A simple error formula

Cruickshank, D. W. J.

doi:10.1107/97809553602060000697

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 410 | 1 | 2 |

Section 18.5.6.2. A simple error formula

D. W. J. Cruickshank^a ^*^‡

^a Chemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: dwj_cruickshank@email.msn.com

18.5.6.2. A simple error formula

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Cruickshank (1960) offered a simple order-of-magnitude formula for $[\sigma (x)]$ in small molecules. It was intended for use in experimental design: how many data of what precision are needed to achieve a given precision in the results? The formula, derived from a very rough estimate of a least-squares diagonal element in non-centrosymmetric space groups, was $[\sigma (x_{i}) = (1 / 2) (N_{i} / p)^{1/2} [R / s_{\rm rms}] \eqno(18.5.6.2)]$ Here p = $[n_{\rm obs} - n_{\rm params}]$ , R is the usual residual $[\sum |\Delta F|/\sum |F|]$ and $[N_{i}]$ is the number of atoms of type i needed to give scattering power at $[s_{\rm rms}]$ equal to that of the asymmetric unit of the structure, i.e., $[\sum_{j}f_{j}^{2} \equiv N_{i}\>f_{i}^{2}]$ . [The formula has also proved very useful in a systematic study of coordinate precision in the many thousands of small-molecule structure analyses recorded in the Cambridge Structural Database (Allen et al., 1995a,b).]

For small molecules, the above definition of $[N_{i}]$ allowed the treatment of different types of atom with not-too-different B's. However, it is not suitable for individual atoms in proteins where there is a very large range of B values and some atoms have B's so large as to possess negligible scattering power at $[s_{\rm rms}]$ .

Often, as in isotropic refinement, $[n_{\rm params} \simeq 4n_{\rm atoms}]$ , where $[n_{\rm atoms}]$ is the total number of atoms in the asymmetric unit. For fully anisotropic refinement, $[n_{\rm params} \simeq 9n_{\rm atoms}]$ .

A first very rough extension of (18.5.6.2) for application in proteins to an atom with $[B = B_{i}]$ is $[\sigma (x_{i}) = k(N_{i} / p)^{1/2} \left[g(B_{i}) / g(B_{\rm avg})\right] C^{-1/3} Rd_{\min}, \eqno(18.5.6.3)]$ where k is about 1.0, $[N_{i} = \sum Z_{j}^{2}/Z_{i}^{2}, B_{\rm avg}]$ is the average B for fully occupied sites and C is the fractional completeness of the data to $[d_{\min}]$ . In deriving (18.5.6.3) from (18.5.6.2), $[1/s_{\rm rms}]$ has been replaced by $[1.3d_{\min}]$ , and the factor [(1/2)(1.3) = 0.65] has been increased to 1.0 as a measure of caution in the replacement of a full matrix by a diagonal approximation. $[g(B) = 1 + a_{1}B+ a_{2}B^{2}]$ is an empirical function to allow for the dependence of $[\sigma (x)]$ on B. However, the results in Section 18.5.4.2 showed that the parameters $[a_{1}]$ and $[a_{2}]$ depend on the structure.

As also mentioned in Section 18.5.4.2, Sheldrick has found that the $[Z_{i}]$ in $[N_{i}]$ is better replaced by $[Z_{i}^{\#}]$ , the scattering factor at $[\sin \theta /\lambda = 0.3\ \hbox{\AA }^{-1}]$ . Hence, $[N_{i}]$ may be taken as $[N_{i} = (\textstyle\sum\displaystyle Z_{j}^{\# 2} / Z_{i}^{\# 2}). \eqno(18.5.6.4)]$

A useful comparison of the relative precision of different structures may be obtained by comparing atoms with the respective $[B = B_{\rm avg}]$ in the different structures. (18.5.6.3) then reduces to $[\sigma (x, B_{\rm avg}) = 1.0 (N_{i} / p)^{1/2} C^{-1/3} Rd_{\min}. \eqno(18.5.6.5)]$ The smaller the $[d_{\min}]$ and the R, the better the precision of the structure. If the difference between oxygen, nitrogen and carbon atoms is ignored, $[N_{i}]$ may be taken simply as the number of fully occupied sites. For heavy atoms, (18.5.6.4) must be used for $[N_{i}]$ .

Equation (18.5.6.5) is not to be regarded as having absolute validity. It is a quick and rough guide for the diffraction-data-only error component for an atom with Debye B equal to the $[B_{\rm avg}]$ for the structure. It is named the diffraction-component precision index, or DPI. It contains none of the restraint data.

References

Allen, F. H., Cole, J. C. & Howard, J. A. K. (1995a). A systematic study of coordinate precision in X-ray structure analyses. I. Descriptive statistics and predictive estimates of e.s.d.'s for C atoms. Acta Cryst. A51, 95–111.Google Scholar

Allen, F. H., Cole, J. C. & Howard, J. A. K. (1995b). A systematic study of coordinate precision in X-ray structure analyses. II. Predictive estimates of e.s.d.'s for the general-atom case. Acta Cryst. A51, 112–121.Google Scholar

Cruickshank, D. W. J. (1960). The required precision of intensity measurements for single-crystal analysis. Acta Cryst. 13, 774–777.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 410