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# Atomic orbitals: 5d 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 5d orbitals)
Table of equations for the 5d orbitals.
Function Equation
Radial wave function, R5d = (1/150√70) × {42 - 14ρ + ρ2}ρ2 × Z3/2 × e-ρ/2
Angular wave functions:
Y5dxy = √(60/4)xy/r2 × (1/4π)1/2
Y5dxz = √(60/4)xz/r2 × (1/4π)1/2
Y5dyz = √(60/4)yz/r2 × (1/4π)1/2
Y5dx2-y2 = √(15/4)(x2 - y2)/r2 × (1/4π)1/2
Y5dz2 = √(5/4){2z2-(x2 + y2)}/r2 × (1/4π)1/2
Wave functions:
ψ5dxy = R5d × Y5dxy
ψ5dxz = R5d × Y5dxz
ψ5dyz = R5d × Y5dyz
ψ5dx2-y2 = R5d × Y5dx2-y2
ψ5dz2 = R5d × Y5dz2
Electron density = ψ5d2

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

Each of the orbitals is named for the expression based upon x, y, and z in the angular wave function, but the "5d2z2-x2-y2" orbital is abbreviated to 5dz2 for simplicity.

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: Saturday 21st July, 2018