  Starting from a compact but general molecular Hamiltonian expressed in the
bases of adiabatic electronic states and position states of nuclei, we make
careful consideration of nonadiabatic derivative coupling (NDC) terms between
adiabatic states. It is clarified that the conventional use of NDC terms
evaluated for an adiabatic electronic state in the textbook expression for the
Fermi's golden rule (FGR) rate implicitly invokes an additional approximation
that ignores non-orthogonality of adiabatic states at different geometries.
Thus, we derive a well-defined FGR rate expression based on a quasi-adiabatic
approximation that explicitly uses the adiabatic states and NDC terms evaluated
at the minimum potential energy state of the initial adiabatic states. We then
clarify conditions and approximations leading to the modeling of all the
nuclear degrees of freedom as a set of harmonic oscillators, and then derive a
closed form FGR rate expression while accounting for the non-Condon effects due
to momenta in NDC terms explicitly. The resulting rate expression includes
terms due to quadratic contribution of NDC terms and also their couplings to
Franck-Condon modes. Model calculations for the case where nuclear vibrations
consist of both a sharp high frequency mode and a broad Ohmic bath spectral
density illustrate new features and implications of the rate expression.
