Automatic Generation of Auxiliary Functions

Automatically generated auxiliary function sets are provided in deMon2k by means of the auxiliary function specification GEN-A$n$, with $n=1,2,3,4$, and GEN-A$n$*, with $n=2,3,4$ [133,156]. The GEN-A1 set possesses only $s$ auxiliary functions and is usually used only for debugging. The GEN-A$n$ sets consist of $s$, $p$, and $d$ Hermite Gaussian functions. In addition, the GEN-A$n$* also have $f$ and $g$ Hermite Gaussians. Because the auxiliary functions are used to fit the electron density they are grouped in $s$, $spd$, and $spdf \! g$ sets. The exponents are shared within each of these sets [53,54]. Therefore, the auxiliary function notation (3,2,2) describes 3 $s$ sets with a total of 3 functions, 2 $spd$ sets with a total of 20 functions and 2 $spdf \! g$ 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, $\beta_{\rm min}$, and largest, $\beta_{\rm max}$, primitive Gaussian exponent of the specified orbital basis set. Therefore, the GEN-A$n$ and GEN-A$n$* auxiliary function sets differ for different orbital basis sets. The number of exponents $N$ (auxiliary function sets) is given by:

$$N = {\rm Int} \, \left ( \, {{\ln ( \, \beta_{\rm max} / \beta_{\rm min} \, ) } \over {\ln (6 - n)}} + 0.5 \, \right )%$$ (1)

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

$$\beta_o = 2 \, \beta_{\rm min} \, (6-n)^{(N-1)}%$$ (2)

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

$$\beta_o = 2 \, \beta_{\rm min} \, (6-n)^N%$$ (3)

Furthermore, only $spd$ and $spdf \! g$ sets are generated on ECP and MCP centers. From $\beta_o$ the two tightest $s$, or $spd$ set exponents, $\beta_1$ and $\beta_2$ are generated according to the formulas:

$$\beta_1 = \left ( 1 + {n \over {12 - 2 n}} \, \right ) \, \beta_o<tex2html_comment_mark>$$ (4)

$$\beta_2 = {\beta_o \over {6 - n}}<tex2html_comment_mark>$$ (5)

The other $s$ or $spd$ set exponents are generated according to the even-tempered progression:

$$\beta_{i+1} = {\beta_i \over {6 - n}}%$$ (6)

The $\beta_o$ exponent of the subsequent $spd$ sets is also generated according to the progression (A..6). Based on this $\beta_o$ exponent, the exponents of the first two $spd$ sets are calculated with the formulas (A..4) and (A..5). The subsequent $spd$ set exponents are calculated again according to the even-tempered progression (A..6). In the same way, the $spdf \! g$ set exponents are calculated. In the case of $3d$ elements, an extra diffuse $s$ auxiliary function set is added.