Convergent partition function for atomic and molecular bound states is derived for weakly non-ideal hydrogen plasmas. Corrections due to charged-neutral particles interactions are also described. These contributions have been realized in the model equation of state of hydrogen plasmas and the HEOS code. The physical model EOS also includes the Coulomb and diffraction corrections, contributions of the scattering states, electron and ion interactions, and the radiation pressure. The calculated dependences of the heat capacity, the ionization degree, and partial pressures are presented.