Modifications: origin removal, sharpening etc.

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 236-237 | 1 | 2 |

Section 2.3.1.3. Modifications: origin removal , sharpening etc.

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.1.3. Modifications: origin removal , sharpening etc.

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A. L. Patterson, in his first in-depth exposition of his newly discovered $[F^{2}]$ series (Patterson, 1935), introduced the major modifications to the Patterson which are still in use today. He illustrated, with one-dimensional Fourier series, the techniques of removing the Patterson origin peak, sharpening the overall function and also removing peaks due to atoms in special positions. Each one of these modifications can improve the interpretability of Pattersons, especially those of simple structures. Whereas the recommended extent of such modifications is controversial (Buerger, 1966), most studies which utilize Patterson functions do incorporate some of these techniques [see, for example, Jacobson et al. (1961), Braun et al. (1969) and Nordman (1980a)]. Since Patterson's original work, other workers have suggested that the Patterson function itself might be modified; Fourier inversion of the modified Patterson then provides a new and perhaps more tractable set of structure factors (McLachlan & Harker, 1951; Simonov, 1965; Raman, 1966; Corfield & Rosenstein, 1966). Karle & Hauptman (1964) suggested that an improved set of structure factors could be obtained from an origin-removed Patterson modified such that it was everywhere non-negative and that Patterson density values less than a bonding distance from the origin were set to zero. Nixon (1978) was successful in solving a structure which had previously resisted solution by using a set of structure factors which had been obtained from a Patterson in which the largest peaks had been attenuated.

The N origin peaks [see expression (2.3.1.3)] may be removed from the Patterson by using coefficients $[|{\bf F}_{{\bf h}, \, {\rm mod}}|^{2} = |{\bf F}_{\bf h}|^{2} - {\textstyle\sum\limits_{i = 1}^{N}}\; f_{i}^{2}.]$ A Patterson function using these modified coefficients will retain all interatomic vectors. However, the observed structure factors $[{\bf F}_{\bf h}]$ must first be placed on an absolute scale (Wilson, 1942) in order to match the scattering-factor term.

Analogous to origin removal, the vector interactions due to atoms in known positions can also be removed from the Patterson function. Patterson showed that non-origin Patterson peaks arising from known atoms 1 and 2 may be removed by using the expression $[|{\bf F}_{{\bf h}, \, {\rm mod}}|^{2} = |{\bf F}_{\bf h}|^{2} - {\textstyle\sum\limits_{i = 1}^{N}}\; f_{i}^{2} t_{i}^{2} - 2f_{1} f_{2} t_{1} t_{2} \cos 2\pi {\bf h} \cdot ({\bf x}_{1} - {\bf x}_{2}),]$ where $[{\bf x}_{1}]$ and $[{\bf x}_{2}]$ are the positions of atoms 1 and 2 and $[t_{1}]$ and $[t_{2}]$ are their respective thermal correction factors. Using one-dimensional Fourier series, Patterson illustrated how interactions due to known atoms can obscure other information.

Patterson also introduced a means by which the peaks in a Patterson function may be artificially sharpened. He considered the effect of thermal motion on the broadening of electron-density peaks and consequently their Patterson peaks. He suggested that the $[F^{2}]$ coefficients could be corrected for thermal effects by simulating the atoms as point scatterers and proposed using a modified set of coefficients $[|{\bf F}_{{\bf h}, \, {\rm sharp}}|^{2} = |{\bf F}_{\bf h}|^{2} / \bar{f}^{2},]$ where $[\bar{f}]$ , the average scattering factor per electron, is given by $[\bar{f} = {\textstyle\sum\limits_{i = 1}^{N}}\; f_{i} \bigg/ {\textstyle\sum\limits_{i = 1}^{N}} Z_{i}.]$ A common formulation for this type of sharpening expresses the atomic scattering factors at a given angle in terms of an overall isotropic thermal parameter B as $[f(s) = f_{0} \exp (- Bs^{2}).]$ The Patterson coefficients then become $[{\bf F}_{{\bf h}, \, {\rm sharp}} = {{Z}_{\rm total} \over \sum_{i = l}^{N} f(s)} {\bf F}_{{\bf h}}.]$ The normalized structure factors , E (see Chapter 2.2 ), which are used in crystallographic direct methods, are also a common source of sharpened Patterson coefficients $[(E^{2} - 1)]$ . Although the centre positions and total contents of Patterson peaks are unaltered by sharpening, the resolution of individual peaks may be enhanced. The degree of sharpening can be controlled by adjusting the size of the assumed B factor; Lipson & Cochran (1966, pp. 165–170) analysed the effect of such a choice on Patterson peak shape.

All methods of sharpening Patterson coefficients aim at producing a point atomic representation of the unit cell. In this quest, the high-resolution terms are enhanced (Fig. 2.3.1.1). Unfortunately, this procedure must also produce a serious Fourier truncation error which will be seen as large ripples about each Patterson peak (Gibbs, 1898). Consequently, various techniques have been used (mostly unsuccessfully) in an attempt to balance the advantages of sharpening with the disadvantages of truncation errors.

Figure 2.3.1.1 | top | pdf |

Effect of `sharpening' Patterson coefficients. (1) shows a mean distribution of $[|{\bf F}|^{2}]$ values with resolution, $[(\sin \theta)/\lambda]$ . The normal decline of this curve is due to increasing destructive interference between different portions within diffuse atoms at larger Bragg angles. (2) shows the distribution of `sharpened' coefficients. (3) shows the theoretical distribution of $[|{\bf F}|^{2}]$ produced by a point-atom structure. To represent such a structure with a Fourier series would require an infinite series in order to avoid large errors caused by truncation.

Schomaker and Shoemaker [unpublished; see Lipson & Cochran (1966, p. 168)] used a function $[|{\bf F}_{{\bf h}, \, {\rm sharp}}|^{2} = {|{\bf F}_{{\bf h}}|^{2} \over \bar{f}^{2}} s^{2} \exp \left[- {\pi^{2} \over p} s^{2}\right],]$ in which s is the length of the scattering vector, to produce a Patterson synthesis which is less sensitive to errors in the low-order terms. Jacobson et al. (1961) used a similar function, $[|{\bf F}_{{\bf h}, \, {\rm sharp}}|^{2} = {|{\bf F}_{{\bf h}}|^{2} \over \bar{f}^{2}} (k + s^{2}) \exp \left[ - {\pi \over p} s^{2}\right],]$ which they rationalize as the sum of a scaled exponentially sharpened Patterson and a gradient Patterson function (the value of k was empirically chosen as $[{2 \over 3}]$ ). This approach was subsequently further developed and generalized by Wunderlich (1965).

References

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 236-237

Section 2.3.1.3. Modifications: origin removal, sharpening etc.

2.3.1.3. Modifications: origin removal, sharpening etc.

References

Section 2.3.1.3. Modifications: origin removal , sharpening etc.

2.3.1.3. Modifications: origin removal , sharpening etc.