Phase-Amplitude Method For Calculating Resonance Energies And Widths For One-Dimensional Potentials

Emil Y. Sidky and Itzik Ben-Itzhak

J. R. Macdonald Laboratory, Physics Department, Kansas State University, Manhattan, KS 66506 USA

We present a method for evaluating accurate resonance energies and widths in any one-dimensional potential. Semi-classical theory (WKB) provides a fast estimate of resonance positions, but it can be off for the lifetimes. Conversely, there is a multitude of numerical techniques, solving the one-dimensional Schrödinger equation that give precise resonance energies and widths, but they are much more difficult to apply. The phase-amplitude method presented here yields accurate resonance energies and widths and is as easy to apply as WKB theory.

Resonances in a one-dimensional potential, which have a local minimum, can be found in many fields. For example, field ionization of atoms and molecules exposed to a strong electrostatic field. Metastable molecules and molecular ions can also decay by tunneling. The tunneling rate is a very sensitive probe of the potential energy curve. Measurements of the resonance widths or their inverse, the mean lifetime, thus provide a stringent test for theoretical treatment of molecular structure of few and many electron systems. Obviously, the WKB approximation is not accurate enough for such a task. Numerically exact treatments are available1; however, these methods are not easy to apply mainly because of the time consuming search for very narrow resonances coupled with the difficulty to define automatic criteria for such searches. Moreover, most methods rely on direct integration of the Schrödinger equation, which is numerically challenging when many nodes in the wave function are present or there is a strong exponential growth caused by large potential barriers.

The efficient calculation of resonances in one-dimensional potentials, presented here, is based on the Milne phase-amplitude solution of the Schrödinger equation2. The wave function is expressed in terms of two real variables

;

is the log of the amplitude, and is the phase of the wave function. Substituting into the wave equation leads to two couple non-linear equations for and

where

Such a parameterization of the wave function lends itself well to numerical solution of the Schrödinger equation, because and are smooth, finite functions over the whole domain of any potential V(r). In tunneling regions of the potential grows monotonically, and does not diverge rapidly because it is the logarithm of the wave function amplitude. In regions where the momentum is high the wave function oscillates rapidly, but the phase increases monotonically. Resonances in V(r) are found by examining ddE as a function of energy, as shown in figure 1(a). Resonances appear in the energy derivative of the phase in the usual Breit-Wigner form, see figure 1(b).

In figure 1 we show the results of a calculation for the metastable electronic ground state of 3He4He2+, using the He22+ potential energy curve computed by Ackermann and Hogreve3. Four vibrational resonances are found and their mean lifetimes range from 378 s for =0 to 2.6 ps for =3.

Figure 1: (a) The phase and the natural logarithm of its derivative as a function of energy. Resonances are associated with a shift of in the former and a peak in the latter. (b) The shape of the ' peak for the =3 resonance. A Lorentzian + constant fit the peak nicely

  1. Emil Y. Sidky and Itzik Ben-Itzhak, submitted for publication in Phys. Rev. A (1999); and references therein.
  2. W. E. Milne, Phys. Rev. 35, 863 (1930).
  3. J. Ackermann and H. Hogreve, J. Phys. B 25, 4069 (1992).

Supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.

Submitted to ICPEAC XXI, Sendai, Japan in July 1999

 
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