Canonical mean shape

Neglect of the last term yields Equation (3.22), which is called (in this context) the zero-order canonical mean-shape (CMS-0) approximation [31]. 

The simplest way to include solvation effects is to calculate the reaction path and tunneling paths of the solute in the gas phase and then add the free energy of solvation at every point along the reaction path and tunneling paths. This is equivalent to treating the Hamiltonian as separable in solute coordinates and solvent coordinates, and we call it separable equilibrium solvation (SES) [74]. Adding tunneling in this method requires a new approximation, namely the canonical mean shape (CMS) approximation [75]. 

In practice, we approximate the exact transmission coefficient by a mean-field-type of approximation that is we replace the ratio of averages by the ratio for an average or effective potential. For gas-phase reactions with small reaction-path curvature, this effective potential would just be the vibrationally adiabatic ground-state potential. In the liquid phase and enzymes we generalize this with the canonical mean-shape approximation. In any event, though, the transmission coefficient should not be thought of as a perturbation. The method used here may be thought of as an approximate full-dimensional quantum treatment of the reaction rate. 

The SES approximation also replaces V by W for the tunneling calculations, which is called the zero-order canonical-mean-shape approximation (CMS-O). Note that the tunneling turning points and hence the tunneling paths may be different in the gas phase and in solution in the SES approximation, even though the reaction path is unaltered. 

At this point one can include optimized multidimensional tunneling in each (i = 1,2,..., 7) of the VTST calculations. The tunneling transmission coefficient of stage 2 for ensemble member i is called and is evaluated by treating the primary zone in the ground-state approximation (see the section titled Quantum Effects on Reaction Coordinate Motion ) and the secondary zone in the zero-order canonical mean shape approximation explained in the section titled Reactions in Liquids , to give an improved transmission coefficient that includes tunneling ... 

For Reaction [328], the Cl, C, and N atoms are collinear. The bond lengths between these three atoms in the gas phase and in solution are listed in Table 1, and the energetics of the stationary points are listed in Table 2. For this reaction, solvent effects are very large for products. The aqueous solution stabilizes the charged products, as shown in Table 3. The gas-phase Vmep and the SES canonical mean-shape potential U(s T) are plotted in Figure 10. Note that... 

Figure 10 Zero-order canonical mean shape potential U for reaction [328] calculated at the HF/6-31 G(d) (gas phase) and SM5.43//FIF/6-31G(d) (SES) levels as functions of the reaction coordinate s for the Menshutkin reaction.

The determinant (= total molecular wavefunction T) just described will lead to (remainder of Section 5.2) n occupied, and a number of unoccupied, component spatial molecular orbitals i//. These orbitals i// from the straightforward Slater determinant are called canonical (in mathematics the word means in simplest or standard form ) molecular orbitals. Since each occupied spatial ip can be thought of as a region of space which accommodates a pair of electrons, we might expect that when the shapes of these orbitals are displayed ( visualized Section 5.5.6) each one would look like a bond or a lone pair. However, this is often not the case for example, we do not find that one of the canonical MOs of water connects the O with one H, and another canonical MO connects the O with another H. Instead most of these MOs are spread over much of a molecule, i.e. delocalized (lone pairs, unlike conventional bonds, do tend to stand out). However, it is possible to combine the canonical MOs to get localized MOs which look like our conventional bonds and lone pairs. This is done by using the columns (or rows) of the Slater T to create a T with modified columns (or rows) if a column/row of a determinant is multiplied by k and added to another column/row, the determinant remains kD (Section 4.3.3). We see that if this is applied to the Slater determinant with k = 1, we will get a new determinant corresponding to exactly the same total wavefunction, i.e. to the same molecule, but built up from different component occupied MOs i//. The new T and the new i// s are no less or more correct than the previous ones, but by appropriate manipulation of the columns/rows the i// s can be made to correspond to our ideas of bonds and lone pairs. These localized MOs are sometimes useful. 

The basic cell employed in a canonical ensemble MC simulations has been chosen in form shown in Fig. 4. The cell has the shape of a rectangular box of dimensions xxlyx z = 30x10x30 (in the units of macroion diameter, D) and contains of both bulk and confined regions. There are two (left and right) bulk regions which are the parts of the same reservoir connected by means of periodic boundary conditions in x direction, i.e., particles that left the cell from the left side enter on the right side and vice versa. 

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