Semiclassical approximations

Given a potential energy curve, it is possible to locate (iteratively) the vibrational energy levels using the semiclassical quantization condition 

The JWKB wavefunction defined by Eq. (5.1.3) is appropriate for either bound (two turning points) or unbound states (one turning point), provided that R is restricted to the region where E V (R). The normalization factor for the unbound yJWKB at energy E is 

In order to take care of the problem that yJWKB is not a good approximation near the turning points, it is necessary to define yusc(f ), USC indicating uniform semiclassical wavefunction (Miller and Good, 1953 Langer, 1937 Miller, 1968), which is uniformly valid close to or far from the turning points 

Thus xusc is constructed so that when R is in the classical region but far from a turning point, xusc(-R) = xJWKB(A). Note also that for a linear potential, 

A typical Airy function, Ai(—Z), is plotted in Fig. 5.3 note, however, that the argument of the Airy function that appears in xUSC is a function of R thus a plot of Ai[—Z(R) vs. R does not behave exactly like Fig. 5.3. 

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Wang H, Sun X and Miller W H 1998 Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems J. Chem. Phys. 108 9726... 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

Heller E J 1981 Frozen Gaussians a very simple semiclassical approximation J. Chem. Phys. 75 2923... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

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-... 

Approximation Property Excluding caustics we can exploit the results of semiclassical approximation theory [19]. This leads to the following statement ... 

This formula resembles (3.32) and, as we shall show in due course, this similarity is not accidental. Note that at n = 0 the short action 1 2 ( q) taken at the ground state energy Eq is not equal to the kink action (3.68). Since in the harmonic approximation for the well Tq = 2n/o)o, this difference should be compensated by the prefactor in (3.74), but, generally speaking, expressions (3.74) and (3.79) are not identical because eq. (3.79) uses the semiclassical approximation for the ground state, while (3.74) does not. 

This simulation performed on the borderline of up-to-date computational capabilities is beyond the framework of the semiclassical approximation, since A is comparable with coq. As far as real systems are concerned, such simulations are often hardly feasible for higher barriers and more degrees of freedom. On the other hand, as tests show (see section 4.1 and sequel), semiclassical methods cost incomparably less, being at the same time quite accurate, even when the barrier is not too high. 

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

Storozhev A. V., Strekalov M. L. Relaxation cross sections for transfer of rotational angular momentum in a semiclassical approximation, Chem. Phys. 153, 99-113 (1991). 

In a recent analysis carried out for a bounded open system with a classically chaotic Hamiltonian, it has been argued that the weak form of the QCT is achieved by two parallel processes (B. Greenbaum et.al., ), explaining earlier numerical results (S. Habib et.al., 1998). First, the semiclassical approximation for quantum dynamics, which breaks down for classically chaotic systems due to overwhelming nonlocal interference, is recovered as the environmental interaction filters these effects. Second, the environmental noise restricts the foliation of the unstable manifold (the set of points which approach a hyperbolic point in reverse time) allowing the semiclassical wavefunction to track this modified classical geometry. 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

Abstract. The Dirac equation is discussed in a semiclassical context, with an emphasis on the separation of particles and anti-particles. Classical spin-orbit dynamics are obtained as the leading contribution to a semiclassical approximation of the quantum dynamics. In a second part the propagation of coherent states in general spin-orbit coupling problems is studied in two different semiclassical scenarios. 

The process of formation of a bubble having a critical radius, can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the well known Wentzel-Kramers-Brillouin (WKB) approximation, the ground state energy Eq and the oscillation frequency //() of the virtual QM drop in the potential well U JV). Then it is possible to calculate in a relativistic framework the probability of tunneling as (Iida Sato 1997)... 

Dunham obtained these eigenvalues using the semiclassical approximation for the potential (1.8) which is an expansion in powers of (r - re)/re. The results for the Morse potential [Eq. (1.14)] can also be written in this form, as can results for other potentials. One therefore often uses Eq. (1.71) as a convenient empirical form. A slightly different form of (1.71) is... 

Since and M" are equivalent in the physical subspace, both Hamiltonians generate the same quantum dynamics in this subspace. However, this is not necessarily true if approximations are employed in the evaluation of the dynamics. For example, adopting a semiclassical approximation, the quantum-mechanically equivalent Hamiltonians H and El may yield different results. Experience shows that it is useful first to transform the Hamiltonian on the... 

To introduce the basic concept of a semiclassical propagator, let us consider an n-dimensional quantum system with Hamiltonian H, which is assumed to possess a well-defined classical analog. In order to obtain the semiclassical approximation to the transition amplitude between the initial... 

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

Within the applicability of the semiclassical approximation, the propagator (108) is rather insensitive to the particular value of the width parameters jj, but this parameter can of course affect the numerical efficiency of the calculation. In the numerical studies presented below, we have chosen the width y as the width of the harmonic ground state of the jth vibrational mode. In the dimensionless units used here, this choice corresponds to y = 1 for all degrees of freedom. 

Subsequently, any of the well-established semiclassical approximations for the quantum propagator can be applied. Employing, for example, the Herman-... 

Kluk propagator, the semiclassical approximation for this transition amplitude is... 

For a nonadiabatic system with spin-orbit interaction, the validity of the semiclassical approximation (based on the spin-coherent state representation) has been discussed in detail in Ref. 147. 

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