Atomic orbitals: 6g 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 (6 for the 6g orbitals)
Table of equations for the 6g orbitals.
Function Equation
Radial wave function, R6g = (1/12960√7) × (8 – ρ)ρ4 × Z3/2 × e-ρ/2
Angular wave functions:
Y6gz4 = √(9/64) × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2
Y6gz3y = √(45/8) × yz(7z2 - 3r2)/r4 × (1/4π)1/2
Y6gz3x = √(45/8) × xz(7z2 - 3r2)/r4 × (1/4π)1/2
Y6gz2xy = √(45/16) × 2xy(7z2 - r2)/r4 × (1/4π)1/2
Y6gz2(x2 - y2) = √(45/16) × (x2-y2)(7z2 - r2)/r4 × (1/4π)1/2
Y6gzy3 = √(315/8) × yz(3x2 - y2)/r4 × (1/4π)1/2
Y6gzx3 = √(315/8) × xz(x2 - 3y2)/r4 × (1/4π)1/2
Y6gxy(x2-y2) = √(315/64) × 4xy(x2 - y2)/r4 × (1/4π)1/2
Y6g(x4 + y4) = √(315/64) × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2
Wave functions:
ψ6gz4 = R6g × Y6gz4
ψ6gz3y = R6g × Y6gz3y
ψ6gz3x = R6g × Y6gz3x
ψ6gz2xy = R6g × Y6gz2xy
ψ6gz2(x2 - y2) = R6g × Y6gz2(x2 - y2)
ψ6gzy3 = R6g × Y6gzy3
ψ6gzx3 = R6g × Y6gzx3
ψ6gxy(x2-y2) = R6g × Y6gxy(x2-y2)
ψ6g(x4 + y4) = R6g × Y6g(x4 + y4)
Electron density = ψ6g2
Radial distribution function = r2R6g2

The radial equations for all the 6g 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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