Electronic Structure of Solids

In any solid, the cohesive forces that hold the solid together are due to the the Coulombic attractions between the electrons and the nuclei

They are manifested in different types of solids as:

Chemical BondsIonic

Metallic

Covalent Network

London dispersion forces, dipole-dipole interactions, hydrogen bondsMolecular

Electronic Structure of Molecular Crystals

Held together by intermolecular forces such as London Dispersion, dipole-dipole, and hydrogen bonds

Electronic Structure of Ionic CrystalsAs with molecular crystals, electronic wavefunction of ionic

crystal is approximated as a product of the wavefunctions of the ions which are

essentially the same as the separated ions

Electronic Structure of Covalent and Metallic CrystalsCrystals of Covalent Network and Metallic solids can be

thought of as Giant Molecules

Both localized and delocalized bonding is possible.

Use band theory to describe electronic structure in covalent Network and Metallic Crystals

Band Theories

Central Aspect of Solids that determines their electrical properties is

the distribution of their electrons.

There are two models for their distribution:

1) Nearly Free Electron Approximation

Valence electrons are assumed trapped in a box with a periodic potential

And

2) The Tight Binding Approximation

the valence electrons are assumed to occupy molecular orbitals delocalized throughout the solid

The Tight-binding Approximationwill be addressed first

In order to do this revisit basic ideas of molecular orbitals

Molecular Orbitals for Diatomic Molecules

These orbitals are formed from linear combinations of the atomic orbitals. They differ from the atomic orbitals in that they are new

orbitals that spread out over the molecule

Can we solve Schrodinger’s eqn analytically for a diatomic many electron molecule?

H

No!

Molecular Orbitals

One electron wave functions obtained from Schrodinger eqn.

- Molecular Orbitals, M.O.s2 gives distribution of the electron in the molecule as a whole

WAYS TO GET MOsH2

+ - could solve the Schrodinger eqn. analytically.

Another way - use approximate orbitals. Make from a linear combination of Atomic Orbitals

Consider H2+

If Electron is:1. Close to H(A) - MO resembles an A.O.! ....... 1s A.O. on H(A).

Schrodinger is the same as that for an isolated H atom.

Lowest energy solution is H 1s orbital 1s(A)

2. Close to B - MO resembles an A.O. ………………….on B.

Solution to Schrodinger is H 1s orbital 1s(B)

Thus an Approximate Overall Molecular Wavefunction is:1 = N1 {1s(A) + 1s(B)} ...... or .......... 2 = N2 {1s(A) -

1s(B)}N1, N2 are defined as the normalization factors!

Approximate Molecular Orbitals formed from Linear Combination of Atomic Orbitals is called

an LCAO-MO.

General Rule:

# of LCAO-MO’s Formed is equal to the number of

AOs used in the linear combination

Could Normalize the LCAO MO’s - Normalize 21 =

Since 1s is normalized and REAL, 1st two terms are:

Define S as the overlap integral. It tells if (A) and (B) are simultaneously large in some region of the defined space.

In the limit that (A) = (B), then

1 =

(A) identical with (B)For two hydrogenic 1s orbital on nuclei separated by a distance R,

OVERLAP is given by:S = {1 + ZR/a0 + 1/3 (ZR/a0)2 exp(-ZR/a0) ...

a0 defined as the Bohr radius = ................................... = 52.9 pmZ = nuclear charge

S = 0.59 for two 1s orbitals at equilibrium bond length in H2+.

SoN2 = 1.10 for 2 LCAO-MO wave function in the case of H2

+

S, the overlap integral also arises in the expression for the probability density of the electron in H2

+

* = ...............................................................

Look at the individual terms:a) 1s(A)2 = probability density if electron is confined to an orbital on atom A.

b) 1s(B)2 = ………………………………………………………………………..

c) 2 1s(A)2s(B) is extra contribution to the density overlap…………………

For one LCAO-MO of H2+ - = N1 {1s(A) + 1s(B)}

see above ................

Bonding MO - the overlap is constructive since there is extra probability that the electron is between the nuclei

For other LCAO-MO of H2+ = N2 {1s(A) - 1s(B)}

Overlap is destructive - get node place where the electron probability is zero between the two nuclei - Antibonding MO.

* = ...............................................................

Relative to separated H atoms - “reduces probability”

Electron in antibond destabilizes molecule

1) Electron is excluded from internuclear region so distributed outside bonding region.

2) In a sense electron pulls nuclei apart!

Structures of DIATOMICS USING MO Theory

Use Aufbau Principle as with many electron atoms - built up Atomic Electronic Configuration From Hydrogen Atom orbitals

COULD!Similarly - use the H2

+ molecular orbitals to build up molecular orbitals - Apply Hunds Rule in the case of degenerate molecular orbitals.

BUTProcedure to Make Molecular MO

“Get the MO’s from A.O.s on atoms bonding:1) Homonuclear Diatomics Each Atom has:

a) Core Electrons - inner closed shellb) Valence Electrons - orbitals of the valence shell

c) Virtual orbitals - orbitals of the atom that are unoccupied in grnd state.

Valence shell electrons most important in bonding - build MOs from these

A) Overall - All orbitals of the appropriate symmetry contribute to a molecular orbital.

( made form s and pz orbitals) ( from px,py)

B) Orbitals with similar symmetry and similar energy may combine to form MO s.

C) Ordering of MO s is: 1) .........

* indicates antibonding MO. (note type orbitals alternate bonding, antibonding, bonding .... etc.

For diatomics heavier than N2, the order becomes 2)

Consider H2 and He2 (M.O. Energy level Diagram)Energy Level of Separated Atoms

Note in case of H2 - both electrons go into M.O. which is stable relative to the separated atoms. (bonding orbital 1s)

In He2 - two electrons are in bonding and two are in an antibonding orbital

Energy of antibonding orbitals is more positive than the bonding orbital is negative ........ so the atoms separate.

Consider C2

orbitals are a result of the fact that the 2px and 2py orbitals of each atom (which are perpendicular to the internuclear axis) overlap broadside.

Thus there is off-axis electron overlap giving rise to angular momentum.

Paramagnetic vs Diamagnetic - If there are unpaired electrons then the molecule is said to be paramagnetic

If all spins are paired the molecule is diamagnetic!

Bond OrderMeasures the net bonding in a diatomic molecule.

Defined by:b = ½ (n-n*) where

n = # of electrons in bonding orbitaln* = # of electrons in .................

b = ? .............. for H

......................... for He2?..................................................... for C2, C2

+, or C2-

Molecular Electronic Configuration (like Atomic Electronic Configuration,but lists the MO for valence electrons)

H2

He2

C2

N2

Give the Molecular Orbital Energy Diagram, the bond order, and determine whetherthe molecule is paramagnetic or diamagnetic.

N2

F2

Heteronuclear Diatomics

HF

Another Approximation Technique of Quantum Mechanics

Variation Principle

Given any Approximate wavefunction satisfying the boundary conditions of a problem, the expectation value of the energy calculated from this

function will always be HIGHER than the true energy of the ground state.

Suggest Procedure For Solving Quantum Problems:

1) Guess several functions - trial functions or have one in mind.

2) Calculate expectation value for each one.

3) Choose the one with the lowest energy and conclude it is the bestfunction that can be obtained from the original guesses

Variation Principle

In practice - start with the trial wavefunction containing arbitrary parameters and minimize the expectation energy with respect to these parameters!

Present application is the building Molecular Orbitals from atomic orbitals:LCAO-MO method.

Trial Function = u cu u

where u are the appropriate atomic orbitals and cu are the parameters to be chosen to minimize the energy.

Use Variation Principle to find best and Energies for H2+ molecule

Huckel Theory: MOs for Polyatomic Systems

Have backbone made out of sigma MO. Consider only conjugated pi orbitals

Make some assumptions about the integrals.Assumptions include:

1) All overlap integrals are set to zero. 2) All resonance integrals between non-neighbors are set equal to zero.

3) All remaining resonance integrals are set equal to

Approximations are very severe but allow calculation of a general picture of the molecular orbital electronic energy levels.

Assumptions result in simplifications to the secular determinant as follows:

1) All diagonal elements become:2) Off-diagonal elements become:3) All other elements become:

Apply to Ethylene: Butadiene Benzene

Now apply the LCAO-MO picture to covalent network and metallic crystals – Tight Binding Approximation.

Consider 1D solid consisting of single infinitely long line of atoms - too simple?

Assumptions:Each atom has one s orbital to form MOConstruct LCAO-MOs by adding N atoms in succession in a line

Results: First atom supplies an s orbital at a certain energy2nd atom brought up overlaps with 1st atom and

forms bonding and antibonding orbitals3rd atom added overlaps it nearest neighbor (and only slightly its next nearest

neighbor andthree molecular orbitals result, one bonding, one antibonding and one in between that is nonbonding

4th atom added leads to the formation of a fourth MO

Successive atoms added essentially just: spread the range of energies covered by the molecular orbitals and fill in the range of energies with more and more orbitals (one more for each atom)

When N atoms have been added to the line there are now:N molecular orbitals covering a band of energies of finite width

and the Huckel secular determinant is:

where is the (s,s) resonance integral.

Evaluation of the secular determinant gives the expression for the roots ER = + 2cos(k/(N+1)) k = 1, 2, … N

When N is infinitely large: the difference between neighboring energy levels is infinitely

small the band has a finite width: EN – E1 = 4 as N ->infinity

The band consists of N different MOs the lowest energy one being fully bonding, and the highest energy orbital (k=N) being fully antibonding between adjacent atoms

The Band Formed from s orbital overlap is: s bandIf the atoms have p orbitals available, then this leads to a

p band

If the atomic p orbitals lie higher in energy than the s orbitals, the the p band lies higher in energy than the s band and there may be a band gap – a range of energies to which no orbital corresponds.

However, in some cases the s and p bands may also be contiguous or even overlap.

There are N atomic orbitals and N molecular orbitals packed into an apparently continuous band. There are N electrons to accommodate.

For T = 0K, only the lowest ½ N MOs will be occupied.

Here the HOMO is called the Fermi Level.

Unlike in molecules, the empty orbitals can be very close in energy to the Fermi level, so:

in this case the empty orbitals are very close in energy to the Fermi level, so it requires hardly any energy to excite the uppermost electrons.

So some of the electrons are very mobile and give rise to electrical conductivity

At Temps. Above absolute zero, electrons can be excited by the thermal motion of the atoms. The population, P, of the orbitals is given by the Fermi-Dirac Distribution

The shape of the Fermi-Dirac distribution is shown

For energies well above the denominator can be neglected and

P = exp(-(E- )/kT)Which is a Boltzmann-like distribution and decays

exponentially with increasing energy. The higher the temp., the longer the exponential tail.

T

Insulators and SemiconductorsEach atom provides two electrons

The 2N electrons fill the N orbitals of the s band, so that the Fermi level lies at the top of the band. (at T = 0K)

As the temp. is increased, the tail of the Fermi-Dirac distribution extends across the band gap and the electrons leave the valence band and populate the empty orbitals of the upper band

This upper band is called, the conduction band and holes are left behind.

The holes and promoted electrons are mobile, and the solid becomes conducting. So this material is a semiconductor.

If the gap is very large, then the distribution no longer extends across the gap and so the material is: an insulator