An analysis of double-quantum coherence ESR in an N-spin system: Analytical expressions and predictions

Electron spin resonance pulsed dipolar spectroscopy (PDS) has become popular in protein 3D structure analysis. PDS studies yield distance distributions between a pair or multiple pairs of spin probes attached to protein molecules, which can be used directly in structural studies or as constraints in theoretical predictions. Double-quantum coherence (DQC) is a highly sensitive and accurate PDS technique to study protein structures in the solid state and under physiologically relevant conditions. In this work, we have derived analytical expressions for the DQC signal for a system with N-dipolar coupled spin-1/2 particles in the solid state. The expressions are integrated over the relevant spatial parameters to obtain closed form DQC signal expressions. These expressions contain the concentration-dependent "instantaneous diffusion" and the background signal. For micromolar and lower concentrations, these effects are negligible. An approximate analysis is provided for cases of finite pulses. The expressions obtained in this work should improve the analysis of DQC experimental data significantly, and the analytical approach could be extended easily to a wide range of magnetic resonance phenomena.

Control of excitation selectivity in pulse EPR on spin-correlated radical pairs with shaped pulses

Spin-correlated radical pairs generated by photoinduced electron transfer are characterised by a distinctive spin polarisation and a unique behaviour in pulse electron paramagnetic resonance (EPR) spectroscopy. Under non-selective excitation, an out-of-phase echo signal modulated by the dipolar and exchange coupling interactions characterising the radical pair is observed and allows extraction of geometric information in the two-pulse out-of-phase electron spin echo envelope modulation (ESEEM) experiment. The investigation of the role of spin-correlated radical pairs in a variety of biological processes and in the fundamental mechanisms underlying device function in optoelectronics, as well as their potential use in quantum information science, relies on the ability to precisely address and manipulate the spins using microwave pulses. Here, we explore the use of shaped pulses for controlled narrowband selective and broadband non-selective excitation of spin-correlated radical pairs in two model donor-bridge-acceptor triads, characterised by different spectral widths, at X- and Q-band frequencies. We demonstrate selective excitation with close to rectangular excitation profiles using BURP (band-selective, uniform response, pure-phase) pulses and complete non-selective excitation of both spins of the radical pair using frequency-swept chirp pulses. The use of frequency-swept pulses in out-of-phase ESEEM experiments enables increased modulation depths and, combined with echo transient detection and Fourier transformation, correlation of the dipolar frequencies with the EPR spectrum and therefore the potential to extract additional information on the donor-acceptor pair geometry.

Intermolecular contributions, filtration effects and signal composition of SIFTER (single-frequency technique for refocusing)

To characterize structure and molecular order in the nanometre range, distances between electron spins and their distributions can be measured via dipolar spin-spin 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 the intermolecular background are predicted in the case of biradicals and in the case of monoradicals. Our theoretical account suggests that the appropriate procedure of extracting the intramolecular dipolar contribution (form factor) requires fitting and subtracting the unmodulated part, followed by division by an intermolecular background function that is different in shape. This scheme differs from the previously used heuristic background division approach. We compare our theoretical derivations 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 for a quantitative analysis of SIFTER data are discussed.

SORDOR pulses: expansion of the Böhlen-Bodenhausen scheme for low-power broadband magnetic resonance

A novel type of efficient broadband pulse, called second-order phase dispersion by optimised rotation (SORDOR), has recently been introduced. In contrast to adiabatic excitation, SORDOR-90 pulses provide effective transverse 90∘ rotations throughout their bandwidth, with a quadratic offset dependence of the phase in the x , y plane. Together with phase-matched SORDOR-180 pulses, this enables the Böhlen-Bodenhausen broadband refocusing approach for linearly frequency-swept pulses to be extended to any type of 90∘ /180∘ pulse-delay sequence. Example pulse shapes are characterised in theory and experiment, and an example application is given with a 19 F -PROJECT experiment for measuring relaxation times with reduced distortions due to J -coupling evolution.

Orientation-Selective and Frequency-Correlated Light-Induced Pulsed Dipolar Spectroscopy

We explore the potential of orientation-resolved pulsed dipolar spectroscopy (PDS) in light-induced versions of the experiment. The use of triplets as spin-active moieties for PDS offers an attractive tool for studying biochemical systems containing optically active cofactors. Cofactors are often rigidly bound within the protein structure, providing an accurate positional marker. The rigidity leads to orientation selection effects in PDS, which can be analyzed to give both distance and mutual orientation information. Herein we present a comprehensive analysis of the orientation selection of a full set of light-induced PDS experiments. We exploit the complementary information provided by the different light-induced techniques to yield atomic-level structural information. For the first time, we measure a 2D frequency-correlated laser-induced magnetic dipolar spectrum, and we are able to monitor the complete orientation dependence of the system in a single experiment. Alternatively, the summed spectrum enables an orientation-independent analysis to determine the distance distribution.

Chirped ordered pulses for ultra-broadband ESR spectroscopy

Recently, applications of swept-frequency pulses proved to be a useful approach to circumvent the problem of limited excitation bandwidth in pulsed ESR posed by conventional pulses. Here, we present a chirped excitation sequence, CHirped ORdered pulses for Ultra-broadband Spectroscopy (CHORUS), for ultra-broadband ESR spectroscopy. It will be demonstrated that the application of this sequence can address the problems of excitation non-uniformity and sensitivity to instrumental instabilities to a greater extent compared to the current state of the art. This sequence is highly promising for finding applications beyond single excitation in many ESR experiments. Theoretical and experimental results for the proposed method are presented along with calibration strategies for experimental implementation.

Spin dynamics during chirped pulses: applications to homonuclear decoupling and broadband excitation

Swept-frequency pulses have found applications in a wide range of areas including spectroscopic techniques where efficient control of spins is required. For many of these applications, a good understanding of the evolution of spin systems during these pulses plays a vital role, not only in describing the mechanism of techniques, but also in enabling new methodologies. In magnetic resonance spectroscopy, broadband inversion, refocusing, and excitation using these pulses are among the most used applications in NMR, ESR, MRI, and in vivo MRS. In the present survey, a general expression for chirped pulses will be introduced, and some numerical approaches to calculate the spin dynamics during chirped pulses via solutions of the well-known Liouville-von Neumann equation and the lesser-explored Wei-Norman Lie algebra along with comprehensive examples are presented. In both cases, spin state trajectories are calculated using the solution of differential equations. Additionally, applications of the proposed methods to study the spin dynamics during the PSYCHE pulse element for broadband homonuclear decoupling and the CHORUS sequence for broadband excitation will be presented.

Optimal background treatment in dipolar spectroscopy

Treatment of the background in dipolar EPR spectroscopy signals is a critical processing step for the recovery of the underlying distance distributions. Here we present new mathematical considerations that show pitfalls of background subtraction and division. In order to overcome these problems we propose an improved background treatment approach. We show, empirically, that this new method outperforms the established ones and analyze the established practice of post-correction signal truncation, as well as the influence of moderate background-fit errors, on accuracy of distance distributions.

Accessing distributions of exchange and dipolar couplings in stiff molecular rulers with Cu(ii) centres

Novel approaches to quantitatively analyse distributed exchange couplings are described and tested on experimental data sets for stiff synthetic molecules.

UWB DEER and RIDME distance measurements in Cu(II)-Cu(II) spin pairs

Distance determination by Electron Paramagnetic Resonance (EPR) based on measurements of the dipolar coupling are technically challenging for electron spin systems with broad spectra due to comparatively narrow microwave pulse excitation bandwidths. With Na₄[{Cu^II(PyMTA)}-(stiff spacer)-{Cu^II(PyMTA)}] as a model compound, we compared DEER and RIDME measurements and investigated the use of frequency-swept pulses. We found very large improvements in sensitivity when substituting the monochromatic pump pulse by a frequency-swept one in DEER experiments with monochromatic observer pulses. This effect was especially strong in X band, where nearly the whole spectrum can be included in the experiment. The RIDME experiment is characterised by a trade-off in signal intensity and modulation depth. Optimal parameters are further influenced by varying steepness of the background decay. A simple 2-point optimization experiment was found to serve as good estimate to identify the mixing time of highest sensitivity. Using frequency-swept pulses in the observer sequences resulted in lower SNR in both the RIDME and the DEER experiment. Orientation selectivity was found to vary in both experiments with the detection position as well as with the settings of the pump pulse in DEER. In RIDME, orientation selection by relaxation anisotropy of the inverted spin appeared to be negligible as form factors remain relatively constant with varying mixing time. This reduces the overall observed orientation selection to the one given by the detection position. Field-averaged data from RIDME and DEER with a shaped pump pulse resulted in the same dipolar spectrum. We found that both methods have their advantages and disadvantages for given instrumental limitations and sample properties. Thus the choice of method depends on the situation at hand and we discuss which parameters should be considered for optimization.

Shaping EPR: Phase and amplitude modulated microwave pulses

The advent of fast arbitrary waveform generators in the sub-nanosecond time regime recently enabled new experimental developments in the field of pulsed EPR. In this article, the new possibilities of such fast phase/amplitude modulated microwave pulses are shortly described with respect to applications in pulsed dipolar spectroscopy. Some of the specific challenges of an accurate creation of such pulses in the field of EPR are outlined. Finally, a short outlook of potential applications is given and some specific experimental conditions are discussed, where shaped pulses might have an especially important impact in the future.

Trityl-Aryl-Nitroxide-Based Genuinely g-Engineered Biradicals, As Studied by Dynamic Nuclear Polarization, Multifrequency ESR/ENDOR, Arbitrary Wave Generator Pulse Microwave Waveform Spectroscopy, and Quantum Chemical Calculations

Trityl and nitroxide radicals are connected by π-topologically controlled aryl linkers, generating genuinely g-engineered biradicals. They serve as a typical model for biradicals in which the exchange (J) and hyperfine interactions compete with the g-difference electronic Zeeman interactions. The magnetic properties underlying the biradical spin Hamiltonian for solution, including J's, have been determined by multifrequency CW-ESR and ¹H ENDOR spectroscopy and compared with those obtained by quantum chemical calculations. The experimental J values were in good agreement with the quantum chemical calculations. The g-engineered biradicals have been tested as a prototype for AWG (Arbitrary Wave Generator)-based spin manipulation techniques, which enable GRAPE (GRAdient Pulse Engineering) microwave control of spins in molecular magnetic resonance spectroscopy for use in molecular spin quantum computers, demonstrating efficient signal enhancement of specific weakened hyperfine signals. Dynamic nuclear polarization (DNP) effects of the biradicals for 400 MHz nuclear magnetic resonance signal enhancement have been examined, giving efficiency factors of 30 for ¹H and 27.8 for ¹³C nuclei. The marked DNP results show the feasibility of these biradicals for hyperpolarization.

Improving the accuracy of Cu(ii)–nitroxide RIDME in the presence of orientation correlation in water-soluble Cu(ii)–nitroxide rulers

Detailed analysis of artefacts in the Cu(ii)–nitroxide RIDME experiments, related to orientation averaging, echo-crossing, ESEEM and background-correction is presented.

Refocused Out-Of-Phase (ROOPh) DEER: A pulse scheme for suppressing an unmodulated background in double electron-electron resonance experiments

EPR pulsed dipolar spectroscopy (PDS) is indispensable for measurements of nm-scale distances between electronic spins in biological and other systems. While several useful modifications and pulse sequences for PDS have been developed in recent years, DEER experiments utilizing pump and observer pulses at two different frequencies remain the most popular for practical applications. One of the major drawbacks of all the available DEER approaches is the presence of a significant unmodulated fraction in the detected signal that arises from an incomplete inversion of the coupled spins by the pump pulse. The latter fraction is perceived as one of the major sources of error for the reconstructed distance distributions. We describe an alternative detection scheme - a Refocused Out-Of-Phase DEER (ROOPh-DEER) - to acquire only the modulated fraction of the dipolar DEER signal. When Zeeman splitting is small compared to the temperature, the out-of-phase magnetization components cancel each other and are not observed in 4-pulse DEER experiment. In ROOPh-DEER these components are refocused by an additional pump pulse while the in-phase component containing an unmodulated background is filtered out by a pulse at the observed frequency applied right at the position of the refocused echo. Experimental implementation of the ROOPh-DEER detection scheme requires at least three additional pulses as was demonstrated on an example of a 7-pulse sequence. The application of 7-pulse ROOPh-DEER sequence to a model biradical yielded the interspin distance of 1.94 ± 0.07 nm identical to the one obtained with the conventional 4-pulse DEER, however, without the unmodulated background present as a dominant fraction in the latter signal.

Artefact suppression in 5-pulse double electron electron resonance for distance distribution measurements

A 5-pulse version of the Double Electron Electron Resonance (DEER) experiment with Carr-Purcell delays and an additional pump pulse has been shown to significantly extend the experimentally accessible distance range in cases where nuclear spin diffusion dominates electron spin phase memory loss [Borbat et al., J. Phys. Chem. Lett., 2013, 4, 170]. We show that the sequence also prolongs coherence decay for spin labels in or near lipid bilayers, where this decay is mono-exponential. Compared to 4-pulse DEER, 5-pulse DEER suffers from additional artefacts that stem from pulse imperfection and excitation band overlap. Only some of these artefacts can be suppressed by phase cycling and the remaining ones have hindered widespread utilization of the method. Here, we report previously unknown additional artefact contributions stemming from overlap between the excitation bands of the microwave pulses that introduce additional dipolar evolution pathways. Experimental conditions are analyzed in detail that suppress these as well as the already known artefacts. Such suppression results in data that contain at most the partial excitation artefact, which can be deliberately shifted in time by a change in pulse timing without affecting the wanted contribution.

Double electron-electron resonance with multiple non-selective chirp refocusing

A new approach to double electron-electron resonance (DEER) for distance determination involving nitroxide spin labels at dilute concentrations is presented. In general, DEER pulse sequences rely on double resonance between pump and observer spins excited by selective pulses at two distinct microwave frequencies. In the new approach abbreviated as nDEER, non-selective chirp pulses that refocus all relevant spin pairs are combined with DEER. This non-selective refocusing results in suppression of unmodulated contributions, such as the constant contribution as well as the background curvature due to inter-molecular spin partners in ordinary DEER data. Due to this dipolar attenuation effect, primary nDEER data are closer to the dipolar modulation of primary interest than ordinary DEER data. Restrictions of nDEER are that secondary information related to these unmodulated contributions becomes difficult to retrieve. Accordingly, incomplete deconvolution of the inter-molecular background prevents the application of nDEER to rigid spin pairs at high concentrations. A key advantage of nDEER is the high fidelity of the chirp refocusing pulses, which is important for nDEER schemes that incorporate dynamical decoupling to access longer distances. In this context, nDEER with Carr-Purcell (CP) pulse trains having N = 2 and N = 4 refocusing pulses are demonstrated. These CP nDEER sequences require a total of N + 2 pulses, which is less than the 2N + 1 pulses required for CP DEER schemes. The pump pulse position is incremented throughout the refocusing pulses, which restricts the minimum time increment to 96 ns on our spectrometer and therefore complicates application to distances below 3 nm. At Q-band frequencies, unwanted modulations related to pulse imperfections contribute only 3.5% relative to the principal nDEER modulation. Accordingly, there is no need for dedicated data reconstruction methods as in CP DEER methods.

High-Resolution Quantum Sensing with Shaped Control Pulses

We investigate the application of amplitude-shaped control pulses for enhancing the time and frequency resolution of multipulse quantum sensing sequences. Using the electronic spin of a single nitrogen-vacancy center in diamond and up to 10 000 coherent microwave pulses with a cosine square envelope, we demonstrate 0.6-ps timing resolution for the interpulse delay. This represents a refinement by over 3 orders of magnitude compared to the 2-ns hardware sampling. We apply the method for the detection of external ac magnetic fields and nuclear magnetic resonance signals of ^{13}C spins with high spectral resolution. Our method is simple to implement and especially useful for quantum applications that require fast phase gates, many control pulses, and high fidelity.

Pulsed triple electron resonance (TRIER) for dipolar correlation spectroscopy

A new pulse sequence is presented for correlating dipolar frequencies in molecules with more than two paramagnetic centers. This triple electron resonance experiment (TRIER) is an extension the double electron-electron resonance (DEER) experiment, which is widely used for distance determination in the nanometer range. We use linear chirp pulses with smoothed edges to create a refocused observer echo, and two hyperbolic secant pulses with distinct excitation windows to excite two other subsets of spins. These pumped spins are coupled to the observed spin through the dipole-dipole interaction. A two-dimensional dipolar modulation pattern is recorded by variation of the position of the two pump pulses. By two-dimensional Fourier transform of the echo integral, a plot is obtained that correlates dipolar frequencies within the same molecule. Such correlation patterns can be used in conjunction with DEER, with which distance distributions are usually determined for several doubly labeled molecules with different spin-labeling sites. In the presence of two conformers, DEER traces give two distances and assignment to an individual conformer is not trivial and usually requires a trial and error approach. TRIER can potentially provide the missing connection between distances as correlations between dipolar frequencies.

Monitoring Complex Formation by Relaxation-Induced Pulse Electron Paramagnetic Resonance Distance Measurements

Biomolecular complexes are often multimers fueling the demand for methods that allow unraveling their composition and geometric arrangement. Pulse electron paramagnetic resonance (EPR) spectroscopy is increasingly applied for retrieving geometric information on the nanometer scale. The emerging RIDME (relaxation‐induced dipolar modulation enhancement) technique offers improved sensitivity in distance experiments involving metal centers (e.g. on metalloproteins or proteins labelled with metal ions). Here, a mixture of a spin labelled ligand with increasing amounts of paramagnetic Cu^II ions allowed accurate quantification of ligand‐metal binding in the model complex formed. The distance measurement was highly accurate and critical aspects for identifying multimerization could be identified. The potential to quantify binding in addition to the high‐precision distance measurement will further increase the scope of EPR applications.

Wideband frequency-swept excitation in pulsed EPR spectroscopy

Excitation of electron spins with monochromatic rectangular pulses is limited to bandwidths that are smaller than the spectral widths of most organic radicals and much smaller than the spectral widths of transition and rare earth metal ions. With frequency-swept pulses, bandwidths of up to 800MHz have previously been attained for excitation and detection of spin packets at frequencies of about 9.6GHz and bandwidths of up to 2.5GHz in a polarization transfer experiment at frequencies of about 34GHz. The remaining limitations, mainly due to resonator bandwidth and due to pulse length restrictions are discussed. Flip angles for state-space rotations on passage of a transition can generally be computed from the critical adiabaticity by the Landau-Zener-Stückelberg-Majorana expression. For hyperbolic secant pulses, the Demkov-Kunike model describes excitation for spin packets within and outside the sweep range. Well within the sweep range, the Bloch-Siegert phase shift is proportional to critical adiabaticity to a very good approximation. Because of the dependence of both flip angle and coherence phase on critical adiabaticity, it is advantageous to use pairs of amplitude and frequency modulation functions that provide such offset-independent adiabaticity. Compensation for the resonator response function should restore offset-independent adiabaticity. Whereas resonance offsets and Bloch-Siegert phase can be refocused at certain pulse length ratios, phase dispersion in coupled spin systems cannot generally be refocused. Based on the bandwidth limitations that arise from spin dynamics, requirements are derived for a spectrometer that achieves precise spin control over wide bands. The design of such a spectrometer and hardware characterization by EPR experiments are discussed.

Perspectives of shaped pulses for EPR spectroscopy

This article describes current uses of shaped pulses, generated by an arbitrary waveform generator, in the field of EPR spectroscopy. We show applications of sech/tanh and WURST pulses to dipolar spectroscopy, including new pulse schemes and procedures, and discuss the more general concept of optimum-control-based pulses for applications in EPR spectroscopy. The article also describes a procedure to correct for experimental imperfections, mostly introduced by the microwave resonator, and discusses further potential applications and limitations of such pulses.

The use of composite pulses for improving DEER signal at 94GHz

The sensitivity of pulsed electron paramagnetic resonance (EPR) measurements on broad-line paramagnetic centers is often limited by the available excitation bandwidth. One way to increase excitation bandwidth is through the use of chirp or composite pulses. However, performance can be limited by cavity or detection bandwidth, which in commercial systems is typically 100-200MHz. Here we demonstrate in a 94GHz spectrometer, with >800MHz system bandwidth, an increase in signal and modulation depth in a 4-pulse DEER experiment through use of composite rather than rectangular π pulses. We show that this leads to an increase in sensitivity by a factor of 3, in line with theoretical predictions, although gains are more limited in nitroxide-nitroxide DEER measurements.

Reliable nanometre-range distance distributions from 5-pulse double electron electron resonance

The partial excitation artefact in 5-pulse DEER data can be eliminated by experimental time shifting and signal processing.

CIDME: Short distances measured with long chirp pulses

Frequency-swept pulses have recently been introduced as pump pulses into double electron-electron resonance (DEER) experiments. A limitation of this approach is that the pump pulses need to be short in comparison to dipolar evolution periods. The "chirp-induced dipolar modulation enhancement" (CIDME) pulse sequence introduced in this work circumvents this limitation by means of longitudinal storage during the application of one single or two consecutive pump pulses. The resulting six-pulse sequence is closely related to the five-pulse "relaxation-induced dipolar modulation enhancement" (RIDME) pulse sequence: While dipolar modulation in RIDME is due to stochastic spin flips during longitudinal storage, modulation in CIDME is due to the pump pulse during longitudinal storage. Experimentally, CIDME is examined for Gd-Gd and nitroxide-nitroxide distance determination using a high-power Q-band spectrometer. Since longitudinal storage results in a 50% signal loss, comparisons between DEER using short chirp pump pulses of 64ns duration and CIDME using longer pump pulses are in favor of DEER. While the lower sensitivity restrains the applicability of CIDME for routine distance determination on high-power spectrometers, this result is not to be generalized to spectrometers having lower power and to specialized "non-routine" applications or different types of spin labels. In particular, the advantage of prolonged CIDME pump pulses is demonstrated for experiments at large frequency offset between the pumped and observed spins. At a frequency separation of 1GHz, where broadening due to dipolar pseudo-secular contributions becomes largely suppressed, a Gd-Gd modulation depth larger than 10% is achieved. Moreover, a CIDME experiment at deliberately reduced power underlines the potential of the new technique for spectrometers with lower power, as often encountered at higher microwave frequencies. With longitudinal storage times T below 10μs, however, CIDME appears rather susceptible to artifacts. For nitroxide-nitroxide experiments, these currently inhibit a faithful data analysis. To facilitate further developments, the artifacts are characterized experimentally. In addition, effects that are specific to the high spin of S=7/2 Gd-centers are examined. Herein, population transfer within the observer spin's multiplet due to the pump pulse as well as excitation of dipolar harmonics are discussed.