Truncation effects

Crystal data and details of data collection, data reduction and final refinement are reported in Table 1. The procedure for data collection and processing, which included a correction for scan-truncation effects, were similar to those recently described for syn-l,6 8,13-biscarbonyl[14]annulene [10] and citrinin [11], Figure 1 shows the numbering scheme adopted in the present analysis. 

These solutions of the idealized problem are a good approximation for the behavior within a time window Xx < t < Xu or the corresponding frequency window 1/2U < co < l/X. Truncation effects can be seen near the edges Xx and Xu. X0 is some material-specific reference time, which has to be specified in each choice of material, and H0r(n) = G0 is the corresponding modulus value. 

Table 8 reports equilibrium binding energies relative to the geometries described above. The differences between MCSCF-MI or SCF-MI energies and the SCF ones are obviously to be ascribed to the different basis set truncation effect and to BSSE. 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

Series truncation effects due to the experimental resolution limit are reduced when the core- or spherical-atom densities are subtracted from the Fourier summation, as in Eq. (6.9). 

For analysis of experimental results, the static model density must be used to eliminate noise, truncation effects, and thermal smearing. Some caution is called for, because the reciprocal space representation of the Laplacian is a function of F(H) H2, and thus has poor convergence properties.2 This difficulty is only partly circumvented by use of the model density, as high-resolution detail may be quite dependent on the nature of the model functions, as is evident in the experimental study of the quartz polymorph coesite discussed in chapter 11. 

When non-decayed FIDs are directly Fourier transformed, truncation-effects resulting in line distortions occurs. This problem is most pronounced with 2D data. 

With 2D data sets non-decayed FIDs in both time domains (Fig. 5.13) are very common and a simple Fourier transformation would give rise to truncation effects in the final spectrum. To circumvent such unwanted effects 2D FIDs are usually rigorously damped, using stringent weighting functions to smoothly bring the last part of the FID close to zero. However this simple procedure severely impairs spectral resolution and should be replaced by LP, followed by suitable weighting, which both improves spectral resolution and excludes any truncation effects. 

Let x(t) and V(t) be the actual solutions to these differential equations. In general a given algorithm will replace these differential equations by a particular set of difference equations. These difference equations will then give approximate values of x(t) and V(t) at discrete, equally spaced points in time tu t2,. .., tn where tJ+x = tj + At. The differences between the solutions to the difference equations at tN and the solutions to the differential equations at t N depend critically on the time step At. If At is too large, the system of difference equations may be unstable or be in error due to truncation effects. On the other hand, if At is too small, the solutions to the difference equations may be in error due to the accumulation of machine rounding of intermediate results. 

Prior to Fourier transformation the time domain data may be modified by multiplication by a function q(t), a process commonly called digital filtering. By suitable choice of q(t), digital filtering may be used in NMR spectroscopy to enhance sensitivity, to improve spectral resolution, or to avoid truncation effects. Conceptually, there is very little difference between filtering of ID and 2D NMR spectra, so the treatment here may later be extended readily to the two-dimensional case. 

The expressions derived in the foregoing text for the behavior of a Gaussian beam tacitly assumed that truncation effects could be ignored that is, the expressions were derived for an infinite aperture system. Any physical system, of course, consists of elements of finite extent. The question to consider, then, is under what conditions those elements may be approximated by ideal elements. 

If truncation effects may not be neglected, the integral may be written as /q dx, which may be evaluated by integration by parts. 

Figure 2-13 Apparent phase errors due to signal-truncation effects, (a) The distorted spectrum caused by an acquisition time that is too short (DT = 0 s). (b) The distortion-free spectrum after the introduction of a relaxation delay time (DT = 1 s).

New NMR methods which should be suitable for fully hydrated membrane samples with better measurement accuracy and less dipolar truncation effects. 

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