Harmonic oscillator reactions

Elements of classical dynamics of unimolecular reactions in particular, the Slater theory for indirect reactions, where the molecule is modeled as a set of uncoupled harmonic oscillators. Reaction is defined to occur when a particular bond length attains a critical value, and the rate constant is given as the frequency with which this occurs. 

Using MMd. calculate A H and. V leading to ATT and t his reaction has been the subject of computational studies (Kar, Len/ and Vaughan, 1994) and experimental studies by Akimoto et al, (Akimoto, Sprung, and Pitts. 1972) and by Kapej n et al, (Kapeijn, van der Steen, and Mol, 198.V), Quantum mechanical systems, including the quantum harmonic oscillator, will be treated in more detail in later chapters. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

The first term is the intrinsic electronic energy of the adsorbate eo is the energy required to take away an electron from the atom. The second term is the potential energy part of the ensemble of harmonic oscillators we do not need the kinetic part since we are interested in static properties only. The last term denotes the interaction of the adsorbate with the solvent the are the coupling constants. By a transformation of coordinates the last two terms can be combined into the same form that was used in Chapter 6 in the theory of electron-transfer reactions. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

For one-electron electrochemical reactions, the harmonic oscillator ("Marcus ) model (21) yields the following predicted dependence of AG upon the electrode potential ... 

It is seen that the "electrochemical estimates of values of AG diverge from the straight line predicted from the harmonic oscillator model to a similar, albeit slightly smaller, extent than the experimental values. Admittedly, there is no particular justification for assuming that the reduction half reactions obey the harmonic oscillator model. However, it turns out that the estimates of AG are relatively insensitive to... 

The harmonic oscillator approximation used to describe the reactants and products in terms of internal reorganization is certainly not suited for describing the broken bond in the product system in the case of dissociative electron transfer. For reactions of the type (41) and (42), where RX does... 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

This Hamiltonian describes a reaction coordinate s in a symmetric double well as - bs that is coupled to a harmonic oscillator Q. The coupling is symmetric for the reaction coordinate and has the form cs Q, which would reduce the barrier height of a quartic double well. The origin of the Q oscillations is taken to be at Q = 0 when the reaction coordinate is at =fso (centers of the the reactant/product wells), which explains the presence of the term —cs Q. This potential has 2 minima at (s, Q) = ( So,0) and one saddle point at (s, Q) = (0, +cs2o/Mi22)... 

However, this simplification obviously does not apply to real electrochemical reactions. This harmonic oscillator model is described here because it is so well known. It will be compared with experiment in Section 9.6.3. 

Fig. 9.25. (e) The hydrogen evolution reaction overthe Tafel relation which is linear over eleven orders of magnitude (experimental points). Curved line Marcus expression with assumption of harmonic oscillators. (Reprinted from J. O M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum 1979, p. 228.)... 

Notice that the only two unknowns remaining are k and In this case, the vibrational frequency should not be thought of as the imaginary frequency that derives from the standard harmonic oscillator analysis, but rather the real inverse time constant associated with motion along the reaction coordinate. However, it is exacdy motion along the reaction coordinate that converts the activated complex into product B. That is, k = (o - Eliminating their ratio of unity from Eq. (15.21) leads to the canonical TST expression... 

The potential energy curve of the dissociating harmonic oscillators is taken to be that of a truncated harmonic oscillator with a finite number of equally spaced energy levels such that level N is the last bound level. The dissociation or activation energy for the reaction is then EN+1 = hv(N + 1). This potential energy curve is shown in Figure 1. 

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