The exact molecular wavefunction as a product of an electronic and a nuclear wavefunction

Abstract

The Born-Oppenheimer approximation is a basic approximation in molecular science. In this approximation, the total molecular wavefunction is written as a product of an electronic and a nuclear wavefunction. Hunter [Int. J. Quantum Chem. 9, 237 (1975)] has argued that the exact total wavefunction can also be factorized as such a product. In the present work, a variational principle is introduced which shows explicitly that the total wavefunction can be exactly written as such a product. To this end, a different electronic Hamiltonian has to be defined. The Schrödinger equation for the electronic wavefunction follows from the variational ansatz and is presented. As in the Born-Oppenheimer approximation, the nuclear motion is shown to proceed in a potential which is the electronic energy. In contrast to the Born-Oppenheimer approximation, the separation of the center of mass can be carried out exactly. The electronic Hamiltonian and the equation of motion of the nuclei resulting after the exact separation of the center of mass motion are explicitly given. A simple exactly solvable model is used to illustrate some aspects of the theory.