The carbon orbitals expanded in Laguerre functions

To illustrate the convergence properties of the Laguerre functions, we shall use these functions to expand the numerical Hartree—Fock orbitals of the ground state of the carbon atom. However, to see how such expansions are obtained, we shall first consider in general terms the expansion of a function /(x) in a set of basis functions Thus, we wish to determine an expansion in 

Returning now to the radial functions, we recall that the overlap integrals in L (K+, r ) are given by 

The 2s and 2p orbitals are fairly well reproduced by Laguerre functions with unit exponent. Errors less than 0.1 are obtained in expansions with 6 and 4 terms, respectively, but an accuracy of 0.01 requires as many as 16 and 13 functions. The l.v function, on the other hand, is much harder to reproduce. Here 12 functions are needed for an accuracy of 0.1, and 21 functions for an accuracy of 0.01. 

It is not difficult to see why the I.a orbital is so hard to reproduce. It is very compacL with an expectation value of r of about 0.3. For comparison, the 2s and 2p expectation values are 1.6 and 1.7. According to (6.5.18), our basis functions with unit exponent have expectation values equal to + and are thus not well suited for describing the core orbital. We conclude that the Litguerrc functions with a fixed exponent are ill suited for describing orbitals with widely different radial distributions and that a large number of such functions would be needed to ensure a uniform description of the core and valence regions of an atomic sy stem. 

Presumably, with a more fortunate choice of exponent, we should be able to describe the l.v orbital quite accurately with only a few terms. Indeed, with = 5.6, only three terms are needed for an accuracy of 0.01. Similarly, we may improve the convergence for the other two orbitals by 

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