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Pierre-Francois Loos 2020-05-26 22:09:30 +02:00
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BSEdyn.tex
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@ -1,4 +1,4 @@
\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
@ -324,7 +324,7 @@ is the BSE kernel that takes into account the self-consistent variation of the H
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 $G$.
In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, 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.
In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron 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 $L_0$ 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 $t_{1'} = t_1^+$, one gets
@ -346,7 +346,7 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\end{subequations}
The $\Oms$'s are the neutral excitation energies of interest.
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 BSE:
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 a modified dynamical BSE, which reads
\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} )
@ -363,9 +363,9 @@ Dropping the (space/spin) variables, the Fourier components with respect to $t_1
\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 \tau^{34} },
e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} },
\end{align}
with $\tau_{34} = t_3 - t_4$ and $\tau^{34} = (t_3 + t_4)/2$.
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, \ie,
\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) },
@ -379,9 +379,9 @@ The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper
%$\hH$ being the exact many-body Hamiltonian.
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?)}
Projecting the $( \omega_1 = \Oms )$ Fourier component $L_0(\bx_1,4;\bx_{1'},3; \Oms)$ onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields
Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Oms )$ onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields
\begin{multline}
\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms)
\iint d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms)
\\
=
\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
@ -395,16 +395,14 @@ For example, we have
\begin{multline}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
\\
= - \qty( e^{ -i \Omega_s t^{56} } ) \sum_{pq} \phi_p(\bx_6) \phi_q^*(\bx_5)
= - \qty( e^{ -i \Omega_s \tau^{56} } ) \sum_{pq} \phi_p(\bx_6) \phi_q^*(\bx_5)
\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
\\
\times \qty[ \theta( \tau_{56} ) e^{- i ( \e{p} - \hOms ) \tau_{56} } + \theta( - \tau_{56} ) e^{ - i ( \e{q} + \hOms) \tau_{56} } ]
\end{multline}
with $t^{56} = (t_5 + t_6)/2$ and $\tau_{56} = t_5 -t_6$.
with $\tau^{56} = (t_5 + t_6)/2$ and $\tau_{56} = t_5 -t_6$.
%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
\xavier{ The $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ are the largest contributions to the
$\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ weight,
while the $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ are much smaller.}
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),
@ -421,51 +419,60 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal
%Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
%Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$,
Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE):
Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE):
\begin{equation} \label{eq:BSE-final}
\begin{split}
( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s}
& + \sum_{jb} \qty[ (ia|jb) - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\
& + \sum_{jb} \qty[ (ia|bj) - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s}
= 0
= 0,
\end{split}
\end{equation}
with
with $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$.
Neglecting the term $Y_{jb}^{s}$ in the dBSE, which is much smaller than $X_{jb}^{s}$, leads to the well-known Tamm-Dancoff approximation (TDA).
In Eq.~\eqref{eq:BSE-final},
\begin{equation}
(pq|rs) = \int d\br d\br' \, \phi_p^*(\br) \phi_q(\br) v(\br -\br') \phi_r^*(\br') \phi_s(\br'),
(pq|rs) = \iint d\br d\br' \, \phi_p^*(\br) \phi_q(\br) v(\br -\br') \phi_r^*(\br') \phi_s(\br'),
\end{equation}
and an effective dynamically-screened Coulomb potential \cite{Romaniello_2009b}
are the bare two-electron integrals, $\phi_p(\br)$ is a spatial orbital, and
\begin{multline}
\widetilde{W}_{ij,ab}(\Oms)
= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)
\\
\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ],
\end{multline}
where $\Om{ib}{s} = \Oms - ( \eGW{b} - \eGW{i} )$, $\Om{ja}{s} = \Oms - ( \eGW{a} - \eGW{j} )$, and
is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Oms - ( \eGW{q} - \eGW{p} )$ and
\begin{equation}
W_{ij,ab}({\omega})
= \int d\br d\br' \, \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
W_{pq,rs}({\omega})
= \iint d\br d\br' \, \phi_p(\br) \phi_q^*(\br) W(\br ,\br'; \omega) \phi_r^*(\br') \phi_s(\br').
\end{equation}
Neglecting the terms $Y_{jb}^{s}$ in Eq.~\eqref{eq:BSE-final} leads to the well-known Tamm-Dancoff approximation (TDA).
\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 : }
In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
\titou{T2: I propose to move what's below in the next section. What do you think Xavier?}
In the present study, we consider the exact spectral representation of $W(\omega)$ at the RPA level, which reads
\begin{multline}
W_{ij,ab}(\omega)
= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
\\
\times \qty( \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } )
\times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ]
\end{multline}
($\Om{m}{\RPA} > 0 $) so that
where
\begin{equation}
\label{eq:sERI}
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
\end{equation}
are the spectral weights.
Because $\Om{m}{\RPA} > 0$, we have
\begin{multline}
\widetilde{W}_{ij,ab}( \Oms )
= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
\\
\times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} )
\end{multline}
\textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that
e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g.
\titou{Due to excitonic effects, the lowest BSE excitation energy, ${\Omega}_1$, stands lower than the lowest RPA excitation energy, $\Omega_m^{RPA}$, so that
e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances.
Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g.
$$
\abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} } < \frac{1}{ \Omega_m^{RPA}}
$$