@@ -286,14 +286,14 @@ On the other hand, if each atom has 0 or 2 electrons in the outermost shell, the
* If two atoms have one orbital and one electron each, the two electrons occupy the bonding orbital.
## Exercises
### Warm-up questions
### Warm-up questions*
1. Is the assumption that the atomic orbitals are orthogonal always a reasonable assumption?
2. What is the effect on the molecular orbitals if the hopping $t$ is chosen to be negative?
3. How does the size of the Hamiltonian matrix change with the number of atoms?
4. How does the size of the Hamiltonian matrix change if each atom now has two orbitals?
5. Assuming that we have two atoms with a single orbital each, what is the size of the Hamiltonian matrix if we also consider the spin of the electron?
### Exercise 1: Shell-filling model of atoms
### Exercise 1*: Shell-filling model of atoms
Here we consider the order in which electrons fill the orbitals on the atoms
What should the electron configurations be if these elements followed Madelung’s rule and the Aufbau principle?
What could be the reason for the deficiency of Madelung's rule?
### Exercise 2: Explicit calculation of the onsite and hopping energies from the wave functions of the atomic orbitals.
### Exercise 2*: Explicit calculation of the onsite and hopping energies from the wave functions of the atomic orbitals.
In the LCAO model, we construct the molecular orbitals from linear combinations of the atomic orbitals. In this exercise, we model the atomic orbital of each individual atom as the bound state of a delta-function potential. To construct the LCAO Hamiltonian, we explicitly calculate the onsite energy and the hopping from these atomic orbitals. Diagonalizing this Hamiltonian yields the molecular orbitals.
Consider an electron moving in 1D between two negative delta-function potential wells.
