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Atomic orbitals: 2p 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 (2 for the 2p orbitals)
Table of equations for the 2p orbital.
Function Equation
Radial wave function, R2p = (1/2√6) × ρ × Z3/2 × e-ρ/2
Angular wave function, Y2px = √(3)x/r × (1/4π)1/2
Wave function, ψ2px = R2p × Y2px
Electron density = ψ2px2
Radial distribution function = r2R2p2

The radial equation for the 2px, 2py, and 2pz orbitals is the same in each case. The angular functions are the same but substitute y and z as appropriate in the formula for Y2px given above. Substitute similarly for the wave equations ψ2py and ψ2pz.

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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