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.