Two-spin approximation

TABLE 1 Second moments (in rad s ) as calculated from simulations of the CT-VPP-REDOR data in Figure lOB and C employing the two-spin approximation and the full five-spin calculation... 

Two-spin approximation Full five-spin calculation ... 

CT-VPP-REDOR) or the pulse duration fp (CT-VPD-REDOR) then produces CT-REDOR curves, from which the second moment may be evaluated with distinctively superior accuracy as compared to the values obtained from a parabolic fit to the conventional REDOR data. When restricting the experiment to short dipolar evolution times, the two-spin approximation may be applied for the data analysis, which proves to be especially attractive for amorphous solids, for which the exact spin geometry is unknovm. The data presented on the model compoimds illustrate the various facets of CT-REDOR NMR spectroscopy. First application examples, namely, the evaluation of the heteronuclear Li-Ti dipolar couplings within the garnet structure of Li5La3Nb20i2, the determi-nation of the intemuclear B- P distance in frustrated Lewis pairs, the analysis of Na- F dipolar interaction in fluormica or Na- P... 

Non-selective Ti and NOE values in bovine Q12C02SOD at 200 MHz [16]. The distances calculated from the NOE values using the two-spin approximation agree with available X-ray distances... 

Fig. 12. Coherence-transfer functions under ideal isotropic-mixing conditions for a system of three coupled spins, which represents a typical H spin system of serine in DjO, with = 4.5 Hz, Jafi = 9 Hz, and = -15 Hz. In the simulations represented by the solid lines, the two-spin approximation is assumed to be valid. The dashed curves result when the exact three-spin case is calculated. The a-/3 transfer function demonstrates the worst case, in which the two-spin approximation predicts a maximum at the same time that the exact three-spin isotropic mixing transfer function reaches a minimum. (Adapted from Remerowski et al., 1989, courtesy of Taylor Francis.)...

Several methods have been described for using 3D NOESY-NOESY cross-peak intensities for structural refinement such as the two-spin approximation (4,5), Taylor series expansion of the NOE-rate equation (6), and direct gradient refinement method (7). The two-spin approximation requires that the NOESY derived distances be obtained from vanishingly short experimental mixing times where the build-up of NOE intensity is linear with respect to interproton distance and the effect of spin diffusion (NOE intensity mediated by multiple relaxation pathways) are minimal. 

In the two spin approximation method the volume of a 3D cross peak,which is derived from interactions between spins i, j and k, is considered to be proportional to the product of the inverse sixth power of the distance between spins i and j and the distance between] and k ... 

Where R is the rate matrix that describes the NOE interactions across the system, and are the two NOE mixing and A(0) is the initial magnetization. To simplify this equation, a Taylor series expansion of the exponential can be made. Usually, only the first few terms in the expansion are kept for the approximation. The first term approximation is equivalent to the two-spin approximation (5,14). At realistic mixing times (50 ms or more), the Taylor series approximation also yields systematic error in determining the inter-proton distances (9). Figure 1 shows comparison of volumes simulated from the two-term Taylor series approximation and an exact rate-matrix calculation for the Dickerson dodecamer... 

Let ns consider a linear chain of N statistical segments, fixed at its extremities. An average orientation is indnced along the chain hy these constrains. To estimate the effect of this average orientation on NMR parameters, the simplest pictnre is to consider that each segment carries an isolated pairs of spins, nsnally protons (two-spin approximation). Within this framework, the time evolntion of the transverse magnetization (t) for a spin pair attached to a chain may he written in the very simple form... 

While the final magnitude of nOe depends, as indicated earlier, on the relaxation pathways Wi, W, and W), the initial rate of buildup of nOe (transient nOe) depends only on the rate of cross-relaxation between the nuclei, and this can provide valuable information about the distance between the nuclei (r). This rate of buildup can be proportional to r" , where r is the distance between the nuclei. Thus, if the proportionality constant is determined, we can calculate an approximate distance between the two nuclei. The best results are obtained in rigid molecules when the nuclei are less than 3 A apart. If only direct nOe s are involved in a two-spin... 

In electrocatalysis, the reactants are in contact with the electrode, and electronic interactions are strong. Therefore, the one-electron approximation is no longer justified at least two spin states on a valence orbital must be considered. Further, the form of the bond Hamiltonian (2.12) is not satisfactory, since it simply switches between two electronic states. This approach becomes impractical with two spin states in one orbital also, it has an ad hoc nature, which is not satisfactory. 

Consider the approximation of four discrete molecular orientations along the axes of a square lattice ( At/4 kBT, AU4 < 0). To conveniently describe orientations, we introduce, at each lattice site, two spin variables, a m = l and s m = 1, which are related to unit vectors nm as follows (Fig. 2.18) ... 

Since the tensor D is strictly proportional to the inverse cube of the distance between the unpaired electrons for two spin systems and is regarded as a measure of the inverse volume of the delocalized spins in higher-spin systems, the absolute Z)-values become smaller as S increases. Since the epr spectral width is approximated by 2D, it is customarily the case that the g = 2 region becomes crowded with spectral peaks with increasing S. 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

For a heteronuclear diatomic molecule, we can take nuclear spin into account simply by multiplying the spatial nuclear wave function (4.28) by two spin functions—one for each nucleus. The overall molecular wave function (4.90) then has the (approximate) form... 

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