ABSTRACT: We study the effect that a statically screened Coulomb potential represented by a Debye-H¨uckel-Yukawa potential has in the electronic structure of the simplest molecule H2+ within the Born-Oppenheimer approximation. The method of solution is based on a two-center partial-wave expansion expressed in confocal elliptic coordinates using B-spline polynomials. General algorithms for the computation of energies, wave functions, and dipole and nonadiabatic radial matrix elements are given in detail. As it occurs in atoms, screening in simple molecules shifts the energies of bound states upwards so that, as screening increases, every bound state eventually crosses the upper ionization threshold at a critical screening value. The loss of long-range Coulomb interactions has its effect in the structure of wave functions, and consequently in the dipole and nonadiabatic matrix elements at intermediate and long internuclear distances, which determine the dynamics in external electromagnetic fields and collisional processes. Other issues related to a practical solution of the arbitrary sign problem, as well as the assignment of angular and radial nodes to the variational eigenfunctions, and the appearance of molecular shape resonances and Borromean states inH2+ as screening increases, are also addressed in this work.