Calix [ n ] arene-based Polyradicals : Enhancing Ferromagnetism by Avoiding Edge Effects

Trough-bond interacting organic polyradicals, rendered by customizable capacities of state-of-the-art synthetic routes, are ideal systems to investigate spin topologies. Relying on Rajca and co-workers synthetic efforts, hereby we investigate the role of borders in the stability of the high-spin ground state in a series of realistic linear and ring-like arylmethyl polyradical derivatives. We show that, compared to the linear counterpart, the absence of borders in a ring-like arrangement of arylmethyl radicals imposes a larger number of spin-alternation rule violations, which strongly stabilizes the high-spin ground state. In addition, the structural flexibility of the investigated compounds translates into the existence of various structural energy minima for which the ferromagnetic ground state is always maintained. In view of the present results we propose these rings as possible candidates for the development of enhanced high spin single molecule toroics. *Corresponding author: daniel.reta@manchester.ac.uk Page 1 of 17 Physical Chemistry Chemical Physics P hy si ca lC he m is tr y C he m ic al P hy si cs A cc ep te d M an us cr ip t Pu bl is he d on 1 1 A ug us t 2 01 7. D ow nl oa de d by A th ab as ca U ni ve rs ity o n 13 /0 8/ 20 17 1 1: 22 :0 0. View Article Online DOI: 10.1039/C7CP04145D


Introduction.
Ever since the synthesis of triphenylmethyl radical by Gomberg, 1 carbon-based πconjugated polyradicals interacting through-bond have been considered as candidates for the practical realization of purely organic magnetism. Among the different coupling schemes enabling one to obtain extended structures from triarylmethyl-based building blocks, only a 1,3 (or meta)-connectivity between the radical centres ensures a preferential high-spin ground state, in agreement with well-known rules derived from topological arguments. [2][3][4][5] This structural constraint in assembling the building blocks results in a limited number of possible structures such as linear, dendritic, star-branched or macrocyclic architectures, which indeed are expected to determine the macroscopic properties of the compounds. [6][7][8] Figure 1 schematically summarizes the most significant experimental achievements obtained by exploiting a 1,3 (or meta)-connectivity using triarylmethyl-based radicals and classifies them in function of the measured multiplicity of the ground state vs relative stability. Here, relative stability is not a well-defined experimental property, but rather a qualitative comparison to the most stable of these radicals. Figure 1 also indicates the relationship between the adopted coupling scheme and the associated deficiencies. One of the main issues here is the lack of chemical stability of the carbonbased radicals, with dimerization, or other type of reactions, likely to result in loss of any magnetic property of interest. To protect the polyradical character of these compounds, two main synthetic strategies emerged. One is based on the idea that larger conjugation would result in a more pronounced delocalization of the unpaired electrons making them less reactive. Rajca and co-workers have led this branch with the refinement of the carbanion method and the development of spin clusters; 9-11 see lower part of Figure 1. The second approach aims at sterically protecting the radical centre by introducing bulky chlorine atoms, resulting in the well-known perchlorotriphenylmethyl (PTM) radical originally synthesized by Ballester et al. 12 On this basis, Veciana and coworkers have managed to synthesize di-13 and triradicals 14 showing respectively stable triplet and quartet states in solution at room temperatures, as well as to synthesize and characterize a large series of multifunctional materials; [15][16][17][18][19] cf. upper part of Figure   1. Unfortunately, one cannot simultaneously take advantage of the two strategies simply because they are mutually exclusive due to steric congestion. 20 As a result, no major experimental advance has been pursued in the field in the last decade. It is at this point when one can envisage exploiting the inherent structural flexibility 21 present in these compounds so as to propose alternative approaches combining both a large conjugation and steric protection of the radical centres.
Relying on the ideas outlined above, it has been recently shown that a linear triarylmethyl-based polyradical molecule adopts a more stable helical conformation, which also promotes larger ferromagnetic interactions. 22 However, the presence of edges in these high-spin ground state linear-like systems play a detrimental role for ferromagnetism. This can be understood by simple arguments derived from the spin alternation rule schematically depicted in Figure 2. Starting from a high spin arrangement with all spins up, the occurrence of a spin-flip in any of the radical centres leads to a situation that disrupts the spin-alternation rule forcing a spin frustration and a concomitant penalizing suppression of spin density in the adjacent phenyl rings. From a theoretical and computational point of view, such a situation with one (or more) spinflip can be represented by an antiferromagnetic solution (AFM). It is clear that the only region of the polyradical chain, either in a linear or helical conformation, where a spin flip imposes a minimum amount of disruptions is precisely at the edge. Consequently, the corresponding AFM solution will be stabilized which, again, is detrimental to an energetically isolated high spin ground state. Consequently, to maintain a stable, ferromagnetic, high spin ground state, one should be able to equally penalize all AFM solutions and this is possible if one can get rid of the borders.
It is obvious that the most straightforward manner to remove edge effects in finite systems is by generating a ring-like structure. Inorganic chemistry has provided extraordinary relevant examples, [23][24][25][26][27][28][29][30][31][32][33] but the localized nature of the magnetic centres and the presence of necessary bridging ligands often results in low-spin ground states and weak to moderate antiferromagnetic interactions. Organic chemistry, on the other hand, has achieved remarkable examples of ring-like molecules [34][35][36][37][38][39][40][41][42][43][44] in the field of hostguest and supramolecular chemistry. However, attempts to obtain organic magnetic rings are scarce. For the purpose of the present work, the most relevant example is precisely the tetraradical that led Rajca and co-workers to coin the spin cluster term. 45 Despite their success, no further attempts have been reported to achieve more extended rings with a larger number of interacting radical centres, as the interest shifted towards the linkage of the tetraradical ring units. 7 Other unrelated approaches consist of planar systems where polyradical character arises from an equilibrium between quinoidal and aromatic forms, 46,47 and of circular covalent organic frameworks (COF) to which stable nitronyl radicals are covalently bound. 48 Of particular significance for the present work is the concept of single molecule toroics (SMT), 32 where on top of exchange interactions between centres, having a significant magnetic anisotropy is a key goal.
Despite the fact that magnetic anisotropy is not expected in purely organic molecules, appropriate coordination with metallic centres appears as an effective manner to introduce such property. 49,50 Bearing in mind the structural flexibility of arylmethyl-derivatives and aiming at further investigate them as the most promising candidates to develop organic magnetism, we study a series of realistic, progressively bigger, circular-like, arylmethylbased polyradicals. We then compare the magnetic features of these ring-like compounds to those of the associated linear and helical counterparts. Through a systematic theoretical study of their structural and magnetic features, we provide compelling evidence that a ring-like arrangement persistently presents comparatively more stable high-spin ground states, which is not affected by the ease to undergo conformational distortions. Additionally, the chemical stability of the investigated examples could be largely increased by promoting a favourable balance between steric protection and the associated strain.

Computational and theoretical details.
All calculations have been carried out with the Gaussian09 suite of programs 51 where ‫ܬ‬ is the exchange coupling constant between the ݅ and ݆ localized spin moments and the 〈݅, ݆〉 symbol indicates that the sum refers to nearest neighbour interactions only. Nevertheless, for the purpose of this work, the important quantity is the total energy difference between the ferro-and lowest in energy antiferromagnetic solutions, which is independent on the adopted model spin Hamiltonian. For completeness, details of a possible mapping to the HDVV is given in the supporting information (SI).

Linear, helical and circular: structure and magnetism.
In order to analyse the impact of edge effects on the relative stability of FM solutions, we compare a series of oligomers in a linear, helical and ring conformation ranging from 4 to 15 radical centres. The linear and helical structures contain exactly the same number and type of atoms and therefore one can make use of absolute energies to discuss relative properties. However, the ring arrangement possesses one less phenyl moiety and the discussion is then referred to relative energy differences per magnetic centre. For completeness, we also investigated steric protection of the radical centres in some of the rings, by substituting the hydrogen atoms in the radical-bearing carbon by phenyl rings and in a subsequent step (for the smallest ring), by substituting all hydrogen atoms by chlorine atoms.
Let us first discuss the results obtained for the linear and helical structures. Table 1 a) presents the energy difference between the FM ground state and the lowest (excited) AFM solution for the linear and helical conformations. As predicted by the simple topological arguments discussed in Figure 2, it is found that the lowest AFM solutions always correspond to the arrangement of consecutive spin down densities at one of the edges (compare ‫ܯܨܣ‬ () and ‫ܯܨܣ‬ () * rows in Table SI2 and SI3 in the SI). This follows from the fact that this type of AFM solutions involves the minimum amount of spin alternation rule violations. Due to technical limitations on the control of the topology of the solution sought for, the solution presenting consecutive spin down densities at the edge was not possible to converge in a few cases (helical ܰ = 8, 10, 13 and 14) despite extensive efforts using different starting densities and convergence control procedures.
In any case, all converged solutions are presented in Table SI 3 thus providing strong support to the conclusions reached in the present work. Table 1 a) also indicates the gain in stability (kcal mol -1 ) per magnetic centre due to the appearance of the helical conformation. As indicated before, the expressions to obtain the magnetic couplings can be found in section 1 of the SI. For the linear molecules (Table SI 4 Using the data in Table 1 b), we focus now on the results for the ring-like structures. Table 1 Figure 3b. From this simple case, the spin densities of all possible AFM solutions for larger systems can be envisioned. As implicit in the previous discussion, these systems present an additional complexity arising from the multiple local minima that a ring structure can adopt. For instance, for ܰ = 8 a total of four, well separated in energy, minima were characterized. Interestingly, for the largest molecular structure investigated (ܰ = 15), the energy minimum structure is reminiscent of a Möbius strip and, despite the different degree of strain found in the different minima, the ground state always remains FM. This strongly support the claim that these systems exhibit a robust FM ground state, a conclusion which, qualitatively speaking, can be considered independent of the DFT based-method used. In fact, the validity of B3LYP functional with the standard Pople-type GTO basis set to describe these type of organic radicals has been extensively validated in previous theoretical 21, 71 and experimental 72,73 works. Taken altogether, one can safely conclude that a more exhaustive investigation of the potential energy surfaces will not reveal a qualitatively different picture in terms of magnetic interactions and that the presented cases are sufficiently representative. To further justify this, Table SI 11 in section 5 of the SI presents the results obtained with TPSS, PBE0, M06-2X and LC-߱PBE functionals together with the 6-31G(d,p) and a triple-ζ polarized quality basis sets, for the 7- Having separately discussed linear-like and ring-like structures we now compare the results obtained for the two types of structures. For a given number of radical centres, the energy difference between the FM and lowest AFM solutions is always larger for the ring, as compared to either linear and helical arrangements; this proves that avoiding edge effects provides a clear strategy to stabilize high spin states in triarylmethyl polyradical derivatives. In fact, a single spin flip anywhere in the ring results in a state approximately 2400 cm -1 above the FM ground state, which is twice the difference found in the linear cases (see ‫ܯܨܣ‬ ଵ rows in Table SI 2 vs Table SI 5 in the SI). Interestingly, due to spin topology, while removing edge effects significantly stabilizes the FM solution, the magnetic coupling constants value J 1 remains similar to the linear counterpart. This can be understood by comparing Eq. (2) and Eq. (6) in the SI. However, J 2 shows a considerable variation along the series. It is worth mentioning that, as in the case of the helical structures, the calculated magnetic coupling constants also show differences depending on which set of equations are used (Table S.I. 7 in section 4 of the SI) which seem to indicate that the HDVV Hamiltonian used is a too crude spin model for this type of systems. In this case, the apparent inconsistency can be traced to the rather irregular distribution of radical centres within the molecule, which makes the distances between first and second nearest neighbours not always constant.
The impact of the adopted conformation on the magnetic coupling constants is further investigated by calculating those values at the different conformers, as presented in Table SI7 of the SI. For all investigated cases, ‫ܬ‬ ଵ remains largely ferromagnetic although in some cases it experiences a noticeable decrease. On the other hand, ‫ܬ‬ ଶ does switch from ferro-to antiferro-character depending on the conformation. Nevertheless, ‫ܬ‬ ଵ is, in all cases, one order of magnitude larger than ‫ܬ‬ ଶ , thus remaining the dominant magnetic interaction. To conclude the study, a similar analysis was carried out on more realistic molecules where the steric protection of the radical centres is increased by i) substituting the hydrogen atom in each carbon-based radical by a phenyl ring for ܰ = 4, 5, 7, 8 or ii) substituting all hydrogen atoms by chlorine atoms for ܰ = 4 (see Table SI 9 in the SI). The general trends are the same but the absolute values of the magnetic coupling constants are smaller, due mainly to a larger delocalization of the unpaired spin density on the rings that do not participate in the exchange coupling. For instance, ‫ܬ‬ ଵ drops from 2333 to 1637 cm -1 for the lowest stationary point found for ܰ = 8 after replacing the hydrogen atoms in the radical centre by phenyl groups.
Additionally, the fully chlorinated ܰ = 4 case also shows a ferromagnetic ground state in both stationary points found, despite presenting a smaller ‫ܬ‬ ଵ value. Finally, the relative energy position of the molecular orbitals does not depend on the size of the molecule and the steric protection of the radical centres does not significantly modify the HOMO-LUMO gap, which remains around 2 eV (see Figures SI2 and SI3 in the SI). This is in contrast with what is predicted for planar arrangements of triarylmethyl-based polyradicals. 74 Nevertheless, the important finding is that, irrespective of the HDVV used to map the different magnetic solution, removing edges significantly stabilizes the FM ground state, strongly suggesting that this may be a potential way to obtain polyradicals with robust ferromagnetism.

Conclusions.
The present study shows that structurally stable, ring-like molecules derived from arylmethyl polyradicals display largely stabilized high-spin ground states as compared to linear-like counterparts. The reason is the absence of borders, as predicted by simple spin topological arguments. Rajca and co-workers 45 exploited the basic calix [4]arene to develop spin clusters through the linkage of these units, but syntheses of larger rings have not been reported.
A meta-connectivity of the radical centres is crucial for two reasons: i) it ensures high-spin ground states and ii) offers a structural freedom, increasing with size, that To summarize, removal of edges by considering purely organic, through-bond interacting, meta-connected ring-like molecules derived from arylmethyl polyradicals appears as a realistic strategy to obtain very high-spin systems and move forward in the field of organic magnetism. Particularly, the investigated rings can be considered as potential candidates for the development of enhanced single-molecule toroics (SMT), 32 owned to the extended conjugation within the molecule and the large ferromagnetic exchange interactions. Taken together, these findings call for efforts to attempt the synthesis of ring-like arylmethyl-based molecules, along a well-defined synthetic route.

3.
Helical: Calculated absolute energies and energy differences of ‫ܯܨ‬ and ‫ܯܨܣ‬ solutions.

4.
Ring: Calculated absolute energies and energy differences of ‫ܯܨ‬ and ‫ܯܨܣ‬ solutions.

5.
Effect of functional and basis set: the case of 7-membered ring. Ar-stands for aromatic ring; Cl-for chlorine atom.