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Atomic orbitals: 5p 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 (5 for the 5p orbitals)
Table of equations for the 5p orbital.
Function Equation
Radial wave function, R5px = (1/150√30) × ρ(120 - 90ρ + 18ρ2 - ρ3) × Z3/2 × e-ρ/2
Angular wave function, Y5px = √(3)x/r × (1/4π)1/2
Wave function, ψ5px = R5px × Y5px
Electron density = ψ5px2
Radial distribution function = r2R5px2

The corresponding radial equations for the 5py and 5pz orbitals are the same. The angular functions are the same but substitute y and z as appropriate in the formula for Y5px given above. Substitute similarly for the wave equations ψ5py and ψ5pz.

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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