Hyperion: A New Computational Tool for Relativistic Ab Initio Hyperfine Coupling

Herein we describe Hyperion, a new program for computing relativistic picture-change-corrected magnetic resonance parameters from scalar relativistic active space wave functions, with or without spin–orbit coupling (SOC) included a posteriori. Hyperion also includes a new orbital decomposition method for assisting active space selection for calculations of hyperfine coupling. For benchmarking purposes, we determine hyperfine coupling constants of selected alkali metal, transition metal, and lanthanide atoms, based on complete active space self-consistent field spin–orbit calculations in OpenMolcas. Our results are in excellent agreement with experimental data from atomic spectroscopy as well as theoretical predictions from four-component relativistic calculations.


S1 Investigation of active space selection using orbital decomposition analysis
To investigate the influence of various orbital optimisation strategies on the computed HFCCs, we choose two small test systems, the 7 Li atom and the 14 N atom, and represent both using an ANO-RCC...8s7p basis set (containing 8 s functions and 7 sets of p functions).
We have already shown that active spaces comprising only s and p shells are sufficient to obtain 7 Li HFCCs in excellent agreement with experiment (Table 1, main text); meanwhile, previous work S1-S3 suggests that our chosen basis set is insufficient to achieve quantitative agreement with the experimental 14 N HFCC, A = 10.4509 MHz. S1 Nevertheless, in this case we are interested not in the accuracy of the computed HFCC, but in the HFC orbital decomposition analysis. We aim to model HFC in the full orbital space, which is achievable via RASCI(3,29) for the chosen test systems. Figures S1 and S2 show orbital decomposition diagrams for the 2 S 1/2 ground state of 7 Li and for the 4 S 3/2 ground state of 14 N, respectively; all diagrams are obtained from RASCI(3,29) calculations using CASSCF-and RASSCF-optimised orbitals.
It is immediately apparent that, as the number of optimised orbitals increases, the diagrams become more sparse and the HFC response is focused into a smaller subset of the orbital space. Surprisingly, CASSCF and RASSCF orbitals obtained using the same active space selection give rise to extremely similar diagrams -the same pairwise couplings are observed in both cases, with only a few small changes in magnitude. The most noticeable difference is the coupling between optimised orbitals, which appears to be slightly stronger when RASSCF is used. The larger HFCCs observed for calculations using RASSCF orbitals are likely a result of this stronger coupling, especially as the variations in HFCC are small.
These observations suggest that, within the multiconfigurational electronic structure frame-S-2 work, large active space optimisations of the orbital space result in the most effective representation of HFC. Compared to active space size, the constraints introduced by a RAS ansatz appear to have a minimal effect on the HFC orbital decomposition and the resulting HFCC. Of course, these conclusions are based on a limited number of small systems and might have limited applicability; further investigation is necessary to establish the validity of such trends in larger, more complex systems.
Finally, it is worth highlighting that orbital energy is not a good indicator of the contribution to HFC. In both 7 Li and 14 N, high-energy diffuse functions couple most strongly with the SOMO and the other valence orbitals, while "mid-range" orbitals (e.g. 2p-dominated functions in 7 Li, 5p-dominated functions in 14 N) have much smaller contributions. In the context of theoretical HFCC determinations, this is not a new observation; Feller and Davidson remarked, in a 1988 study of second-period atomic HFCCs, S4 that "[...] the choice of energy as the selection criterion is probably far from optimal for properties other than energy. However, when an entire group of properties is desired (including the energy) it may be as good a choice as any other." In ab initio studies that target HFCC accuracy, it is therefore essential to explore orbital selection criteria that are not energy-based; this is particularly relevant given the recent developments around automated active space selection methodologies. S5-S7 Figure S1: 7 Li HFCCs and HFC orbital decomposition diagrams determined from RASCI(3,29) wavefunctions using different CASSCF/RASSCF-optimised orbitals. Orbitals are labelled according to the dominant AO basis function contribution. Orange labels correspond to RAS2 orbitals.    Figure S3: Spin-dependent (FC+SD) and spin-independent (PSO) unsigned HFCCs computed by Hyperion for selected energy levels of Cr group and Fe group atoms. S-8