Basis Sets

Equation (1.16) states that the orbitals are made up of a linear combination of basis functions. Obviously, we have restricted the orbitals' flexibility unless the basis functions form a complete set. Each function added to the basis increases the computational cost, so it is vital that the number of functions in a basis is kept as small as possible, while at the same time providing the orbital with as much flexibility as required.

The most convenient way to define a basis set for any nuclear configuration is to define a particular set of functions for each nucleus, depending only on the nuclear charge of that nucleus. There are two main types of basis functions in use today. The first, introduced by Slater in 1930, are termed Slater-Type Atomic Orbitals (STOs) [55]. STOs have exponential radial parts
$$\begin{align}\phi_{a}({\bf r}) = (x-A_{x})^{a_{x}}(y-A_{y})^{a_{y}}(z-A_{z})^{a_{z}}e^{-\alpha\vert{\bf r}-\mathbf{A}\vert} \end{align}$$
with a center A=(A_x,A_y,A_z), angular momentum a=(a_x,a_y,a_z) and nuclei dependent exponent $\alpha$. STOs, like exact wavefunctions, have cusps at the nuclei and decay exponentially. Unfortunately, integrals over STOs are expensive to compute.

In 1950 Boys [56] suggested that basis functions constructed of Gaussian-Type Atomic Orbitals (GTOs) would overcome the computational difficulties of STOs. A GTO has the form
$$\begin{align}\phi_{a}({\bf r}) = (x-A_{x})^{a_{x}}(y-A_{y})^{a_{y}}(z-A_{z})^{a_{z}}e^{-\alpha\vert{\bf r}-\mathbf{A}\vert^{2}}. \end{align}$$
GTOs decay too fast and have incorrect nuclear cusps, so it is not surprising that many more GTOs than STOs are required to achieve the same accuracy [57]. However, the speed with which integrals over GTOs can be calculated more than compensates for this.

If STO properties are desired, they can be approximated by a sum of Gaussians, a philosophy which led to the introduction of the STO-nG basis sets [58]. These are an example of Contracted GTOs [59]
$$\begin{align}\phi_{a}({\bf r}) = \sum_{k}^{K_{A}} D_{ak} (x-A_{x})^{a_{x}}(y-A_{... ...{y}}(z-A_{z})^{a_{z}}e^{-\alpha_{k}\vert{\bf r}-\mathbf{A}\vert^{2}} \end{align}$$
where K_A is referred to as the degree of contraction and the D_ak are the contraction coefficients. The contraction coefficients are not changed during a calculation, reducing the computational overhead.

There are several different basis sets in common use, all offering different trade-offs between accuracy and speed. Some of the more popular are those by Dunning [60] and the `split-valence' sets by Pople and co-workers [61,62,63].