Bonding in Solids

Outline

1. Ionic bonding

2. Partial covalent bonding

3. Metallic bonding

• Ions, ionic radii

• Ionic structures

• Lattice energy of ionic crystal

• Born-Haber cycle

• Coordinated polymeric structures

• Bond valence and bond

• Non-bonding electron effects

• Band theory

• Band structure of metals, insulators and semiconductors

• Band structure of inorganic solids

General idea:

• Mixed bonding type: Ionic, covalent, van der Waals, and metallic e.g. TiO and CdI2

• Ionic bonding: high symmetry with the highest possible coordination number (larger lattice energy)

• Covalent bond: directional, prefer a certain coordination environment

• Position in periodic table… high valency vs. low valency

Bonding

Ionic Bonding

Several models have been established for the calculation of ionic radii e.g. Pauling, Goldschmidt.. Shannon and Prewitt

Ionic BondingWhat’s the size of ions in crystals?

http://www.crystalmaker.com/support/tutorials/crystalmaker/AtomicRadii.html

High quality X-ray diffraction: electron density contour map

• Ions are spherical

• Central core + outer sphere of shell

• Ions are charged, elastic and polarizable

•Assignment of radii is difficult…lead to the variation of reported values

• Flexibility in ionic radii, depends on coordination environment

Ionic Bonding

LiF

• Diagonal ion have similar ionic size

Ionic Radii

Trend in ionic radii:

• Radii increase with atomic number for any vertical group (s and p-block elements)

• Isoelectronic series of cations, radii decrease with increasing charge

• Cation radius decrease with increasing oxidation state

• Cationic radius increase with increasing coordination number

• Lanthanide contraction (d and f block elements)

• Transition vs. main group

• Ions are charged, elastic and polarizable spheres

• Ionic structures are held by electrostatic force

• Highest coordination numbers (to maximize the electrostatic attraction between ions, lattice energy)

• Reduced nearest neighbor repulsive interaction (high symmetry and maximized volume)

• Local electroneutrality

Ionic Structure_General Principles

The charge of a particular ion must be balanced by equal and opposite charge on the immediately surrounding ion

Electrostatic bond strength (ebs) of cation-anion bond:

ebs = m/n

Sum of m/n = ∑ ebs = x

For cations Mm+ surround by n anions :

For anions Ox-:

For example: MgAl2O4, structure

1. Explain the structure with bond strength

2. Predict the polyhedral linkage

Local Electroneutrality

Electrostatic valence rule:

Mg

O

Al

• Cation must be in contact with its anionic neighbors. (lower limit on cation size)

• Neighboring anions may or may not be in contact

Octahedral cation site (6CN):

> 0.414 cation push the anions apart, occupy a site of larger CN

< 0.414, occupy a site of smaller CN (not stable, rattle inside the anion polyhedron)

Ratio of Rcation / Ranion

Radius Ratio Rule

(2rx)2 + (2rx)2 = [2(rM + rx)]2

2rx(√ 2) = 2(rM + rx)

rM/rx = (√ 2) – 1 = 0.414

Cubic (8CN) Tetrahedral (4CN)

Minimum rM/rx of Cubic and Tetrahedral Structures

a = 2rx

2(√ 3)rx = 2(rM + rx)

rM/rx = (√ 3) – 1 = 0.732

2(rM + rx) = cube body diagonal

(2rx)2 + (√ 2rx)2 = [2(rM + rx)]2

2rx = face diagonal

rx(√ 6) = 2(rM + rx)

rM/rx = [(√ 6) – 2]/2 = 0.225

Radius ratio rules: qualitative guide only

Borderline case:

2. Distorted structure: BaTiO3, the reversible displacement of Ti in an applied electric field, lead to Ferroelectrictiy

1. Polymorphs: contains both tetrahedral and octahedral

Predicting Trends in CN

Lattice energy U, is the net potential energy of the arrangement of charges that forms the structure.

NaCl(s) → Na+(g) + Cl-(g) ΔH = U

Ionic structure: hold together by electrostatic forces (electrostatic repulsion and attraction in crystal)

Electrostatic Forces

Coulomb’s Law:

F = Z+Z-e2/r2

Attractive force

Charge on the ions

Electric charge

Distance between the ions

Mz+ Xz-

r

Lattice Energy of Ionic Crystals

Attraction “-” and Repulsion “+”

Coulombic potential energy:

V = -(Z+Z-e2)/r

Charge on the ions

Electric charge

Distance between the ions

Born repulsion energy:

V = B / rn

constant Distance between the ions

In the range of 5 to 12

Determination of Lattice Energy

For example: NaCl

Madelung constant A, depends on crystal structure

V1 = -6(Z+Z-e2)/r

V2 = 12(Z+Z-e2)/√2r

V3 = -8(Z+Z-e2)/√3r

V = [-(Z+Z-e2)/r][6-(12/√2)+(8/√3)-(6/√4)… ]

Lattice energy is calculated by combining the net electrostatic attraction and the Born repulsion energies and finding the internuclear separation, which gives the maximum U value.

V = -(Z+Z-e2)NA/r

Coulombic potential energy

U = -(Z+Z-e2)NA/r + BN/rn

dU/dr = 0, U = [-(Z+Z-e2)NA/re](1-1/n)

Lattice energy:

More precisely: • Born repulsive term is Be(-r/ρ)

• Zero point vibration (2.25Nhν)

• van der Waals dipole interactions (-NC/r6)

Complete equation for U:

U of divalent ions >> monovalent ions (about 4 times)

Determination of Lattice Energy

U = -(Z+Z-e2)NA/re + BNe-r/ρ - NC/r6 + 2.25Nhν

(N, A, B, e, hν are constant for a particular structure)

Kapustinskii’s Equation

When CN increase, Madelung constant A and re increase, the effect are proposed to be auto-compensated

re = rc + ra, V = number of ions per formula

To predict the stable existence of several unknown compounds and the values for ionic radii

U = [-(Z+Z-e2)NA/re](1 - ρ/re)

U = [1200.5V(Z+Z-)/(rc + ra)][1 – 0.345/(rc + ra)]

(substitute ρ=0.345, A=1.745 and values for N and e)

Na+(g) + Cl-(g) → NaCl(s) ΔH = U

Na(s) + ½ Cl2(g) NaCl(s) ΔH = ΔHf

ΔHf

Cl(g) + Na+(g)

Na(g)

Cl-(g)

½ DEA

US

IP

ΔHf = S + ½ D + IP + EA + U

Lattice energy of ionic crystal is equivalent to its heat of formation, which can not be measured experimentally.

However, heat of formation can be measured relative to the reagents in their standard states.

Born-Haber Cycle

Hess’s Law

• Check the internal consistency of data

• Predict the unknown data

• Estimate the stability of unknown compound

• Evaluate the bonding (ionic vs. covalent)

• Crystal field stabilization energies of transition metals

Stability of compound: ionization potentials, lattice energies and the relative stability of elements in different oxidation states

e.g. MgCl vs. MgCl2

Uses of Born-Haber Cycle

Covalent Bonding

Partial Covalent Bond

The outer electronic charge density on an anion is polarized towards and by neighboring cation. There is some overlapping of electron density.

For example:

A transition from ionic to covalent bond as electronegativity difference between two elements decrease

AlF3 > AlCl3 > AlBr3 > AlI3

How to quantify the degree of covalent character?

Sanderson’s Method:

Regards all bonds in non-molecular structures as polar covalent. The value of partial charges on an atom can be estimated using ascale of electronegativity. The charge on the atom is ±1 in pure ionic structure.

The positive charge that would be felt by a foreign electron on arriving at the periphery of the atom.

Valence electron are not effective in screening/shielding the outside world from the positive charge on the nucleus.

Effective Nuclear Charge

Ionization potential, electron affinity and electronegativity increase, atomic radii decrease

Cation: smaller radii (nuclear charge unchange, valence electrons are removed)

Anion: larger radii

r = rc - Bδ

Atomic Radii

constant

Partial charge

Non-polar covalent bond

• Ionic radii is controversial

• Non-polar covalent bond can be measured accurately

The electronegativity of an atom is a measure of the net attractive force experienced by an outermost electron towards nucleus

S = D/Da

Principle of electronegativity equalization:

Sb = (S1S2)1/2

Electronegativity

Electron density of the atom (atomic number/atomic volume)

Electron density that would be expected for the atom by linear interpolation of the D values for the inert gas elements

When two or more atom initially different in electronegativity combine chemically, they adjust to have the same intermediate electronegativitywithin the compound

For example, NaF Sb = (SNaSF)1/2

= (0.7×5.75)1/2

= 2.006

Partial charge δ is the ratio of change in electronegativity undergone by an atom on bond formation to the change it would undergone onbecoming completely ionic with charge + or -1

δ = ΔS/ΔSc

Determination of Partial Charge and Atomic Radii

1. ΔS = S - Sb

2. ΔSc = 2.08(S)1/2

Example BaI2: SBa = 0.78 SI = 3.84

Intermediate electronegativity = Sb = (SBaSI2)1/3 = 2.26

ΔSBa = 2.26-0.78 = 1.48; ΔSI = 3.84-2.26 = 1.58

ΔSc = 1.93 (Ba); 4.08 (I)

δ = ΔS/ΔSc = 0.78 (Ba); -0.39 (I)

rBa = rc – Bδ = 1.70 Å; rI = rc – Bδ = 1.87 Å

Ba-I distance = 1.70 + 1.87 = 3.57 Å

Mooser-Pearson Plots is good for predicting the structure based on covalent character of bonds

•Quantum number n

•Difference in electronegativity

Four structural categories:

Zinc blende, wurtzite, rock salt and CsCl

Bonding Structure

Mooser-Pearson Plots

Ionicity (fi): ionic character in bonds = C2/Eg2

Optical absorption spectra of AB compounds Eg (band gap)

Eg2 = Eh

2 + C2

(C = 0 in homopolar covalent bond, and C = Eg in pure ionic bond)

Philips-Van Vechten Ionicity

For isoelectronic series of compounds, the bandgaps have contributions from: (a) homopolar bandgap, Eh and (b) a charge transfer C

Δx (Mooser-Pearson) ~ C (Philips)

n (Mooser-Pearson) ~ Eh (Philips)

A theoretical support for Mooser-Pearson Plots

Valence Bond Theory

1. Electrostatic bond strength (ebs)

Valence of a atom (Vi) = sum of bond valence between atom and neighboring atom (∑bvij)

Covalent bond

Valence bond theory is developed for molecular materials

Ionic bond

Valence sum rule:

bv inversely correlate with bond length

• To check on the correctness of a proposed structure• To locate hydrogen atom in X-ray crystallography studies• To distinguish isoelectronic atom Al3+ and Si4+ position in aluminosilicate structure

2. Formal charge of anion

Bond valence (bv)

Valence of the atom

http://wwwchem.uwimona.edu.jm:1104/courses/CFT.html

Non-bonding Electron EffectsIn transition metal compounds, the majority of the d electrons on the metal atom do not usually take part in bond formation but doinfluence the coordination environment

Crystal Field Theory

High Spin vs. Low Spin State

The magnitude Δ depends on ligand (strong vs. weak field) and also the metal (which row, 5d > 4d > 3d). It is set equal to 10Dq.

Crystal field stabilizing energies (CFSE), leads to increase lattice energy. d0, d5 and d10 do not exhibit CFSE.

Low spin, Δ > P High spin, Δ < P

Radii in Octahedral Coordination

• d0, d5 and d10: spherically symmetrical

• Electrons in t2g orbital do not shield the bonding electron from this extra charge (geometry), while electron in Eg do shielding the nuclear charge.

• For those ions with not equally occupied Eg orbital, e.g. d9, d7

(LS) and d4 (HS).

• The Eg orbital is no longer degenerate since metal ion is not a free ion but is octahedrally coordinated.

• Doubly occupied Eg orbital experience stronger repulsion will lengthen the metal-ligand bond that leads to a lowering of energy.

Jahn-Teller Distortions

Other Coordination ModesTetrahedral

• None of the d-orbitals point directly toward ligands

• dxy, dxz, dyz are closer to the ligand than other two orbitals

Other coordination structures

e.g. Cr3+ (d3) Octahedral coordination: 1.2 Δoct

Tetrahedral coordination: 0.8 Δtet

The values of CFSE may be used to predict site preferences.

Coordination preference: Spinel structures (AB2O4)

• Normal (A-tetrahedral, B-octahedral; A = M2+, B = M3+)

• Inverse (A-octahedral, B-tetrahedral and octahedral; A = M4+, B = M2+)

• Intermediate between normal and inverse

Octahedral vs. Tetrahedral

For post transition elements

e.g. Pb2+ (4f145d106s2)

Ions are repelled by the lone bond lengthen

Lone-pair Effect

Metallic Bonding

• Metal has delocalized valence electrons

• Metallic bonding is the bonding between atoms within metals. Itinvolves the delocalized sharing of free electrons among a lattice of metal atoms

Chemical approach: molecular orbital theory

Overlap of MOs, delocalized over both atoms

Formation of energy level (delocalized over all atoms in crystal)

Metallic Bonding and Band Theory

For example, Na (1s22s22p63s1):

3s and 3p bands are overlapped at interatomic distance r0, valence electron 3s1 is not confined in 3s band.

Band Structure

Free electron theory

Density of states N(E)

• Fermi level: the highest filled level at absolute zero • Work function: the energy required to remove the uppermost valence electrons from the potential wall

Physical Approach to Band Theory

Refined Band TheoryPotential inside the crystal is regarded as periodic

So the uninterrupted continuum of energy levels does not occur, only certain bands of energies are permitted for electrons

In some materials, band overlap occurs, in others, a forbidden gap exists.

Band Structure of MetalsValence band is part full, electrons in singly occupied states close to Ef are able to move

Overlapping of s and p valence bands

Band structure of insulators

• Valence band is full

• Large forbidden gap from next energy gap, e.g. 6eV

• Very small amount of electron can be promoted to empty conduction band

Band structure of semiconductors

• Valence band is full

• Small forbidden gap from next energy gap, e.g. 0.5 – 3.0eV

• (intrinsic semiconductors) number of electrons in the conduction band is governed by magnitude of bandgap and temperature

Band Structure of Insulators and Semiconductors

Band Structure of Inorganic Structures

1. III-V, II-VI and I-VII compounds

Na+: 1s22s22p6

Cl-: 1s22s22p63s23p6

e.g. NaCl

Empty cation orbitals will form conduction band, and full anion orbitals will form valence band

Promotion of a electron from valence band to conduction band is may be regarded as back transfer of charge from Cl- to Na+. Therefore, the magnitude of the bandgap is correlated to the difference in electronegativity between anion and cation.

Eg2 = Eh

2 + C2

Homopolar bandgap

Ionic energy

2. Transition metal compounds

In some case, partly filled metal d-orbitals could overlap to form d bands

a. The formal charge on the cation is small

b. The cation occurs early in the transition series

c. The cation is in the second or third transition series

d. The anion is reasonably electropositive

General guidelines:

Also related to crystal structure:

Examples please refer to book p. 120

For example: Fe3O4 (inverse spinel) vs. Mn3O4 (normal spinel)

[Fe3+]tet[Fe2+, Fe3+]octO4 [Mn2+]tet[Mn3+2]octO4

Band Structure of Inorganic Structures

3. Fullerenes and graphite

Overlap of π−π orbitals, delocalization of π electrons

Graphite:

• infinite layers of benzene molecules

• π−π* orbitals overlap by about 0.04eV in the 3D graphite structure

Band Structure of Inorganic Structures

• Narrower valence and conduction band

• Bandgap is 2.6eV

• C60 able to form a wide range of intercalation compounds in which C60 act as an electron acceptor

Fullerene (C60)