The truncated conjugate gradient (TCG), a non-iterative/fixed-cost strategy for computing polarization in molecular dynamics: Fast evaluation of analytical forces.

In a recent paper [F. Aviat et al., J. Chem. Theory Comput. 13, 180-190 (2017)], we proposed the Truncated Conjugate Gradient (TCG) approach to compute the polarization energy and forces in polarizable molecular simulations. The method consists in truncating the conjugate gradient algorithm at a fixed predetermined order leading to a fixed computational cost and can thus be considered "non-iterative." This gives the possibility to derive analytical forces avoiding the usual energy conservation (i.e., drifts) issues occurring with iterative approaches. A key point concerns the evaluation of the analytical gradients, which is more complex than that with a usual solver. In this paper, after reviewing the present state of the art of polarization solvers, we detail a viable strategy for the efficient implementation of the TCG calculation. The complete cost of the approach is then measured as it is tested using a multi-time step scheme and compared to timings using usual iterative approaches. We show that the TCG methods are more efficient than traditional techniques, making it a method of choice for future long molecular dynamics simulations using polarizable force fields where energy conservation matters. We detail the various steps required for the implementation of the complete method by software developers.