Atomic Motions in Molecular Crystals from Diffraction Measurements By Jack D. Dunitz,* Emily F. Maverick,* and Kenneth N. Trueblood” 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, 1. What this Article Is About The atoms in a crystal are not stationary; they move appreciably about their mean positions. From diffraction studies one obtains information not only about the mean atomic positions but also about the probability density functions of the individual atoms, a measure of their timeaveraged displacements from their mean positions, averaged again over all the repeating units in the crystal. Both kinds of information are expressed in the innumerable pictures in the chemical and crystallographic literature where atoms are conventionally represented by “vibration ellipsoids”.L’l From such pictures it is not difficult to recognize qualitative features of the atomic motions; in Figure 1 , for example, it is apparent that the atoms vibrate preferentially in definite directions and by different amounts. Ana- simple additional assumptions, it is then in principle possible to derive quantities such as force constants or rotation barriers that are normally associated with the realm of spectroscopic methods. The purpose of this article is to describe to chemists and other scientists what can be learned about atomic motions in crystals from the interpretation of diffraction data. 2. Preliminaries In 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 Fig. l. Vibration ellipsoids for all atoms except hydrogens in the molecule of dimethyl 3,6-dichloro-2,5-dihydroxyterephthalate (see formula 3 in Section 6). 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. (xlr x2,x3), and 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 [*] Prof. J. D. Dunitz Organic Chemistry Laboratory, Swiss Federal Institute of Technology, ETH-Zentrum Universitatsstrasse 16, CH-8092 Ziirich (Swilzerland) Prof. E. F. Maverick Department of Chemistry, Los Angeles City College Los Angeles, CA 90029 (USA) Prof. K. N. Trueblood Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90024 (USA) 880 0 VCH Verlagsgesellschaft mbH. 0-6940 Weinheim. 1988 05 70-0833/88/0707-0880 $! 02.50/0 Angew. Chem. Inr. Ed. Engl. 27 (1988) 880-895 Fig. 2. The one-dimensional electron-density distribution p ( x ) is the convolution of the static density po(x) with the probability distribution function D ( x ) . The second moment of D ( x ) ( u 2 ) corresponds to the MSDA of the atom from its average position. The scattering power (or form factor) of a stationary atom is given by the Fourier transform of its electron density distribution. For a vibrating atom, this form factor has to be multiplied by the Fourier transform of the corresponding pdf. One advantage of the Gaussian description is that the Fourier transform of a Gaussian is also a Gaussian. In three dimensions, this is given by Equation (3), where h ( h , , h,, h3) is the scattering vector (of length 2sin6/L) and U is the second-moment matrix mentioned. The quantity T(h) is traditionally known as the “temperature factor”, here the anisotropic temperature factor since it has different values in different directions. It may, or it may not, however, have anything to d o with the temperature; it will, if the pdf associated with U is due mainly to dynamic disorder. O n the other hand, if the pdf is due mainly to static disorder, some components of U will show little or no temperature dependence. The components of U (in general, six for each atom) are usually included as parameters, together with the atomic positional parameters, in the least-squares refinement stages of a crystal structure analysis. They have sometimes been called “vibration parameters” and sometimes “thermal parameters”, but we prefer a description that is more neutral concerning the possible physical significance of these quantities; we shall therefore refer to them simply as anisotropic displacement parameters (ADPs). Isotropic, one-parameter p d f s are usually assumed in X-ray analysis for hydrogen atoms, and sometimes also for other light atoms in the presence of heavy ones. This is not because the corresponding motions are thought to be really isotropic, but rather because the weak relative scattering power of these atoms calls for extreme economy in the number of parameters involved in the description of their p d f s. For organic crystals at room temperature, typical MSDA’s are around 500 to 1000pm2, corresponding to root-mean-square (rms) displacement amplitudes in the range 25 to 30 pm. At 100 K the MSDA’s would be reduced to about a third of their room-temperature values. Nominal standard deviations in atomic positions obtained from carefully measured diffraction data are often 1 pm o r less. Thus the breadth of a n atomic pdf (the rms displacement) is many times larger than the uncertainty in the position of its centroid and is a n appreciable fraction of a typical interatomic distance. Angen. Chem In!. Ed. Engl. 27 (1988)880-895 A Gaussian atomic pdf would correspond to motion of the atoms in a quadratic potential. In principle, the additional terms (higher cumulants) required for the description of non-Gaussian p d f s could also be determined by including appropriate parameters in the least-squares analysis. In addition to the six parameters describing the second moments, one might include ten cubic terms, fifteen quartic ones, and so on. The main problem is that these higher terms are only likely to be important when the second moments are large. However, the larger the second moments, the more rapidly the scattering from the atom in question falls off with scattering angle. Thus, it is just when the higher terms become important that they become virtually impossible to measure. The “vibration ellipsoids” that adorn so many crystallographic publications are surfaces given by Equation (4) xTU-’x=constant (4) that enclose some specified probability (usually 50%). AIthough one can obtain a vivid impression of the directions and relative magnitudes of the atomic motions from the ellipsoids, a more quantitative interpretation requires a detailed analysis of the numbers from which they are obtained, the A D P s themselves. 3. Problems in the Interpretation of ADP’s Before we discuss the potentialities of A D P s for the study of atomic motion in crystals, we have to mention some difficulties. First, there is the problem that the A D P s associated with a given atomic center in a crystal structure refer to the motion of an averaged atom and not to the motion of any individual one. Thus, although a Gaussian pdf for a n individual atom could be regarded as evidence for a harmonic potential for its motion, it is by no means obvious how far this would apply to the motion of the averaged one, where the averaging has been made over all the unit cells in the crystal. (There is a sense in which we can say that this averaged atom does not exist, any more than the average family with 1.743 children exists.) In the second place, the displacements of the individual atoms are highly correlated, and the ADP‘s provide no information whatsoever about the nature of these correlations; an important part of the information necessary to describe the atomic motions has therefore been lost. The assumption that particular groupings of atoms, e.g., molecules or molecular fragments, are rigid can be very helpful here. It is equivalent to assuming certain types of correlations involving amplitude and phase relationships among the atomic displacements, and we shall make considerable use of it. But even with this assumption, we still have to ignore all forms of correlation between the displacements of atoms belonging to neighboring molecules or between rigid groupings within the same molecule. We can circumvent these problems to some extent by adopting what can be described as a mean-field model. We assume that the motion of a given atom (or molecule) in the crystal is governed by the effective potential imposed by its averaged environment. If a suitable force field is 881 available, this potential can be calculated by summing all relevant interatomic interactions, assuming one atom (or molecule) to be displaced while all neighboring atoms (or molecules) remain fixed in their average Although this model is vastly oversimplified, it nevertheless helps us to visualize important aspects of the real, more complex situation, and, as we shall see later, it often even leads to conclusions that agree remarkably well with the experimental observations. 4. The Rigid-Body Model Most interpretations of ADP's are based on the assumption that the crystal contains more-or-less rigid groupings of atoms. Except for certain kinds of constrained refinement, the U'J parameters, the components of the atomic U matrices, are treated as independent variables. However, it is clear that for a perfectly rigid molecule of known geometry, the values of these U'' parameters would be completely determined by the molecular translational and librational oscillations-under the given conditions, they are the only types of motion possible. Even if the molecule is not perfectly rigid, the amplitudes of the individual atomic motions resulting from its internal vibrations can generally be expected to be much smaller than those from its overall translational and rotational motions. It is therefore often reasonable to use the rigid-body model, at least as a first approximation, and attempt to fit the parameters that describe its translational and rotational motions to the observed U'' values. To describe the translational motion we need six numbers, the components (t' tJ) of a symmetric matrix T, analogous to U but now referring to the molecule as a whole. The translational contribution to the ADP's is necessarily the same for all the atoms in the rigid body. The librational motion is more complicated, but it can also be described by the components (AILJ) of a symmetric matrix L. The fitting of the components of T and L to the observed atomic UIJ values can be accomplished by a straightforward linear least-squares procedure. The pioneering work in this direction was done more than thirty years ago by Crui~kshank'~' who showed how the room-temperature A D P s of the individual carbon atoms in anthracenel5l and naphthalene@' could be analyzed in terms of the overall molecular translational and librational oscillations. Within rather large experimental uncertainty (the data were decidedly poor by present-day standards), the principal axes of the motion were found to coincide with the principal inertial axes of the molecule. Both the largest translational vibration (mean-square amplitude I t 2 ) ca. 380 pm') and the major librational axis ((A') ca. 15(O)') were found to be roughly parallel to the long axis of the molecule, as seems reasonable. Cruickshank assumed that the librational axes intersect at a point (for anthracene and naphthalene, the molecular center of symmetry). However, unless such a molecular center is imposed by the space-group symmetry, there is no reason to suppose that the molecular libration axes intersect at all. The more general case, analyzed in detail by Schomaker and T r u e b l o ~ d ,requires [~~ additional parameters to allow for the quadratic correlation between the pure 882 translational and pure librational motions-the most general motion of a rigid body is rotation about an axis, coupled with a translation parallel to this axis, a screw rotation. The additional parameters, nine in number, are of the type @ I f J ) and form the elements of a new matrix S,which is unsymmetrical, since (I' tJ) is different from (AJ f'). In the general case, the elements of T, L , and S can be found by a linear least-squares fit to the observed A D P s , except that the diagonal elements of S cannot be determined completely in this way; only their differences can be obtained. In the computer program THMALsl (several versions abroad, the latest available being THMAl lc9]),the sum of the diagonal elements is arbitrarily set to zero. Except for very small molecules, the minimum of 20 independent UIJ values required to fix the 20 determinable elements of T,L, and S,is usually comfortably exceeded. However, there are a few special cases where the least-squares equations may be ill-conditioned because of the particular geometrical arrangement of the atoms. During the last thirty years, the rigid-body model has been applied, often rather indiscriminately, to the analysis of ADP's from thousands of crystal structures. Where the assumption of reasonable molecular rigidity seems justified, e.g., carbon skeletons of polycyclic molecules, the model usually gives a very good account of itself. For example, applied to the centrosymmetric isomer of tetramethyltetraasterane 1"01or to the tricyclic ketal 2 at 96 K,"'] the rigid-body analysis yields calculated A D P s in almost perfect agreement with the observed values. Another way of assessing the physical significance of the quantities derived from the rigid-body analysis is to compare them with results of a calculation. In principle, given the equilibrium positions of the atoms in a crystal, together with a suitable set of atom-atom potentials, the elements of the T, L, and S matrices for the oscillatory motions of rigid molecules can be obtained from lattice-dynamical calculations. These are basically similar to the type of calculation employed to obtain the vibrational frequencies and normal vibrations of free molecules, but involve infinite, periodic arrays of molecules. For crystals of several aromatic hydrocarbons (e.g., benzene, naphthalene, anthracene, pyrene)"*] the matrices calculated by lattice dynamics are in at least fair agreement with those obtained by analysis of the experimental A D P s , even when the latter were not of the highest quality. This serves to show that, whatever uncertainties there may be in the interpretation of these quantities, they cannot be entirely devoid of physical significance. 5. The Rigid-Molecule and Rigid-Bond Tests For a perfectly rigid molecule, the interatomic distances d o not change by definition, no matter how the individual atoms may move when the molecule is translated o r rotated. Furthermore, because the individual atoms necessarAngew. Chem. Int. Ed. Engl. 27 (1988) 880-895 cation at 100 K.[I5l The triangles labeled A, B, and C contain A values for the three phenyl rings (Fig. 3); the three rectangular blocks AB, AC, and BC contain A values for pairs of atoms belonging to different rings. The estimated standard deviation (esd) of these A values is about 7 pm‘, and the root-mean-square (rms) A values for the three individual rings are 11, 6, and 8 pm’, respectively. Thus, each ring is essentially rigid (ignoring any out-of-plane deformations, which would not be revealed by this test). On the other hand, there is clear evidence of significant motion of the rings relative to one another, especially of ring A relative to rings B and C; the rms A values for blocks AB and AC are four to five times as large as those within the rings. A calculation of the torsional librational amplitude of each ring about its P-C bond (see Section 6) is consistent with these qualitative conclusions; ring A shows a mean-square libration amplitude of 42(”)*, almost an order of rnagnitude larger than those of rings B and C. With data taken at 150 K for the same crystal, the picture is qualitatively similar, the A values and mean-square libration amplitudes ily move in phase, the mean-square displacement amplitudes (MSDA’s) for every pair of atoms A, B must be equal along the interatomic direction [Eq. (S)], where n is the unit AA.R= u: - u i = nT UAn -nT UBn= 0 (5) vector in the A,B direction. Note that the converse does not necessarily hold; for example, AA,Bwill not differ appreciably from zero for planar or linear molecules with modest vibrations perpendicular to the molecular plane or line. Nevertheless, Equation (5) provides a simple method of testing whether the ADP’s obtained for a given molecule are compatible with the rigid-body model.”31It is also often useful as a test of the overall quality of the ADP’s. Bond-stretching vibrations are generally of much smaller amplitude than other kinds of internal motion; for pairs of bonded atoms, therefore, the A’s should be small, according to H i r ~ h f e l d , ” ~ not ~ greater than 10pm’ for typical bonds in organic molecules, such as C-C, C-N, and C-0, but excluding bonds involving hydrogen atoms. Thus, if the A’s for the bonds in an organic molecule are appreciably larger than 10 pm’, there is ground for suspicion that the A D P s are not of the highest quality. Mere inspection of the matrix of AA,Bvalues can often reveal relative motion of more o r less rigid subgroups in a nonrigid molecule. Within the subgroups, the A values should not differ significantly from zero, while relative motion of the subgroups will be manifested by much larger deviations of some of the intergroup A’s. Table 1 shows A values obtained by X-ray analysis for the triphenylphosphine oxide[*]molecule in its orthorhombic crystal modifi- 9 0 I*] Note added by the editorial staff: Since the systematic name of PH, is phosphane according to IUPAC rule 2.3 (inorganic nomenclature) and IUPAC rule D-5.11 + footnote (organic nomenclature), the editorial staff prefers the use of the -ane nomenclature but has allowed the use o f the name triphenylphosphine oxide for PhlPO at the authors’ request. Indeed, rule 2.3 permits the use of phosphine as an alternative while rule D-5.1 1 retains only this trivial name. Fig. 3. Stereoview of triphenylphosphine oxide molecule. The rings A, B, and C, attached to the phosphorus at atoms 1 , 7 , and 13 respectively, correspond to the triangular blocks in Table 1. Table 1. Matrix o f d values [pm2]for triphenylphosphine oxide at 100 K. For atom numbering see Figure 3. Positived’s mean that the MSDA along the interatomic vector is larger for AZ than for A , ; negative A’s mean the reverse. C18 C17 C16 CIS 8 C14 C13 C12 C11 CIO C9 C8 Cl C6 C5 C4 C3 C2 CI 6 31 -19 10 18 38 8 32 10 22 35 31 32 35 9 22 2 29 -15 25 -22 16 9 24 48 41 -10 13 - 9 - 34 - 20 - 25 - 41 - 30 40 -13 -21 36 -25 44 -89 -31 61 2 -34 0 II - 2 -11 s 5 5 2 10 -6 21 01 PI 31 -15 23 - 33 C1 c2 c3 c4 c5 C6 48 I5 49 56 14 17 28 5 26 12 - 32 - 10 - 2 - 23 - 22 - 53 -18 - 34 -11 -44 -45 -59 -68 -32 - 5 -46 -38 -34 -41 -18 c7 C8 c9 c10 CII CIZ 35 19 19 21 -2 6 42 13 12 44 21 24 1 20 -23 -32 4 - 3 4 - 1 19 -23 -30 - 1 - 5 - 1 C13 c14 CIS C16 c17 7 0 -8 2 -5 16 5 -13 4 0 - 6 25 - -so II 100 - 5 -77 74 12 25 51 -18 -31 -40 -49 - 3 -57 -57 -29 -29 -32 -61 -42 -15 -33 -44 -10 -11 - 7 -22 -16 PI -20 A ~ 23 - 19 - 21 13 13 7 7 2 3 2 - 36 - 36 8 - 12 - I7 5 - 5 - 7 - 1 9 5 8 1 1 0 9 3 - 6 B -11 3 -11 C Angew. Chem. In,. Ed. Engl. 27 (1988) 880-895 883 N PI7 N 3 /@17 3 Fig. 4. Stereoview of 13.1.llpropellane denvative, showing atomic numbering used in Table 2. The propellane moiety comprises the atoms C1, C6, C2, C5, C7, N, and C10. being about 50% larger, as expected for the higher temperature. Our second example involves an internal motion that can be identified as incipient inversion of the pyramidal amino nitrogen atom in the [3.1. llpropellane derivative shown in Figure 4. The A values obtained from an X-ray analysis at 95 K[161 are shown in Table 2 and have esd’s of about 7 pm2 for all bonded pairs and only 6 pm2 for all 15 x 14/2 atom pairs of the hexacyclic skeleton (the atoms Cl-C14 and N in Figure 4), a n indication of how remarkably rigid most of this molecule is. Inspection of Table 2 shows that all A values larger than 18 pm2 involve the Nmethyl carbon atom ((217); the four largest (25-39 pm’) are for the vectors between this atom and the phenyl carbons CllLC14. The signs of these four largest A’s in the column headed C17 are all positive, indicating that this atom has a perceptible motion relative to the others. The pattern of A’s indicates that the excess motion of C17 is almost normal to the N-C17 bond and in the molecular mirror plane, that is, along the path that would ultimately lead to inversion at the nitrogen atom. (This direction of motion is indeed suggested by the ellipsoid of C17 in Figure 4,but the effect is shown more convincingly by the A’s in Table I[”]). Because of the ever-present internal molecular vibrations, the rigid-body model can be expected to apply tolerably well only for groupings composed of atoms of comparable mass. In organic molecules, the hydrogen atoms are expected to have quite substantial motions relative to the carbon skeleton; consequently, they have to be ex- cluded from the scope of a rigid-body model calculation, or else their A D P s have to be corrected for the effect of the internal vibrations. The best way of making such corrections would be on the basis of a complete normal-mode calculation for the molecule in question, but this is seldom possible. The A matrix calculated from the ADP’s taken from a very recent, highly accurate neutron-diffraction study of perdeuterobenzene at 15 Kfl’] clearly shows the relative motion of the deuterium atoms in the molecular plane (Table 3a). At this temperature we are seeing mainly the zeropoint motion. Fortunately, a complete normal-mode calculation has been made for benzene.“’I When the calculated zero-point vibration amplitudes for the molecule are subtracted from the observed UiJ values (expressed in an appropriate molecular coordinate system), the corrected A matrix (Table 3b) is essentially that expected for a rigid body. Table 3. C6D6at 15 K. For explanation of signs see Table 1. a) Matrix of A values [pm2]along interatomic vectors. D3’ D2’ D1’ C3’ C2’ CI‘ D3 D2 D1 C3 C2 5 2 -4 c2 - 1 42 -4 D3 b) Matrix of A values corrected for internal motion. c1 c2 C3 1 -I -3 5 - 1 - 5 -1 2 0 0 -6 I 0 0 0 0 6 5 3 1 1 -1 2 -2 5 - 1 -1 4 -3 7 -3 -. DI D2 D3 0 3 0 4 0 0 0 - 4 - 2 -3 0 I -5 0 - 7 0 -1 Table 2. Matrix of A values [pm21for the (3.1.11propellane derivative (Fig. 4) at 95 K. For atom numbering see Figure 4. For explanation of signs see Table 1. N CI c2 c3 c4 c5 C6 c7 C8 c9 CIO CI1 c12 c13 C14 c15 C16 c17 884 C17 C16 -5 2 0 -3 -3 -4 16 7 10 6 4 8 10 6 12 9 8 7 6 6 1 5 4 - 2 - 18 - 17 - 3 17 21 6 33 39 38 25 -3 -3 CIS C14 C13 C12 1 - 2 6 - 3 -17 4 6 4 8 6 3 7 0 2 8 0 -14 - 5 4 5 7 7 6 3 0 1 3 3 3 6 17 6 8 1 10 6 9 8 7 -11 1 -10 - 2 1 1 3 4 4 1 C11 C10 C9 5 2 0 -1 -3 -3 -I -1 5 2 - 5 0 5 - 2 -14 8 0 1 0 0 5 ~ - 5 -13 -12 - 2 4 0 C8 - 6 - 2 - 6 -11 -10 - 6 - 3 C7 -10 - 6 - 7 - 6 - 3 - 1 C6 -7 -3 C5 C4 C3 C2 5 6 2 0 4 2 4 0 -7 -2 -7 -1 -1 - I Angew. Chem. Int. Ed. Engl. 27 (1988) 880-895 The zero-point vibration amplitude of a n atom depends on its mass and should therefore be sensitive to isotopic substitution.1201The mass effect is largest for the isotopes of hydrogen. From Johnson's values for the internal vibration amplitudes of the atoms in the benzene the difference between the MSDA's of protium and deuterium can be estimated to be about 16, 36, and 60 pm' in the radial, tangential, and perpendicular directions, respectively. The difference in the spherically averaged MSDA's is thus about 37 pm'. Although the X-ray scattering powers of different isotopes are identical, we can consider the possibility of distinguishing between protium and deuterium by X-ray analysis on the basis of the difference in their MSDA's. (This distinction is no problem for neutron diffraction, where the scattering power depends o n the nuclear structure and is very different for different isotopes.) An attempt in this direction has been carried out for isotopically labeled (2s)malic acid obtained by enzymatic addition of D 2 0 to fumaric acid. From parallel X-ray analyses of the monoammonium salts of the unlabeled and labeled acids at 93 K,[22'the isotropic MSDA of one of the two methylene hydrogen atoms was found to decrease by about 4 0 p m Z (as expected for the switch from protium to deuterium), while the other hardly changed. The configuration of the malate was thereby indicated as (2S,3R). Since the esd's of the hydrogen MSDA's in this experiment were about 20 pm', the statistical significance of the result is certainly not overwhelming. Nevertheless, it was encouraging that the relative configuration indicated by the X-ray experiment was in agreement with that established by other methods (including neutron diffraction of the corresponding phenylethylammonium saltrZ3]).At any rate, the experiment shows that the use of X-ray diffraction for distinguishing between deuterium and protium should not be dismissed, especially in cases where a neutron-diffraction analysis seems impracticable. In coordination complexes where the ligand atoms may have an appreciably smaller mass than the metal atom to which they are bonded, the Hirshfeld rigid-bond postulate is again to be interpreted in a somewhat relaxed form; for such ligand-metal bonds, A values of the order of 30 pm2 are to be expected (the MSDA of the lighter atom being the larger). Gross violations of this less stringent test have been exploited to detect spin crossover in crystalline [Fe"'(S,CNR,),] complexes[241and to estimate the magnitude of dynamic Jahn-Teller distortions in crystals of Cu" and Mn"' complexes.[251For example, for five octahedral [CuI'L,] complexes expected to show dynamic distortion the average A along the Cu-L direction was 210pm2, whereas for analogous [Ni"L6] complexes the average was 25 pm2. The larger value in the C u complexes is satisfactorily explained in terms of a statistical distribution of distorted octahedra, each having four short Cu-L bonds of 205 pm and two long ones of 233 pm. The spin-crossover analysisrZs1contains a n interesting correlation of A values against bond distance. The Fe-S distance is about 230 pm2 in low-spin [Fe"'(S2CNR2),] complexes and about 245pm2 in high-spin ones. For a sample of more than 30 crystal structure analyses, it was noticed that complexes with Fe-S distances close to these Angew. Chem. Int. Ed. Engl. 27 (1988) 880-895 extreme values had relatively small A(Fe-S) values (ca. 2030 pm'), whereas intermediate Fe-S distances were associated with A(Fe-S) values of u p to 100 pm2. Indeed, the A(Fe-S) versus d(Fe-S) plot was an approximately parabolic curve with its maximum A at an Fe-S distance of about 238 pm. These observations can be explained by assuming that for the intermediate range of Fe-S distances, the crystals contain disordered arrangements of molecules in both spin states. For fraction p of the molecules in the low-spin state, and fraction (1 - p) in the high-spin one, the apparent Fe-S distance (in pm) would be approximately 230p+245(1 -p) and it would be associated with a A value (in pm') of approximately (15)' p(1- p) over and above the "normal" A(Fe-S) value. This is roughly what was found. 6. Internal Molecular Motion in Crystals Breakdown of the rigid-body model for molecules in crystals may be evident from a casual glance at the displacement ellipsoids (Fig. 1) o r from inspection of the matrix o f A values (Tables 1-3). In general, for free molecules, vibrations involving mainly torsional motions have lower frequencies (and larger amplitudes) than those involving changes in bond distances and angles. For molecules in a crystal, internal and external modes of motion will be strongly coupled, so that no rigorous separation is possible. Nevertheless, as we saw in Section 5, internal motions estimated for free perdeuterobenzene, subtracted from the raw ADP's, were quite adequate to correct for the effect of internal motions in the crystal. For more flexible molecules with reasonably large-amplitude torsional motions of semirigid groups (e.g., triphenylphosphine oxide, Fig. 3, Table l), the magnitudes of such motions can be estimated by a slight modification of the usual T, L, S analysis. In the simplest version, due to Dunitz and White,[261an additional parameter, (@'), a mean-square torsional amplitude, is added for each group suspected of undergoing appreciable torsional motion. In this approach, the torsional axis needs to be specified in advance, the group undergoing the torsional motion is assumed to be rigid, and all correlations between the internal motion and the overall motion are neglected. In spite of its simplicity, the Dunitz-White model has been shown to yield essentially the same (4') values as those derived from more elaborate models for the internal molecular motion.['] Nevertheless, there are problems that cannot be ignored. First, there is the problem of specifying the correct kind of internal motion to be included in the model. There is no rigorous way to d o this, as all information about the relative motions of the different atoms has been lost. Chemical experience and intuition may be valuable guides here, but they can also occasionally lead us astray. A poor guess about the nature of the internal motion is sometimes detectable in that it leads to a (42)value not significantly different from zero, or to a physically unreasonable result, for example, a negative value of (4'). Moreover, the impossibility of distinguishing between types of internal molecular motion that differ only with respect to the relative phases of the atomic displacements leads to unavoidable ambigui885 ties. The torsional and out-of-plane bending motions of a nitro group provide us with an example (Fig. 5). In the torsional motion of such a group about the O N 0 bisector, the two oxygen atoms move by equal amounts in opposite directions. In the corresponding out-of-plane bending distortion (which may be simulated by torsion about an axis perpendicular to the O N 0 bisector and in the plane of the group), the two oxygen atoms move by equal amounts but in the same direction. From the ADP‘s alone, we cannot distinguish between these two types of motion. Fig. 5. Rotation (left) and out-of-plane bending (right) of the nitro group differ only in the relative signs (or phases) associated with the displacements of the oxygen atoms (see text). A second problem concerns the total disregard in the Dunitz-White model of the correlations among the different types of motion. Just as, in the rigid-body model, the components of the S matrix were needed to allow for the quadratic correlation between the pure translational and pure librational motions, so similar terms are called for to allow for the analogous correlations among different types of internal motion and between them and the overall translations and rotations. It is not always easy to decide in advance whether these correlation terms are important o r not; in the computer program THMA11,L8~’71 they can be included if the nature of the problem seems to make this desirable. Correlations between motions of groups that have no atom in common cannot be determined from the U’J’S. Even when the extra correlation terms are included in the least-squares fitting, a n unavoidable indeterminacy remains concerning the value of ($A“),where A” denotes the overall libration about an axis parallel to the internal torsion axis. A mathematical analysis, not given here,’271 shows that only ((,Ill)’) and the sum ( 4 ’ ) + 2 ( $ A ” ) can be determined from the U”’s. This means that when (4’) is determined by the Dunitz-White method, the value is only meaningful when it is very large compared with the parallel component of L. Since, usually, we are interested in the internal molecular motions only when they d o have fairly large amplitudes, this restriction is often fairly innocuous. Similarly, when the quantity (4’) 2(4,l”),as obtained in the model including correlation, is large compared with the parallel component of L, it may be regarded as a good approximation to ( $ 2 ) itself. Correlations between internal and overall molecular motions can sometimes be important. In one of the three known crystal modifications of dimethyl 3,6-dichloro-2,5dihydroxyterephthalate 3, the torsional motion of the carboxyl groups about the exocyclic C-C bonds is extremely large ((4’) = 150(”)’ at room temperature), corresponding to rms displacements of the carboxyl oxygen atoms of about 25 pm from the mean molecular plane.‘281(This is the molecule illustrated in Figure 1 with its vibration ellip- + 886 CI -0 OH L=/ 0 soids). The molecular libration axes in the molecular plane also produce displacements of these oxygen atoms perpendicular to this plane (Fig. 6). As mentioned above, correlation of the torsion with the parallel libration axis is indeterminate, but correlation with the perpendicular libration axis adds to the out-of-plane displacement of one of the two oxygen atoms and subtracts from the other, leading to unequal rms displacements, as observed in the crystal structure analysis. The introduction of these correlation terms into the internal motion analysis gave a dramatic improvement in the agreement between observed and calculated U”’s, the R factor1291 decreasing by a factor of 4, from 0.126 to 0.032. Fig. 6. Torsional rotation of the carboxyl group about the exocyclic C-C bond (left) and libration of the entire molecule about the horizontal axis (right) lead to different relative out-of-plane displacements of the carboxyl oxygen atoms. 7. Relations between Mean-Square Amplitudes and Potential Energy For a particle in a one-dimensional quadratic potential, V(x)= f x 2 / 2 , the classical Boltzmann distribution of displacements from equilibrium is given by Equation (6), p ( x ) =(2nkT/f)-”2exp( - f x Z / 2 k T ) (6) which can be recognized as a Gaussian with second moment (x’) given by Equation (7). For many crystals, the sec- (2) = k T/f (7) ond moments of the atomic p d f s are indeed approximately proportional to the absolute temperature over a considerable range. At sufficiently low and sufficiently high temperatures, however, deviations from this linear dependence are to be expected. At 0 K, it would lead to a zero value of (x’), i.e., to an infinitely sharp pdf, in contradiction to the uncertainty principle. The classical Boltzmann averaging is clearly invalid here. When the averaging is made over the energy levels of a quantized harmonic osAngew. Chem. Int. Ed. Engl. 27 (1988) 880-895 cillator, the expression for the second moment of the distribution is given by Equation (8), where v is the frequency (x’) =(h/8rr2/1~)c~th(hv/2kZ‘) (8) and ,u the reduced mass (or, for a rotational oscillation, the moment of inertia, I ) . For hv>>2 kT, the coth factor is unity, and the expression reduces to Equation (9), correspond(x2)=h/8~’pv (9) ing to the zero-point motion. On the other hand, for h v 4 2 k T , the coth factor becomes 2kT/hv, leading back to the same linear dependence of {x2)on T as was obtained with the classical Boltzmann averaging (withfset equal to 4n2pv’). The temperature dependence of (x’), as given by Equations (7) and (8), is shown in Figure 7. T-Fig. 7. Variation of MSDA (x’) (in units of h/8n2fiv) with temperature (in units of hv/2k) for a one-dimensional harmonic oscillator. The linear dependence depicted by the full line is based on the assumption of a classical Boltzmann distribution, while the dashed curve corresponds to a quantized distribution. For many crystals, the effective mean-field potential in which the atoms move seems to be reasonably quadratic, as judged by the observed temperature dependence of the A D P s or of quantities derived from them, such as T and L components. A fairly thorough study has been made for naphthalene.[301The main deviation from the behavior expected for a quadratic potential is that experimental A D P s tend to increase with temperature rather more steeply than implied by the linear dependence. This can be explained as a consequence of the general softening of the effective potential as the crystal expands on warming. In a more realistic model-the quasi-harmonic rnodelf3’]-the effective force constant of Equation (7) is not constant but decreases with increasing temperature. For a diverse set of inorganic solids, the ADP’s are found to be approximately proportional to T’.5,1321 and a similar relationship has been shown to hold for lithium hydrogen phthalate monohydrate,’331with A D P s estimated by neutron diffraction at 15, 100, and 298 K. As mentioned earlier in Section 3, the experimental A D P s refer to the motion of an averaged atom or moleAngew. Chem. I n [ . Ed. Engl. 27 (1988) 880-895 cule. From experimental values of (x2)and a knowledge of the temperatures at which they were measured, Equations (7) or (8) provide estimates of effective quadratic force constants (or frequencies) for the mean-field motions in question. The mean-field potential associated with a given type of motion can be obtained, at least conceptually, by following the potential energy variation when one atom (or molecule) is gradually displaced from its equilibrium position or orientation, all neighboring atoms (or molecules) being held in their equilibrium positions. This is an oversimplified picture. A more realistic model should contain some means of allowing for the ways in which the instantaneous displacements of different atoms (or molecules) are correlated. This is achievable from a complete latticedynamical treatment, but only at the cost of considerable additional mathematical complexity and loss of conceptual simplicity. In lattice dynamics,[341a suitable force field is used to solve the equations of motion for the whole periodic ensemble of molecules that constitute the crystal structure. The calculation yields the frequencies of all the normal modes of vibration of the crystal. The MSDA’s of the atoms (or molecules) are then obtained by summing contributions from these normal modes. For each type of motion, say, a molecular libration about a particular axis, characterized by a single frequency in the mean-field model, we now have a range of frequencies, a so-called branch of the frequency spectrum. However, the calcuIations show that the frequency dispersion within a given branch is usually not very large. Thus, although the precise physical significance of the effective mean-field force constants and frequencies derived from Equations (7) and (8) may be open to question, the numerical values should generally not differ too much from analogous quantities that appear in more complete dynamical models. In fact, as shown in Table 4, experimental A D P s for a variety of librating rigid groups (such as methyl and tert-butyl) yield quadratic force constants and corresponding cosine-function rotation barriers (see Section 8) that are at least in the same range as those derived by other physical Table 4. Some cosine-function torsional barriers derived from ADPs Group Attached atom Range of barriers [kJ mol -‘I From A D P s Other methods [a] CH3 C, N (trigonally coordinated) C , N,P (tetrahedrally coordinated) 0 1.5-9.6 (s) 1.2-8.5 4-26 (g) 12-22 1-9 CH,OH (g) 4.5 other (s) 15-19 C 14-62 (well correlated with H-bonding) C(CH,), C (trigonally coordinated) C (tetrahedrally coordinated) C , N, P 4-10 (8) 4-7 9-66 (g) 13-25 7-112 [a] s=solid: g=gas. 887 8. Rotation Barriers in Crystals from ADP’s Consider a n individual molecule in crystalline benzene. The molecule, assumed rigid for the moment, will oscillate about its mean orientation, and, as shown by N M R methods, will also from time to time carry out 60” rotational jumps that turn it from one orientation to an equivalent one. The energy barrier hindering such a rotation must have sixfold periodicity in the angular coordinate and, to a first approximation, it can be represented by Equation (lo), where B is the barrier height per mole and n = 6 . For 0.7 0.6 0.5 2 0.L 0.L 0.3 0.2 small excursions from the equilibrium orientation at 4 = O”, we have cos n 4 = 1 - n24’/2, and hence Equation (1 1). In this approximation, the potential energy is a qua- 0.1 0 0 20 LO 60 80 #I”] dratic function of 4. With the help of Equation (7), derived on the basis of a classical Boltzmann distribution, relationship (11) is converted into Equation (12), where @ 2 is re- placed by its mean value. Thus, the barrier may be estimated from a knowledge of {@), which is obtainable from analysis of the ADP’s. This is the approximation used by Trueblood and dun it^'^^] in the study leading to the results shown in Table 4; it is valid only when (nz42) is much smaller than unity, i.e., when the barrier B is at least several times larger than RT. When this condition is not satisfied, V ( 4 ) can no longer be approximated as a quadratic function and the probability distribution for 4 is no longer Gaussian. By analogy with Equation (6), however, the dependence of this probability distribution on B/RT can be obtained from (13), p(@)=Nexp( - B( 1 - cosn@)/2RTJ (13) where N is a normalization factor. This function is shown in Figure 8, where the gradual change from nearly Gaussian for R T < B to nearly flat distribution for R T > B is clearly seen. A more accurate relationship giving (42)as a function of B and RT[Eq. (14)] is obtained by integrating the classical Boltzmann expression (13). Numerical integration of Equation (14) for various values of RT/B leads to the solutions shown graphically (for n = 5 ) in Figure 9, where the linear dependence expected for the quadratic potential described by Equation (11) is also shown. At very low values of R T/B, the classical averaging leading to Figure 9 cannot be expected to hold. The value of (4’) must then be estimated from an expression analogous to Equation (S), based on quantum mechanical averaging. ‘@’) 888 = j@’exp(- B(1- cosn@)/ZRTJd@ j exp( - B( 1 - cos n9)/2RT J d # (14) - 100 120 1LO 160 180 Fig. 8. One-dimensional probability distribution functions [Eq. (13)] for a sinusoidal potential (dashed curve) with barrier height B at various temperatures. The estimation of a rotation barrier with the help of Figure 9 is quite simple. From an observed value of ( I $ ~ )for a given librating group in a crystal, one reads off the corresponding value of RT/B. A knowledge of the temperature T a t which the measurement was carried out then leads directly to the barrier B. The method has been appliedo6]to a series of crystalline metallocenes, for which extensive diffraction data are available over a wide temperature range, and for which barriers for rotation of the cyclopentadienyl rings have also been estimated by NMR spectroscopy (mainly from ‘H spin-lattice relaxation times), by incoherent quasi-elastic neutron scattering (IQENS), and by Raman and IR 300 4 RT/B - Fig. 9. Full curve: Variation of mean-square libration amplitude (4’) with R T / B for a fivefold periodic sinusoidal restricting potential with barrier height B. Dashed curve: (@’) for harmonic oscillator potential with the same quadratic force constant. At very low temperatures, both curves should run parallel to the horizontal axis at the nonzero value of (9’) corresponding to the zero-point motion, as indicated in Figure I . Angew. Chem. Int. E L Engl. 27 (1988) 880-895 spectroscopy. A comparison of the results obtained by the different methods is given in Table 5. Table 5. Apparent barriers to ring rotation [kJ mol-'1 (esd's in parentheses) in crystalline metallocenes by various methods. (Diffraction results yield individual barriers for symmetry-independent rings at the measurement temperature.) Compound Diffraction Ferrocene (triclinic) Other methods: NMR unless mentioned (see [36] for detailed references) 101 K 123 K 148 K 7.6(7) 12(2) 8.4(6) 9(2) 6.4(4) 10.2(5) 6.8(7) 8.5(8) 5.9(5) lO(2) 7.0(6) 8(1) 98 K Ferrocene (orthorhombic) 24.8( 1.0) 23(20) 33(16) Ferrocene (monoclinic) Nickelocene (monoclinic) Ruthenocene (orthorhombic) 7.5(8); 10.3(5), 1 l(2); 8.3; = 9 (IQENS) 173 K 293 K 2.7(5) 4.7(8) 101 K 293 K 6.5(6) 5.2(5) 101 K 293 K 25110) 250 24(7) 38(13) 5.4(5); 4.4(5)(IQENS) 5.0 (Raman); 6.3 (IQENS) 9.618); 18.9 are for triclinic The most extensive diffraction ferrocene, stable or at least metastable below 164 K. There are four symmetry-independent cyclopentadienyl rings, and the crystal structure was determined at three temperatures (101, 123, 148 K). Table 6 shows that the ( I $ ~ ) values increase with temperature and also vary from ring to ring. Table 6. Mean-square libration amplitudes (#*) [(")'I (with esd's in parentheses) in crystalline metallocenes at various temperatures. Compound Triclinic Ferrocene 101 K 123 K 148 K ring IA IB 2A 28 3514) 20(4) 31(3) 26(3) 55(5) 30(3) 47(6) 38(5) Orthorhombic 78(8) 37(5) 63(6) 47(9) ring 1 2 Monoclinic 98 K 173 K 9(7) 7(3) 231(27) 220(30) 101 K 45(4) Nickelocene 293 K 293 K 199(18) Ruthenocene ring 1 2 The first trend may be explained by the thermal expansion of the crystal, which will increase intermolecular distances and thus lower the intermolecular contribution to the barrier. The variation from ring to ring is also probably genuine, arising from differences in packing of the individual rings. This is supported by packing energy calculations,[381 using atom-atom potentials, which reproduce the general From Table 5 it is seen that for triclinic ferrocene, the barrier heights derived from (t~5~)values agree as Angew. Chem. Int. Ed. Engl. 27(1988) 880-895 well as can be expected with those from NMR and IQENS, considering that the latter values refer to averages over a considerable temperature range and that differences among the individual rings are not resolved by the spectroscopic methods. In monoclinic ferrocene,'4'] there is only one symmetryindependent cyclopentadienyl ring. Although the roomtemperature result appears to agree well with the spectroscopic values, this is almost certainly fortuitous. From Table 6, it is seen that (#2) for the monoclinic modification is virtually constant between 173 and 293 K, a strong indication of static disorder in the ring orientations. Indeed, both and neutron diffra~tion'~'' analyses have shown that the centrosymmetric, staggered arrangement of the cyclopentadienyl rings, which appears to be demanded by the space-group symmetry of monoclinic f e r r ~ c e n e , is ~~~' actually a result of static disorder. Since Figure 9 is based on the assumption that the ( # 2 ) value in question is due to genuine librational motion and not to static disorder, little confidence can be placed in the barriers derived by this method for monoclinic ferrocene. Although a rotation barrier can be obtained from a (@) value at a single ternperature, this example illustrates the advisability of multipletemperature measurements, if only to detect the presence of disorder masquerading as vibration. For nickelocene, too, the good agreement indicated in Table 5 between the different methods should perhaps also be regarded with a certain reserve, as there are indications that an order-disorder phase change may possibly occur in this crystal at some intermediate temperature. For orthorhombic ferrocene and ruthenocene, the lowtemperature values of ( # 2 ) are so small that they may be due mainly to zero-point motion, in which case the use of Figure 9 (based on classical averaging) to estimate the barriers may lead to inaccuracies. Besides, the low relative precision of ( # 2 ) leads to further uncertainties in the derived values of the barriers. In view of these difficulties, the agreement (Table 5) between the barriers obtained from ( q j 2 ) and from spectroscopic methods seems eminently satisfactory. The solid-state barriers are considerably larger than those found for the free molecules. In the gas phase, ferrocene is eclipsed, with a barrier to rotation of the cyclopentadienyl rings of about 3.8 kJ mol -' the barrier in nickelocene is probably still lower.Iu1 For free ruthenocene, no experimental value is available, but a theoretical suggests that the barrier is about 2 kJ mol-' higher than that in ferrocene. The increase in the barriers in the solid state is due to intermolecular packing effects and is reproduced nicely by force-field cal~ulations.[~~1 The recent highly accurate neutron-diffraction study of crystalline perdeuterated benzene C6D6[Is1offers another opportunity to test the ( # 2 ) method of obtaining rotation barriers. Here the crystallographically imposed molecular symmetry is a center of inversion. For the libration of the molecule in its own plane, (#2) is estimated to be 2.6(")* at 15 K and 1l.O(o)z at 123 K. The latter value corresponds to a sixfold rotation barrier of 17 kJ mol-*, well within the range (16.5-18.4 kJ mol-') obtained by solid-state NMR The barrier obtained from the 15 K analysis is only 8 kJ mol-', but here again the use of Figure 9 889 (adapted for n = 6) is probably impermissible because of quantum effects at this low temperature; the 17 kJ mol-’ barrier would actually correspond to a zero-point libration of about 4(”)*,not too different from the observed (q52) at 15 K. It seems quite remarkable that barrier heights obtained from (#2) values agree so well on the whole with those estimated by spectroscopic methods. After all, the underlying assumptions behind the two kinds of analysis are so very different. Spectroscopy provides frequencies, relaxation times, and correlation or residence times, which, in turn, can be used to yield rotation barriers by assuming a n Arrhenius type of dependence of rate or time on temperature. Starting from (@), on the other hand, we use classical Boltzmann averaging for a cosine type potential (which can be, at best, the leading term in the Fourier expansion of the periodic potential) to estimate the barrier height at a given temperature. In a sense, one can say that spectroscopy sees the rate at which molecules cross the barrier, while diffraction sees the bottom of the potential well; we derive roughly the same barrier height as long as the potential is approximately sinusoidal. The two methods are thus nicely complementary for studies of molecular motion in the solid state. 9. Vibration or Static Disorder? Among crystallographers, disorder is usually regarded as a nuisance, occasionally as an opportunity to test some potential improvement in refinement techniques, but only seldom as a phenomenon interesting in its own right. Yet disorder is ubiquitous, even in the “ordered” crystalline state, and the physical properties of solids usually depend just as much on the nature and number of the defects from the idealized, perfectly ordered, periodic structure, as on the structure itself. It also has to be admitted that, although diffraction methods are highly successful in establishing the averaged structure of the crystal unit cell, they have much less to say about the nature of the fluctuations from this averaged structure. This is because the influence of all types of disorder on the main Bragg reflections is summed u p in the atomic pdfs. However, as we have seen, it is possible from temperature-dependence studies to distinguish disorder that is mainly dynamic (vibrational) from that which is mainly static in origin. Beyond that, the most we can d o is to enquire whether a given set of atomic p d f s is compatible with a proposed disorder model o r not. Such enquiries sometimes lead to interesting answers. The room-temperature X-ray analysis of tetra-tert-butylcyclobutadiene 4i471 led to a structure in which the sides of the slightly nonplanar four-membered ring were almost equal (146.4 and 148.3 pm), in contrast to results for other cyclobutadiene derivatives where much larger differences 890 (up to 26 pm) between the ring bonds were observed. Ermer and H e i l b r ~ n n e r proposed ~~~’ that the nearly square ring found at room temperature did not correspond to the actual molecular structure but to averaged atomic positions resulting from disorder among two rectangular structures. This was confirmed by a subsequent low-temperature analysis at 123 K,1491where the ring bond lengths were found to differ significantly (144.1 and 152.7 pm), but still not as much as might have been expected. This raises the question whether residual disorder may not still be present at the lower temperature. To answer this, we can postulate a model, calculate a set of A D P s for it, and examine the pattern of discrepancies between the observed A D P s and those calculated for the model. The crystallographically imposed molecular symmetry is here a C, axis (although the effective molecular symmetry in the crystal is D2). An analysis of the A D P s in terms of the T,L,S model with allowance for internal libration of the two independent tert-butyl groups about the exocyclic C-C bondsisoileads to (4’) values of 14 and 23 for the two internal motions, several times larger than the overall molecular librations. Among the discrepancies between observed U’’ components and those calculated for this model, the excess apparent motion of the ring atoms in the tangential directions (by 5 I and 3 1 pm2 for the two symmetry-independent atoms) is striking and is in qualitative agreement with the disorder model proposed by Ermer and Heilbr~nner[~*I (Fig. 10). In the crystal it is not necessary that the two perpendicular orientations of the rectangular ring have a common center, as shown in Figure 10, and, (O)’ I Fig. 10. Disorder model for 4 as proposed by Ermer and Heilbronner [48] and confirmed by analysis of experimental A D P s from the 123 K crystal structure analysis [SO]. In the crystal, one orientation is about four to five times more frequent than the other. indeed, there are indications that their centers d o not quite coincide. The observed excess apparent motion indicated by this analysis is compatible with the presence of two perpendicular orientations of a rectangular ring with sides 134 and 160 pm, with one orientation about four to five times more frequent than the other. On the other hand, a quite different result is obtained from a comparable analysis of the A D P s from the lowtemperature (100 K) crystal structure analysis of 1,3,5,7-tetra-tert-butyl-s-indacene 5 In solution, the twelve C atoms of the ring system give only four I3C-NMR signals that show no perceptible line broadening down to - 130”C;[5’1 this is consistent either with a very low energy Angew. Chem. Int. Ed. Engl. 27 (1988) 880-895 barrier between the two valence isomers o r with a completely delocalized 12-electron n system. In agreement with this, the X-ray analysis at 100 K (and at room temperature) shows effective D2,, symmetry of the carbon skeleton, even though the crystallographically imposed molecular symmetry is only C,. Here again the T,L,S analysis with allowance for torsional motion of the tert-butyl groups leads to a systematic pattern of discrepancies between caiculated and observed CJ‘’ components, but the indications are now such that any additional motion required to improve the agreement must be perpendicular to the molecular plane, as would be produced by a n out-of-plane bending or buckling of the ring skeleton. There is no indication of disorder between two sets of atomic positions for the skeletal atoms in the molecular plane, as found in 4. Either the equilibrium structure of 5 has DZhsymmetry with a delocalized n-eletron system, or, possibly, the set of atomic positions found in this crystal does not correspond to the groundstate molecular structure but rather to the transition state for the valence isomerization. In the crystal, such a transition state could be stabilized relative to the ground state by packing forces. As a precedent, in a somewhat analogous conformational isomerization, we are reminded of the example of biphenyl, planar in its crystal structure, nonplanar as a free m o I e c ~ I e . ~ ~ * ] The low-temperature (99 K) X-ray analysis of crystalline hexaisopropylbenzene 61531provides another carefully field calculations, the two orientations are present in about 2 : 1 ratio, and distances between positions of corresponding atoms are 35.7, 76.9, and 11.5 pm for the aromatic, methine, and methyl carbon atoms, respectively. In the standard least-squares refinement, only the two methine carbons were assigned individual positions and A D P s , the aromatic and methyl atom pairs being treated as single scatterers. The “vibration ellipsoids” for the aromatic carbon atoms, based on this model, clearly show the additional motion in the tangential direction, corresponding to the unresolved disorder between the two atomic sites. Since the homomerization energy of 6 is calculated to be about 150 kJ mol-’, the disorder in this crystal is to be classed as static, in contrast to the previous examples. 10. Corrections to Interatomic Distances The rigid-body model has been applied in the analysis of A D P s from thousands of crystal structures.[541Usually, the main objective has been to obtain libration corrections to interatomic distances. As first noted by Cruick~hank,~”~ rotational oscillations of molecules cause the apparent atomic positions to be slightly displaced from the true positions towards the rotation axes. For a mean-square libration ( # 2 ) , the radial displacement is given approximately by r#’/2, where r is the distance of the atom from the axis (Fig. 12). Errors from librational motions about orthogonal axes are additive. 6 studied example of a disordered structure resulting from the presence of two molecular orientations at crystallographically equivalent sites. In the final structural model (Fig. 1 I), based on a combination of X-ray evidence and force- Fig. I I . Disorder model (stereoview) for 6 as deduced from the crystal structure analysis at 99 K and from force-field calculations [53].The major and minor orientations (thick and thin lines) are present in a ratio of approximately 2 : I . Angew Chem. Inl. Ed. Engl. 27 (1988) 880-895 More generally, X-ray and neutron diffraction analyses locate the centroids of p d f s, and separations computed from these centroids cannot be interpreted directly as average interatomic distances[561(Fig. 13). To determine the corrections required to convert such separations into interatomic distances, we need information about the correlations between the instantaneous atomic displacements, and this information is usually unavailable. Any estimate of the corrections must then depend on assumptions about the Fig. 13. The distance d between the atomic pdf centroids equals the interatomic distance for in-phase motion of the two atoms. For out-of-phase motion the mean interatomic distance is larger than d. 89 1 relative phases of the motions of different atoms.[”] In the rigid-body model, molecular libration corresponds to a correlated motion that makes the molecule appear to shrink slightly. In room-temperature structures, errors in interatomic distances derived from centroids of atomic p d f s without corrections may amount to more than 5 pm, considerably greater than all the other errors put together. Bond distances derived from such structures should be regarded as highly unreliable, because even when corrections are made they are often based on assumptions whose validity is by no means assured. In particular, estimates of libration amplitudes obtained with the T,L,S analysis may be wildly in error when the implicit assumption of molecular rigidity is not warranted. In the riding model, it is assumed that one atom, the “riding” atom, has all the motion of the atom to which it is bonded, plus an additional uncorrelated motion. This model is particularly appropriate for hydrogen atoms (in cases where the A D P s are available from neutron diffraction studies), and X-H bond distance corrections from this source alone can amount to 5 pm.[”l Uncorrected bond distances tend to become shorter with increasing temperature. In reasonably accurately determined structures, however, the motional corrections lead to fairly constant values. For example, the mean apparent (uncorrected) C-C distance in triclinic ferr~cene’~’]decreases from 142.6(1) to 141.5(1) pm as the temperature increases from 101 to 148 K, but the motional corrections lead to much more nearly equal values, 143.3 and 143.0 pm, respectively. Similarly, for naphthalene at five temperatures between 92 and 239 K, the trend towards shorter CC bonds virtually disappears when corrections are made for the effects of rigid-body librational Nevertheless, when bond distances are to be determined with the highest possible accuracy, it is best to reduce libration corrections by performing the crystal structure analysis at as low a temperature as possible. This also has the advantage that all the other types of atomic motions, where motional corrections are more difficult to apply, are reduced as well. But even bond distances obtained by lowtemperature X-ray analysis and corrected for residual libration may be in error if the centroids of the atomic p d f s are displaced from the nuclear positions by contamination with the nonspherical bonding electron density. This depends on the resolving power of the experiment and brings us to the discussion of the experimental problems involved in the measurement of p d f s, a topic that we have avoided so far. 11. Experimental and Computational Aspects The quality of the ADP‘s obtained in a crystal structure analysis obviously depends in the first place on the quality and range of the experimental observations-how accurately the intensities of the Bragg reflections were measured, and how many observations were made. This, in turn, depends o n many factors, the quality of the crystals, their chemical and physical stability under the measurement conditions, the temperature at which the measure892 ments were made, the wavelength of the radiation, the mechanical and electronic stability of the diffractometer, the appropriateness of the measurement protocol to the particular problem at hand, and many other factors besides. Important aspects of these problems, particularly in the context of X-ray analysis, have been recently examined by Seiler,[591 who has discussed strategies for reducing the errors in the measurement of Bragg reflections. The A D P s obtained in a n analysis are also quite sensitive to the way in which the numerical data are treated, especially to details of the least-squares procedures used in the refinement of the structure. Since this article is aimed at the chemist, not at the practicing crystallographer, we mention only a few points that seem particularly relevant. We are often interested in the precision and accuracy of the A D P s as well as in their particular values. The nominal standard deviations associated with the A D P s depend on the extent as well as on the quality of the experimental data.[601With a given set of diffraction data, the A D P s of atoms of stronger scattering power will be the more precisely determined; for atoms of the same type, esd’s of diagonal U“ components appear to be roughly proportional to the U“ values An independent estimate of the precision of the ADP’s can sometimes be made by examining the A values [Eq. ( 5 ) for parts of molecules, such as carbon skeletons of benzene rings, that can reasonably be assumed to behave as rigid bodies; the rms A for a supposedly rigid body should be roughly @ times the average esd of the atomic U components. At the lower end of the quality spectrum, a minimal criterion for the physical significance of the A D P s is that the U matrices must be positive definite; i.e., the constant probability surfaces of Equation (4) must correspond to ellipsoids and not to other quadratic surfaces, such as hyperboloids. Apart from the random errors of measurement, most kinds of experimental error affect the A D P s in a systematic way; i.e., they tend to make the A D P s of all the atoms too large, o r too small, or too large in one direction and too small in another. In the rigid-body model, these systematic errors affect mainly the description of the translational motion; the librational motion, and also the values [Eq. ( S ) ] , are relatively unaffected. This means that estimates of rigid-body librational motion or of internal torsional motion may still be reasonable, even when the experimental data are seriously flawed.[611 In X-ray analysis (not in neutron diffraction), one of the main problems in obtaining accurate A D P s (and atomic positions as well) is the fact that the electron-density distribution in the crystal differs slightly from the superposition of the electron densities of the isolated, spherically averaged atoms. This difference, known as the deformation density o r bonding density, is of interest in the study of chemical bonding and is a subject for itself.L6Z1 The standard, spherically symmetrical, atomic scattering factors used in the analysis of X-ray diffraction data from crystals take n o account of these deformations. This neglect leads to a bias in the atomic positions (and also in the ADPs), the amount of which depends in a complicated way on the nature of the atom, its environment, and the extent of the data set. For example, carbonyl oxygen centroids tend to be drawn towards the lone-pair region, away from the carAngew. Chem. Int. Ed. Engl. 27 (1988) 880-895 bon atom to which they are bonded, and C=O bond lengths tend therefore to be slightly (about 1 pm) too long. Similarly, the p d f s of carbon atoms in benzene rings tend to be slightly too large in the direction perpendicular to the plane of the ring, because the atomic vibration in this direction is not resolved from the n-electron density. One way to reduce the free-atom bias is to use exclusively high-order reflections in the least-squares refineor to assign artificially large weights to these reflection~.~’~] This is because the valence electrons lie mainly in the outer part of the charge distribution around a n atom, so that their contribution to the X-ray scattering falls off more rapidly with increasing scattering angle than that of the inner, core electrons. In fact, not inconsiderable errors in the positional parameters (up to 1-2pm) and A D P s (up to SO pm’) may be introduced by refining only with low-order reflections (say, within the limit of C u K a radiation) o r even with more extensive data sets that include low-order refle~tions.[~’’ Probably the best way of reducing such errors is to model the deformation density by a suitable set of multipole terms centered at each atom. The free-atom scattering factors are augmented by the Fourier transforms of the multipole deformation terms,[661and the coefficients of these additional terms are then obtained, together with the positional parameters and A D P s , in a special kind of least-squares refinement. However, the additional computational effort can be quite formidable and is only justified when the experimental data are of the very highest quality. In spite of all the above-mentioned difficulties, reasonably accurate A D P s have been measured for thousands of crystal structures. The main problem today is that much of the information has disappeared, some of it, we fear, irretrievably. For some time now, nearly all published numerical data describing crystal structures have been restricted to a list of atomic positional coordinates with “isotropic” or “equivalent” displacement parameters; the information expressed by the six components of U has been compressed into a single scalar quantity. The ADP’s have been relegated to supplementary information, which is sometimes deposited in libraries o r other institutions and sometimes not. Deposited A D P s can in principle be retrieved (but often only at the cost of considerable delay, inconvenience, and expense). Numerical data that are not printed are rarely checked. As a result, deposited A D P s , when they can be found at all, are riddled with errors that are difficult to detect and sometimes impossible to correct. Some journals lack a deposition scheme altogether, so that data supplementary to papers published in them are simply lost. One can sympathize with the reluctance of editors of scientific journals to publish long lists of numbers (six per atom for the usual Gaussian ADP‘s), but at the same time steps have to be taken to secure the information and make it more easily accessible. Data bases probably offer the best solution to the problem. At present, the Inorganic Crystal Structure Data Base (ICSD)[671contains A D P s for about half of its 25000 entries. The Cambridge Structural Data Base (CSD),[681dealing with organic crystal structures, has yet to include but we are assured that this omission will be remedied very soon. Angew Chem Int Ed. Engl 27 11988) 880-89s 12. Macromolecules The ideas and applications outlined in this article should be particularly relevant to the study of biological macromolecules, proteins, and nucleic acids. Experimental evidence and theoretical arguments agree that these are fluctuating systems that exist in a very large number of conformational sub state^.[^*^ In the “equilibrium structure” of such biological macromolecules, some atoms should have very diffuse, anisotropic pdfs, while others should be fairly well localized. For an understanding of the detailed mechanisms of biomolecular recognition processes and enzymatic reactions, a knowledge of the relative flexibility and rigidity of different parts of these molecules will certainly be needed, and it is this information that can be provided, in principle at least, by analysis of the ADPs. Unfortunately, however, when it comes to the measurement and interpretation of A D P s from crystalline proteins and oligonucleotides, the problems mentioned in the earlier sections become vastly more severe. In the first place, the experimental resolution attainable for most macromolecular crystals is so limited that refinement of A D P s is usually out of the question. There are simply not enough experimental data. In most analyses of crystalline proteins, only isotropic atomic U’s have been determined, leading in some cases to deductions about the general rigidity or looseness of different parts of the structures. The second main problem concerns the interpretation, which is even less straightforward than for small-molecule structures. In protein crystals, the atomic MSDA’s are compounded of displacements arising from several sources: the multiplicity of conformational substates, diffusion (as in fluid-like regions of the structure), lattice disorder, and genuine vibration. These will all have different kinds of temperature dependence, and it is almost impossible to separate their contributions to the total mean-square displacements. In a few cases, the temperature dependence of the isotropic U’s has been followed; myoglobin is probably the best studied example.[”’ In macromolecules, where the excursions of the atoms from their mean positions are much larger than in small-molecule structures, the potential energy associated with such displacements will not necessarily be quadratic, and the temperature dependence of U expressed in Equation (7) cannot be expected to hold. In myoglobin, different types of temperature dependence were observed for atoms in different regions of the molecule, and attempts were made to interpret these results in terms of the functionality. More recently, A D P s have been obtained by refinement of more extensive diffraction data for several small proteins, including carp muscle calcium binding p r ~ t e i n , ’ ~ ” rubredoxin (for atoms in 30 of its 54 amino and avian pancreatic polypeptide (for atoms in all 36 of its amino No very striking conclusions have been drawn, but many of the results could be interpreted in terms of models involving concerted motions of groups of atoms in distinct regions of each molecule. A model in which biomolecules are regarded as being built from semirigid groups of atoms, each of which moves more o r less independently of the others, is the basis for what has been termedi7’] the “segmented rigid-body model 893 of thermal motion”.1761With this model, in the refinement of the overall structure, the 20 components of T, L , and S are determined for each of a number of assumedly rigid subgroups in the macromolecule, as if each were an independent unit. Refinement of a dodecanucleotide with this model helped to improve the agreement between calculated and observed diffraction intensities and thus reduced the noise level of the difference Fourier map, allowing additional low-occupancy water molecules to be identified. In the analysis of bovine pancreatic ribonuclease A, a model of this kind was applied to the 13 aromatic sidegroups (histidine, tyrosine, phenylalanine).1771 The resulting L components were small and inconclusive, the greater part of the group motions being derived from translation. The pattern of results again suggested collective motions of atoms in regions larger than single residues. Advances in measurement techniques (synchrotoh radiation, area detectors) and in constrained refinement procedures will undoubtedly soon lead to ratios of data to adjustable parameters large enough to permit a determination of anisotropy for at least some of the atoms in many other macromolecules. Together with the kind of information about structure and dynamics that is becoming available from high-resolution two-dimensional NMR these advances should eventually lead to a better understanding of structure-function relationships in biological macromolecules. Much of this article was written while E. F. M . and K . N . T. were on sabbatical leave in Zurich. The work described here has been supported over the years by the Swiss National Science Foundation. Received: October 16, 1987 [A 676 IE] German version: Angew. Chem. 100 (1988) 910 [I] C. K . Johnson: ORTEPII: A FORTRAN Thermal Elliosoid Plot Proaram for Crystal Sfmcture Illustrations (Report ORNL-51381, Oak Ridge National Laboratory, Oak Ridge, TN, USA 1976. J. D. Dunitz, V. Schomaker, K. N. Trueblood, J. Phys. Chern. 92 (1988) 856. U. Shmueli, P. A. Kroon, Acfa Crystallogr. Sect. A30 (1974) 768. D. W. J. Cruickshank, Acta Crystallogr. 9 (1956) 754. D. W. J. Cruickshank, Acfa Crysfallogr.9 (1956) 915. D. W. J. Cruickshank, Acfa CrystalJogr. 10 (1957) 504. V. Schomaker, K. N. Trueblood, Acta Crystallogr. Sect. 8 2 4 (1968) 63. K . N. Trueblood, Acta Crystollogr. Sect. A34 (1978) 950. Available on application to K.N.T., UCLA, or to Dr. W.8. Schweizer, Organic Chemistry Laboratory, Swiss Federal lnstitue of Technology, ETH-Zentrum, CH-8092 Zurich. J. P. Chesick, J. D. Dunitz, U. von Gizycki, H. Musso, Chem. Ber. 106 (1973) 150. K. L. Brown, G. I. Down, I. D. Dunitz, P. Seiler, Acta Crysfallogr.Sect. 8 3 8 (1982) 1241. a) G. Filippini, C. M. Gramaccioli, M. Simonetta, G. B. Suffritti, J. Chem. Phys. 59 (1973) 5088; b) Acta Crystallogr. Sect. A 3 0 (1974) 189; c) C. M. Gramaccioli, G. Filippini, M. Simonetta, ibid. 38 (1982) 350. R. E. Rosenfield, Jr., K. N. Trueblood, J. D. Dunitz, Acta Crysfallogr. Secf. A34 (1978) 828. F. L. Hirshfeld, Acfa Crysfallogr.Sect. A 32 (1976) 239. C . P. Brock, W. B. Schweizer, J. D. Dunitz, J. Am. Chem. SOC.J07(1985) 6964. P. Chakrabarti, P. Seiler, J. D. Dunitz, A.-D. Schliiter, G. Szeimies, J. Am. Chem. Sac. 103 (1981) 7378. An alternative to qualitative inspection of the matrix of A values is to make a half-normal probability plot of the ratios of these A values t o their esd’s against their statistical expectation values. When the slope of this line is significantly smaller than unity, nonrigidity of the group (or serious underestimation of the esd’s) is indicated. See C. Kratky, H. Falk, U. Zrunek, Monatsh. Chem. I16 (1985) 607. 894 G. A. Jeffrey, I. R. Ruble, R. K. McMullan, J. A. Pople, Proc. R. SOC. (London) 414 (1987) 47. S . J. Cyvin: Molecular Vibratronsand Mean Square Amplitudes. Elsevier, Amsterdam 1968, p. 245. At high enough temperature, Equation (7) (Section 7) applies, and the mean-square vibration amplitude IS independent of the mass and depends only on the temperature and the force constant. C. K. Johnson in B. T. M. Willis (Ed.): Thermal Neutron Diffrction. University Press, Oxford 1970, p. 132. P. Seiler, B. Martinoni, J. D. Dunitz, Nature (London) 309 (1984) 435. R. Bau, I. Brewer, M. Y. Chiang, S. Fujita, J. Hoffman, M. I. Watkins, T. F. Koetzle, Biochem. Biophys. Rex Commun. 115 (1983) 1048. K. Chandrasekhar, H. B. Biirgi, Acta Crysfalloyr. Sect. 8 4 0 (1984) 387. a) I. H. Ammeter, H. B. Burgi, E. Gamp, V. Meyer-Sandrin, W. P. Jensen, Inorg. Chem. 18 (1979) 733; b) M. Stebler, H. 8 . Biirgi, J. Am. Chem. Sac. 109 (1987) 1395. J. D. Dunitz, D. N. 1. White, Acta Crystallogr. Sect. A29 (1973) 93. V. Schomaker, K. N. Trueblood, Acla Crysfalloyr. Sect. A40 Suppl. (1984) C-339; a fuller paper is promised and should appear in Acfa Crystallogr. Sect. A . Q.-C. Yang, M. F. Richardson, J. D. Dunitz, J. Am. Chem. SOC. 107 (1985) 5535. Sometimes known as the agreement factor, sometimes as the disagree{Zw[U”(obs)- U’J(calcd)12/ ment factor, here defined as ~ w [ U ’ J ( o b s ) ] z ) with ” 2 , w a weighting factor, usually taken a s inversely proportional to the variance of (1’’. C. P. Brock, J. D. Dunitz, Acta Crysfallogr.Sect. 8 3 8 (1982) 2218. B. T. M. Wiliis, A. W. Pryor: Thermal Vibrations in Crystallography, Cambridge University Press, London 1975, p. 142. T. J. Bastow, S. L. Mair, S. W. Wilkins, J. Appl. Phys. 48 (1977) 494. H. Kiippers, F. Takusagawa, T. F. Koetzle, J. Chem. Phys. 82 (1985) 5636. G. S. Pawley, Phys. Sfafus Solidi 20 (1967) 347. K. N. Trueblood, J. D. Dunitz, Acfa Crystallogr. Sect. 8 3 9 (1983) 120. E. F. Maverick, J. D. Dunitz, Mol. Phys. 62 (1987) 451. P. Seiler, J. D. Dunitz, Actu Crystallogr. Sect. 8 3 5 (1979) 2020. C. P. Brock, E. Maverick, J. D. Dunitz, unpublished results. As these calculations take n o account of cooperative motions, it comes as no surprise that the barriers obtained with them are uniformly too high. P. Seiler, J. D. Dunitz, Acta Cryslallogr. Sect. 8 3 5 (1979) 1068. F. Takusagawa, 7. F. Koetzle, Acta Crystallogr. Secf. 8 3 5 (1979) 1074. a) J. D. Dunitz, L. E. Orgel, Nature (London) 171 (1953) 121; b) J. D. Dunitz, L. E. Orgel, A. Rich, Acta Crysrallogr. 9 (1956) 373. A. Haaland, I. E. Nilsson, Acta Chem. Scand. 22 (1968) 2653. L. Hedberg, K. Hedberg, J Chem. Phys. 53 (1970) 1228. S . Carter, J. N. Murrell, J. Orgnnomet. Chem. 192 (1980) 399. W. E. Sanford, R. K. Boyd, Can. J. Chem. 54 (1976) 2773. H. Irngartinger, N. Riegler, K.-D. Malsch, K.-A. Schneider, G. Maier, Angew. Chem. 92 (1980) 214; Anyew. Chem. I n f . Ed. Engl. 19 (1980) 211. 0. Ermer, E. Heilbronner, Angew. Chem. 95 (1983) 414; Angew. Chem. Int. Ed. Engl. 22 (1983) 402. H. Irngartinger, M. Ntxdorf, Angew. Chem. 95 (1983) 415; Angew. Chem. I n f . Ed. Engl. 22 (1983) 403. 1. D. Dunitz, C. Kriiger, H. Irngartinger, E. F. Maverick, Y. Wang, M. Nixdorf, Angew. Chem. 100 (1988) 415; Angew. Chem. lnf. Ed. Engl. 27 (1988) 387. K. Hafner, B. Stowasser, H.-P. Krimmer, S. Fischer. M. C. Bohm, H. J. Lindner, Angew. Chem. 98 (1986) 646; Angew. Chem. Int. Ed. Engl. 25 (1986) 630. See a) H. Bonadeo, E. Burgos, Actu Crystallogr. Sect. A38 (1982) 29 for a lattice-dynamical treatment of biphenyl and b) W. R. Busing, Acta Crystallogr. Secf. A39 (1983) 340 for an analysis of packing effects in this crystal. 1. Siege!, A. Gutierrez, W. B. Schweizer, 0. Ermer, K. Mislow, J. Am. Chem. Sac. 108 (1986) 1569. _ _including . many where its applicability is highly questionable, to say the least. D. W. J . Cruickshank, Acta Crystallogr. 9 (1956) 757. W. R. Busing, H. A. Levy, Acta Crystallogr. 17 (1964) 142. I n principle, information about correlations among the atomic motions is available from the generalized mean-square displacement matrix computed in a complete lattice-dynamical calculation. See, for example, C. Scheringer, Acta Crystallogr. Sect. A 28 (1972) 512, 616. Corrections to interatomic distances have sometimes been made in this way: for example, C . M. Gramaccioli, G. Filippini, ibrd. A41 (1985) 356 and [52a]. A correction due to anharmonicity in the X - H stretching vibration may amount to half of this, in the opposite direction, and so may cancel out much of the riding correction. See a) J. A. Ibers, Acfa Ciystallogr. I 2 (1959) 25 I ; b) G. A. Jeffrey, I. R. Ruble, Trans. Am. Crystallogr. Assoc. 20 (1984) 129; c) B. M. Craven, S. Swaminathan, ibid. 20 (1984) 133. Angew. Chem. I n l . Ed. Engl. 27 11988) 880-895 [59] a) P. Seiler in A. Domenicano, I. Hargittai, P. Murray-Rust (Eds.): Lecture Notes. International School of Crystallography, Static and Dynamic Implications o/ Precise Structural Information, Erice, Italy 1985, pp- 7994; b) Chimio 41 (1987) 104. [60] When A D P s are transformed from one coordinate system to another, the variances of the transformed coordinates depend not only on the variances in the initial system but also on the covariances. This information is routinely calculated in most least-squares programs but it is usually ignored. I611 For example, Cruickshank's 1956 estimates of rigid-body librational amplitudes in anthracene [5] were based on data that are very poor by present standards although exemplary for the time (visually estimated threedimensional set of reflection intensities from Weissenberg photographs obtained several years earlier by A. McL. Mathieson, J . M. Robertson, V. C. Sinclair, Acta Crystallogr. 3 (1950) 245). [62] a) P. Coppens, M. B. Hall: Electron Distributions and the Chemical Bond. Plenum, New York 1982; b) K. Angermund, K. H. Claus, R. Goddard, C. Kruger, Angew. Chem. 97 (1985) 241; Angew. Chem. i n f . Ed. Engl. 24 (1985) 237. [63] P. Coppens, Isr. J . Chem. 16 (1977) 159. 164) J. D. Dunitz, P. Seiler, Acta Crystullogr. Sect. B29 (1973) 589. 1651 P. Seiler, W. B. Schweizer, J . D. Dunitz, Acta Crystallogr. Sect. 8 4 0 (1984) 319. [66] a) R. F. Stewart, Isr. J . Chem. 16 (1977) 124; b) F. L. Hirshfeld, ibid. 16 (1977) 198. 1671 G . Bergerhoff, R. Hundt, R. Severs, 1. D. Brown, J . Chem. InJ Comput. Sci. 23 (1983) 66. [68] F. H. Allen, S. Bellard, M. D. Brice, B. A. Cartwright, A. Doubleday, H. Higgs, T. Hummelink, B. G. Hummelink-Peters, 0. Kennard, W. D. S. Angew. Chem. Inr. Ed. Engl. 27 (1988) 880-895 Motherwell, J. R. Rodgers, D. G. Watson, Act0 Crystdogr. Sect. 8 3 5 (1979) 2331. [69] As electronic transfer of numerical information becomes more common, direct submission of parameter files containing atomic coordinates and ADP's to databases, such as the ICSD [67] or the CSD [68], should become standard practice. This will avoid the present tedious and errorprone stages of copying from printed records. 1701 G . Careri, P. Fasella, E. Gratton, Crit. Rev. Biochem. 3 (1975) 141. [71] a) H. Frauenfelder, G . A. Petsko, D. Tsernoglou, Nature (London) 280 (1979) 558; b) D. Ringe, J. Kuriyan, G. A. Petsko, M. Karplus, H. Frauenfelder, R. F. Tilton, I. D. Kuntz, Trans. Am. Crysfallogr. Assoc. 20 (1984) 109. I721 J. H. Konnert, W. A. Hendrickson, Acza Cryszallogr. Secf. A36 (1980) 344. 1731 K. D. Watenpaugh, L. C. Sieker, L. H. Jensen, J . Mol. Biol. 128 (1980) 615. [74] 1. Glover, 1. Haneef, J. Pitts, S. Wood, D. S. Moss, I . Tickle, T. Blundell, Biopolymers 22 (1983) 293. 1751 S. R. Holbrook, R. E. Dickerson, S.-H. Kim, Acta CrystaNogr. Sect. 8 4 1 (1985) 255. 1761 An unfortunate turn of phrase, since it will almost inevitably lead to confusion with the similarly named model due to Johnson [21] in which the segments form parts of an overall molecule for which T,L, and S are determined together with the torsional amplitudes about the intersegmental links. 1771 D. S. Moss, I. Haneef, B. Howlin, Trans. Am. Crystallogr. Assoc. 20 (1984) 123. [78] K. Wiithrich: N M R of Proteins and Nuckic Acids, Wiley, New York 1986. 895

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