AMELI: Angular Matrix Elements of Lanthanide Ions
Published May 26, 2026 | Version 3.0.0
Dataset Open

Matrix Elements for the Yb Ion

Authors/Creators

  • 1. Gottfried Wilhelm Leibniz Universität Hannover

Description

This dataset provides angular matrices of spherical tensor operators for the f13 configuration (lanthanide ion Yb3+) in exact arithmetic. The matrices are derived from the software package AMELI (Angular Matrix Elements of Lanthanide Ions). They have been calculated in the determinantal product state space and transformed to $LS$-coupling. Each $LS$-state is classified by a complete set of irreducible representations matching those published by C. W. Nielson and G. F. Koster. The Hamilton matrices are comprehensively compared and verified with the values published by B. R. Judd and W. T. Carnall.

How to Cite

If you use this dataset in your research, please cite the primary publication on which it is based:

R. Caspary (2026): "AMELI: Angular matrix elements of lanthanide ions", J. Chem. Phys. 165, 064308, DOI: 10.1063/5.0335592

Matrix Storage Format

The value $v$ of each non-zero matrix element is represented by three values $s$, $n$, and $d$ as signed square root of a rational number: $$v=(-1)^s\sqrt{\frac{n}{d}}\ .$$ The storage format leverages the sparsity of the matrices. It is closely related to the COO standard format for sparse matrices with three vectors for the row index, column index, and value index of each non-zero matrix element. For symmetric matrices only the lower triangle matrix is stored. The actual values are stored in a separate list of unique values, the length of which is usually much smaller than the total number of non-zero elements. Storage follows the canonical COO order with row as primary and column as secondary index. The list of unique values is sorted in ascending order. If a matrix contains integer values $n$ or $d$ that exceed the 64-bit maximum of HDF5 data types, its values are split bit-wise and stored in multiple parts.

Each matrix is stored in a SciDataContainer file with filename extension .zdc. This container file is a standard ZIP folder with a certain inner structure of metadata in JSON files following the FAIR principles of modern research data management. The matrix data is stored in the HDF5 file data/matrix.hdf5 inside the container. A detailed description of the data structures is given in the JSON file meta.json.

Content of Dataset Record

All tensor operator matrices are grouped with respect to their state spaces. The following files are included in this dataset record:

File Description
product.zip Product state tensor operator matrices in the subfolder product. Contains also the matrix transform.zdc used to transform product states or product state matrices into the $LS$-coupling scheme. Intended for energy level fits and transition calculations for lanthanide ions in crystalline materials.
sljm.zip Tensor operator matrices in $LS$-coupling in the subfolder sljm.
slj.zip Tensor operator matrices of the $J$-multiplets in $LS$-coupling. These are matrices from sljm.zip collapsed to the stretched states $M_J=J$ in the subfolder slj and matrices containing reduced matrix elements in the subfolder sljm_reduced. Intended for energy level fits and Judd-Ofelt calculations for lanthanide ions in amorphous materials.
support.zip This special file is only useful when the AMELI package is extended to generate new custom operator matrices. In this case the included data containers with intermediate results optimize the calculation speed within the AMELI package.
hashes.json The hash codes in this file are used for the automized version management algorithm.

 

Matrix Files

Each set of tensor operator matrices contains the following files:

Filename Description Tensor Operator Radial Integral
U_{k}_{q}.zdc Component $q$ of the total unit tensor operator of rank $k$ in the orbital angular momentum space $\mathrm{{U}}^{{(k)}}_q$  
T_{k}_{q}.zdc Component $q$ of the total unit tensor operator of rank $k$ in the spin space $\mathrm{{T}}^{{(k)}}_q$  
UU_{k}.zdc Squared total unit tensor operator of rank $k$ in the orbital angular momentum space $(\mathrm{{U}}^{{(k)}}\cdot\mathrm{{U}}^{{(k)}})$  
TT_{k}.zdc Squared total unit tensor operator of rank $k$ in the spin space $(\mathrm{{T}}^{{(k)}}\cdot\mathrm{{T}}^{{(k)}})$  
UT_{k}.zdc Scalar product of the total unit tensor operators of rank $k$ in the orbital and spin spaces $(\mathrm{{U}}^{{(k)}}\cdot\mathrm{{T}}^{{(k)}})$  
C_{k}_{q}.zdc Component $q$ of the Coulomb operator of rank $k$ $\mathrm{{C}}^{{(k)}}_q$  
L_{q}.zdc Component $q$ of the total orbital angular momentum operator $\mathrm{{L}}_q$  
S_{q}.zdc Component $q$ of the total spin angular momentum operator $\mathrm{{S}}_q$  
J_{q}.zdc Component $q$ of the total angular momentum operator $\mathrm{{J}}_q$  
LL.zdc Squared total orbital angular momentum operator $(\mathrm{{L}}\cdot\mathrm{{L}})$ $\alpha$
SS.zdc Squared total spin angular momentum operator $(\mathrm{{S}}\cdot\mathrm{{S}})$  
JJ.zdc Squared total angular momentum operator $(\mathrm{{J}}\cdot\mathrm{{J}})$  
LS.zdc Scalar product of the total orbital and spin angular momentum operators $(\mathrm{{L}}\cdot\mathrm{{S}})$  
C2.zdc Casimir operator of the special group $G_2$ $\mathrm{{C}}_2(G_2)$ $\beta$
CR.zdc Casimir operator of the special orthogonal (rotational) group in $2l+1$ dimensions $\mathrm{{C}}_2(SO(2l+1))$ $\gamma$
H1_{k}.zdc Coulomb first order perturbation Hamiltonian of rank $k$ $\mathrm{{f}}_k$ $F^k$
H2.zdc Spin-orbit first order perturbation Hamiltonian $\mathrm{{z}}$ $\zeta$
H4_{c}.zdc Effective Coulomb second order perturbation Hamiltonian $\mathrm{{t}}_c$ $T^c$
Hss_{k}.zdc Spin-spin first order perturbation Hamiltonian of rank $k$ $\mathrm{{m}}_{{k,ss}}$  
Hsoo_{k}.zdc Spin-other-orbit first order perturbation Hamiltonian of rank $k$ $\mathrm{{m}}_{{k,soo}}$  
H5_{k}.zdc Spin-spin and spin-other-orbit first order perturbation Hamiltonian of rank $k$ $\mathrm{{m}}_k$ $M^k$
H6_{k}.zdc Effective electrostatic spin-orbit second order perturbation Hamiltonian of rank $k$ $\mathrm{{p}}_k$ $P^k$
Hcf_{k}_{q}.zdc Component $q$ of the crystal field perturbation Hamiltonian of rank $k$ $\mathrm{{H}}_{{\mathrm{{cf}},q}}^{{(k)}}$ $B^k_q$

 

The component part _{q} is missing in the filenames of the set of matrices containing reduced matrix elements. Reduced matrix elements are an entity introduced by the Wigner-Eckart theorem, which describes the angular behaviour of any spherical tensor operator $\mathrm{A}^{(k)}$: $$ \langle J' M' | \mathrm{A}^{(k)}_{q} | J M \rangle = (-1)^{J'-M'} \begin{pmatrix} J' & k & J \\ -M' & q & M \end{pmatrix} \langle J' || \mathrm{A}^{(k)} || J \rangle $$

State Representation

Each matrix data container includes the quantum numbers of all states in the JSON file data/matrix.json. For product states these are the quantum numbers $n$, $l$, $m_l$, $s$, and $m_s$ of each electron. States in $LS$-coupling are characterised by irreducible representations which in most cases are connected to eigenvalues of certain tensor operators.

Key Irreducible Representation Related Tensor Operator
S2 Multiplicity $2S+1$ derived from the eigenvalue $S(S+1)$ Squared operator of the total spin $(\mathrm{S}\cdot\mathrm{S})$
C7 3-tuple $W=(w_1w_2w_3)$ Casimir operator $\mathrm{C}_2(SO(7))$ of the special orthogonal group in 7 dimensions
C2 2-tuple $U=(u_1u_2)$ Casimir operator $\mathrm{C}_2(G_2)$ of the special group $G_2$
L2 Spectral character of $L$ derived from the eigenvalue $L(L+1)$ Squared operator of the total orbital angular momentum $(\mathrm{L}\cdot\mathrm{L})$
J2 Quantum number $J$ derived from the eigenvalue $J(J+1)$ Squared operator of the total angular momentum $(\mathrm{J}\cdot\mathrm{J})$
Jz Eigenvalue $M_J$ Operator of the axial component of the total angular momentum $\mathrm{J}_0$
sen Seniority number $\nu$ Operators $(\mathrm{S}\cdot\mathrm{S})$ and $\mathrm{C}_2(SO(7))$
num Index of different states with same quantum numbers $L$ and $S$  
tau Index $\tau$ of different states with same set of quantum numbers  

 

Version History

The dataset is published with a semantic version number in the scheme 'major.minor.patch', starting at '1.0.0'. The patch number is incremented for every update of the datasets metadata without any modification of the included files. The minor number identifies a dataset release with modified files. However, these are only non-functional modifications of metadata in the data containers. Modifications may include updated or corrected titles or descriptions in the metadata of data containers, or new metadata attributes. The major number is incremented with functional modifications of data containers or additional tensor operators.

Version 1.0.0

 

  • Initial version

 

Version 2.0.0

 

  • Replaced proprietary sign fix inside $J$-multiplets by ladder operator
  • Coulomb operators added
  • Crystal field operators added

 

Version 3.0.0

 

  • Fixed Hcf/{k},0 matrices (wrong factor 2)

 

Files

hashes.json

Files (3.3 MB)

Name Size
md5:af014307e0eb3b46538286ce7eda79b1
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md5:72ede051ec6169c0e4b45a4fd2fc1c4f
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md5:0351b144342b75607373317963a7eedb
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md5:87fb6ab23a0e380ecc302e6d45e1cfd3
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md5:cadbda6273de60dd38ac2602cf26bf4b
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Additional details

Related works

Is documented by
Journal article: 10.1063/5.0335592 (DOI)