Molecular orbital theory

In chemistry, molecular orbital theory is the theory that deals with the foundation, and routes to computation, of molecular orbitals (MOs). The branch of chemistry that studies MO theory is called quantum chemistry.

Molecular orbitals are wave functions describing the quantum mechanical "motion" of one electron in the screened Coulomb field of all the nuclei of a molecule. In MO theory the nuclear electrostatic field is screened by an average field due to the electrons of the molecule. Different MO theories have different ways to account for this screening, that is, they differ in the (approximate) ways for averaging over the electrons.

The absolute square of an MO (a one-electron density) is usually delocalized, that is, spread out over the whole molecule, hence the adjective "molecular" in their name. This is in contrast to an atomic orbital (AO), which gives rise to a one-electron density localized in the vicinity of a single atom.

In the great majority of MO theories an MO is expanded in a basis χ _i of AOs, centered on the different nuclei of the molecule. Let there be N_nuc nuclei in the molecule, let A run over the nuclei and let there be n_A AOs on the A-th nucleus, then the MO φ of electron 1 has the following LCAO (linear combination of atomic orbitals) form,

$\phi (\mathbf {r} _{1})=\sum _{A=1}^{N_{\mathrm {nuc} }}\sum _{i=1}^{n_{A}}c_{Ai}\chi _{i}(\mathbf {r} _{A1}),\qquad c_{Ai}\in \mathbb {C} ,\qquad \qquad \qquad \qquad (1)$

here $\scriptstyle \mathbf {r} _{A1}$ is the coordinate vector of electron 1 with respect to a Cartesian coordinate system with nucleus A (seen as a point) as origin.

Molecular orbital theory is concerned with the choice of the AOs χ _i and the derivation and solution of the equations for the computation of the expansion coefficients c_Ai. In MO theory the AOs are explicitly known functions (usually algebraic—as opposed to numerical—functions, see this article), and therefore the expansion coefficients determine the molecular orbital unambiguously.