Atomic Orbitals on a Computer

The hydrogen atom is three systems in one: The H-Atom Model Setting up the Schrodinger equation we get new equations to solve for each “motion”   r nucleus distance from the origin azimuthal angle equatorial angle

The Schrodinger equation: Hydrogen Atom Solution: We want to visualize this Radial Wave function Angular Wave function

For the Radial part of the Schrodinger equation: Numerov Procedure Just this again: We can use the Numerov program to solve it! Radial wave function depends on n and l quantum numbers This can be reformed to look like:

The radial wave functions tell you what is happening inside the orbital: Numerov Procedure # radial nodes = n – l – 1

numerov_control.R

For the angular parts of the Schrodinger equation: Doh! We get an equation that we can’t solve with Numerov… So, we will just “look-up” the solutions: Legendre Polynomials

is a probability density. To “look” at a probability density, we can sample from it: Let’s sample from the “bell curve”: Well, … lets try something else

So, let’s try to “look” at the orbitals as a probability density. Let’s sample from it: Well, … lets try something else

Lets do a 3D sample from an orbital probability density: Well, … lets try something else 1s01s0

Lets do a 3D sample form an orbital probability density: Well, … lets try something else 1s01s0

Lets do a 3D sample form an orbital probability density: Well, … lets try something else What orbital is this?

density_control.R