In rotational-vibrational and electronic spectroscopy of diatomic molecules, Hund's coupling cases are idealized descriptions of rotational states in which specific terms in the molecular Hamiltonian and involving couplings between angular momenta are assumed to dominate over all other terms.
There are five cases, proposed by Friedrich Hund in 1926-27[1] and traditionally denoted by the letters (a) through (e).
Most diatomic molecules are somewhere between the idealized cases (a) and (b).
[2] To describe the Hund's coupling cases, we use the following angular momenta (where boldface letters indicate vector quantities): These vector quantities depend on corresponding quantum numbers whose values are shown in molecular term symbols used to identify the states.
For example, the term symbol 2Π3/2 denotes a state with S = 1/2, Λ = 1 and J = 3/2.
Hund's coupling cases are idealizations.
The appropriate case for a given situation can be found by comparing three strengths: the electrostatic coupling of
[3] For other states, Hund proposed five possible idealized modes of coupling.
[4] The last two rows are degenerate because they have the same good quantum numbers.
[5] In practice there are also many molecular states which are intermediate between the above limiting cases.
is electrostatically coupled to the internuclear axis, and
As they are written with the same Greek symbol, the spin component
Combined with the rotational angular momentum of the nuclei
around the nuclear axis is assumed to be much faster than the nutation of
is strongly coupled to the electrostatic field and therefore precesses rapidly around the internuclear axis with an undefined magnitude.
[6] We express the rotational energy operator as
fine-structure states, each with rotational levels having relative energies
[7] The selection rules for allowed spectroscopic transitions depend on which quantum numbers are good.
[8] In addition, symmetrical diatomic molecules have even (g) or odd (u) parity and obey the Laporte rule that only transitions between states of opposite parity are allowed.
precesses quickly around the internuclear axis.
We express the rotational energy operator as
The rotational levels therefore have relative energies
[9] Another example is the 3Σ ground state of dioxygen, which has two unpaired electrons with parallel spins.
[10] For case b) the selection rules for quantum numbers
However for the rotational levels, the rule for quantum number
, which has a projection along the internuclear axis of magnitude
is undefined for this case, the states cannot be described as
and parity are valid as for cases (a) and (b), but there are no rules for
is a good quantum number, the rotational energy is simply
is once again a good quantum number, the rotational energy is