While ProCS15 does not reproduce the DFT results perfectly as discussed above the first two rows of Table \ref{table:vsexp} show that ProCS15 can reproduce experimental chemical shifts with an overall accuracy that is similar to full DFT chemical shielding calculations for Ubiquitin and GB3. The RMSD values predicted with ProCS15 for carbon atoms are 0.1 - 0.6 ppm lower compared to the DFT results, while the RMSD values for hydrogen and nitrogen atoms are 0.0 - 0.1 ppm and 0.2 - 0.4 ppm higher. It is therefore not clear that much is necessarily gained by adding additional terms to ProCS15 without also increasing the underlying level of theory used to compute these terms. For example, it is known that using a larger basis set can significantly improve the prediction of C’ chemical shifts \cite{24403017,Zhu_2012}.

Using structures optimized with CHARMM22/CMAP instead of PM6-D3H+ to predict chemical shifts with ProCS15 does also not seem to lead to overall worse agreement with experiment. In fact the results tend to improve slightly (up to 0.5 ppm) for heavy atoms as judged by the RMSD values. Comparison of ProCS15 to CheShift-2, which has also been parameterized against DFT calculations, show fairly similar accuracy for C\(\alpha\) and slightly worse accuracy for C\(\beta\). The latter observation is perhaps due to the fact that CheShift-2 uses a different (empirical-corrected) reference for each residue type. However, this is also the case for C\(\alpha\) for which ProCS15 predictions give a lower RMSD value.

Comparison of ProCS15 to the empirical methods (CamShift through ShiftX2) generally show considerably lower RMSD of the empirical predictions for all atoms types, except H\(\alpha\) for GB3 where the accuracy is mostly comparable. The \(r\) values are also considerably higher for the empirical methods than for ProCS15 for C\(\alpha\) and, especially, C\(\beta\), while they are comparable for the remaining atom atoms.

As mentioned in the introduction the higher RMSD values generally observed for the DFT-based methods compared to the empirical methods is expected. The important issue in the context of structural refinement against measured chemical shifts is whether the DFT-based methods are more sensitive to relative small differences in structure. While a thorough investigation of this complex issue for ProCS15 will be the subject of future studies, we look at the effect of using different structural ensembles on the accuracy next.