Fig. 2: Method of constructing an equivariant mapping \(\{{{{{{{{\mathcal{R}}}}}}}}\}\,\mapsto\, {\hat{H}}_{{{{{{{{\rm{DFT}}}}}}}}}\). | Nature Communications

Fig. 2: Method of constructing an equivariant mapping \(\{{{{{{{{\mathcal{R}}}}}}}}\}\,\mapsto\, {\hat{H}}_{{{{{{{{\rm{DFT}}}}}}}}}\).

From: General framework for E(3)-equivariant neural network representation of density functional theory Hamiltonian

Fig. 2

Take the Hamiltonian matrix between l = 1 and l = 2 orbitals, for example. a The atomic numbers Zi and interatomic distances ∣rij∣ are used to construct the l = 0 vectors, and the unit vectors of relative positions \({\hat{r}}_{ij}\) are used to construct vectors of l = 1, 2, … . b These vectors are passed to the equivariant neural network. c If neglecting spin–orbit coupling (SOC), the output vectors of the neural network are converted to the Hamiltonian using the rule 1 ⊕ 2 ⊕ 3 = 1 ⊗ 2 via the Wigner–Eckart layer. If including SOC, the output consists of two sets of real vectors which are combined to form complex-valued vectors. These vectors are converted to the spin–orbital DFT Hamiltonian according to a different rule \((1\oplus 2\oplus 3)\oplus (0\oplus 1\oplus 2)\oplus (1\oplus 2\oplus 3)\oplus (2\oplus 3\oplus 4)=(1\otimes \frac{1}{2})\otimes (2\otimes {\frac{1}{2}}^{*})\).

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