Surface jumping in a harmonic model of trans-octatetraene: Franck-Condon factors and accepting vibrational modes in S₁→S₀ non-vertical radiationless transition

A phase-space method for finding the accepting modes in a non-vertical radiationless vibronic transition and for recognizing the final state with the largest Franck-Condon factor is applied to a harmonic model of the S₁ → S₀ relaxation in trans-octatetraene. Input required for the analysis includes the energy gap between S₁ and S₀, normal mode frequencies, reduced masses, and eigenvectors (including the Duschinsky rotation matrix), and the molecule equilibrium configurations (bond lengths and angles) in S₁ and S₀. Some of these data are taken from published experimental results and some are calculated in this work. The energy gap of 0.132 au is much larger than the energy of a vertical transition, which is only 0.047 au. The phase-space method gives a closed-form analytic solution for how to divide the excess energy between the accepting modes. The final distribution includes a large excitation of the two CH₂ end groups, where the motion of the two hydrogen atoms within each quasilocal CH₂ group is antisymmetric; a symmetric stretch of the two central C-H bonds of the molecule; and small totally symmetric bending of the whole molecule. Comparison of Franck-Condon factors (exact within the harmonic model) of the final state obtained by the phase-space analysis and of other similar isoenergetic states shows that the phase-space method indeed chooses the most probable final energy distribution. Possible modifications of these results due to anharmonic effects are discussed.