Normalized Primitive Gaussian Functions

Form the projections of the normalized primitive Gaussian functions on the radial mesh in columns D 11 to I 3011 in the usual manner. Enter the basic formula in cell D 11, with... 

The coefficients specified for the component primitive gaussians are chosen so that the resulting constructed basis functions are normalized. This means that one coefficient in each set is effectively constrained so that this condition is fulfilled. 

Table 1.5 The Gaussian basis sets proposed by Reeves to represent the hydrogenic radial functions. The table entries, for each basis set, are the exponents, a, of the primitive Gaussians and then in the second columns the complete normalized coefficients, d, of the linear combinations.

The examples shown in the table list the primitive Gaussians and the splitting schemes for the case of the lithium atom with added p character in the form of an ip-hybrid and then rfip-hybrid character. Note the symbolism used in the labelling 6-31g), which identifies the core linear combination to be comprised of six primitive Gaussians, while the valence orbital representation, 6-3 Ig ), is a contraction to two linear combinations of three and one primitives. Then, the 6-31g ) basis includes the extra polarization effect of one added d Gaussian. In basis set theory, to provide for the individual symmetry characters of the radial functions being modelled it is customary to define six d functions, the normal set of five in atomic orbital theory and then an additional s-function as + z -... 

These Gaussian sets are very different to the linear combinations determined by Reeves, in that these basis sets were determined to match the equivalent Slater functions [see Section 2.6 page 70]. Like the Reeves sets, though, because of the need, at the time, to calculate integrals as simply as possible, the Gaussian sets in the table for the 2s hydrogen orbital are linear combinations of 1 s Gaussian primitives. If you read the Hehre, Stewart and Pople paper (33) you will see that their normalization criterion is over all the coordinates. Thus, in comparisons with Slater functions, we need to be careful to apply a consistent normalization condition. In this discussion, the choice is to normalize only over the radial coordinate and so to apply equation 1.9, with equation 1.15 multiplied by (1/47t) /2. 

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