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dynker.bib
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dynker.bib
@ -1,13 +1,22 @@
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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 2020-08-22 14:25:29 +0200
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%% Created for Pierre-Francois Loos at 2020-08-27 16:42:19 +0200
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%% Saved with string encoding Unicode (UTF-8)
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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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Date-Modified = {2020-08-27 16:40:16 +0200},
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Journal = {J. Chem. Phys.},
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Pages = {arXiv:2007.13501},
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Title = {Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation},
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Year = {in press}}
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@article{Lowdin_1963,
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Author = {P. L{\"o}wdin},
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Date-Added = {2020-07-20 08:49:22 +0200},
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@ -69,11 +78,13 @@
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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 = {2020-08-22 14:13:11 +0200},
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Date-Modified = {2020-08-27 16:37:40 +0200},
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Doi = {10.1021/acs.jpclett.0c01875},
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Journal = {J. Phys. Chem. Lett.},
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Pages = {7371},
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Title = {The Bethe-Salpeter Equation Formalism: From Physics to Chemistry},
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Year = {in press},
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Volume = {11},
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Year = {2020},
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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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@ -2361,7 +2372,8 @@
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@article{Strinati_1988,
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Author = {Strinati, G.},
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Date-Added = {2020-05-18 21:40:28 +0200},
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Date-Modified = {2020-06-19 14:11:57 +0200},
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Date-Modified = {2020-08-27 16:39:14 +0200},
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Doi = {10.1007/BF02725962},
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Journal = {Riv. Nuovo Cimento},
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Pages = {1--86},
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Title = {Application of the {{Green}}'s Functions Method to the Study of the Optical Properties of Semiconductors},
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@ -5426,7 +5438,8 @@
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@article{Garniron_2018,
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Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
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Date-Added = {2020-01-01 21:36:51 +0100},
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Date-Modified = {2020-01-01 21:36:52 +0100},
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Date-Modified = {2020-08-27 16:41:38 +0200},
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Doi = {10.1063/1.5044503},
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Journal = {J. Chem. Phys.},
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Pages = {064103},
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Title = {Selected Configuration Interaction Dressed by Perturbation},
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357
dynker.tex
357
dynker.tex
@ -228,36 +228,47 @@ with
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and $\Delta\e{} = \e{c} - \e{v}$.
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The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
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For the sake of illustration, we will use the same numerical example throughout this study, and consider the singlet ground state of the \ce{He} atom in Pople's 6-31G basis set.
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The numerical values of the various quantities defined above are
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\begin{subequations}
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\begin{align}
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\e{v} & = -0.914\,127
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&
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\e{c} & = + 1.399\,859
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\\
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\ERI{vv}{vv} & = 1.026\,907
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&
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\ERI{cc}{cc} & = 0.766\,363
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\\
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\ERI{vv}{cc} & = 0.858\,133
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&
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\ERI{vc}{cv} & = 0.227\,670
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\\
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\ERI{vv}{vc} & = 0.316\,490
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&
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\ERI{vc}{cc} & = 0.255\,554
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\end{align}
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\end{subequations}
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This yields the following exact singlet and triplet excitation energies
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\begin{align} \label{eq:exact}
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\omega_{1}^{\updw} & = 1.92145
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&
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\omega_{3}^{\updw} & = 3.47880
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&
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\omega_{1}^{\upup} & = 1.47085
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\end{align}
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where $\omega_{1}^{\updw}$ and $\omega_{3}^{\updw}$ are the singlet single and double excitations (respectively), and $\omega_{1}^{\upup}$ is the triplet single excitation.
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For the sake of illustration, we will use the same numerical examples throughout this study, and consider the singlet ground state of i) the \ce{H2} molecule ($R_{\ce{H-H}} = 1.4$ bohr) in the STO-3G basis, ii) the \ce{HeH+} molecule ($R_{\ce{He-H}} = 1.4632$ bohr) in the STO-3G basis, and iii) the \ce{He} atom in Pople's 6-31G basis set.
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These three systems provide prototypical examples of valence, charge-transfer, and Rydberg excitations, respectively, and will be employed to quantity the performance of the various methods considered in the present study for each type of excited states.
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The numerical values of the various quantities defined above are gathered in Table \ref{tab:params} for each system.
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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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\label{tab:params}
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}
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\begin{ruledtabular}
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\begin{tabular}{llcccccccc}
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System & Method & $\e{v}$ & $\e{c}$ & $\ERI{vv}{vv}$ & $\ERI{cc}{cc}$ & $\ERI{vv}{cc}$ & $\ERI{vc}{cv}$ & $\ERI{vv}{vc}$ & $\ERI{vc}{cc}$ \\
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\hline
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\ce{H2} & HF/STO-3G & $-0.578\,203$ & $+0.670\,268$ & $+0.674\,594$ & $+0.697\,495$ & $+0.663\,564$ & $+0.181\,258$ & $0$ & $0$ \\
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\ce{HeH+} & HF/STO-3G & $-1.632\,802$ & $-0.172\,484$ & $+0.943\,099$ & $+0.752\,526$ & $+0.660\,254$ & $+0.145\,397$ & $-0.172\,968$ & $+0.037\,282$ \\
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\ce{He} & HF/6-31G & $-0.914\,127$ & $+1.399\,859$ & $+1.026\,907$ & $+0.766\,363$ & $+0.858\,133$ & $+0.227\,670$ & $+0.316\,490$ & $+0.255\,554$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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%%% %%% %%% %%%
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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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% \label{tab:gap}
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% }
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% \begin{ruledtabular}
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% \begin{tabular}{llccccccccccccc}
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% & \mc{3}{c}{Hartree-Fock} & \mc{3}{c}{$GW$} & \mc{3}{c}{GF2} & \mc{3}{c}{Exact} \\
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% \cline{2-4} \cline{5-7} \cline{8-10} \cline{11-13}
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% System & $\Delta\e{}$& $-\e{v}$ & $-\e{c}$ & $\Delta\eGW{}$ & $-\eGW{v}$ & $-\eGW{c}$ & $\Delta\eGF{}$ & $-\eGF{v}$ & $-\eGF{c}$ & Gap & IP & EA \\
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% \hline
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% \ce{H2} & $33.97$ & $15.73$ & $-18.24$ & $35.21$ & $16.35$ & $-18.86$ & $34.69$ & $16.09$ & $-18.60$ & & $16.29$ & \\
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% \ce{HeH+} & $39.74$ & $44.43$ & $+4.69$ & $39.49$ & $43.87$ & $+4.38$ & $39.55$ & $44.08$ & $+4.52$ & & $44.04$ & \\
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% \ce{He} & $62.97$ & $24.87$ & $-38.09$ & $60.88$ & $23.50$ & $-37.38$ & $61.66$ & $24.02$ & $-37.64$ & & $23.85$ \\
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% \end{tabular}
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% \end{ruledtabular}
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%\end{table*}
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%%% %%% %%% %%
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The exact values of the singlet single and double excitations, $\omega_{1}^{\updw}$ and $\omega_{3}^{\updw}$, and the triplet single excitation, $\omega_{1}^{\upup}$, are reported, for example, in Table \ref{tab:Maitra}.
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We are going to use these as reference for the remaining of this study.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -304,45 +315,54 @@ In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function
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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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%%% TABLE I %%%
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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 for \ce{He} at the HF/6-31G level.
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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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\label{tab:Maitra}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|cccc|c|}
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Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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\begin{tabular}{lclllll}
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& & \mc{5}{c}{Method} \\
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\cline{3-7}
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System & Excitation & CIS & TDHF & D-CIS & D-TDHF & Exact \\
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\hline
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$\omega_1^{\updw}$ & 52.01 & 51.64 & 51.87[-0.14]& 51.52[-0.12]& 52.29 \\
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$\omega_3^{\updw}$ & & & 93.85 & 93.84 & 94.66 \\
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\hline
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Triplets & & & & & Exact \\
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\hline
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$\omega_1^{\upup}$ & 39.62 & 39.13 & 39.62[+0.00]& 39.13[+0.00]& 40.18 \\
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\ce{H2} & $\omega_1^{\updw}$ & 25.78 & 25.30 & 25.78[+0.00] & 25.30[+0.00] & 26.34 \\
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& $\omega_3^{\updw}$ & & & & & 44.04 \\
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& $\omega_1^{\upup}$ & 15.92 & 15.13 & 15.92[+0.00] & 15.13[+0.00] & 16.48 \\
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\\
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\ce{HeH+} & $\omega_1^{\updw}$ & 29.68 & 29.42 & 27.75[-1.93] & 27.64[-1.78] & 28.05 \\
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& $\omega_3^{\updw}$ & & & 63.59 & 63.52 & 64.09 \\
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& $\omega_1^{\upup}$ & 21.77 & 21.41 & 21.77[+0.00] & 21.41[+0.00] & 22.03 \\
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\\
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\ce{He} & $\omega_1^{\updw}$ & 52.01 & 51.64 & 51.87[-0.14]& 51.52[-0.12]& 52.29 \\
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& $\omega_3^{\updw}$ & & & 93.85 & 93.84 & 94.66 \\
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& $\omega_1^{\upup}$ & 39.62 & 39.13 & 39.62[+0.00]& 39.13[+0.00]& 40.18 \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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%%% FIGURE 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Maitra}
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\caption{
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$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (gray and black) and triplet (orange) manifolds.
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The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered.
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\label{fig:Maitra}
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}
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\end{figure}
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%%% %%% %%% %%%
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Although not particularly accurate for the single excitations, Maitra's dynamical kernel allows to access the double excitation with good accuracy and provides exactly the right number of solutions (two singlets and one triplet).
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Note that this correlation kernel is known to work best in the weak correlation regime (which is the case here) where the true excitations have a clear single and double excitation character, \cite{Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018}
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Its accuracy for the single excitations could be certainly improved in a DFT context.
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However, this is not the point of the present investigation.
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In the case of \ce{H2} in a minimal basis, because $\mel{S}{\hH}{D} = 0$, there is no dynamical correction for both singlets and triplets, and one cannot access the double excitation with Maitra's kernel.
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It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
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Table \ref{tab:Maitra} also reports the slightly-improved (thanks to error compensation) CIS and D-CIS excitation energies.
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In particular, single excitations are greatly improved without altering the accuracy of the double excitation.
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Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra}.
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Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra} for \ce{HeH+}.
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%%% FIGURE 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Maitra}
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\caption{
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$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange) manifolds of \ce{HeH+}.
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The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered.
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\label{fig:Maitra}
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}
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\end{figure}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Dynamical BSE kernel}
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@ -350,11 +370,13 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
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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,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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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,Loos_2020e}
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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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This issue was resolved in the subsequent work of Sangalli \textit{et al.} \cite{Sangalli_2011} via the design of a diagrammatic number-conserving approach based on the folding of the second-RPA Hamiltonian. \cite{Wambach_1988}
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Thanks to a careful diagrammatic analysis of the dynamical kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
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Very recently, Loos and Blase have applied the dynamical correction to the BSE beyond the plasmon-pole approximation within a renormalized first-order perturbative treatment, \cite{Loos_2020e} generalizing the work of Rolhfing and coworkers on biological chromophores \cite{Ma_2009a,Ma_2009b} and dicyanovinyl-substituted oligothiophenes. \cite{Baumeier_2012b}
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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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@ -431,34 +453,45 @@ It can be easily shown that solving the equation
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\begin{equation}
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\det[\bH_{\dBSE}^{\sigma}(\omega) - \omega \bI] = 0
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\end{equation}
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yields 3 solutions per spin manifold (see Fig.~\ref{fig:dBSE}).
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yields 3 solutions per spin manifold (except for \ce{H2} where only 2 roots are observed), as shown in Fig.~\ref{fig:dBSE} for the case of \ce{HeH+}.
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Their numerical values are reported in Table \ref{tab:BSE} alongside other variants discussed below.
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This evidences that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
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This evidences that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV except in the case of \ce{H2} where the agreement is rather satisfactory ($44.30$ eV at the dBSE level compared to the exact value of $44.04$ eV).
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As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
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Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}).
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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 the Newton-Raphson method.
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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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%%% TABLE II %%%
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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 for \ce{He} at the HF/6-31G level.
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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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\label{tab:BSE}
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}
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\begin{ruledtabular}
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\begin{tabular}{|c|ccccccc|c|}
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Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\begin{tabular}{lcllllllll}
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& & \mc{8}{c}{Method} \\
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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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$\omega_1^{\updw}$ & 52.46 & 51.71[-0.75]& 52.12[-0.33]& 51.85[-0.61]& 53.10 & 52.79[-0.31]& 52.79[-0.31]& 52.29 \\
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$\omega_2^{\updw}$ & & & & 75.75 & & & & \\
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$\omega_3^{\updw}$ & & & & 133.37 & & & 133.37 & 94.66 \\
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\hline
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Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
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\hline
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$\omega_1^{\upup}$ & 40.50 & 39.96[-0.53]& 39.80[-0.70]& 39.90[-0.60]& 40.71 & 40.02[-0.69]& 40.02[-0.69]& 40.18 \\
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$\omega_2^{\upup}$ & & & & 75.15 & & & & \\
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$\omega_3^{\upup}$ & & & & 133.76 & & & 133.75 & \\
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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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& $\omega_3^{\updw}$ & & & & 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 \\
|
||||
& $\omega_3^{\upup}$ & & & & 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 \\
|
||||
& $\omega_2^{\updw}$ & & & & 47.85 & & & & \\
|
||||
& $\omega_3^{\updw}$ & & & & 87.47 & & & 87.47 & 64.09 \\
|
||||
& $\omega_1^{\upup}$ & 20.96 & 21.16[+0.20]& 21.07[+0.11]& 21.12[+0.16]& 21.13 & 21.24[+0.11]& 21.24[+0.11]& 22.03 \\
|
||||
& $\omega_2^{\upup}$ & & & & 47.54 & & & & \\
|
||||
& $\omega_3^{\upup}$ & & & & 87.43 & & & 87.43 & \\
|
||||
\\
|
||||
\ce{He} & $\omega_1^{\updw}$ & 52.46 & 51.71[-0.75]& 52.12[-0.33]& 51.85[-0.61]& 53.10 & 52.79[-0.31]& 52.79[-0.31]& 52.29 \\
|
||||
& $\omega_2^{\updw}$ & & & & 75.75 & & & & \\
|
||||
& $\omega_3^{\updw}$ & & & & 133.37 & & & 133.37 & 94.66 \\
|
||||
& $\omega_1^{\upup}$ & 40.50 & 39.96[-0.53]& 39.80[-0.70]& 39.90[-0.60]& 40.71 & 40.02[-0.69]& 40.02[-0.69]& 40.18 \\
|
||||
& $\omega_2^{\upup}$ & & & & 75.15 & & & & \\
|
||||
& $\omega_3^{\upup}$ & & & & 133.76 & & & 133.75 & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
@ -468,7 +501,7 @@ This makes these two latter solutions quite hard to locate with the Newton-Raphs
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{dBSE}
|
||||
\caption{
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (gray and black) and triplet (orange and red) manifolds.
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
|
||||
The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered.
|
||||
\label{fig:dBSE}
|
||||
}
|
||||
@ -480,11 +513,11 @@ There is thus only one spurious excitation in the triplet manifold ($\omega_{3}^
|
||||
The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
|
||||
Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
|
||||
|
||||
%%% FIGURE 2 %%%
|
||||
%%% FIGURE 3 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{dBSE-TDA}
|
||||
\caption{
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (gray and black) and triplet (orange and red) manifolds within the TDA.
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+} within the TDA.
|
||||
The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered.
|
||||
\label{fig:dBSE-TDA}
|
||||
}
|
||||
@ -494,7 +527,7 @@ Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in th
|
||||
In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
|
||||
Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
|
||||
|
||||
Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} a scheme we label as perturbative BSE (pBSE).
|
||||
Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e} a scheme we label as perturbative BSE (pBSE).
|
||||
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
|
||||
\begin{equation}
|
||||
\bH_{\dBSE}^{\sigma}(\omega)
|
||||
@ -549,7 +582,7 @@ Obviously, the TDA can be applied to the dynamical correction as well, a scheme
|
||||
|
||||
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.
|
||||
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.
|
||||
%Note that, although the pBSE(dTDA) value is further from the dBSE value than pBSE, it is quite close to the exact excitation energy.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -628,7 +661,7 @@ The singlet manifold has then the right number of excitations.
|
||||
However, one spurious triplet excitation remains.
|
||||
It is mentioned in Ref.~\onlinecite{Rebolini_2016} that the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
|
||||
|
||||
Numerical results for the two-level model are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
|
||||
Numerical results for the two-level models are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
|
||||
In the case of BSE2, the perturbative partitioning is simply
|
||||
\begin{equation}
|
||||
\bH_{\dBSE2}^{\sigma}(\omega)
|
||||
@ -636,43 +669,54 @@ In the case of BSE2, the perturbative partitioning is simply
|
||||
+ \underbrace{\qty[ \bH_{\dBSE2}^{\sigma}(\omega) - \bH_{\BSE2}^{\sigma} ]}_{\bH_{\pBSE2}^{(1)}(\omega)}
|
||||
\end{equation}
|
||||
|
||||
%%% TABLE III %%%
|
||||
%%% TABLE IV %%%
|
||||
\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.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:BSE2}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\begin{tabular}{lcllllllll}
|
||||
System & Excitation & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 50.31 & 51.96[+1.64]& 51.96[+1.65]& 52.10[+1.79]& 50.69 & 52.34[+1.65]& 52.34[+1.65]& 52.29 \\
|
||||
$\omega_3^{\updw}$ & & & & 121.67 & & & 121.66 & 94.66 \\
|
||||
\hline
|
||||
Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 37.80 & 39.26[+1.46]& 39.27[+1.47]& 39.59[+1.79]& 38.30 & 39.77[+1.47]& 39.77[+1.47]& 40.18 \\
|
||||
$\omega_3^{\upup}$ & & & & 121.85 & & & 121.84 & \\
|
||||
\ce{H2} & $\omega_1^{\updw}$ & 26.03 & 26.03[+0.00]& 26.03[+0.00]& 26.24[+0.21]& 26.49 & 26.49[+0.00]& 26.49[+0.00]& 26.34 \\
|
||||
& $\omega_1^{\upup}$ & 15.88 & 15.88[+0.00]& 15.88[+0.00]& 16.47[+0.59]& 16.63 & 16.63[+0.00]& 16.63[+0.00]& 16.48 \\
|
||||
\\
|
||||
\ce{HeH+} & $\omega_1^{\updw}$ & 29.23 & 28.40[-0.83]& 28.40[-0.83]& 28.56[-0.67]& 29.50 & 28.66[-0.84]& 28.66[-0.84]& 28.05 \\
|
||||
& $\omega_3^{\updw}$ & & & & 79.94 & & & 79.94 & 64.09 \\
|
||||
& $\omega_1^{\upup}$ & 21.22 & 21.63[+0.41]& 21.63[+0.41]& 21.93[+0.71]& 21.59 & 21.99[+0.40]& 21.99[+0.40]& 22.03 \\
|
||||
& $\omega_3^{\upup}$ & & & & 78.70 & & & 78.70 & \\
|
||||
\\
|
||||
\ce{He} & $\omega_1^{\updw}$ & 50.31 & 51.96[+1.64]& 51.96[+1.65]& 52.10[+1.79]& 50.69 & 52.34[+1.65]& 52.34[+1.65]& 52.29 \\
|
||||
& $\omega_3^{\updw}$ & & & & 121.67 & & & 121.66 & 94.66 \\
|
||||
& $\omega_1^{\upup}$ & 37.80 & 39.26[+1.46]& 39.27[+1.47]& 39.59[+1.79]& 38.30 & 39.77[+1.47]& 39.77[+1.47]& 40.18 \\
|
||||
& $\omega_3^{\upup}$ & & & & 121.85 & & & 121.84 & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
As compared to dBSE, dBSE2 produces much larger corrections to the static excitation energies probably due to the poorer quality of its static reference (CIS or TDHF).
|
||||
As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (CIS or TDHF).
|
||||
Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate.
|
||||
Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations, except for \ce{H2} (see below).
|
||||
Overall, the accuracy of dBSE and dBSE2 are comparable (see Tables \ref{tab:BSE} and \ref{tab:BSE2}) for single excitations although their behavior is quite different.
|
||||
For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
|
||||
|
||||
%%% FIGURE 3 %%%
|
||||
%%% FIGURE 4 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{dBSE2}
|
||||
\caption{
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (gray and black) and triplet (orange and red) manifolds.
|
||||
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds.
|
||||
The static BSE2 Hamiltonian (dashed) and dynamic dBSE2 Hamiltonian (solid) are considered.
|
||||
\label{fig:BSE2}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Again, the case of \ce{H2} is a bit particular as the perturbative treatment does not provide dynamical corrections, while its fully-dynamical version does yields sizeable corrections and makes the singlet and triplet excitation energies very accurate.
|
||||
This is to be expected as BSE2 takes into account second-order exchange which is omitted in BSE. \cite{Zhang_2013}
|
||||
However, one cannot access the double excitation.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\subsection{The forgotten kernel: Sangalli's kernel}
|
||||
%\label{sec:Sangalli}
|
||||
@ -731,149 +775,16 @@ 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}
|
||||
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 Zhang \textit{et al.}, \cite{Zhang_2013} 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{
|
||||
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.
|
||||
We 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.}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table}
|
||||
\caption{Singlet and triplet excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:Maitra}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|cccc|c|}
|
||||
Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 29.68 & 29.42 & 27.75[-1.93] & 27.64[-1.78] & 28.05 \\
|
||||
$\omega_3^{\updw}$ & & & 63.59 & 63.52 & 64.09 \\
|
||||
\hline
|
||||
Triplets & & & & & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 21.77 & 21.41 & 21.77[+0.00] & 21.41[+0.00] & 22.03 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
|
||||
%%% TABLE II %%%
|
||||
\begin{table*}
|
||||
\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:BSE}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 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 \\
|
||||
$\omega_2^{\updw}$ & & & & 47.85 & & & & \\
|
||||
$\omega_3^{\updw}$ & & & & 87.47 & & & 87.47 & 64.09 \\
|
||||
\hline
|
||||
Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 20.96 & 21.16[+0.20] & 21.07[+0.11]& 21.12[+0.16]& 21.13 & 21.24[+0.11]& 21.24[+0.11]& 22.03 \\
|
||||
$\omega_2^{\upup}$ & & & & 47.54 & & & & \\
|
||||
$\omega_3^{\upup}$ & & & & 87.43 & & & 87.43 & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% TABLE III %%%
|
||||
\begin{table*}
|
||||
\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{HeH+} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:BSE2}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 29.23 & 28.40[-0.83]& 28.40[-0.83]& 28.56[-0.67]& 29.50 & 28.66[-0.84]& 28.66[-0.84]& 28.05 \\
|
||||
$\omega_3^{\updw}$ & & & & 79.94 & & & 79.94 & 64.09 \\
|
||||
\hline
|
||||
Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 21.22 & 21.63[+0.41]& 21.63[+0.41]& 21.93[+0.71]& 21.59 & 21.99[+0.40]& 21.99[+0.40]& 22.03 \\
|
||||
$\omega_3^{\upup}$ & & & & 78.70 & & & 78.70 & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table}
|
||||
\caption{Singlet and triplet excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:Maitra}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|cccc|c|}
|
||||
Singlets & CIS & TDHF & D-CIS & D-TDHF & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 25.78 & 25.30 & 25.78[+0.00] & 25.30[+0.00] & 26.34 \\
|
||||
$\omega_3^{\updw}$ & & & 63.59 & 63.52 & 44.04 \\
|
||||
\hline
|
||||
Triplets & & & & & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 15.92 & 15.13 & 15.92[+0.00] & 15.13[+0.00] & 16.48 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
|
||||
%%% TABLE II %%%
|
||||
\begin{table*}
|
||||
\caption{Singlet and triplet BSE excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:BSE}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
Singlets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 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 \\
|
||||
$\omega_3^{\updw}$ & & & & 44.30 & & & & 44.04 \\
|
||||
\hline
|
||||
Triplets & BSE & pBSE & pBSE(dTDA) & dBSE & BSE(TDA) & pBSE(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 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 \\
|
||||
$\omega_3^{\upup}$ & & & & 43.61 & & & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
%%% TABLE III %%%
|
||||
\begin{table*}
|
||||
\caption{Singlet and triplet BSE2 excitation energies (in eV) at various levels of theory for \ce{H2} at the HF/STO-3G level.
|
||||
The magnitude of the dynamical correction is reported between square brackets.
|
||||
\label{tab:BSE2}
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
Singlets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\updw}$ & 26.03 & 26.03[+0.00]& 26.03[+0.00]& 26.24[+0.21]& 26.49 & 26.49[+0.00]& 26.49[+0.00]& 26.34 \\
|
||||
\hline
|
||||
Triplets & BSE2 & pBSE2 & pBSE2(dTDA) & dBSE2 & BSE2(TDA) & pBSE2(TDA) & dBSE2(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1^{\upup}$ & 15.88 & 15.88[+0.00]& 15.88[+0.00]& 16.47[+0.59]& 16.63 & 16.63[+0.00]& 16.63[+0.00]& 16.48 \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
% BIBLIOGRAPHY
|
||||
\bibliography{dynker}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user