In this paper, we describe implementation of the semiclassical Liouville method for simulating molecular dynamics on coupled electronic surfaces in the electronic adiabatic representation. We cast the formalism in terms of semiclassical motion on Born-Oppenheimer potential energy surfaces with nonadiabatic coupling arising from the coordinate dependence of the adiabatic electronic eigenstates. Using perturbation theory and asymptotic evaluation of the resulting time integrals, we derive an expression for the probability of transition between adiabatic states which agrees with the result given previously by Miller and George [W. H. Miller and T. F. George, J. Chem. Phys. 56, 5637 (1972)]. We also demonstrate numerically the equivalence of semiclassical trajectory-based calculations in the adiabatic and diabatic representations by performing molecular dynamics simulations on a model two-state system and comparing with exact quantum mechanical results. Excellent agreement between the exact and semiclassical treatments is obtained in both representations. (C) 2000 American Institute of Physics. [S0021-9606(00)00808-4].