Theory of hyperfine and superfine levels in symmetric polyatomic molecules. Trigonal and tetrahedral molecules: Elementary spin-½ cases in vibronic ground states

A simple approximate scheme for treating molecular hyperfine structure is developed by taking account of energy-level clusters. Unitary tableau and frame transformation techniques are reintroduced. Model Hamiltonians for $X{Y}_3$ and $X{Y}_4$ ($X$ spin-zero, $Y$ spin-½) molecules are developed and solved in cluster bases which are appropriate for highly excited rotational states. Two cases emerge: Case (1) for which hyperfine splittings are smaller than the "superfine" cluster splittings and case (2) for which superfine splittings are negligible or zero. The problem of correlating energy levels and states between cases (1) and (2) is solved. Since the $X{Y}_4$ problem in the elementary cluster bases reduces to (2×2) matrices at the worst, the physical interpretation of solutions is not difficult.