Fast coordinates

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

It has been shown that there is a two-dimensional cut of the PES such that the MEP lies completely within it. The coordinates in this cut are 4, and a linear combination of qs-q-j. This cut is presented in fig. 64, along with the MEP. Motion along the reaction path is adiabatic with respect to the fast coordinates q -q and nonadiabatic in the space of the slow coordinates q -qi-Nevertheless, since the MEP has a small curvature, the deviation of the extremal trajectory from it is small. This small curvature approximation has been intensively used earlier [Skodje et al. 1981 Truhlar et al. 1982], in particular for calculating tunneling splittings in (HF)2- The rate constant of reaction (6.45a) found in this way is characterized by the values T<. = 20-25 K, = 10 -10 s , = 1-4 kcal/mol above T, which compare well with the experiment. 

Figure 8. Proposed mechanism for the acid-catalyzed dissociation of Nien2 where a proton adds to an amine N while it is still in the fast coordination sphere of the Ni ion. The species in parentheses may be transition states rather than...

We give in Fig. 2 a schematic plot showing contours on which a vibrational entropy Sv(q,X) is constant, q is a fast coordinate, and the line C is the transition state in the (X, q) space. This Sv(q, X) can be defined as in an equation similar to (2.4) in terms of a kB In p(q,X), p(q,X) being the local density of study, that is, the number per unit energy per unit q and per unit X. While this plot is not used in the derivation, it can be visually helpful. 

Figure 2. Schematic entropic surface as a function of a slow coordinate X and a fast coordinate q. Here, S is a saddle point, the line C is the transition state in this (X, q) space, and Xc lies at its intersection with the X axis.

The identification of fast coordinates can be difficult in simulations of condensed phases. Some of the fast modes are bond or angle vibrations that can be identified and integrated separately. However, other fast modes are transient. They are fast for a short duration of time and slow otherwise. The transient fast modes are collisions for example, two atoms that are close and feel strong repulsive forces due to excluded volume interactions. The relevant degree of freedom (the distance between the atoms) is a fast mode during the collision event and a slow mode before or after the short collision period. The fundamental complication in the treatment of these modes is the identity crisis of these fast/slow coordinates. 

In the above average, only the force depends on the fast coordinates. So we are required to perform averages of the type and These... 

We also note that there is more than one choice of a function to average. Besides the specific choice made in Eq. (27), 5sdet, it is also possible to average (over the fast coordinates) the weight, exp[—5sdet/2g Af], or the force, —dU/ dX. Of the three possibilities the last choice is equivalent to generating a potential of mean force prior to the calculations of dynamics. The direct use of a potential of mean force for peptides and proteins is another direction that we are currently pursuing [18]. 

In this way r is singled out as the fast electronic coordinate. In the absence of prior knowledge we can, equally well, designate R as the fast coordinate. This gives rise to an alternative adiabatic breakdown of the Schrodinger equation, and exemplifies the nonuniqueness associated with the adiabatic separation of bound-state nuclear dynamics. 

For L = CN , the low-spin Fe(III) complex leads to sodium nitroprusside (SNP) as an irreversibly formed product. The proposed mechanism implies a slow associative, reductive step by NO, followed by the fast coordination of NO+ to the [Fen(CN)5H20]3 intermediate.17 Faster reversible reactions are onset with high-spin Fe(III) porphyrins, as shown in Section 7.3.2.5... 

Berezhkovskii A M and Zitserman V Yu 1993 Multi-dimensional Kramers theory of the reaction rate with highly anisotropio friotion. Energy diffusion for the fast coordinate versus overdamped regime for the slow coordinate Chem. Phys. Lett. 212 413-19... 

Although apparently unrelated, some of the redox changes of these complexes are similar. In acetonitrile solution the OS4 complex undergoes a single-step two-electron oxidation [Ep — +0.78 V), which is coupled to fast coordination of solvent molecules according to the mechanismd ... 

Electronic populations Pi x, t) and P2 x, t) of the reactant and the product states, respectively, averaged over the fast coordinate satisfy diffusion equations (9.52) along the slow coordinate (x instead of X) with effective potentials. 

Note that we intentionally used different subscripts than before (1 and 2 instead of s and /) because now separation into slow and fast coordinates is different. Figure 9.24 compares numerically exact rate constant fci2 as a function of rj = T2/T1 against VTST results and our simple spectrum separation procedure. The agreement is quite reasonable, which makes us believe... 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

Although Ru(III) ammine complexes are known to be very inert low-spin d species which only very slowly undergo substitution reactions, their ability to rapidly and efiectively bind nitric oxide seems to be a rather unusual behavior (92). Common characteristics of the Ru(III) nitrosyl complexes, formally Ru NO, studied to date are their octahedral stereochemistry and the presence of an extremely stable Ru—NO mode (93). A broad array of available kinetic and electrochemical data dealing with the formation of Ru(III) nitrosyls clearly shows that the mechanism of unusual fast coordination of nitric oxide to the Ru(III) ammine center cannot be accounted for in terms of a classical ligand substitution process. In this context, the fundamental kinetics of the fast reactions between [Ru (NH3)5X] pC = Cl, ... 

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