A generalization is presented here for a newly developed approach to ab initio molecular dynamics, where the density matrix is propagated with Gaussian orbitals. Including a tensorial fictitious mass facilitates the use of larger time steps for the dynamics process. A rigorous analysis of energy conservation is presented and used to control the deviation of the fictitious dynamics trajectory from the corresponding Born-Oppenheimer dynamics trajectory. These generalizations are tested for the case of the Cl-(H2O)(25) cluster. It is found that, even with hydrogen atoms present in the system, no thermostats are necessary to control the exchange of energy between the nuclear and the fictitious electronic degrees of freedom.