From a set of generalized spheroidal wave equations for a hydrogenic system in half space, a modified comparison method is applied to the case of a bound state at large system-centre surface separation. The asymptotic expansion of the energy eigenvalues E and the separation constant A for the system with respect to R, twice the system-centre- surface separation d, is obtained up to the third order terms in powers of 1/p, where p^2 = -1/4RE The formulae obtained in this work can easily be generalized to the case of the two-centre Coulomb problem.