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Atomic orbitals: 5f 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 5f orbitals)
Table of equations for the 5f orbitals.
Function Equation
Radial wave function, R5f = (1/300√70) × (8 - ρ)ρ3 × Z3/2 × e-ρ/2
Angular wave functions (cubic set):
Y5fx3 = √(7/4) × x(5x2 - 3r2)/r3 × (1/4π)1/2
Y5fy3 = √(7/4) × y(5y2 - 3r2)/r3 × (1/4π)1/2
Y5fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y5fx(z2-y2) = √(105/4) × x(z2-y2)/r3 × (1/4π)1/2
Y5fy(z2-x2) = √(105/4) × y(z2-x2)/r3 × (1/4π)1/2
Y5fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Y5fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Angular wave functions (general set):
Y5fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y5fxz2 = √(42/16) × x(5z2 - r2)/r3 × (1/4π)1/2
Y5fyz2 = √(42/16) × y(5z2 - r2)/r3 × (1/4π)1/2
Y5fy(3x2-y2) = √(70/16) × y(3x2-y2)/r3 × (1/4π)1/2
Y5fx(x2-3y2) = √(70/16) × x(x2-3y2)/r3 × (1/4π)1/2
Y5fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Y5fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Wave functions (cubic set):
ψ5fx3 = R5f × Y5fx3
ψ5fy3 = R5f × Y5fy3
ψ5fz3 = R5f × Y5fz3
ψ5fx(z2-y2) = R5f × Y5fx(z2-y2)
ψ5fy(z2-x2) = R5f × Y5fy(z2-x2)
ψ5fz(x2-y2) = R5f × Y5fz(x2-y2)
ψ5fxyz = R5f × Y5fxyz
Wave functions (general set):
ψ5fz3 = R5f × Y5fz3
ψ5fxz2 = R5f × Y5fxz2
ψ5fyz2 = R5f × Y5fyz2
ψ5fy(3x2-y2) = R5f × Y5fy(3x2-y2)
ψ5fx(x2-3y2) = R5f × Y5fx(x2-3y2)
ψ5fxyz = R5f × Y5fxyz
ψ5fz(x2-y2) = R5f × Y5fz(x2-y2)
Electron density = ψ5f2
Radial distribution function = r2R5f2

For any atom, there are seven 5f 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 5fxyz, 5fz3, and 5fz(x2-y2) orbitals.

The radial equations for all the 5f 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:

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