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Atomic orbitals: 7d 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 (7 for the 7d orbitals)
Table of equations for the 7d orbitals.
Function Equation
Radial wave function, R7d = (1/864√105) × {336 - 168ρ + 24ρ2 - ρ3}ρ2 × Z3/2 × e-ρ/2
Angular wave functions:
Y7dxy = √(60/4)xy/r2 × (1/4π)1/2
Y7dxz = √(60/4)xz/r2 × (1/4π)1/2
Y7dyz = √(60/4)yz/r2 × (1/4π)1/2
Y7dx2-y2 = √(15/4)(x2 - y2)/r2 × (1/4π)1/2
Y7dz2 = √(5/4){2z2-(x2 + y2)}/r2 × (1/4π)1/2
Wave functions:
ψ7dxy = R7d × Y7dxy
ψ7dxz = R7d × Y7dxz
ψ7dyz = R7d × Y7dyz
ψ7dx2-y2 = R7d × Y7dx2-y2
ψ7dz2 = R7d × Y7dz2
Electron density = ψ7d2
Radial distribution function = r2R7d2

There are five real 7d orbitals. The radial equations for all the 7d 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 "7d2z2-x2-y2" orbital is abbreviated to 7dz2 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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