Semiclassical theory for the Maslov-type save packet: Hierarchy below the semiclassical Feynman kernel

A semiclassical theory based on the Maslov-type wave packet is reported. The present approximation constitutes a class of semiclassical theory that brings about a hierarchy of theoretical structures below or equivalent to the standard semiclassical approximation to the Feynman kernel. We first propose a semiclassical wave function; that is, what we call the action-decomposed function (ADF). An ADF is propagated in time semiclassically along a "single action surface," which is characterized in terms of a given initial momentum, with the phase being proportional to the F-2(q,p)-type generating function. An arbitrary wave function can be expanded continuously in the ADF's of different initial momenta, each of which is associated with weighting factors both in configuration and momentum spaces. Depending on the form of the weighting function, the present scheme covers a number of different levels of approximation, ranging from a "single" ADF to a semiclassical kernel. Thus our theory provides a way of connecting the Maslov-type semiclassical wave packet to the WKB theory based on the Feynman kernel. It is generally concluded that the above kernel limit attains the highest accuracy possible within the present scheme, but it requires a large number of classical trajectories to represent it. On the other hand, a single ADF yields a little less accurate results but demands the fewest representing classical trajectories due to the narrowest momentum distribution. In fact, we show numerically that the autocorrelation function represented in terms of a single ADF, from which the energy spectrum is extracted with a Fourier transformation, can be calculated with drastically fewer classical trajectories without losing much accuracy. We also present a Limitation in applying the ADF, and show an interesting symptom arising from a pathological use of the approximation. The present theory is quite promising for a spectral analysis of the vibrational states of relatively large molecules. if applied appropriately, [S1050-2947(99)10005-2].