Empirical electronic polarizabilities in oxides, hydroxides, oxyfluorides, and oxychlorides

Abstract

An extensive set of infinite-wavelength refractive indices recently derived from a single-oscillator Sellmeier equation [J. Phys. Chem. Ref. Data 31, 931 (2002)] was used to obtain mean total polarizabilities for 340 oxides, 3 hydroxides, 46 oxyhydroxides, 10 oxyfluorides, 8 oxychlorides, 80 hydrates, and 51 fluorides. These data, in conjunction with the polarizability additivity rule and a least-squares procedure, were used to obtain electronic polarizabilities for 79 cations, ${\mathrm{H}}_{2}\mathrm{O}$, and 4 anions $({\mathrm{F}}^{\ensuremath{-}},{\mathrm{Cl}}^{\ensuremath{-}},{\mathrm{OH}}^{\ensuremath{-}},{\mathrm{O}}^{{2}_{\ensuremath{-}}})$. Using literature values for free-cation polarizabilities, neglecting cation coordination, and allowing the variation of anion polarizability according to $\mathrm{log}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\alpha}}_{\ensuremath{-}}=\mathrm{log}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\alpha}}_{\ensuremath{-}}^{o}\ensuremath{-}{N}_{o}∕{V}_{\mathrm{an}}^{2∕3}$ where ${\ensuremath{\alpha}}_{\ensuremath{-}}=$anion polarizability, ${\ensuremath{\alpha}}_{\ensuremath{-}}^{o}=$empirical free-anion polarizability, ${V}_{\mathrm{an}}=$anion molar volume and ${N}_{o}=$a constant, the refinement gives agreement $(\ifmmode\pm\else\textpm\fi{}5%)$ in only 92 out of 381 total mean polarizabilities of 252 compounds. Varying cation polarizabilities, but still neglecting dependencies on cation coordination numbers (CN), allowed us to reproduce total polarizability values to within 5% for 611 out of 650 data on 487 oxides, hydrates, oxyfluorides, and oxychlorides. In the next stage we modified a light-scattering (LS) model by Jemmer et al. [J. Phys. Chem. A 102, 8377 (1998)] to give the expression $\ensuremath{\alpha}(\mathrm{CN},R)=[{a}_{1}+{a}_{2}{\mathrm{CN}}_{\mathrm{ca}}{e}^{\ensuremath{-}{a}_{3}R}{]}^{\ensuremath{-}1}$, where ${\mathrm{CN}}_{\mathrm{ca}}=$the number of nearest-neighbor ions (cation-anion interactions); $R=$cation-anion interatomic distance; and ${a}_{1}$, ${a}_{2}$, and ${a}_{3}$ are constants. This expression provides for a smooth decrease in polarizability at low CN's to the free-cation value at infinite CN's $(R=\ensuremath{\infty})$. Fitting polarizability values for Mg, Ca, Sr, Ba, Pb, Y, and La to this relationship provided a fit to within 5% of 601 out of 650 data. The final step in the refinement process, which used 534 total polarizabilities from 387 compounds, excluded compounds with (1) sterically strained (SS) structures, (2) corner-shared octahedral (CSO) network and chain structures such as perovskites, tungsten bronzes, and titanite-related structures, and (3) piezoelectric (PZ) and/or pyroelectric (PY) structures with abnormally high deviations of observed from total calculated polarizabilities. This final refinement, which provides 79 cation polarizabilities with values for Li, Mg, Ca, Sr, Ba, Pb, B, Al, Ga, Sc, Y, $\mathrm{Lu}\ensuremath{\rightarrow}\mathrm{La}$, Ge, and Ti in varying CN's, shows a standard deviation of 0.150 and no discrepancies $>4%$. Systematic comparisons of differences $(\mathrm{\ensuremath{\Delta}}{Z}^{*})$ of Born effective charges $({Z}^{*})$ from formal valence values with deviations of certain ions in $\ensuremath{\alpha}\text{\ensuremath{-}}{r}^{3}$ plots and with differences between empirical and free-ion $\ensuremath{\alpha}$'s indicate good correlations with metal $d$-oxygen $p$-hybridization and covalence. The level of differences increases in the order alkali ions $\ensuremath{\rightarrow}$ alkaline earth ions $\ensuremath{\rightarrow}$ transition metal ions such as ${\mathrm{Ni}}^{2+},\phantom{\rule{0.3em}{0ex}}{\mathrm{Mn}}^{2+},\phantom{\rule{0.3em}{0ex}}{\mathrm{Cd}}^{2+},\phantom{\rule{0.3em}{0ex}}{\mathrm{Pb}}^{2+},\phantom{\rule{0.3em}{0ex}}{\mathrm{Fe}}^{3+},\phantom{\rule{0.3em}{0ex}}\mathrm{and}\phantom{\rule{0.3em}{0ex}}{\mathrm{Cr}}^{3+}\ensuremath{\rightarrow}\mathbit{M}$ ions found in $A\mathbit{M}{\mathrm{O}}_{3}$ perovskites where $\mathbit{M}={\mathrm{Ti}}^{4+},\phantom{\rule{0.3em}{0ex}}{\mathrm{Zr}}^{4+},{\mathrm{Nb}}^{5+},\phantom{\rule{0.3em}{0ex}}\mathrm{and}\phantom{\rule{0.3em}{0ex}}{\mathrm{Ta}}^{5+}$. We ascribe the differences between our empirical polarizabilities and the free-ion values to charge transfer, effectively increasing cation polarizabilities and decreasing anion polarizabilities. Systematic discrepancies are associated with compounds having SS, CSO, and PZ/PY structures. Underbonded cations such as Mg in ${\mathrm{Mg}}_{3}{\mathrm{Al}}_{2}{\mathrm{Si}}_{3}{\mathrm{O}}_{12}$ (garnet) lead to augmented cation polarizabilities that result in increased observed total calculated polarizabilities (up to 6%). Conversely, overbonded cations in the ${\mathrm{KClO}}_{4}$ structure lead to diminished cation polarizabilities and decreased observed total calculated polarizabilities. Deviations, $\ensuremath{\Delta}$, (up to 10%) of observed from total calculated polarizabilities