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Vibrational Energy Levels and Franck-Condon Factors

PGOPHER will simulate the vibrational structure associated with an electronic state or transition, including anharmonic and Renner-Teller effects and vibronic mixing. Ionization can also be simulated, providing spin effects can be ignored. Currently C1, Ci, C2, Cs, D2, C2v, C2h, D2h, C∞v and D∞h symmetry are explicitly supported. See the worked examples listed below for a quick start.

Note that when using this mode, rotational structure associated with the transition is not simulated. For compatibility with the rotational modes, J will often be displayed but is fixed at zero. (The J range is used to select the maximum vibrational angular momentum to include when simulating spectra.)

A harmonic oscillator basis set is used of the form:

|v1l1> |v2l2>…|…ΛΣ…>

where vi are the standard vibrational quantum numbers and li the corresponding vibrational angular momenta (only required for degenerate modes). Λ and Σ are the electronic quantum numbers, normally only required for linear molecules. Anharmonic, Renner-Teller and other vibronic interactions are accounted for by off-diagonal matrix elements (expressed as perturbations in the language of PGOPHER) and a matrix diagonalisation if required.

To calculate the intensities of the vibrational transitions associated with an electronic transition multidimensional Franck-Condon factors are calculated. As the vibrational basis used for each state is different (the basis is determined by the normal modes) the normal modes in the two states (Q, Q') must be related by:

Q' = JQ + K

where K represents the change in equilibrium geometry and J accounts for mixing between modes (The Dushinsky effect). J and K can be input directly, or calculated from the l matrices from the two states. See P. Chen, "Photoelectron Spectroscopy of Reactive Intermediates" (Chapter 8 of Unimolecular and Bimolecular Ion_Molecule Reaction Dynamics, ed C Y Ng, T Baer and I Powis, Wiley, 1994) for a brief review of the theory.

Further Details