Structural control over spin localization in triarylmethyls †

Triarylmethyls (TAMs) are a class of long-lived purely organic radicals discovered at the beginning of the twentieth century. The chemical versatility and high stability of TAMs have lead to their application in many ﬁ elds of science and technology. All compounds of this class are composed of three aryl rings bonded to a central carbon atom, where their unpaired electron mainly resides. Due to the p - conjugated electronic nature of this molecular structure, the possibility arises of controlling the unpaired electron localization ( i

In 1944 Gilbert N. Lewis suggested that TAM radicals are not completely planar but, instead, their three aryl rings are twisted with respect to the sp 2 plane of the central methyl carbon atom (Fig. 1). 668][69][70][71] With some simplifying assumptions (e.g.xing all structural degrees of freedom apart from the torsion angle) one can employ p-orbital overlap arguments to estimate that the spin delocalization through a certain aryl ring is proportional to the cosine squared of its dihedral angle with respect to the sp 2 plane of the radical centre (i.e.cos 2 q). 67,68,70,72Later, experimental works on some specic TAMs also effectively showed that the spin distribution is inuenced by the torsion angles of the aryl rings. 16,18,20,73owever, despite these important early studies, as far as we know, there exists no detailed systematic study of the spin-localization versus structure relation covering a wide range of TAMs.Moreover, the inuence of chemical functionalisation and temperature on this relation is currently unclear.The existence of a general and robust spin versus structure relationship for all TAMs would be extremely useful for the tailored design of TAM-based applications, and could open the possibility of preparing molecular materials with controllable magnetic, optical and electrical properties.
Herein, we have used ab initio density functional theory (DFT) calculations to accurately study the dependence of spin localization in TAMs on their chemical and structural characteristics.We consider the rst order assumption that the spin localisation should entirely depend on the average of the cosine squared of each dihedral angle of the three aryl rings (hcos 2 4 i i).We test this proposal for a wide range of dihedral angle combinations and different chemical functionalisations.More generally, we also study whether this spin localisation versus structure relationship holds at nite temperatures.Our results also thus provide detailed insights into important and subtle dynamic factors that inuence the degree of spin localization under more realistic conditions.We expect our results will be of interest and use for experimental chemists in order to optimize TAMs for future applications.

Methodology
The optimised structures of all TAM derivatives reported herein (see ESI † for chemical structures) were obtained using DFT calculations employing the PBE0 hybrid functional 74 and a 6-311++g(d,p) basis set as implemented in the GAUSSIAN-09 code. 75The PBE0 functional has been demonstrated to provide a very good account of the geometry, electronic structure and spin polarisation in organic radicals. 74Calculated spin densities were atomically partitioned using the Hirshfeld scheme. 76For independently benchmarking our computational methodology we have also tested its capacity to reproduce reported EPR experimental data on a range of TAM derivatives. 16The good match between our calculated spin dipolar coupling constants and the experimental data further validates the level of theory utilised herein.A discussion of this comparison and the corresponding results can be found in the ESI.† For assessing the validity of our proposal at higher temperatures, canonical ab initio molecular dynamics (AIMD) calculations of 10 picoseconds (5 ps of equilibration followed by 5 ps of production) at a temperature of 300 K were also performed employing a 0.5 femtosecond time step and the Bussi-Donadio-Parrinello thermostat. 77The AIMD runs were calculated using the FHI-AIMS code 78 employing the PBE0 functional and a light basis set for all atoms.

Results and discussion
As suggested by Lewis 66 and later corroborated by experimental work, 16,20 the dihedral angle of each aryl ring in TAMs mainly depends on the particular chemical functionalization of the molecule.Thus, for assessing the spin localization dependence on the three aryl rings twists, we rst studied the structural and electronic properties of a set of 27 previously reported TAMs with a wide range of chemical and structural properties.Besides studying some triphenylmethyl derivatives, we also included some perchloro-and peruoro-TAMs with different functionalizations. 16,79Among them, phenyl-, 69,80 terbutyl-, 81 hydroxo-, 81 methoxy-, 20,82 carboxyl-, 31,41,54,83,84 amine-, 85 thioether 41,42,84 and nitro-functional 86 groups have been considered for our study.Some TAMs with direct aryl-aryl bonding that present constrained structures and more extreme angle values were also included. 17,20Further, we also designed two TAMs (see ESI † for structural atomic coordinates) where the three aryl rings are forced by inter-ring linkages to be in perpendicular orientations with respect to the central sp 2 methyl plane, giving rise to congurations that have not yet been reported for any synthesized TAM.The chemical structures of TAMs 1-27 can be found in the ESI.† Additionally, a series of constrained dihedral angle calculations were performed with the triphenylmethyl 1 (herein TAM4) and the perchloro-triarylmethyl 2,22,62 (TAM12).Here, the dihedral angles of the three aryl rings were constrained with equal (i.e. 4 1 ¼ 4 2 ¼ 4 3 ) or different (i.e. 4 1 s 4 2 s 4 3 ) values between 20-90 , while fully optimizing the rest of the molecular structure.In this way we could extract the intrinsic effect of aryl ring twisting on the spin distribution without the additional inuence of chemical substitution (e.g.electro-withdrawing/ donating nature of substituents).These calculations also mimic the hypothetical manipulation of the aryl ring twists by external stimulus, testing the spin-localization/structural relation for out-of-equilibrium congurations.For all cases the spin population on the central carbon atom (aC; see Fig. 1) was used as an indicator of the spin localization/delocalization, since this position is always that with the highest spin population in TAMs.
As a rst order assumption, we consider that the spin delocalisation through a particular aryl ring only depends on the corresponding dihedral angle, regardless of its chemical functionalization.In this scenario, based on p-overlapping ideas, 67,68,70,72 the spin delocalization should be essentially captured by cos 2 4 i , where 4 i is the dihedral angle of the considered aryl ring with the central sp 2 carbon atom plane.Secondly, we consider that the effect of twisting any one aryl ring on the spin localization to be independent of the other two rings torsions.This simplies the complex TAM case to three independent methyl-aryl units and, as a consequence, the average of the three cos 2 4 i (i.e.hcos 2 4 i i ¼ (cos 2 4 1 + cos 2 4 2 + cos 2 4 3 )/3) should become a rst order spin localization descriptor for TAMs.In Fig. 2 we plot the aC spin population against hcos 2 4 i i for our set of more than 40 TAM structures.
As it can be seen in Fig. 2a, hcos 2 4 i i is a good spin localization descriptor for the optimised TAM derivative structures (i.e.TAM1-27, Fig. 2b) where there is little scatter away from the ideal assumed behaviour (dashed line).The fact that the differently functionalized TAMs (black diamonds) nicely follow the main trend means that chemical functionalisation (see Fig. 2b) determines the spin localization by determining the three dihedral angles but other possible effects, such as the electro-donating/withdrawing nature of substituents, clearly play a secondary role.Therefore, for 0 K structures, the dihedral angles of the three aryl rings with aC almost entirely determine the spin localization in TAMs.As a consequence, by means of a proper chemical design, we can have a completely spin delocalized picture induced by a planar structure (TAM1, Fig. 2a-down) or, conversely, a conguration where most of the spin population resides on aC (TAM27) due to a perpendicular orientation of the three aryl rings (Fig. 2a-up).
The aC spin population for the structurally constrained versions of TAM4 and TAM12 structures (circled in Fig. 2b) also linearly varies with hcos 2 4 i i (red spheres in Fig. 2a).This shows that the spin-localization structural relation is also maintained for out-of-equilibrium congurations.In effect, this set of calculations conrm the potential use of this relation for TAM-based materials and devices where the TAM structure could be externally controlled (e.g. by crystal packing, applied pressure) to affect physicochemical properties such as optical absorption bands, magnetic interactions, or electrical conductivities.
However, for real world applications temperature is a critical parameter that must also be taken into account.Thus, for testing the robustness of the spin-localization/structure relation at realistic conditions, we further performed AIMD simulations for radicals TAM1, TAM4, TAM12 and TAM27 at a temperature of 300 K (see computational details in the Methodology section).Fig. 3 shows the time evolution of the three dihedral angles (4 1-3 ) for the four studied TAMs at 300 K.By comparing the structural oscillations of radicals TAM1, TAM12 and TAM27 (Fig. 3a) with that of radical TAM4 (Fig. 3b) we notice important differences due to the correspondingly different chemical functionalisations.Direct aryl-aryl bonding in TAM1 and TAM27 clearly prevents rotation of the three aryl rings and this is shown by the quite constant values of the corresponding dihedral angles over time (red and purple curves in Fig. 3a, respectively).In the case of the fully chlorinated TAM12 it is the strong sterical hindrance between chlorine atoms that inhibits free rotation of the aryl rings, giving rise to similarly dynamically restricted structural behaviour to that of radicals TAM1 and TAM27 (green curves in Fig. 3a).The triphenylmethyl (TAM4) though, presents a completely different scenario (Fig. 3b).Due to the lack of inter-ring bonding plus a low sterical hindrance between each phenyl ring, rotation is not prevented.This greater rotational freedom leads to the oscillations of each separate dihedral angle for TAM4 (blue curves in Fig. 3b) to be much higher than those in Fig. 3a.Therefore, chemical functionality does not only determine the most stable 0 K conguration of each aryl ring, but also their associated rotational freedom and thus their nite temperature dynamic structural behaviour.This also suggests that the ease with which dihedral angles could be manipulated by external means in TAM-based systems will depend on their individual chemical functionalization.Clearly, in addition to such molecular scale considerations, many other environmental factors can affect the degree to which the structure of a TAM can be tailored (e.g.type of solvent, crystal packing, applied external forces).
To test whether the structural dependence of the spinlocalization is maintained at nite temperatures during these structural uctuations, we compared the computed aC spin population over time with that predicted by the relation derived in Fig. 2a.Specically, by using the t equation in the plot of Fig. 2a, we extracted from the dihedral angle uctuations (Fig. 3) a hcos 2 4 i ipredicted aC spin population over time.In Fig. 4 we compare the actual calculated aC spin population (coloured lines), provided by DFT, with the hcos 2 4 i ipredicted aC spin population (black line) for the four studied radicals.
In Fig. 4a, the aC spin population is evaluated 0.5 fs.Even for such a small sampling time the predictions by hcos 2 4 i i (black lines) is approximately followed (with 10% variance) by the calculated spin localization (coloured lines) for each studied TAM over time.This consistent correlation conrms that the spin localization is also dynamically determined by the three aryl rings twists at nite temperatures and, thus, opens up the potential applicability of this dependence under realistic conditions.The computed aC spin populations (coloured curves in 4a) present a fast but fairly small oscillation over time that does not exist in the prediction (black curves in 4a).As already noted by M. Karplus in the 60 s', 72 this faster oscillation is likely to be associated with bond vibrations which are not accounted for by hcos 2 4 i i.To verify this hypothesis the average distance of the three central aCaryl bonds were monitored over time together with the spin population on aC.In the zoomed inset to Fig. 4a it can be seen that, at the femtosecond time scale the rapid aC spin population variation (blue) perfectly correlates with the average of the bond length oscillation of those three bonds (grey), corroborating Karplus' suggestion.Therefore, based on the same p-p overlap ideas, thermal vibrations inuence the spin localization by stretching/shortening the C-C bonds of the three aryl rings with aC.However, by looking at the plots in Fig. 4a it seems that their effect on the aC spin population is to simply add small random deviations away from the ideal behaviour predicted by hcos 2 4 i i.To test this idea we use the Savitzky-Golay ltering procedure 87,88 for removing the high frequency oscillations due to bond vibrations.As it can be seen in Fig. 4b, we then recover a slower aC-spin population variation over time (coloured lines) that matches almost perfectly with the predicted one by hcos 2 4 i i (black lines).This demonstrates that, at sufficiently long times (i.e.hundreds of femto-seconds or higher) the effect of bond vibrations is averaged out and the spin localization is essentially entirely determined by the three dihedral angle values (collected by hcos 2 4 i i).
Finally, since most of laboratory experiments do not measure the physic-chemical properties at the time-scales of femto-or pico-seconds, but from the nano-second time-scale upwards, we wanted to probe the applicability of hcos 2 4 i i as a spin localization predictor at such typically measured time-scales.For this we calculated the average value of hcos 2 4 i i from the dihedral angle oscillations shown in Fig. 3 over the entire AIMD simulation (5 picoseconds).Then, by using the t equation of Fig. 2a, we extract the predicted aC spin population for each TAM radical.In Fig. 4c we compare this predicted value with the computed aC spin population mean value during the entire AIMD.As it can be seen, once again, the matches for all four radicals are excellent.
Overall, our AIMD results conrm the robustness of hcos 2 4 i i as a spin localization descriptor for TAMs even at nite temperatures.Over short time scales bond vibrations also affect spin localization, but their effect simply adds a small random deviation over the value predicted by hcos 2 4 i i. 72 For relatively long times (hundreds of femtoseconds or more) the effect of bond vibrations is averaged out and, then, the spin localization is essentially entirely determined by the dihedral angles of the three aryl rings.Therefore, at nite temperatures dihedral angles are shown to be effective spin localization descriptors in TAMs, opening the possibility of using them for tailoring this important electronic property and all related-characteristics under realistic conditions.

Conclusions
In summary, by using DFT calculations and by examining different TAMs with a range of chemical structures and at nite temperatures we have conrmed that, very generally, the average aC spin localization is well determined by hcos 2 4 i i (where hcos 2 4 i i ¼ (cos 2 4 1 + cos 2 4 2 + cos 2 4 3 )/3 and 4 i are the dihedral angles of each aryl ring with respect to the central sp 2 carbon atom plane).Chemical functionality determines the spin distribution as long as it determines the three dihedral angles but other chemical effects such as the electro-donating/ withdrawing nature of substituents play a secondary role.By performing AIMD simulations at 300 K, the predictive power of hcos 2 4 i i at nite temperatures has been probed.Bond vibrations also affect the spin localization at the femto-second time scale but their effect simply adds a rapid and small random deviation over the predicted value by hcos 2 4 i i.Moreover, at relatively long time-scales (hundreds of femto-seconds or higher) their effect is averaged out and the spin is essentially entirely determined by the three dihedrals.In TAMs, the existence of the unpaired electron gives rise to their most interesting properties. 50,59,64,65Therefore, this general and robust spin-localization/structure relationship also represents a powerful tool for controlling other important characteristics such as magnetic interactions, optical absorption bands, magnetoresistance phenomena or electrical conductivities.Besides the possibility of tunning dihedral angles by chemical design, [89][90][91] some theoretical studies have pointed out the possibility of manipulating this structural feature by external means such as electrical elds, 92 mechanical strain 93 or optical excitations. 94,95Hence, dihedral angles might be nely tuned by external stimulus in properly designed future materials and devices.In that scenario, we strongly believe that TAMs and their spin-localization versus structure dependence could become a key tool for the design and preparation of novel materials and devices with controllable magnetic, optical and electrical properties.

Fig. 1
Fig. 1 Generic TAM structure.Their unpaired electron (green) mainly resides on the central methyl carbon atom (aC) but, due to the pconjugated nature of these molecules, it partially delocalizes to the ortho-and para-positions of each aryl ring.

Fig. 2
Fig.2(a) Middle; partitioned spin population on aC against hcos 2 4 i i, where hcos 2 4 i i ¼ (cos 2 4 1 + cos 2 4 2 + cos 2 4 3 )/3, for each case.Black diamonds represent the fully optimized TAM1-27 (numbers, X, refer to the TAMX, see ESI †) and red spheres correspond to the constrained optimisations of TAM4 and TAM12.Values of 4 lower than 20 were not possible in the constrained optimisations due to too large sterical hindrances between the three aryl rings.Upper/lower; spin population iso-surfaces (green ¼ a; violet ¼ b) for TAM27 (upper) and TAM1 (lower), exhibiting the most perpendicular and most planar aryl ring configurations, respectively.(b) Fully optimized structures of representative TAMs from our study, highlighting the broad range of studied structures.TAM4 (triphenylmethyl) and TAM12 (PTM), used for the constrained optimizations (red spheres in a), are circled.TAM26 and TAM27 have been specially designed for this study (see ESI † for atomic coordinates).Atom colour key: Cgrey, Hwhite, Ored, Syellow, Clgreen, Fviolet.

Fig. 3
Fig. 3 Time dependence of 4 1-3 during the 5 ps AIMD calculations at 300 K for radicals TAM1, TAM12 and TAM27 (red, green and purple, respectively) in (a) and radical TAM4 in (b) (blue).The chemical structures of the radicals are provided to the right of the corresponding plots.