take 1 xavier part

BSEdyn.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-05-21 09:37:18 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-26 10:43:21 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -15,7 +15,8 @@
 	School = {Universit{\'e} Pierre et Marie Curie --- Paris VI},
 	Title = {Range-Separated Density-Functional Theory for Molecular Excitation Energies},
 	Url = {https://tel.archives-ouvertes.fr/tel-01027522},
-	Year = {2014}}
+	Year = {2014},
+	Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
 
 @article{Baumeier_2012a,
 	Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Rohlfing, Michael},

BSEdyn.tex

@@ -309,10 +309,10 @@ can be expressed as a function of the one- and two-body Green's functions
 \begin{subequations}
 \begin{align}
 	\label{eq:G1}
-	G(1,1') & = - i \mel{N}{T \hpsi(1) \hpsi^{\dagger}(1')}{N},
+	G(1,2) & = - i \mel{N}{T [ \hpsi(1) \hpsi^{\dagger}(2) ] }{N},
 	\\
 	\label{eq:G2}
-	G_2(1,2;1',2') & = - \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
+	G_2(1,2;1',2') & = - \mel{N}{T [ \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1') ]}{N},
 \end{align}
 \end{subequations}
 and
@@ -331,17 +331,17 @@ In the optical limit of instantaneous electron-hole creation and destruction, im
 \begin{equation}
 \begin{split}
 	iL(1,2; 1',2') 
-	& = \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_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) 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} },
+	& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms \tau_{12} },
 \end{split}
 \end{equation}
-where $t_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
+where $\tau_{12} = t_1 - t_2$, $\theta$ 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},
+	\chi_s(\bx_1,\bx_{2}) & = \mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N,s},
 	\\
-	\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N}.
+	\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}.
 \end{align}  
 \end{subequations}  
 The $\Oms$'s are the neutral excitation energies of interest. 
@@ -349,100 +349,101 @@ 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:
 \begin{multline} \label{eq:BSE_2}
     \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
-	\theta ( t_{12} )  
+	\theta ( \tau_{12} )  
     \\
 	= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)  
     \\
-	\times \mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s} 
+	\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s} 
 	\theta [\min(t_5,t_6) - t_2].
 \end{multline}
 For the lowest neutral 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?}
+Consequently, special care has to be taken for high-lying excited states (like core or Rydberg excitations) where additional terms have to be taken into account (see Refs.~\onlinecite{Strinati_1982,Strinati_1984}).
 
 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 t_{34} } e^{ i \omega_1 t^{34} }
+	e^{ i \omega \tau_{34} } e^{ i \omega_1 \tau^{34} },
    \end{align}
-with $t_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. 
+with $\tau_{34} = t_3 - t_4$ and $\tau^{34} = (t_3 + t_4)/2$. 
 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) }
+	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 $\mu$ is the chemical potential. 
-The set $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies and the $\phi_p$'s are their associated one-body (spin)orbitals.
+The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\phi_p$'s are their associated one-body (spin)orbitals.
+%where the $\eps_{p}$'s are proper addition/removal energies such that
+%\begin{equation}
+%	e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
+%\end{equation}
+%$\hH$ 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?)}
-After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets
+Projecting $L_0(1,4;1',3)$ 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_1,3;\bx_{1'},4; \Oms) 
+	\int 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 }   
-	\qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) }  ]  
+	\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms \tau_{34}) }  ]. 
 \end{multline}
-with $\tau = t_{34}$. 
-\titou{T2: I think 3 and 4 have been swapped in the previous equation.}
 % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. 
-Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
+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.
-\begin{equation}
-	\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
-\end{equation}
-leads to the following simplified BSE kernel
-\begin{equation}
-	\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.
-\titou{T2: shall we introduce the GW approximation later on?}
-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 of creation/destruction operators.
 For example, we have
 \begin{multline}
-	\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s} 
+	\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( 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
+with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
+
+Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
 \begin{equation}
-	e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
+	\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
 \end{equation}
-$\hH$ being the exact many-body Hamiltonian. 
+leads to the following simplified BSE kernel
-The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies. 
+\begin{equation} \label{eq:Xi_GW}
-Selecting $(p,q)=(j,b)$ yields the largest components
+	\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
-$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker  
+\end{equation}
-$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. 
+where $W$ is the dynamically-screened Coulomb operator.
-Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). 
+The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}. 
-Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ \titou{(where do we need these terms?)}, 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):
+%Selecting $(p,q)=(j,b)$ yields the largest components
-\begin{equation}
+%$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker  
+%$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. 
+%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):
+\begin{equation} \label{eq:BSE-final}
 \begin{split}
-	( \e{a} - \e{i} - \Oms ) X_{ia}^{s}         
+	( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s}         
-	& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s}  \\ 
+	& + \sum_{jb} \qty[ (ia|jb) - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s}  \\ 
-	& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s} 
+	& + \sum_{jb} \qty[ (ia|bj) - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s} 
 	= 0
 \end{split}
 \end{equation}
-with an effective dynamically-screened Coulomb potential \cite{Romaniello_2009b} 
+with 
+\begin{equation}
+	(pq|rs) = \int 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} 
 \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 } ]
+	\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ],
 \end{multline}
-where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.  
+where $\Om{ib}{s} = \Oms - ( \eGW{b} - \eGW{i} )$, $\Om{ja}{s} = \Oms - ( \eGW{a} - \eGW{j} )$, and
-Following Mulliken's notations, the Coulomb matrix elements are defined as
+\begin{equation}
-\begin{align}
-	v_{ai,bj} 
-	& = \int d\br d\br' \; \phi_a(\br) \phi_i^*(\br) v(\br -\br') \phi_b^*(\br') \phi_j(\br'),
-	\\
 	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'),
+	 = \int d\br d\br' \, \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
-\end{align}
+\end{equation}
-where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with orbitals with complex conjugation.
+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 : }