Abstract
Free energy estimation using thermodynamic integration (TI) involves numerically approximating an integral based on a limited set of discrete data points. These discrete data points represent the free energy slope as a function of the switching variable lambda for TI simulations. We present several numerical techniques for generating optimal free energy estimates utilizing polynomials, instead of the often-used quadrature, to fit the data and thus reduce the bias and uncertainty of the resulting estimates. The specific techniques utilized in our current study are Lagrange and Newton interpolation, cubic spline, and polynomial regression. To further improve the overall accuracy of free energy estimates using these techniques, we also investigated the use of non-equidistant lambda values (based on Chebyshev nodes) for thermodynamic integration simulations. Our results demonstrate that the use of non-equidistant lambda values and high degrees of polynomials gives the more accurate and precise free energy estimates compared to that of trapezoidal quadrature. Regression, in particular, offers the greatest flexibility that permits the degree of polynomial to vary for any desired accuracy without imposing any limitations on the number of lambda values.
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