Comparison of approximate quantum simulation methods applied to normal liquid helium at 4 K

The Feynman-Kleinert linearized path integral molecular dynamics (FK-LPI), ring polymer molecular dynamics (RPMD), and centroid molecular dynamics (CMD) methods are applied to the simulation of normal liquid helium. Comparisons of the simulation results at the T = 4 K and rho = 0.01873 A-3 state point are presented. The calculated quantum correlation functions for the three methods show significant differences, both in the short time and in the intermediate regions of the spectrum. Our simulation results are also compared to the recent results of other approximate quantum simulation methods. We find that FK-LPI qualitatively agrees with other approximate quantum simulation results while CMD and RPMD predict a qualitatively different impulsive rebound in the velocity autocorrelation function. Frequency space analysis reveals that RPMD exhibits a broad high-frequency tail similar to that from quantum mode coupling theory and numerical analytic continuation approaches, while FK-LPI provides a somewhat more rapid decay at high frequency than any of these three methods. CMD manifests a high-frequency component that is greatly reduced compared with the other methods.     

New conditions for validity of the centroid molecular dynamics and ring polymer molecular dynamics

Validity of the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) in quantum liquids is studied on an assumption that momenta of liquid particles relax fast. The projection operator method allows one to derive the generalized Langevin equation including a memory effect for the full-quantum canonical (Kubo-transformed) correlation function. Similar equations for the CMD and RPMD correlation functions can be derived too. The comparison of these equations leads to conditions under which the RPMD and CMD correlation functions agree approximately with the full-quantum canonical correlation function. The condition for the RPMD is that the memory effects of the full-quantum and RPMD equations vanish quickly with the same time constants. The CMD correlation function requires additional conditions concerning static correlation.     

On the short-time limit of ring polymer molecular dynamics

We examine the short-time accuracy of a class of approximate quantum dynamical techniques that includes the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) methods. Both of these methods are based on the path integral molecular dynamics (PIMD) technique for calculating the exact static equilibrium properties of quantum mechanical systems. For Kubo-transformed real-time correlation functions involving operators that are linear functions of positions or momenta, the RPMD and (adiabatic) CMD approximations differ only in the choice of the artificial mass matrix of the system of ring polymer beads that is employed in PIMD. The obvious ansatz for a general method of this type is therefore to regard the elements of the PIMD (or Parrinello-Rahman) mass matrix as an adjustable set of parameters that can be chosen to improve the accuracy of the resulting approximation. We show here that this ansatz leads uniquely to the RPMD approximation when the criterion that is used to select the mass matrix is the short-time accuracy of the Kubo-transformed correlation function. In particular, we show that the leading error in the RPMD position autocorrelation function is O(t(8)) and the error in the velocity autocorrelation function is O(t(6)), for a general anharmonic potential. The corresponding errors in the CMD approximation are O(t(6)) and O(t(4)), respectively.     

A centroid molecular dynamics study of liquid para-hydrogen and ortho-deuterium

Centroid molecular dynamics (CMD) is applied to the study of collective and single-particle dynamics in liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point. The CMD results are compared with the results of classical molecular dynamics, quantum mode coupling theory, a maximum entropy analytic continuation approach, pair-product forward- backward semiclassical dynamics, and available experimental results. The self-diffusion constants are in excellent agreement with the experimental measurements for all systems studied. Furthermore, it is shown that the method is able to adequately describe both the single-particle and collective dynamics of quantum liquids.     

Fast centroid molecular dynamics: a force-matching approach for the predetermination of the effective centroid forces

A fast centroid molecular dynamics (CMD) methodology is proposed in which the effective centroid forces are predetermined through a force-matching algorithm applied to a standard path integral molecular dynamics simulation. The resulting method greatly reduces the computational cost of generating centroid trajectories, thus extending the applicability of CMD. The method is applied to the study of liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point. The static and dynamical results are compared to those obtained from full adiabatic CMD simulations and found to be in excellent agreement for all three systems; the transport properties are also compared to experiment and found to have a similar level of agreement.     

Rate Coefficients of Roaming Reaction H+MgH Using Ring Polymer Molecular Dynamics

The ring-polymer molecular dynamics (RPMD) was used to calculate the thermal rate coefficients of the multi-channel roaming reaction H+MgH→Mg+H₂. Two reaction channels, tight and roaming, are explicitly considered. This is a pioneering attempt of exerting RPMD method to multi-channel reactions. With the help of a newly developed optimization-interpolation protocol for preparing the initial structures and adaptive protocol for choosing the force constants, we have successfully obtained the thermal rate coefficients. The results are consistent with those from other theoretical methods, such as variational transition state theory and quantum dynamics. Especially, RPMD results exhibit negative temperature dependence, which is similar to the results from variational transition state theory but different from the ones from ground state quantum dynamics calculations.     

Quantum mechanical correlation functions, maximum entropy analytic continuation, and ring polymer molecular dynamics

The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time betavariant Planck's over 2pi tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where betavariant Planck's over 2pi exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the "nonlinear operator problem" of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.     

Ring-Polymer Molecular Dynamics Studies of Thermal Rate Coefficients for Reaction F+H2O→HF+OH

The prototype tetra-atomic reaction F+H₂O→HF+OH plays a significant role in both atmospheric and astronomical chemistry. In this work, thermal rate coefficients of this reaction are determined with the ring polymer molecular dynamics (RPMD) method on a full-dimensional potential energy surface (PES). This PES is the most accurate one for the title reaction, as demonstrated by the correct barrier height and reaction energy, compared to the benchmark calculations by the focal point analysis and the high accuracy extrapolated ab initio thermochemistry methods. The RPMD rate coefficients are in excellent agreement with those calculated by the semiclassical transition state theory and a two-dimensional master equation technique, and some experimental measurements. As has been found in many RPMD applications, quantum effects, including tunneling and zeropoint energy effects, can be efficiently and effectively captured by the RPMD method. In addition, the convergence of the results with respect to the number of beads is rapid, which is also consistent with previous RPMD applications.     

Thermal Gaussian molecular dynamics for quantum dynamics simulations of many-body systems: application to liquid para-hydrogen

A new method, here called thermal Gaussian molecular dynamics (TGMD), for simulating the dynamics of quantum many-body systems has recently been introduced [I. Georgescu and V. A. Mandelshtam, Phys. Rev. B 82, 094305 (2010)]. As in the centroid molecular dynamics (CMD), in TGMD the N-body quantum system is mapped to an N-body classical system. The associated both effective Hamiltonian and effective force are computed within the variational Gaussian wave-packet approximation. The TGMD is exact for the high-temperature limit, accurate for short times, and preserves the quantum canonical distribution. For a harmonic potential and any form of operator A?, it provides exact time correlation functions C(AB)(t) at least for the case of B, a linear combination of the position, x, and momentum, p, operators. While conceptually similar to CMD and other quantum molecular dynamics approaches, the great advantage of TGMD is its computational efficiency. We introduce the many-body implementation and demonstrate it on the benchmark problem of calculating the velocity time auto-correlation function for liquid para-hydrogen, using a system of up to N = 2592 particles.     

Bimolecular reaction rates from ring polymer molecular dynamics: application to H + CH4 → H2 + CH3

In a recent paper, we have developed an efficient implementation of the ring polymer molecular dynamics (RPMD) method for calculating bimolecular chemical reaction rates in the gas phase, and illustrated it with applications to some benchmark atom-diatom reactions. In this paper, we show that the same methodology can readily be used to treat more complex polyatomic reactions in their full dimensionality, such as the hydrogen abstraction reaction from methane, H + CH(4) → H(2) + CH(3). The present calculations were carried out using a modified and recalibrated version of the Jordan-Gilbert potential energy surface. The thermal rate coefficients obtained between 200 and 2000 K are presented and compared with previous results for the same potential energy surface. Throughout the temperature range that is available for comparison, the RPMD approximation gives better agreement with accurate quantum mechanical (multiconfigurational time-dependent Hartree) calculations than do either the centroid density version of quantum transition state theory (QTST) or the quantum instanton (QI) model. The RPMD rate coefficients are within a factor of 2 of the exact quantum mechanical rate coefficients at temperatures in the deep tunneling regime. These results indicate that our previous assessment of the accuracy of the RPMD approximation for atom-diatom reactions remains valid for more complex polyatomic reactions. They also suggest that the sensitivity of the QTST and QI rate coefficients to the choice of the transition state dividing surface becomes more of an issue as the dimensionality of the reaction increases.     

Ab initio centroid path integral molecular dynamics: application to vibrational dynamics of diatomic molecular systems

An ab initio centroid molecular dynamics (CMD) method is developed by combining the CMD method with the ab initio molecular orbital method. The ab initio CMD method is applied to vibrational dynamics of diatomic molecules, H2 and HF. For the H2 molecule, the temperature dependence of the peak frequency of the vibrational spectral density is investigated. The results are compared with those obtained by the ab initio classical molecular dynamics method and exact quantum mechanical treatment. It is shown that the vibrational frequency obtained from the ab initio CMD approaches the exact first excitation frequency as the temperature lowers. For the HF molecule, the position autocorrelation function is also analyzed in detail. The present CMD method is shown to well reproduce the exact quantum result for the information on the vibrational properties of the system.     

Rate Coefficients and Kinetic Isotope Effects of Cl+XCl→XCl+Cl (X=H,D, Mu) Reactions from Ring Polymer Molecular Dynamics

The ring-polymer molecular dynamics (RPMD) was used to calculate the thermal rate coefficients and kinetic isotope effects of the heavy-light-heavy abstract reaction Cl+XCl\begin{document}$ \rightarrow $\end{document}XCl+Cl (X = H, D, Mu). For the Cl+HCl reaction, the excellent agreement between the RPMD and experimental values provides a strong proof for the accuracy of the RPMD theory. And the RPMD results are also consistent with results from other theoretical methods including improved-canonical-variational-theory and quantum dynamics. The most novel finding is that there is a double peak in Cl+MuCl reaction near the transition state, leaving a free energy well. It comes from the mode softening of the reaction system at the peak of the potential energy surface. Such an explicit free energy well suggests strongly there is an observable resonance. And for the Cl+DCl reaction, the RPMD rate coefficient again gives very accurate results compared with experimental values. The only exception is at the temperature of 312.5 K, results from RPMD and all other theoretical methods are close to each other but slightly lower than the experimental value, which indicates experimental or potential energy surface deficiency.     

A refined ring polymer molecular dynamics theory of chemical reaction rates

We further develop the ring polymer molecular dynamics (RPMD) method for calculating chemical reaction rates [I. R. Craig and D. E. Manolopoulos, J. Chem. Phys. 122, 084106 (2005)]. We begin by showing how the rate coefficient we obtained before can be calculated in a more efficient way by considering the side functions of the ring-polymer centroids, rather than averaging over the side functions of the individual ring-polymer beads. This has two distinct advantages. First, the statistics of the phase-space average over the ring-polymer coordinates and momenta are greatly improved. Second, the resulting flux-side correlation function converges to its long-time limit much more rapidly. Indeed the short-time limit of this flux-side correlation function already provides a "quantum transition state theory" approximation to the final rate coefficient. In cases where transition state recrossing effects are negligible, and the transition state dividing surface is put in the right place, the RPMD rate is therefore obtained almost instantly. We then go on to show that the long-time limit of the new flux-side correlation function, and hence the fully converged RPMD reaction rate, is rigorously independent of the choice of the transition state dividing surface. This is especially significant because the optimum dividing surface can often be very difficult to determine for reactions in complex chemical systems.     

Centroid molecular dynamics approach to the transport properties of liquid para-hydrogen over the wide temperature range

Fundamental transport properties of liquid para-hydrogen (p-H(2)), i.e., diffusion coefficients, thermal conductivity, shear viscosity, and bulk viscosity, have been evaluated by means of the path integral centroid molecular dynamics (CMD) calculations. These transport properties have been obtained over the wide temperature range, 14-32 K. Calculated values of the diffusion coefficients and the shear viscosity are in good agreement with the experimental values at all the investigated temperatures. Although a relatively large deviation is found for the thermal conductivity, the calculated values are less than three times the amount of the experimental values at any temperature. On the other hand, the classical molecular dynamics has led all the transport properties to much larger deviation. For the bulk viscosity of liquid p-H(2), which was never known from experiments, the present CMD has given a clear temperature dependence. In addition, from the comparison based on the principle of corresponding states, it has been shown that the marked deviation of the transport properties of liquid p-H(2) from the feature which is expected from the molecular parameters is due to the quantum effect.     

Formation and evolution of metastable bcc phase during solidification of liquid Ag: a molecular dynamics simulation study

On the basis of the quantum Sutton-Chen potential, the rapid solidification processes of liquid silver have been studied by molecular dynamics simulation for four cooling rates. By means of several analysis methods, the competitions and transitions between microstructures during the cooling processes have been analyzed intensively. It is found that there are two phase transitions in all simulation processes. The first one is from liquid state to metastable (transitional) body-centered cubic (bcc) phase. The initial crystallization temperature T(ic) increases with the decrease of the cooling rate. The second one is from the transitional bcc phase to the final solid phase. This study validates the Ostwald's step rule and provides evidence for the prediction that the metastable bcc phase forms first from liquid. Further analyses reveal that the final solid at 273 K can be a mixture of hexagonal close-packed (hcp) and face-centered cubic (fcc) microstructures with various proportions of the two, and the slower the cooling rate is, the higher proportion the fcc structure occupies.     

Quantum effects in light and heavy liquid water: A rigid-body centroid molecular dynamics study

The centroid molecular dynamics (CMD) method is applied to the study of liquid water in the context of the rigid-body approximation. This rigid-body CMD technique, which is significantly more efficient than the standard CMD method, is implemented on the TIP4P model for water and used to examine isotopic effects in the equilibrium and dynamical properties of liquid H(2)O and D(2)O. The results obtained with this approach compare remarkably well with those determined previously with path integrals simulations as well as those obtained from the standard CMD method employing flexible models. In addition, an examination of the impact of quantization on the rotational and librational motion of the water molecule is also reported.     

Inclusion of inversion symmetry in centroid molecular dynamics: a possible avenue to recover quantum coherence

Inversion symmetry is included in the operator formulation of the centroid molecular dynamics (CMD). This work involves the development of a symmetry-adapted CMD (SA-CMD), here particularly for symmetrization and antisymmetrization projections. A symmetry-adapted quasidensity operator, as defined by Blinov and Roy [J. Chem. Phys. 115, 7822 (2001)], is employed to obtain the centroid representation of quantum mechanical operators. Numerical examples are given for a single particle confined to one-dimensional symmetric quartic and symmetric double-well potentials. Two SA-CMD simulations are performed separately for both projections, and centroid position autocorrelation functions are obtained. For each projection, the quality of the approximation as well as the accuracy are similar to those of regular CMD. It is shown that individual trajectories from two separate SA-CMD simulations can be properly combined to recover trajectories for Boltzmann statistics. Position autocorrelation functions are compared to the exact quantum mechanical ones. This explicit account of inversion symmetry provides a qualitative improvement on the conventional CMD approach and allows the recovery of some quantum coherence.     

Quantum diffusion in liquid para-hydrogen from ring-polymer molecular dynamics

We have used the ring-polymer molecular dynamics method to calculate approximate Kubo-transformed velocity autocorrelation functions and self-diffusion coefficients for low-pressure liquid para-hydrogen at temperatures of 25 and 14 K. The resulting diffusion coefficients are shown to be consistent with experimental shear viscosities and the established finite-size relation D(L) approximately = D(infinity)-2.837k(B)T6pietaL, where k(B) is the Boltzmann constant, T the absolute temperature, eta the shear viscosity, and L the length of the (cubic) simulation cell. The diffusion coefficients D(L) obtained in simulations with finite system sizes are therefore too small. However, the extrapolation to infinite system size corrects this deficiency and leads to excellent agreement with experimental results. This both demonstrates the influence of system-size effects on quantum mechanical diffusion coefficients and provides further evidence that ring-polymer molecular dynamics is an accurate as well as practical way of including quantum effects in condensed phase molecular dynamics.     

Calculation of heat capacities of light and heavy water by path-integral molecular dynamics

As an application of atomistic simulation methods to heat capacities, path-integral molecular dynamics has been used to calculate the constant-volume heat capacities of light and heavy water in the gas, liquid, and solid phases. While the classical simulation based on conventional molecular dynamics has estimated the heat capacities too high, the quantum simulation based on path-integral molecular dynamics has given reasonable results based on the simple point-charge/flexible potential model. The calculated heat capacities (divided by the Boltzmann constant) in the quantum simulation are 3.1 in the vapor H2O at 300 K, 6.9 in the liquid H2O at 300 K, and 4.1 in the ice Ih H2O at 250 K, respectively, which are comparable to the experimental data of 3.04, 8.9, and 4.1, respectively. The quantum simulation also reproduces the isotope effect. The heat capacity in the liquid D2O has been calculated to be 10% higher than that of H2O, while it is 13% higher in the experiment. The results demonstrate that the path-integral simulation is a promising approach to quantitatively evaluate the heat capacities for molecular systems, taking account of quantum-mechanical vibrations as well as strongly anharmonic motions.