Table of Contents of Rotational Energy Levels and Line Intensities in 
Diatomic Molecules

3.3. The Direction Cosine Matrix α

The elements of the 3 × 3 direction cosine matrix α do not involve the electronic or vibrational variables of a diatomic molecule; they involve only the rotational angles (see eqs (2.3) above and [3] (pp. 10-11), [6] (pp. 285-6), [8]). Consequently, matrix elements of elements of the direction cosine matrix are diagonal in the nonrotating-molecule quantum numbers L, Λ, S, Σ. The nonvanishing matrix elements of elements of the direction cosine matrix are conveniently summarized in the form of a table [18] (p. 96). (footnote 7) This table, with some change in notation from [18], is presented here as table 6. Each matrix element of a given element of the direction cosine matrix consists of the product of three factors: one taken from the first line of table 6, one from either the second or third lines, and one from either the fourth or fifth lines; all three factors are taken from the same column of table 6. The choice of rows is determined by the particular element of the matrix α under consideration. The choice of column is determined by the value of ΔJ.

The derivation of table 6 represents a rather elaborate exercise in group theory, operator algebra, or generating functions. Furthermore, the derivation is slightly different for nonlinear molecules (three Eulerian angles) than it is for linear molecules (two Eulerian angles). We shall not consider in this monograph the derivation of table 6.


TABLE 6. Direction cosine matrix elements (after [l8] p. 96)
(Image of Table 6)

The nonvanishing matrix elements $\langle\Omega^\prime~J^\prime~M^\prime|~\alpha_{Rs}~|\Omega~J~M\rangle$, where R = X, Y, or Z and s = x, y, or z, are given by the product of three factors: $f(J^\prime;J)\cdot g_s(J^\prime,\Omega^\prime;J,\Omega)\cdot h_R(J^\prime,M^\prime;J,M)$. The factors f, gs, hR for a given matrix element are taken from different rows of the same column of this table. The choice of columns depends on the value of J′-J. The choice of rows depends on R and s. In all cases, the first factor f is taken from row one; the second factor gs is chosen from rows two and three: and the third factor hR is chosen from rows four and five.
Factor J′ = J+1 J′ = J J′ = J-1
f(JJ) {4(J+1) [(2J+1) (2J+3)]1/2}-1 [4J(J+1)]-1 {4J [(2J+1)(2J-1)]1/2}-1
gz(J′, Ω J, Ω)
gx(J′, Ω ± 1; J, Ω) or
minus plus igy (J′, Ω ± 1; J, Ω)
2[(J+Ω+1) (J-Ω+1)]1/2

minus plus [J ± Ω+1) (J ± Ω+2)]1/2

[(J minus plus Ω) (J ± Ω+1)]1/2
2[(J+Ω) (J-Ω)]1/2

  [(J minus plus Ω) (J minus plus Ω-1)]1/2
hZ(J′, M; J, M)
hX(J′, M ± 1; J, M) or
± i hY(J′, M ± 1; J, M)

minus plus [(J ± M+1) (J ± M+2)]1/2

[(J minus plus M) (J ± M+1)]1/2
2[(J+M) (J-M)]1/2

 ± [(J minus plus M) (J minus plus M-1)]1/2

The elements of the direction cosine matrix transform as follows under the symmetry operations συ and C2; they are invariant under i [see eqs (2.3) and sect. 2.1].

eq3.09 (3.9)

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