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.

Improved methods for Feynman path integral calculations and their application to calculate converged vibrational-rotational partition functions, free energies, enthalpies, entropies, and heat capacities for methane.

We present an improved version of our "path-by-path" enhanced same path extrapolation scheme for Feynman path integral (FPI) calculations that permits rapid convergence with discretization errors ranging from O(P(-6)) to O(P(-12)), where P is the number of path discretization points. We also present two extensions of our importance sampling and stratified sampling schemes for calculating vibrational-rotational partition functions by the FPI method. The first is the use of importance functions for dihedral angles between sets of generalized Jacobi coordinate vectors. The second is an extension of our stratification scheme to allow some strata to be defined based only on coordinate information while other strata are defined based on both the geometry and the energy of the centroid of the Feynman path. These enhanced methods are applied to calculate converged partition functions by FPI methods, and these results are compared to ones obtained earlier by vibrational configuration interaction (VCI) calculations, both calculations being for the Jordan-Gilbert potential energy surface. The earlier VCI calculations are found to agree well (within ∼1.5%) with the new benchmarks. The FPI partition functions presented here are estimated to be converged to within a 2σ statistical uncertainty of between 0.04% and 0.07% for the given potential energy surface for temperatures in the range 300-3000 K and are the most accurately converged partition functions for a given potential energy surface for any molecule with five or more atoms. We also tabulate free energies, enthalpies, entropies, and heat capacities.

Thermal rate constants for the O(3P) + HBr and O(3P) + DBr reactions: transition-state theory and quantum mechanical calculations.

The O((3)P) + HBr → OH + Br and O((3)P) + DBr → OD + Br reactions are studied on a recent high-quality ab initio-based potential energy surface. Thermal rate constants over the 200-1000 K temperature range, calculated using variational transition-state theory (VTST) with the small-curvature tunneling (SCT) correction and quantum mechanical methods with the J-shifting approximation (QM/JS) for zero total angular momentum (J = 0), are reported. These results are compared to the available experimental data, which lie in the ranges of 221-554 and 295-419 K for O + HBr and O + DBr, respectively. The rate constants, in cm(3) molecule(-1) s(-1) and at 298 K, for the O + HBr reaction are 3.66 × 10(-14) for VTST, 3.80 × 10(-14) for QM/JS, and 3.66 × 10(-14) for the average of eight experimental measurements.

Vibrational configuration interaction using a tiered multimode scheme and tests of approximate treatments of vibrational angular momentum coupling: a case study for methane.

We present vibrational configuration interaction calculations employing the Watson Hamiltonian and a multimode expansion. Results for the lowest 36 eigenvalues of the zero total angular momentum rovibrational spectrum of methane agree with the accurate benchmarks of Wang and Carrington to within a mean unsigned deviation of 0.68, 0.033, and 0.014 cm(-1) for 4-mode, 5-mode, and 6-mode representations, respectively. We note that in the case of the 5-mode results, this is a factor of 10 better agreement than for 5-mode calculations reported earlier by Wu, Huang, Carter, and Bowman for the same set of eigenvalues, which indicates that the multimode expansion is even more rapidly convergent than previously demonstrated. Our largest calculations employ a tiered approach with matrix elements treated using a variable-order multimode expansion with orders ranging from 4-mode to 7-mode; strategies for assigning matrix elements to particular multimode tiers are discussed. Improvements of 7-mode coupling over 6-mode coupling are small (averaging 0.002 cm(-1) for the first 36 eigenvalues) suggesting that 7-mode coupling is sufficient to fully converge the results. A number of approximate treatments of the computationally expensive vibrational angular momentum terms are explored. The use of optimized vibrational quadratures allows rapid integration of the matrix elements, especially the vibrational angular momentum terms, which require significantly fewer quadrature points than are required to integrate the potential. We assign the lowest 243 states and compare our results to those of Wang and Carrington, who provided assignments for the same set of states. Excellent agreement is observed for most states, but our results are lower for some of the higher-energy states by as much as 20 cm(-1), with the largest deviations being for the states with six quanta of excitation in the F2 bends, suggesting that the earlier results were not fully converged with respect to the basis set. We also provide corrections to several of the state assignments published previously.

Efficient methods for including quantum effects in Monte Carlo calculations of large systems: extension of the displaced points path integral method and other effective potential methods to calculate properties and distributions.

We present a procedure to calculate ensemble averages, thermodynamic derivatives, and coordinate distributions by effective classical potential methods. In particular, we consider the displaced-points path integral (DPPI) method, which yields exact quantal partition functions and ensemble averages for a harmonic potential and approximate quantal ones for general potentials, and we discuss the implementation of the new procedure in two Monte Carlo simulation codes, one that uses uncorrelated samples to calculate absolute free energies, and another that employs Metropolis sampling to calculate relative free energies. The results of the new DPPI method are compared to those from accurate path integral calculations as well as to results of two other effective classical potential schemes for the case of an isolated water molecule. In addition to the partition function, we consider the heat capacity and expectation values of the energy, the potential energy, the bond angle, and the OH distance. We also consider coordinate distributions. The DPPI scheme performs best among the three effective potential schemes considered and achieves very good accuracy for all of the properties considered. A key advantage of the effective potential schemes is that they display much lower statistical sampling variances than those for accurate path integral calculations. The method presented here shows great promise for including quantum effects in calculations on large systems.

Accelerating the Convergence and Reducing the Variance of Path Integral Calculations of Quantum Mechanical Free Energies by Using Local Reference Potentials.

We present two new methods to accelerate the convergence of Feynman path integral calculations of thermodynamic partition functions. The first enhancement uses information from instantaneous normal mode (INM) calculations to decrease the number of discretized points necessary to represent the Feynman paths and is denoted the local generalized Pitzer-Gwinn (LGPG) scheme. The second enhancement, denoted harmonically guided variance reduction (HGVR), reduces the variance in Monte Carlo (MC) calculations by exploiting the correlation between the sampling error associated with the sum over paths at a particular centroid location for the accurate potential and for the INM approximation of a model potential, the latter of which can be exactly calculated. The LGPG scheme can reduce the number of quadrature points required along the paths by nearly an order of magnitude, and the HGVR scheme can reduce the number of MC samples needed to achieve a target accuracy by more than an order of magnitude. Numerical calculations are presented for H2O2, a very anharmonic system where torsional motion is important, and H2O, a system more amenable to harmonic reference treatment.

Kinetics of the reaction of the heaviest hydrogen atom with H2, the 4Heµ + H2 → 4HeµH + H reaction: experiments, accurate quantal calculations, and variational transition state theory, including kinetic isotope effects for a factor of 36.1 in isotopic mass.

The neutral muonic helium atom (4)Heµ, in which one of the electrons of He is replaced by a negative muon, may be effectively regarded as the heaviest isotope of the hydrogen atom, with a mass of 4.115 amu. We report details of the first muon spin rotation (µSR) measurements of the chemical reaction rate constant of (4)Heµ with molecular hydrogen, (4)Heµ + H(2) → (4)HeµH + H, at temperatures of 295.5, 405, and 500 K, as well as a µSR measurement of the hyperfine coupling constant of muonic He at high pressures. The experimental rate constants, k(Heµ), are compared with the predictions of accurate quantum mechanical (QM) dynamics calculations carried out on a well converged Born-Huang (BH) potential energy surface, based on complete configuration interaction calculations and including a Born-Oppenheimer diagonal correction. At the two highest measured temperatures the agreement between the quantum theory and experiment is good to excellent, well within experimental uncertainties that include an estimate of possible systematic error, but at 295.5 K the quantum calculations for k(Heµ) are below the experimental value by 2.1 times the experimental uncertainty estimates. Possible reasons for this discrepancy are discussed. Variational transition state theory calculations with multidimensional tunneling have also been carried out for k(Heµ) on the BH surface, and they agree with the accurate QM rate constants to within 30% over a wider temperature range of 200-1000 K. Comparisons between theory and experiment are also presented for the rate constants for both the D + H(2) and Mu + H(2) reactions in a novel study of kinetic isotope effects for the H + H(2) reactions over a factor of 36.1 in isotopic mass of the atomic reactant.

Practical methods for including torsional anharmonicity in thermochemical calculations on complex molecules: the internal-coordinate multi-structural approximation.

Many methods for correcting harmonic partition functions for the presence of torsional motions employ some form of one-dimensional torsional treatment to replace the harmonic contribution of a specific normal mode. However, torsions are often strongly coupled to other degrees of freedom, especially other torsions and low-frequency bending motions, and this coupling can make assigning torsions to specific normal modes problematic. Here, we present a new class of methods, called multi-structural (MS) methods, that circumvents the need for such assignments by instead adjusting the harmonic results by torsional correction factors that are determined using internal coordinates. We present three versions of the MS method: (i) MS-AS based on including all structures (AS), i.e., all conformers generated by internal rotations; (ii) MS-ASCB based on all structures augmented with explicit conformational barrier (CB) information, i.e., including explicit calculations of all barrier heights for internal-rotation barriers between the conformers; and (iii) MS-RS based on including all conformers generated from a reference structure (RS) by independent torsions. In the MS-AS scheme, one has two options for obtaining the local periodicity parameters, one based on consideration of the nearly separable limit and one based on strongly coupled torsions. The latter involves assigning the local periodicities on the basis of Voronoi volumes. The methods are illustrated with calculations for ethanol, 1-butanol, and 1-pentyl radical as well as two one-dimensional torsional potentials. The MS-AS method is particularly interesting because it does not require any information about conformational barriers or about the paths that connect the various structures.

Kinetic isotope effects for the reactions of muonic helium and muonium with H2.

The neutral muonic helium atom may be regarded as the heaviest isotope of the hydrogen atom, with a mass of ~4.1 atomic mass units ((4.1)H), because the negative muon almost perfectly screens one proton charge. We report the reaction rate of (4.1)H with (1)H(2) to produce (4.1)H(1)H + (1)H at 295 to 500 kelvin. The experimental rate constants are compared with the predictions of accurate quantum-mechanical dynamics calculations carried out on an accurate Born-Huang potential energy surface and with previously measured rate constants of (0.11)H (where (0.11)H is shorthand for muonium). Kinetic isotope effects can be compared for the unprecedentedly large mass ratio of 36. The agreement with accurate quantum dynamics is quantitative at 500 kelvin, and variational transition-state theory is used to interpret the extremely low (large inverse) kinetic isotope effects in the 10(-4) to 10(-2) range.

Functional representation for the born-oppenheimer diagonal correction and born-huang adiabatic potential energy surfaces for isotopomers of H3.

Multireference configuration interaction (MRCI) calculations of the Born-Oppenheimer diagonal correction (BODC) for H(3) were performed at 1397 symmetry-unique configurations using the Handy-Yamaguchi-Schaefer approach; isotopic substitution leads to 4041 symmetry-unique configurations for the DH(2) mass combination. These results were then fit to a functional form that permits calculation of the BODC for any combination of isotopes. Mean unsigned fitting errors on a test grid of configurations not included in the fitting process were 0.14, 0.12, and 0.65 cm(-1) for the H(3), DH(2), and MuH(2) isotopomers, respectively. This representation can be combined with any Born-Oppenheimer potential energy surface (PES) to yield Born-Huang (BH) PESs; herein, we choose the CCI potential energy surface, the uncertainties of which ( approximately 0.01 kcal/mol) are much smaller than the magnitude of the BODC. Fortran routines to evaluate these BH surfaces are provided. Variational transition state theory calculations are presented comparing thermal rate constants for reactions on the BO and BH surfaces to provide an initial estimate of the significance of the diagonal correction for the dynamics.

Improved methods for Feynman path integral calculations of vibrational-rotational free energies and application to isotopic fractionation of hydrated chloride ions.

We present two enhancements to our methods for calculating vibrational-rotational free energies by Feynman path integrals, namely, a sequential sectioning scheme for efficiently generating random free-particle paths and a stratified sampling scheme that uses the energy of the path centroids. These improved methods are used with three interaction potentials to calculate equilibrium constants for the fractionation behavior of Cl(-) hydration in the presence of a gas-phase mixture of H(2)O, D(2)O, and HDO. Ion cyclotron resonance experiments indicate that the equilibrium constant, K(eq), for the reaction Cl(H(2)O)(-) + D(2)O right harpoon over left harpoon Cl(D(2)O)(-) + H(2)O is 0.76, whereas the three theoretical predictions are 0.946, 0.979, and 1.20. Similarly, the experimental K(eq) for the Cl(H(2)O)(-) + HDO right harpoon over left harpoon Cl(HDO)(-) + H(2)O reaction is 0.64 as compared to theoretical values of 0.972, 0.998, and 1.10. Although Cl(H(2)O)(-) has a large degree of anharmonicity, K(eq) values calculated with the harmonic oscillator rigid rotator (HORR) approximation agree with the accurate treatment to within better than 2% in all cases. Results of a variety of electronic structure calculations, including coupled cluster and multireference configuration interaction calculations, with either the HORR approximation or with anharmonicity estimated via second-order vibrational perturbation theory, all agree well with the equilibrium constants obtained from the analytical surfaces.

Bond angle distributions of carbon dioxide in the gas, supercritical, and solid phases.

Recent work has focused attention on possible shifts in the bond angle distribution of CO(2) as a consequence of intermolecular interactions in the supercritical phase. To investigate the temperature and phase dependence of the intramolecular structure of CO(2), we performed Feynman path integral Monte Carlo calculations based on a spectroscopically derived analytical potential, first principles molecular dynamics simulations using Kohn-Sham density functional theory, and Monte Carlo simulations employing empirical interaction potentials. On the basis of various distributions used to characterize the intramolecular structure, we conclude that the aggregation state has a negligible influence on the intramolecular structure, in particular we find that in the classical limit the distributions are remarkably similar for the ideal gas, supercritical, and solid phases when considered at the same temperature. In contrast, an increase in the temperature from 325 to 673 K or inclusion of nuclear quantum effects leads to a significant broadening of the distributions. With respect to the first C-O bond vector, the second bond vector most prefers a collinear arrangement. However, due to the Jacobian factor the maximum in the bond angle distribution at 325 K is shifted to an angle of about 175.7 degrees in the classical limit or to 173.0 degrees if nuclear quantum effects are included. Nevertheless, an analysis of the temperature dependence of the constant-volume heat capacity demonstrates that carbon dioxide should be viewed as a linear molecule.

Measurements of near-ultimate strength for multiwalled carbon nanotubes and irradiation-induced crosslinking improvements.

The excellent mechanical properties of carbon nanotubes are being exploited in a growing number of applications from ballistic armour to nanoelectronics. However, measurements of these properties have not achieved the values predicted by theory due to a combination of artifacts introduced during sample preparation and inadequate measurements. Here we report multiwalled carbon nanotubes with a mean fracture strength >100 GPa, which exceeds earlier observations by a factor of approximately three. These results are in excellent agreement with quantum-mechanical estimates for nanotubes containing only an occasional vacancy defect, and are approximately 80% of the values expected for defect-free tubes. This performance is made possible by omitting chemical treatments from the sample preparation process, thus avoiding the formation of defects. High-resolution imaging was used to directly determine the number of fractured shells and the chirality of the outer shell. Electron irradiation at 200 keV for 10, 100 and 1,800 s led to improvements in the maximum sustainable loads by factors of 2.4, 7.9 and 11.6 compared with non-irradiated samples of similar diameter. This effect is attributed to crosslinking between the shells. Computer simulations also illustrate the effects of various irradiation-induced crosslinking defects on load sharing between the shells.

Nanoscale fracture mechanics.

Theoretical calculations on undefected nanoscale materials predict impressive mechanical properties. In this review we summarize the status of experimental efforts to directly measure the fracture strengths of inorganic and carbon nanotubes and discuss possible explanations for the deviations between the predicted and observed values. We also summarize the role of theory in understanding the molecular-level origin of these deviations. In particular, we consider the role of defects such as vacancies, Stone-Wales defects, adatoms and ad-dimers, chemical functionalization, and oxidative pitting.

Statistical thermodynamics of bond torsional modes: tests of separable, almost-separable, and improved Pitzer-Gwinn approximations.

Practical approximation schemes for calculating partition functions of torsional modes are tested against accurate quantum mechanical results for H(2)O(2) and six isotopically substituted hydrogen peroxides. The schemes are classified on the basis of the type and amount of information that is required. First, approximate one-dimensional hindered-rotator partition functions are benchmarked against exact one-dimensional torsion results obtained by eigenvalue summation. The approximate one-dimensional methods tested in this stage include schemes that only require the equilibrium geometries and frequencies, schemes that also require the barrier heights of internal rotation, and schemes that require the whole one-dimensional torsional potential. Then, three classes of approximate full-dimensional vibrational-rotational partition functions are calculated and are compared with the accurate full-dimensional path integral partition functions. These three classes are (1) separable approximations combining harmonic oscillator-rigid rotator models with the one-dimensional torsion schemes, (2) almost-separable approximations in which the nonseparable zero-point energy is used to correct the separable approximations, and (3) improved nonseparable Pitzer-Gwinn-type methods in which approaches of type 1 are used as reference methods in the Pitzer-Gwinn approach. The effectiveness of these methods for the calculation of isotope effects is also studied. Based on the results of these studies, the best schemes of each type are recommended for further use on systems where a corresponding amount of information is available.

Benchmark calculations of the complete configuration-interaction limit of Born-Oppenheimer diagonal corrections to the saddle points of isotopomers of the H + H2 reaction.

We present a detailed ab initio study of the effect that the Born-Oppenheimer diagonal correction (BODC) has on the saddle-point properties of the H3 system and its isotopomers. Benchmark values are presented that are estimated to be within 0.1 cm(-1) of the complete configuration-interaction limit. We consider the basis set and correlation treatment requirements for accurate BODC calculations, and both are observed to be more favorable than for the Born-Oppenheimer energies. The BODC raises the H + H2 barrier height by 0.1532 kcal/mol and slightly narrows the barrier--with the imaginary frequency increasing by approximately 2%.

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.

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.

H+H2 thermal reaction: a convergence of theory and experiment.

New experimental and theoretical rate constants for two isotopologs of the simplest chemical reaction, H+H2-->H2+H, are presented. The theoretical results are obtained using accurate quantum dynamics with a converged Born-Oppenheimer potential energy surface and include non-Born-Oppenheimer corrections. The new experiments are carried out using a shock tube and complement earlier investigations over a very large T range, 167 to 2112 K. Experiment and theory now agree perfectly, within experimental error, bringing this 75-year-old scientific problem to completion.