Intermolecular contributions, filtration effects and composition of the SIFTER signal
 ^{1}Laboratory of Physical Chemistry, ETH Zürich, VladimirPrelogWeg 2, CH8093 Zürich, Switzerland
 ^{2}Bielefeld University, Department of Chemistry, Universitätsstrasse 25, D33615 Bielefeld, Germany
 ^{1}Laboratory of Physical Chemistry, ETH Zürich, VladimirPrelogWeg 2, CH8093 Zürich, Switzerland
 ^{2}Bielefeld University, Department of Chemistry, Universitätsstrasse 25, D33615 Bielefeld, Germany
Abstract. To characterise structure and order in the nanometer range, distances between electron spins and their distributions can be measured via dipolar spinspin interactions by different pulsed electron paramagnetic resonance experiments. Here, for the single frequency technique for refocusing dipolar couplings (SIFTER), the buildup of dipolar modulation signal and intermolecular contributions is analysed for a uniform random distribution of monoradicals and biradicals in frozen glassy solvent by using the product operator formalism for electron spin S = 1/2. A dipolar oscillation artefact appearing at both ends of the SIFTER time trace is predicted, which originates from the weak coherence transfer between biradicals. The relative intensity of this artefact is predicted to be temperature independent, but to increase with the spin concentration in the sample. Different compositions of intermolecular backgrounds are predicted in the case of biradicals and in the case of monoradicals. We compare these predictions to experimental SIFTER traces for nitroxide and trityl monoradicals and biradicals. Our analysis demonstrates a good qualitative match with the proposed theoretical description. The resulting perspectives of quantitative analysis of SIFTER data are discussed.
Agathe Vanas et al.
Status: final response (author comments only)

RC1: 'Comment on mr202217', Frédéric MentinkVigier, 21 Oct 2022
Intermolecular contributions, filtration effects and composition of the SIFTER signal
SIFTER is a pulse sequence that can be used to extract the dipolar coupling, and thus the distance, between two unpaired electrons. The pulse sequence uses a single frequency, unlike PELDOR. The article describes a model to explain the SIFTER experimental data to better understand how the presence of inter molecular dipolar couplings affects the evolution of the signal and its decay/background. The work is very interesting and thorough. In general, the article describes in detail the strategy to tackle the problem and I have not found major issues. However, I have found at times the text to be quite dense or unclear, and below are my suggestions to make the reading easier.
Section 2: When starting the description of SIFTER, it would be good to have an example of a SIFTER trace can and how the pulses are changed. This would help the reader get into the pulse sequence without going back to previous articles. Could you also add the Hamiltonians you consider in each section
L105: “In the second parentheses the first term appears due to the time evolution of the first term in the first parentheses and vice versa.” The sentence is a little convoluted. I suggest rephrasing it.
L140 “Such monoradical like signal” is not clear. While you discuss, monoradical before, the “like” is not clear. May be just say: Monoradicallike signal or invert the order of the paragraph describing what was observed first in Doll’s work
L159: wl and wlm are confusing. My understanding is that there is some redundancy here though I do not think it matters. wl refers to the intramolecular dipolar coupling of the biradical l but not the spins themselves, while lm is for the coupling between two spins l and m belonging to two different molecules. I would use upper case Wlm for intramolecular, and lower case wlm for intermolecular. This would ease understanding where intermolecular couplings contribute.
For Eq 9, Since you use the same approach as in 2.1, I would suggest, for consistency, to use sigma(2tau_1deltat) should be used not V.
L170 tan(ωnτ1)·cos(ωnτ1)≈ωnτ1·cos(ωnτ1) is a disturbing approximation. It is far worse than a simple Taylor expansion of sin. I understand why you do this to have a product over all N, I would do it first, then apply the approximation of tan(x) as ~x . Actually you explain this later, L258 “In equation (21) we can add the missing factor cos”
Eq(14), it would be easy to have some numerical values for those terms using a random distribution of radicals and have the corresponding decay. Also in the weak coupling approximation the equation can be further simplified using a Taylor expansion of the cos. Note: for simulations, a good random distribution could be made with packmol.
L195 and subsequent, these sentences could be represented on a figure to simplify the writing.
L215, a graphical representation would help as well.
Eq 15. If B corresponds to decay, then should not we see operators in this equation?
L 243 what do you mean by all the AB coupling terms? This would explain the introduction of Q.
Eq 23, R is not an obvious choice for an operator, as it often represents relaxation.
L367, missing space. “.All”
Could you move the figures where they are first mentioned?

CC1: 'Reply on RC1', Maxim Yulikov, 24 Oct 2022
Dear Frédéric, many thanks for a number of useful suggestions and for the overall positive judgement on our manuscript. While we are discussing within the authors the detailed stepbystep responce and corrections, I would like to ask for some additional clarifications regarding the following three points in your reviewer report.
Your comment for L159: wl and wlm are confusing. My understanding is that there is some redundancy here though I do not think it matters. wl refers to the intramolecular dipolar coupling of the biradical l but not the spins themselves, while lm is for the coupling between two spins l and m belonging to two different molecules. I would use upper case Wlm for intramolecular, and lower case wlm for intermolecular. This would ease understanding where intermolecular couplings contribute.
My question: The frequency wl is written with one index (this is the spin index, not the molecule index) because there are only N/2 different intramolecular frequencies. Summing over all spins (only single index summation is needed here) would just count all these frequencies exactly twice, which is convenient enough to handle. The intermolecular frequencies need to be summed in some cases over two indices, and there are N(N2)/2 different wlm frequencies, i.e. here two indices are really necessary. Thus, I would actually advocate that we stay here with the originally set abbreviations, unless you strongly insist. How do you see this?
Your comment for Eq 9: Since you use the same approach as in 2.1, I would suggest, for consistency, to use sigma(2tau_1deltat) should be used not V.
My question: I would personally prefer to keep sigma in the section 2.1 and V in the section 2.2 and later to discriminate between the twospin density matrix and the multispin density matrix. It is however true that we omitted indicating the time points for the V. I would suggest writing V(2tau1delta) = V1 = equation, and so on for all the equations with V. Would this be acceptable for you?
Your comment for the L170: tan(ωnτ1)·cos(ωnτ1)≈ωnτ1·cos(ωnτ1) is a disturbing approximation. It is far worse than a simple Taylor expansion of sin. I understand why you do this to have a product over all N, I would do it first, then apply the approximation of tan(x) as ~x . Actually you explain this later, L258 “In equation (21) we can add the missing factor cos”
My question: Here, I am not sure I understand the comment. The relation sin(x)=tan(x)*cos(x) is exact, (x = ωnτ1). In the next step we just write the first Taylor term for the tan(x) while keeping the whole cos(x) function unchanged. This is correct up to the linear terms on x, but keeps some terms ~x^3, which is not important within our approximation level. In fact, in the final equation for V we get the most significant term ~x^2, and the next order term, which we partially neglect and partially keep in the cos(x), would be ~x^4. Could you, perhaps, explain in a bit more detail where you see a problem in our approximations, and what is the suggested change?

RC2: 'Reply on CC1', Frédéric MentinkVigier, 24 Oct 2022
Hi Maxim,
All these comments are not essential. The article is dense and this is a lot of work, thus my comments just aim at making the reading easier, and of course this is from my perspective.
Your comment for L159
> you can keep it. I am suggesting to add the range over which the indices run in all the equations?
Your comment for Eq 9:
> I understand the logic of V. Since you start with Greek letters for the density matrix, you could continue with another one? There is just one caveat with V, it looks like a voltage.
Your comment for the L170:
It is all about how you presented it L170 and the questions it left me as a reader. I have found the answer L258 where you better explain better what you are doing: you want to include the missing cosine in the product by rewriting the sin as tancos and then do the approximation. This is clear.
However L170 you do not explain the intent. Instead I read L170 as: we approximate sin(x)=tan(x)cos(x)~ x*cos(x). However sin(x) ~x is a better approximation that sin(x)~xcos(x), which then makes me wonder why use a worse approximation of sin(x).
In fact , I would argue that you could introduce the B2p term starting from line 10., then do the tangent approximation.

RC2: 'Reply on CC1', Frédéric MentinkVigier, 24 Oct 2022

CC1: 'Reply on RC1', Maxim Yulikov, 24 Oct 2022

CC2: 'Comment on mr202217', Mike Bowman, 07 Nov 2022
This is a excellent consideration of the the SIFTER signal under idealized conditions of perfect pulses and limited relaxation. The results are in good agreement with model experiments. They supply a theoretical basis for planning and for analyzing real SIFTER experiments. More realistic treatments will be made as needed to extend these results , for example, to finite width pulses, pulses with nonideal turning angles, partial excitation of the spectrum, mixtures of monoradicals and biradicals. This work certainly deserves publication.
This paper is written in a verbally descriptive style where each formula and term is described verbally in preference to concisely describing it in equations. I find that style rather difficult to follow when the math is as extended as this. However, this style suits some people well and is prefered by them over styles that I would prefer. So I am not advocating a change of style and do not argue the merits of people's taste in styles. However, there are some things that the authors can do to make the paper more easily read and understood.
One of these is concerned with the notation and terms in the equations. EPR has many official and unofficial standards of terminology, but with many conflicts between the different sets. For example, omega<0> is often the microwave frequency of the spectrometer instead of the dipolar splitting of radical pair 0. Also, omega<dd> is often the dipolar frequency at a particular distance r and angle theta, as defined in eq. 1 and used multiple times several pages later. Or it is the maximum frequency for a particular distance but the angular dependence is expressed separately. In addition, the many terms in the paper make it difficult to retain (or even find) their definition and use. So, I would find it a great help in reading this paper if there were a table listing all of the terms and meanings in sufficient detail to understand if omega<dd> is the dipolar splitting with angular dependence included, or if it is the dipolar interaction at a distance r with a seperate angular term. Such a table would also make it easier to see the relevance and role of terms, which is difficults because there are so many used in the paper.
The long verb al descriptions makes it difficult for me to understand the form and significance of terms until I reached the figures at the end of the paper. But then I had to go back and reread the first 3/4 of the paper. In know that when I read it again, I will be constantly refering to the figures as I read section 2 and 4. I would recommend putting some schemes or diagrams in section 2 to illustrate the terms and their relationships and reveal their significance. I think this is a case where each picture would be worth much more than a thousand words.
Finally, I notice some inconsistency in refering to equations and terms in the paper. Many of the equations are refered to later by their eq. number. But others, such as sigma<1> through sigma<3> are refered to by their name. I do not see that those terms ever appear on the right hand side of an equation, although I might have missed it. So, I am not clear why they would be given names in the first place.

AC1: 'Comment on mr202217', Daniel Klose, 10 Nov 2022
Dear Frédéric, dear Michael,
We thank both of you for your appreciation of our work and for your constructive comments!
Based on your comments and suggestions, we will update the manuscript to enhance the readability and include additional figures and table to illustrate the different terms that contribute to the SIFTER background and how they are related. We will include these improvements and the smaller corrections you both had suggested in the revised version. Once again, thank you, and best wishes!
 EC1: 'Comment on mr202217', Stefan Stoll, 24 Nov 2022

EC2: 'A few additional comments', Stefan Stoll, 24 Nov 2022
In addition to the reviewers'comments, I am adding a few remarks as well:
This is an interesting paper that adds theoretical insight about how dividing a SIFTER signal by a SIFTER signal without the second pi/2 pulsecalled SIDRE hereis valid for removing unmodulated signal contributions.
[1]
The main practical implication of this work is that the signal division approach is valid. This should be emphasized more.[2]
This division approach is not new. It was first proposed and succesfully utilized in Spindler et al, J.Magn.Reson. 280, 30 (2017), see Fig. 6 and discussion at the end of section 3.5 around eq.7. The authors do not cite nor disucss this originial prior work. This references should be cited, alongside the similarly early and relevant 2017 Denysenkov reference and the 2018 Bowen reference, in a prominent place such as the introduction.[3]
The entire Section 2 is very difficult to follow, due to notational inconsistencies and omissions, combined with a verbose and at times meandering narrative. It's all probably rigorous and correct, but it is hard to tell. Since this section is the valuable novel contribution of this paper, I strongly encourage to rework it substantially for terseness and clarity.Here are just a few comments on the initial parts of section 2, (comments similar in spirit likely apply to the latter part):
 The section starting at line 127 ("Second, ...") seems to more appropriately be placed with the discussion of interacting biradicals later in the manuscript.
 There is a notational inconsistency between omega_0 (dipolar frequency in biradical with Bspin number 0) and \tilde\omega_l (dipolar frequency in biradical with Bspin number l)  why tilde in one case and not the other?
 The approximation leading to eq.(10) is handy, but its range of validity is not stated. \omega_n\tau_1 must be smaller than about 0.5 for it to be reasonable. This should be mentioned. Does this have any consequence in the context of the experimental data?
 The terms "filtering" and "prefiltering" are used in several places, including the title. I have to admit that I do not completely understand what this term is meant to indicate, so I am not able to judge whether the associated statements are valid. What is the definition of filtering? Can this be clarified?
 The discussion between eqs. (12) to (15) is confusing. It seems to start out with an ensemble average, but then some new notation is introduced, such \tilde{B}_S and B_t(\tau), without defining equations. What does the tilde indicate? What does the S subscript indicate (and the lowercase subscript s that appears elsewhere several times)? And what does t subscript in B_t represent? It's all a bit unpenetrable. Also, to what degree is eq.(15) going beyond just an ensemble average of the detected signal using the density from eq.(12)?
 Eqs.(9) etc. using the symbol V with incrementing subscripts to indicate density operators at various points in the sequence. Since the earliest Hahn papers, or even Bloch, V is universally used to indicate the signal intensity, i.e. something proportional to the trace of density times a detection operator). Why not use the standard notation \sigma for the density operator, as was done in section 2.1? Also, why not use the actual time points instead of the subscripts, like was done in Eqs.(2)(6).
 Eq.(17) is said to have "parabolic shape curved down"  as a function of what? It seems like it must be seen as a function of time t, where tau1 = tau1_0 + t and tau2 = tau2_0  t. The scanned time t should be defined, probably together with the SIFTER sequence in Fig.1. Then, one can write tau1(t)*tau2(t) in Eq.(17), and this is then obviously parabolic in t.
[3]
Equation (40) seems to be the central result. Is it possible to add a figure that plots the various components of the SIFTER signal (and the SIDRE signal) and thereby assists in visualizing the anatomy of the signal?[4]
The experimental figures 2 and 4 show the divisions of the SIFTER and SIDRE signals. While Eq.(40) decsribes the SIFTER signal, what is the expression for the SIDRE signal? Is it \tilde{B}(tau1,tau2)? It would probably be helpful to introduce a notation such as V_SIDRE(tau1,tau2) = ... for clarity.[5]
The "SIDRE" signal is basically the 1D antidiagonal of the 2D signal of the refocused twopulse echo, V(tau1,tau2). 2020 Bahrenberg et al show several plots of V(tau1,tau2), both experimentally and theoretically. It might be good to refer to this work a bit more explicitly in the context of the discussion of the SIDRE sequence and the SIDRE experimental data.
Agathe Vanas et al.
Agathe Vanas et al.
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