Tunneling trajectory quantum mechanics

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

Eyring and Polanyi, and Eyring and Hershfeld tried to calculate the representative trajectories on PES for the H + H2 —> H2 + H reaction but could not succeed due to lack of computational facility at that time. In the early sixties, three different research groups led by Bunker, Karplus and Polanyi developed the quassi-classical trajectory (QCT) approach, which has been proven successful except under circumstances where the quantum mechanical effects like tunneling and resonance are important. 

Alhambra and co-workers adopted a QM/MM strategy to better understand quantum mechanical effects, and particularly the influence of tunneling, on the observed primary kinetic isotope effect of 3.3 in this system (that is, the reaction proceeds 3.3 times more slowly when the hydrogen isotope at C-2 is deuterium instead of protium). In order to carry out their analysis they combined fully classical MD trajectories with QM/MM modeling and analysis using variational transition-state theory. Kinetic isotope effects (KIEs), tunneling, and variational transition state theory are discussed in detail in Chapter 15 - we will not explore these topics in any particular depth in this case study, but will focus primarily on the QM/MM protocol. 

In this case the adjustable parameters of the PES (4.41) are V0 = 18.52 kcal/mol, V = 5.62 kcal/mol, C = 1.07, 11 = 0.91, and w0 = 1.50x 1014s 1 [Bosch et al., 1990], As in the case of malonaldehyde, the PES parameters place this system between the sudden and adiabatic regimes. The PES contour map and the instanton trajectory for this case are shown in Figure 6.11. Benderskii et al. [1993] have utilized the instanton analysis to obtain the prefactor Bt = 54 and the tunneling splitting 1.4 cm-1, which is in excellent agreement with the value 1.30 cm-1 obtained by Bosch et al. [1990] from a quantum mechanical calculation. 

Obviously, purely quantum mechanical effects cannot be described when one replaces the time evolution by classical mechanics. Thus, the quasi-classical trajectory approach exhibits, e.g., the following deficiencies (i) zero-point energies are not conserved properly (they can, e.g., be converted to translational energy), (ii) quantum mechanical tunneling cannot be described. 

Ion-molecule association is seemingly well suited for the application of the quasiclassical trajectory (QCT) method (Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979). Since there is no potential barrier and the centrifugal potential is broad, quantum mechanical tunneling is typically unimportant. Energy transfer from relative translational to vibrational and/or rotational motions of the complex should be reasonably classical because of the... 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

In the previous subsection, we considered the held emitted by one atom and then absorbed by another atom as a superposition of outgoing and incoming spherical waves of multipole photons. This wave picture completely eliminates an inquiry concerning the trajectory of photons between the atoms. In fact, the path of a particle in quantum mechanics is not a well-defined notion. The most that we can state about the path of a quantum particle in many cases is that it is represented by a nondifferentiable, statistically self-similar curve [93]. For example, the path of a tunneling electron and time spending in the barrier are not still defined unambiguously [94]. Moreover, some experiments on photonic tunneling and transmission of information show the possibility of superluminal motion of photons inside an opaque barrier [95]. 

More subtle than the lack of ZPE in bound modes after the collision is the problem of ZPE during the collision. For instance, as a trajectory passes over a saddle point in a reactive collision, all but one of the vibrational (e.g., normal) modes are bound. Each of these bound modes is subject to quantization and should contain ZPE. In classical mechanics, however, there is no such restriction. This has been most clearly shown in model studies of reactive collisions (28,35), in which it could be seen that the classical threshold for reaction occurred at a lower energy than the quantum threshold, since the classical trajectories could pass under the quantum mechanical vibrationally adiabatic barrier to reaction. However, this problem is conspicuous only near threshold, and may even compensate somewhat for the lack of tunneling exhibited by quantum mechanics. One approach in which ZPE for local modes was added to the potential energy (44) has had some success in improving reaction threshold calculations. 

Classical trajectories may either under- or overestimate the rate of IVR. Some relaxation processes are not allowed classically for example, quantum mechanical tunneling through potential energy barriers. Also, in the absence of such a barrier classical mechanics may still not allow an initial zero-order state to relax, even though the state is quantum-mechanically nonstationary. In the other extreme classical mechanics may be more chaotic than quantum dynamics. Quantum mechanics often gives more structured motion with more recurrences among zero-order states than does classical mechanics. Each of these extremes is illustrated in the following. 

Classical mechanically, we know today, that periodic orbits govern the flow of trajectories in a collinear collision, In a sense, one may say that classical collinear collisions are well understood. However the world is quantal and it is of greater interest to also try and understand the quantum mechanics of collinear collisions. In the past decade various numerical techniques have been devised which enable a relatively fast and cheap evaluation of exact quantal collinear reaction probabilities. Here too, a study of classical periodic orbits has provided insight into the quantum mechanics. In section III we show how periodic orbits may be used as an analytic tool for understanding quantal phenomena such as Feshbach resonances and tunneling. ... 

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