Born–Oppenheimer approximation

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be treated separately. 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 is then often used as a starting point for more refined methods. In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be treated separately. 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 is then often used as a starting point for more refined methods. In molecular spectroscopy, using the BO approximation means considering molecular energy as a sum of independent terms, e.g.: E t o t a l = E e l e c t r o n i c + E v i b r a t i o n a l + E r o t a t i o n a l + E n u c l e a r s p i n {displaystyle E_{mathrm {total} }=E_{mathrm {electronic} }+E_{mathrm {vibrational} }+E_{mathrm {rotational} }+E_{mathrm {nuclear,spin} }} . These terms are of different order of magnitude and the nuclear spin energy is so small that it is often omitted. The electronic energies E e l e c t r o n i c {displaystyle E_{mathrm {electronic} }} consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions, which are the terms typically included when computing the electronic structure of moleucles. The benzene molecule consists of 12 nuclei and 42 electrons. 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 + 3×42 = 36 nuclear + 126 electronic = 162 variables for the wave function. The computational complexity, i.e. the computational power required to solve an eigenvalue equation, increases faster than the square of the number of coordinates. 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 the 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 168 2 = 26 , 244 {displaystyle 168^{2}=26,244} hypothetical calculation steps, a series of smaller calculations requiring N 126 2 = N 15 , 876 {displaystyle N126^{2}=N15,876} (with N being the amount of grid points for the potential) and a very small calculation requiring 36 2 = 1296 {displaystyle 36^{2}=1296} steps can be performed. In practice, the scaling of the problem is larger than n 2 {displaystyle n^{2}} 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. The BO approximation recognizes the large difference between the electron mass and the masses of atomic nuclei, and correspondingly the time scales of their motion. Given the same amount of kinetic energy, the nuclei move much more slowly than the electrons. In mathematical terms, the BO approximation consists of expressing the wavefunction ( Ψ t o t a l {displaystyle Psi _{mathrm {total} }} ) of a molecule as the product of an electronic wavefunction and a nuclear (vibrational, rotational) wavefunction. Ψ t o t a l = ψ e l e c t r o n i c ψ n u c l e a r {displaystyle Psi _{mathrm {total} }=psi _{mathrm {electronic} }psi _{mathrm {nuclear} }} . This enables a separation of the Hamiltonian operator into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently. In the first step the nuclear kinetic energy is neglected, 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. (This first step of the BO approximation is therefore often referred to as the clamped-nuclei approximation.) The electronic Schrödinger equation