106
5 What has Quantum Chemistry Got to Do with It?
unimodular number:
ˆ
R
Ψ
k
= exp(iκ R )
Ψ
k
(5.6)
Here, we have dropped the index j since there is only one eigenfunction in this
case. This equation indicates that this eigenfunction must transform as a nondegenerate irrep of the point group, say Γ k , with
exp(iκ R ) = χ
Γ k (R)
(5.7)
2. The electronic state is degenerate (n > 1). In this case the transformed function
may not be proportional to the original one, but in any case it is also an eigenfunction of H with the same eigenvalue. This means that it must be mapped onto
a linear combination of the components of the eigenspace; hence, the eigenspace
itself must form a function space that is invariant under G:
R
Ψ
k
j
=
n
i=1
D ij (R)
Ψ
k
i
(5.8)
Here, the eigenfunctions have again been arranged as a row vector, and the coefficients are gathered in a transformation matrix, D(R). There are now two possibilities:
• The matrix representation is an irrep of the point group.
In this case the electronic degeneracy is equal to the dimension of an irrep of
the point group.
• The matrix representation is reducible.
In this case the eigenspace can always be separated in irreducible blocks by
using projection operators.
This list of possibilities shows that eigenfunctions of the Hamiltonian will also
be (or can be made to be) eigenfunctions of the symmetry group of the Hamiltonian. When there is a perfect match between the eigenfunctions and an irrep, the
presence of degeneracy or nondegeneracy can directly be attributed to the symmetry of the eigenstates. The remaining possibility that the eigenspace may consist of
several irreducible blocks could be described as a case of “accidental degeneracy”,
in the sense that symmetry cannot explain the fact that stationary states are degenerate. When this happens, it could mean that the symmetry of the system exceeds the
apparent spatial symmetry group. This case is referred to as “hidden symmetry”.
A special case of this is Kramers’ degeneracy, which is treated in Sect. 7.6.O ri t
could be that a simplified model Hamiltonian was used, such as, e.g., the nearestneighbor Hamiltonian in Hückel theory, which may give rise to additional degeneracies, as we have explained for the example of triphenylmethyl in the previous
chapter. In addition to these possibilities—in the words of Griffith [2]—experience
tells that “accidents don’t happen, at least in that part of physics which is understood”.
The significance of these observations can hardly be overestimated. The derivation refers to the properties of the exact Hamiltonian and its eigenfunctions. In actual
5 What has Quantum Chemistry Got to Do with It?
unimodular number:
ˆ
R
Ψ
k
= exp(iκ R )
Ψ
k
(5.6)
Here, we have dropped the index j since there is only one eigenfunction in this
case. This equation indicates that this eigenfunction must transform as a nondegenerate irrep of the point group, say Γ k , with
exp(iκ R ) = χ
Γ k (R)
(5.7)
2. The electronic state is degenerate (n > 1). In this case the transformed function
may not be proportional to the original one, but in any case it is also an eigenfunction of H with the same eigenvalue. This means that it must be mapped onto
a linear combination of the components of the eigenspace; hence, the eigenspace
itself must form a function space that is invariant under G:
R
Ψ
k
j
=
n
i=1
D ij (R)
Ψ
k
i
(5.8)
Here, the eigenfunctions have again been arranged as a row vector, and the coefficients are gathered in a transformation matrix, D(R). There are now two possibilities:
• The matrix representation is an irrep of the point group.
In this case the electronic degeneracy is equal to the dimension of an irrep of
the point group.
• The matrix representation is reducible.
In this case the eigenspace can always be separated in irreducible blocks by
using projection operators.
This list of possibilities shows that eigenfunctions of the Hamiltonian will also
be (or can be made to be) eigenfunctions of the symmetry group of the Hamiltonian. When there is a perfect match between the eigenfunctions and an irrep, the
presence of degeneracy or nondegeneracy can directly be attributed to the symmetry of the eigenstates. The remaining possibility that the eigenspace may consist of
several irreducible blocks could be described as a case of “accidental degeneracy”,
in the sense that symmetry cannot explain the fact that stationary states are degenerate. When this happens, it could mean that the symmetry of the system exceeds the
apparent spatial symmetry group. This case is referred to as “hidden symmetry”.
A special case of this is Kramers’ degeneracy, which is treated in Sect. 7.6.O ri t
could be that a simplified model Hamiltonian was used, such as, e.g., the nearestneighbor Hamiltonian in Hückel theory, which may give rise to additional degeneracies, as we have explained for the example of triphenylmethyl in the previous
chapter. In addition to these possibilities—in the words of Griffith [2]—experience
tells that “accidents don’t happen, at least in that part of physics which is understood”.
The significance of these observations can hardly be overestimated. The derivation refers to the properties of the exact Hamiltonian and its eigenfunctions. In actual