This is a very qualitative, highly-non-mathematical description of what is involved in doing a semi-empirical MO calculation. The take-off point is the earlier discussion of LCAO MO theory.

The first change in proceeding from Hückel LCAO theory to semi-empirical calculations is a change in the basis set.

• In Hückel theory, the basis set is a p_z orbital on each atom

• Semi-empirical models add the valence shell s orbital and the p_x and p_y orbitals.
  • The rest of each atom is treated as a "core", with a net or effective charge equal to the atomic number, Z, minus the number of inner shell electrons

Since we are now, at least in part, going to evaluate the integrals, we must use an actual mathematical representation of each of the four basis orbitals in constructing the LCAO, a representation that reflects the spatial distribution of the electrons occupying the orbitals. Two kinds of such representations are available:

• Slater-type orbitals (STO)
  • More accurate

• Gaussian orbitals
  • Easier to deal with mathematically, since any product of two Gaussians can be represented as another Gaussian.

The general difference between the two can be appreciated from the following representations for the 1s orbital of a hydrogen atom:

Slater:

where:

• z is the orbital coefficient

• r is the point in space at which the function is being evaluated

• R is the location of the atom on which the orbital is located.

Gaussian:

In Hückel theory, we divided the integrals into two sets, H and S, according to whether they contained the Hamiltonian operator. Semi-empirical methods represent the H type integrals as a sum of five terms:

• one-center, one-electron integrals, which represent the sum of the kinetic energy of an electron in an AO on atom X and its potential energy due to attraction to its own core;

• one-center, two-electron repulsion integrals;

• two-center, one-electron core resonance integrals;

• two-center, one-electron attractions between an electron on atom X and the core of atom Y; and

• two-center, two-electron repulsion integrals.

The integrals of types (1) and (2) are replaced by numerical values obtained by fitting spectroscopic values of electron energies in various valence states.

• Because these parameters are based on experimental data from real molecules, in which electron correlation is a fact of nature, some correlation is built into semi-empirical methods.

• This makes up, in part, for the failure of semi-empirical methods to consider correlation explicitly.

Integrals of type (3) represent the main contribution to the bonding energy of the molecule; they are effectively the integrals we represented as b in Hückel theory. Semi-empirical methods treat these as proportional to the overlap integral, S:

where f_x is an adjustable parameter.

• None of these terms is set to zero, as in Hückel theory

• However, they do not contain any explicit dependence on interatomic distance.

A modified electrostatic treatment is used to evaluate the integrals of type (4).

• They are taken as proportional to eZ_eff/r, with an adjustable parameter for the proportionality constant,

• They also include a term of the form exp-aR, where a is an adjustable parameter.
  • This function is included to ensure that the net repulsion between neutral atoms vanishes as their separation goes to infinity.

The remaining integrals, type (5), represent the energy of interaction between the charge distribution at atom X and that at atom Y.

• They are calculated as the sum over all multipole interactions, again with adjustable parameters.

• STOs are used to represent the spatial distribution of the charges.

The overlap integrals (those represented by S in Hückel theory) are evaluated analytically.

• However, the orbital exponents are treated as adjustable parameters.

Thus, we see in the way the various integrals are handled part of the reason for labeling these calculations semi-empirical. The use of empirical data, as well as adjustable parameters, justify the designation.

Core repulsions (the core of one atom interacting with the core of the next) are the remaining item to be evaluated.

• This is done with two-center repulsion integrals, the cores being represented as Gaussians.

• In order to reduce excessive repulsions at large atomic separations, which arise in a more classical electrostatic treatment, attractive Gaussians are added to the repulsive ones for most elements.

Finally, the total energy of the molecule is represented as the sum of the electronic energy (net negative) and the core repulsions (net positive).

The enthalpy of formation of the molecule then is obtained from its total energy by subtracting the electronic energies, and adding the experimental heats of formation of the individual atoms.

With all this structuring of the calculations completed, a so-called "training set" of molecules is selected, chosen to cover as many types of bonding situations as possible.

• A non-linear least squares optimization procedure is applied with the values of the various adjustable parameters as variables and a set of measured properties of the training set as constants to be reproduced.

• The measured properties include heats of formation, geometrical variables, dipole moments, and first ionization potentials.

Depending upon the choice of training sets, the exact numbers of types of adjustable parameters, and the mode of fitting to experimental properties, two principal semi-empirical methods have been developed and are incorporated into most semi-empirical software.

• The AM1 (Austin Model 1) method of M. J. S. Dewar

• The PM3 (Parameter Model 3) method of J. J. P. Stewart

Some programs (like Spartan) also retain the older MNDO method, also developed by Dewar. Debate rages in the literature and on the CompChem electronic mailing list as to which of these is the better parameterization.