PTEROSOR/BSEdyn
10
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Intro OK

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-06-06 09:06:45 +02:00
parent 216d5c3ab4
commit 69818de8d8
2 changed files with 19 additions and 7 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

13
BSEdyn.bib
View File

@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-01 08:13:45 +0200
%% Created for Pierre-Francois Loos at 2020-06-06 09:02:31 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Wambach_1988,
Author = {J. Wambach},
Date-Added = {2020-06-06 09:01:23 +0200},
Date-Modified = {2020-06-06 09:02:27 +0200},
Doi = {10.1088/0034-4885/51/7/002},
Journal = {Rep. Prog. Phys.},
Pages = {989},
Title = {Damping of small-amplitude nuclear collective motion},
Volume = {51},
Year = {1988}}
@article{Sakkinen_2012,
Author = {N. Sakkinen and M. Manninen and R. van Leeuwen},
Date-Added = {2020-06-01 08:10:19 +0200},

13
BSEdyn.tex
View File

@ -190,7 +190,8 @@
\begin{document}
\title{ \textcolor{red}{Assessing} Dynamical Corrections to the Bethe-Salpeter Equation}
\title{Dynamical Correction to the Bethe-Salpeter Equation}
%\title{ \textcolor{red}{Assessing} Dynamical Correction to the Bethe-Salpeter Equation}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
@ -240,7 +241,7 @@ where
is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ being the ionization potential and the electron affinity of the $N$-electron system.
Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
Due to the smaller amount of screening in molecules as compared to solids, a faithful description of the excitonic effects is paramount in molecular systems.
Due to the smaller amount of screening in molecules as compared to solids, a faithful description of excitonic effects is paramount in molecular systems.
Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT where the exchange-correlation (xc) kernel is made static, one key consequence of the static approximation within BSE is that double (and higher) excitations are completely absent from the BSE spectrum.
@ -248,7 +249,7 @@ Indeed, a frequency-dependent kernel has the ability to create additional poles
Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms, \cite{Boggio-Pasqua_2007} as they strongly mix with the bright singly-excited states leading to the formation of satellite peaks. \cite{Helbig_2011,Elliott_2011}
They are particularly important in the faithful description of the ground state of open-shell molecules, \cite{Casida_2005,Romaniello_2009a,Huix-Rotllant_2011,Loos_2020c}
and they are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin which is involved in the visual transduction. \cite{Olivucci_2010,Robb_2007,Manathunga_2016}
Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin, a biological pigment found in the rods of the retina and involved in the visual transduction. \cite{Olivucci_2010,Robb_2007,Manathunga_2016}
In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
@ -268,12 +269,12 @@ These higher excitations would be explicitly present in the BSE Hamiltonian by `
Based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations, Romaniello and coworkers \cite{Romaniello_2009b,Sangalli_2011} evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent eigenvalue problem.
For this particular system, it was shown that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations. \cite{Romaniello_2009b}
The appearance of these spurious excitations was attributed to the self-screening problem. \cite{Romaniello_2009a}
This was fixed in a follow-up paper by Sangalli \textit{et al.} \cite{Sangalli_2011} thanks to the design of a number-conserving approach based on the second RPA.
This was fixed in a follow-up paper by Sangalli \textit{et al.} \cite{Sangalli_2011} thanks to the design of a number-conserving approach based on the second RPA. \cite{Wambach_1988}
Finally, let us mention efforts to borrow ingredients from BSE in order to go beyond the adiabatic approximation of TD-DFT.
For example, Huix-Rotllant and Casida \cite{Casida_2005,Huix-Rotllant_2011} proposed a nonadiabatic correction to the xc kernel by using the formalism of superoperators, which includes as a special case the dressed TD-DFT method of Maitra and coworkers. \cite{Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}
For example, Huix-Rotllant and Casida \cite{Casida_2005,Huix-Rotllant_2011} proposed a nonadiabatic correction to the xc kernel using the formalism of superoperators, which includes as a special case the dressed TD-DFT method of Maitra and coworkers. \cite{Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}
Following a similar strategy, Romaniello \textit{et al.} \cite{Romaniello_2009b} took advantages of the dynamically-screened Coulomb potential from BSE to obtain a dynamic TD-DFT kernel.
In this regard, MBPT provides key insights about what is missing in adiabatic TD-DFT, as discussed at length by Casida and Huix-Rotllant in Ref.~\onlinecite{Casida_2016}.
In this regard, MBPT provides key insights about what is missing in adiabatic TD-DFT, as discussed in details by Casida and Huix-Rotllant in Ref.~\onlinecite{Casida_2016}.
In the present study, we extend the work of Rohlfing and coworkers \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} by proposing a renormalized first-order perturbative correction to the static BSE excitation energies.
Importantly, our correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.