RESEARCH NOTE An improved leap-frog rotational algorithm

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Abstract
A new implicit leap-frog algorithm for the integration of rigid body rotational motion is presented. Orientations are represented by quaternions and the algorithm is compared with three existing leap-frog integrators, by solving the classical equations of motion for a (H2O)20 cluster. We find that the present scheme exhibits superior energy conservation properties, especially for integration times of about 10 ps or longer. Contrary to previous algorithms, the present one behaves as a true Verlet integrator, where the degree of energy conservation is independent of the length of the trajectory. The method is similar to the implicit scheme proposed by D. Fincham (1992, Molec. Simulation, 8, 165), with the difference that selfconsistent quaternions, as well as their time derivatives, are obtained by iteration at the mid-timestep instead of after the complete timestep. A slight modification of either the explicit or the implicit leap-frog rotational algorithm in existing molecular dynamics programs may thus lead to significant improvements of energy conservation, as long as this property is not dominated by other sources such as errors due to potential truncation. It is demonstrated that the present algorithm can be used with timesteps as large as 4 fs in water simulations, and still produce stable trajectories of 10 ns duration. 2 20