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 symbol2Π3/2 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 to the internuclear axis, the spin-orbit coupling, and the rotational coupling of and to the total angular momentum. For 1Σ 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.
Hund's case
Electrostatic
Spin-orbit
Rotational
strong
intermediate
weak
strong
weak
intermediate
intermediate
strong
weak
intermediate
weak
strong
weak
intermediate
strong
weak
strong
intermediate
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 in which 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. The spin component is not related to states, which are states with orbital angular component equal to zero. 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 precession of and around the nuclear axis is assumed to be much faster than the nutation of and around. The good quantum numbers in case are,,, and. However is not a good quantum number because the vector 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, where is a rotational constant. There are, ideally, fine-structure states, each with rotational levels having relative energies starting with. For example, a 2Π state has a 2Π1/2 term with rotational levels = 1/2, 3/2, 5/2, 7/2,... and a 2Π3/2 term with levels = 3/2, 5/2, 7/2, 9/2.... Case requires > 0 and so does not apply to any Σ states, and also > 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, the allowed transitions must have and and and and. In addition, symmetrical diatomic molecules have even or odd parity and obey the Laporte rule that only transitions between states of opposite parity are allowed.
Case (b)
In case, the spin-orbit coupling is weak or non-existent. In this case, we take and and assume precesses quickly around the internuclear axis. The good quantum numbers in case are,,, and. We express the rotational energy operator as, where is a rotational constant. The rotational levels therefore have relative energies starting with. For example, a 2Σ state has rotational levels = 0, 1, 2, 3, 4,..., and each level is divided by spin-orbit coupling into two levels = ± 1/2. Another example is the 3Σ 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 =,,. For case b) the selection rules for quantum numbers,, and and for parity are the same as for case a). However for the rotational levels, the rule for quantum number does not apply and is replaced by the rule.
Case (c)
In case, the spin-orbit coupling is stronger than the coupling to the internuclear axis, and and from case cannot be defined. Instead and combine to form, which has a projection along the internuclear axis of magnitude. Then, as in case. The good quantum numbers in case are,, and. Since is undefined for this case, the states cannot be described as, or. An example of Hund's case is the lowest 3Πu state of diiodine, which approximates more closely to case than to case. The selection rules for, and parity are valid as for cases and, but there are no rules for and since these are not good quantum numbers for case.
Case (d)
In case, 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 are,,,, and. Because is a good quantum number, the rotational energy is simply.
Case (e)
In case, we first form and then form by coupling and. This case is rare but has been observed. Rydberg states which converge to ionic states with spin–orbit coupling are best described as case. The good quantum numbers in case are,, and. Because is once again a good quantum number, the rotational energy is.
