Primitive Quasichemical Approximation

Equation (9.50) is the primitive quasichemical approximation that was used to obtain the results of Figs. 9.1 and 9.2. Primitive emphasizes that the equilibrium constants are obtained with initial neglect of the effects of the outer-shell material, as (9.50) 

KriP is readily calculated by considering the inner-shell clustering reactions 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

The final ingredient that enters the calculation is the density factor pw. This is the actual density of water appropriate to the thermodynamic state intended in the calculation. For the usual case of 1 atm. pressure and 298K, this is I gem 3. The reference density in the electronic structure calculations is p° = 1 atm// T, however. Hence to account properly for the entropic cost of sequestering water in the metal-water complexes, the free energies should be adjusted by —mRT In (pi 2o/p ) = —mRTIn (1354). With these inputs the excess chemical potential is readily composed as per (9.50), provided the optimal value of m is known. This is found by composing the excess chemical potential for different assumed m values and identifying the most stable case. For the dication transition metals studied, this is found to be six, consistent with experiment [12]. 

The above procedure readily yields Fig. 9.2. For estimating the excess chemical potential in the protein, again we need to know the ligation state of the metal ion. This is well known for the zinc-finger case. So we followed the above procedure, deciding what clusters to study quantum mechanically, and then composing the free energies as above. 

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The first point from this development and example is that, although the quasichemical approach is directed towards treating strong attractive - chemical - interactions at short range, it can describe traditional packing problems accurately. The second point is that this molecular-field idea permits us to go beyond the primitive quality noted above of the primitive quasichemical approximation, and specifically to account approximately for the influence of the outer-shell material on the equilibrium ratios Km required by the general theory. This might help with cases of delicate structures noted above with anion hydrates. 

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