Chapter II: Molecular Structure
2.6 Angular momentum coupling and Hund’s cases
where
𝐵(2) = ∑︁
𝜂′≠𝜂
⟨𝜂|𝐵 𝐿∓|𝜂′⟩⟨𝜂′|𝐵 𝐿±|𝜂⟩ 𝐸𝜂−𝐸𝜂′
. (2.43)
Notice that ˆ𝐻(2)
𝑅,𝑒 𝑓 𝑓 has the same form as ˆ𝐻(1)
𝑅,𝑒 𝑓 𝑓 so we can write ˆ
𝐻𝑅,𝑒 𝑓 𝑓 =𝐵𝑒 𝑓 𝑓(N2−𝑁2
𝑧) (2.44)
with
𝐵𝑒 𝑓 𝑓 =𝐵(1)+𝐵(2). (2.45)
We have constructed an effective Hamiltionian, ˆ𝐻𝑅,𝑒 𝑓 𝑓, which operates only within the vibronic subspace |𝜂⟩ that incorporates all the effects of the total Hamiltonian
ˆ
𝐻𝑅. The first-order term,𝐵(1), incorporates the effects of ˆ𝐻𝑅that operate only within
|𝜂⟩(e.g., the end-over-end rotation) while the second-order term,𝐵(2), incorporates the effects of ˆ𝐻𝑅 mixing different vibronic states and projects them into the |𝜂⟩ subspace. Higher-order terms can be included in ˆ𝐻𝑒 𝑓 𝑓 and will include higher-order mixings of vibronic states, however the effects of higher-order terms have a smaller and smaller effect on the eigenvalues of ˆ𝐻𝑒 𝑓 𝑓 and therefore can be neglected.
The operator 𝑁2
𝑧 gives a value of Λ2+ 𝑙2 in any vibronic state. Therefore, the term 𝐵𝑒 𝑓 𝑓𝑁2
𝑧 provides a constant offset to the initial energy of the vibronic state.
Therefore, the effective rotational Hamiltonian can be expressed as, ˆ
𝐻𝑅,𝑒 𝑓 𝑓 =𝐵𝑒 𝑓 𝑓N2. (2.46)
The beauty of the effective Hamiltonian approach lies in the fact the energy levels of a vibronic state can be computed without diagonalizing the full Hamiltonian of the molecule. Therefore, to predict the spectrum for a single vibronic band, the energies and wavefunctions of each state can be calculated separately and then the transitions between each set of states computed. If you fit a measured spectrum to the calculated one, the parameters in the effective Hamiltonian can be determined.
These parameters provide insight into each moleuclar state as well as other excited states in the molecule (through higher-order contributions to these parameters, such as 𝐵(2)). In addition to rotation, fine and hyperfine terms are also included in the effective Hamiltionian, and the terms relevant for modeling YbOH and other molecules in doublet states are given in Sections 2.7 and 2.8.
diagonalized in any basis, choosing the one in which ˆ𝐻𝑒 𝑓 𝑓 is most diagonal will result in the eigenfunctions of ˆ𝐻𝑒 𝑓 𝑓 having a large overlap with the basis functions,|𝜂, 𝑖⟩0. The eigenfunctions closely resembling the basis function allow one to develop a good intuition for what the eigenstates look like in terms of the basis functions. The basis in which ˆ𝐻𝑒 𝑓 𝑓 is most diagonal is determined by the relative strength of the interactions in ˆ𝐻𝑒 𝑓 𝑓 and how these interactions couple the angular momentum in the molecule. These angular momentum couplings are described by Hund’s coupling cases [89]. There are five Hund’s coupling cases but we will only describe the first two, as the molecular states of YbOH are best described by these two cases. First we will describe the coupling cases in the absence of hyperfine structure (no nuclear spins) and then introduce the coupling cases in the presence of hyperfine structure.
The relevant angular momenta and their projections on the internuclear axis ˆ𝑛, in the absence of nuclear spins, are shown in Table 2.1.
Hund’s case (a)
For Hund’s case (a) to provide a good description of the molecular state, two conditions must be met. First, the electron orbital angular momentum is strongly coupled to the internuclear axis and, second, that the electron spin S is strongly coupled to L via spin orbit coupling (spin orbit coupling is discussed in Section 2.7). In this caseL andS precess rapidly about the internuclear axis so that their projections on the internuclear axis, Λ and Σ respectively, are well defined [89].
The quantum numberΩ = Λ+Σis also well defined. Additionally, if the molecule is in a bending state,Ω couples to the bending angular momentum G𝑙 to give the quantum number𝑃 = Ω+𝑙 . 𝑃(orΩif𝑙 =0) couples to the end-over-end rotation Rto give the total angular momentumJ. The basis vectors for Hund’s case (a) are
|𝜂,Λ;𝜈𝑏 𝑒𝑛𝑑, 𝑙;𝑆,Σ;𝐽 ,Ω, 𝑀𝐽⟩= |𝜂,Λ⟩|𝜈𝑏 𝑒𝑛𝑑, 𝑙⟩|𝑆,Σ⟩|𝐽 ,Ω, 𝑀𝐽⟩, (2.47) where here𝜂again refers to all the quantum numbers specifying the vibronic state defining the space in which ˆ𝐻𝑒 𝑓 𝑓 operates. 𝑀𝐽is the projection of the total angular momentum J onto the laboratory z-axis. In the absence of external electric or magnetic fields, the different 𝑀𝐽 states are (2𝐽+1)-fold degenerate. The right side of Eq. 2.47 is included to indicate that this basis vector is an uncoupled basis where the kets can be separated. Hund’s case (a) is generally relevant for vibronic states with non-zero projections ofLon the internuclear axis (Λ≥ 0)8.
8The more technical definition for when Hund’s case (a) is relevant is that the spin orbit interaction is much larger than the rotation,𝐴Λ≫𝐵 𝐽.
Table 2.1: Relevant molecular angular momenta, in the absence of nuclear spins, and their projections on the internuclear axis ˆ𝑛[89, 91].
Angular Momentum Projection on ˆ𝑛 Description
L Λ total electron orbital angular
momentum
S Σ total electron spin angular
momentum
G𝑙 𝑙 bending angular momentum
R rotational angular momen-
tum (end-over-end) N=R+L+G𝑙
𝑎 𝐾 = Λ+𝑙 total angular momentum mi- nus spin
J=N+S 𝑃 =𝐾+Σ total angular momentum J𝑒 =L+S Ω = Λ+Σ𝑏 total electronic angular mo-
mentum
𝑎Also given byN=J−S.
𝑏 In diatomic molecules and linear molecules with no bending vibrations Ω is the projection of the total angular momentumJ. The quantum number𝑃is not used in this case since𝑃= Ω.
Hund’s case (b)
In the situation where there is little or no spin orbit coupling (e.g.,Λ =0), the spin is decoupled from the internuclear axis and Hund’s case (b) provides a good description of the molecular state [89]. Since the spin is decoupled from the internuclear axis, N, the total angular momentum minus spin, and it’s projection,𝐾 = Λ+𝑙, are good quantum numbers. Additionally, Σ andΩare not well defined. The electron spin, S, will then couple to the nuclear rotation,N(via spin rotation discussed in Section 2.7), to give the total angular momentumJ. The basis vectors for Hund’s case (b) are
|𝜂,Λ;𝜈𝑏 𝑒𝑛𝑑, 𝑙;𝑁 , 𝐾 , 𝑆, 𝐽 , 𝑀𝐽⟩. (2.48)
Hund’s case (b) provides a good description for molecular states with Λ = 0 or the rare case where the energy scale of the spin orbit coupling is smaller than the rotation [89].
Angular momentum coupling with nuclear spins
If one or more nuclei in the molecule has a non-zero spin, I, the nuclear spin will couple to the other angular momenta present in the molecule. Here we describe two special cases of coupling schemes involving the nuclear spin. The first is Hund’s case (a𝛽 𝐽). Just as in Hund’s case (a), in Hund’s case (a𝛽 𝐽)LandSare coupled to the internuclear axis so thatΛ, Σ, Ω, and 𝑃are well defined and 𝑃couples toRto giveJ. Hyperfine interactions (discussed in Section 2.8) couple the nuclear spin to Jto give the total angular momentum9F=J+I. The basis vectors for Hund’s case (a𝛽 𝐽) are
|𝜂,Λ;𝜈𝑏 𝑒𝑛𝑑, 𝑙;𝑆,Σ;𝐽 ,Ω, 𝐼 , 𝐹 , 𝑀𝐹⟩. (2.49) Hund’s case (a𝛽 𝐽) is a good description of molecular states with Λ ≥ 1 and spin orbit coupling.
In the absence of spin orbit coupling (Λ = 0) and when the hyperfine interactions are strong, Hund’s case (b𝛽𝑆) provides a good description of the molecular state.
As in Hund’s case (b) the electron spin is decoupled from the internuclear axis;
however, now the strong hyperfine interactions cause the electron spin,Sto couple to the nuclear spinIbefore coupling to the rotationN. The coupling ofStoIgives the intermediate quantum numberG = S+Iwhich then couples to the rotation to give the total angular momentum10F. The basis vectors for Hund’s case (b𝛽𝑆) are
|𝜂,Λ;𝜈𝑏 𝑒𝑛𝑑, 𝑙;𝑆, 𝐼 , 𝐺 , 𝑁 , 𝐾 , 𝐹 , 𝑀𝐹⟩. (2.50) Hund’s case (b𝛽𝑆) is a good description of molecular states withΛ = 0 and hyperfine interactions whose energy scales are larger than the rotational energy scale.