13.6 Primitive set definition

A group of basis functions is defined by a data card specifying a set of primitive gaussians, optionally followed by one or more cards specifying particular contractions of primitives to be included in the final basis (see section 13.7 for specification of contractions). When all contraction definitions have been read (delimited by the next data card other than a contraction definition), the remaining primitives in the set which have not been included in any contraction set are added uncontracted to the basis set.

There are four different input forms, as explained below under a) to d). In case that options (e.g. SCALE, NPRIM) are specified, they can be given in any order, but no value without option key must be given after an option.

In all cases type defines the angular symmetry (S, P, D, F, G, H, or I). type can include several types, e.g., SPD or DF. This usually makes sense only with or default library contractions or no contractions.

The basis is loaded for all atoms with tag name atom in the geometry input. If atom is an integer, it refers to a z-matrix row.

a) Library basis sets:

type,atom,key,scale2,nprim;

or

type,atom,key,[SCALE=scale$\vert$SCALE2=scale2],[NPRIM=nprim$\vert$DELETE=ndel];

Load basis named key from the library

If scale or scale2 is present, all exponents are scaled by scale or scale**2, respectively. If nprim is specified, the first nprim exponents only are taken from the library. If nprim is negative or ndel is given, the last $\vert nprim\vert$ ($ndel$) basis functions from the library set are deleted. Associated with the library basis may be a set of default contraction coefficients which may be accessed in subsequent contraction cards. type can include several types, e.g., SPD or DF. This usually makes sense only with default contractions, i.e., such cards should be followed only by ``C'' without any other specifications for contractions.

b) Explicit basis input:

type,atom,exp1,exp2,$\ldots$expn;expn+1,$\ldots$;

General specification of exponents; continuation onto subsequent cards (separated by semicolon) is permitted as shown (the first card can hold up to 19 exponents, each following card 20 exponents.

The exponents (and other numerical parameters described below such as numbers of functions, and contraction coefficients) can be given as general input expressions, possibly involving variables. It is important to note, however, that these expressions are evaluated typically just once, at the same time as the complete basis set is parsed. This generally happens the first time that the basis set is required, perhaps before the first SCF calculation can be done. If the variables on which the basis depends are altered, this will not be noticed by the program, and the new basis set will not be used for subsequent stages of the computation. If, however, a new basis block is presented in the input, then the program marks as outdated any quantities such as integrals that have been calculated with the old basis set; subsequent job steps will then use the new basis.

c) Even tempered basis sets:

type,atom,EVEN,nprim,ratio,centre,dratio

or

type,atom,EVEN,NPRIM=nprim,[RATIO=ratio],[CENTRE=centre],[DRATIO=dratio]

Generates a generalized even tempered set of functions. The number of functions $n$ is specified by nprim, their geometric mean $c$ by centre, the mean ratio of successive exponents $r$ by ratio, and the variation of this ratio, $d$, by dratio. If centre is not given, the previous basis of the same type is extended by diffuse functions. If in this case ratio is not given, $r$ is determined from the exponents of the last two function of the previous basis. If this is not possible, the default $r=2.5$ is adopted. $d=1$ (the default) specifies a true even-tempered set, but otherwise the ratio between successive exponents changes linearly; the exponents are given explicitly by

$$\log e_i = \log c + ((n+1)/2-i) \,\log r + \frac12 ((n+1)/2-i)^2 \,\log d \quad i=1,2,\dots,n$$

Example 1

SP,1,VTZ;C;SP,1,EVEN,1;
generates the generally contracted $s$ and $p$ triple-zeta basis sets for atom 1 and extends these by one diffuse function.

Example 2

SPD,1,VTZ,DELETE=1;C;
SP,1,EVEN,NPRIM=2,RATIO=2.5;
generates the generally contracted $s$, $p$ triple-zeta basis sets for atom 1. Two energy optimized $d$-functions of Dunning are included. The last $s$ and $p$ functions are deleted and replaced by two even tempered functions with ratio 2.5.

d) 3-term tempered basis sets:

type,atom,EVEN3,nprim,$\alpha$, $\beta$, $\gamma$

Generates a 3-parameter set of nprim functions with exponents given by

$$e_i = \alpha ; \qquad e_i=e_{i-1} \beta \left(1+\frac{\gamma i^2}{(\text{\emph{nprim}}+1)^2}\right)$$

e) Regular even tempered basis sets:

type,atom,EVENR,nprim,aa,ap,bb,bp

Generates an even tempered set of nprim functions according to the ``regular'' prescription described in M W Schmidt and K Ruedenberg, J. Chem. Phys. 71 (1970) 3951. If any of the parameters aa, ap, bb, bp is zero or omitted, the values are taken from table III of the above.

f) Even tempered basis set with confined progression:

type,atom,EVENP,nprim,$\alpha$,$\beta$,$\gamma$

Generates an even tempered basis set with nprim functions and a maximal exponent given by $\alpha$. The progression (ratio) between the first and second exponent is adjusted using parameter $\beta$ and the progression between the last but one and the last exponent is adjusted with parameter $\gamma$. In between the progression is linearly interpolated. The explicit values of the progression factors are given by:

$p(\beta)=\frac{\mathrm{exponent^i}}{\mathrm{exponent^{i+1}}}= \frac{5}{\pi}(\arctan(\beta-2.5)+\frac{\pi}{2})+\sqrt{2}$

so that for $\beta \ll 0: p \rightarrow \sqrt{2}$ and for $\beta \gg 0: p \rightarrow 5+\sqrt{2}$ which limits the progression factors in between these two values and enables unconstrained basis set optimisations. For $\beta \approx 0$ the progression has a factor of about 2.

type,atom,EVENP2,nprim,$\alpha$,$\beta$,$\gamma$,$\delta$

Generalises confined progression tempered basis sets by a third paramter (now $\gamma$) which defines the progression as above in the centre. The ratio factors are then determined by interpolating between $p(\beta)\rightarrow p(\gamma)$ and $p(\gamma)\rightarrow p(\delta)$.