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Title: Resonance positions and lifetimes for flexible complex absorbing potentials

Abstract

By adding any complex absorbing potential (CAP) -i{lambda}V(r-vector) to a system Hamiltonian, the corresponding complex eigenvalues are analytical functions of {lambda},E{sub j}({lambda}). It is shown here that for a quite general flexible class of CAP's the real part of lim{sub {lambda}}{sub /0}E{sub j}({lambda}) provides the resonance energy. The imaginary part of that limit is the resonance width (i.e., inverse lifetime) in spite of the fact that Im E{sub j}(0)=0. The need for the Pade approximation within this approach is explained. Application to an illustrative numerical test case model Hamiltonian is given. This method could open a gate for studying systems that could not be studied until now due to the complexity of the numerical computations. In particular, one may in this way obtain resonance energies by a modification, in a straightforward simple manner, of the widely used conventional methods that were developed for the calculations of bound states.

Authors:
; ;  [1]
  1. Department of Chemistry and Minerva Center of Nonlinear Physics in Complex Systems, Technion--Israel Institute of Technology, Haifa 32000 (Israel)
Publication Date:
OSTI Identifier:
20786507
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.72.052704; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; EIGENFUNCTIONS; EIGENVALUES; HAMILTONIANS; LIFETIME; PADE APPROXIMATION; POTENTIALS; RESONANCE; SCHROEDINGER EQUATION; VECTORS

Citation Formats

Lefebvre, Roland, Sindelka, Milan, and Moiseyev, Nimrod. Resonance positions and lifetimes for flexible complex absorbing potentials. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Lefebvre, Roland, Sindelka, Milan, & Moiseyev, Nimrod. Resonance positions and lifetimes for flexible complex absorbing potentials. United States. doi:10.1103/PHYSREVA.72.0.
Lefebvre, Roland, Sindelka, Milan, and Moiseyev, Nimrod. Tue . "Resonance positions and lifetimes for flexible complex absorbing potentials". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786507,
title = {Resonance positions and lifetimes for flexible complex absorbing potentials},
author = {Lefebvre, Roland and Sindelka, Milan and Moiseyev, Nimrod},
abstractNote = {By adding any complex absorbing potential (CAP) -i{lambda}V(r-vector) to a system Hamiltonian, the corresponding complex eigenvalues are analytical functions of {lambda},E{sub j}({lambda}). It is shown here that for a quite general flexible class of CAP's the real part of lim{sub {lambda}}{sub /0}E{sub j}({lambda}) provides the resonance energy. The imaginary part of that limit is the resonance width (i.e., inverse lifetime) in spite of the fact that Im E{sub j}(0)=0. The need for the Pade approximation within this approach is explained. Application to an illustrative numerical test case model Hamiltonian is given. This method could open a gate for studying systems that could not be studied until now due to the complexity of the numerical computations. In particular, one may in this way obtain resonance energies by a modification, in a straightforward simple manner, of the widely used conventional methods that were developed for the calculations of bound states.},
doi = {10.1103/PHYSREVA.72.0},
journal = {Physical Review. A},
number = 5,
volume = 72,
place = {United States},
year = {Tue Nov 15 00:00:00 EST 2005},
month = {Tue Nov 15 00:00:00 EST 2005}
}
  • Analytical expressions for the resonances of the long-range potential (LRP), V(r)=a/r-b/r{sup 2}, as a function of the Hamiltonian parameters were derived by Doolen a long time ago [Int. J. Quant. Chem. 14, 523 (1979)]. Here we show that converged numerical results are obtained by applying the shifted complex scaling and the smooth-exterior scaling (SES) methods rather than the usual complex coordinate method (i.e., complex scaling). The narrow and broad shape-type resonances are shown to be localized inside or over the potential barrier and not inside the potential well. Therefore, the resonances for Doolen LRP's are not associated with the tunnelingmore » through the potential barrier as one might expect. The fact that the SES provides a universal reflection-free absorbing potential is, in particular, important in view of future applications. In particular, it is most convenient to calculate the molecular autoionizing resonances by adding one-electron complex absorbing potentials into the codes of the available quantum molecular electronic packages.« less
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  • Five different forms of complex absorbing potentials are examined and compared. Such potentials are needed to absorb wavepackets near the edges of grids in time-dependent quantum dynamical calculations. The extent to which the different potentials transmit or reflect an incident wavepacket is quantified, and optimal potential parameters to minimize both the reflection and transmission for each type of potential are derived. A rigorously derived scaling procedure, which permits the derivation of optimal potential parameters for use with any chosen mass or kinetic energy from those optimized for different conditions, is described. Tables are also presented which permit the immediate selectionmore » of the parameters for an absorbing potential of a particular form so as to allow a preselected (very small) degree of transmitted plus reflected probability to be attained. It is always desirable to devote a minimal region to the absorbing potential, while at the same time effectively absorbing all of the wavepacket and neither transmitting nor reflecting any of it. The tables presented here enable the use to easily select the potential parameters he will require to attain these goals. 23 refs., 7 figs., 4 tabs.« less
  • No abstract prepared.
  • The conditions for optimal reflection-free complex-absorbing potentials (CAPs) are discussed. It is shown that the CAPs as derived from the smooth-exterior-scaling transformation of the Hamiltonian [J. Phys. B 31, 1431 (1998)] serve as optimal reflection-free CAPs (RF CAPs) in wave-packet propagation calculations of open systems. The initial wave packet, {phi}(t=0), can be located in the interaction region (as in half collision experiments) where the CAPs have vanished or in the asymptote where V{sub CAP}{ne}0. As we show, the optimal CAPs can be introduced also in the region where the physical potential has not vanished. The unavoided reflections due to themore » use of a finite number of grid points (or basis functions) are discussed. A simple way to reduce the 'edge-grid' reflection effect is described.« less