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dBSE2.pdf
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dynker.bib
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%% 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 2020-07-20 08:52:20 +0200 %% Created for Pierre-Francois Loos at 2020-08-22 14:25:29 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -17,7 +17,8 @@
Pages = {12--33}, Pages = {12--33},
Title = {Studies in perturbation theory: Part I. An elementary iteration-variation procedure for solving the Schr{\"o}dinger equation by partitioning technique}, Title = {Studies in perturbation theory: Part I. An elementary iteration-variation procedure for solving the Schr{\"o}dinger equation by partitioning technique},
Volume = {10}, Volume = {10},
Year = {1963}} Year = {1963},
Bdsk-Url-1 = {https://doi.org/10.1016/0022-2852(63)90151-6}}
@article{Petersilka_1996, @article{Petersilka_1996,
Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross}, Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
@ -65,13 +66,15 @@
Year = {2011}, Year = {2011},
Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}} Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
@article{Loos_2020e, @article{Blase_2020,
Author = {X. Blase and Y. Duchemin and D. Jacquemin}, Author = {X. Blase and Y. Duchemin and D. Jacquemin},
Date-Added = {2020-06-22 09:07:38 +0200}, Date-Added = {2020-06-22 09:07:38 +0200},
Date-Modified = {2020-06-22 09:08:39 +0200}, Date-Modified = {2020-08-22 14:13:11 +0200},
Doi = {10.1021/acs.jpclett.0c01875},
Journal = {J. Phys. Chem. Lett.}, Journal = {J. Phys. Chem. Lett.},
Title = {The Bethe-Salpeter Equation Formalism: From Physics to Chemistry}, Title = {The Bethe-Salpeter Equation Formalism: From Physics to Chemistry},
Year = {submitted}} Year = {in press},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c01875}}
@article{Loos_2020d, @article{Loos_2020d,
Author = {P. F. Loos and E. Fromager}, Author = {P. F. Loos and E. Fromager},

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dynker.tex
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-2} \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem} \usepackage[version=4]{mhchem}
@ -20,6 +20,9 @@
\author{Pierre-Fran\c{c}ois \surname{Loos}} \author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Juliette \surname{Authier}}
\affiliation{\LCPQ}
%\author{Friends}
\begin{abstract} \begin{abstract}
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: 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:
@ -94,7 +97,7 @@ As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of
\end{equation} \end{equation}
where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively), and where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively), and
\begin{equation} \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' \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.
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.
@ -110,7 +113,7 @@ To do so, let us consider the usual chemical scenario where one wants to get the
In most cases, this can be done by solving a set of linear equations of the form In most cases, this can be done by solving a set of linear equations of the form
\begin{equation} \begin{equation}
\label{eq:lin_sys} \label{eq:lin_sys}
\bA \cdot \bc = \omega_S \, \bc \bA \cdot \bc = \omega \, \bc
\end{equation} \end{equation}
where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector . where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
If we assume that the matrix $\bA$ is diagonalizable and of size $N \times N$, the \textit{linear} set of equations \eqref{eq:lin_sys} yields $N$ excitation energies. If we assume that the matrix $\bA$ is diagonalizable and of size $N \times N$, the \textit{linear} set of equations \eqref{eq:lin_sys} yields $N$ excitation energies.
@ -346,7 +349,7 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
\label{sec:BSE} \label{sec:BSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As mentioned in Sec.~\ref{sec:dyn}, most of BSE calculations performed nowadays are done within the static approximation. \cite{ReiningBook,Loos_2020e} 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} 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}
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. 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.
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} 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}
@ -607,7 +610,7 @@ with
\begin{gather} \begin{gather}
f_{\dBSE2}^{\co,\updw}(\omega) = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}} f_{\dBSE2}^{\co,\updw}(\omega) = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\ \\
f_{\dBSE2}^{\co,\updw} = - \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}} f_{\dBSE2}^{\co,\updw} = \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and and
@ -615,7 +618,7 @@ and
\begin{gather} \begin{gather}
f_{\dBSE2}^{\co,\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}} f_{\dBSE2}^{\co,\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
\\ \\
f_{\dBSE2}^{\co,\upup} = - \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}} f_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather} \end{gather}
\end{subequations} \end{subequations}
Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets, 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, and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
@ -641,15 +644,15 @@ In the case of BSE2, the perturbative partitioning is simply
\begin{tabular}{|c|ccccccc|c|} \begin{tabular}{|c|ccccccc|c|}
Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\ Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
\hline \hline
$\omega_1^{\updw}$ & 1.84903 & 1.90940 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\ $\omega_1^{\updw}$ & 1.84903 & 1.90940 & 1.90950 & 1.91454 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
$\omega_2^{\updw}$ & & & & & & & & \\ $\omega_2^{\updw}$ & & & & & & & & \\
$\omega_3^{\updw}$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\ $\omega_3^{\updw}$ & & & & 4.47109 & & & 4.47097 & 3.47880 \\
\hline \hline
Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\ Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
\hline \hline
$\omega_1^{\upup}$ & 1.38912 & 1.44285 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\ $\omega_1^{\upup}$ & 1.38912 & 1.44285 & 1.44304 & 1.45489 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
$\omega_2^{\upup}$ & & & & & & & & \\ $\omega_2^{\upup}$ & & & & & & & & \\
$\omega_3^{\upup}$ & & & & 4.47797 & & & 4.47767 & \\ $\omega_3^{\upup}$ & & & & 4.47773 & & & 4.47767 & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
@ -728,10 +731,14 @@ However, they sometimes give too much, and generate spurious excitations, \ie, e
The appearance of these factitious excitations is due to the approximate nature of the dynamical kernel. The appearance of these factitious excitations is due to the approximate nature of the dynamical kernel.
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. 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 Yang and coworkers, and Rebolini and Toulouse. \cite{Rebolini_2016,Rebolini_PhD}
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.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{ \acknowledgements{
The author thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support. PFL would like to thank Xavier Blase, Elisa Rebolini, Pina Romaniello, Arjan Berger, Miquel Huix-Rotllant and Julien Toulouse for insightful discussions on dynamical kernels.
He also thanks Xavier Blase and Juliette Authier for numerous insightful discussions on dynamical kernels.} PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
% BIBLIOGRAPHY % BIBLIOGRAPHY