Implementation and First Evaluation of Strong-Correlation-Corrected Local Hybrid Functionals for the Calculation of NMR Shieldings and Shifts

Local hybrid functionals containing strong-correlation factors (scLHs) and range-separated local hybrids (RSLHs) have been integrated into an efficient coupled-perturbed Kohn–Sham implementation for the calculation of nuclear shielding constants. Several scLHs and the ωLH22t RSLH have then been evaluated for the first time for the extended NS372 benchmark set of main-group shieldings and shifts and the TM70 benchmark of 3d transition-metal shifts. The effects of the strong-correlation corrections have been analyzed with respect to the spatial distribution of the sc-factors, which locally diminish exact-exchange admixture at certain regions in a molecule. The scLH22t, scLH23t-mBR, and scLH23t-mBR-P functionals, which contain a “damped” strong-correlation factor to retain the excellent performance of the underlying LH20t functional for weakly correlated situations, tend to make smaller corrections to shieldings and shifts than the “undamped” scLH22ta functional. While the latter functional can also deteriorate agreement with the reference data in certain weakly correlated cases, it provides overall better performance, in particular for systems where static correlation is appreciable. This pertains only to a minority of systems in the NS372 main-group test set but to many more systems in the TM70 transition-metal test set, in particular for high-oxidation-state complexes, e.g., Cr(+VI) complexes and other systems with stretched bonds. Another undamped scLH, the simpler LDA-based scLH21ct-SVWN-m, also tends to provide significant improvements in many cases. The differences between the functionals and species can be rationalized on the basis of one-dimensional plots of the strong-correlation factors, augmented by isosurface plots of the fractional orbital density (FOD). Position-dependent exact-exchange admixture is thus shown to provide substantial flexibility in treating response properties like NMR shifts for both weakly and strongly correlated systems.


S1
The perturbed potentials The LH functional as given in Equation ( 1 The potential of the RSLH using the NDC/DC contribution as shown in Equation ( 13) in main text is given by

S3
Finally, an scLH with the NDC/DC contribution of Equation ( 16) in main text provides a more involved perturbed potential, due to the presence of q AC (r): • e ex X,σ dr • e ex X,σ dr CCSDT,appr., by using the CCSDT value of the next lower rank of the basis-set hierarchy, corrected by the dierence of CCSD(T) values with the larger and smaller basis as The extrapolated GIAO-CCSDT/pcSseg-3 reference values deviate by only 6.8 ppm (O term.) and 26.8 ppm (O cent.), respectively, from previously computed GIAO-CCSDT/qz2p shielding results.S2 We decided to use the extrapolated values instead of the literature values for consistency with the chosen basis sets for NS372 and for F 3 − (see main text).
O term.
-791.0 -801.5 S5 S3 NS372 relative shifts While the discussion in the main text for NS372 has focused on absolute shielding constants and their deviations from the reference values, here we also provide a short statistical overview on relative shifts (see Equation ( 54) in main text).The latter involve some compensation between computational errors made for the compound in question and that made for the reference compound used to extract shifts for a given nucleus.When evaluating DFAs, some of them provide more eective error compensation than others, and a comparison can be instructive when selecting a DFA for applications.The computational reference standards employed for NS372 are: CH 4 ( 1 H, 13 C), BH 3 ( 11 B), NH 3 ( 15 N), H 2 O ( 17 O), CF 4 ( 19 F), PH 3 ( 31 P) and H 2 S ( 33 S).The error compensation involved can be seen from a narrower distribution of deviations compared to the shieldings discussed in main text (see Figure S1, which uses the same symbols and labels).We do not plot relative standard deviations, as these are almost identical to those for the absolute shieldings.
Starting with the aggregated percentage MAEs, we see that the best-performing functionals are still the same as for the absolute shieldings, but now cscLH21ct-SVWN-m (1.0 %) joins the best-performing double hybrid DSD-PBEP86 (0.7 %) as another functional that achieves the 1 % mark.This is followed by cmPSTS and cB97M-V (1.2 %).cmPSTS has been discussed before as a DFA example with a particularly eective error compensa-S6 tion for the relative shifts.S3 The LH20t-based scLHs (cscLH22t, cscLH22ta, cscLH23-mBR, cscLH23-mBR-P) also all give low aggregate percentage MAEs of 1.3 1.4 %, not improving much over cLH20t (1.4 %).Here the damped cscLH22t switches ranks with the undamped cscLH22ta due to a very systematic behavior for 1 H shieldings (as also reected by the smaller SD, see main text).When excluding 1 H shifts, the undamped model performs slightly better.
Due to the dierent sign convention of shifts compared to shieldings, the percentage mean signed deviations (middle panel) are mostly positive (except for the 11 B subset).The cB97M-V mGGA is an exception with an extremely small aggregate value (+0. 3 .Bottom: one-dimensional plots of q AC with the scLH22t and scLH22ta functionals along the indicated directions. Figure S3: Top: plot of the FOD (isosurface=0.002a.u.) and orientation of the q AC plots for OF 2 .Bottom: one-dimensional plots of q AC with the scLH22t and scLH22ta functionals along the indicated directions.
) results in the perturbed LH potential as derived in ref.S1: CCSDT reference data for the 17 O shieldings in the O 3 molecule We approximate the CCSDT reference value in a larger basis set from the pcSseg-(x) family, σ pcSseg-(x+1)

Figure S1 :
Figure S1: Statistical measures for relative NMR shifts of selected DFAs relative to the GIAO-CCSD(T)/pcSseg-3 reference data for the NS372 benchmark.Percentage deviations are shown.
Figure S2: Top: plot of the FOD (isosurface=0.002a.u.) and orientation of the q AC plots for F