A test for rigid-body vibrations based on a generalization of Hirshfeld's `rigid-bond' postulate

Abstract

The maximum of E is obtained if all 0.~, are -1. The minimum of E is obtained if all 0.k are + 1, which agrees with the result of Kabsch (1976).It has also been shown in Kabsch (1976) that S + L must be positive definite at the minimum of E. Hence, from (2) the determinants of the two matrices, U and R, must have the same signs.In the case that det(R) > 0, the orthogonal matrix U corresponding to the minimum of E will be a proper rotation. In the case that det(FI) < 0, an improper rotation will be obtained at the minimum of E (Nyburg & Yuen, 1977). From (9), the smallest residual E corresponding to a best true rotation is then obtained if 0.~ = 0"2 = + 1 and 0"3 = -1 assuming that/23 is the smallest eigenvalue of [~FI (threedimensional vector space). Note that if the smallest eigenvalue is degenerate a best rotation cannot be determined uniquely in the case det(FI) < 0.Finally, it might be worth mentioning that this procedure can be generalized to find a best unitary matrix to relate two sets of vectors in the complex finite-dimensional vector space.Summarizing the above results, the following procedure for obtaining a best proper rotation in a three-dimensional vector space is suggested. (a) Remove any translation between the two given vector sets x n, yn and determine E 0 = ½ ~,,wn(x 2 + y~) and R.(b) Form ~IFI, determine eigenvalues/2k and the mutually orthogonal eigenvectors a k and sort so that g~ >/22 >/23. Set a 3 = a I × a 2 to be sure to have a right-handed system.(c) Determine Fla k (k = 1, 2, 3), normalize the first two vectors to obtain b l, b 2 and set b 3 = b~ x b 2. This will also take care of the case/22 >/2a = 0.(d) Form U according to (7) A simple test for the validity of the rigid-body model for molecular vibrations in crystals is proposed.Since bond-stretching vibrations for atoms other than hydrogen and deuterium are normally of much smaller amplitude than other vibrations (bond-bending, torsional, rigid-body translational and rotational oscillations), the mean-square vibrational amplitudes of a pair of bonded atoms should be equal along the bond direction, even though they may be widely different in other directions. As Hirshfeld (1976) has pointed out, this provides a necessary (although by no means sufficient) condition that thermal ellipsoids derived by X-ray analysis represent genuine vibrational ellipsoids. If the condition is seriously violated, the Uij values may be suspected of being contaminated by charge-density deformation contributions or absorption or other systematic errors.Hirshfeld's 'rigid-bond' postulate can be expressed in a more general (though somewhat weaker) form as a 'rigidbody' postulate and used as a simple test for the validity of the rigid-body model of any molecule for which Ut/values are available. Since rigidity implies that all distances within a body remain invariant, all pairs of atoms in a rigid molecule can be regarded as being connected by virtual bonds. Hence * Present address:

Structure of and energy calculations for 1,2,9,10,17,18-hexadehydro[2.2.2]paracyclophane

“…1). The generalization of Hirshfeld's (1976) rigid-bond test proposed by Rosenfield, Trueblood & Dunitz (1978) was applied to the vibration parameters to test their self-consistency; the average difference in the mean-square vibration amplitude for pairs of bonded atoms was 0.0030 A 2, comparable to the average e.s.d, expected for this difference (calculated from the average e.s.d, of the U°), 0.0027/~2. Furthermore, the differences in mean-square vibration amplitudes for non-bonded pairs of atoms within each ring were the same as those for bonded pairs of atoms, even in the ring (ring 2, Table 5) with the largest libration amplitude, a finding consistent with the expected rigidity of an aromatic ring.…”

Sodium lanthanum(III) sulfate monohydrate, NaLa(SO4)2.H2O

“…Neither the Na-O nor the La-O polyhedron conformed to rigid-body behavior, but the sulfate ion did; the average magnitude of the differences in the meansquare displacement amplitudes along the interatomic vectors for the ten unique atom pairs of the sulfate ion was 15 (16) x 10 -4 A 2. The Hirshfeld (1976) rigid-bond test, extended as described by Rosenfield, Trueblood & Dunitz (1978), is satisfied and the corrected S--O bond lengths are given in Table 2.…”

Use of TLS parameters to model anisotropic displacements in macromolecular refinement

“…Another approach using re®ned individual ADPs is to use the rigid-body criterion (Rosen®eld et al, 1978) in which a D matrix is built up between all pairs of atoms, with elements equal to the difference in the projected U values along the interatomic vector. Pairs of atoms belonging to the same quasi-rigid group should have a D value close to zero.…”