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Non-normal Lanczos methods for quantum scattering.

Reza Rajaie Khorasani1, Randall S Dumont

  • 1Department of Chemistry, McMaster University, 1280 Main St. W. Hamilton, Ontario L8S 4M1, Canada.

The Journal of Chemical Physics
|July 24, 2008
PubMed
Summary
This summary is machine-generated.

A new complex absorbing potential (CAP) block Lanczos method efficiently computes scattering properties but can be unstable. The Arnoldi algorithm offers a more robust alternative for non-normal systems, preventing breakdown in complex calculations.

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Area of Science:

  • Quantum mechanics
  • Computational chemistry
  • Chemical physics

Background:

  • Calculating scattering eigenfunctions and reaction probabilities is crucial in chemical dynamics.
  • Complex Absorbing Potential (CAP) methods are used to model these processes.
  • The Lanczos method is a common approach for solving such eigenvalue problems.

Purpose of the Study:

  • To introduce a novel Complex Absorbing Potential (CAP) block Lanczos method.
  • To assess the stability and efficiency of this new method for computing scattering eigenfunctions and reaction probabilities.
  • To compare the performance of the CAP-Lanczos method with the Arnoldi algorithm for non-normal systems.

Main Methods:

  • Developed a new Complex Absorbing Potential (CAP) block Lanczos method.
  • Reduced the computation of energy eigenfunctions to solving two energy-dependent systems.
  • Utilized an energy-independent block Lanczos factorization for efficient computation across all energies.

Main Results:

  • The CAP-Lanczos method can exhibit instability and breakdown due to the non-normality of CAP Hamiltonians.
  • Using a Woods-Saxon exponential CAP reduced non-normality but did not guarantee convergence.
  • The Arnoldi algorithm demonstrated superior robustness for non-normal systems, avoiding breakdown.

Conclusions:

  • The CAP-Lanczos method, while efficient, suffers from instabilities for non-normal Hamiltonians.
  • The Arnoldi algorithm is a more reliable choice for systems with non-normal Hamiltonians.
  • An Arnoldi-based approach yielded excellent results for a nonadiabatic tunneling Hamiltonian where Lanczos failed.