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Atomic orbitals: 4f 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 (4 for the 4f orbitals)
Table of equations for the 4f orbitals.
Function Equation
Radial wave function, R4f = (1/96√35) × ρ3 × Z3/2 × e-ρ/2
Angular wave functions (cubic set):
Y4fx3 = √(7/4) × x(5x2 - 3r2)/r3 × (1/4π)1/2
Y4fy3 = √(7/4) × y(5y2 - 3r2)/r3 × (1/4π)1/2
Y4fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y4fx(z2-y2) = √(105/4) × x(z2-y2)/r3 × (1/4π)1/2
Y4fy(z2-x2) = √(105/4) × y(z2-x2)/r3 × (1/4π)1/2
Y4fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Y4fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Angular wave functions (general set):
Y4fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y4fxz2 = √(42/16) × x(5z2 - r2)/r3 × (1/4π)1/2
Y4fyz2 = √(42/16) × y(5z2 - r2)/r3 × (1/4π)1/2
Y4fy(3x2-y2) = √(70/16) × y(3x2-y2)/r3 × (1/4π)1/2
Y4fx(x2-3y2) = √(70/16) × x(x2-3y2)/r3 × (1/4π)1/2
Y4fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Y4fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Wave functions (cubic set):
ψ4fx3 = R4f × Y4fx3
ψ4fy3 = R4f × Y4fy3
ψ4fz3 = R4f × Y4fz3
ψ4fx(z2-y2) = R4f × Y4fx(z2-y2)
ψ4fy(z2-x2) = R4f × Y4fy(z2-x2)
ψ4fz(x2-y2) = R4f × Y4fz(x2-y2)
ψ4fxyz = R4f × Y4fxyz
Wave functions (general set):
ψ4fz3 = R4f × Y4fz3
ψ4fxz2 = R4f × Y4fxz2
ψ4fyz2 = R4f × Y4fyz2
ψ4fy(3x2-y2) = R4f × Y4fy(3x2-y2)
ψ4fx(x2-3y2) = R4f × Y4fx(x2-3y2)
ψ4fxyz = R4f × Y4fxyz
ψ4fz(x2-y2) = R4f × Y4fz(x2-y2)
Electron density = ψ4f2
Radial distribution function = r2R4f2

For any atom, there are seven 4f 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 4fxyz, 4fz3, and 4fz(x2-y2) orbitals.

The radial equations for all the 4f 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:

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