functions. Starting with the next element, scandium, five 3dorbitals are added,
so that scandium to krypton have 13þ 5 ¼18 basis functions. The STO-3G basis is
summarized in Fig.5.13a.
The STO-3G basis introduces us to the concept ofcontraction shellsin con-
structing contracted Gaussians from primitive Gaussians (Section 5.3.2). The
Gaussians of a contraction shell share common exponents. Carbon, for example,
has onesshell and onespshell. This means that the 2sand 2pGaussians (belonging
to the 2spshell) share commonaexponents (which differ from those of the 1s
function). Consider the contracted Gaussians
fð 2 sÞ¼d 1 se#a^1 srþd 2 se#a^2 srþd 3 se#a^3 sr
fð 2 pxÞ¼d 1 pxe#a^1 prþd 2 pxe#a^2 prþd 3 pxe#a^3 pr
fð 2 pyÞ¼d 1 pye#a^1 prþd 2 pye#a^2 prþd 3 pye#a^3 pr
fð 2 pzÞ¼d 1 pze#a^1 prþd 2 pze#a^2 prþd 3 pze#a^3 prThe usual practice is to seta 1 s¼a 1 p,a 2 s¼a 2 p, anda 3 s¼a 3 p. Using commona’s
for thesandpprimitives reduces the number of distinct integrals that must be
calculated. An STO-3G calculation on CH 4 , for example, involves nine basis
functions (five for C and one for each H) in six shells: for C ones(i.e. a 1s) shell,
and onesp(i.e. a 2splus 2p) shell, and for each H ones(i.e. a 1s) shell. The current
view is that the STO-3G basis is not very good, and it would normally be considered
unacceptable for research. Nevertheless, one hesitates to endorse Dewar and
Storch’s assertion that “it must be considered obsolete” [ 43 ]. We do not know
how many publications report work which began with a preliminary and unreported
but valuable investigation using this basis. Its advantages are speed (it is probably
the smallest basis set that would even be considered for an ab initio calculation)
and the ease with which the molecular orbitals can be dissected into atomic orbital
contributions. The STO-3G basis is roughly twice as fast (Table5.3) as the next
larger commonly used one, the 3–21G. Sophisticated semiempirical methods
Table 5.3 Effect of basis set and symmetry on times for single-point, geometry optimization and
geometry optimization + frequencies calculations on acetone, (CH 3 ) 2 CO
Basis set Single point Geometry optimization Geometry optimizationþfrequencies
Time (s) Time (s) Time (s)
C2v C 1 C2v C 1 C2v C 1
STO-3G 0.2 (0.2) 0.3 (0.2) 1 (2) 2 (7) 2 (13) 3 (59)
3–21G() 0.5 (0.3) 0.6 (0.5) 2 (2) 3 (5) 3 (20) 8 (75)
6–31G 1.4 (2) 2 (3) 9 (15) 22 (54) 15 (172) 30 (586)
The starting geometry for the ab initio jobs was a molecular mechanics (MMFF) one. The C2v
geometry is that with two C–H/C¼O eclipsed arrangements (the global minimum). The C 1
symmetry starting geometry was obtained by rotating one C–C bond very slightly (by 1) in the
C2vprecursor molecular mechanics structure (after MM optimization). These calculations were
done with a fairly recent (2006) version of Spartan [ 37 ] on a quadcore 2.66 GHz personal computer
with 4.0 GB of RAM, vintage 2007. For times of ca. 1 s, time differences are scarcely meaningful.
Numbers in parentheses were for calculations done in ca. 2001
242 5 Ab initio Calculations