Molecules
2.2 Effective Hamiltonians .1 Basic Principle.1Basic Principle
2.2.2 Details of the Effective Hamiltonian
We wish to study and model the rotational states belonging to a single vibrational and electronic configuration of the molecule. We generically denote this state
|𝜓(0)
, 𝑖⟩=|𝑃(0)
;𝑣(0)
;𝜂, 𝐽⟩. Here,𝑖is simply an index labeling the quantum numbers defining our state. We use 𝑃(0) to denote the collection of projection quantum numbers relevant to our state. For example, for a diatomic molecule in Hund’s case (a), 𝑃(0) = (Λ,Σ), while for Hund’s case (b), 𝑃(0) = (Λ). Since we are most interested in interactions internal to the molecule frame, it is easier to use case (a) representations, though we will try to keep the discussion as generic as possible. Continuing, we use𝑣(0)
to denote the vibrational state of the molecule, for example𝑣(0) = (𝑣
1, 𝑣
2, 𝑣
3)in a triatomic molecule. Finally, we use𝐽to characterize
the total angular momentum of our state, and 𝜂 to represent all other relevant quantum numbers, such electronic or nuclear spins, which we will often suppress for brevity. Since total angular momentum is conserved by internal interactions, we will consider dynamics that re-orient 𝐽 and change its projections, but never that change the magnitude of𝐽. Of course, when we apply an external electromagnetic field, the total molecule angular momentum will no longer be conserved, as the interaction with the field can exchange angular momenta.
There are many interactions in the molecule that will necessarily couple our state of interest to other vibronic states. We use the index𝛼 to label the space of all states not within our vibronic manifold of interest, characterized by their own values of 𝑃(𝛼) and 𝑣(𝛼). In reality, the true molecular state |𝜓 , 𝑖⟩ is actually a mixture of|𝜓(0), 𝑖⟩and many other contributions|𝜓(𝛼), 𝑘⟩, with potentially different quantum numbers represented by the index𝑘. When the admixtures of other states are comparable to the admixture for𝜓(0)
, it is an indication that we have not used a good basis of approximate quantum numbers. However, it is often the case that the mixtures of other states are perturbative, owing to the large separation of electronic and vibrational energy scales from the rotational energy scale. In such a case, to deal with mixings outside of our vibronic state, we do not have to consider the whole,
“true” wavefunction. Instead, we work only with |𝜓(0), 𝑖⟩, and we incorporate the effects of interactions external to our subspace in a source-agnostic manner by simply adding additional, effective terms to the Hamiltonian. These effective terms must be consistent with the symmetries of our Hamiltonian (rotation, parity), so in free-field they are written as scalar products of possibly many angular momenta. There is also the constraint from the Wigner-Eckart theorem that any operator involving the spin 𝑆can only be allowed if its rank 𝑘 satisfies 2𝑆 ≥ 𝑘 [39]. For example, when working with a state with a single unpaired valence electron spin, we do not need to worry about considering spin-spin interactions. We caution this may not be the case if there is strong configuration mixing (for example, strong mixing with state that has extra valence spin excitation). As a related point, the situation is also quite complicated in multi-electron systems, which must be written in terms of Slater determinants [39]. However, in the single valence electron molecules we consider (often the case for laser coolable molecules), the effective Hamiltonian approach is applicable and quite powerful.
The primary interactions that we would like to “integrate out” are those that involve the electronic angular momentum 𝐿. Later, we will also discuss the very
similar procedure for dealing with operators that couple the vibrational angular momentum𝐺ℓ present in triatomic molecules. Returning to 𝐿, we specifically do not want to deal with the transverse components 𝐿⊥ = 𝐿𝑥 , 𝑦, which can be written in terms of raising and lowering operators𝐿±. Determining the matrix elements of these ladder operators is a hard computational task, as 𝐿 is not well-defined in the molecule due to the breaking of spherical symmetry. We can see that two essential molecular interactions will have 𝐿± operators: spin-orbit (𝐻
SO) and rotation𝐻
Rot. Writing these out in Hund’s case (a), we have:
𝐻Rot =𝐵
𝐽®− ®𝐿− ®𝑆 2
=𝐵
𝐽®2+ ®𝐿2+ ®𝑆2−2𝐽®· ®𝐿−2𝐽®· ®𝑆−2𝐿® · ®𝑆
=𝐵
𝐽®2+ ®𝐿2+ ®𝑆2−2𝐽𝑧(𝐿𝑧+𝑆𝑧) −2𝐿𝑧𝑆𝑧
−𝐽+𝐿−−𝐽−𝐿+−𝐽+𝑆−−𝐽−𝑆+−𝐿+𝑆−−𝐿−𝑆+
(2.32)
𝐻Rot= 𝐴( ®𝐿· ®𝑆)
= 𝐴
𝐿𝑧𝑆𝑧+ 1 2
(𝐿+𝑆−+𝐿−𝑆+)
.
(2.33)
Here, we have written the nuclear rotation as𝑅®=𝐽®− ®𝐿− ®𝑆, and𝐵and 𝐴are “bare”
constants that will not be the parameters we fit in the effective Hamiltonian. We note this can be generalized to a triatomic molecule by writing𝑅®=𝐽®− ®𝐿− ®𝐺ℓ− ®𝑆, as was done in Ref. [162], for example. The dot products have been expanded in the molecular frame, and we emphasize that the matrix elements of𝐽±have anomalous commutation relations, and care must be taken to transform them to the lab frame before evaluation [39, 160]. Keeping this in mind, the operators 𝑆± and 𝐽± can be dealt with using the Wigner-Eckart theorem, but the 𝐿± and 𝐿2 operators, as we mentioned before, are not easy to compute. We note there is an approximate method for estimating their matrix elements, described further in Appendix A.3.2.
The effective Hamiltonian approach actually does not require us to evaluate any matrix elements of 𝐿⊥ or 𝐿2. Instead, we recognize that the operators in eqs. 2.32 and 2.33 can be grouped into three classes. First, we have operators that are well- defined to act among two states𝑖and 𝑗 belonging to𝜓(0), for example the operators 𝐽®2 = 𝐽(𝐽+1) or𝐽+𝑆−. Their matrix elements can be calculated within𝜓(0)
using angular momentum algebra [39]. Then we have operators whose form is identical for all the different states in𝜓(0)
, for example𝑆®2. These operators are simply absorbed into an overall energy offset of the electronic state, known as the “origin.” We note
the distinction between which operators are diagonal shifts and which operators are origin contributions is somewhat arbitrary—for example, 𝐿𝑧𝑆𝑧 can be taken as a diagonal energy shift if our basis of interest contains multipleΩstates, or it can just be taken as a contribution to the origin if we consider just a singleΩstate. Further, here we have shown rotational contributions to the origin, but we note that there are also vibrational contributions, such as the zero point energy.
Continuing, we have operators that contain 𝐿± and connect us to different electronic states |𝜓(0)
, 𝑖⟩ → ∥𝜓(𝛼), 𝑘⟩. These operators only have an impact on the energies of 𝜓(0) state when they connect back, that is when we also consider
|𝜓(𝛼), 𝑘⟩ → |𝜓(0), 𝑗⟩ at some higher order of perturbation theory. For example, separate terms in the spin-orbit and rotational Hamiltonians can take us to another state and back—if we combine 𝐽+𝐿− with 𝐿+𝑆−, we obtain an overall interaction that looks like𝐽+𝑆−. This term looks like a term in the rotational Hamiltonian, and the two effects are actually indistinguishable. We note we have swept under the rug the possibility of combining𝐽+𝐿−and 𝐿−𝑆+. Such a possibility is allowed, but can result instead in parity doubling interactions, where the signs of the projections in 𝑃(0)
are flipped. We discuss parity doubling more later on. In general, once we begin to consider the effective Hamiltonian expansion, the original parameters of the theory, such as 𝐵and 𝐴, are mixed up, and all we can determine are effective parameters that receive contributions at various orders of perturbation theory. In Appendix A.3.3, we provide further information about the mathematical formulation of the effective Hamiltonian.
To gain an intuition, we illustrate the situation diagrammatically in Figure 2.1.
We consider four relevant angular momenta of the molecule,𝐿, the electronic orbit, 𝑆, the electronic spin, 𝑅, the molecular rotation, and 𝐼, the nuclear spin. We also consider the possible coupling of an external magnetic field𝐵, which we elaborate on soon. In the effective Hamiltonian, we are free to write down symmetry conserving terms involving interactions of any of these angular momenta. However, once we reduce the Hamiltonian to act only within the subspace of a given vibronic state by “tracing out” 𝐿, we may end up with different effective interactions, shown at second order in Fig. 2.1. If the traced out interactions involve flips ofΛ, they will result in parity doubling, which we discuss in Sec. 2.2.3.
As a further concrete example, we consider the contributions to the spin-rotation operator,𝛾𝐽®· ®𝑆 in case (a), and 𝛾𝑁® · ®𝑆 in case (b)11. In the effective Hamiltonian
11See Appendix A.3.1 for the use of𝑁vs. 𝑅in effective Hamiltonians.
(a) (b)
(c)
(d)
Figure 2.1: Diagram of second order interactions that arise in the effective Hamil- tonian when we trace out 𝐿, the electronic orbital angular momentum. (a) This diagram describes contributions to an effective spin-rotation interaction. We note that depending on the states involved, this interaction can either generate parity doubling, or be parity preserving, see main text for details. (b) Effective contri- bution to the electron spin-nuclear spin interaction. (c) Effective contribution to the𝑅2rotational interaction, showing how even the rotational constant becomes an effective parameter. (d) An applied magnetic field𝐵 can also couple to the orbital angular momentum 𝐿, and when we form the effective Hamiltonian we can end up with additional interactions between𝐵and𝑆. See main text for further details.
approach, the spin-rotation parameter receives contributions from various orders of perturbation theory, 𝛾 = 𝛾(1) + 𝛾(2) + · · · [39]. The first order term 𝛾(1)
results from the magnetic interaction between the electron spin and the magnetic dipole moment of the rotating molecule [163]. In heavy molecules, the first order term is small compared to the dominant second order contribution 𝛾(2), arising from off-diagonal spin-orbit and rotational perturbations, i.e., combinations of 𝐵 𝐽+𝐿− and𝐴 𝐿+𝑆−. The resulting contribution to𝛾(2) is provided in 7.122 of Ref. [39], and more discussion can be found in Ref. [161].
If we now consider the application of a magnetic field to the molecule, we will
have terms that look like 𝐵®· ®𝐿. The evaluation of this operator requires rotating 𝐵® into the molecule frame, where 𝐿𝑧 is good and 𝐿⊥ is undesirable. We deal with the𝐿⊥terms by using the effective Hamiltonian approach. We can now obtain effective interactions of the form𝐵𝑥𝑆𝑥+𝐵𝑦𝑆𝑦, where𝑥 , 𝑦are defined in the molecule frame. This term encodes electronic mixing of Λ, and the interaction represents the anisotropic response of the electron g-factor in a molecule, compared to the usual isotropic response in an atom. See Ref. [164] for an excellent discussion of magnetic effects in molecules. We can also obtain parity dependent Zeeman effects, which we will return to when discussing parity doubling. Curiously, there are no effective Stark terms in the effective Hamiltonian. However, we can understand why simply; such terms would involve a coupling of the internuclear axis ˆ𝑛, which is 𝑃-odd𝑇-even, to an angular momentum of the molecule, 𝐿®, which 𝑃-even𝑇-odd.
This would be a𝑃, 𝑇 violating effect—exactly what we are searching for!
Finally, as was mentioned earlier, the effective Hamiltonian approach can also be generalized for polyatomic molecules, eliminating the components of𝐺®ℓthat couple outside of our vibronic subspace to other vibrational states. When tracing out𝐺⊥, we obtain additional effective terms that can contribute just like the effective operators obtained by tracing out 𝐿⊥. However, the scale of the contribution from𝐺®ℓ effects is typically smaller compared to spin-orbit effects. For example, there is usually not a strong coupling ofℓ and𝛴. However, this is not always the case, particularly if ℓ is actually mixed withΛ, resulting in effective spin-orbit interactions, as we shall see in the bending mode of YbOH in Ch. 4. Finally, we note different effective Hamiltonians are useful for different molecular states and bases. For example, we will use a different Hamiltonian to describe a Hund’s case (b) molecule with little to no spin-orbit interactions, compared to a Hund’s case (a) molecule with strong spin-orbit. Further, we will have additional terms when dealing with bending modes of polyatomics, or spin-spin interactions in triplet systems.