Atomic orbitals: 3s equations
The symbols used in the following are:
- r = radius expressed in atomic units (1 Bohr radius = 52.9 pm)
- π = 3.14159 approximately
- e = 2.71828 approximately
- Z = effective nuclear charge for that orbital in that atom.
- ρ = 2Zr/n where n is the principal quantum number (3 for the 3s orbital)
The origin of the spherical nodes becomes clearer upon examining the wave equation, which includes a (6 - 6ρ + ρ2) term. When (6 - 6ρ + ρ2) = 0, then we must have nodes. Since for the 3s orbital ρ = 2Zr/3 (Z = effective nuclear charge, r = radius in atomic units), and we can solve the zero values for the quadratic to give ρ = 3 + √3 or 3 - √3 then the nodes are at the radii r = 3(3 + √3)/2Z and 3(3 - √3)/2Z atomic units.