The accuracy of second order perturbation theory for multiply excited vibrational energy levels and partition functions for a symmetric top molecular ion

The vibrational energy levels and partition functions of the nonrotating H3+ molecular ion have been calculated by using second order perturbation theory, including constant, linear, and quadratic terms in the vibrational quantum numbers. Energy levels have been assigned to A′1, A′2, and E′ symmetry species up to 29 244 cm−1, and perturbation theory energy levels have been compared with the results of accurate quantum calculations. The root-mean-square error in 141 energy levels is 4.0 cm−1, as compared to 11.7 cm−1 in the harmonic approximation. Furthermore, perturbation-theory partition functions have errors of 7.4% or less over the factor-of-20 temperature range from 200 to 4000 K. The effect of the constant term in perturbation theory is also discussed; it improves the vibrational partition functions by ∼4% at 200 K.