The hydrogen molecules and are analyzed with electronic correlations taken into account between the 1s electrons in an exact manner. The optimal single-particle Slater orbitals are evaluated in the correlated state of by combining their variational determination with the diagonalization of the full Hamiltonian in the second-quantization language. All electron–ion coupling constants are determined explicitly and their relative importance is discussed. Sizable zero-point motion amplitude and the corresponding energy are then evaluated by taking into account the anharmonic contributions up to the ninth order in the relative displacement of the ions from their static equilibrium value. The applicability of the model to solid molecular hydrogen is briefly analyzed by calculating intermolecular microscopic parameters for the rectangular configuration, as well its ground state energy.