Lecture 2: Bonding in solids

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1 Lecture 2: Bonding in solids Electronegativity Van Arkel-Ketalaar Triangles Atomic and ionic radii Band theory of solids Molecules vs. solids Band structures Analysis of chemical bonds in Reciprocal space Real space Figures: AJK 1

2 Strong chemical bonding The chemical bonds in solids are usually classified as ionic / covalent / metallic Other types: hydrogen bonds / halogen bonds / van der Waals interactions Ionic bonding (e.g. NaCl) Typically high symmetry and high coordination numbers Ref: West p. 125 Covalent bonding (e.g. Si) Typically highly directional bonds and coordination numbers smaller than for ionic structures Figures: AJK Metallic bonding (e.g. Cu) Delocalized valence electrons. Can result in high coordination and close packing of atoms 2

3 Electronegativity The concept of electronegativity is an important tool for estimating how ionic or covalent a chemical bond is The electronegativity is a parameter introduced by Linus Pauling as a measure of the power of an atom to attract electrons to itself when it is part of a compound Pauling defined the difference in electronegativities defined in terms of bond dissociation energies, D 0 : D 0 (AA) and D 0 (BB) are the dissociation energies of A A and B B bonds and D 0 (AB) is the dissociation energy of an A B bond, all in ev units The expression gives differences of electronegativities and to establish an absolute scale Pauling set the electronegativity of fluorine to 3.98 (unitless quantity) Ref: Atkins Physical Chemistry, 9th ed. p

4 Pauling Electronegativities Figure: Wikipedia 4

5 Allen Electronegativities Derived from one-electron energies determined from spectroscopic data Very good correlation with the Pauling Electronegativities Somewhat ambiguous for d- and f-elements! Figure: Wikipedia 5

6 Using electronegativities (χ) The electronegativities can be used to estimate the polarity of a bond There is no clear-cut division between covalent and ionic bonds! Note that χ A χ B = 0 both for fully covalent (C C) and fully metallic bonds (Li Li) To fully understand the nature of the bonding, quantum chemical calculations are required (band structure calculations etc.) Bond A-B χ A χ B Cs F 3.19 Na Cl 2.23 H F 1.78 Fe O 1.61 Si O 1.54 Zn S 0.93 C H

van Arkel-Ketalaar Triangles The electronegativies can be used to

Arkel-Ketalaar Triangles Very illustrative concept for estimating

7 van Arkel-Ketalaar Triangles The electronegativies can be used to arrange binary compounds into Triangles of bonding often called van Arkel-Ketalaar Triangles Very illustrative concept for estimating the nature of a chemical bond Avoid assigning an exact % value for ionic/covalent nature of a bond! Ionic Metallic Covalent Figure: AJK Figure: 7

8 What really determines χ? Pauling determined the χ values from bond dissociation energies Allen used one-electron energies from spectroscopic data The periodic trends of electronegativity (and chemical bonding) can be discussed in terms of effective nuclear charge Z eff experienced by the valence electrons Z eff = Z σ, where Z is the atomic number and σ is shielding by other electrons The shielding can be determined from simple rules such as Slater s rules or from quantum chemical calculations Clementi, E.; Raimondi, D. L., "Atomic Screening Constants from SCF Functions, J. Chem. Phys 1963, 38, Higher the Z eff, the tighter the valence electrons are bound to the atom Element Li Be B C N O F Ne Z Z eff χ (4.8) * * Allen electronegativity 8

1 e - ), while electronegativity decreases Full consideration of orbital shapes etc.

9 χ vs. Z eff for the 2nd period χ and Z eff do actually show a beautiful correlation when moving from left to right in the periodic table However, Z eff of the valence electrons actually increases when moving down in periodic table (e.g. Z eff (Cl) = 6.1 e - ), while electronegativity decreases Full consideration of orbital shapes etc. required The moral of the story: simple explanations of complex manyelectron systems may sound nice, but are probably not right Figure: AJK 9

10 Various atomic radii The size of an atom or ion is not easy to define because there is not clear-cut definition for the border of an atom Various definitions for atomic, ionic, covalent, and van der Waals radii exist, here the following datasets are included: Atomic radii of neutral atoms from quantum chemical calculations (E. Clementi et al. J. Chem. Phys. 1967, 47, 1300). Ionic radii from experimental data (R. D. Shannon, Acta Cryst. 1976, 32, 751) Covalent radii from quantum chemical calculations (P. Pyykkö) van der Waals radii from experimental and quantum chemical data (Bondi, A. J. Phys. Chem. 1964, 68, 441; Truhlar et al. J. Phys. Chem. A, 2009, 113, 5806) 10

11 Atomic radii for neutral atoms Radii decrease when moving from left to right (Z eff increases) Radii increase when moving down in the group (principle quantum number n increases, orbitals become larger) Figure:UC Davis ChemWiki 11

12 Periodic trends of atomic radii Radii decrease when moving from left to right (Z eff increases) Radii increase when moving down in the group (principle quantum number n increases, orbitals become larger) The atomic radii are useful for the illustration of periodic trends, but not so valuable otherwise Figures:UC Davis ChemWiki 12

13 Shannon ionic radii (UC Davis ChemWiki) Ionic Radii (in pm units) of the most common ionic states of the s-, p-, and d-block elements. Gray circles indicate the sizes of the ions shown; colored circles indicate the sizes of the neutral atoms. Source: R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Cryst. 1976, 32, 751. Full radii data available at: 13

14 Applications of the ionic radii The ionic radii have been derived from a large number of experimental data They can be used for example: To check whether a new crystal structure shows ionic bonding To check whether an bond that is expected to be ionic has a reasonable length (even pointing out possible problems with the crystal structure) For example: The Na-Cl distance in solid NaCl is 282 pm, this compares well with the sum of the ionic radii: of Na+ (102 pm) and Cl- (181 pm) = 283 pm Another application is the radius ratio rules for ionic structures 14

contact With simple trigonometry, one can then derive minimum radius ratios for different coordination numbers For

15 Radius ratio rules Cations surround themselves with as many anions as possible, and vice versa. A cation must be in contact with its anionic neighbours (otherwise unstable) Neighbouring anions may or may not be in contact With simple trigonometry, one can then derive minimum radius ratios for different coordination numbers For example, NaCl: 102 pm/181 pm = > octahedral coordination. Nice qualitative tool, but not highly predictive Na Cl Ref: West p. 135 Figure:AJK 15

16 Self-Consistent Covalent Radii The Pyykkö Self-Consistent Covalent radii have been derived from a large number of experimental and computational data Similar to ionic radii, the covalent radii can be used for example: To check whether a new crystal structure shows covalent bonding To check whether an bond that is expected to be covalent has a reasonable length (even pointing out possible problems with the crystal structure) For example: The C-C distance in diamond is 154 pm, this compares well with the sum of the single-bond covalent radii = 150 pm The availability of double and triple bond radii makes the data set useful for interpreting new crystal structures Original papers: P. Pyykkö, M. Atsumi, Chem. Eur. J. 2009, 15, 186. P. Pyykkö, M. Atsumi, Chem. Eur. J. 2009, 15, P. Pyykkö, S. Riedel, M. Patzschke, Chem. Eur. J. 2005, 11, Another (experimental) set of radii: Alvarez et al. Dalton Trans., 2008,

17 17

van der Waals radii Significantly larger than covalent radii Can be used to check for weak interactions / contacts in a crystal structure Rather difficult to determine for d-/f-metals.

18 van der Waals radii Significantly larger than covalent radii Can be used to check for weak interactions / contacts in a crystal structure Rather difficult to determine for d-/f-metals. The values below are a combination of experimental and quantum chemical calues Figure: Ref: Truhlar et al. J. Phys. Chem. A, 2009, 113,

19 Bonding in Extended Structures Short introduction to band structures using two 1D model structures (infinite chains): Equally spaced H atoms Stack of square planar PtH

20 From H 2 to a large ring of H atoms 20

21 Bloch functions for the H atom chain Use translational symmetry and write the wave function ψ of the H atom chain as a linear combination of the H(1s) orbitals χ n The resulting wave functions for two values of k: 21

22 Band width or dispersion The band width is set by inter- unit cell overlap! Band width = dispersion Large band width means that the atoms in a unit cell are interacting with the atoms in neighboring unit cells Small band width (flat band) means that the atoms in a unit cell are not interacting with the neighboring unit cells 22

23 Stack of square planar PtH

24 Band structures in real solids In the 1D chains discussed above, it was enough to consider the band dispersion curves E(k) for one line (0 -> π/a) In 3D solids, k is called the wave vector and has three components (k x, k y, k z ) E(k) needs to be considered for several lines within the first Brillouin zone Primitive cell in reciprocal space, uniquely defined for all Bravais lattices Where do the band energies come from? Quantum chemical calculations (usually density functional theory) They can be measured also experimentally with e.g. electron cyclotron resonance (not that easily, though) Silicon (Fd-3m) Brillouin zone of an FCC lattice (Si) Band structure of silicon 24

25 Molecules vs. solids Molecular orbitals in benzene (highest occupied molecular orbital) For molecules, we only have k=0 (Γ point). For solids, we take the atoms in the neighboring cells into account by using E(k) Energy bands in bulk silicon (band-projected electron density, isovalue 0.02 a.u.) 25

26 Band structure and band gap Empty bands Occupied bands NaCl: insulator, large energy gap between occupied and nonoccupied bands Band gap: 8.75 ev Schematic view: Empty bands: Occupied bands: Silicon: semiconductor, energy gap between occupied and nonoccupied bands. Indirect band gap (1.1 ev at room temperature) Copper: metal, partially filled bands No band gap Fermi level Figures:AJK 26

27 Density of states (1) The band structures are a powerful description of the electronic structure of a solid, but often the spaghetti diagram does not immediately tell much more than just the nature of the band gap A more chemical look at the band structure can be obtained with Density of States diagrams (DOS) DOS(E)dE = number of levels between E and E + de DOS(E) is proportional to the inverse of the slope of E(k) vs. k The flatter the band, the greater the density of states at that energy Molecular bands lead into very sharp features in DOS(E) 27

28 Density of states (2) The DOS is almost like a molecular orbital diagram! 28

Real-space representations There are many ways to convert the reciprocal-space descriptions of the electronic structure back to the real space.

29 Real-space representations There are many ways to convert the reciprocal-space descriptions of the electronic structure back to the real space. Some examples: Band-projected electron densities Electron density difference plots: ρ(solid) ρ(non-interactive isolated atoms). Electron Localization Function (ELF) Figure:AJK Band-projected electron density in silicon (topmost valence band, isovalue 0.02 a.u.) 29 Figure: CRYSTAL tutorial

30 Electron localization function A. D. Becke and K. E. Edgecombe, "A simple measure of electron localization in atomic and molecular systems, 1990 J. Chem. Phys. 92, 5397 where τ is the kinetic energy density and ρ the electron (spin) density The values are scaled with respect to uniform electron gas, resulting in: ELF = 1 corresponds to perfect localization ELF = 0.5 corresponds to the uniform electron gas (completely delocalized) ELF transforms electron density into a chemically more intuitive picture Viable for all kinds of bonding situations Can be calculated for molecules and solids 30

31 Figure: T. F. Fässler Chem. Soc. Rev. 2003, 32, 80 (a) 2D-electron density distribution of Kr atom. (b) Colouring of the electron density distribution of a Kr atom with the values of ELF (colour bar see Fig. 1d). ( c) Full electron 2D-electron density distribution of an ethane molecule. (d) Colouring of the electron density distribution in Fig. 1c with the values of ELF (colour bar see bottom). (e) (g) 2D-valence electron density distribution with ELF colouring of a section through ethane, ethene, and ethine, respectively. (h) (j) 3D-isosurface of ELF with ELF = 0.80 for the valence electron density of ethane, ethene, and ethyne, respectively. 31

32 Some more examples of ELF Benzene Figures: Causa et al. Struct. Bond 2013, 150,

Lecture 4: Band theory

Lecture 4: Band theory Very short introduction to modern computational solid state chemistry Band theory of solids Molecules vs. solids Band structures Analysis of chemical bonding in Reciprocal space