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BSEdyn.tex
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@ -52,7 +52,6 @@
\newcommand{\co}{\text{x}}
%
\newcommand{\Nel}{N}
\newcommand{\Norb}{N_\text{orb}}
\newcommand{\Nocc}{O}
\newcommand{\Nvir}{V}
@ -233,14 +232,14 @@ which is itself a corrected version of the Kohn-Sham (KS) gap
\end{equation}
in order to approximate the optical gap
\begin{equation}
\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
\end{equation}
where
\begin{equation} \label{eq:Egfun}
\EgFun = I^\Nel - A^\Nel
\EgFun = I^N - A^N
\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.
is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ being the ionization potential and the electron affinity of the $N$-electron system.
Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ 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 $\EgOpt < \EgFun$.
Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
@ -275,33 +274,36 @@ Unless otherwise stated, atomic units are used.
%%%%%%%%%%%%%%%%%%%%%%%%
In this Section, following Strinati's seminal work, \cite{Strinati_1988} we first derive in some details 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.
We present, in a second step, the perturbative implementation of the dynamical correction as compared to the standard static approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
More details about this derivation are provided as {\SI}.
%================================
\subsection{General dynamical BSE theory}
%=================================
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}
The two-particle correlation function $L(1,2; 1',2')$ --- a central quantity in the BSE formalism --- relates the variation of the one-body Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, \ie,
\begin{equation}
iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)}
\end{equation}
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 $\chi(1,2) = L(1,2;1^+,2^+)$ at the core of TD-DFT.
(The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$ where $0^+$ is a small positive infinitesimal.)
The two-body 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')
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'),
+ \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'),
\end{multline}
where $\Xi$ is the BSE kernel
where
\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
is the BSE kernel 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^+),
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:
[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 $G$.
$L$ and $L_0$ can be expressed as a function of the one- and two-body ($G_2$) Green's functions as follows:
\begin{gather}
\label{eq:L0}
iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1'),
@ -312,19 +314,19 @@ that takes into account the self-consistent variation of the Hartree potential
\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 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.
where the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
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).
The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $G_2$ and $L$ 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 $t_{1'} = t_1^+$, one gets
\begin{equation}
\begin{split}
iL(1,2; 1',2')
& = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms \tau_{12} }
& = \theta(+t_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms t_{12} }
\\
& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms \tau_{12} },
& - \theta(-t_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms t_{12} },
\end{split}
\end{equation}
with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
where $t_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\begin{subequations}
\begin{align}
\chi_s(\bx_1,\bx_{2}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N,s},
@ -333,43 +335,44 @@ with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\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 $L(6,2;5,2')$ functions, 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:
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 [see Eq.~\eqref{eq: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
\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
\theta ( t_{12} )
\\
\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)
= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)
\\
\times \mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}
\theta [\min(t_5,t_6) - 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} due to excitonic effects, $L_0(1,2;1',2')$ 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?}
For the lowest excitation energies falling in the fundamental gap of the system (\ie, $\Oms < \EgFun$), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response due to excitonic effects since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
\titou{T2: Xavier, should we mention the consequences of this more explicitly?}
The Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads, dropping the (space/spin) variables
Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
\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} )
e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
e^{ i \omega t_{34} } e^{ i \omega_1 t^{34} }
\end{align}
with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
with $t_{34} = t_3 - t_4$ 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, \ie,
\begin{equation}
\begin{equation} \label{eq:G-Lehman}
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 \titou{$\mu$ is the chemical potential}.
The set $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body \titou{(spin)} orbitals, \titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
where $\mu$ is the chemical potential.
The set $\lbrace \e{p} \rbrace$ in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body (spin)orbitals.
In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets
\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)
= e^{i \Oms t^{34} } \times \\
\times
\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) } { \Oms - ( \e{a} - \e{i} ) + i \eta }
\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
\\
=
\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
\qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
\end{multline}
with $\tau = \tau_{34}$ and where
$(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
with $\tau = t_{34}$. % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
\begin{equation}
\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
@ -380,17 +383,15 @@ leads to the following simplified BSE kernel
\end{equation}
where $W$ is its dynamically-screened Coulomb operator.
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)}:
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, 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 \eg,
\begin{multline}
\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
\\
= - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4)
\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
\\
\times \qty[ \theta( \tau_{34} ) e^{- i ( \e{p} - \hOms ) \tau_{34} } + \theta( -\tau_{34} ) e^{ - i ( \e{q} + \hOms) \tau_{34} } ]
\times \qty[ \theta( t_{34} ) e^{- i ( \e{p} - \hOms ) t_{34} } + \theta( - t_{34} ) e^{ - i ( \e{q} + \hOms) t_{34} } ]
\end{multline}
where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that
\begin{equation}