Past, Present and Future of Charge Density and Density Matrix Refinements

Chapter

Abstract

Basic theoretical and some practical aspects of the interpretation of X-ray scattering experiments are described. Our focus is on model building and refinement associated with retrieving information related to electron density matrices from the measured data. The ill-posed nature of this inverse problem is emphasised and the physical significance, reliability and reproducibility of the properties obtained by data fitting are discussed through representative examples taken from recent studies. A special attention is devoted to the pseudoatom formalism widely used to interpret high-resolution single-crystal X-ray diffraction data to map the static electron distribution in solids.

Keywords

Spherical Harmonic Compton Scattering Multipole Expansion Molecular Density Multipole Refinement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

J.-M. Gillet would like to thank D. Sivia for fruitful discussions on combined data treatment. CNRS and Agence Nationale pour la Recherche (CEDA project) are also thanked for financial support. T. Koritsanszky acknowledges the support of the German Science Foundation (SPP 1178: “Experimental Electron Density as the Key to Understand Chemical Bonding”).

References

  1. 1.
    Coppens P (1997) X-ray charge densities and chemical bonding. Oxford University Press, OxfordGoogle Scholar
  2. 2.
    Tsirelson VG, Ozerov RP (1996) Electron density and bonding in crystals. Institute of Physics Publishing, BristolGoogle Scholar
  3. 3.
    Koritsanszky TS, Coppens P (2001) Chemical applications of X-ray charge density analysis. Chem Rev 101:1583–1628CrossRefGoogle Scholar
  4. 4.
    Van Hove L (1954) Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys Rev 95(1):249–262CrossRefGoogle Scholar
  5. 5.
    Schülke W, Schmitz JR, Schulte-Schrepping H, Kaprolat A (1995) Dynamic and structure factor of electrons in Si: inelastic X-ray scattering results. Phys Rev B 52(16):11721–11732CrossRefGoogle Scholar
  6. 6.
    Shukla A (1999) Ab initio Hartree-Fock computation of the electronic static structure factor for crystalline insulators: benchmark results on LiF. Phys Rev B 60(7):4539–4544CrossRefGoogle Scholar
  7. 7.
    Watanabe N, Hayashi H, Udagawa Y, Ten-no S, Iwata S (1998) Static structure factor and electron correlation effects studied by inelastic X-ray scattering spectroscopy. J Chem Phys 108(11):4545–4553CrossRefGoogle Scholar
  8. 8.
    Heisenberger P, Platzman PM (1970) Compton scattering of X-rays from bound electrons. Phys Rev A 2(2):415–423CrossRefGoogle Scholar
  9. 9.
    Chew G (1950) The inelastic scattering of high energy neutrons by deuterons according to the impulse approximation. Phys Rev 80(2):186–202CrossRefGoogle Scholar
  10. 10.
    Chew G, Wick GC (1952) The impulse approximation. Phys Rev 85(4):636–642CrossRefGoogle Scholar
  11. 11.
    Pattison P, Weyrich W, Williams B (1977) Observation of ionic deformation and bonding from Compton profiles. Solid State Commun 21:967–970CrossRefGoogle Scholar
  12. 12.
    Hansen NK (1980) Reports of Hahn-Meitner Institute HMI B342Google Scholar
  13. 13.
    Hansen NK, Pattison P, Schneider J (1987) Analysis of the 3-dimensional electron distribution in silicon using directional Compton profile measurements. Z Phys B 66:305–315CrossRefGoogle Scholar
  14. 14.
    Gillet J-M, Fluteaux C, Becker PJ (1999) Analytical reconstruction of momentum density from directional Compton profiles. Phys Rev B 60(4):2345–2349CrossRefGoogle Scholar
  15. 15.
    Kontrym-Sznajd G (1990) Three dimensional image reconstruction with application in positron annihilation. Phys Stat Solid A 117(1):227–240CrossRefGoogle Scholar
  16. 16.
    Reiter G, Silver R (1985) Measurement of interionic potentials in solids using deep-inelastic neutron scattering. Phys Rev Lett 54(10):1047–1050CrossRefGoogle Scholar
  17. 17.
    Sivia DS, Skilling J (2006) Data analysis. Oxford Science, OxfordGoogle Scholar
  18. 18.
    Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge/New YorkCrossRefGoogle Scholar
  19. 19.
    Jeffreys H (1939) Theory of probability. Clarendon, OxfordGoogle Scholar
  20. 20.
    Schmider H, Edgecombe KE, Smith VH Jr (1992) One-particle matrices along the molecular bonds in linear molecules. J Chem Phys 96(11):8411–8419CrossRefGoogle Scholar
  21. 21.
    Howard S, Hulke JP, Mallinson PR, Frampton CS (1994) Density matrix refinement for molecular crystals. Phys Rev B 49(11):7124–7136CrossRefGoogle Scholar
  22. 22.
    Schmider H, Smith VH Jr, Weyrich W (1992) Reconstruction of the one particle density matrix from expectation values in position and momentum space. J Chem Phys 96(12):8986–8994CrossRefGoogle Scholar
  23. 23.
    Schmider H, Smith VH Jr, Weyrich W (1993) On the inference of the one-particle density matrix from position and momentum-space form factors. Z Naturforsch 48A:211–220Google Scholar
  24. 24.
    Pecora LM (1986) Determination of the quantum density matrix from experiment: an application to positron annihilation. Phys Rev B 33(9):5987–5993CrossRefGoogle Scholar
  25. 25.
    Gillet J-M (2007) Determination of a one-electron reduced density matrix using a coupled pseudoatom model and a set of complementary scattering data. Acta Crystallogr A 63: 234–238CrossRefGoogle Scholar
  26. 26.
    Gillet J-M, Becker PJ, Cortona P (2001) Joint refinement of a local wave-function model from Compton and Bragg scattering data. Phys Rev B 63:235115CrossRefGoogle Scholar
  27. 27.
    Kiang HS (1969) N-representability theorem for reduced density matrices. J Math Phys 10(10):1920–1921CrossRefGoogle Scholar
  28. 28.
    McWeeny R (1959) Hartree-Fock theory with non-orthogonal basis functions. Phys Rev 114(6):1528–1529CrossRefGoogle Scholar
  29. 29.
    McWeeny R (1960) Some recent advances in density matrix theory. Rev Mod Phys 32(2): 335–369CrossRefGoogle Scholar
  30. 30.
    Clinton W, Galli A, Massa L (1969) Direct determination of pure-state density matrices. II. Construction of constrained idempotent one-body densities. Phys Rev 177(1):7–13CrossRefGoogle Scholar
  31. 31.
    Clinton W, Massa L (1972) Determination of the electron density matrix from X-ray diffraction data. Phys Rev Lett 29(20):1363–1366CrossRefGoogle Scholar
  32. 32.
    Weiss AW (1961) Configuration interaction in simple atomic systems. Phys Rev 122: 1826–1836CrossRefGoogle Scholar
  33. 33.
    Stewart RF, Feil D (1980) A theoretical study of elastic X-ray scattering. Acta Crystallogr A 36:503–509CrossRefGoogle Scholar
  34. 34.
    Stewart RF (1997) Vibrational averaging of X-ray-scattering intensities. Isr J Chem 16: 137–143Google Scholar
  35. 35.
    Stewart RF (1969) Generalized X-ray scattering factors. J Chem Phys 51:4569–4576CrossRefGoogle Scholar
  36. 36.
    Stewart RF (1977) One-electron density functions and many-centered finite multipole expansions. Isr J Chem 16:124–131Google Scholar
  37. 37.
    Stewart RF (1976) Electron population analysis with rigid pseudoatoms. Acta Crystallogr A 32:565–574CrossRefGoogle Scholar
  38. 38.
    Hansen NK, Coppens P (1978) Testing aspherical atom refinements on small molecule data sets. Acta Crystallogr A 34:909–921CrossRefGoogle Scholar
  39. 39.
    Fertig HA, Kohn W (2000) Symmetry of atomic electron density in Hartree, Hartree-Fock and density-functional theories. Phys Rev A 62:052511–10CrossRefGoogle Scholar
  40. 40.
    Stewart RF, Bentley J, Goodman B (1975) Generalized X-ray scattering factors in diatomic molecules. J Chem Phys 63:3786–3793CrossRefGoogle Scholar
  41. 41.
    Clementi E, Roetti C (1974) Atom Data Nucl Data Tab 14:177CrossRefGoogle Scholar
  42. 42.
    Spackman MA (1992) Molecular electric moments from X-ray diffraction data. Chem Rev 92:1769–1797CrossRefGoogle Scholar
  43. 43.
    Volkov A, King HF, Coppens P, Farrugia LJ (2006) On the calculation of electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model. Acta Crystallogr A 62:400–408CrossRefGoogle Scholar
  44. 44.
    Spackman MA (2007) Comments on On the calculation of electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model by Volkov, King, Coppens & Farrugia (2006). Acta Crystallogr A 63:198–200CrossRefGoogle Scholar
  45. 45.
    Hirshfeld FL (1977) Charge deformation and vibrational smearing. Isr J Chem 16:168–174Google Scholar
  46. 46.
    Oddershede J, Larsen S (2004) Charge density study of naphthalene based on X-ray diffraction data at four different temperatures and theoretical calculations. J Phys Chem A 108:1057–1063CrossRefGoogle Scholar
  47. 47.
    Kato T (1957) On the eigenfunctions of many-particle systems in quantum mechanics. Commun Pure Appl Math 10:151–177CrossRefGoogle Scholar
  48. 48.
    Katriel J, Davidson ER (1980) Asymptotic behavior of atomic and molecular wave functions. Proc Natl Acad Sci USA 77:4403–4406CrossRefGoogle Scholar
  49. 49.
    Pillet S, Souhassou M, Lecomte C, Schwarz K, Blaha P, Rerat M, Lichanot A, Roversi P (2001) Recovering experimental and theoretical electron densities in corundum using the multipolar model: IUCr multipole refinement project. Acta Crystallogr A 57:290–303CrossRefGoogle Scholar
  50. 50.
    Volkov A, Macchi P, Farrugia LJ, Gatti C, Mallinson P, Richter T, Koritsanszky T (2006) Program manual, XD2006 – a computer program package for multipole refinement, topological analysis of charge densities and evaluation of intermolecular energies from experimental and theoretical structure factors. User’s manual. http://xd.chem.buffalo.edu/docs/xdmanual.pdf
  51. 51.
    Coppens P, Boehme R, Price PF, Stevens ED (1981) Electron population analysis of accurate diffraction data. 10. Joint X-ray and neutron data refinement of structural and charge density parameters. Acta Crystallogr A 37:857–863CrossRefGoogle Scholar
  52. 52.
    Blessing RH (1995) On the differences between X-ray and neutron thermal vibration parameters. Acta Crystallogr B 51:816–823CrossRefGoogle Scholar
  53. 53.
    Schomaker V, Trueblood KN (1968) On the rigid-body motion of molecules in crystals. Acta Crystallogr B 24:63–76CrossRefGoogle Scholar
  54. 54.
    Madsen AO, Sorensen HO, Flensburg C, Stewart RF, Larsen S (2004) Modeling of the nuclear parameters of H atoms in X-ray charge density studies. Acta Crystallogr A 60:550–561CrossRefGoogle Scholar
  55. 55.
    Destro R, Roversi P, Barzaghi M, Marsh RE (2000) Experimental charge density of α-glycine at 23 K. J Phys Chem A 104:1047–1054CrossRefGoogle Scholar
  56. 56.
    Roversi P, Destro R (2004) Approximate anisotropic displacement parameters for H atoms in molecular crystals. Chem Phys Lett 386:472–478CrossRefGoogle Scholar
  57. 57.
    Bürgi HB, Capelli SC, Goeta AE, Howard JAK, Spackman MA, Yufit DS (2002) Electron distribution and molecular motion in crystalline benzene: an accurate experimental study combining CCD X-ray data on C6H6 with multi-temperature neutron-diffraction results on C6D6. Chem Eur J 8:3512–3521CrossRefGoogle Scholar
  58. 58.
    Flaig R, Koritsanszky T, Zobel D, Luger P (1998) Topological analysis of experimental electron densities of amino acids: 1. D,L-Aspartic acid at 20 K. J Am Chem Soc 120:2227–2236CrossRefGoogle Scholar
  59. 59.
    Munshi P, Madsen AO, Spackman MA, Larsen S, Destro R (2008) Estimated H-atom anisotropic displacement parameters: a comparison between different methods and with neutron diffraction results. Acta Crystallogr A 64:465–475CrossRefGoogle Scholar
  60. 60.
    Bader RFW (1990) Atoms in molecules: a quantum theory. Oxford Science, OxfordGoogle Scholar
  61. 61.
    Tsirelson VT (2002) Mapping of electronic energy distributions using experimental electron density. Acta Crystallogr B 58:632–639CrossRefGoogle Scholar
  62. 62.
    Gatti C (2005) Chemical bonding in crystals: new directions. Z Kristallogr 220:399–487CrossRefGoogle Scholar
  63. 63.
    Saunders VR, Dovesi R, Roetti C, Causa M, Harrison NM, Orlando R, Zicovich-Wilson CM (1998) CRYSTAL98 user’s manual. University of Turin, TurinGoogle Scholar
  64. 64.
    Gatti C, Saunders VR, Roetti C (1994) Crystal-field effect on the topological properties of the electron density in molecular-crystals – the case of urea. J Chem Phys 101:10686–10696CrossRefGoogle Scholar
  65. 65.
    Ovegaard J, Hibbs DE (2004) The experimental electron density in polymorphs A and B of the anti-ulcer drug famotidine. Acta Crystallogr A 60:480–487CrossRefGoogle Scholar
  66. 66.
    Hibbs DE, Overgaard J, Platts JA, Waller MP, Hursthouse MB (2004) Experimental and theoretical charge density studies of tetrafluoro-phthalonitrile and tetrafluoro-isophthalonitrile. J Phys Chem B 108:3663–3673CrossRefGoogle Scholar
  67. 67.
    Overgaard J, Waller MP, Platts JA, Hibbs DE (2003) Influence of crystal effects on molecular densities in a study of 9-Ethynyl-9-fluorenol. J Phys Chem A 107:11201–11208CrossRefGoogle Scholar
  68. 68.
    Scheins S, Dittrich B, Messerschmidt M, Paulmann C, Luger P (2004) Atomic volumes and charges in a system with a strong hydrogen bond: L-tryptophan formic acid. Acta Crystallogr B 60:184–190CrossRefGoogle Scholar
  69. 69.
    Munshi P, Guru Row TN (2005) Exploring the lower limit in hydrogen bonds: analysis of weak C–H⋯O and C–H⋯π interactions in substituted coumarins from charge density analysis. J Phys Chem A 109:659–672CrossRefGoogle Scholar
  70. 70.
    Brown AS, Spackman MA (1991) A model study of κ-refinement procedure for fitting valence electron densities. Acta Crystallogr A 47:21–29CrossRefGoogle Scholar
  71. 71.
    Coppens P, Guru Row TN, Leung P, Stevens ED, Becker PJ, Wang YW (1979) Net atomic charges and molecular dipole moments from spherical-atom X-ray refinements, and the relation between atomic charge and shape. Acta Crystallogr A 35:63–72CrossRefGoogle Scholar
  72. 72.
    Spackman MA, Byrom PG (1997) Retrieval of structure-factor phases in non-centrosymmetric space group. Model studies using multipole refinement. Acta Crystallogr B 53:553–564CrossRefGoogle Scholar
  73. 73.
    Haouzi AEl, Hansen NK, Hènass CLe, Protas J (1996) The phase problem in the analysis of X-ray diffraction data in terms of electron-density distributions. Acta Crystallogr A 52:291–301CrossRefGoogle Scholar
  74. 74.
    Howard ST, Hursthouse MB, Lehmann CW (1995) Experimental and theoretical determination of electronic properties in L-dopa. Acta Crystallogr B 51:328–337CrossRefGoogle Scholar
  75. 75.
    Volkov A, Abramov YA, Coppens P (2001) Density optimized radial exponents for X-ray charge density refinement from ab initio calculations. Acta Crystallogr A 57:272–282CrossRefGoogle Scholar
  76. 76.
    Bytheway I, Chandler SG, Figgis BN (2002) Can a multipole analysis faithfully reproduce topological descriptors of a total charge density? Acta Crystallogr A 58:451–459CrossRefGoogle Scholar
  77. 77.
    Spackman MA, Byrom PG, Alfredsson M, Hermansson K (1999) Influence of intermolecular interactions on multipole refined electron densities. Acta Crystallogr A 55:30–47CrossRefGoogle Scholar
  78. 78.
    Pichon-Pesme V, Lecomte C, Lachekar H (1995) On building a data bank of transferable experimental electron density parameters: applications to polypeptides. J Phys Chem 99:6242–6250CrossRefGoogle Scholar
  79. 79.
    Volkov A, Li X, Koritsanszky T, Coppens P (2004) Ab initio quality electro-static atomic and molecular properties from a transferable theoretical pseudoatom databank: comparison of electrostatic moments, topological properties, and interaction energies with theoretical and force-field results. J Phys Chem A 108:4283–4300CrossRefGoogle Scholar
  80. 80.
    Dittrich B, Koritsanszky T, Luger P (2004) A simple approach to molecular densities with invarioms. Angew Chem Int Ed 43:2718–2721CrossRefGoogle Scholar
  81. 81.
    Jelsch C, Pichon-Pesme V, Lecomte C, Aubry A (1998) Transferability of multipole charge-density parameters: application to very high resolution oligopeptide and protein structures. Acta Crystallogr D 54:1306–1318CrossRefGoogle Scholar
  82. 82.
    Pichon-Pesme V, Zarychta B,~Guillot B, Lecomte C, Jelsch C (2007) On the application of an experimental multipolar pseudo-atom library for accurate refinement of small-molecule and protein crystal structures. Acta Crystallogr A 63:108–125CrossRefGoogle Scholar
  83. 83.
    Dominiak PM, Volkov A, Dominiak AP, Jarzembska KN, Coppens P (2009) Combining crystallographic information and an aspherical-atom data bank in the evaluation of the electrostatic interaction energy in an enzyme-substrate complex: influenza neuraminidase inhibition. Acta Crystallogr D 65:485–499CrossRefGoogle Scholar
  84. 84.
    Koritsanszky T, Volkov A (2004) Density radial functions for bonded atoms. Chem Phys Lett 383:431–435CrossRefGoogle Scholar
  85. 85.
    te Velde B, Bickelhaupt FM, van Gisbergen SJA, Fonseca Guerra C, Baerends EJ, Snijders JG, Ziegler TJ (2001) Chemistry with ADF. J Comput Chem 22:931–967CrossRefGoogle Scholar
  86. 86.
    Hirshfeld FL (1977) Spatial partitioning of charge density. Theor Chim Acta 44:129–132CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Structures, Propriétés et Modélisation des Solides, UMR8580Ecole Centrale ParisChatenay-Malabry CedexFrance
  2. 2.Department of Chemistry, Computational Science ProgramMiddle Tennessee State UniversityMurfreesboroUSA

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