A simple approximate solution for the H3+ ion

• Published in: International journal of quantum chemistry 2023-01, Vol.123 (2), p.n/a
• Main Author: Pérez Paz, Alejandro
• Format: Article
• Language: English
• Subjects: Chemistry; electronic structure; Hydrogen bonds; Mathematical analysis; Physical chemistry; quantum Monte Carlo; Quantum physics; trihydrogen cation; variational method; Wave functions
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/qua.27015

Here I derive analytical expressions for the total energy of the H3+ 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 H3+ 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 H3+ ion are in good agreement with high‐level ab initio methods and the experiment, respectively. This Tutorial Review guides the student step‐by‐step toward the variational solution of the trihydrogen cation (H3+) in its equilateral geometry. The trial wavefunction consists of a sum of three 1S Gaussians, each on each nucleus. Despite its simplicity, the calculated equilibrium geometry and breathing vibrational frequency compare well with the experiment and advanced ab initio methods. The preference for the equilateral geometry rather than the linear configuration is discussed.