Computational Physics

4.6 Basis functions 67

are only five d-states! This paradox is solved by noticing that the linear combination

(x^2 +y^2 +z^2 )e−α(r−RA)

2

(4.83)

has the symmetry of an s-orbital, and therefore, instead ofx^2 ,y^2 andz^2 , only the
orbitalsx^2 −y^2 and 3z^2 −r^2 are used (any independent combination is allowed),
thus arriving at five d-states.
GTOs are widely used for molecular calculations, and from now on we shall
restrict ourselves to basis functions of this form. The simplest basis consists of
one GTO per atomic orbital; it is called aminimal basis. The parameterαin the
exponent must somehow be chosen such that the GTO fits the atomic orbital in an
optimal way. However, since there is only one parameter to be fitted in the minimal
basis, this will give poor results, and in general more GTOs per atomic orbital are
used. This means that a 1s basis orbital is now given as a linear combination of
Gaussian functions: ∑

p

Dpe−αpr

2

. (4.84)

As the parametersαpoccur in the exponent, determination of the best combination
(Dp,αp)is a nonlinear optimisation problem. We shall not go into details of solving
suchaproblem(seeRef.[11]), but discuss the criterion according to which the best
values for(Dp,αp)are selected. A first approach is to take Hartree–Fock orbitals
resulting from anatomiccalculation, perhaps determined using Slater type orbitals,
and to fit the form(4.84)to these orbitals. A second way is to perform theatomic
Hartree–Fock calculation using Gaussian primitive basis functions and determine
the optimal set as a solution to the nonlinear variational problem in the space given
by(4.84).
Suppose we have determined the optimal set(Dp,αp), then there are in principle
two options for constructing the basis set. The first option is to incorporate for each
exponential parameterαp, the Gaussian function

e−αp(r−RA)

2

(4.85)

into the basis, that is, the values of theDp-parameters are relaxed since the prefactors
of the Gaussian primitive functions can vary at will with this basis. A second
approach is to take the linear combination (4.84) with the optimal set of(Dp,αp)as
asinglebasis function and add it to the basis set, i.e. keeping theDpfixed as well as
theαp. If we have optimised the solution(4.84)using four primitive functions, the
second option yields a basis four times smaller than the first but, because of its lack
of flexibility (remember theDpare kept fixed), it will result in lower accuracy. The
procedure of taking fixed linear combinations of Gaussian primitive functions as a
single basis function is calledcontraction; the basis set is called acontracted set.
The difference between the GTOs from which the basis functions are constructed