Flow chart depicting the design logic of PYRATE. † Collisional coefficients are calculated according to Eq. 12. †† Einstein coefficients are calculated according to Eq. 11. ‡ Temperature dependence of collisional coefficients. The value of the collisional coefficient used in each grid point (i, j, k) is determined by interpolating based on T_{CJ → J′} and T_i, j, k. ¶ Integration is performed using the QUADPACK library (Piessens et al. 1983). The non-linear system of Eqs. A is solved using the MINPACK library (Moré et al. 1980).