Ylm- and spin-resolved 4f photoemission amplitudes in Quanty
asked by Lukasz Plucinski (2026/06/05 23:18)
Dear All,
I am currently exploring rare-earth 4f photoemission multiplets in Quanty, starting from Ho 4f¹⁰ → 4f⁹. After a few days of work, I have obtained a spectrum that begins to resemble the classic Ho calculation by Gerken (J. Phys. F: Met. Phys. 13, 703 (1983)). My actual goal would be Dy and perhaps Tb.
My main question is whether Quanty can provide photoemission matrix elements resolved into individual Ylm and spin components, i.e. amplitudes of the form
⟨Ψf | a(m,σ) | Ψi⟩
for each final multiplet state, rather than only the total photoemission intensity.
Is this possible within Quanty?
My goal would be to use these Ylm/spin-resolved amplitudes (or equivalently the emitted l+1 and l−1 channels) as input for a photoelectron diffraction calculation (in a separate code).
Best,
Lukasz
PS: I would be happy to share my current Quanty input if needed.
Answers
Dear Lukasz,
Yes, maybe, a bit, or better said, probably Quanty can deliver what you need. I would not suggest to calculate final states. There are just to many of them. We can however calculate the one particle electron removal Green's function as a matrix, which you then can use to do photoelectron diffraction.
The object to calculate would be
$$G_{\tau,\tau'}(\omega) = \left\langle \psi_0 | a^{\dagger}_{\tau} \frac{1}{\omega - (H -E_0) + \mathrm{i} \Gamma/2} a_{\tau'} | \psi_0 \right\rangle$$
This you then can use as an input for the directional and polarisation dependent photo-electron emission.
The code to calculate this object is
Note that we store spectra as spectra objects in S and as response function objects in G. The response functions can be evaluated at a particular energy, or seen as a sum over poles. The poles correspond to eigenstates of a Krylov basis. The Krylov basis only corresponds to the full basis if you make the Krylov basis large enough. That is generally very expensive to calculate.
You can find an example how to get the polarization and directional intensity in this example for NiO. Note that we there generated a polarisation and directional dependent operator. As you want all directions it would be much faster to calculate the one particle electron removal Green's function as a matrix and do the polarisation and directional dependent calculation from this. (Note you can sum and rotate response function objects so this is relatively cheap and straight forward to do in Quanty).
Hope this helps, Best wishes, Maurits
Dear Maurits, dear All,
Thank you, this was very helpful. The Green's function with F2-F6 and zeta works well. However, for Dy I do not yet reproduce the quantitative Gerken-style 4f^9 -> 4f^8 multiplet positions and intensities.
To reproduce the Gerken-style rare-earth multiplet energies, I think the missing ingredient is the “Trees” correction. In Gerken’s notation (used in some other papers too) this is written as
\Delta E = \alpha L(L+1) + \beta G(G_2) + \gamma G(R_7)
I think the (\alpha L(L+1)) part is easy to add approximately, but the \beta and \gamma terms are not just functions of L,S,J, I think they distinguish the Nielson-Koster (W)(U)v terms. Therefore they should be included in the Hamiltonian before spin-orbit diagonalization, not as a post-processing shift of already mixed states. I made some progress with an explicit-pole route for Dy, trying to identify the (W)(U)v terms, but this is rather tedious.
My question is: is there a native way in Quanty to include these Trees corrections for an (f^n) shell? For example, are the G(G_2) and G(R_7) operators available, or can they be generated from existing Quanty operator definitions?
If not, would the correct route be to construct the Trees correction externally in the Nielson-Koster LS basis and then combine it with the Quanty Coulomb + spin-orbit Hamiltonian?
Best,
Lukasz
Dear Lukasz,
Do you have a paper reference that describes these operators? For the first term one can define the operator
and add this to the full Hamiltonian.
For your $G_2$ and $G_7$ I would need to know the equations to be able to write the operators.
Best wishes, Maurits
Dear Maurits,
Thank you for the rapid reply.
The compact modern reference is Eq. (10) of Lizarazo-Ferro et al. ( https://arxiv.org/html/2503.17377v1 ), where the parametric Hamiltonian contains α L^2 + β C(G2) + γ C(SO(7)).
The text below Eq. (10) identifies C(G2) and C(SO(7)) as the Casimir operators of the groups G2 and SO(7). The paper itself says the detailed Racah/fractional-parentage matrix elements are documented in Refs. [2,57], and the configuration-interaction origin of α, β, γ is discussed around the second-order perturbation expression, citing Ref. [59].
Not sure if useful, but Nielson-Koster book is here: https://babel.hathitrust.org/cgi/pt?id=mdp.39015078650028&seq=5 and similar tables are digitized here: https://elsevier.digitalcommonsdata.com/datasets/gs3jt9sshg/1
Classic Gerken paper is here: https://iopscience.iop.org/article/10.1088/0305-4608/13/3/021 More recent calculation is here: https://pubs.acs.org/doi/10.1021/acs.jpclett.2c02203 (see the Supplement, which lists the Trees parameters)
I have been trying to implement these terms externally “by hand”, by constructing the LS/J matrices from Nielson-Koster/ACRY parentage data and then adding the empirical Trees shifts before comparing to Gerken’s multiplet sticks. However, I am not fully confident about the operator conventions for the β and γ terms (so far I was not able to reproduce Gerken in detail), so it would be much cleaner if the standard C(G2) and C(SO(7)) operators could be defined directly in Quanty.
Best, Lukasz
Dear Lukasz
Below you find a Quanty script that defines the Casimir operators $C(SO(3))$, $C(G2)$ and $C(SO(7))$. Your full Hamiltonian then becomes the sum of the Coulomb Hamiltonian that depends of F0, F2, F4 and F6, the spin-orbit Hamiltonian multiplied by $\zeta$, the Casimir operators multiplied by $\alpha$, $\beta$ and $\gamma$. You then diagonalize this Hamiltonian and or calculate spectral functions from this Hamiltonian and optimise the parameters to experiment / literature.
For the Casimir operators: The operator $C(SO(3))$ is proportional to $L^2$ (with proportionality constant $(l(l+1))(2l+1) = 12 * 7$). The Casimir operators are defined based on their commutation and symmetry relations and thus defined up to a constant scaling. You can always replace $\alpha' C'$ by $\alpha C$ and take $\alpha' = c \alpha$, $C' = (1/c) C$ such that the full operator $\alpha' C' = \alpha C$. I do not know the scaling people in the literature took.
The script gives the diagonal elements of the operators on an LSLzSz C(G2) C(SO(7) basis, which might help you to fix the proper scaling.
Best wishes, Maurits