have a 1sorbital represented by an inner 1s^0 and an outer 1s^00 basis function, making
two basis functions. Carbon has a 1sfunction represented by six Gaussians, an inner
2 s,2px,2pyand 2pz(2s^0 ,2px^0 ,2py^0 ,2pz^0 ) function, each composed of three Gaussians,
and an outer 2s,2px,2pyand 2pz(2s^00 ,2px^00 ,2py^00 ,2pz^00 ) function, each composed of
one Gaussian, and six (not five) 3dfunctions, making a total of 15 basis functions.
A 6–31G calculation on CH 2 uses 15þ 2 þ 2 ¼19 basis functions, and generates 19
MO’s. In the closed-shell species the eight electrons occupy four of these MO’s, so
there are 15 unoccupied or virtual MO’s; compare this with a 3–21G()calculation on
CH 2 (above) where there are a total of 13 MO’s with nine of them virtual. The
6–31G basis, also often called 6–31G(d), is summarized in Fig.5.13d.
The 6–31G is probably the most popular basis at present. It gives good
geometries and, often, reasonable relative energies (Section 5.5.2); however,
there seems to be little evidence that it is,in general, much better than the
3–21G()basis for geometry optimizations. Since it is about five times as slow
(Table5.3) as the 3–21G() basis, the general preference for the 6–31G for
geometry optimizations may be due to its better relative energies (Section 5.5.2).
The 3–21G() basisdoes have certain geometry deficiencies compared to the
6–31G, particularly its tendency to overzealously flatten nitrogen atoms (the N
of aniline is wrongly predicted to be planar), and this, along with inferior relative
energies and less consistency, may be responsible for its being neglected in favor of
the 6–31G basis set [ 51 ]. The virtues of the 3–21G()and 6–31G basis sets for
geometry optimizations are discussed further inSection 5.5.1.Note that the geo-
metries and energies referred to here are those from Hartree–Fock-level calcula-
tions. Post-Hartree–Fock (Section 5.4) calculations, which can give significantly
better geometries and much better relative energies (Sections 5.5.1and5.5.2), are
considered to require a basis set of at least the 6–31G size for meaningful results.
The 6–31G basis adds polarization functions only to so-called heavy atoms
(those beyond helium). Sometimes it is helpful to have polarization functions on the
hydrogens as well; a 6–31G* basis with three 2pfunctions on each H and He atom
(in addition to their 1s^0 and 1s^00 functions) is called the 6–31G* (or 6–31(d,p))
basis. The 6–31G and 6–31G bases are the same except that in the 6–31G
each H and He has five, rather than two, functions. The 6–31G* basis probably
offers little advantage over the 6–31G unless the hydrogens are engaged in some
special activity like hydrogen bonding or bridging [ 52 ]. In high-level calculations
on hydrogen bonding or on boron hydrides, for example, polarization functions are
placed on hydrogen. For calculations on and references to the hydrogen bonded
water dimer, see Sections 5.4.3.1 and 5.4.3.3.
5.3.3.4 Diffuse Functions
Core electrons and electrons engaged in bonding are relatively tightly bound to the
molecular nuclear framework. Lone-pair electrons or electrons in a (previously)
virtual orbital, are relatively loosely held, and are on the average at a larger distance
from the nuclei than core or bonding electrons. These “expanded” electron clouds
5.3 Basis Sets 247