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Atomic orbitals: 6f 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 6f orbitals)
Table of equations for the 6f orbitals.
Function Equation
Radial wave function, R6f = (1/300√70) × (8 - ρ)ρ3 × Z3/2 × e-ρ/2
Angular wave functions (cubic set):
Y6fx3 = √(7/4) × x(5x2 - 3r2)/r3 × (1/4π)1/2
Y6fy3 = √(7/4) × y(5y2 - 3r2)/r3 × (1/4π)1/2
Y6fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y6fx(z2-y2) = √(105/4) × x(z2-y2)/r3 × (1/4π)1/2
Y6fy(z2-x2) = √(105/4) × y(z2-x2)/r3 × (1/4π)1/2
Y6fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Y6fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Angular wave functions (general set):
Y6fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y6fxz2 = √(42/16) × x(5z2 - r2)/r3 × (1/4π)1/2
Y6fyz2 = √(42/16) × y(5z2 - r2)/r3 × (1/4π)1/2
Y6fy(3x2-y2) = √(70/16) × y(3x2-y2)/r3 × (1/4π)1/2
Y6fx(x2-3y2) = √(70/16) × x(x2-3y2)/r3 × (1/4π)1/2
Y6fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Y6fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Wave functions (cubic set):
ψ6fx3 = R6f × Y6fx3
ψ6fy3 = R6f × Y6fy3
ψ6fz3 = R6f × Y6fz3
ψ6fx(z2-y2) = R6f × Y6fx(z2-y2)
ψ6fy(z2-x2) = R6f × Y6fy(z2-x2)
ψ6fz(x2-y2) = R6f × Y6fz(x2-y2)
ψ6fxyz = R6f × Y6fxyz
Wave functions (general set):
ψ6fz3 = R6f × Y6fz3
ψ6fxz2 = R6f × Y6fxz2
ψ6fyz2 = R6f × Y6fyz2
ψ6fy(3x2-y2) = R6f × Y6fy(3x2-y2)
ψ6fx(x2-3y2) = R6f × Y6fx(x2-3y2)
ψ6fxyz = R6f × Y6fxyz
ψ6fz(x2-y2) = R6f × Y6fz(x2-y2)
Electron density = ψ6f2
Radial distribution function = r2R6f2

For any atom, there are seven 6f orbitals. The f-orbitals are unusual in that there are two sets of orbitals in common use. The cubic set is appropriate to use if the atom is in a cubic environment. The general set is used at other times. Three of the orbitals are common to both sets. These are are the 6fxyz, 6fz3, and 6fz(x2-y2) orbitals.

The radial equations for all the 6f 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 some abbreviated names are useful for simplicity. These are:

  • 6fx3 used for 6fx(5x2 - 3r2) (=6fx(2x2 - 3y2 - 3z2) since r2 = x2 + y2 + z2)
  • 6fy3 used for 6fy(5y2 - 3r2) (=6fy(2y2 - 3x2 - 3z2) since r2 = x2 + y2 + z2)
  • 6fz3 used for 6fz(5z2 - 3r2) (=6fz(2z2 - 3x2 - 3y2) since r2 = x2 + y2 + z2)
  • 6fxz2 used for 6fx(5z2 - r2) (=6fx(4z2 - x2 - y2) since r2 = x2 + y2 + z2)
  • 6fyz2 used for 6fy(5z2 - r2) (=6fy(4z2 - x2 - y2) since r2 = x2 + y2 + z2)

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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