Matrix Elements for the Er Ion

This dataset provides angular matrices of spherical tensor operators for the f11 configuration (lanthanide ion Er3+) 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. 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: FileDescription 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}.zdcComponent $q$ of the total unit tensor operator of rank $k$ in the orbital angular momentum space$\mathrm{{U}}^{{(k)}}_q$ T_{k}_{q}.zdcComponent $q$ of the total unit tensor operator of rank $k$ in the spin space$\mathrm{{T}}^{{(k)}}_q$ UU_{k}.zdcSquared total unit tensor operator of rank $k$ in the orbital angular momentum space$(\mathrm{{U}}^{{(k)}}\cdot\mathrm{{U}}^{{(k)}})$ TT_{k}.zdcSquared total unit tensor operator of rank $k$ in the spin space$(\mathrm{{T}}^{{(k)}}\cdot\mathrm{{T}}^{{(k)}})$ UT_{k}.zdcScalar 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}.zdcComponent $q$ of the Coulomb operator of rank $k$$\mathrm{{C}}^{{(k)}}_q$ L_{q}.zdcComponent $q$ of the total orbital angular momentum operator$\mathrm{{L}}_q$ S_{q}.zdcComponent $q$ of the total spin angular momentum operator$\mathrm{{S}}_q$ J_{q}.zdcComponent $q$ of the total angular momentum operator$\mathrm{{J}}_q$ LL.zdcSquared total orbital angular momentum operator$(\mathrm{{L}}\cdot\mathrm{{L}})$$\alpha$ SS.zdcSquared total spin angular momentum operator$(\mathrm{{S}}\cdot\mathrm{{S}})$ JJ.zdcSquared total angular momentum operator$(\mathrm{{J}}\cdot\mathrm{{J}})$ LS.zdcScalar product of the total orbital and spin angular momentum operators$(\mathrm{{L}}\cdot\mathrm{{S}})$ C2.zdcCasimir operator of the special group $G_2$$\mathrm{{C}}_2(G_2)$$\beta$ CR.zdcCasimir operator of the special orthogonal (rotational) group in $2l+1$ dimensions$\mathrm{{C}}_2(SO(2l+1))$$\gamma$ H1_{k}.zdcCoulomb first order perturbation Hamiltonian of rank $k$$\mathrm{{f}}_k$$F^k$ H2.zdcSpin-orbit first order perturbation Hamiltonian$\mathrm{{z}}$$\zeta$ H4_{c}.zdcEffective Coulomb second order perturbation Hamiltonian$\mathrm{{t}}_c$$T^c$ Hss_{k}.zdcSpin-spin first order perturbation Hamiltonian of rank $k$$\mathrm{{m}}_{{k,ss}}$ Hsoo_{k}.zdcSpin-other-orbit first order perturbation Hamiltonian of rank $k$$\mathrm{{m}}_{{k,soo}}$ H5_{k}.zdcSpin-spin and spin-other-orbit first order perturbation Hamiltonian of rank $k$$\mathrm{{m}}_k$$M^k$ H6_{k}.zdcEffective electrostatic spin-orbit second order perturbation Hamiltonian of rank $k$$\mathrm{{p}}_k$$P^k$ Hcf_{k}_{q}.zdcComponent $q$ of the crystal field perturbation Hamiltonian of rank $k$$\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 S2Multiplicity $2S+1$ derived from the eigenvalue $S(S+1)$Squared operator of the total spin $(\mathrm{S}\cdot\mathrm{S})$ C73-tuple $W=(w_1w_2w_3)$Casimir operator $\mathrm{C}_2(SO(7))$ of the special orthogonal group in 7 dimensions C22-tuple $U=(u_1u_2)$Casimir operator $\mathrm{C}_2(G_2)$ of the special group $G_2$ L2Spectral character of $L$ derived from the eigenvalue $L(L+1)$Squared operator of the total orbital angular momentum $(\mathrm{L}\cdot\mathrm{L})$ J2Quantum number $J$ derived from the eigenvalue $J(J+1)$Squared operator of the total angular momentum $(\mathrm{J}\cdot\mathrm{J})$ JzEigenvalue $M_J$Operator of the axial component of the total angular momentum $\mathrm{J}_0$ senSeniority number $\nu$Operators $(\mathrm{S}\cdot\mathrm{S})$ and $\mathrm{C}_2(SO(7))$ numIndex of different states with same quantum numbers $L$ and $S$ tauIndex $\tau$ of different states with same set of quantum numbers Version Policy 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.本数据集提供了针对$f^{11}$组态（镧系离子$mathrm{Er}^{3+}$）的球张量算子角矩阵的精确算术形式。该矩阵源自AMELI（镧系离子角矩阵元，Angular Matrix Elements of Lanthanide Ions）软件包，在行列式乘积态空间中计算得到，并转换至LS耦合（LS-coupling）表象。每个LS耦合态均通过一套完整的不可约表示（irreducible representation）进行分类，该分类与C. W. Nielson和G. F. Koster发表的标准一致。哈密顿矩阵（Hamilton matrix）已与B. R. Judd及W. T. Carnall发表的数值进行了全面比对与验证。 ## 矩阵存储格式 每个非零矩阵元的数值$v$可通过三个参数$s$、$n$、$d$表示为有理数的带符号平方根： $$v=(-1)^ssqrt{frac{n}{d}}$$ 该存储格式利用了矩阵的稀疏性，与稀疏矩阵的COO标准格式（Coordinate List format）高度相似：使用三个向量分别存储每个非零矩阵元的行索引、列索引与值索引。对于对称矩阵，仅存储其下三角部分。实际数值存储于独立的唯一值列表中，该列表的长度通常远小于非零元总数量。存储遵循规范的COO顺序，以行作为主索引、列作为次索引，且唯一值列表按升序排列。若矩阵中的整数$n$或$d$超出HDF5数据类型的64位最大值，则将其按位拆分并分多段存储。 每个矩阵均存储于SciDataContainer文件中，文件扩展名为`.zdc`。该容器文件为标准ZIP文件夹，其内部包含遵循FAIR原则（可发现、可访问、可互操作、可复用，Findable, Accessible, Interoperable, Reusable）的JSON元数据文件。矩阵数据存储于容器内的HDF5文件`data/matrix.hdf5`中。数据结构的详细说明见JSON文件`meta.json`。 ## 数据集记录内容 所有张量算子矩阵均按其态空间进行分组。本数据集记录包含以下文件： - `product.zip`：子文件夹`product`内存储乘积态张量算子矩阵，同时包含用于将乘积态或乘积态矩阵转换至LS耦合表象的`transform.zdc`文件，适用于晶体材料中镧系离子的能级拟合与跃迁计算。 - `sljm.zip`：子文件夹`sljm`内存储LS耦合表象下的张量算子矩阵。 - `slj.zip`：包含LS耦合表象下$J$多重态的张量算子矩阵。其中，子文件夹`slj`存储了`sljm.zip`中坍缩至拉伸态$M_J=J$的矩阵；子文件夹`sljm_reduced`存储约化矩阵元矩阵，适用于非晶态材料中镧系离子的能级拟合与Judd-Ofelt计算。 - `support.zip`：该特殊文件仅在扩展AMELI软件包以生成自定义算子矩阵时可用，其中包含的中间结果数据容器可优化AMELI软件包内的计算速度。 - `hashes.json`：该文件中的哈希码用于自动化版本管理算法。 ## 矩阵文件 每组张量算子矩阵均包含以下文件： | 文件名 | 说明 | |-----------------------|----------------------------------------------------------------------| | $U_{k,q}$.zdc | 轨道角动量空间中$k$阶单位张量算子的$q$分量$mathrm{U}^{(k)}_q$ | | $T_{k,q}$.zdc | 自旋空间中$k$阶单位张量算子的$q$分量$mathrm{T}^{(k)}_q$ | | $UU_{k}$.zdc | 轨道角动量空间中$k$阶单位张量算子的平方$(mathrm{U}^{(k)}cdotmathrm{U}^{(k)})$ | | $TT_{k}$.zdc | 自旋空间中$k$阶单位张量算子的平方$(mathrm{T}^{(k)}cdotmathrm{T}^{(k)})$ | | $UT_{k}$.zdc | 轨道与自旋空间中$k$阶单位张量算子的标量积$(mathrm{U}^{(k)}cdotmathrm{T}^{(k)})$ | | $C_{k,q}$.zdc | $k$阶库仑算子的$q$分量$mathrm{C}^{(k)}_q$ | | $L_{q}$.zdc | 总轨道角动量算子的$q$分量$mathrm{L}_q$ | | $S_{q}$.zdc | 总自旋角动量算子的$q$分量$mathrm{S}_q$ | | $J_{q}$.zdc | 总角动量算子的$q$分量$mathrm{J}_q$ | | $LL$.zdc | 总轨道角动量算子的平方$(mathrm{L}cdotmathrm{L})$ | | $SS$.zdc | 总自旋角动量算子的平方$(mathrm{S}cdotmathrm{S})$ | | $JJ$.zdc | 总角动量算子的平方$(mathrm{J}cdotmathrm{J})$ | | $LS$.zdc | 总轨道与自旋角动量算子的标量积$(mathrm{L}cdotmathrm{S})$ | | $C2$.zdc | 特殊群$G_2$的卡西米尔算子$mathrm{C}_2(G_2)$ | | $CR$.zdc | $2l+1$维特殊正交（旋转）群的卡西米尔算子$mathrm{C}_2(SO(2l+1))$ | | $H1_{k}$.zdc | $k$阶库仑一阶微扰哈密顿量$mathrm{f}_k$（即$F^k$） | | $H2$.zdc | 自旋轨道一阶微扰哈密顿量$mathrm{z}$（即$zeta$） | | $H4_{c}$.zdc | $k$阶有效库仑二阶微扰哈密顿量$mathrm{t}_c$（即$T^c$） | | $Hss_{k}$.zdc | $k$阶自旋-自旋一阶微扰哈密顿量$mathrm{m}_{k,ss}$ | | $Hsoo_{k}$.zdc | $k$阶自旋-其他轨道一阶微扰哈密顿量$mathrm{m}_{k,soo}$ | | $H5_{k}$.zdc | $k$阶自旋-自旋与自旋-其他轨道一阶微扰哈密顿量$mathrm{m}_k$（即$M^k$） | | $H6_{k}$.zdc | $k$阶有效静电自旋轨道二阶微扰哈密顿量$mathrm{p}_k$（即$P^k$） | | $Hcf_{k,q}$.zdc | $k$阶晶体场微扰哈密顿量的$q$分量$mathrm{H}_{mathrm{cf},q}^{(k)}$（即$B^k_q$） | 约化矩阵元是由维格纳-埃克特定理（Wigner-Eckart theorem）引入的概念，用于描述任意球张量算子$mathrm{A}^{(k)}$的角向行为： $$ langle J' M' | mathrm{A}^{(k)}_{q} | J M angle = (-1)^{J'-M'} egin{pmatrix} J' & k & J \ -M' & q & M end{pmatrix} langle J' || mathrm{A}^{(k)} || J angle $$ 对于包含约化矩阵元的矩阵组，其文件名中省略了分量下标$q$。 ## 态表示 每个矩阵数据容器均在JSON文件`data/matrix.json`中包含所有态的量子数信息。对于乘积态，其量子数为每个电子的$n$、$l$、$m_l$、$s$、$m_s$。LS耦合态通过不可约表示进行表征，多数情况下这些不可约表示与特定张量算子的本征值相关联。 | 关键不可约表示相关张量算子 | 说明 | |----------------------------|----------------------------------------------------------------------| | $S^2$ | 由本征值$S(S+1)$推导得到的多重度$2S+1$，对应总自旋平方算子$(mathrm{S}cdotmathrm{S})$ | | $C_7$ 3元组$W=(w_1w_2w_3)$ | 7维特殊正交群的卡西米尔算子$mathrm{C}_2(SO(7))$ | | $C_2$ 2元组$U=(u_1u_2)$ | 特殊群$G_2$的卡西米尔算子$mathrm{C}_2(G_2)$ | | $L^2$ | 由本征值$L(L+1)$推导得到的$L$光谱特征，对应总轨道角动量平方算子$(mathrm{L}cdotmathrm{L})$ | | $J^2$ | 由本征值$J(J+1)$推导得到的量子数$J$，对应总角动量平方算子$(mathrm{J}cdotmathrm{J})$ | | $J_z$ | 本征值$M_J$，对应总角动量轴向分量算子$mathrm{J}_0$ | | $ ext{sen}$ | Seniority数$ u$，对应算子$(mathrm{S}cdotmathrm{S})$与$mathrm{C}_2(SO(7))$ | | $ ext{num}$ | 具有相同量子数$L$和$S$的不同态的索引 | | $ au$ | 具有相同量子数集合的不同态的索引$ au$ | ## 版本政策 本数据集采用语义化版本号规范，版本格式为「主版本号.次版本号.修订号」，起始版本为「1.0.0」。当仅更新数据集元数据且未修改包含的文件时，修订号递增；次版本号用于标识包含文件修改的数据集发布，此类修改仅为数据容器元数据的非功能性调整，例如更新或修正数据容器元数据中的标题与描述，或新增元数据属性；主版本号则在数据容器发生功能性修改或新增张量算子时递增。