Disentangling global and local ring currents

Magnetic field-induced ring currents in aromatic and antiaromatic molecules cause characteristic shielding and deshielding effects in the molecules' NMR spectra. However, it is difficult to analyze (anti)aromaticity directly from experimental NMR data if a molecule has multiple ring current pathways. Here we present a method for using the Biot-Savart law to deconvolute the contributions of different ring currents to the experimental NMR spectra of polycyclic compounds. This method accurately quantifies local and global ring current susceptibilities in porphyrin nanorings, as well as in a bicyclic dithienothiophene-bridged [34]octaphyrin. There is excellent agreement between ring current susceptibilities derived from both experimental and computationally-predicted chemical shifts, and with ring currents calculated by the GIMIC method. Our method can be applied to any polycyclic system, with any number of ring currents, provided that appropriate NMR data are available.

Electronic Supplementary Material (ESI) for Chemical Science. This journal is © The Royal Society of Chemistry 2023 The RCGFs (in μT/nA) represent the magnetic field (in μT) induced by a ring current of 1 nA. For the comparison of RCGF with experimental NMR data, the experimental chemical shift needs to be referenced to an analogous "non-aromatic chemical shift", i.e. the shift that would be observed in absence of the studied ring current. For the nanorings, all references were used as in our previous studies. For the bicyclic [34]octaphyrin, we used a default value for a non-aromatic olefin of 5.8 ppm for all protons, as no direct reference is available. When we fit the RCGFs to chemical shift differences derived from DFT, we used the calculated chemical shift of cyclohexa-1,3-diene (5.87 ppm, BLYP35/6-31G*) as our non-aromatic reference.
The RCGFs from the MCL model (also called composite RCGFs) are defined as: Figure S1: Experimental and DFT model structures of c-P6·T6, T6 and benzene in this study.

S2. Choice of ring current pathway for MCL analysis
There are multiple potential current pathways in porphyrin nanorings ( Figure S2). The choice of pathway affects RCGF values, so we investigated how equation 2 fits our data for several different models. The global ring current can pass through all four nitrogen atoms of the porphyrin (the inner path), or it can pass through the outside of the porphyrin (the outer path). The true current pathway is probably a mixture of these options, so the inner and outer pathways can be weighted at 50% each, creating a mixed path. The ring current model can also be constructed assuming that current either passes through the π-system, which adds complexity, or through the σ-system, which is simpler to calculate. We compare all variations in Table S1 and S2. There seems to be no significant difference between the results from the σ and π ring current models (compare Tables S1 and S2), suggesting that the simpler σ-model can provide meaningful results. Our method does not include RCGFs arising from pyrrole ring currents.
The choice of ring current pathway (local, global, or mixed) affects I/Blocal by up to a factor of 2. Consistent with our previous efforts, and to avoid introducing bias into to the model, we used the mixed ring current pathway, which is equivalent to the ring current splitting equally at each fork in its path. Table S1: Ring current susceptibilities for C-P6·T6 6+ with different ring current pathways (mix, inner, and outer, defined below). RCGFs were calculated according to the procedure described in S1 -in these data the ring current passes through the π system. See Figure S2 for ring current pathways.  Table S2: Ring current susceptibilities for C-P6·T6 6+ with different ring current pathways (mix, inner, and outer, defined below). RCGFs were calculated according to the procedure described in S1 -in these data the ring current passes through the σ system. See Figure S2 for ring current pathways.   Figure S2: Representative ring current pathways in porphyrin nanorings for the repeating unit of c-P6 as used in Table S1 and   Table S2. Accurate quantification of multiple ring currents requires good orthogonality between RCGFs for different circuits. The local and global RCGF values for β, γ, and δ are linearly dependent ( Figure S3; a straight line can be drawn between the three points corresponding to these atoms) -these protons alone could not be used to deconvolute the magnetic shielding into local and global ring current components. Other protons, such as o(in), o(out) and α, are not linearly dependent. The o(in) and o(out) protons are sensitive only to shielding arising from global ring currents, whereas including the α proton adds some sensitivity to local ring currents.

S3. Orthogonality of RCGFs
The fits of different combinations of RCGFs for c-P6·T6 6+ are shown in Figure S4.

S4. GIMIC: Optimizing the size of the integration plane
Template-bound porphyrin nanorings are complex structures, and so the extraction of bond-currents from GIMIC calculations requires that the size of the ring current integration plane (red in Fig S6) is chosen carefully. The current integration plane sits perpendicular to the bond axis, and bisects the bond. The plane is required to capture as much of the bond current as possible, without including contamination from other nearby bonds.
We first optimized the width of the integration plane for intra-porphyrin bonds (i.e. Cb-Cb, N-Ca, and Ca-Cm in Fig. S5), by extending the plane inside and outside the nanoring, maintaining a constant height of 3.0 Bohr. The integrated ring current strengths do not change significantly for planes wider than 8.0 Bohr, therefore we adopted this plane width for all of our measurements (Fig. S7). We then set out to optimize the height of the integration plane (Figs S8-S9). Increasing the height of the plane for the porphyrin Cb-Cb and N-Ca bonds does not have a clear "best" value. At heights of greater than 4.0 Bohr, which is roughly the center of the pyrrole ring, the integrated current becomes contaminated by the bonds on the opposite side of the pyrrole, and the plane eventually reaches the pyrrole N atom when its height reaches 7.0 Bohr. For the Ca-Cm bonds, the integrated current has a clearer dependence on the height of the integration plane: current strengths increase with plane height from 2.0 -4.0 Bohr, before decreasing for larger planes. We therefore used a plane with dimensions 8 × 4 Bohr (w × h) for the intra-porphyrin bonds.
The central butadiyne C-C is further from any possible contaminating influences, and so a larger integration plane with dimensions 8 × 8 Bohr could be used (Fig. S10).    In c-P6·T6 4+ the global ring current primarily flows through the inner pathway of each porphyrin, while 30% of the current takes the outer pathway.
As in c-P6·T6 2+ , there are two porphyrin environments in c-P6·T6 4+ . When the magnetic field is orientated perpendicular to a porphyrin, two porphyrins have paratropic ring currents and four porphyrins have diatropic ring currents. The diatropic currents appear to flow mostly through the outer pathway, which has a strength of 10 nA/T compared to 1 nA/T tracing the inner path. The two porphyrins with paratropic ring currents seem to have two "levels" of aromaticity: the porphyrin aromaticity, and the pyrrole aromaticity. In this case, the porphyrin is weakly antiaromatic and each pyrrole subunit aromatic. This has the interesting effect of the porphyrin ring current being reinforced over the pyrrole Ca -N -Ca bridge, hence the dark red color ( Figure S15).

S5.4. Ring current pathways in c-P6·T6 6+
Figure S17  While c-P6·T6 12+ does not sustain a global ring current, its warped geometry means some pyrrole subunits have weak ring currents when the magnetic field is applied perpendicular to the plane of the nanoring.
The six local porphyrin ring currents are equivalent, with 90% of its ring current travelling through the inner pathway (i.e. across the pyrrole N atom) and the remaining 10% through the outside of the pyrrole units.      Figure      In c-P6·T6 and c-P6·T6 12+ , the MCL model offers little improvement over the SCLlocal model, on account of the lack of global ring current in these oxidation states. Using DFT and experimental data, the RMSE values are mostly the same between these two current loop models.

[34]Octaphyrin
The two ring current models for the [34]octaphyrin were based on the two viable conjugated paths. The RCM paths were split into two, above and below the normal plane (plane defined by the local atoms).

S7. Statistical analyses
We used the adjusted R 2 and corrected Akaike information criterion (AICc) to evaluate the goodness of fit for different current-loop models. These statistical methods evaluate the fit between a model and data, with a penalty for additional unnecessary parameters (i.e. they reward parsimony). These statistics can be used in conjunction with external knowledge (e.g. that there may be multiple ring current pathways in a molecule) to select the most appropriate model(s).
Since we have a relatively small number of chemical shifts for each species (n in Tables S9 and S10), the statistics cannot be relied upon in isolation. Addition of extra observations would have a large effect on both AIC and R 2 adj. For our present purposes we consider that a AIC of >5 or a -adj-R 2 of >0.01 ought to raise appreciable doubts about the statistical suitability of any model in a pairwise comparison.  Taken together, the fit statistics do not provide a strong argument against the MCL model, or for/against the addition of an intercept parameter. In many cases the MCL model provides a similar or superior improvement to the SCLglobal fit as adding an intercept term to the latter model. The inclusion of an intercept term in the MCL model is not justified because it offers only a minor (and inconsistent) improvement to the fit, while delivering no extra chemical insight. Its inclusion would risk overparameterizing the system, especially where n is low. Generally, the addition of an intercept term has a negligible effect on the fitted values for the ring current susceptibilities. The fit could be improved, in the absence of an intercept term, by extending the MCL model to define more ring current cycles, or by correcting the  values to remove contributions not attributable to ring current effects.