the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 07 Oct 2024
Electron-spin decoherence in trityl radicals in the absence and presence of microwave irradiation
Abstract. Trityl radicals feature prominently as polarizing agents in solid-state dynamic nuclear polarization experiments and as spin labels in distance distribution measurements by pulsed dipolar EPR spectroscopy techniques. Electron-spin coherence lifetime is a main determinant of performance in these applications. We show that protons in these radicals contribute substantially to decoherence, although the radicals were designed with the aim of reducing proton hyperfine interaction. By spin dynamics simulations, we can trace back the nearly complete Hahn echo decay for a Finland trityl radical variant within 7 μs to the contribution from tunnelling of the 36 methyl protons in the radical core. This contribution, as well as the contribution of methylene protons in OX063 and OX071 trityl radicals, to Hahn echo decay can be predicted rather well by the previously introduced analytical pair product approximation. In contrast, predicting decoherence of electron spins dressed by a microwave field proves to be a hard problem where correlations between more than two protons contribute substantially. Cluster correlation expansion (CCE) becomes borderline numerically unstable already at order 3 at times comparable to the decoherence time T2ρ and cannot be applied at order 4. We introduce partial CCE that alleviates this problem and reduces computational effort at the expense of treating only part of the correlations at a particular order. Nevertheless, dressed-spin decoherence simulations for systems with more than 100 protons remain out of reach, whereas they provide only semi-quantitative predictions for 24 to 48 protons. Our experimental and simulation results indicate that solid-state magnetic resonance experiments with trityl radicals will profit from perdeuteration of the compounds.
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CC1: 'Comment on mr-2024-17', Stefan Stoll, 19 Oct 2024
This is a really interesting and well-written manuscript!
It convincingly demonstrates that methyl groups in trityl radicals are not innocent bystanders when it comes to decoherence. This insight has important practical implications. Also, I find the detailed investigation of dressed-spin decoherence dynamics illuminating.
The topological partitioning used in the pCCE (partial CCE) approach is interesting and appears to work well for the trityl methyl and methylene groups, but I wonder whether the partitioning can be automated for general structures. It reminds me somewhat of Kuprov's work on Liouville space reductions, see e.g. Fig.5 in his 2007 JMR paper https://doi.org/10.1016/j.jmr.2007.09.014. On a more general level, I would expect a partitioning or clustering criterion based on interaction energies to be more efficient and generalizable than a geometry-based one.
I am wondering whether it is possible to pinpoint the specific clusters responsible for the impressive decoherence slowdown of OX071 when going from H to D matrix (see Fig. 5). That could lead some additional valuable physical insight.
Regarding the methylene hyperfine couplings in OX071 and OX063 (mentioned in line 391), how strong is the isotropic contribution? Dihedral angle variations could modulate the isotropic part substantially.
In line 470 it is stated that it is unrealistic to apply CCE and pCCE to systems with much larger number of protons (compared to 36 presumably). My lab has run CCE on systems with up to about 1000 protons (see Canarie et al, JPCL 2020; Bahrenberg et al, MR 2022; Jahn et al, JPCL 2022, all cited in this manuscript; Jahn et al, JCP 2024 in press). We even have managed to run CCE-6, although only on our HPC facility.
Line 494: The vision that decoherence can be calculated from a structural model in a short amount of time has been realized with CCE-2 by Kanai et al in PNAS 2022 https://doi.org/10.1073/pnas.2121808119 on 12000 structures, see also the Python package PyCCE by Onizhuk et al.
Small comments:
136: It appears that instead of "factorize" it would be better to write "decompose". A product of 2^N two-level spaces would give 2^(2^N) states.
164: Just to clarify here: the coupling between the bystander spin and the two other nuclear spins is neglected here, correct?
330: The distinction between outer and inner methyl groups is not immediately clear from the figure.Citation: https://doi.org/10.5194/mr-2024-17-CC1 -
AC1: 'Reply on CC1', Gunnar Jeschke, 31 Oct 2024
Thanks a lot for all comments that we will consider and answer upon revision. Meanwhile I have a question on the following comment:
"In line 470 it is stated that it is unrealistic to apply CCE and pCCE to systems with much larger number of protons (compared to 36 presumably). My lab has run CCE on systems with up to about 1000 protons (see Canarie et al, JPCL 2020; Bahrenberg et al, MR 2022; Jahn et al, JPCL 2022, all cited in this manuscript; Jahn et al, JCP 2024 in press). We even have managed to run CCE-6, although only on our HPC facility."
We see that we should qualify our statement, but need to establish how exactly. Our statement is based on the following timing considerations: On a single core of an AMD Ryzen Threadripper 3950X 2.9-4.3 GHz we have run CCE-3 (and now CCE-6) for a single orientation (we use parallel execution for powder averaging). Our code appears to be reasonably effeicient; 80% of total time is spent for the matrix multiplications that are unavoidable for propagating the density matrix. For CCE-3 with 36 protons (7140 possible three-proton clusters) the computation takes about 540 s. A system with 1000 protons has 1.6617·108 possible three-proton clusters, corresponding to about 145 days of computation time. For CCE-6 with 12 protons (924 six-spin clusters) our computation time was about 2278 s. For 1000 protons, there exist 1.3682·1015 possible six-spin clusters. This scales to 449 million years of computation time.
Do you possibly mean something that we would call "local CCE", where only clusters are considered in which all proton distances are below a certain threshold?
Citation: https://doi.org/10.5194/mr-2024-17-AC1 -
CC2: 'Reply on AC1', Stefan Stoll, 01 Nov 2024
Indeed, in the past, we used geometric constraints to make higher-order CCE calculations manageable by only including clusters with nuclei within a certain distance of each other. Our current approach, which appears to be more efficient., is to disregard clusters that are expected to make negligible contributions to the actual echo decay. In fact, this was inspired by your 2023 paper in J.Magn.Reson.Open. The 2-cluster signal is V(2*tau) = 1 - k*sin(omega*2*tau)^4 = 1 - k*omega^4*(2*tau)^4 + O(tau^6). A particular 2-cluster is included if both k and k*omega^4 are above chosen thresholds. For clusters with three or more nuclei, we require the nuclei to form a connected graph with the nuclei as vertices and with an edge defined between a pair of nuclei only if both k and k*omega^4 are above thresholds. This choice assumes that important k-clusters contain at least k−1 important 2-clusters. Here is SI Fig.1 from our J.Chem.Phys. paper that is in press. This shows that the combined k & k*omega^4 criterion is effective in identifying the top contributing 2-clusters.
Citation: https://doi.org/10.5194/mr-2024-17-CC2 -
AC2: 'Reply on CC2', Gunnar Jeschke, 01 Nov 2024
Thanks a lot. This helps me to revise this part.
Citation: https://doi.org/10.5194/mr-2024-17-AC2
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AC2: 'Reply on CC2', Gunnar Jeschke, 01 Nov 2024
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CC2: 'Reply on AC1', Stefan Stoll, 01 Nov 2024
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AC1: 'Reply on CC1', Gunnar Jeschke, 31 Oct 2024
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RC1: 'Comment on mr-2024-17', Anonymous Referee #1, 21 Oct 2024
The article by Jeschke et al. discusses the dephasing mechanism in the bare and dressed state. It was a great read, and the article fits perfectly MR. While I have enjoyed it, I still have found it at times difficult to read. I have a few comments to mostly improve the readability of manuscript.
The dephasing models discussed are non-trivial, but the authors assume a familiarity with them. Although the spin system and Hamiltonian are thoroughly described, the text introduces the CCE model abruptly, assuming the APPA model is well-known to the reader. I suggest a minor revision to the theory section by briefly reintroducing the CCE model and providing a clearer explanation of Equation 17, without adding excessive detail. Including a figure to visually explain the pCCE approach would also be beneficial, especially since the criteria for determining cluster size are not clearly defined. Similarly, the APPA model could be reintroduced with a brief explanation, helping readers quickly grasp the distinctions between models. It would also be valuable to discuss the assumptions behind these models and how they may fail under certain conditions. Overall, I recommend adding a figure or table to summarize and compare the key aspects of the models for clarity.
L204: negelct -> neglect
Figure 7 is difficult hard to "read". I would either use lines for the theoretical model or various symbol shapes for the different theoretical models
L375: why where the simulations done in vacuum? A solvation model like COSMO may have improved the results and give more accurate tunnel splittings. More importantly, the basis is rather small which may induce errors. PBEh-3c would have been a better approach (yet fast) see 10.1002/anie.202205735 This would justify the scaling mentioned subsequently.
Citation: https://doi.org/10.5194/mr-2024-17-RC1 -
RC2: 'Comment on mr-2024-17', Anonymous Referee #2, 22 Oct 2024
This paper is a significant milestone on the road to understanding spin decoherence in organic matrices with many protons and even methyl groups. Spin decoherence has a significant impact on many forms of magnetic resonance including pulse dipolar spectroscopies such as DEER and RIDME, pulse hyperfine spectroscopy like ENDOR, EPR-based quantum computing, and dynamic nuclear polarization, etc. Spin decoherence limits the ability to control and manipulate spins which directly impacts sensitivity and resolution in techniques that are used in many different fields. Being able to tune and control decoherence will make existing techniques more useful and can enable development of capabilities that do not exist at present.
This paper considers decoherence in a set of three trityl radicals with a range of potential decoherence routes in both proton-containing and deuterated matrices. Perhaps more significantly, it considers both the decoherence of 'bare' spins in the absence of a microwave field and of 'dressed' spins in the presence of a large resonant microwave field.
Decoherence of bare spins is simpler to treat. As the paper notes in more than one place, the different decoherence routes should be independent of each other, basically parallel kinetic pathways. When two pathways are operative in a sample, the decoherence should be the product of the decoherence of the individual paths. This expectation might be confirmed by the bare spin experiments summarized in Fig. 5. FTR has a rapid decoherence, almost entirely due to methyl group tunneling. That route is absent in OX063 and OX071 which have much slower decoherence dominated by matrix protons in the (H) samples. When the matrix proton route is removed by deuteration of the matrix, decoherence again slows to values dominated by the methylene groups in OX071 (D) or the hydroxy-ethyl groups in OX063 (D). Fig. 5 provides an excellent opportunity to demonstrate the multiplicativity of decoherence via the different kinetic pathways for decoherence of bare spins. It can be done by taking simple ratios of decays curves to get the contributions from different routes. Such a demonstration would emphasize the understanding of bare state decoherence and the difference between bare and dressed states.
Some readers may not expect solvent protons to have a larger effect on decoherence than protons on the radical in OX063 and OX071, but Hyperfine interactions of narrow-line trityl radical with solvent molecules, J. Magn. Reson., 233 (2013) 29-36. https://doi.org/10.1016/j.jmr.2013.04.017 showed that the matrix protons in Finland trityl had hyperfine couplings and spin-flip satellite line intensities comparable to those of the radical protons, consistent with a significant role for matrix protons in decoherence. The detailed dynamics of the nuclear spins determines the relative impact of matrix or radical protons on decoherence for bare spins.
In the case of dressed states, the total decoherence is not the result of kinetic paths operating in parallel, but of a more complicated mechanism involving all possible routes without simple relationships between the decoherence of different samples. The unfortunate result is that you can't see if you predict or calculate decoherence from a single route without also correctly calculating decoherence from all routes. This seems to be illustrated in Fig. 6 where the seems to be little relation between decoherences of all of the samples and appears to reappear frequently in the rest of the paper. The rest of the paper is focused on identifying practical methods for numerically calculating decoherence. Some computational approaches clearly diverge and do not appear promising. However, no method clearly succeeds in predicting the kinetics and timescale of decoherence in a consistent manner. There are several mentions about 'cancellation of errors' and uncertain convergence. But this is not surprising at this stage of a complicated project and is related to one character of dressed spin decoherence: you can't correctly calculate the contribution to decoherence of one route until you can correctly calculate every route.
This paper is missing a potentially important type of information relevant to the dressed spins. That is the details of the sample composition. These are not so important for the bare spins because each bare spin at this concentration undergoes decoherence independent of other spins. But in the dressed spin experiments, the dressed spins have essentially the same frequency and cross relaxation can occur more rapidly between the degenerate spins. Consequently, a small population of rapidly decohering spins can cause the entire population of spins to decohere at a faster rate. So, in future experiments, several aspects of the samples need to be carefully considered. Fortunately, most can be checked to see if they are significant and then ignored if they have no impact. Some of these aspects of the sample include:
⦁ Isotopic composition of the solvent. Exchange with humidity in the atmosphere by the hygroscopic glycerol/water can result in impactful amounts of protons in the deuterated solvent samples. Contaminating protons can be measured in the solvent by NMR. Whether it affects decoherence can be checked by intentionally spiking a deuterated sample to double or triple its concentration of protons.
⦁ Similarly, the isotopic purity of deuteration of the OX063 and OX071 was not noted. It is readily measured by high resolution mass spectrometry. Its relevance can be probed by adding a small amount of OX063 to a OX071 sample. Another test could be to exploit the convergent synthesis of the trityls to make a chimeric trityl with one OX063-like ring and two OX071-like rings.
⦁ Roughly 10% of the radicals have a C-13 in natural abundance having a hyperfine coupling comparable to twice the C-13 nuclear Zeeman frequency. This provides an opportunity for strong mixing of the electron and C-13 spin states (reminiscent of the 'matched pulse' for ESEEM). The matched pulse may offer a new route for decoherence of dressed spins and could operate in samples with contaminating protons whose nuclear Zeeman frequency is roughly half of the B1 field used to dress the spins.
⦁ In the absence of any protons, there must be some decoherence route. It may involve deuterons and their tunneling, or instantaneous diffusion, or the influence of random 'pairs' of radicals in the sample (or pairs formed with trace amounts of divalent metal ions), or dissolved molecular oxygen, or the very low-frequency phonon and vibrational modes contributing to the anomalous low-temperature properties of glassy matrices, or some other unexpected route. Its magnitude could be checked by a perdeuterated version of OX061 in deuterated solvent.
However, these are considerations for future work building on the base reported here. I see little need to go back at this point and characterize samples that will likely not be used in future work. I do recommend that the multiplicative nature of bare spin decoherence and its lack for dressed spins, be exposed better by additions to Figs. 5 and 6 with an explanation in the text. A short table or graphic showing the relations and differences between the calculational methods would be extremely useful for readers who are not numerically inclined or that have not been following computational spin dynamics very carefully.
Minor Points:
⦁ Wouldn't the omega-dd in eq. 8 be time dependent from tunnelling if the protons were on different methyl groups?
⦁ The 2nd line in eq. 14 seems to have a + or - sign missing.
⦁ It would help to refer reader at line 220 back to where it was mentioned in the paper.Citation: https://doi.org/10.5194/mr-2024-17-RC2 -
EC1: 'Comment on mr-2024-17', Malcolm Levitt, 28 Oct 2024
The description of the experiment could be clearer.
Please specify the magnetic field and the microwave frequency (not just as the code word "Q-band").
The pulse sequence in Fig.2a is fully specified since the the relevant pulse lengths etc. are given in the Materials and Methods section. However for Fig.2b, some aspects are unclear. The spin-lock field amplitude is specified as a nutation frequency \omega_1 which is given as 2pi*100MHz in the Materials and Methods. However the caption states that the additional PM pulses have a frequency that matches \omega_1. This is misleading. Presumably the caption does not mean that the pulses have a frequency \omega_1, but that the +modulation frequency+ of the microwave field is equal to \omega_1. Furthermore, the precise form and timing of the modulation should be specified. In addition, the amplitude of those pulses does not seem to be specified, at least not very clearly. It may be there somewhere, but it should be made clearer.
Since the paper describes attempts to match simulations and experiment, it is essential that the experimental procedure is clear and unambiguous.
Citation: https://doi.org/10.5194/mr-2024-17-EC1
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Electron-spin decoherence in trityl radicals in the absence and presence of microwave irradiation G. Jeschke https://doi.org/10.5281/zenodo.13850793
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