Computational modeling of isotropic electron paramagnetic resonance spectra of doublet state main group radicals †

The combined use of theoretical and mathematical methods in the analysis of electron paramagnetic resonance data has greatly increased the ability to interpret even the most complex spectra reported for doublet state inorganic main group radicals. This personal account summarizes the theoretical basis of such an approach and provides an in-depth discussion of some recent illustrative examples of the utilization of this methodology in practical applications. The emphasis is on displaying the enormous potential embodied within the approach.


Introduction
Isotropic (solution state) electron paramagnetic resonance (EPR) spectra of doublet state organic radicals are usually simple to interpret.There are two main arguments which promote the above statement: from the different nuclei typically present in organic systems, only hydrogen and nitrogen have spin-active isotopes with significant (approx.100%) natural abundances, and the values of nuclear spin for both 1 H and 14 N nuclei are low, I = 1/2 and 1, respectively [1].In the majority of cases, this leads to simple and easily detectable splitting patterns in the experimental spectrum [2].
Conversely, isotropic EPR spectra of inorganic main group radicals including other s-and p-block elements than hydrogen and nitrogen are often poorly resolved due to the hyperfine coupling (hfc) of the unpaired electron to magnetically active nuclei with large nuclear spin values (I > 1) and more than one naturally abundant spin-active isotope (see Figure 1) [1].
The complexity in the EPR spectra of main group radicals containing multiple spinactive nuclei generally renders it difficult to extract accurate values of hyperfine coupling (hfc) constants from the experimental spectrum.Since such data is used to gather information of the spin distribution within a paramagnetic molecule, this is quite problematic, as it can impede researchers from gaining a thorough understanding of a particular radical system.In addition, without any knowledge of the hfc constants, it is impossible to produce a simulation of the experimental spectrum which will most likely prevent the identification of the observed radical species.Thus, it is evident that accurate hyperfine parameters play a prominent role in the study of paramagnetic systems.One possible and very often used method to overcome the above difficulties in spectral interpretation is to employ theoretical first principles methods to calculate the magnitudes of the hfcs and then compare these results to data extracted from the experimental spectrum [3].In fact, there is a long history of using theoretical calculations in the interpretation of EPR spectra as perturbation molecular orbital and semi-empirical methods have been used as early as 50's and 60's [4].However, an even more powerful approach is to employ the calculated hfc constants as initial estimates of the true spectral parameters and then use iterative least-squares fit based methods to automatically refine the simulation with respect to the experimental EPR spectrum.We have recently shown that this technique can be an extremely successful tool in the assignment and interpretation of complex EPR spectra reported for inorganic main group radicals [5].This short personal account briefly reviews the theoretical basis of the approach and summarizes the results from its application to the analysis of some paramagnetic systems.

Theoretical methods for calculation of isotropic hyperfine coupling constants
The 3  3 hyperfine coupling tensors A(i) describe the interaction of the unpaired electron with the spin-active nuclei i [6].They can be separated into isotropic and anisotropic components of which only the former is discussed herein.A good (first-order) approximation of the isotropic hfc Aiso(i) for nucleus i is given by the Fermi contact interaction term iso 88 ( ) ( ), 33 () where n is the nuclear magneton, ge is the electronic g-factor, and gi and (ri) are the nuclear g-factor and electron spin density at the nucleus, respectively, for nucleus i.
Simply put, hfc constants are obtained by multiplying the electronic spin density evaluated exactly at each nucleus i with the appropriate physical constants.As suggested by the form of the Fermi contact term, the connection between spin density and hfcs can also be used in the opposite order i.e. experimentally determined hfc constants provide a practical means for evaluation of nuclear spin densities.
From the form of the Fermi contact interaction term, it follows that the hfc constants are extremely difficult to calculate theoretically because of their high sensitivity to the quality of the wave function (spin density) at one point in space; the Dirac delta function (ri) evaluates the wave function only at the nucleus thus making the property very local and unlikely to benefit from error cancellations.Although alternative formulations which use more global operators than the Dirac delta function in calculation of Fermi contact interaction have been introduced [7], none of them have yet found widespread use.
The delta function-based formulation also implies that the Gaussian-type (GTO) basis sets employed in the majority of molecular orbital methods are fundamentally flawed to be used in calculation of Fermi contact interactions [8].However, it has been shown that, when augmented with tight s-functions, the standard Gaussian-type basis sets can indeed overcome the nuclear cusp problem [9].An exhaustive number of calculations have also demonstrated that the EPR-III basis sets perform well in density functional theory (DFT) calculations [10], as do the more conventional TZVP [10,11], while the IGLO basis sets generally give good magnetic properties with a reasonable basis set size in ab initio calculations [12].It should also be pointed out that specifically tailored GTO basis sets for the calculation of hfc constants of main group radicals exist only for Period 1 and 2 nuclei (EPR-II and EPR-III) [10a].In addition, the performance of the TZVP basis sets (and some of their variants) in DFT based calculations has been statistically analyzed only up to Period 3 nuclei silicon, phosphorus and sulfur [10c]; there exists only a limited number of main group radicals in which significant spin density is found on nuclei from fourth (or higher) Period and whose hfc constants are known experimentally with sufficient accuracy to facilitate comparison with calculated results [13].
For systems with heavier s-and p-block elements (Period 4 and beyond), relativistic effects need to be treated appropriately [14].In principle, this can be done in both wave function and DFT based approaches.Relativistic effects can also be approximated in theoretical calculations by using pseudopotential basis sets for heavy elements.In this quasi-relativistic approach, the replacement of core electrons of the heavy element with an effective potential allows modeling of the effects of relativity only to the nearby lighter nuclei and actually prevents the determination of hfc constants to the heavy element itself as the inner s-electrons are not treated implicitly in the calculation.
In order to obtain accurate results, ab initio calculations of hfc constants need to be conducted using correlated levels of theory [15].On purely theoretical grounds, it is, however, not immediately obvious which of the many modern density functionals is most appropriate to use in DFT calculations of hfc constants [16].The choice of a functional is therefore strongly influenced by its reported performance.In general, generalized gradient approximation (GGA) functionals depending also on the kinetic energy density (meta-GGA) lead usually to small or no improvements over the basic GGA approximation [17], and hybrid functionals tend to yield the best results for a wide variety of main group systems [10,18].It should also be noted that reproducing trends in hfc constants as functions of structural parameters is a much easier task than matching the absolute values with experimental data.Such correlations are often achieved with almost any number of functionals [19].

Mathematical methods for spectral simulation
The accurate extraction of relevant physical parameters from experimental data comprises a major problem in all kinds of spectroscopic methods.The analysis of an experimental spectrum is generally a deceivingly straightforward procedure for simple molecular systems, but can become a very challenging problem when the system size is increased or the identity of even some of the atoms in a molecule is changed.As exemplified in the introduction, EPR spectroscopy of doublet state main group radicals is no exception to this rule: the presence of multiple low-spin or even a few high-spin (I > 1) nuclei can lead to extremely complex and, therefore, incompletely resolved hyperfine coupling patterns in which the majority of the information describing the paramagnetic species is lost under the broad lineshape.In such cases, simple trial-and-error based simulation attempts are clearly inadequate, since minor errors in the parameters can alter the appearance of a simulated spectrum significantly.Therefore, efficient spectral fitting algorithms augmented with an automated refinement of the parameters used in the simulation are prerequisites for successful simulation of complex EPR spectra [20].
All spectral fitting programs are based on the repeated utilization of three basic steps, simulation, comparison and refinement, which are iterated in an ordered fashion until either the maximum number of steps is reached or the procedure converges to a solution for which the calculated parameter indicating the goodness of the fit is lower than a pre-defined threshold.In the context of isotropic EPR spectra of doublet radicals, the above is nothing more complex than performing a nonlinear least-squares fit in a multidimensional parameter space in which the parameters are the number and identity of spin-active nuclei i, the isotropic hyperfine coupling constants Aiso(i), the g-value of the radical, as well as both the spectral line-width and amplitude [21,22].As this problem is very general in nature, several algorithms designed to perform such fitting have been described in the mathematical literature, e.g.simplex, Monte-Carlo, Hooke-Jeeves and Levenberg-Marquardt [23], and are readily available in several of the more advanced simulation software designed for the needs of EPR spectroscopists (e.g.XSophe, EasySpin and EPR-Winsim) [24].We note here that the simplex algorithm is especially well suited to the least-squares analysis of highly complex EPR spectra [25] and has, therefore, also been used in the selected examples described in the next section.
Despite the immense advantages that the use of iterative least-square based fitting algorithms offer, their use in spectral analysis also introduces some problems: it is in general impossible to build an automated simulation program which, in all possible cases, would lead to an unambiguous set of chemically relevant hfcs and other spectral parameters that would reproduce the experimental spectrum with great precision.This arises in part from problems associated with the fitting algorithms themselves (failure of convergence or convergence to an undesired local minimum), but also from the limitations imposed by the quality of the experimental spectrum (poorly resolved lines due to spectral overlap or imperfect measuring conditions).
Even though the experimental EPR spectrum would be of perfect quality, there exists no such least-squares fit algorithm which would be able to converge to an unambiguous solution in a reasonable amount of time without good initial estimates of the individual parameters used in the iterative optimization.Whereas it is relatively straightforward to determine the proper linewidth, g-value and amplitude from the experimental spectrum, it is in most cases rather difficult to come up with reasonable estimates for the individual hyperfine coupling constants.After all, these are the key parameters that the iterative procedure is supposed to determine in the first place!For a long time there were only two practical methods available to produce the required estimates of hfcs: sophisticated guesses based on chemical knowledge of the system and the use of hfc constants determined via electron-nuclear double resonance (ENDOR) spectroscopy [26].A third, much more modern, approach is to employ high-frequencyhigh-field EPR spectroscopy (W-and Q-bands).The advent of high-performance computers has, however, introduced another, more easily applicable, possibility to determine a suitable set of initial hfc constants by performing theoretical calculations for radical species in question.Statistical analyses of results have shown that the accuracy of the most common methods in predicting hfc constants is as good as 5-10% and 10-20% for Period 1 and 2 nuclei [10,15,16] respectively, which in the majority of cases facilitates smooth convergence of the iterative least-squares fit procedure to a global minimum.Despite the long-term general availability of efficient quantum chemical codes capable of performing hfc constant calculations, an approach that combines computational results with mathematical methods has not yet, however, found widespread usage in practical applications.

Illustrative examples
In the following sections we present three examples of studies in which theoretical calculations have been used in combination with mathematical spectral optimization methods to aid in the analysis of some very complex EPR spectra reported for inorganic main group radicals.
All spectra were recorded using Bruker EMX113 and ESP300E spectrometers operating in the X-band.Spectral simulations were carried out with the XEMR [27] and WINEPR SimFonia programs [28].Hyperfine coupling constant calculations were done at the PBE0 (non-relativistic) and PBEPBE (relativistic) levels of theory [29] using Gaussian 03 [30]  An intriguing observation in the synthesis of systems 1 was the formation of intensely colored solutions when the reaction mixtures were briefly exposed to air [32].
The formation of radical species was confirmed by EPR spectroscopy and the paramagnetic nature of the solutions was thought to arise from the formation of an anion radical {[PhB(N t Bu)2]} −• .As the resulting EPR spectra were found to be extremely complex, the oxidation reactions of compounds 1 were performed in a controlled manner using half an equivalent of iodine and the EPR spectra were re-determined.Although the new spectra showed more details in their fine structure, the overall shapes of the signals in the two sets of spectra remained identical.confirmed that the two complexes are indeed isostructural and exist in the delocalized spirocyclic geometries in the solid state.An examination of the experimental (optimized) hfc constants listed in Table 2 reveals that the couplings to the 14 N and 11 B nuclei in the two boraamidinate moieties are similar in magnitude: the difference is less than 0.5 G in all four spectra.This coincidence creates spectral splitting patterns in which most of the expected fine structure becomes disguised under the broad lineshape.For example, the combination of four and two equivalent 14 N and 11 B nuclei, respectively, should theoretically give rise to 63 individual lines.In the case of compounds 2, the majority of the signals overlap, creating a splitting pattern which consists only of fifteen broad lines.In the case of the aluminum and gallium spirocycles 2b and 2c, the hyperfine coupling of the unpaired electron to the central metal is also approximately a multiple of the smaller hfcs to nitrogen and boron nuclei, which brings about the deceivingly simple splitting patterns in the experimental spectra.However, any simulation attempts using hfcs which are exact multiples of each other are doomed to fail.Although they are able to predict the correct number of individual lines, they do not reproduce the correct lineshapes which are extremely sensitive to even smallest changes in hfcs.

Cubic tetraimidophosphate dianion radicals
Polyimido anions of the p-block elements have in recent years been of topical interest in main group chemistry [35].They are related to the more common oxoanions as the imido group [NR] 2− is isoelectronic with the oxo [O] 2− substituent.Second only to silicon, phosphorus forms the largest number of oxoanions, many of which are of significant industrial importance [36].It is therefore not surprising that numerous imidoanalogues of phosphorus oxoanions have been prepared, including the butoxide [42].Support for this hypothesis came from studies of the iodide derivative 4c, in which the solvated complex (THF)3LiI was isolated and characterized by X-ray crystallography [41].The THF-solvated LiO t Bu salt was, however, not detected in solutions of 4d [42], raising doubts about the correctness of the dissociation hypothesis.
Hence, the systems 4a and 4d were subjected to a more thorough study in order to reveal the true identity of these radical species in solution.
Upon closer inspection of the spectra in Figure 3a, it becomes evident that both of them show a distinct doublet feature corresponding to a large (25 G) hfc to 31 P nucleus (I = 1/2) [1].The slowly decaying shape of the signals present in the outer ends of the spectra indicate that hfcs to several magnetically equivalent nuclei contribute to the spectral fine structure [43].Therefore, the solution structures of the paramagnetic species 4a and 4d should be highly symmetric.Indeed, it may well be that neither of the cubic radicals dissociates to the proposed monocycle upon solvation.It is however extremely difficult to uncover any unambiguous coupling patterns from the spectra as numerous repeating peak differences falling within the range of 1-10 G can be found.Hence, theoretical calculations for radicals 4 using isostructural solid-state geometries were carried out in order to facilitate spectral simulation and either confirm or refute the hypothesis that both 4a and 4d share similar cubic structures also in solution [5b].only minimal spin density (via spin polarization) on the phosphorus and lithium atoms, significant 31 P and 7 Li hfcs are anticipated due to the large gyromagnetic ratios of these nuclei [1].Conversely, the hfcs to the lone trimethylsilyl nitrogen atom and the remote halogen atoms are calculated to be very small (less than 0.5 G) and should, thus, have only a small contribution to the width of experimental EPR spectra.
The calculated hfcs for compounds 4a and 4d are listed in Table 2 along with the values obtained with iterative least-squares fit methods [5b]; there is an excellent agreement between the two set of numbers.Figure 3b shows that simulations created by using the least-squares optimized hfc constants are able to reproduce even the minutest features present in the experimental spectra.It is therefore safe to conclude that the spectral and computational analyses are able to prove the identity of these paramagnetic species in solution conclusively.The retention of cubic geometry in solution for 4a and 4d is also consistent with the fact that the expected by products of solvation, LiCl or LiO t Bu, could not be isolated from the paramagnetic solutions.The hfc pattern in the EPR spectra of 4a and 4d is primarily a doublet due to the large dominant coupling of the unpaired electron to the central 31 P nucleus.The signal is further split into septets by the three equivalent 14 N nuclei and the (mostly hidden) decet structure arises from the smaller hfcs to the three 7 Li centers.The numerical values listed in Table 2 illustrate the sensitivity of the spectral appearance to very small changes in the parameters: the different fine structure in the two spectra arises entirely from a variation of less than 0.3 G in both the spectral linewidth and the hfcs to the three equivalent 7 Li nuclei as well as to the remote 14 N and 35,37 Cl centers.
In contrast to the results obtained for 4a and 4d, it proved impossible to simulate the EPR spectra of the corresponding bromide and iodide derivatives accurately by assuming retention of the cubic framework [5b,41].This is attributed to the fact that the  polarization effects and relatively high gyromagnetic ratios of the two isotopes of gallium [1].Conversely, only small hfcs are calculated to the pnictogen atoms both in 7 and 8.
These couplings vary roughly with the relative magnitudes of the gyromagnetic ratios of 14 N, 31 P, and 75 As nuclei [1], which implies a relatively constant spin density on the remote pnictogen centres.In comparison to radicals 8, significantly smaller hfcs to the gallium metal are calculated for systems 7. Instead, considerable hyperfine interactions to the 127 I nuclei in 7 are calculated which should also be visible in the experimental spectra.
Excellent simulations of the experimental EPR spectra recorded for mono-(Figure 4b) and dipnictogen (Figure 5b) complexes were obtained using the calculated hfc constants as initial estimates of the true couplings, and optimizing the sets of values using iterative methods [5e].The slight differences between the left-and right-hand sides of the experimental spectra of compounds 7 (i.e. the lack of perfect inversion symmetry with respect to signal intensities) is believed to be caused by higher-order hyperfine interactions due to the presence of large hfcs to the heavy nucleus 127 I [48].As expected (see above), the experimental EPR spectrum of 7a does not display a quartet pattern characteristic of a large coupling to gallium (see Figure 4).Instead, the spectrum shows a partially resolved sextet pattern due to hfc to one 127 I nucleus (I = 5/2) which dominates the spectrum [1]; the sextet pattern, although present, is hidden in the spectrum of 7b by the significantly larger hfc to the remote pnictogen nucleus (7 G for 31 P vs. 1 G for 14 N).In the experimental EPR spectra of radicals 8b and 8c (see Figure 5), the quartet pattern due to the hfc to the gallium nucleus is only visible in the spectrum recorded for the arsenic species: in case of 8c, the hyperfine coupling to the two remote pnictogen nuclei is not a multiple of the hfcs to the 1 H and 14 N nuclei in the DAB backbone which leads to better resolved splitting patterns (cf.data for 8b).

Conclusions
Theoretical calculations have been used in combination with mathematical methods to aid in the analysis of some very complex experimental EPR spectra of main group radicals.The results and methodologies reviewed in this paper clearly demonstrate the capability of density functional methods to predict hyperfine coupling constants that are in semi-quantitative agreement with the experimental values even for complex multinuclei compounds including heavy main group elements.Thus, calculated coupling constants provide essential initial estimates of the true hyperfine couplings present in main group radical systems.Their subsequent use in least-squares fit based spectral simulation methods yields an accurate description of the magnetic hyperfine interactions, provided that the experimental data contains enough characteristic information to allow unambiguous convergence of the iterative methods and that the spectrum truly corresponds to the purported radical species.Through this combination of experimental and computational methodologies, the research described in this account has provided fundamental information about many new stable and persistent main group radicals that is not accessible by other methods.We hope that the examples reviewed here illustrate the enormous potential of this methodology and that the high quality of the reported results will encourage other chemists working in the field of EPR spectroscopy to make use of this approach.

Fig 1 .
Fig 1.The 42 different s-and p-block elements.Lower triangle denotes an element for which nuclear spin is greater than one and upper triangle denotes an element with more than one spin-active isotope.
trisimidometaphosphate [(P(NR)3] −[37], bisimidophosphinate [R2P(NR)2] 2−[38], and the tetraimidophosphate [P(NR)4] 3− anions[39].The synthesis of the unsymmetrical tetraimidophosphates Li3[P(NR)3(NSiMe3)] (R = i Pr, t Bu, Cy, Ad) (3) has been reported recently[39b].During the course of this research it was noted that the initially colorless THF solutions of 3 became deep blue upon exposure to oxygen.This observation is reminiscent of the behavior reported earlier for the chalcogen-centered polyimido anions in {Li2[E(N t Bu)3]}2 (E = S, Se) which form deeply colored persistent radicals upon oxidation[40].These results provided impetus for a more detailed study of the radical species accessible from controlled reactions of asymmetric tetraimidophosphates with common oxidizing agents.The one-electron oxidation of Li3[P(N t Bu)3(NSiMe3)] (3, R = t Bu) was performed with an half an equivalent of SO2Cl2, bromine and iodine.This resulted in the successful isolation and EPR spectroscopic characterization of the stable neutral radical complexes{[Me3SiNP(μ 3-N t Bu)3][μ 3-Li(THF)]3X} • (4a, X = Cl; 4b, X = Br; 4c, X = I) [5b,41], of which the iodide salt was the only one analyzed with X-ray crystallography[41].In the solid state, the complex 4c was shown to adopt a distorted cubic structure (approx.C3 symmetry) in which a tetraimidophosphate dianion radical {[Me3SiNP(N t Bu)3]} 2−• is capped by two Li + ions and a molecule of lithium iodide; each of the three Li + cations is also coordinated to one solvent molecule.Recently, it was shown that the dilithiated tetraimidophosphate radical can also trap a monomeric lithium alkoxide, namely LiO t Bu, forming a cluster (4d, X = O t Bu) which was found to be isostructural with the iodide derivative in the solid state[42].The experimental EPR spectra of THF solutions of 4a and 4d at room temperature are shown in Figure3a[5b,42].At first glance, it is surprising how little the spectra resemble each other, despite the fact that the two radical systems share similar cubic structures in the solid state.Initially, this was thought to arise from the dissociation of the cubic framework in 4d to a monocyclic anion radical {[(Me3SiN)( t BuN)P(μ-N t Bu)2]Li(THF)2} −• (with a [Li(THF)4] + counterion) and a THF-solvated lithium tert-
Figures 4a and 5a, proved to be extremely complex which prevented the authors from
i = isotope; n = number of equivalent spin-active nuclei b experimental values are obtained from least-squares fit optimizations of the spectra c values taken from Ref. [5e] d relativistic spin-orbit ZORA calculation As shown above, the complexity in the EPR spectra of 7 and 8 does not arise from different isotropic g-values for the 69 Ga and 71 Ga isotopomers as previously suggested [47g], but from the presence of higher-order splitting effects and hfcs which are approximately multiples of each other [5e].Taken in concert with some previous experimental work [47], the spectral analyses described herein suggested that the published experimental EPR results of some related paramagnetic Group 13-DAB complexes might also have been misinterpreted [47h].Detailed theoretical investigations of these systems have recently been reported [49].

Table 1
Experimental and calculated hyperfine coupling constants (in Gauss) of radicals 2 a, b, c a i = isotope; n = number of equivalent spin-active nuclei b experimental values are obtained from least-squares fit optimizations of the spectra c values taken from Ref. [5c,d]

Table 2
Experimental and calculated hyperfine coupling constants (in Gauss) of radicals 4a and a i = isotope; n = number of equivalent spin-active nuclei b experimental values are obtained from least-squares fit optimizations of the spectra c values taken from Ref. [5b]

Table 3 [
5e].The calculated values of both 1 H and 14 N hfcs show only minor variation between the different systems, as expected, since both 7 and 8 are ligand-centered radicals; the numerical values of the couplings are also consistent with the hfcs observed in other related DAB-centered radical systems [47].In radicals 8, the dominating 69,71 Ga couplings (25-30 G) arise from a combination of spin

Table 3
Experimental and calculated hyperfine coupling constants (in Gauss) of radicals 7 and 8 a, b, c