Numerical Solutions of the Convolution‐Hypernetted Chain Integral Equation for the Pair Correlation Function of a Fluid. I. The Lennard‐Jones (12, 6) Potential

1963 ◽  
Vol 39 (6) ◽  
pp. 1367-1387 ◽  
Author(s):  
Max Klein ◽  
M. S. Green
Keyword(s):  
Integral Equation ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Numerical Solutions ◽  
Pair Correlation ◽  
Lennard Jones ◽  
Hypernetted Chain

Monte Carlo Simulation of the Interfaces Liquid-Crystal and Liquid-Rigid Wall

1978 ◽  
Vol 33 (12) ◽  
pp. 1557-1561 ◽  
Author(s):  
B. Borštnik ◽  
A. Ažman
Keyword(s):  
Monte Carlo ◽  
Correlation Function ◽  
Liquid Crystal ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Rigid Wall ◽  
Bulk Liquid ◽  
Lennard Jones ◽  
The Monte Carlo Method

Abstract The structure of liquids at liquid-crystal and liquid-rigid wall interfaces was studied by the Monte Carlo method on systems consisting of either 128 Lennard-Jones atom s or 128 hard spheres. The resulting density profile can serve as a reference for the approximative methods based on the BGYB hierarchy of integral equations. The pair correlation function close to the rigid wall is found to deviate appreciably from the bulk liquid pair correlation function. The maxima and minima of g(r) are more pronounced in the first two layers of atom s close to the rigid wall.

EFFECTIVE POTENTIAL FOR TWO-BODY INTERACTIONS

1990 ◽  
Vol 04 (13) ◽  
pp. 2005-2023 ◽  
Author(s):  
RUGGERO VAIA ◽  
VALERIO TOGNETTI
Keyword(s):  
Correlation Function ◽  
Effective Potential ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Body System ◽  
Characteristic Parameters ◽  
Helium Atoms ◽  
Lennard Jones ◽  
Hydrogen Molecules ◽  
Exact Data

A new kind of effective potential, which permits the calculation of the quantum equilibrium averages of configuration dependent observables in a classical-like way, is used for calculating the quantum pair correlation function of a two-body system. The main feature of this effective potential is the capability to fully account for the quantum harmonic effects, so it proves much more efficient than the analogous one defined by the Wigner expansion. Applications and comparisons with exact data are made for the Lennard-Jones interaction, with the characteristic parameters of helium atoms and hydrogen molecules.

scholarly journals The Kirkwood Superposition Approximation, Revisited and reexamined

2013 ◽  
Vol 1 (1) ◽  
pp. 27-35 ◽  
Author(s):  
Arieh Ben Naim
Keyword(s):  
Integral Equation ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Potential Of Mean Force ◽  
Superposition Approximation ◽  
Potential Of Mean Forces

The Kirkwood superposition approximation (KSA) was originally suggested to obtain a closure to an integral equation for the pair correlation function. It states that the potential of mean force of say, three particles may be approximated by sum of potential of mean forces of pairs of particles. Nowadays, this approximation is widely used, explicitly or implicitly, in many fields unrelated to the problem for which it was suggested.It is argued that the KSA is neither a good approximation nor a bad approximation; it is simply not an approximation at all.

Pair correlation function in a fluid with density inhomogeneities: results of the Percus‒Yevick and hypernetted chain approximations for hard spheres near a hard wall

1986 ◽  
Vol 404 (1827) ◽  
pp. 323-337 ◽  
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Bulk Fluid ◽  
Pair Correlations ◽  
Hypernetted Chain ◽  
Density Inhomogeneities ◽  
Bulk Case

Solutions for the pair correlation function and density profile of a system of hard spheres near a hard wall are obtained by using the Percus‒Yevick and hypernetted chain approximations, generalized for inhomogeneous fluids. The Percus‒Yevick (PY) results are similar in accuracy to those obtained for the bulk fluid. The PY pair correlation function is generally too small near contact but quite good overall. The hypernetted chain (h. n. c.) results are difficult to obtain numerically and are poorer than in the bulk. Often the h. n. c. pair correlations are too small at contact, in contrast to the bulk case where they are too large, although there are configurations where the contact values of the pair correlation function are too large. Nearly always the error in the h. n. c. results is much worse than is the case for the bulk. Both approximations are qualitatively satisfactory in that they predict the correct asymmetries between the values of the pair correlation functions for pairs of hard spheres whose line of centres is parallel or normal to the surface of the wall.

Integral Equation Methods

1984 ◽  
Author(s):  
C. G. Gray ◽  
K. E. Gubbins
Keyword(s):  
Integral Equation ◽  
Perturbation Theory ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Infinite Order ◽  
Ad Hoc ◽  
Pair Correlation ◽  
Integral Equation Methods ◽  
Atomic Liquids ◽  
Integral Equation Approach

In this chapter we describe some of the integral equation methods which have been devised for calculating the angular pair correlation function g(rω1ω2) and the site-site pair correlation function gαβ( r ) for molecular liquids. These methods are in the main natural extensions of methods devised for calculating the pair correlation function g(r) for atomic liquids. They can be derived from infinite-order perturbation theory (an example is given in § 5.4.8), whereby one partially sums the perturbation series of Chapter 4 to infinite order usually with the help of diagrams, or graphs, but alternative methods of derivation are also available, e.g. functional expansions. The original integral equation theories are in a certain sense more complete than perturbation theories, in that the full correlation function g (or gαβ) is calculated, whereas in perturbation theory one calculates the correction g — g0 to the reference fluid value g0. On the other hand the perturbation theory approximations are controlled; one can estimate the error by calculating the next term. I t is extremely difficult to estimate a priori the error in integral equation approximations, since certain terms are neglected almost ad hoc. Their validity must therefore be a posteriori, according to agreement with computer simulation results (or, less satisfactorily, with experiment). Of particular interest are theories which are a combination of perturbation theory and an integral equation, which tend to have some of the advantages of both approaches (see also §5.3.1). An example is the GMF/SSC theory of §5.4.7. The structure of the integral equation approach for calculating g(r ω1 ω2) is as follows. One starts with the Ornstein-Zernike (OZ) integral equation (3.117) between the total correlation function h = g — 1 and the direct correlation function c, which we write here schematically as . . . h = c + pch (5.1) . . . or, even more schematically, as . . . h = h[c], (OZ) (5.2) . . . where h[c] denotes a functional of c.

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