Cartesian-3j and Cartesian-nj Symbols

• Cartesian-3j and Cartesian-nj symbols are tensor-coupling coefficients that decompose irreducible Cartesian tensors (ICTs) for machine learning, providing a clear alternative to spherical approaches.
• They systematically extract angular-momentum blocks from tensor products using precomputed ICTD matrices, ensuring symmetry and orthonormality in the coupling process.
• Their implementation in the Python package cartnn enhances computational efficiency in low-rank ICT operations, optimizing performance in equivariant neural network architectures.

The Cartesian-3j and Cartesian-nj symbols are tensor-coupling coefficients that generalize the landscape of equivariant computations in atomistic machine learning, providing a direct analogue to the widely used Wigner-3j and Wigner-nj symbols from spherical tensor theory. These symbols enable the combination of irreducible Cartesian tensors—ICTs—into new irreducible representations, thereby furnishing an alternative to spherical-based approaches. This formalism underpins tensor product and contraction operations in Cartesian space, facilitates rigorous comparisons between Cartesian and spherical architectures, and is implemented in the Python package cartnn as an extension of e3nn (Xu et al., 18 Dec 2025).

1. Definitions and Structural Role

An irreducible Cartesian tensor (ICT) of rank $\nu$ is a representation of the orthogonal group $O(3)$ of dimensionality $3^\nu$, which is generally reducible. Decomposition yields irreducible blocks:

$$T^{(\nu)} = \bigoplus_{\ell=0}^{\nu} T^{(\nu;\,\ell;q)},$$

where each block with weight $\ell$ has $2\ell+1$ components; $q$ distinguishes multiplicities. The tensor product of two ICTs with weights $\ell_1,\,\ell_2$ gives components of weight $\ell_3$ for $|\ell_1-\ell_2| \leq \ell_3 \leq \ell_1+\ell_2$.

The Cartesian-3j symbol, denoted $Z(\ell_1, \ell_2, \ell_3)$, is the linear map extracting the irreducible component of weight $\ell_3$ from the tensor product:

$$Z(\ell_1,\ell_2,\ell_3) : \mathbb{R}^{3^{\nu_1}} \otimes \mathbb{R}^{3^{\nu_2}} \to \mathbb{R}^{3^{\nu_3}},$$

or, vectorized:

$$Z(\ell_1,\ell_2,\ell_3) \in \mathbb{R}^{(2\ell_3+1) \times (3^{\nu_1}\cdot 3^{\nu_2})}.$$

The Cartesian-nj symbol generalizes this coupling to $n$ ICTs via a product of 3j symbols, constructing

$$\mathcal{Z}^{L_n M_n}_{\ell_1 m_1, ..., \ell_n m_n} = Z^{L_2 M_2}_{\ell_1 m_1, \ell_2 m_2} \cdot Z^{L_3 M_3}_{L_2 M_2, \ell_3 m_3} \cdots Z^{L_n M_n}_{L_{n-1} M_{n-1}, \ell_n m_n},$$

with the chain $(\ell_1, \ell_2) \to L_2$, $(L_2, \ell_3) \to L_3$, up to $n$-fold coupling.

2. Mathematical Formulation, Symmetry, and Orthogonality

The construction of the Cartesian-3j symbol leverages the ICTD “path” matrix $C^{(\nu;\ell;1)}$ to select angular-momentum blocks and re-project into the irreducible space:

$$Z(\ell_1,\ell_2,\ell_3) = C^{(\nu_1+\nu_2;\, \ell_3;\, 1)} \cdot C^{(\ell_3;\, \ell_3;\, 1)\, T}.$$

The full $n$-fold symbol is correspondingly:

$$\mathcal{Z}^{L M}_{\{\ell_i m_i\}} = \prod_{i=2}^n Z^{L_i M_i}_{L_{i-1} M_{i-1},\, \ell_i m_i},$$

using $L_1 \equiv \ell_1$, $M_1 \equiv m_1$.

Permutation symmetry is inherited from the orthogonality of ICTD matrices:

$$Z(\ell_2,\ell_1,\ell_3) = (-1)^{\ell_1+\ell_2-\ell_3}\, Z(\ell_1,\ell_2,\ell_3),$$

with orthonormality on input indices as

$$\sum_{I_1, I_2} Z_{I_3; I_1, I_2}(\ell_1, \ell_2, \ell_3) Z_{I_3'; I_1, I_2}(\ell_1, \ell_2, \ell_3') = \delta_{\ell_3,\ell_3'}\delta_{I_3,I_3'}.$$

3. Algorithmic Realization and Computational Recursion

Computation proceeds as follows:

1. ICTD Matrix Construction: For ranks $\nu_1$, $\nu_2$, build
   • $$C^{(\nu_1+\nu_2;\,\ell_3;1)} \in \mathbb{R}^{3^{\nu_1+\nu_2} \times (2\ell_3+1)}$$;
   • $$C^{(\ell_3;\,\ell_3;1)} \in \mathbb{R}^{3^{\ell_3} \times (2\ell_3+1)}$$.
2. Tensor Product Flattening: Take $T^1 \otimes T^2$, flatten to $x \in \mathbb{R}^{3^{\nu_1+\nu_2}}$.
3. Projection: Project $x$ to weight $\ell_3$ using $[C^{(\nu_1+\nu_2;\,\ell_3;1)}]^T$:

$$y = [C^{(\nu_1+\nu_2;\,\ell_3;1)}]^T x \in \mathbb{R}^{2\ell_3+1}.$$

1. Post-processing: If required, recover Cartesian representation with $C^{(\ell_3;\,\ell_3;1)}$.

For $n$-fold coupling, steps 2–4 are iterated recursively, treating each new projected tensor as a factor for further coupling.

4. Relation to Spherical Wigner Symbols

The role of $Z(\ell_1, \ell_2, \ell_3)$ precisely mirrors the Wigner 3-j symbol in spherical tensor coupling. Both facilitate angular-momentum block selection and abide by identical orthogonality and permutation-phase relations. The principal distinction is the ambient space dimensionality and decomposition requirements:

• Spherical basis: Each irreducible representation has dimension $2\ell+1$. Coupling is optimized by construction.
• Cartesian basis: ICTs are embedded in larger $3^\nu$ spaces, requiring ICTD decomposition to isolate angular-momentum blocks.

Both coupling diagrams yield the same admissible $\ell_3$ values, $|\ell_1-\ell_2| \leq \ell_3 \leq \ell_1+\ell_2$.

5. Implementation Aspects in cartnn and e3nn Ecosystem

The cartnn Python module integrates these symbols via the ICTD routines in cartnn.o3.ICTD. C-matrices and corresponding $Z(\ell_1,\ell_2,\ell_3)$ are precomputed for all $\ell \leq \ell_{\mathrm{max}}$ on module import, typically up to $\ell_{\mathrm{max}} = 4$. Runtime costs for coupling are

• Initial computation: $\mathcal{O}(3^{\nu_1+\nu_2}\cdot(2\ell_3+1))$, amortized over multiple uses.
• Each coupling: $\mathcal{O}((2\ell_1+1)(2\ell_2+1)(2\ell_3+1))$.

ICTD matrices are orthonormal, ensuring numerical stability up to machine precision. Memory requirements for storage scale as $$\mathcal{O}(\ell_{\mathrm{max}}^3\cdot(2\ell_{\mathrm{max}}+1)\cdot 3^{2\ell_{\mathrm{max}}})$$; for $\ell_{\mathrm{max}} \leq 4$, this remains feasible. For high $\ell$ or correlation order $n$, the size of Cartesian-nj tensors can become prohibitive.

6. Explicit Example: Rank-1 ICT Coupling

Consider coupling two rank-1 ICTs $T^1_i$ and $T^2_j$, with weights $\ell_1=\ell_2=1$. Possible couplings are:

$\ell_3$	Symbol	Expression
$0$	$Z(1,1,0)_{;ij}$	$\frac{1}{\sqrt{3}}\delta_{ij}$, yields $y_0 = \frac{1}{\sqrt{3}}\ T^1\cdot T^2$
$1$	$Z(1,1,1)_{k;ij}$	$\frac{1}{\sqrt{2}}\epsilon_{kij}$, yields $y_k = \frac{1}{\sqrt{2}}(T^1\times T^2)_k$
$2$	$Z(1,1,2)_{ij;i'j'}$	$$\tfrac{1}{2}(T^1_iT^2_j + T^1_jT^2_i) - \frac{1}{3}\delta_{ij}(T^1\cdot T^2)$$

Given $T^1=(1,0,0), T^2=(0,1,0)$:

• $y_0=0$
• $y^{(\ell_3=1)}=(0,0,\tfrac{1}{\sqrt{2}})$
• $y^{(\ell_3=2)}_{12}=y_{21} = \tfrac{1}{2}$, other components zero.

7. Regimes and Architectures Benefiting from Cartesian Coupling

Cartesian coupling is advantageous in specific computational regimes:

• For low-rank ICTs ($\ell \leq 2$ or $3$), Cartesian formulas avoid spherical-harmonic evaluation and incur fewer transcendental operations.
• In edge-feature architectures where $\ell_{\mathrm{max}} \leq 2$, Cartesian packaging decreases memory footprint and speeds up convolutions in architectures such as TensorNet.
• At high correlation order or large $\ell$, Cartesian-nj storage costs scale combinatorially, and spherical tensors remain preferred for compactness.
• Hybrid designs may utilize Cartesian coupling for low-$\ell$ (local) filters, spherical for high-$\ell$ (global) features; this mixing offers flexibility for application-specific optimization.

The direct implementation of Cartesian-3j and Cartesian-nj in the cartnn extension of e3nn, based on precomputed orthonormal ICTD matrices, enables comprehensive support for irreducible Cartesian tensor-product operations. While low-rank cases are speed- and simplicity-optimized, high-rank or extended correlation scenarios often favor the established compactness of spherical tensor approaches (Xu et al., 18 Dec 2025).