A ‘path-by-path’ monotone extrapolation sequence for Feynman path integral calculations of quantum mechanical free energies

Direct Computation of the Quantum Partition Function by Path-Integral Nested Sampling

In the present work we introduce a computational approach to the absolute rovibrational quantum partition function using the path-integral formalism of quantum mechanics in combination with the nested sampling technique. The numerical applicability of path-integral nested sampling is demonstrated for small molecules of spectroscopic interest. The computational cost of the method is determined by the evaluation time of a point on the potential energy surface (PES). For efficient PES implementations, the path-integral nested sampling method can be a viable alternative to the direct-Boltzmann-summation technique of variationally computed rovibrational energies, especially for medium-sized molecules and at elevated temperatures.

A whole-path importance-sampling scheme for Feynman path integral calculations of absolute partition functions and free energies

Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function.

An Integrated Path Integral and Free-Energy Perturbation-Umbrella Sampling Method for Computing Kinetic Isotope Effects of Chemical Reactions in Solution and in Enzymes

An integrated centroid path integral and free-energy perturbation-umbrella sampling (PI-FEP/UM) method for computing kinetic isotope effects (KIEs) for chemical reactions in solution and in enzymes is presented. The method is based on the bisection sampling in centroid path integral simulations to include nuclear quantum effects to the classical potential of mean force. The required accuracy for computing kinetic isotope effects is achieved by coupled free-energy perturbation and umbrella sampling for reactions involving different isotopes. The use of FEP with respect to different masses results in relatively small statistical uncertainties, whereas if KIEs are computed directly by the difference in free energies obtained from the quantum mechanical potentials of mean force for different isotopes, the statistical errors are significantly greater. The PI-FEP/UM method is illustrated in two applications. The first reaction is the decarboxylation of N-methyl picolinate in water, for which the primary (13)C and secondary (15)N KIEs have been determined. The second reaction is the proton-transfer reaction between nitroethane and an acetate ion in water. In both cases, the computational results are in accord with experimental data, and the findings provide further insight into the mechanism of these reactions in water.

Path Integral Simulations of Proton Transfer Reactions in Aqueous Solution Using Combined QM/MM Potentials

A bisection sampling method was implemented in path integral simulations of chemical reactions in solution in the framework of the quantized classical path approach. In the present study, we employ a combined quantum mechanical and molecular mechanical (QM/MM) potential to describe the potential energy surface and the path integral method to incorporate nuclear quantum effects. We examine the convergence of the bisection method for two proton-transfer reactions in aqueous solution at room temperature. The first reaction involves the symmetrical proton transfer between an ammonium ion and an ammonia molecule. The second reaction is the ionization of nitroethane by an acetate ion. To account for nuclear quantum mechanical corrections, it is sufficient to quantize the transferring light atom in the ammonium ion-ammonia reaction, while it is necessary to also quantize the donor and acceptor atoms in the nitroethane-acetate ion reaction. Kinetic isotope effects have been computed for isotopic substitution of the transferring proton by a deuteron in the nitroethane-acetate reaction. In all computations, it is important to employ a sufficient number of polymer beads along with a large number of configurations to achieve convergence in these simulations.

High-precision quantum thermochemistry on nonquasiharmonic potentials: converged path-integral free energies and a systematically convergent family of generalized Pitzer-Gwinn approximations

Accurate quantum mechanical (QM) vibrational-rotational partition functions for HOOD, D(2)O(2), H(18)OOH, H(2)(18)O(2), D(18)OOH, and H(18)OOD are determined using a realistic potential energy surface for temperatures ranging from 300 to 2400 K by using the TT-FPI-ESPE path-integral Monte Carlo method. These data, together with our prior results for H(2)O(2), provide benchmarks for testing approximate methods of estimating isotope effects for systems with torsional motions. Harmonic approximations yield poor accuracy for these systems, and although the well-known Pitzer-Gwinn (PG) approximation provides better results for absolute partition functions, it yields the same results as the harmonic approximation for isotope effects because these are intrinsically quantal phenomena. We present QM generalizations of the PG approximation that can provide high accuracy for both isotope effects and absolute partition functions. These approximations can be systematically improved until they approach the accurate result and converge rapidly. These methods can also be used to obtain affordable estimates of zero-point energies from accurate partition functions-even those at relatively high temperatures.

Quantum mechanical single molecule partition function from path integral Monte Carlo simulations

An algorithm for calculating the partition function of a molecule with the path integral Monte Carlo method is presented. Staged thermodynamic perturbation with respect to a reference harmonic potential is utilized to evaluate the ratio of partition functions. Parallel tempering and a new Monte Carlo estimator for the ratio of partition functions are implemented here to achieve well converged simulations that give an accuracy of 0.04 kcal/mol in the reported free energies. The method is applied to various test systems, including a catalytic system composed of 18 atoms. Absolute free energies calculated by this method lead to corrections as large as 2.6 kcal/mol at 300 K for some of the examples presented.

Implementation of the bisection sampling method in path integral simulations

A bisection sampling method is implemented in the framework of a quantized classical path algorithm to include nuclear quantum effects in path integral simulations. The present study examines the convergence of these calculations on two model systems with respect to the number of beads used in the polymer chain and the number of configurations both in free-particle sampling and in classical configuration sampling. The results will be useful for future studies of kinetic isotope effects in enzymatic reactions.

Accurate vibrational-rotational partition functions and standard-state free energy values for H2O2 from Monte Carlo path-integral calculations

Accurate quantum mechanical partition functions and absolute free energies of H(2)O(2) are determined using a realistic potential energy surface [J. Koput, S. Carter, and N. C. Handy, J. Phys. Chem. A 102, 6325 (1998)] for temperatures ranging from 300 to 2,400 K by using Monte Carlo path integral calculations with new, efficient polyatomic importance sampling methods. The path centroids are sampled in Jacobi coordinates via a set of independent ziggurat schemes. The calculations employed enhanced-same-path extrapolation of trapezoidal Trotter Fourier path integrals, and the paths were constructed using fast Fourier sine transforms. Importance sampling was also used in Fourier coefficient space, and adaptively optimized stratified sampling was used in configuration space. The free energy values obtained from the path-integral calculations are compared to separable-mode approximations, to the Pitzer-Gwinn approximation, and to values in thermodynamic tables. Our calculations support the recently proposed revisions to the JANAF tables.