Quantized Nuclear Motion

The theory advanced so far assumes that the nuclei move by classical mechanics, on Born-Oppenheimer (BO) energy surfaces. This is an approximation, since nuclear motion should be treated quantum mechanically like electronic motion. One may think that as long as the temperature is as high as room temperature, a classical treatment of the nuclei should not cause any problem, except possibly for the very lightest one, the proton. But at low temperatures, even heavy nuclei show quantum correction. This is particularly the case for ET problems, where we know that the coupling between electrons and nuclei is particularly important. 

FIGURE 10.19 Typical In k = E j/kT plotforET reactions. Circles are experimental numbers. 

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The techniques of discretized Feynman path integrals make the use of Eq. 41 practical for the more general case of quantized nuclear motion which is not restricted to harmonic behavior [36, 94, 99b[. Applications of this approach are discussed in Section 1.5 of this chapter. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

The calculated [using a quantized classical path (QCP) approach] and observed isotope effects and rate constants are in good agreement for the proton-transfer step in the catalytic reaction of carbonic anhydrase. This approach takes account of the role of quantum mechanical nuclear motions in enzyme reactions.208... 

Essential features of the nuclear motion associated with chemical reactions can be described by classical mechanics. The special features of quantum mechanics cannot, of course, be properly described but some aspects like quantization can, in part, be taken into account by a simple procedure which basically amounts to a proper assignment of... 

At first sight it might be surprising that the Hessian matrix, which after all in oh initio molecular orbital theory is inherently quantum mechanical, is amenable to a purely classical treatment. This is because the Born-Oppenheimer approximation (Born and Oppenheimer 1927 Wikipedia 2010) allows for a pretty good separation of the electronic and nuclear motion, allowing the latter to be treated classically. A quantum mechanical description of the simple harmonic oscillator leads to quantized energy levels given by... 

The essential problem of the Born-Oppenheimer approximation lies in the fact, that initially the electronic states are quantized whereas the motion of nuclei remains in classical form. Then the transition from the Cartesian to the normal coordinates is carried out on the basis of Newton mechanics, and finally the nuclear motion is quantized as the system of independent harmonic oscillators. This procedure represents the hierarchical type of quantization, which is a complete contradiction of the fundamental requirement of the second quantization procedure of the total Hamiltonian that must be simultaneous. 

It is necessary to notice that the crude representation (28.30) is the first and the last one where the quantization of nuclear motion can be accomplished by means of classical Newton mechanical separations of the degrees of freedom. All other representations will mix the vibrational, rotational and translational modes, and they will not be separable any more. 

The electron excitation process is caused by the interaction with the incoming atom or molecule. Since we have seen that some charge transfer takes place from the metal to the incoming atom/molecule the simplest model assumes that this charged particle can induce the e-h excitation through its interaction with the electrons. In this approach the e-h processes are viewed as additional inelastic (nonadiabatic) processes taking place on top of the adiabatic electronic adjustment connected with the charge transfer and the Bom-Oppenheimer construction of the adiabatic surfaces used for the nuclear motion. In order to facilitate the theoretical treatment of these inelastic processes, it is convenient to introduce the concept of second quantization. 

The first application of quantum theory to a problem in chemistry was to account for the emission spectrum of hydrogen and at the same time explain the stability of the nuclear atom, which seemed to require accelerated electrons in orbital motion. This planetary model is rendered unstable by continuous radiation of energy. The Bohr postulate that electronic angular momentum should be quantized in order to stabilize unique orbits solved both problems in principle. The Bohr condition requires that... 

The basic idea of the slow-motion theory is to treat the electron spin as a part of the lattice and limit the spin part of the problem to the nuclear spin rather than the IS system. The difficult part of the problem is to treat, in an appropriate way, the combined lattice, now containing the classical degrees of freedom (such as rotation in condensed matter) as well as quantized degrees of freedom (such as the electron Zeeman interaction). The Liouville superoperator formalism is very well suited for treating this type of problems. 

In terms of quantum mechanics, a system with zero energy is impossible. A quantum system must possess a minimum energy of Ev. This postulate is due to the irrepressible zero-point motion imposed on microscopic systems by the uncertainty principle and by quantization. Thus, the classical concept of nuclei in space and associated motion is replaced by the concept of a nuclear or vibrational... 

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