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The valence quadrupole moment

  • Kendrick Shepherd , Matthew C.F. Wander , Barry R. Bickmore EMAIL logo , W. Joel Johansen , Tyler Goodell , Matthew Davis , Charles Andros , Larissa Lind and Kathleen Robertson
Published/Copyright: February 18, 2016
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American Mineralogist
From the journal American Mineralogist Volume 101 Issue 2

Abstract

The bond-valence model has recently been expanded to include a directional component, the vectorial bond-valence sum, which is useful for characterizing non-centrosymmetric distortions involving lone-pair and second-order Jahn-Teller effects. Here we show that the bond-valence sum and vectorial bond-valence sum are equivalent to monopole and dipole terms in a multipole expansion of the bond valence incident to an atom. We then extend the multipole expansion to include a quadrupole term, which describes the ellipsoidal deviation from spherical symmetry of the bonding environment, and is useful for characterizing centrosymmetric distortions, such as those caused by first-order Jahn-Teller effects. These distortions follow characteristic patterns in valence space, which depend upon factors that include the d-orbital configuration and size of the transition metal involved. This extended approach, called the Valence Multipole Model, should prove useful for modeling molecular and crystal structures, including those associated with spin transitions.

Keywords: Bond valence; vectorial bond-valence model; valence multipole model; bond angle; molecular mechanics; Jahn-Teller effect; crystal field effects

Acknowledgments

We gratefully acknowledge the National Science Foundation (EAR-1227215 and EAR-1424682) and the Brigham Young University College of Physical and Mathematical Sciences undergraduate mentoring program for funding this project. We also thank I.D. Brown and Frank Hawthorne for providing helpful reviews of the initial manuscript.

References cited

Adams, S. (2001) Relationship between bond valence and bond softness of alkali halides and chalcogenides. Acta Crystallographica, B57, 278–287.10.1107/S0108768101003068Search in Google Scholar

Adams, S., and Rao, R.P (2009) Transport pathways for mobile ions in disordered solids from the analysis of energy-scaled bond-valence mismatch landscapes. Physical Chemistry Chemical Physics, 11, 3210–3216.10.1039/b901753dSearch in Google Scholar PubMed

Adams, S., and Rao, R.P (2014) Understanding ionic conduction and energy storage materials with bond-valence-based methods. Structure and Bonding, 158, 129–160.10.1007/430_2013_137Search in Google Scholar

Adams, S., and Swenson, J. (2002) Bond valence analysis of transport pathways in RMC models of fast ion conducting glasses. Physical Chemistry Chemical Physics, 4, 3179–3184.10.1039/b111310kSearch in Google Scholar

Adams, S., Moretzki, O., and Canadell, E. (2004) Global instability index optimizations for the localization of mobile protons. Solid State Ionics, 168, 281–290.10.1016/j.ssi.2003.04.002Search in Google Scholar

Bambi, C. (2011) Constraint on the quadrupole moment of super-massive black hole candidates from the estimate of the mean radiative efficiency of AGN. Physical Review D, 83, 103003.10.1103/PhysRevD.83.103003Search in Google Scholar

Bickmore, B.R. (2014) Structure and acidity in aqueous solutions and oxide-water interfaces. Structure and Bonding, 158, 191–203.10.1007/430_2012_84Search in Google Scholar

Bickmore, B.R., Wander, M.C.F., Edwards, J., Maurer, J., Shepherd, K., Meyer, E., Johansen, W.J., Frank, R.A., Andros, C., and Davis, M. (2013) Electronic structure effects in the vectorial bond-valence model. American Mineralogist, 98, 340–349.10.2138/am.2013.4329Search in Google Scholar

Brown, I.D. (2002) The Chemical Bond in Inorganic Chemistry: The bond valence model, 278 p. Oxford University Press, New York.Search in Google Scholar

Brown, I.D. (2009) Recent developments in the methods and applications of the bond valence model. Chemical Reviews, 109, 6858–6919.10.1021/cr900053kSearch in Google Scholar PubMed PubMed Central

Brown, I.D. (2014) Bond valence theory. Structure and Bonding, 158, 11–58.10.1007/430_2012_89Search in Google Scholar

Burns, R.G. (1993) Mineralogical Applications of Crystal Field Theory, 551 p. Cambridge University Press, Cambridge.10.1017/CBO9780511524899Search in Google Scholar

Comba, P, Hambley, T.W., and Martin, B. (2009) Molecular modeling of inorganic compounds, 326 p. Wiley-VCH, Weinheim.10.1002/9783527628124Search in Google Scholar

Cooper, V.R., Grinberg, I., and Rappe, A.M. (2003) Extending first principles modeling with crystal chemistry: A bond-valence based classical potential. In PK. Davies and D.J. Singh, Eds., Fundamental Physics of Ferroelectrics. American Institute of Physics, Melville, New York.10.1063/1.1609957Search in Google Scholar

Cramer, C.J. (2004) Essentials of Computational Chemistry, 596 p. Wiley, Chichester, U.K.Search in Google Scholar

Cygan, R.T. (2001) Molecular modeling in mineralogy and geochemistry. Reviews in Mineralogy and Geochemistry, 42, 1–35.10.1515/9781501508721-004Search in Google Scholar

Downs, R.T., and Hall-Wallace, M. (2003) The American Mineralogist Crystal Structure Database. American Mineralogist, 88, 247–250.Search in Google Scholar

Farfan, G., Wang, S., Ma, H., Caracas, R., and Mao, W.L. (2012) Bonding and structural changes in siderite at high pressure. American Mineralogist, 97, 1421–1426.10.2138/am.2012.4001Search in Google Scholar

Gillespie, R.J., and Hargittai, I. (1991) The VSEPR Model of Molecular Geometry, 248 p. Allyn and Bacon, Boston.Search in Google Scholar

Golub, G.H. (2013) Matrix Computations, 728 p. Johns Hopkins University Press, Baltimore.10.56021/9781421407944Search in Google Scholar

Grinberg, I., Cooper, V.R., and Rappe, A.M. (2002) Relationship between local structure and phase transitions of a disordered solid solution. Nature, 419, 909–911.10.1038/nature01115Search in Google Scholar PubMed

Grinberg, I., Cooper, V.R., and Rappe, A.M. (2004) Oxide chemistry and local structure of PbZrxTf-O studied by density-functional theory supercell calculations. Physical Review B, 69, 144118.10.1103/PhysRevB.69.144118Search in Google Scholar

Grinberg, I., Shin, Y.-H., and Rappé, A.M. (2009) Molecular dynamics study of dielectric response in a relaxor ferroelectric. Physical Review Letters, 103, 197601.10.1103/PhysRevLett.103.197601Search in Google Scholar PubMed

Harvey, M.A., Baggio, S., and Baggio, R. (2006) A new simplifying approach to molecular geometry description: the vectorial bond-valence model. Acta Crystallographica, B62, 1038–1042.10.1107/S0108768106026553Search in Google Scholar PubMed

Hinchliffe, A. (2003) Molecular Modelling for Beginners, 410 p. Wiley, Chichester, U.K.Search in Google Scholar

Horn, R.A., and Johnson, C.R. (2012) Matrix Analysis, 655 p. Cambridge University Press, Cambridge.10.1017/CBO9781139020411Search in Google Scholar

Jackson, J.D. (1998) Classical Electrodynamics, 808 p. Wiley, New York.Search in Google Scholar

Lavina, B., Dera, P, Downs, R.T, Yang, W., Sinogeikin, S., Meng, Y., Shen, G., and Schiferl, D. (2010) Structure of siderite FeCO3 to 56 GPa and hysteresis of its spin-pairing. Physical Review B, 82, 064110.10.1103/PhysRevB.82.064110Search in Google Scholar

Liu, S., Grinberg, I., and Rappé, A.M. (2013a) Development of a bond-valence based interatomic potential for BiFeO3 for accurate molecular dynamics simulations. Journal of Physics: Condensed Matter, 25, 102202.10.1088/0953-8984/25/10/102202Search in Google Scholar PubMed

Liu, S., Grinberg, I., Takenaka, H., and Rappé, A.M. (2013b) Reinterpretation of the bond-valence model with bond-order formalismml: An improved bond-valence-based interatomic potential for PbTiO3. Physical Review B, 88, 104102.10.1103/PhysRevB.88.104102Search in Google Scholar

Lufaso, M.W., and Woodward, PM. (2001) Prediction of the crystal structures of perovskites using the software program SPuDS. Acta Crystallographica, B57, 725–738.10.1107/S0108768101015282Search in Google Scholar PubMed

Müller, U. (2007) Inorganic Structural Chemistry, 268 p. Wiley, Chichester, U.K.Search in Google Scholar

O’Keeffe, M., and Brese, N.E. (1992) Bond-valence parameters for anion-anion bonds in solids. Acta Crystallographica, B48, 152–154.10.1107/S0108768191013083Search in Google Scholar

Perez-Mato, J.M., Withers, R.L., Larsson, A.-K., Orobengoa, D., and Liu, Y. (2009) Distortion modes and related ferroic properties of the stuffed tridymite-type compounds SrAl2O4 and BaAE2O4. Physical Review B, 79, 064lll.10.1103/PhysRevB.79.064111Search in Google Scholar

Pijpers, F.P. (1998) Helioseismic determination of the solar gravitational quadrupole moment. Monthly Notices of the Royal Astronomical Society, 297, L76–L80.10.1046/j.1365-8711.1998.01801.xSearch in Google Scholar

Preiser, C., Lösel, J., Brown, I.D., Kunz, M., and Skowron, A. (1999) Long-range Coulomb forces and localized bonds. Acta Crystallographica, B55, 698–711.10.1107/S0108768199003961Search in Google Scholar PubMed

Rappé, A.K., and Casewit, C.J. (1997) Molecular Mechanics Across Chemistry, 444 p. University Science Books, Sausalito, California.Search in Google Scholar

Shin, Y.-H., Cooper, V.R., Grinberg, I., and Rappe, A.M. (2005) Development of a bond-valence molecular-dynamics model for complex oxides. Physical Review B, 71, 054104.10.1103/PhysRevB.71.054104Search in Google Scholar

Shin, Y.-H., Grinberg, I., Chen, I.-W., and Rappé, A.M. (2007) Nucleation and growth mechanism of ferroelectric domain-wall motion. Nature, 449, 881–884.10.1038/nature06165Search in Google Scholar PubMed

Shin, Y.-H., Son, J.-Y., Lee, B.-J., Grinberg, I., and Rappé, A.M. (2008) Order-dis-order character of PbTiO3. Journal of Physics: Condensed Matter, 20, 015224.Search in Google Scholar

Stixrude, L. (200i) First principles theory of mantle and core phases. Reviews in Mineralogy and Geochemistry, 42, 319–343.10.1515/9781501508721-012Search in Google Scholar

Takenaka, H., Grinberg, I., and Rappé, A.M. (2013) Anisotropic local correlations and dynamics in a relaxor ferroelectric. Physical Review Letters, 110, 147602.10.1103/PhysRevLett.110.147602Search in Google Scholar PubMed

Wander, M.C.F., Bickmore, B.R., Davis, M., Johansen, W.J., Andros, C., and Lind, L. (2015a) The use of cation-cation and anion-anion bonds to augment the bond-valence model. American Mineralogist, 100, 148–159.10.2138/am-2015-4810Search in Google Scholar

Wander, M.C.F., Bickmore, B.R., Lind, L., Andros, C., Hunt, J., Checketts, H., and Goodell, T. (2015b) AIM analysis and the form of the bond-valence equation. American Mineralogist, 100, 160–161.10.2138/am-2015-5037Search in Google Scholar

  1. Manuscript handled by Alejandro Fernandez-Martinez

Received: 2015-1-14
Accepted: 2015-9-1
Published Online: 2016-2-18
Published in Print: 2016-2-1

© 2016 by Walter de Gruyter Berlin/Boston

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https://doi.org/10.2138/am-2016-5304
Keywords for this article
Bond valence; vectorial bond-valence model; valence multipole model; bond angle; molecular mechanics; Jahn-Teller effect; crystal field effects

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