The Correlation Function Approach

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. 

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Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

Unlike case 1, the correlation function approaches zero when t —> oo this is because if t is large, we expect to find the process in state off. Using Eq. (16), the asymptotic behavior of the normalized correlation function Eq. (11) is... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

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. 

Haymet and co-workers have calculated the mole fraction of dimers (associated ions) in electrolytic solutions, and some of their results are shown in Fig. 3.51. Use the equations of the Bjerrum theory applied to NajP04 and compare the results with those of the correlation function approach used by Haymet et al. The essential difference between the Haymet approach and that of Bjerrum is that... 

The advantage of the correlation function approach is that only the storage of scalar quantities, rather than wave packets, is needed. Thus, the memory requirement is significantly reduced, an issue that may become more important for large systems. The implementation with the Chebyshev propagator takes further advantage of its numerical properties discussed above. In cases where resonances are dominant, the LSFD approach can be used to further reduce computational costs. We note in passing this approach can be extended to the calculation of thermal rate constants. 

There are several geometrical variables that one can extract from the correlation function approach. First, a correlation... 

The diffuse scattering from such polymers may be treated by the correlation function approach discussed in Section 3.3.2. The correlation distance may be related to the size of the void and the value of to the... 

At r = 0 the correlation function approaches imity. A common expression is... 

Equations (7) and (10) generalize the Kirkwood-Frohlich equilibrium theory to the dynamic situation. The correlation function approach to dielectric relaxation was first made by Glarum (37) and was extended by Cole (38) and by Steele (39). 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

The feature of the correlation function approach is that it provides a field-free method for deducing the dielectric behaviour [using equations (44)-(46)] if the model for motion is specified. Williamshas shown how this may be done for a variety of rotational models including free rotation, and rotations in different barrier systems. 

Figure 5 displays the counter- and coion correlation functions (equivalent to the concentration profile per bulk concentration) according to the Gouy-Chapman, Dehye-Hiickel, and apparent Dehye-Hiickel solutions (Eqs. [23], [32], and [93]) for a bulk symmetric electrolyte at 0.1 M near a planar surface with charge density ct = 0.01 ealK. Far from the surface, where the potential is small, the correlation functions approach unity and the amounts of counter- and coions are equal close to the surface the number of counterions increases and the number of coions decreases as a result of the electrostatic attraction to and repulsion from the surface, respectively. The apparent DH counterion correlation function is, as can be seen, a marked improvement over the standard DH result. 

The density functional approach of Refs. 91, 92 introduces a correction to the wall-particle direct correlation function resulting from the HNCl approximation (see Eqs. (32)-(34)). A correction to Eq. (34) reads (we drop the species label because the model is one-component)... 

For At = 0 the function is equal to the variance a = X2 (equation 5.1), but for At - oo its value approaches zero because of the increasing probability of products of both positive and negative values, the summation of which becoming zero. Normalization of eqn. 5.2 by dividing both members by a yields the correlation function ... 

A number of papers are devoted to the effect of dissipation on tunneling.81"83,103,104 Wolynes81 was one of the first to consider this problem using the Feynman path integral approach to calculate the correlation function of the reactive flux involved in the expression for the rate constant,... 

Recent progress in protein dynamics studies by NMR was greatly facilitated by the invention of the model-free formalism [28, 32]. In this approach, the local dynamics of a protein are characterized by an order parameter, S, measuring the amplitude of local motion on a scale from 0 to 1, and the correlation time of the motion, T oc. The model-free expression for the correlation function of local motion reads... 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

In essence, it is the probability density of the two nuclei to have relative separation Since the orientation of the molecule is not fixed (nuclei are not fixed any more if we deal with an non-BO approach), gi( ) is a spherically symmetric function. The plots of gi ( ) are presented in Figs. 1-4-. It should be noted that all the correlation functions shown are normalized in such a way that... 

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

We defer the discussion of the effects of (r) until Section VII.C and begin with the special case, referred as a force-free diffusion, with a uniform distribution of electron spins outside the distance of closest approach with respect to the nuclear spin. Under the assumption of the reflecting-wall boundary condition at rjs = d, Hwang and Freed found the closed analytical form of the correlation function for translation diffusion (138) ... 

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