In this paper we consider the nonradiative intramolecular decay of a large molecule utilizing BornOppenheimer wavefunctions as a zero order basis, and bypassing the conventional Condon approximation for the calculation of the electronic coupling matrix matrix elements. The electronic adiabatic wavefunctions are expanded in terms of the Wigner-Brillouin perturbation series in the weak electronic-vibrational coupling limit. We have applied a generalized version of Feynman's operator calculus to derive general expressions for the nonradiative decay probability of a statistical harmonic molecule characterized by displaced potential surfaces. Numerical calculations were performed for the decay of the vibrationless excited electronic state in the "non-Condon" scheme. The numerical data for the decay rate in a two electronic level system in the weak electronic vibrational coupling limit exceed the results obtained invoking the Condon approximation by 2-3 orders of magnitude; the exact correction factor depends on the molecular parameters, and, in particular, is roughly proportional to the square of the (frequency normalized) electronic energy gap. Finally, the relevant off resonance coupling terms in the adiabatic representation are shown to be appreciably smaller than the near resonance coupling terms, demonstrating the superiority of the adiabatic basis over the crude adiabatic basis in describing electronic relaxation processes.