Error cancellation in the molecular dynamics method for total energy calculations

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Abstract
A detailed analysis of Car and Parrinello's molecular dynamics method (1985) is presented. It is shown that the degrees of freedom associated with the electronic wavefunctions do not behave as classical degrees of freedom because their motions are damped by the constraint of normalisation of the wavefunctions. Therefore, the accuracy to which the electronic configuration remains on the Born-Oppenheimer surface and the ionic configuration evolves at constant energy during a dynamical simulation is not a result of treating the electronic degrees of freedom as classical degrees of freedom. Instead it is shown that the accuracy of the constant energy evolution of the ionic system in a molecular dynamics calculation is explained by the tendency for errors in the Hellmann-Feynman forces to cancel when the molecular dynamics equations of motion are used to evolve the electronic degrees of freedom. A quantitative analysis of this error cancellation is presented. By analysing the magnitude of the error in the electronic wavefunction, a criterion is developed for the maximum velocity of propagation of the ions at which the evolution of the electronic configuration remains stable.