Atomic orbitals: 2p 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 (2 for the 2p orbitals)
The radial equation for the 2px, 2py, and 2pz orbitals is the same in each case. The angular functions are the same but substitute y and z as appropriate in the formula for Y2px given above. Substitute similarly for the wave equations ψ2py and ψ2pz.
For s-orbitals the radial distribution function is given by 4πr2ψ2, but for non-spherical orbitals (where the orbital angular momentum quantum number l > 0) the expression is as above. See D.F. Shriver and P.W. Atkins, Inorganic Chemistry, 3rd edition, Oxford, 1999, page 15.