The numerical evaluations of the density of states weighted Franck–Condon factor in large molecules. Effects of frequency changes

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Abstract
The numerical integration method (NIM) is applied to calculations of the density of states weighted Franck–Condon (DWFC) factor which plays an important role in the theory of nonradiative transitions. The 200 point Legendre–Gauss quadrature formula is used to save computing time. Values of the integrand are given with the quadruple precision to avoid accumulated errors occuring in the numerical integration. The illustrative calculations are carried out for the T1→S0 intersystem crossing in benzene. If frequency changes of all vibrational modes are taken into account, the energy gap law applies very well, but it fails without frequency changes. This fact suggests that the ordinary saddle point method based on the harmonic displaced potential surface model is inaccurate. The DWFC factor is sensitive to frequency changes of a high frequency mode which plays an important role in nonradiative transition. When low and medium frequency modes are added, as a group, to high frequency ones, they give a significant effect upon the DWFC factor. If a frequency change of a low frequency mode is hypothetically reduced to an extremely low value, the DWFC factor rapidly increases. This relates to the proximity effect by Wassam and Lim in the nonradiative transition in N‐heterocyclic aromatic hydrocarbons. Finally, Metz’s method is discussed.