# Atomic orbitals: 2*p* 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*= 2*Zr*/*n*where*n*is the principal quantum number (2 for the 2*p*orbitals)

Function | Equation |
---|---|

Radial wave function, R_{2p} |
= (1/2√6) × ρ × Z^{3/2} × e^{-ρ/2} |

Angular wave function, Y_{2px} |
= √(3)x/r × (1/4π)^{1/2} |

Wave function, ψ_{2px} |
= R_{2p} × Y_{2px} |

Electron density | = ψ_{2px}^{2} |

Radial distribution function | = r^{2}R_{2p}^{2} |

The radial equation for the 2*p*_{x}, 2*p*_{y}, and 2*p*_{z} orbitals is the same in each case. The angular functions are the same but substitute *y* and *z* as appropriate in the formula for *Y*_{2px} given above. Substitute similarly for the wave equations *ψ*_{2py} and *ψ*_{2pz}.

For *s*-orbitals the radial distribution function is given by 4π*r*^{2}*ψ*^{2}, 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

^{TM}, 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.