Separating nuclear and electronic variables

Electrons are very light particles and cannot be described by classical mechanics, while nuclei are sufficiently heavy that they display only small quantum effects. The large mass difference indicates that the nuclear velocities are much smaller than the electron velocities, and the electrons therefore adjust very fast to a change in the nuclear geometry. 

For a general M-partide system, the Hamiltonian operator contains kinetic (T) and potential (V) energy for all particles. 

The potential energy operator is the Coulomb potential (eq. (1.3)). Denoting nuclear coordinates with R and subscript n, and electron coordinates with r and subscript e, this can be expressed as follows. 

The above approximation corresponds to neglecting the coupling between the nuclear and electronic velocities, i.e. the nuclei are stationary from the electronic point of view. The electronic wave function thus depends parametrically on the nuclear coordinates, since it only depends on the position of the nuclei, not on their momentum. To a good approximation, the electronic wave function thus provides a potential energy surface upon which the nuclei move, and this separation is known as the Born-Oppenheimer approximation. 

The Bom-Oppenheimer approximation is usually very good. For the hydrogen molecule (H2) the error is of the order of 10 au, and for systems with heavier nuclei the approximation becomes better. As we shall see later, it is possible only in a few cases to solve the electronic part of the Schrodinger equation to an accuracy of 10 au, i.e. neglect of the nuclear-electron coupling is usually only a minor approximation compared with other errors. 

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