The following table contains the normalized real spherical harmonics of orders 0 to 3 (s to f).
| l | Symbol |
Trigonometric form |
Cartesian form |
| s-orbital | |||
| 0 | s | √((1/2π)(1/2)) | √(1/4π) |
|
p-orbitals |
|||
| 1 | pz | √((1/2π)(3/2))cos(θ) | √(3/4π)(z/r) |
| px | √((1/2π)(3/2))sin(θ)cos(φ) | √(3/4π)(x/r) | |
| py | √((1/2π)(3/2))sin(θ)sin(φ) | √(3/4π)(y/r) | |
|
d-orbitals |
|||
| 2 | dz2 | √((1/2π)(5/8))(2cos2(θ) -sin2(φ)) | √(5/16π)(3z2-r2)/r2 |
| dxz | √((1/2π)(15/4))(cos(θ)sin(θ))cos(φ) | √(15/4π)(xz)/r2 | |
| dyz | √((1/2π)(15/4))(cos(θ)sin(θ))sin(φ) | √(15/4π)(yz)/r2 | |
| dx2-y2 | √((1/2π)(15/16))sin2(φ)cos(2φ) | √(15/16π)(x2-y2)/r2 | |
| dxy | √((1/2π)(15/16))sin2(φ)sin(2φ) | √(15/4π)(xy)/r2 | |
|
f-orbitals, axial set |
|||
| 3 | fz3 | √((1/2π)(7/8))(2cos3(θ) -3cos(θ)sin2(φ)) | √(7/16π)z(5z2-3r2)/r3 |
| fxz2 | √((1/2π)(21/32))(4cos2(θ)sin(θ) -sin3(θ))cos(φ) | √(21/32π)x(5z2-r2)/r3 | |
| fyz2 | √((1/2π)(21/32))(4cos2(θ)sin(θ) -sin3(θ))sin(φ) | √(21/32π)y(5z2-r2)/r3 | |
| fz(x2-y2) | √((1/2π)(105/16))(cos(θ)sin2(θ))cos(2φ) | √(105/16π)z(x2-y2)/r3 | |
| fxyz | √((1/2π)(105/16))(cos(θ)sin2(θ))sin(2φ) | √(105/16π)xyz/r3 | |
| fx(x2-3y2) | √((1/2π)(35/32))sin3(φ)cos(3φ) | √(35/32π)x(x2-3y2)/r3 | |
| f-y(y2-3x2) | √((1/2π)(35/32))sin3(φ)sin(3φ) | -√(35/32π)y(y2-3x2)/r3 | |
|
f-orbitals, cubic set |
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| fx3 | √((1/2π)(7/16))(2sin3(θ)cos3(φ)-3(sin3(θ)cos(φ)sin2(φ)+sin(θ)cos2(θ)cos(φ))) | √(7/16π)x(5x2-3r2)/r3 | |
| fy3 | √((1/2π)(7/16))(2sin3(θ)sin3(φ)-3(sin3(θ)sin(φ)cos2(φ)+sin(θ)cos2(θ)sin(φ))) | √(7/16π)y(5y2-3r2)/r3 | |
| fz3 | √((1/2π)(7/8))(2cos3(θ) -3cos(θ)sin2(φ)) | √(7/16π)z(5z2-3r2)/r3 | |
| fx(z2-y2) | √((1/2π)(105/8))(sin(θ)cos2(θ)cos(φ)-sin3(θ)cos(φ)sin2(φ)) | √(105/16π)x(z2-y2)/r3 | |
| fy(z2-x2) | √((1/2π)(105/8))(sin(θ)cos2(θ)sin(φ)-sin3(θ)sin(φ)cos2(φ)) | √(105/16π)y(z2-x2)/r3 | |
| fz(x2-y2) | √((1/2π)(105/16))(cos(θ)sin2(θ))cos(2φ) | √(105/16π)z(x2-y2)/r3 | |
| fxyz | √((1/2π)(105/16))(cos(θ)sin2(θ))sin(2φ) | √(105/16π)xyz/r3 |