Shape resonances as poles of the semiclassical Green's function obtained from path-integral theory: application to the autodissociation of the He2++ 1sigma(g)+ state.
It is known that one-dimensional potentials, V(R), with a local minimum and a finite barrier towards tunneling to a free particle continuum, can support a finite number of shape resonance states. Recently, we reported a formal derivation of the semiclassical Green's function, G(SC)(E), for such V(R), with one and two local minima, which was carried out in the framework of the theory of path integrals [Th. G. Douvropoulos and C. A. Nicolaides, J. Phys. B 35, 4453 (2002); J. Chem. Phys. 119, 8235 (2003)]. The complex poles of G(SC)(E) represent the energies and the tunneling rates of the unstable states of V(R). By analyzing the structure of G(SC)(E), here it is shown how one can compute the energy, E(nu), and the radiation-less width, gamma(nu), of each resonance state beyond the Wentzel-Kramers-Brillouin approximation. In addition, the energy shift, delta(nu), due to the interaction with the continuum, is given explicitly and computed numerically. The dependence of the accuracy of the semiclassical calculation of E(nu) and of gamma(nu) on the distance from the top of the barrier is demonstrated explicitly. As an application to a real system, we computed the vibrational energies, E(nu), and the lifetimes, tau(nu), of the 4He2++, nu = 0, 1, 2, 3, 4, and 4He3He++ nu = 0, 1, 2, 3, 1sigma(g)+ states, which autodissociate to the He(+)+He+ continuum. We employed the V(R) that was computed by Wolniewicz [J. Phys. B 32, 2257 (1999)], which was reported as being accurate, over a large range of values of R, to a fraction of cm(-1). For example, for J = 0, the results for the lowest and highest vibrational levels for the 4He2+ 1sigma(g)+ state are nu = 0 level, E0 = 10,309 cm(-1) below the barrier top, tau0 = 6400 s; nu = 4 level, E4 = 96.6 cm(-1) below the barrier top, tau4 = 31 x 10(-11) s. A brief presentation is also given of the quantal methods (and their results) that were applied previously for these shape resonances, such as the amplitude, the exterior complex scaling, and the lifetime matrix methods.