We have investigated the feasibility of using higher derivatives of the free energy, with respect to a coupling parameter lambda, for predicting free energy changes between different states from a single simulation corresponding to the initial state, i.e., lambda=0 in the normal thermodynamic integration approach. Terms up to and including the fifth derivative have been used. For a diatomic with point charges (dipole moment=2.4 D) in a box of water, the free energy corresponding to charge rearrangements of the order of +/-0.25e could be predicted accurately after 750 ps of simulation and inclusion of the third derivative. Predictions were best for small perturbations. After 1000 ps of simulation, the first, second, third, and probably the fourth derivatives had converged, while the fifth was still oscillating. The possibility of predicting a multitude of free energy changes from a single simulation is discussed.