Special Choices of Basis Functions

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

The expansion of four-component one-electron functions into a set of global basis functions can be done in several ways independently of the particular choice of the type of the basis functions. For instance, four independent expansions may be used for the four components as sketched above. However, one might also relate the expansion coefficients of the four components to each other. In contrast to these expansions, the molecular spinors can also be expressed in terms of 2-spinor expansions. This latter ansatz reflects the structure of the one-electron Fock operator and is therefore very efficient (for a detailed discussion compare Ref. [565]) The expansion in terms of 2-spinor 

We will later in this chapter understand that the use of STO exponential functions produces two-electron integrals that cannot be calculated analytically. It turned out that this problem can be avoided if Gauss-type orbitals (GTOs) are used whose radial components are of the form 

the small component s radial function has been fixed according to the kinetic balance condition. 

As an alternative to the spherical Gaussian basis sets introduced so far, Cartesian Gaussian functions. 

---