molspecutils.molecule.tme_sum

tme_sum(pair, mode)

Unweighted sum of transition matrix elements.

The sum goes over all degenerate states corresponding to the energy levels in pair. The degeneracy may include spatial degeneracy (magnetic states), hyperfine degeneracy or other kinds. It is related to the Einstein A coefficient for spontaneous emission by:

\[\sum_{m_i,m_f,d_if} |\langle \alpha_f J_f m_f d_f|\vec{mu}| J_i m_i d_i \rangle |^2 = \frac{3 A_{fi} \epsilon_0 h c^3 g_f}{16\pi^3\nu^3}\]

where \(d_x\) are labels for degenerate states other than spatial states, \(g_f=(2J_f+1)D_f\) is the total upper state degeneracy (statistical weight). For rovibrational transitions, this can be expressed as:

See Eq. (32) in M. Simečková, M., Jacquemart, D., Rothman, L. S., Gamache, R. R. & Goldman, A. Einstein a-coefficients and statistical weights for molecular absorption transitions in the hitran database. Jqsrt 98, 130–155 (2006) and Gamache, R. R. & Rothman, L. S. Extension of the HITRAN database to non-LTE applications. Journal of quantitative spectroscopy and radiative transfer 48, 519–525 (1992).