A new. general formula that comtects the derivatives of the free energy along the selected,

In free energy calculations based on thermodynamic integration, it is necessary to compute the derivatives of the free energy as a function of one (scalar case) or several (vector case) order parameters. We derive in a compact way a general formulation for evaluating these derivatives as the average of a mean force acting on the order parameters, which involves first derivatives with respect to both Cartesian coordinates and time. This is in contrast with the previously derived formulas, which require first and second derivatives of the order parameter with respect to Cartesian coordinates. As illustrated in a concrete example, the main advantage of this new formulation is the simplicity of its use, especially for complicated order parameters. It is also straightforward to implement in a molecular dynamics code, as can be seen from the pseudocode given at the end. We further discuss how the approach based on time derivatives can be combined with the adaptive biasing force method, an enhanced sampling technique that rapidly yields uniform sampling of the order parameters, and by doing so greatly improves the efficiency of free energy calculations. Using the backbone dihedral angles Phi and Psi in N-acetylalanyl-N'-methylamide as a numerical example, we present a technique to reconstruct the free energy from its derivatives, a calculation that presents some difficulties in the vector case because of the statistical errors affecting the derivatives.

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We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix-vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix-vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CN iter N log N floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM.

The efficiencies of two recently developed methods for calculating free energy changes along a generalized coordinate in a system are discussed in the context of other, related approaches. One method is based on Jarzynski's identity [Phys. Rev. Lett. 78, 2690 (1997)]. The second method relies on thermodynamic integration of the average force and is called the adaptive biasing force method [Darve and Pohorille, J. Chem. Phys. 115, 9169 (2001)]. Both methods are designed such that the system evolves along the chosen coordinate(s) without experiencing free energy barriers and they require calculating the instantaneous, unconstrained force acting on this coordinate using the formula derived by Darve and Pohorille. Efficiencies are analyzed by comparing analytical estimates of statistical errors and by considering two numerical examples-internal rotation of hydrated 1,2-dichloroethane and transfer of fluoromethane across a water-hexane interface. The efficiencies of both methods are approximately equal in the first but not in the second case. During transfer of fluoromethane the system is easily driven away from equilibrium and, therefore, the performance of the method based on Jarzynski's identity is poor.