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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 (5 for the 6g orbitals)
• k = various constants
Table of equations for the 6g orbitals.
Function Equation
Radial wave function, R6g = (1/900√70) × (8 - ρ)ρ4 × Z3/2 × e-ρ/2
Angular wave functions:
Y6gz4 = k × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2
Y6gz3x = k × xz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2
Y6gz3y = k × yz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2
Y6gz2xy = k × xy(6z2 - x2 - y2)/r4 × (1/4π)1/2
Y6gz2(x2 - y2) = k × (x2-y2)(6z2 - x2 - y2)/r4 × (1/4π)1/2
Y6gzx3 = k × xz(x2 - 3y2)/r4 × (1/4π)1/2
Y6gzy3 = k × yz(3x2 - y2)/r4 × (1/4π)1/2
Y6gxy(x2-y2) = k × xy(x2 - y2)/r4 × (1/4π)1/2
Y6gx4 + y4 = k × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2
Wave functions:
ψ6gz4 = R6g × Y6gz4
ψ6gz3x = R6g × Y6gz3x
ψ6gz3y = R6g × Y6gz3y
ψ6gz2xy = R6g × Y6gz2xy
ψ6gz2(x2 - y2) = R6g × Y6gz2(x2 - y2)
ψ6gzx3 = R6g × Y6gzx3
ψ6gzy3 = R6g × Y6gzy3
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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 The Orbitron is a gallery of orbitals on the WWW The OrbitronTM, a gallery of orbitals on the WWW, URL: http://winter.group.shef.ac.uk/orbitron/ Copyright 2002-2015 Prof Mark Winter [The University of Sheffield]. All rights reserved. Document served: Tuesday 17th July, 2018