High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2

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Abstract
Experimental and theoretical methodologies have been developed to determine the hyperfine structure of molecular ions from detailed studies of the Rydberg spectrum and have been tested on molecular hydrogen. The hyperfine structure in l=0–3 Rydberg states of H2 located below the X 2Σg+(v+=0,N+=1) ground state of ortho H2+ has been measured in the range of principal quantum number n=50–65 at sub-MHz resolution by millimeter wave spectroscopy following laser excitation to np and nd Rydberg states using a variety of single-photon and multiphoton excitation sequences. The np11, nd11, and the nf10–3 Rydberg states were found to be metastable and to have lifetimes of more than 5 μs beyond n=50. Members of other series, such as the nd12, nd13, and the np10 series, were found to have lifetimes of more than 1 μs. Local perturbations induced by low-n Rydberg states belonging to series converging on rovibrationally excited levels of H2+ reduce the lifetimes in narrow ranges of n values. The hyperfine structure is strongly dependent on the value of the orbital angular momentum ℓ. In the penetrating s and p states at n≈50 the exchange interaction dominates over the hyperfine interaction and the levels can be labeled by the total electron spin angular momentum quantum number S (S=0 or 1). In the less penetrating d and f Rydberg states, the hyperfine interaction between the core nuclear and electron spins is larger than the exchange interaction and the Rydberg states are of mixed singlet and triplet character. A procedure based on the Stark effect and on the systematic analysis of selection rules and combination differences was developed to determine the orbital and the total angular momentum quantum numbers ℓ and F and to construct an energy map of p and f Rydberg levels between n=54 and 64 with relative positions of an accuracy of better than 1 MHz. Multichannel quantum defect theory (MQDT) was extended to treat the hyperfine structure in molecular Rydberg states and was used to analyze the observed hyperfine structure of the p and f Rydberg states of H2. The frame transformation between the Born-Oppenheimer channels described by the angular momentum coupling scheme (aβJ) and the asymptotic channels described by the (e[bβS+]) coupling scheme was derived and enables an elegant treatment of all intermediate coupling cases. Purely ab initio quantum defect theory reproduced the experimentally determined positions to within 40 MHz for the p levels and 13 MHz for the f levels. By slight adjustments of the quantum defect functions and their energy dependences and by consideration of the p-f interaction, of the singlet-triplet splittings of the f levels, and of the departure of the ionic levels from pure coupling case (bβS+), the agreement between theory and experiment could be improved to 600 kHz. By comparing the results of MQDT calculations of the hyperfine structure of f Rydberg levels with those of coupled equations calculations, the frame transformation approximation of MQDT was shown to be accurate to within 300 kHz. The extrapolated ionic hyperfine structure of the X 2Σg+(v+=0,N+=1) ionic level corresponds to the ab initio prediciton of Babb and Dalgarno [Phys. Rev. A 46, R5317 (1992)] within the experimental error.