Discrepancy factors for use in crystal structure analysis

Abstract

Theoretical expressions for two types of discrepancy factors have been obtained for crystals containing both heavy and light atoms in the unit cell. One of the discrepancy factors is defined in terms of the structure amplitude and is called the Booth's reliability index (RD. The other discrepancy factor (denoted by R2) is based on intensities. While the expressions for R2 can be used for crystals of any space group and for crystals containing any number and type of atoms in the unit cell, those for Rn can be used only for crystals in the triclinic and a few monoclinic space groups.

Some statistical results for structures with pseudosymmetry

“…From the work of Parthasarathy & Parthasarathi (1972), it can be seen that the maximum probable value of RB for a completely 'wrong' structure is 0.429 for the acentric distribution and 0.727 for the centric distribution while the values of R 2 for the corresponding cases' are 1.0 and 1.333 respectively. Thus structures with an acentric distribution having values* of RB--0.25 and R 2 ~-0"6 might be expected to be essentially correct while the corresponding values for structures with a centric distribution could be taken as RB -----0.44 and R2"~0"8.…”

“…index RB [Booth, 1945;Parthasarathy & Parthasarathi, 1972;see equation (11) below for the definition] for the various cases.…”

On the evaluation of root-mean-square errors in atomic coordinates in protein crystallography

“…Parthasarathy & Parthasarathi (1972) have shown that, provided Wilson's (1949)distribution functions apply, it is possible, with the value of R 3, to calculate I~]-rl even if an incomplete model has been used. They derived expressions for R 3 as a function of D and 02; the latter is a measure of the incompleteness of the model, and may be easily calculated (10) if we know the number of atoms the structure contains (N) and the number of atoms in the model (C).…”

“…It is interesting that various reciprocal-space residuals may serve both as correlation functions in rotation and translation function searches (Nixon & North, 1976) and as a basis of IArl evaluation (Luzzati, 1952;Parthasarathy & Parthasarathi, 1972;Nixon & North, 1976). Attention has also been paid to the physical interpretation of refinement based on the minimization of some of these residuals (Wilson, 1976).…”

Discrepancy indices for use in crystal structure analysis. IV. Theoretical expressions for the indicesBR1(I) andBR(I) for any incomplete model of a complex structure

“…8R~ (1), is the best during the structure completion stage. It is therefore useful to obtain general expressions for this index which can be used to test the correctness of any type of incomplete model of a complex structure.t Explicit expressions for the indices aRt(I) and ~R(I) for the related and unrelated cases have been derived earlier for a few special cases (Parthasarathy & Parthasarathi, 1972;; -the former is hereafter briefly referred to as PP). Wilson (1969) has studied the effect of a single wrong atom in an otherwise correct and complete model structure on the index BR(I) and Lenstra (1974) has recently carried out similar studies for incomplete models.…”