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Electron Domains and the VSEPR Model of Molecular Geometry.

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REVIEWS
Electron Domains and the VSEPR Model of Molecular Geometry
Ronald J. Gillespie" and Edward A. Robinson
The valence shell electron pair repulsion
(VSEPR) model-also
known as the
Gillespie-Nyholm rules-has for many
years provided a useful basis for understanding and rationalizing molecular geometry, and because of its simplicity it
has gained widespread acceptance as a
pedagogical tool. In its original formulation the model was based on the concept that the valence shell electron pairs
behave as if they repel each other and
thus keep as far apart as possible. But in
recent years more emphasis has been
placed on the space occupied by a valence shell electron pair, called the domain of the electron pair, and on the
relative sizes and shapes of these domains. This reformulated version of the
model is simpler to apply, and it shows
more clearly that the Pauli principle provides the physical basis of the model.
Moreover, Bader and his co-workers'
analysis of the electron density distribution of many covalent molecules have
shown that the local concentrations of
electron density (charge concentrations)
in the valence shells of the atoms in a
molecule have the same relative locations and sizes as have been assumed for
the electron pair domains in the VSEPR
model, thus providing further support
for the model. This increased understanding of the model has inspired efforts to examine the electron density distribution in molecules that have long
been regarded as exceptions to the
VSEPR model to try to understand
these exceptions better. This work has
shown that it is often important to consider not only the relative locations and
sizes, but also the shapes, of both bonding and lone pair domains in accounting
1. Introduction
The valence shell electron pair repulsion (VSEPR) model has
for many years provided a useful basis for understanding and
rationalizing molecular geometry, and its simplicity has led to its
widespread acceptance as a pedagogical tool.[' 31 Nevertheless
it has often been regarded as a purely empirical theory having no
real physical basis, or as a classical electrostatic theory. It is the
purpose of this review to show that 1) the model continues to
provide a simple explanation of the structures of many new
molecules that have been prepared in recent years, including
many detailed features of their structures that often cannot be
easily accounted for in any other way; 2) the formulation of the
model in terms of electron domains is simpler to understand and
to apply than the original version of the model; 3) the Pauli
~
[*] Prof. R. J. Gillespie
Department of Chemistry, McMaster University
12x0 Main Street West
Hamilton, Ontario, LXS 4M1 (Canada)
Fan. Int. code + (905) 522-2509
e-mail : gillespi(a mcmai1.cis.mcmaster.ca
Prof. E. A. Robinson
Department of Chemistry, University of Toronto (Canada)
Angew. Chem. In!. Ed. Engl. 1996. 3.5, 495-514
for the details of molecular geometry. It
has also been shown that a basic assumption of the VSEPR model, namely
that the core of an atom underlying its
valence shell is spherical and has no influence on the geometry of a molecule, is
normally valid for the nonmetals but often not valid for the metals, including
the transition metals. The cores of polarizable metal atoms may be nonspherical
because they include nonbonding electrons or because they are distorted by
the ligands, and these nonspherical
cores may have an important influence
on the geometry of a molecule.
Keywords: electron density distribution
electron pair domains . molecular geometry . VSEPR model
principle is the physical basis of the electron pair domain concept and therefore of the VSEPR model; 4) the domain version
of the VSEPR model provides a complement to the conventional orbital models that is useful in introductory chemistry for
discussing not only the topic of molecular geometry but also the
concept of the chemical bond itself; 5 ) the domain version of the
model is supported by the analysis of the total electron density
of a molecule in terms of both electron density deformation
maps and the Laplacian of the electron density; 6) the analysis
of the Laplacian of the electron density leads to a better understanding of the structures of metal -nonmetal type molecules,
including those of the transition metals and shows how the
VSEPR model can be modified so that it applies to such
molecules.
2. The Pauli Principle and Electron Pair Domains
2.1. The Pauli Principle
The Pauli principle provides the physical basis for the VSEPR
model. In its most general form it states the following: The total
wave functiom for a system must be antisymmetric to the exchange
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R. J. Gillespie and E. A. Robinson
REVIEWS
of’any tvvo electrons.[41A direct conclusion is that two electrons
of the same spin have a zero probability of being at the same
point in space and a maximum probability of being as far apart
as possible, i n other words, electrons with the same spin tend to
keep apart in space.[’] The Pauli principle places no restrictions
on electrons of opposite spin, however, which may indeed be
close together or even at the same location.
The importance of the Pauli principle in determining molecular geometry can be clearly seen if we consider the valence shell
of eight electrons (the octet) that is commonly found for many
atoms in their molecules. In such a valence shell there are four
electrons of a spin and four of spin. As a consequence of the
Pauli principle there is a high probability that the four electrons
of c( spin will have a mutually tetrahedral arrangement, as will
the four fi electrons (Fig. 1a, b, d, e).[’] From the point of view
of the Pauli principle the two tetrahedral sets of electrons are not
correlated. However, electrostatic repulsion not only reinforces
the tendency of each set of electrons to adopt a tetrahedral
arrangement but also tends to keep the two tetrahedra apart. In
a free atom such as the neon atom no external force restrains the
free movement of the two tetrahedral sets of electrons, which
results in an overall spherical total electron density. In contrast,
in a molecule the positive cores of one or more ligand atoms
attract the valence shell electrons of the central atom. In accordance with the Pauli principle a ligand core may attract two
electrons of opposite spin into the same region to form a bond.
If A is the central atom with tetrahedral arrangements of both
a electrons and p electrons in its valence shell, the formation of
two A-H (or two A-X bonds where X is a singly bonded
ligand) brings the x spin tetrahedron into approximate coincidence with the fi spin tetrahedron, thus forming four electron
H
a)
bl
C)
@
d)
e)
f)
Fig. 1. Most probable arrangement of four valence shell electrons with spin 1 (a)
and spin fl (b), and of eight valence shell electrons in an AH, molecule ( c ) . Two
dimensional representations of the domains of four valence shell electrons with
spin CI (d) and spin fl (e), and of eight valence shell electrons (0in an AH, molecule.
pairs-two bonding pairs and two nonbonding (lone) pairs
(Fig. 1 c, f). Each pair of electrons is approximately localized in
a part of the valence shell of A-one pair in each of two bonding
regions and one pair in each of two nonbonding regions.
2.2. Electron Pair Domains
We call the region of space in the valence shell in which an
electron pair is most probably to be found an electron pair
domain. DaudelC6]has previously called such a region of space
a loge. Thus a valence shell containing four electron pairs may
Prof. Ronald J. Gillespie graduated from University College London, obtaining his Ph.D. degree in 1949 and DSc. in 1957. He was
a Lecturer at University College London from 1948 to 1958 when
he moved to McMaster University as Associate Professor, becoming Professor in 1960 and Emeritus Professor in 1989. After his
Ph.D. research on aromatic nitration in sulphuric acid with Sir
Christopher Ingold, he did extensive work on the chemistry of
superacid media such as HJO,, HSO& and HF including the
preparation of noble gasjluorocations and a large number of novel
polyatomic cations of the nonmetals. He has received many
awards ,for his work, including election to the Fellowship of the
E. A. Robinson
R, J. Ci,,espie
Royal Society of London in 1977 and the American Chemical
Society awards for Creative Work in Fluorine Chemistry and for Distinguished Service to Inorganic Chemistry. He has had a
long standing interest in Chemical Education and in the use of simple models in the teaching of chemistry. He first proposed the
widely used VSEPR model with Ronald S. Nyholm in 1957.
Born in London, England, in 1933, E. A . (Peter) Robinson obtainedhis Ph.D. under RonaldJ. Gillespie in 1958 at the University
College, London. Then he moved to McMaster University and worked as an original member of the Gillespie research group in
Canada. From 1961 to 1994 he was at the University of Toronto, where he is now) Professor Emeritus. In 1965 he became a
founding member of the University of Toronto’s Erindale Campus in Mississaugu. As the first Associate Dean, the second Dean,
and the third Principal, heplayeda major role in the organization and construction of the newest Toronto campus, which is now
equally recognized for the quality of its teaching and research. Ongoing research interests include the understanding of both
lengths in terms of the Lewis octet rule and expanded octets, and hydrogen bonding involving C - H bonds. He was awarded a
D.Sc. by the University of London in 1969.
496
Angew. Chrm. I n [ . Ed Engl. 1996, 35,495-514
The X E P RRIodel
be thought of .IS being divided into four tetrahedrally arranged
regions or domains in each of which the probability of finding
an electron pair is high. We will see later that it is important to
consider the shapes of domains but as a rough first approximation wc may consider that a domain has a spherical shape, as
first proposed by Bent in his tangent-sphere model.[’] Four
spheres pack as closely as possible around a point (the core of
the atom) by adopting a tetrahedral arrangement, which
demonstrates in a simple way that four equivalent domains are
expected to have a tetrahedral arrangement (Fig. 2). Similarly
REVIEWS
drally, and the molecule has a tetrahedral geometry. In an AX,E
molecule, where E is a lone pair, there are three single bond
domains giving a triangular-pyramidal molecule with an approximately tetrahedral bond angle. In an AX,E, molecule two
single bond domains and two lone pair domains give a molecule
with an angular geometry and a bond angle of approximately
109” (Fig. 3).
a)
X
Fig. 2 Arr.ingeincnh of two to six spherical domains surrounding the core.
two spheres representing two electron pair domains adopt a
linear arrangement, three spheres a triangular arrangement, five
spheres a trigonal-bipyramidal arrangement, and six spheres an
octahedral arrangement (Fig. 2). In the arrangements of two,
three, four, and six electron pairs all the domains are equivalent.
However, it is not possible to have an arrangement of five equivalent domains, except by arranging them in a plane. But this
arrangement does not keep the domains as far apart as
possible. In the trigonal-bipyramidal arrangement there is a set
of three equivalent (equatorial) domains and a set of two
equivalent (axial) domains (Fig. 2). As a result AX,, molecules
with five identical ligands d o not have five equal AX bonds,
but have two longer axial bonds and three shorter equatorial
bonds.
The basic assumption of the VSEPR model in its original
form was that a given number of valence shell electron pairs,
considered as points, adopt that arrangement that keeps them as
far apart a s possible (the points-on-a-sphere model). This formulation of the model has sometimes led to the misunderstanding that i t is just electrostatic repulsion that keeps electron pairs
apart. But the formulation of the model in terms ofelectron pair
domains makes it clear that the formation ofelectron pairs that
keep as far apart as possible is a consequence of the Pauli exclusion principle. electron-nucleus attraction, and electron-electron repulsion.
2.3. Single Bond and Lone Pair Domains
In the valence shell of an atom that obeys the octet rule there
are four electron pair domains. When these domains are equivalent, as in an AX, molecuie where X is a singly bonded ligand,
the four equivalent single bond domains are arranged tetrahe-
Fig. 3 AX,. AX,E. and AX,E, molecules. a) Domain model\: S is a single bond
domain and L a lone pair domain b) Conventional bond line sti-uctures.c ) Approximate relative shapes and sizes of the doinains i n an AX,E molecule
Deviations in bond angles from the ideal angles corresponding to the various most probable electron pair arrangements can
be accounted for by recognizing that bonding and nonbonding
domains are not equivalent, and in particular, by assuming that
they have different shapes and sizes. A nonbonding valence shell
electron pair is attracted by just one positive core and so its
domain tends to surround the core and thus take up as much
space as possible in the valence shell. However, a single bond
domain is restricted in size and has a different shape because
the two electrons are attracted by two positive cores. Thus in
any valence shell a nonbonding domain is larger and is closer
to the core than a bonding domain (Fig. 3c). The larger size
of a lone pair domain accounts for the observation that in
very many molecules in which an atom has one or more lone
pairs the angles between the bonds are smaller than the ideal
values. For example, they are smaller than 109.5’ for AX,E
and AX,E, molecules (Tables 1 and 2) and smaller than 90”
for an AX,E molecule (Table 3). Although electron pair domains are not as localized as shown in Figures 2 and 3 it is
very convenient, as we shall see, to think of them as nonoverlapping. So the representations of the domains in Figures2
and 3 can be regarded as approximately denoting those regions
where there is a relatively high probability of finding an electron pair.
In its original formLs1the VSEPR model postulated that the
strength of the repulsions between electron pairs decreases in
the order lone-pair-lone-pair > lone-pair- bond-pair > bondpair-bond-pair. This assumption leads to the same conclusions
about bond angles as the domain model, although the conclu497
R. J. Gillespie and E. A. Robinson
REVIEWS
Table I . Bond angles 3 ['I in trigonal-pyramidal AX,E molecules and ions.
Molecule
z
Ref.
107.3
93.8
91.8
91.7
102.2
97.8
96 1
87.3
107.1
100.3
98.6
97.2
100
101.1
99.8
98.2
lOO(4)
1441
[45]
1461
1461
[47]
1481
[49, 501
[51]
1521
[53]
[50]
[54]
1551
[56]
[57]
[54]
1551
Table 2 Bond angles z
Molecule
1
Ref.
102
100 2
99.1
110.9
98.6
96.0
94.2
96.7
110.9
97.2
100.0
100.0
97.5
94.5
YO(2)
103.3
1'1 in angular AX&,
that time. However, even though it has several advantages over
the electron-pair repulsion formulation, the domain concept has
not often been used in the discussion of molecular shapes in
terms of the VSEPR model. These advantages are particularly
evident for AX,E, AX,E, and AX,E, molecules. It is simple to
understand, for example, why a lone pair always occupies an
equatorial position in these types of molecules. An equatorial
position is less crowded than an axial position because it has two
neighboring axial positions at 90" and two equatorial positions
further away at 120", whereas an axial position has three neighboring equatorial positions at 90". Because they are larger than
bond pair domains, lone pair domains always occupy the less
crowded equatorial positions of these molecules, so that AX,E
molecules such as SF, always have a disphenoidal shape,
AX,E, molecules such as CIF, always have a T-shape, and
AX,E, molecules such as XeF, always have a linear shape
(Fig. 4). Another advantage of the domain model is that it allows us to understand how a lone pair domain may exert direc-
molecules and ions
~~~~
Molecule
x
Ref.
Molecule
OH,
SH,
SeH
TeH,
0 F,
SF,
SeF,
OCI,
SCI,
SeCI,
TeC1,
O(CH,),
S(CH,),
Se(CH,),
S(CF,),
Se(CF,),
104.5
92.1
90.6
90.3
103.1
98.2
94
111.2
102.8
99.6
97.0
111.7
99.1
96.3
97.3
95.6
[71]
(721
1731
[74]
[75]
[76]
[771
[78]
[79]
[SO]
[Sl]
[82]
1831
[841
[85]
[86]
CIF:
BrFf
la;
HOF
HOCl
HOBr
CF,OF
CH,OH
CH,SH
CH,SeH
CF,SH
CH,SCI
CF,SF
CF,SCI
,
Table 3 Bond angles z(ax-eq)
ions
Compound
ao;
NH;
NF;
['I
1
Ref.
96
92
93
97 2
102.5
110
104.8
108.6
96.5
95.5
111.2
99.4
97.1
98.9
108.6
99.4
96.7
(87. 881
[881
I891
1901
1911
[921
[93]
(941
(951
[96]
[97]
[98]
[85]
[SS]
[99]
[2]
[2]
and bond lengths d[pm] in AX& molecules and
4,
4,
179.3(9)
182.6(9)
181.0
157(1)
168.9(8)
184.4(25)
186 2(4)
191.6(4)
236
184.5(8)
185.2(19)
184.3
167(2)
177.4(3)
186.9(5)
195.2(4)
207.5(2)
258 - 269
1
Ref
~
77.7(3)
79
W2)
84.8(1)
81.9(1)
87 8
88.0
85
sions are not always reached so simply. Because electrons
of the same spin have a low probability of being close together,
electron pair domains resist overlap. It is in this sense that domains may be said to repel each other, and so larger domains
appear to repel other domains more strongly than smaller domains.
The importance of considering the relative sizes and shapes of
electron pair domains was pointed o u t by Gillespie many years
ago,"' although the term electron domain was not introduced at
498
F
F
F
1+
F
F
Fig. 4. Examples of AX,E. AX,E,, and AX,E, molecules
always occupy equatorial positions
F
in
which lone pairs
tional repulsive effects. The angles between the domains in the
equatorial plane of AX,E, AX,E,, and AX,E, molecules are
larger than the angles between the axial and equatorial domains.
Accordingly a lone pair domain in an equatorial position
spreads out more in the equatorial direction than in the axial
direction giving it a noncylindrically symmetrical shape, and
distorting a larger equatorial bond angle more than a smaller
axial-equatorial bond angle. Thus in SF,, for example, the
more easily distorted F,,SF,, angle is reduced by 19" from the
ideal value of 120". while the F,,SF,, angles are reduced by
only 1.2" from the ideal angle of 90". Other similar data
are given in Table 4. A recent analysis of the electron density of
the SF, molecule through its L a p l a ~ i a n [ (discussed
~]
in Section 3.2) has confirmed the unsymmetrical shape of its lone pair
domain.
We can similarly understand the shape of the planar XeF;
ion, the unique example of an AX,E, molecule (Fig. 5).['01 Although the best arrangement of seven equivalent domains cannot be predicted with certainty,I2]it is reasonable to suppose that
the geometry of XeF; would be based on that of the closely
related IF, and TeF; molecules, which have a pentagonalbipyramidal geometry." '. * In the pentagonal-bipyramidal arrangement of seven domains in a valence shell more space is
available in the axial positions, which make angles of 90" with
the neighboring equatorial positions, than in an equatorial position, which makes two angles of 72" and two of 90" with neighboring positions. Hence, large lone pair domains are always
expected to occupy the axial positions in a pentagonal bipyramidal arrangement of seven domains leading to the observed pen-
''
tagonal-planar shape of the XeF; ion.
Angeir.. Chem. Ini. Ed. Engl. 1996. 35, 495-514
Thc VSFFR
Idodel
- - ._
.
.-
REVIEWS
Tahlc 4. I3oiid .inJle, 1 [ ] and bond lengths d [pm] in disphenoidal AX,E and
inolecoles and ions.
T-sl1.ipc.d .\X
Molecule
4,.
4"
174
160
164.6(3)
167.3(7)
168.9(1)
167.9(4)
168.1(3)
167.0(7)
177.0(2)
188.9(4)
226-232
177.1(4)
182.7
230(5)
238
2 5 3 1)
255
200.6
251
250
250.6
268.2(3)
285 - 299
292.8( 1 )
186
184
171 3(3)
200
189.9(3)
169.8(5)
181.0
245
190.5(15)
258
F,,PF,,
F,PF,
FJF,,
154.5(3)
F,,SF,,
F,,SF,,
156.9(8)
F,,SS,,
FJF,,
157.5(5)
F,,SC,,
159.6(11) FAXSF,,
F,,SC,,
188.8(4)
F,,SF,,
C,,SC,,
FJF,,
156 3(9)
F=,,SN,,
FJF,,
164.8(2)
F,,SN,,
180.7(7)
O,,SO,,
C,,SC,,
CI,,SCI,,
I78
C,,SC,,
FJeF,,
168.2(4)
F,,SeF,,
202.2
F,,SeF,,
C,,SeC,,
CI,,SeCI,,
C,,SeC.,
CI,,SeCI,,
C,,SeC,,
BrJeBr,,
191(3)
C,,SeC,,
Br,,SeBr,,
C,,SeC,,
F,,TeF,,
21 1 3 3 )
C,,TeC,,
CI,,TeC1,,
210
C,,TeC,,
215
CI,,TeCI,,
C,,TeC,,
210.7
CI,,TeCI,,
C,,TeC,,
Br,,TeBr.,
218(3)
C,,TeC,,
IJxTeI4\
210-216
C,,TeC,,
ldrTeIdX
213.3(13)
C,,TeC,,
F,,,IF,,
177
F,,IF,,
177
FJF,,
F,,IF,,
160.3(4)
F,.ClF,,
140.5(3) [a] F,,ClO,,
193
F,,XeF,,
O.,IO.,
F,,XeF,,
171.4(4)
O,,XeO,,
-
-
159.X(5 )
172.1
200
183.5(10)
174
Ref.
3
F,,CIF,,
F,,BrF,,
CIJCldx
F,,XeF,,
Br,,SeBr,,
168.3
99.9
173.1( 5 )
101.6(5)
167.0
104.9(14)
174.6(8)
102.9(8)
165.2(25)
100.7(16)
173 9(8)
97.3(8)
174.0(12)
104.6(10)
174.7(1)
102.3(1)
175.1(2)
104.4(3)
174311)
91.4(3)
169.2(7)
100.6(7)
1572
118.7
180(5)
118
177.5
106.5
lXO(3)
1 lO(10)
177
106:
175.3(1)
96.9(1)
172 3(3)
98.2(11)
169
98
175 5(1)
99.0(3)
177.9(2)
95.4
177- 180
91 -97
17535)
95.3(5)
173.5
92.4
160
92
170.5
108.9
180
100
1X3.2(2)
105.7(3)
175(4)
172.4
172
160.9(5)
179.2
1-
F
.!
Fig. 5. a) Structure of the XeF, ion (bond lengths in pm). b) Comparison of the
structure of IF, (an AX, molecule) and XeF: (an AX,E, molecule).
less than a lone pair as exemplified by the three molecules
NO:, NO,. and NO;, which have bond angles of 180",
134", 115" respectively (Fig. 6 a). The difference in size
between an electron pair domain and a single electron domain
is also seen clearly in the calculated structure of the SF, radical,
which has C,, symmetry. This structure is consistent with a
trans arrangement of the lone pair and the single electron and
with the bending of the SF bonds, away from the lone pair
towards the single electron (Fig. 6b).[13J SF, may be said to
have an AX,Ee geometry, where e represents a single electron
domain.
Fig 6. Single electron domains. a) The bond angle decreases in the series NO;,
NO,. NO,. h) The SF, radical has a C,, structure consistent with the smaller
domain for a single nonbonding electron and a larger domain for the nonbonding
lone pair. c) Bond angles in the singlet (left) and triplet (right) States of CH,.
[a] Cl=O bond length
2.4. Single Electron Domains
In some free radicals the unpaired electron is located mainly
on a single atom. A single electron can be considered to occupy
a single electron domain that is expected to be smaller than an
electron pair domain, so it affects the geometry of a molecule
The different stereochemical influence of single electron domains and those of lone pairs is further exemplified by the singlet and triplet states of carbene (Fig. 6c). In the singlet state
CH, may be described as an AX,E molecule and, as expected,
has a bonding angle small than 120". The triplet state, which
may be described as an AX,E, molecule, has a bond angle larger
than the tetrahedral angle, showing the smaller stereochemical
effect of the single electron domains.['3b1
499
R. J. Gillcspie and E.
A. Robinson
_____
REVIEWS
2.5. Single Bond Domains and Electronegativity
The space occupied by a single bond domain in the valence
shell of A depends on the relative electronegativities of the central atom A and the ligand X. The larger the electronegativity of
X and the smaller the electronegativity of A, the less space the
A-X bond domain occupies in the valence shell of the central
atom A and the smaller the angle between the A-X bond and
other neighboring bonds, as can be seen in many of the examples
in Tables 1-4. If we consider a lone pair as a bond pair that is
bonding a ligand of zero electronegativity we can again see why
a lone pair has a domain that is larger than that of any bond
pair.
The bond angles of the hydrides of the elements of groups V
(that is, Group 15) and VI (that is, Group 16), however, appear
to be exceptions to the rule that the X-A-X bond angle decreases
with increasing electronegativity of X. The bond angles in PH,
and H,S, for example are smaller than in PF, and SF,, although
hydrogen i s less electronegative than fluorine (Tables 1 and 2).
However, hydrogen is a unique element in several ways.
One important difference between hydrogen and any other
ligand is that the electron pair of an A - H bond has to play
the role of both a bonding pair and a nonbonding pair in
that it also shields the proton on the opposite side to the bond,
in other words the domain of this pair of electrons is not confined to the bonding region but extends into the nonbonding
region of the hydrogen atom. Hence the electron density in an
A-H bond is smaller than in other single bonds. It follows that
A-H bonds can be squeezed together by lone pairs on A much
more easily than any A -X bonds, thereby giving a smaller bond
angle.
The existence of SF,, SeF,, and BrF;, but not of CIF, and
BrF, , suggests that the elements of periods 3 and 4 can accommodate up to, but no more than, six electron pair domains with
an octahedral arrangement in their valence ~ h e l l s . 1 ~A~ 1few
molecules such as SeCIi- that appear to be exceptions to this
rule are discussed later (Section 3). Hence it is reasonable to
suppose that the repulsions between bonding domains in the
valence shells of these elements only become significant as the
angle between them approaches 90"---the bond angle in an octahedral arrangement of six domains. Hence the bond angles in
PH,, ASH,, H,S, and H,Se approach this limiting value of 9 0 .
However, for all other ligands, repulsions between the bond pair
domains become significantly large before the bond angle is
close to 90".
It has long been recognized that more electronegative ligands
have a preference for the axial sites and less electronegative
ligands a preference for the equatorial sites of a trigonalbipyramidal structure, as in the examples in Figure 7. The
electron pair domains in the axial positions are more crowded
than those in the equatorial positions, so the smaller domains bonding the more electronegative ligands favor the
axial sites, while the larger domains bonding the less
electronegative ligands favor the equatorial sites. In contrast in a pentagonal-bipyramidal AX, structure, the axial
sites are less crowded and the equatorial sites are more
crowded, and the less electronegative ligands favor the axial
sites (Fig. 8).
P
n
Fig. 8. Structures of the pentagonal-bipyramidal molecules TeF,(OMe) (left) and
TeFJOMe), (right) in which the less electronegative OMe group preferentially
occupies an axial position.
2.6. Multiple Bond Domains
F
F
F
F
F
F
F
F
F
F
CF3
Fig. 7. Examples of PX, molecules in which the less electronegative ligands always
occupy the equatorial positions.
500
A double bond is formed by the sharing of two electron pair
domains between two atoms and a triple bond by the sharing of
three electron pair domains between two atoms (Fig. 9 a, b).
These representations of multiple bonds correspond to the classical bent bond descriptions (Fig. 9c, d), which account for the
planar shape of the ethene molecule and the linear shape of the
ethyne molecule, and for the decreased C C bond lengths in these
However, we would not expect the HCH bond
angle in ethene to retain the ideal tetrahedral angle. The two
bonding pairs are attracted toward the same two positive carbon cores and are therefore squeezed together, which allows the
repulsion between the C H bond domains to open the HCH
bond angle to 117.4" in ethene. In C,F4 the FCF bond angle of
112.4" i s still smaller and is closer to the tetrahedral angle
(Table 5).
Alternatively, and often more conveniently, the two single
bond electron pair domains forming a double bond may be
considered to be overlapped to such an extent that they form
A n p i s . Chrm In!. Ed E q / . 1996, 35. 495-514
Table 5. Examples of bond lengths d [pml and single bond-double bond angles in
X,C=CY, and X,C=Z molecules.
v
Molecule
Bond
rl
Bond angles
Ref.
x - c - x Y-c-Y x - c - c Y - c- c
H,C=CH,
C-H
1174
117.4
121.3
121.3
[139]
F,C=CH,
C-F
C-H
110.6
119.3
124.7
120.3
[140]
112.4
112.4
123.8
123.8
[141]
112.1
119.0
1240
120.5
[142]
115.6
115.6
171.2
122.2
[143]
115.2
115.2
122.4
122.4
[144]
114.2
114.2
122 9
122.9
[143]
113.2
113.2
12.14
123.4
[145]
115.6
117.4
122.2
121.3
[146]
c=c
108.7(1)
133.9(1)
131.5(3)
109.1(1 0)
134.0(6)
131.9(1)
131 l(3)
131.5(15)
170.6(8)
134.5(25)
171.9(3)
13533)
188.2(3)
136 3(9)
210.6(5)
136.3(15)
151.1(2)
1 35.1(3)
150.7(1)
109.5(20)
134.2(1)
C-H
-
c=c
c=c
F,C=CF,
F,C=CCI
C-F
,
c=c
C-F
c-CI
c=c
CI,C=CCI,
Br,C=CBr,
Fig. 9. Representations of the structures ofethene and ethyne. a), b) Electron pair
domain models c). d ) Bent bond structural diagrams. e) Prolate ellipsoidal double
bond domain formed from two electron pair domains. f ) Oblate ellipsoidal triple
hond domain formed from three electron pair domains. g) Domain model ofethene
showing the AX, triangular arrangement of two single bond domains S and a
prolate ellipsoidal double-bond domain D at both carbon atoms. h j Domain model
of ethyne showing the linear AX, arrangement of a single-bond domain S and an
ohlate ellipsoidal triple-bond domain T at both carbon atoms.
one double bond domain containing four electrons in which the
two separate pairs cannot be distinguished (Fig. 9e). The total
electron density distribution has an elliptical cross section with
a maximum on the CC axis. Figure 9 g shows the arrangement
of the single and double bond domains giving a planar ethene
molecule. Similarly the three single electron pair domains forming a triple bond may be considered to be overlapped to form
one six electron triple bond domain (Fig. 9 f). Figure 9 h shows
the arrangement of single and triple bond domains giving a
linear ethyne molecule.
The bent bond picture of a double or a triple bond is sometimes criticized, because it appears to suggest that there is an
absence of electron density along the bond axis. However, this
is an incorrect interpretation of the bent bond model, because
bond lines give no indication of the size of the corresponding
electron pair domain; in particular they d o not show that there
is considerable overlap of the bent bond domains to give a
double or triple bond domain that has an electron density along
the CC axis that is greater than along any other line between the
two nuclei. The bent bond model is also sometimes criticized on
the grounds that the photoelectron spectrum of ethene shows
two ionization energies associated with the double bond, whereas the bent bond model implies that all four of the bond electrons are equivalent. The photoelectron spectrum does not,
however, tell us anything about the ground state of ethene but
only that, when one electron is removed from the double bond,
two states of the ion may be obtained, which can be described
as O'T' and as d~'.
Clearly it is convenient when discussing
such spectra to represent the ground state as d n ' , but it is not
essential to d o so, because a bent bond description, for example
one based on sp3 hybrid orbitals, is essentially e q ~ i v a l e n t . " ~ ~
The bent bond description is, however, not convenient for the
ionized state because the unpaired electron is more delocalized
than this description permits.
c-CI
c=c
C-Br
c=c
I,C==CI,
c-I
(CH,),C=C(CH,),
C-C
(CH,),C=CH,
c=c
c=c
c-c
C-H
H,C=O
c=o
F,C=O
C-F
CI,C=O
c-CI
Br,C=O
C-Br
c=o
c=o
(CF,),C=NH
c=o
c-c
c=o
c-c
c=o
c-c
H,C=S
C=N
C-H
F,C=S
C-F
CI,C=S
c-CI
Br,C=S
C-Br
(CH,j,C=O
(CF,),C=O
c=s
c=s
c=s
c=s
120.8(3)
131.6(1)
117.2(1)
173.8(2)
117.6(3)
192.3(5)
117.8(9)
151.7(2)
121.1(2)
154.9(8)
124.6(14)
155.1(7)
l29.6(29)
109.2(2)
161.1(2)
I3 1.6(2)
158.9(2)
172.9(3)
1 6 0 43j
189.4(9)
159.7(5)
x-c-x
x-c-Y
116.5
121 8
[I 471
107.7
126.2
~1481
111 8
124.1
[1491
112.3
123 8
IljOI
116.7
121 7
11511
121.4
119 3
11521
121.6
119 2
[I 521
116.9
121 6
[1531
107 1
126 5
11541
111.2
I24 4
[1551
111.6
124 2
[1561
Only the most electronegative atoms attract electrons strongly enough to pull two or three electron pairs into the bonding
region against their mutual repulsion so that in the vast majority
of multiple bonds at least one of the atoms is C,N, or 0.Only
a few examples of molecules containing double bonds between
third and subsequent period elements exist, because their
smaller electronegativity than that of period 2 elements allows
the repulsion between the four electrons to produce a larger,
more diffuse and hence more reactive domain that is very readily
attacked by electrophilic reagents. The only stable examples of
such molecules have large bulky groups protecting the double
bond from attack, as for example in tetramesityldisilene
(Mes),Si = Si(Mes), .
The general shapes of molecules containing multiple bonds
can be readily predicted simply on the basis of the number of
single, double, triple, and lone-pair domains in the valence shell
of any given atom by using the basic rule that the domains share
out the space in the valence shell of an atom so as to keep as far
apart as possible. So three domains (single, double, triple, or
lone-pair) always have a trigonal planar arrangement, four do501
R. J. Gillespie and E. A. Robinson
REVIEWS
mains a tetrahedral arrangement, five domains a trigonalbipyramidal arrangement, and six domains an octahedral
arrangement. Examples of shapes of molecules containing multiple bonds are given in Figure 10.
o=c=o
A X2
Table 6. Examples of bond lengths d [pm] and bond angles [ ]
molecules.
Bond
d
F P ,
S-F
153.0(3)
140.5(3)
201 2(4)
141.8(3)
157.4(10)
142.2(10)
177 114)
143.5(3)
176.9(4)
143.8(3)
162.3(5)
144.515)
179.312)
153 5(2)
S=O
CI,SO,
CI
>c=o
S-CI
S=O
-0
-o ) L o
">S=0
(HO)zSO,
CI
(CHJW2
(CH2=CH),S0,
AX2E
0
CI
N
S-OH
S=O
S-C
S=O
S-C
S=O
[(CH,),N],SO,
S-N
S=O
(CH,),S(NH),
S-C
S=N
AX,E
X2S(=Y)?
Ref.
Bond angles
x-s-x
Y-s-Y
Molecule
H-CrN
in
96.1(2)
124.0(2)
[I571
100.3(2)
123 512)
IlSSI
101.3(10)
123.3(10)
[159]
102.6(9)
119.7(11)
11601
-
119.5112)
[I611
112.6(4)
119.714)
11621
101.2(14)
135.0(12)
[163]
F
I
F
I
F
F\
!
'
I
F
==O
F
F
F
Fig. 10. Shapes of molecules containing multiple bonds.
Qualitative predictions about the relative magnitudes of bond
angles in molecules containing multiple bonds can be made on
the basis of the relative sizes of the single, double. and triple
bond domains. It is clear that a double bond domain must be
larger than a single bond domain, and a triple bond domain
larger still. For example, in ethene the H-C-C angles that involve
the double bond domain are larger (121.3') than the H-C-H
bond angle (1 17.4"); in SO,Cl, the angle between the two S - 0
double bonds (123.5') is much larger than that between the two
S-C1 single bonds (100.3"). Other examples are given in Tables 5 and 6.
The larger size of a multiple bond domain compared to a
single bond domain also explains the observation that a doubly
bonded ligand Y, like a lone pair, always occupies an equatorial
site in trigonal-bipyramidal and related molecules, as can be
seen from the examples given in Figure 1 1 . Similarly in the case
of the pentagonal-bipyramidal seven-coordinated molecule
IOF; the doubly bonded oxygen atom occupies a less crowded
axial site (Fig. 12).["' Unlike the IF; molecule that is nonrigid
and fluxional'".121 IOF, has a rigid structure, which can be
attributed to the resistance of the 1 - 0 double bond domain
against moving into a more crowded position.["]
502
F
F
Fig. 11. a ) Shapes of some AX, = Y molecules in which the doubly bonded ligdnd Y
always occupies a less crowded equatorial position. b) Bent bond model of
H,C=SF, showing that the CH, group lies in the axial plane of the SF, group. c)
Model showing the ellipsoidal double bond domain with its long axis in the equatorial plane of the SF, group. d) Structure of XeO,F,. e) Illustration of the Xe0,F2
molecule showing the approximate shape of the xenon lone pair domain.
In interpreting details of molecular geometry it is often important to take into account not only the size of a multiple bond
domain but also its shape, which depends on the directions of
the other bonds formed by the two
bonded atoms. In the common case
of a carbon atom forming one double and two single bonds, the doubie
bond domain has a prolate ellipsoidal shape; its long axis is constrained to lie perpendicular to the
CX,_plane
to minimize its interac.
tion with the single bond domains. In
the ethene molecule the tWO CH,
groups are thereby constrained to lie
in the same plane, and so the
molecule overall is planar (Fig. 9 g).
0
1-
F
Fig, 12, The double bond
domain occupies a less
crowded axial Posit*on in an
AX, = Y pentagonal bipyram l d a l molecule such as
IOF,
Angris. Chem 1711.Ed Engl 1996, 35,495-514
REVIEWS
__ VQEPR
The
-.
Vodel
.h o t her exmiple of the importance of taking into account the
shape o f a multiple bond domain is seen in the molecules OSF,
and HICSF,. Here the large size of the SO and SC double bond
domains compared to the SF single bond domains causes all the
F-S-F angles to be smaller than the ideal angles of 120" and 90"
(Fig. I I a). The SC double bond domain is forced by the CH,
group to have a prolate ellipsoidal shape with its long axis perpendicular to the CH, plane (Fig. 11 c). To minimize its interaction with the axial S F bond domains, the double bond domain
lies in the equatorial plane (with the CH, group perpendicular
to this plane) so that it decreases the F,,-S-F,, angle from 120"
to 96.4 . while it only decreases the Fa,-S-F,, angle from 180" to
170.
Here the bent bond model of the double bond is particularly appropriate (Fig. 11 b), as all the angles approach the
ideal angle of 90 for an octahedral arrangement of six domains.
In contrast the 0s double bond domain is not constrained to
have an ellipsoidal shape, by any bonds on the oxygen atom so
it has an almost cylindrically symmetrical shape. Therefore the
SO double bond domain in SOF, affects the F,,-S-F,, angle,
which is decreased to 114.7", much less than does the CS double
bond in H,CSF,, while it has a larger effect on the Fa,-S-Fa,
angle, which is reduced to 160.4".[91These conclusions about
the shapes of the SO and SC double bonds domains have
been confirmed in a study of the Laplacian of the electron density of these molecules (Section 3.2). The shape of the
SC double bond domain is also shown to be elliptical in an
electron density deformation map of the related molecule
(CF,)(CH,) (C=SF,.""]
It is interesting to note that the F-Xe-F angle in the AX,Y,E
molecule XeO,F, is less than 180" (174.7') rather than greater
than 180 as is normally the case for AX,E molecules such as
SF, (Fig. 9 0. Because of the large size of the xenon core the lone
pair domain is expected to be much more spread out in the
equatorial plane than in the axial plane; indeed, while it exerts
a strong repulsion on the Xe = 0 bonds it exerts a weaker repulsion on the Xe-F bond domains than d o the X e = O bond
domains. Consequently, the Xe-F bonds bend towards the lone
pair and away from the X e = O bonds.
2.7. Linear Molecules
In linear (cylindrically symmetrical) molecules such as H F
and CO, there is no requirement that the domains of electrons
of opposite spin are brought into coincidence. This was recognized many years ago by Linnett when he formulated the double
quartet
For example, in the H F molecule two electrons
of opposite spin are attracted into the bonding region, but the
two tetrahedra of electrons of opposite spin are still free to
rotate around the bond axis so that the remaining electrons d o
not form pairs but keep apart due to electrostatic repulsion (see
Fig. 13a, b). Similarly in the CO, molecule the tetrahedra of
electrons of opposite spin are not brought into coincidence and
so d o not form close pairs (Fig. 13c, d). Rather there are four
noncoincident single electron domains forming a cylindrically
symmetric double bond (Fig. 13e). Thus the CO, molecule can
be described as a linear AX, molecule in which there is a linear
arrangement of two cylindrically symmetric double bond domains. As the electron -electron repulsion in this kind of double
d)
C)
e)
Fig. 13. Domain version of the Linnett double quartet model for the HF and CO,
molecules Domains of the cx electrons (a) and the jelectrons (h) of HE Domains
of the cx electrons (c) and the jelectrons (d) I n CO, e) Cross-section of the C - 0
double bond showing the two 1 electron domains and the two /$electron domains
which lead to overall cylindrical symmetry.
bond domain should be less than in an elliptical double bond
domain in which the electrons are more closely paired, it is not
surprising that the C = 0 bond in CO, (116 pm) is shorter than
the standard C = O bond distance (1 18 pm).
3. The Analysis of Electron Density Distributions
The domain model is a very approximate description of the
electron density distribution in a molecule and yet a useful tool
for the understanding of molecular geometry. But an increasing
amount of accurate information about the total electron density
in molecuIes is now forthcoming from X-ray crystallographic
studies and from a b initio calculations, and increasing attention
is being paid to the analysis of these electron density distributions in an attempt to extract from them the concepts of bonding
and nonbonding electron pairs or domains and to improve our
understanding of molecular geometry. However, because the
electron density changes accompanying bond formation are relatively very small, very accurate low-temperature X-ray crystallographic studies or high-level a b initio calculations are needed
to obtain sufficiently accurate electron density distributions to
justify a detailed analysis.
At first sight the total electron density distribution provides
little information of interest to the chemist, because there is
apparently no evidence of the localized pairs of electrons that
chemists associate with bonds and lone pairs, as we see, for
example, in the illustration of the total electron density of the
water molecule in Figure 14. The electron density distribution of
a molecule is dominated by the very large electron density close
to each nucleus while the electron density in the interatomic
regions in which changes in the electron density distribution due
to bond formation occur are relatively very small and correspondingly difficult to detect. Bader et a1.ri71
has shown that the
total electron density of a molecule can be partitioned into
atomic densities by a rigorous quantum mechanical method and
that between any two bonded atoms there is a line, called the
bond path, along which the electron density is greater than
503
R. J. Gillespie and E. A. Robinson
REVIEWS
Fig. 14. Electron density distribution in the water molecule. a) Contour map
in the plane in which all atoms lie. b) Relief map of the electron density in the
plane depicted in (a). c) Relief map of the electron density in the plane through
the oxygen atom, perpendicular to the plane depicted in (a) and bisecting the
HOH angle.
....
along any other line between the two nuclei. The bond path
appears as a ridge connecting two bonded nuclei in a relief map
of the electron density in a plane containing all atoms of the
molecule (Fig. 14b). However, there is no indication of bonding
or nonbonding electron pairs in these plots of the electron density. There have been several approaches to the problem of
analyzing the electron density distribution to provide support
for the concept of localized bonding and nonbonding electron
pairs. We next consider two of these: the calculation of electron
density deformation maps and the analysis of the total electron
density in terms of its Laplacian.
3.1. Electron Density Deformation Maps
Electron density deformation maps are obtained by subtracting from the experimental or calculated total electron density of
a molecule the density of a promolecule constructed from spherical atoms with the same geometrical arrangement as the actual
molecule. In principle such a deformation map should show the
changes in electron density that occur on bond formation, and
in many cases such maps d o show concentrations of electron
density in bonding regions and in lone pair regions that correspond to the bond pair and lone pair domains of the VSEPR
model. Thus these studies provide some support for the model.
An early electron density deformation map is that obtained for
the AX,E molecule (CH3),TeCl, (Fig. 15a), which clearly
shows electron density corresponding to the lone pair.['*]
A recent example shows the noncylindrically symmetrical
(ellipsoidal) shape of the C = S bond in the molecule
(CF,)(CH,)C = SF, (Fig. 15 b).[16b1 However, such deformation
maps do not always show the presence of the expected bonding
or lone pair densities and there are serious difficulties associated
with the construction of an appropriate p r o m ~ l e c u l e . [ 'In~~
creasing attention is therefore being paid to other methods for
the analysis of the electron density distribution, such as the
topological analysis of its Laplacian.
504
....
.
.....
- ............
....................
. .. .
.............
., .
..,.. ...... .- ..;. ...........
. . .. .
Fig. 15. a) Electron density deformation maps of Te(CH,),CI, in the equatorial
plane (left) and the axial plane bisecting the angle between the CH, ligands (right)
showing the deformation density associated wrth the lone pair. b) Deformation
maps of the (CF,)(CH,)C=SF, molecule In the FSF equatorial plane (left) and in
a plane bisecting the C = S bond showing the anisotropy of the bond corresponding
to an ellipsoidal shaped double bond domain.
3.2. The Topological Analysis of the Electron Density
and its Laplacian
Bader et al.I1'] have shown that the total electron density of a
molecule can be usefully analyzed by studying the Laplacian
V 2p of the electron density p , which is given in Equation (a).
0
2
~=
a2p/ax2
+ a2p/ay2 + azP/az2
(a)
This function has the property of magnifying very small variations in the electron density and revealing small local electron
density concentrations (charge concentrations) that are not apparent in the total electron density distribution (see appendix).
For example, in a free atom there is a continuous decrease in the
total density with increasing distance from the nucleus, and
there are no maxima and minima in the total electron density
corresponding to the successive electron shells. But these shells
are revealed clearly in the Laplacian as a local concentration of
electron density in the form of a spherical shell surrounded by
a spherical region of density depletion (Fig. 16). Although the
electron shells in a free atom are also apparent in the radial
distribution function, the radial distribution function has no
meaning in a molecule. However, the Laplacian of a molecule
still reveals the electron shells as regions of concentration and
depletion of electron density but distorted by bond formation.
In a large number ofcovalent molecules the valence shell charge
concentration exhibits maxima corresponding to each of the
bonds and each of the lone pairs expected from the conventional
Lewis diagram for the molecule as shown in Figures 17- 19 for
Angebi Chem In/ Ed. En@ 1996,35, 495-514
_ _W E P R_Thc
Model
.-
REVIEWS
Fig. 16. a) Relief map of the
electron density ( p ) in a plane
through the nucleus of the argon atom b) Relief map of the
negative Laplacian ( - V ' p ) of
the electron density in the
same plane.
the H,O. CIF, and SF, molecule^.^'^^^ Moreover, these local
charge concentrations have the same locations and the same
relative sizes as the electron pair domains of the VSEPR model.
In the valence shell of the oxygen atom in the water molecule
there are charge concentrations in an approximately tetrahedral
arrangement that correspond to two bonding and two nonbonding (lone pair) domains of the VSEPR model. Similarly the
valence shell of the chlorine atom in CIF, contains three bondIn the valence shell of the oxygen atom in the water molecule
there are charge concentrations in an approximately tetrahedral
arrangement that correspond to two bonding and two nonbonding (lonc pair) domains of the VSEPR model. Similarly the
valence shell of the chlorine atom in ClF, contains three bonding and two nonbonding charge concentrations in a trigonalbipyramidal xrangement. And in the valence shell of sulfur in
SF, thc four bonding and one lone pair charge concentration
Fig. 18. Relief map of -V'p
for the mirror planes u: (a)
and uv (b) of the CIF,
molecule. In a) the three
maxima in the valence shell
of CI correspond to two
nonbonding charge concentrations and a nonbonding
(lone pair) charge concentration. in b) three bonding
charge concentrations can
be seen The fourth apparent
maximum represents the
saddle between the two nonbonding maxima in a).
Fig. 19 lsosurface of the Laplacian
distribution of the sulfur atom in SF,,
showing a large charge concentration
for the lone pair two small charge concentrations for the bonding equatorial
electrons. and two still smaller charge
concentrations for the axial bonding
electrons surrounding a spherical core.
Parts of the valence shell charge concentrations of the equatorial fluorine
atoms are seen i n the lower corners.
are in a trigonal bipyramidal arrangement. Moreover, the
Laplacian provides confirmation (Fig. 20) of the increasingly
SF,O
bonding charge concentrations. The third apparent maximum in a) is the saddle
hetwecn the two iliaxima observed in b) that are nonbonding (lone pair) charge
conccnlr;ttions
SF,NH
SF,CH,
Fig. 20. Contour plot of the electron density p (top) and of -V'p (bottom) i n a
plane through the maximum of - V ' p Tor the double bond in O=SF,. HN=SF,.
and H,C=SF, showing the increasing ellipticity of the double bond
R. J. Gillespie and E. A. Robinson
REVIEWS
elliptical shape of the double bond domain in the series 0 = SF,,
HN=SF,, and H,C=SF, that we discussed in Sections 2.2
and 2.5.l91
The Laplacian therefore provides evidence that the domain
version of the VSEPR model has a real physical basis. Although
electrons are not as localized into pairs as the domain model
depicts them, local concentrations of electron density nevertheless correspond to and have the same properties of relative size
and shape as, both bonding and nonbonding electron pair domains.
4. Exceptions to the VSEPR Model
As we have seen in the preceding sections the VSEPR model
accounts very successfully for the general shapes of the
molecules of the main group nonmetals and also in a qualitative
way for deviations of the bond angles from the “ideal” values.
However, certain exceptions to the predictions of the model
have been recognized, particularly among metal - nonmetal
molecules. The rest of this review is concerned with these exceptions, real o r apparent. Because the model is so successful in
many cases, it is worthwhile examining the exceptions in an
attempt to discover why the model breaks down in some cases.
In this way we can hope to improve our understanding of bonding and to see if the VSEPR model can be appropriately modified to cover the apparent exceptions. First we discuss exceptions among the main group elements.
4.1. AX, Molecules
Although the trigonal bipyramid with longer axial than equatorial bonds is the preferred structure, the energy of the square
pyramid is only slightly higher, as is shown, for example, by the
fact that it is the transition state in the Berry pseudorotation of
a trigonal-bipyramidal molecule. PX, and AsX, molecules all
have the expected trigonal-bipyramidal structures with longer
axial than equatorial bonds, as d o most SbX, molecules. Examples are SbCI,, Sb(CH,),,[Zoal and also Bi(CH3)j.[20bfIn contrast, Sb(C,H,),[Z’ ’] and Bi(C,H,),[2’ b1 in the solid state have
structures that can best be described as square pyramids, although they both are somewhat distorted towards trigonalbipyramidal geometry. However, the closely related molecules
Sb(p-CH,C,H,), and Bi(o-FC,H,), are trigonal-bipyramidal
in the solid state, although the latter exists as an equilibrium
mixture of both trigonal-bipyramidal and square-pyramidal
molecules in solution.[z21It is clear that the energy difference
between the trigonal-bipyramidal and square-pyramidal structures is very small; moreover, it is to be expected that the
difference in energy between the two structures will decrease
with the decreasing interaction between the bonding domains
as the size of the central atom increases, so it is not surprising
that it is essentially zero for Bi(o-FC,H,),. It seems clear
therefore that because of the very small energy difference between the trigonal-bipyramidal and square-pyramidal structures
of AX molecules, solid-state effects cause the square pyramid to
be the preferred structure in a few molecules of antimony and
bismuth.
506
4.2. Stereochemically Inactive Lone Pairs
As we have mentioned, SeC1;- and the other AX,E molecules
listed in Table 7 are exceptions to the duodecet rule: they appear
to have seven electron pairs in the valence shell of the central
atom whereas six is the maximum number normally observed.
Table 7. Bond lengths in some AX& ions with octahedral geometry.
Ion
SnCl2SnBr:PbCl;tPbBr2Pbl;tSbCI2SbBriBiCl2BiBr2Sir:SeCIzSeBriTea:TeBriTeIiBrF;
obs.
Bond length [pm]
calcd [a]
difference [%] [b]
283
308
289
313
322
265
219
266
285
307
240
258
254
269
293
185
239
254
252
267
286
242
257
248
263
282
216
23 1
234
249
268
168 [c]
18.4
21.3
14.7
17.2
12.6
9.5
8.6
7.3
8.4
8.9
11.1
11.7
8.5
8.0
9.3
10.1
Ref.
I1641
[I651
11651
11661
[I671
[I681
~691
~701
11711
~1721
[I731
[I741
[I751
[1761
11771
~ 5 1
[a] Sum of covalent radii r,,,. [b] Observed/calculated ratio. [c] r Z + rLoL=
(1 14 54) = 168 pm. where the covalent radius of fluorine has the value of 54 pm
as we suggested recently [178]
+
They are also exceptional in another sense: they appear to have
regular octahedral structures, so the lone pair is said to be stereochemically inactive o r to be an “inert air''.['^] However, the
experimental evidence on the structures of these molecules is not
always clear: For solid state structures disorder in the crystal
may lead to an average octahedral structure and in solution the
spectroscopic evidence for an octahedral structure is not always
In solution some of the molecules may appear
to be octahedral because they have fluxional non-octahedral
structures. However, it seems certain that some of these
molecules have octahedral structures. Apparently in these cases
either the valence shell electron pairs and/or the ligands would
be too crowded if the valence shell contained seven electron pair
domains. So the lone pair is forced inside the six bond domains
to occupy a spherical domain surrounding the core. This explanation is given some support by the observation that the bonds
in these molecules are unusually long (Table 7).
That crowding of the electron pairs in the valence shell of the
central atom is important in determining the octahedral structures of some AX,E molecules is supported by the observations
that, although the BrF; ion has an octahedral s t r u c t ~ r e , [ ~ ~ ~
both IF; [261 and XeF6[”] with larger cores and therefore less
crowding of the electron pair domains have monocapped octahedral structures with the lone pair in the capping position. But
if the crowding of the electron pairs in the valence shell of xenon
is increased as in the XeFi- ion,[”] which is an AX,E molecule,
the lone pair again becomes “inert”, because this ion has the
square antiprismatic structure expected for an AX, molecule,
just like IF;. That the Xe-F distances are longer than the I - F
distances in IF, again suggests that the lone pair occupies an
inner spherical domain.
A n g m . Chcn?. Int. Ed. Engl.
1996. 35. 495- 514
The VSEPR Model
4.3. "Ionic" iblolecules
Molecules of the type OX,E, are expected to be angular with
bond angles smaller than the tetrahedral angle. Although this is
the ciise in H,O and F,O, for example (Table2), in other
molecules of this type such as (SiH,),O and (GeH,),O the bond
angles are 144 and 126", respectively, and in Li,O and Na,O
the bond angles are 180".The VSEPR model rests on the assumption that the ligands X in an AX, molecule attract the electrons
of the central atom A sufficiently strongly to form relatively well
localized pairs. This is the case when X has an electronegativity
that is comparable to, or greater than, the electronegativity of
A, for example when the ligands are F, =0,or OH, and so such
molecules always have the VSEPR-predicted geometry. However, when the ligands are less electronegative than the central
atom. the electrons on A are not well localized into pairs. In an
extreme case such as Na,O, there is little localization of the
electrons in the oxygen valence shell; consequently the molecule
can be described as consisting of a slightly distorted oxide ion
and two sodium ions. In such a case ligand-ligand repulsions
dominate to give a linear molecule. Many ligands are less electronegative than oxygen, and in such cases we expect the bonds
to have considerable ionic character and the oxygen electrons to
be only poorly localized giving bond angles larger than the
tetrahedral angle. For example, the charges on oxygen and silicon in (SiH,),O have been found from a b initio calculations[28a]
to be - 1.7 and 3.1, respectively, the molecule has considerable ionic character consistent with the large bond angle. The
analogous (CH,),O molecule has a bond angle of 112", which is
slightly larger than 109", consistent with the smaller electronegativity difference between oxygen and carbon than between oxygen and silicon and the smaller ionic character of the C - 0 bonds.
In general. the VSEPR rules only apply when the ligands have an
electronegativity that is comparable to that of the central atom
with the result that the bonds are predominately covalent.
REVIEWS
the alkaline earth dihalides in the gas phase, although a simple
electrostatic model for an ionic molecule and the VSEPR model
for a covalent molecule would both predict a linear geometry.
For CaF, it was later
that, if the molecule is considered to be predominately ionic. the interaction of the negatively charged fluorine ligands with the polarizable Ca2+ core
would cause the eight electrons of the outer shell of the core to
become partially localized to form four tetrahedrally oriented
pairs, hence giving the core a nonspherical shape. According to
this model the negatively charged ligands tend to adopt the
arrangement that places them opposite two of the faces of this
tetrahedron, so as to minimize their interaction with these four
partially localized electron pairs (Fig. 21 c). Thus, if the core is
sufficiently polarized an MX, molecule will have an angular
rather than a linear geometry (Fig. 21 a, b). According to this
model the bond angle would be expected to decrease from 180"
with increasing polarizability of the metal, that is from Be to Ba,
and with increasing polarizing power of the ligand from I to F,
as is observed (Table 8).
+
5. Metal-Nonmetal Molecules
The largest number of apparent exceptions to the VSEPR
model are found among molecules of the metals, including the
transition metals. For this reason and because the bonds in
many metal -nonmetal molecules are predominantly ionic, it is
widely believed that the model does not apply to these
molecules, although it was shown over twenty years ago that, if
distortion of the core of the metal atom to a nonspherical shape
is taken into account, many of these exceptions can be accounted for."' Recent studies of the Laplacian of the core electron
density in several metal -nonmetal molecules have given support to this view and provided important information on the
nature of the core distortion. We review this recent work in this
section.
5.1. The Alkaline Earth Dihalides and Related Molecules
that polarization of the
It was suggested some time
metal atom core in the alkaline earth dihalide molecules by the
halide ligands could account for the angular shape of some of
Fig. 21. Structures of the alkaline earth dihalides a) Contour maps of - V z p for
SrF, in the plane of the molecule (top) and perpendicular to this plane and bisecting
the F-Sr-F angle (bottom). b) Contour maps of V ' p for BaH, i n the plane of the
molecule (top) and perpendicular to this plane and bisecting the H-Ba-H angle
(bottom). c) Model originally proposed to explain the angular shape of the CaF,
molecule showing four tetrahedrally arranged partially localized electron pairs.
d) Three-dimensional plot of an isosurface of -Vzp for BaH, showing the four
charge concentrations in the outer shell of the core surrounding the spherical inner
core.
Table 8. Bond angles
["I
of halides of Group I1 in the gaseous phase [2]
MXi
X=F
c1
Br
I
BeX,
MgX*
CaX,
SrX,
BaX,
180
180
133-155
108-135
100-117
180
180
180
120-143
100-127
180
180
173-180
133-180
95-135
180
180
I80
161 180
102-105
Ab initio calculations have likewise shown that the dihalides
and the dihydrides of Ca, Sr, and Ba should have an angular
~ h a p e , ~ ~and
' ~ .the
~ ] total electron densities obtained from these
calculations have been analyzed in terms of their Laplacians.
Although the Laplacian of the calculated electron density of
each of the fluorides and hydrides of Ca, Sr, and Ba, and of
Ca(CH,),, does not display a valence shell charge concentration, presumably because of the ionic nature of the molecules, in
501
R. J. Gillespie and E. A. Robinson
REVIEWS
each case it does show that the outer shell of the core is nonspherical with four approximately tetrahedrally arranged
b1 Two of these core charge
charge concentrations (Fig.
concentrations (CCs) are on the opposite side of the core from
the ligands. These CCs arise from the repulsion of the core
electrons by the ligands, while the other two charge concentrations arise from the operation of the Pauli principle. Thus the
model proposed earlier is fully confirmed by the Laplacian of
the electron density. The formation of four charge concentrations in the outer shell of the core of the metal in these molecules
is analogous to the formation of four charge concentrations in
the valence shell of the oxygen atom in the water molecule. In
the water molecule it is the attraction of the hydrogen nuclei for
electrons combined with the Pauli principle that determines the
formation of two bonding charge concentrations and two nonbonding charge concentrations. In contrast in the BaF,
molecule, for example, it is the repulsion of the core electrons by
the fluoride ion ligands that causes the formation of four charge
concentrations in the outer shell of the core; the fluoride ions
adopt positions opposite the “holes” between the charge concentrations thus giving an angular molecule.
It is reasonable to suppose that the bent structure of bis(pentamethylcyclopentadieny1)calcium is similarly due to distortion of the calcium core by the ligands.
X
Kxx*x
x*
X
X
x
Fig. 22 Diagrammatic representation of the effect of four tetrahedral core
charge concentrations on the structures of MX,. MX,, and MX, molecules (M is
a polarizable metal atom) giving angular, pyramidal, and tetrahedral molecules
respectively.
5.2.2. MX, , MX, Y, M X , Y z , and MX,Z Molecules
Four ligands are expected to produce four core CCs. If the
ligands have a tetrahedra1 arrangement they will be opposite the
holes between these CCs, and so no distortion of the VSEPRpredicted tetrahedral geometry is expected (Fig. 22). All MX,
molecules with a central do transition metal atom have been
found to have the regular tetrahedral structure, including the
tetrahalides of Ti, Zr, and Hf, all the MO2- ions, and the tetraoxides RuO, and OsO,. The tetrahedral structure of TiH,
has been shown by calculation to be a local minimum, and it is
also very probably a global minimum.[341All MX,Y, MX,Y,,
5.2. Compounds of do Transition Metals
It has usually been assumed that the VSEPR model does not
apply to compounds with transition metals and most recent
discussions of the shapes of such molecules have been based on
the molecular orbital (MO) model. Nevertheless, the VSEPR
model does apply to many molecules containing transition
metals, although the number of exceptions is rather large. The
application of the VSEPR model to transition metal atoms is, at
first sight, straightforward as the lack of nonbonding electrons
in the valence shell of the metal yields no other predicted shapes
than the AX,, AX,, .. . AX,, shapes. As a consequence, the predicted shapes are completely independent of the ionic character
of the bonds; the packing of negative ions around a central
positive ion and the packing of electron pair domains in the
valence shell of the central atom give the same shapes. However,
even some do metal molecules have nonVSEPR shapes. We will
now see that these shapes may be accounted for by the distortion
of the core by the ligands (as we have seen for the group2
halides and hydrides) and that this core distortion has been
confirmed in some cases by studies of the Laplacian of electron
density of the core of the metal atom.
/A;
F
5.2.1. MX, Molecules
The calculated structures of ScH,, TiH: and Ti(CH,): are
pyramidal [ 3 2 . 331 rather than planar. This pyramidal structure is
consistent with the formation of a tetrahedral arrangement of
charge concentrations as in BaH, . The pyramidal geometry of
these molecules then results from the ligands’ occupying positions opposite the “holes” between these CCs (Fig. 22). This
conclusion awaits conformation by the study of the Laplacian
of the electron density.
SO8
Fig 23. a) Structural parameters of the SO,F, and CrO,F, molecules. b)-e) Details for CrO,F,: Contour map of - V 2 p in the OCrO plane (b) and the FCrF plane
(c). Three-dirnenslonal plot of an isosurface of -Vzp for the core of the Cr atom (d)
showing four charge concentrations, two larger concentrations opposite the
0 atoms and !wo smaller concentrations opposite the F atoms, surrounding a
spherical inner core DiaErammdtic representation of CrO,F, molecule (e) showing
the hgdnds and the core charge concentrations.
The '\ISEPF. blodel
and MX,Z in0 ecules such as VOF,, VOCl,, MoNF,, CrO,F,,
CrO,Cl,, and WNF, also have a tetrahedral geometry, although the bond angles are distorted in an unexpected way from
the ideal tetrahedral angle.['] For example, in CrO,F, and
CrO,CI, the 0 = Cr = 0 angles are smaller than the tetrahedral
angle, whereas in an analogous molecule with a central main
group element such as SO,F, this angle is, as expected, larger
than the tetrahedral angle (Fig. 23 a). The Laplacian of the electron density[351
shows no valence shell charge concentration but
charge concentrations in the outer shell of the core opposite
each ligand; those opposite the oxygen ligands are larger than
those opposite the halogen ligands (Fig. 23 b-d). Interaction of
the double bond domains with these larger charge concentrations reduces the 0 = Cr = 0 bond angle allowing the F-Cr-F
angle to increase (Fig. 23e). Similarly in VOC1, the 0-V-Cl
bond angles are smaller than tetrahedral, whereas in POC1, the
0-P-CI angles are larger than tetrahedral as expected because of
the large size of the P - 0 double bond domain. In this case also,
a large charge Concentration is found in the outer shell of the
core opposite the oxygen atom, which tends to open up the
C1-V-C1 angles and thereby reduce the 0-V-C1 angles
(Fig. 24).[361A similar situation is observed in the calculated
structures of MoNF, and WNF,, where the N-Mo-F (104.5")
and N-W-F (105.8") angles are smaller than the tetrahedral
angle, and large CCs are observed opposite the triply bonded
nitrogen
CI
Fig. 24. a) Structural parameters and b) location of the core charge concentrations
relative to the ligands in the VOCI, molecule.
5.2.3. M X , and MX, Y Molecules
The known pentahalides of V, Nb, and Ta all have the geometry predicted by the VSEPR model: trigonal-bipyramidal with
longer axial than equatorial bonds. However, Ta(CH,), has
square-pyramidal geometry,[" 1' and the calculated structures of
Ta(CH,), and the hypothetical molecules VH,, V(CH,), , and
TaH, are also sq~are-pyramidal.[~*]
The Laplacian of the electron density has been studied for
VF, , VH, , and V(CH,), . In each case there is no valence shell
charge concentration but there are five ligand opposed charge
concentrations in the outer shell of the core (Fig. 25).13,1 In the
trigonal-bipyramidal VF, model the equatorial fluorine ligands
are opposite holes in the core but the axial fluorine ligands are
opposite charge concentrations, whereas in the square-pyramidal VH, and V(CH,), molecules all the ligands are opposite
holes in the core. The interaction of the ligands with the core
charge concentrations would be expected to increase the energy
of the trigonal-bipyramidal structure relative to that of the
square pyramid. So if the interaction of the ligands with the core
is sufficiently great, the square pyramidal structure will be stabilized with respect to the trigonal bipyramid. Thus the more
Angebi ('lwm Ini Ed Engl 1996, 35,495 -514
REVIEWS
Fig. 25. a) Contour map of - V * p in the equatorial plane of the trigonal-hipyramidal VF, molecule. b) Isosurface of -V'p for the core of the vanadium atom in VF,,
showing five ligand opposed charge concentrations surrounding the inner spherical
core. c) Contour map of -V2p for the plane through the C , axis of the square planar
V(CH,), molecule containing two basal and the axial CH, groups.
covalently bound ligands such as -H and -CH3 stabilize the
square pyramid structure as in VH,, TaH,, and Ta(CH,),,
whereas the less covalently bound halide ligands, such as the
halogens, do not perturb the core sufficiently to make this the
preferred structure, and the VSEPR-predicted trigonal-bipyramidal structure is retained in molecules such as VF, .
All the known (do-M)X,Y molecules such as CrOF,,
MoOF,, and WSCl, have square-pyramidal structures['] rather
than the expected trigonal-bipyramidal structure of the
analogous SOF, molecule. The Laplacian of the electron density of CrOF, and M O O F , ' ~shows
~ ~ that, as in VF,, there is no
valence shell charge concentration but in each case the core has
five CCs in a square-pyramidal arrangement. The CC produced
by the doubly bonded oxygen is larger than those produced by
the singly bonded F ligands (Fig. 26a-c). This large charge concentration produced by oxygen is responsible for stabilizing the
square pyramidal structure with respect to the trigonal bipyramidal structure (Fig. 26). The structures of these square-pyramidal MX,Y molecules of transition metals bear a noteworthy
similarity to the square pyramidal AX,E molecules of the main
group elements, such as BrF,, in which the nonbonding electron
pair E occupies a position in the base of the square pyramid
completing an octahedral arrangement of six valence shell domains. The large charge concentration opposite the oxygen in
MoOF, and CrOF, behaves like a lone pair in that it distorts the
trigonal bipyramid to a square pyramid. But as we might expect,
the distortion is less than that caused by the lone pair in an
AX,E molecule; the X,-A-X, angles in the square-pyramidal
509
R. J. Gillespie and E. A. Robinson
REVIEWS
0
I
Fig. 26. a) Contour map of -V2p in a vertical plane containing two basal atoms of
the square-pyramidal CrOF, molecule. b) Structural parameters and c) arrangement of the Crcore charge concentrations relative to the ligands in CrOF,. d) Shape
of the BrF, molecule.
molecules of transition metals are greater than 90" whereas in
the AX,E molecules they are smaller than 90" (Fig. 26d).
In all cases that have been studied so far a doubly bonded
ligand such as oxygen distorts the core more than a singly bonded ligand such as fluorine or chlorine. Moreover, it appears
from the Laplacian of the calculated electron densities of nitrido
complexes such as MoNF, and MoNF, that the triply bonded
nitrogen atom perturbs the metal atom core even more strongly
than a doubly bonded oxygen atom.[371So MoNF, has a square
pyramidal structure like MoOF,.
5.2.4. MX, Molecules
Two structures have been found for MX, molecules of do
transition metals-the VSEPR-predicted octahedron for CrF, ,
MoF,, WF,, WCI,, WBr,, TiFi-, ZrCli-, W(NMe,),, and
W(OMe),[381 and the trigonal prism for WMe,,I3'I and
ZrMei-.[401The Laplacian of the electron density of the metal
atom core in CrF, has six LOCCs in an octahedral arrangement,
so each ligand is opposite a charge concentration (Fig. 27).[351
The Laplacian of the electron density has not been calculated for
either WMe, or ZrMei-, but it has been calculated for an assumed trigonal prismatic structure for CrMe, (Fig. 27 b, c).[,~]
Five LOCCs were found in a trigonal bipyramidal arrangement
with each ligand opposite one of the six faces of the trigonal
bipyramid--that is, opposite holes in the charge density of the
core. Apparently, interaction between the fluorine ligands and
the core is not strong enough to modify the VSEPR-predicted
octahedral structure of CrF,, but for CrMe, the stronger interaction of the methyl ligand distorts the Cr core more strongly
and causes the change to the trigonal prism in which the interaction between the ligands and the LOCCs is minimized.'351
On the basis of an accumulating number of examples it seems
reasonable to conclude that the cores of metal atoms are distorted by the ligands acting in conjunction with the Pauli principle
to form a number of charge concentrations. These charge concentrations have an arrangement that minimizes their mutual
510
Fig 27 Contour map of -Vzp in an equatorial plane of the octahedral CrF,
molecule b) Contour map of -Vzp in a vertical plane Containing d C , axis of the
trigonal-prismatic Cr(CH,), molecule c) Contour map of - V 2 p for the horizontdl
symmetry plane of Cr(CH,),
interaction: four CCs have a tetrahedral arrangement, five a
trigonal-bipyramidal or square-pyramidal arrangement, and six
an octahedral arrangement. Ligands that interact sufficiently
strongly with the core of the metal atom (in particular -H,
-CH, , and =O in the case of transition metals, and F, H, and
CH, in the case of the more polarizable group 2 metals) adopt
a geometry that in some cases differs from the VSEPR-predicted
geometry, in which they occupy sites opposite the "holes" in the
core between the CCs and which therefore minimizes their interaction with these CCs while it does not greatly increase the
ligand- ligand interactions. The geometry of molecules containing ligands that interact only weakly with the core, for example
F in the case of the transition metals, is dominated by the interactions between the bonding electron pairs or the ligands, so
they have the expected VSEPR geometry. Moreover, it appears
that the more strongly covalently bonded ligands induce a larger
distortion of the core electrons. For example, in the series
ScH;-, TiHg-, VH;, CrH,, MnH;, ab initio calculations1331
have shown that for the more ionic molecules ScHZ- and TiHg
the octahedral structure has a lower energy than the trigonal
prismatic structure, whereas the trigonal prism has the lower
energy for the more covalent VH;, CrH,, and MnH:
molecules. Further studies are needed before these hypotheses
can be fully confirmed.
~
5.3. Compounds with d'-d'O
Transition Metals
It seems clear that the shapes of molecules containing nontransition metal and do transition metal centers are distorted
from the VSEPR-predicted shapes by the interaction between
Arigebt. Chem. Inr. Ed. Engl. 1996, 35,495- 514
The VSEPR Model
the ligands and a distorted (nonspherical) core of the metal’s
electrons. So it is reasonable to assume that the shapes of compounds with d’-dIo transition metal centers will also be determined by the shape of the metal atom’s core. Indeed, if it is
assumed, as a rough first approximation, that an incompletely
filled d shell (d’ -d9) has either a prolate or oblate ellipsoidal
shape. the geometry of many such molecules can be accounted
for.[21For example, a prolate ellipsoidal core can result in
a square-planar AX, molecule, a square-pyramidal AX,
molecule, and a square-bipyramidal molecule (Ddh).
A preliminary study of the Laplacian of the electron density
of molecules with transition metal centers with incomplete
d shells has been reported by MacDougall and Ha11.[41JAs yet
there have been too few similar studies to enable any general
conclusions to be drawn about the influence of the core on the
shapes of these molecules. Laingl4’’ has drawn attention to the
striking fact that molecules with transition metal centers that
obey Sidgwick’s effective atomic number
(in which,
therefore, the metal atom has a total of 18 bonding and nonbonding electrons in its outer shell) always have the predicted
VSEPR geometry. N o simple and general explanation of
Sidgwick’s rule has yet been proposed, but it seems reasonable
to suppose that further studies of the electron density distribution of the core of the metal atom will eventually lead to a much
better understanding of how the shape of the core modifies the
VSEPR-predicted geometry for all molecules containing transition metals. And it seems reasonable to hope that a modified
VSEPR model can be developed that will prove as useful for
transition metal molecules as the original model has been for
nonmetal molecules.
6. Conclusions and Pedagogical Considerations
The VSEPR model, particularly when formulated in terms of
electron pair domains, continues to be the most useful and most
easily used model for the qualitative prediction of molecular
shape. It enables us to understand in a qualitative way many
features of the structures of molecules that are not readily accounted for In other ways, and it continues to form the basis for
the discussion and understanding of the structures of many new
molecules. Its physical basis rests on the Pauli principle, and it
is independent of any orbital model. It thus provides a useful
alternative to the familiar valence bond and molecular orbital
methods that can also give us useful new insights into molecular
geometry.
Elementary treatments of bonding and molecular geometry
are almost always based on hydrogen-like atomic orbitals. It is
commonly stated at this level that “an orbital may be thought
of as a region of space where there is a high probability of
finding an electron”. An orbital is often pictured as a charge
cloud of varying density. The shapes of s, p. and sometimes
d orbitals are then described-usually
in a very approximate
manner- but are not derived, explained, o r otherwise accounted for. A bond in this model is considered to be a “merging or
overlapping of the orbitals of the bonded atoms to create a
charge cloud that spreads over and between the nuclei and binds
them together.” Because atomic orbitals are not suitable as a
basis for this type of description of many molecules including
REVIEWS
those of carbon, the concept of hybrid orbitals is introduced.
and the hybrid orbitals are chosen to match the observed geometry. Again the mathematical operation of forming hybrid
orbitals cannot be explained at the introductory level, and the
conventional diagrams of these hybrid orbitals give the incorrect impression that hybridization of a set of atomic orbitals to
give a set of hybrid orbitals is a physical phenomenon that
results in a change in the total electron density of the atom. Not
only does this conventional introductory treatment give no explanation of molecular geometry, it introduces incorrect, or at
least very misleading, ideas. In contrast the domain version of
the VSEPR model provides a straightforward and readily understandable picture of the chemical bond. while at the same
time leading directly to an explanation of molecular geometry
without giving incorrect o r misleading descriptions of quantum
mechanical concepts that cannot be adequately dealt with at the
introductory level.
Good quantitative predictions of molecular shape can be
made by a b initio molecular orbital calculations for many
molecules, but because of the complexity of the large set of basis
functions (modified atomic orbitals) used in these calculations
they provide little understanding of the factors that determine
molecular shape. However, from these calculations, as well as
from very careful, low-temperature X-ray crystallographic studies we can obtain a rather accurate map of the electron density
distribution in a molecule; from its analysis we may hope to
obtain a better understanding of bonding. It is particularly important to derive more electron density distributions from lowtemperature X-ray crystallographic studies to confirm the results of a b initio calculations. It is also important to note that
the electron density distribution of a molecule is an observable
property of the molecule that is independent of the particular
experimental or theoretical method used to determine it, provided that it is determined with sufficient accuracy. It is therefore
incorrect to claim, as has sometimes been done. that the analysis
of an electron density distribution obtained by a b initio calculations using the molecular orbital method cannot yield any more
information than is provided by a molecular orbital description
of the molecule.
The analysis of the electron density distribution in terms of its
Laplacian has been particularly fruitful. The Laplacian of the
electron density of molecules containing the main group nonmetals shows that the local concentrations of electron density
correspond rather closely in location, size, and shape to the
electron pair domains of the VSEPR model, thus giving substantial support to the model, which can be regarded as giving
a very crude but nevertheless useful picture of the valence shell
electron density distribution in a molecule. Moreover, the study
of the Laplacian of the electron density provides considerable
insight into the reasons why certain molecules appear to be
exceptions to the VSEPR model. In particular it has been shown
that it is important to take into account not only the relative
sizes of lone pairs and multiple bonds but also their shapes. A
basic assumption of the VSEPR model, namely that the core of
an atom is spherical and therefore has no effect on the arrangement of the surrounding ligands appears to be true for the compounds of nonmetals, but recent studies of the Laplacian of the
cores of metal atoms in their molecules show that they often
have nonspherical cores, which d o affect the geometry of the
51 1
R. J. Gillespie and E. A. Robinson
REVIEWS
molecule. Even though we cannot yet predict the shapes of
molecules with transition metal centers with certainty in all cases, it is clear that the nonspherical shape of the core is an important factor that must be taken into account. Further studies of
the shapes of the cores of molecules containing transition metals
are needed to give us a better understanding of their geometries
and to enable us to extend the VSEPR model to such molecules.
R. J. G. thanks Trevellyan College, University of Durham
( U K ),for the award of a fellowship in 1994, which enabled him to
write the $rst draft of this article. and Professor K. Wade for
fruitful discussions and useful comments.
Appendix: The Laplacian of the Electron Density
The Laplacian of a function [ V ff(x.y.z)] is the second derivative of this function of x, y , and z.To understand, in an approximate way, how the Laplacian of the electron density distribution of a molecule ( O ' p ) reveals features that are not readily
apparent in the total electron density, we consider the derivatives o f a one-dimensional function such as that shown in Figure 28. This function might represent the radial behavior of the
b)
dfo
dx
X
Fig. 28 a) The function ~.
f(.x) = 8e '' + e - ''I' o s ' 2 , b) its first derivattvef(.x), c) its
second derivative.f'(r), and d) the negative of the second derivative - f " ( r )
electron density of an atom although in real cases the shoulder
a t x l , where f(x) is greater than the average of Ax dx) and
f(x - dx), would not be so apparent. The first derivative df(x)/
d x (Fig. 28 b) is negative for all values of x, showing the absence
of any maximum infix), but it has a point of inflection at x I .At
this point the second derivative (Fig. 28c) has a pronounced
minimum and therefore -d2f(.x)/dx2 (Fig. 28d) has a maximum. So the shoulder at x i at which ,Ax) > [f(x dx)
+Ax dx)]/2 is made much more apparent in -d'f(x). If this
function represents the radial behavior of the electron density,
we could say that the electron density is locally concentrated in
the radial direction. Similarly a maximum in - V ' p indicates a
local concentration in the three-dimensional electron density
distribution. Maxwell (Treatise on Electricity and Magnetism,
Vol. 1, p. 31, Dover reprint, New York, 1954) coined the phrase
Locally concentratedto signify that the value of a function of r (in
this case p ( v ) ) at a particular r is larger than the average value
+
+
~
512
of p on the surface of an infinitesimal sphere centered at r. Thus
in the regions in which - V * p is positive (or V ' p negative) the
electron density is locally concentrated. Analogously, regions
where - V ' p has negative values correspond to regions of local
depletion of electron density. A more detailed discussion of the
Laplacian of the electron density is given in Ref. [17].
Received: July 15, 1994
Revised: June 12, 1995 [A 75 IE]
German version: Angew. Chem. 1996, 108. 539-560
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