saving work: still working on Xav part

BSEdyn.tex

@@ -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.
+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, $E_s^{\Nel}$ 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.
+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:
+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,
-$$
+\begin{equation}
-iL(1,2; 1',2') =   \frac{ \partial  G(1,1') }{ \partial  U(2',2) }
+	iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)}
-$$
+\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 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}  
+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(1,2; 1',2') = L_0(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 }
+    \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
-	\theta ( \tau_{12} )  = \int d3456 \times
+	\theta ( t_{12} )  
     \\
-      \times  L_0(1,4;1',3) \Xi(3,5;4,6)  
+	= \int d3456 \, 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)
+	\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.
+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: I think we should specify at which level of theory this quasiparticle gap is computed. What do you think?}
+\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}. 
+where $\mu$ 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?)}
+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.
-After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component  
+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 }   
+	\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4)  e^{i \Oms t^{34} }} {   \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) }  ]  
+	\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 
+with $\tau = t_{34}$. % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. 
- $(i,j)$/$(a,b)$ 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 
+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.  
-electron-hole product space, with 
+This is done by  expanding the field operators  over a complete orbital basis creation/destruction operators, with \eg, 
-$(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ 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)}:
 \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}