Born-Oppenheimer approximation
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The Born-Oppenheimer approximation, also known as the adiabatic approximation, is a technique used in quantum chemistry and condensed matter physics in order to de-couple the motion of nuclei and electrons (i.e. to separate the variables corresponding to the nuclear and electronic coordinates in the Schrödinger equation associated to the molecular Hamiltonian). It is based upon the fact that typical electronic velocities far exceed those of nuclei.
The Born-Oppenheimer approximation is commonly confused with the Born-Oppenheimer representation. This representation is the use of a particular basis set to solve the molecular problem described by the molecular Hamiltonian. This basis set is defined as the direct product of the eigenfunctions of the electronic molecular Hamiltonian times functions depending on the molecular geometry only and describing the vibration and dissociation of the molecule. Considering that the eigenfunction of the electronic molecular Hamiltonian are slowly depending on the geometry is called the Born-Oppenheimer approximation.
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Hand-waving derivation of the approximation
Since the mass of atomic nuclei are far greater than the mass of those electrons orbiting them (by a factor of about 2000), for a given energy, the electrons move much faster than the nuclei. To get an idea of what kinds of numbers we are talking, it is instructive to note that a typical electron velocity is about <math>10^6 ms^{-1}</math> (the Fermi velocity), while those of nuclei are about <math>10^3ms^{-1}</math> (the speed of sound). The much swifter electronic system can always respond quickly to changes in the configuration of nuclei, thus allowing the electronic system to remain in its ground state (for that particular configuration of nuclei).
The motion of the electrons can therefore be considered decoupled from the motion of the nuclei, which leads to the elimination of several terms from the Schrödinger equation - in practice one goes ahead and solves the quantum mechanical problem only for the system of electrons, and treating the nuclei either as entities fixed in a lattice, or perhaps as having some phononic degrees of freedom. One solves the Schrödinger equation for the electronic molecular Hamiltonian only. The neglected terms of the molecular Hamiltonian are taken into account in a subsequent step. The Born-Oppenheimer approximation consists in replacing the electronic molecular Hamiltonian in the molecular Hamiltonian by its eigenvalues (which are adiabatically dependent of the molecular geometry) called in this context potential energy surfaces. The neglected terms are called vibronic coupling.
The Born-Oppenheimer approximation is a good one, and has become a routine foundation stone for the physical study of solid and molecular systems. It is implicitly used in most computational chemical studies.
Beyond the Born-Oppenheimer Approximation
The so-called 'diagonal Born-Oppenheimer correction' (DBOC) can be obtained as
<math><\psi_e(r_e;R_e)|T_n|\psi_e(r_e;R_e)></math>
where <math>T_n</math> is the nuclear kinetic energy operator and the electronic wavefunction <math>\psi_e</math> is parametrically (not explicitly) dependent on the nuclear coordinates (See, e.g., Nicholas C. Handy, Yukio Yamaguchi, and Henry F. Schaefer, III, J. Chem. Phys. 84, 4481 (1986) [1]).
The explicit consideration of the coupling of electronic and nuclear (vibrational) movement is known as electron-phonon coupling in extended systems such as solid state systems. In non-extended systems such as complex isolated molecules, it is known as vibronic coupling which is important in the case of avoided crossing or conical intersection.
Original paper formulation
In the original paper written to adress the Born-Oppenheimer approximation (M.Born and R.Oppenheimer, Ann. d. Phys. 84, 457 (1927)) the approach used to arrive at the modern interpretation of the approximation is a pertubative one, where the pertubation parameter is identified to be
- <math>\kappa = \left ( {m \over M_0} \right )^{1 \over 4}</math>
where m is the mass of the electron and M0 is the reduced mass of the nuclei.
See also
es:Aproximación de Born-Oppenheimer fr:Approximation de Born-Oppenheimer it:Approssimazione di Born - Oppenheimer ja:断熱近似 zh:玻恩-奥本海默近似