108
5 What has Quantum Chemistry Got to Do with It?
conventional choice of the real d-orbitals is thus based on a tetragonal splitting field.
Once the splitting field has been applied, the eigenspace is already nearly fixed;
the only freedom that remains is the relative phases of the components. To freeze
these phases, one can sometimes use an extra ladder operator, which moves from
one component to the other, thereby imposing a phase convention. For the E-state,
one uses a threefold axis to connect the two components, and the corresponding
representation matrix is defined by
ˆ
C 3
|Eθ| Eǫ
=
|Eθ| Eǫ
−1/2 −
√
3/2
√
3/2 −1/2
(5.11)
This operator connects the two components, and, therefore, if we require that the
matrix be of the specified form, then the relative phase freedom is lifted. With such
a connecting element, one can indeed easily construct a proper ladder operator:
2
√
3
ˆ
C 3 +
1
2
ˆ
E
| ˆ
Eθ=| ˆ
Eǫ
−
2
√
3
ˆ
C 3 +
1
2
ˆ
E
| ˆ
Eǫ=| ˆ
Eθ
(5.12)
In Appendix D we list standard conventions that are frequently used to define canonical basis sets for degenerate irreps.
For the octahedral T 1 state, the standard basis complies with the transformation
properties of the real p-orbitals and is marked as |T 1x , |T 1y , |T 1z . If we diagonalize this set under a fourfold splitting field, we obtain the complex p-orbitals. In
applications where real functions are preferred, the ˆ
C 4 axis is represented as
ˆ
C 4
|T 1x | T 1y | T 1z
=
|T 1x | T 1y | T 1z
⎛
⎝
0 −10
100
001
⎞
⎠
(5.13)
Here, |T 1z is recognized as a totally symmetric eigenfunction, and it is uniquely
defined by this eigenvalue since the other eigenvalues are ±i. The splitting field is
thus operating only partially; nonetheless, it uniquely picks the z-component. Then
the ˆ
C 3 axis is a perfect ladder operator, effecting a cyclic permutation of z to x, and
further to y.
5.4 The Molecular Symmetry Group
So far, the symmetry of the Hamiltonian was defined as the set of all operations that
leave the Hamiltonian invariant. This invariance group was assumed to coincide with
the point group of the nuclear frame of the molecule, but it is now time to provide
a clear explanation of this connection. This section relies on the definition of the
5 What has Quantum Chemistry Got to Do with It?
conventional choice of the real d-orbitals is thus based on a tetragonal splitting field.
Once the splitting field has been applied, the eigenspace is already nearly fixed;
the only freedom that remains is the relative phases of the components. To freeze
these phases, one can sometimes use an extra ladder operator, which moves from
one component to the other, thereby imposing a phase convention. For the E-state,
one uses a threefold axis to connect the two components, and the corresponding
representation matrix is defined by
ˆ
C 3
|Eθ| Eǫ
=
|Eθ| Eǫ
−1/2 −
√
3/2
√
3/2 −1/2
(5.11)
This operator connects the two components, and, therefore, if we require that the
matrix be of the specified form, then the relative phase freedom is lifted. With such
a connecting element, one can indeed easily construct a proper ladder operator:
2
√
3
ˆ
C 3 +
1
2
ˆ
E
| ˆ
Eθ=| ˆ
Eǫ
−
2
√
3
ˆ
C 3 +
1
2
ˆ
E
| ˆ
Eǫ=| ˆ
Eθ
(5.12)
In Appendix D we list standard conventions that are frequently used to define canonical basis sets for degenerate irreps.
For the octahedral T 1 state, the standard basis complies with the transformation
properties of the real p-orbitals and is marked as |T 1x , |T 1y , |T 1z . If we diagonalize this set under a fourfold splitting field, we obtain the complex p-orbitals. In
applications where real functions are preferred, the ˆ
C 4 axis is represented as
ˆ
C 4
|T 1x | T 1y | T 1z
=
|T 1x | T 1y | T 1z
⎛
⎝
0 −10
100
001
⎞
⎠
(5.13)
Here, |T 1z is recognized as a totally symmetric eigenfunction, and it is uniquely
defined by this eigenvalue since the other eigenvalues are ±i. The splitting field is
thus operating only partially; nonetheless, it uniquely picks the z-component. Then
the ˆ
C 3 axis is a perfect ladder operator, effecting a cyclic permutation of z to x, and
further to y.
5.4 The Molecular Symmetry Group
So far, the symmetry of the Hamiltonian was defined as the set of all operations that
leave the Hamiltonian invariant. This invariance group was assumed to coincide with
the point group of the nuclear frame of the molecule, but it is now time to provide
a clear explanation of this connection. This section relies on the definition of the