# Automatic Generation of Auxiliary Functions

Automatically generated auxiliary function sets are provided in deMon2k by means of the auxiliary function specification GEN-A, with , and GEN-A*, with [133,156]. The GEN-A1 set possesses only auxiliary functions and is usually used only for debugging. The GEN-A sets consist of , , and Hermite Gaussian functions. In addition, the GEN-A* also have and Hermite Gaussians. Because the auxiliary functions are used to fit the electron density they are grouped in , , and sets. The exponents are shared within each of these sets [53,54]. Therefore, the auxiliary function notation (3,2,2) describes 3 sets with a total of 3 functions, 2 sets with a total of 20 functions and 2 sets with a total of 70 functions (see also 4.3.3). The range of exponents of all automatically generated auxiliary functions is determined by the smallest, , and largest, , primitive Gaussian exponent of the specified orbital basis set. Therefore, the GEN-A and GEN-A* auxiliary function sets differ for different orbital basis sets. The number of exponents (auxiliary function sets) is given by:
 (1)

Here is 1, 2, 3, or 4 according to the index of the selected GEN-A or GEN-A* set. The exponents are generated in almost even-tempered form (see details below) and split into , and, if a GEN-A* set is specified, sets. The tightest (largest) exponents are assigned to the sets, followed by the , and, if specified, sets. The basic exponent from which the generation starts is defined as:
 (2)

In case ECPs or MCPs are used, equation (A..2) changes to:
 (3)

Furthermore, only and sets are generated on ECP and MCP centers. From the two tightest , or set exponents, and are generated according to the formulas:
 (4) (5)

The other or set exponents are generated according to the even-tempered progression:
 (6)

The exponent of the subsequent sets is also generated according to the progression (A..6). Based on this exponent, the exponents of the first two sets are calculated with the formulas (A..4) and (A..5). The subsequent set exponents are calculated again according to the even-tempered progression (A..6). In the same way, the set exponents are calculated. In the case of elements, an extra diffuse auxiliary function set is added.