# Atomic orbitals: 6*f* 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 (6 for the 6*f*orbitals)

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

Radial wave function, R_{6f} |
= (1/2592√35) × (72 - 18ρ + ρ^{2})ρ^{3} × Z^{3/2} × e^{-ρ/2} |

Angular wave functions (general set): | |

Y_{6fz3} |
= √(7/4) × z(5z^{2} - 3r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fyz2} |
= √(42/16) × y(5z^{2} - r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fxz2} |
= √(42/16) × x(5z^{2} - r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fxyz} |
= √(105/4) × 2xyz/r^{3} × (1/4π)^{1/2} |

Y_{6fz(x2-y2)} |
= √(105/4) × z(x^{2}-y^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fy(3x2-y2)} |
= √(70/16) × y(3x^{2}-y^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fx(x2-3y2)} |
= √(70/16) × x(x^{2}-3y^{2})/r^{3} × (1/4π)^{1/2} |

Angular wave functions (cubic set): | |

Y_{6fy3} |
= √(7/4) × y(5y^{2} - 3r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fz3} |
= √(7/4) × z(5z^{2} - 3r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fx3} |
= √(7/4) × x(5x^{2} - 3r^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fy(z2-x2)} |
= √(105/4) × y(z^{2}-x^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fz(x2-y2)} |
= √(105/4) × z(x^{2}-y^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fx(z2-y2)} |
= √(105/4) × x(z^{2}-y^{2})/r^{3} × (1/4π)^{1/2} |

Y_{6fxyz} |
= √(105/4) × 2xyz/r^{3} × (1/4π)^{1/2} |

Wave functions (general set): | |

ψ_{6fz3} |
= R_{6f} × Y_{6fz3} |

ψ_{6fyz2} |
= R_{6f} × Y_{6fyz2} |

ψ_{6fxz2} |
= R_{6f} × Y_{6fxz2} |

ψ_{6fxyz} |
= R_{6f} × Y_{6fxyz} |

ψ_{6fz(x2-y2)} |
= R_{6f} × Y_{6fz(x2-y2)} |

ψ_{6fy(3x2-y2)} |
= R_{6f} × Y_{6fy(3x2-y2)} |

ψ_{6fx(x2-3y2)} |
= R_{6f} × Y_{6fx(x2-3y2)} |

Wave functions (cubic set): | |

ψ_{6fy3} |
= R_{6f} × Y_{6fy3} |

ψ_{6fz3} |
= R_{6f} × Y_{6fz3} |

ψ_{6fx3} |
= R_{6f} × Y_{6fx3} |

ψ_{6fy(z2-x2)} |
= R_{6f} × Y_{6fy(z2-x2)} |

ψ_{6fz(x2-y2)} |
= R_{6f} × Y_{6fz(x2-y2)} |

ψ_{6fx(z2-y2)} |
= R_{6f} × Y_{6fx(z2-y2)} |

ψ_{6fxyz} |
= R_{6f} × Y_{6fxyz} |

Electron density | = ψ_{6f}^{2} |

Radial distribution function | = r^{2}R_{6f}^{2} |

For any atom, there are seven 6*f* orbitals. The *f*-orbitals are unusual in that there are two sets of orbitals in common use. The *cubic set* is appropriate to use if the atom is in a cubic environment. The *general set* is used at other times. Three of the orbitals are common to both sets. These are are the 6*f*_{xyz}, 6*f*_{z3}, and 6*f*_{z(x2-y2)} orbitals.

The radial equations for all the 6*f* 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 some abbreviated names are useful for simplicity. These are:

- 6
*f*_{x3}used for 6*f*_{x(5x2 - 3r2)} - 6
*f*_{y3}used for 6*f*_{y(5y2 - 3r2)} - 6
*f*_{z3}used for 6*f*_{z(5z2 - 3r2)} - 6
*f*_{xz2}used for 6*f*_{x(5z2 - r2)} - 6
*f*_{yz2}used for 6*f*_{y(5z2 - r2)}

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 2002-2021 Prof. Mark Winter [The University of Sheffield]. All rights reserved.