Gaussian-type orbitals spherical GTOs

An alternative solution to the gauge-independence problem in molecular calculations is to attach the complex phase factors directly to the atomic basis functions or atomic orbitals (AOs) rather than to the MOs. Thus, each basis function—which in modern calculations usually corresponds to a Gaussian-type orbital (GTO)—is equipped with a complex phase factor according to Eq. 87. A spherical-harmonic GTO may then be written in the from... 

The normalized Is spherically symmetric Gaussian Type Orbital (GTO) centered in the point shown by the vector Rp reads as... 

The essential features of this approach is this. Let each unit cell be divided into molecules (real ones or not), and we devise a basis set to represent the MO s of these molecules. The primitive basis set used are spherical-harmonic Gaussian Type Orbitals (GTO s) centered about different origins, and have the form... 

Fig. 8.10. Two spherically symmetric GTOs of the "Is-type C r 1,0) (a) These are used to form the difference orbital (b) G(r 1, —0.5/)—G(r 1, +0.5/), where / is the unit vector along the j -axis. For comparison (c), the Gaussian-type p, orbital is shown xG r 1.01. It can be seen that the spherical orbitals may indeed simulate the 2p ones. Similarly, they can model the spatial functions of arbitrary complexity.

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