Essentials of Computational Chemistry

6.2.1 Functional Forms
6.2 BASIS SETS 167
Slater-type orbitals were introduced in Section 5.2 (Eq. (5.2)) as the basis functions used
in extended Hückel theory. As noted in that discussion, STOs have a number of attractive
features primarily associated with the degree to which they closely resemble hydrogenic
atomic orbitals. In ab initio HF theory, however, they suffer from a fairly significant limitation.
There is no analytical solution available for the general four-index integral (Eq.
(4.56)) when the basis functions are STOs. The requirement that such integrals be solved by
numerical methods severely limits their utility in molecular systems of any significant size.
Nevertheless, high quality STO basis sets have been developed for atomic and diatomic
calculations, where such limitations do not arise (Ema et al. 2003).
Boys (1950) proposed an alternative to the use of STOs. All that is required for there to be
an analytical solution of the general four-index integral formed from such functions is that
the radial decay of the STOs be changed from e −r to e −r2
. That is, the AO-like functions are
chosen to have the form of a Gaussian function. The general functional form of a normalized
Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is
φ (x,y,z; α, i, j, k) =
2α
π
3/4 i+j+k
1/2
(8α) i!j!k!
x
(2i)!(2j)!(2k)!
i y j z k e −α(x2 +y2 +z2 )
(6.2)
where α is an exponent controlling the width of the GTO, and i, j, andk are non-negative
integers that dictate the nature of the orbital in a Cartesian sense.
In particular, when all three of these indices are zero, the GTO has spherical symmetry,
and is called an s-type GTO. When exactly one of the indices is one, the function has axial
symmetry about a single Cartesian axis and is called a p-type GTO. There are three possible
choices for which index is one, corresponding to the px, py, andpz orbitals.
When the sum of the indices is equal to two, the orbital is called a d-type GTO. Note
that there are six possible combinations of index values (i, j, k) thatcansumtotwo.In
Eq. (6.2), this leads to possible Cartesian prefactors of x 2 ,y 2 ,z 2 , xy, xz, andyz. These six
functions are called the Cartesian d functions. In the solution of the Schrödinger equation for
the hydrogen atom, only five functions of d-type are required to span all possible values of
the z component of the orbital angular momentum for l = 2. These five functions are usually
referred to as xy, xz, yz, x 2 − y 2 ,and3z 2 − r 2 . Note that the first three of these canonical
d functions are common with the Cartesian d functions, while the latter two can be derived
as linear combinations of the Cartesian d functions. A remaining linear combination that
can be formed from the Cartesian d functions is x 2 + y 2 + z 2 , which, insofar as it has
spherical symmetry, is actually an s-type GTO. Different Gaussian basis sets adopt different
conventions with respect to their d functions: some use all six Cartesian d functions, others
prefer to reduce the total basis set size and use the five linear combinations. [Note that if
the extra function is kept, the linear combination having s-like symmetry still has the same
exponent α governing its decay as the rest of the d set. As d orbitals are more diffuse than s
orbitals having the same principal quantum number (which is to say the magnitude of α for
the nd GTOs will be smaller than that for the α of the ns GTOs), the extra s orbital does not
really contribute at the same principal quantum level, as discussed in more detail below.]