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BSEdyn.bib
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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 2020-06-19 15:03:36 +0200
%% Created for Pierre-Francois Loos at 2020-06-19 21:54:18 +0200
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@ -1242,15 +1242,10 @@
@article{Huix-Rotllant_2011,
Author = {{Huix-Rotllant}, Miquel and Ipatov, Andrei and Rubio, Angel and Casida, Mark E.},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-06-19 21:34:49 +0200},
Doi = {10.1016/j.chemphys.2011.03.019},
File = {/Users/loos/Zotero/storage/A4JUV4M4/Huix-Rotllant et al. - 2011 - Assessment of dressed time-dependent density-funct.pdf},
Issn = {03010104},
Journal = {Chem. Phys.},
Language = {en},
Month = nov,
Number = {1},
Pages = {120-129},
Pages = {120--129},
Title = {Assessment of Dressed Time-Dependent Density-Functional Theory for the Low-Lying Valence States of 28 Organic Chromophores},
Volume = {391},
Year = {2011},
@ -3539,14 +3534,10 @@
@article{Boggio-Pasqua_2007,
Author = {{Boggio-Pasqua}, Martial and Bearpark, Michael J. and Robb, Michael A.},
Date-Added = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-06-19 21:54:14 +0200},
Doi = {10.1021/jo070452v},
Issn = {0022-3263, 1520-6904},
Journal = {J. Org. Chem.},
Language = {en},
Month = jun,
Number = {12},
Pages = {4497-4503},
Pages = {4497--4503},
Title = {Toward a {{Mechanistic Understanding}} of the {{Photochromism}} of {{Dimethyldihydropyrenes}}},
Volume = {72},
Year = {2007},
@ -4149,15 +4140,10 @@
@article{Cave_2004,
Author = {Cave, Robert J. and Zhang, Fan and Maitra, Neepa T. and Burke, Kieron},
Date-Added = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-06-19 21:18:58 +0200},
Doi = {10.1016/j.cplett.2004.03.051},
File = {/Users/loos/Zotero/storage/6L9X6HT4/Cave et al. - 2004 - A dressed TDDFT treatment of the 21Ag states of bu.pdf},
Issn = {00092614},
Journal = {Chem. Phys. Lett.},
Language = {en},
Month = may,
Number = {1-3},
Pages = {39-42},
Pages = {39--42},
Title = {A Dressed {{TDDFT}} Treatment of the {{21Ag}} States of Butadiene and Hexatriene},
Volume = {389},
Year = {2004},

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Notes/BSEdyn-notes.tex
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@ -22,11 +22,11 @@
\affiliation{\LCPQ}
\begin{abstract}
We discuss the physical properties and accuracy of three distinct dynamical (\ie, frequency-dependent) kernels for the computation of excitation energies within linear response theory:
i) an \textit{a priori} built kernel inspired by the dressed time-dependent density-functional theory (TD-DFT) kernel proposed by Maitra and coworkers [\href{https://doi.org/10.1063/1.1651060}{J.~Chem.~Phys.~120, 5932 (2004)}],
We discuss the physical properties and accuracy of three distinct dynamical (\ie, frequency-dependent) kernels for the computation of optical excitations within linear response theory:
i) an \textit{a priori} built kernel inspired by the dressed time-dependent density-functional theory (TDDFT) kernel proposed by Maitra and coworkers [\href{https://doi.org/10.1063/1.1651060}{J.~Chem.~Phys.~120, 5932 (2004)}],
ii) the dynamical kernel stemming from the Bethe-Salpeter equation (BSE) formalism derived originally by Strinati [\href{https://doi.org/10.1007/BF02725962}{Riv.~Nuovo Cimento 11, 1--86 (1988)}], and
iii) the second-order BSE kernel derived by Yang and coworkers [\href{https://doi.org/10.1063/1.4824907}{J.~Chem.~Phys.~139, 154109 (2013)}].
In particular, using a simple two-level model, we analyze the appearance of spurious excitations, as first evidenced by Romaniello and collaborators [\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}], due to the approximate nature of the kernels.
In particular, using a simple two-level model, we analyze, for each kernel, the appearance of spurious excitations, as first evidenced by Romaniello and collaborators [\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}], due to the approximate nature of the kernels.
%\\
%\bigskip
%\begin{center}
@ -39,8 +39,9 @@ In particular, using a simple two-level model, we analyze the appearance of spur
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear response theory}
\label{sec:LR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Linear response 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{Casida_1995}
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{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{pmatrix}
@ -62,22 +63,40 @@ From a practical point of view, these quantities are obtained by solving non-lin
\end{pmatrix}
\end{equation}
where the explicit expressions of the resonant and coupling blocks, $\bR(\omega)$ and $\bC(\omega)$, depend on the level of approximation that one employs.
Neglecting the coupling block between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega_s)$, is known as the Tamm-Dancoff approximation (TDA).
Neglecting the coupling block between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega)$, is known as the Tamm-Dancoff approximation (TDA).
The non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$, and, thanks to its non-linear nature stemming from its frequency dependence, it potentially generates more than just single excitations.
In a wave function context, introducing a spatial orbital basis $\lbrace \MO{p} \rbrace$, we assume here that the elements of the matrices defined in Eq.~\eqref{eq:LR} read
In a wave function context, introducing a spatial orbital basis $\lbrace \MO{p} \rbrace$, we assume here that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic form:
\begin{subequations}
\begin{gather}
R_{ia,jb}(\omega) = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + 2 \sigma \ERI{ia}{jb} - \ERI{ib}{ja} + f_{ia,jb}^\sigma(\omega)
\begin{split}
R_{ia,jb}(\omega)
& = \iint \MO{i}(\br) \MO{a}(\br) \bR(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\\
& = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + 2 \sigma \ERI{ia}{jb} - \ERI{ib}{ja} + f_{ia,jb}^\sigma(\omega)
\end{split}
\\
C_{ia,jb}(\omega) = 2 \sigma \ERI{ia}{bj} - \ERI{ij}{ba} + f_{ia,bj}^\sigma(\omega)
\begin{split}
C_{ia,jb}(\omega)
& = \iint \MO{i}(\br) \MO{a}(\br) \bC(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\\
& = 2 \sigma \ERI{ia}{bj} - \ERI{ij}{ba} + f_{ia,bj}^\sigma(\omega)
\end{split}
\end{gather}
where $\sigma = 1 $ or $0$ for singlet ($\updw$) and triplet ($\upup$) excited states (respectively), $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $f(\omega)^{\sigma}$ is the correlation part of the spin-resolved kernel.
\end{subequations}
where $\sigma = 1 $ or $0$ for singlet ($\updw$) and triplet ($\upup$) excited states (respectively), and
\begin{equation}
\ERI{ia}{jb} = \iint \MO{i}(\br) \MO{a}(\br) \frac{1}{\abs{\br - \br'}} \MO{j}(\br') \MO{b}(\br') d\br d\br'
\end{equation}
are the usual (bare) two-electron integrals.
Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $f^{\sigma}(\omega)$ is the correlation part of the spin-resolved kernel.
(Note that, usually, only the correlation part of the kernel is frequency dependent.)
In the case of a spin-independent kernel, we will drop the superscrit $\sigma$.
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The concept of dynamical quantities}
\label{sec:dyn}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%s
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009b,Sangalli_2011,ReiningBook}
@ -89,7 +108,7 @@ In most cases, this can be done by solving a set of linear equations of the form
\end{equation}
where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
If we assume that the operator $\bA$ has a matrix representation of size $N \times N$, this \textit{linear} set of equations yields $N$ excitation energies.
However, in practice, $N$ might be very large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
However, in practice, $N$ might be (very) large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
\begin{equation}
\label{eq:lin_sys_split}
\begin{pmatrix}
@ -129,28 +148,34 @@ with
\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb
\end{equation}
which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{ReiningBook}
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the single-excitation basis can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{ReiningBook}
How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension.
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Garniron_2018,QP2}
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
However, because there is usually no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
For example, assuming that $\bA_2$ is a diagonal matrix is of common practice (see, for example, Ref.~\onlinecite{Garniron_2018} and references therein).
Another of these approximations is the so-called \textit{static} approximation, which corresponds to fixing the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism, \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
Another of these approximations is the so-called \textit{static} approximation, where one sets the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory (MBPT), \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
A similar example in the context of time-dependent density-functional theory (TD-DFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making the exchange-correlation (xc) kernel static (\ie, frequency-independent). \cite{Maitra_2016}
A similar example in the context of time-dependent density-functional theory (TDDFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making static the exchange-correlation (xc) kernel (\ie, frequency-independent). \cite{Maitra_2016}
These approximations come with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $N$ to $N_1$.
Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $N_1$ excitation energies are associated with single excitations.
Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the single-excitation basis cannot provide double excitations anymore, and the $N_1$ excitation energies are associated with single excitations.
All additional solutions associated with higher excitations have been forever lost.
In the next section, we illustrate these concepts and the various levels of approximation that can be used to recover some of these dynamical effects.
In the next section, we illustrate these concepts and the various tricks that can be used to recover some of these dynamical effects starting from the static eigenproblem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A two-level model}
\section{Dynamical kernels}
\label{sec:kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Exact Hamiltonian}
\label{sec:exact}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b} in its singlet ground state (\ie, the lowest orbital is doubly occupied).
@ -161,7 +186,7 @@ There is then only one single excitation possible which corresponds to the trans
As usual, this can produce a singlet singly-excited state $\ket{S} = (\ket{v\bar{c}} + \ket{c\bar{v}})/\sqrt{2}$, and a triplet singly-excited state $\ket{T} = (\ket{v\bar{c}} - \ket{c\bar{v}})/\sqrt{2}$. \cite{SzaboBook}
For the singlet manifold, the exact Hamiltonian in the basis of the (spin-adapted) configuration state functions reads
\begin{equation}
\begin{equation} \label{eq:H-exact}
\bH^{\updw} =
\begin{pmatrix}
\mel{0}{\hH}{0} & \mel{0}{\hH}{S} & \mel{0}{\hH}{D} \\
@ -170,32 +195,110 @@ For the singlet manifold, the exact Hamiltonian in the basis of the (spin-adapte
\end{pmatrix}
\end{equation}
with
\begin{gather}
\mel{0}{\hH}{0} \equiv \EHF = 2\e{v} - \ERI{vv}{vv}
\begin{subequations}
\begin{align}
\mel{0}{\hH}{0} & = 2\e{v} - \ERI{vv}{vv} = \EHF
\\
\mel{1}{\hH - \EHF}{1} = \Delta\e{} + \ERI{vc}{cv} - \ERI{vv}{cc}
\mel{S}{\hH - \EHF}{S} & = \Delta\e{} + \ERI{vc}{cv} - \ERI{vv}{cc}
\\
\mel{1}{\hH - \EHF}{1} = 2\Delta\e{} + \ERI{vv}{vv} + \ERI{cc}{cc} + 2\ERI{vc}{cv} - 4\ERI{vv}{cc}
\begin{split}
\mel{D}{\hH - \EHF}{D}
& = 2\Delta\e{} + \ERI{vv}{vv} + \ERI{cc}{cc}
\\
& + 2\ERI{vc}{cv} - 4\ERI{vv}{cc}
\end{split}
\\
\mel{0}{\hH - \EHF}{1} = 0
\mel{0}{\hH}{S} & = 0
\\
\mel{1}{\hH - \EHF}{2} = \sqrt{2}[\ERI{vc}{cc} - \ERI{cv}{vv}]
\mel{S}{\hH}{D} & = \sqrt{2}[\ERI{vc}{cc} - \ERI{cv}{vv}]
\\
\mel{0}{\hH - \EHF}{2} = \ERI{vc}{cv}
\end{gather}
\mel{0}{\hH}{D} & = \ERI{vc}{cv}
\end{align}
\end{subequations}
and $\Delta\e{} = \e{c} - \e{v}$.
The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
For the sake of illustration, we will use the same numerical example throughout this study, and consider the singlet ground state of the \ce{He} atom in Pople's 6-31G basis set.
This system contains two orbitals and the numerical values of the various quantities defined above are
\begin{subequations}
\begin{align}
\e{v} & = -0.914\,127
&
\e{c} & = + 1.399\,859
\\
\ERI{vv}{vv} & = 1.026\,907
&
\ERI{cc}{cc} & = 0.766\,363
\\
\ERI{vv}{cc} & = 0.858\,133
&
\ERI{vc}{cv} & = 0.227\,670
\\
\ERI{vv}{vc} & = 0.316\,490
&
\ERI{vc}{cc} & = 0.255\,554
\end{align}
\end{subequations}
This yields the following exact singlet and triplet excitation energies
\begin{align} \label{sec:exact}
\omega_{1}^{\updw} & = 1.92145
&
\omega_{3}^{\updw} & = 3.47880
&
\omega_{1}^{\upup} & = 1.47085
\end{align}
that we are going to use a reference for the remaining of this study.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Maitra's kernel}
\subsection{Maitra's dynamical kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The kernel proposed by Maitra in the context of dressed TD-DFT corresponds to a static kernel to which
where a frequency-dependent kernel is build \textit{a priori} and manually for a particular excitation
The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the context of dressed TDDFT (D-TDDFT) corresponds to an \textit{ad hoc} many-body theory correction to TDDFT.
More specifically, D-TDDFT adds manually to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian \eqref{eq:H-exact}.
The very same idea was taking further by Huix-Rotllant, Casida and coworkers. \cite{Huix-Rotllant_2011}
For the singlet states, we have
\begin{equation} \label{eq:f-Maitra}
f_M^{\updw}(\omega) = \frac{\abs*{\mel{S}{\hH}{D}}^2}{\omega - (\mel{D}{\hH}{D} - \mel{0}{\hH}{0}) }
\end{equation}
while $f_M^{\upup}(\omega) = 0$.
The expression \eqref{eq:f-Maitra} can be easily obtained by folding the double excitation onto the single excitation starting from the Hamiltonian \eqref{eq:H-exact}, as explained in Sec.~\ref{sec:dyn}.
It is clear that one must know \textit{a priori} the structure of the Hamiltonian to construct such dynamical kernel, and this obviously hampers its applicability to realistic photochemical systems where it is sometimes hard to get a clear picture of the interplay between excited states. \cite{Boggio-Pasqua_2007}
For the two-level model, the non-linear equations defined in Eq.~\eqref{eq:LR} provides the following effective Hamiltonian
\begin{equation} \label{eq:H-M}
\bH_{M}(\omega) =
\begin{pmatrix}
R_M(\omega) & C_M(\omega)
\\
-C_M(-\omega) & -R_M(-\omega)
\end{pmatrix}
\end{equation}
with
\begin{subequations}
\begin{gather}
R_M(\omega) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - \ERI{vc}{vc} + f_M^{\sigma}(\omega)
\\
C_M(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_M^{\sigma}(\omega)
\end{gather}
\end{subequations}
which provides the following excitation energies when diagonalized:
\begin{align} \label{sec:M}
\omega_{1}^{\updw} & = 1.89314
&
\omega_{3}^{\updw} & = 3.44865
&
\omega_{1}^{\upup} & = 1.43794
\end{align}
Although not particularly accurate, this kernel provides exactly the right number of solutions (2 singlets and 1 triplet).
Its accuracy could be certainly improved in a DFT context.
However, this is not the point of the present investigation.
Because $f_M^{\upup}(\omega) = 0$, the triplet excitation energy is equivalent to the TDHF excitation energy.
In the static approximation where $f_M^{\updw}(\omega) = 0$, the singlet excitations are also TDHF excitation energies.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dynamical BSE kernel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Within many-body perturbation theory (MBPT), one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Within MBPT, one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is
\begin{equation}
\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
@ -209,11 +312,7 @@ are the correlation parts of the self-energy associated with wither the valence
\begin{equation}
Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
\end{equation}
is the renormalization factor, and
\begin{equation}
\ERI{pq}{rs} = \iint \MO{p}(\br) \MO{q}(\br) \frac{1}{\abs{\br - \br'}} \MO{r}(\br') \MO{s}(\br') d\br d\br'
\end{equation}
are the usual (bare) two-electron integrals.
is the renormalization factor.
In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\eGW{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
@ -298,25 +397,7 @@ are the eigenvectors of $\bH^{\BSE}$, and
\end{equation}
This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following.
We now take a numerical example by considering the singlet ground state of the \ce{He} atom in the 6-31G basis set.
This system contains two orbitals and the numerical values of the various quantities defined above are
\begin{align}
\e{v} & = -0.914\,127
&
\e{c} & = + 1.399\,859
\\
\ERI{vv}{vv} & = 1.026\,907
&
\ERI{cc}{cc} & = 0.766\,363
\\
\ERI{vv}{cc} & = 0.858\,133
&
\ERI{vc}{cv} & = 0.227\,670
\\
\ERI{vv}{vc} & = 0.316\,490
&
\ERI{vc}{cc} & = 0.255\,554
\end{align}
which yields
\begin{align}
\Omega & = 2.769\,327
@ -395,17 +476,9 @@ while the static values are
\omega_{1,\upup}^{\TDABSE} & = 1.49603
\end{align}
It is now instructive to provide the exact results, \ie, the excitation energies obtained by diagonalizing the exact Hamiltonian in the same basis set.
A quick configuration interaction with singles and doubles (CISD) calculation provide the following excitation energies:
\begin{align}
\omega_{1}^{\updw} & = 1.92145
&
\omega_{1}^{\upup} & = 1.47085
&
\omega_{3}^{\updw} & = 3.47880
\end{align}
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
All these numerical results are gathered in Table \ref{tab:BSE}.
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
A
The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value.
Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.

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