# Atomic orbitals: 5*g* 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 (5 for the 5*g*orbitals)*k*= various constants

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

Radial wave function, R_{5g} |
= (1/900√70) × ρ^{4} × Z^{3/2} × e^{-ρ/2} |

Angular wave functions: | |

Y_{5gz4} |
= √(9/64) × (35z^{4} - 30z^{2}r^{2} + 3r^{4})/r^{4} × (1/4π)^{1/2} |

Y_{5gz3y} |
= √(45/8) × yz(7z^{2} - 3r^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gz3x} |
= √(45/8) × xz(7z^{2} - 3r^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gz2xy} |
= √(45/16) × 2xy(7z^{2} - r^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gz2(x2 - y2)} |
= √(45/16) × (x^{2}-y^{2})(7z^{2} - r^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gzy3} |
= √(315/8) × yz(3x^{2} - y^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gzx3} |
= √(315/8) × xz(x^{2} - 3y^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5gxy(x2-y2)} |
= √(315/64) × 4xy(x^{2} - y^{2})/r^{4} × (1/4π)^{1/2} |

Y_{5g(x4 + y4)} |
= √(315/64) × (x^{4} + y^{4} - 6x^{2}y^{2})/r^{4} × (1/4π)^{1/2} |

Wave functions: | |

ψ_{5gz4} |
= R_{5g} × Y_{5gz4} |

ψ_{5gz3y} |
= R_{5g} × Y_{5gz3y} |

ψ_{5gz3x} |
= R_{5g} × Y_{5gz3x} |

ψ_{5gz2xy} |
= R_{5g} × Y_{5gz2xy} |

ψ_{5gz2(x2 - y2)} |
= R_{5g} × Y_{5gz2(x2 - y2)} |

ψ_{5gzy3} |
= R_{5g} × Y_{5gzy3} |

ψ_{5gzx3} |
= R_{5g} × Y_{5gzx3} |

ψ_{5gxy(x2-y2)} |
= R_{5g} × Y_{5gxy(x2-y2)} |

ψ_{5g(x4 + y4)} |
= R_{5g} × Y_{5g(x4 + y4)} |

Electron density | = ψ_{5g}^{2} |

Radial distribution function | = r^{2}R_{5g}^{2} |

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

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.