3.1 Nitroxyl Bi-, Tri- and Tetra-radicals

Nitroxyl biradicals furnished the model systems on which most of the theoretical principles of the PELDOR method were experimentally verified. The most commonly-used biradicals have linear molecular linkers terminated by two piperidine, pyrrolidine, pyrroline, or 3-imidazoline radicals. The structure and properties of such biradicals are extensively studied by CW EPR and are detailed in several monographs [3, 5, 6]. Structures of some of the nitroxyl biradicals used to establish the measurement of distances between >N-O· fragments are given in Table 3.1.

Table 3.1 Some nitroxyl biradicals studied by PELDOR
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3.1.1 Origins of PELDOR

The PELDOR modulation due to dipolar interactions was first detected in frozen glassy solutions of nitroxyl biradical 3-1 [12]. The distance between unpaired electrons was found either from direct measurement of the modulation period, Fig. 3.1, or from the Pake pattern in its Fourier spectrum. The first PELDOR work with magnetic field jumps also used this biradical [13].

Fig. 3.1
figure 1

Reprinted from Milov et al. [12]

PELDOR time trace of frozen solutions of biradicals 3-1 in toluene

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Biradicals 3-1 with R = H and 3-2 helped explore how the duration of pulses in PELDOR affects the distance distribution \(F(r)\) Sect. 1.6 [14]. This work improved the reliability of \(F(r)\) and the determination of the magnitude and sign of the exchange integral \(J\) for biradicals with less than 2 nm between >N-O· fragments. Two conformations of 3-2 were found with similar distances of 1.62 and 1.70 nm between >N-O· groups.

A distance of 1.97 ± 0.01 nm between the >NO· groups of 3-3 was determined from 3pPELDOR Fourier spectra [15]. The same value was also found for the closely-related biradical 3-4 by 3pPELDOR [16] and 4pPELDOR [17], indicating virtually identical dimensions and conformations for 3-3 and 3-4. In fact, these two biradicals are often confused with each other in the literature, but Gunnar Jeschke indicated to us that they are indeed different. Different groups consistently measure 1.97 nm between the >N-O· groups of 3-4 and report quite narrow lines, ~3–4 MHz, in the PELDOR Fourier spectrum, which agrees with molecular modelling [16] and indicates a single conformation for 3-4 in a glassy matrix at 80 K. The first test of 4pPELDOR was on 3-4 [17].

Nitroxyl biradicals were the molecular ruler to compare distances measured by PELDOR with those known from molecular structures, and to determine the experimental distance limits of PELDOR [16, 18, 21, 23, 24]. The bridging groups connecting the radical fragments provided precise control of distances. Biradicals 3-5 with aliphatic bridges were studied at 80 K in a polystyrene matrix [16]. Although peaks in the Fourier spectra are fairly broad, Fig. 3.2a, their maxima progressively shift to lower frequencies as the chain length increases. Distances from 2.04 nm at \(n = 8\) to 3.28 nm at n = 20 are obtained [16]. Each link in the chain increases the distance between >NO· fragments in 3-5 by ~0.1 nm. The linewidths in the Fourier spectrum in these experiments result from the set of conformations of the aliphatic chain. The distances determined by PELDOR agree with statistical calculations assuming a completely stretched conformation of the aliphatic chain, Fig. 3.2b.

Fig. 3.2
figure 2

Reprinted from Pfannebecker et al. [16]

a PELDOR Fourier spectra of biradicals 3-5; b distances r between >N-O· groups depend on the number of CH2 groups in 3-5: points denote the experiment for n = 8, 12, 14, and 20; the dotted curve shows a statistical calculation of conformations; the solid line shows a completely stretched trans-structure.

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The PELDOR spectra of 3-5 have unusually good resolution. Biradicals 3-9 and 3-10, with similar molecular structures, lack modulation beats in their 3pPELDOR time traces and the Fourier spectra lack distinct peaks [20]. In our opinion, the aliphatic bridge in these frozen biradical solutions adopt a more diverse set of conformations than for 3-5 in polystyrene. Subsequent PELDOR studies focus on biradicals with rigid, conformationally-fixed bridging groups.

3.1.2 The Outer Limits

The upper boundary of distances that can be measured by PELDOR using nitroxyl radicals was explored with biradicals 3-11 to 3-15 having a rigid molecular bridge composed of triple bonds and benzene rings [21, 23]. The largest distance of 7.44 ± 0.82 nm was measured for 3-15 in deuterated ortho-terphenyl at 50 K [23]. Good accuracy at a distance of 5.0 nm was achieved with 3-14 by PELDOR [19, 23] and by another dipolar spectroscopy known as SIFTER. Both methods gave 5.10 ± 0.12 nm [21], while theoretical MD modelling predicted a distance between 5.15 and 5.23 nm for 3-14. The excellent correlation between MD and PELDOR methods for the distances between >N-O· fragments also holds for other shorter, but more rigid, biradicals: 3-6 to 3-8, 3-11, and 3-12 [18, 19, 21, 25].

Spin relaxation seems to limit the maximum distances that can be measured by PELDOR using nitroxyl labels, while the shortest distances are limited more by instrumental parameters.

3.1.3 Small Clusters of Radicals

Nitroxyl radicals specially linked as pairs, triples, and quadruples provided the most rigorous test of PELDOR theory [26]. The goal was to experimentally verify the theoretical basis for determining the number of spins in a group and their pair distribution function for fixed values of N = 2, 3, and 4.

The rather rigid 3-18 to 3-22 containing precise numbers of spins were used, Table 3.2. The 4pPELDOR time traces of frozen solutions in deuterated toluene show distinct dipole modulation, Fig. 3.3. The limiting signal amplitude \(V_{p}\) was measured after background subtraction. The pump probability \(p_{B}\) was determined to be 0.43 and 0.12 for pump pulse durations of 12 and 92 ns, respectively, based on PELDOR time traces of 3-16 with n = 2. With an accuracy of ~5%, Eq. 1.26 gave experimental values of N = 2.1, for biradicals 3-18, and 3-19; N = 3.0 for triradicals 3-20 and 3-21, and N = 3.9 for tetraradical 3-22, corresponding completely to the values expected for these polyradicals.

Table 3.2 Bi-, tri-, and tetra-nitroxyls studied by the PELDOR method in [26, 27]
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Fig. 3.3
figure 3

Reprinted from Bode et al. [26]

a Experimental PELDOR time traces used to determine the number of radicals N in biradicals 1—3-18, 2—3-19; triradicals 3—3-20 and 4—3-21; and 5—tetraradical 3-22; b-d PELDOR time traces and \(F(r){\text{d}r}\) for 3-19, 3-21, and 3-22, respectively.

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The distance distribution \(F(r)\) for polyradicals has some noteworthy features. For biradicals 3-18, 3-19 and the symmetric triradical 3-20, \(F(r)\) has only one maximum, as expected for structures having the same distance between their nitroxyl groups. However, \(F(r)\) for triradical 3-21 contains two lines with an intensity ratio of 2:1, Fig. 3.3c, as expected from the inequivalent positions of nitroxyl groups. For tetraradical 3-22, \(F(r)\) has three maxima, Fig. 3.3d, corresponding to the three possible distances for a rectangle with nitroxyl groups at the vertices. The distances between spins in these polyradicals agree well with MD results given in parentheses, Table 3.2.

A triradical can have up to three dipolar frequencies corresponding to the distances between pairs of spins in the triradical. The PELDOR time trace or its Fourier spectrum contain these dipolar frequencies, but in addition, combinations in the form of sums and differences of these frequencies can be present. If the combination frequencies fall in the bandwidth of the PELDOR measurements, they appear as additional lines in both the Fourier spectrum and \(F(r)\), or broaden the line in \(F(r)\). It is possible to separate combination from dipolar frequencies in polyradicals from measurements for several values of \(p_{B}\) [27]. Because combination frequencies arise from dipolar frequencies in a single polyradical, they could provide a new way to correlate spins within the same polyradical.

4pPELDOR measurements of bi- and triradicals 3-23 to 3-27, Table 3.2 [28], were made in glassy solutions of perdeuterated ortho-terphenyl at 50 K [27]. The dependence of the modulation on \(p_{B}\) was used to eliminate the combination frequencies. The \(F(r)\) for triradicals substantially sharpened, and the PELDOR Fourier spectra became sharper without the combination frequencies. It turns out that even in the chemically-symmetric triradical 3-23, the distances between the three >N-O· fragments are not equal because each fragment has several possible conformations [27].

Mixtures of bi- and triradicals provided a good model for small clusters of radicals. The wide range of nitroxyl polyradicals developed over the years played a key part in establishing the basis for reliable determination of the number of spins in studies of complexation and aggregation in complicated biological systems.

3.2 Oligomers and Supramolecules with Nitroxyls

In addition to verifying the theoretical foundation of PELDOR and testing the reliability of PELDOR measurements, nitroxyl radicals enabled PELDOR studies of oligomers and supramolecules. That work focused on the structure and flexibility of the large molecular building blocks used in nanostructures and nanomachines. The real size and flexibility of a large, complicated molecule depend on all of its conformations and the energy of each conformation, which are reflected in the histograms of distances and the distance distribution \(F(r)\). PELDOR measurements of \(F(r)\) for terminal nitroxyl groups attached to oligomers and supramolecules provide a critical comparison of end-to-end distances and persistence lengths obtained from MD calculations and other physical measurements. The conformations and structures, that are present in room temperature solutions, usually are preserved when the solvent is flash frozen as a glass. This allows the full distribution of conformations to be characterized by PELDOR measurements.

3.2.1 p-Phenylene-Ethynylene Nanowires

Long π-conjugated oligomers, so-called molecular wires, are used in organic conductors, sensors, and photoconversion devices. In these applications, the conformational and molecular flexibility of these oligomers is important for structural integrity and for conductivity. Because PELDOR measures distances in a nanometer range and the distance distribution function, it is one of the few methods to determine molecular rigidity.

An important class of nanowires is the p-phenylene + ethynylene (PPE) system of carbon-carbon triple bonds interspersed with phenyl groups. PPE oligomers were used as linkers in biradicals 3-12 to 3-15 used to test MD methods for predicting the length and flexibility of such oligomers [29]. This set of nitroxyl-terminated nanowires was extended to 3-28 to 3-31, Table 3.3, with additional PPE nanowires of varying length [29, 30].

Table 3.3 Structure of PPE oligomers, Hex can be either hexyl or 6-methoxyhexyl [29, 30]
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Total flexibility can be split into contributions from conjugated segments, based on MD calculations [29, 30]. This was tested by measurements made at 50 K in a series of glassy solvents with different glass-transition temperatures \({\text{T}}_{g}\). The harmonic segmented chain (HSC) model predicts that \(F(r)\) for linear molecules is asymmetric. The shape and position of peaks in \(F(r)\) measured by PELDOR changes systematically with length, Fig. 3.4. and agrees with predictions. The width of the distribution increases substantially as the length increases. All these features are seen for each nanowire type, Fig. 3.4 [29, 30]. The structure of the nitroxyl spin label has only a weak effect, ~10% on the experimental \(F(r)\),

Fig. 3.4
figure 4

Reprinted from Godt et al. [28]

The distance distribution \(F(r)\) between >N-O· groups from Tikhonov regularization of PELDOR data for 3-30, n = 1–4.

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3.2.2 Molecular Ladders

Another class nanostructures whose flexibility was studied by PELDOR are molecular ladders. They are constructed from helical blocks and are labeled by pyrroline nitroxyl radicals at both ends for PELDOR measurements [31]. Blocks of bis-peptide monomers were formed into ladders 3-32 with n = 4–8, Table 3.4 [31]. These biradical ladders are water-soluble, unlike the biradicals in Table 3.1 which, as a rule, are soluble only in organic solvents. Aqueous solutions of 3-32 were studied at 80 K in buffer with glycerin added for glassing.

Table 3.4 Supramolecular systems studied by the PELDOR method
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The 4pPELDOR Fourier spectra and the \(F(r)\) were analysed. As n increases, the average distance between >N-O· groups grows from 2.37 nm at n = 4 to 3.46 nm at n = 8, with each additional monomer block adding 0.27 nm, on average, to the distance. It is worth noting that the width or standard deviation of the \(F(r)\) peak increases with n from 0.18 nm at n = 4 to 0.58 nm at n = 8. This width characterizes the molecular flexibility of this structure and is consistent with MD calculations.

Bis-peptide ladder structures similar to 3-32, but with slightly different monomer blocks were synthesized and examined [32]. Depending on the monomers and their placement in the ladder chain, the shape of the ladder could be controlled, from linear to S-shaped or even circular. Both the distance between >N-O· groups and the width of the \(F(r)\) peak reflect the conformation spread or flexibility of these ladders. Directed synthesis of such supramolecular ladder structures, especially with monomers having different functional groups, enables the targeted design of useful nanosystems.

PELDOR provides an important, objective method to measure sizes and shapes of these ladders. Theoretical calculations based on a rigid segment model for these bis-peptide ladders agree much better with the \(F(r)\) measured by PELDOR than did conventional MD calculations [35].

3.2.3 Porphyrin Oligomers

Porphyrin oligomers can serve as convenient building blocks for supramolecular structures potentially suitable for nanoelectronics and nanorobotics. Nitroxyl radicals are readily attached as spin labels to the ends, Table 3.4, for PELDOR studies of structure and flexibility. Linear supramolecules of butadiyne-linked Zn-porphyrin rings were labeled by a nitroxyl group at each end of 3-33 to 3-36 [33].

These molecules were studied at 50 K in frozen solutions of deuterated toluene and ortho-terphenyl with pyridine-d5 and 4-benzyl pyridine, respectively, added to prevent aggregation. PELDOR time traces were analyzed after averaging to remove orientation selectivity [36], see 3.3. The \(F(r)\) from a series of 3-33 with n = 1−4 show skewed peaks, Fig. 3.5a, b, like other oligomeric structures and are well described by the worm-like chain (WLC) theory [37], and yield flexibility parameters such as the persistent length. The distances found by PELDOR are in excellent agreement with those obtained from X-ray diffraction crystal structures and from MD simulations, Table 3.4 [37].

Fig. 3.5
figure 5

Reprinted from Lovett et al. [33]

a PELDOR time traces for porphyrin chains 3-33 with n = 1–4; b the corresponding distance distributions F(r).

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Fourier spectra of linear oligomers 3-33 with n = 1–4 reveal J = 0 and the absence of spin exchange. This result was unexpected because porphyrin oligomers are considered to have a highly-delocalized electron system and high polarizability [33], which should produce significant spin exchange. These properties had made porphyrins seem ideal units for the design of molecular conductors.

Some of the properties of supramolecular Zn-porphyrin systems can be tuned by extending the linear geometry into three dimensions by coordination of the Zn, as demonstrated by 3-34 to 3-36 [33]. In 3-34, two porphyrin centers are coordinated, while four are coordinated in 3-35. Conformational variations appear, in these complicated, non-linear structures, caused by flexibility of groups containing the nitroxyl group. The conformations produce Gaussian distributions in \(F(r)\) rather than the skewed shapes seen in linear systems. For 3-34, the distance between >N-O· groups is 4.15 nm with a Gaussian width at half maximum of 0.68 nm, while for 3-35, these values are 2.49 and 0.87 nm. These results agree well with MD calculations.

A more sophisticated situation arises for the four nitroxyl labels in complex 3-36. Here the geometry of the complex is characterized by three distances: the sides and the diagonal of a rectangle. Distances from PELDOR experiments and MD simulations are given in Table 3.4. The experimental diagonal of 4.76 nm is much less than the 5.08 nm from MD simulations or values for a rectangular structure with the measured sides. This discrepancy is evidence of a twisted conformation for 3-36 in solution. A calculation based on the probable non-planar geometry suggests a twist angle of 44°.

3.2.4 Inclusion Complexes

One goal of molecular design is to control the rigidity of elements in molecular machines and mechanisms by understanding the flexibility of supramolecular aggregates. Mobility was studied at 77 K in frozen aqueous glycerol solution for biradical 3-38 and its supramolecular complex with cucurbit [6] uril-based [3] rotaxane 3-39, Table 3.5 [38]. By itself, 3-38 has a very unconstrained structure, but after it is threaded through a pair of tube-like curcurbit molecules, the conformational flexibility is largely frozen out. This is shown by the lack of modulation in the PELDOR time trace of 3-38 and a very broad distance distribution peaking at \(r \ge 2.5\) nm which is supported by MD simulations. In stark contrast, the PELDOR time trace of 3-39 has deep modulation giving \(r = 3.07\) nm with a remarkably narrow width of 0.07 nm. The PELDOR method provides experimental estimates of the flexibility of supramolecules based, most importantly, on frozen solutions rather than single crystals [38].

Table 3.5 Supramolecular inclusion complexes
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Broad distributions of distances provide important information in some situations, as seen with 3-38 and 3-39. The [2] catenane supramolecule 3-37, Table 3.4 [34] provides another example. This supramolecular structure consists of a pair of interlocked rings. Each ring is quite large and is composed of molecular groups that do not have strong interactions with each other, so it is not obvious what its structure might be. The problem of how the rings are oriented relative to each other is easily solved from PELDOR measurements of the \(F(r)\) between >N-O· groups.

The PELDOR time traces were different at 15 K for frozen solutions of 3-37 in three solvents: chloroform, o-terphenyl, and 2-methyltetrahydrofuran. A simulated \(F(r)\) was calculated with the >NO groups randomly arranged on two circular rings with an effective radius \(r_{eff}\) [34]. Only in chloroform did the experimental data correspond to this simulation. The [2] catenanes aggregated in 2-methyltetrahydrofuran instead of being dispersed as individual supramolecular complexes. There was a substantial distortion of the ring geometry in o-terphenyl, likely due to nonlinear conformations of the alkyl part of the rings. The experimental distance distributions in the chloroform solutions, Table 3.4, have widths that reach 4 nm. The predicted \(r_{eff}\) values are close to the experimental values seen in chloroform, indicating that the rings do move freely around each other. This simple model describes the properties of molecules several nanometers in diameter and may also be useful even for large supramolecular systems [34].

3.3 Nitroxyl Orientation Selection

Orientation selection in PELDOR means that the PELDOR signal represents some orientations of a radical pair more than others. Orientation selection can provide valuable information on the geometry and dynamics of the pair of spins. At the same time, information about some orientations of the pair of spins may be missing from the PELDOR time trace \(V(T)\) when there is orientation selection, information that helps measure distances or \(F(r)\). Consequently, standard analysis methods, particularly Tikhonov regularization, can report differing r or \(F(r)\) from \(V(T)\) measured at different parts of the EPR spectrum or for different values of \(\Delta\omega_{AB}\). When orientation selection is not desired, methods have been developed for averaging \(V(T)\) obtained for different \(\Delta\omega_{AB}\) [29, 39, 40], see Sect. 1.4. This averaging works well with nitroxyl radicals and allows standard PELDOR analysis methods to be used.

3.3.1 Flexible Biradicals

The first studies of orientation selection in PELDOR were performed on 3-3 [15]. If measurements are made by changing the position of the observe or pump pulses in the EPR spectrum, or by changing \(\Delta\omega_{AB}\), then the \(V(T)\) contain information on the mutual orientation of radicals in the pair [41,42,43,44]. These experiments are outlined for a typical nitroxyl EPR spectrum at X-band in Fig. 1.9. Orientation selection for nitroxyl radicals becomes stronger at high frequencies, e.g., Q-, W-bands, where g-factor anisotropy begins to dominate the shape of the EPR spectrum.

The \(V(T)\) of biradicals 3-18 and 3-19, Table 3.2, were measured at X-band for \(\Delta \omega_{\text{AB}} /2\pi = 40 - 80\)  MHz at 40 K in frozen toluene and terphenyl solutions [41]. The modulation frequency and depth does change with \(\Delta\omega_{AB}\), as expected when there is orientation selection. A simple model readily accounts for the range of conformations, Fig. 3.6 [41]. Radicals at the ends of the linear 3-18 and bent 3-19 are not oriented randomly, but rotate around the triple bonds so that the N-O· bond lies on a cone with \(\alpha = 22^\circ\) for a five-member nitroxyl ring. The linker connecting the radical pairs bends with a random amplitude in another cone with an angle β, characterizing the conformational flexibility of the biradical. The experimental \(V(T)\) were best fit with \(\beta = 40^{ \circ }\) for 3-18, while for 3-19, β = 20° in addition to its intrinsic bend of 60° from linearity. PELDOR time traces were calculated numerically and compared to experimental data. The model seems to describe the experimental data well and makes it possible to determine conformational properties of molecules, such as structural flexibility [41].

Fig. 3.6
figure 6

The geometric model to describe the conformers of biradicals 3-18 and 3-19 [41]

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A more detailed treatment of orientation selection for 3-18 and 3-19 at X-band was attempted [43]. The PELDOR signal was modeled as the convolution of a kernel function containing \(F(r)\) with an angular intensity \(\lambda (\cos (\theta ))\), where θ is the angle between \(\vec{r}\) and the vector \(\vec{{B_{0} }}\) connecting the spins. The function λ is a constant for randomly-oriented pairs without orientation selection. But, with orientation selection, \(\lambda (\cos (\theta ))\) is the relative contribution from radical pairs at the angle θ. In order to obtain \(\lambda (\cos (\theta ))\) from experimental data, \(V(T)\) must be measured for several \(\Delta\omega_{AB}\). Next, it is essential to find \(F(r)\) by Tikhonov regularization from a time trace averaged to remove orientation selection. After that, the function \(\lambda (\cos (\theta ))\) can be found from the measured \(V(T)\) for each \(\Delta\omega_{AB}\).

The \(F(r)\) for 3-18 and 3-19 have a maximum at ~ 3.3 nm, Fig. 3.7. The width of \(F(r)\) is much greater for 3-19 than for 3-18 [43] because flexibility changes the distance between spins directly for 3-19, but only in second order for 3-18. The asymmetry of \(F(r)\) indicates the conformational flexibility in 3-18 and 3-19 [41]. The \(\lambda (\cos (\theta ))\) lack any distinctive features and it is not clear how to interpret them. This is likely the result of relatively weak orientation selection for nitroxyl radicals at X-band and the spectral overlap of the observe and pump pulses at small \(\Delta\omega_{AB}\).

Fig. 3.7
figure 7

Reprinted from Marko et al. [43]

\(F(r)\) of nitroxyls 3-18 and 3-19.

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3.3.2 Rigid Biradicals

Orientation selection was used to study the geometry and dynamics of planar, highly-conjugated biradicals 3-40 to 3-42, Table 3.6 [45, 46]. The nitroxyls are rigidly oriented so that the largest \(A_{ZZ}\) component of the hyperfine tensor is perpendicular to the plane of the biradical. The PELDOR time traces and their Fourier spectra varied strongly with \(\Delta\omega_{AB}\), an indication of strong orientation selection. The distances between >NO groups were found either from Fourier spectra [45, 46] or by Tikhonov regularization of averaged PELDOR time traces [36]. Examples of orientation selection with other radicals and at higher EPR frequencies are discussed in Sects. 7.2.2 and 8.2.2.

Table 3.6 Planar complexes with strong PELDOR orientation selection [45,46,47], the z axis of the hyperfine tensor is perpendicular to the plane
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3.4 Exchange Interaction

As a general rule, the exchange interaction is negligible for PELDOR, i.e., \(\left| J \right|\) <1 MHz, for distances greater than 1.5 nm between unpaired spins. The situation changes if the distance is smaller or if the biradical is part of a rigid system of conjugated bonds and delocalized electrons, in such cases, \(J \ne 0\). Moreover, the spin system has antiferromagnetic properties if J < 0, but ferromagnetic properties if \(J \, < \, 0\). Correct determination of the sign of J is important for establishing the magnetic properties of a multi-spin system. The value and sign of the exchange integral can be found from PELDOR Fourier spectra, Fig. 1.4 [24].

A detailed investigation of the value and sign of the exchange integral was performed at X-band on nitroxyl biradicals 3-40 to 3-42 [45, 46]. Their rigid planar structures result in substantial orientation selection. The PELDOR time traces and their Fourier spectra were obtained for different \(\omega_{B}\) values. The pump pulse at \(\omega_{B}\) was applied at the maximum of the EPR spectrum and the observe pulse at \(\omega_{A}\) was applied so that \({\Delta\omega_{AB}} /{2\pi}\) ranged from 40 to 80 MHz in 10 MHz steps.

The large \(A_{Z}\) hyperfine component in 3-40 to 3-42 is perpendicular to the biradical plane while the small \(A_{x}\) and \(A_{y}\) lie in the plane. In this situation, measurements at different \(\Delta\omega_{AB}\) provide very clear \(\omega_{ \bot }\) and \(\omega_{\parallel }\) features in the PELDOR Fourier spectra, and hence, very accurate r and J Table 3.7. Fourier spectra from the initial PELDOR time traces, as a rule, depend strongly on the sign of J. The sign was verified by comparing calculated Fourier spectra for +J and −J with the experimental spectra. Subsequent W-band PELDOR measurements with even better orientation selection [47] confirmed the X-band results.

Table 3.7 PELDOR results for planar biradicals [45,46,47]
Full size table

The larger distance between nitroxyl groups in 3-40 relative to 3-41 and 3-42 causes the exchange interaction to disappear, \(J \approx 0\), while a seemingly trivial structural change from 3-41 to 3-42 reverses the sign of J, changing the spin system from ferromagnetic to antiferromagnetic.

In a detailed study of 3-1, the value and sign of the exchange integral was estimated to be J =  +0.7 ± 0.4 MHz [14]. X- and S-band studies of 3-4, 3-12, 3-16, and 3-17 in frozen toluene-d8 observed overlapped Fourier spectral lines from dipole or hyperfine interactions [18]. It was difficult to distinguish dipolar frequencies from nuclear modulation in a measurement. However, the nuclear modulation frequencies were identified because only they changed with magnetic field between X and S-band, allowing unambiguous assignment. All biradicals gave frequencies corresponding to \(v_{ \bot }\) and \(v_{\parallel }\), but only for 3-17 is \(\nu_{ \bot } \ne 2\nu_{\parallel }\), yielding J = −11.0 MHz.

It is interesting that no modulation of the PELDOR time trace was found in either X- or S-bands for a biradical similar to 3-17 but with only one benzene ring in the bridging group [18]. This linker provides a very short, direct conjugation path between the nitroxyl groups, and an exchange integral of \(\left| J \right| = 73\) MHz is required to simulate the EPR spectrum. The exchange is stronger than the hyperfine interactions (or g-factorm differences), so this biradical is better described in terms of triplet and singlet states rather than weakly interacting doublet states [18]. Differences in the selection rules for EPR transitions for these sets of states will destroy the PELDOR modulation when there is strong exchange.

The link between the two spins has some effect on the magnitude and the sign of the exchange J, even for biradicals with very different spins, e.g., a Cu(II) ion paired with a nitroxyl radical in Sect. 4.2. Methods to analyze PELDOR data from 4-8 [48] were employed to determine the value and sign of J in the closely-related 4-9 [49]. The copper-porphyrin complex 4-9 has a completely conjugated linker to the nitroxyl radical, unlike 4-8 where an ester bond breaks the conjugation. The \(V(T)\) from 4-9 at different \(\Delta\omega_{AB}\) could be analyzed only by including a weak \(J = +4\) MHz [49].

Despite experimental and theoretical difficulties, the PELDOR method will find more and more applications in studies of ion–radical and ion–ion systems that are of particular interest in biophysics and biocatalysis.

3.5 High-Frequency PELDOR

The main advantage for PELDOR spectroscopy in Q-, W-, and higher-frequency bands is the high \(g\)-factor resolution. There is a great advantage when only one anisotropic interaction dominates the shape of the spectrum rather than a mixture of \(g\)-factor anisotropy and hyperfine interactions from multiple nuclei. Very selective orientation selection measurements become possible simply by changing the position of the observe and pump pulses relative to the gx, gy, and gz features of the EPR spectrum, Fig. 3.8. Higher frequency PELDOR offers greater opportunities to determine the spatial orientation of nitroxyl groups than at X-band.

Fig. 3.8
figure 8

Typical EPR absorption spectrum of a nitroxyl radical at X-, W-, and mm- bands; notice the different magnetic field scales

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Methods to measure dipolar interactions by W-band PELDOR were illustrated with 3-1 with R = H and 3-12 containing 14N and 15N [22]. The gz axes of both >N-O· fragments in 3-1 are parallel to each other and perpendicular to the vector \(\vec{r}\) between spins, which are 2.90 ± 0.02 nm apart. Thus, the two nitroxyl rings are coplanar and \(\vec{r}\) makes a 26 ± 2° angle with the nitroxyl x axis lying along the N–O bond. The same spatial parameters were found for 3-12, except the angle between the x axis and \(\vec{r}\) is 44°. The authors noted that it is not possible to distinguish cis- from trans-orientations of the nitroxyl rings for either biradical.

MW power is usually limited at high frequencies, resulting in excitation that is rather selective, with a narrow bandwidth relative to the EPR spectrum, and is well suited for orientation selection measurements, Sect. 3.3. Unfortunately, the low power causes only a slight inversion of the spin magnetization which limits sensitivity and the modulation depth of the PELDOR time trace. These limitations were removed, to a large extent, by using a W-band source with ~1 kW power at 94 GHz and up to 14 ns pulse duration [47].

Six high-power, W-band PELDOR time traces were measured for 3-16 with \(n =\) 2, 3-41, and 3-42; observing at each gx, gy, and gz spectral line while pumping at each of the other two principal \(g\)-values. Experimental \(V(T)\) were simulated using spin Hamiltonian parameters from W-band CW EPR spectra and the flexibility angle β, Fig. 3.6. The better resolution of gx, gy, and gz at W-band produced better information on orientation and flexibility in these biradicals while the high power produced more intense modulation and better sensitivity [47].

Very impressive results were achieved at Q-band with 150 W and pulse durations up to 80 ns, with more than an order of magnitude gain in sensitivity compare X-band [40]. Such high power can produce broadband excitation to suppress orientation selection in most cases with nitroxyl radicals, making it easier to determine \(F(r)\). On the other hand, orientation selection measurements are still possible. Nuclear modulation from matrix protons and deuterons is strongly suppressed at Q-band, which simplifies analysis of the Q-band time traces and improves signal intensity [40].