Transitional configuration

V-methylpiperidine oxide [17206-00-7] does not undergo the reaction because of its inabihty to achieve the highly strained transition configuration, whereas the... 

Fig. 60. Configuration and relevant coordinates of the planar HF dimer in stable and transition configurations. The angles and intermolecular distance are = 9°, 6 = 116°, R = 2.673 A in the stable configuration 0, = 02 = 54.9°, R = 2.S61 k in the transition configuration. The HF bond lengths are constant within an accuracy of 0.003 A.

Important parameters involved in the Volmer-Butler equation are the transfer coefficients a and (1. They are closely related to the Bronsted relation [Eq. (14.5)] and can be rationalized in terms of the slopes of the potential energy surfaces [Eq. (14.9)]. Due to the latter, the transfer coefficients a and P are also called symmetry factors since they are related to the symmetry of the transitional configuration with respect to the initial and final configurations. 

At a small value of this parameter, the system can perform multiple passages through the transitional configuration before the electron transfer occurs. The exact expression for the transition probability taking into account these multiple passages has the form... 

Thus, overcoming the activation barrier is performed here by fluctuation of the solvent polarization to the transitional configuration P, whereas electron-proton transmission coefficient is determined by the overlap of the electron-proton wave-functions of the initial and final states. 

Equation (28) gives the physical meaning of the activation factor H as the free energy required for the system to reach the transitional configuration. 

It has been shown16 that in the Condon approximation the value of the polarization in the transitional configuration P is equal to... 

The interactions V P and involved in Eqs. (13) determining the zeroth-order electron states in the transitional configuration represent the interactions of the electron with the polarizations P and PB, respectively. At long transfer distances the perturbation leading to the electron transfer has the form... 

Equations (50) and (51) show that for 0 < 6 < 1 the potential well for the electron near the donor site is more shallow than that in the initial equilibrium configuration. This leads to the fact that the radius of the electron density distribution in the transitional configuration is greater than in the initial equilibrium one (Fig. 3). A similar situation exists for the electron density distribution near the acceptor site. This leads to an increased transmission coefficient as compared to that calculated in the approximation of constant electron density (ACED). 

The results for the symmetric system are given in Table 2. A comparison of Tables 1 and 2 shows that the dependence of nAnB on and rj influences the position of the transitional configuration and this effect increases with increase in the transfer distance. The physical reason for the change of the path of the transition in this case is that the system prefers to shift from the saddle point to the... 

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

When the system approaches the transitional configuration two effects take place (1) the coefficient CL increases and at Q = Q it may be of the order of, or even greater than, the coefficient CA (CL CA), and (2) the distance between the ligand L and atom B decreases with increasing Q at a fixed distance R between atoms A and B, leading to an exponential increase of the overlap integral (0l 0b)-... 

In the classical limit, the transitional configuration q, Q in the Condon approximation is determined by the equations... 

Figure 7. Potential energy surfaces for the proton at the transitional configuration for the medium molecules.

The physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

If at the optimum distance R, the frequency flp is considerably smaller than at large distances (f fiClp R )/4 — 1), then along with the shift of the proton equilibrium position when the system goes to the transitional configuration, the contribution of transitions between the excited vibrational states of the proton increases and the proton transition may occur from the levels located near the top of the potential barrier. This case corresponds to the Kreevoy type of transition.49 In the limit /3fiflp(R )/4 1 we have the case of the overbarrier proton transition. However, the formulas of the nonadiabatic theory become inapplicable in this case and the reaction is an adiabatic one. 

The quantity A< h in Eq. (132) is the shift of the equilibrium positions of the protons (at fixed transitional configurations of the other nuclei). 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

The activation factor in the first case is determined by the free energy of the system in the transitional configuration Fa, whereas in the second case it involves the energy of the reactive oscillator U(q ) = (l/2)fi(oq 2 in the transitional configuration. The contrast due to the fact that in the first case the transition probability is determined by the equilibrium probability of finding the system in the transitional configuration, whereas in the second case the process is essentially a nonequilibrium one, and a Newtonian motion of the reactive oscillator in the field of external random forces in the potential U(q) from the point q = 0 to the point q takes place. The result in Eqs. (171) and (172) corresponds to that obtained from Kramers theory73 in the case of small friction (T 0) but differs from the latter in the initial conditions. 

BD/St-copolymers were also prepared by the use of the catalyst system Nd(OCOCCl3)3/TIBA/DEAC [503]. The copolymer exhibited 79% cis-1,4-structure in the BD units at a content of incorporated styrene of 23 mol%. In this study diades were also determined. According to the authors BD moieties which are adjacent to St moieties predominantly exhibit a transit-configuration whereas BD moieties in BD-BD-diades exhibit a cis-1,4-structure [503]. It therefore can be concluded that the microstructure of an entering BD monomer is controlled by a penultimate effect. This effect can be best described by a model in which backbiting coordination of a penultimate BD unit to Nd is involved [177,367]. 

M = niA + mg + me + mg and mAB = mA + mg where mx is the mass of atom X. The transformation Eq. (3) is nothing but the identity transformation. The only reason it does not look like one is that the (primed) coordinates used to describe the products, Eq. (2), are not the same as the coordinates used to describe the reactants. That Eq. (3) is really the identity matrix follows from the assumption of the model that the motion of the reactants is unperturbed, up to the transition configuration and that beyond this configuration the products recede without any intermolecular coupling. 

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