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BSEdyn.tex
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BSEdyn.tex
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@ -34,7 +34,7 @@
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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\newcommand{\SI}{\textcolor{blue}{supplementary material}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\T}[1]{#1^{\intercal}}
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@ -398,9 +398,7 @@ Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )
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\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta }
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\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} } ].
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\end{multline}
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% and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
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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.
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% with $(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively.
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This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
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For example, we have
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\begin{multline} \label{eq:spectral65}
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@ -412,7 +410,6 @@ For example, we have
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\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} } ],
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\end{multline}
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with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
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%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
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%================================
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\subsection{Dynamical BSE within the $GW$ approximation}
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@ -428,12 +425,7 @@ leads to the following simplified BSE kernel
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\end{equation}
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where $W$ is the dynamically-screened Coulomb operator.
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The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}.
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%Selecting $(p,q)=(j,b)$ yields the largest components
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%$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
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%$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions.
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%Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
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%Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$,
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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):
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\begin{equation} \label{eq:BSE-final}
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\begin{split}
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@ -467,6 +459,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
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%================================
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\subsection{Dynamical screening}
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\label{sec:dynW}
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%=================================
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In the present study, we consider the exact spectral representation of $W$ at the RPA level:
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@ -566,16 +559,6 @@ Accordingly to Eq.~\eqref{eq:BSE-final}, the BSE matrix elements in Eq.~\eqref{e
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\B{ia,jb}{}(\Om{s}{}) & = \kappa \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}).
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\end{align}
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\end{subequations}
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%\begin{equation}
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% \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
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%\end{equation}
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%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
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%\begin{multline}
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%\label{eq:W}
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% \W{ij,ab}{}(\omega) = \ERI{ij}{ab} + 2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
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% \\
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% \times \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}).
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%\end{multline}
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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
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\begin{multline}
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@ -773,12 +756,51 @@ All the static and dynamic BSE calculations have been performed with the softwar
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\end{ruledtabular}
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\end{table*}
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%%%% TABLE I %%%
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%\begin{table*}
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% \caption{
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% Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
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% \label{tab:N2}
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% }
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% \begin{ruledtabular}
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% \begin{tabular}{lddddddddd}
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% & \mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}
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% & \mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
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% & \mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)} \\
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% \cline{2-4} \cline{5-7} \cline{8-10}
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% State & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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% & \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
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% \hline
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% $^1\Pi_g(n \ra \pis)$ & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
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% $^1\Sigma_u^-(\pi \ra \pis)$ & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
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% $^1\Delta_u(\pi \ra \pis)$ & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
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% $^1\Sigma_g^+$(R) & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
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% $^1\Pi_u$(R) & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
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% $^1\Sigma_u^+$(R) & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
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% $^1\Pi_u$(R) & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
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% \\
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% $^3\Sigma_u^+(\pi \ra \pis)$ & 8.69 & -0.80 & -0.72 & 8.91 & -0.97 & -0.53 & 9.06 & -1.01 & -0.80 \\
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% $^3\Pi_g(n \ra \pis)$ & 9.09 & -0.41 & -0.29 & 9.31 & -0.54 & -0.14 & 9.43 & -0.57 & -0.34 \\
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% $^3\Delta_u(\pi \ra \pis)$ & 9.49 & -0.73 & -0.62 & 9.62 & -0.89 & -0.59 & 9.74 & -0.93 & -0.99 \\
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% $^3\Sigma_u^-(\pi \ra \pis)$ & 10.29 & -0.65 & -0.54 & 10.34 & -0.79 & -0.43 & 10.45 & -0.82 & -0.51 \\
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% \end{tabular}
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% \end{ruledtabular}
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% \fnt[1]{Excitation energy larger than the fundamental gap.}
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%\end{table*}
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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).
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Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
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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.
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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.
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As we shall see later, the magnitude of the dynamical correction is characteristic of the type of transitions.
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As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions.
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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.
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This is quite a nice feature as one does not need to compute this more expensive correction in a very large basis.
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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.
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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.
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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.
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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.
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In accordance with the success of the dTDA, the remaining calculations of the present study are performed within this approximation.
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%%% TABLE I %%%
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%\begin{squeezetable}
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@ -928,6 +950,13 @@ As we shall see later, the magnitude of the dynamical correction is characterist
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\end{table*}
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%\end{squeezetable}
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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.
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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.
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Finally, the highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference.
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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}.
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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.
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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.
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%%% TABLE III %%%
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%\begin{table}
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% \caption{
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@ -953,6 +982,11 @@ As we shall see later, the magnitude of the dynamical correction is characterist
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%%%%%%%%%%%%%%%%%%%%%%%%
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This is the conclusion
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supplementary material}
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%%%%%%%%%%%%%%%%%%%%%%%%
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See the {\SI} for a detailed derivation of the dynamical BSE equation.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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%%%%%%%%%%%%%%%%%%%%%%%%
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@ -962,9 +996,9 @@ Funding from the \textit{``Centre National de la Recherche Scientifique''} is ac
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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''.}}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%\section*{Supporting Information}
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\section*{Data availability}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%See {\SI} for plenty of stuff
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The data that support the findings of this study are available within the article and its {\SI}.
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\bibliography{BSEdyn}
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