2.2 The Eigenstates of Many-Electron Atoms
39
called orbital angular moment quenching. On the other hand, if the external potential
destabilizes p x with respect to p y,z , the orbital angular moment of the p 1 electronic
configuration is defined by the lower-right 2 × 2 sub-block of matrix 2.10 and results
in non-zero expectation values. Now, the orbital angular moment is not quenched.
The general condition for non-zero orbital angular moment for a given RussellSaunders term 2S+1 Γ is that the direct product Γ × Γ contains irreducible representations of the orbital moment operators ˆ
L x,y,z . Since these operators have identical
transformation properties as the rotation operator ˆ
R x,y,z (which is usually listed in
the character tables of the symmetry point groups), it is easier to work with the rotation operator. For example, the ground state of the d 1 electronic configuration in an
octahedral surrounding is 2 T 2g . The rotation operator transforms as T 1g in the O h
point group. Since the direct product T 2g × T 2g = A 1g + E g + T 1g + T 2g contains
the irreducible representation of the rotation operator, one expects a non-zero orbital
angular moment for this system. Note, however, that the d 1 electronic configuration
is Jahn-Teller active and the geometry spontaneously distorts to a lower symmetry
group accompanied by a (partial) quenching of the orbital angular moment.
2.4 Predict the (non-)existence of a net orbital angular moment for the highspin d 2 electronic configuration in complexes with tetrahedral, octahedral and
C 2v symmetry.
2.3 Further Removal of the Degeneracy of the N-electron
States
The first two columns of Fig. 2.1 show how the free atom levels of a d 7 configuration
are split by a distorted tetrahedral ligand-field. In this example, the states are labeled
by the IR’s of the D 2d subgroup of T d and the 4 F (with a degeneracy of (2S + 1) ×
(2L + 1) = 4 × 7 = 28) is split in five energy levels. Based on the discussion in
the previous section, one only expects a non-zero orbital moment for the 4 E states.
The inset of the figure zooms in on the levels of the 4 B 1 state and shows how the
degeneracy is removed under the influence of spin-orbit coupling and when the
system is placed in an external magnetic field. In the following two subsections we
will discuss these two effects.
2.3.1 Zero Field Splitting
In 3d transition metal complexes, the splitting of the Russell–Saunders terms due to
spin-orbit coupling is in general more important than the one caused by the external
magnetic field typically used in EPR experiments. Therefore, the description of the
39
called orbital angular moment quenching. On the other hand, if the external potential
destabilizes p x with respect to p y,z , the orbital angular moment of the p 1 electronic
configuration is defined by the lower-right 2 × 2 sub-block of matrix 2.10 and results
in non-zero expectation values. Now, the orbital angular moment is not quenched.
The general condition for non-zero orbital angular moment for a given RussellSaunders term 2S+1 Γ is that the direct product Γ × Γ contains irreducible representations of the orbital moment operators ˆ
L x,y,z . Since these operators have identical
transformation properties as the rotation operator ˆ
R x,y,z (which is usually listed in
the character tables of the symmetry point groups), it is easier to work with the rotation operator. For example, the ground state of the d 1 electronic configuration in an
octahedral surrounding is 2 T 2g . The rotation operator transforms as T 1g in the O h
point group. Since the direct product T 2g × T 2g = A 1g + E g + T 1g + T 2g contains
the irreducible representation of the rotation operator, one expects a non-zero orbital
angular moment for this system. Note, however, that the d 1 electronic configuration
is Jahn-Teller active and the geometry spontaneously distorts to a lower symmetry
group accompanied by a (partial) quenching of the orbital angular moment.
2.4 Predict the (non-)existence of a net orbital angular moment for the highspin d 2 electronic configuration in complexes with tetrahedral, octahedral and
C 2v symmetry.
2.3 Further Removal of the Degeneracy of the N-electron
States
The first two columns of Fig. 2.1 show how the free atom levels of a d 7 configuration
are split by a distorted tetrahedral ligand-field. In this example, the states are labeled
by the IR’s of the D 2d subgroup of T d and the 4 F (with a degeneracy of (2S + 1) ×
(2L + 1) = 4 × 7 = 28) is split in five energy levels. Based on the discussion in
the previous section, one only expects a non-zero orbital moment for the 4 E states.
The inset of the figure zooms in on the levels of the 4 B 1 state and shows how the
degeneracy is removed under the influence of spin-orbit coupling and when the
system is placed in an external magnetic field. In the following two subsections we
will discuss these two effects.
2.3.1 Zero Field Splitting
In 3d transition metal complexes, the splitting of the Russell–Saunders terms due to
spin-orbit coupling is in general more important than the one caused by the external
magnetic field typically used in EPR experiments. Therefore, the description of the