diff --git a/BSEdyn.bib b/BSEdyn.bib index 8e39645..4125427 100644 --- a/BSEdyn.bib +++ b/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}, -journal = {J. Chem. Theory Comput. }, +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}, +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} } diff --git a/BSEdyn.tex b/BSEdyn.tex index ebd21a0..350fcf5 100644 --- a/BSEdyn.tex +++ b/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{ -\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 induced by dynamical corrections. } +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. +Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements brought 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} - [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) +\begin{equation} + [L_0](x_1,3;x_{1'},4 \;| \; \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} -With the change of variable $\omega \rightarrow \omega + {\omega_1}/2$ one obtains readily -\begin{align} - [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} ) \; + \end{equation} +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} ) \; 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*} - \int & \frac{ d \omega }{2i\pi} \; G(x_1,x_3; \omega + \homu ) G(x_4,x_{1'}; \omega - \homu ) e^{ i \omega \tau } - = \sum_{bj} \Bigg\{ \\ - & \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 } - \left[ \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } \right] \\ - - & \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 } - \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 } -\end{align*} + \begin{equation} + \begin{split} + \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 } + & = \sum_{bj} + \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 } + \qty[ \theta(\tau) e^{i ( \varepsilon_j + \homu ) \tau } + \theta(-\tau) e^{i ( \varepsilon_b - \homu ) \tau } ] + \\ + & - \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 } + \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^{\dagger}(5) \hpsi(6) }{N,s} - \end{align*} -we use the relation between operators in their Eisenberg and Schr\"{o}dinger representations (Eq.~\ref{Eisenberg}) to obtain: - \begin{align*} - \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^{\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*} + = \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 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 } + - \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{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^{\dagger}(x_5) = \sum_q \phi_q^{*}(x_5) - {\hat a}^{\dagger}_q + \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 \end{align*} one obtains - \begin{align*} - \langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = - \sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \; \times\\ - & \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] - \end{align*} -We now act on the N-electron ground-state with + \begin{equation} + \langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = + \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 } \\ + - \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{equation} +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: - \begin{align*} - \langle N | T [\hpsi(6) & \hpsi^{\dagger}(5)] | N,s \rangle = - \sum_{pq} \phi_p(x_6) \phi_q^{*}(x_5) \; \times\\ - & \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] - \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: + \begin{equation} + \langle N | T [\hpsi(6) \hpsi^{\dagger}(5)] | N,s \rangle = + \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 } + - \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{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}