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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Coppens P (1997) X-ray charge densities and chemical bonding. Oxford University Press, Oxford
Tsirelson VG, Ozerov RP (1996) Electron density and bonding in crystals. Institute of Physics Publishing, Bristol
Koritsanszky TS, Coppens P (2001) Chemical applications of X-ray charge density analysis. Chem Rev 101:1583–1628
Van Hove L (1954) Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys Rev 95(1):249–262
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):11721–11732
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):4539–4544
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):4545–4553
Heisenberger P, Platzman PM (1970) Compton scattering of X-rays from bound electrons. Phys Rev A 2(2):415–423
Chew G (1950) The inelastic scattering of high energy neutrons by deuterons according to the impulse approximation. Phys Rev 80(2):186–202
Chew G, Wick GC (1952) The impulse approximation. Phys Rev 85(4):636–642
Pattison P, Weyrich W, Williams B (1977) Observation of ionic deformation and bonding from Compton profiles. Solid State Commun 21:967–970
Hansen NK (1980) Reports of Hahn-Meitner Institute HMI B342
Hansen NK, Pattison P, Schneider J (1987) Analysis of the 3-dimensional electron distribution in silicon using directional Compton profile measurements. Z Phys B 66:305–315
Gillet J-M, Fluteaux C, Becker PJ (1999) Analytical reconstruction of momentum density from directional Compton profiles. Phys Rev B 60(4):2345–2349
Kontrym-Sznajd G (1990) Three dimensional image reconstruction with application in positron annihilation. Phys Stat Solid A 117(1):227–240
Reiter G, Silver R (1985) Measurement of interionic potentials in solids using deep-inelastic neutron scattering. Phys Rev Lett 54(10):1047–1050
Sivia DS, Skilling J (2006) Data analysis. Oxford Science, Oxford
Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge/New York
Jeffreys H (1939) Theory of probability. Clarendon, Oxford
Schmider H, Edgecombe KE, Smith VH Jr (1992) One-particle matrices along the molecular bonds in linear molecules. J Chem Phys 96(11):8411–8419
Howard S, Hulke JP, Mallinson PR, Frampton CS (1994) Density matrix refinement for molecular crystals. Phys Rev B 49(11):7124–7136
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):8986–8994
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:211–220
Pecora LM (1986) Determination of the quantum density matrix from experiment: an application to positron annihilation. Phys Rev B 33(9):5987–5993
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: 234–238
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 63:235115
Kiang HS (1969) N-representability theorem for reduced density matrices. J Math Phys 10(10):1920–1921
McWeeny R (1959) Hartree-Fock theory with non-orthogonal basis functions. Phys Rev 114(6):1528–1529
McWeeny R (1960) Some recent advances in density matrix theory. Rev Mod Phys 32(2): 335–369
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):7–13
Clinton W, Massa L (1972) Determination of the electron density matrix from X-ray diffraction data. Phys Rev Lett 29(20):1363–1366
Weiss AW (1961) Configuration interaction in simple atomic systems. Phys Rev 122: 1826–1836
Stewart RF, Feil D (1980) A theoretical study of elastic X-ray scattering. Acta Crystallogr A 36:503–509
Stewart RF (1997) Vibrational averaging of X-ray-scattering intensities. Isr J Chem 16: 137–143
Stewart RF (1969) Generalized X-ray scattering factors. J Chem Phys 51:4569–4576
Stewart RF (1977) One-electron density functions and many-centered finite multipole expansions. Isr J Chem 16:124–131
Stewart RF (1976) Electron population analysis with rigid pseudoatoms. Acta Crystallogr A 32:565–574
Hansen NK, Coppens P (1978) Testing aspherical atom refinements on small molecule data sets. Acta Crystallogr A 34:909–921
Fertig HA, Kohn W (2000) Symmetry of atomic electron density in Hartree, Hartree-Fock and density-functional theories. Phys Rev A 62:052511–10
Stewart RF, Bentley J, Goodman B (1975) Generalized X-ray scattering factors in diatomic molecules. J Chem Phys 63:3786–3793
Clementi E, Roetti C (1974) Atom Data Nucl Data Tab 14:177
Spackman MA (1992) Molecular electric moments from X-ray diffraction data. Chem Rev 92:1769–1797
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 62:400–408
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 63:198–200
Hirshfeld FL (1977) Charge deformation and vibrational smearing. Isr J Chem 16:168–174
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 108:1057–1063
Kato T (1957) On the eigenfunctions of many-particle systems in quantum mechanics. Commun Pure Appl Math 10:151–177
Katriel J, Davidson ER (1980) Asymptotic behavior of atomic and molecular wave functions. Proc Natl Acad Sci USA 77:4403–4406
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 57:290–303
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
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 37:857–863
Blessing RH (1995) On the differences between X-ray and neutron thermal vibration parameters. Acta Crystallogr B 51:816–823
Schomaker V, Trueblood KN (1968) On the rigid-body motion of molecules in crystals. Acta Crystallogr B 24:63–76
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 60:550–561
Destro R, Roversi P, Barzaghi M, Marsh RE (2000) Experimental charge density of α-glycine at 23 K. J Phys Chem A 104:1047–1054
Roversi P, Destro R (2004) Approximate anisotropic displacement parameters for H atoms in molecular crystals. Chem Phys Lett 386:472–478
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 8:3512–3521
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 120:2227–2236
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 64:465–475
Bader RFW (1990) Atoms in molecules: a quantum theory. Oxford Science, Oxford
Tsirelson VT (2002) Mapping of electronic energy distributions using experimental electron density. Acta Crystallogr B 58:632–639
Gatti C (2005) Chemical bonding in crystals: new directions. Z Kristallogr 220:399–487
Saunders VR, Dovesi R, Roetti C, Causa M, Harrison NM, Orlando R, Zicovich-Wilson CM (1998) CRYSTAL98 user’s manual. University of Turin, Turin
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 101:10686–10696
Ovegaard J, Hibbs DE (2004) The experimental electron density in polymorphs A and B of the anti-ulcer drug famotidine. Acta Crystallogr A 60:480–487
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 108:3663–3673
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 107:11201–11208
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 60:184–190
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 109:659–672
Brown AS, Spackman MA (1991) A model study of κ-refinement procedure for fitting valence electron densities. Acta Crystallogr A 47:21–29
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 35:63–72
Spackman MA, Byrom PG (1997) Retrieval of structure-factor phases in non-centrosymmetric space group. Model studies using multipole refinement. Acta Crystallogr B 53:553–564
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 52:291–301
Howard ST, Hursthouse MB, Lehmann CW (1995) Experimental and theoretical determination of electronic properties in L-dopa. Acta Crystallogr B 51:328–337
Volkov A, Abramov YA, Coppens P (2001) Density optimized radial exponents for X-ray charge density refinement from ab initio calculations. Acta Crystallogr A 57:272–282
Bytheway I, Chandler SG, Figgis BN (2002) Can a multipole analysis faithfully reproduce topological descriptors of a total charge density? Acta Crystallogr A 58:451–459
Spackman MA, Byrom PG, Alfredsson M, Hermansson K (1999) Influence of intermolecular interactions on multipole refined electron densities. Acta Crystallogr A 55:30–47
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 99:6242–6250
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 108:4283–4300
Dittrich B, Koritsanszky T, Luger P (2004) A simple approach to molecular densities with invarioms. Angew Chem Int Ed 43:2718–2721
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 54:1306–1318
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 63:108–125
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 65:485–499
Koritsanszky T, Volkov A (2004) Density radial functions for bonded atoms. Chem Phys Lett 383:431–435
te Velde B, Bickelhaupt FM, van Gisbergen SJA, Fonseca Guerra C, Baerends EJ, Snijders JG, Ziegler TJ (2001) Chemistry with ADF. J Comput Chem 22:931–967
Hirshfeld FL (1977) Spatial partitioning of charge density. Theor Chim Acta 44:129–132
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”).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Gillet, JM., Koritsanszky, T. (2011). Past, Present and Future of Charge Density and Density Matrix Refinements. In: Gatti, C., Macchi, P. (eds) Modern Charge-Density Analysis. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3836-4_5
Download citation
DOI: https://doi.org/10.1007/978-90-481-3836-4_5
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3835-7
Online ISBN: 978-90-481-3836-4
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)