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A Lanczos-based algorithm for sum-over-states calculations of NMR spin--spin coupling constants at the RPA level of theory: The Fermi-contact term

Sarah L. V. Zahn, Luna Zamok, Sonia Coriani, Stephan P. A. Sauer

Abstract

The analysis of nuclear magnetic resonance parameters, such as the indirect nuclear spin-spin coupling constants, in terms of contributions from localised molecular orbitals is a commonly used approach for gaining a deeper understanding of experimentally observed trends in these parameters. In the vast majority of these studies, contributions from pairs of one occupied and one virtual orbital are calculated and analyzed. Analyses in terms of two pairs of an occupied and a virtual orbital, that would allow for the study of coupling pathways, are much more seldom, as they require calculating the coupling constants as a sum over all excited states. Previous studies have shown that, for the often dominating Fermi-contact contribution to the coupling constants, more or less all excited states have to be calculated when employing a Davidson algorithm, because the most high-lying excited states can also make a significant contribution to the Fermi-contact term. In this study we investigated therefore, whether by employing a Lanczos algorithm one can obtain converged values of the Fermi-contact contribution to the indirect nuclear spin-spin coupling constants already with a significantly smaller percentage of the total number of excited states included in the sum-over-states expression. To this purpose we have extended the recent implementation of a Lanczos algorithm for the RPA/TDHF or TDDFT eigenvalue problem in the Dalton program (L. Zamok et al. J. Chem. Phys. 156, 014102 (2022)). The new procedure was tested on 17 molecules containing first, second and third row atoms. We find that, for most coupling constants, less than 50% of the excited pseudo states are necessary for converging the Fermi-contact term with an error of less than 0.5 Hz. For the few exceptions, typically for molecules with third-row atoms, around 60% were necessary.

A Lanczos-based algorithm for sum-over-states calculations of NMR spin--spin coupling constants at the RPA level of theory: The Fermi-contact term

Abstract

The analysis of nuclear magnetic resonance parameters, such as the indirect nuclear spin-spin coupling constants, in terms of contributions from localised molecular orbitals is a commonly used approach for gaining a deeper understanding of experimentally observed trends in these parameters. In the vast majority of these studies, contributions from pairs of one occupied and one virtual orbital are calculated and analyzed. Analyses in terms of two pairs of an occupied and a virtual orbital, that would allow for the study of coupling pathways, are much more seldom, as they require calculating the coupling constants as a sum over all excited states. Previous studies have shown that, for the often dominating Fermi-contact contribution to the coupling constants, more or less all excited states have to be calculated when employing a Davidson algorithm, because the most high-lying excited states can also make a significant contribution to the Fermi-contact term. In this study we investigated therefore, whether by employing a Lanczos algorithm one can obtain converged values of the Fermi-contact contribution to the indirect nuclear spin-spin coupling constants already with a significantly smaller percentage of the total number of excited states included in the sum-over-states expression. To this purpose we have extended the recent implementation of a Lanczos algorithm for the RPA/TDHF or TDDFT eigenvalue problem in the Dalton program (L. Zamok et al. J. Chem. Phys. 156, 014102 (2022)). The new procedure was tested on 17 molecules containing first, second and third row atoms. We find that, for most coupling constants, less than 50% of the excited pseudo states are necessary for converging the Fermi-contact term with an error of less than 0.5 Hz. For the few exceptions, typically for molecules with third-row atoms, around 60% were necessary.
Paper Structure (11 sections, 39 equations, 22 figures, 1 algorithm)

This paper contains 11 sections, 39 equations, 22 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theory
  3. The NMR spin--spin coupling constant as a sum-over-states property
  4. RPA excitation energies from a Lanczos algorithm
  5. Computational details
  6. Results and Discussion
  7. One-bond couplings
  8. Two-bond couplings
  9. Geminal couplings
  10. Vicinal couplings
  11. Conclusion

Figures (22)

  • Figure 1: The fraction fr$_N$ of the full space of excitations at which the Fermi-contact terms, $^1J^\textrm{FC}(M,N)$), converge with an allowed deviation of 0.5 Hz for all one-bond couplings.
  • Figure 2: The fraction of the full space fr$_N$ at which the Fermi-contact terms converge with an allowed deviation of 0.5 Hz for all X-H one-bond couplings.
  • Figure 3: The fraction of the full space fr$_N$ at which the Fermi-contact terms converges with an allowed deviation of 0.5 Hz for all X-Y one-bond couplings.
  • Figure 4: C$_2$H$_2$: The Fermi-contact term for the one-bond coupling between C1 and H1, $^1J^{\textrm{FC}}$(C1-H1), as a function of the Lanczos chain length. The calculations were carried out with the property gradient for the FC operator of C1 as start vector. The reference value of the FC term calculated as linear response function is shown as solid line.
  • Figure 5: C$_2$H$_2$: The Fermi-contact term for the one-bond coupling between C2 and H2, $^1J^{\textrm{FC}}$(C2-H2), as a function of the Lanczos chain length. The calculations were carried out with the property gradient for the FC operator of C1 as start vector. The value of the FC term calculated as linear response function is shown as solid line.
  • ...and 17 more figures