Fixed Spin–Orbit Operators in an LCAO Basis

This chapter defines the density-independent spin–orbit operator supplied by a fully relativistic norm-conserving pseudopotential and derives its matrix in a localized atomic-orbital basis. It also fixes the decomposition used when ELFES learns the remainder of a self-consistent Hamiltonian. Hartree atomic units are used unless stated otherwise.

Operator Boundary

For fixed ionic positions, numerical atomic orbitals, and a fully relativistic pseudopotential, write the Kohn–Sham Hamiltonian schematically as

\[ \hat H[n,\boldsymbol m] = \hat H_{\mathrm{fixed}} + \hat V_{\mathrm H}[n] + \hat{\mathbf V}_{\mathrm{xc}}[n,\boldsymbol m]. \]

The fully relativistic ionic nonlocal operator belongs to \(\hat H_{\mathrm{fixed}}\). Its matrix changes when the structure, basis, or pseudopotential changes, but not when the self-consistent density changes at fixed geometry. It nevertheless affects self-consistency through the sequence

\[ \hat V_{\mathrm{NL}}^{\mathrm{FR}} \longrightarrow \{\psi_{a\boldsymbol k},f_{a\boldsymbol k}\} \longrightarrow \{n,\boldsymbol m\} \longrightarrow \hat V_{\mathrm H}+\hat{\mathbf V}_{\mathrm{xc}}. \]

The corresponding expectation value \(\operatorname{tr}(\mathbf D\mathbf V_{\mathrm{NL}}^{\mathrm{FR}})\) therefore depends on the density matrix even though the operator does not. Density-dependent noncollinear exchange, DFT+\(U\), and other occupation-dependent spin terms are not part of the fixed spin–orbit operator defined here.

Fully Relativistic Separable Pseudopotential

Let \(\kappa\) label a projector center and \(q\) a radial projector channel. For angular momentum \(\ell>0\), a fully relativistic channel resolves the coupled total angular momentum \(j=\ell\pm\tfrac12\). Its spin-angular projector is

\[ \lvert\beta_{\kappa q\ell j m_j}\rangle = \sum_{m s} C^{j m_j}_{\ell m,\frac12 s} \lvert\beta_{\kappa q\ell m};s\rangle, \]

where \(C^{j m_j}_{\ell m,\frac12 s}\) is a Clebsch–Gordan coefficient and \(\beta_{\kappa q\ell m}\) is a spatial projector. A general separable fully relativistic nonlocal operator is

\[ \hat V_{\mathrm{NL}}^{\mathrm{FR}} = \sum_{\kappa} \sum_{q q'\ell j m_j} \lvert\beta_{\kappa q\ell j m_j}\rangle d^{\kappa\ell j}_{q q'} \langle\beta_{\kappa q'\ell j m_j}\rvert. \]

The coefficient matrices \(d^{\kappa\ell j}\) and the radial projectors are pseudopotential data. They are not reconstructed from the self-consistent density.

ELFES defines the fixed SOC operator as the spin-traceless part of this physical fully relativistic operator:

\[ \hat V_{\mathrm{SOC}} \equiv \hat V_{\mathrm{NL}}^{\mathrm{FR}} - \frac12 \operatorname{tr}_s \left( \hat V_{\mathrm{NL}}^{\mathrm{FR}} \right) \otimes\sigma_0. \]

This definition is unique. It does not require constructing a second scalar-relativistic pseudopotential or choosing how two different radial projector sets should be subtracted.

Pauli Coupling in a Spatial Projector Basis

Within the product space of fixed \(\ell\) and spin \(\tfrac12\), the projectors onto the two \(j\) subspaces are

\[ \hat P_{\ell+\frac12} = \frac{(\ell+1)\sigma_0+\hat{\boldsymbol L}\cdot\boldsymbol\sigma}{2\ell+1}, \qquad \hat P_{\ell-\frac12} = \frac{\ell\sigma_0-\hat{\boldsymbol L}\cdot\boldsymbol\sigma}{2\ell+1}. \]

Consequently, the spin-traceless coupling between spatial projectors \(a=(q,\ell,m)\) and \(b=(q',\ell,m')\) has Pauli components

\[ D_{ab}^{\kappa\alpha} = \sum_{j=\ell\pm\frac12} \eta_{\ell j} d^{\kappa\ell j}_{q q'} \frac{(L_\alpha^{(\ell)})_{m m'}}{2\ell+1}, \qquad \eta_{\ell,\ell+\frac12}=+1, \quad \eta_{\ell,\ell-\frac12}=-1. \]

Here \(\mathsf L_\alpha^{(\ell)}\) is the angular-momentum matrix in the same spatial spherical-harmonic convention as the projectors. For real spherical harmonics it is purely imaginary and antisymmetric. Therefore \(\mathsf D^{\kappa\alpha}\) is also purely imaginary and can be stored exactly as its real imaginary amplitude.

This projector identity is equivalent to an explicit Clebsch–Gordan sum, while exposing the unique Pauli-vector part directly. It also makes clear that \(\ell=0\) contributes no spin-traceless SOC term.

LCAO Matrix Elements

Use the periodic LCAO convention of Localized Atomic-Orbital Basis,

\[ H_{\mu\nu}^{\mathrm{SOC},\alpha}(\boldsymbol R) = \langle\phi_{\mu\boldsymbol 0}\vert \hat V_{\mathrm{SOC}}^\alpha \vert\phi_{\nu\boldsymbol R}\rangle. \]

For a translated projector center \(\kappa\boldsymbol T\), define the scalar AO–projector leg

\[ P_{\mu a}^{\kappa\boldsymbol T} \equiv \langle \phi_{\mu\boldsymbol 0} \vert \beta_{\kappa a\boldsymbol T} \rangle. \]

The fixed SOC matrix is the projector-centered contraction

\[ H_{\mu\nu}^{\mathrm{SOC},\alpha}(\boldsymbol R) = \sum_{\kappa\boldsymbol T} \sum_{a b} P_{\mu a}^{\kappa\boldsymbol T} D_{a b}^{\kappa\alpha} P_{\nu b}^{\kappa,\boldsymbol T-\boldsymbol R\,*}. \]

Both AO–projector legs may connect different atoms to the same projector center, so this is a local three-center contraction rather than an onsite atomic approximation. Finite AO and projector cutoffs make each center couple only a bounded neighborhood at fixed basis, density, and cutoff radii.

For real AOs and projectors, \(P\) is real and the spin–orbit Pauli components are purely imaginary. Hermiticity gives

\[ H_{\nu\mu}^{\mathrm{SOC},\alpha}(-\boldsymbol R) = H_{\mu\nu}^{\mathrm{SOC},\alpha}(\boldsymbol R)^* = -H_{\mu\nu}^{\mathrm{SOC},\alpha}(\boldsymbol R). \]

The complete spinor contribution is

\[ \mathbf H_{\mathrm{SOC}}(\boldsymbol R) = \sum_{\alpha=x,y,z} \mathsf H^{\mathrm{SOC},\alpha}(\boldsymbol R) \otimes\sigma_\alpha. \]

Learning Decomposition

For the Hamiltonian content specified above, ELFES uses

\[ \mathbf H_{\mathrm{learn}}[n,\boldsymbol m] \equiv \mathbf H_{\mathrm{KS}}[n,\boldsymbol m] - \mathbf H_{\mathrm{SOC}}, \qquad \mathbf H_{\mathrm{KS}}^{\mathrm{pred}} = \mathbf H_{\mathrm{learn}}^{\mathrm{pred}} + \mathbf H_{\mathrm{SOC}}^{\mathrm{calc}}. \]

The model therefore need not reproduce a fixed pseudopotential operator from training data. The calculated term is evaluated from the current geometry, the actual numerical AO basis, and the same fully relativistic projector dataset that defines the target calculation.

In the minimal real-AO Hamiltonian described in Localized Atomic-Orbital Basis, the fixed SOC term equals \(\mathrm i\operatorname{Im}\mathsf H^\alpha\) for \(\alpha=x,y,z\). This equality is a validation identity for that Hamiltonian content; it is not a declaration that every imaginary spin-dependent term in an extended Hamiltonian must originate from the pseudopotential.

Locality and Complexity

Let \(z_\kappa\) be the number of AO functions whose support overlaps projectors on center \(\kappa\), and let \(p_\kappa\) be the number of spatial projector functions on that center. A direct local implementation costs schematically

\[ O\!\left( \sum_\kappa z_\kappa p_\kappa + \sum_\kappa z_\kappa^2\,\operatorname{nnz}(\mathsf D^\kappa) \right). \]

At fixed species set, basis size per atom, cutoff radii, and atomic density, \(z_\kappa\) and \(p_\kappa\) are bounded, so time and sparse output storage scale as \(O(N)\) with atom count. Species-level radial transforms and two-center tables can be cached; only neighbor enumeration, table evaluation, and local contractions are geometry dependent.

Implementation Conventions

The theory above is independent of a source code's array ordering. ELFES implementations use Å for lengths, eV for Hamiltonian matrices, the project real-spherical-harmonic convention, and the cell-shift direction fixed in Localized Atomic-Orbital Basis. A source adapter is responsible for converting radial normalization, units, angular ordering, spin frame, and periodic matrix orientation before values enter the calculator.

The current ABACUS path converts the fully relativistic norm-conserving UPF into an ordered UniformNumericalAtomicBasis of KB projectors and a projector-space HermOrbMatrix(pauli="xyz") containing \(\mathsf D_{\mathrm{SO}}\). Its numerical validation compares the calculated three Pauli components directly with the purely imaginary Pauli-vector sector extracted from a converged ABACUS real-space Hamiltonian.