Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.     

Linearized semiclassical initial value time correlation functions using the thermal Gaussian approximation: applications to condensed phase systems

The linearized approximation to the semiclassical initial value representation (LSC-IVR) has been used together with the thermal Gaussian approximation (TGA) (TGA/LSC-IVR) [J. Liu and W. H. Miller, J. Chem. Phys. 125, 224104 (2006)] to simulate quantum dynamical effects in realistic models of two condensed phase systems. This represents the first study of dynamical properties of the Ne(13) Lennard-Jones cluster in its liquid-solid phase transition region (temperature from 4 to 14 K). Calculation of the force autocorrelation function shows considerable differences from that given by classical mechanics, namely that the cluster is much more mobile (liquidlike) than in the classical case. Liquid para-hydrogen at two thermodynamic state points (25 and 14 K under nearly zero external pressure) has also been studied. The momentum autocorrelation function obtained from the TGA/LSC-IVR approach shows very good agreement with recent accurate path integral Monte Carlo results at 25 K [A. Nakayama and N. Makri, J. Chem. Phys. 125, 024503 (2006)]. The self-diffusion constants calculated by the TGA/LSC-IVR are in reasonable agreement with those from experiment and from other theoretical calculations. These applications demonstrate the TGA/LSC-IVR to be a practical and versatile method for quantum dynamics simulations of condensed phase systems.     

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.     

A new determination of the structure of water at 25°C

The results of a new neutron diffraction experiment to measure the structure of water are presented. The data, measured at the McMaster Nuclear Reactor, are of a high quality and are analysed to yield the hydrogen-hydrogen pair correlation function using a subtraction procedure which has been used in previous experiments of this kind. This procedure circumvents the necessity of applying inelasticity corrections. The results are in good agreement with earlier work and serve to establish the general correctness of the subtraction procedure when used to determine hydrogen correlations. The data are further analysed to yield separate oxygen-hydrogen and oxygen-oxygen partial structure factors for liquid water. For the second part of the analysis an effective mass model of the dynamic scattering law is used, with the model parameter, the effective mass of the scattering particle, chosen by a least-squares fit to the measured differential cross sections. The final pair correlation functions are obtained using a maximum entropy analysis of the structure functions.     

Nuclear momentum distribution in solid and liquid HF from ab initio calculation

A calculation of nuclear momentum distribution of liquid and solid hydrogen fluoride was performed. In both systems, density functional theory generalized gradient approximation functional of Perdew, Burke, and Ernzerhof was used for the calculation: for liquid hydrogen fluoride, using an atom centered basis set for an isolated molecule with optimized geometry, and for solid hydrogen fluoride using plane-wave basis sets on optimized orthorhombic crystal cell. For liquid hydrogen fluoride, a semiclassical approach was adopted with the vibrational contribution to momentum distribution obtained from the density functional theory calculation and translational and rotational contributions calculated classically. Nuclear momentum distribution in the solid hydrogen fluoride was calculated entirely quantum mechanically using phonon dispersion and vibrational density of states calculated in the framework of plane-wave density functional theory. Theoretical results were contrasted with recently obtained results of Compton (deep inelastic) neutron scattering on liquid and solid hydrogen fluoride. In case of liquid hydrogen fluoride, almost a perfect agreement between theory and experiment was achieved within the harmonic Born-Oppenheimer approximation. For the solid system under investigation, the harmonic approximation leads to small (4%) overestimation of the square root of the second moment indicating that neutron Compton scattering technique is sensitive to proton delocalization due to hydrogen bonding in solid hydrogen fluoride.     

Determination of pair correlation functions of dense colloidal systems by means of indirect Fourier transformation

The interparticle pair correlation gr function is determined by the small angle scattering data of interacting, monodisperse, globular colloidal particles. The new approach to this problem requires that the form factor and the particle number density must be known. Instrumental broadening effects and experimental imperfections are included in a way that makes it unnecessary to deconvolute the scattering function from beam profiles. The applicability of the approach is demonstrated by the simulated data of a charged colloidal dispersion and by way of the neutron scattering data of nonionic micellar solutions. In both cases, gr functions are obtained that are in good agreement with the known behavior of the systems.     

Configuration interaction study of X-ray and fast electron scattering factors for light atomic systems

A study of X-ray and fast electron scattering by light atoms and ions has been carried out in the first Born approximation. Coherent and incoherent scattering factors calculated with configuration interaction wave functions are compared with those obtained with Hartree–Fock wave functions. These configuration interaction wave functions involve only L-shell correlation. It is shown that the changes in the coherent scattering factors due to configuration interaction are not negligible and that the electron correlation effects on the incoherent scattering factors are important. Tables of coherent and incoherent scattering factors for light atomic systems are given.     

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.     

A hierarchical algorithm for fast debye summation with applications to small angle scattering

Debye summation, which involves the summation of sinc functions of distances between all pair of atoms in three‐dimensional space, arises in computations performed in crystallography, small/wide angle X‐ray scattering (SAXS/WAXS), and small angle neutron scattering (SANS). Direct evaluation of Debye summation has quadratic complexity, which results in computational bottleneck when determining crystal properties, or running structure refinement protocols that involve SAXS or SANS, even for moderately sized molecules. We present a fast approximation algorithm that efficiently computes the summation to any prescribed accuracy ? in linear time. The algorithm is similar to the fast multipole method (FMM), and is based on a hierarchical spatial decomposition of the molecule coupled with local harmonic expansions and translation of these expansions. An even more efficient implementation is possible when the scattering profile is all that is required, as in small angle scattering reconstruction (SAS) of macromolecules. We examine the relationship of the proposed algorithm to existing approximate methods for profile computations, and show that these methods may result in inaccurate profile computations, unless an error‐bound derived in this article is used. Our theoretical and computational results show orders of magnitude improvement in computation complexity over existing methods, while maintaining prescribed accuracy. © 2012 Wiley Periodicals, Inc.     

Ab initio molecular-dynamics study of supercritical carbon dioxide

Car-Parrinello molecular-dynamics simulations of supercritical carbon dioxide (scCO(2)) have been performed at the temperature of 318.15 K and at the density of 0.703 g/cc in order to understand its microscopic structure and dynamics. Atomic pair correlation functions and structure factors have been obtained and good agreement has been found with experiments. In the supercritical state the CO(2) molecule is marginally nonlinear, and thus possesses a dipole moment. Analyses of angle distributions between near neighbor molecules reveal the existence of configurations with pairs of molecules in the distorted T-shaped geometry. The reorientational dynamics of carbon dioxide molecules, investigated through first- and second-order time correlation functions, exhibit time constants of 620 and 268 fs, respectively, in good agreement with nuclear magnetic resonance experiments. The intramolecular vibrations of CO(2) have been examined through an analysis of the velocity autocorrelation function of the atoms. These reveal a red shift in the frequency spectrum relative to that of an isolated molecule, consistent with experiments on scCO(2). The results have also been compared to classical molecular-dynamics calculations employing an empirical potential.     

Insights in quantum dynamical effects in the infrared spectroscopy of liquid water from a semiclassical study with an ab initio-based flexible and polarizable force field

The dynamical properties of liquid water play an important role in many processes in nature. In this paper, we focus on the infrared (IR) absorption spectrum of liquid water based on the linearized semiclassical initial value representation (LSC-IVR) with the local Gaussian approximation (LGA) [J. Liu and W. H. Miller, J. Chem. Phys. 131, 074113 (2009)] and an ab initio based, flexible, polarizable Thole-type model (TTM3-F) [G. S. Fanourgakis and S. S. Xantheas, J. Chem. Phys. 128, 074506 (2008)]. Although the LSC-IVR (LGA) gives the exact result for the isolated three-dimensional shifted harmonic stretching model, it yields a blueshifted peak position for the more realistic anharmonic stretching potential. By using the short-time information of the LSC-IVR correlation function; however, it is shown how one can obtain more accurate results for the position of the stretching peak. Due to the physical decay in the condensed phase system, the LSC-IVR (LGA) is a good and practical approximate quantum approach for the IR spectrum of liquid water. The present results offer valuable insight into future attempts to improve the accuracy of the TTM3-F potential or other ab initio-based models in reproducing the IR spectrum of liquid water.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.     

Properties of polymer blends and triblock copolymers obtained by neutron scattering

Static and dynamic scattering properties of polymer blends and block copolymers are examined within the random phase approximation (RPA). A self-consistent theoretical scheme for a simultaneous analysis of elastic and quasielastic scattering data is presented. The case of a triblock copolymer made of an ordinary central block and two deuterated lateral blocks in a matrix of deuterated homopolymers is considered in detail. The theoretical predictions of the RPA are compared with the experimental data obtained by elastic neutron scattering experiments using mixtures of deuterated poly(dimethylsiloxane) homopolymers and copolymers made of three blocks of approximately equal sizes. The lateral blocks are deuterated poly(dimethylsiloxane) and the central one is an ordinary poly(dimethylsiloxane). A good agreement is found in the whole range of wavevectors covered by the experiments. An extension of the RPA to the analysis of the dynamical scattering data for the same systems is put forward. It is shown how the time relaxations of the bare response functions obtained from the single chain dynamics are used to extract the intermediate scattering function characterizing the system of interacting chains. © 1996 John Wiley & Sons, Inc.     

The structure of liquid water at room temperature

Full details are given of a recent time-of-flight neutron diffraction experiment in which partial pair correlation functions for liquid water are extracted by isotope substitution. Measurements of differential cross sections of mixtures of heavy and light water are made, and by performing the experiment at the high neutron energies available at a pulsed neutron source (Los Alamos) the magnitude of dynamic corrections to the data is reduced. The problematic hydrogen incoherent scattering is removed by a subtraction technique which avoids reference to dynamic models of the liquid. The results, although they show good agreement with computer simulations, show serious qualitative differences with other neutron experiments on water, and it is suggested these discrepancies are a result of lack of attention to the unusual properties of hydrogen as a scatterer of neutrons.     

Density functional theory of realistic models of polyethylene liquids in slit pores: comparison with monte carlo simulations

Density functional theory is applied to study properties of fully detailed, realistic models of polyethylene liquids near surfaces and compared to results from Monte Carlo simulations. When the direct correlation functions from polymer reference interaction site model (PRISM) theory are used as input, the theory somewhat underpredicts the density oscillations near the surface. However, good agreement with simulation is obtained with empirical scaling of the PRISM-predicted direct correlation functions. Effects of attractive interactions are treated using the random-phase approximation. The results of theoretical predictions for the attractive system are also in reasonable agreement with simulation results. In general, the theory performs best when the wall-polymer interaction strength is comparable to polymer-polymer interactions.     

Neutron scattering studies of the structure of amorphous and crystalline polymers and of polymer blends

The technique of neutron scattering for studying both the structure and dynamics of polymer systems is now well established. In the case of amorphous flexible polymers in most cases the bulk chain dimensions agree rather well with those of the unperturbed coil. Some newer results concerning for example side-chain liquid-crystal polymers and segmented polyurethane elastomers are described. Neutron small-angle scattering can also be used for the investigation of the molecular deformation mechanism during the drawing process of polymers. In the case of semicrystalline polymers the way in which a single macromolecule traverses the crystalline and amorphous phases can also be evaluated by neutron scattering. A method has been proposed for the evaluation of the neutron scattering data without introducing detailed structural models. The only assumption made is that the molecular structure can be described as consisting of “clusters” of crystalline stems which belong to the same molecule. It is shown that this cluster model can be verified experimentally for the cases of poly(ethylene oxide), polypropylene and polyethylene. The structure factor S(q) in compatible polymer blends is usually treated by the random phase approximation according to de Gennes. The temperature dependence of S(q) displayed by some systems, however, appears to be anomalous within this approximation. A new method is presented for evaluating the neutron scattering data which takes into account that the Flory-Huggins interaction parameter χ is not a point-function χδ(τ), but can be interpreted in terms of a structure model χ(τ) which consists of spatially separated positive and negative contributions within a 1–10 å range. Finally the application of neutron scattering to diblock copolymers is discussed and it is shown that the results for the case of polystyrene-poly (p-methylstyrene) are in good agreement with the theoretically expected behaviour.     

Hydrodynamic radius of polyethylene glycol in solution obtained by dynamic light scattering

In order to compare the size characterizations in poly(ethylene glycol) (PEG) obtained by dynamic light scattering (DLS) and small angle neutron scattering (SANS), DLS experiments were performed in various PEG solutions to ascertain the hydrodynamic radius. Data from the experiments were analyzed by using a method to eliminate effects of PEG aggregation on dynamic correlation functions. The results of the analysis were then compared to the radii of gyration reported from SANS experiments. The relation between the hydrodynamic radius, obtained by DLS, and the radius of gyration, obtained by SANS, in PEG in solution was found to be in agreement with a previously obtained relation for PEG, where the radius of gyration was found by static light scattering.     

Recent developments in polymer dynamics investigations of architecturally complex systems

The tube model for linear and branched architectures is nowadays able to predict in high precision the linear viscoelastic relaxation time spectrum. For linear chains, the involved time scales fit to the commonly accessible dynamic scattering techniques. This makes it possible to microscopically investigate the correlation between structures and relaxation processes. In branched systems, however, the hierarchical nature of relaxations limits direct investigation via these microscopic methods as the dynamic processes are prolongated to much longer relaxation times that are no more accessible to usual dynamic scattering methods. A way to overcome this difficulty is offered by the use of static small angle neutron scattering. Here, the combination of annealing and quenching steps after a step deformation provides unique information of the structure at particular times along the relaxation spectrum. This, however, necessitates the availability of architecturally clean and specifically deuterium labelled model polymers due to the sensitivity of the scattering method. Therefore, we outline in this contribution first the current status on the synthesis and analysis of such compounds with relation to neutron scattering. Secondly, we present exemplary neutron scattering results from in situ stress relaxation studies inside the neutron beam on linear and H-shaped branched polymers which were molecularly designed to highlight specific relaxation processes. We discuss the relevance of the tube model parameters in linear and non-linear studies.     

A+BC共线碰撞能量传递的动力学李代数结合中间绘景研究

在半经典近似下将动力学李代数方法和中间绘景结合应用于原子-双原子分子(非谐振子)共线碰撞中的平动-振动传能的研究。在群参量的一级近似下求解群参量的运动方程,进而确定时间演化算符。并以散射体系H₂+He为例,计算了H₂分子的振动跃迁几率,计算结果与精确量子力学计算结果符合得较好。