4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in Ammonia
55
combination of generators and, hence, to the whole group. So, in the case of C 3v ,
we have to examine the behavior of the |ψ m functions under a vertical symmetry
plane as well, say ˆ
σ 1 . This plane will leave |1s A unchanged and will interchange
|1s B and |1s C . Its effect on the trigonal eigenfunctions is thus given by
ˆ
σ 1 |ψ 0 =|ψ 0
ˆ
σ 1 |ψ +1 =|ψ −1
(4.20)
ˆ
σ 1 |ψ −1 =|ψ +1
What does this result tell us? The SALC |ψ 0 is simultaneously an eigenfunction
of both ˆ
C 3 and ˆ
σ 1 ; hence, it forms in itself a one-dimensional function space that
is completely adapted to the full group. This symmetry characteristic will be denoted by the totally symmetric representation A 1 . However, the other two SALCs
are transformed into each other. We can easily turn them into eigenfunctions of ˆ
σ 1
and in this way obtain alternative eigenfunctions, one of which is symmetric under reflection and one of which is antisymmetric. These will be labeled as x and
y, respectively, since their symmetry under reflection mimics the symmetries of p x
and p y :
|ψ x =
1
√
2
|ψ +1 +|ψ −1
=
1
√
6
2|1s A −|1s B −|1s C
|ψ y =
i
√
2
|ψ +1 −|ψ −1
=
1
√
2
|1s B −|1s C
(4.21)
A schematic drawing of these eigenfunctions is shown in Fig. 4.1. The downside
of this symmetry adaptation is that it has destroyed the diagonalization along the
trigonal axis. Indeed, one has:
ˆ
C 3
|ψ x | ψ y
=
|ψ x | ψ y
cos(2π/3) − sin(2π/3)
sin(2π/3) cos(2π/3)
(4.22)
Hence, it is impossible to resolve the function space formed by |ψ ±1 into simultaneous eigenfunctions of both generators. The action of the symmetry group ties
these functions together into a two-dimensional space, which is thus irreducible.
This symmetry characteristic is denoted by the degenerate irreducible representation E. We have learned form this simple example the following. The construction
of SALCs of a symmetry group is based on simultaneous diagonalization of the
representation matrices of the group generators. This will resolve the function space
into separate blocks, which may consist of one function, or which may form a subspace that cannot be further reduced. The results are functions that transform as
irreducible representations (irreps). Why is such a resolution important? The irreducible subspaces into which the function space has been separated are invariant
under the actions of the actual symmetry group. This means that there are no operators that send SALCs from one irreducible subspace into SALCs from another
irreducible subspace. This also implies that the eigenenergies associated with these
55
combination of generators and, hence, to the whole group. So, in the case of C 3v ,
we have to examine the behavior of the |ψ m functions under a vertical symmetry
plane as well, say ˆ
σ 1 . This plane will leave |1s A unchanged and will interchange
|1s B and |1s C . Its effect on the trigonal eigenfunctions is thus given by
ˆ
σ 1 |ψ 0 =|ψ 0
ˆ
σ 1 |ψ +1 =|ψ −1
(4.20)
ˆ
σ 1 |ψ −1 =|ψ +1
What does this result tell us? The SALC |ψ 0 is simultaneously an eigenfunction
of both ˆ
C 3 and ˆ
σ 1 ; hence, it forms in itself a one-dimensional function space that
is completely adapted to the full group. This symmetry characteristic will be denoted by the totally symmetric representation A 1 . However, the other two SALCs
are transformed into each other. We can easily turn them into eigenfunctions of ˆ
σ 1
and in this way obtain alternative eigenfunctions, one of which is symmetric under reflection and one of which is antisymmetric. These will be labeled as x and
y, respectively, since their symmetry under reflection mimics the symmetries of p x
and p y :
|ψ x =
1
√
2
|ψ +1 +|ψ −1
=
1
√
6
2|1s A −|1s B −|1s C
|ψ y =
i
√
2
|ψ +1 −|ψ −1
=
1
√
2
|1s B −|1s C
(4.21)
A schematic drawing of these eigenfunctions is shown in Fig. 4.1. The downside
of this symmetry adaptation is that it has destroyed the diagonalization along the
trigonal axis. Indeed, one has:
ˆ
C 3
|ψ x | ψ y
=
|ψ x | ψ y
cos(2π/3) − sin(2π/3)
sin(2π/3) cos(2π/3)
(4.22)
Hence, it is impossible to resolve the function space formed by |ψ ±1 into simultaneous eigenfunctions of both generators. The action of the symmetry group ties
these functions together into a two-dimensional space, which is thus irreducible.
This symmetry characteristic is denoted by the degenerate irreducible representation E. We have learned form this simple example the following. The construction
of SALCs of a symmetry group is based on simultaneous diagonalization of the
representation matrices of the group generators. This will resolve the function space
into separate blocks, which may consist of one function, or which may form a subspace that cannot be further reduced. The results are functions that transform as
irreducible representations (irreps). Why is such a resolution important? The irreducible subspaces into which the function space has been separated are invariant
under the actions of the actual symmetry group. This means that there are no operators that send SALCs from one irreducible subspace into SALCs from another
irreducible subspace. This also implies that the eigenenergies associated with these