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 and in N-acetylalanyl--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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Research Article| April 11 2008
Adaptive biasing force method for scalar and vector free energy calculations
Andrew Pohorille b)
2Exobiology Branch, MS 239-4,
NASA Ames Research Center, Moffett Field, California 94035,
3Department of Pharmaceutical Chemistry,
University of California San Francisco, 600 16th Street, San Francisco, California 94143-2280,
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Eric Darve, David Rodríguez-Gómez, Andrew Pohorille; Adaptive biasing force method for scalar and vector free energy calculations. J. Chem. Phys. 14 April 2008; 128 (14): 144120. https://doi.org/10.1063/1.2829861
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