probes in an organic semiconductor: S R and DFT calculations of the μ Mu adducts of Alq3 and 8-hydroxyquinoline.

It has been claimed that longitudinal ﬁeld muon spin relaxation (LF-μ SR) experiments on the organic semiconductor (OSC) tris-(8-hydroxyquinoline)aluminum(III) (Alq 3 ) have measured electron hopping rates of ∼ 10 12 s − 1 , while density functional theory (DFT) calculations suggest that electron hopping between a muoniated radical and a neighboring molecule is energetically unfavorable and that the LF-μ SR experiments were probing muoniated radicals with localized spin density. We have performed avoided level crossing muon spin resonance (ALC-μ SR) and transverse ﬁeld muon spin rotation (TF-μ SR) measurements on Alq 3 and 8-hydroxyquinoline (8hq), which is meant to model the muoniated radicals present in Alq 3 when they are not in an OSC. These are supplemented by benchmarked DFT calculations. The ALC-μ SR and TF-μ SR spectra of 8hq and Alq 3 are best explained by Mu adding to all six secondary carbons of the quinolate rings with roughly equal yields and localized spin density. There is no evidence in the TF-μ SR spectrum of Alq 3 for the formation of radicals with muon hyperﬁne coupling constants of 23 or 91 MHz as reported earlier by others. Our measurements support the view that there is localized spin density on the molecule to which Mu is covalently bound and the muon is not a passive probe in organic systems as it can be incorporated into radicals that have different electronic structures to the parent compounds. The muoniated radicals in Alq 3 are more short-lived than in 8hq, which could be due to interactions with mobile electrons in the OSC, but with electron spin ﬂip rates on the order of ∼ 10 7 s − 1 .


I. INTRODUCTION
Throughout the history of the muon spin rotation, relaxation, and resonance (μSR) technique, there have been questions concerning whether it is accurately probing the material or whether the perturbations caused by the muon stopping in the sample give rise to distorted views of the local environment.Whether it is an "innocent" probe or not depends on what material is being studied.The most well-known example of muons being a perturbation is in semiconductors such as GaAs where the muon is not probing the pristine local environment around it but instead forms bondcentered muonium (Mu), where there is hyperfine coupling to the unpaired electron it has brought with it. 1Another example is muons in Pr 2 Sn 2 O7, a rare earth pyrochlore, where the presence of the The Journal of Chemical Physics ARTICLE scitation.org/journal/jcpmuon changes the crystal field around the rare earth ion and results in the muon probing something very different from the pristine crystal. 2 In many chemical systems, muons are undoubtedly a perturbation because they form Mu, which can react with unsaturated bonds to produce a muoniated radical.Muoniated radicals have a very different electronic structure to the parent diamagnetic compound.These radicals can be used as local probes of materials when studying dynamics of a particular component such as a cosurfactant in bilayers or micelles, [3][4][5] but whether they can be used to study electron hopping is controversial due to the different electronic structure.The μSR technique and its applications in the study of free radicals have been recently reviewed. 6,7In this paper, we consider whether muons are innocent probes in organic semiconductors (OSCs).
OSCs are materials having tremendous technological applications.Tris(8-hydroxyquinoline)aluminum(III) (Alq 3 ) is an OSC that is one of the most frequently used low-molecular weight materials for organic light-emitting devices (OLEDs). 8Alq 3 consists of an aluminum atom with three bidentate 8-hydroxyquinolate (8hq) ligands in an octahedral geometry (Fig. 1).In the crystal, there is good π-π orbital overlap between ligands of different molecules along the c axis, and this results in highly one-dimensional motion of charge carriers.μSR measurements on Alq 3 were reported in 2008. 9here, it was assumed that Mu adds to Alq 3 to give multiple types of muoniated radicals (Mu-Alq 3 ) with an average isotropic muon hyperfine coupling constant (hfcc) or Aμ of 425 ± 20 MHz, based on the repolarization curve.The signals due to these radicals were not observed in the transverse field muon spin rotation (TF-μSR) spectrum at 0.25 T. Instead, two types of radical with much smaller muon hfccs (23 ± 1 and 91 ± 1 MHz) were observed at low temperature (10 K) but not at 290 K.The inset in Fig. 3 of Ref. 9 shows the 10 K FFT spectrum and compares it to the 290 K longitudinal field (LF) decoupling behavior.These signals were not assigned.Longitudinal field muon spin relaxation (LF-μSR) spectra were obtained as a function of an applied longitudinal magnetic field.The muon spin polarization relaxed slowly and it was not obvious which relaxation function should be used to fit the spectra.The authors hypothesized that the relaxation of the muon spin in the longitudinal field was due to long-range diffusion of the unpaired electron, which was introduced by the reaction of Alq 3 with Mu.The LF-μSR spectra were then analyzed using the Risch-Kehr (RK) function, 10 which was developed for systems where there is long-range diffusion of the unpaired electron, such as in trans-polyacetylene. 11,12 It is important to note that other relaxation functions could also have resulted

ARTICLE
scitation.org/journal/jcp in comparable fits of the spectra.The magnetic field dependence of the RK relaxation parameter led to the conclusion that the unpaired electron was hopping at the rate of (1.4 ± 0.2) ×10 12 s −1 at 290 K.The interest in these measurements stems from the claim that μSR could be used to measure the intrinsic hopping rate of an electron in an OSC with very low concentrations of carriers due to the feature of self-generation of carriers by muon implantation. 13here is also the additional advantage that these measurements are not susceptible to disorder-induced bottlenecks present in conventional mobility measurements due to the muon being a local probe.
One can only obtain sensible information from an LF-μSR spectrum if one uses a relaxation function based on a physically realistic model. 14The problem is that sometimes different relaxation functions, which imply different microscopic behavior in the material, can fit spectra equally well.One can often not distinguish between different models based on the quality of the fit.This is the Achilles heel of the LF-μSR technique and is worst when one is analyzing spectra with slowly relaxing signals as in Alq 3 .The preceding analysis assumed that the muon just supplies the excess electron and is an innocent observer as the electron diffuses through the sample [Fig.2(a)].
A different model has been proposed to explain the μSR experiments on Alq 3 ; rather than being a passive probe, the muon is a significant perturbation and the muoniated radicals that are formed have different electronic structures to Alq 3 .The structures of the muoniated radicals formed by Mu addition to Alq 3 are shown in Fig. 1.Addition of Mu preferentially occurs at secondary unsaturated carbons rather than tertiary carbons, 15 so it was assumed that Mu addition to each ligand of Alq 3 should produce six types of substituted muoniated cyclohexadienyl radical.Mu is distinguishable from H, so Mu addition leads to a center of chirality at the methylene carbon of the radical formed. 16Enantiomers give rise to identical spectra in magnetic resonance, but the chirality of Alq 3 breaks the degeneracy and makes it possible to distinguish between the diastereomers formed by Mu addition of different sides of the ligand rings.The result is that there are 36 distinguishable radicals (3 ligands × 6 addition site × 2 orientations).Density functional theory (DFT) calculations at the UB3LYP/6-31G(d,p) level of theory showed that the muoniated radicals have a very different electronic structure to the parent compound Alq 3 and it is energetically unfavorable for electron transfer in either direction between Alq 3 and the muoniated radical (ranging from 4.09 to 5.68 eV) [Fig.2(b)]. 17This implies that the unpaired electron is localized on the molecule where Mu is covalently bound.
As a way to distinguish between the "innocent" muon and strongly interacting muon models, the muon and methylene proton hfccs for the Mu adducts of Alq 3 , which are proportional to the unpaired electron spin density at the nucleus ∥|ψ(0)| 2 ∥, were obtained from DFT calculations reported in 2010. 17  The Gibbs energy for electron transfer, ΔG 0 , is 0 and the reorganization energy for electron transport, λ, was calculated by Lin et al. 18 (b) Muon as a perturbation; the energy levels of the muoniated radical are different from those of the parent compound.ΔG 0 and λ for electron transfer to and from the muoniated molecule calculated by McKenzie [UB3LYP/6-31G(d,p)]. 17 The singly occupied molecular orbital lies between the HOMO and LUMO of Alq where γ μ and γ e and the muon and electron gyromagnetic ratios, respectively.The Δ 1 resonance is only observed when the radical is undergoing anisotropic motion or very slow isotropic reorientation, such as in a solid.The Δ 0 resonance field is given by where γ X is the nuclear gyromagnetic ratio.Δ 0 resonances in the solid state tend to be much broader than Δ 1 resonances and, thus, smaller in amplitude.An example of this would be the Mu adducts of [2.2]paracyclophane where Δ 0 resonances were not observed even though the Δ 1 resonances were very strong. 19Based on the DFT calculations in Ref. 17, multiple Δ 1 resonances were predicted in the ALC-μSR spectrum of Alq 3 between ∼0.8 and 1.7 T. The observation of Δ 1 resonances in this field range with a full-widthat-half-maximum of ∼0.1 T would be incompatible with an electron hopping at the rate of (1.4 ± 0.2) × 10 12 s −1 over a number of molecules.This assertion is supported by simulations performed using the Quantum Monte Carlo simulation program (Fig. 3). 20The simulations were performed on 10 molecules, one of which has Mu covalently bound; when the unpaired electron is on this molecule, there is a nonzero muon hfcc (isotropic muon hfcc ≙ 250 MHz and dipolar muon hfcc ≙ 10 MHz), and when it is on the other molecules, Aμ is zero.The Δ 1 resonance broadens substantially for electron hop rates of 10 6 -10 7 s −1 and disappears when the electron hop rate is comparable to the isotropic muon hfcc.A new resonance forms at 1/10th the magnetic field of the original resonance when the electron hop rate is much larger than Aμ and is due to the unpaired spin density being equally distributed over the ten molecules.Electron hopping at ∼10 12 s −1 in a system containing a larger number of molecules would lead to the resonance going to zero field and disappearing as the unpaired electron spin density is distributed over a large number of molecules and would tend to zero at the muon.
In 2013 ALC-μSR measurements were performed on Alq 3 at 10 and 300 K. 21 The ALC-μSR spectra were modeled using the Quantum Monte Carlo simulation program. 20It was claimed that both the isotropic and dipolar muon hfccs as well as electron spin relaxation rates of five types of muoniated radical could be determined from modeling the spectra. 21We contend that it was not possible to determine these 20 parameters from overlapping resonance with only ∼70 data points, especially with the errors that were reported.Nevertheless, the observation of multiple overlapping resonances between ∼1.0 and 1.6 T appears to validate the calculations of McKenzie. 17wever, in several recent reviews, Nuccio et al. 22 and Wang et al. 23 have cast doubt on the calculations of McKenzie and extolled calculations that were reported in the Ph.D. thesis of Willis. 24Wang et al. note that "only two of the eight known radical states are in good agreement with experimentally derived values.The remaining six FIG. 3. Simulated ALC-μSR spectra for electron hopping between 10 molecules where Mu is bound to one.There is a nonzero muon hfcc (isotropic muon hfcc Aμ = 250 MHz and dipolar muon hfcc Dμ = 10 MHz) when the unpaired electron is on the same molecule as the muon and zero hyperfine coupling when the electron is on one of the other nine molecules.The curves correspond to different electron hopping rates between adjacent molecules.The resonance is a Δ 1 resonance.
are either not predicted to be present or there are significant differences (of up to 40%) in the experimentally derived and theoretically predicted hfccs" and "benchmarking of the DFT methodology used, and it could be that these discrepancies could be minimized if a more thorough investigation was carried out."We contend that it is more important to look at the pattern of the predicted and measured hyperfine couplings rather than whether there is exact agreement in the values.
It was claimed that the calculations reported by Willis were closer to the experimental values; however, these calculations used a demonstrably inferior computational method, specifically the semiempirical PM3 for geometry optimization.Curioni et al. found the PM3 method significantly overestimated the Al-N bond lengths and underestimated the Al-O bond lengths while calculations with the BLYP density functional were much closer to the experimental values. 25The calculations reported by Willis did not account for the light mass of the muon on the hfccs, which increases the muon hfcc by ∼28% for the C 6 H 6 Mu radical. 26The claim of better agreement with experimental values can be considered accidental at best, especially given the over-parameterized fits.
The goal of this paper is to characterize the muoniated radical states in Alq 3 and determine whether the observed spectra are consistent with localized spin density or rapid electron hopping.We have chosen to study the muoniated radicals formed by Mu addition to the quinolate ring when they are not in an OSC, and so would not expect rapid electron hopping.Ideally, we would study isolated Alq 3 molecules, but Alq 3 is not sufficiently soluble in any suitable solvent for μSR measurements.Instead, we studied the 8hq ligand on its own and obtained μSR spectra and performed DFT calculations.8hq is not an OSC like Alq 3 , but the resulting radicals formed by Mu addition to 8hq will have very similar structures to the Mu adducts of Alq 3 (Fig. 4).We obtained an ALC-μSR spectrum of 8hq from 0 to 2.5 T and a high-statistics TF-μSR spectrum at 1.45 T. We calculated the hyperfine coupling constants of the possible muoniated radicals with a larger basis set [UB3LYP/6-311+G(d,p)], accounted for vibrational averaging effects, and extensively benchmarked the calculations against similar muoniated radicals.We have compared the results on 8hq with improved μSR spectra and DFT calculations of Alq 3 .We have obtained a higher-quality ALC-μSR spectrum of Alq 3 at 298 K over a wider magnetic field range (0-3.4T) than previously reported and high-statistics TF-μSR spectra at 0.25, 1.45, and 3.0 T. We have also calculated the isotropic and dipolar hfccs of the Mu adducts of Alq 3 in the same manner as the Mu adducts of 8hq.
The ALC-μSR and TF-μSR spectra of 8hq and Alq 3 are consistent with the radicals produced by addition to the six secondary carbons on the quinolate ring being formed in approximately equal amounts.This should be contrasted with radicals formed in the liquid state, where there is a clear preference for the lower energy products. 15The radicals were observed in the TF-μSR spectra, although this generally required short time windows for the Fourier transform as the radicals, particularly in Alq 3 , have large spin relaxation rates.The muoniated radicals with muon hfccs of 23 ± 1 and 91 ± 1 MHz reported by Drew et al. were not observed.
This paper conclusively demonstrates that the muoniated radicals formed in Alq 3 have localized spin density and suggest that the model proposed by Drew et al. involving electron hopping at rates of (1.4 ± 0.2) × 10 12 s −1 is incorrect.This is likely the case as well for other OSCs studied by μSR, such as Spiro-DPO. 27This does not mean that muoniated radicals do not provide information about OSC.The broadening of the ALC resonances in Alq 3 due to electron spin flips of ∼10 7 s −1 could result from spin-exchange reactions between the muoniated radical and mobile carriers in the OSC and this should be investigated further.

II. EXPERIMENTAL
8hq and Alq 3 were purchased from Sigma-Aldrich and used without further purification.Alq 3 was encapsulated in a packet made of 25 μm silver foil measuring 1 × 1 cm 2 .This packet was placed on a silver backing with Apiezon N grease and held in place with a thin film of x-ray Mylar.8hq was encapsulated in an Al sample cell with a thin Ti foil window.
ALC-μSR measurements on 8hq and Alq 3 at 298 K were performed using the HELIOS spectrometer on the M15 beamline at TRIUMF.The spin polarization was antiparallel to the muon momentum.Both samples were mounted in a cold-finger cryostat, which was inserted axially in the HELIOS spectrometer.The ALC-μSR spectra were obtained by scanning the magnetic field with steps of 20 mT and are the average of multiple scans.TF-μSR measurements were made with the spin polarization rotated by 90 ○ to the muon momentum.The magnetic field was calibrated at several points within the range of the ALC-μSR spectrum by measuring the precession frequency of diamagnetic muons.ALC resonances were observed on a nonlinear background that is due to the changes in the beam spot with applied field and field-dependent positron trajectories.The main background was accounted for by running the silver backing plate and fitting this spectrum with a sixth-order polynomial.Least-squares fitting of multiple Lorentzians and third-order polynomial background was The Journal of Chemical Physics ARTICLE scitation.org/journal/jcpapplied to the corrected data using a procedure based on the Minuit function minimization library.The fits to the background-corrected and raw data were visually compared in addition to obtaining an acceptable minimized χ 2 value for the least-squares fit.The resonances width and amplitude were strongly correlated with the background.TF-μSR measurements on 8hq at 298 K were performed using the HELIOS spectrometer in a magnetic field of 1.45 T. TF-μSR measurements of Alq 3 at 298 K were performed on the HELIOS spectrometer in a magnetic field of 1.45 T. TF-μSR measurements of Alq 3 at 10 and 298 K were performed on the HAL-9500 spectrometer at Swiss Muon Source (Paul Scherrer Institute, Villigen, Switzerland) in magnetic fields of 0.25 and 3.0 T.
DFT calculations were performed using the Gaussian 09 package of programs.The structures were optimized using the unrestricted B3LYP functional and the 6-311+G(d,p) basis set.Muonium was treated as an isotope of hydrogen with a magnetic moment of 8.890 597 μ N .The light mass of the muon was treated in a manner suggested by Roduner. 28The structure was first optimized with no constraints.The structure was then reoptimized with the bond corresponding to the C-Mu bond constrained at a length 4.9% longer than the optimized value.
ALC-μSR spectra were simulated using the program Quantum. 20Only the Δ 1 and methylene proton Δ 0 resonances were simulated for each radical.

A. Benchmarking DFT calculations for related muoniated radicals
A criticism made about the DFT calculations in Ref. 17 was that the calculated hyperfine coupling constants did not match the values reported by Nuccio et al. 21This could be due to errors either in the calculations or in the previous μSR measurements.The first step to resolving this controversy is to validate the DFT calculations by comparing the calculated values for several muoniated radicals with the experimentally measured muon and methylene proton hfccs.These are reported in Table I.There is very good agreement between the experimental and calculated values; the average magnitude of the % difference is 2.2% for Aμ and 3.4% for Ap.The results of the benchmarking calculations suggests that the computational method used in this paper is appropriate for the study of muoniated radicals formed by Mu addition to aromatic systems and containing heteroatoms.

B. DFT calculations of the Mu adducts of 8hq and simulated ALC-μSR spectra
The ALC-μSR spectra of the Mu adducts of 8hq can be modeled using the calculated muon and methylene proton hyperfine parameters listed in Table II and the supplementary material.The overall ALC-μSR spectrum depends on the relative yield of each of the where k j M is the second-order rate constant for reaction j and ∥Rj∥ is the concentration of the jth reaction partner.This can be the concentration of different molecules or different sites on the same molecule.This is observed for Mu addition in the liquid state. 33Since we are considering different sites on the same molecule and each site is present in the same amount, Eq. (3) reduces to The relative yield just depends on the rate constants for addition at the different sites.We will also assume that the rate constants can be described by the Arrhenius equation, where Ea,j is the activation energy, k B is the Boltzmann constant, and T is the temperature.
According the Bell-Evans-Polanyi hypothesis, the activation energy is linearly related to the reaction enthalpy (ΔHj) for a series of related single-step reactions, 34 Ea,j ≙ E 0 + αΔHj, (6)   where α is a measure of "lateness" of the transition state (0 > α > 1).This indicates that the activation energy is lower for more exothermic reactions (i.e., more negative ΔH).The corollary to this is the reaction is fastest for the pathways that generate the most stable free radical.
In order to simulate the overall spectrum, we have made several assumptions for addition of Mu at the different sites; (1) the prefactor, A, is the same; (2) the "lateness" of the transition state, α, FIG. 5. Simulated ALC-μSR spectra of 8hq powder with an electron spin flip rate of 1 μs −1 (a) with the relative yield of each type of muoniated radical is given by the Boltzmann weighting with respect to the relative energy of the radical at 298 K ( 7) and (b) every muoniated radical formed in equal amount.
is the same; and (3) ΔHi ∼ ΔH 0 + ΔEi, where ΔH 0 is the reaction enthalpy for the most exothermic reaction and ΔEi is the difference in the internal energy of the muoniated radical Ri and that of the lowest energy product.Combining the two previous equations and applying our assumptions gives The relative yields, PR i , were used to produce the spectrum shown in Fig. 5(a).We also considered the possibility that the six types FIG. 6. Background-subtracted ALC-μSR spectra of 8hq powder at 298 K. of muoniated cyclohexadienyl radical formed by addition to the secondary carbons of the aromatic ring were formed with equal probability, i.e., PR i ≙ 1/6.The simulated ALC-μSR spectrum for this scenario is shown in Fig. 5(b).In both cases, the amplitude of the Δ 0 resonances were negligible compared with the Δ 1 resonances.

C. μSR measurements of 8hq
The ALC-μSR and TF-μSR spectra of 8hq powder at 298 K are shown in Figs. 6 and 7, respectively.There are four distinct  III).The width of the resonances indicates that there is no significant electron hopping (Fig. 2).The resonances were assigned comparing the experimental ALC-μSR spectrum with simulated spectrum.The calculated hfccs are about 20% larger than the measured values, but there are clear similarities between the experimental spectrum and simulated spectrum in all radicals are formed with equal yield.Our assignment is based on the pattern of the resonances and a comparison with the DFT calculations.These are given in Table III and shown in Figs. 6 and 7.The smaller measured Aμ values could indicate a small amount of delocalization of the unpaired   V.There exist much larger differences between the Aμ values of diastereomers in this case than there are in the muoniated cyclohexadienyl radicals formed by Mu addition to l-phenyl-ethylamine, 1-phenyl-ethanol, and cumene. 16pectra were simulated for each of the 36 muoniated radicals listed in Table IV using the isotropic and dipolar muon and methylene proton hfccs (supplementary material).Spectra were simulated first by assuming the relative yield of each type of radical is given by a Boltzmann average of the calculated relative energies [Eq.( 7)] and then assuming each radical is formed with equal yield (Fig. 8).

E. μSR measurements of Alq 3
The background-subtracted ALC-μSR spectrum of Alq 3 at 298 K is shown in Fig. 9.There are several overlapping resonances that are assumed to be Δ 1 resonances based on their amplitude.There are two major resonances at ∼1.08 and 1.41 T with shoulders at ∼0.85 and 1.62 T. The major resonances correspond to Aμ values of ∼294 and 384 MHz.The shoulders correspond to Aμ values of ∼232 and 441 MHz.As with 8hq, the ALC-μSR spectrum of Alq 3 most resembles the simulated spectrum where the radicals are formed with equal yield.The situation where the muoniated radicals are formed with a relative yield depending on their relative energy results in a spectrum where the resonances are at much lower fields than observed experimentally.We have not assigned the peaks in the ALC-μSR spectrum of Alq 3 to individual muoniated radicals as the overlap makes this impractical.
TF-μSR spectra of Alq 3 at 298 and 10 K are shown in Fig. 10.Fourier transforming over short time windows makes it possible to detect signals due to short-lived muoniated radicals.There are overlapping powder spectra of several muoniated radicals with muon hfccs on the order of 250-450 MHz.This is consistent with the ALC-μSR spectra.
Drew et al. reported two types of muoniated radical with |Aμ| of 23 ± 1 and 91 ± 1 MHz at 10 K and not, as implied, at 290 K. 9 We found no evidence for muoniated radicals with hfccs in this range in the TF-μSR spectrum of Alq 3 at either 10 or 298 K.This was true for the TF-μSR measurements on HAL-9500 at 3.0 and 0.25 T and HELIOS at 1.45 T (supplementary material).
The simulated resonances in the ALC-μSR spectrum are comparable in width to the experimental resonances when including an electron spin flip rate of tens of μs −1 (Fig. 2).We suggest that this means that there are dynamic processes that lead to the relaxation of the muon spin in the OSC and this does not occur in 8hq.Grecu et al. performed EPR measurements on non-doped Alq 3 and found several paramagnetic defect centers corresponding to 1/2, 1, and 3/2 spin at room temperature. 36 assumed to be the radical anion where the unpaired electron is mobile, which could cause spin relaxation by the Heisenberg spinexchange reaction.The rate for this process is on the order of tens of μs −1 , which is approximately five orders of magnitude slower than that suggested by Drew et al.

IV. CONCLUSIONS
The magnitude of the Aμ values of the Mu adducts of 8hq and Alq 3 clearly shows that the unpaired electron spin density is localized on the same Alq 3 molecule to which Mu is covalently bound.No muoniated radicals with |Aμ| in the range of 23 ± 1 and 91 ± 1 MHz were observed in the TF-μSR spectrum.The observed muoniated radicals substantially different energy levels compared with the parent molecules this results in the unpaired electron being pinned.This indicates that μSR cannot be used to measure the intrinsic hop rate of electrons Alq 3 .ALC resonances were observed in 37 other organic based on polyaromatic so it is likely that a similar pinning of the charge is occurring in these systems as well, but further investigation is required.This is not to say that μSR studies of the Mu adducts of Alq 3 and other OSC could not provide information about organic semiconductors.The resonance widths are broader than expected in Alq 3 and this could be due to interactions between the muoniated radicals and other paramagnetic species in the OSC.Further measurements are needed to see if the broadening is related to the concentration of defects in Alq 3 .

SUPPLEMENTARY MATERIAL
See the supplementary material for calculated muon and methylene proton isotropic and dipolar hyperfine coupling constants of 8-hydroxyquinoline and Alq 3 and for additional TF-μSR spectra of Alq 3 .

FIG. 1 .
FIG. 1.Structure of the meridional isomer of tris(8-hydroxyquinoline)aluminum (III) (Alq 3 ) with the numbering of the positions on the A ligand and the structures of the most likely Mu adducts of the A ring.Radicals with similar structures will form by Mu addition to the B and C rings of Alq 3 .
Fig.1.Addition of Mu preferentially occurs at secondary unsaturated carbons rather than tertiary carbons,15 so it was assumed that Mu addition to each ligand of Alq 3 should produce six types of substituted muoniated cyclohexadienyl radical.Mu is distinguishable from H, so Mu addition leads to a center of chirality at the methylene carbon of the radical formed.16Enantiomers give rise to identical spectra in magnetic resonance, but the chirality of Alq 3 breaks the degeneracy and makes it possible to distinguish between the diastereomers formed by Mu addition of different sides of the ligand rings.The result is that there are 36 distinguishable radicals (3 ligands × 6 addition site × 2 orientations).Density functional theory (DFT) calculations at the UB3LYP/6-31G(d,p) level of theory showed that the muoniated radicals have a very different electronic structure to the parent compound Alq 3 and it is energetically unfavorable for electron transfer in either direction between Alq 3 and the muoniated radical (ranging from 4.09 to 5.68 eV) [Fig.2(b)].17This implies that the unpaired electron is localized on the molecule where Mu is covalently bound.As a way to distinguish between the "innocent" muon and strongly interacting muon models, the muon and methylene proton hfccs for the Mu adducts of Alq 3 , which are proportional to the unpaired electron spin density at the nucleus ∥|ψ(0)| 2 ∥, were obtained from DFT calculations reported in 2010.17Resonances can be observed in the avoided level crossing resonance spectrum (ALC-μSR) and the resonance fields (Bres) are related to the hfccs.Two types of resonances are typically observed; they are characterized by the selection rule ΔM = 0 and ±1, where M is the sum of the

FIG. 2 .
FIG. 2. (a) Muon as a passive observer.The Gibbs energy for electron transfer, ΔG 0 , is 0 and the reorganization energy for electron transport, λ, was calculated by Lin et al.18 (b) Muon as a perturbation; the energy levels of the muoniated radical are different from those of the parent compound.ΔG 0 and λ for electron transfer to and from the muoniated molecule calculated by McKenzie [UB3LYP/6-31G(d,p)].17The singly occupied molecular orbital lies between the HOMO and LUMO of Alq 3 .The large ΔG 0 and λ values indicate the unpaired electron is localized on the molecule bound to the muon.

FIG. 8 .
FIG. 8.Simulated ALC-μSR spectra of 3 powder with an electron spin flip rate of 20 μs −1 (a) with the relative yield of each type of muoniated radical given by Boltzmann weighting with respect to the relative energy of the radical at 298 K(7)  and (b) every muoniated radical formed in equal amount.

FIG. 10 .
FIG.10.TF-μSR spectra of Alq 3 powder at 298 K (top) and 10 K (bottom) in a magnetic field of 3.0 T with different time windows for the Fourier transformation (fftw3 as implemented in musrfit35 ).The horizontal lines denote the range of frequencies corresponding to the muon hyperfine couplings determined from the ALC-μSR measurement.

The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp mz
quantum numbers of the muon, electron, and nuclear spins.The resonances are referred to as Δ 0 and Δ 1 resonances, respectively.The Δ 1 resonance field is given by

TABLE I .
Benchmarking DFT calculations [UB3LYP/6-311+G(d,p)] on muoniated radicals by comparing the calculated muon and methylene proton hyperfine coupling constants with the corresponding experimental values.The numbering is based on the IUPAC nomenclature.The calculated hfccs were obtained by fixing the C-Mu bond to be 4.9% longer than the optimized value and partially optimizing all other parameters.Muonium was treated as an isotope of hydrogen with a magnetic moment of 8.890 597 μ N .
a Experimental values from Ref. 26. b Experimental values from Ref. 29. c Boltzmann average of two structures at 298 K assuming rapid interconversion.d Experimental muon hfcc from Ref. 30 and methylene proton hfcc from Ref. 31.e Experimental values from Ref. 32. f Experimental values from Ref. 15. g Experimental values from Ref. 33.

TABLE II .
Calculated [UB3LYP/6-311+G(d,p)] muon hfccs and relative energies of the Mu adducts of 8-hydroxyquinoline.The calculated hfccs were obtained by fixing the C-Mu bond to be 4.9% longer than the optimized value and partially optimizing all other parameters.Muonium was treated as an isotope of hydrogen with a magnetic moment of 8.890 597 μ N .we have accounted for this by summing up the individual spectra multiplied by a factor representing the probability that it is formed.We considered two possibilities in simulating the overall ALC-μSR spectrum.The first was that the relative yield of each type of muoniated radical depends on the rates that the radicals are formed.The amount of each type of radical that could form is the result of competition kinetics.The relative yield (PR i ) of the muoniated radical Ri is given by

TABLE III .
Muon hyperfine coupling constants of the Mu adducts of 8-hydroxyquinoline powder at 298 K determined by ALC-μSR and TF-μSR spectra.ALC-μSR spectrum with a shoulder to the highest field peak, suggesting additional unresolved resonances.The experimental spectrum greatly resembles the simulated spectrum where all of the radicals are formed in equal amounts.This could indicate that Mu is not thermalized prior to addition.There are also four radical frequencies in the TF-μSR spectrum and the peak at ∼40 MHz is considerably broader than the other radical lines, which could indicate unresolved signals.The resonances in the ALC-μSR spectrum can be identified as Δ 1 resonances due to the resulting Aμ values matching those obtained from the radical lines in the TF-μSR spectrum (Table

TABLE IV .
Calculated [UB3LYP/6-311+G(d,p)] muon hfccs and relative energies of the Mu adducts of Alq 3 .The labeling of the muoniated radicals is based on the numbering scheme in 1 with the final number indicating different orientations of the muon with respect to the molecule.The calculated hfccs were obtained by fixing the C-Mu bond to be 4.9% longer than the optimized value and partially optimizing all other parameters.Muonium was treated as an isotope of hydrogen with a magnetic moment of 8.890 597 μ N .

TABLE V .
Calculated Δ 1 resonance field ranges of the Mu adducts of Alq 3 determined using the hyperfine parameters in TableIV.The calculated Aμ values and relative energies of the Mu adducts of Alq 3 are listed in TableIV.The calculated Δ 1 resonance field ranges are listed in Table