You are at: University of Sheffield » Chemistry » Mark Winter » Orbitron (atomic orbitals and molecular orbitals)
Chemistry books (USA)
Chemistry books (UK) WebElements Chemdex Chemputer
Introduction Wave function Electron density Dots! Radial distribution Equations

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.

Orbitron logo
Google
 
Web winter.group.shef.ac.uk
www.shef.ac.uk
Copyright Feedback The images Acknowledgments Problems? References

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: Thursday 24th May, 2018