Conventional Correlation Function Research Articles

Evaluation of thermal rates for reactions with intermediate wells: Removal of bound state contributions to quantum flux correlation functions

We consider the calculation of rates of chemical reactions that have bound intermediate states (i.e., wells along the reaction path) using flux correlation function methods. When time-dependent wave packets are used to evaluate the propagator matrix elements, and the dividing surface is located at a point where bound states have nonzero probability density, the standard expression for the flux correlation function shows infinitely long lived oscillations due to these bound states, making the evaluation of rate constants numerically ill-behaved. However, if the bound part of the initial wave packet is projected out, the resulting continuum-only propagators produce rapidly decaying correlation functions, and numerically well-behaved rate constants. We illustrate this projection operator approach by considering a one-dimensional reaction path model in which the potential is taken to be an Eckart well, and the dividing surface is located at the minimum. In another application, we consider a two degree of freedom model of H2 dissociative chemisorption on a rigid metal surface. This application is sufficiently complex that it is impractical to calculate the chemisorption rate using conventional flux correlation function methods, but with the projected wavepacket approach, the problem is made relatively easy. We also consider a second approach to the treatment of bound states in which the flux correlation function is altered to remove implicitly the bound state contributions to the propagator at long times. This second expression can be used with the full propagator, eliminating the need to construct and project explicitly the bound states. This should be advantageous when many bound states are present.

Integral equations for conventional correlation functions of simple dimerizing fluids

An integral equation theory is proposed for the atom-atom correlation functions of a simple reacting fluid, namely one in which a dimerization reaction is occurring. For a model fluid composed of hard sphere monomers and rigid dumbbell dimers the integral equations are solved exactly in the Percus-Yevick approximation, and an explicit expression for the equilibrium constant is derived. Comparison is made with the relevant Monte Carlo simulation data.

Correlation functions revisited

Correlation functions (auto-correlations, cross-correlations, rotational correlations) are used extensively in general signal processing, in Fourier optics, and particularly in electron microscopical image processing. In spite of their widespread and successful use in a broad spectrum of pattern-matching procedures, we believe there is reason to reconsider their general use. Correlation functions have the intrinsic limitation that they are calculated by squaring or multiplication operations in Fourier space. This fact causes the strong frequency components in the data (typically the low-frequency ones) to become overwhelmingly stronger relative to the weaker components, typically the high-frequency components associated with the fine details in the data in which we are especially interested. By dividing Fourier components by the square roots of their amplitudes, these unfavorable effects can be avoided resulting in substantially better correlation functions. Our new mutual-correlation function (MCF) and self-correlation function (SCF), which relate to each other the way the conventional cross-correlation function (CCF) relates to the auto-correlation function (ACF), can supersede conventional correlation functions for most purposes. We have subjected the novel correlation functions to practical tests in routine image-alignment procedures and found the results satisfactory. The problems we experienced with conventional squared correlation functions are likely to hinder triple-correlation function applications to an even greater extent.

Angular pair correlations of observed and simulated galaxy distributions

The main body of quantitative information about galaxy statistics is obtained from correlation studies. It has recently turned out that a modified correlation formalism can provide details about large-scale structure in the galaxy distribution, which are obscured by artefacts of the conventional correlation function. The modified pair correlation function, as applied to the Zwicky catalogue of galaxies, shows two distinct power-law regimes at small scales (< 1°) and large scales (around 10°). Based on the comparison of simulated bubblelike large-scale structures with the Zwicky sample, these regimes are interpreted to correspond to the distribution of galaxies within the shells of the bubbles (small scale), and the distribution of the bubbles themselves (large scale).

Level crossing statistics and Hausdorff‐Besicovitch similarity dimension of generalized rough surfaces

Generalized rough surfaces can exhibit a rich variety of relief that cannot be easily described as either periodic or Gaussian. For such surfaces, one can explore the spatial distribution per unit length of occurrence of the mean height (=0 by definition), the rms heights, or the crossing of the relief with a chosen level. An extension of this concept is also possible with joint probability distribution characterization of the distribution of successive level crossing known as Palm distributions. Evaluation of these measures for some test functions and stochastically generated surface relief is presented and the benefits of their use is contrasted with conventional statistical and correlation function techniques. Surfaces that contain a possible infinite hierarchy of scales, e.g., fractal surfaces, are shown to best be characterized by a parameter known as the Hausdorff‐Besicovitch similarity (HBS) dimension. The use of this parameter in describing generalized rough surfaces possessing self‐similar characteristics or multiple scales of roughness is described and the reflection characteristics of such surfaces in optical and acoustical applications are discussed.

Crystal scattering of two coherent neutron beams

The crystal scattering of two coherent neutron beams is discussed. It is found that the conventional pair correlation function used in single-beam scattering is insufficient to describe the scattering of two coherent neutron beams. A generalized pair correlation function with unequal momentum transfer is introduced. It is found that the energy exchange between vibrational quanta of the crystal and incident neutrons leads to a modification of the scattering amplitude. Through the scattering of two coherent neutron beams one is able to measure directly the phase of the scattering amplitude.