Atomic orbitals: 4p 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 (4 for the 4p orbitals)

**Table of equations for the 4p orbital.**
Function	Equation
Radial wave function, R_{4p_x}	= (1/32√15) × ρ(20 - 10ρ + ρ²) × Z^3/2 × e^-ρ/2
Angular wave function, Y_{4p_x}	= √(3)x/r × (1/4π)^1/2
Wave function, ψ_{4p_x}	= R_{4p_x} × Y_{4p_x}
Electron density	= ψ_{4p_x}²
Radial distribution function	= r²R_{4p_x}²

The corresponding radial equations for the 4p_y and 4p_z orbitals are the same. The angular functions are the same but substitute y and z as appropriate in the formula for Y_{4p_x} given above. Substitute similarly for the wave equations ψ_{4p_y} and ψ_{4p_z}.

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.