Charge correlation

D.D. Johnson and F.J. Pinski, Including charge correlations in calculations of the energetics and electronic structure for random substitutional alloys, Phys. Rev. B 48 11553 (1993). 

Theoretical Calculations of 3 and Charge Correlated tt -Electron States —... 

The frontier orbital approach (Fukui et al., 1962, 1954b) has met with considerable success in so far as frontier orbital charges correlate well with experimental data. The performance of these indices is often superior to that of others, with the possible exception of localization energies. It is, however, difficult to give meaning to the correlation since physical interpretations of the role of the frontier electrons in reaction mechanisms are often obscure, and attempts to give substance to Fukui s hypothesis have frequently embodied questionable procedures or models. 

From Tables 6.3 and 6.4 it seems that the size and charge correlations can be extended to complex ions. This observation is very important because it indicates a possibility to estimate the ion interaction coefficients for complexes by using such correlations. It is, of course, always preferable to use experimental ion interaction coefficient data. However, the efforts needed to obtain these data for complexes will be so great that it is unlikely that they will be available for more than a few complex species. It is even less likely that one will have data for the Pitzer parameters for these species. Hence, the specific ion interaction approach may have a practical advantage over the inherently more precise Pitzer approach. 

The specific ion interaction approach is simple to use and gives a fairly good estimate of activity factors. By using size/charge correlations, it seems possible to estimate unknown ion interaction coefficients. The specific ion interaction model has therefore been adopted as a standard procedure in the NEA Thermochemical Data Base review for the extrapolation and correction of equilibrium data to the infinite dilution standard state. For more details on methods for calculating activity coefficients and the ionic medium/ ionic strength dependence of equilibrium constants, the reader is referred to Ref. 40, Chapter IX. 

Under the Born-Oppenheimer approximation, two major methods exist to determine the electronic structure of molecules The valence bond (VB) and the molecular orbital (MO) methods (Atkins, 1986). In the valence bond method, the chemical bond is assumed to be an electron pair at the onset. Thus, bonds are viewed to be distinct atom-atom interactions, and upon dissociation molecules always lead to neutral species. In contrast, in the MO method the individual electrons are assumed to occupy an orbital that spreads the entire nuclear framework, and upon dissociation, neutral and ionic species form with equal probabilities. Consequently, the charge correlation, or the avoidance of one electron by others based on electrostatic repulsion, is overestimated by the VB method and is underestimated by the MO method (Atkins, 1986). The MO method turned out to be easier to apply to complex systems, and with the advent of computers it became a powerful computational tool in chemistry. Consequently, we shall concentrate on the MO method for the remainder of this section. 

Let us first recall that standard DH theory presumes constant ion density, so that the pair correlation function cannot say anything about density fluctuations. In contrast, simple DH theory describes charge fluctuations via the well-known screening decay as exp(—rDr). Note, however, that this result does not satisfy a rigorous condition for the second moment of the charge-charge correlation function first derived by Stillinger and Lovett (SL) [39] ... 

Also, the decay of ha r) is monotonous, ignoring the need for an oscillatory decay at high ion densities, anticipated from various other treatments [274, 275]. Thus, also with respect to charge correlations, DH theory is quite insufficient. 

We note that, in contrast to the simple DH expression, the GDH result for the charge-charge correlation function [276, 278] h r) satisfies the SL second-moment condition (24). When ion pairs are introduced, the SL second-moment condition is, however, only satisfied up to order p2. 

A crucial assumption for such a scenario is that the charge-charge correlation function ha(r) decays as exp (—IV), where the inverse charge-charge correlation length T = 1/ does not vanish at the critical point. At low ion densities, where T —> T0, this has meanwhile been proved for DH theory. Then, at the critical point, only the density fluctuations become... 

Second, there is a line of charge-ordering in the T -p plane, where the charge-charge correlation function begins to oscillate. This line, as established from GDH theory, passes close to the critical point and may generate a virtual tricritical state. A charge-density wave scenario also arises from r-dependent cavity interactions. 

Figure 7. Contour diagram of the wave functions for the MNA ground state (bottom) and a principal excited state (top) showing charge correlations from Garito et. al.

Z. G. Soos, L. R. Ducasse, and R. M. Metzger, Partial ionicity, cohesion, and charge correlations in narrow-band solids, J. Chem. Phys. 77 3036-3045 (1982). 

All the complex ions studied are regarded as hydrophobic stnicture-makers, since the dln( X °° 7] o)/d7 values are positive, similarly to the tetraalkylammonium ions.5 The hydrophobicity of the trivalent [Co(phen)3]3 ion is inferior to that of the divalent [Fe(phen)3]2+ ion since the dln( A °° Tt ldT value of the former complex is smaller than that of the latter complex. This suggests that the hydrophobic atmosphere around the complex ion is weakened with increasing ionic charge, correlated to the reduction of the partial molar volume (1, °°) which is explicable as the increase in the electrostriction by Glueckauf s equation. 

ELECTRONIC ORIGIN OF SECOND ORDER NONLINEAR OPTICAL RESPONSES IN CONJUGATED LINEAR CHAIN STRUCTURES AND HIGHLY CHARGE CORRELATED tt-ELECTRON STATES... 

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