856
J . Phys. Chem. 1988, 92, 856-867
Simonetta, M.; Gamba, A.; Rusconi, E. “Interpretation of the Electron Spectra of Azines by the Molecules in Molecules Method”. In Quantum Aspects of the Heterocyclic Compounds in Chemistry and Biochemistry, The Jerusalem Symposia on Quantum Chemistry and Biochemistry, 11. The Jerusalem Academy of Sciences and Humanities: Jerusalem, 1970. Simonetta, M. “Empirical and Semiempirical Calculations for Reaction Paths”. XXIII International Congress of Pure and Applied Chemistry, London, 1971; Vol. 1. Morosi, G.; Simonetta, M . “The Geometry of the Complex in an Electrophilic Aromatic Substitution Reaction”. In Chemical and Biochemical Reactivity; The Jerusalem Symposia on Quantum Chemistry and Biochemistry, VI. The Jerusalem Academy of Sciences and Humanities: Jerusalem, 1974.
Simonetta, M.; Gavezzotti, A. “The Application of the Extended Hiickel Method to Surface Chemistry and Crystallography: Chemisorption of C2H2on Pt( 111)”. Proceedings IV International Conference on Computers in Chemistry; Novosibirsk, 1979, p 645. Simonetta, M . “Ab Initio Valence Bond Theory and Applications”. Proceedings of an International Conference and Workshop, Barcelona (1981), In Studies in Physical and Theoretical Chemistry; Carbo, R. Ed.; Elsevier: Amsterdam, 1982; Vol. 21, p 239. Simonetta, M.; Gavezzotti, A. “Extended Hiickel and Empirical Force Field Calculations for Chemisorption on Metal Surfaces. XIV Congress of Theoretical Chemists of Latin Expression, Louvain-la-Neuve”. J . Mol. Struct. THEOCHEM. 1984, 107, 75.
FEATURE ARTICLE Interpretation of Atomic Displacement Parameters from Dlffraction Studies of Crystals Jack D. Dunitz,* Organic Chemistry Laboratory, Swiss Federal Institute of Technology, ETH- Zentrum. CH-8092 Zurich, Switzerland
Verner Schomaker,* Division of Chemistry and Chemical Engineering,’ California Institute of Technology, Pasadena, California 91 I25
and Kenneth N. Trueblood* Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: April 2, 1987)
Anisotropic Gaussian displacement parameters (ADP’s), routinely obtained from crystal structure analyses, define the second moments of atomic probability distribution functions and hence provide information about averaged displacements of atoms from their mean positions in crystals. From the ADP’s alone it is not possible to derive rigorous conclusions about crystal or molecular vibrations, but simple models involving correlated motions of groups of atoms are often capable of reproducing the observed ADP’s and their temperature dependence. The possibilities and limitations of such models are discussed. Judiciously exercised and interpreted, the analysis of ADP’s can yield quantities such as mean-square libration amplitudes that are not easily obtainable by other physical methods.
Introduction The idea that atoms and molecules move in crystals-indeed, move sometimes with large amplitude-would have struck most chemists of earlier generations as outlandish. Professor Leopold Ruzicka once opined to one of us that “a crystal is a chemical cemetery”. We know what he meant: long rows of molecules, interred in a rigid geometrical arrangement, lifeless compared with the molecular mazurkas that can be imagined to occur in solution. The view that Ruzicka expressed in his characteristically vivid and outspoken manner is perhaps still widely shared among chemists and even (we suspect) among some crystallographers. It should not be. In 1913, only a few months after the discovery of X-ray diffraction, Debye2 showed in detail how increasing the temperature should reduce the diffraction intensities, especially (1) Contribution No. 7229 from the A. A. Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, CA. (2) Debye, P. Verh. Dtsch. Phys. Ges. 1913, I S , 738-752.
at large scattering angles, by increasing the extent of vibration of the atoms about their average positions. Crystallographers have been confronted by anisotropic “vibration parameters” since they began to use three-dimensional diffraction data in the early 1950s. In 1956, Cruickshank showed how such quantities for anthracene and naphthalene could be interpreted in terms of molecular rigid-body motion3 and thus related to the spectroscopic and thermodynamic properties4 of the crystals. Nowadays, with improvements in the precision and accuracy of intensity measurement, it has become possible to detect quite subtle effects arising from the motion of molecules, and portions of molecules, in crystals. What were originally termed anisotropic “vibration parameters” or “thermal parameters” are more precisely described as anisotropic Gaussian displacement parameters, v’j, called here ADP’s. (3) Cruickshank, D. W. J. Acta Crystallogr. 1956, 9, 754-756. (4) Cruickshank, D. W. J. Acta Crystallogr. 1956, 9, 1005-101 1 .
0022-3654/88/2092-0856%01.50/0 0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 4, 1988 857
Feature Article We have chosen the word displacement in this connection because, ,~ as is now known, there are numerous kinds of d i ~ o r d e rsome static, some dynamic, some temperature dependent, some not, that all affect the intensities of diffraction in proportion to the mean-square displacements of the atom perpendicular to the Bragg plane-not just the authentic thermal vibrations considered by Debye. We shall stay carefully with the term ADP while Yeferring generally to atomic and molecular “motions”, whether or not the actual nature of the displacements in question has been established. ADP’s are obtained now by the thousands from routine crystal structure determinations. Even for analyses with only modest pretensions to accuracy, these parameters can often provide information about rigid-body in crystals and also about internal motions of supposedly rigid groupings of atoms within molec~les.~-’ISince these kinds of information are important in many areas of chemistry, physics, and biology and are not easily available elsewhere, we can expect an increased interest in the use of ADP’s and demand for providing them. However, the measurement and interpretation of these quantities are not free from difficulties. The experimental problems are essentially those inherent in the measurement of a sufficiently large and representative set of Bragg intensities. For the X-ray case, they have been discussed recently by Seiler.Iz In this paper we discuss some of the ways in which ADP’s can be used to provide information about the motion of atoms and groups of atoms in crystals and draw attention to some outstanding problems of interpretation.
Preliminaries The atoms in a crystal are not stationary, and it is not a simple matter to describe the details of their motion. From diffraction studies we can obtain information about the probability density function (pdf) resulting from the time-averaged motion, averaged again over all unit cells in the crystal. In X-ray and neutron crystallographic studies of small-molecule structures, the atomic pdf is usually taken as a Gaussian probability function
where U-’is the inverse of the second-moment matrix U p (uu) of the pdf. If U is positive definite, the surfaces of the set of equations xTU-’x = constant
(2)
are ellipsoids. The ORTEP’~“vibration ellipsoids” that adorn so many illustrations of crystal structures are such surfaces that enclose some specified probability, usually 50%. In X-ray studies, an isotropic Gaussian distribution is usually assumed for hydrogen atoms, not because the motion is believed to be more nearly (5) The perfectly ordered crystal would have every atom firmly fixed to its own perfectly defined site in each unit cell for the entire period of observation. If the atom jumps to a different site, that is one kind of disorder; if it moves to and fro, that is another kind of disorder; if it is forever in one site in a certain unit cell and in a different site in another cell, that is still another kind. Some of these disorders can be described by ADP’s. All of them give rise to diffuse scattering, besides attenuating or altering the Bragg intensity. (6) Schomaker, V.; Trueblood, K.N. Acta Crystallogr., Sect. B 1968, 24, 63-76. (7) Johnson, C. K In Thermal Neutron Diffraction; Willis, B. T. M., Ed.; Oxford University Press: London, 1970; pp 132-160. (8) Dunitz, J. D.; White, D. N. J. Acta Crystallogr., Sect A 1973, 29, 93-94. (9) Trueblood, K. N. Acra Crystalogr., Sect A 1978, 34, 950-954. (10) Trueblood, K. N.; Dunitz, J. D. Acta Crystallogr., Sect. B 1983, 39, 120-1 33. (11) He, X. M.; Craven, B. M. Acta Crystallogr., Sect. A 1985, 41, 244-25 1. (12) (a) Seiler, P. In Lecture Notes, International School of Crystallography, Static and Dynamic Implications of Precise Structural Information; Domenicano, A., Hargittai, I., Murray-Rust, P., Eds.; CNR: Rome, Italy, 1985; pp 79-94. (b) Seiler, P. Chimia 1987, 41, 104-116. (13) Johnson, C. K. “ORTEP A FORTRAN Thermal-Ellipsoid Plot Program for Crystal Structure Illustrations”; Report ORNL-3794; Oak Ridge National Laboratory: Oak Ridge, TN, 1965.
isotropic for hydrogen atoms than for other atoms but because the weak scattering of X-rays by hydrogen atoms calls for extreme economy in the description of their pdf s. The Gaussian description is especially convenient in diffraction studies because the Fourier transform of a Gaussian is also a Gaussian. The atomic “temperature factor^'','^ the Fourier transforms of the corresponding pdfs, then take the form isotropic: anisotropic:
Th = eXp(-2??( U2)hTh) Th = e x p ( - 2 ~ ~ h ~ U h )
(3) (4)
where ( u 2 ) is the (isotropic) mean-square displacement from the equilibrium position, h is a reciprocal-lattice vector, and U is the symmetric matrix or tensor mentioned above. Although many other notations for Th are possible, involving such quantities as B = 8 r 2 ( u 2 )or piJ = 27r2aidLPJ(no summation implied), the interpretation of the corresponding displacement parameters is greatly facilitated by the LPJ n0tati0n.l~ Several different terms have been used for the LfJ’s, but as already discussed, we call them anisotropic Gaussian displacement parameters or ADP’s for short, a description that is essentially neutral regarding the possible physical significance of the quantities concerned. They are measures of mean-square displacements, Le., of the second moments of the pdfs. The mean-square displacement amplitude (MSDA) of an atom in the direction defined by a unit vector n is given by ( u 2 ) , = nTUn, where n is referred to unit vectors parallel to the reciprocal-lattice axes. Note the difference between the two quadratic forms, xTU-’x in (2) and nTUn. Whereas the former quantity is dimensionless, the latter has the dimension (length)2. Moreover, the surface whose radius along n is proportional to nTUn is not an ellipsoid; it is peanut-shaped.16J7 The matrix U is the matrix of second moments of the pdf, whatever the form of the latter. If the pdf is Gaussian, corresponding to motion of the atom in a harmonic potential, it is completely defined by U. Provided that sufficiently definitive diffraction data are available, the higher cumulants7 of nonGaussian pdf s can also be determined, but in practice, for X-ray studies at least, having sufficient data is the exception rather than the rule. Since ADP’s have been obtained by least-squares analysis of X-ray or neutron diffraction data for many thousands of crystal structures, we have available, in principle if not always in practice,18 an enormous amount of information about the pdfs of atoms (14) “Temperature factor” is often a misnomer, because it may have nothing to do, for a given atom in a given crystal, with the temperature. It does not, if U represents zero-point vibrations or static displacements exclusively. For the same reason, reference to the ADP’s as “thermal parameters” is hardly to be condoned. (15) The advantage of the U‘J notation is that the values are independent of the accident of the particular unit-cell edge lengths and so can be compared, from crystal to crystal, more readily than p values. Differences in cell obliquities still somewhat obscure such comparisons, so that for some purposes it is advantageous to report components of U in a Cartesian coordinate system or provide the eigenvalues and eigenvector directions. (16) Nelmes, R. J. Acta Crystallogr., Sect. A 1969, 25, 523-526. (17) Johnson, C. K. In Crystallographic Computing Ahmed, F. R., Ed.; Munksgaard: Copenhagen, 1970; pp 207-219. (1 8) The qualification is necessary because many journals publishing results of crystal structure analyses, including Acta Crystallographica, relegate ADPs to supplementary material, having decreed that for publication all (or almost all) U’s be contracted into scalars, variously described as U,aotrop,c or Ucqulvalcnt or otherwise. This indiscriminate degradation of the information about atomic pdfs in crystals seems a great pity. At present, retrieval of the missing information is often uncertain. Many sets of ADP’s are, in prinicple, available in microfiche in libraries or by request from various depositories (often with considerable delay, inconvenience, or expense). However, numerical data that are not printed are usually not checked at any stage. As a result, the deposited ADP’s are riddled with errors that are difficult to detect and sometimes impossible to correct, besides often being at least in part illegible. A further problem is that some journals lack a deposition scheme altogether, so that data supplementary to papers in such journals are simply consigned to oblivion. As electronic transfer of information and parameter files becomes more common, direct submission of atomic position parameters and ADP’s to databases such as the Cambridge Structural Database19 (CSD) and the Inorganic Crystal Structure Data BaseB (ICSD) will become standard practice, without the present tedious and error-prone intermediate steps of copying (and checking) from printed records. The ICSD already has ADP’s for about half of its 24000 entries; the CSD does not yet include them.
858
The Journal of Physical Chemistry, Vol. 92, No. 4, 1988
in crystals. Although X-ray and neutron diffraction studies, especially at low temperature, can yield U components with nominal estimated standard deviations (esd‘s for short) well under 10 pm2 (1 pmz = lo4 A2),systematic errors arising from inadequate correction for experimental errors (e.g., absorption, thermal diffuse scattering, premature scan termination) may be much larger than this. Indeed, ADP’s at the lower end of the quality spectrum are so tainted by experimental error as to be of dubious physical significance. For example, if an atomic U has one or more negative eigenvalues, its interpretation in terms of a pdf is clearly untenable. Moreover, very large U components (implying a very diffuse pdf) may sometimes arise not only from errors in measurement but also from an incorrect structural model-a grossly misplaced or misidentified atom in the proposed structure. Indeed, the only way a least-squares refinement can eliminate a spurious atom from a structural model is to assign it an extremely high “temperature factor”, corresponding to an almost infinitely diffuse pdf. Not only the U’s are important; so too are their measures of precision. But even when the U’s are published, or otherwise made available, they are accompanied at most by their esd’s,zl whereas the full variance-covariance matrix is needed in order that they can be followed through the transformations among various coordinate systems-oblique crystal axes to Cartesian crystal axes, molecular axes, rigid-group axes, etc.-that are typical of calculations involving ADPs. (It is sad, ironic, and inexcusable that although the covariance matrix is routinely computed in most least-squares programs, it is usually ignored or even suppressed.) Moreover, the assumption of zero covariance is quite unreasonable: even for transformations among orthonormal coordinate systems it will lead to inconsistent resultsz3 for the transformed esd’s. In the absence of covariance values, reasonable assumptions about them have to be made, and suggestions for this have been proposed by Hirshfeld and S h m ~ e l i . ~ ~
Harmonic Oscillator Model The time-averaged motion of a harmonic oscillator in thermal equilibrium with its surroundings is describedZSby a Gaussian pdf. Conversely, a Gaussian pdf for an atom can be regarded as evidence for a harmonic potential for its motion. However, it is an open question how far this model can be applied to the averaged motion of an averaged atom or molecule in a crystal. In the first place, the U’s provide no direct information about the correlations with the motions of other atoms. These correlations are usually ignored in the work we are discussing, and we have ignored them so far here, as we shall again after this reminder. However, they are often important and sometimes subtle. Consider, for example, the pdfs of the atoms of a crystal mounted on a vibrating tuning fork (or, for that matter, on a moving-crystal camera or diffractometer); they have vastly diffuse pdf s as expressed in lab(19) Allen, F. H.; Bellard, S.;Brice, M. D.; Cartwright, B. A.; Doubleday, A.; Higgs, H.; Hummelink, T.; Hummelink-Peters, B. G.; Kennard, 0.; Motherwell, W. D. S.;Rcdgers, J. R.; Watson, D. G. Acta Crystallogr., Sect. B 1979, 35, 2331-2339. (20) Bergerhoff, G.; Hundt, R.; Sievers, R.; Brown, I. D. J . Chem. Znf. Comput. Sci. 1983, 23, 66-69. (21) For a given set of diffraction data and a given atom type, esd’s of diagonal U components, U’, appear to be roughly linear in the V ivalues themselves.z2 (22) Seiler, P.; Martinoni, B.; Dunitz, J. D. Nature (London) 1984, 309, 435-4 3 8. (23) For example, consider the case of isotropic variance with orthogonal axes in two dimensions: two independent variances, which we term V l l i (equal to P2 but not so constrained by symmetry) and V212, and two independent covariances, C1122and C1112 (equal to @212 but not so constrained), the second of which must vanish. If C1Iz2is set equal to zero in one coordinate system, neither it nor some of the other v a r i a n m v a r i a n c e terms will in general be invariant to arbitrary rotation of the coordinate axes. This violates the assumption of isotropic variance, which requires all terms of the v a r i a n m v a r i a n c e matrix to remain unchanged upon rotation. Neglecting the covariances can thus lead to quite incorrect values for the esd’s of the transformed ADPs; the distortion will be even larger with oblique axes. (24) Hirshfeld, F. L.; Shmueli, U. Acta Crystallogr., Sect. A 1972, 28, 648-652. (25) Blwh, F. Z . Phys. 1932, 74, 295-335.
Dunitz et al. oratory-fixed coordinates, yet contribute normally to the intensities of Bragg reflections because the motions of the atoms are almost perfectly correlated. The pdf s relevant to diffraction effects are referred to a coordinate system moving with the crystal and based on the (for this purpose supposedly) rigid crystal lattice. Further, any disorder whatsoever, whether dynamic or static (or even the intrinsic quantum mechanical disorder in the positions of the electrons in the crystal), while attenuating the Bragg intensity, also gives rise to diffuse scattering, which may or may not, depending on the correlations among the excursions of the atoms, have an informative content. For long-range correlations, the corresponding disorder scattering merges with the Bragg scattering, as, e.g., in thermal diffuse scattering. Such correlations could even be of such a nature as to invalidate the usual calculation of the Bragg intensity in terms of the square of the absolute value of the disorder-averaged structure factor. (Imagine a perfectly periodic, rigid, three-dimensional zeolite framework occupied by a perfectly matching, equally rigid and periodic, but loosely fitting polymer molecule, vibrating as a whole against the zeolite framework. Each Bragg intensity is the sum of a zeolite part, a polymer part, and a zeolite-polymer part. The last part is attenuated by the vibration, whereas the first two parts are not; the sum cannot be represented in terms of an averaged structure factor .) In any case, vibrational correlations are the stuff of lattice dynamics, and lattice-dynamical calculations based on atom-atom pair potentials are often capable of producing Gaussian pdf s in good agreement with However, the results are not particularly transparent nor are they easily expressible in terms of simple models. There is therefore an obvious motivation for discussing experimental pdfs (and quantities derived from them) in as simple a way as possible, independently of the results of lattice-dynamical calculations and, indeed, as far as possible, independently of the entire lattice-dynamical approach, Le., ignoring all forms of correlation between the motions of atoms belonging to neighboring molecules or even between rigid groupings within the same molecule. Instead of working from the lattice-dynamical model to the pdfs, we choose to see how far we can go with a knowledge of the pdfs alone. For this purpose a view with some advantages is that the motion of an averaged atom (or molecule or rigid grouping) in a crystal is determined by the effective mean force field exerted on the individual atom (or molecule or rigid grouping) by the crystal environment. Although vastly oversimplified and hence unrealistic in fine detail, it gives a picture of the situation that is capable of faithfully reproducing the U data, and it sometimes leads to reasonable conclusions of wider scope. This is essentially the view taken by Shmueli and K r o ~ n . ~ ~ For many crystals this effective potential appears to be reasonably harmonic, as judged by the temperature dependence of the experimental ADP’s or of quantities directly derived from them, e.g., for n a ~ h t h a l e n e . ~For ~ a particle in a one-dimensional harmonic potential, V(u) = fu2/2, the classical Boltzmann distribution of displacements from equilibrium is P(U)
cc
exP(-fU2/2kT,
(5)
a Gaussian with second moment (u2) =
kT/f
(6)
(26) Pawley, G. S.Phys. Status Solidi 1967, 20, 347-360. (27) Pawley, G. S . Phys. Status Solidi B 1972, 49, 475-488. (28) Filippini, G.; Gramaccioli, C. M.; Simonetta, M.; Suffriti, G. B. J . Chem. Phys. 1973, 59, 5088-5101. (29) Filippini, G.; Gramaccioli, C. M.; Simonetta, M.; Suffriti, G. B. Acta Crystallogr., Sect. A 1974, 30, 189-196. (30) Gramaccioli, C . M.; Filippini, G.; Simonetta, M. Acta Crystallogr., Sect. A 1982, 38, 350-356. (31) Gramaccioli, C. M.; Filippini, G. Acta Crystallogr., Sect. A 1983,39, 784-19 1. (32) Shmueli, U.; Kroon, P. A. Acta Crystallogr., Sect. A 1974, 30, 768-77 1. (33) Brock, C. P.; Dunitz, J. D. Acta Crystallogr., Sect. B 1982, 38, 2218-2228.
The Journal of Physical Chemistry, Vol. 92, No. 4, 1988 859
Feature Article
TABLE I: Some Torsional Potential Barriers” Derived1ofrom ADP’s range of barriers, kJ mol-’ attached from ADP’s other methods group atom = 1.5-9.6 E3 4-26 E3 = 1-9
trig C,N tetr C, N, P 0
-CH3
E3
E3 = 14-62b E3 = 4-10 E3 = 9-66 E3 = 7-1 12
(s), 1.2-8.5 (g), 12-22 CH3OH(g), 4.5 other (s), 15-19 (B), 4-7 (g), 13-25 (g), 2-27
‘These are equivalent cosine function barriers derived from the force constant in (6) or (implicit in) (8). B3 denotes a 3-fold barrier: in the classical approximation,I0 it is = 2 R T / ( 9 ( a Z ) ) . Well-correlated with H bonding.
The Gaussian approximation should hold for small-amplitude vibrations (low temperature, large force constants). For largeamplitude vibrations, it would be desirable to replace (4) by an expression involving higher cumulants of the probability distribution in order to allow for anharmonicity in the potential. However, large-amplitude vibrations lead to a pronounced falloff in the scattering power of the atoms concerned at high scattering angle, so just where the additional parameters in the temperature factor expression are called for they become especially difficult to determine experimentally. Consequently, in what follows, we restrict ourselves to results obtained with the harmonic approximation, which has turned out to be more widely applicable than at first expected. According to ( 6 ) , the MSDA is proportional to the absolute temperature. The Boltzmann distribution over energy levels of a quantized harmonic oscillator leads to (u2) =
h
-
0
4
5
6
atomic-groupingsestimated from experimental ADPS in this way10 and the equivalent cosine function potential barriers (Table I) are found to be in reasonable agreement with values measured by other techniques. These results show that even if the model of the mean-field potential is oversimplified, it is certainly not completely wrong, and we can learn from it.
The Rigid-Molecule and Rigid-Bond Tests If is the MSDA of atom A in the direction of atom B, Le., ( u * ~ , ~then ) , for a rigid molecule, by definition
where v is the frequency and fi the reduced mass (or moment of inertia for a rotational oscillation). For hv >> 2kT, this reduces to (u2)
3
Figure 1. The dashed line is the function y = coth (l/x), which shows the variation of mean-square amplitude with temperature for a harmonic oscillator with x = 2kT/hcw (see eq 7). When the frequency w is expressed in cm-’ and the temperature in kelvin, the mean-square amplitude is essentially proportional to T for T / w greater than about 5 and becomes almost independent of T (see eq 8) when T / w is smaller than about 0.5.
(7)
8rZp
2 T/U
1
AA,B I z’A,B
- z’B,A = 0
(9)
holds for all pairs of atoms within the molecule. In special cases, the converse does not hold A h B may be zero for all atom pairs in a nonrigid group of atoms, e.g., for a planar or linear model with modest “perpendicular” vibrations. Relative motion of rigid subgroups within a molecule can usually be recognized easily by inspection of the experimental A values for all intramolecular pairs of atoms;34within the rigid subgroups’the A values should not
h =81r’pv
the zero-point motion. For hv
