Adiabatic-potential-energy surfaces of the self-trapped exciton (STE) of both the spin-singlet and -triplet states in NaF, NaCl, and NaBr are calculated with use of ab initio Hartree-Fock cluster methods. In all cases, there is clear evidence of adiabatic instability in the on-center geometry. The STE undergoes a spontaneous symmetry-breaking relaxation, resulting in a structure which is equivalent to an F-H center pair (a Frenkel defect pair in the anion sublattice). The potential energy is remarkably flat, to within about 0.2 eV, for further separation of the H center from the F center up to a distance of about 3 \AA{}, the largest distance attained in this study. In view of the flat nature of the excited-state energy surface, and given the difficulties inherent in an ab initio computation, the calculated recombination energies are found to be within an acceptable range of values when compared with experiment for both the \ensuremath{\sigma} and \ensuremath{\pi} bands. From this work, the initial state of the \ensuremath{\sigma} emission band is attributed to the singlet STE of the lowest orbital state. On the basis of the adiabatic potential energies obtained in this work, a mechanism of \ensuremath{\pi}-band quenching is proposed.
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