Thank you for this comment! I am glad to see that quantum-chemical methods yield an almost equal orientation of the CS tensor main frame as our experimental data suggest (see also my E-mail to you from May 25th.) 

We thank Anonymous Referee #1 for comments which opens the opportunity to improve the way of explaining our results

and to remove some errors from the first version of our manuscript:

Comment : Even though the samples and experimental conditions are not the major concern of the

study, and the information is given somewhere in the manuscript, it is still easier to read if

summarized in a dedicated section.

Response: In the revised version a paragraph "Experimental" will be added.

Comment: In line 487 reads "The second to last column in Table 3 is the noise-related sum

squared uncertainty of 10 SSB intensities considered" and in Table 3 the 2nd last column

is 10ΔI2. Does this supposed to mean 10-time the integration uncertainty of some sort of

"average SSB", assuming each SSB has similar line width?

Response: In fact, we assumed and observed that all SSB of one species have almost equal line width.

(The factor 10 arises from the χ² procedure: We sum up over the mean-square differences between experimental

data and polynomial values for both C_-2 to C₂ and S _-2 to S₂ (10 instances). The resulting χ² is then

compared with the summed squaric uncertainties of the experimental data.)

Therefore we will add this explanation to the related paragraph.

Comment: Following #2 in line 488 reads "Only for the C1+C2+C5 combined signal does χ2 from

the best fit exceed this value significantly". From Table 3, χ2 of C3 and C4 is even bigger

compared to the uncertainty, as discussed in the later section. Did I misunderstand

something?

Response: These were typing errors. In fact, the χ² values for C3 and C4 must have

the exponents -6 (instead of -5 as displayed in the table.) This will be corrected in the revised version.

Comment: Besides, there are a few typos to be corrected:

1. Line 36, "different order are separated"

2. In Fig. 2 caption, should be "for η=1" instead of "for q=1"

3. Line 389, "the intensity of this valuable signal"

4. In eqn. 40, a missing second half parentheses after ω2.

Response: The typos will be removed in the revised version.

We thank Anonymous Referee #2 for comments and questions which helped to improve the way of explaining our results, add some necessary details and remove some errors.

Comment: This paper introduces an exact polynomial representation of SSB intensities to overcome

the "far too complicated to be applied", in the words of the authors, Herzfeld-Berger

relation.

Response: Here we refer to the powder average over a threefold sum from - ∞ to + ∞ over products of Bessel functions as analytical expression (equation (25) of this paper) which is of course an important result, but a polynomial seems to have a simpler structure. Furthermore, if χ² has to be calculated in dependence on two, three, .., parameters where the multiply repeated calculation of SSB is required, the polynomial should be clearly advantageous.

However, we will weaken this phrase to "rather complicated equation (25)".

Comment: The first question that comes up is the justification that the polynomial evaluation will be appreciably faster than a numerical powder average of the expression that is routinely used.

Response:

• Calculation of powder average in a numerical simulation requires one or two loops around the calculation of SSB (for example using a REPULSION dataset with say 1000 points, or a independent variation of polar and azimuthal angle with a sufficient number of steps). In the polynomial formulae introduced here, the powder average is already included!
• Multidimensional χ² calculation for a multitude of n-tuples {par1, par2, ...} is expected to be simpler and faster if a polynomial is used instead a program or a subroutine; furthermore
• a simple polynomial can be better included into a fitting routine than a program. For example Newton procedure or Levenberg-Marquardt procedure demand calculation of first derivatives.

This comment was helpful in the sense that we will add to the paper a rough back-of-the-envelope estimation of the improvement in calculation efficiency.

Comment: Secondly, it is not clear how the polynomial expression that is obtained is exact, as is claimed; it is an approximation to some order.

Response: The polynomials for the SSB of the isotropic case are exact if they would be summed up to infinity. This becomes clear from the derivation of the formula: All included series (binomial) and expressions are used without any approximation. Of course, practically one has to restrict to a maximum power. However this can be chosen to get arbitrary accuracy.

Comment: That said, there are some interesting aspects to the paper. Taking sum and difference of positive and negative sidebands of a

particular order is one such. Is this only true for polynomial expansion or is it general? So also is the use of symbolic manipulation programs for evaluating the coefficients.

Response: The qualitative issue does not depend on using polynomial description or another. But it seems that it can be expressed advantageously in a mathematical fashion if the polynomial representation ist used. Especially the fact that the odd powers change the sign between positive and negative sidebands, the even powers not, imply an easy possibility for the representation of sum and difference.

We will add an explanation of this issue.

Question: It is understandable that in successive rotations the last rotation of the previous rotation and the first rotation of the present can be combined because they are about the same axis. However, wouldn't the first rotation of the first and the last rotation of the last

remain?

Response: Of course, the last rotation remain. Yet this is a rotation around B₀ (after the magic-angle flip) which has no meaning here and is never performed. The first rotation is member of the first R_z - R_y doubly rotation.

Comment: Because of performing rotations in the vector space spanned by the tensors, rotations are treated as a left multiplication with a 5 x 5 matrix. The advantage that this affords over bilinear matrix operations would depend on the number of multiplication and addition

operations involved, and not on the size (storage is a minor factor).

Response: Yes, we agree concerning the importance of number of operations and will clarify the statement. (This is related probably to lines 95 ... 106. But concerning the last part of this comment, it was not clear to which position of the manuscript the referee refers to.)

Question: In equation 25 (and may be a few others), the coefficients vary over orders of magnitude with some of the coefficients being extremely small. Are these small coefficients really significant?

Response: Whether or not they are significant, depends on the variable w. For example the realistic case of parameters η = 1

and (ω₀δ / ω_r) = 3 gives w = 6;  and 6¹² > 2x10⁹ leads to significance also of coefficients in the order 10^-10.

Comment: In Figure 2, one find that the I₊ intensity for the 12th order polynomial lies between 6

and 12. Increase the order would make it go further towards 2? That looks counterinutitive.

Response: This might appear if you observe the position w = 6 only. However if we consider the whole curve: the second-order polynomial is simply a parabola which will never be fitted by the 12-order polynomial. Moreover, there MUST be a crossing at some point because the parabola grows unlimited but not I₊. For an example, please imagine the Taylor series of sin or cos functions: Because of alternating signs

within the series we get an up and down of the function during enhancing the order.

Question: In Figure 4, what are the triangles and what is the red curve?

Response: Thank you - it must be added for the revised version that the triangles are experimantal data and the red curve is the fit on the  polynomial formula.

Comment: Something is missing from the sentence starting at the end of line 321.

Response: This sentence will be corrected in the revised version.

Question: Why is the \delta of C1 and C2 in Table 2 different from that on line 374?

Response: Did I misundersrand your question? For my opinion, the δ values in line 374 are EQUAL to that in table 2.

Comment: Equation 38 assumes that C2 and C5 are polarized identically despite the protons around them being two bonds apart in one case and three bonds apart in the other.

Response: For both carbons, the nearest protons are those two which are bound to the neighbouring aromatic carbons (C3 and C4, resp.), i.e. two bonds apart. This is equal distance, and therefore we assumed that the polarization of both C2 and C5 are very similar.

Question: What is the origin of equation 39?

Response: A note on the origin of this equation is missing and will be added in the revised version. For obtaining this equation,

we have played with the coefficients of I _-2, I _-1, I ₁ and I ₂ until the oscillation along t₁ vanishes. The aim was as explained in the text

a check if the CP efficiency was stable over the long-lasting experiment.