Perturbation theory of the pair correlation function in molecular fluids

1978 ◽  
Vol 56 (5) ◽  
pp. 571-580 ◽  
Author(s):  
R. L. Henderson ◽  
C. G. Gray
Keyword(s):  
Perturbation Theory ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Mean Field ◽  
Pair Potential ◽  
Harmonic Series ◽  
Harmonic Space ◽  
Potential Symmetries ◽  
Harmonic Components

We study the perturbation theory of the angular pair correlation function g(rω1ω2)in a molecular fluid. We consider an anisotropic pair potential of the form u = u0 + ua, where u0 is an isotropic 'reference' potential, and for simplicity in this paper we assume the perturbation potential ua to be 'multipole-like', i.e., to contain no l = 0 spherical harmonics. We expand g in powers of ua about g0, the radial distribution function appropriate to u0. This series is examined by expanding ha = h−h0 (where h = g−1) and its corresponding direct correlation function ca in spherical harmonic components. We consider approximate summations of the series in ua that automatically truncate the corresponding harmonic series, so that the Ornstein–Zernike (OZ) equation relating ha and ca can be solved in closed form. We first expand ca = c1 + c2 + … where cn includes all terms in ca of order (ua)n. Taking ua to be a quadrupole–quadrupole interaction, we find that a 'mean field' (MF) approximation ca = c1 gives rise to only three nonvanishing harmonic components in ha, so that OZ is solved explicitly in Fourier space. The MF solution for multipoles of general order is given in an appendix. Graphical methods are then used to identify the class of all terms in the ua series that are restricted to the harmonic space defined by MF. A portion of this class can be summed by solving OZ with the closure ca = −βg0ua + h0(ha−ca), where β = (kT)−1, h0 = g0−1 This system is designated as generalized MF (GMF), and solved by numerical iteration. Numerical results from MF and GMF are presented for quadrupolar ua, taking u0 to be a Lennard-Jones potential. Symmetries imposed by the restricted harmonic space are foreign to the full g, yet harmonics within this space are sufficient for evaluation of many macroscopic properties. The results are therefore evaluated in harmonic form by comparison with the corresponding harmonic components of the 'correct' g as evaluated by Monte Carlo simulation.

Temperature dependence of the pair correlation function in an inhomogeneous liquid: application to C24Cs

1982 ◽  
Vol 60 (8) ◽  
pp. 1139-1144 ◽  
Author(s):  
Michael Plischke ◽  
W. D. Leckie
Keyword(s):  
Correlation Function ◽  
Temperature Dependence ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Pair Potential ◽  
Intercalation Compounds ◽  
Effective Pair Potential ◽  
Graphite Intercalation ◽  
Effective Pair ◽  
Graphite Lattice

We study the properties of a model appropriate to stage 2 graphite intercalation compounds such as C24Cs and C24K. The Hamiltonian consists of an effective pair potential for the intercalate and a periodic background potential due to the graphite lattice. We obtain the pair correlation function over the range of temperatures 0–300 K. We find orientational effects and compare our results with data on C24Cs.

Computer simulations of polyatomic molecules - I. Monte Carlo studies of hard diatomics

1976 ◽  
Vol 348 (1655) ◽  
pp. 485-510 ◽  
Keyword(s):  
Monte Carlo ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Harmonic Series ◽  
The Monte Carlo Method ◽  
Monte Carlo Studies ◽  
High Degree ◽  
Orientational Structure

The Monte Carlo method has been used to study a model system of 256 hard diatomic molecules, each consisting of two fused hard spheres of diameter σ with centres separated by reduced distance L = L/σ of 0.2, 0.4 and 0.6, at densities typical of the liquid state. The orientational structure of dense, hard diatomic fluids has been studied by calculating up to sixteen terms in the expansion of the total pair correlation function, g ( r 12 , ω 1 , ω 2 ), in spherical harmonics. The coefficients g u'm ( r 12) the series have been calculated as ensemble averages in the simulation. At short distances, the system exhibits a high degree of angular correlation, which increases with increasing density and elongation; however, this correlation is relatively short ranged at all densities and elongations, and in no case is there significant angular structure at distances greater than twice the major diameter of the molecule. In the nearest neighbour shell there is a strong preference for 'T-shaped’ pair orientations. At low elongations and densities the spherical harmonic coefficients are in close agreement with those predicted both by the ‘blip function’ theory and the solution of the Percus-Yevick equation for hard diatomics. The harmonic series for the total pair correlation function, is rapidly convergent at distances greater than L + σ , but slowly convergent at smaller distances. The results are suitable for use as a non-spherical reference system for perturbation calculations.

scholarly journals Entropy Pair Functional Theory: Direct Entropy Evaluation Spanning Phase Transitions

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 234
Author(s):  
Donald M. Nicholson ◽  
C. Y. Gao ◽  
Marshall T. McDonnell ◽  
Clifton C. Sluss ◽  
David J. Keffer
Keyword(s):  
Free Energy ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Excess Entropy ◽  
Pair Correlation ◽  
Pair Potential ◽  
Thermodynamic Integration ◽  
First Principles Calculation ◽  
Simple Modification ◽  
Single Temperature

We prove that, within the class of pair potential Hamiltonians, the excess entropy is a universal, temperature-independent functional of the density and pair correlation function. This result extends Henderson’s theorem, which states that the free energy is a temperature dependent functional of the density and pair correlation. The stationarity and concavity of the excess entropy functional are discussed and related to the Gibbs–Bugoliubov inequality and to the free energy. We apply the Kirkwood approximation, which is commonly used for fluids, to both fluids and solids. Approximate excess entropy functionals are developed and compared to results from thermodynamic integration. The pair functional approach gives the absolute entropy and free energy based on simulation output at a single temperature without thermodynamic integration. We argue that a functional of the type, which is strictly applicable to pair potentials, is also suitable for first principles calculation of free energies from Born–Oppenheimer molecular dynamics performed at a single temperature. This advancement has the potential to reduce the evaluation the free energy to a simple modification to any procedure that evaluates the energy and the pair correlation function.

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.

Sign in / Sign up

Export Citation Format

Share Document