Mechanics on a Potential Energy Surface

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent  

In this context, E is the total classical energy including kinetic energy. You can then investigate the potential energy surface in a purely classical way using the positions (Rj) and velocities (Vj = dRj/dt) of the constituent atoms. 

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In discussing reactivity in previous sections, we have more or less assumed that the nuclei move around as described by classical mechanics, on a potential energy surface derived from the (quantum mechanical) electronic Schrodinger equation. We discuss here two areas in which quantum mechanical effects need to be taken into account even when considering motion of nuclei. 

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