A modified perturbation approach for the calculation of the vibrational wave function of polyatomic molecules is discussed. It is demonstrated that if the expansion point of the potential is determined variationally, the leading first-order term in the perturbation expansion of the vibrational wave function vanishes. Furthermore, the new expansion point is a very good approximation to the vibrationally averaged molecular geometry. The required third derivatives of the potential energy with respect to geometrical distortions have been calculated by numerical differentiation. Two approaches are discussed, one based on the differentiation of the molecular Hessian and the other on the molecular gradient. Results are presented for the averaged molecular geometry of a large set of molecules, including studies of electronically excited states and effects of electron correlation. The largest molecule included is butane with a total of 14 atoms. (C) 2000 American Institute of Physics. [S0021-9606(00)30905-9].