In this lab, we wish to help you to understand how basis sets are formed. Basis sets are mathematical functions that are used to approximate the size and shape of atomic orbitals, usually one-electron atomic orbitals. These functions are then combined, using the linear combination of atomic orbitals approximation, to describe molecular orbitals.
Basis sets are generally constructed by mathematically combining some number of gaussian functions, as indicated by their name. We will be working primarily with split-valence basis sets. Minimal basis sets, examples of which are STO-3G and STO-6G, are constructed by combining three and six gaussian functions, respectively. Split-valence sets, such as 3-21G and 3-21G*, are constructed of a number of gaussians distributed differently over the orbital. Split-valence basis sets more accurately describe the atomic orbital, and thus give better results!
A general mathematical representation of a basis function is:
where G is the basis function, the next seven values in the parenthesis are independent variables. N is known as a normalization factor. X, y, and z are the coordinates of the electron. The Greek symbol alpha is a contraction coefficient, and r is the radius of the electron from the nucleus. The variables l, m, and n are simply power coefficients.
In this lab, we wish to visualize what happens when two gaussian functions overlap to form a new gaussian. It is the gaussian formed that is used to represent the atomic orbital. In this case, we are representing the orbital with a 2G basis set. The functions used in this lab approximate the types of functions used in creating basis sets, but are simplified:
where a0 is the Bohr radius (0.529 Angstroms) and the values of r are the radius of the electron from the nucleus in Angstroms.
For this model, we have two gaussians. Gaussian A begins -5.0 Angstroms from the nucleus, going to 5 Angstroms from the nucleus, in increments of 0.1. Gaussian B starts at -3.5 and stops at 6.5.
As is the case in most combinations of gaussians, the resultant gaussian is simply the product of Gaussians A and B:
Resultant Gaussian = Gaussian A * Gaussian B
Prepare a spreadsheet such as shown below, then plot the three gaussians as a function of Radius A.
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