conclusion
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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 2020-07-20 08:52:20 +0200
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%% Created for Pierre-Francois Loos at 2020-08-22 14:25:29 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -17,7 +17,8 @@
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Pages = {12--33},
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Title = {Studies in perturbation theory: Part I. An elementary iteration-variation procedure for solving the Schr{\"o}dinger equation by partitioning technique},
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Volume = {10},
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Year = {1963}}
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Year = {1963},
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Bdsk-Url-1 = {https://doi.org/10.1016/0022-2852(63)90151-6}}
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@article{Petersilka_1996,
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Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
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@ -65,13 +66,15 @@
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Year = {2011},
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Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
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@article{Loos_2020e,
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@article{Blase_2020,
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Author = {X. Blase and Y. Duchemin and D. Jacquemin},
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Date-Added = {2020-06-22 09:07:38 +0200},
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Date-Modified = {2020-06-22 09:08:39 +0200},
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Date-Modified = {2020-08-22 14:13:11 +0200},
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Doi = {10.1021/acs.jpclett.0c01875},
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Journal = {J. Phys. Chem. Lett.},
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Title = {The Bethe-Salpeter Equation Formalism: From Physics to Chemistry},
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Year = {submitted}}
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Year = {in press},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.0c01875}}
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@article{Loos_2020d,
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Author = {P. F. Loos and E. Fromager},
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dynker.tex
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dynker.tex
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-2}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
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\usepackage[version=4]{mhchem}
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@ -20,6 +20,9 @@
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Juliette \surname{Authier}}
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\affiliation{\LCPQ}
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%\author{Friends}
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\begin{abstract}
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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:
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@ -94,7 +97,7 @@ As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of
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\end{equation}
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where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively), and
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\begin{equation}
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\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'
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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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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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@ -110,7 +113,7 @@ To do so, let us consider the usual chemical scenario where one wants to get the
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In most cases, this can be done by solving a set of linear equations of the form
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\begin{equation}
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\label{eq:lin_sys}
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\bA \cdot \bc = \omega_S \, \bc
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\bA \cdot \bc = \omega \, \bc
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\end{equation}
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where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
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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.
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@ -346,7 +349,7 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
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\label{sec:BSE}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As mentioned in Sec.~\ref{sec:dyn}, most of BSE calculations performed nowadays are done within the static approximation. \cite{ReiningBook,Loos_2020e}
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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}
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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}
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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.
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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}
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@ -607,7 +610,7 @@ with
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\begin{gather}
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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{}}
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\\
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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{}}
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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{}}
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\end{gather}
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\end{subequations}
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and
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@ -615,7 +618,7 @@ and
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\begin{gather}
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f_{\dBSE2}^{\co,\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
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\\
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f_{\dBSE2}^{\co,\upup} = - \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
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f_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
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\end{gather}
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\end{subequations}
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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.
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@ -641,15 +644,15 @@ In the case of BSE2, the perturbative partitioning is simply
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\updw}$ & 1.84903 & 1.90940 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
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$\omega_1^{\updw}$ & 1.84903 & 1.90940 & 1.90950 & 1.91454 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
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$\omega_2^{\updw}$ & & & & & & & & \\
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$\omega_3^{\updw}$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
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$\omega_3^{\updw}$ & & & & 4.47109 & & & 4.47097 & 3.47880 \\
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\hline
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Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 1.38912 & 1.44285 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
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$\omega_1^{\upup}$ & 1.38912 & 1.44285 & 1.44304 & 1.45489 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
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$\omega_2^{\upup}$ & & & & & & & & \\
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$\omega_3^{\upup}$ & & & & 4.47797 & & & 4.47767 & \\
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$\omega_3^{\upup}$ & & & & 4.47773 & & & 4.47767 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -728,10 +731,14 @@ However, they sometimes give too much, and generate spurious excitations, \ie, e
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The appearance of these factitious excitations is due to the approximate nature of the dynamical kernel.
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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.
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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}
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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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.
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He also thanks Xavier Blase and Juliette Authier for numerous insightful discussions on dynamical kernels.}
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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.
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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.}
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%%%%%%%%%%%%%%%%%%%%%%%%
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% BIBLIOGRAPHY
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