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BSEdyn.tex
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@ -235,7 +235,11 @@ in order to approximate the optical gap
\begin{equation}
\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
\end{equation}
where $\EgFun = I^\Nel - A^\Nel$ is the the fundamental gap, \cite{Bredas_2014} $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ being the ionization potential and the electron affinity of the $\Nel$-electron system.
where
\begin{equation} \label{eq:Egfun}
\EgFun = I^\Nel - A^\Nel
\end{equation}
is the the fundamental gap, \cite{Bredas_2014} $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ being the ionization potential and the electron affinity of the $\Nel$-electron system.
Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ 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 $\EgOpt < \EgFun$.
@ -270,9 +274,9 @@ Unless otherwise stated, atomic units are used.
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
In this Section, following the seminal work by Strinati, \cite{Strinati_1988} we describe, first, the theoretical foundations leading to the dynamical Bethe-Salpeter equation.
We present, in a second step, the perturbative implementation \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} of the dynamical correction as compared to the standard static approximation.
More details of the derivation are provided as {\SI}.
In this Section, following the seminal work of Strinati, \cite{Strinati_1988} we describe, first, the theoretical foundations leading to the dynamical Bethe-Salpeter equation.
We present, in a second step, the perturbative implementation of the dynamical correction \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} as compared to the standard static approximation.
More details about this derivation are provided as {\SI}.
%================================
\subsection{General dynamical BSE theory}
@ -287,22 +291,26 @@ The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation
\end{multline}
with
\begin{gather}
iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')
\label{eq:L0}
iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1'),
\\
\label{eq:L}
iL(1,2; 1',2') = - G_2(1,2;1',2') + G(1,1') G(2,2'),
\\
\label{eq:G2}
i^2 G_2(1,2;1',2') = \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
\end{gather}
where, \eg, $1 = (\bx_1,t_1)$ a space-spin plus time variable, starts with the expansion of the two-body Green's function $G_2$ and the response function $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (where $\ket{N} \equiv \ket{N,0}$ corresponds to the ground state).
where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable, starts with the expansion of the two-body Green's function $G_2$ and the response function $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (where $\ket{N} \equiv \ket{N,0}$ corresponds to the ground state).
In Eq.~\eqref{eq:BSE}, the BSE kernel
\begin{equation}
\Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)},
\Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)}
\end{equation}
takes into account the self-consistent variation of the Hartree potential
\begin{equation}
v_\text{H}(1) = - i \int d2 v(1,2) G(2,2^+),
\end{equation}
[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of the one-body Green's function $G$.
\titou{The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ in Eq.~\eqref{eq:G2} remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.}
In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with, \eg, $t_2^+ = t_2 + 0^+$ where $0^+$ is a positive infinitesimal, one gets
\begin{equation}
@ -316,14 +324,17 @@ In the optical limit of instantaneous electron-hole creation and destruction, im
with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\begin{subequations}
\begin{align}
\chi_s(\bx_1,\bx_{1'}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s},
\chi_s(\bx_1,\bx_{2}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N,s},
\\
\tchi_s(\bx_2,\bx_{2'}) & = \mel{N,s}{T \hpsi(\bx_2) \hpsi^{\dagger}(\bx_{2'})}{N}.
\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N}.
\end{align}
\end{subequations}
The $\Oms$'s are the neutral excitation energies of interest.
\titou{T2: shall we specify the physical meaning of $\chi_s$ and $\tchi_s$?}
Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE.
For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, $L_0(1,2;1',2')$ cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
\titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system}, $L_0(1,2;1',2')$ cannot contribute \titou{to?} since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
\titou{T2: I think we should specify at which level of theory this quasiparticle gap is computed. What do you think?}
The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables:
\begin{align} \label{eq:iL0}
@ -332,11 +343,11 @@ The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space
e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
\end{align}
with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation :
We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation, \ie,
\begin{equation}
G(x_1,x_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }
G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }
\end{equation}
where the $\lbrace \varepsilon_p \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ the associated one-body molecular orbitals, namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
where the $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ the associated one-body molecular orbitals, namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
\begin{multline}
\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,3;\bx_{1'},4; \Oms)
\\