From a solution of a Schrodinger-type wave equation with a nonradiative boundary condition based on Maxwell's equations, Mills solves the hydrogen atom, the hydrogen molecular ion, the hydrogen molecule and predicts corresponding species having fractional principal quantum numbers. Atomic hydrogen may undergo a catalytic reaction with certain atomized elements and ions which singly or multiply ionize at integer multiples of the potential energy of atomic hydrogen, m27.2 eV wherein in is an integer. The reaction involves a nonradiative energy transfer to form a hydrogen atom H(1/p) that is lower in energy than unreacted atomic hydrogen that corresponds to a fractional principal quantum number (n = 1/p = 1/integer replaces the well known parameter n = integer in the Rydberg equation for hydrogen excited states). One such atomic catalytic system involves argon ions. The reaction Ar+ to Ar2+ has a net enthalpy of reaction of 27.63 eV, which is equivalent to m = 1. Thus, it may serve as a catalyst to form H(1/2). Also, the second ionization energy of helium is 54.4 eV; thus, the ionization reaction of He+ to He2+ has a net enthalpy of reaction of 54.4 eV which is equivalent to 2 x 27.2 eV. The products of the catalysis reaction H(1/3) may further serve as catalysts to form H(1/4) and H(1/2). H(1/p) may react with a proton to form an excited state molecular ion H-2*(1/p)(+) that has a bond energy and vibrational levels that are p(2) times those of the molecular ion comprising uncatalyzed atomic hydrogen where p is an integer. Thus, the excited state spectrum of H-2*[n = 1/4; n* = 2](+), was predicted to comprise rotationally 4 broadened vibrational transitions at 1.185 eV increments to the dissociation limit of H-2[n =1/4](+), E-D= 42.88 eV (28.92 nm). Extreme ultraviolet spectroscopy was recorded on microwave discharges of argon or helium with 10% hydrogen in the range 10-65 nm. Novel emission lines assigned to vibrational transitions of H-2*[n = 1/4; n* = 2](+) were observed in this range with energies of v1.185 eV, v = 17-38 that terminated at about 28.9 nm. In addition, fractional molecular hydrogen rotational transitions were assigned to previously unidentified lines in the Solar coronal spectrum that matched theoretical predictions to five figures. (C) 2002 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved.