Molecules to solid state materials
Today:
Bonding in molecules
Quantum theory of solids
FINAL EXAM is Monday, Dec. 15 10:30A-1P HERE
Duane G1B20.
EXTRA CREDIT HWK 14: Practice questions for the
Final Exam. Available on the website. Due Mon. 10AM
Please fill out the online participation survey. Worth
10points on HWK 13.
Electronic structure of atom determines its form
(metal, semi-metal, non-metal):
- related to electrons in outermost shell
- how these atoms bond to each other
Semiconductors
Chemical Bonding
- Main ideas:
1. involves outermost electrons and their wave
functions
2. interference of wave functions
(one wave function from each atom) that produces
situation where atoms want to stick together.
3. degree of sharing of an electron across 2 or more
atoms determines the type of bond
Degree of sharing of electron
Ionic or Inert
electron completely
transferred from one atom to
the other, or not at all.
Li+ F- or Helium
Covalent
electron equally shared
between adjacent atoms
H2
Metallic
electron shared between
all atoms
in solid
Solid Copper
Ionic Bond (NaCl)
Na (outer shell 3s1)
Has one weakly bound
electron
Low ionization energy
Cl (outer shell 3s23p5)
Needs one electron to fill
shell
Strong electron affinity
Na+ ClV(r)
Attracted by coulomb
attraction
Separation
of ions
Energy
Na+ Cl-
Repulsion of electrons
Cl-
Na+
Coulomb attraction
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
Protons far apart …
1
Wave function if electron
bound to proton 1
Proton 1
Potential energy curve
Proton 2
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
Protons far apart …
1
Wave function if electron
bound to proton 1
Proton 1
Proton 2
2
Wave function if electron
bound to proton 2
Proton 1
Proton 2
Covalent Bond
Sharing of an electron… look at example H2+
(2 protons (H nuclei), 1 electron)
If 1 and 2 are both valid solutions,
then any combination is also valid solution.
+ = 1 + 2
1
(molecular orbitals)
2
- = 1-2
-2
Add solutions
(symmetric):
+ = 1 + 2 and
Subtract solutions
(antisymmetric):
- = 1-2
Look at what happens to these wave functions as you
bring the protons closer…
Visualize how electron cloud is distributed… for
which wave function would this cloud distribution
tend to keep protons together? (bind atoms?) …
what is your reasoning?
a. S or +
b. A or -
Look at what happens to these wave functions as you
bring the protons closer…
+ puts electron density
between protons .. glues
together protons.
Bonding Orbital
- … no electron density
between protons … protons
repel (less / not stable)
Antibonding Orbital
+ = 1 + 2
1
2
(molecular orbitals)
Smaller proton-proton
repulsion.
Smaller electron KE.
- = 1-2
-2
Larger proton-proton
repulsion.
Larger electron KE.
Energy (molecule)
V(r)
Energy of - as distance decreases
Separation of protons
Energy of + as distance decreases
(more of electron cloud between them)
Quantum Bound State Sim
Now FIX the protons: what does the electron energy look like
What would you expect for
two square wells?
“Degenerate”
energy levels
Energy levels
split apart
For two atoms?
If protons far away, symmetric and antisymmetric state both have same energy as
ground state of electron bound to single proton:
Eatom
As protons get closer together, symmetric and antisymmetric state become more
distinct and energy levels split:
Eatom + 
Eatom – 
As separation decreases, energy
splitting  increases
Same idea with p-orbital bonding … need constructive
interference of wave functions between 2 nuclei.
Sign of wave function matters!
Determines how wave functions interfere.
Why doesn’t He-He molecule form?
Not exact same molecular orbitals as H2+, but similar.
With He2, have 4 electrons …
fill both bonding and anti-bonding orbitals. Not stable.
So doesn’t form.
Big Picture.
Now almost infinite power!
Know how to predict everything about behavior of atoms and electrons or
anything made out of them:
1. Write down all contributions to potential energy,
includes e-e, nuc.-nuc., nuc.-e for all electrons and nuclei.
q1q2/r1-2 + q2q3/r1-3 + qnuc1qnuc2/rqnuc1-qnuc2 +q1qnuc1/r1-nuc1 +
one spin up and one down electron per state req....
(plus little terms involving spin, magnetism, applied voltage)
2. Plug potential energy into Schrod. eq., add boundary. cond.
3. Solve for wave function elec1,(r1, r2, rnuc1, ...)
elec2,
nuc1,
nuc2, ...
get energy levels
for system
calculate/predict everything there is to know!!
almost
why "almost"...one little problem...
Limitations of Schrodinger
• With three objects (1 nuclei + 2 electrons) solving
eq. very hard.
• Gets much harder with each increment in number of
electrons and nuclei !!
Don’t need to always solve S. E. exactly-Use various models and approximations.
Not perfect but very useful, tell a lot.
(lots of room for cleverness, creativity, intuition)
How does atom-atom interaction lead to band structure?
1. Energy levels and spacings in atoms  molecules  solids
2. How energy levels determine how electrons move.
Insulators, conductors, semiconductors.
3. Using this physics for nifty stuff like copying machines,
diodes and transistors (all electronics), light-emitting diodes.
What happens to energy levels as we
put a bunch of atoms together?
Quantum Bound State Sim
Now FIX the protons: what does the electron energy look like
What would you expect
for two square wells?
For two atoms?
Bound State Sim.. Many Wells
In solid, `1022 atoms/cm3, many!! electrons, and levels
countless levels smeared together, individual levels
indistinguishable. "bands" of levels. Each level filled with 2
electrons until run out.
empty
empty
“conduction
band”
Energy
“band gap” ~ few eV
3
filled with electrons
2
filled with electrons
1
bands
atom level
more atoms
“valence
band”
Which band structure goes with
which material?
1. Diamond
2. copper
empty
full
3. germanium (poor conductor)
a. 1=w, 2=x, 3=y b. 1=z, 2=w, 3=y c. 1=z, 2=y, 3=x
d. 1=y, 2= w, 3=y. e. 1=w, 2=x, 3=y
Energy
25 eV
element w
x
0
only top 2 filled and lowest 2 empty bands shown
y
z
And so much more…
• Quantum 1, 2, and electives!
• Quantum theory of statistical mechanics!
• Relativistic quantum field theory!
Thanks for a great semester!
Lecture ended here.
QM of electrical conduction
energy levels of atoms  molecules  solids
Energy
top energy wave functions spread waaaay out
at 2
at 1
many levels!
at 3
at 4
QM of electrical conduction
multielectron atoms
energy levels of atoms  molecules  solids
Energy
inner electrons stick close
to nuclei. Outer e’s get
shared.
at 1
at1-at2 molec
at 2
Quantum Mechanics to understand (predict, control, etc.)
flow of electricity through materials.
The foundation of modern technology
insulators, conductors,
QM  control current flow in semiconductors
results: transistors, cell phones, iPods,…
Where to start in understanding flow of electrons in object
at QM level?
V
V
What is important for flow of current from QM perspective?
a. electrons move through material as classical particles, so QM effects are only a
minor effect.
b. spacing of electron energy levels is important because big spacing between levels
means electrons can move easily.
c. spacing of electron energy levels is important because small spacing between
levels means electrons can move easily.
d. QM is important because the shape of the wave function determines the direction
in which electron can move.
small
to what?
e. some other
QMcompared
effect
from class 20 months ago : )
Nanotechnology: how small does a wire have to be
before movement of electrons starts to depend on size
and shape due to quantum effects?
How to start?
Need to look at
Energy level spacing compared to thermal energy, kT.
Almost always focus on energies in QM.
Electrons, atoms, etc. hopping around with random energy kT.
Larger than spacing, spacing irrelevant. Smaller, spacing big deal.
So need to calculate energy levels.
pit depth compared
to kT?
+ = 1 + 2
1
2
(molecular orbitals)
- = 1-2
-2
V(r)
Energy
Energy of - as distance decreases
Separation of protons
Energy of + as distance decreases
(more of electron cloud between them)
V = -ke2/r
Potential energy of electron due to single proton:
(r) ~ e-r
Ground state wave function of
electron in this potential:
Eatom
Potential energy of electron due to two protons:
+
=
Ground state wave function of electron (symmetric/bonding):
+
=
1st excited state wave function (antisymmetric/antibonding):
+
=
For every energy level for 1 proton, 2 energy levels for 2 protons.
Look at what happens to these wave functions as bring protons closer…
+ puts electron density between protons ..  … no electron density between protons
glues together protons.
… protons repel (not stable)
Bonding Orbital
Antibonding Orbital