Homework 7
(due Friday, November, 13)
There is a 20% bonus for handing the homework on time and a penalty of 10% per day
for handing the homework in late.
This assignment is an attempt to cover the quantum mechanics of covalent bonds at the
level of a first year graduate course in quantum mechanics. Usually, this material is treated
either at the elementary level or at a very advanced level where one is trying to get the
energies accurately using sophisticated computer codes. Here, I am trying to do something
in between.
1. sp2 hybridization: The n = 2 orbital levels of the hydrogen atom are degenerate and
may be written as
1
(1)
ψ0 = R20 (r) √
4π
s
3 x
(2)
ψ1 = R21 (r)
4π r
s
ψ2 = R21 (r)
s
ψ3 = R21 (r)
3 y
4π r
(3)
3 z
,
4π r
(4)
where
1 1
r
R20 (r) = √ 3/2 1 −
e−r/2a
(5)
2a
2a
1 1 r −r/2a
R21 (r) = √
e
.
(6)
24 a3/2 a
Since the states are degenerate, any linear combination of the states has the same
energy. Create linear combinations of ψ0 , ψ1 , and ψ2 to get three new orthonormal
√
3
1
x̂
+
ŷ,
states such that
the
expectation
value
of
~
r
for
the
three
states
is
in
the
x̂,
−
2
2
√
1
3
and − 2 x̂ − 2 ŷ directions. This is called sp2 hybrization. Make a contour plot of the
resulting wave functions in the x-y plane.
2. Bonus: sp3 hybridization: Make linear combinations of ψ0 , ψ1 , ψ2 , and ψ3 to create
a new orthonormal basis for which the expectation values of ~r lie on the vertices of a
tetrahedron.
3. Covalent bond: In Chapter 10 we covered the tight binding model where one had
localized atomic levels that were not the full eigenstates of the system once the atoms
were put in a lattice. Consider two atoms, A and B. Let ψa be an atomic eigenstate of
atom A, and φb be an atomic eigenstate of atom B.
h̄2 2
a ψa (r) = −
∇ ψa (r) + UA (r)ψb (r)
2m
h̄2 2
b ψb (r) = −
∇ ψb (r) + UB (r)ψa (r)
2m
(7)
(8)
Now we wish to find a single particle solution when the two atoms are a distance R
apart:
h̄2 2
∇ ψ(r) + UA (r)ψ(r) + UB (r − R)ψ(r).
(9)
ψ(r) = −
2m
Both ψa (r) and ψb (r − R) are approximate solutions when R is large. Do a variational
calculation with a wave function of the form ca ψa (r) + cb ψb (r − R) also taking a = b .
What is the energy and by how much is it lower than a ? This is the major contribution
to the bonding energy. Why do we say that the electron is shared between atoms A
and B?
4. The cohesive energy for diamond is 7.37 eV per atom. Assuming that this energy is
due to the covalent bonds, what would you estimate as the energy of a single CarbonCarbon covalent bond? Convert this to kJoule/mole. A typical C-C bond energy is
347 kJ/mol. How does your answer compare to this?