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BSEdyn.tex
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@ -266,11 +266,13 @@ Unless otherwise stated, atomic units are used.
\section{Theory} \section{Theory}
\label{sec:theory} \label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
In this Section, we describe the theoretical foundations leading to the dynamical Bethe-Salpeter equation, following the seminal work by Strinati, \cite{Strinati_1988} presenting in a second step the perturbative implementation \cite{Rohlfing_2000,Ma_2009} of the dynamical correction as compared to the standard adiabatic calculations. More details of the derivation are provided in ...
%================================ %================================
\subsection{Theory for physicists} \subsection{General dynamical BSE theory}
%================================= %=================================
The resolution of the Bethe-Salpeter equation [Strinati] The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation:
\begin{align*} \begin{align*}
L(1,2; & 1',2') = L_0(1,2;1',2') + \\ L(1,2; & 1',2') = L_0(1,2;1',2') + \\
&+ \int d3456 \; &+ \int d3456 \;
@ -281,13 +283,13 @@ The resolution of the Bethe-Salpeter equation [Strinati]
iL(1,2; 1',2') &= -G_2(1,2;1',2') + G(1,1')G(2,2') \\ iL(1,2; 1',2') &= -G_2(1,2;1',2') + G(1,1')G(2,2') \\
i^2 G_2(1,2;1',2') &= \langle N | T {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') | N \rangle i^2 G_2(1,2;1',2') &= \langle N | T {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') | N \rangle
\end{align*} \end{align*}
where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $ |N,s \rangle $ of the N-electron excited state with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $ |N,s \rangle $ of the N-electron system excited states, with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing
$t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one obtains: $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with e.g. $t_2^+ = t_2 + 0^+$ where $0^+$ is a small positive infinitesimal, one obtains:
\begin{align*} \begin{align*}
iL(1,2;1',2') &= \theta(\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) {\tilde \chi}_s(x_2,x_{2'}) iL(1,2;1',2') &= \theta(\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) {\tilde \chi}_s(x_2,x_{2'})
e^{ +i \Oms \tau_{12} } \\ e^{ -i \Oms \tau_{12} } \\
&- \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) {\tilde \chi}_s(x_1,x_{1'}) &- \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) {\tilde \chi}_s(x_1,x_{1'})
e^{ - i \Oms \tau_{12} } e^{ + i \Oms \tau_{12} }
\end{align*} \end{align*}
with $\tau_{12} = t_1 - t_2$ and with $\tau_{12} = t_1 - t_2$ and
\begin{align*} \begin{align*}
@ -298,19 +300,19 @@ The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+
The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables: The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables:
\begin{align*} \begin{align*}
[iL_0]( \omega_1 ) = \frac{ 1 }{ 2\pi } \int d \omega \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} } [iL_0]( \omega_1 ) = \int \frac{ d \omega }{ 2\pi } \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
\end{align*} \end{align*}
with $\tau_{34} = t_3 - t_4$ and with $\tau_{34} = t_3 - t_4$ and
$t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g. $t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g.
$$ $$
G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) } G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn} \times (\varepsilon_n - \mu) }
$$ $$
and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component
\begin{align*} \begin{align*}
\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times \\ \int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times \\
& \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta } & \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta }
\Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau } \left[ \theta( \tau ) e^{i ( \vari + \hOms) \tau }
+ \theta( - \tau ) e^{i (\vara - \hOms \tau } \Big) + \theta( - \tau ) e^{i (\vara - \hOms \tau } \right]
\end{align*} \end{align*}
with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel: with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel:
$$ $$
@ -392,7 +394,7 @@ This leads to reduced electron-hole screening, namely larger electron-hole stabi
%Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$. %Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
%================================ %================================
\subsection{Theory for chemists} \subsection{Perturbative dynamical correction}
%================================= %=================================
For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988} For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}