# Hund's cases

In rotational-vibrational and electronic spectroscopy of diatomic molecules, **Hund's coupling cases** are idealized cases where specific terms appearing in the molecular Hamiltonian and involving couplings between angular momenta are assumed to dominate over all other terms. There are five cases, traditionally notated with the letters (a) through (e). Most diatomic molecules are somewhere between the idealized cases (a) and (b).^{[1]}

## Angular momenta

To describe the Hund's coupling cases, we use the following angular momenta:

- , the electronic orbital angular momentum
- , the electronic spin angular momentum
- , the total electronic angular momentum
- , the total angular momentum of the system
- , the total angular momentum minus the electron spin
- , the rotational angular momentum of the nuclei

## Choosing the applicable Hund's case

Hund's coupling cases are idealizations. The appropriate case for a given situation can be found by comparing three strengths: the electrostatic coupling of to the internuclear axis, the spin-orbit coupling, and the rotational coupling of and to the total angular momentum .

Hund's case | Electrostatic | Spin-orbit | Rotational |
---|---|---|---|

(a) | strong | intermediate | weak |

(b) | strong | weak | intermediate |

(c) | intermediate | strong | weak |

(d) | intermediate | weak | strong |

(e) | weak | intermediate | strong |

strong | intermediate |

The last two rows are degenerate because they have the same good quantum numbers.^{[2]}

## Case (a)

In case (a), is electrostatically coupled to the internuclear axis, and is coupled to by spin-orbit coupling. Then both and have well-defined axial components and , respectively. defines a vector of magnitude pointing along the internuclear axis. Combined with the rotational angular momentum of the nuclei , we have . In this case, the procession of and around the nuclear axis is assumed to be much faster than the nutation of and around .

The good quantum numbers in case (a) are , , , and . We express the rotational energy operator as , where is a rotational constant. There are, ideally, fine-structure states, each with rotational levels having relative energies starting with .^{[1]}

## Case (b)

In case (b), the spin-orbit coupling is weak or non-existent (in the case ). In this case, we take and and assume precesses quickly around the internuclear axis.

The good quantum numbers in case (b) are , , , and . We express the rotational energy operator as , where is a rotational constant. The rotational levels therefore have relative energies starting with .^{[1]}

## Case (c)

In case (c), the spin-orbit coupling is stronger than the coupling to the internuclear axis, and and from case (a) cannot be defined. Instead and combine to form , which has a projection along the internuclear axis of magnitude . Then , as in case (a).

The good quantum numbers in case (c) are , , and .^{[1]}

## Case (d)

In case (d), the rotational coupling between and is much stronger than the electrostatic coupling of to the internuclear axis. Thus we form by coupling and and the form by coupling and .

The good quantum numbers in case (d) are , , , , and . Because is a good quantum number, the rotational energy is simply .^{[1]}

## Case (e)

In case (e), we first form and then form by coupling and . This case is rare but has been observed.^{[3]}

The good quantum numbers in case (e) are , , and . Because is once again a good quantum number, the rotational energy is .^{[1]}

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}^{1.5}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ Template:Cite doi
- ↑ Template:Cite doi