An alternative Liouville formulation of mixed quantum-classical dynamics outlined recently [K. Ando, Chem. Phys. Lett. 360, 240 (2002)] is expanded in detail by taking an explicit account of the parametric dependence of the electronic (adiabatic) basis on the nuclear coordinates. As a consequence of the different operational order of the partial Wigner transformation for the nuclear coordinates and the calculation of the matrix elements in the adiabatic electronic basis, the present formula differs from the previously proposed one, slightly in the appearance but significantly in the treatment of nonadiabatic transitions in the trajectory implementation in that the former does not contain the " off- diagonal Hellmann -Feynman forces'' representing the so- called " momentum- jump'' associated with the nonadiabatic transitions. Because of this, the present formula is free from the numerical instability intrinsically coming from the momentum- jump operation at around the classical turning points of the nuclear motion. It is also shown that the density matrices from the two approaches coincide when the electronic basis is independent of the nuclear coordinates ( R), and hence the momentum- jump approximation stems from the R- dependence of the adiabatic electronic basis. Improved stability and comparable to better reproduction of the quantum reference calculations are demonstrated by applications to one and three dimensional spin-boson models and a two- state three- mode model of the S-2 --> S-1 internal conversion of pyrazine. Also discussed is the importance of electronic coherence for the proper treatment of nonadiabatic transition rates which is naturally described by the Liouville methods compared to the conventional independent trajectory approaches. (C) 2003 American Institute of Physics.