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Determination of Reaction Paths for Pentacoordinate Metal Complexes with the Structure Correlation Method.

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Determination of Reaction Paths for Pentacoordinate Metal Complexes with
the Structure Correlation Method
Thomas Auf der Heyde”
Dedicated to Professor Hans-Beat Biirgi
Crystal structures potentially deliver far
more information than is present in the
average structural communication-if
sufficient structural data on closely related molecules or molecular fragments
are available, it may be possible to infer
dgtails of geometric changes occurring
along certain reaction pathways for the
species of interest. This geometric information is extrapolated from an analysis
of the similarities between the structures
of the fragment in the various crystalline
environments, by a method that is now
known as structure correlation analysis.
Since it was first proposed twenty years
ago, the method has been applied to a
large variety of chemical systems, but
none have received as much attention as
the class of five-coordinate compounds.
Comparative analyses of the structures
of pentacoordinate complexes have yielded information about the intimate mechanisms of substitution and addition/elimination reactions at tetrahedral and
square-planar complexes, and about intramolecular isomerizations of five-co-
1. Introduction
One of the most irksome and paradoxical shortcomings of the
molecular approach to chemistry that is currently prevalent in the
discipline“1 is the lack of a suitable technique for directly observing changes occurring at the molecular level during the course of
a chemical reaction. In this context, the geometric characteristics of a particular reaction are usually inferred from dynamic
NMR or kinetic studies, or from volume of activation measurements, for example. In some cases they may be obtained from
the results of a b initio calculations, but even the unprecedented
development of computational technology in the last few years
has not yet enabled a quantum chemical solution for most of the
real systems that the discipiine has turned its attention to.[*]
Although direct observation of a chemical molecule along a
given reaction coordinate does not seem feasible, its visualization at least does. The structure correlation
41 assumes that the geometric changes that a given molecular fragment would undergo along a specific reaction coordinate are
mirrored by the gradual distortion or static deformation manifested by that fragment collectively over a large variety of crystalline environments. The various crystal or molecular structures
[*I Dr. T. Auf der
Department of Chemistry, University of Cape Town
Rondehosch. Cape Town 7700 (South Africa)
Tdrfdx: Int. code + (213650-3788
AIIFM.. C h m . In!. Ed Engl. 1994,33, 823 -839
$3 VCH
ordinate compounds. Since its inception,
the structure correlation method has
gradually adapted techniques from other
branches of science, in particular grouptheoretical and multivariate statistical
techniques, which have been shown to
be enormously powerful tools for probing geometrically complex systems. This
review traces the development of the
method of structure correlation and the
evolution of these co-opted techniques,
with a specific emphasis on studies of
five-coordinate metal complexes.
are considered to constitute a series of “frozen-in’’ points, or
snapshots, taken along the reaction pathway, which, when viewed
in the correct order, yield a cinematic film of the reaction.
The rationale behind the hypothesis is enticingly simple. Crystalline structures represent stable atomic arrangements-their
representative points on the Born -0ppenheimer potential energy surface will consequently tend to aggregate around local
potential minima, either a “well”, a “dip”, or a point in a “valley”. Crystal structures of closely related molecular fragments
will be represented by points variously displaced from the potential minima, along pathways oj’minimum energy. Arranged in
the correct sequence, then, these fragments can be assumed to
map such pathways, their static deformation mirroring that expected along the coordinate. This idea has found expression in
the structure correlation hypothesis: If a correlation can be
found between two or more independent parameters describing
the structure of a given structural fragment in a variety of environments, then the correlation function maps a minimum energy path in the corresponding parameter
Essentially, the method of structure correlation involves choosing a suitable data set comprising crystal or molecular structuresr5’that are closely related to the fragment of interest. and
then searching for correlations (similarities) between them. In
the twenty years since the first paper explicitly outlining the idea
of structure correlation was p ~ b l i s h e d , ‘ ~the
] method has been
applied to a wide variety of chemical systems, including organic,
Vcrlugs~esellschufim h H , D-6945fWemheim. 1994
0570-0833/94:0H#R-0823 S 10.00+ .25:0
T. Auf der Heyde
inorganic, organometallic, and cluster compounds. During this
time it has undergone considerable development involving the
application of sophisticated multivariate statistical techniques,
and group-theoretical methods. Some of the earlier studies have
already been
b1 and a book covering later work,
with a comprehensive summary of the methods that have been
developed during this time, appeared in late 1993.[61
Undoubtedly, the class of compounds that has been most
frequently and intensively studied by the technique of structure
correlation is that of pentacoordinate complexes, which contains
both main group elements and transition metals; indeed, the
first paper in this area examined five-coordinate cadmium comp l e ~ e s . [Because
these compounds have been so widely studied,
the various papers offer interesting historical comparisons: 1)
collectively they trace the various stages in the development of
the structure correlation method (SCM), and 2) they illustrate
how successively more complex methods of correlation analysis
have yielded significantly greater insight into the stereochemistry of pentacoordination. This paper reviews structure correlation studies of pentacoordinate metal[’] complexes with these
two aspects in mind. Papers dealing with nonmetal complexes
will not be discussed here since they have received attention
elsewhere[EJand in any event, d o not add significant information, nor do they shed more light on the development of the
SCM. We begin with a brief overview of the techniques that have
proved useful in structure correlation studies.[g1
2. The Method of Structure Correlation
2.1. Representing Molecular Structures
Chemists represent molecules in many ways;[”] for the purposes of the SCM, their “depiction” in terms of a set of numbers
is the most convenient for the subsequent statistical correlation
analysis. This requirement necessitates an important conceptual
leap in that the molecules comprising a given data set are no
longer regarded as three-dimensional objects embedded in Cartesian space; they are, instead, represented by points in a multidimensional space spanned by a set of parameters (or variables)
that are capable of defining the molecular geometry. In general,
3N - 6 such parameters are necessary for the complete description of an N-atom fragment.[”] Both internal coordinates (for
example, bond angles and distances) and external coordinates
such as fractional (crystallographic) coordinates can serve this
function, but in almost all cases published thus far, the former
have been used to represent the molecular geometry.[”’ Bond
angles and distances and other internal coordinates like them
have the advantage of being easy to use and to visualize, and of
appealing readily to the chemist’s intuition, but they d o suffer
from several disadvantages that may introduce bias in comparative studies. Firstly, in general, no unique set of 3 N - 6
parameters can be chosen to describe the molecular structure, so
the selection will be biased by an arbitrary choice of parameters.
But this choice is not always easy to make, since the larger the
molecule, the greater the number of alternative parameters,[’31
and the more complicated and opaque the geometric relations
between them become ; this complicates the identification of
redundant or dependent coordinates. Secondly, in crystal structure analyses the internal coordinates are not the original refined parameters, and their errors are not directly estimated: an
error in one coordinate may be coupled with errors in others,
and there is no guarantee that such an effect will be obvious.
Finally, the parameters are usually not all of the same type or
have the same units (scale), and this can severely distort an
analysis of covariance.
Where observed molecular geometries need to be referred to
an idealized or a reference molecule, deformation coordinates or
symmetry coordinates (SCs)-originally developed for use in
vibrational spectroscopy-are very useful.[’41 SCs are, for example, linear combinations of internal coordinates that transform according to the irreducible representations of the point
group G of the chosen reference molecule; their derivation is
described in most textbooks on applied group theory.[I5]They
are closely related to normal coordinates, and thus to the vibrational modes of the molecule.
As an example, consider a n analysis of triatomic molecules
that attempts to describe their deformations in relation to a reference molecule XYX with G = CZv;any such molecule is fully
described by two bond-length increments ( r , and r,) and the
increment in the bond angle (A@, relative to the respective bond
lengths and the bond angle in the reference structure. These
(internal) coordinates transform as 2 A , B , under C,, symmetry, and the corresponding SCs are given in Scheme 1. Any given
molecule would be represented by a particular point in the
(three-dimensional) space spanned by S , , S,, and S , . A dis-
Born in South Africa in 1958, Thomas Aufder Heyde studied chemistry at the University of
Cape Town ( U C T ) . As an undergraduate he worked,for two years as a research assistant in
coordination chemistry with D. A . Thornton. After obtaining his Honours degree (1981j he
took up a lectureship at the University ?f the Western Cape (UWC) , completing his Masters
and Ph.D. degrees with L. R. Nassimbeni 11988). During this titne he spent some months in
Berne, Switzerland, with H.-B. Biirgi us pre-doctoral,felloM~.He returned fo Berne after completing his studies, and suhsequentlji worked with F A . Allen in Cambridge at the Crystallographic Data Centre and with K. Mislow in Princeton. At the end of 1992 he resigned as
associate professor ar U WC and hus since been visiting ar UCT H i s main research interests
revolve around issues o f molecular structure, incorporating areas such us the stereochemistry of
coordination compounds, extraction qf informution about the dynamic behavior of molecules
,from their solid-state structure, and the quantification of chirality.
An,ycii Clwni. Inr
Ed Engl 1994, 33, 823-839
Structure Correlation Method
placement along the positive S , coordinate would correspond to an
opening of the X-Y-X angle relative
S , ( B J = (I:\,'?) ( r , - r 2 )
to its value in the reference strucScheme I
ture, a displacement along the S ,
axis corresponds to a symmetric
stretch --both r 1 and r2 increase or decrease simultaneouslywhile a displacement along S , represents an antisymmetric
In general, the geometry of an observed triatomic molecule
would not correspond to that of the reference molecule, and its
representative point in the space spanned by the symmetry coordinates would not coincide with the origin, which represents the
undistorted reference. However, the coordinates of the representative point, given by its displacements (dJ along the SCs
(SJ, can be used to define both the type of distortion and its
extent: 1 ) A displacement along an Sitransforming as under
G would indicate that the corresponding molecule has retained
the 7symmetry of the reference molecule;[16'2) the total molecular distortion can be described in terms of a displacement vector D = d J i , which measures the overall displacement of the
point representing the molecule from the origin, that is, the
deformation of the given molecule away from the reference geometry.
Although SCs have proved extremely useful in structure correlation studies, their use does have limitations. The main restriction is that a symmetrical reference geometry needs to be
chosen. This condition is not always possible to achieve, especially for large molecules, and often the choice is not straightforward, as for example when dealing with highly flexible molecules.[' ']
S , ( A , )= AH
s , ( A ,=
) (I/)'?)
+ r2)
observation. A comparison of these individual e.s.d.'s with the
sample variance can indicate the presence either of experimental
error or environmental factors such as crystal lattice effects, or
of geometric constraints on the fragment under investigation
that may emanate from some residual part of the molecule.'231
Thus, if a particular squared e.s.d. is much larger than the
sample variance for a given parameter, this could indicate a
gross experimental error in that measurement only. On the other
hand, a large sample variance relative to the e.s.d.'s indicates
structural variance that may warrant further investigation. A
second criterion for absence of environmental effects hinges on
the calculation of the weighted sum of squares of deviations,
which should approximately follow a xz distribution if environmental effects are negligible.[241(Effectively, this test evaluates
whether the observations agree with one another as closely as
might be expected from their e.s.d.'s.)
Histograms and two-dimensional scatter plots may be very
useful in identifying clusters of data points or outliers, which
may arise from experimental error, unusual structural features,
or unique molecular environments. Possible linear correlations
between pairs of observed parameters can be investigated
through an analysis of covariance: a strong correlation would
suggest that a reduction in dimensionality of the problem is
possible, since some linear combination of the two parameters
could adequately represent them. But since in most cases
3 N - 6 parameters are required for a molecular representation,
multivariate statistical analysis should usually be performed.
2.2.2. Principal Component
Reduction in dimensionality of the problem is generally possible only after a principal component analysis (PCA) has been
Instead of breaking the data set down
perf~rrned.~'. 22a.
into two-dimensional subsets--as in covariance analysis or scatterplots-PCA searches for correlations among all 3 N - 6
parameters simultaneously, extracting linear combinations of
highly correlated variables that describe, in turn, the greatest
sample variance, the next greatest (orthogonal to the first), and
so on. The consecutive linear combinations are called the principal components (PCs), and they appear as linear combinations
of the original variables (a, b, c,. . .) [Eq. (I)].
2.2. Data Analysis: Statistical Methods"']
Once a coordinate system has been chosen and a suitable
data set comprising the fragment of interest and closely
related ones in a variety of crystalline environments has been
There may be a need to standardize or scale
the data.[".
The subsequent data analysis can be conveniently broken down into two stages. In the first, the behavior of individual geometric parameters, or pairs of them, is
analyzed in order to identify systematic effects on the distribution of observed values, that is, to detect the existence of
environmental effects. For this analysis univariate and
bivariate statistics are generally sufficient. In the second
stage the distributions of the molecular geometries as a
whole are evaluated, that is, the types of conformation
present in the data set are identified. This stage requires
multivariate statistical techniques, since at this point one is
dealing with complete representations of the molecules in
(3 N - 6)-dimensional parameter space.
2.2.1. L'nivariate and Bivariate Statistics
In addition to positional coordinates or bond lengths and
angles and derived parameters, X-ray or electron diffraction
studies yield estimated standard deviations (e.s.d.'s) that give an
indication of the experimental error associated with a particular
A i r ~ e Chrm
Inr Ed Engl 1994. 33. 823-839
= ).,a
+ h,b + i.,c + . . .
(1 1
n = I , 2, ..., 3 N - 6
The variables a, b, c,. . . that appear in any particular PC are
called its components, while the coefficients .?,,A,, A,, . . . are
called the loadings of the PC, and they indicate the relative
importance of the corresponding variable in the PC. In addition,
the technique yields the percentage of total sample variance
accounted for by each PC, so it is possible to reduce the dimensionality of the problem to less than 3 N - 6 by choosing only
the first k PCs (k < 3 N - 6 ) that together account for, say,
90 YOof the variance, and ignoring the others.
Mathematically, PCA is accomplished through an eigenanalysis of the covariance or correlation matrix.[271This process
corresponds to a rotation of the original 3 N - 6 orthogonal
parameter axes into 3 N - 6 new axes, the PCs, which lie along
directions of maximal variance.
T. Auf der Heyde
2.2.3. Cluster Analysis
The aim of cluster analysis (CA) is to cluster together similar
points (observations) and/or clusters in the multidimensional
parameter space, thereby yielding clusters of data points that
have something in common.r9* 22a, "I Clustering algorithms
differ in the criteria used for establishing similarity and in the
rationale according to which clusters are fused together. They
may be divided into two types: hierarchical and nonhierarchical. In hierarchical clustering each observation is initially considered either as a cluster on its own, and the most similar
clusters are successively merged together in subsequent steps
until all of the data points are in one large cluster; or, conversely, each point is initially considered part of the same large cluster, which is then progressively subdivided into smaller ones
until, finally, each observation is again in a cluster of its own.
Nonhierarchical techniques partition a data set either into some
pre-set number of disjoint clusters, or into clusters situated at
some user-specified locations in the multidimensional parameter space; in both cases the partitioning is done in a single step,
rather than a number of hierarchical stages.
All these methods initially require the calculation of a similarity matrix whose elements are numbers indicating the similarity
between each pair of observations (molecular structures) in the
data set. The similarity between any two structures k and 1 is
generally measured by a distance dk,between their representative points [Eq. (211, where xkjis the value of thej-th parameter
' ' 9
for observation (structure) k. For n = 2 Equation (2) yields the
Euclidian distance in 3 N - 6 dimensions, while for n = 1 the
resulting measure is called the "city-block" metric. It is important that the parameters used in a clustering analysis are suitably
scaled, if not standardized, for the distance measure to be a good
reflection of the relation between representative points in the
multidimensional data
CA is less objective than PCA, since the user needs to make
a number of decisions, each of which could affect the outcome
of a CA. Thus, once the similarity criterion has been defined,
there are a large number of different linkage criteria-at what
stage to link together clusters?-from which to choose. Because
of the large number of user-defined variables in CA, it is good
practice to apply different clustering techniques to the same
body of data wherever possible.
2.3. Symmetry Considerations
For a given N-atom fragment there are, in general, N ! permutations of the atom labels on the molecular skeleton. This means
that for each such molecule, N! possible representative points
exist in the parameter space.[301Which to choose? Murray-Rust
examined this problem in detail, and suggested two ways in
which to deal with
In the first a unique system of nomenclature is developed to ensure that every individual molecule is
labeled in exactly the same way. In practice this may not always
be possible; moreover this approach has implications for the
statistical evaluation in that the distribution of the observed
parameters, artificially skewed by the nomenclature, will not
reflect the true distribution.
A second approach that has been widely applied in structure
correlation studies makes explicit use of the symmetry of the
reference molecule in generating all symmetry-equivalent permutations of the atomic labels on the molecular skeleton. This
permutation has the effect of generating all symmetry-related
siblings for each representative point, and it is tantamount to a
symmetry expansion of the data set, in which each representative point is transformed according to the operations of the
reference point group G.I3I1 While the method of symmetry
expansion multiplies the number of data points by the order of
G , it does not add any new data, but merely brings into prominence the symmetry that is implicitly assumed through the use
of the group G as a reference group. Moreover, the explicit
symmetry of the data expansion must reappear in the results of
PCA and CA, thereby yielding an indicator as to the correctness
of the results.
3. Proposed Reaction Paths for
Five-Coordinate Metal Complexes
Complexes with the ML, composition (M = metal atom,
L = any ligating atom) have been postulated and demonstrated
to be intermediates or transition states for many ligand exchange reactions of tetracoordinate metal complexe~.[~~1
Historically these reactions have been divided into three main
groups: nucleophilic substitution, electrophilic substitution,
and oxidative addition followed by reductive elimination. It has
become fairly clear, though, that a relationship exists between
many types of reactions previously classified quite separately.[33]Often this relationship results from geometrically similar
reaction paths involving the formation of a five-coordinate species, which could be either a true intermediate in an associative
nucleophilic substitution, an early transition state in an oxidative addition, or a solvent0 species in a dissociative substitution,
for example. In this sense then, many if not most reactions of
four-coordinate complexes can be deemed at some stage to involve the formation of a five-coordinate species (Scheme 2).
+ Y + [XML,YI + X + ML3Y
Scheme 2. X and L represent any coordinated ligands, and Y is either X, L, or a
solvent molecule.
Figure 1 depicts the reaction paths that have most commonly
been invoked in mechanistic explanations of these ligand exchange processes at tetracoordinate metals. From a purely geometric perspective, step A maps the addition of a fifth donor to
a square-planar complex-or the reverse (elimination) reaction-such as might occur for the d8 metals. The incipient stages
of this path are characterized by pentacoordinate complexes
whose metal atom lies close to the base of a square pyramid
(SQP); such conformers will be dubbed flattened square pyramids (fSQP). As the atom of the fifth ligand becomes more
firmly bonded to the metal, the latter is drawn away from the
Angew. Chem. Int. Ed. Engl. 1994,33,823-839
Structure Correlation Method
Fig. 1 . Deformation and reaction pathways of five-coordinate complexes. fSQP
= flattened square pyramid, eSQP = elevated square pyramid, TBP = trigonal
bipyramid.[”’I Path A maps a reversible addition/elimination reaction at a squareplanar metal complex, path B represents half of the Berry coordinate (Fig. 2), while
path C depicts a reversible nucleophilic substitution at a tetrahedral center.
basal plane of the B Q P to form what is termed an elevated SQP
(eSQP).[34a1Step B maps half of the Berry intramolecular rearrangement1351whereby a trigonal bipyramid (TBP) deforms
into a SQP by concerted bends in the axial angle and one of the
equatorial angles of the TBP.[34b1
The Berry mechanism[361was
originally proposed in an attempt to explain the NMR equivalence of axial and equatorial fluorine atoms in PF, at ambient
temperature, and it offers a pathway[37.381 for the continuous
deformation of one TBP into another, through a SQP intermediate (Fig. 2). Finally, step C maps the addition of a fifth ligand
correlation method,[31Muetterties and Guggenberger similarly
applied the new idea to a study of five-coordination, but focused
on a different deformation path.[391In his paper, Biirgi viewed
a series of five-coordinate metal complexes as essentially adopting
TBP configuration, and then extracted an SN2reaction pathway
(path C in Fig. 1) from correlations between them. Muetterties
and Guggenberger, on the other hand, examined deformations
away from TBP configuration, establishing a Berry pathway in
the process (path B). Between them, the methods developed in
these two papers laid the foundations for many of the subsequent structure correlation studies of pentacoordinate metal
(and nonmetal) complexes, and because of their seminal importance, we shall dwell on them at some length below.
4.1.1. Burgi’s Cadmium Study-the
The original paper examined eleven approximately trigonal
bipyramidal cadmium complexes with equatorial thioglycolate
ligands, and iodine, sulphur, or oxygen as axial ligands. The
coordination polyhedra were simply described by internal coordinates, and correlations were mainly analyzed by means of
scatterplots. The most telling correlations were those between
the out-of-plane displacement (Az) of the central Cd atom, and
the distance
of the two axial bonds (Ax, Ay). This
correlation is depicted in Figure 3. The results were interpreted
Fig. 2. The Berry intramolecular exchange mechanism. The atom labeling shown is
used throughout this paper.
to a tetrahedral complex, yielding a TBP, or the reverse elimination; this path is equivalent to the SN2 reaction coordinate of
organic chemistry, and it might be expected for those metals
whose tetracoordinate complexes adopt tetrahedral configurations. The two structures of highest symmetry, usually chosen as
reference geometries for ML, complexes, are the TBP (D3,,symmetry) and the SQP (C4J; while the angular geometry of the
TBP is fixed by its symmetry, the same is not true for the SQP,
which can adopt any value for the angles between trans ligands
in the eSQP, so long as these angles are equal to each other.
4. Reaction Paths Derived from Crystal Structures
4.1. Laying the Foundations
Soon after Biirgi published the first paper outlining and explicitly employing what has since come to be known as the structure
Angew. Chem. Int. Ed. Engl. 1994, 33, 823-839
Fig. 3. The correlation between axial distance increments (Ax or Ay) and out-ofplane displacement (Az) of the Cd atom. The distortion maps an S,2 coordinate
equivalent to step C in Figure 1.
in terms of an SN2 distortion coordinate equivalent to step C
(Fig. 1): beginning with a TBP distorted toward C,, symmetry-that is, a TBP that could be viewed as a tetrahedron with
a fifth ligand at one of the tetrahedral faces-ne
axial bond
(Ax) becomes longer, as the other (Ay) shortens, while the metal
is concomitantly drawn into the equatorial plane of the TBP. In
the intermediate or transition state the two axial bonds are
lengthened from their respective “standard” lengths by about
0.32 8,each, and the metal atom is situated exactly in the equatorial plane. An analysis of the variation in the internal angles
T. Auf der Heyde
that accompanies the bond length changes, reveals the typical
“umbrella-type” motion whereby a tetrahedron inverts during a
classic organic S,2 reaction.
An analytic expression [Eq. (3)], which parallels that proposed by Pauling [Eq. (4)],t411was used to describe a geometric
distortion coordinate along which changes in interatomic distance Ad are related to bond numbers n. On the basis of the
A ~ , A y = g ( k A z ) =-1.5Ig[(+A~+O.84)/1.68]
analytic expression Burgi was able to show that bond number n
is linearly dependent on Az, and that the sum of the axial bond
numbers (nx,ny)equals unity at each stage of the reaction, which
implies that the bond between the incoming ligand and cadmium is strengthened at the expense of that between the departing
group and the metal. The bond variations observed by Biirgi
have since been approximated by Burdett in an angular-overlap
molecular orbital study, in which he modeled distortions with
C,, symmetry for ML, fragments with D,, symmetry (corresponding to step C in Fig. I), as well as coupled distortions
along paths A and B.I4’] Despite discrepancies between his calculated curves and those of Burgi and others, B ~ r d e t t [reports
a fairly good qualitative reproduction of the observed bond
shortenings and elongations.
Apart from the proposal of mapping reaction paths from
crystal structures, Burgi’s initial paper also for the first time
introduced the idea of using symmetry coordinates to describe
molecular deformations in the solid state. He suggested that an
alternative way of looking at the nucleophilic addition was to
view it as the reverse of the decay of the TBP, mapped by the SCs
collated in Scheme 3, where Actnx,Act,, represents the deviation
Fig. 4. Top: correlations between the SCs S, (XCdY, antisymmetric stretch) and S.,
(CdS3,out-of-plane bend); bottom: correlations between S , (XCdY, antisymmetric
stretch) and S, (XCdY, symmetric stretch). The solid curves were obtained by
ellipsoid, and making certain simplifying assumptions. Although this approach has since been criticized,[431it suggested
the beguiling notion of extracting data on the topography of the
potential energy surface of the ML, fragment from the experimental (crystallographic!) information at a time when such data
would have been very difficult to come by computationally.
Recently this idea has again been given some attention,[44-451
though there has as yet been no further attempt to obtain empirical information on the potential energy surface for pentacoordinate metal complexes.
4.1.2. Muetterties and Guggenberger-the Berry Rearrangement
of the observed angle between the n-th equatorial ligand and the
axial ligands, either X or Y, from their values in the reference
structure (the TBP). Figure 4 depicts the correlations between
these coordinates for the eleven structures of the study. The
smooth curves were obtained by regression from Equation (2)
and the corresponding expressions for the SCs, and they were
interpreted as follows: The decay of the TBP is initiated by
antisymmetric stretching of the axial bonds ( S , , Fig. 4 top)
coupled to out-of-plane bending of the CdS, moiety (S,) . Initially-for values of S , smaller than about 0.2A-the complex hardly deforms along the symmetric stretch coordinate ( S , ,
Fig. 4 bottom), but as the decay proceeds this stretch becomes
more pronounced.
Another novel idea in the paper was explored in the attempt
to estimate the relative magnitude of quadratic and cubic potential (force) constants for the displacement reaction, by modeling
the reaction coordinate-the smooth curves in Figure &by
equations describing a principal axis of the potential energy
Although Burgi’s study revealed convincing correlations, it
ignored certain deformations of the coordination polyhedron,
by assuming that the complexes all approached at least C,,
symmetry. A careful analysis of the data reveals, however, that
some of the complexes are quite significantly distorted away
from this symmetry; the angle between axial ligands can be as
low as 161”. Muetterties and G~ggenberger~~’]
explicitly set out
to deal with this problem of approximate configuration~.~~~1
From a structural examination of a large number of coordination complexes they identified a “surprising consistency of
shape”, in that for compounds of the type ML,, the n ligating
atoms generally describe a polytope in which all faces are equilateral triangles or nearly so. They further found that while the
fully triangulated form is always the favored one, there are
alternate polyhedra (polytopal isomerst4’]) for all commonly
observed classes of complexes containing n ligating atoms that
may be generated from the favored forms by minor bendingstretching modes. The favored form of ML, complexes is the
TBP; the SQP is formed when two of the adjacent triangular
faces of the TBP become coplanar in a square face during the
(Berry) polyhedral rearrangement. As shape parameters Muetterties and Guggenberger proposed the dihedral angles between
Angew. Chem. Inr. Ed. Engl. 1994,33,823-839
Structure Correlation Method
4.2. Applications of the Early Methods
the normals to adjacent polytopal faces, coupled with associated
angles subtended at the central metal. With reference to Figure 2, the angle
between the normals to the faces 124 and
524, which share the common edge 24, changes from 53.1" to
zero as the TBP deforms to the SQP along the Berry coordinate.
This change is accompanied by a concomitant change in the
angles O , , (from 180" to about 155") and Oz4 (from 120" to also
about 155"). By correlating these three angles, Muetterties and
Guggenberger were able to arrange seven pentacoordinate complexes containing six different central
into a series
convincingly mimicking the Berry path. The corresponding diagram (Fig. 5 ) beautifully illustrates how the individual molecular
4.2.1. Nickel Complexes
The structures of 78 five-coordinate nickel complexes have
been examined along similar lines to those discussed in Section 4.1.rszl The molecular geometries were described by internal coordinates (bond distances being expressed as increments
relative to the respective standard bond lengths[401),and no
special account was taken of symmetry considerations of the
kind mentioned in Section 2.3. The data comprised all five-coordinate nickel entries deposited in the Cambridge Structural
Database at that time,['g. "1 but excluded allylic complexes.[531
Scatterplots revealed that 34 of the compounds mapped an association reaction for tetrahedral nickel complexes (path C in
Fig. 1), while the remaining 44 structures lay along a reaction
coordinate depicting an addition reaction at square-planar centers (path A). This finding is, of course, reconcilable with the
fact that four-coordinate nickel complexes can adopt either tetrahedral or square-planar configuration, and ligand substitutions at such centers must therefore necessarily proceed at least
partly by different pathways.
The reaction coordinate for square-planar complexes can be
suitably described by a correlation between the angular parameters defined on the right of Figure 6; the corresponding scatter-
Fig. 5 . Muetterties and Guggenberger's arrangement of five-coordinate complexes,
which mimic the Berry mechanism, that is, the progression from D,, to C,, symmetry.
structures yield a cinematic film of the reaction, when they are
strung together in the right order.[491
4.1.3. Extension of the Dihedral Angle Technique
For a number of years after the two papers discussed in Sections 4.1.1. and 4.1.2. were published, little or no further attention was given to pentacoordinate metal complexes. Nonmetal
compounds, however, were fairly intensively investigated by
Holmes, Deiters, and co-workers (see ref. [8]), who developed
an extension of Muetterties and Guggenberger's dihedral angle
method, which allows a quantification of the extent to which a
given molecule has traversed the TBP + SQP divide.[501Instead
of (subjectively) focusing on just two or three of the internal
coordinates, Holmes and Deiters included all nine dihedral
angles (between normals) in their shape ~ a r a m e t e r . ~In
~ ' ]their
method the sum of the differences between the angles Sij(C) for
each edge i j in the observed structures C and the corresponding
angles in the reference polyhedra [B,,(SQP) and hij(TBP), respectively]is determined. For the concerted axial and equatorial
bends accompanying the Berry rearrangement, a plot of
217.9" --xijl~ij(C)- hij(SQP)I versus CijISij(C)- Gij(TBP)I,
where 217.9" = ZijIGij(TBP) - aij(SQP)I, should then theoretically yield a straight line with unit slope and length (217.9")1/2.
The percentage displacement of a given representative point
along this coordinate-that is, the percentage displacement of
the particular structure along the TBP -+ SQP pathway-is then
easily expressed as a proportion of the length of this line.
Angew. Chem. Int. Ed. En,$ 1994, 33, 823-839
1 0
Fig. 6. Deformation of nickel complexes mapped by the bond angles Ols,OZ4 and
the parameter 6 (defined on the right of the diagram). Points on the lower right of
the plot represent fSQPs, those in the center eSQPs, and those at the top right and
left TBP conformers. The solid line maps the theoretical coordinate for the deformation described on the right of the scatterplot. The numbers give the oxidation state
of the metal.
plot is shown on the left. The solid line represents the theoretical
deformation pathway for the scheme shown on the right of the
figure. In the scatterplot, points on the lower right represent
compounds mapping the incipient stages of step A, that is, flattened SQPs, those at the center represent elevated SQPs, while
points in the upper right and left represent TBP complexes; the
T. Auf der Heyde
split part of the coordinate corresponds to the Berry pathway.
The data are fairly evenly spread along the theoretical trajectory, which suggests that there may only be a very small energy
difference between the B Q P and the eSQP, and that the activition barrier for the transformation of the one into the other is
very small. However, it is not clear how to interpret the scatter
about the theoretical coordinate: the phenomenon may arise
from structural variations that are not adequately modeled by
the method, which includes only three geometric parameters
and may therefore be biased by the choice of variables.
The S,2 pathway was mapped in reverse-that is, an axial
ligand in a TBP was removed along the C, axis-by correlating
the normalized shift parameter s = (deq- 3z)/degwith the sum
of the angles at the metal atom along [CO,,] and around [Ce,,]
the threefold axis, according to Equation ( 5 ) and Equation ( 6 ) ,
respectively.t541The parameter deq is the average equatorial
bond length, z the displacement of the nickel atom out of the
equatorial plane. Figure 7 depicts the corresponding scatterplot
t 3601
ze- t
sFig. 7. Adherence of nickel compounds to the coordinate mapping the transition
from C,, to Td symmetry (solid line), that is, the reverse of path C. The numbers
is the sum of the angles ["I at the
indicate the oxidation states of the metal.
the sum of the equatorial angles
metal atom along the threefold axis [Eq. ( 5 ) ] , 2Qee
[Eq. ( 6 ) ] ,and s a normalized shift parameter (see text). The numbers indicate the
oxidation state of the metal.
for the 34 complexes that mapped this pathway; the solid line
represents the theoretical coordinate. Complexes containing Ni'
atoms are indicated by the numbers in the diagram, and lie
furthest along the dissociation pathway. This may indicate that
in those cases where homolytic bond cleavage results in Ni'
species, it will be an axial bond in a TBP that will cleave.
= 32'
+ '34 + '24
Finally, the Berry mechanism was found to be a viable pathway for intramolecular exchange in these complexes: a plot of
Fig. 8. Adherence of nickel complexes to the Berry coordinate (solid line). Filled
circles correspond to structures mapping the association reaction at square-planar
nickel, empty circles to those that do not. The numbers indicate oxidation state of
the metal. NIHNPB and DMPADP are outliers.
270 1
3 Y&C) - 8,CrBP)I 1
217.9" - C i j l S i j ( C ) Gij(SQP)I versus C i j l S i j ( C) 6,,(TBP)(
revealed a reasonable adherence of the structures to the Berry
coordinate depicted by the solid line in Figure 8, in which the
origin corresponds to a TBP and the upper end of the straight
line at (217.9 O ; 217.9") to a SQP. The fairly even clustering of the
data around both ends of the coordinate suggests a small energy
difference between the TBP and SQP, since a large difference
would presumably result in preferred clustering of observed geometries about the lower energy conformer, and a relative
paucity of points around the higher energy one. The two outliers
could easily be accounted for by unique, highly restrictive structural features that would be expectell to distort them away from
a minimum energy pathway.
A number of different explanations could account for the
scatter in Figure 8. In the paper it was suggested that the complexes that do not map the association reaction for squareplanar centers (empty circles) possibly exhibit slightly less scatter than those that do (filled circles). This would make sense,
since from a geometric perspective one could argue that structures distorted toward C3, symmetry-that is, those lying along
path C (see Fig. 1) and closer to the origin in Figure 8-would
more readily undergo Berry exchange than would BQPs, which
lie along path A and at the upper right of Figure 8. Indeed, the
fact that the BQPs lie at the bottom right of Figure 6, displaced
from the Berry coordinate, substantiates this suggestion. A certain amount of the scatter is also likely to arise randomly from
the different crystalline environments. But another factor that
complicates all studies of five-coordinate Complexes is that the
geometry of the SQP is not uniquely fixed by its symmetry, as is
the case for the TBP. In the SQP one degree of freedom remains:
the trans-basal angles (Q15and OZ4, Fig. 6) can adopt any value
under C4, symmetry, provided that they are equal. This implies
that an arbitrary choice has to be made about the constitution
of the "reference SQP" and the values of its angles. Clearly, this
affects the calculation of Holmes and Deiters's shape parameters; specifically, the length of the Berry coordinate in plots like
Angew. Chem. Int. Ed. Engl. 1994, 33, 823-839
Structure Correlation Method
that of Figure 8 and the coordinates of the representative points
in these diagrams would be affected.[551
4.2.2. Zinc, Molybdenum, and Iron Complexes
Scatterplots and the application of the dihedral angle method
to a sample of 33 pentacoordinate zinc compounds (represented
by their internal coordinates) has enabled a mapping of steps B
Results are similar to those for the cadmiand C in Figure I
um complexes and the 78 nickel complexes described earlier,
with the exception that in the S,2 pathway the TBP intermediate
was shown to be distorted (with an axial angle of about 163"),
as a consequence of the directional preferences of the substitution pathway. There are two extreme approach trajectories of
the incoming nucleophile N with respect to the ZnL, tetrahedron: for an ideal 'face' approach (F) one N-Zn-L angle would
be 180" and three 70.5",while an ideal 'edge' attack (E) gives rise
to two pairs of angles of 125.3" and 54.7".Figure9 reveals
that the nucleophile approaches
the tetraheF
oblique anQ
gle, midway between F
and E, which may well
cause the observed distortion of the intermediate. An absence of
correlations for the
more square pyramidal
configurations of the
data set suggests that in
the case of Zn" complexes any dissociation
must occur from the axial positions of a TBP
(path C), as opposed to
the apical position of
an SQP (path A):
hence, for the substitution reaction, Berry deformation of a complex
towards an SQP leads
Fig. 9. Scatterplot showing the dependence of
into a cul-de-sac.
the angle of nucleophilic attack on the distance
A sample of 44
increment [40] of the Zn-to-nucleophilebond.
Filled circles plot OlS, open circles OZs, O,, ,and
five-coordinate molybO,, (see Fig. 2 for nomenclature).F = face atdenum complexes has
tack, E = edge attack.
been very similarly
treated, and coordinates for an S,2 reaction and a Berry exchange were obA Berry coordinate has also been mapped from an
analysis of nine structures of the type [Fe(CO),P,], where P2
represents either a bidentate or two monodentate phosphane
l i g a n d ~ . [In
~ ~this
] case, the data were symmetry-expanded according to C, symmetry. The distribution of the resulting 18
data points was fairly even along the TBP + SQP pathway,
again suggesting a low energy barrier for this deformation; indeed, two of the compounds are fluxional even at temperatures
as low as - 80 "C. This result was not reproduced in a study of
twelve [Fe(CO),L] complexes (L = monodentate ligand), which
found that most of them clustered around the TBP configuraAngew. Chem. Inf. Ed. Engl. 1994,33, 823-839
tion, though some displacement along the Berry coordinate was
The aim of the latter study was to establish correlations between the geometric deformations of these complexes
and certain Mossbauer parameters, in an attempt to relate their
electronic nature to their structure. The results of the study led
the authors to predict that axial ligands whose n-acceptor qualities are similar to those of CO would facilitate the Berry mechanism, whereas strong o-donor ligands would present barriers
to the intramolecular exchange. Kinetic and dynamic NMR
data to verify this speculation are yet to be forthcoming.
4.2.3. Copper Complexes
Complexes of copper(n) exhibit a wide variety of regular and
nonregular stereochemistries, partially as a result of the lack of
spherical symmetry of this d9 ion. In particular, it is often impossible to distinguish between penta- and hexacoordination.
This feature has led to attempts[601to classify these complexes
as (4 2) when there are four short and two long bonds, or as
(4 1 1*) when there are four short bonds, one long bond,
and one (1*) of intermediate length. These configurations could
alternatively also be considered as a perturbed ML, geometry
(4 2 or 4 + 1 + l*), perturbed ML, (4 1 + 1*), or distorted
ML, (4 + 2 or 4 1 + l*). Such a structural spectrum invariably invites attempts to map deformation pathways. But it
might also frustrate those attempts, since the broad range of
configurations could be indicative of a flat potential energy
surface, in which case correlations may not be easily identifiable.
In the following we concentrate only on those complexes that
are readily identifiable as pentacoordinate.
The geometry of the [Cu(bpy),Cl]+ ion (bpy = 2,2' bipyridyl)
has been shown to be closer to TBP, though distorted towards
SQP, in a series of five complexes with various anion^.[^'-^^]
Correlations between some of the internal parameters of the
CuN,CI chromophore led Hathaway et al. to suggest that these
structures map out a TBP + SQP kansformation. In terms of
the Berry mechanism there are three possible deformations of a
trigonal bipyramidal CuN,Cl polyhedron, of which two are
equivalent (Fig. 10). The data for the five [Cu(bpy),Cl]+ complexes seemed to map pathway E (or the symmetry-equivalent
path F ) , but not path D. Since the distortion along path E does
+ +
Fig. 10. Possible deformation modes of trigonal bipyramidal complex
[Cu(bpy),X]+. Path D maintains C,, symmetry, while the symmetry-equivalent
paths E and F do not.
T. Auf der Heyde
not maintain C, or C,, symmetry, Hathaway rationalized the
deformation as a linear combination of the normal modes of
vibration of a trigonal bipyramidal [CuN4C1]+ ion, rather than
as a Berry mechanism. Path D, however, retains C, symmetry,
and can hence “be considered as a Berry twist-type mechanistic
pathway”; the latter path is mapped by three out of four
Extended Hiickel molecular orbital calculations were then
used to estimate the relative total energies for a [Cu(NH,),CI]+
ion as a function of one of the N-Cu-N angles, defined as clj
(equivalent to the angle N2-Cu-N4 in Fig. 10) .L6’, 631 The resulting energy profile is shown in Figure 11. Plotted onto the profile
Path E
Path D
Fig. 11. The extended Hiickel energy profile for the model complex
[Cu(NH,),Cl]+, as a function of the N2-Cu-N4 angle ( a 3 )in Figure 10. The crosses
represent the observed values of u3 for complexes [Cu(bpy),X]+ in which X = CI,
the circles those in which X = OH,.
are the values of c13 for the chloro complexes (crosses) and the
aqua complexes (circles). The diagram should not be misconstrued as the correlation between the observed values of a3 and
the energy of the complex, but it does serve to illustrate one
possible approach to empirical potential energy surfaces : if a
large enough sample pertains, it might be possible to model the
data distribution density along the reaction coordinate and
thereby perhaps obtain information about the topography of
the energy well (or minimum energy path). However, in this case
the calculations tellingly revealed the TBP to be of slightly
higher energy than the SQP, even though most of the structures
approximate more closely to the TBP than the SQP; perhaps
this illustrates the point that the potential energy surfaces for the
various observed structures and the model complex are significantly different, even though the complexes have similar compos i t i o n ~ . [In
~ ~a]subsequent paper the results of variable temperature NMR experiments on the [Cu(bpy),X]+ ions (X = C1 or
I) are reported, and it is shown that they exhibit no fluxionality;[641in view of this, it is not clear how to interpret the structural correlations obtained for these molecules in the earlier
reports. In another study, Hathaway et al. have proposed that
three related complexes the CuN,-containing chromophore
may be viewed as stationary points along a Berry pathway, but
All of the
no further reports have emerged for this sy~tem.[~’l
studies discussed so far in this section suffer from the drawback
that they included only very limited numbers of structures, and
the conclusions-as far as there are any-must therefore be
treated with a certain amount of caution.
In a separate report,[661brief mention is made of a principal
component analysis (PCA) of 26 [Cu(L),X] complexes (L = bpy
or 1,lO-phenanthroline (phen)), which revealed that the structural variance for these compounds essentially lies in a plane in
the 15-dimensional data spacet6’*that was analyzed. The plane
is described by two axes that correspond to the eigenvectors for
the two deformations depicted in Figure 10, that is, the symmetric (Berry) pathway (path D), and the asymmetric (non-Berry)
deformation (path E). Hathaway’s conclusion that the structures of this type of complex can be discussed in terms of these
two deformation paths has therefore been borne out by the
PCA. A more recent studyt681suggests, however, that the true
deformation pathway for complexes of this kind is intermediate
between the symmetry-equivalent paths E and F. In the latter
case, 36 bpy and phen complexes were analyzed on the basis of
their internal coordinates, specifically the equatorial angles
around the metal atom. The conclusion about the nature of the
deformation path may be doubtful though, because of certain
problems in the geometric treatment.[691InJa slight contradiction of the above suggestion, the study also concluded that
[Cu(L),X] complexes do not readily distort toward SQP
through the symmetric deformation, whereas they do through
path E. This observation was rationalized by the argument that
the Cu(L), chromophores cannot be planar-as they would be
along path D-because of steric hindrance between the hydrogen atoms near the pyridinic nitrogen atoms.
The studies on five-coordinate copper complexes that have
been mentioned here barely scratch the surface of the data that
are available : less than forty, highly select complexes have been
examined. It seems that the complexity of the problem+mphasized by the confused results and conclusions from the different
for the application of far more sophisticated
methods of analysis than have thus far been brought to bear on
these structures.
4.3. The Application of Multivariate Techniques:
d8 Metal Complexes
Perhaps the most comprehensive analysis of an ML, system
to date can be found in a study involving the multivariate analysis of a data set comprising 196 pentacoordinate d8 complexes
containing the metals nickel, palladium, platinum, rhodium,
and iridium.170.711 Bond distances were expressed as increment~,[~’l
and angles were all scaled to the approximate dimension of the average bond distance in the data set.[”’ The coordination geometries were expressed by means of a complete set of
symmetry coordinates for both TBP and SQP configurations.
Each ML, fragment was thus represented by a point in two
different 12-dimensional parameter spaces : T-space, where each
structure is related to a TBP, and S-space, where each is related
to a SQP. A novel feature of this analysis is that no “ideal” value
of the trans-basal angle (6, and OZ4, Fig. 2) in the reference SQP
had to be assumed, since the evaluation of the SCs does not
require a value for this angle: even though it is the increments
r or deviations A of the observed values from the reference
values that appear in the SC expressions (see Section 2.1), the
Angew. Chem. Inr. Ed. Engl. 1994, 33, 823-839
Structure Correlation Method
reference values cancel for all but the totally symmetric reprein others peculiar structural features of the compounds desentation~.[’~~
731 The atom labeling problem (Section 2.3) was
formed them away from the minimum energy path.
explicitly discussed: the 120 (= 5!) possible permutations of the
An interesting comparison can be made between the scatterlabels give rise to ten and fifteen groups, respectively, of twelve
plots for the addition reaction at a nickel center in a squareand eight so-called permutationally equivalent isomers-strucplanar environment in the earlier
(Fig. 6) and the corresponding diagram for the study discussed here (Fig. 12, right).
tures that are represented by identical lengths, but different
In the former, where only three geometric parameters are emsenses, of the total distortion vector (Section 2.1), that is, they
are equally deformed from the reference TBP or SQP, but their
ployed in the analysis, a distinction between the eSQP and the
atom labels are permuted according to the symmetry of the
fSQP is hardly possible, whereas the clustering algorithm has
been able to subdivide the group of square-pyramidally disreference point groups.
posed structures into the two classes automatically. This speaks
For both spaces the data were symmetry-expanded (DShand
C,, symmetry), leading to 2352 and 1568 data points in T- and
volumes for the power of this form of analysis, since the distincS-space, respectively. Univariate and bivariate statistics (analytion between these two forms would appear to be a totally arbisis of variance and covariance) suggested the importance of
trary one on the basis of a graphical analysis only: the represendeformations along paths B and C (Fig. 1) for the data in Ttative points for the eSQP and the fSQP are likely to be
separated by only a very small potential barrier on the Bornspace, while in S-space deformations along paths A and B predominated.
Oppenheimer potential energy surface for the ML, fragment,
The resultant twelve-dimensional data distributions were then
and the spread of observed molecular structures around these
analyzed by both hierarchical and nonhierarchical cluster analpoints can be expected to be large and diffuse, with no clear
ysis (CA), and the Euclidian distance was chosen as a measure
boundaries between the two distributions. Purely graphical
of similarity. The method of nonhierarchical CA classified the
techniques would consequently be unlikely to suffice as a means
data set into a user-specified number of clusters, while hierarchiof differentiation. This point becomes clear when one compares
cal CA employed a method called “Ward’s method”.[741The
the average sizes[761of S 2 and S 4 (0.697 and 0.932 A) with the
search for the “ideal” number of clusters was fairly random in
distance between them (0.833 A t p o s s i b l y the clouds of data
the initial stages, but a solution was soon found by use of the
points diffuse into each other. Despite this, the statistical techsymmetry criterion, namely that the symmetry of the data distrinique evidently succeeded in fixing the positions of the archetybution made explicit by the symmetry expansion must reappear
pal configurations somewhere in the center of the cloud of data
points surrounding them.
in the results of CA. An additional criterion, implementable
Principal component analysis did not enable a dramatic rebecause two different clustering algorithms were employed, was
that the classification of data points had to be identical, or very
duction in dimensionality, but it did reveal chemically meaningnearly so, for both methods.[751Four clusters
were identified in both spaces (see Fig. 12): in
T-space a central cluster (cluster T3) repres1
senting an ideal TBP is surrounded by three
identical clusters (TI, T2, T4) whose median
configurations (archetypes) are distorted
SQPs. Interestingly, the three SQPs reprec .’* :
sented by these clusters correspond to the alt
ternate SQPs that could be obtained from the
central TBP (cluster T3) by means of the
2 .... :
Berry mechanism. In S-space, the data cluster
about an fSQP and an eSQP archetype (S2
and S4), as well as two permutationally
equivalent TBP archetypes (S 1, S 3)-the lat.,.a
ter again correspond to the two TBPs that
could ideally be formed from the eSQP. Note
how the symmetry of the data distribution
reflects the symmetry of the reference point
The study interestingly revealed a strong
homogeneity of the clusters. Histograms displaying the distance of each observation from
the cluster center in most cases revealed no
more than two or three outliers, and for all of
these it was possible to rationalize why they
appeared on the fringes of the clusters: in
some cases the points corresponded to strucFig. 12. Results of cluster analysis of d* metal complexes. The scatterplot on the left shows the clusters
tures that were intermediate between SQP
formed in T-space, that on the right those formed in S-space. Below each scatterplot is a qualitative
interpretation of the cluster archetypes, and the deformation coordinates relating them.
and TBP, or between fSQP and eSQP, while
I ” !
- ...
Angew. Chem. Int. Ed. Engl. 1994.33,823-839
T. Auf der Heyde
ful results. For the TBP the two most important distortions are
the SN2and the Berry deformations (steps C and B in Fig. l),
accounting for 15 and 24% of the variance, respectively. The
eSQP was shown to distort along both the addition/elimination
pathway (step A, 14%) and the Berry coordinate (12%). In an
interesting contrast, the E Q P maps only the incipient stages of
the addition/elimination reaction (14 YOof variance), and was
not seen to distort along the Berry path. In this analysis, symmetry considerations also played an important role, in that only
SCs of identical symmetry appeared together in the same PC. As
an example, consider the first two PCs for the eSQP [Eq. (7a, b)]
PC1 = 0.8888,-0.8798,
PC2 = 0.8378,
+ 0.87933
(7 a)
+ 0.8378,
where the SCs (Si)
are defined as in Figure 13. The component
SCs of PC1, mapping the addition/elimination deformation, are
all of A , symmetry, while those of PC2, describing the Berry
coordinate, all have B , symmetry, that is, no mixing of symmetries is observed.
type of theoretical approach, and the results of the structural
analyses. However, studies involving simpler theoretical models
are available. Kepert et al. have developed a simple repulsion
model in which the ligand donor atoms, regarded as point
charges, are assumed to be distributed on the surface of a hard
sphere at the center of which the metal is sited.[”] The main
purpose of this model is to classify, rationalize, and predict the
stereochemistries of [MLg complexes, where L represents either
unidentate (x = l), bidentate (x = 2), or less commonly, tridentate ligands (x = 3). For the composition [ML;], for instance,
the data, which include both transition and nontransition
metals, appear to fall into two categories: those that map the
TBP -+ SQP transition, and those that do not; the latter to a large
extent map the departure of an apical ligand from a SQP instead.[77*
781 This finding was latert52]corroborated in the analysis
of 78 fivecoordinate nickel complexes discussed in Section 4.2.1,
where just under half the complexes were found to lie along
paths B and C of Figure 1, while the rest map paths A and B.
In their study of [ML:] complexes, Favas and Kepertt781defined an axis in the TBP, such that the angles between this axis
and each of the ligand atoms A, B, and C, are all equal; angles
&, and +E are then measured between this axis and the bonds to
D and E. Figure 14 depicts the potential energy surface (PES)
Fig. 13. Examples of some of the SCs used to represent the structures of observed
pentacoordinate complexes. SCs shown are those derived for the SQP reference
geometry; the atom numbering scheme is indicated. Their symmetries are given in
the brackets.
Chemically, the results suggested interesting new possibilities :
1) the fact that the TBP cluster distorts along the SN2coordinate
suggests that five-coordinate Rh and Ir compounds, which are
significantly represented in this cluster, may undergo dissociation by an S,2 mechanism via a (short-lived) tetrahedral intermediate, even though this may not be evident from the chemistry of their (square-planar) tetracoordinate complexes;
2) since the E Q P is not seen to map the Berry coordinate, other
mechanisms may be. operable in fluxional pentacoordinate Pt
and Pd complexes (which are principally represented in this
cluster), or at least, the complexes may have to undergo a considerable distortion of their coordination sphere to an eSQP
before this mechanism can operate. This observation may be the
reason for the lack of unambiguous experimental results that
indicate Berry fluxionality in complexes of these metals.[70c]
5. Comparison of Structural Results
with Theoretical Models
Ab initio calculations on compbunds of the type dealt with in
this paper are-arguably-only
just becoming computationally
feasible, and no real comparisons are yet possible between that
Fig. 14. Potential energy surface for [ML:]complexes. To,T,, T2,and T3represent
TBP conformers that may be interconverted through SQP intermediates, represented by s,, s,, and s,. The angular parameters &, and & are described in the text.
for [ML:] molecules with C, symmetry, projected onto the 4D/
q5E plane. The positions of the TBPs and SQPs are shown by T
and S, respectively, and the PES is symmetrical about the line
& =18O0-~,. The topography of this PES is mirrored in a
scatterplot (Fig. 15, left) of the structural data for 33 zinc complexes (Section 4.2.2), in which the data have been projected
/ ~ ~
but is
onto a plane that is almost equivalent to the c # J ~ plane,
defined in terms of bond angles in the structures.t561
For complexes of the type [MA,B3], where A and B are
unidentate, the PES has similar general features to those shown
in Figure 14, but the relative energies of stereochemically distinct TBPs and SQPs, that is, of the potential minima (To, T I ,
Angew. Chem. Int. Ed. Engl. 1994.33, 823-839
Structure Correlation Method
80 -
e B d O 120
180 -
. .*’
analyzed.[881In the case of the multivariate
analysis discussed in Section 4.3, the average
axial bond in the TBP was found to be
0.030 8,longer than its usual reference value,
while the equatorial bonds were 0.025 A
shorter, in contradiction to both the earlier
tions. In the
case and
of the
results for the larger data set corresponded
with those of Holmes, and with those of the
theoretical prediction: for the eSQP the apical
bond will be 0.030 A longer, the basal bonds
0.008 A shorter than the references, while
the fSQP will have apical bonds that are on
Angew. Chem. Int. Ed. Engl. 1994.33, 823-839
T. Auf der Heyde
Johnson have put forward three hypotheses concerning possible
mechanisms for such polyhedral rearrangement^.^^^] They postulate that there are only two possible mechanisms[g11for site
exchange in pentaatomic clusters: the first involves the stepwise
cleavage and subsequent formation of a polyhedral edge, with
an edge-bridged tetrahedral intermediate, while the second involves a concerted process and a SQP intermediate, akin to the
Berry process. These alternative mechanisms are depicted in
Figure 16.
7. Summary and Outlook
Fig. 16. Possible mechanisms for site exchange in Au,Ru, clusters. Top: stepwise
mechanism with edge-bridged intermediate; bottom: Berry process. The molecular
fragments were defined by the lengths of the nine edges indicated for the TBP on the
left, plus the distance Aul-Ru2.
An examination[921of the structures of 19 Au,Ru, fragments
contained in clusters with a variety of carbonyl, phosphane, and
p,-ligands (e.g. S, COMe) has revealed that site exchange most
likely occurs by the Berry process. It proved possible to rule out
tetrahedral edge-bridged intermediates and the alternative turnstile mechanism[37-381 on the basis of the structural evidence.
The data, composed of interatomic distances and selected bond
and dihedral angles, were symmetry-expanded according to the
C, symmetry group common to the idealized TBP and SQP
clusters, and the analysis included multivariate techniques and
scatterplots. A dihedral angle plot, in which the dihedral angle
between the planes Aul-Rul-Au2 and Ru2-Rul-Au2 (Fig. 16)
-quivalent to the dihedral angle a, of Section 4.1.2. and Fig-
Fig. 17. Dihedral angle plot for Au,Ru, clusters. The angles b,, and OlS, 024are
described in the text.
ure 2-is plotted against the corresponding bond angles QI5 and
OZ4, reveals the typical ‘V-shaped‘ Berry coordinate (Fig. 17).
Corroborating evidence comes from the PCA, which yielded
one PC accounting for 92% of the variance in the ten metalmetal distances chosen to represent the fragments (Fig. 16); the
major components of this PC are the symmetry equivalent distances Aul-Ru2 and Au2-Rul. The proposed mechanism is
fully consistent with the results of NMR studies on the dynamic
behavior of these fragments in solution.[891
In retrospect, by and large the various structure correlation
studies on pentacoordinate complexes have served to substantiate earlier hypotheses (made on the strength of (indirect) evidence from solution studies) about the likely geometric route
that reaction pathways of ML, or (ML, + L) systems might
take. Whereas those studies employing simpler two- or three-dimensional methods of analysis may not have significantly deepened our understanding of the mechanisms even though they
have lent support to the structure correlation hypothesis, the
multidimensional analysis revealed geometric details which suggest that these reaction mechanisms require more careful study
with techniques that extend to shorter time-scales. Questions
remain, for example, about the viability of the Berry pathway
for isomerization reactions of platinum and palladium complexes, or about the possible existence of short-lived tetrahedral
rhodium and iridium intermediates in substitution reactions of
(square-planar) compounds of these metals. There may also be
a need to reinvestigate the possible correspondence (or lack of
it) between theoretical predictions[83] about ratios between
axial/equatorial and apical/basal bond lengths and the empirical
Even the application of the struFture correlation method to
ML, complexes has not yet been exhausted, since pentacoordinate copper complexes still seem to offer a fruitful field of research, for example. Another area of great interest would be an
analysis of site preferences for various classes of l i g a n d ~ , [ ~ ~ ~
particularly in complexes of the d8 metals; this could yield information about the trans effect in square-planar complexes, since
five-coordinate intermediates are often invoked in mechanistic
explanations of this phenomenon. Future work might focus
more on obtaining energetic information along with geometric
detail by combining kinetic and structural data to construct
individual reaction-path and energy profiles.t4s1It might also
take advantage of the recent dramatic increases in availibility of
computers with high performance to compare structural results
with ab initio calculations on some of the systems discussed
here. Connected to this is the use of results from structure correlation studies in the optimization of molecular modeling software: the averaged geometries obtained from structure correlation studies represent, at least, local potential minima, and can
therefore be used in the process of rninimi~ation!~~]
Finally, without being able to offer an answer but in the hope
that it may spur more work in the area, I wish to close with what
appears to me a fundamental paradox. No single molecular
Hamiltonian exists for the series of [ML,] complexes, and hence
there can be no potential energy surface for this “system”. Why
Angew. Chem. Inr. Ed. Engl. 1994, 33, 823-839
Structure Correlation Method
is it, then, that these complexes exhibit such consistent correlations, when at most what they have in common is a metal atom
surrounded by n coordinated ligand atoms?
I wish to thank the Chemistry Department, University of Cape
Town, for hosting me so generously, thereby affording me the
opportunity to do what most academic chemists can only dream
of-completing unfinished manuscripts.
Received: June 7,1993 [AZIE]
German version: Angew. Chem. 1994, 106,
[l] This approach allows us to represent a molecule in terms of classical concepts
of shape and structure. that is, as an ensemble of nuclei that vibrate with small
amplitudes around well-defined positions in space. From this notion arises the
possibility of depicting a molecule by a model in which the time-averaged
nuclear positions are made to coincide with well-defined points in space.
Notwithstanding its success, it should not be forgotten that this molecular
theory has its origin in the Born-Oppenheimer approximation to the molecular Hamiltonian, according to which electronic and nuclear motions may be
separated-and the latter largely ignored-r
that there may also be other
molecular theories that give rise to nonclassical interpretations and models; see
S. J. Weininger, J. Chem. Educ. 1984,61,939-944; R. G. Wooley, ibid. 1985,62,
1083-1084; R. G. Wooley, New Sci. 1988, 22, 53-57; A. Amann, W. Gans,
Angew. Chem. 1989, 101, 277-285; Angew. Chem. Int. Ed. Engl. 1989, 28,
268-276; A. Amann, S. Afr. J. Chem. 1992, 45. 29-38; J. C. A. Boeyens,
Struct. Bonding (Berlin) 1985, 63. 65-101.
(21 a) D. G. Truhlar, R. Steckler, M. S. Gordon, Chem. Rev. 1987,87, 217-236;
b) W. F. van Gunsteren, H. J. C. Berendsen, Angew. Chem. 1990, 102, 10201052; Angew. Chem. Int. Ed. Engl. 1990,29, 992-1023.
[3] H.-B. Biirgi, Inorg. Chem. 1973, 12, 2321-2326.
[4] a) H.-B. Burgi, Angew. Chem. 1975,87,461-474; Angew. Chem. Int. Ed. Engl.
1975, 14, 460-473; b) H.-B. Burgi, J. D. Dunitz, Acc. Chem. Res. 1983, 16,
153-161; c) J. Am. Chem. SOC.1975,97,921-922.
[5] For the remainder of this paper, such structures will collectively be referred to
as "molecular structures".
[6] Structure Correlation (Eds.: H.-B. Biirgi, J. D. Dunitz), VCH, Weinheim, 1993.
171 Transition metals, including zinc and cadmium.
[8] Antimony compounds: R. R. Holmes, R. 0. Day, V. Chandrasekhar, J. M.
Holmes, Inorg. Chem. 1987,26,163- 168; germanium compounds: A. 0.Mozzchukhin, A. A. Machardshvili. V. E. Shklover, Yu. T. Struchkov, A. G.
Shipov, V. N. Sergeev, S. A. Artamkin, S. V. Pestunovich, Yu. I. Baukov, J.
Orgunomet. Chem. 1991, 408, 305-322; silicon compounds: M. J. Barrow,
E. A. V. Ebsworth, M. M. Harding, J. Chem. SOC.Dalton Trans. 1980, 18381844; F. Liebau, Inorg. Chim. Acta 1984,89, 1-7; R. R. Holmes, R. 0. Day,
V. Chandrasekhar, J. J. Harland, J. M. Holmes, Inorg. Chem. 1985, 24, 20162020; G . Klebe, J. Organomef.Chem. 1987,332,35-46; A. A. Macharashvili,
V. E. Shklover. Yu. T. Struchkov, G. I. Oleneva, E. P. Kramarova, A. G.
Shipov, Yu. I. Baukov, J. Chem. SOC.Chem. Commun. 1988, 683-685; G .
Klebe, Struct. Chem. 1990, 1,597-616; R. R. Holmes, Chem. Rev. 1990, 90,
17-31; V. F. Sidorkin, V. V. Vladimirov, M. G. Voronkov, V. A. Pestunovich,
J. Mol. Struct. (THEOCHEM) 1991,228,l-9; K. Tamao. T. Hayashi, Y. Ito,
M. Shiro, Organometallics 1992, 11, 2099-2114; phosphorus compounds:
R. R. Holmes, Acc. Chem. Res. 1979,12,257-276; tincompounds: D. Britton,
J. D. Dunitz, J. Am. Chem. Soc. 1981,103,2971 -2979; U. Kolb. M. Driiger, B.
Jousseaume, Organometallics 1991, 10, 2737-2742; S. A. Jackson, 0. Eisenstein, J. D. Martin, A. A. Albeniz, R. H. Crabtree, ibid. 1991, 10, 30623069; see also ref. [6], Chapter 8. for a further review of silicon and germanium
[9] For more comprehensive reviews: a) ref. [6], Chapters 1.2, and 4; b) T. Auf der
Heyde, S. Afr. J. Chem., in press.
[lo] For example, structural diagrams, chemical formulas, bond distances, and
angles. For a descriptive review, see R. Hoffmann, P. Laszlo, Angew. Chem.
1991, 103, 1-17; Angew. Chem. Int. Ed. Engl. 1991, 30, 1-16.
[ ll ] Under specific constraints, such as those imposed by symmetry considerations
or those resulting from limitations on certain geometrical parameters, it may be
possible to reduce the number of degrees of freedom from 3 N-6. For example,
the imposition of Tdsymmetry on a five-atom fragment reduces the number of
parameters needed from nine to only one (the bond length); see J. D. Dunitz,
X-Ray Analysis and the Strucfure of Organic Molecules, Cornell University
Press, Ithaca, 1979, Chapter 9.
[12] For one example of a structure correlation analysis involving positional coordinates see P. Murray-Rust, J. Raftery, J. Mol. Graphics 1985, 3, 50-59.
[13] For example, the number ofinternal distances of N(N-1)/2 is much larger than
the number of degrees of freedom (3N-6).
[14] a) P. Murray-Rust, H.-B. Biirgi, J. D. Dunitz, Acta Crystallogr. Sect. B 1978,
34, 1787-1793; b) P. Murray-Rust, H.-B. Biirgi, J. D. Dunitz, ibid., 17931803; c) Acta Crysfullogr. Sect. A 1979, 35, 703-713.
Angew. Chem. Int. Ed. Engl. 1994, 33, 823-839
[15] a) F. A. Cotton, Chemical Applications of Group Theory, 3rd ed., Wiley, New
York, 1990; b) S. F. A. Kettle, Symmetry and Sfructure, Wiley, Chichester,
1985; c) E. B. Wilson, J. C. Decius, P. C. Cross, Molecular Vibrations-The
Theory of Infrared and Raman Vibrational Specfra, McGraw-Hill, London,
1955;d) D. Wald, Gruppentheorie fur Chemiker, 1st ed., VCH, Weinheim, 1985.
[16] With reference to the case cited in the text, for example, any molecules whose
representative points fall exactly onto the S, or S, coordinates would exhibit
full C,, symmetry, since these coordinates transform as A , . Expressed differently, any such molecules would differ from the reference structure only in their
bond angle (S,), or in the length of the two X- Y bonds (each being shortened
or lengthened by the same amount; S2);both of these deformations maintain
C,, symmetry.
[17] In such cases it may be necessary to refer to the group theory of "nonrigid"
molecules, see H. C. Longuet-Higgins, Mol. Phys. 1963, 6,445-460; H. Frei,
A. Bauder, H. H. Gunthard, Top. Curr. Chem. 1977,81,1-97; R. L. Flurry, J.
Chem. Educ. 1984,61,663--665.
[18] Statistical methods suitable for structure correlation studies have been reviewed in T. P. E. Auf der Heyde, J. Chem. Educ. 1990,67,461-469 and in ref.
[9a, bl .
[19] Crucial tools for this stage are the Cambridge Structural Database [20], which
contains crystallographic data for a large number of organic and organometallic compounds, or the Inorganic Crystal Structure Database [G. Bergerhoff, R.
Hundt, R. Severs, I. D. Brown, J. Chem. In5 Comput. Sci. 1983, 23, 66-69],
housing data on inorganic structures. The databases allow the user to define a
search fragment, and then retrieve all crystallographic structures that contain
that fragment. The choice of a search fragment needs to be a judicious compromise between a definition that is too narrow in its scope, thereby precluding the
retrieval of entries whose distortions might be of extreme interest and importance, and one that is so broadly stated as to include fragments which add
nothing to the analysis-apart, that is, from time and effort spent on it.
[20] a) F. H. Allen, 0. Kennard, R. Taylor, Acc. Chem. Res. 1983, 16, 146-153;
b) Cambridge Structural Database ( C S D ) User Manual, Version 4.2, Crystallographic Data Center, Cambridge, 1990.
[21] Angular and distance measurements, for example, are not directly comparable.
Angles may be scaled to about the same range as distances, by expressing them
as radial displacements a on a circle whose radius r is equal to the average bond
distance in the data set: a = 27tr (0/360").Alternatively, the data may be sfandardized to unit variance and zero mean, by using the r-transformation; see ref.
[22] for other suitable scaling and standardization techniques. For an example
of the r-transformation applied to crystallographic data in a structure correlation study, see R. Taylor, 1 Mol. Graphics 1986, 4, 123-131.
[22] a) D. L. Massart, B. G. M. Vandeginste, S. N. Deming, Y. Michotte, L. Kaufman, Chemometrics, Elsevier, Amsterdam, 1988; b) R. L. Anderson, Pracfical
Statisticsfor Analytical Chemists, Reinhold, New York, 1987; c) R. M. Bethea,
B. S. Duran, T. L. Boullion, Statistical Methodr for Engineers and Scientists,
Dekker, New York, 1975; d) R. Caulcutt, R. Boddy, Statistics for Analytical
Chemists, Chapman and Hall, London, 1983.
[23] a) P. Murray-Rust, W. D. S. Motherwell, Acta Crystallogr. Secf. B 1978, 34,
2518-2526; b) P. Murray-Rust, R. Blafid, ibid. 1978, 34, 2527-2533; c) P.
Murray-Rust, W D. S. Motherwell, ibid. 1978,34,2534-2546; d) J. Am. Chem.
SOC.1979,101,4374-4376; e) P. Murray-Rust, Acta Crystallogr. Secf.B 1982,
38,2765-2771 ;f) P. Murray-Rust, J. Raftery, J. Mol. Graphics 1985,3,60-68.
[24] R. Taylor, 0. Kennard, Acta CrystaNogr. Sect. B 1983.39, 517-525.
[25] Principal component analysis is often referred to as factor analysis, a generically related, but distinct, method: C. Chatfield, A. J. Collins, Infroduction to
Multivariate Analysis, Chapman and Hall, London, 1980.
[26] E. R. Malinowski, D. G. Howery, Factor Analysis in Chemisfry,Wiley, New
York, 1980.
[27] The matrix (E) composed of the normalized eigenvectors is multiplied by the
diagonal matrix
of the square roots of the eigenvalues: P = E.A'/'. The
PCs are then obtained from the columns of the resulting (3 N-6) x (3 N-6)
matrix (P), while the eigenvalue associated witheach PC represents the proportion of the total sample variance accounted for by that PC.
[28] D. L. Massart, L. Kaufman, The Interpretation of Analyfical Chemical Data by
the Use of Cluster Analysis, Wiley, New York, 1983; J. Zupan, Clustering of
Large Data Sets, Research Studies Press (Wiley), Chichester, 1982.
[29] For instance, a pair of otherwise identical structures may differ by 0.5 8, in a
particular bond distance, while another pair differs only by lo" in a given bond
angle. Without suitable scaling or standardization, the representative points
for the first pair will be closer to each other than those of the second pair, even
though the molecules of the first pair differ more than those of the second in
a chemical sense.
[30] In crystallographic terms this is equivalent to the problem of ensuring that
wherever possible all the atoms listed for a given molecule are found within the
same asymmetric unit and not distributed over symmetry-equivalent positions
throughout the crystal lattice, which would give rise to a disjoint molecule.
[31] In the case of the triatomic molecules discussed in Section 2.1, for example, this
means that four points would be generated for any given structure, one for each
of the four operations of the point group (C,").However, in this instance, the
permutations for the operations E and a(xy) will generate the same coordinates
for the representative point because the molecules are planar; the same goes for
the permutations brought about by the operations C, and a’bz). The points
will all be equidistant from the origin, that is, equally distorted with respect to
the reference molecule, since they represent symmetry-equivalent permutations
of the labeling of the atoms on the same molecular skeleton.
[32] See, for example, a) F. Basolo, R. G. Pearson, Mechanisms oflnorganic Reactions,2nd ed., Wiley, New York, 1967; b) G. K. Anderson, R. J. Cross, Chem.
SOC.Rev. 1980, 9, 185-215; c) Mechanisms of Inorganic and Organometallic
Reactions, Vol. 2 (Ed.: M. V. Twigg), Plenum, New York, 1984; d) R. J. Cross,
Adv. Inorg. Chem. 1989,34,219-292.
[33] G. K. Anderson, R. J. Cross, Ace. Chem. Res. 1984, 17, 67-74; R. J. Cross,
Chem. SOC.Rev. 1985, 14, 197-223; R. J. Cross in ref. [32c], Chapter 5.
[34] a) Clearly, when the ligands are not all identical in the tetravalent complex the
symmetry of the SQP will be lower (than C4J. The term SQP then implies that
all apical equatorial angles are only approximately equal. b) The SQP + TBP
rearrangement implies that pairs of opposite apical-equatorial angles remain
approximately equal during the process.
1351 R. S. Berry, J. Chem. Phys. 1960.32, 933-938.
136) The Berry deformation has been inappropriately called a “pseudorotation”;
the latter term was borrowed from work on the fluxional puckering motion of
cyclopentane (see J. E. Kilpdtrick, K. S. Pitzer, R. Spitzer, J. Am. Chem. SOC.
1947, 69, 2483-2488). where it denotes an equipotential process in which the
motion of the atoms is perpendieular to the direction of pseudorotation. As
early as 1970 Muetterties pointed out that “the terminology is imprecise and
should be dropped’ (see footnote 1441 in E. L. Muetterties, Ace. Chem. Res.
1970, 3, 266-273).
[37] Another mechanism proposed for such an intramolecular exchange process is
the turnstile rearrangement [38]. However, two independent ab initio examinations, one on the model compound PH, (J. A. Altmann, K. Yates, I. G. Csizmadia, J. Am. Chem. SOC.1976,98,1450-1454), and the other on PF, (P. Russegger, J. Brickmann. Chem. Phys. Lett. 1975,30,276-278). have shown that this
mechanism requires aconsiderably higher activation energy and is not a strictly
different pathway, but can he regarded as a distortion away from the minimum
energy path of the Berry coordinate. Both studies conclude that a turnstile
mechanism will only be realized for structurally restricted systems, and that the
true saddle point corresponds to the square-pyramidal intermediate with C.,,
symmetry. For the remainder of this paper we shall focus on the Berry process,
referring to the turnstile mechanism only where explicit mention is made of it
in the study under review.
[38] I. Ugi, D. Marquarding, H. Klusacek, P. Gillespie, F. Ramirez, Acc. Chem.
Res. 1971, 4, 288-296.
[39] E. L. Muetterties. L. J. Guggenberger, J. Am. Chem. Soe. 1974,96,1748-1756.
1401 The increment in the observed bond distance between two atoms A and B is
established relative to the length of a “standard” covalent A-B bond length.
Standard bond lengths can be obtained from any number of tables, such as
those compiled by L. Pauling, The Nature of’The Chemical Bond, 3rd ed.,
Cornell University Press, Ithaca, 1960. A comprehensive set of bond distances
has recently been collected in A. G. Orpen, L. Brammer, F. H. Allen, 0. Kennard, D. G . Watson, R. Taylor, J. Chem. SOC.Dalton Trans. 1989, Sl-SS3;
F. H. Allen, 0. Kennard, D. G. Watson, L. Brammer, A. G. Orpen, R. Taylor,
J. Chem. SOC.Perkin Trans. 2 1987, Sl-Sl9.
[41] L. Pauling, J. Am. Chem. SOC.1947.69, 542-553.
1421 J. K. Burdett, Inorg. Chem. 1979, 18, 1024-1030.
[43] Insofar as the reaction coordinates are obtained from a correlation analysis of
all the structures comprising the data set, the underlying assumption behind
this kind of approach (see also ref. [44]) is that the potential energy surface for
each observed structure has exactly the same topography. This is clearly an
untenable assumption, given the widely divergent compounds involved; the
shortcomings of this approach have since been discussed in H.-B. Biirgi, J. D.
Dunitz, Acta Crysfallogr.Sect. B 1988, 44,445-448.
[44] F. Pavelcik, E. Luptakova, Collect. Czech. Chem. Commun. 1990, 55, 14271434.
(451 H.-B. Biirgi. K. C. Dubler-Steudle, J. Am. Chem. Soc. 1988, 1f0,4953-4957.
1461 They complained of the “irksome phrase. , .’the molecule is a distorted trigonal
bipyramid‘. . .encumbered with imprecise English and often encountered in
structural papers’’ [author’s emphasis], while admitting to using such language
[47] The term “polytopal isomer” was first coined by Muetterties in 1969 (E. L.
Muetterties, J. Am. Chem. SOC.1969,9f, 1636-1643), and it refers to aclass of
stereoisomers “of idealized polygons or polyhedra whose vertices are defined
by the ligand atom positions in coordination compounds”.
[48] Cadmium, phosphorus, cobalt, nickel, niobium, and antimony.
[49] This diagram has even found its way into an inorganic chemistry textbook; see
J. E. Huheey, Inorganic Chemisrry, 2nd ed., Harper and Row, New York, 1979,
p. 196.
[SO] R. R. Holmes, J. A. Deiters, J. Am. Chem. Soc. 1977, 99, 3318-3326.
[51] A critique that might be raised here is that the nine dihedral angles described
by the shape of the TBP polyhedron are not all independent.
1521 T. P. E. Auf der Heyde, L. R. Nassimbeni, Inorg. Chem. 1984,23,4525-4532.
[53] Allylic compounds were excluded from all the papers reviewed here, because it is difficult to identify a definite atom or point of ligation in such
T. Auf der Heyde
1541 A. Gleizes, A. Kerkeni, M. Dartiguenave, Y. Dartiguenave, H. F. Klein, Inorg.
Chem. 1981, 20,2372-2391.
[55] In this case the value of the trans-basal angle was chosen as 105”, which is the
value obtained when placing the metal atom at the center of mass of an L,
square pyramid.
[56] T. P. E. Auf der Heyde, L. R. Nassimbeni, Acta Crystallogr. Sect. B 1984, 40,
[57] D. Liao, H. Fu, Y. Tang, Wuli Huaxue Xuebao 1987,3,449-452; Chem. Abstr.
1988, 108, 119883.
[58] C. P. Casey, G. T. Whiteker, C. E Campana, D. R. Powell, Inorg. Chem. 1990,
29, 3376-3381.
[59] G. H. M. Dias, M. K. Morigaki, Polyhedron 1992, ff, 1629-1636.
[60] B. J. Hathaway, Coord. Chem. Rev. 1982,41,423-487.
(611 a) W. D. Harrison, D. M. Kennedy, M. Power, R. Sheahan, B. J. Hathaway, J.
Chem. SOC.Dalton Trans. 1981, 1556-1564; b) S. Tyagi, B. J. Hathaway, ibid.
[62) B. J. Hathaway, Struct. Bonding (Berlin) 1984, 57, 55-118.
[63] W. D. Harrison, D. M. Kennedy, B. J. Hathaway, Inorg. Nucl. Chem. Lett.
1981, 17, 87-90.
[64] S . Tyagi, B. J. Hathaway, S. Kremer, H. Stratemeier, D. Reinen, J. Chem. SOC.
Dalton Trans. 1984, 2087-2091.
[65] N. J. Ray, L. Hulett, R. Sheahan, B. J. Hathaway, J. Chem. Soc. Dalton Trans.
1981, 1463-1469.
1661 E. Muller, C. Piguet, G. Berndrdinelli, A. F. Williams, Inorg. Chem. 1988, 27,
[67] This representation included 15 parameters (five distances and ten angles), of
which only twelve are independent.
[68] 0. Carugo, C. B. Castellani, J. Chem. SOC.Dalton Trans. 1990, 2895-2902.
[69] The main problem in this study lies with the conceptualization of the parameter
space, which is defined in terms of “three coplanar axes at 120””, and yet is
treated as an orthogonal “three-dimensional” space. The three axes are defined
by the differences between the observed angles and the values they would adopt
in a presumed SQP, each axis corresponding to a different permutation of the
ligand labels in the complex. It is not clear that the three coplanar axes represent truly independent coordinates, and, moreover, in the hypothetical case of
an observed complex having exactly the same geometry as the reference SQP,
the denominator in the shape parameter becomes zero, rendenng the parameter space ill-defined.
[70] a) T. P. E. Auf der Heyde, H.-B. Biirgi, Inorg. Chem. 1989, 28, 3960-3969;
b) ibid. 1989, 28, 3970-3981; c) ibid. 1989, 28, 3982-3989.
1711 The data were retrieved from the July 1984 version of the CSD [20] and comprised 113 nickel, 15 palladium, 9 platinum, 28 rhodium, and 31 iridium structures.
[72] With regard to the SCs of Section 2.1, for instance, the length of the reference
bond (in the calculation of the distance increment [40]) would cancel in S , ,
which transforms as B,, but not in S,, which is totally symmetric ( A , ) .
[73] The fact that no “ideal” SQP needs to be defined also means, however, that
there is no single representative point for this conformer in the corresponding
12-dimensional parameter space. Instead, all allowed SQPs with C,, symmetry
will lie along a continuous line, which reflects the one degree of freedom that
this symmetry allows the SQP (see Section 3).
[74] L. Nsrskov-Lauritsen, H.-B. Biirgi, J. Comput. Chem. 19856,216-228.
[75] A given structure whose representative point is clustered together with the
points for all the other SQPs in the data set should, in general, be classified in
this way, irrespective of the kind of algorithm employed.
[76] The size is defined as the average distance [A] of cluster members from the
center of the cluster.
[77] D. L. Kepert, Inorganic Stereochemistry, Heidelberg, Springer, Berlin, 1982,
and references therein.
[78] M. C. Favas, D. L. Kepert,Prog. Inorg. Chem. 1980,27, 325-463.
[79] The angles BME and BMD are defined within the same plane in the structures
as the angles bD and 41E; the two pairs of angular parameters are therefore
geometrically related.
[SO] D. L. Kepert, E. S. Kucharski, A. H. White, J. Chem. SOC.Dalton Trans. 1980,
1932- 1938.
[81] J. L. Mesa, M. I. Arriortua, L. Lezama, J. L. Pizarro, T. Rojo, D. Beltran,
Polyhedron 1988, 7, 1383-1388.
1821 a) R. J. Gillespie, J. Chem. SOC.1%3,4679-4685; b) J. K. Burdett, Adv. Inorg.
Chem. Radiochem. 1978,2f, 113-134.
[83] A. R. Rossi, R. Hoffmann, Inorg. Chem. 1975.14, 365-374.
[84] In line with our use of these terms in earlier papers [70], we refer to the apical
bond in a SQP, and the axial bonds in a TBP.
[SS] Rossi and Hoffmann also include n-donor and -acceptor effects in their analysis, but insofar as most of the complexes in the studies considered here would
have minimal n-bonding ligands (see ref. [S3]), these effects will not be discussed at this point.
1861 In these papers [87], Holmes presents a comprehensive overview of his structural work specifically with compounds containing pentacoordinate phosphorus and other main-group elements, as well as a compilation of studies on
five-coordinate transition metal complexes. He compares the observed structures to those predicted by Rossi and Hoffmann (831 for a variety of different
Angew. Chem. h t . Ed. Engl. 1994.33, 823-839
Structure Correlation Method
d-orbital configurations, and interprets the observed conformations in terms of
repulsions between nonbonding electrons in d-orbitals and electrons in bonding orbitals.
[87] a) R. R. Holmes, Prog. Inorg. Chem. 1984, 32, 119-235; b) R. R. Holmes, J.
Am. Chem. SOC.1984, 106, 3745-3750.
[88] Twenty-eight nickel, four platinum, five iridium, and seven rhodium complexes.
[89] See, for example, C. P. Blaxill, S. S. D. Brown, J. C. Frankland, I. D. Salter, V.
Sik, J. Chem. SOC.Dalton Trans. 1989, 2039-2047, and references therein;
S. S. D. Brown, I. D. Salter, V. Sik, I. J. Colquhouu, W McFarlane, P. A.
Bates, M. B. Hursthouse, M. Murray, ibid. 1988, 2177-2185; L. J. Farrugia,
M. J. Freeman, M. Green, A. G. Orpen, F. G.A. Stone, I. D. Salter, J.
Organomet. Chem. 1983,249,273-288.
[90] A. Rodger, B. F. G. Johnson, Polyhedron 1988, 7, 1107-1120,
[91] See also D. Britton, J. D. Dunitz, 1 Am. Chem. Soc. 1975, 97, 3836.
[921 A. G. Orpen, I. D. Salter, Organometallics 1991, 10, 111-117.
[93] How do the R and u properties of ligands affect their positions in the TBP and
SQP, and how do they correlate with the stereochemistry of the complexes?
Such an analysis would require algorithms capable of handling categorical
variables (for the different sites or the different kinds of ligands) as opposed to
continuous variables. For example, the influence of o-donor capability
on the dynamic stereochemistry of [Ni(CH,),A,], where A is a weaker u
donor than CH,, has been examined in K. Tatsumi, A. Nakamura, S. Komiya,
A. Yamamoto, T. Yamamoto, 1 Am. Chem. SOC. 1984, 106, 81818188.
1941 A. Vedani, D. W Huhta, d Am. Chem. Soc. 1990, 112, 4759-4767.
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correlation, structure, reaction, metali, method, determination, complexes, pentacoordinate, path
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