are found in molecules with heteroatoms, in anions, and in electronically excited
molecules. To simulate well the behaviour of such speciesdiffuse functionsare
used. These are Gaussian functions with small values ofa; this causes exp(#ar^2 ) to
fall off very slowly with the distancerfrom the nucleus, so that by giving enough
weight to the coefficients of diffuse functions the SCF process can generate
significant electron density at relatively large distances from the nucleus. Typically
a basis set with diffuse functions has one such function, composed of a single
Gaussian, for each valence atomic orbital of the “heavy atoms”. The 3# 21 þG basis
set for carbon (¼ 3 # 21 þG()for this element) is
1 s
2 s^02 p^02 p^02 p^0
2 s^002 p^002 p^002 p^00
2sþ, 2pþ, 2pþ, 2pþ
13 Basis functions
and the 6–31þG basis for carbon is
1 s
2 s^02 p^02 p^02 p^0
2 s^002 p^002 p^002 p^00
3 d 3 d 3 d 3 d 3 d 3 d
2 sþ,2pþ,2pþ,2pþ
19 Basis functions
Sometimes diffuse functions are added to hydrogen and helium as well as to the
heavy atoms; such a basis set is indicated byþþ. The 3–21þþG and 6–31þþG
basis for hydrogen and helium is
1 s
1 s^0
1 sþ
Three basis functions
A 3–21þþG calculation on CH 2 would use 13þ 3 þ 3 ¼19 basis functions, a
6–31þþG* calculation 19þ 3 þ 3 ¼25 basis functions, and a 6–31þþG*
calculation 19þ 6 þ 6 ¼31 basis functions.
There is some disagreement over when diffuse functions should be used. Cer-
tainly most workers employ them routinely in studying anions and excited states,
but not ordinary lone pair molecules (molecules with heteroatoms, like ethers and
amines). A reasonable recommendation is to study with and without diffuse func-
tions species representative of the problem at hand, for which experimental results
are known, and see if these functions help. A paper by Warner [ 52 ] gives useful
references and a good account of the efficacy of diffuse functions in treating certain
molecules with heteroatoms. He settles on the 6–31þG, i.e. 6–31þG(d), basis.
5.3.3.5 Large Basis Sets
The 3–21G()is a small basis set and the 6–31G and 6–31G* are moderate-size
basis sets. Of those we have discussed, only the 6–31G and 6–31G** with diffuse
248 5 Ab initio Calculations