(MW 9:30am -10:50pm, Mayer Hall 5301) Course Syllabus
pdf file
Lecture notes
Elements
Lecture 1 : The symmetry principle
Lecture 2 : Basic concepts and examples of group
Lecture 3 : Representations
Lecture 4 : Orthogonality and Character
Lecture 5 : Character tables and representations for finite groups
Lecture 6 : Point groups
Lecture 7 : Decomposition of direct product representations, projection operators, crystal harmonics
Lecture 8 : Crystal field splitting, cystalline tensors
Lecture 9 : Molecular vibration
Lecture 10 : Crystal symmetry (I) - Crystal system, Bravis lattice
Lecture 11 : Crystal symmetry (II) - non-symmorphic operations
Lecture 12 : Crystal symmetry (III) - Representations of space group
Lecture 13 : Crystal symmetry (IV) - more examples of space group
Galois theory
Lecture 1 : Number field and field extension
Lecture 2 : Automorphism and Galois groups
Lecture 3 : More on Galois groups
Lecture 4 : Normal and composite group series, solvability
Lecture 5 : Sufficient and necessary conditions of algebraic solutions
Lecture 6 : Compass-straightedge construction problem
howework assignment
HW1 : Due on Oct 15 Monday class.
HW2 : Due on Oct 22 Monday class.
HW3 : Due on Oct 30 Wednesday class.
Final exam
- Due on Dec 16 (Sunday) midnight 12:00am
Midterm Project: Icosahedral symmetry -- Electronic and vibrational modes
of C60 - Due on Nov 16, Friday afternoon 3:00pm
Solution to Midterm posted on Dec. 7.
You need to apply group theory to guide the study of electronic and vibrational modes of a C60 modelcule. S
uggested topics include but are not limited to the following contents.
If you can go beyond the list below, you can receive extra credit.
You may do literature search on line, but are not allowed to discuss
with others before you turn in your solutions to me.
Please note that below we use the full symmetry group of icosahedral, which
includes inversion.
This group is denoted as Yh, which has 120 symmetry elements.
It is just Y \times Z2, and the Z2 includes identity and inversion.
1. Approximate C60 as a spherically symmetric shell.
Solve the electron orbital problem on the sphere
The radius of the shell is R, and the thickness of the shell is neglected.
Load 60 electrons in these orbitals.
What is the filling configuration of the ground state of 60 electrons?
(Hint: For fully filled orbitals, each state is filled by a pair of spin up
and down electrons.
The HOMO (highest ocupied molecular orbital) is partially filled, and you
can use Hund's rule to determin its configuration.
)
Certainly this picture is oversimplified: You would get that C60 is
gapless and magnetic in contradiction to experimental facts.
In fact, C60 is non-magnetic and there is an excitation gap
between HOMO and LUMO (lowest unoccupied molecular orbital).
2. To improve, let us break the spherical symmetry to the icosehedral
symmetry by adding potential distributions on the sphere.
Figure out each orbital with angular momentum $l$ in the original
spherical symmetry splits into what irreducible representations
of the icosahedral group. (You can do for states up to $l$=5.
Assume that the potential is not strong enough to change the
Sequence of energies of orbitals with different $l$, but only
splits the degeneracy of states within the same $l$.)
Based on the fact of a gap between HOMO and LUMO,
can you figure about what is the representation of the HOMO?
3. Now let us work out the same problem from the discrete side.
Suppose each site contribute an atomic orbital. The C60 molecular
orbitals form a 60 dimensional representation of icosahedral group.
Please decompose it into direct sum of irreducible ones.
4. Write down a tight-binding model with the neareast neighboring
hopping amplitude -t. Diagonalize the Hamiltonian matrix. You may use either
diagonalize numerically, or, first use group theory to reduce the size
of the matrix, and then diagonalize.
Sketch a level diagram with labeling the degeneracy and the representation
of each orbital solved from the tight-binding model.
Compare your result here with that obtained in part 2.
What is the representation of LUMO and what is its degenreacy?
5. The vibrational modes of the C60 forms a 180 dimensional
representation of the icosahedral group Yh. Please decompose it into
representation of Yh.
6. If you can design a model of elasticity, say, connecting
atoms with springs, to solve the vibrational models of C60. You will
get extra credits. This is quite challenging.
Back to home
Last modified: Jan 7, 2010.
Galois theory
howework assignment
HW1 : Due on Oct 15 Monday class.
HW2 : Due on Oct 22 Monday class.
HW3 : Due on Oct 30 Wednesday class.
Final exam
- Due on Dec 16 (Sunday) midnight 12:00am
Midterm Project: Icosahedral symmetry -- Electronic and vibrational modes
of C60 - Due on Nov 16, Friday afternoon 3:00pm
Solution to Midterm posted on Dec. 7.
You need to apply group theory to guide the study of electronic and vibrational modes of a C60 modelcule. S
uggested topics include but are not limited to the following contents.
If you can go beyond the list below, you can receive extra credit.
You may do literature search on line, but are not allowed to discuss
with others before you turn in your solutions to me.
Please note that below we use the full symmetry group of icosahedral, which
includes inversion.
This group is denoted as Yh, which has 120 symmetry elements.
It is just Y \times Z2, and the Z2 includes identity and inversion.
1. Approximate C60 as a spherically symmetric shell.
Solve the electron orbital problem on the sphere
The radius of the shell is R, and the thickness of the shell is neglected.
Load 60 electrons in these orbitals.
What is the filling configuration of the ground state of 60 electrons?
(Hint: For fully filled orbitals, each state is filled by a pair of spin up
and down electrons.
The HOMO (highest ocupied molecular orbital) is partially filled, and you
can use Hund's rule to determin its configuration.
)
Certainly this picture is oversimplified: You would get that C60 is
gapless and magnetic in contradiction to experimental facts.
In fact, C60 is non-magnetic and there is an excitation gap
between HOMO and LUMO (lowest unoccupied molecular orbital).
2. To improve, let us break the spherical symmetry to the icosehedral
symmetry by adding potential distributions on the sphere.
Figure out each orbital with angular momentum $l$ in the original
spherical symmetry splits into what irreducible representations
of the icosahedral group. (You can do for states up to $l$=5.
Assume that the potential is not strong enough to change the
Sequence of energies of orbitals with different $l$, but only
splits the degeneracy of states within the same $l$.)
Based on the fact of a gap between HOMO and LUMO,
can you figure about what is the representation of the HOMO?
3. Now let us work out the same problem from the discrete side.
Suppose each site contribute an atomic orbital. The C60 molecular
orbitals form a 60 dimensional representation of icosahedral group.
Please decompose it into direct sum of irreducible ones.
4. Write down a tight-binding model with the neareast neighboring
hopping amplitude -t. Diagonalize the Hamiltonian matrix. You may use either
diagonalize numerically, or, first use group theory to reduce the size
of the matrix, and then diagonalize.
Sketch a level diagram with labeling the degeneracy and the representation
of each orbital solved from the tight-binding model.
Compare your result here with that obtained in part 2.
What is the representation of LUMO and what is its degenreacy?
5. The vibrational modes of the C60 forms a 180 dimensional
representation of the icosahedral group Yh. Please decompose it into
representation of Yh.
6. If you can design a model of elasticity, say, connecting
atoms with springs, to solve the vibrational models of C60. You will
get extra credits. This is quite challenging.
Back to home
Last modified: Jan 7, 2010.
Final exam
- Due on Dec 16 (Sunday) midnight 12:00am
Midterm Project: Icosahedral symmetry -- Electronic and vibrational modes
of C60 - Due on Nov 16, Friday afternoon 3:00pm
Solution to Midterm posted on Dec. 7.
You need to apply group theory to guide the study of electronic and vibrational modes of a C60 modelcule. S
uggested topics include but are not limited to the following contents.
If you can go beyond the list below, you can receive extra credit.
You may do literature search on line, but are not allowed to discuss
with others before you turn in your solutions to me.
Please note that below we use the full symmetry group of icosahedral, which
includes inversion.
This group is denoted as Yh, which has 120 symmetry elements.
It is just Y \times Z2, and the Z2 includes identity and inversion.
1. Approximate C60 as a spherically symmetric shell.
Solve the electron orbital problem on the sphere
The radius of the shell is R, and the thickness of the shell is neglected.
Load 60 electrons in these orbitals.
What is the filling configuration of the ground state of 60 electrons?
(Hint: For fully filled orbitals, each state is filled by a pair of spin up
and down electrons.
The HOMO (highest ocupied molecular orbital) is partially filled, and you
can use Hund's rule to determin its configuration.
)
Certainly this picture is oversimplified: You would get that C60 is
gapless and magnetic in contradiction to experimental facts.
In fact, C60 is non-magnetic and there is an excitation gap
between HOMO and LUMO (lowest unoccupied molecular orbital).
2. To improve, let us break the spherical symmetry to the icosehedral
symmetry by adding potential distributions on the sphere.
Figure out each orbital with angular momentum $l$ in the original
spherical symmetry splits into what irreducible representations
of the icosahedral group. (You can do for states up to $l$=5.
Assume that the potential is not strong enough to change the
Sequence of energies of orbitals with different $l$, but only
splits the degeneracy of states within the same $l$.)
Based on the fact of a gap between HOMO and LUMO,
can you figure about what is the representation of the HOMO?
3. Now let us work out the same problem from the discrete side.
Suppose each site contribute an atomic orbital. The C60 molecular
orbitals form a 60 dimensional representation of icosahedral group.
Please decompose it into direct sum of irreducible ones.
4. Write down a tight-binding model with the neareast neighboring
hopping amplitude -t. Diagonalize the Hamiltonian matrix. You may use either
diagonalize numerically, or, first use group theory to reduce the size
of the matrix, and then diagonalize.
Sketch a level diagram with labeling the degeneracy and the representation
of each orbital solved from the tight-binding model.
Compare your result here with that obtained in part 2.
What is the representation of LUMO and what is its degenreacy?
5. The vibrational modes of the C60 forms a 180 dimensional
representation of the icosahedral group Yh. Please decompose it into
representation of Yh.
6. If you can design a model of elasticity, say, connecting
atoms with springs, to solve the vibrational models of C60. You will
get extra credits. This is quite challenging.
Back to home
Last modified: Jan 7, 2010.
Midterm Project: Icosahedral symmetry -- Electronic and vibrational modes
of C60 - Due on Nov 16, Friday afternoon 3:00pm Solution to Midterm posted on Dec. 7.
You need to apply group theory to guide the study of electronic and vibrational modes of a C60 modelcule. S uggested topics include but are not limited to the following contents. If you can go beyond the list below, you can receive extra credit. You may do literature search on line, but are not allowed to discuss with others before you turn in your solutions to me.
Please note that below we use the full symmetry group of icosahedral, which includes inversion. This group is denoted as Yh, which has 120 symmetry elements. It is just Y \times Z2, and the Z2 includes identity and inversion.
Load 60 electrons in these orbitals. What is the filling configuration of the ground state of 60 electrons? (Hint: For fully filled orbitals, each state is filled by a pair of spin up and down electrons. The HOMO (highest ocupied molecular orbital) is partially filled, and you can use Hund's rule to determin its configuration. )
Certainly this picture is oversimplified: You would get that C60 is gapless and magnetic in contradiction to experimental facts. In fact, C60 is non-magnetic and there is an excitation gap between HOMO and LUMO (lowest unoccupied molecular orbital).
Figure out each orbital with angular momentum $l$ in the original spherical symmetry splits into what irreducible representations of the icosahedral group. (You can do for states up to $l$=5. Assume that the potential is not strong enough to change the Sequence of energies of orbitals with different $l$, but only splits the degeneracy of states within the same $l$.)
Based on the fact of a gap between HOMO and LUMO, can you figure about what is the representation of the HOMO?
Sketch a level diagram with labeling the degeneracy and the representation of each orbital solved from the tight-binding model. Compare your result here with that obtained in part 2. What is the representation of LUMO and what is its degenreacy?