As apparent in Fig. 4, both the longitudinal relaxation rate constant R1=1/T1 and the long-lived relaxation rate constant RLLS=1/TLLS depend linearly on the concentration of TEMPOL (in units of M or mol L-1): 1R1=R10+r1[TEMPOL],RLLS=RLLS0+rLLSTEMPOL. The slopes rLLS and r1 are known as relaxivities (in units of M-1 s-1); the intercepts R10 and RLLS0 are the rate constants determined in the absence of TEMPOL. Figure 4 shows that variations of R1 between neighboring CH2 groups within each molecule are much smaller than variations from one molecule to another. Whereas the T1 values of small molecules correlate with the molecular mass – the larger the molecule, the shorter T1 – this is not true for TLLS. In the absence of TEMPOL, the longest TLLS of ca. 15 s was observed for compound III, whereas the shortest TLLS of ca. 5 s was found for compound IV, although their T1 relaxation times and molecular masses are roughly the same, so their correlation times should be similar. The difference in TLLS may be explained by the presence of 12 methyl protons in compound IV, which cause faster relaxation of LLSs.

Wokaun and Ernst famously demonstrated that PRE is less efficient for relaxation of zero-quantum coherences than for single- and double-quantum coherences (Wokaun and Ernst, 1978). Tayler and Levitt demonstrated that a similar logic also applies to LLSs: whereas longitudinal relaxation is enhanced by fluctuations of external local fields induced by unpaired electrons of radicals, a LLS involving two spins I1 and I2 is only relaxed by fluctuating external fields if these are not correlated. In general, the extent of correlation of the two fluctuating fields at the locations of the two spins I1 and I2 can be characterized by the correlation coefficient C=〈B1⋅B2〉/B1B2, where Bi=〈B1⋅B2〉 is the mean (time-averaged) amplitude. Only the uncorrelated part of the two fluctuating fields given by 〈B1-B2〉2=B12+B22-2〈B1⋅B2〉 contributes effectively to LLS relaxation (Tayler and Levitt, 2011). The smaller the radical, the closer it can approach one of the two geminal protons and hence the smaller the correlation coefficient C. It has been shown (Tayler and Levitt, 2011) that the ratio of relaxivities, 2κ=rLLS/r1, is a characteristic measure of the correlation coefficient C; the smaller κ, the larger C. The experimental ratio κ for the (chemically inequivalent) protons of the CH2 group in the (chiral) dipeptide alanine–glycine varied in the range 0.5<κ<0.3, depending on the size of the paramagnetic agent (Tayler and Levitt, 2011). A similar ratio κ=0.36 was observed for the CH2 group in the terminal glycine residue of the tripeptide Ala–Gly–Gly for PRE caused by triplet oxygen (Erriah and Elliott, 2019).

In CH2 chains with chemically equivalent pairs of protons in achiral molecules excited by exploiting magnetic inequivalence, the LLSs can be delocalized over several CH2 groups. Relaxation of a LLS localized within an individual CH2 group will contribute to the decay of a delocalized LLS, so one may expect the relaxivity of delocalized LLS to be more strongly affected by PRE than the relaxivity of a (hypothetical) localized LLS. We must however remain cautious, since one cannot assume that all CH2 groups are equally accessible to radicals. If some of the CH2 groups are more accessible, one may expect delocalized LLSs to have averaged relaxivities. As we shall discuss below, the variations in the observed relaxivities rLLS are not very large for different combinations of excitation and reconversion methods, and intramolecular variations are much smaller than differences between distinct compounds, so one can estimate an average ratio of relaxivities 〈κ〉=〈r1〉/〈rLLS〉 for all CH2 groups in a given molecule. Compounds I–IV feature average ratios 〈κI〉≈0.22, 〈κII〉≈0.23, 〈κIII〉≈0.18, and 〈κIV〉≈0.32 (see Table 1). To assess the effects of LLS delocalization on the relaxivity, it is useful to compare molecules II and III, as they differ by only one CH2 group. The mean relaxivity 〈rLLS〉 of compound II averaged over three CH2 groups is slightly higher than the relaxivity 〈rLLS〉 measured for compound III (see Table 1). This suggests that a higher degree of delocalization leads to a higher relaxivity. The LLS can be delocalized to a variable extent between all three CH2 groups in I and II, but they are always equally distributed between the two CH2 groups in compounds III and IV.

Even though delocalized LLSs are less affected by TEMPOL than longitudinal magnetization, the observed decrease in TLLS is undesirable in the context of d-DNP. Since the use of TEMPOL or other polarizing agents is mandatory for d-DNP experiments, the question arises if it is worth scavenging TEMPOL after dissolution by addition of a reducing agent such as sodium ascorbate (vitamin C) to extend TLLS after dissolution (Miéville et al., 2010, 2011). Note that the preparation of samples comprising two types of beads is rather cumbersome, in particular for bullet DNP. According to Miéville et al. (2010, 2011), the rate of the reduction of TEMPOL by sodium ascorbate may be slow on the timescale of the transfer of the dissolved sample from the polarizer to the NMR magnet. Hence, the reaction may not be entirely completed by the time the sample arrives in the spectrometer. Scavenging by sodium ascorbate may be accelerated ca. 100 times if one uses Frémy's salt instead of TEMPOL (Negroni et al., 2022). Several alternative approaches have been developed to remove radicals once DNP has been achieved. One approach is to use radicals obtained by UV irradiation of frozen pyruvic acid. These radicals are quenched as soon as the temperature increases (Eichhorn et al., 2013). One may also use radicals grafted onto mesostructured silica materials (Gajan et al., 2014) or microporous polymers (Ji et al., 2017; El Daraï et al., 2021). However, the small relaxivities presented in Table 1 suggest that scavenging may not be necessary when using LLSs to preserve the hyperpolarization.

It was shown (Sonnefeld et al., 2022b) that for the excitation of LLSs in systems with n=3 neighboring CH2 groups, i.e., with 2n=6 spins, there are seven orthogonal LLS product operators that can be created, with seven coefficients λi that depend on the excitation scheme: 3σ^LLS=-λAA′I^A⋅I^A′-λMM′I^M⋅I^M′-λXX′I^X⋅I^X′-λAA′MM′I^A⋅I^A′I^M⋅I^M′-λAA′XX′I^A⋅I^A′I^X⋅I^X′-λMM′XX′I^M⋅I^M′I^X⋅I^X′-λAA′MM′XX′I^A⋅I^A′I^M⋅I^M′I^X⋅I^X′. Here A and A′ denote the two protons of the CH2(1) group, M and M′ denote those of the middle CH2(2) group, and X and X′ correspond to the terminal CH2(3) group. This equation gives a general form of the density operator obtained after poly-SLIC, containing all long-lived terms found by numerical solution of the Liouville–von-Neumann equation. In addition to three bilinear terms, one encounters four higher terms that contain products of four and six spin operators. In principle, each term in Eq. (3) can decay with a different rate constant, so one could distinguish up to seven distinct rate constants RLLS(μ) with μ=AA′, MM′, XX′, AA′MM′, AA′XX′, MM′XX′, and AA′MM′XX′. Each term can be excited with a different amplitude and can contribute with a different weight to the observed signal.

In systems such as compounds III and IV with only two CH2 groups, only one LLS can be excited: 4σ^LLS=-λAA′I^A⋅I^A′-λXX′I^X⋅I^X′-λAA′XX′I^A⋅I^A′I^X⋅I^X′. The coefficients of the first two bilinear terms are always equal, i.e., λAA′=λXX′, while the four-spin term is always proportional to the leading bilinear terms, with a weight λAA′XX′=8/3λAA′ (Sonnefeld et al., 2022a). This state corresponds to the imbalance between the singlet–singlet state and the triplet–triplet manifold and is therefore expected to decay monoexponentially. In two sets of complementary experiments performed for compound IV, the experimental relaxation rate constants were indeed found to be indistinguishable, as can be seen by comparing the orange triangles and the green inverted triangles in Fig. 4d.

In compounds I and II however, which contain three adjacent CH2 groups, different SLIC excitation schemes lead to populate different LLSs, with different coefficients λLLS(μ) in Eq. (3). There are 9 different ways of exciting miscellaneous LLSs and 9 different ways of reconverting them, giving 81 possible experimental combinations. In order to investigate the relaxivities of these different LLSs which may have different decay rate constants RLLS(μ) and different relaxivities rLLS(μ), we performed six different poly-SLIC experiments with different SLIC pulses for excitation and reconversion, and we indeed found different LLS lifetimes (Fig. 5). Depending on the excitation and reconversion scheme used, there are pronounced differences between the relaxivities rLLS within one and the same molecule.

We calculated the contributions of each of the seven terms to the observable LLS-derived signals, after two consecutive transformations: I^zin→σ^LLS→I^xobs (see Fig. 6). For each excitation scheme used in this work, we considered all seven coefficients λμM→LLS corresponding to the seven terms in Eq. (3), as well as all seven reconversion coefficients λ̃μLLS→M. The coefficients were calculated as follows: 5λμM→LLS(I^zin→σ^LLS)=TrP^μ†σ^LLSTrP^μ†P^μ,λ̃μLLS→M(P^µ→I^x,µobs)=TrI^x†I^x,μobsTrI^x†I^x, where Tr stands for trace and where the index μ corresponds to one of the seven LLS terms in Eq. (3), the operator P^μ represents the μth LLS term, I^zin is the initial magnetization of the excited spins, I^x is the transverse magnetization of the observed spins after reconversion, and I^x,μobs is the transverse magnetization obtained after reconversion of only the μth term P^μ instead of the full σ^LLS. The observed signal Sμ stemming from the μth term is determined by the product of two coefficients, λμM→LLS and λ̃μLLS→M, for a given combination of excitation and reconversion SLIC pulses. These contributions are shown in Fig. 6. The sum of all seven amplitudes for each panel in Fig. 6 was normalized to one. These graphs show how the LLSs are delocalized across spin systems comprising n=3 neighboring CH2 groups. We only consider coherent spin dynamics during excitation and reconversion, neglecting possible redistributions of LLSs due to Overhauser-type cross-relaxation effects and neglecting zero-quantum coherences.

Note that a single SLIC pulse applied at the chemical shift of any of the three CH2 groups results in the excitation of a delocalized state, which is predominantly (but not exclusively) associated with the irradiated pair. By using triple SLIC excitation and single SLIC reconversion applied to the middle CH2 group, one can excite a fairly even distribution of LLSs involving all 2n=6 coupled spins. For compound II, the most strongly delocalized state features the largest relaxivity rLLS. For compound I, however, the largest relaxivities were obtained for experiments where the largest contribution to the observed signal came from the terminal group CH2(3) that is closest to the trimethylsilane group. This group has also the largest longitudinal relaxivity r1, as can be seen in Fig. 4a. Detailed calculations of the relaxation superoperator might help to rationalize the experimental results obtained here.