Helium atom using Gaussian basis sets

The use of polarisation basis functions is indicated by an asterisk ( ). Thus, 6-31G refers to a 6-31G basis set with polarisation functions on the heavy (i.e. non-hydrogen) atoms. Two asterisks (e.g. 6-31G ) indicate the use of polarisation (i.e. p) functions on hydrogen and helium. The 6-31G basis set is particularly useful where hydrogen acts as a bridging atom. Partial polarisation basis sets have also been developed. For example, the 3-21G basis set has the same set of Gaussians as the 3-21G basis set (i.e. three functions for the inner shell, two contracted functions and one diffuse function for the valence shell) supplemented by six d-type Gaussians for the second-row elements. This basis set therefore attempts to account for d-orbital effects in molecules containing second-row elements. There are no special polarisation functions on first-row elements, which are described by the 3-21G basis set. 

Exercise 5.3. Calculation of the energy of the helium atom using a Gaussian basis set. 

We will illustrate the stages involved in the Roothaan-Hall approach using the helium hydrogen molecular ion, HeH, as an example. This is a two-electron system. Our objective here is to show how the Roothaan-Hall method can be used to derive the wavefunction, for a fixed internuclear distance of 1 A. We use HeH rather than H2 as our system as the lack of symmetry in HeH makes the procedure more informative. There are two basis functions, Isa (centred on the helium atom) and Isg (on the hydrogen). The numerical values of the integrals that we shall use in our calculation were obtained using a Gaussian series approximation to the Slater orbitals (the STO-3G basis set, which is described in Section 2.6). This detail need not concern us here. Each wavefunction is expressed as a linear combination of the two Is atomic orbitals centred on the nuclei A and B ... 

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