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Atomic orbitals: 7f 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 7f orbitals)
Table of equations for the 7f orbitals.
Function Equation
Radial wave function, R7f = (1/300√70) × (8 - ρ)ρ3 × Z3/2 × e-ρ/2
Angular wave functions (cubic set):
Y7fx3 = √(7/4) × x(5x2 - 3r2)/r3 × (1/4π)1/2
Y7fy3 = √(7/4) × y(5y2 - 3r2)/r3 × (1/4π)1/2
Y7fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y7fx(z2-y2) = √(105/4) × x(z2-y2)/r3 × (1/4π)1/2
Y7fy(z2-x2) = √(105/4) × y(z2-x2)/r3 × (1/4π)1/2
Y7fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Y7fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Angular wave functions (general set):
Y7fz3 = √(7/4) × z(5z2 - 3r2)/r3 × (1/4π)1/2
Y7fxz2 = √(42/16) × x(5z2 - r2)/r3 × (1/4π)1/2
Y7fyz2 = √(42/16) × y(5z2 - r2)/r3 × (1/4π)1/2
Y7fy(3x2-y2) = √(70/16) × y(3x2-y2)/r3 × (1/4π)1/2
Y7fx(x2-3y2) = √(70/16) × x(x2-3y2)/r3 × (1/4π)1/2
Y7fxyz = √(105/4) × xyz/r3 × (1/4π)1/2
Y7fz(x2-y2) = √(105/4) × z(x2-y2)/r3 × (1/4π)1/2
Wave functions (cubic set):
ψ7fx3 = R7f × Y7fx3
ψ7fy3 = R7f × Y7fy3
ψ7fz3 = R7f × Y7fz3
ψ7fx(z2-y2) = R7f × Y7fx(z2-y2)
ψ7fy(z2-x2) = R7f × Y7fy(z2-x2)
ψ7fz(x2-y2) = R7f × Y7fz(x2-y2)
ψ7fxyz = R7f × Y7fxyz
Wave functions (general set):
ψ7fz3 = R7f × Y7fz3
ψ7fxz2 = R7f × Y7fxz2
ψ7fyz2 = R7f × Y7fyz2
ψ7fy(3x2-y2) = R7f × Y7fy(3x2-y2)
ψ7fx(x2-3y2) = R7f × Y7fx(x2-3y2)
ψ7fxyz = R7f × Y7fxyz
ψ7fz(x2-y2) = R7f × Y7fz(x2-y2)
Electron density = ψ7f2
Radial distribution function = r2R7f2

For any atom, there are seven 7f 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 7fxyz, 7fz3, and 7fz(x2-y2) orbitals.

The radial equations for all the 7f 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:

  • 7fx3 used for 7fx(5x2 - 3r2) (=7fx(2x2 - 3y2 - 3z2) since r2 = x2 + y2 + z2)
  • 7fy3 used for 7fy(5y2 - 3r2) (=7fy(2y2 - 3x2 - 3z2) since r2 = x2 + y2 + z2)
  • 7fz3 used for 7fz(5z2 - 3r2) (=7fz(2z2 - 3x2 - 3y2) since r2 = x2 + y2 + z2)
  • 7fxz2 used for 7fx(5z2 - r2) (=7fx(4z2 - x2 - y2) since r2 = x2 + y2 + z2)
  • 7fyz2 used for 7fy(5z2 - r2) (=7fy(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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