Dimensional scaling for quasistationary states

Abstract

Complex energy eigenvalues which specify the location and width of quasibound or resonant states are computed to good approximation by a simple dimensional scaling method. As applied to bound states, the method involves minimizing an effective potential function in appropriately scaled coordinates to obtain exact energies in the D→∞ limit, then computing approximate results for D=3 by a perturbation expansion in 1/D about this limit. For resonant states, the same procedure is used, with the radial coordinate now allowed to be complex. Five examples are treated: the repulsive exponential potential (e−r); a squelched harmonic oscillator (r2e−r); the inverted Kratzer potential (r−1 repulsion plus r−2 attraction); the Lennard-Jones potential (r−12 repulsion, r−6 attraction); and quasibound states for the rotational spectrum of the hydrogen molecule (X 1∑g+, v=0, J=0 to 50). Comparisons with numerical integrations and other methods show that the much simpler dimensional scaling method, carried to second-order (terms in 1/D2), yields good results over an extremely wide range of the ratio of level widths to spacings. Other methods have not yet evaluated the very broad H2 rotational resonances reported here (J≳39), which lie far above the centrifugal barrier.