Centroid Molecular Dynamics Method Research Articles

Quantum diffusion of H/D on Ni(111)-A partially adiabatic centroid MD study.

We present the results of a theoretical study of H/D diffusion on a Ni(111) surface at a range of temperatures, from 250 K to 75 K. The diffusion is studied using both classical molecular dynamics and the partially adiabatic centroid molecular dynamics method. The calculations are performed with the hydrogen (or deuterium) moving in 3D across a static nickel surface using a novel Fourier interpolated potential energy surface which has been parameterized to density functional theory calculations. The results of the classical simulations are that the calculated diffusion coefficients are far too small and with too large a variation with temperature compared with experiment. By contrast, the quantum simulations are in much better agreement with experiment and show that quantum effects in the diffusion of hydrogen are significant at all temperatures studied. There is also a crossover to a quantum-dominated diffusive regime for temperatures below ∼150 K for hydrogen and ∼85 K for deuterium. The quantum diffusion coefficients are found to accurately reproduce the spread in values with temperature, but with an absolute value that is a little high compared with experiment.

A molecular dynamics study of nuclear quantum effect on diffusivity of hydrogen molecule.

In this paper, the nuclear quantum effect of the hydrogen molecule on its diffusivity was analyzed using the molecular dynamics (MD) method. The centroid MD (CMD) method was applied to reproduce the time evolution of the molecules. The diffusion coefficient of hydrogen was calculated using the Green-Kubo method over a wide temperature region, and the temperature dependence of the quantum effect of the hydrogen molecule on its diffusivity was addressed. The calculated results were compared with classical MD results based on the principle of corresponding state (PCS). It was confirmed that the difference in the diffusion coefficient calculated in the CMD and classical MD methods was small, and the PCS appears to be satisfied on the temperature dependence of the diffusion coefficient, even though the quantum effect of the hydrogen molecules was taken into account. It was clarified that this result did not suggest that the quantum effect on the diffusivity of the hydrogen molecule was small but that the two changes in the intermolecular interaction of hydrogen due to the quantum effect offset each other. Moreover, it was found that this tendency was related to the temperature dependence of the ratio of the kinetic energy of the quantum fluctuational motion to the classical kinetic energy.

Formulation of state projected centroid molecular dynamics: Microcanonical ensemble and connection to the Wigner distribution.

A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.

Velocity autocorrelation in liquid parahydrogen by quantum simulations for direct parameter-free computations of neutron cross sections

Accurate knowledge of the single-molecule (self) translational dynamics of liquid para-H2 is an essential requirement for the calculation of the neutron scattering properties of this important quantum liquid. We show that, by using Centroid Molecular Dynamics (CMD) quantum simulations of the velocity autocorrelation function, calculations of the total neutron cross section (TCS) remarkably agree with experimental data at the thermal and epithermal incident neutron energies where para-H2 dynamics is actually dominated by the self contributions. This result shows that a proper account of the quantum nature of the fluid, as provided by CMD, is a necessary and very effective condition to obtain the correct absolute-scale cross section values directly from first-principle computations of the double differential cross section, and without the need of introducing any empirically adjusted quantity. At subthermal incident energies, appropriate modeling of the para-H2 intermolecular (distinct) dynamics also becomes crucial, but quantum simulations are not yet able to cope with it. Existing simple models which account for the distinct part provide an appropriate correction of self-only calculations and bring the computed results in reasonable accord with TCS experimental data available until very recently. However, if just published cross section measurements in the cold range are considered, the agreement turns out to be by far superior and very satisfactory. The possible origin of slight residual differences will be commented and suggest further computational and experimental efforts. Nonetheless, the ability to reproduce the total cross section in the wide range between 1 and 900 meV represents an encouraging and important validation step of the CMD method and of the present simple algorithm.

Can the ring polymer molecular dynamics method be interpreted as real time quantum dynamics?

The ring polymer molecular dynamics (RPMD) method has gained popularity in recent years as a simple approximation for calculating real time quantum correlation functions in condensed media. However, the extent to which RPMD captures real dynamical quantum effects and why it fails under certain situations have not been clearly understood. Addressing this issue has been difficult in the absence of a genuine justification for the RPMD algorithm starting from the quantum Liouville equation. To this end, a new and exact path integral formalism for the calculation of real time quantum correlation functions is presented in this work, which can serve as a rigorous foundation for the analysis of the RPMD method as well as providing an alternative derivation of the well established centroid molecular dynamics method. The new formalism utilizes the cyclic symmetry of the imaginary time path integral in the most general sense and enables the expression of Kubo-transformed quantum time correlation functions as that of physical observables pre-averaged over the imaginary time path. Upon filtering with a centroid constraint function, the formulation results in the centroid dynamics formalism. Upon filtering with the position representation of the imaginary time path integral, we obtain an exact quantum dynamics formalism involving the same variables as the RPMD method. The analysis of the RPMD approximation based on this approach clarifies that an explicit quantum dynamical justification does not exist for the use of the ring polymer harmonic potential term (imaginary time kinetic energy) as implemented in the RPMD method. It is analyzed why this can cause substantial errors in nonlinear correlation functions of harmonic oscillators. Such errors can be significant for general correlation functions of anharmonic systems. We also demonstrate that the short time accuracy of the exact path integral limit of RPMD is of lower order than those for finite discretization of path. The present quantum dynamics formulation also serves as the basis for developing new quantum dynamical methods that utilize the cyclic nature of the imaginary time path integral.

Quantum–Thermal Crossover of Hydrogen and Tritium Diffusion in α-Iron

The diffusion coefficients of hydrogen (H) and tritium (T) in α-Fe have been computed using two approximate quantum dynamical techniques, that is, centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD), in the temperature range of T = 100–1000 K using the embedded atom method (EAM) potential. It has been found that the RPMD and CMD methods give very similar results. From a further analysis based on quantum transition-state theory (centroid density QTST) combined with path integral molecular dynamics (PIMD), it has been clear that there is a crossover between thermal and quantum mechanisms at about T = 500 and 300 K for H and T diffusions, respectively. The importance of nuclear quantum effects at low temperatures has been illustrated in terms of the effective free-energy surface map.

Centroid molecular dynamics: Comparison with exact results for model systems

The relation between the accuracy of centroid molecular dynamics correlation functions, and the geometry of the centroid potential is investigated. It is shown that, depending on the temperature, there exist several regimes, and in each of them certain features of the exact Kubo correlation functions are reproduced. The change of regimes is related to the emergence of barriers in the centroid potential. In order to clarify how the above described picture of regimes is modified in real systems when dissipation is important, a methodology is developed to test the accuracy of centroid correlation functions for the model of a particle coupled to a harmonic heat bath. A modification of the centroid molecular dynamics method to include the influence of the heat bath is introduced. Preliminary results of comparison of centroid molecular dynamics with the numerically exact results of filtered propagator functional method are presented.

Finite temperature infrared spectroscopy of polycyclic aromatic hydrocarbon molecules: Path-integral molecular dynamics

The vibrational spectra of the naphthalene, pyrene, and coronene molecules have been computed in the 0-3500 cm(-1) infrared range using classical and quantum molecular dynamics simulations based on a dedicated tight-binding potential energy surface. The ring-polymer molecular dynamics (RPMD) and partially adiabatic centroid molecular dynamics (CMD) methods have been employed to account for quantum nuclear effects. The contributions of quantum delocalization to the line shift and broadening are significant in the entire spectral range and of comparable magnitude as pure thermal effects. While the two methods generally produce similar results, the CMD method may converge slower at low temperature with increasing Trotter discretization number. However, and contrary to the CMD method, the RPMD approach suffers from serious resonance problems at high frequencies and low temperatures.

Communications: On artificial frequency shifts in infrared spectra obtained from centroid molecular dynamics: Quantum liquid water

Centroid molecular dynamics (CMD) is a popular method to extract approximate quantum dynamics from path integral simulations. Very recently we have shown that CMD gas phase infrared spectra exhibit significant artificial redshifts of stretching peaks, due to the so-called "curvature problem" imprinted by the effective centroid potential. Here we provide evidence that for condensed phases, and in particular for liquid water, CMD produces pronounced artificial redshifts for high-frequency vibrations such as the OH stretching band. This peculiar behavior intrinsic to the CMD method explains part of the unexpectedly large quantum redshifts of the stretching band of liquid water compared to classical frequencies, which is improved after applying a simple and rough "harmonic curvature correction."

Quantum effects in liquid water from an ab initio-based polarizable force field

The importance of quantum effects as well as the accuracy of the ab initio-based polarizable TTM2.1-F force field in describing liquid water are quantitatively assessed by a detailed analysis of the temperature dependence of several thermodynamic and dynamical properties computed using the path-integral molecular dynamics and centroid molecular dynamics methods. The results show that quantum effects are not negligible even at relatively high temperatures, and their inclusion in simulations with the TTM2.1-F water model is necessary to achieve a more accurate description of the liquid properties. Comparison with the results reported in the literature for empirical, nonpolarizable force fields demonstrates that the effects of the nuclear quantization on the dielectric constant are dependent in part on how the electronic polarization is described in the underlying water model, while comparison with other ab initio-based force fields shows that the TTM2.1-F model provides an overall accurate description of liquid water. Analysis of the isotope effect on the dynamical properties does not display significant temperature dependence. This suggests that the contribution of quantum tunneling, which has been proposed as a possible cause for the different orientational dynamics observed for the HDO:H(2)O and HDO:D(2)O systems, appears to be small.

An accurate and simple quantum model for liquid water

The path-integral molecular dynamics and centroid molecular dynamics methods have been applied to investigate the behavior of liquid water at ambient conditions starting from a recently developed simple point charge/flexible (SPC/Fw) model. Several quantum structural, thermodynamic, and dynamical properties have been computed and compared to the corresponding classical values, as well as to the available experimental data. The path-integral molecular dynamics simulations show that the inclusion of quantum effects results in a less structured liquid with a reduced amount of hydrogen bonding in comparison to its classical analog. The nuclear quantization also leads to a smaller dielectric constant and a larger diffusion coefficient relative to the corresponding classical values. Collective and single molecule time correlation functions show a faster decay than their classical counterparts. Good agreement with the experimental measurements in the low-frequency region is obtained for the quantum infrared spectrum, which also shows a higher intensity and a redshift relative to its classical analog. A modification of the original parametrization of the SPC/Fw model is suggested and tested in order to construct an accurate quantum model, called q-SPC/Fw, for liquid water. The quantum results for several thermodynamic and dynamical properties computed with the new model are shown to be in a significantly better agreement with the experimental data. Finally, a force-matching approach was applied to the q-SPC/Fw model to derive an effective quantum force field for liquid water in which the effects due to the nuclear quantization are explicitly distinguished from those due to the underlying molecular interactions. Thermodynamic and dynamical properties computed using standard classical simulations with this effective quantum potential are found in excellent agreement with those obtained from significantly more computationally demanding full centroid molecular dynamics simulations. The present results suggest that the inclusion of nuclear quantum effects into an empirical model for water enhances the ability of such model to faithfully represent experimental data, presumably through an increased ability of the model itself to capture realistic physical effects.

Evaluation of quantum correlation functions from classical data: Anharmonic models

The previously introduced method of evaluating quantum mechanical time correlation functions using as input only classical simulation data is generalized and applied to two anharmonic model systems, as a further test. The quantum correction approach utilizes the relation between a general quantum correlation function and its classical analog. For the tested models, we obtain numerical results of nonlinear correlation functions with comparable accuracy to that of the centroid molecular dynamics method, although the present method is much simpler to implement and not limited to real valued quantum correlation functions.

Evaluation of Nonlinear Quantum Time Correlation Functions within the Centroid Dynamics Formulation

A method to evaluate nonlinear centroid correlation functions is presented that is amenable to simple numerical computation. It can be implemented with the centroid molecular dynamics method for approximate quantum dynamics with no additional assumptions. Two nonlinear correlation functions are evaluated for a model potential using this scheme and compared with results from exact quantum calculations.

A comparative study of imaginary time path integral based methods for quantum dynamics

The recently introduced approximate many-body quantum simulation method, ring polymer molecular dynamics (RPMD), is compared to the centroid molecular dynamics method (CMD). Comparisons of simulation results for liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point are presented. The calculated quantum correlation functions for the two methods are shown to be in good agreement with one another for a large portion of the time spectrum. However, as the quantum mechanical nature of the system increases, RPMD is less accurate in predicting the kinetic energy of the system than is CMD. A simplified and highly efficient algorithm is proposed which largely corrects this deficiency.

Path-integral centroid dynamics for general initial conditions: A nonequilibrium projection operator formulation

The formulation of path-integral centroid dynamics is extended to the quantum dynamics of density operators evolving from general initial states by means of the nonequilibrium projection operator technique. It is shown that the new formulation provides a basis for applying the method of centroid dynamics to nonequilibrium situations and that it allows the derivation of new formal relations, which can be useful in improving current equilibrium centroid dynamics methods. A simple approximation of uniform relaxation for the unprojected portion of the Liouville space propagator leads to a class of practically solvable equations of motion for the centroid variables, but with an undetermined parameter of relaxation. This new class of equations encompasses the centroid molecular-dynamics (CMD) method as a limiting case, and can be applied to both equilibrium and nonequilibrium situations. Tests for the equilibrium dynamics of one-dimensional model systems demonstrate that the new equations with appropriate choice of the relaxation parameter are comparable to the CMD method.

Quantum effects in ice Ih

Quantum and classical simulations are carried out on ice Ih over a range of temperatures utilizing the TIP4P water model. The rigid-body centroid molecular dynamics method employed allows for the investigation of equilibrium and dynamical properties of the quantum system. The impact of quantization on the local structure, as measured by the radial and spatial distribution functions, as well as the energy is presented. The effects of quantization on the lattice vibrations, associated with the molecular translations and librations, are also reported. Comparison of quantum and classical simulation results indicates that shifts in the average potential energy are equivalent to rising the temperature about 80 K and are therefore non-negligible. The energy shifts due to quantization and the quantum mechanical uncertainties observed in ice are smaller than the values previously reported for liquid water. Additionally, we carry out a comparative study of melting in our classical and quantum simulations and show that there are significant differences between classical and quantum ice.

Chemical reaction rates from ring polymer molecular dynamics

We show how the ring-polymer molecular dynamics method can be adapted to calculate approximate Kubo-transformed flux-side correlation functions, and hence rate coefficients for condensed phase reactions. An application of the method to the standard model for a chemical reaction in solution--a quartic double-well potential linearly coupled to a bath of harmonic oscillators--is found to give results of comparable accuracy to those of the classical Wigner model and the centroid molecular dynamics method. However, since the present method does not require that one evaluate the Wigner transform of a thermal flux operator or that one perform a separate path integral calculation for each molecular dynamics time step, we believe it will prove easier to apply to more general problems than either of these alternative techniques. We also present a (logarithmic) discretization scheme for the Ohmic bath in the system-bath model that gives converged results with just nine bath modes--a surprisingly small number for a model of a condensed phase reaction. Finally, we present some calculations of the transmission through an Eckart barrier which show that the present method provides a satisfactory (although not perfect) description of the deep quantum tunneling regime. Part of the reason for the success of the method is that it gives the exact quantum-mechanical rate constant for the transmission through a parabolic barrier, as we demonstrate analytically in the Appendix.

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.

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.

Theoretical simulations on photoexcitation dynamics of the silver atom embedded in helium clusters

Quantum molecular dynamics simulations have been performed to study the photoexcitation Ag(5p 2PJ)←Ag(5s 2S1/2) in size-selected helium clusters considering electronically nonadiabatic transitions. We employed the hybrid method in which the electronic degree of freedom of Ag(2P) was treated quantum mechanically while the motions of helium atoms were described by the semiclassical path integral centroid molecular dynamics method in order to take the quantum fluctuation effect into account. It has been found that the dynamics after photoexcitation is dominantly nonadiabatic in all cluster sizes studied, and that nonadiabatic transitions are enhanced by quantum fluctuation of helium motions. Most of the photoexcited AgHen clusters decompose into an isolated Ag atom and free helium atoms within several picoseconds. However, Ag*Hen (n=1–5) exciplex formation was also found to occur for all helium cluster sizes studied, although this process is found to be minor.