Introduction

Hartree-Fock (HF) theory belongs to a class of methods coined ab initio, because all molecular integrals that are required are analytically or numerically evaluated. Semiempirical methods use parametric expressions for some integrals so that the calculation of electronic energies is significantly simplified. There is however not a clear distinction between semiempirical and ab initio methods because, e.g., basis set parameters can be parametrised for specific ends. Ab initio methods require large basis sets so the calculations converge to reasonable values (there are recent minimal basis methods developed in the group of Stefan Grimme that are corrected to predict accurately some properties). This means that basis sets are at least double zeta in size (at least two functions are used to describe each atomic orbital). Semiempirical methods use minimal basis sets instead. In the end, due to all approximations these methods introduce, they lie somewhere between ab initio (or DFT) and molecular mechanics: they are based on a quantum mechanical treatment of the system; the integral evaluation is quick. Here we focus on semiempirical methods.

Perhaps the main reason why so much effort was placed in the development of semiempirical methods is the fact that reasonable results are obtained at a much lower computational cost. Integral evaluation is enormously simplified to say the least. There is to my knowledge no semiempirical method that considers 3- or 4-center electron repulsion integrals and even a great deal of the calculation of 2-center electron repulsion integrals is very much simplified. This was an essential feature in the success of semiempirical quantum chemical methods because solving most of molecular integrals using Slater type orbitals is a very arduous task that most people avoid. By appropriately parametrising integrals, semiempirical methods approximate to some good degree ab initio calculations or experimental results. However, not even the best parametrisation can fix faults from the underlying theory these methods are based upon, which can be Hartree-Fock or Density Functional Theory. Despite the great advances of modern quantum chemistry, semiempirical methods still enjoy some degree of success because people will always want to calculate larger systems, and lets face it, we are all impatient to get results quickly.