[1] The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.
[2][3] The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules.
These terms are of different orders of magnitude and the nuclear spin energy is so small that it is often omitted.
The Schrödinger equation, which must be solved to obtain the energy levels and wavefunction of this molecule, is a partial differential eigenvalue equation in the three-dimensional coordinates of the nuclei and electrons, giving 3 × 12 = 36 nuclear plus 3 × 42 = 126 electronic, totalling 162 variables for the wave function.
, and more approximations are applied in computational chemistry to further reduce the number of variables and dimensions.
The slope of the potential energy surface can be used to simulate molecular dynamics, using it to express the mean force on the nuclei caused by the electrons and thereby skipping the calculation of the nuclear Schrödinger equation.
In the first step, the nuclear kinetic energy is neglected,[note 1] that is, the corresponding operator Tn is subtracted from the total molecular Hamiltonian.
In the remaining electronic Hamiltonian He the nuclear positions are no longer variable, but are constant parameters (they enter the equation "parametrically").
The electron–nucleus interactions are not removed, i.e., the electrons still "feel" the Coulomb potential of the nuclei clamped at certain positions in space.
The electronic energy eigenvalue Ee depends on the chosen positions R of the nuclei.
Varying these positions R in small steps and repeatedly solving the electronic Schrödinger equation, one obtains Ee as a function of R. This is the potential energy surface (PES):
[note 3] In the second step of the BO approximation, the nuclear kinetic energy Tn (containing partial derivatives with respect to the components of R) is reintroduced, and the Schrödinger equation for the nuclear motion[note 4] is solved.
This second step of the BO approximation involves separation of vibrational, translational, and rotational motions.
At the same time we will show how the BO approximation may be improved by including vibronic coupling.
To that end the second step of the BO approximation is generalized to a set of coupled eigenvalue equations depending on nuclear coordinates only.
Off-diagonal elements in these equations are shown to be nuclear kinetic energy terms.
It will be shown that the BO approximation can be trusted whenever the PESs, obtained from the solution of the electronic Schrödinger equation, are well separated: We start from the exact non-relativistic, time-independent molecular Hamiltonian: with The position vectors
The only constants explicitly entering the formula are ZA and MA – the atomic number and mass of nucleus A.
, the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO
We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider which in general will not be zero.
is diagonal, and the nuclear Hamilton matrix is non-diagonal; its off-diagonal (vibronic coupling) terms
In order to show when this neglect is justified, we suppress the coordinates in the notation and write, by applying the Leibniz rule for differentiation, the matrix elements of
We reiterate that when two or more potential energy surfaces approach each other, or even cross, the Born–Oppenheimer approximation breaks down, and one must fall back on the coupled equations.
To include the correct symmetry within the Born–Oppenheimer (BO) approximation,[2][5] a molecular system presented in terms of (mass-dependent) nuclear coordinates
and formed by the two lowest BO adiabatic potential energy surfaces (PES)
To ensure the validity of the BO approximation, the energy E of the system is assumed to be low enough so that
becomes a closed PES in the region of interest, with the exception of sporadic infinitesimal sites surrounding degeneracy points formed by
The starting point is the nuclear adiabatic BO (matrix) equation written in the form[6] where
Next a new function is introduced:[7] and the corresponding rearrangements are made: In order for this equation to yield a solution with the correct symmetry, it is suggested to apply a perturbation approach based on an elastic potential
is an arbitrary contour, and the exponential function contains the relevant symmetry as created while moving along