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Pierre-Francois Loos 2020-06-06 22:40:10 +02:00
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BSEdyn.tex
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@ -34,7 +34,7 @@
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SI}{\textcolor{blue}{supporting information}}
\newcommand{\SI}{\textcolor{blue}{supplementary material}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}
@ -398,9 +398,7 @@ Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )
\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta }
\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \frac{\Om{s}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \frac{\Om{s}{}}{2}) \tau_{34} } ].
\end{multline}
% and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,s}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
% with $(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively.
This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have
\begin{multline} \label{eq:spectral65}
@ -412,7 +410,6 @@ For example, we have
\times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ],
\end{multline}
with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
%================================
\subsection{Dynamical BSE within the $GW$ approximation}
@ -428,12 +425,7 @@ leads to the following simplified BSE kernel
\end{equation}
where $W$ is the dynamically-screened Coulomb operator.
The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}.
%Selecting $(p,q)=(j,b)$ yields the largest components
%$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
%$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions.
%Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
%Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$,
Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE):
\begin{equation} \label{eq:BSE-final}
\begin{split}
@ -467,6 +459,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
%================================
\subsection{Dynamical screening}
\label{sec:dynW}
%=================================
In the present study, we consider the exact spectral representation of $W$ at the RPA level:
@ -566,16 +559,6 @@ Accordingly to Eq.~\eqref{eq:BSE-final}, the BSE matrix elements in Eq.~\eqref{e
\B{ia,jb}{}(\Om{s}{}) & = \kappa \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}).
\end{align}
\end{subequations}
%\begin{equation}
% \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
%\end{equation}
%are the bare two-electron integrals in the molecular orbital basis $\lbrace \MO{p}(\br{}) \rbrace_{1 \le p \le \Norb}$, and the dynamically-screened Coulomb potential reads
%\begin{multline}
%\label{eq:W}
% \W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
% \\
% \times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}).
%\end{multline}
Now, let us decompose, using basic perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that
\begin{multline}
@ -773,12 +756,51 @@ All the static and dynamic BSE calculations have been performed with the softwar
\end{ruledtabular}
\end{table*}
%%%% TABLE I %%%
%\begin{table*}
% \caption{
% Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
% \label{tab:N2}
% }
% \begin{ruledtabular}
% \begin{tabular}{lddddddddd}
% & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
% & \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
% & \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
% \cline{2-4} \cline{5-7} \cline{8-10}
% State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
% \hline
% $^1\Pi_g(n \ra \pis)$ & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
% $^1\Sigma_u^-(\pi \ra \pis)$ & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
% $^1\Delta_u(\pi \ra \pis)$ & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
% $^1\Sigma_g^+$(R) & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
% $^1\Pi_u$(R) & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
% $^1\Sigma_u^+$(R) & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
% $^1\Pi_u$(R) & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
% \\
% $^3\Sigma_u^+(\pi \ra \pis)$ & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
% $^3\Pi_g(n \ra \pis)$ & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
% $^3\Delta_u(\pi \ra \pis)$ & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
% $^3\Sigma_u^-(\pi \ra \pis)$ & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
% \end{tabular}
% \end{ruledtabular}
% \fnt[1]{Excitation energy larger than the fundamental gap.}
%\end{table*}
First, we investigate the basis set dependency of the dynamical correction as well as the validity of the dTDA (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian).
Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that each corrected transition is well below this gap.
The \ce{N2} molecule is a very convenient example as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions.
As we shall see later, the magnitude of the dynamical correction is characteristic of the type of transitions.
As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions.
One key result of the present investigation is that the dynamical correction is quite basis set insensitive with a maximum variation of $0.03$ eV between in smallest (aug-cc-pVDZ) and largest (aug-cc-pVQZ) basis sets.
This is quite a nice feature as one does not need to compute this more expensive correction in a very large basis.
The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies.
The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate.
Indeed, although the dynamical correction systematically red-shift the excitation energies (as anticipated in Sec.~\ref{sec:dynW}), taking into account the coupling between the resonant and anti-resonant parts of the BSE Hamiltonian [see Eq.~\eqref{eq:BSE-final}] yields a systematic blue-shift of the correction, the overall correction remaining negative but by a smaller amount.
This outcome is similar to the conclusions of several benchmark studies \cite{Jacquemin_2017b,Rangel_2017} which clearly concluded that static BSE triplet excitations are notably too low in energy and that the use of the TDA is able to partly reduce this error.
In accordance with the success of the dTDA, the remaining calculations of the present study are performed within this approximation.
%%% TABLE I %%%
%\begin{squeezetable}
@ -928,6 +950,13 @@ As we shall see later, the magnitude of the dynamical correction is characterist
\end{table*}
%\end{squeezetable}
Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported.
Finally, the highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference.
For each excitation, we report the static and dynamic excitation energies, $\Om{s}{\stat}$ and $\Om{s}{\dyn}$, as well as the value of the renormalization factor $Z_s$ defined in Eq.~\eqref{eq:Z}.
As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_s$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq.~\eqref{eq:Om1}] yields basically the same results as its (cheaper) renormalized version.
%%% TABLE III %%%
%\begin{table}
% \caption{
@ -953,6 +982,11 @@ As we shall see later, the magnitude of the dynamical correction is characterist
%%%%%%%%%%%%%%%%%%%%%%%%
This is the conclusion
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supplementary material}
%%%%%%%%%%%%%%%%%%%%%%%%
See the {\SI} for a detailed derivation of the dynamical BSE equation.
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
%%%%%%%%%%%%%%%%%%%%%%%%
@ -962,9 +996,9 @@ Funding from the \textit{``Centre National de la Recherche Scientifique''} is ac
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}
%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supporting Information}
\section*{Data availability}
%%%%%%%%%%%%%%%%%%%%%%%%
%See {\SI} for plenty of stuff
The data that support the findings of this study are available within the article and its {\SI}.
\bibliography{BSEdyn}