Fourier truncation errors

The electron density function is given in the form of a Fourier summation (Equation 4.2). This means that electron density and hence the atoms in a structure are represented by a number of sine-waves, which are added up. The higher the number of sine-waves, the smoother and more accurate the electron density becomes. As with every Fourier summation, if terms are missing, ripples appear. Especially when some strong reflections are missing from the dataset (e.g. incomplete dataset or some reflections hidden behind the beamstop) artefactual electron density— negative or positive—can appear near heavy atom sites. The same effect can be observed with low-resolution data. An excellent description of the theory behind this effect can be found in an article by Cochran and Lipson (1966). 

A famous example is the structure of Nitrogenase MoFe-Protein, a protein that contains a Fe7MoS9 cluster. The inside of this cluster is about 4 A wide with six iron atoms closest to the centre, and older crystal stmctures had been determined at resolutions of about 2 A. Termination of the Fourier summation at that resolution creates an artefactual miiumum in the electron density of about —0.2 electrons about 2 A away from each iron atom. These spurious minima from all heavy atoms in the 

2 When it is vital to determine die hydrogen positions with high accuracy, neutrons should be used instead of X-rays in the diffraction experiment 

Besides artefacts, there are other systematic errors that can have negative effects on a crystal structure global pseudo-symmetry for example, or inaccurate scaling. And then there is the group of avoidable errors. Common avoidable errors are  

The German crystallographer Roland Boese (1999) made an interesting point in distinguishing between avoidable errors and really avoidable errors . As examples for the latter kind he listed  

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Within the computational scheme described in the course of this work, the available information about the atomic substructure (core+valence) can be taken into account explicitly. In the simplest possible calculation, a fragment of atomic cores is used, and a MaxEnt distribution for valence electrons is computed by modulation of a uniform prior prejudice. As we have shown in the noise-free calculations on l-alanine described in Section 3.1.1, the method will yield a better representation of bonding and non-bonding valence charge concentration regions, but bias will still be present because of Fourier truncation ripples and aliasing errors ... 

An alternative approach is to use gexp(r) as the target function in the RMC simulations. The analysis of the experimental diffraction data to obtain Sexp(q) involves a sequence of corrections that are generally well understood. However, in order to obtain gexp(r), it is necessary to Fourier transform Sexp(q). This operation is particularly vulnerable to the limitations of the experimental data [10-12]. For example, Sexp(q) is obtained up to a maximum value of q, at which there may still be oscillations. Since the Fourier transform involves an integral from q equals zero to infinity, this limitation yields to truncation errors... 

Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [135, 174, 136]. This method is used to study the effects of numerical diffusion on the solution. 

I FIGURE 3.4 Truncation error in the digitized FID (a) manifests itself as bumps, ridges, or corduroy around the base of narrow peaks in the Fourier-... 

All Fourier series have to be made finite when performed numerically the choice of the number of waves used in any calculation is a compromise between the computational effort and the errors caused by the truncation they are difficult to estimate and one usually resorts to numerical testing. An example of a convergence test for a, B - calculated from p(a) as described above (Fig. 3.1) - is s iown in Tab. 3.1 the behavior of truncation errors is typical for many similar situations. Whereas the absolute values of both pressure and energy vary considerably with increasing number of waves, the a, B calculated from them evolve only slowly. Apparently a large part of the truncation error is systematic. Detailed convergence tests for the different potentials used can be found e.g. in Ref. 24 so far the most detailed study... 

The truncation error associated with convection/advection schemes can be analyzed by using the modified equation method [254]. By use of Taylor series all the time derivatives except the 1. order one are replaced by space derivatives. When the modified equation is compared with the basic advection equation, the right-hand side can be recognized as the error. The presence of Ax in the leading error term indicate the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. Another method for analyzing the truncation error of advection schemes is the Fourier (or von Neumann method) [157, 158, 215]. This method is used to study the effects of numerical diffusion on the solution. 

The exact amount of error introduced cannot immediately be inferred from the strength of the amplitudes of the neglected Fourier coefficients, because errors will pile up in different points in the crystal depending on the structure factors phases as well to investigate the errors, a direct comparison can be made in real space between the MaxEnt map, and a map computed from exponentiation of a resolution-truncated perfect m -map, whose Fourier coefficients are known up to any order by analysing log(<7 (x) tm (x)). 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

Figure 2 shows the constraint of minimum negativity applied to the same deconvolution as that shown in Fig. 1 but with different truncation points for the Fourier spectrum. Figure 2(a) shows restoration to the inverse-filtered estimate with seven (complex) coefficients retained, and illustrates the distortion occurring when too many noise-laden coefficients are retained in the Fourier spectrum. From Fig. 3(b) of Chapter 9 it is evident that the seventh coefficient contains a large amount of noise error. Figure 2(b) shows restoration to the inverse-filtered result with only five complex coefficients retained in the Fourier spectrum. It differs little from the restoration with only six coefficients retained in the inverse-filtered estimate shown in Fig. 1. For both cases shown in Fig. 2, 16 complex coefficients were restored.

The complete Fourier transform requires measurement of V(u) over an infinite range. Even though V(m) may be small outside the range that can be measured, the truncation will introduce errors. 

FIGURE 9.12. Series-termination errors, (a) A normal atomic scattering factor curve and (b) the atomic peak obtained by Fourier transformation, (c) A truncated atomic scattering factor curve, such as that used for data that are measured to a lower sin 6/ value than advisable. The missing portion of the scattering curve is indicated, (d) The atomic peak obtained by Fourier transformation. Note the ripples caused by loss of the missing portion of the atomic scattering curve. 

Series-termination errors Errors that result from a limitation in the number of terms in a Fourier series. Ideally an infinite number of data is required to calculate a Fourier series. In practice, the number of data depends on the resolution (reciprocal radius or sin0/A) to which the data have been measured. Because of truncation of the Fourier series at the highest value of sin 0/X of the data, peaks in the resulting Fourier syntheses are surrounded by a series of ripples. These are especially noticeable around a heavy atom because its scattering factor is still appreciable at the highest values of sin 9/X measured. Difference maps (q-v.) can be used to obviate most of the effects of series-termination errors. 

The principal errors in numerically approximating the transform integral to infinite time by a discrete Fourier transform (DFT), or series of a finite number of values at discrete points, are from aliasing and truncation. The magnitudes of the errors for the functions G(t) usually encountered can be readily assessed and kept... 

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