The effect of spin polarization on double electron–electron resonance (DEER) spectroscopy

Abstract Double electron–electron resonance (DEER) spectroscopy measures the distribution of distances between two electron spins in the nanometer range, often on doubly spin-labeled proteins, via the modulation of a refocused spin echo by the dipolar interaction between the spins. DEER is commonly conducted under conditions where the polarization of the spins is small. Here, we examine the DEER signal under conditions of high spin polarization, thermally obtainable at low temperatures and high magnetic fields, and show that the signal acquires a polarization-dependent out-of-phase component both for the intramolecular and intermolecular contributions. For the latter, this corresponds to a phase shift of the spin echo that is linear in the pump pulse position. We derive a compact analytical form of this phase shift and show experimental measurements using monoradical and biradical nitroxides at several fields and temperatures. The effect highlights a novel aspect of the fundamental spin physics underlying DEER spectroscopy.


S1 Theoretical DEER signal from a single A-B spin pair
Here we derive the 3-pulse DEER signal from a single A-B spin pair, without making the hightemperature approximation. The recapitulates the derivation by Marko et al (Marko et al., 2013), with adjusted notation. Assuming that the effect of the hyperfine interactions can be folded into the resonance frequencies of the individual spins, the spin Hamiltonian contains terms for the Zeeman and dipolar interactionsĤ lab =Ĥ Z +Ĥ D (S1) In the high-field limit, the Zeeman interaction term expands tô whereh is the reduced Planck constant, ω A and ω B are the resonance angular frequencies of the two spins, andŜ Az andŜ Bz are the spin operators. In the high-field limit, the dipolar interaction term iŝ where ω AB is the dipolar coupling angular frequency with the dipolar constant where µ 0 is the magnetic constant, µ B is the Bohr magneton, g e is g-value of the free electron, r AB is the length of the interspin vector, and θ AB is the orientation of interspin vector with respect to the external magnetic field, B 0 . In order to calculate the DEER signal, it is convenient to transform the Hamiltonian from the laboratory frame to a rotating frame that is rotating with angular frequency ω around the z-axis, where ω is the detection angular frequencŷ H = exp(iωt(Ŝ Az +Ŝ Bz ))Ĥ lab exp(−iωt(Ŝ Az +Ŝ Bz )) −hω(Ŝ Az +Ŝ Bz ) This givesĤ =h(∆ω AŜAz + ∆ω BŜBz + ω ABŜAzŜBz ) where ∆ω i = ω i − ω are the angular frequency offsets of the two spins (i = A, B). Deriving the DEER signal requires propagating the density operator through the course of the DEER experiment, beginning from the equilibrium densitŷ where k B is the Boltzmann constant, T is the temperature, and Z = tr(exp(−Ĥ Z /k B T )) is the partition function. In Eq. (S8), we have neglected the dipolar coupling termĤ D in the Hamiltonian, which is much smaller than the two Zeeman terms. Next, we use where1 is the identity operator, and c i and s i are abbreviations for the cosh and sinh factors. With this, we getσ is the thermal spin polarization of spin i. The evolution in time of the density matrix is governed by the Liouville-von Neumann equation.
whereÛ is the total propagator of the DEER sequence and t is the pump pulse position. In the rotating frame and with ideal pulses, the Hamiltonian is piece-wise constant, and therefore the propagator is a composition of simple exponential operators: For describing the pulses, we assume ideal pulses that are selective for A or B spins, and we neglect fast oscillating terms and internal interactions during the pulses. All pulses inÛ (t) are set to have y phase. A more visual way to represent the propagator is Before applying the propagator to the starting density based on Eq. (S12), we simplify the propagator. This is possible due the simplicity of the rotating-frame HamiltonianĤ, which only contains terms withŜ Az andŜ Bz that all mutually commute.
First, inserting the identity operator 1 = e +iπŜ By e −iπŜ By to the right of theĤt propagator gives Next, the bracketed propagator combination can be written as e −iĤ B t , whereĤ B is the transformed HamiltonianĤ This is possible since, in general,Û e −iÂÛ † = e −iÛÂÛ † . SinceĤ andĤ B commute (as both contain only z spin operators), we combine them into a single propagator exponent. The total propagator now readŝ U (t) = e −îHτ e −iπŜ Ay e −i(Ĥ(τ −t)+Ĥ B t) e −iπŜ By e −i(π/2)Ŝ Ay (S17) Inserting the identity operator 1 = e +iπŜ Ay e −iπŜ Ay to the right of theĤ/Ĥ B propagator yieldŝ U (t) = e −iĤτ e −iπŜ Ay e −i(Ĥ(τ −t)+Ĥ B t e +iπŜ Ay e −iπŜ Ay e −iπŜ By e −i(π/2)Ŝ Ay (S18) The propagator combination of the right bracket simplifies to e −iπŜ By e −i(3π/2)Ŝ Ay , sinceŜ Ay andŜ By commute. The propagator combination of the left bracket can again be rewritten with the transformed Hamiltonians:Û (t) = e −iĤτ e −i(Ĥ A (τ −t)+Ĥ AB t e −iπŜ By e −i(3π/2)Ŝ Ay (S19) whereĤ A = e −iπŜ AyĤ e +iπŜ Ay = −∆ω AŜAz + ∆ω BŜBz − ω ABŜAzŜBz (S20) After combining the adjacent exponential operators with the Hamiltonians in the exponent, we get The sum of the Hamiltonian terms iŝ SinceŜ Bz and 2Ŝ AzŜBz commute, we can separate the propagator into two, one for each term. With this, the final expression for the total propagator iŝ Next, to get the final density σ(t), we apply this propagator to the starting density from Eq. (S10): (Note that theŜ Bz propagator has left the density unchanged.) The echo signal is The only two terms from the final density in Eq. (S26) that give a non-zero trace withŜ A+ are those containingŜ Ax andŜ Ay , since tr(Ŝ A+ŜAx ) = 1 and tr(Ŝ A+ŜAy ) = i. Therefore, only these terms survive: Dropping the prefactor (which is absorbed into the overall amplitude factor V 0 ), we get the final result In a thermally equilibrated sample, the polarization of pump and probe spins are identical, therefore we can set ϵ = ϵ a = ϵ B .

S2 Theoretical DEER signal from A-spins in a homogeneous distribution of B-spins
For a sample with an A-spin in a uniform, random, spatial distribution of B-spins, the total signal is a product of all individual V AB signals from Eq. (S29), additionally averaged over all B-spin configurations Here, N B is the number of B-spins in a configuration, r b is the vector from the A-spin to the bth B-spin, and the angled brackets indicate an average over B-spin configurations. To arrive at this product form, the dipolar couplings among B spins are neglected.
With the assumption that the positions of the B-spins are uncorrelated, the averaged product can be replaced with the product of the averages.
Additionally, with the assumption that all B-spins are equally distributed, all averages are equal and Defining a (large) spherical region S R with a radius R, V inter (t) is obtained by calculating the signal for all N B B-spins within S R and then taking the limit R → ∞ based on Eq. (S29). The averages over the oscillatory terms are where V R = 4π 3 R 3 is the volume of the sphere. The integrals C R (t) and S R (t) cannot be evaluated analytically, but if we take the limit R → ∞, the integrals give The derivations are given separately below in sections S2.1 and S2.2 below. Note that C(t) is proportional to |t|, whereas S(t) is proportional to t. Using C(t) and S(t) instead of C R (t) and S R (t), Eq. (S34) becomes with a small error for finite R that will vanish once we take the limit R → ∞. Given a volume concentration of B-spins of c B , the number of B-spins within S R is N B = c B V R , and the volume is Combining this with Eq. (S39) and inserting into Eq. (S33) gives The limit R → ∞ corresponds to V R → ∞ and N B → ∞, so that we get Inserting the expressions for C(t) and S(t) from Eqs. (S37) and (S38) yields the final expression

S2.1 The cos integral
Here we solve the C(t) integral from Eq. (S37). Its explicit form is Since the integrand is independent of ϕ, the ϕ integral reduces to a prefactor of 2π. Next, we simplify the notation using the substitution z = cos θ Since the integrand is symmetric about z = 0, we can reduce the z integration interval to [0, 1] and prepend a factor of 2. The integral now is Inserting this substitution gives The integral of (cos(ax)−1)/x 2 over x can be solved using integration by parts with u(x) = cos(ax)−1 and v ′ (x) = 1/x 2 . This gives The first term is zero for both x = 0 and x = ∞, so that we get This gives Evaluating the remaining z integral gives so that we finally get where we have also used the fact that D is positive.

S2.2 The sin integral
Next, we solve the integral S(t) from Eq. (S38). This turned out to be somewhat involved. The explicit form is Again, the ϕ integral adds a prefactor of 2π. Applying the same cos θ → z and 1/r 3 → x substitutions as for C(t) gives As a result of the r → x substitution, the integrand now diverges for x → 0. Therefore, we need to be cautious and write the overall integral in terms of a limit: (In principle, we could already write the initial integral expression with a limit r max → ∞.) Depending on t and z, the argument of the sin function can be positive or negative. To solve the x integral, we need a non-negative argument. Utilizing sin(ξ) = sgn(ξ) sin |ξ|, we get The integral of sin(ax)/x 2 over x can be solved using integration by parts with u(x) = sin (ax) and v ′ (x) = 1/x 2 . For a ≥ 0, this gives dx sin(ax) with the cosine integral function Ci(·), defined only for non-negative arguments. For x → ∞, both terms evaluate to 0, so that the overall integral can now be written as We can pull |Dt| out of the integral and replace |Dt|u → x. We get The divergence of the integrand is due to the cosine integral function. To deal with this, we rewrite it using Ci(ξ) = γ + ln(ξ) + Cin(ξ) where γ is Euler's constant and Cin(·) is the modified cosine integral function which satisfies Cin(0) = 0. The full integral is now The only term in the integrand that diverges for x → 0 is (1−3z 2 ) ln x. However, the z integral over this term is zero, since the z integral over (1 − 3z 2 ) is zero. Therefore, we can drop the ln x term from the integrand. The divergent term along x disappears due to symmetry along z! A similar argument allows us to drop the (1 − 3z 2 )γ term as well. All remaining terms in the integrand are finite everywhere and converge for x → 0, so we can swap the x limit and the z integral. (In more technical terms, the integrand converges uniformly over the entire z interval, and we can apply the Lebesgue dominated convergence theorem to interchange the x limit and the z integral.) For x = 0, the first term becomes (1−3z 2 ), which integrates to zero. The third term equals zero for x = 0 (since Cin(0) = 0). We are left with Evaluation of this with Mathematica yields S3 Numerical calculations

S3.1 Monte Carlo simulation of intermolecular signal
To verify the analytical expression of the polarized background signal in Eq. (S42) and to investigate how it depends on neighboring spins, we performed Monte Carlo simulations. We start with Eq. (S30), and use an average over a finite number N conf of configurations and a product over a finite number N B of B-spins: with the analytical signal Eq. (S29) for V AB (r b , t j ). t j is a set of time points over which V inter is evaluated.
The input parameters to this Monte Carlo model are the polarization ϵ, the pump efficiency p B , the B-spin concentration c B , the number N B of B-spins per configuration, and the number N conf of configurations. For each configuration, an A-spin is placed at the origin of a spherical volume of radius R = (3N B /4πc B ) 1/3 , and N B B-spins are placed at random uniformly distributed positions r b within the sphere using where u r is drawn from the standard uniform distribution U(0, 1), and n x , n y , and n z are drawn from the standard normal distribution N (0, 1). Figure S1 shows Monte Carlo simulations as a function of the number of excited B-spins in a configuration. The signal predicted analytically is captured perfectly. These simulations provide interesting insight into B-spin contributions to the decay. Remarkably, it is sufficient to consider configurations with only a single excited B-spin (p B N B = 1) in order to accurately recapture the initial 20% drop of the signal from its value at time zero. Both in-phase and out-of-phase signals are close to converged with 5 excited B-spins (p B N B = 5). With 50 excited B-spins, the entire decay curve is visually indistinguishable from the analytical model. These simulations show that the decay is dominated by just a few excited B-spins that are closest to the A-spin.

S4 Fit parameters
This section contains all of the fit parameters for the Q-band and G-band monoradical data (Table S1) and the G-band biradical data (Table S2) at various temperatures. Fitting was done using Eqs. (14)-(16), defined in the main text, with MATLAB's lsqcurvefit. All fit parameters are listed with the their 95% confidence intervals.   (16), where the fit parameters are the inversion efficiency p B , the decay rate constant k, overall signal amplitude V 0 , the zerotime shift t 0 , additional phenomenological fit factor for the intramolecular signal q F and intermolecular signal q B , the peak position r 0 and the standard deviation w.

S5 Experimental validation
This section contains the results of the various experiments conducted towards the aim of verifying the observed signal and resolving any discrepancies with the theoretical predictions. Among these experiments are pump-probe pulse excitation band overlap (Fig. S3), gain imbalance between the inphase and out-of-phase detectors (Fig. S4), signals recorded on and off the echo to observe resonator background and phase instability (Fig. S5), and saturation recovery data (Fig. S7).
If there is a significant amount of pulse overlap, the pump pulse could partially excite A spins (and vice versa). This could lead to additional contributions to the DEER signal from secondary dipolar pathways (Fábregas-Ibáñez et al., 2022). The spectrum and pulse excitation profiles for the experimental conditions in the monoradical experiments (6.42 T, observer freuqency 180.000 GHz, pump frequency 180.050 GHz, pulse widths 58 ns) are shown in Fig. S3. Total excitation by the probe pulse is approx. 3.7% and the pump pulse is approx. 4.3%. The overlap between pulse excitation profiles is approx. 2.7%. This is too small to create significant amplitudes in secondary dipolar pathways. Experiments with varying pump-probe offsets between 40 MHz and 120 MHz showed the same behavior (data not shown). One possibility to get an overly intense out-of-phase component is that the second channel (out-ofphase with the echo) has a higher gain than the first channel. To test for gain imbalance, multiple traces of the monoradical data were recorded (Fig. S4). The first was run in the same manner as the experiments shown in the main text (blue), the second with the detector phase rotated by 90 degrees (green). Although not completely identical, there is no substantial difference in the relative amplitudes of the out-of-phase signals.
To verify that the observed out-of-phase signal is entirely due to the refocusing spins and not due to instrumental offsets, two traces of the monoradical sample were recorded, the first being run in the same manner as the experiments shown in the main text, the second recorded with the detector window offset in time from the echo (Fig. S5. The data show that there is no slope in the out-of-phase signal when recording off-echo, confirming that there is no instrumental offset or phase drift as a function of pump pulse position. The data also reveal that the actual out-of-phase signal is much noisier than the instrumental baseline obtained off-echo. This likely arises from some small phase  measured at 180 GHz and 6.42 T at 5 K, run in the forward direction (blue) and the reverse direction (green). Experimental conditions are identical to those shown in the main text for the monoradical data at 6.42 T and 5 K. The data were fit with Eq. (14) and are shown with their 95% confidence interval in parentheses.
jitter in the instrument. The detector phase drifts slowly over the (wall clock) course of the experiments. This could be a possible contributor to the observed mismatch between experiment and theory. However, it can be excluded because backwards and forward sweeps of t give approximately the same result (Fig. S6), indicated by the identical fit factors (within 95% confidence). Regardless, the experiments were run with the shortest feasible acquisition times while still obtaining sufficient signal-to-noise to eliminate as much drift as possible.
To ensure that spin saturation was not occurring during experiments, a saturation recovery experiment was conducted to select the repetition time (Fig. S7). From the data shown, a 500 ms repetition time was used for all 5 K data shown.
The G-band setup typically utilizes a slower freezing method where the sample freezes after being placed inside the cold resonator. To ensure that the freezing procedure was not producing any unwanted effects on the experimental signal, i.e. aggregation upon freezing, two field sweeps (normalized) were obtained at 50 K for 250 µM biradical in d 8 toluene with different freezing procedures. The resulting spectra are shown in Fig. S8. The black trace shows the data for a sample that was frozen inside the resonator and the red trace shows data for a sample plunge frozen in liquid nitrogen outside the resonator. There is no significant difference.  Figure S8: Field sweeps for a 0.25 mM solution of the biradical in deuterated toluene, measured at 180 GHz and 6.42 T at 50 K, for a sample frozen inside the resonator (black) and plunge frozen in liquid nitrogen (red).