Solution coordination

V 1s n) is the normalized thermal distribution of configurations of the distinguished molecule in isolation [10], i.e., the required marginal distribution. The remaining set of brackets here indicates the average over solvent coordinates. The second set of brackets are not written on the right here because the averaging over solute coordinates is explicitly written out. This last formula is... 

First, consider the solvent. The characterization of the solute-solvent coupling by a relaxation time is based on analogy to Brownian motion, and the relaxation time is called the frictional relaxational time Xp. It is the relaxation time for momentum decay of a Brownian motion in the solute coordinate of interest when it interacts with the solvent under consideration. If we call the subject solute coordinate s, then the component of frictional force along this coordinate may be written as... 

The Statistical Perturbation Theory should be applied allowing a complete sampling of the solute coordinates (and, if possible, of the solvent coordinates). This way no solvent equilibrium hypothesis would be introduced at all. 

The time dependent friction coefficient, per solute mass p, is related to the fluctuating forces exerted by the solvent on the solute coordinate x through their time correlation function ... 

On short time scales where the solvent has not moved while the solute coordinate x crosses the barrier, (2.9) indicates, with (2.1), that the x motion is... 

Here cos is the solvent frequency and we have assumed the solvent s equilibrium position to vary linearly with the solute coordinate x in the neighborhood of the barrier top ... 

If the solvent were to adjust rapidly (adiabatically) and equilibrate to the solute coordinate x so that there is no force on s, then we would have the equilibrium condition... 

The content of (3.11) can be clarified by considering the time correlation function of the solvent coordinate itself, when the solute coordinate is fixed at its Transition State value x=0. It is then a straighforward exercise to show from (3.10) that... 

Metal Ion Adsorption in Mixtures of Multiple Solid Phases. One of the arguments put forth for extending the concepts of solution coordination chemistry to heterogeneous systems is the hypothesis that the mineral components of soils or sediments can be considered as ligands which compete for complexation of adsorbates. To this end, it is important to know the relative ability of different mineral surfaces to complex solutes. 

All of the interaction mechanisms described above are expected to produce electric fields in the solute cavity. In the case of specific interactions and reaction field effects these electric fields are expected to have some specific orientation with respect to the solute coordinate system. Dispersion forces and Stark effects are not expected to have any specific orientation with respect to the solute. Magnetic field effects seem unlikely to be important in light of the well-known invariance of coupling constants to changes of the external magnetic field. However, it is conceivable that a solvent magnetic reaction field might... 

The second hydration sphere may be considered as "outer-sphere complexing , a phenomenon which is extremely important in all reactions inaqueoiis solution. Coordination of a third hydration sphere will... 

In alkaline solution coordination of Cu(II) to a branched pentammine ligand... 

At the next level of approximation, we continue to imagine the solvent to be fully equilibrated to tlie reacting system at every point, but instead of working with the solvated MEP from the gas-phase surface, we find tlie equilibrium solvation patli (ESP) which is the MEP on the fully solvated surface (see Figure 11.1). While both die gas-phase and solvated surfaces are defined entirely in terms of solute coordinates, tlie I iSP may be quite different from the gas-phase MEP because solvation effects may push the patli in directions orthogonal to the gas-phase reaction coordinate (see Figure 11.5). With die ESP in hand, TST (or VTST) analysis may be carried out in the usual way lo obtain a condensed-phase rate constant. 

The beauty of the prior approximations is that by assuming a mean-field influence of solvation we can continue to work in a phase space having the same dimensionality as that for the gas phase that being the case, analysis using the tools of TST is mechanically identical for the two phases. When the solvent is not fully equilibrated with the complete reaction path, however, the reacting system can no longer legitimately be described exclusively in terms of solute coordinates. 

In the above treatments only the solute coordinates R appear explicitly and therefore the definition of the transition state does not depend on solvent coordinates. The NES approximation [60,61] provides a way to include solvent in the reaction coordinate while retaining a continuum description of the solvent by adding a coupling Hamiltonian for a collective solvent coordinate [60-70] (or more than one) to the Hamiltonian for the... 

R degrees of freedom. (Sometimes the collective solvent coordinate is assumed to be the reaction coordinate itself [70] rather than, as here, finding a reaction coordinate in the space obtained by augmenting the solute coordinates by the collective solvent coordinate.) As in ESP theory, in NES theory the equilibrium solvent effects are included by replacing V(R) by W(R) in nontunneling parts of the calculation and by t/(R) in tunneling algorithms. 

We now introduce the density operator p(r), defined in terms of the electronic (e) and nuclear (n) solute coordinates ... 

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