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 Δ*J*.

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.

The nonvanishing matrix elements ,
where R = X, Y, or Z and s = x, y,
or z, are given by the product of three factors: .
The factors f, g, _{s}h 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 _{R}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 g is
chosen from rows two and three: and the third factor _{s}h is
chosen from rows four and five._{R} |
|||

Factor | J′ = J+1 |
J′ = J |
J′ = J-1 |
---|---|---|---|

f(J′ J) |
{4(J+1) [(2J+1) (2J+3)]^{1/2}}^{-1} |
[4J(J+1)]^{-1} |
{4J [(2J+1)(2J-1)]^{1/2}}^{-1} |

g(_{z}J′, Ω J, Ω)g(_{x}J′,
Ω ± 1; J,
Ω)ig
(_{y}J′, Ω ± 1; J, Ω) |
2[(J+Ω+1) (J-Ω+1)]^{1/2}[ J ± Ω+1)
(J ± Ω+2)]^{1/2} |
2ΩJ Ω)
(J ± Ω+1)]^{1/2} |
2[(J+Ω) (J-Ω)]^{1/2}± [( J
Ω) (J
Ω-1)]^{1/2} |

h(_{Z}J′, M; J, M)h(_{X}J′,
M ± 1;
J, M)i h(_{Y}J′,
M ± 1; J, M) |
2[(J+M+1)(J-M+1)]^{1/2}[( J ± M+1)
(J ± M+2)]^{1/2} |
2M[( J M)
(J ± M+1)]^{1/2} |
2[(J+M) (J-M)]^{1/2}± [( J M)
(J M-1)]^{1/2} |

The elements of the direction cosine matrix transform as follows under the
symmetry operations σ_{υ} and *C*_{2}; they are invariant under
*i* [see eqs (2.3) and
sect. 2.1].

(3.9) |