Molecular one-electron functions

The expansion of four-component one-electron functions into a set of global basis functions can be done in several ways independent of the particular choice of the type of the basis functions. For instance, four independent expansions may be used for the four components. However, we 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 appears to be quite common and will be described in greater detail now. Obviously, analogous thoughts also apply for the first two possibilities (for a detailed discussion compare Dyall etal. 1991a). 

The most frequently used ansatz for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors (where we have adopted the notation used in Quiney et al. (1998b)) 

Quiney 1988). The small component s radial function has been fixed according to the kinetic balance condition (Stanton and Havriliak 1984), which has its origin in the coupled nature of Dirac s first-order differential equations and is introduced to keep the method variationally stable. The index A denotes the coordinates of the nucleus s centre RA of atom A, to which the basis function is attached, i.e. rA = r — RA. As an alternative, Cartesian Gaussians, 

Apart from the expansion into Gauss-type functions the use of Slater-type functions has been discussed (Grant and Quiney 1988), although the analytic evaluation of integrals becomes as hopeless as in the nonrelativistic theory. Therefore, these STFs are only a good choice for atoms, linear molecules, or for four-component density functional calculations, where integrals over the total electron density are evaluated numerically. 

---