The electronic states of AlF
In diatomic molecules, the angular momenta can be coupled in various ways. The coupling of electronic and rotational motion can be understood by considering several idealised situations and described by ''Hund's coupling cases'' [1].
In particular, the description of the couplings involves the following angular momenta [1]:
The electronic orbital angular momentum L,
The electronic spin angular momentum S,
The total angular momentum J,
The total angular momentum excluding electron spin N=J-S,
The rotational angular momentum of the nuclei R=N-L.
There are several interactions being considered [1]. For example, the interaction between the orbital motion of electrons and the electrostatic field of the nuclei gives rise to the precession of L about the internuclear axis. The Hund's coupling cases describe different relative strengths of these couplings. In the end, they can be used to define the ''good'' quantum numbers for diatomic molecules.
Hund's coupling case (a) describes the strong coupling of the orbital angular momentum L and the spin angular momentum S to the internuclear axis of the molecule, when \(A\Lambda\) is much greater than \(BJ\). As a result, Λ and Σ are well defined, which are the axial components of L and S, respectively. Their sum can be used to define the total electronic angular momentum along the internuclear axis Ω = Λ + Σ, which couples with the rotational angular momentum R. The total angular momentum J=Ω+R. The rotational Hamiltonian can be expressed with a ''decoupled'' basis set of L and S:
For AlF, the \(a^3\Pi\) state can be well described by Hund's case (a), with a large \(A_0/B_0 \approx 85\) [2]. The \(d^3\Pi\) state can also described by Hund's case (a), with a much smaller \(A_0/B_0\) ratio [3]. When J becomes higher, the Hund's case (b) becomes the the better choice.
On the other hand, Hund's coupling case (b) can be applied when the spin angular momentum S is uncoupled from the internuclear axis. In this case, the spin-orbit coupling is very weak, and Σ is not defined. The L projection on the internuclear axis, Λ, is well-defined. The coupling between Λ and R forms N. The total angular momentum J is formed by the coupling between N and S. The rotational Hamiltonian can be expressed as
The singlet states of AlF (\(X^1\Sigma^+, A^1\Pi,\)...), with S=0 by definition, are always well-described by both Hund's case (a) and (b) basis, as they become the same when S=0. For the triplet states (S=1) of AlF, the \(^3\Sigma^+\) states (\(b^3\Sigma^+, c^3\Sigma^+, f^3\Sigma^+\)) can be well-described by Hund's case (b) [3][4]. The \(e^3\Delta\) state is also found to be close to Hund's case (b) [3].
References
[1] John M. Brown and Alan Carrington. Rotational spectroscopy of diatomic molecules. Cambridge university press, 2003.
[2] Nicole Walter, Maximilian Doppelbauer, Silvio Marx, Johannes Seifert, Xiangyue Liu, Jesús Pérez-Ríos, Boris G. Sartakov, Stefan Truppe, and Gerard Meijer. Spectroscopic characterization of the a3Π state of aluminum monofluoride. The Journal of Chemical Physics, 156(12), 2022.
[3] Nicole Walter, Maximilian Doppelbauer, Sascha Schaller, Xiangyue Liu, Russell Thomas, Sidney Wright, Boris G. Sartakov, and Gerard Meijer. Triplet rydberg states of aluminum monofluoride. The Journal of Physical Chemistry A, 128(14):2752-2762, 2024.
[4] Maximilian Doppelbauer, Nicole Walter, Simon Hofsäss, Silvio Marx, Christian H. Schewe, Sebastian Kray, Jesús Pérez-Ríos, Boris G. Sartakov, Stefan Truppe, and Gerard Meijer. Characterisation of the b3Σ+, v=0 state and its interaction with the A1Π state in aluminium monofluoride. Molecular Physics, 119(1-2):e1810351, 2021.
[5] István Kovács and William Lichten. Rotational structure in the spectra of diatomic molecules, 1972.