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# Atomic orbitals: 7g 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 7g orbitals)
• k = various constants
Table of equations for the 7g orbitals.
Function Equation
Radial wave function, R7g = (1/900√70) × (= - =ρ + =ρ2)ρ4 × Z3/2 × e-ρ/2
Angular wave functions:
Y7gz4 = k × (35z4 - 30z2r2 + 3r4)/r4 × (1/4π)1/2
Y7gz3x = k × xz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2
Y7gz3y = k × yz(4z2 - 3x2 - 3y2)/r4 × (1/4π)1/2
Y7gz2xy = k × xy(6z2 - x2 - y2)/r4 × (1/4π)1/2
Y7gz2(x2 - y2) = k × (x2-y2)(6z2 - x2 - y2)/r4 × (1/4π)1/2
Y7gzx3 = k × xz(x2 - 3y2)/r4 × (1/4π)1/2
Y7gzy3 = k × yz(3x2 - y2)/r4 × (1/4π)1/2
Y7gxy(x2-y2) = k × xy(x2 - y2)/r4 × (1/4π)1/2
Y7gx4 + y4 = k × (x4 + y4 - 6x2y2)/r4 × (1/4π)1/2
Wave functions:
ψ7gz4 = R7g × Y7gz4
ψ7gz3x = R7g × Y7gz3x
ψ7gz3y = R7g × Y7gz3y
ψ7gz2xy = R7g × Y7gz2xy
ψ7gz2(x2 - y2) = R7g × Y7gz2(x2 - y2)
ψ7gzx3 = R7g × Y7gzx3
ψ7gzy3 = R7g × Y7gzy3
Electron density = ψ7g2
Radial distribution function = r2R7g2

The radial equations for all the 7g 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.

 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: Wednesday 21st October, 2020