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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-27 18:05:25 +0200
%% Created for Pierre-Francois Loos at 2020-08-27 21:46:59 +0200
%% Saved with string encoding Unicode (UTF-8)
@ -17,16 +17,17 @@
Pages = {3426--3432},
Title = {The Simplest Possible Approach for Simulating S0−S1 Conical Intersections with DFT/TDDFT: Adding One Doubly Excited Configuration},
Volume = {10},
Year = {2019}}
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b00981}}
@article{Loos_2020e,
Author = {P. F. Loos and X. Blase},
Date-Added = {2020-08-27 16:03:48 +0200},
Date-Modified = {2020-08-27 16:40:16 +0200},
Journal = {J. Chem. Phys.},
Pages = {arXiv:2007.13501},
Title = {Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation},
Year = {in press}}
@misc{Loos_2020e,
Archiveprefix = {arXiv},
Author = {Pierre-Fran{\c c}ois Loos and Xavier Blase},
Date-Modified = {2020-08-27 21:23:51 +0200},
Eprint = {2007.13501},
Primaryclass = {physics.chem-ph},
Title = {{{Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation}}},
Year = {2020}}
@article{Lowdin_1963,
Author = {P. L{\"o}wdin},

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@ -235,7 +235,7 @@ The numerical values of the various quantities defined above are gathered in Tab
%%% TABLE I %%%
\begin{table*}
\caption{Numerical values (in hartree) of the valence and conduction orbital energies, $\e{v}$ and $\e{c}$, and various two-electron integrals in the orbital basis for various two-level systems.
\caption{Numerical values (in hartree) of the valence and conduction orbital energies, $\e{v}$ and $\e{c}$, and two-electron integrals in the orbital basis for various two-level systems.
\label{tab:params}
}
\begin{ruledtabular}
@ -317,6 +317,7 @@ The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies.
Because, there is nothing to dress for the triplet state, only the TDHF and D-TDHF triplet excitation energies are equal.
%%% TABLE II %%%
%\begin{squeezetable}
\begin{table}
\caption{Singlet and triplet excitation energies (in eV) for various levels of theory and two-level systems.
The magnitude of the dynamical correction is reported in square brackets.
@ -328,25 +329,26 @@ Because, there is nothing to dress for the triplet state, only the TDHF and D-TD
\cline{3-7}
System & Excitation & CIS & TDHF & D-CIS & D-TDHF & Exact \\
\hline
\ce{H2} & $\omega_1^{\updw}$ & 25.78 & 25.30 & 25.78[+0.00] & 25.30[+0.00] & 26.34 \\
& $\omega_3^{\updw}$ & & & & & 44.04 \\
& $\omega_1^{\upup}$ & 15.92 & 15.13 & 15.92[+0.00] & 15.13[+0.00] & 16.48 \\
\ce{H2} & $\omega_1^{\updw}$ & $25.78$ & $25.30$ & $25.78[+0.00]$ & $25.30[+0.00]$ & $26.34$ \\
& $\omega_3^{\updw}$ & & & & & $44.04$ \\
& $\omega_1^{\upup}$ & $15.92$ & $15.13$ & $15.92[+0.00]$ & $15.13[+0.00]$ & $16.48$ \\
\\
\ce{HeH+} & $\omega_1^{\updw}$ & 29.68 & 29.42 & 27.75[-1.93] & 27.64[-1.78] & 28.05 \\
& $\omega_3^{\updw}$ & & & 63.59 & 63.52 & 64.09 \\
& $\omega_1^{\upup}$ & 21.77 & 21.41 & 21.77[+0.00] & 21.41[+0.00] & 22.03 \\
\ce{HeH+} & $\omega_1^{\updw}$ & $29.68$ & $29.42$ & $27.75[-1.93]$ & $27.64[-1.78]$ & $28.05$ \\
& $\omega_3^{\updw}$ & & & $63.59$ & $63.52$ & $64.09$ \\
& $\omega_1^{\upup}$ & $21.77$ & $21.41$ & $21.77[+0.00]$ & $21.41[+0.00]$ & $22.03$ \\
\\
\ce{He} & $\omega_1^{\updw}$ & 52.01 & 51.64 & 51.87[-0.14]& 51.52[-0.12]& 52.29 \\
& $\omega_3^{\updw}$ & & & 93.85 & 93.84 & 94.66 \\
& $\omega_1^{\upup}$ & 39.62 & 39.13 & 39.62[+0.00]& 39.13[+0.00]& 40.18 \\
\ce{He} & $\omega_1^{\updw}$ & $52.01$ & $51.64$ & $51.87[-0.14]$ & $51.52[-0.12]$ & $52.29$ \\
& $\omega_3^{\updw}$ & & & $93.85$ & $93.84$ & $94.66$ \\
& $\omega_1^{\upup}$ & $39.62$ & $39.13$ & $39.62[+0.00]$ & $39.13[+0.00]$ & $40.18$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
%\end{squeezetable}
%%% %%% %%% %%%
Although not particularly accurate for the single excitations, Maitra's dynamical kernel allows to access the double excitation with good accuracy and provides exactly the right number of solutions (two singlets and one triplet).
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 DFT 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.
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.
@ -371,13 +373,13 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As mentioned in Sec.~\ref{sec:dyn}, most of BSE calculations performed nowadays are done within the static approximation. \cite{ReiningBook,Onida_2002,Blase_2018,Blase_2020}
However, following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored this formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the dynamically screened Coulomb potential $W$ \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e}
Based on the very same two-level model that we employ here, Romaniello and coworkers \cite{Romaniello_2009b} clearly evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent BSE eigenvalue problem.
However, following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored this formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the screened Coulomb potential $W$ \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e}
Based on the very same two-level model that we employ here, Romaniello \textit{et al.} \cite{Romaniello_2009b} clearly evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent BSE eigenvalue problem.
For this particular system, they showed that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations, \cite{Romaniello_2009b} attributed to the self-screening problem. \cite{Romaniello_2009a}
This issue was resolved in the subsequent work of Sangalli \textit{et al.} \cite{Sangalli_2011} via the design of a diagrammatic number-conserving approach based on the folding of the second-RPA Hamiltonian. \cite{Wambach_1988}
Thanks to a careful diagrammatic analysis of the dynamical kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
Very recently, Loos and Blase have applied the dynamical correction to the BSE beyond the plasmon-pole approximation within a renormalized first-order perturbative treatment, \cite{Loos_2020e} generalizing the work of Rolhfing and coworkers on biological chromophores \cite{Ma_2009a,Ma_2009b} and dicyanovinyl-substituted oligothiophenes. \cite{Baumeier_2012b}
They compiled a comprehensive set of transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be quiet sizable and improve the static BSE excitations quite considerably. \cite{Loos_2020e}
They compiled a comprehensive set of vertical transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be sizable and improve the static BSE excitations considerably. \cite{Loos_2020e}
Within the so-called $GW$ approximation of MBPT, \cite{Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Assuming that $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is simply
@ -395,14 +397,6 @@ is the correlation part of the self-energy $\Sig{}$, and
\end{equation}
is the renormalization factor.
In Eq.~\eqref{eq:SigGW}, $\Omega = \Delta\e{} + 2 \ERI{vc}{cv}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
Numerically, we get
\begin{align}
\Omega & = 2.769\,327
&
\eGW{v} & = -0.863\,700
&
\eGW{c} & = +1.373\,640
\end{align}
One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988,Romaniello_2009b}
\begin{equation} \label{eq:HBSE}
@ -456,13 +450,13 @@ It can be easily shown that solving the equation
\end{equation}
yields 3 solutions per spin manifold (except for \ce{H2} where only 2 roots are observed), 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.
This evidences that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV except in the case of \ce{H2} where the agreement is rather satisfactory ($44.30$ eV at the dBSE level compared to the exact value of $44.04$ 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 except in the case of \ce{H2} where the agreement is rather satisfactory ($44.30$ eV at the dBSE level compared to the exact value of $44.04$ eV).
As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}).
Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^{\dBSE,\updw}$) and two spurious solutions for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$ and $\omega_{3}^{\dBSE,\upup}$).
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 two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
This makes these two latter solutions quite hard to locate with a method like Newton-Raphson (for example).
It is interesting to note 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 produce a systematic red-shifting of the static excitations, here we observe both blue- and red-shifted transitions.
%%% TABLE III %%%
\begin{table*}
@ -476,24 +470,24 @@ It is interesting to note that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
\cline{3-10}
System & Excitation & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
\hline
\ce{H2} & $\omega_1^{\updw}$ & 26.06 & 25.52[-0.54]& 26.06[+0.00]& 25.78[-0.28]& 27.02 & 27.02[+0.00]& 27.02[+0.00]& 26.34 \\
& $\omega_3^{\updw}$ & & & & 44.30 & & & & 44.04 \\
& $\omega_1^{\upup}$ & 16.94 & 17.10[+0.16]& 16.94[+0.00]& 17.03[+0.09]& 17.16 & 17.16[+0.00]& 17.16[+0.00]& 16.48 \\
& $\omega_3^{\upup}$ & & & & 43.61 & & & & \\
\ce{H2} & $\omega_1^{\updw}$ & $26.06$ & $25.52[-0.54]$ & $26.06[+0.00]$ & $25.78[-0.28]$ & $27.02$ & $27.02[+0.00]$ & $27.02[+0.00]$ & $26.34$ \\
& $\omega_3^{\updw}$ & & & & $44.30$ & & & & $44.04$ \\
& $\omega_1^{\upup}$ & $16.94$ & $17.10[+0.16]$ & $16.94[+0.00]$ & $17.03[+0.09]$ & $17.16$ & $17.16[+0.00]$ & $17.16[+0.00]$ & $16.48$ \\
& $\omega_3^{\upup}$ & & & & $43.61$ & & & & \\
\\
\ce{HeH+} & $\omega_1^{\updw}$ & 28.56 & 28.41[-0.15]& 28.63[+0.07]& 28.52[-0.04]& 29.04 & 29.11[+0.07]& 29.11[+0.07]& 28.05 \\
& $\omega_2^{\updw}$ & & & & 47.85 & & & & \\
& $\omega_3^{\updw}$ & & & & 87.47 & & & 87.47 & 64.09 \\
& $\omega_1^{\upup}$ & 20.96 & 21.16[+0.20]& 21.07[+0.11]& 21.12[+0.16]& 21.13 & 21.24[+0.11]& 21.24[+0.11]& 22.03 \\
& $\omega_2^{\upup}$ & & & & 47.54 & & & & \\
& $\omega_3^{\upup}$ & & & & 87.43 & & & 87.43 & \\
\ce{HeH+} & $\omega_1^{\updw}$ & $28.56$ & $28.41[-0.15]$ & $28.63[+0.07]$ & $28.52[-0.04]$ & $29.04$ & $29.11[+0.07]$ & $29.11[+0.07]$ & $28.05$ \\
& $\omega_2^{\updw}$ & & & & $47.85$ & & & & \\
& $\omega_3^{\updw}$ & & & & $87.47$ & & & $87.47$ & $64.09$ \\
& $\omega_1^{\upup}$ & $20.96$ & $21.16[+0.20]$ & $21.07[+0.11]$ & $21.12[+0.16]$ & $21.13$ & $21.24[+0.11]$ & $21.24[+0.11]$ & $22.03$ \\
& $\omega_2^{\upup}$ & & & & $47.54$ & & & & \\
& $\omega_3^{\upup}$ & & & & $87.43$ & & & $87.43$ & \\
\\
\ce{He} & $\omega_1^{\updw}$ & 52.46 & 51.71[-0.75]& 52.12[-0.33]& 51.85[-0.61]& 53.10 & 52.79[-0.31]& 52.79[-0.31]& 52.29 \\
& $\omega_2^{\updw}$ & & & & 75.75 & & & & \\
& $\omega_3^{\updw}$ & & & & 133.37 & & & 133.37 & 94.66 \\
& $\omega_1^{\upup}$ & 40.50 & 39.96[-0.53]& 39.80[-0.70]& 39.90[-0.60]& 40.71 & 40.02[-0.69]& 40.02[-0.69]& 40.18 \\
& $\omega_2^{\upup}$ & & & & 75.15 & & & & \\
& $\omega_3^{\upup}$ & & & & 133.76 & & & 133.75 & \\
\ce{He} & $\omega_1^{\updw}$ & $52.46$ & $51.71[-0.75]$ & $52.12[-0.33]$ & $51.85[-0.61]$ & $53.10$ & $52.79[-0.31]$ & $52.79[-0.31]$ & $52.29$ \\
& $\omega_2^{\updw}$ & & & & $75.75$ & & & & \\
& $\omega_3^{\updw}$ & & & & $133.37$ & & & $133.37$ & $94.66$ \\
& $\omega_1^{\upup}$ & $40.50$ & $39.96[-0.53]$ & $39.80[-0.70]$ & $39.90[-0.60]$ & $40.71$ & $40.02[-0.69]$ & $40.02[-0.69]$ & $40.18$ \\
& $\omega_2^{\upup}$ & & & & $75.15$ & & & & \\
& $\omega_3^{\upup}$ & & & & $133.76$ & & & $133.75$ & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
@ -527,7 +521,7 @@ Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in th
%%% %%% %%% %%%
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 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 $\omega_1^{\dBSE,\upup}$.
Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e} a scheme we label as perturbative BSE (pBSE).
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
@ -591,7 +585,7 @@ However, because the perturbative treatment is ultimately static, one cannot acc
\label{sec:BSE2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The third and final dynamical kernel that we consider here is the second-order BSE (BSE2) kernel as derived by Yang and collaborators in the TDA, \cite{Zhang_2013} and by Rebolini and Toulouse in a range-separated context \cite{Rebolini_2016,Rebolini_PhD} (see also Refs.~\onlinecite{Myohanen_2008,Sakkinen_2012}).
The third and final dynamical kernel that we consider here is the second-order BSE (BSE2) kernel derived by Yang and collaborators in the TDA, \cite{Zhang_2013} and by Rebolini and Toulouse in a range-separated context \cite{Rebolini_2016,Rebolini_PhD} (see also Refs.~\onlinecite{Myohanen_2008,Sakkinen_2012}).
Note that a beyond-TDA BSE2 kernel was also derived in Ref.~\onlinecite{Rebolini_2016}, but was not tested.
In a nutshell, the BSE2 scheme applies second-order perturbation theory to optical excitations within the Green's function framework by taking the functional derivative of the second-order self-energy $\SigGF{}$ with respect to the one-body Green's function.
Because $\SigGF{}$ is a proper functional derivative, it was claimed in Ref.~\onlinecite{Zhang_2013} that BSE2 does not produce spurious excitations.
@ -610,7 +604,7 @@ and
\begin{equation}
Z_{p}^{\GF} = \qty( 1 - \left. \pdv{\SigGF{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation}
The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$.
The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$, which shows that there is no screening within GF2 and BSE2.
The static Hamiltonian of BSE2 is just the usual TDHF Hamiltonian where one substitutes the HF orbital energies by the GF2 quasiparticle energies, \ie,
\begin{equation}
@ -656,7 +650,7 @@ and
f_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather}
\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, and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
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} and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
This latter point has an important consequence as this lack of frequency dependence removes one of the spurious pole (see Fig.~\ref{fig:BSE2}).
The singlet manifold has then the right number of excitations.
However, one spurious triplet excitation remains.
@ -672,7 +666,7 @@ In the case of BSE2, the perturbative partitioning is simply
%%% TABLE IV %%%
\begin{table*}
\caption{Singlet and triplet BSE2 excitation energies (in eV) for various levels of theory for \ce{He} at the HF/6-31G level.
\caption{Singlet and triplet BSE2 excitation energies (in eV) for various levels of theory and two-level systems.
The magnitude of the dynamical correction is reported in square brackets.
\label{tab:BSE2}
}
@ -680,42 +674,41 @@ In the case of BSE2, the perturbative partitioning is simply
\begin{tabular}{lcllllllll}
System & Excitation & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
\hline
\ce{H2} & $\omega_1^{\updw}$ & 26.03 & 26.03[+0.00]& 26.03[+0.00]& 26.24[+0.21]& 26.49 & 26.49[+0.00]& 26.49[+0.00]& 26.34 \\
& $\omega_1^{\upup}$ & 15.88 & 15.88[+0.00]& 15.88[+0.00]& 16.47[+0.59]& 16.63 & 16.63[+0.00]& 16.63[+0.00]& 16.48 \\
\ce{H2} & $\omega_1^{\updw}$ & $26.03$ & $26.03[+0.00$]& $26.03[+0.00]$& $26.24[+0.21]$& $26.49$ & $26.49[+0.00]$& $26.49[+0.00]$& $26.34$ \\
& $\omega_1^{\upup}$ & $15.88$ & $15.88[+0.00]$& $15.88[+0.00]$& $16.47[+0.59]$& $16.63$ & $16.63[+0.00]$& $16.63[+0.00]$& $16.48$ \\
\\
\ce{HeH+} & $\omega_1^{\updw}$ & 29.23 & 28.40[-0.83]& 28.40[-0.83]& 28.56[-0.67]& 29.50 & 28.66[-0.84]& 28.66[-0.84]& 28.05 \\
& $\omega_3^{\updw}$ & & & & 79.94 & & & 79.94 & 64.09 \\
& $\omega_1^{\upup}$ & 21.22 & 21.63[+0.41]& 21.63[+0.41]& 21.93[+0.71]& 21.59 & 21.99[+0.40]& 21.99[+0.40]& 22.03 \\
& $\omega_3^{\upup}$ & & & & 78.70 & & & 78.70 & \\
\ce{HeH+} & $\omega_1^{\updw}$ & $29.23$ & $28.40[-0.83]$& $28.40[-0.83]$& $28.56[-0.67]$& $29.50$ & $28.66[-0.84]$& $28.66[-0.84]$& $28.05$ \\
& $\omega_3^{\updw}$ & & & & $79.94$ & & & $79.94$ & $64.09$ \\
& $\omega_1^{\upup}$ & $21.22$ & $21.63[+0.41]$& $21.63[+0.41]$& $21.93[+0.71]$& $21.59$ & $21.99[+0.40]$& $21.99[+0.40]$& $22.03$ \\
& $\omega_3^{\upup}$ & & & & $78.70$ & & & $78.70$ & \\
\\
\ce{He} & $\omega_1^{\updw}$ & 50.31 & 51.96[+1.64]& 51.96[+1.65]& 52.10[+1.79]& 50.69 & 52.34[+1.65]& 52.34[+1.65]& 52.29 \\
& $\omega_3^{\updw}$ & & & & 121.67 & & & 121.66 & 94.66 \\
& $\omega_1^{\upup}$ & 37.80 & 39.26[+1.46]& 39.27[+1.47]& 39.59[+1.79]& 38.30 & 39.77[+1.47]& 39.77[+1.47]& 40.18 \\
& $\omega_3^{\upup}$ & & & & 121.85 & & & 121.84 & \\
\ce{He} & $\omega_1^{\updw}$ & $50.31$ & $51.96[+1.64]$& $51.96[+1.65]$& $52.10[+1.79]$& $50.69$ & $52.34[+1.65]$& $52.34[+1.65]$& $52.29$ \\
& $\omega_3^{\updw}$ & & & & $121.67$ & & & $121.66$ & $94.66$ \\
& $\omega_1^{\upup}$ & $37.80$ & $39.26[+1.46]$& $39.27[+1.47]$& $39.59[+1.79]$& $38.30$ & $39.77[+1.47]$& $39.77[+1.47]$& $40.18$ \\
& $\omega_3^{\upup}$ & & & & $121.85$ & & & $121.84$ & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
%%% %%% %%% %%%
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 (CIS or TDHF).
Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate.
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.
Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations, except for \ce{H2} (see below).
Overall, the accuracy of dBSE and dBSE2 are comparable (see Tables \ref{tab:BSE} and \ref{tab:BSE2}) for single excitations although their behavior is quite different.
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.
%%% FIGURE 4 %%%
\begin{figure}
\includegraphics[width=\linewidth]{dBSE2}
\caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds.
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
The static BSE2 Hamiltonian (dashed) and dynamic dBSE2 Hamiltonian (solid) are considered.
\label{fig:BSE2}
}
\end{figure}
%%% %%% %%% %%%
Again, the case of \ce{H2} is a bit particular as the perturbative treatment does not provide dynamical corrections, while its fully-dynamical version does yields sizable corrections and makes the singlet and triplet excitation energies very accurate.
This is to be expected as BSE2 takes into account second-order exchange which is omitted in BSE. \cite{Zhang_2013}
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 yields sizable corrections and makes the singlet and triplet excitation energies very accurate.
However, one cannot access the double excitation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -768,7 +761,7 @@ However, one cannot access the double excitation.
%\end{gather}
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\section{Take-home message}
\section{Take-home messages}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The take-home message of the present paper is that dynamical kernels have much more to give that one would think.
In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations).
@ -777,6 +770,8 @@ The appearance of these factitious excitations is due to the approximate nature
Moreover, because of the non-linear character of the linear response problem when one employs a dynamical kernel, it is computationally more involved to access these extra excitations.
Using a simple two-model system, we have explored the physics of three dynamical kernels: i) a kernel based on the dressed TDDFT method introduced by Maitra and coworkers, \cite{Maitra_2004} ii) the dynamical kernel from the BSE formalism derived by Strinati in his hallmark 1988 paper, \cite{Strinati_1988} as well as the second-order BSE kernel derived by Zhang \textit{et al.}, \cite{Zhang_2013} and Rebolini and Toulouse. \cite{Rebolini_2016,Rebolini_PhD}
Prototypical examples of valence, charge-transfer, and Rydberg excited states have been considered.
From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, if one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
We hope that the present contribution will foster new developments around dynamical kernels for optical excitations, in particular to access double excitations in molecular systems.