Fundamental Limitation on Size of MTS Methods

One obvious remedy for this problem is to choose time-step lengths so as to avoid small integer multiples of half-periods of any oscillatory motion. However, it has been demonstrated that the molecular dynamics potential gives rise to motion with a continuum of periods greater than or equal to 10 fs. Furthermore, the energy instability of impulse MTS methods becomes exponentially worse at larger multiples of the half-periods. This rules out the possibility that a fortuitously chosen assortment of impulse multiple time steps longer than 5 fs could yield stable trajectories. 

A number of methods have been proposed to overcome the MTS barrier, including averaging methods that mollify the impulse, allowing time steps of up to 6fs while maintaining the favorably small energy drift attained by impulse MTS methods with 4fs time steps. We will omit here the details of these time-stepping algorithms but point to a reference that explicitly provides implementation details. 

Langevin stabilization as an approach to multiple time-step numerical integration was introduced by Barth and Schlick in 1998. One particularly simple method described in that work can be written as a modification of Eq. [19] subject to Eq. [7]  

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