Semiclassical Consideration

We consider first the one-dimensional problem of two adiabatic potential curves and (Pig.3) denoting by and 

X = r is a function of time, x = x(t) therefore, time t can be introduced as an adiabatic parameter instead of the nuclear coordinate. In this way we can calculate the transition probability W by solving the time-dependent Schrbdlnger equation (1.1). For this purpose 

Using (148cII) to solve (1.1) under the initial conditions 

It should be noted that the adiabatic approximation may be violated during the passage of the region x x. if the nuclear velocity dx/dt is high. However, this approximation is certainly valid outside it, i.e., in the initial and final states of the system. Under these conditions can be evaluated using the quasiclassical method of 

In this situation the probability of a non-adiabatic transition from the adiabatic state a to the adiabatic state b is given by the Landau expression /36/ 

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The full multiple spawning (FMS) method has been developed as a genuine quantum mechanical method based on semiclassical considerations. The FMS method can be seen as an extension of semiclassical methods that brings back quantum character to the nuclear motion. Indeed, the nuclear wave function is not reduced to a product of delta functions centered on the nuclear positions but retains a minimum uncertainty relationship. The nuclear wave function is expressed as a sum of Born-Oppenheimer states ... 

V(x) along the reaction coordinate (presiamed to be separable) is excluded so that only transitions over the barrier >0) are allowed in either a classical and semiclassical consideration According to (80 III) or (63.Ill), from (89.Ill) it follows that ... 

A second recent development has been the application 46 of the initial value representation 47 to semiclassically calculate A3.8.13 (and/or the equivalent time integral of the flux-flux correlation fiinction). While this approach has to date only been applied to problems with simplified hannonic baths, it shows considerable promise for applications to realistic systems, particularly those in which the real solvent bath may be adequately treated by a fiirther classical or quasiclassical approximation. 

Transition state theory, as embodied in Eq. 10.3, or implicitly in Arrhenius theory, is inherently semiclassical. Quantum mechanics plays a role only in consideration of the quantized nature of molecular vibrations, etc., in a statistical fashion. But, a critical assumption is that only those molecules with energies exceeding that of the transition state barrier may undergo reaction. In reality, however, the quantum nature of the nuclei themselves permits reaction by some fraction of molecules possessing less than the energy required to surmount the barrier. This phenomenon forms the basis for QMT. ... 

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

Due to the development of efficient initial-value representations of the semiclassical propagator, recently there has been considerable progress in the semiclassical description of multidimensional quantum processes [104—111,... 

Error bars of the experimental data were not specified. However, Hunt provided measurements at two different ortho- to para-H2 concentration ratios, 3 1 (solid squares and dots) and 1 1 (open squares and circles). According to the theory developed above, variation of this ratio should not affect the results. While symmetry considerations of the interacting H2 molecules are important at low temperatures (T < 40 K), the semiclassical approach does not distinguish para- and ortho-H2 we think that the differences of the data taken at different ortho-para ratios may, in essence, reflect the uncertainties of the measurement. We note that earlier works by Chisholm and Welsh [121] and by Hare and Welsh [175] gave values... 

For exoergic channels, there is often no accessible avoided crossing, in which case the trajectory assumptions underlying the LZS theory are violated. The nonadiabatic coupling region may extend over a considerable range of internuclear distance, and semiclassical methods using exact classical trajectories represent the minimal necessary improvement over LZS. 

A linear approximation of the potential is certainly too sweeping a simplification. In reality, Vr varies with the internuclear separation and usually rises considerably at short distances. This disturbs the perfect (mirror) reflection in such a way that the blue side of the spectrum (E > Ve) is amplified at the expense of the red side (E < 14).t For a general, nonlinear potential one should use Equations (6.3) and (6.4) instead of (6.6) for an accurate calculation of the spectrum. The reflection principle is well known in spectroscopy (Herzberg 1950 ch.VII Tellinghuisen 1987) the review article of Tellinghuisen (1985) provides a comprehensive list of references. For a semiclassical analysis of bound-free transition matrix elements see Child (1980, 1991 ch.5), for example. 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

Considerable use continues to be made of classical trajectory calculations in relating the experimentally determined attributes of electronically adiabatic reactions to the features in the potential energy surface that determine these properties. However, over the past 3 or 4 years, considerable progress has been made with semiclassical and quantum mechanical calculations with the result that it is now possible to predict with some degree of confidence the situations in which a purely classical approach to the collision dynamics will give acceptable results. Application of the semiclassical method, which utilises classical dynamics plus the superposition of probability amplitudes [456], has been pioneered by Marcus [457-466] and by Miller [456, 467-476],... 

However, a multidimensional system is generally classically nonintegrable, and so the existence of classical chaos, which more or less appears in the (complex) phase space, introduces some intrinsic difficulties to applying the semiclassical method to multidimensional tunneling. Even if we restrict ourselves to the real domain, which means that we don t take into account tunneling phenomena, the existence of chaos is a real obstacle to endowing the semiclassical method with the rigorously mathematical basis, while some practical applications of the semiclassical method work well in prediction of quantal quantities which are used to characterize the quantum chaotic nature of a system under consideration [9,10]. The extension of the phase space to the complex domain will introduce further complexities and difficulties, and there is... 

As to future directions, the problem of the canonical density matrix, or equivalently the Feynman propagator, for hydrogen-like atoms in intense external fields remain an unsolved problem of major interest. Not unrelated, differential equations for the diagonal element of the canonical density matrix, the important Slater sum, are going to be worthy of further research, some progress having already been made in (a) intense electric fields and (b) in central field problems. Finally, further analytical work on semiclassical time-dependent theory seems of considerable interest for the future. 

The motion of electrons in a magnetic field in a situation in which inhomogeneity of some kind exists remains of considerable interest at the time of writing. Therefore, in this Appendix, we shall first summarize some results of Freeman and March [49] for the current density in a simple model of independent harmonically confined electrons in a constant magnetic field. Then we shall go on to discuss the semiclassical theory of current density in atoms. 

A. This is the Bohr radius of the semiclassical approach (1913) and is often adopted as the atomic unit of length. The electron has a maximum probability of being at this distance from the nucleus but a good chance of being anywhere within a very considerable volume. 

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