overall screening OK
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dynker.bib
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dynker.bib
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-08-27 16:42:19 +0200
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%% Created for Pierre-Francois Loos at 2020-08-27 18:05:25 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Teh_2019,
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Author = {H.-H. Teh and J. E. Subotnik},
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Date-Added = {2020-08-27 17:31:44 +0200},
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Date-Modified = {2020-08-27 17:33:14 +0200},
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Doi = {10.1021/acs.jpclett.9b00981},
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Journal = {J. Phys. Chem. Lett.},
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Pages = {3426--3432},
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Title = {The Simplest Possible Approach for Simulating S0−S1 Conical Intersections with DFT/TDDFT: Adding One Doubly Excited Configuration},
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Volume = {10},
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Year = {2019}}
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@article{Loos_2020e,
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Author = {P. F. Loos and X. Blase},
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Date-Added = {2020-08-27 16:03:48 +0200},
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@ -5444,7 +5455,8 @@
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Pages = {064103},
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Title = {Selected Configuration Interaction Dressed by Perturbation},
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Volume = {149},
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Year = {2018}}
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Year = {2018},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
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@article{Garziano_2016,
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Author = {Garziano, Luigi and Macr\`i, Vincenzo and Stassi, Roberto and Di Stefano, Omar and Nori, Franco and Savasta, Salvatore},
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45
dynker.tex
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dynker.tex
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@ -29,6 +29,7 @@ i) an \textit{a priori} built kernel inspired by the dressed time-dependent dens
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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
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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)}].
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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.
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Prototypical examples of valence, charge-transfer, and Rydberg excited states are considered.
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%\\
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%\bigskip
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%\begin{center}
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@ -43,7 +44,7 @@ In particular, using a simple two-level model, we analyze, for each kernel, the
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\section{Linear response theory}
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\label{sec:LR}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
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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}
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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}
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\begin{equation} \label{eq:LR}
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\begin{pmatrix}
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@ -83,7 +84,7 @@ where $\delta_{pq}$ is the Kronecker delta, $\e{p}$ is the one-electron (or quas
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f_{ia,jb}^{\Hxc,\sigma}(\omega)
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= \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
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\end{equation}
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Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $p$ and $q$ indicate arbitrary orbitals.
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Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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In Eq.~\eqref{eq:kernel},
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\begin{equation} \label{eq:kernel-Hxc}
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f^{\Hxc,\sigma}(\omega) = f^{\Hx,\sigma} + f^{\co,\sigma}(\omega)
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@ -98,7 +99,7 @@ where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively)
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\begin{equation}
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\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'
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\end{equation}
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are the usual two-electron integrals.
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are the usual two-electron integrals. \cite{Gill_1994}
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The launchpad of the present study is that, thanks to its non-linear nature stemming from its frequency dependence, a dynamical kernel potentially generates more than just single excitations.
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Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
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@ -195,7 +196,7 @@ The ground state $\ket{0}$ has a one-electron configuration $\ket{v\bar{v}}$, wh
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There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations.
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As usual, this produces 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}
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For the singlet manifold, the exact Hamiltonian in the basis of these (spin-adapted) configuration state functions reads
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For the singlet manifold, the exact Hamiltonian in the basis of these (spin-adapted) configuration state functions reads \cite{Teh_2019}
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\begin{equation} \label{eq:H-exact}
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\bH^{\updw} =
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\begin{pmatrix}
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@ -234,7 +235,7 @@ The numerical values of the various quantities defined above are gathered in Tab
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%%% TABLE I %%%
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\begin{table*}
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\caption{Numerical values (in hartree) of the energy of the valence and conduction orbitals, $\e{v}$ and $\e{c}$, and various two-electron integrals in the orbital basis for various two-level systems.
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\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.
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\label{tab:params}
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}
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\begin{ruledtabular}
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@ -280,7 +281,7 @@ The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the
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More specifically, D-TDDFT adds to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian: a single and double excitations, assumed to be strongly coupled, are isolated from among the spectrum and added manually to the static kernel.
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The very same idea was taking further by Huix-Rotllant, Casida and coworkers, \cite{Huix-Rotllant_2011} and tested on a large set of molecules.
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Here, we start instead from a HF reference.
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The static problem corresponds then to the TDHF Hamiltonian, while in the TDA, it reduces to CIS.
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The static problem corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS).
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For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields
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\begin{equation} \label{eq:f-Maitra}
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@ -309,16 +310,16 @@ with
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C_\text{M}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_\text{M}^{\co,\sigma}(\omega)
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\end{gather}
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\end{subequations}
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yielding the excitation energies reported in Table \ref{tab:Maitra} when diagonalized.
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yielding, for our three two-electron systems, the excitation energies reported in Table \ref{tab:Maitra} when diagonalized.
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The TDHF Hamiltonian is obtained from Eq.~\eqref{eq:H-M} by setting $f_\text{M}^{\co,\sigma}(\omega) = 0$ in Eqs.~\eqref{eq:R_M} and \eqref{eq:C_M}.
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In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds.
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In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds in \ce{HeH+}.
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The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies.
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Because, there is nothing to dress for the triplet state, only the static TDHF excitation energy is reported.
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Because, there is nothing to dress for the triplet state, only the TDHF and D-TDHF triplet excitation energies are equal.
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%%% TABLE II %%%
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\begin{table}
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\caption{Singlet and triplet excitation energies (in eV) at various levels of theory and two-level systems.
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The magnitude of the dynamical correction is reported between square brackets.
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\caption{Singlet and triplet excitation energies (in eV) for various levels of theory and two-level systems.
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The magnitude of the dynamical correction is reported in square brackets.
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\label{tab:Maitra}
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}
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\begin{ruledtabular}
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@ -379,7 +380,7 @@ Very recently, Loos and Blase have applied the dynamical correction to the BSE b
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They compiled a comprehensive set of transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be quiet sizeable and improve the static BSE excitations quite considerably. \cite{Loos_2020e}
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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.
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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
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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
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\begin{equation}
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\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
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\end{equation}
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@ -420,7 +421,7 @@ with
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C_{\dBSE}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vc}{cv} - W^{\co}_C(\omega)
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\end{gather}
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\end{subequations}
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and
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and where
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\begin{subequations}
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\begin{gather}
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W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
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@ -461,11 +462,12 @@ Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two
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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 $\omega_{3}^{\dBSE,\upup}$).
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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.
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This makes these two latter solutions quite hard to locate with a method like Newton-Raphson (for example).
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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.
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%%% TABLE III %%%
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\begin{table*}
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\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory and two-level systems.
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The magnitude of the dynamical correction is reported between square brackets.
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\caption{Singlet and triplet BSE excitation energies (in eV) for various levels of theory and two-level systems.
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The magnitude of the dynamical correction is reported in square brackets.
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\label{tab:BSE}
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}
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\begin{ruledtabular}
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@ -474,9 +476,9 @@ This makes these two latter solutions quite hard to locate with a method like Ne
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\cline{3-10}
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System & Excitation & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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\ce{H2} & $\omega_1^{\updw}$ & 26.06 & 25.52[-0.54]& 26.06[+0.00]& 25.78[-0.04]& 27.02 & 27.02[+0.00]& 27.02[+0.00]& 26.34 \\
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\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 \\
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& $\omega_3^{\updw}$ & & & & 44.30 & & & & 44.04 \\
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& $\omega_1^{\upup}$ & 16.94 & 17.10[+0.16]& 16.94[+0.00]& 17.03[+0.16]& 17.16 & 17.16[+0.00]& 17.16[+0.00]& 16.48 \\
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& $\omega_1^{\upup}$ & 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 \\
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& $\omega_3^{\upup}$ & & & & 43.61 & & & & \\
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\\
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\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 \\
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@ -534,7 +536,7 @@ To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static p
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= \underbrace{\bH_{\BSE}^{\sigma}}_{\bH_{\pBSE}^{(0)}}
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+ \underbrace{\qty[ \bH_{\dBSE}^{\sigma}(\omega) - \bH_{\BSE}^{\sigma} ]}_{\bH_{\pBSE}^{(1)}(\omega)}
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\end{equation}
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Thanks to (renormalized) first-order perturbation theory, one gets
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Thanks to (renormalized) first-order perturbation theory, \cite{Loos_2020e} one gets
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\begin{equation}
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\begin{split}
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\omega_{1}^{\pBSE,\sigma}
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@ -580,9 +582,8 @@ and the renormalization factor is
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This corresponds to a dynamical perturbative correction to the static excitations.
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Obviously, the TDA can be applied to the dynamical correction as well, a scheme we label as dTDA in the following.
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The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very efficient at reproducing the dynamical value for the single excitations.
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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, especially in the TDA where pBSE(TDA) is an excellent approximation of dBSE(TDA).
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However, because the perturbative treatment is ultimately static, one cannot access double excitations with such a scheme.
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%Note that, although the pBSE(dTDA) value is further from the dBSE value than pBSE, it is quite close to the exact excitation energy.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -671,8 +672,8 @@ In the case of BSE2, the perturbative partitioning is simply
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%%% TABLE IV %%%
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\begin{table*}
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\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{He} at the HF/6-31G level.
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The magnitude of the dynamical correction is reported between square brackets.
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\caption{Singlet and triplet BSE2 excitation energies (in eV) for various levels of theory for \ce{He} at the HF/6-31G level.
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The magnitude of the dynamical correction is reported in square brackets.
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\label{tab:BSE2}
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}
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\begin{ruledtabular}
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