Galerkin method for unsplit 3-D Dirac equation using atomically/kinetically balanced B-spline basis

A Galerkin method is developed to solve the time-dependent Dirac equation in prolate spheroidal coordinates for an electron-molecular two-center system. The initial state is evaluated from a variational principle using a kinetic/atomic balanced basis, which allows for an efficient and accurate determination of the Dirac spectrum and eigenfunctions. B-spline basis functions are used to obtain high accuracy. This numerical method is used to compute the energy spectrum of the two-center problem and then the evolution of eigenstate wavefunctions in an external electromagnetic field.

💡 Research Summary

The paper presents a comprehensive numerical framework for solving the three‑dimensional Dirac equation without operator splitting, using a Galerkin discretization based on atomically and kinetically balanced B‑spline basis functions in prolate spheroidal coordinates. The authors focus on a two‑center electron‑molecular system, where the nuclei are fixed at positions ±R along the symmetry axis, and the external electromagnetic field is prescribed (e.g., a strong laser pulse).

The motivation stems from the limitations of existing operator‑splitting schemes, which, while efficient for time propagation, suffer from slow convergence when computing bound and continuum initial states and are prone to spurious (non‑physical) eigenvalues due to the unbounded Dirac spectrum. To overcome these issues, the authors adopt a variational Rayleigh‑Ritz approach for the time‑independent Dirac equation (TIDE) and enforce a balance between the large and small spinor components. Two balance operators are introduced: kinetic balance (L_{KB}= \frac{1}{2mc^{2}}\alpha!\cdot!p) and atomic balance (L_{AB}= \frac{1}{2mc^{2}-V_c}\alpha!\cdot!p). By constructing the basis such that the small component is expressed through the large component via these operators, the method eliminates the spurious spectrum, as rigorously proved in Theorems 3.1 and 3.2 (citing Ref.