Blush
This commit is contained in:
parent
6dffbead0e
commit
5e9f11a00e
53
BSEdyn.tex
53
BSEdyn.tex
@ -282,14 +282,26 @@ More details about this derivation are provided as {\SI}.
|
||||
\subsection{General dynamical BSE theory}
|
||||
%=================================
|
||||
|
||||
The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation
|
||||
The two-particle correlation function $L(1,2; 1',2')$ central to the BSE formalism relates the variation of the one-particle Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, namely:
|
||||
$$
|
||||
iL(1,2; 1',2') = \frac{ \partial G(1,1') }{ \partial U(2',2) }
|
||||
$$
|
||||
where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable. The relation between $G$ and the charge density $\; \rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility at the core of TD-DFT with $\chi(1,2) = L(1,2;1^+,2^+)$. The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$ where $0^+$ is a small positive infitesimal. This two-particle correlation function $L$ satisfies the self-consistent Bethe-Salpeter equation\cite{Strinati_1988}
|
||||
\begin{multline} \label{eq:BSE}
|
||||
L(1,2; 1',2')
|
||||
= L_0(1,2;1',2')
|
||||
\\
|
||||
+ \int d3456 \; L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'),
|
||||
\end{multline}
|
||||
with
|
||||
where $\Xi$ is 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)}
|
||||
\end{equation}
|
||||
that 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 $G$. $L$ and $L_0$ can be expressed as a function of the one-body and two-body ($G_2$) Green's functions:
|
||||
\begin{gather}
|
||||
\label{eq:L0}
|
||||
iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1'),
|
||||
@ -300,19 +312,10 @@ with
|
||||
\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, $1 \equiv (\bx_1 t_1)$ 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)}
|
||||
\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.}
|
||||
where the field 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
|
||||
The resolution of the dynamical BSE equation\cite{Strinati_1988} 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 the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one gets
|
||||
\begin{equation}
|
||||
\begin{split}
|
||||
iL(1,2; 1',2')
|
||||
@ -332,11 +335,19 @@ with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
|
||||
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.
|
||||
\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.
|
||||
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 modified dynamical Bethe-Salpeter equation:
|
||||
\begin{multline} \label{eq:BSE_2}
|
||||
\mel{N}{T \hpsi(\bx_1) &\hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
|
||||
\theta ( \tau_{12} ) = \int d3456 \times
|
||||
\\
|
||||
\times L_0(1,4;1',3) \Xi(3,5;4,6)
|
||||
\mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}
|
||||
\theta (t^{56}_m - t_2)
|
||||
\end{multline}
|
||||
with $t^{56}_m = \min(t_5,t_6)$. \titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system}, $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response 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
|
||||
The Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads, dropping the (space/spin) variables
|
||||
\begin{align} \label{eq:iL0}
|
||||
[iL_0]( \omega_1 )
|
||||
= \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} )
|
||||
@ -358,7 +369,7 @@ After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains th
|
||||
\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
|
||||
\end{multline}
|
||||
with $\tau = \tau_{34}$.
|
||||
\titou{We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. (T2: I wouldn't call that chemist notations...)}
|
||||
and where we adopt the iWe used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. (T2: I wouldn't call that chemist notations...)}
|
||||
|
||||
Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
|
||||
\begin{equation}
|
||||
@ -369,7 +380,11 @@ leads to the following simplified BSE kernel
|
||||
\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
|
||||
\end{equation}
|
||||
where $W$ is its dynamically-screened Coulomb operator.
|
||||
We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators over a complete orbital basis creation/destruction operators \titou{(T2: I don't understand why we need this spectral representation)}:
|
||||
|
||||
As a final step, we express the $\mel{N}{T \hpsi(\bx_1) &\hpsi^{\dagger}(\bx_{1}')}{N,s}$ and $\mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}$ weights present in Eq.~\ref{eq:BSE_2} in the standard
|
||||
electron-hole product 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 creation/destruction operators, with e.g. \titou{(T2: I don't understand why we need this spectral representation)}:
|
||||
\begin{multline}
|
||||
\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
|
||||
\\
|
||||
|
||||
Loading…
Reference in New Issue
Block a user