Some effects of deviations from the Lorentz-Berthelot (LB) combining rules for Lennard-Jones binary mixtures are studied with a model in which both constituents have the same size and energy interaction parameters but the cross interaction deviates substantially from the LB rules. A single temperature and density and two compositions are studied by molecular dynamics simulations and the recently proposed Duh-Haymet-Henderson (DHH) intergral equation theory. The agreement is satisfactory in both cases. The excess volume and enthalpy of mixing are also studied systematically with the DHH theory. Some interesting phenomena are observed for these no-LB mixtures.