The Energy of the Triatomic Hydrogen Molecule and Ion, V

Abstract

In previous papers, the energy of H₃ and of H₃⁺ has been obtained by the variational method for linear configurations. In this treatment we are able to evaluate the difficult three center integrals for nonlinear configurations with the aid of the differential analyzer and to compute the energy of H₃ and of H₃⁺ as a function of the angle between the nuclei. The excited states as well as the ground states are considered. Direct comparison of calculated energy values for the equilateral triangle show that the molecular orbital approximation is inferior to the method of homopolar bond functions. The triatomic hydrogen molecule has its lowest energy for linear configurations. The angle dependence calculated by the variational method agrees well with that calculated on the basis of the Eyring semi‐empirical scheme. By the theorem of Jahn and Teller there could not be a minimum in the energy for the equilateral triangle, as here the lowest electronic state is doubly degenerate. The triatomic hydrogen ion, H₃⁺, is very stable (when left to itself) and has an energy lower by more than 184. kcal. than two separated hydrogen atoms and a proton. Thus the chemical reaction: $${\mathrm{H}}_2+{\mathrm{H}}_2^{+}\to {\mathrm{H}}_3^{+}+\mathrm{H}$$ is certainly exothermic by more than 11 kcal. and probably is exothermic by 38 kcal. The triatomic hydrogen ion has a stable configuration corresponding to separation between the nuclei of about 1.79A with the nuclei lying intermediate between a right and an equilateral triangle. The vibration frequencies of H₃⁺ are estimated but their exact value as well as the exact configuration of the stable state are somewhat in doubt. Two of these frequencies should be infra‐red active and susceptible to direct experimental measurement.