Essentials of Computational Chemistry

184 6 AB INITIO HARTREE–FOCK MO THEORY
O1s
O2s
H a 1s
H b 1s
O2p x
O2p y
O2p z
Unsymmetrized
basis set
H a
H a
H a
H a
H a
H a
O
O
O
O
H b
H b
H b
H b
H b
H b
H a
O
Fock
Matrix
a 1
Dense
Symmetric
transform
0
b 2
b 1
H b
O1s
O2s
H a 1s + H b 1s
O2p z
H a 1s − H b 1s
O2p x
O2p y
C 2n basis set
Figure 6.7 Transformation of the minimal water AO basis set to one appropriate for the C2v point
group. The effect on the form of the Fock matrix is also illustrated
In particular, if the totally symmetric representation is not included in the product of the
irreducible representations of basis functions µ, ν, λ, andσ ,then(µν|λσ ) = 0. The analogous
rule holds for the one-electron integrals. In general, of course, the atomic basis functions
do not belong to any irreducible representation, since they themselves do not transform
with all the symmetry elements of the molecule. Thus, as a first step to taking advantage
of symmetry, linear combinations of the various basis functions must be formed that do
belong to irreps of the molecular point group. This process is illustrated in Figure 6.7 for a
HF/STO-3G calculation on water, which belongs to the C2v point group. The seven atomic
basis functions can be linearly transformed to four, two, and one functions belonging to the
0
a 1
H a
H a
H a
b 2
H a
b 1
H a
O
O
O
O
H b
H b
H b
H b
H b