Merge pull request #3 from pfloos/overleaf-2020-07-01-0750

Overleaf 2020 07 01 0750

BSEdyn.bib

@@ -12796,21 +12796,6 @@
 	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},
@@ -14439,3 +14424,16 @@
 	Year = {2016},
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113},
 	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
+
+@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},
+journal = {J. Chem. Theory Comput.},
+volume = {10},
+number = {3},
+pages = {1212--1218},
+year = {2014},
+doi = {10.1021/ct401101u},
+}
+
+

BSEdyn.tex

@@ -1100,7 +1100,7 @@ with $\tau_{13} = (t_1-t_3)$ to obtain:
 	\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{equation}
-With the change of variable $\omega \\to \omega + {\omega_1}/2$ one obtains readily
+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} )  \;
@@ -1134,7 +1134,7 @@ with $ (\omega_1 \rightarrow \Omega_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}
-we use the   relation between operators in their HeEisenberg 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{equation}
      \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 }
@@ -1143,9 +1143,8 @@ we use the   relation between operators in their HeEisenberg and Schr\"{o}dinger
 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^{\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{equation}
@@ -1157,7 +1156,7 @@ one obtains
 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*}