Hund's cases
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 and traditionally denoted by the letters (a) through (e). Most diatomic molecules are somewhere between the idealized cases (a) and (b). Angular momenta. 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 Π denotes a state with S = 1/2, Λ = 1 and J = 3/2. 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 formula_1 to the internuclear axis, the spin-orbit coupling, and the rotational coupling of formula_1 and formula_2 to the total angular momentum formula_10. For Σ states the orbital and spin angular momenta  are zero and the total angular momentum is just the nuclear rotational angular momentum. For other states, Hund proposed five possible idealized modes of coupling. The last two rows are degenerate because they have the same good quantum numbers. In practice there are also many molecular states which are intermediate between the above limiting cases. Case (a). The most common case is case (a) in which formula_1 is electrostatically coupled to the internuclear axis, and formula_2 is coupled to formula_1 by spin-orbit coupling. Then both formula_1 and formula_2 have well-defined axial components, formula_16 and formula_17 respectively. The spin  component formula_17 is not related to formula_17 states, which are states with orbital angular component formula_16 equal to zero. formula_21 defines a vector of magnitude formula_22 pointing along the internuclear axis. Combined with the rotational angular momentum of the nuclei formula_4, we have formula_24. In this case, the precession of formula_1 and formula_2 around the nuclear axis is assumed to be much faster than the nutation of formula_21 and formula_4 around formula_10. The good quantum numbers in case (a) are formula_16, formula_31, formula_17, formula_33 and formula_34. However formula_35 is not a good quantum number because the vector formula_1 is strongly  coupled to the electrostatic field and therefore precesses rapidly around the internuclear axis with an undefined magnitude. We express the rotational energy operator as formula_37, where formula_38 is a rotational constant. There are, ideally, formula_39 fine-structure states, each with rotational levels having relative energies formula_40 starting with formula_41. For example, a Π state has a Π term (or fine structure state) with rotational levels formula_10 = 1/2, 3/2, 5/2, 7/2, ... and a Π term with levels formula_10 = 3/2, 5/2, 7/2, 9/2... Case (a) requires formula_16 > 0 and so does not apply to any Σ states, and also  formula_31 > 0 so that it does not apply to any singlet states. The selection rules for allowed spectroscopic transitions depend on which quantum numbers are good. For Hund's case (a), the allowed transitions must have formula_46 and formula_47 and formula_48 and formula_49 and formula_50. 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. Case (b). In case (b), the spin-orbit coupling is weak or non-existent (in the case formula_51). In this case, we take formula_52 and formula_53 and  assume formula_1 precesses quickly around the internuclear axis. The good quantum numbers in case (b) are formula_16, formula_56, formula_31, and formula_33. We express the rotational energy operator as formula_59, where formula_38 is a rotational constant. The rotational levels therefore have relative energies formula_61 starting with formula_62. For example, a Σ state has rotational levels formula_56 = 0, 1, 2, 3, 4, ..., and each level is divided by spin-orbit coupling into two levels formula_33 = formula_56 ± 1/2 (except for formula_56 = 0 which corresponds only to formula_33 = 1/2 because formula_33 cannot be negative). Another example is the  Σ ground state of dioxygen, which has two unpaired electrons with parallel spins. The coupling type is Hund's case b), and each rotational level N is divided into three levels formula_33 = formula_70, formula_56, formula_72. For case b) the selection rules for quantum numbers formula_16, formula_31, formula_17 and formula_34 and for parity are the same as for case a). However for the rotational levels, the rule for quantum number formula_33 does not apply and is replaced by the rule formula_78. Case (c). In case (c), the spin-orbit coupling is stronger than the coupling to the internuclear axis, and  formula_16 and formula_17 from case (a) cannot be defined. Instead formula_1 and formula_2 combine to form formula_83, which has a projection along the internuclear axis of magnitude formula_34. Then formula_24, as in case (a). The good quantum numbers in case (c) are formula_86, formula_33, and formula_34. Since formula_16 is undefined for this case, the states cannot be described as formula_17, formula_91 or formula_92. An example of Hund's case (c) is the lowest Π state of diiodine (I), which approximates more closely to case (c) than to case (a). The selection rules for formula_31, formula_34 and parity are  valid as for cases (a) and (b), but there are no rules for formula_16 and formula_17 since these are not good quantum numbers for case (c). Case (d). In case (d), the rotational coupling between formula_1 and formula_4 is much stronger than the electrostatic coupling of formula_1 to the internuclear axis. Thus we form formula_100 by coupling formula_1 and formula_4 and the form formula_10 by coupling formula_100 and formula_2. The good quantum numbers in case (d) are formula_35, formula_107, formula_56, formula_31, and formula_33. Because formula_107 is a good quantum number, the rotational energy is simply formula_112. Case (e). In case  (e), we first form formula_83 and then form formula_10 by coupling formula_83 and formula_4. This case is rare but has been observed. Rydberg states which converge to ionic states with spin–orbit coupling (such as Π) are best described as case (e). The good quantum numbers in case (e) are formula_86, formula_107, and formula_33. Because formula_107 is once again a good quantum number, the rotational energy is formula_112.