Primer on Atomic Orbital Basis Sets

To introduce the concepts of a basis set and basis functions, we begin with a simple (unknown) function of a single variable, f x). A variety of procedures can be used to fit (or estimate) this function. For example, a simple power series could be used to approximate this simple function of the variable x  

Here we are using a basis set to approximate the unknown function f x). The basis functions are od i = 0,1,2,3. The expansion coefficients, c are determined by some sort of procedure that adjusts their values in order to obtain the best fit to the function f x). The approximation can generally be improved by using a larger basis set. 

Because of the systematic nature of the correlation-consistent family of basis sets, it is possible to use extrapolation techniques to estimate the CBS limit. For example, the three-parameter exponential function introduced by Feller nicely describes the convergence of SCF energy to the CBS limit with 

For the fitting of correlation energies obtained with extremely large hextuple-and heptuple-zeta (6Z and 7Z) basis sets, the following revision to Eq. [43] has been suggested based on MP2 pair energies  

Martin has proposed a two-parameter fit to a quartic polynomial (Schwartz4) and a three-parameter fit to a sixth-degree polynomial (Schwartz6)  

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