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.     

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.     

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.     

Orientational dynamics in the isotropic phase of a calamitic liquid-crystal model

We report a molecular dynamics simulation study on the isotropic phase of an idealized calamitic liquid crystal model with a length-to-width ratio of approximately 5-6. The study focuses on the characterization of single-particle and collective orientational dynamics on approaching the phase transition to the nematic phase. Recent experimental and simulation works have suggested that a power law behavior exists at relatively short times in the decay of the time derivative of the orientational correlation functions. Qualitatively, our simulation data are consistent with these findings. Both single-particle and collective time correlation function derivatives possess, in their respective log-log plots, a linear region at very short times, whose slope is essentially independent from the thermodynamic state. Nevertheless, the single-particle orientational correlation functions are better described by a function which is the sum of a fast exponential, an intermediate stretched-exponential and a slow exponential, while the collective orientational correlation functions are satisfactorily described by a sum of two exponentials, at higher density, or by just one exponential, at lower density.     

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.     

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.     

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.     

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.     

On the calculation of single-particle time correlation functions from Bose-Einstein centroid dynamics

The calculation of single-particle time correlation functions using the Bose-Einstein centroid dynamics formalism is discussed. A new definition of the quasidensity operator is used to calculate the centroid force on a given particle for an anharmonic system. The force includes correlation effects due to quantum statistics and is used for the calculation of the classical-like dynamics of phase-space centroid variables within the centroid molecular dynamics approximation. Time correlation functions are then obtained for single-particle quantities. These correspond to the double-Kubo transform of exact quantum-mechanical correlation functions. The centroid dynamics results are compared to those of exact basis-set calculations and a good agreement is found. The level of accuracy is in fact the same as what was observed earlier for the calculation of center-of-mass correlation functions for Fermi-Dirac and Bose-Einstein statistics, and for any correlation function for Boltzmann statistics. These results show that it is now possible to use Bose-Einstein centroid molecular dynamics to calculate single-particle correlation functions for systems where quantum exchange effects are present.     

Molecular dynamics simulation of liquid sulfur dioxide

A previously proposed model for molecular dynamics (MD) simulation of liquid sulfur dioxide, SO(2), has been reviewed. Thermodynamic, structural, and dynamical properties were calculated for a large range of thermodynamic states. Predicted (P,V,T) of simulated system agrees with an elaborated equation of state recently proposed for liquid SO(2). Calculated heat capacity, expansion coefficient, and isothermal compressibility are also in good agreement with experimental data. Calculated equilibrium structure agrees with X-ray and neutron scattering measurements on liquid SO(2). The model also predicts the same (SO(2))(2) dimer structure as previously determined by ab initio calculations. Detailed analysis of equilibrium structure of liquid SO(2) is provided, indicating that, despite the rather large dipole moment of the SO(2) molecule, the structure is mainly determined by the Lennard-Jones interactions. Both single-particle and collective dynamics are investigated. Temperature dependency of dynamical properties is given. The MD results are compared with previous findings obtained from the analysis of inelastic neutron scattering spectra of liquid SO(2), including wave-vector dependent structural relaxation, tau(k), and viscosity, eta(k).     

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.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.     

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.     

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.     

Electronic excitations in narrow quantum wells via intersubband Raman scattering: Theoretical considerations

In this work a generalized self-consistent field theory was applied to investigate the elementary excitations of two-dimensional electron gas formed from narrow quantum wells via resonant intersubband Raman scattering. The developed model considers the existence of equally coupled and degenerated excitations of the electron gas and allows to observe that in extreme resonance regime the plasma oscillations splits into two contributions: a set of renormalized collective excitations (plasmons) and unrenormalized electronic transitions (single-particle excitations). Our results show that the asymmetries which appear in the Raman profile of doped narrow quantum wells can be interpreted as the entrance or exit of resonance of collective modes overlapped with single-particle transitions.     

Molecular dynamics simulation of liquid trimethylphosphine

Structural and dynamical properties of liquid trimethylphosphine (TMP), (CH(3))(3)P, as a function of temperature is investigated by molecular dynamics (MD) simulations. The force field used in the MD simulations, which has been proposed from molecular mechanics and quantum chemistry calculations, is able to reproduce the experimental density of liquid TMP at room temperature. Equilibrium structure is investigated by the usual radial distribution function, g(r), and also in the reciprocal space by the static structure factor, S(k). On the basis of center of mass distances, liquid TMP behaves like a simple liquid of almost spherical particles, but orientational correlation due to dipole-dipole interactions is revealed at short-range distances. Single particle and collective dynamics are investigated by several time correlation functions. At high temperatures, diffusion and reorientation occur at the same time range as relaxation of the liquid structure. Decoupling of these dynamic properties starts below ca. 220 K, when rattling dynamics of a given TMP molecules due to the cage effect of neighbouring molecules becomes important.     

A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces

The multiconfigurational time-dependent Hartree (MCTDH) approach facilitates multidimensional quantum dynamics calculations by representing the wavepacket in an optimal set of time-dependent basis functions, called single-particle functions. Choosing these single-particle functions to be themselves multidimensional wavefunctions which are represented using a MCTDH representation, a multilayer MCTDH scheme has been constructed and used for quantum dynamics calculations treating up to 1000 degrees of freedom rigorously [Wang and Thoss, J. Chem. Phys. 199, 1289 (2003)]. The present work gives a practical scheme which facilitates the application of the multilayer MCTDH approach, which previously has only been employed to study systems described by model-type Hamiltonians, to molecular systems described by more complicated Hamiltonians and general potential energy surfaces. A multilayer extension of the correlation discrete variable representation (CDVR) scheme employed in MCTDH calculations studying quantum dynamics on general potential energy surfaces is developed and tested in a simple numerical application. The resulting multilayer MCTDH/CDVR approach might offer a perspective to rigorously describe the quantum dynamics of larger polyatomic systems.     

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.     

Quantum effects in liquid water and ice: model dependence

This paper explores the influence of choice of potential model on the quantum effects observed in liquid water and ice. This study utilizes standard rigid models and a more formal context for the rigid-body centroid molecular dynamics methodology used to perform the quantum simulations is provided. Quantum and classical molecular dynamics simulations are carried out for liquid water and ice Ih at 298 and 220 K, respectively, with the simple point charge/extended and TIP4P-Ew water models. The results obtained for equilibrium and dynamical properties are compared with those recently reported on TIP4P [L. Hernandez de la Pena and P. G. Kusalik, J. Chem. Phys. 121, 5992 (2004); L. Hernandez de la Pena et al., J. Chem. Phys 123, 144506 (2005)]. For the liquid, an energy shift of about 8% and an average molecular uncertainty of about 11 degrees were found independently of the water model. The self-diffusion coefficient consistently increases by more than 50% when going from the classical to the quantum system and quantum dynamics are found to reproduce the experimental isotopic shifts with the models examined. The ice results compare remarkably well with those previously reported for the TIP4P water model; they confirm that quantum effects are considerable and that the quantum mechanical uncertainty and the energy shifts due to quantization are smaller in ice than in liquid water. The relevance of these findings in the context of the construction of water models is briefly discussed.