Quantum mechanics collisions

Kaye J A and Kuppermann A 1988 Mass effect in quantum-mechanical collision-induced dissociation in collinear reactive atom diatomic molecule collisions Chem. Phys. 125 279-91... 

G. C. Schatz and M. A. Ratner (1993) Quantum Mechanics in Chemistry (Prentice-Hall, Englewood Cliffs, NJ). An advanced text emphasizing molecular symmetry and rotations, time-dependent quantum mechanics, collisions and rate processes, correlation functions, and density matrices. 

Here, i and stand for initial states, k and I for final states and a i(gfi,4>) dQ, defined for a rearrangement collision in Eq. (161) has been substituted for the classical 2nb db. Equation (230) may be derived from the quantum mechanical Boltzmann equation of Uehling and Uhlenbeck with the assumption of Boltzmann statistics, i.e., that the low-temperature symmetry effects of Fermi-Dirac or Bose-Einstein statistics are negligible, but that the quantum mechanical collision cross-section should be retained. We should note that although one may heuristically introduce the quantum mechanical cross-section as in Eq. (230), the Uehling-Uhlenbeck or a similar equation is strictly derived for special interactions only. In this connection it is interesting that the method given later in Section V-C yields the same result [Eq. (330)] as that of this section [Eq. (251)] only when an approximation equivalent to the Bom approximation is made. 

The rate equation (70.Ill) represents the full classical (semiclassical) analogue of the exact quantum-mechanical collision theory expression (51. HI). 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Miller W H 1971 Semiclassical nature of atomic and molecular collisions Accounts Chem. Res. 4 161-7 Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. Phys. 25 69-177... 

There are two basic physical phenomena which govern atomic collisions in the keV range. First, repulsive interatomic interactions, described by the laws of classical mechanics, control the scattering and recoiling trajectories. Second, electronic transition probabilities, described by the laws of quantum mechanics, control the ion-surface charge exchange process. 

Marcus R A 1966 On the analytical mechanics of chemical reactions. Quantum mechanics of linear collisions J. Chem. Phys. 45 4500... 

Miller W H 1974 Classical-limit quantum mechanics and the theory of molecular collisions Adv. Chem. 

By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

How might the interaction between two discrete particles be described by a finite-information based physics Unlike classical mechanics, in which a collision redistributes the particles momentum, or quantum mechanics, which effectively distributes their probability amplitudes, finite physics presumably distributes the two particles information content. How can we make sense of the process A scatters J5, if B s momentum information is dispersed halfway across the galaxy [minsky82]. Minsky s answer is that the universe must do some careful bookkeeping, ... 

If a particle A must know B s total information content before colliding, the collision process must be delayed until A has full access to that information. However, such a delay is consistent neither with classical nor quantum mechanics, Minsky instead suggests that the collision proceeds immediately, but with the particles both working with less than all the information that is classically required i.e, the incoming particles momenta are estimated. Outgoing momenta are determined via conventional classical rules, but, because of the estimation errors, each scattered particle leaves behind a receipt recording how much momentum was really taken away in the process. Receipts not only mark prospective event-locations at which future collisions might take place, but harbor information that can be used to estimate new real momenta. 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

Kinetics on the level of individual molecules is often referred to as reaction dynamics. Subtle details are taken into account, such as the effect of the orientation of molecules in a collision that may result in a reaction, and the distribution of energy over a molecule s various degrees of freedom. This is the fundamental level of study needed if we want to link reactivity to quantum mechanics, which is really what rules the game at this fundamental level. This is the domain of molecular beam experiments, laser spectroscopy, ah initio theoretical chemistry and transition state theory. It is at this level that we can learn what determines whether a chemical reaction is feasible. 

N. F. Mott and H. S. W. Massey (1965) The Theory of Atomic Collisions, 3rd edition (Oxford University Press, Oxford). The standard reference for the quantum-mechanical treatment of collisions between atoms. 

The results of this section can be summarized by comparison with those of the previous one. Thus, the corresponding quantities in the classical and quantum-mechanical treatments of the collision problem are given in Table 1. 

Perhaps the first clear observation of a reactive resonance in a collision experiment was recently made for the F + HD —> HF + D reaction.65-67 This reaction was one isotopomer of the F + H2 system studied in the landmark molecular beam experiments of Lee and co-workers in 1985.58 Unlike the F + H2 case, no anomalous forward peaking of the product states was reported, and results for F + HD were described as the most classical-like of the isotopes considered. Furthermore, a detailed quantum mechanical study68 of F + HD —> HF + D reaction on the accurate Stark-Werner (SW)-PES69 failed to locate resonance states. Therefore, it was surprising that the unmistakable resonance fingerprints emerged so clearly upon re-examination of this reaction. 

Finally, quantum mechanical trapping at the resonance energy can be verified using a time-delay analysis on the quantum S-matrix. In Fig. 8, the average time delay for the J = 0 partial wave of the F + HD — HF + D reaction, defined using Eq. (22), is plotted versus collision energy. A clear... 

Since chemical reactions involve the making and breaking of chemical bonds with their associated energy effects and geometric requirements, it is not unreasonable to assume that these factors play an important role in determining the probability that a bimolecular collision will lead to chemical reaction. In addition to these factors there are restrictions on bimolecular combination or association reactions and quantum mechanical requirements that can influence this probability. 

Quantitative estimates of E are obtained the same way as for the collision theory, from measurements, or from quantum mechanical calculations, or by comparison with known systems. Quantitative estimates of the A factor require the use of statistical mechanics, the subject that provides the link between thermodynamic properties, such as heat capacities and entropy, and molecular properties (bond lengths, vibrational frequencies, etc.). The transition state theory was originally formulated using statistical mechanics. The following treatment of this advanced subject indicates how such estimates of rate constants are made. For more detailed discussion, see Steinfeld et al. (1989). 

---