Three-body bound-state calculations by the Lagrange-mesh method: selection of a coordinate system

Four coordinate systems adapted to three-body problems, the relative, Jacobi, perimetric, and renormalized Hylleraas coordinates, are compared in bound-state Lagrange-mesh calculations. The convergence of the energy with respect to the Lagrange basis size and the filling rate of the Hamiltonian matrix are analyzed. Three kinds of potentials are considered: harmonic, Gaussian, and Coulomb-like potentials. When at most one interaction potential presents a singularity at the origin, Jacobi coordinates represent the best choice for three-body Lagrange-mesh calculations. When all three potentials contain 1/r singularities, Jacobi coordinates provide only a limited accuracy, and perimetric coordinates take over. In all cases, with a good choice of coordinates, the Lagrange-mesh method provides very good accuracies on the three-body ground-state energy with small numbers of mesh points.