Chemistry Reference
In-Depth Information
molecular point group is hence C
3v
, whose character table was given in
Table 12.3. The minimal basis set for the MO calculation is given, in an
obvious brief notation, by the row matrix of the m
¼
8 STOs:
x ¼ð
kszxyh
1
h
2
h
3
Þ
ð
12
:
37
Þ
which must be combined linearly to give eight MOs, the first n
¼
5 being
doubly occupied by electrons with opposite spin, so accommodating the
N
¼
2n
¼
10 electrons of the molecule in its totally symmetric singlet
1
A
1
ground state.
Even in this case, the construction of the symmetry-adapted basis for the
calculation can be done at once by simple inspection, since it is imme-
diately evident that the functions belonging to the different irreps of the
point group C
3v
are
1
k
;
s
;
z
;
h
z
¼
p ð
h
1
þ
h
2
þ
h
3
Þ)
A
1
ð
12
:
38
Þ
<
:
1
x
;
h
x
¼
p ð
2h
1
h
2
h
3
Þ
)
E
ð
12
:
39
Þ
1
y
;
h
y
¼
p ð
h
2
h
3
Þ
the last being a doubly degenerate irrep whose basic vectors transform as
(xy).
Turning to the group theoretical techniques, following what
was done before for H
2
O, Table 12.7 gives the dimensions of the
representative matrices for the different operations R in the reducible
representation
G
in the original basis and those of the irreducible
j
corresponding to the symmetry-adapted functions
of the last column. The latter are obtained by letting the full projec-
tor (12.29) act on the original AO basis (12.37). Table 12.8 is the
transformation table of the original AO basis (recall that in the table
c
¼
1
representations
G
p
3
2).
In the case of NH
3
, because of the presence of the doubly degenerate
irrep E, the simple projector (12.30) based on characters would yield
nonorthogonal linearly dependent symmetry functions, which should
then be Schmidt orthogonalized to give a linearly independent set. It is
=
2,
s
¼
=
