Section 3.2.1. Introduction

By way of introduction, we remark that in earlier days of crystal structure analysis, before the advent of high-speed computers and routine three-dimensional analyses, molecular planarity was often assumed so that atom coordinates along the direction of projection could be estimated from two-dimensional data [see, e.g., Robertson (1948)]. Today, the usual aim in deriving the coefficients of a plane is to investigate the degree of planarity of a group of atoms as found in a full, three-dimensional structure determination. We further note that, for such purposes, a crystallographer will often be served just as well by establishing the plane in an almost arbitrary fashion as by resorting to the most elaborate, nit-picking and pretentious least-squares treatment. The approximate plane and the associated perpendicular distances of the atoms from it will be all he needs as scaffolding for his geometrical and structural imagination; reasonable common sense will take the place of explicit attention to error estimates.

Nevertheless, we think it appropriate to lay out in some detail the derivation of the `best' plane, in a least-squares sense, through a group of atoms and of the standard uncertainties associated with this plane. We see two cases: (1) The weights of the atoms in question are considered to be isotropic and uncorrelated (i.e. the weight matrix for the positions of all the atoms is diagonal, when written in terms of Cartesian axes, and for each atom the three diagonal elements are equal). In such cases the weights may have little or nothing to do with estimates of random error in the atom positions (they may have been assigned merely for convenience or convention), and, therefore, no one should feel that the treatment is proper in respect to the theory of errors. Nevertheless, it may be desired to incorporate the error estimates (variances) of the atom positions into the results of such calculations, whereupon these variances (which may be anisotropic, with correlation between atoms) need to be propagated. In this case the distinction between weights (or their inverses) and variances must be kept very clear. (2) The weights are anisotropic and are presumably derived from a variance–covariance matrix, which may include correlation terms between different atoms; the objective is to achieve a truly proper Gaussian least-squares result.