two-level models are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
In the case of BSE2, the perturbative partitioning is simply In the case of BSE2, the perturbative partitioning (pBSE2) is simply
\begin{equation} \begin{equation}
\bH_{\dBSE2}^{\sigma}(\omega) \bH_{\dBSE2}^{\sigma}(\omega)
= \underbrace{\bH_{\BSE2}^{\sigma}}_{\bH_{\pBSE2}^{(0)}} = \underbrace{\bH_{\BSE2}^{\sigma}}_{\bH_{\pBSE2}^{(0)}}
@ -692,11 +697,12 @@ In the case of BSE2, the perturbative partitioning is simply
As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS). As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS).
Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate due to error compensations. Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate due to error compensations.
Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations (except for \ce{H2}, see below), especially in the TDA which justifies the use of the perturbative treatment in Ref.~\onlinecite{Zhang_2013,Rebolini_2016}. Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations (except for \ce{H2}, see below), especially in the TDA.
This justifies the use of the perturbative treatment in Refs.~\onlinecite{Zhang_2013,Rebolini_2016}.
Overall, the accuracy of dBSE and dBSE2 are comparable for single excitations although their behavior is quite different (see Tables \ref{tab:BSE} and \ref{tab:BSE2}). Overall, the accuracy of dBSE and dBSE2 are comparable for single excitations although their behavior is quite different (see Tables \ref{tab:BSE} and \ref{tab:BSE2}).
For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value. For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
Again, the case of \ce{H2} is a bit peculiar as the perturbative treatment (pBSE2) does not provide dynamical corrections, while its dynamical version (dBSE2) does yield sizable corrections originating from the coupling term $f_{\dBSE2}^{\co,\sigma}$ which is non-zero in the case of dBSE2. Again, the case of \ce{H2} is a bit peculiar as the perturbative treatment (pBSE2) does not provide any dynamical corrections, while its dynamical version (dBSE2) does yield sizable corrections originating from the coupling term $C_{\dBSE2}^{\co,\sigma}$ which is non-zero in the case of dBSE2.
Although frequency-independent, this additional term makes the singlet and triplet excitation energies very accurate. Although frequency-independent, this additional term makes the singlet and triplet excitation energies very accurate.
However, one cannot access the double excitation. However, one cannot access the double excitation.