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Does the Death Knell Toll for the Metallic Bond.

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Does the Death Knell Toll for the Metallic Bond?**
J. Christian Schon*
“There is an appointed time for everything under the heavens--a time to be born, and a time to die.” Judging from two
recent publications,[” the latter fate awaits the concept of the
metallic bond. Based on two somewhat different trains of
thought. the authors conclude that the metallic bond is a disposable quantity in descriptions of bonding in solids, and should be
subsumed into the supposedly better established concepts of the
covalent and ionic bond.
Analyzing their arguments in more detail, we notice that Allen et al.[’] base their conclusion on an attempt to quantify the
so-called van Arkel -Ketelaar triangle.[3s41They suggest that a
configuration energy (CE) should be defined as an average “valence shell energy”, in other words, the average energy needed to
remove a single valence electron. Here, it is assumed that the
remaining valence electrons remain in their original states during this “ionization” process. Thus, according to Koopman’s
theorem. these valence shell energies correspond to the one-particle energies in the Hartree-Fock a p p r ~ x i r n a t i o n . [ ~ ~ [For
* * *a~
compound composed of A and B, Allen et al. then identify CE,
and CE, with the diagonal elements of the interaction matrix
within the approximation of the extended Hiickel theory
(EHT) .I1,21 Using a common approximation for the off-diagonal
matrix elements,[61 Hij =1/2(Hi + H j ) , they find that
H A , = (CE, + CE,)/2 holds. Since the off-diagonal matrix elements explain (within the context of the EHT, at least) the
Dr. J. C. Schon
Institut fur Anorganische Chemie der Universitat
Gerhard-Domagk-Strasse 1 . D-53121 Bonn (Germany)
Teletdx: Int. code + (228)73-5660
e-mail. tlnc419(
[**I I would like to thank Prof. Dr. M. Jansen. Bonn. for many valuable comments
and discussions.
[***I The interpretation of the one-particle energies is subtle: the total energy ofthe
solid IS not the sum of all one-particle energies, since we are dealing with an
interacting system. In particular. a one-particle energy is in general not identical with the standard ionization energy.
A n g w Chcrn. I n f . Ed. Engl. 1995. 34, No. 10
Peierls distortion,[7.s1 which accompanies the change from
“metallic” to “covalent” band structures, Allen et al. conclude
that (CE, + CE,)/2 is a natural coordinate along the metallic +covalent edge of the van Arkel-Ketelaar triangle.
The second coordinate in the description of the triangle would
have to reflect the degree of ionicity in the bonds. Here, Allen
et al.[91 refer to the classical concept of the difference in electronegativities for the characterization of the polarity of the
bond. Within their model, this concept is most easily implemented by employing the difference in configuration energies,
(CE, - CE,)/2.[*] From these considerations it follows that the
three traditional bonding concepts should be expressed in a
natural way through the two “coordinates” derived from the
configuration energy.[**]
At the same time, Anderson, Burdett, and Czechf2’( = ABC)
have addressed the issue that certain materials should not necessarily be described in terms of typical metallic bonding, although they show macroscopic metallic behavior. Examples are
doped molecular crystals and ionic systems, for instance fullerenes and compounds related to the high-temperature superconductors, respectively. The band structures of such “metals” can
be determined within the tight-binding (TB) approach,[5, and
it is not necessary to invoke the metallic bond with its somewhat
vague concept of delocalized vs. localized electrons to explain
the metallike behavior. Thus, ABC suggest that one should
reevaluate the concept of metallic bonding in light of the current
understanding of theoretical models of the bonding in solids.
They point out that most of the properties associated with
metallic behavior are related to the position of the Fermi surface
in the band approximation[”] for the theoretical analysis of the
Based on these considerations, Allen draws the conclusion that CE should be
used as a general characteristic quantity for the chemical behavior of an atom
on the same level as its position in the periodic table.
For an alternative approach to go beyond the van Arkel-Ketelaar triangle by
applying quantum mechanical concepts, see the work by Urland [lo].
VCH Verlugsgesrllsdiu/t mhH, 0-69451 Weinheim, 1995
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electronic degrees of freedom in the solid. In metals, the bands
are filled only partly, and thus no energy gap exists for the
promotion of a (quasi-)electron into a conducting state. With
statistical mechanics it can be shown that this lack of a gap
explains many of the characteristic properties of a metal.[’21
It is known[’.1L1that within the context of the one-electron
band approximation the eigenfunctions of the corresponding
one-particle Hamiltonian for a periodic lattice are Bloch funcu,(R; 4. The function u,
tions of the form Y,,(g;?) = exp
reflects the periodicity of the crystal lattice, such that u,,(R;
?) = u,(z; F + di),
where diis any translation vector of the
lattice. Since these Bloch functions are obviously “delocalized”,
any description of the electronic degrees of freedom of the crystal relying on this approximation refers to delocalized (quasi)particles. On the other hand, it is always possible to choose any
complete orthogonal basis of functions, including rather localized
bases, and express the Bloch functions in this basis. One should
notice here that nothing has been said about the actual method
used to calculate the band structure. The reason is that planewave expansions (NFE, OPW, etc.),[’. ’’] expansions using atomic/molecular orbitals including ionization states (TB, LCAO,
etc.),[’% and also the “cell-methods” (APW, KKR, etc.)[’%‘‘I
are in principle applicable to all band structures and all types of
chemical bonding.[’] Only the ease of computation for a given
system and property favors one method over the others.
Next, ABC point out that because of the high degree of degeneracy a Fermi surface is subject to instabilities, which in a partly
filled band can lead to the opening of an energy gap in the
density of states.[*]Slight distortions of the arrangement of the
ion cores away from a high-symmetry configuration can remove
this degeneracy (= Peierls distortion in analogy to the JahnTeller effect,[’] its strength should be related to the electronphonon interaction). Similar effects can be produced by a modulation of the electron distribution, in other words, charge density waves.[’] This removal of the degeneracy can result in the
opening of an energy gap at the Fermi surface. This would be of
importance in a metal, since the band gap would cause a change
to an insulator/semiconductor.Whether this actually occurs depends on whether the ground state energy of the symmetric
(undistorted) structure lies above or below that of the distorted
structure, a fact that can be influenced, for instance, by the
application of external pressure.[’31 As was pointed out earlier,
within the EHT one can estimate the relative stability of the
Peierls distortion in compounds by considering the off-diagonal
elements. Thus the considerations of ABC lead to the same
classification of the metallic +covalent bond transition as those
of Allen et al.,”] who started from the calculation of the configuration energy.
Finally, ABC suggest that one might define the metallic bond
to apply to those systems within the one-electron approximation
for which the interpretation of Wannier functions[51as singleelectron wave functions is inappropriate. Since the (relatively
localized) Wannier functions are not eigenfunctions of the oneparticle Hamiltonian (in contrast to the Bloch functions, which
can be transformed into Wannier functions by a type of Fourier
transform), their Slater determinant may be interpreted as a
many-electron state only if all the Bloch states within a band are
filled and present in the determinant. Recalling that metals are
essentially defined by the property of having a partly filled ener-
,i: VCH Verlug.rjirs~~lls~hufi
nrhH, 0.69451 W(+z.inliairn. 1995
gy band, one can conclude that metallic bonds are just bonds in
metals; no special attributes belong to the metallic bond itself.
Adding to this their analysis of the transition from metallic to
covalent bonds. ABC draw the conclusion that the traditional
concept of a metallic bond as such, as an individual type essentially different from other bonds, is no longer necessary. It might
even be disadvantageous to maintain this concept, since it hides
the more fundamental aspects of the bonding situation as reflected in the (calculated) band structures.
Now, while the arguments of Allen and ABC against the
metallic bond as an entity completely separate and distinct from
other types of bonds are very interesting, one should not overlook that the concept of metallic bonding is first and foremost
a qualitative one based on the prominence of certain features of
the wave function(s) describing a solid. In this sense, the metallic
bond is not different from other types of bonds that stress other
properties of the wave function. They all serve the same purpose
of allowing a first qualitative characterization of a solid and its
behavior. Similarly, as pointed out above, in the context of
quantum mechanical band structure calculations, covalent or
ionic bonds are no more special than metallic ones (and neither
are the use of localized or delocalized bases for the expansion of
the Hamiltonian). Thus, the arguments of both ABC and Allen
et al. carried to their final conclusion actually imply that not
only the concept of the metallic bond should vanish, but so
should all the traditional concepts of bonds in solids!
Has a major change of paradigm thus occurred in solid-state
chemistry? Not really. The seemingly radical conclusion we
have just reached should come as no surprise, since it is known
that any solid is composed of strongly interacting particles,
namely ion cores and electrons. Thus any more detailed understanding of its properties will always involve studying the collective behavior of its constituents, which implies that local concepts like bonds will only be of limited value.[*] However, the
necessity of having to solve the full quantum mechanical problem
first is a certain disadvantage over more simple classification
procedures, although it seems quite likely that with the increasing availability of powerful computers and sophisticated algorithms the concepts of metallic or ionic bonds will become
more quantitative.[**]
As a consequence, the boundaries between bonding regimes
will become less sharp than a more qualitative heuristic picture
of bonds in solids usually implies. Nevertheless, the extreme
cases of the ideal covalent, ionic, or metallic bond will retain
their usefulness, as long as one is always aware of their limitations. Thus, we expect the different concepts of bonds in solids,
including the metallic one, to keep their place in the tool box of
the solid-state scientist. The time has not yet come for the metallic bond to pass into oblivion.
German version: A n y w . Cliem. 1995, i07, 11x3
Keywords: bond theory . metallic bond . solid-state chemistry
Compare, however, the fruitful concept of quasi-particles introduced by
Landau [S].
Such a direct classification method has been recently proposed by Silvi and
Savin [14]. They suggest employing the electron localization function(s) (ELF)
115. 161 for a more rigorous definition of quantities like ionicity or metallicity
of a compound. Clearly, their procedure points the way towards a quantum
mechanical classification of the types of bonds prevalent in a given chemical
compound, thus providing us with definitions that would allow us to speak
with confidence about the metallic. covalent, and/or ionic character of bonds.
0570-0833/95/1010-1082$ 10.00 + .25/0
Anyew. Chrm. Int. Ed. Enyl. 1995, 34, No. I0
[ I ] L. c'. Allen. J. F. Capitani, J. Am. C'hiw. Xoc. 1994, 116. 8810.
121 W. P. Anderson. J. K . Burdett, P. T. Czech, J. A m . C17en7. Soc. 1994.116. 8808-
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