Classical Mechanical Treatment

The classical mechanical problem of two coupled identical harmonic oscillators is described by the classical mechanical Hamiltonian 

The G (mass and geometry dependent) and F (mass and geometry independent) matrices are discussed by Wilson et al, (1955). For two bond stretches coupled by a shared atom, the relevant F and G matrix elements are defined by 

The MD approach has also been extensively used in gas-surface calculations and the first calculations already appeared in the sixties [71]. Thus the atoms are divided in two groups 

The interaction potential is, as mentioned, often assumed to be a pairwize additive potential containing terms such as Vn(Rij), VniRik) and Viiifi-ki), where the indices refer to the group given above. In the harmonic approximation the latter interaction will be 

The coupled equations (6.3)-(6.6) can be solved using standard integrators such as Runge-Kutta, Predictor-Corrector, and so on. However, using eqs. (6.3) and (6.5) we can obtain equations of the type 

Higher-order expansions involving At, At, and so on may easily be derived. 

The potential for the solid, i.e., V22. is usually just taken in its harmonic approximation including, say, nearest and next-nearest neighbor interaction. The potential may then be expanded to second order in the displacement coordinates as 

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A further advance occurred when Chesnavich et al. (1980) applied variational transition state theory (Chesnavich and Bowers 1982 Garrett and Truhlar 1979a,b,c,d Horiuti 1938 Keck 1967 Wigner 1937) to calculate the thermal rate coefficient for capture in a noncentral field. Under the assumptions that a classical mechanical treatment is valid and that the reactants are in equilibrium, this treatment provides an upper bound to the true rate coefficient. The upper bound was then compared to calculations by the classical trajectory method (Bunker 1971 Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979) of the true thermal rate coefficient for capture on the ion-dipole potential energy surface and to experimental data (Bohme 1979) on thermal ion-polar molecule rate coefficients. The results showed that the variational bound, the trajectory results, and the experimental upper bound were all in excellent agreement. Some time later, Su and Chesnavich (Su 1985 Su and Chesnavich 1982) parameterized the collision rate coefficient by using trajectory calculations. 

Thus, this completely wave mechanical development leads to a picture in which pi is a vector with fixed z component but x and y components that vary sinusoidally, 90° out of phase—i.e., executing a circular motion, as shown in Fig. 2.2. Overall, then, the motion is a precession about the magnetic field axis and is similar to that which would have been obtained from a classical mechanical treatment... 

The kind of quantum mechanical studies described in the previous sections can be performed only for tri- or at most tetratomics. For larger systems, a classical mechanics treatment is the only alternative beyond statistical approaches. The general aspects of classical dynamics studies and several examples will be discussed in this section. 

A quantum-mechanical treatment has been given for the coherent excitation and detection of excited-state molecular vibrations by optical absorption of ultrashort excitation and probe pulses [66]. Here we present a simplified classical-mechanical treatment that is sufficient to explain the central experimental observations. The excited-state vibrations are described as damped harmonic oscillations [i.e., by Eq. (11) with no driving term but with initial condition Q(0) < 0.] We consider the effects of coherent vibrational oscillations in Si on the optical density OD i at a single wavelength k within the Sq -> Si absorption spectrum. Due to absorption from Sq to Si and stimulated emission from Si and Sq,... 

Thus, the quantum mechanical and classical mechanical treatments of nuclear magnetic resonance closely correspond, as has been demonstrated mathematically [ ] ... 

The name random phase approximation comes from a classical mechanical treatment of the collective properties of the electron gas by Bohm and Pines in the eaxly 1950s. 

Consider the classical-mechanical treatment of two interacting particles of masses mi and m2. We specify their positions by the radius vectors ri and r2 drawn from the origin... 

So-called ab initio molecular dynamics techniques in which the DFT (usually just in its LDA approximation) is combined with a classical mechanical treatment of the nuclear (ion) motion have been a very popular way of studying condensed matter, i.e., the dynamics of liquids and solids. These techniques may, for instance, be used to study dynamic processes such as binding and atom diffusion [240] at surfaces, and in principle also reactivity at surfaces, without resorting to the usual procedure, namely that of determining the potential energy surface first and then doing the nuclear dynamics afterwards. 

Molecule-surface scattering calculations have been carried out treating all six dimensions of the diatomic (hydrogen) quantum mechanically [270, 271]. Comparison [270, 272] with calculations using a classical mechanical treatment of the dynamics confirmed results known from gas-phase dynamics, namely that average quantitites often are well predicted using classical dynamics, whereas state-resolved quantities are often poorly predicted by classical mechanics (for an early study see ref. [273]). 

Finally, we note that although all of the discussion in this section has applied to a completely classical mechanical treatment of the reaction, the expression for the cumulative reaction probability can be quantized in the usual ad hoc fasion in statistical theory by replacing the classical flux of equation (21) by the quantum mechanical integral density of states... 

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