Atomic orbitals: 4f 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 4f orbitals)
Table of equations for the 4f orbitals.
Function 
Equation 
Radial wave function, R_{4f} 
= (1/96√35) × ρ^{3} × Z^{3/2} × e^{ρ/2} 
Angular wave functions (cubic set): 

Y_{4fx3} 
= √(7/4) × x(5x^{2}  3r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fy3} 
= √(7/4) × y(5y^{2}  3r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fz3} 
= √(7/4) × z(5z^{2}  3r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fx(z2y2)} 
= √(105/4) × x(z^{2}y^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fy(z2x2)} 
= √(105/4) × y(z^{2}x^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fz(x2y2)} 
= √(105/4) × z(x^{2}y^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fxyz} 
= √(105/4) × xyz/r^{3} × (1/4π)^{1/2} 
Angular wave functions (general set): 

Y_{4fz3} 
= √(7/4) × z(5z^{2}  3r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fxz2} 
= √(42/16) × x(5z^{2}  r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fyz2} 
= √(42/16) × y(5z^{2}  r^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fy(3x2y2)} 
= √(70/16) × y(3x^{2}y^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fx(x23y2)} 
= √(70/16) × x(x^{2}3y^{2})/r^{3} × (1/4π)^{1/2} 
Y_{4fxyz} 
= √(105/4) × xyz/r^{3} × (1/4π)^{1/2} 
Y_{4fz(x2y2)} 
= √(105/4) × z(x^{2}y^{2})/r^{3} × (1/4π)^{1/2} 
Wave functions (cubic set): 

ψ_{4fx3} 
= R_{4f} × Y_{4fx3} 
ψ_{4fy3} 
= R_{4f} × Y_{4fy3} 
ψ_{4fz3} 
= R_{4f} × Y_{4fz3} 
ψ_{4fx(z2y2)} 
= R_{4f} × Y_{4fx(z2y2)} 
ψ_{4fy(z2x2)} 
= R_{4f} × Y_{4fy(z2x2)} 
ψ_{4fz(x2y2)} 
= R_{4f} × Y_{4fz(x2y2)} 
ψ_{4fxyz} 
= R_{4f} × Y_{4fxyz} 
Wave functions (general set): 

ψ_{4fz3} 
= R_{4f} × Y_{4fz3} 
ψ_{4fxz2} 
= R_{4f} × Y_{4fxz2} 
ψ_{4fyz2} 
= R_{4f} × Y_{4fyz2} 
ψ_{4fy(3x2y2)} 
= R_{4f} × Y_{4fy(3x2y2)} 
ψ_{4fx(x23y2)} 
= R_{4f} × Y_{4fx(x23y2)} 
ψ_{4fxyz} 
= R_{4f} × Y_{4fxyz} 
ψ_{4fz(x2y2)} 
= R_{4f} × Y_{4fz(x2y2)} 
Electron density 
= ψ_{4f}^{2} 
Radial distribution function 
= r^{2}R_{4f}^{2} 
For any atom, there are seven 4f orbitals. The forbitals are unusual in that there are two sets of orbitals in common use. The cubic set is appropriate to use if the atom is in a cubic environment. The general set is used at other times. Three of the orbitals are common to both sets. These are are the 4f_{xyz}, 4f_{z3}, and 4f_{z(x2y2)} orbitals.
The radial equations for all the 4f orbitals are the same. The real angular functions differ for each and these are listed above.
Each of the orbitals is named for the expression based upon x, y, and z in the angular wave function, but some abbreviated names are useful for simplicity. These are:
 4f_{x3} used for 4f_{x(5x2  3r2)} (=4f_{x(2x2  3y2  3z2)} since r^{2} = x^{2} + y^{2} + z^{2})
 4f_{y3} used for 4f_{y(5y2  3r2)} (=4f_{y(2y2  3x2  3z2)} since r^{2} = x^{2} + y^{2} + z^{2})
 4f_{z3} used for 4f_{z(5z2  3r2)} (=4f_{z(2z2  3x2  3y2)} since r^{2} = x^{2} + y^{2} + z^{2})
 4f_{xz2} used for 4f_{x(5z2  r2)} (=4f_{x(4z2  x2  y2)} since r^{2} = x^{2} + y^{2} + z^{2})
 4f_{yz2} used for 4f_{y(5z2  r2)} (=4f_{y(4z2  x2  y2)} since r^{2} = x^{2} + y^{2} + z^{2})
For sorbitals the radial distribution function is given by 4πr^{2}ψ^{2}, but for nonspherical 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.
