SPONTANEOUS EMISSION BETWEEN ORTHO- AND PARA-LEVELS OF WATER-ION, H$_2$O$^+$

Abstract

Nuclear spin conversion interaction of water ion, H$_2$O$^+$, has been studied to derive spontaneous emission lifetime between {\em ortho}- and {\em para}-levels. H$_2$O$^+$ is a radical ion with the $^2B_1$ electronic ground state. Its off-diagonal electron spin-nuclear spin interaction term, $T_{ab}(S_a\Delta I_b + S_b\Delta I_a)$, connects {\em para} and {\em ortho} levels, because $ \Delta \mbox{\boldmath $I$} = \mbox{\boldmath $I$}_1 - \mbox{\boldmath $I$}_2$ has nonvanishing matrix elements between $I = 0$ and 1. The mixing by this term with $T_{ab}$ = 72 MHz predicted by $ab~initio$ theory in the MRD-CI/Bk level, \textbf{80}, 1485 (1993)} is many orders of magnitude larger than for closed shell molecules because of the large magnetic interaction due to the un-paired electron. Using the molecular constants reported by Murtz et al. by FIR-LMR \textbf{109}, 9744 (1998).}, we searched for {\em ortho} and {\em para} coupling channels below 1000 cm$^{-1}$ with accidental near degeneracy between {\em para} and {\em ortho} levels. For example, hyperfine components of the 4$_{2,2}$($ortho$) and 3$_{3,0}$($para$) levels mix by 1.2 $\times$ 10$^{-3}$ due to their near degeneracy ($\Delta E$ = 0.417 cm$^{-1}$), and give the {\em ortho}-{\em para} spontaneous emission lifetime of about 0.63 year. The most significant low lying 1$_{0,1}$($para$) and 1$_{1,1}$($ortho$) levels, on the contrary, mix only by 8.7 $\times$ 10$^{-5}$ because of their large separation ($\Delta E$ = 16.267 cm$^{-1}$) and give the spontaneous emission lifetime from 1$_{0,1}$($para$) to 0$_{0,0}$($ortho$) of about 100 year.These results qualitatively help to understand the observed high {\em ortho-} to {\em para-} H$_2$O$^+$ ratio of 4.8 $\pm$ 0.5 \textbf{521}, L11 (2010).} toward Sgr B2 but they are too slow to compete with the conversion by collision unless the number density of the region is very low ($n \sim$ 1 cm$^{-3}$) or radiative temperature is very high ($T_r$ $>$ 100 K).