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Pierre-Francois Loos 2020-07-26 23:03:55 +02:00
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BSEdyn.tex
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@ -1121,9 +1121,10 @@ To provide further insight into the magnitude of the dynamical correction to val
The dynamical correction associated with the HOMO-LUMO transition reads The dynamical correction associated with the HOMO-LUMO transition reads
\begin{equation*} \begin{equation*}
\W{hh,ll}{\text{stat}} - \widetilde{W}_{hh,ll}( \Om{1}{} ) \W{hh,ll}{\text{stat}} - \widetilde{W}_{hh,ll}( \Om{1}{} )
= 4 \sERI{hh}{hl} \sERI{ll}{hl} \frac{ \Om{hl}{1} - 2 \OmRPA{hl}{}}{\OmRPA{hl}{} ( \OmRPA{hl}{} - \Om{hl}{1} ) }, % = 4 \sERI{hh}{hl} \sERI{ll}{hl} \frac{ \Om{hl}{1} - 2 \OmRPA{hl}{}}{\OmRPA{hl} ( \OmRPA{hl}{} - \Om{hl}{1} ) },
= - 4 \sERI{hh}{hl} \sERI{ll}{hl} \qty( \frac{1}{\OmRPA{hl}} - \frac{1}{\Om{hl}{1} - \OmRPA{hl}} ),
\end{equation*} \end{equation*}
where the only RPA excitation energy, $\Om{1}{}$, is again the HOMO-LUMO transition, \ie, $m=hl$ [see Eq.~\eqref{eq:sERI}]. where the only RPA excitation energy, $\OmRPA{hl} = \e{l} - \e{h} + 2 \ERI{hl}{lh}$, is again the HOMO-LUMO transition, \ie, $m=hl$ [see Eq.~\eqref{eq:sERI}].
For CT excitations with vanishing HOMO-LUMO overlap [\ie, $\ERI{h}{l} \approx 0$], $\sERI{hh}{hl} \approx 0$ and $\sERI{ll}{hl} \approx 0$, so that one can expect the dynamical correction to be weak. For CT excitations with vanishing HOMO-LUMO overlap [\ie, $\ERI{h}{l} \approx 0$], $\sERI{hh}{hl} \approx 0$ and $\sERI{ll}{hl} \approx 0$, so that one can expect the dynamical correction to be weak.
Likewise, Rydberg transitions which are characterized by a delocalized LUMO state, that is, a small HOMO-LUMO overlap, are expected to undergo weak dynamical corrections. Likewise, Rydberg transitions which are characterized by a delocalized LUMO state, that is, a small HOMO-LUMO overlap, are expected to undergo weak dynamical corrections.
The discussion for $\pi \ra \pis$ and $n \ra \pis$ transitions is certainly more complex and molecule-specific symmetry arguments must be invoked to understand the magnitude of the $\sERI{hh}{hl}$ and $\sERI{ll}{hl}$ terms. The discussion for $\pi \ra \pis$ and $n \ra \pis$ transitions is certainly more complex and molecule-specific symmetry arguments must be invoked to understand the magnitude of the $\sERI{hh}{hl}$ and $\sERI{ll}{hl}$ terms.