Semiclassical wave function

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 WKB theory we developed and briefly described below is formally theoretically equivalent to the instanton theory [56-58, 71-76], but is more straightforward and practical, and probably easier to understand. Let us start with the ID case in order to comprehend the basic ideas. The semiclassical wave function is given as usual by... 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 39 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e., =... 

Figure 42. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full tine is the quantum result, and the dashed line depicts the semiclassical mapping result. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave function.

The semiclassical wave function (A. 10) in the barrier region may be written in the form... 

When we write the semiclassical wave function on the adiabatic potential En(R) as... 

For actual molecules with three or more vibrational modes methods more sophisticated than the Poincare surface of section approach must be used to solve the semi-classical quantization conditions, and several approaches have been advanced (Baker et al. 1984 Martens and Ezra, 1985 Skodje et al., 1985 Johnson, 1985 Duchovic and Schatz, 1981 Martens and Ezra, 1987 Pickett and Shirts, 1991). Semiclassical vibrational energy levels have been determined for SO2, H2O, HJ, CO2, Arj, and l2Ne . Semiclassical wave functions have also been determined for vibrational energy levels of molecules (DeLeon and Heller, 1984). 

However, one may immediately observe that the entire action eikonal may be explicitly written once the successive iterative equations are solved moreover there is noted that the zero-th order in action corresponds entirely to the classical Hamilton-Jacobi equation the restriction to the first order in h makes nonetheless the Wentzel, Kramers and Brillouin (WKB) framework for semiclassical wave-function approximation ... 

The data for the B state ) were taken from those reported by Steinfeld et al. (1965). Child (1973) has shown that this potential curve is comparable to that by a direct inversion procedure by using semiclassical wave-functions. 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 16 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e. pnorm P2/ Pi + P2)]. The normalized results for the population are seen to match the quantum reference data quantitatively. It should be emphasized that the deviation of the norm shown in Fig. 16 is not a numerical problem, but rather confirms the common wisdom that a two-level system as well as its bosonic representation is a prime example of a quantum system and therefore difficult to describe within a semiclassical theory. Nevertheless, besides the well-known problem of norm conservation, the semiclassical mapping approach clearly reproduces the nonadiabatic quantum dynamics of the system. It is noted that the semiclassical results displayed in Fig. 16 have been obtained without using filtering techniques. Due to the highly chaotic classical dynamics of the system, therefore, a very large number of trajectories ( 2 x 10 ) is needed to achieve convergence, even over... 

Various one-dimensional processes can be expressed by conneeting these diagrams and can be described by combining the appropriate semiclassical matrices. This technique is called diagrammatic technique [48,52]. When we write the semiclassical wave function on the adiabatic potential E x) with a as a reference point as... 

This semiclassical wave function does not require any special matching condition at the boundary between classically allowed and classically forbidden regions, which represents the main obstacle in the ordinary WKB theory. The tunneling splitting Aq can be calculated from the Herring formula [54],... 

The first task is to construct semiclassical wave function in the classically allowed region. As is well known, the KAM torus exists according to the Kolmogorov-Arnold-Moser (KAM) [57,58] and this integral system can be quantized by the Einstein-Brillouin-Keller (EBK) quantization rule [58] as... 

In this case the semiclassical wave function Equation (6.130) in the vicinity of the potential minimum (-Sm,0) becomes... 

As mentioned before, the longitudinal excitation always promotes tunneling, A , =i > A =o- The effect of transversal excitation, however, depends on the behavior of the effective frequency 9 x). For instance, for monotonically growing (decreasing) 9 r) the excitation of the transversal mode suppresses (promotes) the tunneling splitting. We see below that the effective frequency is generally determined as a solution of auxiliary differential equation that describes the nodal structure of the semiclassical wave function. This corresponds to the discussion of Takada and Nakamura [30,31] (see Chapter 4). 

The vector U characterizes the direction along which the semiclassical wave function has a node. At the potential minimum (t = -oo) U coincides with one of the normal modes while 0(-oo) is the corresponding normal mode frequency (excitation energy). Equation (6.191) has A - 1 independent solutions corresponding to A - 1 possible types of the transversal excitation. The fact that U indeed has the vector transformation properties can be deduced from Equation (6.191). Note also that similarly to Equation (6.164), this equation is pertinent only to the transversal excitation for which Equation (6.163) holds. 

Equations (6.161), (6.191), and (6.193) completely determine the semiclassical wave function and the tunneling splitting can now be calculated by the Herring formula [see Equation (6.84)]... 

This is readily checked by making the coordinate transformation Equation (8.12) and using the properties of The quantity Wb has the simple physical meaning as the main exponent factor of the semiclassical wave function exp[-Wb(q)], which coincides with the ground state of the harmonic oscillator in the potential well. In the same way as before, we introduce the second derivative matrix... 

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