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Potential Energy Surfaces and Conical Intersections

Molecular Electronic Structure

The many-particle, three-dimensional time-independent Schrödinger (equation (2.20)) can be adapted for molecular systems. In this case, the molecular Hamiltonian becomes:

Equation 3.1 (3.1)

i and j refer to electrons (of which there are n), and A and B refer to nuclei (of which there are N).
The first term represents the electronic kinetic energy operator, the second term the nuclear kinetic energy operator. The third term is the potential energy of the system due to coulombic repulsions between the electrons; rij being the distance between electrons i and j. The fourth term represents the potential energy of the system due to coulombic repulsions between the nuclei; RAB being the distance between the nuclei A and B, with atomic numbers ZA and ZB, respectively.

3.1 The Born-Oppenheimer approximation

Solving the time-independent Schrödinger equation for a molecular system (equation (3.1)) quickly becomes very complex. As an example, benzene includes 861 electron-electron distances alone.
Because of this, in 1927, physicists Max Born and J. Robert Oppenheimer suggested what is now known as the Born-Oppenheimer approximation (also the Born-Oppenheimer separation or the electronic adiabatic approximation)[9] – that, since the nuclei are very much heavier than the electrons in a molecular system, their motion can be separated. This means that, to an approximation, the nuclei are effectively stationary on the timescale of electron movement; or, conversely, that as the nuclei move, the electrons move infinitely faster, that is, they react to nuclear changes instantaneously.
Under these conditions, the Schrödinger wavefunction can be factorised into electronic and nuclear components:

Equation 3.2 (3.2)

Thus, to obtain the electronic Schrödinger equation, the nuclear kinetic energy terms are omitted:

Equation 3.3 (3.3)

…where the electronic Hamiltonian is given by equation (3.4):

Equation 3.4 (3.4)

The nuclear repulsion term in the electronic Schrödinger equation, VNN, is given by equation (3.5):

Equation 3.5 (3.5)

The energy U in (3.3) is the electronic energy including nuclear repulsion. The term VNN is a constant, in that the wavefunction does not depend on VNN, but there is a different wavefunction for each value of VNN. The electronic wavefunction is said to be parametrically dependent on the nuclear coordinates; this is shown by the underlined nuclear coordinate, R, in equation (3.2).
If VNN is omitted from equation (3.3), the energy obtained is purely the electronic energy, with no nuclear contribution:

Equation 3.6 (3.6)

This is related to the electronic energy including nuclear repulsion simply by the sum of the nuclear repulsion term, VNN:

Equation 3.7 (3.7)

Therefore, the electronic energy for a particular configuration of nuclei can be found by solving equation (3.6), and then U can be found using (3.7) and (3.5), with knowledge of the assumed nuclear coordinates.

The nuclear Hamiltonian is given by equation (3.8):

Equation 3.8 (3.8)

The first term in this equation is the kinetic energy of the nuclei; the second term is their potential energy, and is often referred to as the inter-atomic potential energy surface (PES).

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[9] M. Born and J.R. Oppenheimer, Ann. d. Physik, 84, 457, 1927