Titolo presentazione The connubiumsottotitolo between quantum mechanicsMilano, XX andmese 20XXcrystallography Piero Macchi Department of Chemistry, Materials, Chemical Engineering, «Giulio Natta», Politecnico di Milano, Milano, Italy The connubium between quantum mechanics and crystallography Macchi P. Cryst. Rev. 2020, 26, 209-268 https://www.tandfonline.com/doi/full/10.1080/0889311X.2020.1853712 1. Historical Background: Who, What, When, Where, Why? 1. the atomic model and the electronic structure 2. the chemical bonding 3. supramolecular interactions 4. wavefunction from experiments 5. modelling electron charge and spin densities 2. State of the Art 1. standard models and beyond 2. chemical bonding analysis 3. molecules and beyond Historical Background The Solvay Conference 1927: Crystallographers & Quantum Physicists AH Comton WL Bragg L. Brillouin L De Broglie 1. The atomic model 1803 1897 1911 1913 1926 Rontgen 1896 Laue 1912 It seems to me that the experimental study of the scattered radiation, in particular from light atoms, should get more attention, since along this way it should be possible to determine the arrangement of the electrons in the atoms. Debye P. Zerstreuung von Röntgenstrahlen. Ann. Phys. 1915; 351: 809–823. 1 The atomic model. Was X-ray diffraction able to shed light? The elastic scattering (Thomson) A distribution which fits Bragg's data acceptably is an arrangement of the electrons in equally-spaced, concentric rings, each ring having the same 4휋푠푖푛휗 퐤 퐤 = number of electrons, and the diameter of the outer ring being about 0.7 of the 휆 distance between the successive planes of atoms. Compton AH. The Distribution of the Electrons in Atoms. Nature. 1915; 95: 343-344. 퐤퐬 퐤ퟎ 휗 휗 2휗 electron r dhkl 2휋 퐤 = 퐤 = 퐬 휆 hkl Bragg WH, Bragg WL. The Reflexion of X-rays by Crystals. Proc. R. Soc. Lond. A. 1913; 88: 428–38 1 The atomic model. What we gain with a crystal? No second spectacular crystallographic discovery has since been made of Diffraction image consequences comparable with those of the Lens original discovery of X-ray diffraction. Crystal structure Ewald PP. Editorial Preface. Acta Cryst. 1948; 1: 1-2. Crystal X-ray 1 nm 100 nm the crystal is a periodically homogeneous object, a special tool that enables the magnification of a given functionality 1 The atomic model. What we gain with a crystal? …although calculation, and not a lens, is used in the final stages of its Diffraction image formation, and although Lens Crystal structure certain essential data, used automatically by a lens, Crystal namely the relative phases of the spectra, have to be inferred by means other than optical before the X-ray calculation can be carried out, the [Fourier] 1 nm projection is nevertheless formally an optical image. James RW. The optical principles of the diffraction of X-rays, G. Bell and 100 nm Sons, London, 1958, page 385. the crystal is a periodically homogeneous object, a special tool that enables the magnification of a given functionality 1 The atomic model. Bragg experiment and the radial electron density distribution 4휋푟휌 푟 Bragg WL, James RW, Bosanquet CH. The distributions of electrons around the nucleus in the sodium and chlorine atoms. Phil Mag. 1922; 44: 433-449. 1 The atomic model. Bragg experiment and the radial electron density distribution From experiment (1921) From experiment (1921) From theory (>1930) From theory (>1930) 1 The atomic model. Bragg experiment and the radial electron density distribution 25 K Na form factor Na 8 20 L 7푓 ] 6 15 M 5 4 푟10 [푒Å 휌 3 5 2 4휋푟 1 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 -1 푠푖푛휗 푟(Å) Å 휆 First reflection Resolution limit for Any attempt to determine the state of ionization of atoms in a crystal is likely to fail, since of NaCl scattering factor curves will differ appreciably only at angles for which no spectra exist Bragg experiment James RW. The optical principles of the diffraction of X-rays, G. Bell and Sons, London, 1958 (Pt Lα) 1. The Electronic structures of atoms in crystals Electronic configurations in metals Crystals of elemental metals are analyzed by Bragg diffraction technique to conclude that most of the valence electrons reside in 3d-states (Cu, Ni and Co) or in hybrid states (Cr and Fe). A very difficult task, strongly affected by the so called extinction Weiss RJ. A Physicist remembers. World Scientific Publishing Co. Pte. Ltd., Singapore, 2007. 1 The atomic model. Relativistic form factors Clementi E, Roetti C. Roothaan-Hartree-Fock atomic wavefunctions: Basis functions and their coefficients for ground and certain excited states of neutral and ionized Form factor Xe atoms, Z≤54 Atomic Data and Nuclear Data Tables. 1974; 14: 177-478. 3.5% Atomic Data and Nuclear Data Tables 50 Roothan Hartree-Fock (non relativistic, non correlated) 3.0% 푓 40 Macchi P, Coppens P. Relativistic analytical wave functions and scattering 2.5% factors for neutral atoms beyond Kr and for all chemically important ions up to I-. Acta Cryst., 2001, A57, 656-662. 30 2.0% Dirac-Fock method (relativistic & correlation) 1.5% 20 Volkov A., Macchi P. Zero-order relativistic analytical wave functions for all 1.0% 13 neutral atoms. Unpublished 10 12 Zora-Kohn-Sham method (relativistic & DFT) 0.5% 11 0 0.0% 10 0 0.5 1 1.5 2 % difference between non relativistic and relativistic 9 푠푖푛휗 Å 8 휆 1.5 1.7 1.9 1. The atomic model Crystallography reveals the nature of the electrons 2. The Chemical Bonding X-ray structures paved the way toward the development of modern chemical bonding theories A small part only of the body of contributions of quantum mechanics to chemistry has been purely quantum-mechanical in character; only in a few cases, for example, have results of direct chemical interest been obtained by the accurate solution of the Schrödinger wave equation Pauling L. The Nature of the Chemical Bonding. First edition. Cornell University Press. New York, 1939 2. The chemical bonding Visualization through the deformation density 1 퐤퐫 ∆휌 퐫 = 퐅(퐤)− 퐅,(퐤) 푒 푉 퐤 Coppens, P. (1967) Science, 158, 1577–1579. Based on a model refined from neutron diffraction 2. The chemical bonding Topological Analysis of the Electron Density Based on the gradient field of the electron density 휌(퐫) Obtained from a multipole expansion refined against X-ray diffraction data 훻휌(퐫) 3. Supramolecular interactions The Hydrogen bonding Deformation density approach QTAIM approach ELF approach Tsirelson V, Stash A. Determination of the electron Wang Y, Tsai CJ, Liu WL. Temperature-Dependence Madsen GKH, Iversen BB, Larsen FK, Kapon M, localization function from electron density. Chem. Studies of α-Oxalic acid Dihydrate. Acta Cryst. 1985; Reisner GM, Herbstein FH. Topological Analysis of Phys. Lett. 2002; 142: 142-148. B41:131-135 the Charge Density in Short Intramolecular O-H---O Hydrogen Bonds. Very Low Temperature X-ray and Neutron Diffraction Study of Benzoylacetone. J. Am. Chem. Soc. 1998; 120:10040-10045. 3. Supramolecular interactions The Hydrogen bonding A kind of structure Same results using a correlation with promolecular charge density results approach (spherical (from multipolar atom densities) model expansions) Spackman M. Hydrogen bond Espinosa E, Molins E, energetics from topological Lecomte C. Hydrogen bond analysis of strengths revealed by experimental electron topological analyses of densities: Recognising the experimentally observed importance of electron densities. Chem. the promolecule. Chem. Phys. Phys. Lett. 1998; 285:170–173 Lett. 1999; 301:425-429 4. Wavefunction from experiment A dream? Wavefunction models: a molecular wavefunctions adjusted to reproduce the measured X-ray diffraction 2 Hˆ E ()r () r Approximated Hamiltonian ˆ ˆ ˆ ˆ H E H H H err Weiss RJ. X-ray determination of Electron Distributions. North-Holland Publishing Company. Amsterdam. 1966. Weiss RJ. Charge and Spin Density. Physics Today 1965; 18: 43-44. 4. Wavefunction from experiment 4. Wavefunction from experiment Wavefunction models: a molecular wavefunctions Density matrix from X-ray diffraction experiments. adjusted to reproduce the measured X-ray diffraction N-representability problem Jayatilaka D. Wave Function for Beryllium from X-Ray Diffraction Data. Phys. Rev. Lett. 1998; 80: 798–801. Clinton W, Massa L. Determination of the Electron Density Matrix from X-Ray Diffraction Data Phys. Rev. Lett. 1972; 29: 1363-1366. ˆ Jˆ H ˆ 2 JXC E XC XC Density matrix models: Refinement of atomic 2 2 expansion against Compton profiles Fcalc F target 1 H H 2 (,)rr (,) rr (,) rr 2 a ab target Nr N p H F a a, b H Gillet J-M. Determination of a one-electron reduced density matrix using a coupled pseudo-atom model and a set of complementary scattering data. Wavefunction models: Hirshfeld atom refinement Acta Cryst. 2007; A63: 234-238. ˆ 2 ()r () r H E ()r () r i Wavefunction models: Stewart atom refinement i ) ikr i Stewart atoms are the unique nuclear-centered 푓(퐤 Fk() fi ()() kk T i e i spherical functions whose sum best fits a molecular Jayatilaka D, Dittrich B. X-ray structure refinement using aspherical atomic density functions obtained from quantum-mechanical calculations. Acta Cryst. 2008; A64: 383-393 electron density in a least-squares sense. 4. Wavefunction from experiment First attempt of X-ray wavefunction refinement 휑 = 푐,휒 MO-LCAO 휓 = det 휑 , 휌 퐫 = 푛푐,푐,휒휒 = 푃휒휒 Population coefficients Atomic orbital products Coppens P, Csonka LN, Willoughby TV. Electron Population Parameters from Least-Squares Refinement of X-ray Diffraction Data. Science. 1970; 167: 1126-1128 Coppens P, Willoughby TV, Csonka LN. Electron Population Analysis of Accurate Diffraction Data. I. Formalisms and Restrictions. Acta Cryst. 1971; A27: 248-256 Coppens P, Pautler D, Griffin JF.