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:

• $\mathbf L$, the electronic orbital angular momentum
• $\mathbf S$, the electronic spin angular momentum
• $\mathbf J_a=\mathbf L+\mathbf S$, the total electronic angular momentum
• $\mathbf J$, the total angular momentum of the system
• $\mathbf N=\mathbf J-\mathbf S$, the total angular momentum minus the electron spin
• $\mathbf R=\mathbf N-\mathbf L$, 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 $\mathbf L$ to the internuclear axis, the spin-orbit coupling, and the rotational coupling of $\mathbf L$ and $\mathbf S$ to the total angular momentum $\mathbf J$.

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), $\mathbf L$ is electrostatically coupled to the internuclear axis, and $\mathbf S$ is coupled to $\mathbf L$ by spin-orbit coupling. Then both $\mathbf L$ and $\mathbf S$ have well-defined axial components $\Lambda$ and $\Sigma$, respectively. $\boldsymbol\Omega$ defines a vector of magnitude $\Omega=\Lambda+\Sigma$ pointing along the internuclear axis. Combined with the rotational angular momentum of the nuclei $\mathbf R$, we have $\mathbf J=\boldsymbol\Omega+\mathbf R$. In this case, the precession of $\mathbf L$ and $\mathbf S$ around the nuclear axis is assumed to be much faster than the nutation of $\boldsymbol\Omega$ and $\mathbf R$ around $\mathbf J$.

The good quantum numbers in case (a) are $\Lambda$, $S$, $\Sigma$, $J$ and $\Omega$. We express the rotational energy operator as $H_{rot}=B\mathbf R^2=B(\mathbf J-\mathbf L-\mathbf S)^2$, where $B$ is a rotational constant. There are, ideally, $2S+1$ fine-structure states, each with rotational levels having relative energies $BJ(J+1)$ starting with $J=\Omega$.^[1]

Case (b)

In case (b), the spin-orbit coupling is weak or non-existent (in the case $\Lambda=0$). In this case, we take $\mathbf N=\boldsymbol\lambda+\mathbf R$ and $\mathbf J=\mathbf N+\mathbf S$ and assume $\mathbf L$ precesses quickly around the internuclear axis.

The good quantum numbers in case (b) are $\Lambda$, $N$, $S$, and $J$. We express the rotational energy operator as $H_{rot}=B\mathbf R^2=B(\mathbf N-\mathbf L)^2$, where $B$ is a rotational constant. The rotational levels therefore have relative energies $BN(N+1)$ starting with $N=\Lambda$.^[1]

Case (c)

In case (c), the spin-orbit coupling is stronger than the coupling to the internuclear axis, and $\Lambda$ and $\Sigma$ from case (a) cannot be defined. Instead$\mathbf L$ and $\mathbf S$ combine to form $\mathbf J_a$, which has a projection along the internuclear axis of magnitude $\Omega$. Then $\mathbf J=\boldsymbol\Omega+\mathbf R$, as in case (a).

The good quantum numbers in case (c) are $J_a$, $J$, and $\Omega$.^[1]

Case (d)

In case (d), the rotational coupling between $\mathbf L$ and $\mathbf R$ is much stronger than the electrostatic coupling of $\mathbf L$ to the internuclear axis. Thus we form $\mathbf N$ by coupling $\mathbf L$ and $\mathbf R$ and the form $\mathbf J$ by coupling $\mathbf N$ and $\mathbf S$.

The good quantum numbers in case (d) are $L$, $R$, $N$, $S$, and $J$. Because $R$ is a good quantum number, the rotational energy is simply $H_{rot}=B\mathbf R^2=BR(R+1)$.^[1]

Case (e)

In case (e), we first form $\mathbf J_a$ and then form $\mathbf J$ by coupling $\mathbf J_a$ and $\mathbf R$. This case is rare but has been observed.^[3]

The good quantum numbers in case (e) are $J_a$, $R$, and $J$. Because $R$ is once again a good quantum number, the rotational energy is $H_{rot}=B\mathbf R^2=BR(R+1)$.^[1]

References

1. Brown, John M.; Carrington, Alan (2003). Rotational Spectroscopy of Diatomic Molecules. Cambridge University Press. ISBN 0521530784.
2. Nikitin, E. E.; Zare, R. N. (1994). "Correlation diagrams for Hund's coupling cases in diatomic molecules with high rotational angular momentum". Molecular Physics 82: 85. doi:10.1080/00268979400100074.
3. Carrington, A.; Pyne, C. H.; Shaw, A. M.; Taylor, S. M.; Hutson, J. M.; Law, M. M. (1996). "Microwave spectroscopy and interaction potential of the long-range He⋯Kr+ ion: An example of Hund's case (e)". The Journal of Chemical Physics 105 (19): 8602. Bibcode:1996JChPh.105.8602C. doi:10.1063/1.472999.