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

TL;DR

This paper introduces a Lanczos-based algorithm to efficiently compute the Fermi-contact contribution to NMR spin--spin coupling constants at the RPA level, significantly reducing the number of excited states needed for convergence.

Contribution

The study extends a Lanczos algorithm implementation to calculate sum-over-states NMR parameters, enabling faster convergence with fewer excited states compared to traditional Davidson methods.

Findings

Less than 50% of excited states needed for convergence in most cases.

Around 60% of states required for molecules with third-row atoms.

Achieved error less than 0.5 Hz in coupling constants.

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.