finer-tuning of the electron distribution and a lower energy, than would be possible
with unsplit basis functions.
A still more malleable basis set would be one withallthe basis functions, not just
those of the valence AO’s but the core ones too, split; this is called adouble zeta
(doublez) basis set (perhaps from the days before Gaussians, with exp(#ar^2 ), had
almost completely displaced Slater functions with exp(#zr) for molecular calcula-
tions). Double zeta basis sets are much less widely used thansplit valencesets,
since the former are computationally more demanding and for many purposes only
the contributions of the “chemically active” valence functions to the MO’s need to
be fine-tuned, and in fact “double zeta” is sometimes used to refer to split valence
basis sets.
Returning to the 3–21G basis: here lithium to neon have a 1sfunction and inner
and outer 2s,2px,2pyand 2pz(2s^0 ,2s^00 ,...,2pz^00 ) functions, for a total of 9 basis
functions. These inhabit three contraction shells (see the STO-3G discussion): a 1s,
anspinner and anspouter contraction shell. Sodium to argon have a 1s,a2sand
three 2pfunctions, and an inner and outer shell of 3sand 3pfunctions, for a total of
1 þ 4 þ 8 ¼13 basis functions. These are in four shells: a 1s, ansp(2s,2p), ansp
inner and anspouter (3sand 3pinner, 3sand 3pouter). Potassium and calcium have
a1s,a2sand three 2p, and a 3sand three 3pfunctions, plus inner and outer 4sand
4 pfunctions, for a total of 1þ 4 þ 4 þ 8 ¼17 basis functions. The 3–21G basis set
is summarized in Fig.5.13b.
For molecules with atoms beyond the first row (beyond neon), this “simple” 3–21G
basis set tends to give poor geometries. This problem is largely overcome for second-
row elements (sodium to argon) by supplementing this basis withdfunctions, called
polarization functions. The term arises from the fact thatdfunctions permit the
electron distribution to be polarized (displaced along a particular direction), as
shown in Fig.5.16. Polarization functions enable the SCF process to establish a
more anisotropic electron distribution (where this is appropriate) than would other-
wise be possible (cf. the use of split valence basis sets to permit more flexibility in
adjusting the inner and outer regions of electron density). The 3–21G basis set
augmented where appropriate (beyond neon) with sixdfunctions is in some compu-
tational programs designated 3–21G(), where the asterisk indicates polarization
functions (din this case) and the parentheses emphasize that these extra (compared
to the “simple” 3–21G basis) functions are present only beyond the first row. For H to
Ne, the 3–21G and the 3–21G basis sets are identical. The simple 3–21G basis,
withoutthe possibility of invoking polarization functions, is probably obsolete, and
when we see “3–21G” we can usually take it to mean, really, the 3–21G()basis
summarized in Fig.5.13c; for precision, the 3–21G()designation will be preferred
here from now on.p-Polarization functions can also be added not only to heavy atoms
(recall that in computational chemistry atoms beyond hydrogen and helium in the
periodic table are called heavy atoms), but to hydrogen and helium also (below).
Examples of geometries calculated with the simple and augmented 3–21G basis
sets are shown in Fig.5.14. The 3–21G(*)gives remarkably good geometries for
such a small set, and in fact it is used for the geometry optimization step of some
high-accuracy energy methods (Section 5.5.2). Since it is roughly five times as fast
5.3 Basis Sets 245