Bound modes

The photoelectron spectrum of FH,is shown in figure A3.7.6 [54]. The spectrum is highly structured, showing a group of closely spaced peaks centred around 1 eV, and a smaller peak at 0.5 eV. We expect to see vibrational structure corresponding to the bound modes of the transition state perpendicular to the reaction coordinate. For this reaction with its entrance chaimel barrier, the reaction coordinate at the transition state is... 

Note that the imaginary frequency associated with such a barrier is zoo. For a bound mode with frequency 0)vli, an analogous approximation would be... 

Table I shows examples of the steady-state and time-resolved emission characteristics of [Ru(phen)2(dppz)]2+ upon binding to various DNAs. The time-resolved luminescence of DNA-bound Ru(II) is characterized by a biexponential decay, consistent with the presence of at least two binding modes for the complex (47, 48). Previous photophysical studies conducted with tris(phenanthroline)ruthenium(II) also showed biexponential decays in emission and led to the proposal of two non-covalent binding modes for the complex (i) a surface-bound mode in which the ancillary ligands of the metal complex rest against the minor groove of DNA and (ii) an intercalative stacking mode in which one of the ligands inserts partially between adjacent base pairs in the double helix (36, 37). In contrast, quenching studies using both cationic quenchers such as [Ru(NH3)6]3+ and anionic quenchers such as [Fe(CN)6]4 have indicated that for the dppz complex both binding modes...

Two mechanisms which can be discounted at this time are pure dephasing and resonant energy transfer to other covalent modes of the molecule. In addition to coupling strengths both of these mechanisms depend on Che "bath" density of states of modes not directly involved as dissociation channels. The dependence of pure dephasing rates on occupation numbers is such that the low temperatures used in these experiments rule out such a relaxation mechanism. The mechanism whereby energy is redistributed among bound modes is not viable for the ethylene complexes based on Che dependence of rate on molecular structure (15). 

The need to include quantum mechanical effects in reaction rate constants was realized early in the development of rate theories. Wigner [8] considered the lowest order terms in an -expansion of the phase-space probability distribution function around the saddle point, resulting in a separable approximation, in which bound modes are quantized and a correction is included for quantum motion along the reaction coordinate - the so-called Wigner tunneling correction. This separable approximation was adopted in the standard ad hoc procedure for quan-... 

Equations (27.1) and (27.2) are hybrid quantized expressions in which the bound modes orthogonal to the reaction coordinate are treated quantum mechanically, that is, the partition functions (T, s) and 0 T) are computed quantum mechanically... 

Treating bound modes quantum mechanically, the adiabatic separation between s and u is equivalent to assuming that quantum states in bound modes orthogonal to s do not change throughout the reaction (as s progresses from reactants to products). The reaction dynamics is then described by motion on a one-mathematical-dimensional vibrationally and rotationally adiabatic potential... 

Like Eq. (27.2), Eqs. (27.11) and (27.12) are also hybrid quantized expressions in which the bound modes are treated quantum mechanically but the reaction coordinate motion is treated classically. Whereas it is difficult to see how quantum mechanical effects on reaction coordinate motion can be included in VTST, the path forward is straightforward in the adiabatic theory, since the one-dimensional scattering problem can be treated quantum mechanically. Since Eq. (27.12) is equivalent to the expression for the rate constant obtained from microcanonical variational theory [7, 15], the quantum correction factor obtained for the adiabatic theory of reactions can also be used in VTST. 

The ratio of partition functions for bound modes at the saddle point is determined by the frequencies for those modes. 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

The quantized generalized transition state rate constant equation (17) is a hybrid expression in which the bound modes are treated quantum mechanically but the reaction... 

Another, potentially far more devastating, weakness in classical mechanical methods is the problem of adiabatic leak (32-38). The difficulty arises from the fact that in quantum mechanics, bound modes are constrained to contain at least the ZPE for that mode. Classical mechanics has no such restrictions. Even if ZPE is initially put into a mode in classical mechanics, it can leak out during the course of a trajectory. Thus it is quite possible to have (in the simplest case) a diatomic molecule as the product of a reactive trajectory containing less energy than the vibrational ZPE. This is clearly unphysical. 

More subtle than the lack of ZPE in bound modes after the collision is the problem of ZPE during the collision. For instance, as a trajectory passes over a saddle point in a reactive collision, all but one of the vibrational (e.g., normal) modes are bound. Each of these bound modes is subject to quantization and should contain ZPE. In classical mechanics, however, there is no such restriction. This has been most clearly shown in model studies of reactive collisions (28,35), in which it could be seen that the classical threshold for reaction occurred at a lower energy than the quantum threshold, since the classical trajectories could pass under the quantum mechanical vibrationally adiabatic barrier to reaction. However, this problem is conspicuous only near threshold, and may even compensate somewhat for the lack of tunneling exhibited by quantum mechanics. One approach in which ZPE for local modes was added to the potential energy (44) has had some success in improving reaction threshold calculations. 

The primary surface reaction steps which include the activation of ethylene to vinyl and the subsequent coupling of vinyl with acetate surface intermediates have been cited as potential rate-determining steps in the Nakamura and Yasui route. Moiseev and Vargaftik carried out experiments over giant palladium clusters comprised of 561 atoms and arrived at a similar set of pathways They suggested, however, that the rate-controlling step for this process involves the shift of ethylene from the 7r-bound mode to a di-cr-bound mode. 

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