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In this work, we introduce an algorithm to compute the derivatives ofphysical observables along the constrained subspace when flexible constraintsare imposed on the system (i.e., constraints in which the hard coordinates arefixed to configuration-dependent values). The presented scheme is exact, itdoes not contain any tunable parameter, and it only requires the calculationand inversion of a sub-block of the Hessian matrix of second derivatives of thefunction through which the constraints are defined. We also present a practicalapplication to the case in which the sought observables are the Euclideancoordinates of complex molecular systems, and the function whose minimizationdefines the constraints is the potential energy. Finally, and in order tovalidate the method, which, as far as we are aware, is the first of its kind inthe literature, we compare it to the natural and straightforwardfinite-differences approach in three molecules of biological relevance:methanol, N-methyl-acetamide and a tri-glycine peptide

An exact expression to calculate the derivatives of position-dependent observables in molecular simulations with flexible constraints