Requirements on one-electron basis functions

The question then arises as to what functions Xpix) are suitable for orbital expansions. Ideally, a suitable set of basis fiuictions should fulfil the following requirements  

The basis should be designed such that it allows for an orderly and systematic extension towards completeness with respect to one-electron square-integrable functions. 

The basis should allow for a rapid convergence to any atomic or molecular electronic state, requiring only a few terms for a reasonably accurate description of molecular electron distributions. 

The functions should have an analytical form that allows for easy manipulation. In particular, all molecular integrals over these functions should be easy to evaluate. It is also desirable that the basis functions are orthogonal or at least that their nonorthogonality does not present problems related to numerical instability. 

In practice, it has proved difficult to construct basis sets that combine all these properties. For example, as we shall see in Section 6.2, it is not difficult to set up a complete orthonormal set of basis functions that may be applied to any molecular system in an unambiguous fashion. The problem is that such a set in general does not converge rapidly to the exact solution and therefore becomes intractable because of the laige number of terms that must be included. A useful set of one-electron functions must therefore represent a compromise between the three requirements. In particular, we shall see that the Gaussian basis functions, introduced in Section 6.6, are veiy successful in this respect, combining reasonably short expansions with a fast integral evaluation. 

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