Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization.

Area of Science:

• Quantum mechanics
• Computational physics
• Theoretical chemistry

Background:

• Solving scattering problems in quantum mechanics is computationally intensive.
• Existing methods like the Kohn variational principle (KVP) can be slow and numerically unstable.
• Boundary condition errors are a significant challenge in scattering calculations.

Purpose of the Study:

• To implement and test a Minimum Error Method (MEM) for quantum mechanical scattering problems.
• To compare the efficiency and stability of MEM with the Kohn variational principle (KVP).
• To explore the utility of Chebyshev polynomials as a basis set for scattering calculations.

Main Methods:

• Utilized a variational method based on minimizing the least-squares error in (H Psi - E Psi).
• Employed Chebyshev polynomials as the basis set for representing the wave function.
• Used a preconditioned conjugate gradient (CG) method for error minimization.
• Matched wave functions to asymptotic states to avoid boundary scattering errors.

Main Results:

• The MEM, implemented with Chebyshev polynomials, proved faster and more stable than KVP.
• The method accurately handled boundary conditions by matching wave functions to asymptotic states.
• Chebyshev polynomials allowed for efficient kinetic energy evaluation and avoided artificial periodicity.
• The MEM is suitable for problems in surface science and molecular electronics where periodicity is problematic.

Conclusions:

• The Minimum Error Method with Chebyshev polynomials offers an efficient and stable alternative for solving quantum mechanical scattering problems.
• This approach overcomes limitations of traditional methods, particularly in handling boundary conditions and avoiding artificial periodicity.
• The Chebyshev basis set provides a valuable tool for complex systems in surface science and molecular electronics.