Semi-Empirical Techniques
Historically these approximate methods were pursued first. As ab initio methods became better, the semi-empirical methods were used for larger and larger systems.
Semi-empirical techniques compute chemically accurate results on 'large' systems at reasonable cost (time, computer resources)
These methods reduce the N4 step to N2 in computational complexity. This reduces the CPU time and the intermediate storage requirements.
Semi-empirical techniques
Review of the approximations for ab initio calculations, (see the appendix: Derivation of the Hartree-Fock Equations)
There are additional approximations for semi-empirical techniques,
To begin, consider the two electron integrals. The complexity of these integrals is at worst four center, orbitals on four different atomic centers.
In semi-empirical methods, replace the analytic solution of these integrals by a set of parameters to approximate the value of the repulsion integrals.
Flavors of semi-empirical hamiltonians:
MNDO Heat of Formation
where
is the experimental heat of formation for atom A,
is the calculated energy of formation.
In PM3, up to 18 independent parameters are used for each element. (11 for hydrogen)
In AM1, 10 - 19 parameters.
All the parameters are obtained via a least square fitting process.