# J. Chem. Theory Comput., 2014, 10 (4), pp 1440–1453

#### Efficient calculation of free energy differences associated with isotopic substitution using Path Integral Molecular Dynamics

Marsalek, Ondrej ; Chen, Pei-Yang ; Dupuis, Romain ; Benoit, Magali ; Méheut, Merlin ; Bačić, Zlatko ; Tuckerman, Mark E.

http://dx.doi.org/10.1021/ct400911m

### Abstract

The problem of computing free energy differences due to isotopic substitution in chemical systems is discussed. The shift in the equilibrium properties of a system upon isotopic substitution is a purely quantum mechanical effect that can be quantified using the Feynman path integral approach. In this paper, we explore two developments that lead to a highly efficient path integral scheme. First, we employ a mass switching function inspired by the work of Ceriotti and Markland [ *J. Chem. Phys.* **2013**, *138*, 014112] that is based on the inverse square root of the mass and which leads to a perfectly constant free energy derivative with respect to the switching parameter in the harmonic limit. We show that even for anharmonic systems, this scheme allows a single-point thermodynamic integration approach to be used in the construction of free energy differences. In order to improve the efficiency of the calculations even further, however, we derive a set of free energy derivative estimators based on the fourth-order scheme of Takahashi and Imada [ *J. Phys. Soc. Jpn.* **1984**, *53*, 3765]. The Takahashi–Imada procedure generates a primitive fourth-order estimator that allows the number of imaginary time slices in the path-integral approach to be reduced substantially. However, as with all primitive estimators, its convergence is plagued by numerical noise. In order to alleviate this problem, we derive a fourth-order virial estimator based on a transferring of the difference between second- and fourth-order primitive estimators, which remains relatively constant as a function of the number of configuration samples, to the second-order virial estimator. We show that this new estimator converges as smoothly as the second-order virial estimator but requires significantly fewer imaginary time points.

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