An Adaptive Log-Laguerre Spectral Method for the Radial Dirac Equation: Resolving Asymptotic Decay and Core Singularities in Atomic Calculations

Abstract

The high-precision solution of the radial Dirac equation is fundamental to relativistic quantum chemistry, essential for reliable pseudopotential generation and all-electron electronic structure methods. Capturing both the non-polynomial singularities at the origin and the state-dependent asymptotic decay on semi-infinite domains presents a significant computational challenge. In this work, we propose the Adaptive Log-Laguerre Spectral Method (ALLSM), a novel coupled spectral-element solver that seamlessly integrates three advanced mathematical methodologies into a unified framework. Specifically, Generalized Log-Orthogonal Functions (GLOFs) are deployed in the near-core region to intrinsically approximate complex $r^s$ singular behaviors without requiring prior knowledge of the exact analytical exponent $s$. Concurrently, an adaptive Laguerre spectral method is employed to dynamically capture diverse exponential tails on $[0, \infty)$, avoiding artificial domain truncation. To structurally guarantee spectral purity across this bipartite basis, the framework rigorously incorporates the Inverse Dirac Operator Method (IDOM), effectively eliminating variational collapse and spurious states. Validated across diverse physical regimes, including Coulomb, finite-nucleus, and screened potentials, the proposed solver restores exponential convergence and consistently achieves relative accuracies of $10^{-10}$. This work provides a robust, pollution-free computational kernel for atomic structure calculations, establishing a highly reliable numerical standard for complex molecular simulations.