Harmonic oscillator expression, quantum

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... 

I he bawis of tho alternate method mentioned in Sec. 7-3 is the use of polar rather than cartesian degenerate normal coordinates. It is shown in treatises on quantum mechanics that the solution of the wave equation for the doubly degenerate harmonic oscillator expressed in terms of coordinates p and (j> defined by... 

To find q brationai for a diatomic molecule, we shall use the idealization of a harmonic oscillator. The quantum mechanical energy level expression for a harmonic oscillator developed earlier in Equahon 7.34 yields... 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

We will use the harmonic oscillator approximation to derive an equation for the vibrational partition function. The quantum mechanical expression gives the vibrational energies as... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

When the electron transfer process is coupled to classical reorientation modes and to only one harmonic oscillator whose energy quantum h( is high enough for only the ground vibrational level to be populated, the expression of the electron transfer rate is given by [4, 9] ... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

Note that if j = 1, (9.12) is formally identical with the classical expression (9.7) the classical multiple oscillator model, which will be discussed in Section 9.2, is even more closely analogous to (9.12). However, the interpretations of the terms in the quantum and classical expressions are quite different. Classically, o30 is the resonance frequency of the simple harmonic oscillator quantum mechanically 03 is the energy difference (divided by h) between the initial or ground state / and excited state j. Classically, y is a damping factor such as that caused by drag on an object moving in a viscous fluid quantum mechanically, y/... 

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.)... 

Since we have abandoned the individual quantum numbers of the degenerate modes Qx and Qy, we will replace the summation over the 3N — 6 normal modes in energy expressions by a summation over the distinct vibrational frequencies. For a linear triatomic molecule there are four normal modes, but only three distinct vibrational frequencies. In the harmonic-oscillator approximation, the energy contribution of the doubly degenerate modes is... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

In this expression, according to the theory of the quantum harmonic oscillator, the operator q appearing on the right-hand side, may couple two successive eigenstates /c ) of the Hamiltonian of the harmonic oscillator. Consequently, by ignoring the scalar term p(0,0), which does not couple these states, we may write the dipole moment operator according to... 

Consider the effective Hamiltonian (47) of a driven quantum harmonic oscillator. Since it is not diagonal, it may be suitable to diagonalize it with the aid of a canonical transformation that will affect it or its equivalent form (50), but not that of (46) or its equivalent expression (49), which is yet to be diagonal [13]. 

Then, owing to the orthonormality (n m) = 8 m of the eigenstates of the quantum harmonic oscillator Hamiltonian, the above expression reduces to... 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

Fig. 5 Levels of the two-dimensional harmonic oscillator up to tiy = ng + iig = 5, expressed in terms of ng, tig, the quantum numbers of the components Qg and Qg...

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