Quantum complex rotation and uniform semiclassical calculations of complex energy eigenvalues

Abstract

Articles you may be interested inUniform semiclassical selfconsistent field and adiabatic calculations of complex energy eigenvalues for nonseparable systems J. Chem. Phys. 84, 6285 (1986); 10.1063/1.450773The Prüfer phase function calculation of complex energy eigenvalues for cubic anharmonic oscillator Quantum and semiclassical calculations of complex energy eigenvalues have been carried out for an exponential potential of the form Vor' exp(-r) and Lennard-Jones (12,6) potential. A straightforward method, based on the complex coordinate rotation technique, is described for the quantum calculation of complex eigenenergies. For singular potentials, the method involves an inward and outward integration of the radial SchrOdinger equation, followed by matching of the logarithmic derivatives of the wave functions at an intermediate point. For regular potentials, the method is simpler, as only an inward integration is required. Attention is drawn to the World War II researches of Hartree and co-workers who anticipated later quantum mechanical work on the complex rotation method. Complex eigenenergies are also calculated from a uniform semiclassical three turning point quantization formula, which allows for the proximity of the outer pair of complex turning points. Limiting cases of this formula, which are valid for very narrow or very broad widths, are also used in the calculations. We obtain good agreement between the semiclassical and quantum results. For the Lennard-Jones (12,6) potential, we compare resonance energies and widths from the complex energy definition of a resonance with those obtained from the time delay definition.In Sec. V, we present quantum and semiclassical complex eigenenergies for the regular potential De = 60 cm-t, re = 3.56 A, /l = 1. 92 u, and with a conversionfactor of 1i 2 /2=16.858056ucm-1 A2. These param-

Jost function for singular potentials

“…The third column of these tables, contains the results obtained in Ref. [8] using a Complex Rotation (CR) method which in some aspects is similar to ours. The authors of that reference perform the rotation directly in the Schrödinger equation and integrate it from r = 0 outwards and from a large r max inwards.…”

“…Combined with a rotational barrier, this potential supports narrow as well as broad resonant states (see, for example, Ref. [8]). To locate them, any method employed must be pushed to the extreme, thus exhibiting its advantages and drawbacks.…”

“…The use of more complicated boundary conditions at r = 0 does not change the condition (9) for the existence of the limit (8). Indeed, in deriving this condition we used only the behavior of the potential and the Riccati-Hankel functions at large distances [1,4].…”

“…(22) it is clear that the angular momentum appears only in the phase factor of the asymptotic behavior of the Riccati-Hankel functions and hence of the regular solution. Therefore, the use of any complex ℓ cannot change the domain of the k-plane where the limit (8) exists. This means that the Jost function can be calculated, for any complex angular momentum, using the same equations.…”

“…Usually a is taken to be the inner turning point [8], but it is obvious from Eq. (15) that an additional integration from a to the turning point can only change the overall normalization of the solution which is not our concern at the moment.…”