Atomic and ionic radii: a comparison with radii derived from electron density distributions
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.
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.
To download the article click on the link below:
https://sci-hub.cc/https://link.springer.com/article/10.1007/s002690050057
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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