Atomic Orbitals on a Computer

http://www.askamathematician.com

• 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”

q

f

r

nucleusdistance 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

1s0

• Lets do a 3D sample form an orbital probability density:

Well, … lets try something else

1s0

• Lets do a 3D sample form an orbital probability density:

Well, … lets try something else

What orbital is this?

density_control.R