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Modern Theoretical Views on the Chemical Bond.

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for some time at room temperature, the N2 atmosphere
is removed and the salt is filtered off. It is extracted with
boiling benzene until the extract gives no reaction for triphenylphosphine [241, and is finally dried in a high vacuum
at 100 "C over P205. The phosphonium salt (142) obtained
is very sparingly soluble in water, alcohol, or chloroform;
m.p. 223-224 O C , yield 87 %.
9,IO-Dihydrophenanthrene (144)
Five millimoles of the phosphonium salt (140)is dissolved in
1000 ml of hot water, 50 mmole of NaOH is added, and the
mixture is heated to boiling. Two substances separate out, one
being crystalline, and the other an oil which solidifies on cooling. The solids are filtered off and washed with water, and the
crystals are dried in air. The hydrocarbon (144) is sublimed
from the crystal mixture o n the water bath at 90-100°C/
1 mm Hg, and recrystallized from methanol, yield 8996. The
sublimation residue is pure triphenylphosphine oxide (m.p.
153 "C).
The phosphonium salt (140) (10 mmole) is converted under
nitrogen by the sodamide method (see Part I, Section D 2) to
a suspension of the ylide in anhydrous benzene. The ylide is
only sparingly soluble; for this reason, the sodium bromide
formed is not filtered off. To the red suspension of the ylide
10 mmole of freshly distilled benzaldehyde is added, and the
reaction mixture is stirred magnetically at room temperature
until it becomes colorless (about 10 h). The Nz atmosphere
can now be dispensed with. The sodium bromide is filtered off,
and the filtrate is evaporated to dryness.The residue is extracted twice with 25 ml portions of boiling methanol to remove
the triphenylphosphine oxide. The benzylidene compound,
which is practically insoluble in methanol, has m.p. 152 to
156 O C in the crude form, and can be purified by recrystallization from n-hexane (m.p. 158 'C, yield 72%).
Received: November 2nd, 1964 [A 443b/231b IE]
German version: Angew. Chem. 77, 651 (1965)
Translated by Express Translation Service, London
Modern Theoretical Views on the Chemical Bond
Starting from classical concepts, semi-empirical approaches to an understanding of the
chemical bond are critically discussed, particularly with respect to their limitations. Rules
such as the additivity of the bond constants and the octet rule are reviewed in relation to new
chemical compounds. It is shown that these rules are not sufficient to explain chemical
bonding; this is also true of the simple concept of hybridization, the basis of which is discussed. Some generally valid statements regarding the chemical bond can be propounded on
the basis of wave-mechanical considerations. It is shown that the hybridization concept and
a number of other rules are too primitive to provide generally valid information or a basis for
predictions about the chemical bond, and even that they are par fly based on false premises.
A better foundation for an understanding of the chemical bond can only be laid with the aid
of wave mechanics.
1. Introduction
When chemists speak about the chemical bond, they
differentiate between the ionic bond, the covalent bond,
or the semipolar bond. The first two are often termed
polar and non-polar (homopolar) bonds.
In ionic bonding, it is assumed that electrons are transferred from one of the bonded atoms to the other, so that
oppositely charged ions result which attract one another.
This picture was first proposed by Kossel(l917). It explained certain properties of a limited number of ionic
compounds, but was found to be inadequate in many
cases, since the formation of ionic molecules does not
necessarily proceed by initial ionization of the atoms
being linked, and since many ionic compounds do not
dissociate to form ions. In this picture, the number of
bonds drawn in a structural formula is identified with
the number of electrons transferred to the other atom.
For example, if we assume that CaO, HF, and CO wntain the ions Ca2+, 02-, F-, C2+,and the proton, we obtain
Ca=O ( l a ) , F-H ( l b ) , or C=O (Ic).
It could even be assumed that CH4 and NH3 contained
the ions C4- and N3-. This would lead to the formulae
(Id) and ( l e ) , which are generally drawn in two dimensions, although experience has shown that they are
tetrahedral and pyramidal, respectively (see Fig. I).
Fig. 1.
According to this view, all the atoms of a compound
have the "rare gas configuration", which was formerly
Angew. Chem. internat. Edit. / Vol. 4(1965) 1 No. 8
thought to be particulariy stable, and hence chemically
inert. In the above examples, we have the following
correspondences :
Caz+ 2 Ar,
2 Ne,
2 Ne,
4 Be,
2 Ne,
2 Ne.
This concept has been extended somewhat in putting
C2f in correspondence to Be. Although the electronic
configuration of Be does not correspond to that of any
of the rare gases (He, Ne, Ar, Kr, Xe, Rn), it can be
regarded as being similar to that of the rare gases.
Almost simultaneously with the rise of the electrostatic
theory, Lewis developed his “octet rule”. It was essentially intended to apply to molecules consisting of atoms
from the first and second periods. According to this rule,
the atoms do not become ionized; instead, the bonding
electrons are shared by the atoms in such a way that
each atom “fills up” its electron shell to “rare gas configuration” [two or ten (84-2) electrons]. It is usual to
represent the electrons by dots, so obtaining, for
example, Figures 2 and 3.
bond, and is represented by a line; all bonds are thus
two-electron bonds.
Thus the valence bond has different meanings in the
Kossel picture and in the Lewis picture, although both
concepts often give the same valence-bond diagram, as
is shown by (Ib), ( I d ) , ( l e ) and Figures 2 and 4. Even
when the valence-bond representations are identical,
however, they are generally not chemically equivalent
(cf. the C-C bond in CH3-CHO and CCI3-CHO).
From the valence-bond representation it would be assumed that the same bond in different molecules has the
same properties. The chemical nature of a molecule
would then be determined by the sum of the properties
of the bonds in the molecule. Such additivity actually
does exist to a close approximation in many classes of
compounds. There are other compounds, however, in
which the molecular properties cannot be even roughly
described in this way. There is always some interaction
between bonds, often leading to completely new properties in a molecule. Thus the valence-bond representation gives only a rough indication. It shows the
atoms between which the most important interaction,
leading to chemical bonding, exists.
In pursuance of Lewis’ ideas, the outer-shell electron
pairs in an atom which are not involved in a bond are
referred to as ‘‘lone’’ pairs of electrons, which appear to
be outwardly saturated, so to speak, by an “internal
bond”. In such a case, a line is often drawn beside the
atom in question, as shown in ( 2 4 and (2b). This
Fig. 2
concept also leads to the structures (34 and (36), since
the octet rule can be applied to ions, too. By the same
Fig. 3.
In these illustrations, the shaded circle represents the
configuration of a closed shell containing two electrons,
as found in helium. The Lewis model is also applicable
to bonds between similar atoms (see e.g. Fig. 4). In Clz,
token, we have not ( 4 4 , but (461, a representation
which agrees with experience. Thus there is an ionic
Fig. 4
the innei circles represent the shells containing two (He)
and eight electrons (Ne).
The Lewis rule presupposes some knowledge of the
electronic configurations of the atoms involved, and can
be extended to elements with nuclear charges of up to
Z = 36 by assuming the formation of a krypton shell.
The remaining elements can be included in a similar
manner. In every case, the configurations assumed to be
stable have “magic” numbers of electrons, e.g. 2 or 10,
which are correlated with the inert gas configurations.
Every pair of electrons between atoms corresponds to a
Angew. Chem. internat. Edit. 1 Vol. 411965)
/ No. 8
bond between the [NH4]@and the chloride ion, whereas
the four NO-H bonds are covalent.
In the Kossel scheme, the bond between [NH4] and C1
would also be denoted by a line. However, it has become
common practice to use the line for a two-electron bond
only, and to indicate an ionic bond by the signs of the
charges, as shown in (46). Thus amine oxide should be
written as (5a). This compound contains a “semipolar”
10 - 0
H - 7 -01
H - N*O
bond and is therefore frequently represented by (5b),
the arrow indicating that negative charge has been
transferred from the nitrogen to the oxygen.
These methods of bringing order to the maze of chemical
experience have not lost their value. In manly cases,
chemical bonds are formed by sharing electrons, but an
increasing number of new chemical compounds fit into
this scheme only with great difficulty, or not at all. This
group includes the simplest molecule HzQ, or e.g. the
molecule NO, which is stabilized by an odd number of
electrons. New structural representations must be found
for free radicals, in order to take into account the
unsaturated electron. The van der WaaIs molecules or
molecules containing ion-dipole bonds also fail to fit
into this scheme. The group further includes the boron
hydrides and their derivatives, as well as the “sandwich”
complexes. Compounds of the inert gases should be
incapable of existence, since it is the inert gas configuration which is considered to be the most stable one.
Although schemes of this type bring a certain order into
experimental observations, this order is formal, and
offers no possibility of quantitative calculation. This is
also true for most symmetry considerations, unless they
exclude certain atomic constellations.
Using the general valence-bond representation (including atomic charges and lone pairs of electrons), it is
possible to write several “structures” for many compounds. This ambiguity, which arises from the definition
of the valence-bond structure, has often been erroneously
correlated with energy principles, by assuming that the
stability of a compound increases with the number of
valence-bond structures which can be used to represent
it. In reality, tnis is an incorrect application of a wavemechanical approximation method, as we shall show
(Section 4,Eq. 44).
2. Semi-empirical Approaches
The ability of atoms to form molecules can be explained
by the electronic structure of the atoms. Two important
properties of the electron systems are the first ionization
energy Iz and the electron affinity AZ of the atom Z . If
energy supplied to a system is regarded as positive, the
energy which must be expended to transfer an electron
from the atom Z to the atom Y is
Similarly, if the electron is transferred from Y to Z , the
energy is
I -Az.
If the two atoms approach each other, the behavior of
the electrons will depend on which of these two energies
is greater, unless they are equal, as is the case when
I=-A Y < 1Y -AZ
Rearrangement of (c) gives
The atomic parameter
is known as the electronegativity, and has been used by
Pauling and by Mulliken in the discussion of chemical
bonds. Using this parameter, Pauling introduced some
order into the diversity of bonds. The important quantity here is the difference in the electronegativities of the
two atoms A and B:
It was found that the dissociation energies D(A-€3) of
single bonds approximately satisfy the relationship
where a and
+ p Xz
p are constants. When X = 0, Equation (g)
D(A-B) = a
Since ci is independent of X, it represents the dissociation energy of a bond between two identical atoms
when A = B. Therefore, Equations (i) and (k) can be
written, both of which satisfy Equation (g) reasonably
+ D(B-B))
a = I/D(AlA)D(B-B)
This ambiguity shows that Equation (g) is a somewhat
arbitrary formulation using the quantity X; if (8) is
expressed in the form (1) it is seen that in some cases the
left-hand side of Equation (I)
becomes negative, which
is meaningless, since is always positive. The question
therefore arises of to what extent bond relationships can
be described by the simple parameters The expression
(c) can have meaning only for single bonds. Second ionization energies (and second electron affinities) are not
considered at all.
Thus we have a greatly simplified picture of the bond,
although originally this represented a considerable advance. The main value of expression (c) is that, in contrast to the octet rule and the valence-bond scheme, it
points to a continous transition between ionic and covalent bonding. The line representing a bond is correlated with the ionization energy and electron affinity,
both of which can be calculated only with the aid of
wave mechanics.
According to the octet rule, the xenon atom (like all
inert gas atoms) has a full shell of electrons (octet), i. e.
it cannot be expected to combine with another atom in
order to share further electrons. With the concept of
electronegativity, however, we must ask whether the
transfer of an electron to another atom might not be
energetically favored. Thus the inert gases are distin-
Angew. Chem. internat. Edit. I Vol. 4(1965)
No. 8
guished by their first ionization energies. A possible
bonding partner must have an electron affinity which is
comparable with these ionization energies, since only
then can an electron be transferred from the inert gas
atom. However, it cannot be decided on the basis of these
semi-empirical ruIes whether this process in fact takes
place, since it remains unknown whether or not the four
pairs of electrons in an octet are equivalent. Moreover,
the behavior of the inner electrons (and other effects) are
not taken into account. The semi-empirical rules must
therefore be regarded as a primitive stage of theory. In
fact, examples show that Equation (g) is only of limited
applicability. Thus, with p = 0.00137, expression (m)
that the X” values can be compared with the electronegativities [see Equation (e’)] introduced by MuIIiken :
= 2.53
= 2.23
= 4.06
80.6 kcal/mole.
According to Equation (l), these values should be equal
to 1x1. Equations (e) and (f) give the values
126.5 kcal/mole.
x’= (I + A)/ 130
This leads to the values (n‘) and (03,since p in Equation
(m) must now be replaced by the constant 8’ = 23.11.
I .02
There is no .agreementbetween the sets of values (n‘) and
(0’).It should be remembered, however, that a in Equation (g) is large in comparison with the “correction”
pX2, which for the above examples is
x:’= 2.5
xii = 1.0
= 3.01
X: = 2.43
xLi = 0.95
D(A-B) = a + p X2
+ D(B-B)
x”= xi-x
p‘ x ” 2
xz = 2.5
x;= 4.0
xp = 2.1
xi = 2.5
This is particularly true of the ionic character AAB of a bond,
which is defined in terms of X [see Equation (u)]. This
property is applicable only in the case of the hydrogen
halides, for which it was first derived. The assumption that
A in Eq. (t) should be a measure of the polarity of a bond,
+ D(B-B))
because A = 0 when X = 0, appears to have little justification if, for example, the expression A’ is compared with A
(see Table 1). As expected, there is no relation between A
and A’.
A’= D(A-B)-(D(A--A)D(B-B)) ‘ l Z
Table 1 . Dissociation energies and energy values A andA’ for a number
of hydrogen compounds AH.
These values of X” were calculated by averaging the
values of X obtained for the same bond in various compounds. The value of p‘ was also found in this way, so
Angew. Chem. internat. Edit. / Vol. 4 (1965) 1 No. 8
+ y X4
From this and with an arbitrary reference value, e.g.
xg =4.0, we obtain 1
and fixing the coefficients x , p, and y.The choice of a in
Equation (g), using Equation (i), gives negative values
for the left-hand side of Equation (1) for some bonds.
Thus the values found for As-H, C-I, and Se-H are
-12.0. -1.8, and -7.5 kcal/mole, respectively.
Clearly, the electronegativity is of some importance, but
it has not yet been possible to find completely satisfactory relationships between electronegativity and other
measurable quantities. Thus X is an important, but
insufficient quantity for the description of the bonding
processes, so that all semi-empirical relationships involving X must be incomplete to some extent.
A =D(A-B)--
These values are small in comparison with D(A-B), so
that the expression (g) does not appear to be unjustified.
It is obvious that corrections must be applied to the
electronegativity values introduced in Equation (e).
Pauling therefore calcuIated another scale of X” values,
using Equation (p).
xkb = 0.8
xkb= 0.14
It should be noted that the accuracy of the approximation (g) is greatly affected by the differences between X”
and x‘. We shall prefer the definition of electronegativity
given by Equation (e) or (e’), which is unambiguous and
consistent. Equation (g) might be improved by taking
~ ’ into
account, as in Equation (g’),
9.1 kcal/rnole.
The electronegativity
as given by Equation (e) is
usually divided by 130 to facilitate comparison with
Pauling’s scale of electronegativities (cf. below). Thus
It is thus possible to construct different electronegativity
scales, one based on Equation (e) or (e’) and the other
on Equation (p). Comparison of the two sets of values,
(r) and (s), shows that the results correspond to within
certain lilnits. A similar agreement is found in other
examples :
yields the following values for P-H, C-S, H-Br, and
0-F, respectively :
xb = 3.08
& = 2.21 &, = 2.16
For this reason, equations such as (u), which relate X to
the ionic character [l] should be viewed with suspicion. I t can
1x1 + 0.035 1x1 2
111 L. Puuling: The Nature of the Chemical Bond. 3rd Edit.,
Cornell University Press, Ithaca (N.Y.) 1960.
be shown with the aid of wave mechanics that the polarity
of a bond need not be directly related t o the dipole moment.
Consequently the so-called effective charges introduced in
this connection suffer greatly from their arbiti-ary nature.
The definition
p.4BleR1B = ’A0
in which F A 0 is the dipole moment of the single bond and
R i O is t h e bond length (e = electronic charge), is therefore
applicable only in the case of a pure single bond, i.e. where
bonding is effected by one pair of electrons, so that each of
the atomic residues carries a positive charge; this condition
is adequately fulfilled in only a few cases. It is desirable not to
identify A.40 in Equation (v) with that in Equation (u), since
1x1 can only be a measure of the approximate charge separation in a bond, if n o other effects are involved. Polarization
effects are also disregarded.
Let us repeat, thefi, that although a large 1
x1may indicate
the existence of charge displacement in the single bond,
e.g. as in H(+)Cl(-) or Li(+)H(-), any quantitative
data derived from the semi-empirical rules should be
regarded with caution, and the value of such data is at
best orientative.
However, the symbols (+) and (-) go further than + and
- or
and 0,since partial charge transfers can now be
included. Thus ionic bonding and covalent bonding (as
well as semipolar bonding) become limiting cases of the
same bonding mechanism. This view, which is very close
to the wave-mechanical picture, has unfortunately been
over-simplified in the semi-empirical representation.
Here a superposition of ionic and non-polar structures,
e.g. (6a), (6b) and ( 6 j , is assumed. The same is true of
the resonance picture, which is based on the fact that one
molecule can be represented by several structural formulae. The best-known example is that of benzene (7). If
ionic structures are also admitted, the number of for-
mulae can be greatly increased as, for example, in the
case of phenol (8). Here it is assumed that the bonding
behavior of the atom ions is essentially the same as that
of neutral atoms containing the same number of electrons, i.e.
C+ & B(III),
B- -2 N+ & C(IV),
C- & 0-& N(III),
0-2 F(1).
N- A O(II),
In this picture, the valence is a purely empirical quantity, and
the properties of the atoms are characterized only by the
number of electrons. With the introduction of the concept
of electronegativity it becomes possible to distinguish, at
least in one property, the most weakly bound electron and
the elctron which can be taken up. I n some cases lone pairs of
electrons occur.
It is widely believed that the structures which make the
greatest contributions to a molecule are those in which
the most electronegative atoms are negative with respect
to their neighbors; however, the situation is not always
quite so simple. The stability of a compound cannot be
determined by the number of possible structures, since
on the basis of the electronegativity concept, certain
statistical weights must be assigned to the various structures, although these weights cannot be given quantitatively, particularly to excited states. In other words, the
resonance treatment cannot lead to a quantitative theory, since it is incapable of yielding information regarding the actual contribution of a given structure. It is
therefore very misleading to suppose that certain structures are responsible for given aspects of the behavior
of the molecule. For this reason, an interpretation of
this type is generally given afler a reaction has been
found to occur.
An important point which is almost always overlooked
in the resonance picture is the interatomic distance.
Taking this into account, it is even less plausible to attribute resonance energy to a structure whose significance must remain unknown.
In writing structures, care is taken to ensure that all the
valences are saturated, i. e. the only structures written
are those in which the number of bonds leaving an atom
corresponds to its valence; this is also borne in mind in
the case of ionic structures. Structures in which valences
remain unsaturated represent free radicals.
It might be asked, however, whether free-radical structures
should also be considered for molecules in which each atom
is using all its valences. It is generally believed that such
structures have only small statistical weights, the doubtful
(and sometimes false) argument being that these “molecular
states” are energetically “unfavorable”. This question will be
examined in connection with the wave-mechanical arguments.
Thus structural formulae are made to satisfy the octet
rule as far as possible. The principle of saturation of
valence is taken to mean that the most stable compound
is obtained when all valences are saturated, and when
no further atoms will be bound, i. e. when the approach
of other atoms does not lead to a further decrease in the
energy of the system.
According to this principle, the H2 molecule cannot bind
a further H atom, and the sum of the energies of He and
H is smaller the energy of HeH. The energy of NH3 + H
should also be smaller than the energy of NH4, and the
approach of a hydrogen atom to an HF molecule should
lead to repulsion. Recent wave-mechanical calculations
have shown, however, that the energy of NH4 is lower
by about 5 kcal/moIe than that of NH3 + H, and that
the four H atoms are situated about 1.0 A from the
nitrogen, and form the corners of a regular tetrahedron
[ 2 ] . This is an infringement of the octet rule, although
NH4 probably cannot exist as a stable molecule, and
almost certainly decomposes rapidly [Reaction (w)].
[2] D. M. Biskop, J. chem. Physics 40, 432 (1964).
Angew. Chem. internat. Edit. / Vol. 4(1965)
1 No. 8
Who can guarantee that there are no stable molzcules
which should, according to the octet rule, be unstable or
metastable. How is the NH4 radical stabilized, f. e. why
does NH4 contain less energy than NH3 f H?
One may adopt the view that the octet rule is only applicable to molecules, a,id not to molecular complexes.
How then are we to explain e.g. the simplest of the boron
hydrides, namely diborane (9), or the sandwich compounds derived from cyclopentadiene (e.g . ferrocene)?
In order to satisfy the “valence rule”, B2H6 must be
assumed to contain the structures (9a) and (9b) ; these
representations seem very artificial, since what are we to
H I H/ ‘H
H\ /H
H/ \H
understand by o-bonds which oscillate back and forth ?
The difficulty is not resolved by assuming that the structures (9a) and (96) are present in diborane, since there
are in fact two BH3 units simultaneously exchanging
two H atoms. Since the electronegativities of B and H
are essentially the same, ionic structures such as (9c) and
(9d) can be ruled out.
H’ ‘H
B@ B‘
,B -BO
H’ ‘H
On the other hand, formulae such as ( l o ) and (11) are
often written. So why should we not represent NH4 by
formula (12) (there are six possible structures)? Thus
an H i unit would be indicated in NH4 as BH3 is in diborane by formula (9a). It can be seen that the octet rule
is satisfied in formula (12). In (9a) or (96) it would be
necessary to include “inner” electrons in order to obtain formula (13). In (II), on the other hand, only ClH,
has a closed shell, whereas two electrons of B+ participate in bonding, and the K she11 forms the atomic
If it is still impossible to satisfy the octet rule using considerations of this kind, it is usual to turn to completely
ionic formulae, as in the case of Be(C5Hs)z (14), or
Angew. Chem. internat. Edit. f Vol. 4 (1965) f No. 8
XeF2 (15). Is it then permissible to write the structure
(16) ? Here things become doubtful, particularly since
Be in Be(C5H5)2 is not centrally positioned between the
rings as is postulated in structure (14) [3].
FFF- X e 2 + F-
We can see from all this that such an “interpretation”
of bonding relationships is unsatisfactory. On the one
hand, more and more knowledge of the electronic structures of the atoms involved is required, and on the other,
we encounter ambiguities, so that it is no longer possible
to speak of a true understanding. Here it is felt that the
relative importance of the individual structures of the
real molecule is not known. What does it mean to say a
certain structure of a molecule is more important than
the others? Can this conclusion be drawn with certainty
from the physical and chemical properties of the compound? Should it not be possible to predict, and possibly understand, the behavior of the molecule from
theoretical considerations?
Clearly, many compounds can be decribed by the octet
rule and the usual structural formulae. Although these
descriptions are not accurate, they can at least be used
by the chemist as a temporary measure, without which
all practical experience would remain unrelated. However, many of the compounds discovered in recent years
can be reconciled with these rules only with difficulty,
or fall entirely outside the scope of previous chemical
views. It is clear, moreover, that false information may
be obtained for these new compounds if the old scheme
is used indiscriminately.
Normally a rule is applicable only to certain classes of
compounds. For example, the additivity of the mean
dissociation energies is valid only in the case of localized
bonds, i.e. for molecules which can be essentially described by a single valence structure. The bond constants remain additive as long as the interactions between the various bonds are of the same type. However,
it is difficult to determine the range of validity, particularly since the significance of the individual valence
structures can only be discussed on the basis of wave
mechanics. All molecules for which the mean dissociation energies are approximately additive are compounds
for which the octet rule is approximately valid.
The rule of the additivity of dissociation energies gives
only a rough representation of the situation; this can be
seen in the case of the ideal disosciation energies DT
introduced by Szabo [4], for which there is no additivity
rule. Assuming, however, that the perturbations on a
given bond are due only to the adjacent atoms and
bonds, it is possible to determine the true dissociation
energies D, such that a useful rule is obtained. The diY
[3] A. Almenningen, 0. Bastiansen, and A . Haaland, J. chem.
Physics 40,3434 (1964).
[4] 2. G. Szabo, Z . Elektrochem., Ber. Bunsenges. physik. Chem.
61, 1183 (1957). - &-refers to a bond which is unperturbed by
neighboring atoms and bonds.
sturbancesdue to the neighbors aregiven by a quantitivity
d, which should be constant for a given atom and a given
bond, irrespective of the nature of the molecule. Thus in
(I) the wave functions $1, $2 . . . $M of the valence
electrons of the central atom are mutually orthogonal,
(2) the surrounding atoms arrange in those directions
in which the wave functions $1, $2 . . . have their
The $j functions are linear combinations of the atomic
functions 'ps and 'pD,as shown in Equation (y), where A
the case of a compound of the type (17), the expression
for the bond h - p is:
-DT(1-p) + d(-X) i-d(=Y) f d(-2) t d(=Z)
This rule can be extended to cover unsaturated and
aromatic systems and is important for the estimation of
activation energies. The form of this rule comes close to
the theoretical views of the valence bond and group
function methods. It fails when there is a change in the
bonding relationships.
3. Hybridization
No concept is so frequently used in valence theory and
none is so controversial as that of hybridization.
The problem of directed valence, i. e . the spatial structure of a molecule, was essentially explained by Slater
and Pauling [S] in 1931. However, the usual elementary
picture involves more theory than is supposed, and has
therefore not been free from serious objections [67.
The logical application of the hybridization concept
requires the use of wave-mechanical concepts such as
that of the atomic state (0)and the atomic functions
('ps, 'pp), as well as the introduction of electron spin
(f , 4). This is generally illustrated graphically, e.g. for
the atoms Be, C, N, and 0, as shown in Figure 5. In this
illustration, C* denotes an excited state.
is the hybridization parameter. The form of Equation (y)
is so chosen that Qiis normalized:
The p functions in Equation ( y ) at first lie in any direction in space; the number of # j functions involved
depends on the number of valence electrons, which are
defined here as the electrons of the singly occupied
energy levels of the free atoms. The direction of the p1
functions and the value of h are given by assumptions
1 and 2. When d and f functions are involved, Equation
(y) expands to give (2). However, the most important
points can be studied with the aid of Equation (y). Ac-
cording to Figure 5, four $i functions must be formed
in the excited state C* of carbon. According to assumption 1, Equation (a*) must be satisfied. This together
c)j d r = 0;(i f j)
with Equation Q leads to Equation (b*), where Oii is
cos Oij = -1/A2; (i, j = 1
M; i
the angle between the two functionsYpiandcppj.In order
to satisfy postulate (2) we introduce the bond strength,
as defined in Equation (c*), which represents the angular
dependence of a Jlj function in the direction of maximum
Fig. 5.
Aceording to this concept, the directed valence is determined by the electronic structure of the central atom;
this assumption is by no means satisfactorily fulfilled in
every case, as we shall see.
The elementary picture of hybridization is based on two
[S] J. C. Slater, Physic. Rev. 37, 481 (1930); 38, 1109 (1930);
L. Pading, J. Amer. chem. SOC.53, 1367 (1930); Physic. Re\.. 37,
I185 (1930); The Nature of the Chemical Bond [l].
[6JK . Artmann, Z.Naturforsch. 1, 426 (1946); Z.Physik 137,
137 (1954); 141, 445 (1955); 142, 518 (1955); D. Y. Loessl, Doctorate Dissertation, Universitat Hamburg, 1952.
extension [*].Thus, with(d*), Equation(c*)leadsto(e*):
This value of A leads to Equation (f *), which shows that
the four bonds of C* are directed towards the corners of
0, = - 1/3, i.e. 0..
U = 109.5
a tetrahedron, according to postulate (2). It should be
remembered here that the four atoms bound by a carbon atom are assumed to combine with the carbon in
The factor
13 in Eq. (c*) is contributed by the p-function.
Angew. Chern. internat. Edit. 1 Vol. 4(1965) 1 No. 8
these directions, in which the $i functions have maximum values. This assumption is not, as often asserted,
self-evident since there are systems in which carbon has
five neighbors. This brings us back to the NH4 radical or
compounds in which phosphorus could be assumed to be
tetravalent. According to Figure 5 , N (like P) has three
valence electrons in p states, and no s functions. Thus A
in Equation (y) must be infinitely large, and there is no
hybridization [@ = 90 according to Equation (b*)].
Pentavalent phosphorus might be thought to contain
states such as those shown in Figure 6 . However, which
tion energy (cp-zs), with the result that the p-companent in Equation (y) is smaller than predicted by Equation (d*); the transitition to the sp3 state also requires
an excitation energy. This gives rise to an energy principle which opposes postulate (2). At the same time, some
uncertainty is introduced into the determination of the
valence angle, since it is necessary to know the relationship between A and the excitation energy. Estimates
made in theoretical studies show [7] that this relationship can be represented by Equation (g*), where zD is
3 P 0 0 0
Fig. 6 .
valence state is closest to the true situation evidently
depends on energetic factors and on the number of different states (e.g. five in the case of d states), which must
be found from wave-mechanical considerations. Can
nitrogen also exist in a pentavalent state, as has sometimes been assumed, e.g. for the azide ion (18) ? Would
it not also be possible to write (18a) ?
Whereas (18) should involve hybridization with 3d or 3s
functions, the electron shell of the central nitrogen atomion in (18a) is the same as that of carbon, so that we
can again use Equation (y). This does not enable us to
decide whether (18) or (18a) is closer to reality, since
the energy required for the participation of d states in
(18) is offset by the need to form charges in (18a).
If we consider the molecules BeX2 and BX3 (Figure 7),
the result obtained using Equations (d*) and (b*),
= (E,-CJ/E~
the energy of dissociation of a bond partner from the
central atom. Equation (g*) is obviously not connected
with Equation (b*). The criterion of maximum overlap,
which is involved in Equation (d*), is therefore not sufficient for a calculation of valence angles, as has already
been pointed out by Van Vleck [8]. In agreement with
the above considerations, it was found that both postulates (1) and (2) cannot be satisfied simultaneously even
for CH4, when small deviations of the H atoms from
their equilibrium positions are considered.
The above discussion shows that, in addition to its ambiguity,
the usual elementary form of the hybridization concept cannot be satisfactorily substantiated, and leads to contradictions. The value of this concept in chemical bonding is therefore doubtful. It is generally called upon only as an aid to the
interpretation of experimental findings.
4. Wave-Mechanical Aspects
The determining factor here is the wave function y,
which depends on all the electronic coordinates (including the two possible electron spin directions, which
are described by spin co-ordinates), and the square of
which, Y * y , is proportional to the probability W of
finding the electrons at the positions given by the chosen
values of the electronic coordinates:
W a Y*Y
which led to A = 13,is in conflict with observation,
since BeX2 is linear, whereas BX3 is planar, with bond
angles of 120 O . It is usual in these cases to assume A = 1
and A = VT, respectively; Equation (b*) then gives
@ = 180" and @ = 120". This is contrary to postulate
(2), which gave such good agreement with experiment in
the case of carbon. These examples (and many others)
show that postulates 1 and 2 are not sufficient to lead to
directed valence. It is the empirical assumption that
A = 1 and A = 12 which ensures that the desiredresult
will be obtained. Artmann [6] was probably the first to
point out this discrepancy; he gave a better interpretation which did not make use of hybrization, and which
we shall discuss in Section 4.
The lower values of A are often explained by assuming
that the production of the sp (or sp*) configuration from
the ground state of the central atom requires an excitaAngew. Chem. internat. Edit. / Vol. 4(1965)
/ No. 8
Each electron distribution yk*yk
is associated with a
certain energy &k (k = 0,1, . . .) of the system as a whole,
which is a function of the positions of the atoms in space.
In fact
E~ =
+ A,
( k = 0, 1, Z...)
where E, is the electronic energy and A is the repulsive
energy of the positive atomic nuclei (charge ZAe;
A = 1 . . . N), which is of the form (1*) for N atoms,
where RAGis the distance between the nuclei of atoms A
and p. Ykand the associated energy Ek (or Ek) can in
principle be determined as accurately as desired from
the Schrodinger equation. Generally, however, it is
sufficient to know the approximate valuesYk and rk for
[7] K . Artmann, Z. Physik 138, 640 (1954).
[S] J. H . van Vleck, J. chern. Physics I , 209 (1933).
a discussion of bonding. Every conceivable fact about
an atom or molecule can theoretically be found f r o m y
and E.
According to Equation (k*), a molecule generally has an
infinite number of energy states E ~ which
should be
regarded as a function of the bond length R (see Figure
Sa). The electron energies E, (Figure 8b) are described
which is valid for all bonds, and which states that the
kinetic energy of the electrons is higher in a bond. The
total energy E is equal to the sum of the kinetic energy
K and the potential energy P of the total system, as
expressed in Equation (s*). In general, therefore, the
potential energy of the entire system is SD greatly reduced
in any bond, that it more than compensates for the
increase in the kinetic energy of the electrons.
An important factor in chemical bonding is the change
in the energies E as a function of the internuclear distances. Energy curves such as those shown in Figure 8a
can intersect. It can now be shown that the overlapping
of certain energy states is forbidden [lo]. In this case,
we may obtain Figure 9a, where the energy corresponds to an excited state, since when R-t co,E, is greater
Fig. 8. Total energy Ek and electronic energy Ek of a diatomic moiecule, as functions of the interatomic distance R. Curves I, 11,111, IV and
V correspond to different values of k.
for R-0 as the energy E$L of a united atom, and for
R-tm as the energy EFL of the separated atoms. For
example, as far as the electronic energy is concerned, the
united atom of the Hz molecule is the He atom. CH4
corresponds to Ne when all the C-H distances tend to
zero; if only one C-H distance tends to zero, the result
corresponds to the NH3 system.
It can be seen from Figure 8 that the repulsion of two
atoms at short interatomic distances is not due to interpenetration of the “electron clouds” and the resulting
mutual repulsion of the electrons. The repulsion is due
only to the A term in Equation (k*), since E, is always
smaller than EsA.
The kinetic energy K of the electrons is found [9] with
the aid of the virial theorem to be
Since in the case of bond formation there is an equilibrium distance R = Ro such that, for 0 < R < a,
~ R oG
) dR)
When R+m, on the other hand, the second term in (m*)
disappears to give:
= -ER+m=
(q *)
Because of the relationship (n*), a comparison of Equations (p*) and (q*) gives
K(Ro) > KR+ffl
than the energy of the ground state EO. At the “intersection”, the curves follow the broken lines. Thus if we
were to examine only the ground state, disregarding E ~
there would be no bonding, since EO = f(R) does not have
a minimum.
A similar situation is found e.g. in C h , when R is the
C-H distance (all the C-H distances being regarded as
equal). When R+w, E, corresponds to the energy of
the system C* + 4 H, whereas EO is the energy of the
system C + 4 H. Similar bonding relationships probably
also prevail in the xenon fluorides: E, corresponds to
Xe@-Fe, and Q to Xe-F. If the
curve is higher,
while still intersecting EO, metastable states can be formed, giving a lowest energy curve as shown in Figure 9b.
This is the case in He? or in the metastable systems
formed by the rare gases Ne and Ar with fluorine or
Knowing the potential energy curves (or the poteniial
energy surfaces when more than two atoms are under
consideration), we can calculate the vibrational and
rotational states of the systems with the aid of wave
mechanics. Reactions can also be examined, since these
can be regarded as movements of a system on the
energy surfaces. Thus the theories of reaction kinetics
and spectroscopy are based on a knowledge of E as a
function of the distances between the atomic nuclei. The
difference between a stable and a transient molecule can
also be seen from the energy curves.
[9] E.g. P. 0. Lowdin, Molecular Spectroscopy 3, 46 (1959).
Fig. 9. (a): Change in the total energy for a bonding and a nonbonding state; overlapping forbidden.
(b): Energy curve for metastable systems (e.g. XeF).
[lo] E.g. H . Hellmann: Einfiihrung in die Quantenchemie. Deuticke, Wien 1937, pp. 285-290.
Angew. Chem. infernat. Edit. [ Vol. 4 (1965) No. 8
For the molecule M(CsH&, it would be feasible as an
approximation to express the energy as the sum of the
curves of the potential energy between the metal and
each ring.
o ( H ~ C ~ - M - C ~ H%~~(Hgc5-M)
The E values on the right-hand side of Equation (t*)
change as shown in Figure Sa, whereas the left-hand
side is of the form shown in Figure 10 if the
D = j x1(1){;, x,(2){; ,...., xn(n) {;
(i = 1,2 . . .). It has been assumed in the expression (v*)
that the system in question contains n electrons. M and p
represent the spin functions for the two possible spin
directions, which we shall indicate by f (N) and 4 (p).
Equation (v*) satisfies one requirement imposed on '%'",
i.e. that only the sings of the wave functions should
change when the coordinates of two electrons are interchanged (Pauli's principle).
functions that they
It is demanded of the chosen
should represent a complete system of functions, and
that the integrals formed from them be defined.
Fig. 10. Change in the course of the energy curve for a dicyclopentadienylmetal M(CsHs)>, according to the masses of the partners.
(b) mM > mC5H5.
(a) mM < mCsHs
metal can move. We find that, under certain circumstances, the asymmetric position of the metal
becomes understandable if the "potential peak" in the
middle (Figure lOa) is sufficiently high and the vibrations of the molecule consist preferentially in the vibration of the metal. This can be the case only if the mass
of the metal is small in comparison with that of the fivemembered rings [e.g. in (14)]. If the metal atom is
heavy, on the other hand, the vibrations will be essentially those of the rings, thus leading to the potential
curve of Figure lob; in this case the molecule should
not be asymmetric, even if the two rings are identical in
The chemical significance of potential energy surfaces
has gained increasing recognition in recent years. Liquefaction and solidification are also connected with molecular interaction energies, the temperatures at which
these processes occur being determined by the position
and depth of the minima. The slight minimum in the
potential energy curve of He. . .He, for example, causes
the formation of condensation nuclei at very low temperatures.
There is no place for the hybridization concept in considerations involving the use of energy surfaces, unless
several energy curves corresponding to different states
of the central atom are being studied. More can be said
about hybridization if we introduce the '%'" functions.
In theoretical work, the approximation Y is generally
used instead of yi', constructed by a linear combination
[Equation (u*)] of the Qj functions; the sum can repreN
. .
sent the accurate Y function when the number of @ j
functions becomes very large (M+m). The coefficients
Cj are given by the requirement that (for a fixed M)
should be as close as possible to Y. (This requirement
can be used, although'%'"itself is unknown). In this case,
E is also an approximation to E.
The functions
are generally expressed as determinants, which depend on one-electron functions xj
Let us again consider BeX2 as an example. This molecule may be regarded as a four-electron system, since
for each bond there are two electrons siruated inside the
field of the remaining electrons. According to Figure 5,
the two valence electrons of the Be occupy a 2s and a 2p
state (cp,, cp,),
so that two functions (iil and $2 must be
The electrons of the
formed according to Equation 0).
two atoms X should occupy the orbitals cpx and yX.. As
an approximation to the wave function of BeX2, we find:
(the derivation cannot be discussed in the present paper).
According to Equation (y), A has not yet been fixed, and
is determined so that Y is the best possible approximation to Y. Similarly the directions of cppl and qp2are
still undecided, but the molecule is assumed in the first
place to be linear, in order to simplify the calculations.
One obtains X = 0.61, and it is found, in accordance
with Equation (x*), that
and (ii2 are not mutually
orthogonal, as assumed in the hybridization theory
[postulate (1) and Equation (a*)].
Equation (x*) means that postulates (1) and (2) cannot
be imposed if all the bonding electrons are taken into
consideration. The same has been found for BX3 and
CH4. More generally, we can say that the orthogonality
requirement (1) is almost always a violation of the wavemechanical treatment of the molecule. There are therefore no reasons for imposing condition (a*) on the
functions (y). A wave-mechanical discussion of the
valence angle shows that, e.g. in the case of BeX2, an
angular molecule is obtained if we admit only the A
values which agree with Equation (a*) [6,7].
If, as a simplification, we assume the interatomic distances to be increased, so that to a close approximation
only the central atom need be considered in relation to
the valence angle, this atom is only slightly influenced
by the outer atoms (separated atom model). For example, if we consider the C* atom of Figure 5, we can write
Angew. Chem. internat. Edit.
!Vol. 4 (1965) 1 No. 8
where px, py, and pz denote the p functions in the x, y,
and z directions.
If we now examine the probability W [cf. Equation(h*)I
of finding electrons on a spherical surface surrounding
the C atom, we obtain1thezremarkable result that the
maximum probabilities for >he four electrons occur at
the corners of a tetrahedron, so that the outer atoms
can- best be accommodated in these directions [6,7].
This result follows from the simple assumption that the
four valence electrons are present in the sp3 configuration, and is not restricted to the C atom. This procedure also gives the correct angles for the sp2 and sp
configurations. The maximum probabilities for the four
electrons in the case of the spzd configuration occur at
the corners of a square; this agrees with experimental
results, e.g. for [PdCl#-.
It must be emphasized that the notation of Equation
(y*) follows purely from Pauli's principle, and that
no electrostatic effects among the electrons are involved ; moreover, the concept of hybridization has
not been used. The wave-mechanical procedure is therefore already superior to the elementary hybridization
theory, particularly since it permits an unambiguous
determination of the bond directions. If there are several configurations to be discussed, the poteatla1 energy
curve for each configuration should be derived from
these calcuhtions, paying attention to forbidden overlapping. One reason why the usual concept of hybridization is unsatisfactory is that partial wave functions are
The procedure indicated in Equation (v*) may be carried
further, in order to obtain a better approximation t o y .
For example, the Y function of He is known with very
great accuracy. The most probable distribution of the
electrons on the surface of a sphere surrounding the
helium atom, as shown in Figure 11, is that of case (a).
(a I
attract a p-electron to itself, so that a bond is formed.
The bonding in XeF2, XeF4, and XeF6, which are reported to have linear, square, and octahedral structures respectively, could be of this type.
These greatly simplified considerations on the xenon fluorides
show that the electronegativity as defined in Equation (e) can
be used for a rough treatment. However, the detailed situation can only be correctly represented by the wave-mechanical calculations involving!I' and E. A number of wavemechanical points have already been taken into account i n
Figures 11 and 12.
As was pointed out above, we must seek a Y which
closely approaches y. When M is small the expressions
corresponding to Equation (u*) are somewhat arbitrary as regards the choice of the Qj functions. In the
valence-bond method, the Qj functions are sometimes
identified with particular structures. Such a procedure is
dubious, however, since we do not always know the part
played by the structures which have been omitted. It is
only when a large number of Qj functions (all, if possible) are considered, that we can assign a certain significance to the Cj values obtained. It must be remembered
that the Qj functions do not occur in the molecule, since
they do not describe a stationary state of the system. In
this interpretation, therefore, the @ j functions have an
imaginary character; consequently, the resonance energies which are often calculated in this connection are
also without absolute significance, since these depend on
the Qj functions chosen. Since the only important point
is Y, it is possible to use entirely different Qj systems,
provided that they represent the exact '?" functions when
M-tco. In other words, a picture of chemical bonding
is only admissible if it is based on adequate wave-mechariical foundations. An example of what we mean is
the remark which is sometimes made, that antiparallel
electron spins in different atoms lead to bonding. This
is not, however, the reason for chemical bonding, but
the fact that the spins are antiparallel is a characteristic
of a special type of bond.
Finally, from 9 and the integral
Fig. 1 1 .
In the case of the Xe atom, the most probable distribution is that of Figure 12, if we consider the p6 configuration of the ground state. On the approach of another
atom whose electron affinity is comparable with the
ionization energy of Xe (e.g. F or 0),this new atom can
Fig. 12. Most probable distribution of electrons in the xenon atom
(p6 state).
for a system containing n electrons, the coordinates of
the center of negative charge can be determined, which
can provide information on dipole moments. It should
be noted, however, that this center of charge must be
correlated with that of the positive nuclear charges. Thus,
the relationship between the dipole moment and'?"is not
mathematically simple. There is no relationship between the dipole moment and the difference in the
electronegativicies of the bonded atoms. An example is
carbon monoxide, for which there is a strong tendency
to attach considerable importance to the structure
Ce=O@,particularly since CO is very similar to Nz;
however, the dipole moment cf CO is very small.
Received: January 4th, 1965
[A 445/235 IE]
German version: Angew. Chem. 77, 666 (1965)
Translated by Express Translation Service, 1.ondon
Angew. Chem. internat. Edit.
Vol. 4 (1965) No. 8
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