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Atomic orbitals: 3d 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, R3d = (1/9√30) × ρ2 × Z3/2 × e-ρ/2
Angular wave functions:
Y3dxy = √(60/4)xy/r2 × (1/4π)1/2
Y3dxz = √(60/4)xz/r2 × (1/4π)1/2
Y3dyz = √(60/4)yz/r2 × (1/4π)1/2
Y3dx2-y2 = √(15/4)(x2 - y2)/r2 × (1/4π)1/2
Y3dz2 = √(5/4){2z2-(x2 + y2)}/r2 × (1/4π)1/2
Wave functions:
ψ3dxy = R3d × Y3dxy
ψ3dxz = R3d × Y3dxz
ψ3dyz = R3d × Y3dyz
ψ3dx2-y2 = R3d × Y3dx2-y2
ψ3dz2 = R3d × Y3dz2
Electron density = ψ3d2
Radial distribution function = r2R3d2

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 "3d2z2-x2-y2" orbital is abbreviated to 3dz2 for simplicity.

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.

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