3d atomic orbitals 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 3d orbitals)

**Table of equations for the 3d orbitals.**
Function	Equation
Radial wave function, R_3d	= (1/9√30) × ρ² × Z^3/2 × e^-ρ/2
Angular wave functions:
Y_{3d_z²}	= √(5/4) × (3z² – r²)/r² × (1/4π)^1/2
Y_{3d_yz}	= √(60/4) × yz/r² × (1/4π)^1/2
Y_{3d_xz}	= √(60/4) × xz/r² × (1/4π)^1/2
Y_{3d_xy}	= √(15/4) × 2xy/r² × (1/4π)^1/2
Y_{3d_x²-y²}	= √(15/4) × (x² - y²)/r² × (1/4π)^1/2
Wave functions:
ψ_{3d_z²}	= R_3d × Y_{3d_z²}
ψ_{3d_yz}	= R_3d × Y_{3d_yz}
ψ_{3d_xz}	= R_3d × Y_{3d_xz}
ψ_{3d_xy}	= R_3d × Y_{3d_xy}
ψ_{3d_x²-y²}	= R_3d × Y_{3d_x²-y²}
Electron density	= ψ_3d²
Radial distribution function	= r²R_3d²

There are five real 3d orbitals. The radial equations for all the 3d 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 the 3d_{3z² – r²} orbital is abbreviated to 3d_z².

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.