Chemistry Reference
In-Depth Information
Fig. 5.1
Dynamic symmetry group of ammonia, with permutations of nuclei, and inversion of
all particles (indicated by
an asterisk
).
The plus
(
minus
)
sign
indicates the position of an electron
above (below) the plane of the hydrogen atoms
Ta b l e 5 . 1
Embedding of the
C
3
v
point group in the
Longuet-Higgins group. The
symmetry elements of the
point group act on the
electrons. They are identified
as the product of nuclear
permutations, inversion of all
particles (star operation), and
bodily rotations of all
particles (
Q
operators) along
particular directions
C
3
v
Rotation
×
Inversion
×
Permutation
E
E
E
E
C
3
Q
z
3
E
(ABC)
C
3
(
Q
z
3
)
−
1
E
(ACB)
Q
⊥
σ
1
2
E
∗
σ
1
(A)(BC)
Q
⊥
σ
2
2
E
∗
σ
2
(B)(AC)
Q
⊥
σ
3
2
E
∗
σ
3
(C)(AB)
an axis that is perpendicular to the
σ
1
reflection plane. The net result is that the elec-
tron, marked by the little circle in the figure, has been reflected in this plane. This
operation thus leaves the nuclei in place, and only the positions of the electrons are
changed. This is precisely the definition of symmetry operators that we have been
using all along. The operators are displacing the electrons and in this way lead to
transformations of the electronic wavefunctions. The complete embedding of
C
3
v
in the Longuet-Higgins group is given in Table
5.1
.
By contrast, an operation such as (A)(BC), not followed by spatial inversion of
all particles, gives rise to an alternative arrangement of the nuclei, which cannot be
brought into coincidence with the original positions by mere spatial rotations. As a
result, this operation is not compatible with the Born
−
Oppenheimer boundary con-
