Zero quantum

The transitions between energy levels in an AX spin system are shown in Fig. 1.44. There are four single-quantum transitions (these are the normal transitions A, A, Xi, and X2 in which changes in quantum number of 1 occur), one double-quantum transition 1% between the aa and j8 8 states involving a change in quantum number of 2, and a zero-quantum transition 1% between the a)3 and fia states in which no change in quantum number occurs. The double-quantum and zero-quantum transitions are not allowed as excitation processes under the quantum mechanical selection rules, but their involvement may be considered in relaxation processes. 

Figure 1.44 Transitions between various energy levels of an AX spin system. A, and Aj represent the single-quantum relaxations of nucleus A, while Xi and Xj represent the single-quantum relaxations of nucleus X. W2 and are double- and zero-quantum transitions, respectively.

Since the equilibrium state has been disturbed, the system tries to restore equilibrium. For this it can use as the predominant relaxation pathways the double-quantum process (in fast-tumbling, smaller molecules), leading to a positive nOe, or the zero-quantum process 1% (in slower-tumbling macromolecules), leading to a negative nOe. 

Is it possible to predict the predominant mode of relaxation (zero-quantum or double-quantum) by observing the sign of nOe (negative or positive) ... 

Why do we need to involve zero-quantum (VI()) or double-quantum Wi) processes to explain the origin of the nuclear Overhauser enhancement ... 

The positive nOe observed in small molecules in nonviscous solution is mainly due to double-quantum relaxation, whereas the negative nOe observed for macromolecules in viscous solution is due to the predominance of the zero-quantum 1% cross-relaxation pathway. 

When the zero-quantum 1% transition is greater than double-quantum Wi, the nOe enhancements will be negative. Similarly, when is greater than 1%, the resultant nOe will have a positive sign. The predominance of Wi and over one another depends on the molecular motion. It is known that the Wo transition is maximal when the molecule tumbles at a rate of about 1 KHz, while the Wi transition is fastest at a tumbling rate of about 800 MHz. On this basis, a rough idea of the sign of nOe can be obtained. For example, small molecules in nonviscous... 

Coherence A condition in which nuclei precess with a given phase relationship and can exchange spin states via transitions between two eigenstates. Coherence may be zero-quantum, single-quantum, double-quantum, etc., depending on the AM of the transition corresponding to the coherence. Only single-quantum coherence can be detected directly. 

Zero-filling A procedure used to improve the digital resolution of the transformed spectrum (e.g., in the tj domain of a 2D spectrum) by adding zeros to the FID so that the size of the data set is adjusted to a power of 2. Zero-quantum coherence The coherence between states with the same quantum number. It is not observable directly. 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

Vega, and coworkers as a sequence of well-placed ideal (i.e., infinitely strong) 7r-pulses serving to reintroduce the zero-quantum (ZQ) part of the homonuclear dipole-dipole coupling operator in a homonuclear two-spin system. The coherent averaging due to MAS is in the toggling frame of the n-pulses partially disrupted by a differential chemical shift term and thereby recoupling takes place. 

Here, we present the example of the trans hydrogen bond coupling between the C of the acceptor and the N of the donor h/(N, C ) that is measured by excitation of double-quantum and zero-quantum coherence between the HN and the C nuclei [12] in a protein. Thus, the double-quantum coherence is split by h /(N, C )+ /(N, H) while the zero-quantum coher-... 

HN(CO) experiment. In this experiment doublequantum and zero-quantum coherence between Hn and the C bound to the proton via a hydro-... 

The cross-correlated relaxation rate observed for double- or zero-quantum coherence involving A1 or A2 and B1 or B2 for two dipolar interactions therefore takes the following form ... 

To illustrate how cross-correlated relaxation can be used to measure the angle between two bond vectors, we will use the example of the generation of double and zero quantum coherence between spins A1 and B1 and call the angle between the Ax-A2 and B1-B2 vectors 8 (Fig. 7.18). 

Fig. 7.18 Schematic representation of cross-corre- two involved internuclear vectors, the differential lated relaxation of double and zero quantum co- relaxation affects the multiplet in the given way. herences. Depending on the relative angle of the...

The method relies on the measurement of cross-correlated relaxation rates in a constant time period such that the cross-correlated relaxation rate evolves during a fixed time r. In order to resolve the cross-correlated relaxation rate, however, the couplings need to evolve during an evolution time, e.g. tt. The first pulse sequence published for the measurement of the cross-correlated relaxation rate between the HNn and the Ca j,Ha i vector relied on an HN(CO)CA experiment, in which the Ca chemical shift evolution period was replaced by evolution of 15N,13C double and zero quantum coherences (Fig. 7.20). 

Therein, cross-correlated relaxation T qHj c h °f the double and zero quantum coherence (DQ/ZQ) 4HizCixCjj generated at time point a creates the DQ/ZQ operator 4HjzCjJCiy. In the second part of the experiment, the operator 4HJZCjxQy is transferred via a 90° y-pulse applied to 13C nuclei to give rise to a cross peak at an(i... 

As an example of the measurement of cross-correlated relaxation between CSA and dipolar couplings, we choose the J-resolved constant time experiment [30] (Fig. 7.26 a) that measures the cross-correlated relaxation of 1H,13C-dipolar coupling and 31P-chemical shift anisotropy to determine the phosphodiester backbone angles a and in RNA. Since 31P is not bound to NMR-active nuclei, NOE information for the backbone of RNA is sparse, and vicinal scalar coupling constants cannot be exploited. The cross-correlated relaxation rates can be obtained from the relative scaling (shown schematically in Fig. 7.19d) of the two submultiplet intensities derived from an H-coupled constant time spectrum of 13C,31P double- and zero-quantum coherence [DQC (double-quantum coherence) and ZQC (zero-quantum coherence), respectively]. These traces are shown in Fig. 7.26c. The desired cross-correlated relaxation rate can be extracted from the intensities of the cross peaks according to ... 

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