Exact quantum mechanical treatment

Secondly, Secrest and Johnson showed that for the problem of vibrational energy transfer with soft repulsive forces, it is valid not only to close channels which are forbidden classically (negative kinetic energy) but even channels which are open may be disregarded in a particular calculation. In setting up equations (10) and (11) to examine the problem of excitation from o 0 - = 1 all 

There are presently two main difficulties which handicap attempts at exact calculation. The first concerns the intermolecular potential, and the hazards of extrapolation from models derived from viscosity measurements have been discussed. Furthermore, such a method is of dubious validity for polyatomic molecules, because the intermolecular repulsive potential will generally appear to become progressively shallower with increasing molecular dimensions if the viscosity data are cast, for example, in the Lennard-Jones form. Energy transfer depends 

The second main difficulty concerns simultaneous rotational excitation, which does play a role, though probably a minor one, in the vibrational relaxation of hydrides. Presently, there is no satisfactory theory, and little direct experimental evidence, to show that rotational excitation is strongly coupled to vibrational relaxation. 

A number of applications of the theory may be found in the various sections which follow. 

So far only 1 0 transfers have been considered. For a molecule excited to a 

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It is true that the structure, energy, and many properties ofa molecule can be described by the Schrodingcr equation. However, this equation quite often cannot be solved in a straightforward manner, or its solution would require large amounts of computation time that are at present beyond reach, This is even more true for chemical reactions. Only the simplest reactions can be calculated in a rigorous manner, others require a scries of approximations, and most arc still beyond an exact quantum mechanical treatment, particularly as concerns the influence of reaction conditions such as solvent, temperature, or catalyst. 

The last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Molecular mechanics The accurate predictions of the structure and physical properties for a molecule can be made from an exact quantum mechanical treatment of every atom within the molecular system. However, a simpler molecular mechanical treatment is applied to solve a complex macromolecular system. 

The simplified model described in this paper should be compared with a formal and exact quantum mechanical treatment of the photofragmentation of triatomic molecules. According to the theory the helicity X (corresponding to M ) is restricted by X=min(J, J). This is consistent with the assignment M =0 in our experiment as a result of the specific conditions J>J. and J. V/O. 

For not too many spins this basis is often small enough to allow for exact diago-nalization of the SH and therefore exact quantum mechanical treatments of the spin-physics in the SH framework. For high-dimensional SH problems, both brute-force [7] and a variety of perturbation flieoretieal methods ean be employed in order to arrive at exact or good approximate solutions. 

Although the theory of photodissociation has not yet reached the level of sophistication of experiment, major advances have been made in recent years by many research groups. This concerns the calculation of accurate multi-dimensional potential energy surfaces for excited electronic states and the dynamical treatment of the nuclear motion on these surfaces. The exact quantum mechanical modelling of the dissociation of a triatomic molecule is nowadays practicable without severe technical problems. Moreover, simple but nevertheless realistic models have been developed and compared against exact calculations which are very useful for understanding the interrelation between the potential and the nuclear dynamics on one hand and the experimental observables on the other hand. 

The exact shape of the potential energy curve is different for each possible pair of atoms, and can only be calculated by a detailed quantum mechanical treatment. One convenient approximate potential for covalent bonds is the Lennard-Jones 6-12 potential, shown as the dashed line in Figure 3.9 and in more detail in Figure 3.10. 

In the quantum mechanical treatment of vibrational normal modes, the vibrational Schrddinger equation is separated into individual harmonic oscillator equations by exactly the same transformation of variables [28]. 

To answer these and other questions, in this appendix we describe the quantum mechanical treatment for two spins. The description provides one of the few examples of an exact solution of the Schrodinger equation. 

One of these was written by Nikitin (1968) another one by Light (1971), emphasizing quantum calculations with coupled equations and related approximations. An overview of the quantum theory has recently been presented by Kouri (1973) who concentrated on exact quantum mechanical methods. In the present review we wish to include computational and analytical developments of the quantum theory, and both approximate and formally exact treatments. We have covered the literature since 1970 and have referred to previous articles only occasionally, when they have been a basic source of information. The coverage extends to articles that had appeared or were scheduled to appear by the end of 1973. 

Nevertheless, as we saw in the preceding section, semiclassical formulas give exact enough results even in the quantum region I > h. More exact relations can be easily derived by means of a quantum-mechanical treatment (9 10). 

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