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Methane Symmetry Operations - Irreducible Tensorial Sets

16.   Irreducible Tensorial Sets

The mathematical formalism developed for coupling and recoupling angular momenta in atoms [22, 29, 62] and in molecules [63] can be used advantageously when the molecular point group contains doubly and triply degenerate representations. Unfortunately, irreducible-tensor formalism is quite complicated notationally, and somewhat complicated conceptually. Furthermore, it is not particularly useful when the molecular point group contains only singly and doubly degenerate representations. Thus, many molecular spectroscopists remain unfamiliar with this extensive formalism.

If only a purely mathematical familiarity is acquired, the formalism can actually be rather easily misused. One need only observe that the point groups C2h, C2v, and D2 are isomorphic, for example, to conclude that all purely group theoretical algebra must be identical for these three point groups. Nevertheless, in spite of the fact that purely mathematical statements must be identical, it is well known [2, 4] that for molecules of the point group C2h, rotations and translations (and therefore Raman and infrared intensity operators) never belong to the same symmetry species, for D2 they always belong to the same species in pairs, and for C2v the situation is mixed. These differences arise because the same group theoretical formalism must be applied to the molecules in three rather different ways depending on the physics of the situation.

One should thus resist the temptation to apply the formalism of vector coupling and recoupling developed for atoms to problems in methane without sufficient regard for the physical distinction between laboratory-fixed rotations and point-group rotations, or between the laboratory-fixed inversion operation and the molecule-fixed inversion operation. Such application can give rise to a sequence of absolutely correct mathematical operations followed by a false physical interpretation. (Note that difficulties of this nature do not arise in atomic problems, for which the formalism of angular momentum coupling was first developed and systematized, since atom-fixed coordinate axes are never considered.)

In the remainder of this section we shall discuss in general some of the ways in which irreducible tensor formalism can be applied to methane, without attempting to illustrate or to choose between the entire variety of notational and phase-factor conventions found in the group-theoretical literature and textbooks.

16.1 Vector coupling coefficients

Vector coupling coefficients are also called Wigner coefficients and Clebsch-Gordan coefficients. Two common symbol [2964] for the same vector coupling coefficient are

$$(j_1 j_2 m_1 m_2 \, | \, j_1 j_2\,j\,m) \equiv C (j_1 j_2 j\,;\,m_1 m_2 m) ~ . $$

(eq. 58)

There are further, not always obvious, modifications of this notation in the literature. The meaning of the quantity in (eq. 58) can be deduced almost immediately from the following equation

$$| j_1 j_2\,j\,m \rangle = \sum_{m_1 m_2} ~ | j_1 m_1\rangle \, | j_2 m_2\rangle \, (j_1 j_2 m_1 m_2 \, | \, j_1 j_2\,j\,m) ~ . $$

(eq. 59)

From one point of view, (eq. 59) gives a relationship between eigenfunctions | jm1⟩ of the angular momentum operators j12,  j1, eigenfunctions | jm2⟩ of   j22,  j2, and eigenfunctions | j1 j2 jm⟩ of  j12,  j22,  j2  ≡ (j1 + j2)2,  j. From another point of view, (eq. 59) gives a relationship between the (2j1 + 1) functions | j1m1⟩ which transform according to the irreducible representation D(j1) if the variables occurring in | j1m1⟩ are subjected to the rotational operations of the continuous three-dimensional rotation group, the (2j2 + 1) functions | j2m2⟩, which transform according to D(j2) if the variables occurring in | j2m2⟩ are subjected to the rotational operations, and the (2j + 1) functions | j1 j2 jm⟩ which transform according to D(j) if all variables are subjected to the rotational operations. Physical invariance of a system, of course, is normally maintained only if all variables are subjected simultaneously to the same rotational operation.


Created September 28, 2016, Updated June 2, 2021