Chemistry Reference
In-Depth Information
and zx), their intersection determining the twofold symmetry axis C
2
. The
molecular point group is hence C
2v
, whose character table was given in
Table 12.1. Theminimal set for theMOcalculation is given, in an obvious
brief notation, by the row matrix of the m
¼
7 STOs:
x ¼ð
kszxyh
1
h
2
Þ
ð
12
:
31
Þ
which must be combined linearly to give seven MOs, the first n
¼
5
being doubly occupied by electrons with opposite spin, so accomodat-
ing the N
¼
2n
¼
10 electrons of the molecule in its totally symmetric
singlet
1
A
1
ground state. The LCAO coefficients of the Roothaan
equations are obtained by the Ritz method through the iterative solu-
tion of a (7
7) secular equation, but for our illustrative purposes it will
be sufficient to consider the simple H
uckel approximation in the same
basis. The construction of the symmetry-adapted basis for the calcula-
tion can be done at once by simple inspection, since it is immediately
evident that the functions belonging to the different irreps of the point
group C
2v
are
€
1
k
;
s
;
z
;
h
z
¼
p ð
h
1
þ
h
2
Þ)
A
1
ð
12
:
32
Þ
x
)
B
1
ð
12
:
33
Þ
1
y
;
h
y
¼
p ð
h
1
h
2
Þ)
B
2
ð
12
:
34
Þ
Since functions of different symmetry cannot mix under the totally
symmetric H
uckel representative
H
over
the minimum set will be factorized into three blocks of different symme-
tries: A
1
(4
4), B
1
(1
1) and B
2
(2
2).
Turning to the more formal group theoretical techniques, we give in
Table 12.5 the dimensions of the representative matrices for the
different operations R in the reducible representation
€
uckel operator H, the (7
7) H
€
G
in the original
j
corresponding to
the symmetry-adapted functions of the last column. The latter are easily
obtained by letting the projector (12.30) act on the original AO
basis (12.31). It is left to the reader to verify in this case all the
basis, and those of the irreducible representations
G
