# Hund's rules

In atomic spectroscopy, **Hund's rules** predict the order of atomic energy levels with quantum numbers *L* (orbital), *S* (spin) and *J* (orbital plus spin). The rules are called after Friedrich Hund who formulated them in 1925.^{[1]}

## LS coupling

A group of atomic energy levels, obtained by Russell-Saunders coupling, is concisely indicated by a term symbol. As discussed in the article Russell-Saunders coupling, closed shells and closed subshells have *L* = *S* = 0 and hence can be ignored in the coupling. A *term* (also known as *multiplet*) is a set of simultaneous eigenfunctions of **L**^{2} (total orbital angular momentum squared) and **S**^{2} (total spin angular momentum squared) with given quantum numbers *L* and *S*, respectively. That is, the respective eigenvalues are *L(L+1)*ℏ^{2}
and *S(S+1)*ℏ^{2}.

If there is no spin-orbit coupling, the functions of one term (fixed *L* and *S*) are (2*L*+1)×(2*S*+1)-fold degenerate (have the same energy). If there is weak spin-orbit coupling it is useful to diagonalize the matrix of the corresponding spin-orbit operator within the *L-S* basis [consisting of the (2*L*+1)×(2*S*+1) functions of the term] in the spirit of first-order perturbation theory. This lifts partially the degeneracy and introduces the new conserved quantum number *J*, with |*L*-*S*| ≤ *J* ≤ *L*+*S*, that labels a (2*J*+1)-fold degenerate energy level.

## Formulation of the rules

Hund's rules are:^{[2]}

- Of the Russell-Saunders states arising from a given electron configuration those with the largest spin quantum number
*S*lie lowest, those with the next largest next, and so on; in other words, the states with largest spin multiplicity are the most stable. - Of the group of terms with a given value of
*S*, that with the largest value of*L*lies lowest. - Of the states with given values of
*S*and*L*in an electronic configuration consisting of less than half the electrons in a closed subshell, the state with the smallest value of*J*is usually the most stable, and for a configuration consisting of more than half the electrons in a closed subshell the state with largest*J*is the most stable.

The levels of the second sort, largest *J* most stable, can be seen as arising from holes in a closed subshell.

## Examples

- The ground state carbon atom, (1
*s*)^{2}(2*s*)^{2}(2*p*)^{2}, gives by Russell-Saunders coupling a set of energy levels labeled by term symbols. Hund's rules predict the following order of the energies:

- The ground state oxygen atom, (1
*s*)^{2}(2*s*)^{2}(2*p*)^{4}, (a two-hole state) gives by Russell-Saunders coupling a set of energy levels labeled by term symbols. Hund's rules predict the following order of the energies:

## References

- ↑ F. Hund,
*Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel.*[On the interpretation of complicated spectra, in particular the elements scandium through nickel]. Zeitschrift für Physik, vol.**33**, pp. 345-371 (1925). - ↑ L. Pauling,
*The Nature of the Chemical Bond*, Cornell University Press, Ithaca, 3rd edition (1960)