Within this scenario two centres and the saddle in between, corresponding to
three different fixed points, reduce to a single centre, see the right column of
Fig. 7.6. Also in this case the Gomes theorem is satisfied, as shown by adding the
indices, (+1 + 1 − 1 = +1).
7.5 Magnetic Symmetry of a Molecule in an External
Magnetic Field
The point group symmetry of a molecule in a field B and the local symmetry at the
site of nucleus I, carrying the permanent magnetic dipole m I , determine the essential
features of J
B
ðrÞ and J
m I ðrÞ fields and of their SGs. In the absence of magnetic
perturbations, one can take into account the 32 point groups describing the symmetry
of the time-averaged charge density .ðrÞ of a molecule in the equilibrium state. In
the presence of magnetic field, or intramolecular magnetic dipoles at the nuclei, the
analysis of stationary electron current densities J
B and J
m I ðrÞ becomes an essential
tool. Since the equilibrium state is unchanged if the sign of these vectors is reversed,
it is expedient to introduce the time-reversal operator T [97], which changes the sign
of the current at each point in space, but does not act on the spatial coordinates.
T commutes with the spatial rotations and reflections and satisfies the cyclic condition T
2
¼ E, but it cannot itself be regarded as an element of a group. It always
appears as a combination TG i for any operator G i in a group G, but TC 3 cannot occur
as a separate symmetry transformation, since ðTC 3 Þ
3 ¼ TE ¼ T.
By introducing T in the 32 finite point symmetry groups, 58 new symmetry
groups can be constructed via the Tavger-Zaitsev algorithm [58, 98, 99]. The
procedure can be summarized in the following terms. For any of the 32 groups,
G G i
f g; i ¼ 1; 2. . .n, a subgroup H of index 2 in G is selected. The elements A i
of the set G À H, such that G ¼ H þ A i H, are used to obtain a magnetic group G
0 ,
isomorphic to G, by the recipe
G
0
¼ H þ TA i H:
ð7:71Þ
In the Schönflies notation we denote this magnetic group as G
0
ðHÞ. The corresponding notations in the Shubnikov system and in the international system are
also used [58].
The magnetic groups of a few relevant molecules in the presence of a magnetic
field are considered hereafter to discuss the main features of the J
B
field.
Hamermesh [58] and Mulliken conventions [100, 101] are used.
Orthorombic system
D 2 E C 2 ðzÞ C 2 ðyÞ C 2 ðxÞ
f
g 222
C 2 E C 2 ðzÞ
f
g 2
D 2 ðC 2 Þ E C 2 ðzÞ TC 2 ðyÞ TC 2 ðxÞ
f
g 222
ð7:72Þ
178
P. Lazzeretti
three different fixed points, reduce to a single centre, see the right column of
Fig. 7.6. Also in this case the Gomes theorem is satisfied, as shown by adding the
indices, (+1 + 1 − 1 = +1).
7.5 Magnetic Symmetry of a Molecule in an External
Magnetic Field
The point group symmetry of a molecule in a field B and the local symmetry at the
site of nucleus I, carrying the permanent magnetic dipole m I , determine the essential
features of J
B
ðrÞ and J
m I ðrÞ fields and of their SGs. In the absence of magnetic
perturbations, one can take into account the 32 point groups describing the symmetry
of the time-averaged charge density .ðrÞ of a molecule in the equilibrium state. In
the presence of magnetic field, or intramolecular magnetic dipoles at the nuclei, the
analysis of stationary electron current densities J
B and J
m I ðrÞ becomes an essential
tool. Since the equilibrium state is unchanged if the sign of these vectors is reversed,
it is expedient to introduce the time-reversal operator T [97], which changes the sign
of the current at each point in space, but does not act on the spatial coordinates.
T commutes with the spatial rotations and reflections and satisfies the cyclic condition T
2
¼ E, but it cannot itself be regarded as an element of a group. It always
appears as a combination TG i for any operator G i in a group G, but TC 3 cannot occur
as a separate symmetry transformation, since ðTC 3 Þ
3 ¼ TE ¼ T.
By introducing T in the 32 finite point symmetry groups, 58 new symmetry
groups can be constructed via the Tavger-Zaitsev algorithm [58, 98, 99]. The
procedure can be summarized in the following terms. For any of the 32 groups,
G G i
f g; i ¼ 1; 2. . .n, a subgroup H of index 2 in G is selected. The elements A i
of the set G À H, such that G ¼ H þ A i H, are used to obtain a magnetic group G
0 ,
isomorphic to G, by the recipe
G
0
¼ H þ TA i H:
ð7:71Þ
In the Schönflies notation we denote this magnetic group as G
0
ðHÞ. The corresponding notations in the Shubnikov system and in the international system are
also used [58].
The magnetic groups of a few relevant molecules in the presence of a magnetic
field are considered hereafter to discuss the main features of the J
B
field.
Hamermesh [58] and Mulliken conventions [100, 101] are used.
Orthorombic system
D 2 E C 2 ðzÞ C 2 ðyÞ C 2 ðxÞ
f
g 222
C 2 E C 2 ðzÞ
f
g 2
D 2 ðC 2 Þ E C 2 ðzÞ TC 2 ðyÞ TC 2 ðxÞ
f
g 222
ð7:72Þ
178
P. Lazzeretti