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Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is 'very small.' In theory 'very small' implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments. [ABSTRACT FROM AUTHOR]

Methods, such as the continued fractions that are based on the exact moments (Taylor coefficients) provide powerful analytic techniques for calculating approximate dynamic correlation functions of physical properties. Owing to the practical difficulties associated with obtaining higher order exact moments these methods have been largely eclipsed by molecular dynamics. In this paper we develop the formalism for extracting trajectory moments from a molecular dynamics trajectory and compare the performance of the two methods. We begin by using the classical phase space analogs of quantum wave functions to obtain matrix representations (transition matrices) of the Lie group of time displacement operators. The group elements for small time displacements, which correspond to a step of the trajectory, are approximated by using Trotter’s theorem. The method is applied to the linear harmonic oscillator, the Morse oscillator and the Lennard-Jones potential and the resulting moments compared to those obtained by the Taylor expansion of a truncated continued fraction. We find that in the case of the harmonic oscillator the results from the continued fraction are much better than those from molecular dynamics. In the latter two cases, while the truncated fraction performs better than molecular dynamics both methods produce poor quality higher order moments. However, we are able to show the criterion whereby the overall error introduced by the truncated continued fraction is smaller than that introduced by molecular dynamics. From this work we conclude that the moment based methods produce good results, they are analytic in nature rather than numerical and should not be rejected but can be used to complement molecular dynamics. [ABSTRACT FROM AUTHOR]

The original formulation of the virial in the isotropic periodic sum (IPS) method assumes that the sphere defining the local region has a constant radius (the cutoff) independent of the system size. This assumption neglects a virial term originating from the separation between the local sphere and its periodic images. When comparing the IPS virial with that calculated from the cutoff plus long range correction method, the difference observed can be erroneously attributed to the representation of the infinite region. We show that when the two virials are calculated consistently the observed difference is significantly reduced. Additionally, the correct virial that includes the previously missing term is much simpler to calculate. We prove that in the IPS method the virial can be obtained as n/3 times the potential energy for the case of 1/rn type potentials. [ABSTRACT FROM AUTHOR]

Proton transport in perfluorosulfonic acid (PFSA) membranes is investigated through a statistical mechanical model that includes the effects of the interaction of the tethered sulfonate groups with both the water and solvated protons. We first derive a potential that describes the electrostatic field due to the dissociated sulfonic acid groups by extending the work of Gro\nbech-Jensen et al. [ Mol. Phys. 92, 941 (1997)] to a finite array of point charges. A highly convergent series is obtained which includes the effects of screening due to the protons. We then investigate the effects of both dielectric saturation and two distinct formulations of ionic screening on the proton self-diffusion coefficient in Nafion membranes over a range of water contents. Our computations show that the two phenomena (i.e., dielectric saturation and ionic screening) under constant temperature conditions result in canceling affects. Our calculations provide a radial dependence of the proton mobility suggesting that the dominant self-diffusion occurs in the central region of the pores, well separated from the sulfonate groups. Through comparison of our calculated diffusion coefficients with the experimental values we derived a slightly smaller average separation distance of the hydronium ion from the sulfonate ions than suggested by either electronic structure calculations or multistate empirical valence bond molecular-dynamics simulations. [ABSTRACT FROM AUTHOR]

A recently derived mathematical model to compute the effective friction and diffusion coefficients of hydronium ions in hydrated polymer electrolyte membranes is described and tested for dependence on membrane-specific parameters. Contributions to the friction coefficient due to water–polymer, water–hydronium, and hydronium–polymer interactions are determined through computation of force–force correlation functions. The conventional Stokes law friction coefficient of the hydronium ion in bulk water is then “corrected” with these statistically derived contributions and the corresponding diffusion coefficient calculated. For a Nafion® membrane pore with an hydration level of six water molecules per sulfonic acid functional, the model was used to compute friction coefficients for various distributions of the fixed sites, and for different side chain lengths. The model showed substantial sensitivity to these parameters and predicted that for pores of fixed volume and a constant total number of sulfonate groups, the friction on the hydrated proton is the greatest for distributions with high local anionic charge density. In a second series of computations where the radius and length of the pore were varied, the model demonstrated that the proton diffusion increases with increasing channel diameter. These calculations, therefore, demonstrate the important predictive capability of this molecular-based, nonequilibrium statistical mechanical model. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An equilibrium statistical mechanical model is derived to compute the spatial variation in the permittivity of water within the hydrated pores of ion-containing polymeric membranes. The fixed anionic groups within the pore are modeled as periodic arrays of point charges. The Helmholtz free energy is calculated from a total Hamiltonian of the pore that includes energy from (1) interactions between the fields generated by the fixed charge groups and the dipoles of the water molecules, (2) “hard core” interactions between the water molecules, and (3) dipole–dipole interactions between the water molecules. The free energy is divided into two parts: (a) a reference free energy associated with five water molecules in a cluster interacting with each other through the hard core potentials and with the fixed charge groups and (b) an excess free energy due to the dipolar interactions between the water molecules in two cluster units. In the present work we calculate the polarization and corresponding permittivity from this reference free energy. We first show that our calculations, even at this level of sophistication, go beyond all the traditional approaches. Furthermore, with our model we compute radial profiles of the permittivity in the pores of the sulfonic acid–based Nafion® and 65% sulfonated poly ether ether ketone ketone polymer electrolyte membranes at several different hydration levels. These numerical results and predictions are in agreement with known experimental measurements. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We report the effect of donor-doped perovskite-type BaCeO3 on the chemical stability in CO2 and boiling H2O and electrical transport properties in various gas atmospheres that include ambient air, N2, H2, and wet and dry H2. Formation of perovskite-like BaCe1-xNbxO3±δ and BaCe0.9-xZrxNb0.1O3±δ (x = 0.1; 0.2) was confirmed using powder X-ray diffraction (XRD) and electron diffraction (ED). The lattice constant was found to decrease with increasing Nb in BaCe1-xNbxO3±δ, which is consistent with Shannon's ionic radius trend. Like BaCeO3, BaCe1-xNbxO3±δ was found to be chemically unstable in 50% CO2 at 700 °C, while Zr doping for Ce improves the structural stability of BaCe1-xNbxO3±δ. AC impedance spectroscopy was used to estimate electrical conductivity, and it was found to vary with the atmospheric conditions and showed mixed ionic and electronic conduction in H2-containing atmosphere. Arrhenius-like behavior was observed for BaCe0.9-xZrxNb0.1O3±δ at 400-700 °C, while Zr-free BaCe1-xNbxO3±δ exhibits non-Arrhenius behavior at the same temperature range. Among the perovskite-type oxides investigated in the present work, BaCe0.8Zr0.1Nb0.1O3±δ showed the highest bulk electrical conductivity of 1.3 x 10-3 S cm-1 in wet H2 at 500 °C, which is comparable to CO2 and H2O unstable high-temperature Y-doped BaCeO3 proton conductors. [ABSTRACT FROM AUTHOR]