Rotational uncoupling, with application to the singlet hydrogen bands

The general theory of uncoupling has been discussed by Kronig and Fujioka, and by MacDonald, but before considering the hydrogen bands it will be convenient to establish some general results. We consider first the integral ∫ Pv KA. B. Pv KA' . ρ2 d ρ, which appears in the formulæ, and which may be called Bvv AA'. In the hydrogen levels we shall only consider the first one or two values ofv and the first four or five values of K ; thus the unperturbed levels can be written as WA ° = C(v )A + B(v )A(K +1/2)2 - D(v )A(K + 1/2)4 + F(v )A(K + 1/2)6 -. The F term is exceedingly small at our K’s. We consider first the value of (1) in the hypothetical case where the Un (p)’s of the two states concerned differ only by a constant, so that the two P eigen­functions are identical, and in (2) it is only the C’s which differ. The ordinary vibrational equation (MacDonald’s (5) for example) may be regarded as con­taining K (K +1) or (K +1/2)2 as a parameter ; calling either of them ε we have the general formula (WA0 )ε+∆ε = (WA 0 )ε + (∂WA 0 /∂ε)ε ∆ε + term in (∆ε)2 + But the perturbation theory (for variation of a parameter) shows at once that in the case we are considering the coefficient of ∆ε in (3) is exactly the integral (1), whose value is thus found by differentiating (2). It is B(v) -2D(v) (K+1/2)2 +3F(v) (K+1/2)4 -.