8/16/12 Real Spherical Harmonics

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Real Spherical Harmonics

The following table contains the normalized real spherical harmonics of orders 0 to 4 (s to f).

l SymbolTrigonometric form Cartesian form

s-orbital

0 s√((1/2p)(1/2)) √(1/4p)

p-orbitals

1 pz√((1/2p)(3/2))cos(q) √(3/4p)(z/r)

px√((1/2p)(3/2))sin(q)cos(f) √(3/4p)(x/r)

py√((1/2p)(3/2))sin(q)sin(f) √(3/4p)(y/r)

d-orbitals

2 dz2√((1/2p)(5/8))(2cos2(q) -sin2(q)) √(5/16p)(3z2-r2)/r2

dxz√((1/2p)(15/4))(cos(q)sin(q))cos(f) √(15/4p)(xz)/r2

dyz√((1/2p)(15/4))(cos(q)sin(q))sin(f) √(15/4p)(yz)/r2

dx2-y2√((1/2p)(15/16))sin2(q)cos2(f) √(15/16p)(x2-y2)/r2

dxy√((1/2p)(15/16))sin2(q)sin2(f) √(15/4p)(xy)/r2

f-orbitals, axial set

3 fz3√((1/2p)(7/8))(2cos3(q) -3cos(q)sin2(q)) √(7/16p)z(5z2-3r2)/r3

fxz2√((1/2p)(21/32))(4cos2(q)sin(q) -sin3(q))cos(f) √(21/32p)x(5z2-r2)/r3

fyz2√((1/2p)(21/32))(4cos2(q)sin(q) -sin3(q))sin(f) √(21/32p)y(5z2-r2)/r3

fz(x2-y2)√((1/2p)(105/16))(cos(q)sin2(q))cos2(f) √(105/16p)z(x2-y2)/r3

fxyz√((1/2p)(105/16))(cos(q)sin2(q))sin2(f) √(105/16p)xyz/r3

fx(x2-3y2)√((1/2p)(35/32))sin3(q)cos3(f) √(35/32p)x(x2-3y2)/r3

f-y(y2-

3x2)√((1/2p)(35/32))sin3(q)sin3(f) -√(35/32p)y(y2-3x2)/r3

f-orbitals, cubic set

fx3√((1/2p)(7/16))

(2sin3(q)cos3(f)-3(sin3(q)cos(f)sin2(f)+sin(q)cos2(q)cos(f)))√(7/16p)x(5x2-3r2)/r3

fy3√((1/2p)(7/16))(2sin3(q)sin3(f)-3(sin3(q)sin(f)cos2(f)+sin(q)cos2(q)sin(f))) √(7/16p)y(5y2-3r2)/r3

fz3√((1/2p)(7/8))(2cos3(q) -3cos(q)sin2(q)) √(7/16p)z(5z2-3r2)/r3

fx(z2-y2)√((1/2p)(105/8))(sin(q)cos2(q)cos(f)-sin3(q)cos(f)sin2(f)) √(105/16p)x(z2-y2)/r3

fy(z2-x2)√((1/2p)(105/8))(sin(q)cos2(q)sin(f)-sin3(q)sin(f)cos2(f)) √(105/16p)y(z2-x2)/r3

8/16/12 Real Spherical Harmonics

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fz(x2-y2)√((1/2p)(105/16))(cos(q)sin2(q))cos2(f) √(105/16p)z(x2-y2)/r3

fxyz√((1/2p)(105/16))(cos(q)sin2(q))sin2(f) √(105/16p)xyz/r3

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