Electronic Structure of Materials Flashcards
Magnitude of electron spin
h(bar)/2
General Hamiltonian expression.
H = T + V
What the Hamiltonian for an electron in a hydrogen atom
Check
Write the momentum operator
Check
Use the momentum operator to give an expression for the Hamiltonian operator.
Check
State the time dependent Schrödinger equation.
Check
State the time independent Schrödinger equation,
Check
Show that the position and momentum operators do not commute.
Show
Show that the components of the angular momentum operator do not commute.
Check
Show that the angular momentum squared commutes with an angular momentum operator in a single dimension.
Check
State the eigenvalue equations with the spherical harmonics for the angular momentum operators.
Check
What values can the quantum numbers l and m take?
0<=l
How do you get from the Hamiltonian for hydrogen to the Hamiltonian operator for hydrogen?
Insert the p operator.
When solving the Hamiltonian operator for the hydrogen atom, do the angular parts of the wavefunction depend on the potential?
No.
The solutions of the TISE for any spherically symmetric potential have the same angular dependence.
Expression for energy of each level in a hydrogen atom.
Check
Define a Ry in terms of constants
Check
Define Bohr radius in terms of constants
Check
l=0,1,2,3 can be denoted by which letters?
s,p,d,f
Give the Hamiltonian for the electrons in a He atom.
Check
What is the mean field approach to the problem of many electrons in a system?
All the atoms are said to have the same potential which is an average of all the interactions in the system.
What is a central field approxiamtion?
The use of a mean field however it is spherically symmetric.
What does the Aufbau principle state?
That electrons will occupy the lowest energy states available.
What are the empirical rules that the Aufbau principle follows?
Fill the states with the lowest value of n + l first.
If there are multiple states with equal n + l then fill the states with the lowest n first.
State the Hamiltonian operator for a single electron in a hydrogen molecule.
Check
Derive the bonding and antibonding solutions for a single electron in a hydrogen molecule.
Check derivation
Sketch the probability densities for bonding and antibonding in a hydrogen molecule.
Sketch and check
How is the bond integral usually expressed?
In a modulus sign.
What do HOMO and LUMO stand for?
Highest occupied molecular orbital
Lowest unoccupied molecular orbital
What is the HOMO-LUMO gap?
Minimum energy required to promote an electron to a higher energy level.
What is a basis function for molecular orbitals?
It is a building block function that we can make molecular orbitals from.
The number of molecular orbitals calculated equals the number of basis functions (due to bonding and antibonding).
When extending the LCAO for a general molecule, what does the Hamiltonian become?
A matrix.
How can we use density of states and energy levels to show degeneracy?
Plot DoS against energy and the relative heights of the peaks show the relative degeneracies.
What is the area under a single peak on a plot of DoS against energy equal to?
1
What is a perturbation?
A stimulus provided by experiment to the system.
State the Kronecker delta function.
= 1 where I=j
=0 otherwise
State the Dirac delta function
delta(x) = 0 for all x except when x=0 int(delta(x)) = 1 between -inf and +inf