PHYS 7440
Assignment 6
Due Fri. Feb. 28, 2003
Reading:
1. Marder Chapter 7 and 8.
Comments:
Let's start using the tight binding and nearly free electron pictures to look at a variety
of problems.
Problems:
1. In Assignment 5 (Problem 1) we considered a one dimensional chain in which atoms
paired at low temperatures. In this problem, we use a tight binding model to consider
why such pairing occurs and how it affects properties of the system. The argument
that you will follow is due originally to Rudolph Pierels.
Consider a one-dimensional chain of hydrogen atoms with the possible structures
shown below:
a
Symmetric chain
2a
a-2u
Paired chain
a. Use a nearest neighbor tight binding approach based on the hydrogen 1S-state (do
not evaluate the integrals, just name them) to write down a form for the band
structure for the symmetric chain. Sketch the density of states and show where the
highest occupied level is at T=0. Is this material an insulator or metal?
b. Now consider the paired chain where the atoms move off the symmetric sites by
alternating small amounts, +/-u, to produce a chain of pairs of separation, a-2u.
This introduces a new periodicity to the problem. Sketch the form of the new band
structure. Sketch the associated density of states and the position of the highest
filled level. Based on the change in band structure, argue why such a distortion
might be favored energetically over the symmetric chain. What are the conduction
properties now?
c. This problem has immediate application to real systems, especially long chain
polymers, where dimerization of this type occurs in essentially all cases. Find an
example of a polymer where the molecular structure is known, sketch a picture of
the structure and indicate on your sketch where the dimerizaton occurs. Comment
on the likely conduction properties of most polymeric materials. Comment on any
approaches you can think of to make plastics into metals.
Physics 7440
H6.1
Spring 2003
2. Practice with tight binding: Molecular binding in the H2 molecular ion.
When we study the behavior of electrons in solids, we will be considering an
approximation in which the electrons move in the field of a collection of fixed
ions. We will continue to treat the electrons as non-interacting among themselves,
but will include the periodic electron-ion interaction. The single electron
Hamiltonian for this type of system looks like this:

2
2m


2  V R  r  Vion - ion
R
where the attractive electron-ion potential is written as a sum of the interactions
between the electron at some position, r, and the ions located at positions, R, and
the constant (because they are at fixed positions) energy of ion-ion repulsion is the
third term. To determine the behavior of solids in this approximation, we must
solve for the single electron energy levels in the presence of the ionic potentials,
construct many-electron wavefunctions from products of the single particle states,
and then ask for ground state and finite temperature properties. This is the same
process we used in the Sommerfeld Model, but now the Hamiltonian is tougher.
The singly ionized hydrogen molecule is representative of this type of problem:
There is a single electron that is moving in the attractive potential of two protons.
Assuming that the two protons are at fixed positions, say at +/-R/2 on the z-axis,
the Hamiltonian is:

2
 
2
2m
e2
4 0 r  12 R

e2
4 0 r  12 R

e2
4 0 R
Here, I have placed one nucleus to one side of the origin and the other on the
opposite side of the origin such that the two nuclei are separated by a fixed
distance, R. There are two attractive potential terms and a fixed proton-proton
repulsion. Your job in this problem, is to construct a variational wavefunction and
then calculate the predicted properties of this system, including the ground state
energy, the expected equilibrium nuclear separation, and the molecular binding
potential.
Variational wavefunction.
We will construct a wave function based upon a linear combination of atomic
orbitals. As I have pointed out previously, in the variational technique, you guess
a wavefunction and attempt to be savvy regarding the physics.
In this case, if you consider the behavior for very large nuclear separations, you
expect that the electron is either in the ground state, 1S, orbital around one
Physics 7440
H6.2
Spring 2003
nucleus, or in the 1S orbital around the other. Of course, we know the functional
form of the 1S orbital. For the general nucleus of charge number , Z, it is:
1 Z 
 1S  r  
 
  a0 
32
 Zr 
exp   
 a0 
where the Bohr radius determines the natural length scale. In this problem, we
have two protons, so that at least one of them is off the origin. Therefore, we must
consider a more general form of the 1S state. Here’s a picture of the system:
Z
p+
Y
R/2
r1
X
r
-R/2
p+
e-
r2
In this picture, I have shown the two proton positions, a possible location for the
electron, and three different position vectors for the electronic position: There is
the traditional vector, r, and vectors that measure position from each proton. The
hydrogenic wavefunctions for the two 1S states are most simply written in terms
of the vectors r1 and r2. I will refer to these two states as:
1 Z 
 1S ,1  r1  
 
  a0 
32
1 Z 
 1S ,2  r2  
 
  a0 
32
 Zr 
exp   1 
 a0 
 Zr 
exp   2 
 a0 
Of course, we really need to choose a consistent coordinate system in which to
write both wavefunctions. In Cartesian coordinates, they are:
Physics 7440
H6.3
Spring 2003
1 Z 
 1S ,1  x, y, z  
 
  a0 
32
 Z
exp  
 a0
2 
x2  y 2   z  R 2 

32
 Z
exp  
 a0
2 
x2  y 2   z  R 2 

1 Z 
 1S ,2  x, y, z  
 
  a0 
As you will see below, part of the complexity of this problem is that there is no
simple coordinate transformation that gets all the coordinates out of the root.
Therefore, the necessary normalization and average value integrals are a
nightmare.
OK, so pressing on with the variational guess: There is no a priori way to know
which orbital has the electron, so a natural thing to do is to give a 50% probability
that the electron is in one or the other. Thus, a reasonable guess for the
approximate wavefunction at large separation is:
 G . S .  Z , R, r  
1
2

1S ,1
  1S ,2 
Here, I have tried to explicitly remind you that this guess for the ground state
wavefunction is a function of the nuclear charge, Z, nuclear separation, R, as well
as the electronic position, r.
The parameter R
In our variational calculation, we will look for the values of R that minimizes the
expectation value of the energy and thereby determine the equilibrium nuclear
separation in the ground state and the ground state energy. Also, the energy as a
function of R serves as an approximation to the effective nuclear binding
potential, the equivalent of the Lennard-Jones potential for the molecule.
The parameter Z
We will also use the apparent nuclear charge, Z, as a parameter. Although it may
not seem natural to let Z vary (after all, the proton charges are known), you can
physically see why it is reasonable if you consider the short nuclear separation
case:
a) Think about very short separation between the protons and use what you know
about the ground state of singly ionized Helium to argue that Z should be
expected to vary. What is the range you expect for Z as a function of R?
Normalization of the wavefunction
Physics 7440
H6.4
Spring 2003
If you insist that this wavefunction is normalized at each possible nuclear
separation, you will see that our guess for large separation needs to be slightly
modified.
b) From the normalization condition, show that our wavefuntion must become:
1
 G . S .  Z , R, r  
2 1  S 

1S ,1
 1S ,2 
where (remember, the wavefunctions here are real-valued functions)
S Z, R 

dxdydz 1S ,1 1S ,2
all space
The quantity, S, is referred to as an overlap integral. Qualitatively, you can see that
this integral becomes very small at large nuclear separation because the two
exponential wavefunctions cut off the integrand so strongly. It is the evaluation of
integrals like this one that makes the problems in quantum chemistry and in solid
state physics so difficult.
As it happens, a miracle occurs and this integral AND all the other integrals we
need for this problem can actually be done analytically! We find that [1]:
2
 ZR   ZR 1  ZR  
S  Z , R   exp  
 
 1 
 
a
a
3
a
0 
0

 0  
In this case, Z and R enter together in the same functional way.
c) Show that the behavior of our normalized wavefunction matches our
expectations for very small and very large R under the further assumption that
Z is unity for large separation and Z=2 for short separation.
Expectation value of the energy
Now that we have a normalized variational wavefunction, which seems to do
sensible things, we take the next step in a variational calculation, namely we take
the expectation value of the energy:
E   G .S . H  G .S .


dxdydz G* .S . H G.S .
all space
d) Substitute in the variational wavefunction and show that the energy
expectation value can be written as:
Physics 7440
H6.5
Spring 2003
 2  Z 2
1
E Z, R 

   I1  I 2  
1  S  2m  a0 

2

e Z

 I3  I 4  2I5 
4 0 a0

where the following dimensionless integrals have been defined:
I1  
1
d  exp    exp   
 
3
I2  
1
3
2
d  exp    exp   
 
2
1
2
I3 
1
d  exp    exp   
 

3
I4 
1
3
I5 
1
1
d  exp   
 

1
1
1
d  exp   
 

3
exp  1 
2
1
exp   2 
2
When you do all the terms, you will see several integrals that are not on the list,
but they will be symmetric versions where the subscripts 1 and 2 are interchanged.
You can convince yourself that pairs of integrals are equal and can be combined
as above. Notice also, that you have already evaluated the first and third integrals
when you did the variational calculation of the hydrogen ground state.
As it happens, each of these integrals is actually solvable in closed form. They
are:1
I1  1


 ZR  
ZR 
I 2    S  2 1 
 exp  
 
a0 

 a0  

I3  1
1
Quantum Chemistry Methods and Applications, by R. Daudel, R. Lefebure, and C. Moser,
Interscience Publishers, Inc., New York, (1959).
Physics 7440
H6.6
Spring 2003
 a
 ZR  
a0  
I 4     0  exp  2

 1 
a0  
ZR  

 ZR
 ZR  ZR 
I5  exp  
1 

a0 
 a0 
I have written each explicitly in terms of the Bohr radius. However, it is clear that
there is a natural dimensionless quantity shared by each function.
e) Substitute in these forms for the integrals to get a function for the expected
ground state electronic energy. Use Mathematica or whatever package you like
to plot this energy function versus Z and R. Choose some appropriate energy
scale to make the energy function dimensionless; make sure you state the
scale. Similarly, scale R by the Bohr radius. Your discussion in part a) should
give you an idea of a reasonable range for Z. What are the minimum energy
values (here only Z is a parameter) and minimizing values of Z for large R
and very small R? What does it mean?
f) Now add the repulsive proton-proton repulsion to the electronic energy to get
the total system ground state energy. Graphically find the combination of Z
and R that minimizes this total energy. Quote your numerical results for the
values of Z, R, and energy at the minimum in some appropriate physical units.
g) Experimentally, we define the ‘dissociation energy’ as the energy required to

take the H2 ion from its ground state to a state where you have a neutral
hydrogen atom infinitely removed from an isolated proton. Your calculation
determines approximate energies for these two states. How does your
calculation compare with the experimental result that the ground state
separation of the protons is 0.106 nm and the dissociation energy is 2.65 eV?
Kittel 7.1 Square lattice, free electron energies.
(a) Show for a simple square lattice (two dimensions) that the kinetic energy of a
free electron at a corner of the first zone is higher than that of an electron at
midpoint of a side face of the zone by a factor of 2. (b) What is the corresponding
factor for a simple cubic lattice (3D)? (c) What bearing might the result of (b)
have on the conductivity of divalent metals?
Kittel 7.2 Free electron energies in reduced zone.
Consider the free electron energy bands of an fcc crystal lattice in the
approximation of an empty lattice, but in the reduced zone scheme in which all kvectors are transformed to lie in the first Brillouin zone. Plot in the [111] direction
the energies of all bands up to six times the lowest band energy at the zone
boundary at k=(2/a)(1/2,1/2,1/2). Let this be the unit of energy. This problem
Physics 7440
H6.7
Spring 2003
shows why band edges need not necessarily be at the zone center. Several of the
degeneracies (band crossings) will be removed when account is taken of the
crystal potential.
Physics 7440
H6.8
Spring 2003