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Efficient methods and practical guidelines for simulating isotope effects

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10.1063/1.4772676

### Abstract

The shift in chemical equilibria due to isotope substitution is frequently exploited to obtain insight into a wide variety of chemical and physical processes. It is a purely quantum mechanical effect, which can be computed exactly using simulations based on the path integral formalism. Here we discuss how these techniques can be made dramatically more efficient, and how they ultimately outperform quasi-harmonic approximations to treat quantum liquids not only in terms of accuracy, but also in terms of computational cost. To achieve this goal we introduce path integral quantum mechanics estimators based on free energy perturbation, which enable the evaluation of isotope effects using only a single path integral molecular dynamics trajectory of the naturally abundant isotope. We use as an example the calculation of the free energy change associated with H/D and ^{16}O/^{18}O substitutions in liquid water, and of the fractionation of those isotopes between the liquid and the vapor phase. In doing so, we demonstrate and discuss quantitatively the relative benefits of each approach, thereby providing a set of guidelines that should facilitate the choice of the most appropriate method in different, commonly encountered scenarios. The efficiency of the estimators we introduce and the analysis that we perform should in particular facilitate accurate *ab initio* calculation of isotope effects in condensed phase systems.

© 2013 American Institute of Physics

Received 26 October 2012
Accepted 05 December 2012
Published online 07 January 2013

Acknowledgments: We would like to thank David Manolopoulos for insightful discussion and many precious suggestions and Timothy Berkelbach for a thorough reading of the manuscript. M.C. acknowledges funds from the EU Marie Curie IEF No. PIEFGA-2010-272402. T.E.M acknowledges funding from a Terman fellowship and Stanford start-up funds.

Article outline:

I. INTRODUCTION

II. THEORY

A. Discretizing the free energy change

B. The pitfalls of harmonic approximations

C. Imaginary time path integrals

D. Isotope substitution by free energy perturbation

III. RESULTS

A. Generating path integral configurations

B. Direct computation of ⟨*T* _{ CV }⟩_{μ}

C. “Thermodynamic” free energy perturbation

D. Scaled coordinates free energy perturbation

E. Recommendations

IV. CONCLUSIONS

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2013-01-07

2014-04-16

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