In quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer (BO) approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons. The time independent Schrödinger equation, which must be solved to obtain the energy and molecular wavefunction of this molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei. The BO approximation makes it possible to compute the wavefunction in two less formidable, consecutive steps. This approximation was proposed in the early days of quantum mechanics by Born and Oppenheimer (1927) and is still indispensable in quantum chemistry.
In basic terms, it allows the wavefunction of a molecule to be broken into its electronic and nuclear (vibrational, rotational) components.
In the first step of the BO approximation the electronic Schrödinger equation is solved, yielding the wavefunction ψelectronic depending on electrons only. For benzene this wavefunction depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be repeated for many different nuclear configurations. The set of electronic energies thus computed becomes a function of the nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables.
The success of the BO approximation is due to the high ratio between nuclear and electronic masses. The approximation is an important tool of quantum chemistry; without it only the lightest molecule, H2, could be handled, and all computations of molecular wavefunctions for larger molecules make use of it. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations.
The electronic energies, constituting the nuclear potential, consist of kinetic energies, interelectronic repulsions and electron–nuclear attractions. In a handwaving manner the nuclear potential is taken to be an averaged electron–nuclear attraction. The BO approximation follows from the inertia of electrons being considered to be negligible in comparison to the atom to which they are bound.
The Born–Oppenheimer (BO) approximation is ubiquitous in quantum chemical calculations of molecular wavefunctions. It consists of two steps.
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