Cartesian function sets, spurious components

ADF employs Slater-type exponential basis functions centered on the atoms. Such a function consists of an exponential part exp(-ar) and a polynomial pre-factor r^krx^kxy^kyz^kz. A function set is characterized by its radial behavior (the exponential part and the power of r, kr) and by its angular momentum quantum number l. The functions in such a set consist of all possible combinations x^kxy^kyz^kz, such that kx+kx+kz=l. These are denoted the Cartesian spherical harmonics.

The Cartesian function sets are very suitable for computational manipulations, but they have a drawback. By inspection it is easily verified that a d-set consists of 6 Cartesian functions, while there can of course be only 5 true d-type functions among them: one (linear combination) of them is in fact an s-type function (x²+y²+z²). Similarly, there are 10 f-type Cartesian functions, 3 of which are in fact p-functions. And so on. In ADF all such lower-l (combinations of) functions are projected out of the basis and not employed. As a consequence the basis set size in the sense of the number of degrees of freedom and hence the number of possible eigenfunctions of the Fock operator is smaller than the number of expansion coefficients that refer to the primitive (Cartesian) basis functions.

The abbreviation bas is used for references to the elementary Cartesian basis functions.