Editing Born–Oppenheimer approximation (section)

== Example ==

The [[benzene]] molecule consists of 12 nuclei and 42 electrons. The [[Schrödinger equation]], which must be solved to obtain the [[energy level]]s and wavefunction of this molecule, is a [[partial differential equation|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. The [[Computational complexity of mathematical operations|computational complexity]], i.e., the computational power required to solve an eigenvalue equation, increases faster than the square of the number of coordinates.<ref>T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, ''Introduction to Algorithms'', 3rd ed., MIT Press, Cambridge, MA, 2009, §&nbsp;28.2.</ref>

When applying the BO approximation, two smaller, consecutive steps can be used:
For a given position of the nuclei, the ''electronic'' Schrödinger equation is solved, while treating the nuclei as stationary (not "coupled" with the dynamics of the electrons). This corresponding [[eigenvalue]] problem then consists only of the 126 electronic coordinates. This electronic computation is then repeated for other possible positions of the nuclei, i.e. deformations of the molecule. For benzene, this could be done using a grid of 36 possible nuclear position coordinates. The electronic energies on this grid are then connected to give a [[potential energy surface]] for the nuclei. This potential is then used for a second Schrödinger equation containing only the 36 coordinates of the nuclei.

So, taking the most optimistic estimate for the complexity, instead of a large equation requiring at least <math>162^2 = 26\,244</math> hypothetical calculation steps, a series of smaller calculations requiring <math>126^2 N = 15\,876 \,N</math> (with ''N'' being the number of grid points for the potential) and a very small calculation requiring <math>36^2 = 1296</math> steps can be performed. In practice, the scaling of the problem is larger than <math>n^2</math>, 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.