Coherence transfer in two-pulse double quantum (DQ) and five-pulse double- quantum modulation (DQM) sequences in EPR: Orientation selectivity, structural sensitivity and distance measurements


 Double-quantum (DQ) coherence transfers in two-pulse DQ and five-pulse DQM (double quantum modulation) EPR pulse sequences, utilized for orientation selectivity and distance measurements in biological systems using nitroxide biradicals, are investigated. Analytical expressions, along with numerical algorithms, for EPR signals are given in full details. Please see manuscript .pdf for full abstract.


Introduction
It is important to study coherence transfer from one coherent state, = 0, described by the difference in magnetic quantum numbers = Δ , corresponding to the matrix element ρ of the density matrix, to the forbidden one, = +2, which is two quantum numbers different, in pulsed-EPR (Electron Paramagnetic Resonance) experiments [1]. This coherence transfer is known as double quantum (DQ) coherence and has been investigated frequently in NMR (nuclear magnetic resonance) [2][3][4]. However, unless = −1, it does not lead to observable magnetization. It can, on the other hand, be observed indirectly by transferring the = +2 coherence to = −1 coherence state by a subsequent coherence transfer, similar to that done in NMR [5,6].
The advantage of DQ coherence, is that it depends on the dipolar interaction, so that the intensity of the signal decreases rather slowly with distance, since the dipolar interaction is inversely proportional to the cube of the distance, enabling measurement of larger distances [7]. Another advantage of the DQ coherence technique is that the measured signal has a preferential sensitivity to the dipolar interaction, since the DQ coherence is generated by it. As a consequence, the analysis of DQ coherence data to extract distances is cleaner that those from other techniques. For example, in the frequently used DEER (Double Electron Resonance) technique, one has to extract the weak dipolar echo modulation from the large echo decay background. As well, mono-radical impurities do not affect the DQ signal, unlike that with other techniques. The DQ method offers another advantage in that using it one can measure directly the double-quantum relaxation rate, 2 , the knowledge of which is very important to interpret motional dynamics.
Multi-pulse EPR has been frequently exploited for distance measurements in biological systems [8][9][10][11][12][13][14][15][16][17][18][19]. It is useful to have analytical expressions for the resulting pulsed EPR signals to keep track of the evolution of the various elements of the density matrix during the free evolution, as well as during the action of pulses, to follow how different coherence transfers are generated. On the other hand, it is easier to calculate pulsed EPR signals by numerical techniques, using the eigenvalues and eigenvectors of the pulse and static spin-Hamiltonian matrices, since the analytical expressions become lengthier and unwieldy to manipulate 4 numerically as the number of pulses increases. Besides, using them for calculation is susceptible to human error when transcribing the analytical expressions to numerical code.
This paper deals with a detailed study of coherence transfer for two-pulse DQ, as well as five-pulse DQM, which is an elaboration of the 2-pulse DQ sequence by introducing a refocusing π -pulse to enhance the signal, in samples containing nitroxide biradicals as spin probes, considering the dipolar interaction between the two nitroxide dipoles of the nitroxide biradical, as well as the fully asymmetric g and hyperfine matrices and the angular geometry of the biradical. Among others, it is focused on showing how to make the = 0 to = 2 DQ transition possible, which is forbidden for an infinite pulse, by using a finite pulse. In addition, we will discuss the important role the DQ transition plays in two-dimensional (2D) EPR, demonstrating the high sensitivity of the 2D-DQ EPR signals to the strength of the dipolar interaction and how to exploit it to measure orientational selectivity, as a result of constraints on the structural geometry, i.e., on the orientations of the dipolar axis of the two nitroxides of the biradicals. A full treatment of the problem will here be carried out, i.e., the calculation of Pake doublets in polycrystalline averages for the two-pulse DQ and five-pulse DQM signals, which are direct measures of the dipolar interaction, from which the distance between the two nitroxide dipoles in the nitroxide biradical used as spin probe, can be determined, being inversely proportional to the cube of the distance as = 10�51.9/ ( )�
DQ transitions, which are forbidden for an infinite pulse and only possible by a finite pulse in conjunction with the dipolar interaction between the two nitroxides of a biradical, also provide orientational selectivity for smaller amplitudes of the irradiation field, B1. This implies, as the analysis in this paper confirms, that the signal arises predominantly from those dipolar vectors, which are preferentially oriented symmetrically about the magic angle 54.74 ○ at which (3 2 θ − 1) = 0.. Furthermore, the signal is also found to be sensitive to the orientations of the two nitroxide dipoles, useful for structural studies in biomolecules.
It is the purpose of this paper to carry out a thorough and complete treatment, performing all required numerical simulations, which can now be carried out on laptops equipped with very fast processors. Accordingly, we will derive (i) analytical expressions using Mathematica for two-pulse DQ and five-pulse DQM signals for an arbitrary orientation of the external magnetic field with respect to the dipolar axis required for studying orientational 5 selection, and (ii) to calculate, using Matlab, the polycrystalline (powder) signals using Monte-Carlo averaging over random orientations of the respective Euler angles of the two nitroxide dipoles with respect to the dipolar axis, needed to calculate the Pake doublets from their Fourier transforms. To this end, full numerical diagonalizations of the pulse-and spin-Hamiltonian matrices will be carried out, and the two-pulse DQ and five-pulse DQM signals will be calculated exploiting the algorithm of Misra et al. [8]. In addition, five-pulse DQM EPR signal will be simulated to fit the experimental data reported in [20]. The effect of relaxation on the powder averages will be considered by multiplying the final signal with a stretched exponential [21,22]. In addition, it will be shown here, from general considerations, that a finite, rather than infinite, pulse, in conjunction with the dipolar interaction, is needed to produce non-zero 02 and 2-1 coherence transfers in the DQ experiment.
The organization of this paper is as follows. Section 2 deals with the required theoretical details, including the spin-and pulse-Hamiltonians for the nitroxide biradical and the solution of the Liouville von Neumann equation to calculate the signal, Derivation of full analytical expressions for two-pulse one-dimensional DQ and five-pulse DQM echo signals is given in detail in Sec. 3, including coherence transfer for the DQ and DQM signals. A discussion of the orientational selectivity and structural sensitivity of the coherence transfer, T0 2 , is provided in Sec. 4. The numerical algorithm to simulate the two-pulse DQ and fivepulse DQM EPR signals for a polycrystalline sample, in the absence of relaxation, is provided in Sec 5. Calculation of the effect of relaxation on a polycrystalline sample, using a stretched exponential as described in [21,22], is discussed in Sec. 6. In Sec. 7 a discussion of the numerical results of the various simulations is presented. The conclusions are summarized in Sec. 8.

Theoretical Details
This section deals with the theory and the procedure to calculate the twodimensional (2D) EPR signal for a coupled nitroxides biradical. One considers two dipolarinteraction coupled nitroxides, each characterized by an electron with spin = 1/2 and a nucleus with spin = 1. The dimension of the Hilbert space, in which the present calculations will be carried out, for such a system is 36x36, since (2 1 + 1)(2 2 + 1)(2 1 + 1) (2 2 + 1)=36, where the indices 1,2 refer to the two nitroxide. The effect of relaxation will be, for a polycrystalline 6 average, to multiply the calculated signal by a stretched exponential factor; more details are given in Sec. 4 below.

Spin Hamiltonian
For the coupled nitroxides the spin Hamiltonian is [7,8] where 0 ; k=1,2, denote the static Hamiltonians of the two nitroxide radicals, including the Zeeman and hyperfine interactions: In Eq. (2.2), , , + and − are the spin operators for the two nitroxides, and the expressions for the coefficients , and are given in Appendix A below. 12 in Eq. (2.1) includes the dipolar and exchange coupling between the two nitroxide radicals, expressed as where θ is the polar angle of the orientation of the static magnetic field with respect to the dipolar axis that connects the magnetic dipoles of the two nitroxides, J is the exchange-interaction constant between the two electrons, and is the dipolar-interaction constant, expressed in terms of , the distance between nitroxides, as [7,8].
where γ is the gyromagnetic ration of the electron and ℏ = ℎ/2 is the reduced Planck's constant. The constant = 2/3 will be used hereafter, referred to as the "dipolar constant".
The Hamiltonian for the pulse of the radiation microwave magnetic field is expressed as where 1 is the amplitude of the pulse, ± are the raising/lowering operators of the total electronic spin of the coupled nitroxide system in the 36 × 36 direct product Hilbert space: where ⨂ stands for the direct product, σ ± are expressed in terms of the Pauli matrices σ and σ as σ ± = σ ± σ , , and 1 ; = 1,2 are identity matrices in the electronic 2 × 2 and nuclear 9 × 9 spaces, respectively, of the two nitroxides. The magnetic basis described by the basisvectors � 1 , 2 , 1 , 2 � is used hereafter, where 1 , 2 , 1 , 2 denote the two electronic and the two nuclear magnetic quantum numbers, respectively, for the two nitroxides.

Initial density matrix.
To calculate the signal for a multi-pulse sequence, one starts with the initial density matrix, 0 , governed by the Boltzmann distribution for two electrons each with spin ½ in thermal equilibrium. Using the high-temperature approximation and neglecting the energy-level modification by the hyperfine interaction, which is much less than the electronic Zeeman interaction, one can write: Since the final signal is obtained by taking the trace: ( + ) and during the evolution of 0 to the term remains invariant, it does not contribute to the signal as ( + ) = 0. One can then replace 0 , as follows:

Calculation of pulsed EPR signal
The pulsed EPR signal for the system of coupled nitroxides, undergoing spin relaxation, is calculated by solving Liouville von-Neumann (LVN) equation that governs the time evolution of the density matrix during free evolution, i.e., in the absence of a pulse. It is expressed as [7][8][9][10][11][12][13][14][15][16][17][18][19] where = − 0 is the reduced density matrix, with 0 ∝ 1 + 2 , being the initial density matrix, as discussed in Sec. 2.2, and 0 is given by Eq. (2.1). In Eq. (2.7), Γ � � is the relaxation superoperator in Liouville space, whose matrix elements are: where ( 1 ) are the spin-lattice relaxation times between the populations ii to kk, which are operative on the coherent pathway = 0, and 2 , are the spin-spin relaxation times operative along the = ±1(index S), and = ±2 (indexD) pathways, respectively, as shown in Fig. 1. It is noted that, in general, the spin-spin relaxation times, � 2 , � . are different for different transitions ij, but these relaxation times are only slightly different from each other as shown in [23]. As a consequence, they are all approximated here in Eq. (2.8) to have the same average spinspin relaxation time 2 , .
In the two-pulse DQ sequence, shown in Fig. 1, the pathway = 0 is excluded, so that the relaxation times ( 1 ) , with i = k, affecting the populations, that appear in Eq. (2.8) for the = 0 pathway, have no effect on the signals. In the two-pulse sequence considered here, only the coherent pathways = ±1 and = ±2 participate, so that in the relaxation only the second term on the right-hand side of Eq. (2.8) which corresponds to ≠ elements of the reduced density matrix, affect the two-pulse DQ signal. Then, the solution of Eq. (2.7) after time , expressing the evolution of χ due to the relaxation along the = ±1 , ±2 pathways, is obtained as Appendix B below lists the elements of the matrix for − 0 used in Eq. (2.9) for the coupled nitroxides system.
As for the five-pulse DQM signal, which includes the coherence pathway = 0 ( Fig. 1), one also needs to consider the relaxation between the populations. To do that rigorously, one first needs to diagonalize the non-diagonal part of the relaxation matrix in Liouville space, as given by larger than both the spin-spin relaxation times and the duration of the experiment by two order of magnitudes [24]; hence, they do not have any significant effect on the final signal and can thus be neglected. Keeping now only the diagonal elements of the relaxation matrix and assuming that they are all equal to each other [23,24], the time evolution of the reduced density matrix on the coherence pathway = 0 is During the application of a pulse, the spin relaxation is here neglected, since the duration of the pulses are much smaller than the relaxation time. In that case, the evolution of the density matrix is described in Hilbert space, as follows: with being expressed by Eq. (2.5).
. The solution of Eq. (2.11), after the application of a pulse of duration , neglecting relaxation during the pulse, is given as [7]: After the application of a pulse, the density matrix is projected onto the coherence pathway of interest, which are: = 2, −1 for DQ and 0, ±1,0, ±2, −1 for DQM experiments, respectively, as given in Fig. 1. This is achieved by applying a projection operator that retains only the relevant elements of the density matrix which correspond to a particular pathway, , putting all the other elements equal to zero. The projection operators for the various coherence pathways are given in Appendix C.
For the calculation of two-pulse DQ signals, the final density matrix ( 1 , 2 ), where 1 is the time between the two pulses and 2 is the time after the second pulse, at which the signal is recorded, as shown in Fig. 1, is obtained as follows. is obtained after free evolution with relaxation of the density matrix obtained in step (iv) over the For the five-pulse DQM, signals, the final density matrix ( 1 , . . . , 5 ) with ; = 1, … ,5 being the time between the th and ( + 1)th pulse is obtained by successive applications of the 5 pulses to the initial density matrix, using Eq. (2.12), followed by the application of the relevant coherence pathway projection operator and then free evolutions over the coherence pathways as shown in Fig. 1, using Eqs. (2.9) and (2.10) for the coherence pathways = ±1, ±2 and = 0, respectively.
Calculation of Pake doublets. For this, one needs to average over the Euler angles λ 1 , λ 2 , the orientations of the dipoles of the two nitroxides. This is an enormous task as there are infinite many such possibilities. However, one can, instead, use Monte-Carlo averaging, wherein one varies λ 1 , λ 2 randomly as follows: Twenty such averaging were found to be sufficient, because another set of twenty Monte-Carlo averaging gave almost identical results.

Number of simulations.
Thus, a total of 90 -values and 90 -values were used over a unit sphere, along with 20 sets of five Euler angles (λ 1 , λ 2 ). This amounts to an average over 90 × 90 × 20 = 1.62 × 10 5 simulations. The procedure to calculate the two-pulse DQ or fivepulse DQM signal is described in the flowchart in Appendix D.

General procedure to derive analytical expressions for multi-pulse-EPR Signals.
In this section, a general algorithm to calculate the analytical expression for any pulse sequence is described. The calculation of the analytical expressions here is presented in the 4 × 4 electronic subspace, of which each element is a 9 × 9 matrix in the hyperfine subspace. To calculate the signal for an n-pulse sequence, one starts with the initial density matrix, 0 , which can be effectively reduced, for calculating the signal to as discussed in Sec. 2.2, as follows: The density matrix, , after the application of the first pulse, is obtained by using Eq. (2.12). After the application of a pulse, the density matrix on the coherence pathway of interest is obtained by applying the projection operator, , defined in Appendix C, to the resulting density matrix, .
This is done by the Hadamard product, i.e. ρ ( ) = ○ , where ; = 0, ±1, ±2 is the projection operator for the coherence pathway = and ○ denotes the Hadamard product of two matrices, where each element i, j of the resulting density matrix is the product of the i, j of the projection operator matrix, , and the element i, j of the l density matrix, . In the absence of relaxation, the density matrix, after free evolution over time on a coherence pathway, p, is found, by using Eqs. (2.9), (2.10) and (B.11), to be as follows: where ρ (0) , ρ (+1) and ρ (2) are the density matrices before the free evolutions, just after the application of the respective pulses, on the pathways = 0, +1, +2, respectively, achieved by the application of the relevant projection operator. (Experimentally, it is achieved by phase cycling.) The ( ) ( ); = 0,1,2 terms for free evolution over the various coherence pathways, used in Eqs.
(3.2)-(3.4), are listed in Table I. The evolutions of the density matrix over the coherence pathways = −1 and = −2 are Hermitian adjoints of those for = +1 and = +2, respectively, as given above. The resulting density matrix at the end of a coherence pathway will serve as the starting density matrix to which the next pulse is applied. The same procedure is repeated for all other pulses in turn to calculate the final density matrix. The signal for a chosen orientations of the two nitroxide dipoles with respect to the dipolar axis, oriented at an angle with respect to the lab axis, is obtained using Eq. (2.14).

Analytical Expression for Two-pulse DQ signal
In this section, the DQ signal will be calculated analytically using the procedure outlined in Sec. 2 above. The pulse sequence for a DQ experiment is shown in Fig. 1(a). It consists of two finite arbitrary pulses each with the duration . The phase cycling required for the pathways = 2 and = −1 in the two-pulse DQ sequence is given in Table II of [7]. Finite pulses, as opposed to infinite pulses, are needed to produce non-zero coherence transfers 02 and 2−1 as shown below, in general, in Sec. 3.3.2. For theoretical calculation, after the application of the first pulse, only the matrix elements corresponding to the double quantum coherence pathway = +2 are retained in the density matrix, removing all the other matrix elements in it, using the appropriate projection operator. The system is then allowed to evolve freely with relaxation over time 1 before the application of the second finite pulse, after which the coherence pathway = −1 is chosen. wherein only the elements of the density matrix corresponding to the pathway = −1 are retained, by using the appropriate projection operator. The signal is measured after time 2 after the second pulse as shown in Fig. 1. It is noted that the tip angle = γ 1 for the pulses is a function of both the amplitude of the microwave field 1 and the duration of the pulse. , which determine the intensity of coherence transfer as shown in Sec. 3.3.2. Furthermore, it is the presence of the dipolar interaction only during a pulse that makes the non-zero coherence transfers possible.
Following the procedure described in Sec. 2, the DQ signal is calculated for a chosen orientation of the two nitroxide dipoles with respect to the dipolar axis, oriented at an angle with respect to the lab axis, to be as follows: where the superscript (2) on the terms refer to the second pulse and the superscript * denotes the complex conjugate of the terms, given in Table I. In Eq. (3.5), and the trace is taken over the 9 × 9 hyperfine space. In Eq. The 9 × 9 ( ) and terms in Eq. (3.5) are given by Eq. (2.13) above.

Coherence transfer
It is important to have an estimate of the coherence transfer, since the intensity of the signal increases with increasing coherence transfer. The efficiency of the coherence transfer,  , from the coherence pathway m to the coherence pathway n, is calculated as follows. The resulting density matrix for the pathway n by the application of a pulse to the density matrix is proportional to the spin operator corresponding to the coherence pathway , which is for the coherence pathway = 0 for the initial density matrix, ± for the coherence pathways = ±1 and ± . ± for the coherence pathways = ±2. The density matrix for the pathway n is then obtained by taking the trace of the density matrix resulting by the action of the pulse with the projection operator for the coherence pathway , listed in Appendix C.

Coherence transfer efficiencies for 2-pulse DQ sequence for a finite pulse
The coherence transfer efficiency for the transitions 02 and 2-1 in a DQ experiment is zero for an infinite pulse [7]. With the application of a finite pulse, however, one can obtain a nonzero 02 and 2−1 .Using a rigorous analytical treatment, it is shown here that, indeed, 02 and 2−1 are non-zero for the system of coupled nitroxides as effected by a pulse of finite duration, when there is present a non-zero dipolar interaction, seen as follows.
In the static Hamiltonian given by Eq. (2.1), 01 and 02 terms do not contribute to coherence transfer, because they contain the spin operator , so only the dipolar-interaction term 12 need to be taken into consideration for the calculation of coherence transfer. In the magnetic basis of the two electrons, the matrix for 0 in the direct-product space (see Appendix B), is then given as In the magnetic basis, the matrix of the pulse Hamiltonian together with the static Hamiltonian in the direct-product space is: whose eigenvalues, and the eigenvectors, , are given as Then the pulse propagator, including the static Hamiltonian, is expressed as Following the procedure to calculate the coherence transfer as given in Sec. 3.2 below and using the pulse propagator, given by Eq. (3.11), the coherence transfer for the transition 0 2, 02 is found to be: and for the transition 2  − 1, the coherence transfer is: It is noted from Eqs. (3.12) and (3.13) that when either the dipolar coupling, or the pulse, is zero both the coherence transition 02 and 2−1 vanish, since is proportional to the dipolar constant and is proportional to 1 .

One-dimensional Signals for DQ and DQM sequences
It is shown in Secs. 3.3.1 and 3.3.2 below, that the signal is maximum for 2 = 2 1 for twopulse DQ sequence, whereas it is maximum for 5 = 1 for five-pulse DQM sequence, respectively.
Thus, one can use one-dimensional (1D) measurements along these equal times, instead of varying them independently, to measure Pake doublets to obtain the value of the dipolar interaction constant, which is inversely proportional to the cube of the distance between the two nitroxides of the biradical.

One-dimensional-two-pulse DQ signal
It is seen from the analytical expressions in Eq. (3.5) that the time-dependent parts of the DQ signals at times 1 and 2 , specified by ( 1 ), ( 2 ) * , = 1, … ,6, given in Table I, contain the terms − ω ij 1 and ω ij 2 , respectively, where ω ij = ( − )/ℏ; where the , and are defined by Eq. (B.10) in Appendix B below. In the rotating frame, the value of | 1 − 2 | 2 and After substituting the various ω terms given by Eq. (3.15) and ( ); = 1, … ,6; = 1,2 , given in Table I, the expressions for the DQ signal, as given by Eq. (3.5), can be written as It is seen from Eqs. B.6 and B.8 in Appendix B that, when the dipolar interaction is large, i.e., one can then make the approximation 22 ≈ 33 for which yields Ξ ≈ π/4. The − factor in Eq.
(3.17) then becomes zero, and the reduced 1D DQ signal( 2 = 2 1 ) can now be expressed as whose Fourier transform as a function of 1 would yield a peak at the frequencies implies that the peak of the signal, i.e., the Pake doublet, will occur at ± 3 2 when averaged over the five independent Euler angles defining the orientations of the two nitroxide dipoles.

One-dimensional Five-pulse DQM signal
The five-pulse DQM pulse sequence is shown in Fig 1(b). In this sequence, the first pulse (π/2) moves the density matrix to the single-quantum coherence pathway ( = +1), over which it evolves for a period 1 . The second finite (π) pulse transfers this magnetization to = 0 coherence pathway and the density matrix evolves on it over the period 2 . Thereafter, the third pulse transfers the density matrix to the double quantum ( = ±2) coherence pathways, over which the density matrix undergoes free evolution for the time interval 3 . It is then subjected to the fourth, refocussing pulse (π) . After the time interval 4 on the coherence pathways = ±2, the fifth (π/2) pulse finally transfers the density matrix to the single quantum coherence pathway = −1 , on which the signal is detected after the time interval 5 . In the experiment, 1 = 2 , which are stepped. As well, 3 = 4 which will here be denoted for this pulse sequence as ( ) ; it is kept fixed. The echo in this five-pulse DQM sequence occurs at the time 5 = 1 .
The five-pulse DQM sequence, shown in Fig. 1(b), is equivalent to two coherence pathways, both of which lead to the formation of an echo at 5 = 1 . These are: In the above, for the partial pathways 0 → +2 and 0 → −2 in (i) and (ii), respectively, finite pulses are used for the indicated coherence transfers, otherwise for the remaining pathways infinite pulses are used.
Following the procedure described in Secs. 2 and 3, the 1D five-pulse DQM signals for the two coherence pathways in Eq. (3.19) are calculated for 5 = 1 , for chosen orientations of the two nitroxide dipoles with respect to the dipolar axis, oriented at an angle with respect to the lab axis, respectively, to be as follows: The five-pulse DQM signal, combining the ones due to the two simultaneous coherence pathways (i) and (ii), is then: It is seen from Eq. (3.20) that the main dipolar peaks for a chosen orientation of the two nitroxide dipoles with respect to the dipolar axis, oriented at an angle with respect to the lab axis, will occur in the Fourier transform at ± × (3cos 2 − 1). This shows that similar to the 1D two-pulse DQ case in Sec. 3.3.1, the Pake doublet for the 1D five-pulse DQM signal, will occur at ± when averaged over the five independent Euler angles defining the orientations of the two nitroxide dipoles.

3(c) and 3(d)
, giving rise to orientational selectivity. implying that these spins will be preferentially pumped from = 0 to = 2 coherence state. This is a first-ever novel result, as far as orientational sensitivity is concerned, found with the help of extensive quantitative simulations for the first time. This orientational selectivity of the forbidden DQ signal occurs for d > 10 MHz up to a maximum value of B 1 that depends on d.. As for 20 ≤ ≤ 40 , it occurs for B 1 ≤ 3.0 G for both two-pulse DQ and five-pulse DQM sequences. It is noted that there is no orientational selectivity possible for ≤ 10 MHz.

Elucidation of structure of biomolecules.
As seen from the simulations made here for different sets of Euler angles, defining the orientations of the two magnetic dipoles of the nitroxide bilabel, in Sec 7 below, the signals from two-pulse DQ and five-pulse DQM sequences are found to be sensitive to the orientations of the two nitroxide dipoles, as described by the Euler angles (α 1 , β 1 , γ 1 ); (α 2 , β 2 , γ 2 ). This provides structural sensitivity to the two-pulse DQ and fivepulse DQM signals, useful for understanding details of the configuration of biomolecules.

Numerical simulations in the absence of relaxation
Although analytical expressions are useful in deducing important features of the signal and its Fourier transform, they are susceptible to human error when transcribing them into a code for numerical calculation. Therefore, the best way to calculate pulsed EPR signals is by numerical techniques using the eigenvalues and eigenvectors of the Hamiltonian matrix. The details of the calculations presented in this section, based on the algorithm developed by Misra to calculate the six-pulse DQC signal [8], are quite general, applicable to both one-and two-dimensional twopulse DQ and five-pulse DQM signals.
In the numerical calculations performed here, the magnetic basis with the basis vectors � 1 , 2 , 1 , 2 � is used to calculate the various matrix elements. Here 1 , 2 , 1 , 2 are the two electronic and the two nuclear magnetic quantum numbers, respectively, for the two nitroxides. In this magnetic basis, the static Hamiltonian 0 is not diagonal; the eigenvalues of 0 are obtained by the diagonalization † 0 = , where E is the eigenvalue matrix, whose diagonal elements are the eigenvalues, whereas the columns of matrix are the corresponding eigenvectors.
To calculate the -pulse signal, one starts with the initial density matrix, which is as discussed in Sec. 3, in the direct-product space: 0 → = � σ 1 2 � ⨂ 2 ⨂ 1 ⨂ 2 + 1 ⨂ � σ 2 2 � ⨂ 1 ⨂ 1 , where σ ; = 1,2 are the Pauli spin matrices for the electron spin. The final density matrix ({ }) with k=2,5 for two-and five-pulse signals, respectively, is obtained by successive applications of the pulses to it using Eq. (2.12), followed by free evolutions over the coherence pathways as shown in Fig. 1 for each pulse sequence using Eqs. (2.9) and (2.10) without the exponential factor that considers the effect of relaxation. The complex signal is then obtained using Eq. (2.14). The polycrystalline average and Pake doublets are calculated using Eqs.
(2.15) and (2.16), respectively. Thereafter, the relevant Gaussian inhomogeneous broadening factor is multiplied to Eq. (2.16) for Pake doublets, as discussed in Sec.2, The flow chart for simulations is given in Appendix D.

Relaxation in a polycrystalline sample
In Sec. 4. above, the signal for a polycrystalline sample is calculated in the absence of relaxation. To consider the effect of the relaxation for a powder average, the stretched exponential approach is used here, following the discussion in [21,22], which considers the average effect of different relaxation times for different orientations of the dipolar axis with respect to the magnetic field by a single exponential with the exponent β. For the present cases this is discussed as follows.
Averaging over relaxation times , and . According to Eqs. (2.9) and (2.10), after time , for a single-orientation of the dipolar axis with respect to the external magnetic field, the effect of relaxation on the signal is considered by multiplying the calculated signal by the exponential factors (− / 2 ), (− / 2 ) and (− / 1 ) for the coherence pathways = ±1, = ±2 and = 0, respectively, with the time constants 2 , 2 , 1 , appropriate for that orientation. Then the cumulative effect of the relaxation on the multi-pulse signals, considering all coherence pathways as shown in Fig. 1, is calculated by multiplying the signal with two and five decaying exponential functions for two-and five-pulse sequences, respectively. For a polycrystalline sample, the signal is averaged over different values of (θ, ϕ), each characterized by different relaxation times 1 (η, λ 1 , λ 2 ) 2 (η, λ 1 , λ 2 ) , 2 (η, λ 1 , λ 2 ). The effect of relaxation at the top of the echo, i.e., at 2 = 2 1 for two-pulse DQ and at 5 = 1 for five-pulse DQM, is expressed as where 0 ( , η, λ 1 , λ 2 ) is the EPR signal calculated without relaxation as given by Es. (2.14).
(6.2a) and (6.2b) ranges between zero and one [21,22]. Equations (6.2a) and (6.2b) reduce to a system with orientation-independent relaxation times in the limit when β1. It is noted that, in general, there are two different stretching parameters, β (S) and β ( ) , characterizing the orientational distribution of the single and double quantum relaxation times which are to be found by fitting the simulation to the experimental data [21,22]. In this paper, since the experimental values for β (S) and β ( ) are not available, and β ( ) affects only the intensity of the signal, the value β (S) =β ( ) = β = 0.8 is used, being the average of the two values 0.78 and 0.85 used in [21,22].

Discussion of results of numerical simulations
For reference, the values and definitions of the constants used in the numerical simulations are listed in Table I  This results in a considerable saving of time in experimental measurements. As well, the simulations for a polycrystalline sample can be carried out using a much larger grid over the unit sphere because only a single time variable is needed.

Two-pulse DQ signal.
The maint features of the simulation for the coherence transfer for the DQ signal are as follows.
(i) Figure 2  2 has more dominant contribution as compared to that of 2 , as its value is almost half that of 2 as determined experimentally in [20], since the relaxations factors are inversely proportional to the exponential of the relaxation time. specifically, the efficiency of the coherence transfer in the transition 02 increases from 0.06 for = 10 MHz to 0.12 for = 30 MHz. This is in accordance with the predictions of Eqs. (3.12) and (3.13), giving the theoretical expressions for these two coherence transfers, which imply that with larger d there is greater coherence transfer.
(iii) The effect of relaxation on the DQM signal in shown in Fig. 10, displaying the Fourier transform of the DQM signal without and with relaxation, calculated for the relaxation times 1 = 10μ (over the pathway = 0), 2 = 500 ns (over the pathways = ±1) and 2 = 200 ns (over the pathways = ±2), using the stretching parameter =0.8, = 40 MHz and 1 = 17.8 . As expected, the Pake doublets become broadened by relaxation. The relaxation times 1 and 2 contribute to this broadening whereas 2 does not have any effect on the broadening of the peaks, since it operates over a constant time, reducing the intensity but not contributing to the broadening.
Furthermore, 2 has more dominant contribution as compared to that of 1 , as its value is shorter than 1 , since the relaxations factors are inversely proportional to the exponential of the relaxation time.
(iv) Figure 11 shows the simulation of the five-pulse DQM spectrum of the nitroxide biradical to fit the experimental data obtained by Saxena and Freed [20]. The simulations are carried out, using the same parameters as listed in [20], employing the numerical algorithm as given above in Sec. 4. The experimental data shown here is a profile of the maximum, i.e., that occurs for 5 = 1 of the three-dimensional experimental data (intensity versus 5 , 1 ), reported in [20], The parameters used for the simulation are the same as that used in [20], which are specifically: 1 =17.8 G; d=12.3 MHz, 2 = 500 , 2 = 300 ; � � 1 = � � 3 = � � 5 = 5 and � � 2 = � � 4 = 10 . The other parameters are the same as those listed in Table I. The simulation shows a reasonably good agreement, within experimental errors, to the experiment [20].
(v) It is found from Figs. 2 and 8, showing the simulations of the Pake doublets for the twopulse and five-pulse sequences, respectively, that the five-pulse DQM sequence produces much cleaner Pake doublets than does the two-pulse DQ sequence, as far as the side peaks are concerned.

Conclusions
The salient features of the present study are as follows.

(i)
It is shown here from general considerations that a finite, rather than an infinite, pulse is needed, in conjunction with the dipolar interaction, to produce a non-zero coherence transfer in the transitions 02 and 2-1 for both the two-pulse DQ and five-pulse DQM sequences.
(ii) The simulations show that the coherence transfer, T 02 , as effected by a finite pulse in conjunction with the dipolar interaction, is found to increases as the amplitude of the irradiation field (B 1 ) decreases. Furthermore, it is maximum for those coupled nitroxides, whose dipolar axes are oriented symmetrically about ±10 ○ away from the magic angle 0 ~ 54.74 ○ , at which (3 2 θ − 1) = 0, for 0 ○ ≤ ≤ 90 ○ , being symmetric about = 90 ○ in the range 0 ○ ≤ ≤ 180 ○ , implying that these spins willl be preferentially pumped from This is a first-ever novel result, as far  (iii) A full derivation of the analytical expressions and a complete algorithm for the numerical simulations for two-pulse DQ and five-pulse DQM sequences, using a finite pulse, are given in this paper.
(iv) The Fourier transforms of the DQM signals depend upon the orientations of the two nitroxide magnetic dipoles as described by the respective sets of Euler angles. This can be exploied to study the structural geometry of the nitroxide biradical. This is a cutting-edge topic in the study of biomolecules.
(v) The Pake doublets occur at ± (vi) By comparing the Pake doublets, it is found that the five-pulse DQM sequence produces much cleaner Pake doublets than does the two-pulse DQ sequence.
(vii) For the purpose of distance measurement, it is shown here that one needs to perform only one-dimensional time-dependent experiments, i.e., involving only 2 = 2 1 for two-pulse DQ and 5 = 1 forfive-pulse DQM sequences.
(viii) The simulation of the five-pulse DQM signal calculated using the algorithm described here shows a good agreement with the experimental data.

Appendix A. The static spin Hamiltonian for nitroxide biradical
In this appendix, the coefficients , and of the static spin Hamiltonian for the two nitroxides of the biradical, given by Eq. (2.2), are defined; more details can be found in [7]. It can be expressed in terms of the irreducible spherical tensor operators (ISTO) as [7,8] where determines type of the interaction and takes two values: and for Zeeman and hyperfine interactions, respectively; (=1,2) specifies the two nitroxides; is the rank of the tensor; and takes integer values between − and + for a given L. Here stands for the laboratory frame, defined to be such that the z-axis is parallel to the static magnetic field [7]. In where the coefficients , , are defined in the rotating frame as It is noted that the isotropic part of the Zeeman term is put equal to zero in the rotating frame and the Zeeman term, as given by Eq. (A.4), is the resonant offset term as calculated here quantitively.
for one of the nitroxide biradicals, can be expressed as [26] The Hamiltonian in Eq. (B.1) is diagonalized by the unitary transformation 1 = (1) †

Appendix C. Projection operators for the various coherence pathways
The projection operators to project the density matrix on to the coherence pathway of interest after the application of a pulse are listed in this appendix for the coherence pathways = ±1, = ±2 and = 0 as used in the calculation of DQ and DQM signals and discussed in Sec.
3. After the application of a projection operator, only those 9 × 9 hyperfine blocks of the density matrix which correspond to the non-zero elements of the projection operator are retained in the 4x4 electronic-spin space, putting all the other elements of the density matrix equal to zero. Choose the desired value of the Euler angles (β 1 , γ 1 , α 2 , β 2 , γ 2 ) Monte Carlo Simulation?
Choose a set of five random Euler angles (β 1 , γ 1 , α 2 , β 2 , γ 2 ) Start the DO loop for the orientation over the grid of (θ, ) Calculate the initial density matrix 0 (∝ 1 + 2 ) Calculate ( ) after the application a pulse using Eq. (2.12) Apply the projection operator as given in Appendix C to select those matrix elements of the density which correspond to the relevant pathway.
Calculate χ( ) after free evolution over time using Eqs. (2.9) and (2.10) for the relevant coherence pathway without any relaxation.      Table II. Relaxation is not considered in these simulations. All Pake doublets appear at ±3 /2, in agreement with those calculated analytically as given by Eq. (3.18).   Table II.
The Fourier transforms of the DQ signal are quite sensitive to the relative orientations of the nitroxide biradicals, making the DQ experiment feasible for structural studies.
47  Table II. Due to relaxation, the peaks are broadened, and the intensity of the calculated Fourier transform of the DQ signal is reduced by a factor of two.
48   Table II. Relaxation is not considered in these simulations. All Pake doublets appear at ± , in agreement with those calculated analytically as given by Eq. (3.20).  Table II Table II. Due to relaxation, the peaks are broadened, and the intensity of the calculated Fourier transform of the DQM signal is reduced by a factor of three.
53  Table II. The simulation shows a reasonably good agreement with the experimental data, within experimental error.