blush

BSEdyn.bib

@@ -12783,21 +12783,6 @@
 	Year = {2016},
 	Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4940139}}
 
-@article{Boulanger_2014,
-	Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
-	Doi = {10.1021/ct401101u},
-	File = {/Users/loos/Zotero/storage/KTW3SS9F/Boulanger_2014.pdf},
-	Issn = {1549-9618, 1549-9626},
-	Journal = {J. Chem. Theory Comput.},
-	Language = {en},
-	Month = mar,
-	Number = {3},
-	Pages = {1212--1218},
-	Title = {Fast and {{Accurate Electronic Excitations}} in {{Cyanines}} with the {{Many}}-{{Body Bethe}}\textendash{}{{Salpeter Approach}}},
-	Volume = {10},
-	Year = {2014},
-	Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct401101u}}
-
 @article{Bruneval_2009,
 	Author = {Bruneval, Fabien},
 	Doi = {10.1103/PhysRevLett.103.176403},
@@ -14429,16 +14414,13 @@
 
 @article{Boulanger_2014,
 author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
-title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe–Salpeter Approach},
+title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe-Salpeter Approach},
-journal = {J. Chem. Theory Comput. },
+journal = {J. Chem. Theory Comput.},
 volume = {10},
 number = {3},
-pages = {1212-1218},
+pages = {1212--1218},
 year = {2014},
 doi = {10.1021/ct401101u},
-    note ={PMID: 26580191},
-URL = {  https://doi.org/10.1021/ct401101u},
-eprint = {   https://doi.org/10.1021/ct401101u}
 }
 
 

BSEdyn.tex

@@ -207,9 +207,8 @@
 \begin{abstract}
 Similar to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
 Here, going beyond the static approximation, we compute the dynamical correction in the electron-hole screening for molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
-The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random phase approximation. \xavier{
+The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random phase approximation.
-\sout{Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approximation by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.}
+Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements brought by dynamical corrections.
-Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements induced by dynamical corrections. }
 %\\
 %\bigskip
 %\begin{center}
@@ -1078,6 +1077,7 @@ 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}
@@ -1095,28 +1095,32 @@ $(t_{1'} = t_1^{+})$
      G(1,3) =  \int \frac{ d\omega }{ 2\pi } G(x_1,x_3;\omega) e^{-i \omega \tau_{13} }
 \end{align*}
 with $\tau_{13} = (t_1-t_3)$ to obtain:
-\begin{align}  
+\begin{equation}  
-	[L_0](x_1,3;x_{1'},4 & \;| \;  \omega_1 )   =
+	[L_0](x_1,3;x_{1'},4 \;| \;  \omega_1 )   =
-	\int \frac{ d\omega }{ 2i\pi } \;  G(x_1,x_3;\omega)  \; \times   \\ & \times \; G(x_4,x_{1'};\omega-\omega_1)  
+	\int \frac{ d\omega }{ 2i\pi } \;  G(x_1,x_3;\omega) \; G(x_4,x_{1'};\omega-\omega_1)  
 	  e^{ i \omega t_3 }	e^{-i (\omega-\omega_1) t_4 } \nonumber
-   \end{align}
+   \end{equation}
-With the change of variable $\omega \rightarrow \omega + {\omega_1}/2$ one obtains readily
+With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
-\begin{align}  
+\begin{equation}  
-	[L_0](x_1,3;x_{1'},4 &\; | \; \omega_1 )     = e^{ i  \omega_1  t^{34} }
+	[L_0](x_1,3;x_{1'},4 \; | \; \omega_1 )     = e^{ i  \omega_1  t^{34} }
-	\int \frac{ d\omega }{ 2i\pi } \;   G(x_1,x_3;\omega+ \frac{\omega_1}{2} )  \times  \nonumber \\ &  \times G(x_4,x_{1'};\omega-\frac{\omega_1}{2} )  \;
+	\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} )  \;
   e^{ i \omega \tau_{34} }	
-   \end{align}
+   \end{equation}
    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.~\ref{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{align*}
+    \begin{equation}
- \int & \frac{ d \omega }{2i\pi}    \; G(x_1,x_3; \omega +  \homu  )    G(x_4,x_{1'}; \omega - \homu  )  e^{  i    \omega  \tau    } 
+    \begin{split}
-         = \sum_{bj} \Bigg\{   \\
+    \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    } 
-    &   \frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'})} { \omega_1   - (\varepsilon_b - \varepsilon_j) + i\eta    }  
+         & = \sum_{bj}   
-   \left[  \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau }  + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } \right] \\
+     \frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'})} { \omega_1   - (\varepsilon_b - \varepsilon_j) + i\eta    }  
-  -  &  \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'})} {  \omega_1   + (\varepsilon_b - \varepsilon_j ) -i\eta } 
+   \qty[  \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau }  + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } ] 
-     \left[  \theta(\tau)  e^{i ( \varepsilon_j - \homu ) \tau } + \theta(-\tau)  e^{i ( \varepsilon_b + \homu ) \tau }    \right] \\
+  \\
- & \Bigg\} +  \sum_{ab} \text{ pp terms }  + \sum_{ij} \text{ hh terms }
+  & -  \sum_{bj} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'})} {  \omega_1   + (\varepsilon_b - \varepsilon_j ) -i\eta } 
-\end{align*}
+     \qty[  \theta(\tau)  e^{i ( \varepsilon_j - \homu ) \tau } + \theta(-\tau)  e^{i ( \varepsilon_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.~\ref{eq:iL0bis}
 with $ (\omega_1 \rightarrow \Omega_s )$.
@@ -1125,48 +1129,48 @@ with $ (\omega_1 \rightarrow \Omega_s )$.
    
   We now derive in some more details Eq.~\ref{eq:spectral65}. 
    Starting with:
-   \begin{align*}
+   \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(6) \hpsi^{\dagger}(5)  }{N,s} 
-     & - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
+     - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(5) \hpsi(6) }{N,s}
-   \end{align*}
+   \end{equation}
-we use the   relation between operators in their Eisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain:
+we use the   relation between operators in their Heisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain:
-   \begin{align*}
+   \begin{equation}
-     \langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = \\
+     \langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = \\
-     & +  \theta(\tau_{65}) \mel{N}{ \hpsi(x_6) e^{-i{\hat H} \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(x_6) e^{-i{\hat H} \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{\hat H} \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
+     - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(x_5) e^{ i{\hat H} \tau_{65}} \hpsi(x_6) }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 }
-   \end{align*}
+   \end{equation}
 with $E^N_0$ the N-electron ground-state energy and $E^N_s$ the enrgy of the s-th excited state $| N,s \rangle$. Expanding now the field operators with creation/destruction operators in the MO basis   
 \begin{align*}
-    \hpsi(x_6)  = \sum_p \phi_p(x_6) {\hat a}_p \;\;\; \text{and} \;\;\;
+    \hpsi(x_6)  & = \sum_p \phi_p(x_6) {\hat a}_p 
-    \hpsi^{\dagger}(x_5)  = \sum_q \phi_q^{*}(x_5) 
+    & 
-    {\hat a}^{\dagger}_q
+    \hpsi^{\dagger}(x_5)  & = \sum_q \phi_q^{*}(x_5) {\hat a}^{\dagger}_q
 \end{align*}
 one obtains
-   \begin{align*}
+   \begin{equation}
-     \langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = 
+     \langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = 
-     \sum_{pq} \phi_p(x_6)  \phi_q^{*}(x_5) \; \times\\
+     \sum_{pq} \phi_p(x_6)  \phi_q^{*}(x_5) \; 
-     & \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p e^{-i{\hat H} \tau_{65}} {\hat a}^{\dagger}_q  }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }  \\
+     \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p e^{-i{\hat H} \tau_{65}} {\hat a}^{\dagger}_q  }{N,s} e^{ i E^N_0 t_6 } e^{ - i E^N_s t_5 }  \\
-     &   - \theta(-\tau_{65}) \mel{N}{  {\hat a}^{\dagger}_q e^{ i{\hat H} \tau_{65}} {\hat a}_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } \; \big]
+      - \theta(-\tau_{65}) \mel{N}{  {\hat a}^{\dagger}_q e^{ i{\hat H} \tau_{65}} {\hat a}_p }{N,s} e^{ i E^N_0 t_5 } e^{ - i E^N_s t_6 } \; \big]
-   \end{align*}
+   \end{equation}
-We now act on the N-electron ground-state with
+We now act on the $N$-electron ground-state with
   \begin{align*}
   e^{i{\hat H} \tau_{65} } {\hat a}^{\dagger}_p | N \rangle &=
-    e^{i ( E^N_0 + \varepsilon_p ) \tau_{65} } | N \rangle \\
+    e^{i ( E^N_0 + \varepsilon_p ) \tau_{65} } | N \rangle &
       e^{ -i{\hat H} \tau_{65} } {\hat a}_q | N \rangle &=
     e^{-i ( E^N_0 - \varepsilon_q ) \tau_{65} } | N \rangle
   \end{align*}
-  where $\lbrace \varepsilon_{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. Taking the associated  bras that we plug into the MOs product basis expansion of $\langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle $  one obtains:
+  where $\lbrace \varepsilon_{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. Taking the associated  bras that we plug into the MOs product basis expansion of $\langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle $  one obtains:
-   \begin{align*}
+   \begin{equation}
-     \langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = 
+     \langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = 
-     \sum_{pq} \phi_p(x_6)  \phi_q^{*}(x_5) \; \times\\
+     \sum_{pq} \phi_p(x_6)  \phi_q^{*}(x_5) \; 
-     & \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p   {\hat a}^{\dagger}_q  }{N,s} e^{ -i \varepsilon_p \tau_{65} } e^{ - i \Omega_s t_5 }  \\
+     \big[ \; \theta(\tau_{65}) \mel{N}{ {\hat a}_p   {\hat a}^{\dagger}_q  }{N,s} e^{ -i \varepsilon_p \tau_{65} } e^{ - i \Omega_s t_5 } 
-     &   - \theta(-\tau_{65}) \mel{N}{  {\hat a}^{\dagger}_q {\hat a}_p }{N,s} e^{ -i \varepsilon_q \tau_{65}  } e^{ - i \Omega_s t_6 } \; \big]
+     - \theta(-\tau_{65}) \mel{N}{  {\hat a}^{\dagger}_q {\hat a}_p }{N,s} e^{ -i \varepsilon_q \tau_{65}  } e^{ - i \Omega_s t_6 } \; \big]
-   \end{align*}
+   \end{equation}
  leading to  Eq.~\ref{eq:spectral65} with $\Omega_s = (E^N_s - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$.  \\
   
-  \center{ \rule{3cm}{1} }
+\end{widetext}
-  
+
 \bibliography{BSEdyn}
 
 \end{document}