# 3*d* atomic orbitals 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 (3 for the 3*d*orbitals)

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

Radial wave function, R_{3d} |
= (1/9√30) × ρ^{2} × Z^{3/2} × e^{-ρ/2} |

Angular wave functions: | |

Y_{3dz2} |
= √(5/4) × (3z^{2} – r^{2})/r^{2} × (1/4π)^{1/2} |

Y_{3dyz} |
= √(60/4) × yz/r^{2} × (1/4π)^{1/2} |

Y_{3dxz} |
= √(60/4) × xz/r^{2} × (1/4π)^{1/2} |

Y_{3dxy} |
= √(15/4) × 2xy/r^{2} × (1/4π)^{1/2} |

Y_{3dx2-y2} |
= √(15/4) × (x^{2} - y^{2})/r^{2} × (1/4π)^{1/2} |

Wave functions: | |

ψ_{3dz2} |
= R_{3d} × Y_{3dz2} |

ψ_{3dyz} |
= R_{3d} × Y_{3dyz} |

ψ_{3dxz} |
= R_{3d} × Y_{3dxz} |

ψ_{3dxy} |
= R_{3d} × Y_{3dxy} |

ψ_{3dx2-y2} |
= R_{3d} × Y_{3dx2-y2} |

Electron density | = ψ_{3d}^{2} |

Radial distribution function | = r^{2}R_{3d}^{2} |

There are five real 3d orbitals. The radial equations for all the 3*d* 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 3*d*_{3z2 – r2} orbital is abbreviated to 3*d*_{z2}.

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 Winter [The University of Sheffield]. All rights reserved.