5_atoms_and_lcao.md

from matplotlib import pyplot

import numpy as np

from common import draw_classic_axes, configure_plotting

configure_plotting()

pi = np.pi

Atoms and bonds

(based on chapters 5 and 6.2 of the book)

!!! success "Expected prior knowledge"

Before the start of this lecture, you should be able to:

- Write down the Schrödinger equation
- Compute eigenvectors and eigenvalues of a matrix
- Solve the Schrödinger equation of a bound state with a $δ$-function potential in 1D (for the exercises)
- Write down the quantum numbers of the hydrogen atom
- Describe the orbitals of the hydrogen atom using the quantum numbers

!!! summary "Learning goals"

After this lecture you will be able to:

- Describe the shell-filling model of atoms
- Derive the LCAO model
- Obtain the energy spectrum of the LCAO model of several orbitals

Looking back

So far we have:

• Introduced the k-space (reciprocal space)
• Postulated the dispersion relation of free electrons and phonons
• Calculated the heat capacity of free electrons and phonons

As a result we:

• Understood how phonons store heat (Debye model)
• Understood how free electrons conduct (Drude model) and store heat/energy (Sommerfeld model)

We made several approximations and postulations through these models. However, there are still several mysteries:

• Why is there a phonon cutoff frequency? Why are there no more phonon modes beyond this cutoff frequency?
• Why don't electrons scatter off from every single atom in the Drude model? Atoms are charged and should provide a lot of scattering.
• Why are some materials not metals? (Think if you know a crystal that isn't a metal)

To answer these questions we will need to study atoms in more detail.

A quick review of atoms

Why chemistry is not physics

Everything is described by the Schrödinger equation:

Hψ = Eψ,

with H the sum of kinetic energy and the potential energy. In the hydrogen atom, the potential energy is due to the Coulomb interaction between the electron and the nucleus: