Lagrange-mesh calculations of three-body atoms and molecules

The Lagrange-mesh numerical method has the simplicity of a mesh calculation and the accuracy of a variational calculation. A three-dimensional Lagrange-mesh method based on zeros of Laguerre polynomials is applied to the study of the ground states of the helium atom, the hydrogen and positronium negative ions and the hydrogen molecular ion. The calculations lead to big sparse matrices. Highly accurate energies are obtained, which essentially agree with recently published results or improve them. The accuracy is estimated by varying the mesh size and two nonlinear scale parameters. Different mean values of observables are obtained very simply with the Gauss quadrature approximation associated with the mesh.