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 + ρ^{2}) term. When (6  6ρ + ρ^{2}) = 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.
