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 Bonds
Ionic
Metallic
Covalent Network
London dispersion forces, dipole-dipole interactions, hydrogen bonds
Molecular
Electronic Structure of Molecular Crystals
Held together by intermolecular forces such as London Dispersion,
dipole-dipole, and hydrogen bonds
Electronic Structure of Ionic Crystals
As 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 Crystals
Crystals 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 Approximation
will 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?
Hy=Ey
No!
Molecular Orbitals
One electron wave functions obtained from Schrodinger eqn.
- Molecular Orbitals, M.O.s
Y2 gives distribution of the electron in the molecule as a whole
H2+
WAYS TO GET MOs
- 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 y1s(A)
2. Close to B - MO resembles an A.O. ………………….on B.
Solution to Schrodinger is H 1s orbital y1s(B)
Thus an Approximate Overall Molecular Wavefunction is:
Y1 = N1 {y1s(A) + y1s(B)} ...... or .......... Y2 = N2 {y1s(A) - y1s(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 y2
1=
Since y1s is normalized and REAL, 1st two terms are:
Define S as the overlap integral. It tells if y(A) and y(B) are
simultaneously large in some region of the defined space.
In the limit that y(A) = y(B), then
1=
y(A) identical with y(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 pm
Z = nuclear charge
S = 0.59 for two 1s orbitals at equilibrium bond length in H2+.
So
N2 = 1.10 for y2 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+
y*y = ...............................................................
Look at the individual terms:
a) y1s(A)2 = probability density if electron is confined to an orbital on atom A.
b) y1s(B)2 = ………………………………………………………………………..
c) 2 y1s(A)y2s(B) is extra contribution to the density overlap…………………
For one LCAO-MO of H2+ - y = N1 {y1s(A) + y1s(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+ y = N2 {y1s(A) - y1s(B)}
Overlap is destructive - get node place where the electron probability is zero
between the two nuclei - Antibonding MO.
y*y = ...............................................................
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.
BUT
Procedure 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 shell
b) 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.
(s made form s and pz orbitals)
(p from px,py)
B) Orbitals with similar symmetry and similar energy may combine to form MO s.
C) Ordering of MO s is:
1) 1s, 2s*, 1p, 3s, 2p*, 4s* .........
* indicates antibonding MO. (note s type orbitals
alternate bonding, antibonding, bonding .... etc.
For diatomics heavier than N2, the order becomes
2) 1s, 2s*, 3s, 1p, 2p*, 4s*......................
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
p 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 Order
Measures the net bonding in a diatomic molecule.
Defined by:
b = ½ (n-n*) where
n = # of electrons in bonding orbital
n* = # of electrons in .................
b = ? .............. for H2
......................... for He2?
..................................................... for C2, C2+, or C2Molecular 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 best
function 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 y = Su cu fu
where fu are the appropriate atomic orbitals and
cu are the parameters to be chosen to minimize the energy.
Use Variation Principle to find y 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 b.
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 MO
Construct LCAO-MOs by adding N atoms in succession in a line
Results:
First atom supplies an s orbital at a certain energy
2nd atom brought up overlaps with 1st atom and
forms bonding and antibonding orbitals
3rd atom added overlaps it nearest neighbor (and only slightly its next nearest
neighbor and
three 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 b is the (s,s) resonance integral.
Evaluation of the secular determinant gives the expression for the roots
ER = a + 2bcos(kp/(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 = 4b 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 band
If 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 m the denominator can be
neglected and
P = exp(-(E- m)/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 Semiconductors
Each 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