Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 19-22   | 1 | 2 |

Section 1.2.11. Rigid-body analysis

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail:

1.2.11. Rigid-body analysis

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The treatment of rigid-body motion of molecules or molecular fragments was developed by Cruickshank (1956)[link] and expanded into a general theory by Schomaker & Trueblood (1968)[link]. The theory has been described by Johnson (1970b)[link] and by Dunitz (1979)[link]. The latter reference forms the basis for the following treatment.

The most general motions of a rigid body consist of rotations about three axes, coupled with translations parallel to each of the axes. Such motions correspond to screw rotations. A libration around a vector [{\boldlambda}\ (\lambda_{1},\ \lambda_{2},\ \lambda_{3})], with length corresponding to the magnitude of the rotation, results in a displacement [\delta {\bf r}], such that [\specialfonts \delta {\bf r} = ({\boldlambda} \times {\bf r}) = {{\bsf D}}{\bf r} \eqno(] with [\specialfonts{{\bsf D}} = \left[\matrix{0 &-\lambda_{3} &\lambda_{2}\cr \lambda_{3} &0 &-\lambda_{1}\cr -\lambda_{2} &\lambda_{1} &0\cr}\right], \eqno(] or in tensor notation, assuming summation over repeated indices, [\delta r_{i} = D_{ij}r_{j} = - \varepsilon_{ijk} \lambda_{k} r_{j} \eqno(] where the permutation operator [\varepsilon_{ijk}] equals +1 for i, j, k a cyclic permutation of the indices 1, 2, 3, or −1 for a non-cyclic permutation, and zero if two or more indices are equal. For [i = 1], for example, only the [\varepsilon_{123}] and [\varepsilon_{132}] terms occur. Addition of a translational displacement gives [\delta r_{i} = D_{ij}r_{j} + t_{i}. \eqno(]

When a rigid body undergoes vibrations the displacements vary with time, so suitable averages must be taken to derive the mean-square displacements. If the librational and translational motions are independent, the cross products between the two terms in ([link] average to zero and the elements of the mean-square displacement tensor of atom n, [{U^{n}_{ij}}], are given by [\eqalign{U_{11}^{n} &= + L_{22} r_{3}^{2} + L_{33} r_{2}^{2} - 2L_{23} r_{2} r_{3} + T_{11}\cr U_{22}^{n} &= + L_{33} r_{1}^{2} + L_{11} r_{3}^{2} - 2L_{13} r_{1} r_{3} + T_{22}\cr U_{33}^{n} &= + L_{11} r_{2}^{2} + L_{22} r_{1}^{2} - 2L_{12} r_{1} r_{2} + T_{33}\cr U_{12}^{n} &= - L_{33} r_{1} r_{2} - L_{12} r_{3}^{2} + L_{13} r_{2} r_{3} + L_{23} r_{1} r_{3} + T_{12}\cr U_{13}^{n} &= - L_{22} r_{1} r_{3} + L_{12} r_{2} r_{3} - L_{13} r_{2}^{2} + L_{23} r_{1} r_{2} + T_{13}\cr U_{23}^{n} &= - L_{11} r_{2} r_{3} + L_{12} r_{1} r_{3} - L_{13} r_{1} r_{2} - L_{23} r_{1}^{2} + T_{23},\cr} \eqno(] where the coefficients [L_{ij} = \langle \lambda_{i} \lambda_{j}\rangle] and [T_{ij} = \langle t_{i} t_{j}\rangle] are the elements of the [3 \times 3] libration tensor [ \specialfonts{{\bsf L}}] and the [3 \times 3] translation tensor [ \specialfonts{{\bsf T}}], respectively. Since pairs of terms such as [\langle t_{i} t_{j}\rangle] and [\langle t_{j} t_{i}\rangle] correspond to averages over the same two scalar quantities, the [\specialfonts {{\bsf T}}] and [\specialfonts {{\bsf L}}] tensors are symmetrical.

If a rotation axis is correctly oriented, but incorrectly positioned, an additional translation component perpendicular to the rotation axes is introduced. The rotation angle and the parallel component of the translation are invariant to the position of the axis, but the perpendicular component is not. This implies that the [\specialfonts {{\bsf L}}] tensor is unaffected by any assumptions about the position of the libration axes, whereas the [\specialfonts {{\bsf T}}] tensor depends on the assumptions made concerning the location of the axes.

The quadratic correlation between librational and translational motions can be allowed for by including in ([link] cross terms of the type [\langle D_{ik} t_{j}\rangle], or, with ([link], [\eqalignno{U_{ij} &= \langle D_{ik} D_{jl}\rangle r_{k} r_{l} + \langle D_{ik} t_{j} + D_{ji} t_{i}\rangle r_{k} + \langle t_{i} t_{j}\rangle \cr &= A_{ijkl} r_{k} r_{l} + B_{ijk} r_{k} + \langle t_{i} t_{j}\rangle,&(}] which leads to the explicit expressions such as [\eqalignno{U_{11} = \langle \delta r_{1}\rangle^{2} &= \langle \lambda_{3}^{2}\rangle r_{2}^{2} + \langle \lambda_{2}^{2}\rangle r_{3}^{2} - 2 \langle \lambda_{2} \lambda_{3}\rangle r_{2} r_{3}&\cr &\quad - 2 \langle \lambda_{3} t_{1}\rangle r_{2} - 2 \langle \lambda_{2} t_{1}\rangle r_{3} + \langle t_{1}^{2}\rangle,&\cr U_{12} = \langle \delta r_{1} \delta r_{2}\rangle &= -\langle \lambda_{3}^{2}\rangle r_{1} r_{2} + \langle \lambda_{1} \lambda_{3}\rangle r_{2} r_{3} + \langle \lambda_{2} \lambda_{3}\rangle r_{1} r_{3}&\cr &\quad - \langle \lambda_{1} \lambda_{2}\rangle r_{3}^{2} + \langle \lambda_{3} t_{1}\rangle r_{1} - \langle \lambda_{1} t_{1}\rangle r_{3} &\cr &\quad - \langle \lambda_{3} t_{2}\rangle + r_{2} \langle \lambda_{2} t_{2}\rangle r_{3} + \langle t_{1} t_{2}\rangle.&\cr&&(}]

The products of the type [\langle \lambda_{i}t_{j}\rangle] are the components of an additional tensor, [\specialfonts {{\bsf S}}], which unlike the tensors [\specialfonts {{\bsf T}}] and [ \specialfonts{{\bsf L}}] is unsymmetrical, since [\langle \lambda_{i}t_{j}\rangle] is different from [\langle \lambda_{j}t_{i}\rangle]. The terms involving elements of [\specialfonts {{\bsf S}}] may be grouped as [\langle \lambda_{3}t_{1}\rangle r_{1} - \langle \lambda_{3}t_{2}\rangle r_{2} + (\langle \lambda_{2}t_{2}\rangle - \langle \lambda_{1}t_{1}\rangle) r_{3} \eqno(] or [S_{31}r_{1} - S_{32}r_{2} + (S_{22} - S_{11}) r_{3}.] As the diagonal elements occur as differences in this expression, a constant may be added to each of the diagonal terms without changing the observational equations. In other words, the trace of [\specialfonts {{\bsf S}}] is indeterminate.

In terms of the [ \specialfonts{{\bsf L},{\bsf T}}] and [\specialfonts {{\bsf S}}] tensors, the observational equations are [U_{ij} = G_{ijkl}L_{kl} + H_{ijkl}S_{kl} + T_{ij}. \eqno(] The arrays [G_{ijkl}] and [H_{ijkl}] involve the atomic coordinates [(x,\ y,\ z) = (r_{1},\ r_{2},\ r_{3})], and are listed in Table[link] Equations ([link] for each of the atoms in the rigid body form the observational equations, from which the elements of [\specialfonts {{\bsf T},{\bsf L}}] and [\specialfonts {{\bsf S}}] can be derived by a linear least-squares procedure. One of the diagonal elements of [\specialfonts {{\bsf S}}] must be fixed in advance or some other suitable constraint applied because of the indeterminacy of [\specialfonts \hbox{Tr}({{\bsf S}})]. It is common practice to set [\specialfonts \hbox{Tr}({{\bsf S}})] equal to zero. There are thus eight elements of [\specialfonts {{\bsf S}}] to be determined, as well as the six each of [\specialfonts {{\bsf L}}] and [ \specialfonts{{\bsf T}}], for a total of 20 variables. A shift of origin leaves [\specialfonts {{\bsf L}}] invariant, but it intermixes [ \specialfonts{{\bsf T}}] and [\specialfonts {{\bsf S}}].

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The arrays [G_{ijkl}] and [H_{ijkl}] to be used in the observational equations [U_{ij} = G_{ijkl} L_{kl} + H_{ijkl} S_{kl} + T_{ij}] [equation ([link]]


ij kl
11 0 [z^{2}] [y^{2}] [-2yz] 0 0
22 [z^{2}] 0 [x^{2}] 0 [-2xz] 0
33 [y^{2}] [x^{2}] 0 0 0 [-2xy]
23 yz 0 0 [-x^{2}] xy xz
31 0 xz 0 xy [-y^{2}] yz
12 0 0 xy xz yz [-z^{2}]


ij kl
11 0 0 0 0 [-2y] 0 0 0 2z
22 0 0 0 0 0 [-2z] 2x 0 0
33 0 0 0 [-2x] 0 0 0 2y 0
23 0 x x 0 0 y 0 z 0
31 y 0 y z 0 0 0 0 x
12 z z 0 0 x 0 y 0 0

If the origin is located at a centre of symmetry, for each atom at r with vibration tensor [\specialfonts {{\bsf U}}^{n}] there will be an equivalent atom at −r with the same vibration tensor. When the observational equations for these two atoms are added, the terms involving elements of [\specialfonts {{\bsf S}}] disappear since they are linear in the components of r. The other terms, involving elements of the [\specialfonts {{\bsf T}}] and [ \specialfonts{{\bsf L}}] tensors, are simply doubled, like the [\specialfonts {{\bsf U}}^{n}] components.

The physical meaning of the [\specialfonts {{\bsf T}}] and [\specialfonts {{\bsf L}}] tensor elements is as follows. [T_{ij}l_{i}l_{j}] is the mean-square amplitude of translational vibration in the direction of the unit vector l with components [l_{1},\ l_{2},\ l_{3}] along the Cartesian axes and [L_{ij} l_{i} l_{j}] is the mean-square amplitude of libration about an axis in this direction. The quantity [S_{ij} l_{i} l_{j}] represents the mean correlation between libration about the axis l and translation parallel to this axis. This quantity, like [T_{ij}l_{i}l_{j}], depends on the choice of origin, although the sum of the two quantities is independent of the origin.

The non-symmetrical tensor [\specialfonts {{\bsf S}}] can be written as the sum of a symmetric tensor with elements [{S^{S}_{ij}} = (S_{ij} + S_{ji})/2] and a skew-symmetric tensor with elements [{S^{A}_{ij}} = (S_{ij} - S_{ji})/2]. Expressed in terms of principal axes, [\specialfonts {{\bsf S}}^{S}] consists of three principal screw correlations [\langle \lambda_{I}t_{I}\rangle]. Positive and negative screw correlations correspond to opposite senses of helicity. Since an arbitrary constant may be added to all three correlation terms, only the differences between them can be determined from the data.

The skew-symmetric part [\specialfonts {{\bsf S}}^{A}] is equivalent to a vector [({\boldlambda} \times {\bf t})/2] with components [({\boldlambda} \times {\bf t})_{i}/2 = (\lambda_{j}t_{k} - \lambda_{k}t_{j})/2], involving correlations between a libration and a perpendicular translation. The components of [\specialfonts {{\bsf S}}^{A}] can be reduced to zero, and [\specialfonts {{\bsf S}}] made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes [\specialfonts {{\bsf S}}] also minimizes the trace of [\specialfonts {{\bsf T}}]. In terms of the coordinate system based on the principal axes of [\specialfonts {{\bsf L}}], the required origin shifts [\widehat{\rho}_{i}] are [\widehat{\rho}_{1} = {\widehat{S}_{23} - \widehat{S}_{32} \over \widehat{L}_{22} + \widehat{L}_{33}} \quad \widehat{\rho}_{2} = {\widehat{S}_{31} - \widehat{S}_{13} \over \widehat{L}_{11} + \widehat{L}_{33}} \quad \widehat{\rho}_{3} = {\widehat{S}_{12} - \widehat{S}_{21} \over \widehat{L}_{11} + \widehat{L}_{22}}, \eqno(] in which the carets indicate quantities referred to the principal axis system.

The description of the averaged motion can be simplified further by shifting to three generally non-intersecting libration axes, one each for each principal axis of [\specialfonts {{\bsf L}}]. Shifts of the [\specialfonts {{\bsf L}}_{1}] axis in the [\specialfonts {{\bsf L}}_{2}] and [\specialfonts {{\bsf L}}_{3}] directions by [^{1}\widehat{\rho}_{2} = - \widehat{S}_{13}/\widehat{L}_{11} \hbox{ and } ^{1}\widehat{\rho}_{3} = \widehat{S}_{12}/\widehat{L}_{11}, \eqno(] respectively, annihilate the [S_{12}] and [S_{13}] terms of the symmetrized [\specialfonts {{\bsf S}}] tensor and simultaneously effect a further reduction in [\specialfonts \hbox{Tr}({{\bsf T}})] (the presuperscript denotes the axis that is shifted, the subscript the direction of the shift component). Analogous equations for displacements of the [\specialfonts {{\bsf L}}_{2}] and [\specialfonts {{\bsf L}}_{3}] axes are obtained by permutation of the indices. If all three axes are appropriately displaced, only the diagonal terms of [\specialfonts {{\bsf S}}] remain. Referred to the principal axes of [\specialfonts {{\bsf L}}], they represent screw correlations along these axes and are independent of origin shifts.

The elements of the reduced [\specialfonts {{\bsf T}}] are [\eqalignno{^{r}T_{II} &= \widehat{T}_{II} - {\textstyle\sum\limits_{K \neq I}} (\widehat{S}_{KI})^{2}/\widehat{L}_{KK}\cr ^{r}T_{IJ} &= \widehat{T}_{IJ} - {\textstyle\sum\limits_{K}} \widehat{S}_{KI} \widehat{S}_{KJ}/\widehat{L}_{KK},\quad J \neq I. &(}]

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions – three screw librations about non-intersecting axes (with screw pitches given by [\widehat{S}_{11}/\widehat{L}_{11}] etc.) and three translations. The parameter set consists of three libration and three translation amplitudes, six angles of orientation for the principal axes of [\specialfonts {{\bsf L}}] and [\specialfonts {{\bsf T}}], six coordinates of axis displacement, and three screw pitches, one of which has to be chosen arbitrarily, again for a total of 20 variables.

Since diagonal elements of [\specialfonts {{\bsf S}}] enter into the expression for [^{r}T_{IJ}], the indeterminacy of [\specialfonts \hbox{Tr}({{\bsf S}})] introduces a corresponding indeterminacy in [\specialfonts^{r} {{\bsf T}}]. The constraint [\specialfonts \hbox{Tr}({{\bsf S}}) = 0] is unaffected by the various rotations and translations of the coordinate systems used in the course of the analysis.


Cruickshank, D. W. J. (1956). The analysis of the anisotropic thermal motion of molecules in crystals. Acta Cryst. 9, 754–756.
Dunitz, J. D. (1979). X-ray analysis and the structure of organic molecules. Ithaca and London: Cornell University Press.
Johnson, C. K. (1970b). An introduction to thermal-motion analysis. In Crystallographic computing, edited by F. R. Ahmed, S. R. Hall & C. P. Huber, pp. 207–219. Copenhagen: Munksgaard.
Schomaker, V. & Trueblood, K. N. (1968). On the rigid-body motion of molecules in crystals. Acta Cryst. B24, 63–76.

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