The hydrogen molecule in the adiabatic representation

The simplest four-body system with Coulomb interaction is investigated using the adiabatic basis. The wavefunctions of this basis are the solutions of the quantum mechanical two-centre problem (one electron in the field of two fixed nuclei). An infinite set of ordinary differential equations, describing the hydrogen molecule in the adiabatic representation, has been constructed. An approximate finite system of equations has been solved and the first three vibrational states of the hydrogen molecule were obtained. Some advantages and the convergence of the proposed method are discussed.