Atomic orbitals: 5g 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 5g orbitals)
  • k = various constants
Table of equations for the 5g orbitals.
Function Equation
Radial wave function, R5g = (1/900√70) × ρ4 × Z3/2 × e-ρ/2
Angular wave functions:
Y5gz4 = √(9/64) × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2
Y5gz3y = √(45/8) × yz(7z2 - 3r2)/r4 × (1/4π)1/2
Y5gz3x = √(45/8) × xz(7z2 - 3r2)/r4 × (1/4π)1/2
Y5gz2xy = √(45/16) × 2xy(7z2 - r2)/r4 × (1/4π)1/2
Y5gz2(x2 - y2) = √(45/16) × (x2-y2)(7z2 - r2)/r4 × (1/4π)1/2
Y5gzy3 = √(315/8) × yz(3x2 - y2)/r4 × (1/4π)1/2
Y5gzx3 = √(315/8) × xz(x2 - 3y2)/r4 × (1/4π)1/2
Y5gxy(x2-y2) = √(315/64) × 4xy(x2 - y2)/r4 × (1/4π)1/2
Y5g(x4 + y4) = √(315/64) × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2
Wave functions:
ψ5gz4 = R5g × Y5gz4
ψ5gz3y = R5g × Y5gz3y
ψ5gz3x = R5g × Y5gz3x
ψ5gz2xy = R5g × Y5gz2xy
ψ5gz2(x2 - y2) = R5g × Y5gz2(x2 - y2)
ψ5gzy3 = R5g × Y5gzy3
ψ5gzx3 = R5g × Y5gzx3
ψ5gxy(x2-y2) = R5g × Y5gxy(x2-y2)
ψ5g(x4 + y4) = R5g × Y5g(x4 + y4)
Electron density = ψ5g2
Radial distribution function = r2R5g2

The radial equations for all the 5g orbitals are the same. The real angular functions differ for each and these are listed above.

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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