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BSEdyn.tex

@@ -266,11 +266,13 @@ Unless otherwise stated, atomic units are used.
 \section{Theory}
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
+
+In this Section, we describe the theoretical foundations leading to the dynamical Bethe-Salpeter equation, following the seminal work by Strinati, \cite{Strinati_1988} presenting in a second step the perturbative implementation \cite{Rohlfing_2000,Ma_2009} of the dynamical correction as compared to the standard adiabatic calculations. More details of the derivation are provided in ...
 %================================
-\subsection{Theory for physicists}
+\subsection{General dynamical BSE theory}
 %=================================
 
-The resolution of the  Bethe-Salpeter equation  [Strinati]
+The resolution \cite{Strinati_1988} of the  Bethe-Salpeter equation:
 \begin{align*}
     L(1,2; & 1',2')  = L_0(1,2;1',2') + \\
     &+ \int d3456 \;
@@ -281,14 +283,14 @@ The resolution of the  Bethe-Salpeter equation  [Strinati]
  iL(1,2;   1',2') &= -G_2(1,2;1',2') + G(1,1')G(2,2')    \\
  i^2 G_2(1,2;1',2') &= \langle N | T {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') | N \rangle
 \end{align*}
-where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set  $  |N,s \rangle   $   of the N-electron excited state with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing 
-$t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one obtains:
+where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set  $  |N,s \rangle   $   of the N-electron system excited states, with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing 
+$t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with e.g. $t_2^+ = t_2 + 0^+$ where $0^+$ is a small positive infinitesimal, one obtains:
 \begin{align*}
  iL(1,2;1',2') &= \theta(\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) {\tilde \chi}_s(x_2,x_{2'})
- e^{ +i \Oms  \tau_{12}  } \\
+ e^{ -i \Oms  \tau_{12}  } \\
  &- \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) {\tilde \chi}_s(x_1,x_{1'})
- e^{ - i \Oms \tau_{12} }  
-     \end{align*}
+ e^{ + i \Oms \tau_{12} }  
+\end{align*}
      with $\tau_{12} = t_1 - t_2$ and
 \begin{align*}
  \chi_s(x_1,x_{1'}) = \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle \\
@@ -298,19 +300,19 @@ The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+
  
  The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$   reads, dropping the (space/spin)-variables:  
 \begin{align*}
-[iL_0]( \omega_1 )  = \frac{ 1 }{  2\pi } \int d \omega   \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2}  )   e^{  i  \omega  \tau_{34}  } e^{  i  \omega_1   t^{34}   }   
+[iL_0]( \omega_1 )  =  \int \frac{ d \omega }{  2\pi }   \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2}  )   e^{  i  \omega  \tau_{34}  } e^{  i  \omega_1   t^{34}   }   
    \end{align*}
 with $\tau_{34} = t_3 - t_4$  and 
 $t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g.
 $$
-G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) }
+G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn} \times (\varepsilon_n - \mu) }
 $$
 and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component  
 \begin{align*}
 \int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'})  L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times    \\
      &       \frac{ \phi_a^*(x_3) \phi_i(x_4)  } {   \Oms - (  \vara - \vari )  + i \eta }   
-       \Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau }   
-     +   \theta( - \tau  )   e^{i (\vara -  \hOms \tau }  \Big)  
+       \left[ \theta( \tau ) e^{i ( \vari + \hOms) \tau }   
+     +   \theta( - \tau  )   e^{i (\vara -  \hOms \tau }  \right]  
 \end{align*}
 with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel:
 $$
@@ -392,7 +394,7 @@ This leads to reduced electron-hole screening, namely larger electron-hole stabi
 %Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\}) \delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
 
 %================================
-\subsection{Theory for chemists}
+\subsection{Perturbative dynamical correction}
 %=================================
 
 For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}