Calculation of the multimode Franck-Condon factors based on the coherent state method

The equivalence of two ways for the calculation of overlap integrals, i.e. the Sharp Rosenstock generating function method and the Doktorov coherent state method, has been proved. On the basis of the generating function of the overlap integrals, a new closed form expression for the Franck - Condon integrals for overlap multidimensional harmonic oscillators has been exactly derived. In addition, some useful analytical expressions for the calculations of the multimode Franck - Condon factors have been given.