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clean equations in appendices

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Pierre-Francois Loos 2020-07-02 22:37:15 +02:00
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BSEdyn.tex
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@ -524,8 +524,7 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\\
\times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ].
\end{multline}
\titou{One can verify that in the static limit, that can be obtained with
$\Om{m}{\RPA} \rightarrow +\infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, and the dynamical BSE formalism recovers the form of the standard BSE formalism.}
One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit.
Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances.
Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations.
Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
@ -542,11 +541,12 @@ The analysis of the (off-diagonal) screened Coulomb potential matrix elements mu
\\
\times \qty[ \frac{1}{\Om{ij}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ba}{s} - \Om{m}{\RPA} + i\eta} ].
\end{multline}
reveals on the contrary strong divergences even for low-lying excitations with e.g.
reveals on the contrary strong divergences even for low-lying excitations with, \eg,
$$
\Om{ba}{s} - \Om{m}{\RPA} = \Om{s}{} - \Om{m}{\RPA} - ( \eGW{a} - \eGW{b} )
$$
Since $( \eGW{a} - \eGW{b})$ can take small to large positive or negative values, (a,b) indexes such that $(\Om{ba}{s} - \Om{m}{\RPA})$ cancels can always occur, even for low lying $\Om{s}{}$ excitations, namely negative $( \Om{s}{} - \Om{m}{\RPA} )$ energies. Such divergences may explain that in previous calculations dynamical effects were only accounted for at the TDA level. Going beyond the TDA stands beyond the present study.
Since $( \eGW{a} - \eGW{b})$ can take small to large positive or negative values, $(a,b)$ indexes such that $(\Om{ba}{s} - \Om{m}{\RPA})$ cancels can always occur, even for low lying $\Om{s}{}$ excitations, namely negative $( \Om{s}{} - \Om{m}{\RPA} )$ energies. Such divergences may explain that in previous calculations dynamical effects were only accounted for at the TDA level.
Going beyond the TDA is beyond the present study.
}
%=================================
@ -743,6 +743,7 @@ The $GW$ calculations performed to obtain the screened Coulomb operator and the
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
\titou{Comment on evGW.}
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
@ -1079,7 +1080,6 @@ This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 i
%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}.
\begin{widetext}
\appendix
\section{$L_0(1,3; 1',4)$ $(t_1)$-time Fourier transform}
@ -1089,91 +1089,113 @@ In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
$(t_{1'} = t_1^{+})$
\begin{align}
[L_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) = -i
[L_0](\bx_1,3;\bx_{1'},4 \; | \; \omega_1 ) = -i
\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
\end{align}
we plug-in the Fourier expansion of the Green's function, e.g.
\begin{align*}
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(x_1,x_3;\omega) e^{-i \omega \tau_{13} }
G(1,3) = \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} }
\end{align*}
with $\tau_{13} = (t_1-t_3)$ to obtain:
\begin{equation}
[L_0](x_1,3;x_{1'},4 \;| \; \omega_1 ) =
\int \frac{ d\omega }{ 2i\pi } \; G(x_1,x_3;\omega) \; G(x_4,x_{1'};\omega-\omega_1)
\begin{multline}
[L_0](\bx_1,3;\bx_{1'},4 \;| \; \omega_1 ) =
\\
\int \frac{ d\omega }{ 2i\pi } \; G(\bx_1,\bx_3;\omega) \; G(\bx_4,\bx_{1'};\omega-\omega_1)
e^{ i \omega t_3 } e^{-i (\omega-\omega_1) t_4 } \nonumber
\end{equation}
\end{multline}
With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
\begin{equation}
[L_0](x_1,3;x_{1'},4 \; | \; \omega_1 ) = e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2i\pi } \; G\qty(x_1,x_3;\omega+ \frac{\omega_1}{2} ) G\qty(x_4,x_{1'};\omega-\frac{\omega_1}{2} ) \;
\begin{multline}
[L_0](\bx_1,3;\bx_{1'},4 \; | \; \omega_1 ) =
\\
e^{ i \omega_1 t^{34} }
\int \frac{ d\omega }{ 2i\pi } \; G\qty(\bx_1,\bx_3;\omega+ \frac{\omega_1}{2} ) G\qty(\bx_4,\bx_{1'};\omega-\frac{\omega_1}{2} ) \;
e^{ i \omega \tau_{34} }
\end{equation}
\end{multline}
with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
Using now the Lehman representation of the Green's functions (Eq.~\eqref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower) half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
\begin{equation}
\begin{split}
\int \frac{ d \omega }{2i\pi} \; G\qty(x_1,x_3; \omega + \homu ) G\qty(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau }
\begin{equation}
\begin{split}
& \int \frac{ d \omega }{2i\pi} \; G\qty(\bx_1,\bx_3; \omega + \homu ) G\qty(\bx_4,\bx_{1'}; \omega - \homu ) e^{ i \omega \tau }
\\
& = \sum_{bj}
\frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'})} { \omega_1 - (\e{b} - \e{j}) + i\eta }
\frac{ \phi_b(\bx_1) \phi_b^*(\bx_3) \phi_j(\bx_4) \phi_j^*(\bx_{1'})} { \omega_1 - (\e{b} - \e{j}) + i\eta }
\qty[ \theta(\tau) e^{i ( \e{j} + \homu ) \tau } + \theta(-\tau) e^{i ( \e{b} - \homu ) \tau } ]
\\
& - \sum_{bj} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'})} { \omega_1 + (\e{b} - \e{j} ) -i\eta }
& - \sum_{bj} \frac{ \phi_j(\bx_1) \phi_j^*(\bx_3) \phi_b(\bx_4) \phi_b^*(\bx_{1'})} { \omega_1 + (\e{b} - \e{j} ) -i\eta }
\qty[ \theta(\tau) e^{i ( \e{j} - \homu ) \tau } + \theta(-\tau) e^{i ( \e{b} + \homu ) \tau } ]
\\
& + \sum_{ab} \text{ pp terms } + \sum_{ij} \text{ hh terms }
\end{split}
\end{equation}
where (pp) and (hh) labels particle-particle and hole-hole channels neglected here.
Projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$ selects the first line of the RHS, leading to Eq.~\eqref{eq:iL0bis}
Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ selects the first line of the RHS, leading to Eq.~\eqref{eq:iL0bis}
with $ (\omega_1 \to \Omega_s )$.
\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ in the electron/hole product basis }
We now derive in some more details Eq.~\eqref{eq:spectral65}.
Starting with
\begin{equation}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
= \theta(\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,s}
- \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
\end{equation}
\begin{equation}
\begin{split}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
& = \theta(+\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5) }{N,s}
\\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
\end{split}
\end{equation}
we use the relation between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain
\begin{equation}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s} = \\
+ \theta(\tau_{65}) \mel{N}{ \hpsi(x_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(x_5) }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }
- \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(x_5) e^{ i\hH \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
\end{equation}
\begin{equation}
\begin{split}
& \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s} = \\
& + \theta(\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5) }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }
\\
& - \theta(-\tau_{65}) \mel{N}{ \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
\end{split}
\end{equation}
with $E^N_0$ the $N$-electron ground-state energy and $E^N_s$ the energy of the $s$th excited state $\ket{N,s}$.
Expanding now the field operators with creation/destruction operators in the orbital basis
\begin{align*}
\hpsi(x_6) & = \sum_p \phi_p(x_6) \ha_p
&
\hpsi^{\dagger}(x_5) & = \sum_q \phi_q^{*}(x_5) \ha^{\dagger}_q
\end{align*}
\begin{subequations}
\begin{align}
\hpsi(\bx_6) & = \sum_p \phi_p(\bx_6) \ha_p
\\
\hpsi^{\dagger}(\bx_5) & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q
\end{align}
\end{subequations}
one gets
\begin{equation}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s} =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5)
\qty[ \theta(\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }
- \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } ]
\end{equation}
\begin{equation}
\begin{split}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s}
\\
= \sum_{pq} \phi_p(\bx_6) \phi_q^{*}(\bx_5)
[ & \theta(+\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }
\\
- & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } ]
\end{split}
\end{equation}
We now act on the $N$-electron ground-state with
\begin{align*}
e^{i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=
e^{i ( E^N_0 + \e{p} ) \tau_{65} } \ket{N} &
\begin{subequations}
\begin{align}
e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=
e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N}
\\
e^{ -i\hH \tau_{65} } \ha_q \ket{N} &=
e^{-i ( E^N_0 - \e{q} ) \tau_{65} } \ket{N}
\end{align*}
e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N}
\end{align}
\end{subequations}
where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies.
Taking the associated bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s}$ one obtains:
\begin{equation}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s} =
\sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5)
\qty[ \theta(\tau_{65}) \mel{N}{ \ha_p \ha^{\dagger}_q }{N,s} e^{ -i \e{p} \tau_{65} } e^{ - i \Omega_s t_5 }
- \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q \ha_p }{N,s} e^{ -i \e{q} \tau_{65} } e^{ - i \Omega_s t_6 } ]
\end{equation}
\begin{equation}
\begin{split}
\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,s}
\\
= \sum_{pq} \phi_p(\bx_6) \phi_q^{*}(\bx_5)
[ & \theta(+ \tau_{65}) \mel{N}{ \ha_p \ha^{\dagger}_q }{N,s} e^{ -i \e{p} \tau_{65} } e^{ - i \Omega_s t_5 }
\\
- & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q \ha_p }{N,s} e^{ -i \e{q} \tau_{65} } e^{ - i \Omega_s t_6 } ]
\end{split}
\end{equation}
leading to Eq.~\eqref{eq:spectral65} with $\Omega_s = (E^N_s - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$.
\end{widetext}
\bibliography{BSEdyn}