We present a study of the band structure of liquid Carbon and Silicon modelled in a Tight-Binding Hamiltonian approximation by means of an integral equation approximation that includes non-linear corrections. The theoretical predictions are contrasted with Tight Binding Molecular Dynamics simulations in which the energy bands are obtained by direct diagonalization of the Hamiltonian matrix. The results for Silicon an excellent, whereas in liquid Carbon only some of the qualitative features of the band structure are captured by the non-linear corrections. We find that this can largely be understood as an effect of missing three-body correlation functions in the theoretical treatment of the energy bands. This is particularly crucial in the case of strongly directional and short range bonding, as it occurs in Carbon.