The authors propose a regularization technique which allows Lagrange-mesh calculations to retain their accuracy, efficiency and simplicity when the Hamiltonian is singular at finite distance. Compact analytical expressions of kinetic-energy matrix elements are an essential ingredient of the method. The authors present general procedures for deriving them. Their approach is tested on the hydrogen atom and the squared orbital-momentum operator. They demonstrate its accuracy on the non-separable problem of the hydrogen atom in a magnetic field. This method opens the way to numerous new applications of Lagrange-mesh calculations in atomic and molecular physics.

The Lagrange-mesh method is an approximate variational calculation which resembles a mesh calculation because of the use of a Gauss quadrature. In order to analyze its accuracy, four different Lagrange-mesh calculations based on the zeros of Laguerre polynomials are compared with exact variational calculations based on the corresponding Laguerre basis. The comparison is performed for three solvable radial potentials: the Morse, harmonic-oscillator, and Coulomb potentials. The results show that the accuracies of the energies obtained for different partial waves with the different mesh approximations are very close to the variational accuracy, even in the presence of the centrifugal singularity. The same property holds for the approximate wave functions. This striking accuracy remains unexplained.

The hydrogen molecular ion in an aligned strong magnetic field is studied in prolate spheroidal coordinates with the Lagrange-mesh method. Different variants of the regularization of singularities are described in detail and tested. The simple resulting equations provide a high accuracy with small computing times. At fixed fields and basis sizes, the accuracy on energies decreases with increasing |m| values. At γ = 1, accuracies from 10−12 for m = 0 to 10−9 for m = −4 in positive parity and 10−7 in negative parity are obtained with various bases. Energies at equilibrium for m = 0 can be determined with at least 12 significant digits up to γ = 1000. Equilibrium distances are calculated with at least six significant digits.