Obtaining Solution Properties from Correlation Functions

Obtaining Solution Properties from Correlation Functions 

To understand this section, it is best to review briefly the idea of a correlation function (see Section 2.11.3). Consider a hypothetical photograph of a number of ions in a solution. One looks for one ionic species i in a small volume dV at a series of distances r from a molecule of species j. The chance of finding the ion i in dVis 

There are two kinds of ways to obtain g values, i.e., gy, and g j. One is a computational approach using the Monte Carlo method or MD. In Fig. 3.49 one has seen the example of P. Bopp s results of such a determination using MD. 

On the other hand, one can experimentally determine the various g s. Such determinations are made by a combination of X-ray and neutron diffraction measurements (see Section 2.11.3 and 5.2.3). 

One now has to use the correlation functions to calculate a known quantity. One way of doing this (Friedman, 1962) is to use the following equation  

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Obtaining Solution Properties from Correlation Functions... 

The theory reflects the solvent properties through the thermody-namic/hydrodynamic input parameters obtained from the accurate equations of state for the two solvents. However, the theory employs a hard sphere solute-solvent direct correlation function (C12), which is a measure of the spatial distribution of the particles. Therefore, the agreement between theory and experiment does not depend on a solute-solvent spatial distribution determined by attractive solute-solvent interactions. In particular, it is not necessary to invoke local density augmentation (solute-solvent clustering) (31,112,113) in the vicinity of the critical point arising from significant attractive solute-solvent interactions to theoretically replicate the data. 

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

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. 

It is evident that this is a way of rewriting the exact solution of Eq. (47). However, it is interesting to recover the fluctuation-dissipation prediction from a perspective that might lead to a free diffusion with no upper limit if an error is made that does not take into account the statistical properties of the fluctuation E,(f). Let us evaluate the correlation function of E,(f). Using the property of Eq. (48) and moving to the asymptotic time limit reflecting the microscopic equilibrium condition, we obtain... 

However, there is no doubt that from the 1980s on, a very hopeful type of development has been taking place in ionic solution theories. It is the correlation function approach, not a theory or a model, but an open-ended way to obtain a realistic idea of how an ionic solution works (Fig. 3.58). In this approach, pair correlation functions that are experimentally determined from neutron diffraction measurements represent the truth, without the obstructions sometimes introduced by a model. From a knowledge of the pair correlation function, it is possible to calculate properties (osmotic pressure, activities). The pair correlation function acts as an ever-ready test for new models, for the models no longer have to be asked to re-replicate specific properties of solutions, but can be asked to what degree they can replicate the known pair correlation functions. 

Describe the correlation function as applied to ionic solutions. Sketch out a schematic of such a quantity for a hypothetical solution of FeCla. There are two entirely different types of methods of obtaining (r) state them. Finally, describe the point of knowing this quantity and comment on the meaning of the statement From a knowledge of the correlation function, it is possible to calculate solution properties (which may, in turn, be compared with the results of experiment). Hence the calculation of the correlation function is the aim of all new theoretical work on solutions. ... 

Studies of the thermodynamic properties obtained from solutions of the SSOZ equation have been much more extensive but the success has been mixed. The first such calculations were those of Lowden and Chandler who obtained the pressure of hard diatomic fluids from the RISM (SSOZ-PY) equation. They used two routes to the equation of state a compressibility equation of state in which they integrated the bulk modulus calculated from the site-site correlation functions via... 

As can be seen from Fig. 4.44, there are considerable discrepancies between results obtained by different simulations and by theoretical calculations. At present, it is difficult to claim that one simulated result is better than another. This is also true for any theoretical calculations based on a specific pair potential for water. It is a fortiori true if the theory uses as an input the orientational averaged pair correlation function of water, as was done by Pratt and Chandler (1977). In my opinion, using such an input into the theory will inevitably fail to reproduce any property of water and aqueous solutions that is sensitive to the angular dependence of the pair potential — or, equivalently, that depends on the principle. For further details, see Ben-Naim (1989), Guarino and Madden (1982), and Tani (1984). 

The spectral representations above are not computationally efficient, as they would require knowledge of all intermediate excited states. Computationally tractable formulas for the response functions within the various approximate methods are obtained instead through the following steps (1) choose a time-independent reference wavefunction (2) choose a parametrization of its time-development, for instance an exponential parametrization (3) set up the appropriate equations for the time development of the chosen wavefunction parameters (4) solve these equations in orders of the perturbation to obtain the wavefunction (parameters) (5) insert the solutions of these equations into the expectation value expression and obtain the RTFs and (6) identify the excited-state properties from the poles and residues. The computationally tractable formulas for the response functions therefore differ depending on the electronic structure method at hand, and the true spectral representations given above are only valid in the limit of a frill-configuration interaction (FCI) wavefunction. For approximate methods (i.e., where electron correlation is only partially included), matrix equations appear instead of the SOS expressions, for example. 

The session on Theories of Liquid Structures treated the equilibrium properties of simple liquids from a quantum-statistical and statistical-thermodynamics point-of-view. The tenor in these presentations was the Pair-Correlation Function. The various equations that have been proposed to describe its behavior, and techniques for obtaining their solutions (in special cases) were reviewed. These concepts became more "concrete with a discussion of electrolytic solutions. 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

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