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.

The purpose of molecular orbital theory is to obtain approximate solutions of the time-independent Schrödinger equations of molecules. From the solutions, also known as wave functions, all kinds of molecular properties of chemical interest can be computed. The Schrödinger equation contains a Hamilton operator (quantum mechanical energy operator) from which certain terms of lesser importance are omitted, see molecular Hamiltonian for the details.

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.

Types of MO theory

A crucial part of MO theory is concerned with the computation of molecular integrals. These are integrals that contain products of AOs, centered on different nuclei, and operators arising from the molecular hamiltonian that entered the Schrödinger equation. Before 1970, when computers were still in their infancy, the computation of these integrals formed a major hurdle. This is why approximations were introduced in their computation. Many of these approximations were based on the fact that certain groups of integrals could be seen to represent some empirical (experimentally observable) quantity, usually ionization potentials, electron affinity, or electron energy differences. Replacement of groups of integrals by their empirical counterpart (rather than calculation) leads to methods known as semi-empirical MO methods.

When computers and quantum chemical software developed from the 1970s onward, the computation of all necessary molecular integrals became possible. Methods in which all integrals are computed are known as ab initio MO methods. The Latin phrase ab initio stands for from the beginning and implies that no empirical data enter the computation.

An important error in MO calculations is the neglect of the electronic correlation. By the averaging inherent to MO theory, the correlation between the electronic motions is lost. If electron 1 is at point P, the chance that electron 2 will be near P is smaller than that it will be far away from P, due to electrostatic repulsion beween the electrons, wich falls off with the inverse distance. Neither ab initio nor semi-empirical MO theory account for correlation. However, there is a third variant of MO theory, density functional theory (DFT) that somehow accounts for part of the correlation. This method requires a choice of AOs, as do the other MO methods, but also a choice of density functional. Many different density functionals have been proposed, and in that sense DFT is reminiscent of semi-empirical theory where there is the freedom of choice in empirical parameters.