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Publication# Monte Carlo evaluation of the equilibrium isotope effects using the Takahashi-Imada factorization of the Feynman path integral

Abstract

The Feynman path integral approach for computing equilibrium isotope effects and isotope fractionation corrects the approximations made in standard methods, although at significantly increased computational cost. We describe an accelerated path integral approach based on three ingredients: the fourth- order Takahashi-Imada factorization of the path integral, thermodynamic integration with respect to mass, and centroid virial estimators for relevant free energy derivatives. While the frst ingredient speeds up convergence to the quantum limit, the second and third improve statistical convergence. The combined method is applied to compute the equilibrium constants for isotope exchange reactions H2+D=H+HD and H2+D2=2HD.

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Related concepts

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Related concepts (6)

Monte Carlo method

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use rando

Monte Carlo integration

In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While othe

Kinetic isotope effect

In physical organic chemistry, a kinetic isotope effect (KIE) is the change in the reaction rate of a chemical reaction when one of the atoms in the reactants is replaced by one of its isotopes.

Related publications (3)

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Investigating the effect of isotope substitution on equilibrium and kinetic properties of molecules has become an important tool for estimating the importance of nuclear quantum effects. In this work, we discuss calculating both equilibrium and kinetic isotope effects, i.e., the isotope effects on a system's partition function and a reaction's rate constant. With the help of Feynman's path integral formalism, both quantities can be estimated using standard Monte Carlo methods that scale favorably with system's dimensionality; improving efficiency of such approaches is the main focus of this work.
First of all, we developed a novel procedure for changing mass stochastically during an equilibrium isotope effect calculation, and evaluated the numerical benefits of combining it with two popular approaches for calculating isotope effects, using either direct estimators or thermodynamic integration. We demonstrate that the modification improves statistical convergence of both methods, and that it additionally allows to eliminate integration error of thermodynamic integration. The improved methods are tested on equilibrium isotope effects in a model harmonic system and in methane.
Then we turn our attention to kinetic isotope effect calculations with the quantum instanton approximation, a method whose path integral implementation belongs among the most accurate approaches for evaluating reaction rate constants in polyatomic systems. To accelerate quantum instanton calculations of kinetic isotope effects, we combine higher-order Boltzmann operator factorization with virial estimators, allowing us to speed up both the convergence to the quantum limit and statistical convergence of the calculation. We estimate the overall resulting acceleration using H+H2/D+D2 as a benchmark system, and then apply the accelerated method to several kinetic isotope effects associated with the H+CH4=H2+CH3 exchange.
Last but not least, we explored ways to improve on the quantum instanton approximation for reaction rate constants. To that end, we review quantum instanton and Hansen-Andersen approximations, and propose a combined method, which, as the Hansen-Andersen approximation, has the correct high-temperature behavior, and at the same time, as the quantum instanton approximation, has more flexibility by allowing the dividing surface for the reaction to split into two surfaces at low temperatures. The properties of the combined method are tested on symmetric and asymmetric Eckart barrier.

A general method for computing kinetic isotope effects is described. The method uses the quantum-instanton approxn. and is based on the thermodn. integration with respect to the mass of the isotopes and on the path-integral Monte- A general method for computing kinetic isotope effects is described. The method uses the quantum-instanton approximation and is based on the thermodynamic integration with respect to the mass of the isotopes and on the path- integral Monte-Carlo evaluation of relevant thermodynamic quantities. The central ingredients of the method are the Monte-Carlo estimators for the logarithmic derivatives of the partition function and the delta-delta correlation function. Several alternative estimators for these quantities are described here and their merits are compared on the benchmark hydrogen-exchange reaction, H+H_2->H_2+H on the Truhlar-Kuppermann potential energy surface. Finally, a qualitative discussion of issues arising in many- dimensional systems is provided.

A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis–Monte Carlo evaluation of the Boltzmann operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction H+H2H2+H. In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than ~10%.

2005