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<p>Lambda functions constitute a Laguerre-type basis that supports highly accurate atomic structure calculations. While 150-term quadruple-precision expansions yield 30-digit Hartree–Fock energies for atoms up to the third period, heavier atoms require larger expansions, placing the computational bottleneck on two-electron integral evaluation. The algebraic framework developed by Zamastil et al. [<CitationRef CitationID="CR18">18</CitationRef>] for Coulomb Sturmians and the su(1,1)-based recurrence formalism for Lambda functions of McCoy and Caprio [<CitationRef CitationID="CR26">26</CitationRef>] were extended to formulate two-electron integrals over Lambda functions and to implement a quadruple-precision computer program. However, when the number of expansion terms exceeds 100, deterioration in numerical accuracy became significant. This problem was overcome by substituting multiple-precision arithmetic for quadruple-precision arithmetic in the numerically unstable step. The revised implementation achieves a significant speedup over our previous Gauss–Laguerre quadrature-based program and has enabled a 250-term Hartree–Fock calculation for radon with 25-digit accuracy. The theoretical derivation and computational results are presented.</p> Graphical abstract <p></p>

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Highly accurate atomic Hartree–Fock calculations by algebraic evaluation of two-electron integrals over Lambda functions

  • 期刊论文

  • 发表日期：

    2026年6月20日

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  • Shigeyoshi Yamamoto,
  • Yasuyo Hatano

摘要

Lambda functions constitute a Laguerre-type basis that supports highly accurate atomic structure calculations. While 150-term quadruple-precision expansions yield 30-digit Hartree–Fock energies for atoms up to the third period, heavier atoms require larger expansions, placing the computational bottleneck on two-electron integral evaluation. The algebraic framework developed by Zamastil et al. [18] for Coulomb Sturmians and the su(1,1)-based recurrence formalism for Lambda functions of McCoy and Caprio [26] were extended to formulate two-electron integrals over Lambda functions and to implement a quadruple-precision computer program. However, when the number of expansion terms exceeds 100, deterioration in numerical accuracy became significant. This problem was overcome by substituting multiple-precision arithmetic for quadruple-precision arithmetic in the numerically unstable step. The revised implementation achieves a significant speedup over our previous Gauss–Laguerre quadrature-based program and has enabled a 250-term Hartree–Fock calculation for radon with 25-digit accuracy. The theoretical derivation and computational results are presented.

Graphical abstract