Molecular Orbital Method

(MO method), the most important method in quantum chemistry. The method is based on the assumption that each electron in a molecule is described by its own wave function, called a molecular orbital (MO). Because of the impossibility of an exact solution of the Schrödinger equation for systems with two or more electrons, the means of obtaining expressions for molecular orbitals are ambiguous.

In practice, each molecular orbital ψ i is usually represented as a linear combination of atomic orbitals (LCAO) X_p(LCAO MO approximation) of the form ψ i =Σ_pC_iPX_p, where i is the index number of the molecular orbital, p is the index number of an atomic orbital, and C_ip are algebraic coefficients indicating the contributions of the individual atomic orbitals to the molecular orbital. This approximation is based on the assumption that, in the vicinity of any atomic nucleus, a molecular orbital ψi should be similar to its component atomic orbitals XP of the atom. Since the change in the state of electrons relative to the original state when atoms combine to form a molecule may be regarded as not very radical, atomic wave functions continue to be used in this approximation (although not necessarily with the same parameters as for the free atoms). However, an LCAO description of an electron reflects the qualitative changes in the state of an electron that take place upon formation of a molecule: it is no longer possible to state that any particular electron in a molecule is located on a specific atom. Similarly, as in the hydrogen atom an electron may be found with varying probabilities at different points in the near-nuclear space, in a molecule the electron is “spread out” over the entire molecule.

In the general case, the MO method regards the formation of chemical bonds as the result of the movement of all the electrons in the total field produced by all the electrons and nuclei of the original atoms. However, since the major contribution to the formation of bonds is made by the electrons of the outer (valence) shells, consideration is usually limited to these electrons. The total wave function Ψ of the molecule is constructed from single-electron molecular orbitals ψi, taking into account the requirement of antisymmetry of the wave function Ψ that follows from the Pauli principle. The functions Ψ, ψi, and X_P are found by solving the Schrodinger equation by the variation method, usually according to the Hartree-Fock self-consistent field (SCF) scheme.

Quantitative calculations for multielectron molecules entail serious mathematical and technical difficulties. Complete nonempirical calculations by the MO method that reach the Hartree-Fock precision limit (which sometimes is also inadequate for quantitative comparison with experiment) have been carried out for molecules with a number of electrons on the order of 50. Thus, most of the calculations are semiempirical in nature, and additional approximations are made. There are many versions of the LCAO MO SCF method, which differ in the extent to which the electron interactions and the procedures for self-consistency are taken into account; the success of such methods depends on the objects of study and on their properties. It is significant that the molecular orbital method in any form, even in its most simple versions, is organically related to molecular spatial symmetry. This makes it possible to obtain completely unequivocal qualitative information on many properties of molecules (the degree of degeneracy of energy levels, the magnitude of magnetic moment, and the intensity of spectral lines), regardless of the nature of the approximation selected.

A new version of the MO method that does not use the LCAO approximation has been developing intensively since 1965. In this version, the statistical atomic model is combined with some models from solid-state theory. As a result, it is possible to construct special molecular orbitals that are easy to define by means of a numerical (not analytical) solution of the Schrodinger equation, also according to the SCF scheme. Calculations by this new method, which are only slightly inferior in accuracy to nonempirical LCAO MO SCF calculations, usually require 100 to 1,000 times less computer time (minutes instead of tens of hours). This new method is especially promising for qualitative calculations of large molecules.

The MO method (especially the LCAO MO method) is important in chemistry, since it makes it possible to obtain data on the structure and properties of molecules on the basis of the corresponding atomic characteristics. Thus, almost all modern concepts of the chemical bond and chemical reactivity are based on concepts of the MO method.