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BSEdyn.tex
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@ -192,7 +192,7 @@
\begin{document} \begin{document}
\title{Dynamical Correction to the Bethe-Salpeter Equation} \title{ \textcolor{red}{Assessing} Dynamical Corrections to the Bethe-Salpeter Equation}
\author{Pierre-Fran\c{c}ois \surname{Loos}} \author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
@ -329,15 +329,14 @@ The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space
\begin{align} \label{eq:iL0} \begin{align} \label{eq:iL0}
[iL_0]( \omega_1 ) [iL_0]( \omega_1 )
= \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} ) = \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 \tau_{34} } e^{ i \omega_1 t^{34} }
\end{align} \end{align}
with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
Plugging now the Lehman representation of the one-body Green's function We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation :
\begin{equation} \begin{equation}
G(x_1,x_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \text{sgn} \times (\e{p} - \mu) } 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) }
\end{equation} \end{equation}
\titou{T2: did you introduce a new basis here?} 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 $\eta$ is a positive infinitesimal) into Eq.~\eqref{eq:iL0} and projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
\begin{multline} \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) \int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,3;\bx_{1'},4; \Oms)
\\ \\
@ -367,20 +366,20 @@ We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hps
\end{multline} \end{multline}
with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
\begin{equation} \begin{equation}
e^{i H \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N} e^{i {\hat H} \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
\end{equation} \end{equation}
\titou{T2: what is $H$?} with ${\hat H}$ the exact many-body Hamiltonian.
The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies. The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies.
Selecting $(p,q)=(j,b)$ yields the largest components Selecting $(p,q)=(j,b)$ yields the largest components
$A_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
$B_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones. $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones.
Neglecting the $B_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) : Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) :
\begin{equation} \begin{equation}
\begin{split} \begin{split}
( \e{a} - \e{i} - \Oms ) A_{ia}^{s} ( \e{a} - \e{i} - \Oms ) X_{ia}^{s}
& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] A_{jb}^{s} \\ & + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\
& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] B_{jb}^{s} & + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s}
= 0 = 0
\end{split} \end{split}
\end{equation} \end{equation}
@ -391,7 +390,17 @@ with an effective dynamically screened Coulomb potential (see Pina eq. 24):
\\ \\
\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ] \times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ]
\end{multline} \end{multline}
where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$. where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$. The Coulomb matrix elements are defined following the Mulliken notations :
\begin{align*}
v_{ai,bj} &= \int d{\bf r} d{\bf r}' \; \phi_a({\bf r}) \phi_i^*({\bf r}) v({\bf r} -{\bf r}')
\phi_b^*({\bf r}') \phi_j({\bf r}') \\
W_{ij,ab}({\omega}) &= \int d{\bf r} d{\bf r}' \; \phi_i({\bf r}) \phi_j^*({\bf r}) W({\bf r} ,{\bf r}'; \omega)
\phi_a^*({\bf r}') \phi_b({\bf r}')
\end{align*}
where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with MO with complex conjugation.
\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: In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
\begin{multline} \begin{multline}
W_{ij,ab}(\omega) W_{ij,ab}(\omega)