### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1440-1453 |

Number of pages | 14 |

Journal | Journal of Chemical Theory and Computation |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Apr 8 2014 |

### Fingerprint

### ASJC Scopus subject areas

- Physical and Theoretical Chemistry
- Computer Science Applications

### Cite this

*Journal of Chemical Theory and Computation*,

*10*(4), 1440-1453. https://doi.org/10.1021/ct400911m

**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; Bacic, Zlatko; Tuckerman, Mark.

Research output: Contribution to journal › Article

*Journal of Chemical Theory and Computation*, vol. 10, no. 4, pp. 1440-1453. https://doi.org/10.1021/ct400911m

}

TY - JOUR

T1 - Efficient calculation of free energy differences associated with isotopic substitution using path-integral molecular dynamics

AU - Marsalek, Ondrej

AU - Chen, Pei Yang

AU - Dupuis, Romain

AU - Benoit, Magali

AU - Méheut, Merlin

AU - Bacic, Zlatko

AU - Tuckerman, Mark

PY - 2014/4/8

Y1 - 2014/4/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84898410430&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898410430&partnerID=8YFLogxK

U2 - 10.1021/ct400911m

DO - 10.1021/ct400911m

M3 - Article

AN - SCOPUS:84898410430

VL - 10

SP - 1440

EP - 1453

JO - Journal of Chemical Theory and Computation

JF - Journal of Chemical Theory and Computation

SN - 1549-9618

IS - 4

ER -