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In the simulation of biological molecules, it is customary to imposeconstraints on the fastest degrees of freedom to increase the time step. Theevaluation of the involved constraint forces must be performed in an efficientmanner, for otherwise it would be a bottleneck in the calculations; for thisreason, linearly-scaling calculation methods have become widely used. Ifintegrators of order higher than 2 (e.g. Gear predictor-corrector methods) areused to find the trajectories of atoms, the derivatives of the forces on atomswith respect to the time also need to be calculated, which includes thederivatives of constraint forces. In this letter we prove that such calculationcan be analytically performed with linearly scaling numerical complexity(O(Nc), being Nc the number of constraints). This ensures the feasibility ofconstrained molecular dynamics calculations with high-order integrators.

Exact and efficient calculation of derivatives of Lagrange multipliers for molecular dynamic simulations of biological molecules