Closed-form spin-relativistic corrections from the Dirac equation enabling a modified Schrödinger solver

Area of Science:

• Quantum Mechanics
• Computational Physics
• Atomic Physics

Background:

• The Dirac equation describes relativistic electrons but is computationally intensive.
• Non-relativistic approximations often neglect crucial spin-dependent effects.

Purpose of the Study:

• To derive a Schrödinger-like equation incorporating leading spin-relativistic corrections.
• To develop a practical computational framework bridging non-relativistic and relativistic quantum mechanics.

Main Methods:

• Derivation of a closed-form Schrödinger-like equation.
• Finite-difference discretization of the radial equation.
• Formulation as a quadratic eigenvalue problem (QEP).
• Development of an open-source QEP solver.
• Application of perturbation theory for energy and wavefunction corrections.

Main Results:

• A novel Schrödinger-like equation retaining spin-relativistic effects.
• An efficient open-source solver for the derived QEP.
• Accurate calculations for Coulomb, harmonic oscillator, Woods-Saxon, and Yukawa potentials.
• First-order corrections for 3D isotropic harmonic oscillator and Coulomb potentials.

Conclusions:

• The developed framework accurately quantifies relativistic effects.
• It provides a computationally efficient alternative to full four-component Dirac calculations.
• This method facilitates a deeper understanding of relativistic phenomena in atomic and molecular systems.