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In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics. The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but even then the approximation is usually used as a starting point for more refined methods. In molecular infrared spectroscopy, using the BO approximation means considering molecular energy as a sum of independent terms, e.g.: [math]\displaystyle{ E_\text{total} = E_\text{electronic} + E_\text{vibrational} + E_\text{rotational} + E_\text{nuclear spin}. }[/math] These terms are of different orders of magnitude and the nuclear spin energy is so small that it is often omitted. The electronic energies [math]\displaystyle{ E_\text{electronic} }[/math] consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions, which are the terms typically included when computing the electronic structure of molecules.
References
- T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, § 28.2.
- Authors often justify this step by stating that "the heavy nuclei move more slowly than the light electrons". Classically this statement makes sense only if the momentum p of electrons and nuclei is of the same order of magnitude. In that case mn ≫ me implies p2/(2mn) ≪ p2/(2me). It is easy to show that for two bodies in circular orbits around their center of mass (regardless of individual masses), the momenta of the two bodies are equal and opposite, and that for any collection of particles in the center-of-mass frame, the net momentum is zero. Given that the center-of-mass frame is the lab frame (where the molecule is stationary), the momentum of the nuclei must be equal and opposite to that of the electrons. A hand-waving justification can be derived from quantum mechanics as well. The corresponding operators do not contain mass and the molecule can be treated as a box containing the electrons and nuclei. Since the kinetic energy is p2/(2m), it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 104).
- Typically, the Schrödinger equation for molecules cannot be solved exactly. Approximation methods include the Hartree-Fock method
- It is assumed, in accordance with the adiabatic theorem, that the same electronic state (for instance, the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a discontinuity (jump) in the PES if electronic state switching would occur.
- This equation is time-independent, and stationary wavefunctions for the nuclei are obtained; nevertheless, it is traditional to use the word "motion" in this context, although classically motion implies time dependence.
- Max Born; J. Robert Oppenheimer (1927). "Zur Quantentheorie der Molekeln" (in de). Annalen der Physik 389 (20): 457–484. doi:10.1002/andp.19273892002. Bibcode: 1927AnP...389..457B. https://dx.doi.org/10.1002%2Fandp.19273892002
- Born, M.; Huang, K. (1954). "IV". Dynamical Theory of Crystal Lattices. New York: Oxford University Press.
- "Born-Oppenheimer Approach: Diabatization and Topological Matrix". Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections. Hoboken, NJ, USA: John Wiley & Sons, Inc.. 28 March 2006. pp. 26–57. doi:10.1002/0471780081.ch2. ISBN 978-0-471-78008-3. https://dx.doi.org/10.1002%2F0471780081.ch2
- Baer, Michael; Englman, Robert (1997). "A modified Born-Oppenheimer equation: application to conical intersections and other types of singularities". Chemical Physics Letters (Elsevier BV) 265 (1–2): 105–108. doi:10.1016/s0009-2614(96)01411-x. ISSN 0009-2614. Bibcode: 1997CPL...265..105B. https://dx.doi.org/10.1016%2Fs0009-2614%2896%2901411-x
- Baer, Roi; Charutz, David M.; Kosloff, Ronnie; Baer, Michael (22 November 1996). "A study of conical intersection effects on scattering processes: The validity of adiabatic single‐surface approximations within a quasi‐Jahn–Teller model". The Journal of Chemical Physics (AIP Publishing) 105 (20): 9141–9152. doi:10.1063/1.472748. ISSN 0021-9606. Bibcode: 1996JChPh.105.9141B. https://dx.doi.org/10.1063%2F1.472748
- Adhikari, Satrajit; Billing, Gert D. (1999). "The conical intersection effects and adiabatic single-surface approximations on scattering processes: A time-dependent wave packet approach". The Journal of Chemical Physics (AIP Publishing) 111 (1): 40–47. doi:10.1063/1.479360. ISSN 0021-9606. Bibcode: 1999JChPh.111...40A. https://dx.doi.org/10.1063%2F1.479360
- Charutz, David M.; Baer, Roi; Baer, Michael (1997). "A study of degenerate vibronic coupling effects on scattering processes: are resonances affected by degenerate vibronic coupling?". Chemical Physics Letters (Elsevier BV) 265 (6): 629–637. doi:10.1016/s0009-2614(96)01494-7. ISSN 0009-2614. Bibcode: 1997CPL...265..629C. https://dx.doi.org/10.1016%2Fs0009-2614%2896%2901494-7
