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<p>We present closed-form expressions for the exchange integral between general hydrogenic orbitals and its derivatives with respect to effective decay parameters. This work is a sequel to our earlier Coulomb integral study, but here the structural difficulty is inverted: the Legendre expansion that simplified the Coulomb case becomes cumbersome due to surviving phase couplings, while the Laplace route is comparatively more tractable. The results enable fully analytic screening optimization incorporating both Coulomb and exchange contributions.</p>

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Closed-form representations of the exchange integral over hydrogenic orbitals

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Balakrishnan Viswanathan,
  • Darien DeWolf

摘要

We present closed-form expressions for the exchange integral between general hydrogenic orbitals and its derivatives with respect to effective decay parameters. This work is a sequel to our earlier Coulomb integral study, but here the structural difficulty is inverted: the Legendre expansion that simplified the Coulomb case becomes cumbersome due to surviving phase couplings, while the Laplace route is comparatively more tractable. The results enable fully analytic screening optimization incorporating both Coulomb and exchange contributions.