Summary of 3d atomic orbitals

	Symmetry	Orbitals
Free atom		E1 m=2, Ψ3,2,2 = (1 / 81√(2π)) (Z/a)7/2 r2 e-Zr/3a sin2θ ei2φ m=1, Ψ3,2,1 = (√2 / 81√π) (Z/a)7/2 r2 e-Zr/3a sinθ cosθ eiφ m=0, Ψ3,2,0 = (1 / 81√(6π)) (Z/a)7/2 r2 e-Zr/3a (3cos2θ -1) m=-1, Ψ3,2,-1 = (√2 / 81√π) (Z/a)7/2 r2 e-Zr/3a sinθ cosθ e-iφ m=-2, Ψ3,2,-2 = (1 / 81√(2π)) (Z/a)7/2r2 e-Zr/3a sin2θ e-i2φ
Octahedral / Tetrahedral	Oh / Td	E1 (eg) x2-y2=(3,2,2)+(3,2,-2) sin2θ cos2φ z2=(3,2,0) (3cos2θ-1) E2 (t2g) xy=(3,2,2)-(3,2,-2) sin2θ sin2φ yz=(3,2,1)-(3,2,-1) sinθ cosθ sinφ xz=(3,2,1)+(3,2,-1) sinθ cosθ cosφ
Triangular bipyramid symmetry		E1 x2-y2, xy E2 xz, yz E3 z2

Angular momentum operator:
L_z=(ħ/i)∂/∂φ