The information provided by modern crystal structure analyses is not limited to the atomic arrangement. It also includes, for each atom, a set of quantities known as anisotropic Gaussian displacement parameters (ADPs), which provide information about averaged displacements of atoms from their mean positions. From analysis of these quantities, conclusions can be drawn about the rigid-body motion of molecules, about large-amplitude internal molecular motions, and about the identification of any disorder present in the crystals as being mainly dynamic or mainly static in nature. For some crystals, such analyses yield energy barriers to rotation of rigid molecules or molecular fragments that are in good agreement with values obtained by other physical methods,
What this Article Is AboutThe atoms in a crystal are not stationary; they move ap- lysis of the numerical parameters on which such pictures are based often makes it possible to obtain quantitative information about the rigidity of molecules in crystals, about the nature and degree of rigid-body molecular motions, and even about internal motions of supposedly rigid fragments in nonrigid molecules. With the aid of a few rather
PreliminariesIn most modern X-ray or neutron crystallographic studies of small-molecule structures, it is assumed that the probability density function (pdf) of each individual atom can be represented by a Gaussian. In one dimension, this would be Equation (l), where u2 is the second moment (sometimes known as the variance or dispersion) of the pdf. In three dimensions, the corresponding equation looks more complicated, but it is exactly analogous [Eq. (2)]. Here x is now a vector with three components (xlr x2, x3), and U -I is the inverse of the symmetric secondmoment matrix U. The equiprobability surfaces of this pdf are ellipsoids, and its second moment in an arbitrary direction defined by a unit vector n ( n , , n2, n3) is u2 =nT Un, corresponding to the mean-square displacement amplitude (MSDA) in that direction."] It must be emphasized that these p d f s are not the functions that describe the electron density of the stationary atoms; rather, they approximate the ways in which these electron densities are further spread out (Fig. 2) by the diffuseness in the nuclear positions resulting from lack of perfect periodicity in the crystal. In the real crystal, the atoms are vibrating about their equilibrium positions (dynamic disorder), and they may also be distributed at random over different sets of equilibrium positions from one unit cell to another (static disorder). The p d f s approximate the distributions obtained by averaging the instantaneous atomic positions over time and over all unit cells in the crystal.
