Central barrier

See text. The first two columns give the numbers of metal atoms at which electronic shell closings have been observed in experiment for Cs-covered C o and for pure alkali metal clusters, respectively. The columns on the right list the number of electrons required for shell closings in an infinitely deep potential well with and without a central barrier. The numbers in the different columns are mainly arranged in a manner to show correlations. 

Recognizing that at steady state the net rate over each barrier is the same, the simplest expression for the current is obtained for the central barrier i.e. 

As defined in Fig. BA, the quantity 2d in Equation 4 is the total length from one side to the other of the channel and at is the distance from the binding site to the central barrier. 

K = 63 M 1, Kb = 1.4M-1)47 lithium-7 (K = 14 M 1 K" = 0.5 M 1) 49) and for cesium-133 (K, st 50 M-1, K = 4M 1)S0). In the case of sodium-23, transverse relaxation times could also be utilized to determine off-rate constants k ff = 3 x 105/sec k"ff = 2x 107/sec47,51). Therefore for sodium ion four of the five rate constants have been independently determined. What has not been obtained for sodium ion is the rate constant for the central barrier, kcb. By means of dielectric relaxation studies a rate constant considered to be for passage over the central barrier, i.e. for jumping between sites, has been determined for Tl+ to be approximately 4 x 106/sec 52). If we make the assumption that the binding process functions as a normalization of free energies, recognize that the contribution of the lipid to the central barrier is independent of the ion and note that the channel is quite uniform, then it is reasonable to utilize the value of 4x 106/sec for the sodium ion. 

At R > 400 pm the orientation of the reactants looses its importance and the energy level of the educts is calculated (ethene + nonclassical ethyl cation). For smaller values of R and a the potential energy increases rapidly. At R = 278 pm and a = 68° one finds a saddle point of the potential energy surface lying on the central barrier, which can be connected with the activated complex of the reaction (21). This connection can be derived from a vibration analysis which has already been discussed in part 2.3.3. With the assistance of the above, the movement of atoms during so-called imaginary vibrations can be calculated. It has been attempted in Fig. 14 to clarify the movement of the atoms during this vibration (the size of the components of the movement vector... 

Figure 11. The minimum energy path of the OH + CH3F reaction, not including zero-point energy. The four labeled structures are (A), the central barrier TS (B), the nearly collinear backside well complex [HOCH3 F] (C) the transition of the F atom toward the OH moiety (D) the hydrogen-bonded [CH3OH F ] structure. Reprinted from [63] with permission from the American Association for the Advancement of Science.

Direct dynamics trajectory calculations at the MP2/6-31-FG level of theory were then used to explore the reaction dynamics of this system [63]. Sixty-four trajectories were started from the central barrier shown at A in Fig. 11, with initial conditions sampled from a 300 K Boltzmann distribution. Of the 31 trajectories that moved in the direction of products, four trajectories followed the MEP and became trapped in the hydrogen-bonded [CH3OH ... 

E. Dynamical Model for Sn2 Substitution and Central Barrier Recrossing. 152... 

Calculations have identified three transition states (TS) for an SN2 reaction.4"6 Two are variational, one of which is located along the X + RY association reaction path, and the other along the XR + Y" association reaction path i.e. see Figure 1. Variational transition state theory (VTST) calculations show that the third TS is located at the central barrier.4... 

Statistical rate theories have been used to calculate rate constants for gas-phase Sn2 reactions.1,7 For a SN2 reaction like Cl" + CH3Clb, which has a central barrier higher than the reactant asymptotic limit (see Figure 1), transition state theory (TST) assumes that the crossing of the central barrier is rate-limiting. Thus, the TST expression for the SN2 rate constant is simply,... 

For highly exothermic SN2 reactions, which have a central barrier significantly lower in energy than that of the reactants, association of the reactants may be the rate controlling step in TST.1 The SN2 rate constant can then be modeled by a capture theory9 such as VTST,10 average dipole orientation (ADO) theory,11 the statistical adiabatic channel model (SACM),12 or the trajectory capture model.13... 

The most accurate energies and geometries for the Cl" + CH,Clb system are those calculated at the CEPA-l/avtz and G2(+) levels of theory. Without zero-point energies included, the CEPA- 1/avtz calculations give a complex well depth of-10.6 kcal/mol and a central barrier height of 2.8 kcal/mol. The G2(+) values for these energies are -10.7 and 3.0 kcal/mol. The most recent experimental value for the 0 K complex well depth is 12.2 2 kcal/mol.23... 

CH3C1 and CHjBr, are 1.06 and 1.01 times larger than the respective experimental harmonic vibrational frequencies. One of the most interesting properties of the vibrational frequencies is the increase in the C-H stretch frequencies in going from the reactants (or products) of the SN2 reaction to the central barrier. 

Scale factors determined from the CH3Br and CH,C1 ab initio and experimental frequencies were used to scale ab initio frequencies for the complexes and central barrier see text... 

Figure 4. Probability of different trajectory events versus the Cla-C-Clb angle 0, which is evaluated at the first inner turning point (ITP) for complex formation and at the central barrier for the trajectories which attain this configuration ( ), association probability (o), probability of attaining the central barrier. fre = 0.5 kcal/mol and nc-cib = 6. Trot = 0 K in (a) and 300 K in (b) (from ref. 38).

Figure 5. Reaction probabilities vs. impact parameter for frei = 0.5 kcal/mol and Trot = 300 K. Part (a) is for nc-cib = 0. The solid lines are for FVib = 0 and the dotted lines are for Fvib = zpe. The squares are for total complex formation and the triangles for long-range complex formation. Part (b) is for nc-clh = 6 and fVib = 0 with ( , —) for total complex formation, (a, —) for long-range complex formation, and (o, ) for attaining the central barrier with one or no ITP (from ref. 38).

Adding quanta to the C-Cl bond promotes bond extension, so that the central barrier can be reached as Cl- approaches. This dynamical effect is in accord with the role of vibrational energy in A + BC -> AB + C triatomic displacement reactions.15 The plot in Figure 5 of the probability of directly attaining the central barrier versus Cl + CH3Clb collision impact parameter shows that direct substitution occurs at small impact parameters. In contrast, association extends to larger impact parameters. 

A dynamical model for SN2 nucleophilic substitution that emerges from the trajectory simulations is depicted in Figure 9. The complex formed by a collision between the reactants is an intermolecular complex CinterR. To cross the central barrier, this complex has to undergo a unimolecular transition in which energy is... 

Figure 9. Dynamical model for Sn2 nucleophilic substitution. The labels R and P denote the reactant and product sides of the central barrier, respectively.

The dynamical model described in Figure 9 indicates that the trajectories may recross the central barrier several times if the Cintra R Cintra p transition is faster... 

If crossing the central barrier is not rate-controlling in TST, then trapping in the ion-dipole complex must be incorporated into the statistical model and it is more difficult to represent the effect of central barrier recrossings correcting TST with the K factor is not sufficient. The recrossings and presence of both intermolecular and intramolecular complexes are expected to affect the k, kisom, and k rate constants in equation 6. The value for k should be smaller than that of a capture model, and kisom and k 8 should disagree with the predictions of RRKM theory. 

Additional experimental, theoretical, and computational work is needed to acquire a complete understanding of the microscopic dynamics of gas-phase SN2 nucleophilic substitution reactions. Experimental measurements of the SN2 reaction rate versus excitation of specific vibrational modes of RY (equation 1) are needed, as are experimental studies of the dissociation and isomerization rates of the X--RY complex versus specific excitations of the complex s intermolecular and intramolecular modes. Experimental studies that probe the molecular dynamics of the [X-. r - Y]- central barrier region would also be extremely useful. 

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