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Lecture 2: Bonding in solids
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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
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Strong chemical bonding
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The chemical bonds in solids are usually classified as ionic / covalent / metallic
Other types: hydrogen bonds / halogen bonds / van der Waals interactions
Figures: AJK
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
Metallic bonding (e.g. Cu)
Delocalized valence
electrons. Can result in
high coordination and close
packing of atoms
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Electronegativity
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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, D0:
D0(AA) and D0(BB) are the dissociation energies of A–A and B–B bonds and D0(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. 389
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Pauling Electronegativities
Figure: Wikipedia
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Allen Electronegativities
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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
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Using electronegativities (χ)
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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 |χA – χB| = 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
|χA – χB|
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
0.35
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van Arkel-Ketalaar Triangles
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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: http://www.meta-synthesis.com/webbook/37_ak/triangles.html
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What really determines χ?
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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 Zeff experienced by the valence electrons
Zeff = 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, 2686–2689
Higher the Zeff, the tighter the valence electrons are “bound” to the atom
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Element
Li
Be
B
C
N
O
F
Ne
Z
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5
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Zeff
1.28
1.91
2.42
3.14
3.83
4.45
5.10
5.76
χ
0.98
1.57
2.04
2.55
3.04
3.44
3.98
(4.8)*
* Allen electronegativity
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χ vs. Zeff for the 2nd period
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χ and Zeff do actually show a
beautiful correlation when
moving from left to right in the
periodic table
However, Zeff of the valence
electrons actually increases
when moving down in periodic
table (e.g. Zeff (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
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Various atomic radii
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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)
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Atomic radii for neutral atoms
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Radii decrease when moving from left to right (Zeff increases)
Radii increase when moving down in the group (principle quantum number n
increases, orbitals become larger)
Figure:UC Davis ChemWiki
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Periodic trends of atomic radii
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Radii decrease when moving from left to right (Zeff 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
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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,
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751. Full radii data available at: http://abulafia.mt.ic.ac.uk/shannon/ptable.php).
Applications of the ionic radii
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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
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Radius ratio rules
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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 = 0.56 -> octahedral coordination.
Nice qualitative tool, but not highly predictive
Na
Cl
Figure:AJK
Ref: West p. 135
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Self-Consistent Covalent Radii
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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 75 + 75 = 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, 12770.
– P. Pyykkö, S. Riedel, M. Patzschke, Chem. Eur. J. 2005, 11, 3511.
Another (experimental) set of radii: Alvarez et al. Dalton Trans., 2008, 2832.
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van der Waals radii
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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
Ref: Truhlar et al. J. Phys. Chem. A, 2009, 113, 5806
Figure: http://www.webassign.net/question_assets/wertzcams3/ch_8/manual.html
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Bonding in Extended Structures
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Short introduction to band structures using
two 1D model structures (infinite chains):
– Equally spaced H atoms
– Stack of square planar PtH42-
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From H2 to a large ring of H atoms
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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 χn
The resulting wave functions for two
values of k:
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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
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Stack of square planar PtH42-
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Band structures in real solids
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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 (kx, ky, kz)
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
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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.)
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Band structure and band gap
Empty bands
Occupied bands
NaCl: insulator, large
energy gap between
occupied and nonoccupied bands
Band gap: 8.75 eV
Empty bands:
Schematic
view:
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
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Density of states (1)
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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)
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Density of states (2)
The DOS is almost like a molecular
orbital diagram!
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Real-space representations
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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.)
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Figure: CRYSTAL tutorial
Electron localization function
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A. D. Becke and K. E. Edgecombe, "A simple measure of electron localization in
atomic and molecular systems“, 1990 J. Chem. Phys. 92, 5397
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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
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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.
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Some more examples of ELF
Benzene
Figures: Causa et al. Struct. Bond 2013, 150, 119
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