Abstract
The bonded radii of anions obtained in topological
analyses of theoretical and experimental electron
density distributions differ from atomic, ionic and crystal
radii in that oxide-, fluoride-, nitride- and sulfide-anion
radii are not constant for a given coordination number.
They vary in a regular way with bond length and the
electronegativity of the cation to which they are bonded,
exhibiting radii close to atomic radii when bonded to a
highly electronegative cation and radii close to ionic radii
when bonded to a highly electropositive cation. The electron
density distributions show that anions are not spherical
but exhibit several different radii in different bonded
directions. The bonded radii of cations correlate with
ionic and atomic radii. But unlike ionic radii, the bonded
radius of a cation shows a relatively small increase in
value with an increase in coordination number. In contrast
to atomic and ionic radii, the bonded radius of an ion
in a crystal or molecule can be used as a reliable and
well-defined estimate of its radius in the direction of its
bonds.
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A little history
Atomic radii
To our knowledge, the first set of atomic radii was
derived in 1920 by Sir Lawrence Bragg who assumed
that the atoms in crystals and molecules can be treated as
hard, incompressible spheres and that the radii of two
bonded atoms can be added to generate an approximate
bond length. Bragg began his derivation by making the
simple yet reasonable assumption that the radius of the
sulfur atom, r(S)51.02 A˚ , can be taken as equal to onehalf
the observed SS separation, R(SS)52.05 A˚ , of the
S2 molecular dimer in pyrite, FeS2. The radius of the zinc
atom, r(Zn)51.33 A˚ , was obtained by simply subtracting
r(S) from the bond length, R(ZnS)52.35 A˚ , reported
for zincblende. The radius of the oxygen atom,
r(O)50.65 A˚ , was determined by subtracting r(Zn) from
the bond length, R(ZnO)51.97 A˚ , observed for zincite.
He also observed that 23r(Zn)52.66 A˚ matches the average
separation (2.65 A˚ ) between the bonded Zn atoms
in Zn metal. Continuing in this fashion, he obtained a
table of empirical atomic radii for ¥40 elements. In spite
of the well-known fact that atoms in molecules and crystals
are neither strictly spherical nor hard, his radii, when
added together, were found to reproduce the bond
lengths in ¥50 oxides, sulfides, halides and metallic
crystals to within ¥0.06 A˚ , on average. Nearly half a
century later, Slater (1964) used Bragg’s strategy to extend
the table to more than 85 elements and found that the
radii reproduce ¥1200 bond lengths recorded for all
types of crystals and molecules, including oxides, sulfides,
nitrides, halides, metals and intermetallic compounds,
to within ¥0.12 A˚ , on average (Slater 1965).
These radii were considered to be universal in their application
because they reproduce bond lengths reasonably
well regardless of the bond type, the coordination
number or the oxidation state of the cation or whether the
bonded atoms comprise a crystal or a molecule. Nonetheless,
he was not as interested in the ability of his radii to
reproduce bond lengths accurately as he was in answering
such questions as ‘What is the connection between
the radius of an atom and the wave functions and the
electron density distribution of the atom which ultimately
must determine its radius?’ and ‘Why do atomic and
ionic radii reproduce bond lengths equally well yet the
cation radius for a given electropositive element is typically
¥0.85 A˚ smaller than its atomic radius?’ Another
question that might be asked is ‘Can anything be said
about the nature of the long-range forces that govern the
dimensions of a coordination polyhedron in a crystal
given that its average bond length can be reproduced
moderately well by simply adding the radius of a cation
and an anion?’
With accurate SCF wave functions corrected for relativistic
effects, Slater (1964, 1965) determined the positions
of the stationary points of the distributions of the
outermost valence electrons for nearly all of the atoms of
the periodic table to obtain a set of calculated radii for
their shells of maximum radial charge densities. A comparison
of this set of radii with his empirical set shows a
strong correlation when the two are plotted one against
the other. He argued that maximal overlap may be expected
to be largely achieved when the maxima of the
outermost shells of two bonded atoms coincide. For example,
in the case of rock salt, the Na and Cl atoms are
observed to adopt a minimum energy NaCl distance of
2.82 A˚ compared with the sum of the atomic radii
r(Na)1r(Cl)51.8011.0052.80 A˚ where the overlap of
the valence shells of the two atoms can be argued to be
largely maximal. This is the general significance that
Slater (1964) attached to Bragg’s atomic radii and why
they work.
Ionic radii
One of the first sets of ionic radii was derived by
Wasatjerne (1923), who from ionic refraction measurements
and observed unit cell dimensions, determined the
radii of the oxide and fluoride ions to be 1.32 A˚ and
1.33 A˚ , respectively. With these values, Goldschmidt
et al. (1926) used Bragg’s strategy to derive a set of radii
for a large number of elements by subtracting r(O22
) and
434
r(F21
) from bond length data measured for MX and MX2
compounds (M5metal cation, X5O22
, F21
). In applying
these radii, Goldschmidt et al. (1926) observed that the
number of anions surrounding a cation in an ionic crystal
tends to be maximal subject to the constraint that the
cation-anion contact is preserved and that adjacent anions
in the coordinated polyhedra tend to be in contact.
With the geometrical constraints imposed on regular
polyhedra by this rule, it was suggested that the coordination
number of a cation in an ionic crystal tends to be
governed by the radius ratio of the cation and anion. It
was also observed that the radius of a cation depends on
its coordination number and oxidation state, the larger
the coordination number and the smaller the oxidation
number, the larger the cation.
The following year, Pauling (1927) derived a set of
six-coordinate ionic radii using quantum mechanically
derived screening constants, the Born-Lande equation
and a set of observed bond length data used to scale the
radii. The agreement between Pauling’s semiempirical
radii (when corrected for coordination number) and
Goldschmidt’s empirical radii was taken to be a confirmation
of the ionic model and Pauling’s strategy for
deriving ionic radii (see also Zachariasen 1931). In
1952, Ahrens used ionization potential data to modify
Pauling’s radii by deriving a slightly different but improved
set of radii together with a number of previously
undetermined radii (see also Politzer et al. 1983 and Rosseinsky
1994). These radii were used rather extensively
during the fifties and sixties (Whittaker and Muntus
1970) until sets of tailor-made empirical radii were
derived that reproduce bond lengths rather accurately
when the factors of the local chemical environment of an
ion are taken into account.
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