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dynker.bib
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@ -1,13 +1,24 @@
%% 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 2020-08-27 16:42:19 +0200 %% Created for Pierre-Francois Loos at 2020-08-27 18:05:25 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Teh_2019,
Author = {H.-H. Teh and J. E. Subotnik},
Date-Added = {2020-08-27 17:31:44 +0200},
Date-Modified = {2020-08-27 17:33:14 +0200},
Doi = {10.1021/acs.jpclett.9b00981},
Journal = {J. Phys. Chem. Lett.},
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}}
@article{Loos_2020e, @article{Loos_2020e,
Author = {P. F. Loos and X. Blase}, Author = {P. F. Loos and X. Blase},
Date-Added = {2020-08-27 16:03:48 +0200}, Date-Added = {2020-08-27 16:03:48 +0200},
@ -5444,7 +5455,8 @@
Pages = {064103}, Pages = {064103},
Title = {Selected Configuration Interaction Dressed by Perturbation}, Title = {Selected Configuration Interaction Dressed by Perturbation},
Volume = {149}, Volume = {149},
Year = {2018}} Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
@article{Garziano_2016, @article{Garziano_2016,
Author = {Garziano, Luigi and Macr\`i, Vincenzo and Stassi, Roberto and Di Stefano, Omar and Nori, Franco and Savasta, Salvatore}, Author = {Garziano, Luigi and Macr\`i, Vincenzo and Stassi, Roberto and Di Stefano, Omar and Nori, Franco and Savasta, Salvatore},

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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
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 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)}]. 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, 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. 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.
Prototypical examples of valence, charge-transfer, and Rydberg excited states are considered.
%\\ %\\
%\bigskip %\bigskip
%\begin{center} %\begin{center}
@ -43,7 +44,7 @@ In particular, using a simple two-level model, we analyze, for each kernel, the
\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 $f_S$ [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}
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}
@ -83,7 +84,7 @@ where $\delta_{pq}$ is the Kronecker delta, $\e{p}$ is the one-electron (or quas
f_{ia,jb}^{\Hxc,\sigma}(\omega) f_{ia,jb}^{\Hxc,\sigma}(\omega)
= \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br' = \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
\end{equation} \end{equation}
Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $p$ and $q$ indicate arbitrary orbitals. Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
In Eq.~\eqref{eq:kernel}, In Eq.~\eqref{eq:kernel},
\begin{equation} \label{eq:kernel-Hxc} \begin{equation} \label{eq:kernel-Hxc}
f^{\Hxc,\sigma}(\omega) = f^{\Hx,\sigma} + f^{\co,\sigma}(\omega) f^{\Hxc,\sigma}(\omega) = f^{\Hx,\sigma} + f^{\co,\sigma}(\omega)
@ -98,7 +99,7 @@ where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively)
\begin{equation} \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' \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} \end{equation}
are the usual two-electron integrals. are the usual two-electron integrals. \cite{Gill_1994}
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. 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.
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript. Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
@ -195,7 +196,7 @@ The ground state $\ket{0}$ has a one-electron configuration $\ket{v\bar{v}}$, wh
There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations. There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations.
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} 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}
For the singlet manifold, the exact Hamiltonian in the basis of these (spin-adapted) configuration state functions reads For the singlet manifold, the exact Hamiltonian in the basis of these (spin-adapted) configuration state functions reads \cite{Teh_2019}
\begin{equation} \label{eq:H-exact} \begin{equation} \label{eq:H-exact}
\bH^{\updw} = \bH^{\updw} =
\begin{pmatrix} \begin{pmatrix}
@ -234,7 +235,7 @@ The numerical values of the various quantities defined above are gathered in Tab
%%% TABLE I %%% %%% TABLE I %%%
\begin{table*} \begin{table*}
\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. \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.
\label{tab:params} \label{tab:params}
} }
\begin{ruledtabular} \begin{ruledtabular}
@ -280,7 +281,7 @@ The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the
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. 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.
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. 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.
Here, we start instead from a HF reference. Here, we start instead from a HF reference.
The static problem corresponds then to the TDHF Hamiltonian, while in the TDA, it reduces to CIS. The static problem corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS).
For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields
\begin{equation} \label{eq:f-Maitra} \begin{equation} \label{eq:f-Maitra}
@ -309,16 +310,16 @@ with
C_\text{M}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_\text{M}^{\co,\sigma}(\omega) C_\text{M}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_\text{M}^{\co,\sigma}(\omega)
\end{gather} \end{gather}
\end{subequations} \end{subequations}
yielding the excitation energies reported in Table \ref{tab:Maitra} when diagonalized. yielding, for our three two-electron systems, the excitation energies reported in Table \ref{tab:Maitra} when diagonalized.
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}. 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}.
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 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+}.
The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies. The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies.
Because, there is nothing to dress for the triplet state, only the static TDHF excitation energy is reported. Because, there is nothing to dress for the triplet state, only the TDHF and D-TDHF triplet excitation energies are equal.
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{Singlet and triplet excitation energies (in eV) at various levels of theory and two-level systems. \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 between square brackets. The magnitude of the dynamical correction is reported in square brackets.
\label{tab:Maitra} \label{tab:Maitra}
} }
\begin{ruledtabular} \begin{ruledtabular}
@ -379,7 +380,7 @@ Very recently, Loos and Blase have applied the dynamical correction to the BSE b
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} 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}
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. 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 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
\begin{equation} \begin{equation}
\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p}) \e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
\end{equation} \end{equation}
@ -420,7 +421,7 @@ with
C_{\dBSE}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vc}{cv} - W^{\co}_C(\omega) C_{\dBSE}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vc}{cv} - W^{\co}_C(\omega)
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and and where
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}} W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
@ -461,11 +462,12 @@ Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two
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 $\omega_{3}^{\dBSE,\upup}$). 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. 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). 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.
%%% TABLE III %%% %%% TABLE III %%%
\begin{table*} \begin{table*}
\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory and two-level systems. \caption{Singlet and triplet BSE excitation energies (in eV) for various levels of theory and two-level systems.
The magnitude of the dynamical correction is reported between square brackets. The magnitude of the dynamical correction is reported in square brackets.
\label{tab:BSE} \label{tab:BSE}
} }
\begin{ruledtabular} \begin{ruledtabular}
@ -474,9 +476,9 @@ This makes these two latter solutions quite hard to locate with a method like Ne
\cline{3-10} \cline{3-10}
System & Excitation & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\ System & Excitation & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
\hline \hline
\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 \\ \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 \\ & $\omega_3^{\updw}$ & & & & 44.30 & & & & 44.04 \\
& $\omega_1^{\upup}$ & 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 \\ & $\omega_1^{\upup}$ & 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 & & & & \\ & $\omega_3^{\upup}$ & & & & 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 \\ \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 \\
@ -534,7 +536,7 @@ To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static p
= \underbrace{\bH_{\BSE}^{\sigma}}_{\bH_{\pBSE}^{(0)}} = \underbrace{\bH_{\BSE}^{\sigma}}_{\bH_{\pBSE}^{(0)}}
+ \underbrace{\qty[ \bH_{\dBSE}^{\sigma}(\omega) - \bH_{\BSE}^{\sigma} ]}_{\bH_{\pBSE}^{(1)}(\omega)} + \underbrace{\qty[ \bH_{\dBSE}^{\sigma}(\omega) - \bH_{\BSE}^{\sigma} ]}_{\bH_{\pBSE}^{(1)}(\omega)}
\end{equation} \end{equation}
Thanks to (renormalized) first-order perturbation theory, one gets Thanks to (renormalized) first-order perturbation theory, \cite{Loos_2020e} one gets
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\omega_{1}^{\pBSE,\sigma} \omega_{1}^{\pBSE,\sigma}
@ -580,9 +582,8 @@ and the renormalization factor is
This corresponds to a dynamical perturbative correction to the static excitations. This corresponds to a dynamical perturbative correction to the static excitations.
Obviously, the TDA can be applied to the dynamical correction as well, a scheme we label as dTDA in the following. Obviously, the TDA can be applied to the dynamical correction as well, a scheme we label as dTDA in the following.
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. 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).
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.
%Note that, although the pBSE(dTDA) value is further from the dBSE value than pBSE, it is quite close to the exact excitation energy.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -671,8 +672,8 @@ In the case of BSE2, the perturbative partitioning is simply
%%% TABLE IV %%% %%% TABLE IV %%%
\begin{table*} \begin{table*}
\caption{Singlet and triplet BSE2 excitation energies (in eV) at 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 for \ce{He} at the HF/6-31G level.
The magnitude of the dynamical correction is reported between square brackets. The magnitude of the dynamical correction is reported in square brackets.
\label{tab:BSE2} \label{tab:BSE2}
} }
\begin{ruledtabular} \begin{ruledtabular}