diff --git a/dBSE2.pdf b/dBSE2.pdf index dc251ad..d5ae500 100644 Binary files a/dBSE2.pdf and b/dBSE2.pdf differ diff --git a/dynker.bib b/dynker.bib index 75dd1a9..98dac08 100644 --- a/dynker.bib +++ b/dynker.bib @@ -1,7 +1,7 @@ %% This BibTeX bibliography file was created using BibDesk. %% 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) @@ -17,7 +17,8 @@ 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}, Volume = {10}, - Year = {1963}} + Year = {1963}, + Bdsk-Url-1 = {https://doi.org/10.1016/0022-2852(63)90151-6}} @article{Petersilka_1996, Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross}, @@ -65,13 +66,15 @@ Year = {2011}, 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}, 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.}, 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, Author = {P. F. Loos and E. Fromager}, diff --git a/dynker.tex b/dynker.tex index fff9bec..aa9e4eb 100644 --- a/dynker.tex +++ b/dynker.tex @@ -1,4 +1,4 @@ -\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[version=4]{mhchem} @@ -20,6 +20,9 @@ \author{Pierre-Fran\c{c}ois \surname{Loos}} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} +\author{Juliette \surname{Authier}} + \affiliation{\LCPQ} +%\author{Friends} \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: @@ -94,7 +97,7 @@ As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of \end{equation} where $\sigma = 1 $ or $0$ for singlet and triplet 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' + \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 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. @@ -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 \begin{equation} \label{eq:lin_sys} - \bA \cdot \bc = \omega_S \, \bc + \bA \cdot \bc = \omega \, \bc \end{equation} 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. @@ -346,7 +349,7 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the \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} 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} @@ -607,7 +610,7 @@ with \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} = - \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{subequations} and @@ -615,7 +618,7 @@ and \begin{gather} 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{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. @@ -641,15 +644,15 @@ In the case of BSE2, the perturbative partitioning is simply \begin{tabular}{|c|ccccccc|c|} Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\ \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 \\ $\omega_2^{\updw}$ & & & & & & & & \\ - $\omega_3^{\updw}$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\ + $\omega_3^{\updw}$ & & & & 4.47109 & & & 4.47097 & 3.47880 \\ \hline Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\ \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 \\ $\omega_2^{\upup}$ & & & & & & & & \\ - $\omega_3^{\upup}$ & & & & 4.47797 & & & 4.47767 & \\ + $\omega_3^{\upup}$ & & & & 4.47773 & & & 4.47767 & \\ \end{tabular} \end{ruledtabular} \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. 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{ -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. -He also thanks Xavier Blase and Juliette Authier for numerous insightful discussions on dynamical kernels.} +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. +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