Bonding in solids

� Many different types of interaction are important– electrostatic (ionic)– covalent– Van de Waals

� Ionic bonding favors high symmetry structures with high coordination numbers

� Covalent bonding favors low coordination numbers

Bonding solids overview� Ionic model

– Ionic radii– Electrostatic bond strength– Radius ratio rules– Lattice energies

» Born-Harber cycles� More realistic approaches

– Mooser-Pearson plots– Bond valence bond strength correlations– Crystal Field Stabilization Energy (CFSE)– Inert pair effect

� Band theory

Ionic model� In its simplest form it treats ions as hard spheres of

well defined size– More sophisticated treatments allow that the ions are not

hard spheres and that they do not have a precisely defined size

� In reality, ions such as Al3+ and O2- do not have +3 and –2 charges in the solid state

� While the ionic model is not a very realistic picture it is simple and it can provide a useful guide to structural and thermodynamic trends

Ionic radii

� The distances between nearest neighbor anions and cations can be considered to be the sum of an anion and cation radius

� How do we determine the radii?– many different methods used– preferred method uses a crystallographic determination of

electron density

� Different methods give different answers– never mix values of radii from different sources

The experimental electron density distribution in LiF

Electron density variation between Li+ and F-. Note the variation has a very flat bottom. M, G and P indicate the true minimum and the Goldshmidt and Pauling ionic radii.

The variation of ionic size with coordination number

From Shannon and Prewitt, Acta Cryt. B25, 725 (1969) andB26, 1046 (1970). Data based on rF- = 1.19Å and rO2- = 1.26Å

Trends in ionic radii� Ionic radii increase going down a group� Ionic radii decrease with increasing charge for any isoelectronic

series of ions• Na+, Mg2+, Al3+, Si4+

� Ionic radii increase with increasing coordination number � Ionic radii decrease with increasing oxidation state� For lanthanide 3+ ions with a given coordination number there is

a steady decrease in size on going across the series left to right• Lanthanide contraction

• A similar decrease in size is seen on going across transition metals series but it is not always a smooth decrease

General principles of ionic bonding� Ions are charged elastic spheres� Held together by electrostatic forces so cations are surrounded

by anions and vice versa� In order to maximize the attractions, cations are surrounded by

as many anions as possible provided that the cation maintains contact with all the anions

� Next nearest neighbor interactions are repulsive. So ionic structures tend to have a high symmetry and the maximum volume possible– This minimizes the repulsions

� Structures are locally electrically neutral– The valence of an ions is equal to the sum of the electrostatic bond

strengths between it and all of its opposite charge neighbors

Electrostatic bond strength

� For a cation Mm+ surrounded by n anions, Xx-, the e.b.s. is given by– e.b.s. = m / n

� For the anion, the cation e.b.s. must balance the charge on the anion– Σ (m / n) = x (sum over nearest neighbors)

� This rule precludes certain structures– you can never have three SiO4 units sharing a common

corner

Use of ebs� ebs can be used to rationalize why some types of

polyhedral linkage do not occur

What determines a crystal structure?

� We are not currently able to predict the structure of new complex materials with certainty

� There are some tools available to help us make an educated guess for simple materials

� There is a large amount of structural data that can be used as a guide

Structure prediction

� The so called “Radius ratio rules” are often used to make predictions of preferred structure type or to rationalize an observed change in structure type� Just about the simplest approach possible

� Radius ratio rules make use of idea that cation will have as many anions around it as possible as long as the cation can still touch all of the anions

Radius ratio rules

� It is possible to predict the type of ion coordination that you will get if you know the ratio of the cation to anion size

r+/r- values Preferred coordination number

> 0.732 8 – cubic coordination

0.414 – 0.732 6 – octahedral coordination

0.225 – 0.414 4 – tetrahedral coordination

How the limiting values are calculated

Use of radius ratio rules

� Can be used to explain trends

Failure of radius ratio rules

� Not reliable for absolute prediction- Ions are not hard spheres,

ion size varies with coordination number, radius ratio varies depending whose ionic radii you use

Distorted structures

� The radius ratio rules were based on the notion that structures were unstable if the cations could rattle around inside their coordination polyhedra– this is not universally valid

� BaTiO3, PbTiO3, LiNbO3, KTiOPO4 (KTP) etc. have ions that can rattle

Lattice energies and the prediction of structural stability

� It is possible to calculate the thermodynamic stability (∆Hf not ∆Gf) of an ionic solid using relatively simple thermodynamic arguments– As part of this calculation we have to now the

materials lattice energy» Related to forces holding solid together

Lattice energy

� For an ionic compound the lattice energy is defined as the energy needed to break up the solid into its constituent ions in the gas phase– MX(s) -----> M+

(g) + X-(g)

� Determined by a combination of long range electrostatic interactions and short range repulsions

Attractive and repulsive interactions� There are electrostatic interactions between

every pair of ions in the solid– electrostatic energy = -Z1Z2e2/4πε0d

» overall electrostatic interaction energy for an ionic solid is always favorable

� The repulsive interactions are short range in nature– repulsive energy = b/dn

» n is usually quite large ~10

The balance between repulsive and attractive forces

Madelung constant� The exact value of the electrostatic

component of the energy depends upon the crystal structure– For NaCl structure energy of one ion

» PE = -6e2/4πε0d + 12e2/1.41x4πε0d -8e2/1.71x4πε0d + 6e2/2x4πε0d - ...

» PE = -Ae2/4πε0d � A is the Madelung constant and depends upon the crystal

structure

The NaCl structure

Madelung constants

Total lattice energy

� PE = -Ae2Z1Z2/4πε0d + B/dn

� For a crystal at equilibrium the distance between neighboring ions, d0, will be the one that gives the lowest PE

� U0 = NAZ1Z2e2(1 -1/n)/4πε0d0

– n is readily estimated so lattice energies can be easily calculated using simple arguments

The Born - Lande equation

� Total interaction energy between ions– U = - e2 Z+ Z- N A / r + BN / rn

– to get equilibrium value differentiate with respect to r and set dU/dr = 0

– other functional forms of repulsive part are sometimes used

� Lattice energy is– U = - [e2 Z+ Z- NA / re ] (1 - 1/n)– n can be experimentally determined

The Born -Meyer equation

� Born Meyer equation is obtained when the repulsive potential takes the form,

– V = B exp (-r/ ρ)

� Born - Meyer equation

– U = - [e2 Z+ Z- NA / re ] (1 - ρ / re)

The Kapustinskii equation

� Kapustinskii noticed that A / ν, is almost constant for all structures

– ν is the number of ions in the formula unit

� Variation in A / ν with structure is partially canceled by change in ionic radii with coordination number

� U = [1200 ν Z+ Z- / (r+ + r-)][1 - 0.345/(r+ + r-)]

Extended calculations

� Include zero point vibrational motion

� Heat capacity of the solid

� Van der Waals forces

� Total correction ~ 10 kJ mol-1

Thermodynamic data for the alkali metal halides

∆Uc = coulomb term, ∆uB Born repulsive term, ∆ULdd = London dipole-dipole term,∆ULdq = London dipole-quadrupole term ∆UZ = zero point term

Trends in lattice energies� Lattice energies go up as the charge on the ions go up� Lattice energies go up as the size of ions decreases

Effect of covalency�The experimental lattice energies for

compounds that have a significant covalent contribution to their bonding are often in poor agreement with those calculated using the ionic model

The use of lattice energies

� Can be used to estimate heats of formation for compounds– is an unknown compound likely to be stable?– will a compound disproportionate?

� Can be used to estimate electron affinities� Can be used to estimate thermochemical radii� All these applications make use of a Born-Harber

cycle

The formation of ionic compounds� Energies of formation can be calculated by

considering the process of formation to occur in a distinct series of steps

� Consider forming NaCl(s)– atomize the metal– dissociate chlorine molecules– ionize the sodium– form ions from the chlorine atoms– bring the ions together to form solid NaCl

Born-Harber cycles� This step wise approach is often shown diagramatically

The stability of compounds

� Born-Harber cycles along with lattice energy calculation and experimentally measured quantities such as ionization energies allow the calculation of enthalpies of formation for compounds that have never been made

� This allows you to rationalize why some compounds form and others do not

Magnesium fluorides� Why is MgF2 the only stable magnesium fluoride ?

+3430-1040-260∆Hf

-5900-2880-900Lattice energy-990-660-330F electron affinity

+9930+2190

+740Mg ionization+240+160+80F-F bond energy+150+150+150Mg atomization

MgF3MgF2MgFEnthalpy contributions (kJmol-1)

So 2MgF(s) � MgF2(s) + Mg(s) ∆H = -520 kJmol-1

Enthalpy of formation for MgF(s)

�Mg(s) � Mg(g) ∆H atomization�0.5F2(g) � F(g) 0.5 bond enthalpy�Mg(g) � Mg(g)

+ + e- 1st ionization enthalpy�F(g) + e- � F(g)

- electron affinity�Mg(g)

+ + F(g)- � MgF(s) minus lattice energy

�Add these up– Mg(s) + 0.5F2(g) � MgF(s) Enthalpy of formation

Making compounds containing ions with unusually low or high oxidation states

� Compounds containing cations in usually low oxidation states are often unstable with respect to disproporationation– This tendency can be minimized by reducing the lattice

energy of the compound with the cation in the higher oxidation state

» Use large anion

� Compounds containing cations in usually high oxidation states are often unstable with respect to decomposition giving a compound in a lower oxidation state– Maximize lattice energy of high oxidation state compound

» Use small high charge anion

Thermochemical radii

� How do we obtain an ionic radius for an ion such as CO3

2- ?� Measure heat of formation of carbonate compound� Estimate lattice energy using Born-Harber cycle� Calculate ionic radius using Kapustinskii equation

Electron affinities

� We can not measure some of the required electron affinity data directly

� However, we can use a Born-Harber cycle to estimate the electron affinity if we know all of the other terms

2e-(g) + S(g) --------> S2-

(g)

2e-(g) + O(g) --------> O2-

(g)

Prediction of thermal stability

� MCO3(s) -------> MO(s) + CO2(g)

� The decomposition temperature depends upon T= ∆H0 / ∆S0

� ∆H0 can be calculated with the help of some lattice energy data

Solid State Metathesis Reactions can be Very Exothermic

� MoCl5(s) + 5/2Na2S(s) ---> MoS2(s) + 5NaCl(s) + 1/2S(s)

Reaction reaches 1050 ºC and is over in 300 ms

Empirical structure prediction

� Radius ratio rules do not work very well

� Find some other simple way of predicting structure– search database of known compounds looking for

features that allow us to predict structure

– can create stability field diagrams based on:» ion size

» electronegativity and average principle quantum number

Stability field diagram for MX structures

Stability field diagrams for MX2 structures

Stability field diagram for A2BO4 structures

Stability field diagram for AIIIBIIIO3structures (size only)

Stability fields for AIIIBIIIO3 structures (size and ionicity)

Bond valence sum rules

� Pauling’s e.b.s concept was a first step towards associating a bond (cation anion distance) with a valence

� Other workers (particularly Brown) expanded the concept– calculate a valence associated with every cation

anion distance in a structure. – For a particular anion or cation these should

sum to give the formal valence of the species

Bond valence and bond length

� It seems reasonable that bond strength should correlate with bond length

The form of the bond strength bond length relationship

� Two commonly used forms– s = (r / ro)-N

– s = exp [(ro - r)/ B]

� The second functional form is superior as B is roughly the same for all cation anion pairs– only one parameter, ro, for a given anion cation pair

see Brown and Altermat, Acta Cryst. B41, 244 (1985)

Applications of bond-valence bond-length relationships

� Can be used to check new structures

� Can be used to locate missing atoms– good for things like hydrogen that are not easily

located using X-ray techniques

� Can be used to examine site occupancies– aluminosilicates

Non-bonding electron effects� The structures and stability of many transition metal

containing solids are effected by the d-electron configuration of the metal ion– Preference for a particular site geometry due to Crystal

Field Stabilization Energy (CFSE) effects– Distortions due to Jahn-Teller effect

� The structures and properties of many compounds containing heavy post transition metal ions are effects by the presence of a stereochemically active lone pair

Crystal Field Theory� Consider the ligands are point negative

charges or as dipoles. How do these charges interact with the electrons in the d-orbitals?

Octahedral complexes� Two of the d-orbitals point towards the ligands

– Repulsion between the ligand electrons and electrons in these two d-orbitals destabilizes them

Crystal field splitting

� The crystal field splitting depends upon the oxidation state of the metal, which row the metal is from, and the ligand type� High oxidation state favors large ∆� Trend in ∆ is usually 5d > 4d > 3d� Effect of ligand is given by the spectrochemical

series» I-< Br-< Cl-< F-< OH-< OH2< NH3< en< CN-< CO

High spin and low spin complexes� HS versus LS is determined by the relative size of

the ligand field splitting and the pairing energy

� If ∆ is bigger than the pairing energy the complex will be low spin

Tetrahedral complexes� Three of the d-orbitals point almost towards the

ligands. The other two point between the ligands– Repulsion between the ligand electrons and electrons in

the three d-orbitals that almost point at the ligandsdestabilizes them

Square planar complexes

Magnetic properties

� The loss of degeneracy of the d-orbitals due to crystal field splitting explains why some complexes are diamagnetic and others are paramagnetic– e.g. Ni(CN)4

2- (square planar) is diamagnetic – but NiCl42- (tetrahedral) is paramagnetic

Hydration energies� The double humped trend that is seen in the hydration

enthalpies of TM ions can be explained using the Crystal Field Stabilization Energy

CFSE for high spin d4 is= (+3/5 – 2/3 – 2/3 – 2/3)∆

Ionic radii for 3d metals

�For high spin ions there is a “double humped” trend in ionic radii– Due to crystal field

stabilization effects

Lattice energies of 3d oxides MO

�Double humped trend due to CSFE and high spin ions

CFSE and coordination preferences� The CFSE for octahedral and tetrahedral sites is

different and the magnitude of the difference varies with d-electron configuration– Some metal ions show a strong preference for octahedral

coordination due to CFSE effects

Degree of inversion in Spinels� AB2O4 materials with the Spinel structure have one tetrahedral

and two octahedral sites per formula unit� The fraction of the A cations that are found in the octahedral sites

is referred to as the degree of inversion γ– If all A cations are octahedral the material is an inverse Spinel, and if all A

cations are tetrahedral the Spinel is said to be normal– Degree of inversion can often be rationalized using CFSE arguments

Inert pair effect

� Many heavy main group cations that are in an oxidation state two less than that normally displayed by other group members show highly distorted coordination environments– Tl+, Pb2+, Sn2+, Sb3+, Bi3+

– Ions have a lone pair that can be stereochemically active

» Lone pair occupies space around ion just as lone pair on ammonia does Coordination environment

of Pb2+ in PbO

Jahn-Teller effect� Jahn-Teller theorem states that any species with an

electronically degenerate ground state will distort to remove the degeneracy– Compounds containing approximately octahedral Cu2+ (d9-

t2g6eg

3), Mn3+ (d4 - t2g3eg

1) and L.S. Ni3+ (d7 - t2g6eg

1) often display distorted coordination environments as the distortion breaks the degeneracy of the octahedral ground state

Jahn-Teller effect 2� Typical distortion of an octahedron leads to 4 + 2

coordination with either 2 short or 2 long bonds� JT effect important in copper oxide superconductors

and manganese CMR materials� JT effect can also occur for tetrahedrally coordinated

species� JT effect not very strong for “octahedral” compounds

with degenerate ground state involving incomplete occupancy of t2g orbitals

Examples of the Jahn-Teller effect

�CuF2 has a distorted rutile structure�CuO shows almost square planar

coordination of Cu2+

�Cs2CuCl4 shows a flattened tetrahedral coordination of Cu2+ due to JT effect