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Real Spherical Harmonics

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

 
  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

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J. J. P. Stewart
Stewart Computational Chemistry, 2007