In this paper, we have studied the vibronic transitions between two symmetric double-well potentials by proposing a model Hamiltonian consisting of a harmonic oscillator and a parturition described by a Gaussian function that leads to a double minima potential with a barrier between the two energy minima. Making use of the contour integral form of Hermite polynomials, we present a new formula that can calculate Franck Condon factors of the system rigorously. The simulated vibronic spectra of ammonia and the negatively charged nitrogen-vacancy center in diamond are presented as an application of the formula.