Long-range Rydberg molecules represent rather exotic excited bound states of systems formed by an atom in its ground state located within the range of the Rydberg electron of another atom in its highly excited state. Internuclear separations at which their vibrational states are located, are typically 400 - 3000 atomic units. A more general theoretical research of these states is proposed here. The applicant proposes a formulation and application of a computational method to treat these states that takes into account the effects of the polarization of the neutral atom by the other ionic core and interaction of the Rydberg electron with the neutral atom in higher partial waves than those considered in previously published theoretical works. An application to Rb2, LiRb, Cs2 and Fr2 is proposed. The proposed research pays attention to the range of energies and internuclear separations where the application of more simple methods, developed previously, becomes questionable. The role of fine and hyperfine structure is studied in the context of recent experimental works.