A simple approximate solution for the H3+ ion

Here I derive analytical expressions for the total energy of the (Formula presented.) cation in its equilateral and linear geometries. The theoretical model consists of a simple variational trial wavefunction made of the sum of three 1S Gaussian functions, each centered on each nucleus. Detailed derivations are presented and the advantages and limitations of this simple model are discussed. The correctness of the results was verified independently via Monte Carlo integration. This simple model correctly predicts and rationalizes the preference of (Formula presented.) for the equilateral geometry rather than the linear configuration. Despite its simplicity, the calculated H-H bond length (R = 0.9088 Å) and breathing vibrational frequency (ν₁ = 3276.59 cm⁻¹) for equilateral (Formula presented.) ion are in good agreement with high-level ab initio methods and the experiment, respectively.