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

TL;DR

This paper introduces an adaptive spectral-element method combining Laguerre and Log-Orthogonal functions to accurately solve the radial Dirac equation across different physical regimes, overcoming traditional basis-set limitations.

Contribution

It develops a novel high-precision adaptive spectral framework that effectively captures singularities and asymptotic decay without domain truncation or prior knowledge of singular behavior.

Findings

Achieves exponential convergence with relative errors of 10^{-10} in atomic calculations.

Successfully handles Coulomb, finite-nucleus, and screened potentials.

Restores spectral accuracy and eliminates spurious states in solutions.

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. However, standard basis-set approaches struggle to simultaneously capture two distinct physical regimes: the non-polynomial singularities at the origin and the state-dependent, multi-scale asymptotic decay of wavefunctions on semi-infinite domains. In this work, we propose a high-precision adaptive spectral-element framework designed to rigorously resolve these spatial challenges. To capture the diverse exponential decay behavior on $[0, \infty)$ without arbitrary domain truncation, an adaptive generalized Laguerre spectral method is introduced, dynamically optimizing the basis scaling factors. Concurrently, near-origin non-polynomial {$r^s$} singularities are resolved utilizing Log-Orthogonal Functions, a basis that intrinsically approximates complex singular behaviors without requiring prior knowledge of the exact analytical exponent {$s$}. Furthermore, the framework incorporates an inverse operator formulation to guarantee spectral purity and eliminate spurious states. Validated across diverse physical regimes, including Coulomb, finite-nucleus, and screened potentials, the proposed method restores exponential convergence and consistently achieves relative accuracies of $10^{-10}$ {in Hartree atomic units or electron volts}. This work provides a robust, non-pollution computational kernel for atomic structure calculations, establishing a numerical standard for generating high-precision atomic data in complex molecular simulations.