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) × ρ³ × Z^3/2 × e^-ρ/2
Angular wave functions (general set):
Y_{4f_z³}	= √(7/4) × z(5z² - 3r²)/r³ × (1/4π)^1/2
Y_{4f_yz²}	= √(42/16) × y(5z² - r²)/r³ × (1/4π)^1/2
Y_{4f_xz²}	= √(42/16) × x(5z² - r²)/r³ × (1/4π)^1/2
Y_{4f_xyz}	= √(105/4) × 2xyz/r³ × (1/4π)^1/2
Y_{4f_z(x²-y²)}	= √(105/4) × z(x²-y²)/r³ × (1/4π)^1/2
Y_{4f_y(3x²-y²)}	= √(70/16) × y(3x²-y²)/r³ × (1/4π)^1/2
Y_{4f_x(x²-3y²)}	= √(70/16) × x(x²-3y²)/r³ × (1/4π)^1/2
Angular wave functions (cubic set):
Y_{4f_y³}	= √(7/4) × y(5y² - 3r²)/r³ × (1/4π)^1/2
Y_{4f_z³}	= √(7/4) × z(5z² - 3r²)/r³ × (1/4π)^1/2
Y_{4f_x³}	= √(7/4) × x(5x² - 3r²)/r³ × (1/4π)^1/2
Y_{4f_y(z²-x²)}	= √(105/4) × y(z²-x²)/r³ × (1/4π)^1/2
Y_{4f_z(x²-y²)}	= √(105/4) × z(x²-y²)/r³ × (1/4π)^1/2
Y_{4f_x(z²-y²)}	= √(105/4) × x(z²-y²)/r³ × (1/4π)^1/2
Y_{4f_xyz}	= √(105/4) × 2xyz/r³ × (1/4π)^1/2
Wave functions (general set):
ψ_{4f_z³}	= R_4f × Y_{4f_z³}
ψ_{4f_yz²}	= R_4f × Y_{4f_yz²}
ψ_{4f_xz²}	= R_4f × Y_{4f_xz²}
ψ_{4f_xyz}	= R_4f × Y_{4f_xyz}
ψ_{4f_z(x²-y²)}	= R_4f × Y_{4f_z(x²-y²)}
ψ_{4f_y(3x²-y²)}	= R_4f × Y_{4f_y(3x²-y²)}
ψ_{4f_x(x²-3y²)}	= R_4f × Y_{4f_x(x²-3y²)}
Wave functions (cubic set):
ψ_{4f_y³}	= R_4f × Y_{4f_y³}
ψ_{4f_z³}	= R_4f × Y_{4f_z³}
ψ_{4f_x³}	= R_4f × Y_{4f_x³}
ψ_{4f_y(z²-x²)}	= R_4f × Y_{4f_y(z²-x²)}
ψ_{4f_z(x²-y²)}	= R_4f × Y_{4f_z(x²-y²)}
ψ_{4f_x(z²-y²)}	= R_4f × Y_{4f_x(z²-y²)}
ψ_{4f_xyz}	= R_4f × Y_{4f_xyz}
Electron density	= ψ_4f²
Radial distribution function	= r²R_4f²

For any atom, there are seven 4f orbitals. The f-orbitals 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(x²-y²) 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_y³ used for 4f_{y(5y² - 3r²)}
4f_z³ used for 4f_{z(5z² - 3r²)}
4f_xz² used for 4f_{x(5z² - r²)}
4f_yz² used for 4f_{y(5z² - r²)}

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.