Essentials of Computational Chemistry

6.2 BASIS SETS 173
in each. The general contraction scheme has some technical advantages over the segmented
one. One advantage in terms of efficiency is that integrals involving the same primitives,
i.e., those occurring in the final line of Eq. (6.4), need in principle be calculated only once,
and the value can be stored for later reuse as needed. Examples of split-valence basis sets
using general contractions are the cc-pVDZ, cc-pVTZ, etc. sets of Dunning and co-workers,
where the acronym stands for ‘correlation-consistent polarized Valence (Double/Triple/etc.)
Zeta’ (Dunning 1989; Woon and Dunning 1993). The ‘correlation-consistent’ part of the
name implies that the exponents and contraction coefficients were variationally optimized
not only for HF calculations, but also for calculations including electron correlation, methods
for which are described in Chapter 7. The subject of polarization is what we turn to next.
6.2.4 Polarization Functions
The distinction between atomic orbitals and basis functions in molecular calculations has
been emphasized several times now. An illustrative example of why the two should not
necessarily be thought of as equivalent is offered by ammonia, NH3. The inversion barrier
for interconversion between equivalent pyramidal minima in ammonia has been measured
to be 5.8 kcal mol −1 . However, a HF calculation with the equivalent of an infinite, atomcentered
basis set of s and p functions predicts the planar geometry of ammonia to be a
minimum-energy structure!
The problem with the calculation is that s and p functions centered on the atoms do
not provide sufficient mathematical flexibility to adequately describe the wave function for
the pyramidal geometry. This is true even though the atoms nitrogen and hydrogen can
individually be reasonably well described entirely by s and p functions. The molecular
orbitals, which are eigenfunctions of a Schrödinger equation involving multiple nuclei at
various positions in space, require more mathematical flexibility than do the atoms.
Because of the utility of AO-like GTOs, this flexibility is almost always added in the form
of basis functions corresponding to one quantum number of higher angular momentum than
the valence orbitals. Thus, for a first-row atom, the most useful polarization functions are
d GTOs, and for hydrogen, p GTOs. Figure 6.3 illustrates how a d function on oxygen can
polarize a p function to improve the description of the O–H bonds in the water molecule.
The use of p functions to polarize hydrogen s functions has already been mentioned in
Section 4.3.1. [An alternative way to introduce polarization is to allow basis functions not
to be centered on atoms. Such floating Gaussian orbitals (FLOGOs) are illustrated on the
left-hand side of Figure 4.1. While the use of FLOGOs reduces the need to work with integrals
involving high-angular-momentum functions, the process of geometry optimization is
rendered considerably more complicated, so they are rarely employed in modern calculations.]
Adding d functions to the nitrogen basis set causes HF theory to predict correctly a
pyramidal minimum for ammonia, although some error in prediction of the inversion barrier
still exists even at the HF limit because of the failure to account for electron correlation.
A variety of other molecular properties prove to be sensitive to the presence of polarization
functions. While a more complete discussion occurs in Section 6.4, we note here
that d functions on second-row atoms are absolutely required in order to make reasonable