By reasonable, we mean that the results are adequate for the application in hand. A method and basis set that is quite adequate for one application may be totally inadequate for another application. We also have to take into account the cost of doing calculations and the total amount of computer time required. If an answer is needed today, there is no point in doing a calculation that will take two days of processor time with the results coming back after three days. However, if the results are not adequate for the purpose, there is no point in doing the calculation, however cheap it may be.

A wide range of ab initio methods have been employed, but we will restrict ourselves to the sub class of methods that is employed in the vast majority of all calculations carried out today. This is the sub class that uses the molecular orbital method, possibly followed by a post molecular orbital method that uses the molecular orbital wave function as the reference function. The molecular orbital method is generally referred to as the Hartree-Fock method.

To give us freedom to vary the molecular orbitals to best suit the molecule
in question, we expand each molecular orbital in terms of a set of basis
functions which are normally centred on the atoms in the molecule. This
gives:

Here each molecular orbital

Our aim is to find the value of the coefficients C_{µi}
that gives the best molecular orbitals. The sum is over *n* basis
functions. *n* is the number of basis functions chosen for the system.
We call this "the basis set size". The question of basis set selection
is taken up in the next page.

You may now wish to explore the details of Hartree-Fock theory in more detail.

A simple introduction to the situation for open shell systems (ones with unpaired electrons - often due to an odd number of electrons) is also appropriate here.

First, why is the Hartree-Fock method not capable of giving the correct
solution to the Schrödinger equation if a very large and flexible basis
set is selected? In passing, we note that the very best Hartree-Fock wave
function, obtained with just such a large and flexible basis set, is called
the "Hartree-Fock limit". The problem is that electrons are not paired
up in the way that the Hartree-Fock method supposes. It suggests that the
two electrons have the same probability of being in the same region of
space as being in separate symmetry equivalent regions of space. For example,
in H_{2} it would give the same probablity of both electrons being
near one atom as one being near one atom and the other near the second
atom. This is clearly wrong. The Hartree-Fock method also only evaluates
the repulsion energy as an average over the whole molecular orbital.

The two electrons in a molecular orbital are in reality moving in such a way that they keep more apart from each other than being close. We call this effect "correlation". The difference in energy between the exact result and the Hartree-Fock limit energy is called the "correlation energy".

There are three distinct classes of method used to deal with the correlation problem (and here we start peeling off the layers of the onion - you might want to go only one page down at first reading). The methods can be classified as: