Gaussian Orbital - Using Tables of Gaussian Orbitals

Using Tables of Gaussian Orbitals

There is a large database of Gaussian orbitals EMSL portal. The database determines shapes of (atom-centered) basis functions (in fact radial part of these function) optimised for one or the other criteria. The shapes are described as tables of factors and coefficients. The factors and coefficient define Gaussian primitives, which needs to be contracted (summed over) to determine a radial orbital. The basis functions possess a usual radial-angular decomposition

,

where is spherical harmonic functions, and are angular momentum and projection of angular momentum quantum numbers, are spherical coordinates. The function is the radial orbital we want to determine. Radial orbital is a sum of Gaussian primitives with an angular momentum factor in front of expression

,

where and are listed in the tables of EMSL portal, the radial coordinate must be in atomic units (see Bohr radius) and are normalisation factors to ensure the norm of Gaussian primitives is one

A closed form expression for normalisation factors can be obtained using Gaussian integrals

Particular values of the normalisation factor for different are