Reviews in Computational Chemistry Volume 18
Reviews in Computational Chemistry Volume 18
Reviews in Computational Chemistry Volume 18
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194 Charge-Transfer Reactions <strong>in</strong> Condensed Phases<br />
of the gas-phase vibronic envelope FCWD v ðnÞ with the normalized solvent<strong>in</strong>duced<br />
band shape<br />
FCWD s ðnÞ ¼hdð EðqÞ hnÞi<br />
½130Š<br />
where the average is taken over the solvent configurations statistically weighted<br />
with the Boltzmann factor expð bF Þ with ‘‘ ’’ for absorption and ‘‘þ’’ for<br />
emission.<br />
In Eqs. [129] and [130], FCWD v ðnÞ and FCWD s ðnÞ refer to the normalized<br />
Franck–Condon weighted density of the vibrational excitations of the<br />
solute (<strong>in</strong>clud<strong>in</strong>g quantum overlap <strong>in</strong>tegrals of the vibrational normal modes<br />
of the solute coupled to the transferred electron 17 ) and the normalized solvent<strong>in</strong>duced<br />
spectral distribution function, respectively. The gap, EðqÞ ¼<br />
EþðqÞ E ðqÞ, <strong>in</strong> Eq. [130] is def<strong>in</strong>ed between the upper adiabatic surface<br />
EþðqÞ and the lower adiabatic surface E ðqÞ depend<strong>in</strong>g on a set of nuclear solvent<br />
modes q. Because the transitions occur between the adiabatic free energy<br />
surfaces E ðqÞ, the unperturbed basis set <strong>in</strong> the quantum mechanical perturbation<br />
theory is built on the wave functions f ~ f 1ðqÞ; ~ f 2ðqÞg diagonaliz<strong>in</strong>g the correspond<strong>in</strong>g<br />
two-state Hamiltonian matrix (Eq. [114]). The dependence on the<br />
nuclear solvent configuration comes <strong>in</strong>to the transition dipole moment (as calculated<br />
with<strong>in</strong> the two-state model, TSM)<br />
j ~m12ðqÞj ¼ jh ~ f 1ðqÞj ^m0j ~ f 2ðqÞij<br />
¼jm12j E12<br />
EðqÞ<br />
½131Š<br />
only through the energy gap EðqÞ, which is equal to hn accord<strong>in</strong>g to<br />
Eq. [130]. This relationship is the reason for the dependence of the transition<br />
dipole on the light frequency <strong>in</strong> Eq. [129]. Coupl<strong>in</strong>g to higher ly<strong>in</strong>g excited<br />
states modifies Eq. [131], but if the dependence on the solvent field comes<br />
<strong>in</strong>to ~m12ðqÞ only through the <strong>in</strong>stantaneous energy gap, the transition dipole<br />
can still be taken out of the solvent average with, however, a more complicated<br />
dependence on the frequency of the <strong>in</strong>cident light. 17,93 In the TSM,<br />
one has, accord<strong>in</strong>g to Eq. [131]<br />
~m12ðnÞ ¼m12 E12=hn ½132Š<br />
where m12 is the gas-phase adiabatic transition dipole moment.<br />
The vibronic envelope FCWD v ðnÞ <strong>in</strong> Eq. [129] can be an arbitrary gasphase<br />
spectral profile. In condensed-phase spectral model<strong>in</strong>g, one often simplifies<br />
the analysis by adopt<strong>in</strong>g the approximation of a s<strong>in</strong>gle effective vibrational<br />
mode (E<strong>in</strong>ste<strong>in</strong> model) with the frequency nv and the vibrational reorganization<br />
energy lv. The vibronic envelope is then a Poisson distribution of