3-body harmonic molecule

• Published in: arXiv.org 2022-09
• Main Authors: Olivares-Pilón, H, Escobar-Ruiz, A M, Montoya, F
• Format: Article
• Language: English
• Subjects: Angular momentum; Chaos theory; Degeneration; Eigenvectors; Exact solutions; Excitation spectra; Finite element method; Molecular physics
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2208.08947

In this study, the quantum 3-body harmonic system with finite rest length \(R\) and zero total angular momentum \(L=0\) is explored. It governs the near-equilibrium \(S\)-states eigenfunctions \(\psi(r_{12},r_{13},r_{23})\) of three identical point particles interacting by means of any pairwise confining potential \(V(r_{12},r_{13},r_{23})\) that entirely depends on the relative distances \(r_{ij}=|{\mathbf r}_i-{\mathbf r}_j|\) between particles. At \(R=0\), the system admits a complete separation of variables in Jacobi-coordinates, it is (maximally) superintegrable and exactly-solvable. The whole spectra of excited states is degenerate, and to analyze it a detailed comparison between two relevant Lie-algebraic representations of the corresponding reduced Hamiltonian is carried out. At \(R>0\), the problem is not even integrable nor exactly-solvable and the degeneration is partially removed. In this case, no exact solutions of the Schr\"odinger equation have been found so far whilst its classical counterpart turns out to be a chaotic system. For \(R>0\), accurate values for the total energy \(E\) of the lowest quantum states are obtained using the Lagrange-mesh method. Concrete explicit results with not less than eleven significant digits for the states \(N=0,1,2,3\) are presented in the range \(0\leq R \leq 4.0\)~a.u. . In particular, it is shown that (I) the energy curve \(E=E(R)\) develops a global minimum as a function of the rest length \(R\), and it tends asymptotically to a finite value at large \(R\), and (II) the degenerate states split into sub-levels. For the ground state, perturbative (small-\(R\)) and two-parametric variational results (arbitrary \(R\)) are displayed as well. An extension of the model with applications in molecular physics is briefly discussed.