Ab initio methods

The term ab initio means from first principles. It does not mean that we are solving the Schrödinger equation exactly. It means that we are selecting a method that in principle can lead to a reasonable approximation to the solution of the Schrödinger equation and then selecting a basis set that will implement that method in a reasonable way.

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.

Hartree-Fock method

The essential idea of the Hartree-Fock or molecular orbital method is that, for a closed shell system, the electrons are assigned two at a time to a set of molecular orbitals. This can be represented by the simple picture:
 
[M.O. picture]
 
Here we see a system of 8 electrons occupying the 4 molecular orbitals of lowest energy. Only one unoccupied molecular orbital is shown. Note that the unoccupied molecular orbitals are often called virtual orbitals.

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:
 

[LCAO equation]
 
Here each molecular orbital [psi]i is now expanded as a linear combination of basis functions, [phi]µ:

Our aim is to find the value of the coefficients Cµi 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.

Electron correlation methods

A large number of methods have been used to improve the Hartree-Fock method.

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 H2 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:


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