Pair correlations

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Figure Al.3.29. Pair correlation and structure factor for liquid silicon from experiment [41],...

G(/) is also called the pair correlation fiinction and is sometimes denoted by h(/). Integration over /and / tln-ough the domain of system volume gives, on the one hand. 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

The pair correlation fiinction has a simple physical interpretation as the potential of mean force between two particles separated by a distance r... 

The direct correlation fimction c(r) of a homogeneous fluid is related to the pair correlation fimction tiirough the Omstein-Zemike relation... 

It follows that the exact expression for the pair correlation function is... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

This is Kirkwood s expression for the chemical potential. To use it, one needs the pair correlation fimction as a fimction of the coupling parameter A as well as its spatial dependence. For instance, if A is the charge on a selected ion in an electrolyte, the excess chemical potential follows from a theory that provides the dependence of g(i 2, A) on the charge and the distance r 2- This method of calculating the chemical potential is known as the Gimtelburg charging process, after Guntelburg who applied it to electrolytes. 

By analogy with the correlation function for the ftilly coupled system, the pair correlation ftmction g(r A) for an intennediate values of A is given by... 

Theories based on the solution to integral equations for the pair correlation fiinctions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Furtlier improvements for simple fluids would require better approximations for the bridge fiinctions B(r). It has been suggested that these fiinctions can be scaled to the same fiinctional fomi for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation fiinction c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

The thennodynamic properties are calculated from the ion-ion pair correlation fimctions by generalizing the expressions derived earlier for one-component systems to multicomponent ionic mixtures. For ionic solutions it is also necessary to note that the interionic potentials are solvent averaged ionic potentials of average force ... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

The pair distribution fiinction clearly has dimensions (density), and it is nomial to introduce the pair correlation fiinction g(X, X2) defined by... 

Given that Wj. r) we finally obtain for the pair correlation coefficient... 

The pair correlation fiinctions qq, gQ and have been obtained for water on an uncharged... 

Figure A2.4.11. Water pair correlation functions near the Pt(lOO) surface. In each panel, the frill curve is for water molecules in the first layer, and the broken curve is for water molecules in the second layer. From [30].

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

Table 3.1.1 Pair correlation energies for the four electrons in Be.

For these reasons, in the MCSCF method the number of CSFs is usually kept to a small to moderate number (e.g. a few to several thousand) chosen to describe essential correlations (i.e. configuration crossings, near degeneracies, proper dissociation, etc, all of which are often tenned non-dynamicaI correlations) and important dynamical correlations (those electron-pair correlations of angular, radial, left-right, etc nature that are important when low-lying virtual orbitals are present). 

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