Semiclassical initial value calculations of the collinear helium atom.

Semiclassical calculations using the Herman-Kluk initial value treatment are performed to determine energy eigenvalues of bound and resonance states of the collinear helium atom. Both the eZe configuration (where the classical motion is fully chaotic) and the Zee configuration (where the classical dynamics is nearly integrable) are treated. The classical motion is regularized to remove singularities that occur when the electrons collide with the nucleus. Very good agreement is obtained with quantum energies for bound and resonance states calculated by the complex rotation method.

Dynamical singularities for complex initial conditions and the motion at a real separatrix.

This work investigates singularities occurring at finite real times in the classical dynamics of one-dimensional double-well systems with complex initial conditions. The objective is to understand the relationship between these singularities and the behavior of the systems for real initial conditions. An analytical treatment establishes that the dynamics of a quartic double well system possesses a doubly infinite sequence of singularities. These are associated with initial conditions that converge to those for the real separatrix as the singularity time becomes infinite. This confluence of singularities is shown to lead to the unstable behavior that characterizes the real motion at the separatrix. Numerical calculations confirm the existence of a large number of singularities converging to the separatrix for this and two additional double-well systems. The approach of singularities to the real axis is of particular interest since such behavior has been related to the formation of chaos in nonintegrable systems. The properties of the singular trajectories which cause this convergence to the separatrix are identified. The hyperbolic fixed point corresponding to the potential energy maximum, responsible for the characteristic motion at a separatrix, also plays a critical role in the formation of the complex singularities by delaying trajectories and then deflecting them into asymptotic regions of space from where they are directly repelled to infinity in a finite time.

Semiclassical initial value treatment of correlation functions.

Two semiclassical, initial value representation (IVR) treatments are presented for the correlation function psi(f) e-iHt/h psi(i), where psi(i) and psi(f), are energy eigenfunctions of a "zero-order" Hamiltonian describing an arbitrary, integrable, vibrational system. These wave functions are treated semiclassically so that quantum calculations and numerical integrations over these states are unnecessary. While one of the new approximations describes the correlation function as an integral over all phase space variables of the system, in a manner similar to most existing IVR treatments, the second approximation describes the correlation function as an integral over only half of the phase space variables (i.e., the angle variables for the initial system). The relationship of these treatments to the conventional Herman-Kluk approximation for correlation functions is discussed. The accuracy and convergence of these treatments are tested by calculations of absorption spectra for model systems having up to 18 degrees of freedom, using Monte Carlo techniques to perform the multidimensional phase space integrations. Both treatments are found to be capable of producing spectra of excited, anharmonic states that agree well with quantum results. Although generally less accurate than full phase space or Herman-Kluk treatments, the half phase space method is found to require far fewer trajectories to achieve convergence. In addition, this number is observed to increase much more slowly with the system size than it does for the former methods, making the half-phase space technique a very promising method for the treatment of large systems.