Matrix Elements for the Gd Ion

This dataset provides angular matrices of spherical tensor operators for the f7 configuration (lanthanide ion Gd3+) 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 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)本数据集提供了f⁷电子组态（镧系离子Gd³⁺）下球张量算符的角动量矩阵，采用精确算术运算生成。该矩阵源自软件包AMELI（镧系离子角动量矩阵元，Angular Matrix Elements of Lanthanide Ions）。矩阵在行列式乘积态空间中计算，并转换为LS耦合（LS-coupling）表象。每个LS耦合态通过一套完整的不可约表示（irreducible representation）进行分类，该套表示与C. W. Nielson和G. F. Koster发表的标准一致。哈密顿矩阵已与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$以三个参数$s$、$n$、$d$表示为带符号的有理数平方根，形式如下： $$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 Format，坐标格式）高度相似：通过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。实际数值存储于单独的唯一值列表中，该列表的长度通常远小于非零元素的总数量。存储遵循标准COO顺序，以行作为主索引、列作为次索引。唯一值列表按升序排列。若矩阵中整数$n$或$d$超出HDF5数据类型的64位最大值，则将数值按位拆分，分多段存储。 每个矩阵均存储于扩展名为.zdc的SciDataContainer文件中。该容器文件为标准ZIP压缩包，内部包含遵循现代科研数据管理FAIR原则（可发现、可访问、可互操作、可复用，Findable, Accessible, Interoperable, Reusable）的JSON格式元数据。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元数据文件`meta.json`。 ## 数据集记录内容 所有张量算符矩阵按其所处的态空间进行分组。本数据集记录包含以下文件： 1. `product.zip`：位于子文件夹`product`中，包含乘积态张量算符矩阵，同时附带用于将乘积态或乘积态矩阵转换为LS耦合表象的`matrix 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。 2. `sljm.zip`：位于子文件夹`sljm`中，包含LS耦合表象下的张量算符矩阵。 3. `slj.zip`：包含LS耦合表象下J多重态（J-multiplet）的张量算符矩阵。该数据集由`sljm.zip`中的矩阵精简而来：子文件夹`slj`中存储仅包含拉伸态$M_J=J$的矩阵，子文件夹`sljm_reduced`中存储包含约化矩阵元（reduced matrix element）的矩阵。适用于非晶态材料中镧系离子的能级拟合与Judd-Ofelt计算。 4. `support.zip`：该特殊文件仅在扩展AMELI软件包以生成自定义算符矩阵时可用，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。 5. `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 每套张量算符矩阵均包含以下文件： | 文件名 | 描述与对应张量算符 | | --- | --- | | $U_{k,q}$.zdc | 轨道角动量空间中秩为$k$的单位张量算符的第$q$分量$mathrm{U}^{(k)}_q$ | | $T_{k,q}$.zdc | 自旋空间中秩为$k$的单位张量算符的第$q$分量$mathrm{T}^{(k)}_q$ | | $UU_{k}$.zdc | 轨道角动量空间中秩为$k$的单位张量算符的平方$(mathrm{U}^{(k)}cdotmathrm{U}^{(k)})$ | | $TT_{k}$.zdc | 自旋空间中秩为$k$的单位张量算符的平方$(mathrm{T}^{(k)}cdotmathrm{T}^{(k)})$ | | $UT_{k}$.zdc | 轨道与自旋空间中秩为$k$的单位张量算符的标量积$(mathrm{U}^{(k)}cdotmathrm{T}^{(k)})$ | | $C_{k,q}$.zdc | 秩为$k$的库仑算符（Coulomb operator）的第$q$分量$mathrm{C}^{(k)}_q$ | | $L_{q}$.zdc | 总轨道角动量算符（total orbital angular momentum operator）的第$q$分量$mathrm{L}_q$ | | $S_{q}$.zdc | 总自旋角动量算符（total spin angular momentum operator）的第$q$分量$mathrm{S}_q$ | | $J_{q}$.zdc | 总角动量算符（total angular momentum operator）的第$q$分量$mathrm{J}_q$ | | $LL$.zdc | 总轨道角动量算符的平方$(mathrm{L}cdotmathrm{L})$ | | $SS$.zdc | 总自旋角动量算符的平方$(mathrm{S}cdotmathrm{S})$ | | $JJ$.zdc | 总角动量算符的平方$(mathrm{J}cdotmathrm{J})$ | | $LS$.zdc | 总轨道与自旋角动量算符的标量积$(mathrm{L}cdotmathrm{S})$ | | $C2$.zdc | 特殊群$G_2$的卡西米尔算符（Casimir operator）$mathrm{C}_2(G_2)$ | | $CR$.zdc | $2l+1$维特殊正交（转动）群$mathrm{SO}(2l+1)$的卡西米尔算符$mathrm{C}_2(mathrm{SO}(2l+1))$ | | $H1_{k}$.zdc | 秩为$k$的库仑一级微扰哈密顿量$mathrm{f}_k$（即$F^k$） | | $H2$.zdc | 自旋-轨道一级微扰哈密顿量$mathrm{z}$（即$zeta$） | | $H4_{c}$.zdc | 有效库仑二级微扰哈密顿量$mathrm{t}_c$（即$T^c$） | | $Hss_{k}$.zdc | 秩为$k$的自旋-自旋一级微扰哈密顿量$mathrm{m}_{k,ss}$ | | $Hsoo_{k}$.zdc | 秩为$k$的自旋-其他轨道一级微扰哈密顿量$mathrm{m}_{k,soo}$ | | $H5_{k}$.zdc | 秩为$k$的自旋-自旋与自旋-其他轨道一级微扰哈密顿量$mathrm{m}_k$（即$M^k$） | | $H6_{k}$.zdc | 秩为$k$的有效静电自旋-轨道二级微扰哈密顿量$mathrm{p}_k$（即$P^k$） | | $Hcf_{k,q}$.zdc | 秩为$k$的晶体场微扰哈密顿量（crystal field perturbation Hamiltonian）的第$q$分量$mathrm{H}^{(k)}_{mathrm{cf},q}$（即$B^k_q$） | 包含约化矩阵元的矩阵集合中，其文件名不含下标$_{q}$分量。约化矩阵元是维格纳-埃克特定理（Wigner-Eckart theorem）引入的概念，用于描述任意球张量算符$mathrm{A}^{(k)}$的角向行为，其形式如下： $$ 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 $$ ## 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。对于乘积态，量子数为每个电子的$n$、$l$、$m_l$、$s$和$m_s$。LS耦合态通过不可约表示进行表征，在大多数情况下，这些表示与特定张量算符的本征值相关联。 ### 核心不可约表示与对应张量算符 | 标识 | 含义与对应算符 | | --- | --- | | $S^2$ | 多重度$2S+1$，由总自旋算符平方$(mathrm{S}cdotmathrm{S})$的本征值$S(S+1)$推导而来 | | $C7$ | 3元组$W=(w_1,w_2,w_3)$，对应7维特殊正交群的卡西米尔算符$mathrm{C}_2(mathrm{SO}(7))$ | | $C2$ | 2元组$U=(u_1,u_2)$，对应特殊群$G_2$的卡西米尔算符$mathrm{C}_2(G_2)$ | | $L^2$ | $L$的谱特征，由总轨道角动量算符平方$(mathrm{L}cdotmathrm{L})$的本征值$L(L+1)$推导而来 | | $J^2$ | 量子数$J$，由总角动量算符平方$(mathrm{J}cdotmathrm{J})$的本征值$J(J+1)$推导而来 | | $J_z$ | 轴向分量本征值$M_J$，对应总角动量算符的轴向分量$mathrm{J}_0$ | | $ ext{sen}$ | 资历数（seniority number）$ u$，由算符$(mathrm{S}cdotmathrm{S})$与$mathrm{C}_2(mathrm{SO}(7))$表征 | | $ 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 | ## 版本历史 本数据集采用语义化版本号规范`主版本号.次版本号.修订号`进行发布，初始版本为`1.0.0`。修订号递增对应仅修改数据集元数据但不改动包含文件的更新；次版本号递增对应修改了包含文件的数据集发布，但这些修改仅为数据容器中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；主版本号递增对应数据容器的功能性修改或新增张量算符。 - 版本1.0.0：初始版本 - 版本2.0.0：将J多重态内的专有符号修正替换为阶梯算符；新增库仑算符；新增晶体场算符 - 版本3.0.0：修复了$Hcf_{k,0}$矩阵（存在错误的因子2）