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Figure 1
(a) A schematic representation of the elements of a VCV matrix S of the positional parameters x1, y1, z1 and x2, y2, z2 of two atoms. The diagonal elements marked in dark green, such as [S\left({{y_1},{y_1}} \right)], are the variances of the coordinates equal to the squares of the individual parameter e.s.u.'s quoted in cif files. The covariances of the positional parameters of a single atom are coloured in pale green. If the atom occupies a g.e.p. these elements are zero, unless the full VCV matrix is available from the least-squares refinement. If it occupies a s.e.p., for which the rotational part of the stabilizer is R, then they are the off-diagonal elements of S(x)RT (Section 2.2[link]). The covariances between the coordinates of the two atoms appear in the two yellow blocks. In the absence of the covariance from the least-squares refinement they are only non-zero if the atoms are related by symmetry, in which case their values are S(x1)RT where R is the rotational part of the symmetry operator relating the two positions. (b) A diagram showing the extension of the principle of construction of S as a 12 × 12 matrix for a calculation involving four atom positions, x1, x2, x3, x4, as required for calculating the volume of a tetrahedron. Each block in this diagram is a 3 × 3 matrix; the 3 × 3 blocks on the diagonal are the VCV matrices of the individual positions, and the off-diagonal 3 × 3 blocks are the covariances of individual pairs of atom positions. The area outlined in blue is the matrix elements shown in part (a).

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ISSN: 2053-2733
Volume 81| Part 3| May 2025| Pages 202-210
https://doi.org/10.1107/S2053273325002682
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