Atomic orbitals: 2p 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 (2 for the 2p orbitals)
|Radial wave function, R2p||= (1/2√6) × ρ × Z3/2 × e-ρ/2|
|Angular wave function, Y2px||= √(3)x/r × (1/4π)1/2|
|Wave function, ψ2px||= R2p × Y2px|
|Electron density||= ψ2px2|
|Radial distribution function||= r2R2p2|
The radial equation for the 2px, 2py, and 2pz orbitals is the same in each case. The angular functions are the same but substitute y and z as appropriate in the formula for Y2px given above. Substitute similarly for the wave equations ψ2py and ψ2pz.
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 OrbitronTM, a gallery of orbitals on the WWW: https://winter.group.shef.ac.uk/orbitron/
Copyright 2002-2023 Prof Mark J. Winter [Department of Chemistry, The University of Sheffield]. All rights reserved.