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Atomic Motions in Molecular Crystals from Diffraction Measurements.

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Atomic Motions in Molecular Crystals from Diffraction
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
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)
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)
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
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
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
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 , ” ~
~ 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-
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
is larger for AZ than for A , ; negative A’s mean the reverse.
- 9
- 34
- 20
- 25
- 41
- 30
- 2
- 32
- 10
- 2
- 23
- 22
- 53
- 34
- 5
- 3
- 1
- 1
- 5
- 1
- 5
- 3
- 7
- 19
- 21
- 36
- 36
- 12
- I7
- 5
- 7
- 6
Angew. Chem. In,. Ed. Engl. 27 (1988) 880-895
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
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
Table 3. C6D6at 15 K. For explanation of signs see Table 1.
a) Matrix of A values [pm2]along interatomic vectors.
D3’ D2’
CI‘ D3
- 1
b) Matrix of A values corrected for internal motion.
1 -I
5 - 1 - 5
2 -2
5 - 1
0 - 4 - 2
0 - 7
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.
- 2
- 18
- 17
- 3
- 2
- 3
- 5
- 2
- 5
- 2
0 0
- 5
- 2
- 6
- 2
- 6
- 6
- 3
- 6
- 7
- 6
- 3
- 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
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-
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 )
which can be recognized as a Gaussian with second moment (x’) given by Equation (7). For many crystals, the sec-
= k T/f
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
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
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
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
Attached atom
Range of barriers [kJ mol -‘I
From A D P s
Other methods [a]
C, N
C , N,P
(s) 1.2-8.5
(g) 12-22
CH,OH (g) 4.5
other (s) 15-19
(well correlated
with H-bonding)
C , N, P
(8) 4-7
(g) 13-25
[a] s=solid: g=gas.
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
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-
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
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.
j@’exp(- B(1- cosn@)/ZRTJd@
j exp( - B( 1 - cos n9)/2RT J d #
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
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.)
Other methods: NMR
unless mentioned (see [36]
for detailed references)
101 K
123 K
148 K
98 K
24.8( 1.0)
7.5(8); 10.3(5), 1 l(2); 8.3;
= 9 (IQENS)
173 K
293 K
101 K
293 K
101 K
293 K
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.
101 K 123 K 148 K
ring IA
ring 1
98 K
173 K
231(27) 220(30)
101 K
293 K
293 K
ring 1
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
(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
(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
(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,
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.
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
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
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
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
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
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An alternative to qualitative inspection of the matrix of A values is to
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Sometimes known as the agreement factor, sometimes as the disagree{Zw[U”(obs)- U’J(calcd)12/
~ w [ U ’ J ( o b s ) ] z ) with
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C. P. Brock, J. D. Dunitz, Acta Crysfallogr.Sect. 8 3 8 (1982) 2218.
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P. Seiler, J. D. Dunitz, Actu Crystallogr. Sect. 8 3 5 (1979) 2020.
C. P. Brock, E. Maverick, J. D. Dunitz, unpublished results.
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lattice-dynamical treatment of biphenyl and b) W. R. Busing, Acta Crystallogr. Secf. A39 (1983) 340 for an analysis of packing effects in this
1. Siege!, A. Gutierrez, W. B. Schweizer, 0. Ermer, K. Mislow, J. Am.
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_ _including
many where its applicability is highly questionable, to say
the least.
D. W. J . Cruickshank, Acta Crystallogr. 9 (1956) 757.
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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
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(1959) 25 I ; b) G. A. Jeffrey, I. R. Ruble, Trans. Am. Crystallogr. Assoc.
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Angew. Chem. I n l . Ed. Engl. 27 11988) 880-895
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[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).
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[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.
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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
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[78] K. Wiithrich: N M R of Proteins and Nuckic Acids, Wiley, New York
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