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overall screening OK

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Pierre-Francois Loos 2020-08-27 18:13:54 +02:00
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@ -95,7 +95,7 @@ As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of
\begin{equation}
f_{ia,jb}^{\Hx,\sigma} = 2\sigma \ERI{ia}{jb} - \ERI{ib}{ja}
\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}
\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}
@ -350,7 +350,7 @@ Its accuracy for the single excitations could be certainly improved in a DFT con
However, this is not the point of the present investigation.
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.
It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
Table \ref{tab:Maitra} also reports the slightly-improved (thanks to error compensation) CIS and D-CIS excitation energies.
Table \ref{tab:Maitra} also reports the slightly improved (thanks to error compensation) CIS and D-CIS excitation energies.
In particular, single excitations are greatly improved without altering the accuracy of the double excitation.
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+}.
@ -371,13 +371,13 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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,Loos_2020e}
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}
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}
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}
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}).
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}
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 sizable 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.
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
@ -714,7 +714,7 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
\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.
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 sizable 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.
@ -783,8 +783,7 @@ We hope that the present contribution will foster new developments around dynami
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
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.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481)}
%%%%%%%%%%%%%%%%%%%%%%%%
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{dynker}