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BSEdyn.tex
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@ -90,6 +90,10 @@
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
\newcommand{\tOm}[2]{\Tilde{\Omega}_{#1}^{#2}}
\newcommand{\homu}{\frac{{\omega}_1}{2}}
% Matrix elements
\newcommand{\A}[2]{A_{#1}^{#2}}
\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
@ -203,8 +207,9 @@
\begin{abstract}
Similar to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
Here, going beyond the static approximation, we compute the dynamical correction in the electron-hole screening for molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random phase approximation. \xavier{
\sout{Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.}
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements induced by dynamical corrections. }
%\\
%\bigskip
%\begin{center}
@ -342,7 +347,7 @@ In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$
The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$).
In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, and using the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, \eg,
\begin{equation}
\begin{equation} \label{Eisenberg}
\hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 },
\end{equation}
($\hH$ being the exact many-body Hamiltonian), one gets
@ -399,9 +404,9 @@ 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}
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.
More details are provided in the Appendix. 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.
This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have
For example, we have (see derivation in the Appendix)
\begin{multline} \label{eq:spectral65}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
\\
@ -410,7 +415,8 @@ 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 $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
%================================
\subsection{Dynamical BSE within the $GW$ approximation}
@ -455,7 +461,8 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
= \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br').
\end{equation}
\xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : }
\xavier{\sout{ A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of
$\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
%================================
@ -521,6 +528,25 @@ Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities f
Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies.
%================================
\subsection{Tamm-Dancoff (TDA) approximation }
\xavier{
The analysis of the (off-diagonal) screened Coulomb potential matrix elements multiplying the $Y_{jb,s}$ coefficients
\begin{multline}
\widetilde{W}_{ib,aj}( \Om{s}{} )
= \ERI{ib}{aj} + 2 \sum_m \sERI{ib}{m} \sERI{aj}{m}
\\
\times \qty[ \frac{1}{\Om{ij}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ba}{s} - \Om{m}{\RPA} + i\eta} ].
\end{multline}
reveals on the contrary strong divergences even for low-lying excitations with e.g.
$$
\Om{ba}{s} - \Om{m}{\RPA} = \Om{s}{} - \Om{m}{\RPA} - ( \eGW{a} - \eGW{b} )
$$
Since $( \eGW{a} - \eGW{b})$ can take small to large positive or negative values, (a,b) indexes such that $(\Om{ba}{s} - \Om{m}{\RPA})$ cancels can always occur, even for low lying $\Om{s}{}$ excitations, namely negative $( \Om{s}{} - \Om{m}{\RPA} )$ energies. Such divergences may explain that in previous calculations dynamical effects were only accounted for at the TDA level. Going beyond the TDA stands beyond the present study.
}
%=================================
%================================
\subsection{Perturbative dynamical correction}
%=================================
@ -1050,6 +1076,85 @@ This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 i
%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}.
\appendix
\section{$L_0(1,3; 1',4)$ $(t_1)$-time Fourier transform}
\label{appendixA}
We derive in this Appendix Eqs.~\ref{eq:iL0} to ~\ref{eq:iL0bis}.
Defining the $t_1$-time Fourier transform of $iL_0(1,3;4,1')$ with
$(t_{1'} = t_1^{+})$
\begin{align}
[iL_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) =
\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
\end{align}
we plug-in the Fourier expansion of the Green's function, e.g.
\begin{align*}
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(x_1,x_3;\omega) e^{-i \omega \tau_{13} }
\end{align*}
with $\tau_{13} = (t_1-t_3)$ to obtain:
\begin{align}
[iL_0](x_1,3;x_{1'},4 & \;| \; \omega_1 ) =
\int \frac{ d\omega }{ 2\pi } \; G(x_1,x_3;\omega) \; \times \\ & \times \; G(x_4,x_{1'};\omega-\omega_1)
e^{ i \omega t_3 } e^{-i (\omega-\omega_1) t_4 } \nonumber
\end{align}
With the change of variable $\omega \rightarrow \omega + {\omega_1}/2$ one obtains readily
\begin{align}
[iL_0](x_1,3;x_{1'},4 &\; | \; \omega_1 ) = e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2\pi } \; G(x_1,x_3;\omega+ \frac{\omega_1}{2} ) \times \nonumber \\ & \times G(x_4,x_{1'};\omega-\frac{\omega_1}{2} ) \;
e^{ i \omega \tau_{34} }
\end{align}
with $\tau_{34} = ( t_3 - t_4 )$ and $t^{34}= (t_3+t_4)/2$.
Using now the Lehman representation of the Green's functions (Eq.~\ref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
\begin{align*}
\int \frac{ d \omega }{2i\pi} & \; G(x_1,x_3; \omega + \homu ) G(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau } = \\
& \theta( \tau ) \sum_{nj} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'}) } { \varepsilon_j + \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j + \homu ) \tau } \\
+ & \theta( \tau_ ) \sum_{jn} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_j - \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j - \homu ) \tau } \\
- & \theta(- \tau ) \sum_{nb} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'}) } { \varepsilon_b + \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b + \homu ) \tau } \\
- & \theta(- \tau ) \sum_{bn} \frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_b - \omega_1 - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b - \homu ) \tau }
\end{align*}
Projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$ selects the first and fourth lines of the right-hand side, leading to Eq.~\ref{eq:iL0bis}
with $ (\omega_1 \rightarrow \Omega_s )$.
\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ in the electron/hole product basis }
We now derive in some more details Eq.~\ref{eq:spectral65}.
Starting with:
\begin{align*}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
& = \theta(\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,s} \\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
\end{align*}
we use the relation between operators in their Eisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain:
\begin{align*}
\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = \\
& + \theta(\tau_{65}) \mel{N}{ \hpsi(x_6) e^{-i{\hat H} \tau_{65}} \hpsi^{\dagger}(x_5) }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }\\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(x_5) e^{ i{\hat H} \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
\end{align*}
with $E^N_0$ the N-electron ground-state energy and $E^N_s$ the enrgy of the s-th excited state $| N,s \rangle$. Expanding now the field operators with creation/destruction operators in the MO basis
\begin{align*}
\hpsi(x_6) = \sum_p \phi_p(x_6) {\hat a}_p \;\;\; \text{and} \;\;\;
\hpsi^{\dagger}(x_5) = \sum_q \phi_q^{*}(x_5)
{\hat a}^{\dagger}_q
\end{align*}
one obtains
\begin{align*}
\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \; \times\\
& \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p e^{-i{\hat H} \tau_{65}} {\hat a}^{\dagger}_q }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 } \\
& - \theta(-\tau_{65}) \mel{N}{ {\hat a}^{\dagger}_q e^{ i{\hat H} \tau_{65}} {\hat a}_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } \; \big]
\end{align*}
We now act on the N-electron ground-state with
\begin{align*}
e^{i{\hat H} \tau_{65} } {\hat a}^{\dagger}_p | N \rangle &=
e^{i ( E^N_0 + \varepsilon_p ) \tau_{65} } | N \rangle \\
e^{ -i{\hat H} \tau_{65} } {\hat a}_q | N \rangle &=
e^{-i ( E^N_0 - \varepsilon_q ) \tau_{65} } | N \rangle
\end{align*}
where $\lbrace \varepsilon_{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. Taking the associated bras that we plug into the MOs product basis expansion of $\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle $ one obtains Eq.~\ref{eq:spectral65}. \\
\center{ \rule{3cm}{1} }
\bibliography{BSEdyn}
\end{document}