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Effective Core Potentials for calculations of continuum spectra of molecules using the molecular R-matrix method

Zdenek Masin, Jakub Benda, Martin Crhan, Gregory S. J. Armstrong, Anna Nelson, Sebastian Mohr, Jonathan Tennyson

TL;DR

This work introduces a robust implementation of Effective Core Potentials (ECPs) into the molecular R-matrix framework UKRmol+ and the QEC interface, enabling accurate continuum-state calculations for molecules containing heavy elements. By deriving and implementing general expressions for ECP integrals over both Gaussian-type and B-spline orbitals, and by translating B-spline orbitals in momentum space, the approach handles 1-, 2-, and 3-centre integrals essential for scattering and photoionization in the presence of ECPs. The authors demonstrate the method on Br$_2$, SiBr$_4$, WH, and CH$_3$I, showing close agreement with all-electron benchmarks and experiment while enabling calculations that would be impractical with all-electron cores. The work broadens the applicability of molecular scattering calculations to heavy-atom targets, supports energy ranges up to ~100 eV with BTOs, and lays groundwork for future inclusion of spin-orbit ECPs and improved quadrature strategies. Overall, this represents a significant step toward efficient and relativistically informed continuum simulations in molecular systems.

Abstract

Implementation of Effective Core Potentials (ECPs) into the molecular scattering suite UKRmol+ is presented together with a set of calculations for a range of targets relevant for plasma modeling. Continuum description in scattering and photoionization calculations for large targets or high-energy electrons often requires the use of numerical continuum functions and the associated molecular integrals. We derive expressions for ECP integrals over B-spline type orbitals using their momentum-space representation and describe their implementation. Sample calculations are presented for electron collision from bromine molecule (Br$_2$), silicon tetrabromide (SiBr$_4$) and tungsten hydride (WH) as well as photoionisation of methyl iodide (CH$_3$I).

Effective Core Potentials for calculations of continuum spectra of molecules using the molecular R-matrix method

TL;DR

This work introduces a robust implementation of Effective Core Potentials (ECPs) into the molecular R-matrix framework UKRmol+ and the QEC interface, enabling accurate continuum-state calculations for molecules containing heavy elements. By deriving and implementing general expressions for ECP integrals over both Gaussian-type and B-spline orbitals, and by translating B-spline orbitals in momentum space, the approach handles 1-, 2-, and 3-centre integrals essential for scattering and photoionization in the presence of ECPs. The authors demonstrate the method on Br, SiBr, WH, and CHI, showing close agreement with all-electron benchmarks and experiment while enabling calculations that would be impractical with all-electron cores. The work broadens the applicability of molecular scattering calculations to heavy-atom targets, supports energy ranges up to ~100 eV with BTOs, and lays groundwork for future inclusion of spin-orbit ECPs and improved quadrature strategies. Overall, this represents a significant step toward efficient and relativistically informed continuum simulations in molecular systems.

Abstract

Implementation of Effective Core Potentials (ECPs) into the molecular scattering suite UKRmol+ is presented together with a set of calculations for a range of targets relevant for plasma modeling. Continuum description in scattering and photoionization calculations for large targets or high-energy electrons often requires the use of numerical continuum functions and the associated molecular integrals. We derive expressions for ECP integrals over B-spline type orbitals using their momentum-space representation and describe their implementation. Sample calculations are presented for electron collision from bromine molecule (Br), silicon tetrabromide (SiBr) and tungsten hydride (WH) as well as photoionisation of methyl iodide (CHI).

Paper Structure

This paper contains 18 sections, 17 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Implementing ECPs in UKRMol+
  4. One electron basis functions
  5. Translation of the BTO orbitals
  6. Semi-local term
  7. BTO - BTO class
  8. GTO - BTO class
  9. Local term
  10. BTO - BTO class
  11. GTO - BTO class
  12. QEC implementation
  13. Results and discussion
  14. Br$_2$
  15. SiBr$_4$
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Elastic scattering and momentum transfer cross sections for Br$_2$.
  • Figure 2: Electronic excitation cross sections for Br$_2$$\Pi$ states (left) and $\Sigma$ and $\Delta$ states (right). Dashed lines are the cross sections obtained using the all-electron cc-pVTZ basis set, solid lines are the cross sections obtained using an ECP.
  • Figure 3: Eigenphases for Br$_2$. Solid lines are the results using the 28-electron ECP for each Br atom, dashed lines are the results from the all-electron calculation.
  • Figure 4: Elastic scattering and momentum transfer cross sections for SiBr$_4$.
  • Figure 5: Vibrational excitation cross sections for SiBr$_4$.
  • ...and 5 more figures