3-body harmonic molecule

Abstract

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.