Coupled Two-spin System Research Articles

An analytical treatment of electron spectral saturation for dynamic nuclear polarization NMR of rotating solids.

Saturation of electron magnetization by microwave irradiation under magic-angle spinning (MAS) is studied theoretically. The saturation is essential for dynamic nuclear polarization (DNP) enhancement of nuclear magnetic resonance signals. For a spin with a large g-anisotropy and a long T1 relative to the rotor period, the sample rotation distributes saturation to the whole powder sample spectrum. Analytical expressions for the saturation and frequency profiles are obtained. For a pair of coupled electrons such as those in bis-nitroxides, which are commonly used for MAS DNP, an el-er model (where el and er stand for electrons on the left and the right, respectively, in their spectral positions) is introduced to simplify the analysis of a coupled two-spin system under MAS. For such a system, strong electron couplings exchange magnetization during dipolar/J rotor events when the two electron frequencies cross each other. The exchange is equivalent to a swap of the el and er electrons. This allows for the treatment of a coupled spin pair as two independent spins such that an analytical solution can be obtained for the steady-state magnetization and the difference between the two electrons. The theoretical study with its analytical result provides a simple physical picture of electron saturation under MAS and of how radical properties and experimental parameters affect cross-effect DNP. The effects of depolarization and the extension to more coupled electron spins are also discussed using this approach.

Experimental observation of saddle points over the quantum control landscape of a two-spin system

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have predicted the existence of critical points over the landscapes, including saddle points with indefinite Hessians. This paper presents a systematic experimental study of quantum control landscape saddle points. Nuclear magnetic resonance control experiments are performed on a coupled two-spin system in a $^{13}\mathrm{C}$-labeled chloroform $({}^{13}{\mathrm{CHCl}}_{3})$ sample. We address the saddles with a combined theoretical and experimental approach, measure the Hessian at each identified saddle point, and study how their presence can influence the search effort utilizing a gradient algorithm to seek an optimal control outcome. The results have significance beyond spin systems, as landscape saddles are expected to be present for the control of broad classes of quantum systems.

Dynamics of a resonantly driven two-spin system

Dynamics of a coupled two-spin system in a static magnetic field are investigated. An analysis is presented of resonance transitions driven by a circularly polarized radio-frequency (RF) field orthogonal to the static field. When the RF field amplitude is modulated at a certain frequency depending on the field strength, the system exhibits parametric resonance behavior. The periodicity of transitions breaks down, and the Shannon entropy of the recurrence probability density for the system’s states increases by more than an order of magnitude.

Control of a coupled two-spin system without hard pulses

Constructive techniques for preparing a specific unitary generator from a coupled two-spin system, via bounded amplitude radio frequency selective single spin pulses, are presented. The frequency of any pulse in the sequence is one of the two Larmor frequencies, while the phase takes one of two values. The fact that the amplitude is bounded implies that the Larmor frequency separation between the two spins need not be large to maintain selectivity. In addition, the contribution of the J coupling term to single spin evolution is not neglected, but instead plays a crucial role. The procedure is based on a certain decomposition of SU(4) available in the literature. A method for determining the parameters entering this factorization, in terms of the entries of the target unitary generator, is also provided.

Separating structure and dynamics in CSA/DD cross-correlated relaxation: a case study on trehalose and ubiquitin.

We present two new sensitivity enhanced gradient NMR experiments for measuring interference effects between chemical shift anisotropy (CSA) and dipolar coupling interactions in a scalar coupled two-spin system in both the laboratory and rotating frames. We apply these methods for quantitative measurement of longitudinal and transverse cross-correlation rates involving interference of (13)C CSA and (13)C-(1)H dipolar coupling in a disaccharide, alpha,alpha-D-trehalose, at natural abundance of (13)C as well as interference of amide (15)N CSA and (15)N-(1)H dipolar coupling in uniformly (15)N-labeled ubiquitin. We demonstrate that the standard heteronuclear T(1), T(2), and steady-state NOE autocorrelation experiments augmented by cross-correlation measurements provide sufficient experimental data to quantitatively separate the structural and dynamic contributions to these relaxation rates when the simplifying assumptions of isotropic overall tumbling and an axially symmetric chemical shift tensor are valid.

Efficient scalar spin relaxation in the rotating frame for matched radio-frequency fields

A theoretical analysis is presented for liquid-state T1ρ relaxation in a coupled two-spin system I=1/2, S=1 in the presence of two radio-frequency fields applied to each of the spins individually. It is demonstrated that the relaxation rate constant T1ρ−1 of the spin I due to scalar relaxation sharply increases when the two radio-frequency fields are matched according to the Hartmann–Hahn condition. Relaxation measurements on the amino-protons of 3-nitroaniline show good agreement with theory.

Response of a coupled two-spin system to on-resonance amplitude modulated RF pulses

A weakly scalar-coupled two-spin system subjected to two amplitude modulated RF pulses on exact resonance is treated by means of the rotation operator approach. The theory presented here enables coherence evolution to be evaluated by the routine procedure and to be expressed in analytical form. The evolution behaviour from the equilibrium state is discussed in some detail. It is shown that the application of rotation matrix and quaternion elements clarifies evolution expressions. The numerical calculation is performed by way of quaternions. Examples of BURP (band-selective, uniform response, purephase) and sinc-shaped RF pulses are given and the case of time-symmetrical RF pulses is analysed further.

Response of a coupled two-spin system to on-resonance amplitude modulated RF pulses

Response of a coupled two-spin system to on-resonance amplitude modulated RF pulses

Rapid recording of solvent-suppressed 2D COSY spectra with inherent quadrature detection using pulsed field gradients

Multidimensional NMR experiments have drawn much attention, particularly due to their ability to elucidate the structure of complex mo lecules. One of the ma jor drawbacks of many mu ltidimensional experiments is the long time required to record a dataset. In many cases, the m inimal experimental time is not lim ited by sensitivity, but rather by the number of repetitions needed to complete a phase-cycl ing protocol. Phase cycling may be necessary to select the desired coherence pathway ( 2 ) , to obtain absorptive l ineshapes (2-4, and to eliminate undesired signals such as axial peaks or quadrature artifacts (5). In order to reduce experimental time, efforts have been undertaken to design reduced phase-cycl ing schemes (6, 7). Spectral artifacts may also appear if the spin system is not in a constant state at the beginning of each pulse sequence, for example, in the case of a fast repetition rate. This problem may be solved by the use of a saturation-recovery sequence (8). Coherence pathways can also be selected in a single scan with the application of field gradients (9-11). Sotak et al. ( 12) have applied this principle to obtain in vivo 2D double-quantum spectra, and Hurd recently reported gradient-enhanced doublequantum-fi ltered COSY and gradient-enhanced TOCSY ( 13). In this contribution, we discuss how standard COSY spectra can be recorded without phase cycling if pulsed field gradients are available. Furthermore, it will be shown that quadrature detection in thef; doma in is attained inherently. F inally, solvent suppression is accomplished by using an alternative pulse sequence, and an improved deconvolut ion method for postprocessing. A similar approach, applied to in vivo 2D spectroscopy, has independently been reported by Z iegler et al. ( 14). F igure 1 a shows the two-pulse sequence for the basic COSY experiment (IS). Using the product-operator formalism to describe the evolution of a weakly coupled twospin system, the observable terms of the density operator after the second 90” pulse at the beginning of the detection period t2 are (16)

Diagonal-peak patterns in double-quantum-filtered COSY

Two-dimensional correlation spectroscopy with double-quantum filtration (DQCOSY) is now widely used by NMR spectroscopists because it facilitates examination of cros peaks close to the diagonal (1, 2). For a coupled two-spin system, the double-quantum-filtered spectrum resembles the conventional COSY spectrum except that the diagonal peaks are in the (antiphase) absorption mode and responses from uncoupled spins have been suppressed

Experimental design optimization and confidence interval estimation in multiexponential NMR relaxation measurements

Numerical simulations were used to estimate the precision of parameters responsible for NMR spin-lattice relaxation in a coupled two-spin system where two relaxation mechanisms, dipole-dipole and chemical shift anisotropy, are present. Confidence intervals in nonlinear least-squares fitting, necessary for multiexponential relaxation, can be obtained with restrictive assumptions. The simulations allow one to investigate the effect of experimental variables on parameter precision without performing many lengthy experiments. Precision depends upon may factors including the nucleus observed, how the recovery curves are sampled, the sample temperature stability, and the number of simultaneously estimated parameters.

Effects of thermal denaturation on the longitudinal relaxation time ( T1) of water protons in protein solutions: Study of the factors determining the T1 of water protons

The factors determining the longitudinal relaxation time ( T1) of water protons in protein solutions were investigated by analyzing the effects of thermal denaturation on the T1 of the water protons. We treated the water protons and the protein protons “on a protein surface” as a dipole-dipole coupled two-spin system where relative translational diffusion is the dominant mechanism, and measured the change in the time development of the nuclear Overhauser effect (NOE) factors of the water protons. The T1 of the water protons was shortened markedly when the proteins were thermally denatured. Our analysis indicates that this relaxation enhancement is due to an increase in the value of the translational correlation time as well as the fraction of hydration water molecules, though the influence of “proton exchange” between the water protons and the labile protein protons cannot be completely neglected.

Spin-lattice relaxation of ligand nuclei in slowly reorienting paramagnetic complexes in the electronic doublet spin state ( [formula omitted]). A theoretical approach for strongly coupled two-spin systems

In this paper a general theory for treating the spin-lattice relaxation of a ligand nucleus (denoted by I) is derived for a metal complex in a doublet electron spin state ( S = 1 2 ). The dipole-dipole SI interaction is treated for the case where the electron spin is also strongly coupled to the metal nucleus K. The SK interaction considered here is the hyperfine coupling, both scalar (SC) and dipolar (DD). The present theory is valid for slowly reorienting complexes in solution and can, furthermore, incorporate relaxation effects of the electron spin S, and the metal nucleus K due to processes which are faster than, and independent of, reorientation, i.e., for processes that fulfil the strong narrowing conditions. The effects of chemical exchange of the ligands and of anisotropic reorientation of the complex are also studied. Together with our previous studies of paramagnetic complexes with electron spin S ≧ 1, that have been recently reviewed by J. Kowalewski, L. Nordenskiöld, N. Benetis, and P. O. Westlund, ( Prog. NMR Spectrosc. 17, 141 (1985)), the present work completes the elementary relaxation features of ligand nuclei of metal complexes in the slow motional regime. The present theory is shown to be more general than the theory of Bertini and co-workers ( J. Magn. Reson. 59 , 213 (1984)), which can be obtained as a limit of the present approach by decoupling the reorientation from the motions of the S-K two spin system. The treatment of a strongly coupled two-spin system is emphasized since it provides a necessary step to the treatment of the relaxation of paramagnetic doublets.

Longitudinal relaxation in a homonuclear coupled two-spin system. The case of ( 13C, 2H) glycine

The longitudinal relaxation of the two carbon-13 nuclei of doubly enriched glycine ( 13C and 2H) has been investigated by the conventional nonselective [180°-τ-90°] pulse sequence. Some approximations are proposed which lead to a straightforward interpretation of the experimental results, a refined analysis being carried out by means of a general computer program previously described. These results together with those of other isotopomers (normal glycine NH 2CH 2000H and NH 2 12CH 2 13COOD) allow the determination of random field rates at the two carbons and of the spectral densities relative to the carbon-carbon and methylene carbon-proton dipolar interactions. These spectral densities yield the following correlation times: τ CC = 17 ± 4 psec and τ CH = 5.5 ± 0.3 psec. In addition, the effect of a dipolar chemical shift anisotropy interference term has been observed by applying the sequence [180°-τ-30°]. Its sign is in agreement with a positive carbon-carbon indirect coupling and a positive carbonyl chemical shift anisotropy.

NMR Spin-Echo Trains for a Coupled Two-Spin System

Abstract : The effect of pulse separation on NMR CarrPurcell spin echo trains is considered for pulse experiments on coupled pairs of spin-1/2 nuclei in the liquid phase. Previous discussions have assumed that in the homonuclear weakly coupled AX limit the modulation of the echo amplitude envelope depends only on the magnitude of the spin coupling and is independent of the pulse separation. Theoretical and experimental results are presented which show that in both AB and AX systems the modulation decreases in frequency and effectively disappears as the rf pulse separation decreases. (Author)

Nuclear Magnetic Double-Resonance Spectrum of a Strongly Coupled Two-Spin System

The density-matrix theory of double resonance is applied for a detailed analysis of the frequency-sweep nuclear magnetic double-resonance spectra of the strongly coupled two-proton system in 2-bromo-5-chlorothiophene under various conditions of irradiation. These spectra show significant relaxation and coherence effects. The results of the theory are very sensitive to the relaxation mechanisms assumed for the system, indicating the possibility of deriving significant information about the relaxation of the system by this method. Comparison of the observed and theoretical double-resonance spectra shows that the proton relaxation in this molecule is due, in large measure, to external interactions with a high degree of correlation between the interactions with the two protons. An approximation proposed by Bloch for simplifying the calculations is found to be applicable even when the irradiation frequency is close to some single-resonance transition frequencies.