Can the Dirac equation be reduced to an exact two-component spinorial Weyl form?

SWORD propagates half the spinor without approximating the Hamiltonian

The paper's answer to the title question is structural rather than approximate. In the Weyl representation the kinetic operator sits in the diagonal blocks of the Hamiltonian and the off-diagonal coupling is just the scalar mass term mc^2. That arrangement lets the authors write an exact relation between the two chiral components of the Dirac spinor, so the four-component problem is reorganized rather than replaced by an approximate two-component Hamiltonian.

The practical consequence is that only one chirality is propagated explicitly. The opposite chirality is reconstructed self-consistently from the exact coupling relation. The authors stress that this is not an energy-decoupling approximation: no inverse-potential operators appear, which is the feature that distinguishes SWORD from conventional two-component reductions.

The propagated chirality obeys an inhomogeneous Helmholtz equation

Eliminating the right-handed component from the coupled Weyl equations turns the left-handed equation into a Helmholtz-type differential operator, with screening parameter mu^2 = (m^2 c^4 - E^2)/c^2. The formal inverse is applied by convolution with the Helmholtz Green's function kernel exp(-mu r)/(4 pi r), which is exactly the operation the multiwavelet framework is built to perform.

For a static external potential the commutator term simplifies to i c (sigma . grad V) psi, giving a final iterative update whose structure mirrors the integral form of the one-electron Schrodinger equation, except that it is a two-component equation and its right-hand side is energy-dependent. Separating the total energy as E = mc^2 + epsilon shows that all relativistic corrections vanish in the c -> infinity limit, recovering the usual non-relativistic equation.

Variational collapse is removed only in the one-electron Coulomb case

For a single electron in a homogeneous, spherically symmetric Coulomb potential, the authors rescale coordinates by r -> (1 + E/mc^2) r. This removes the energy dependence and produces a two-component Hamiltonian H_w whose spectrum is bounded from below: the positive- and negative-energy continua of the Dirac Hamiltonian map onto its positive spectrum, and the bound states map onto its discrete negative spectrum. Because the physically relevant bound states sit in a part of the spectrum bounded from below, ordinary variational minimization can be applied without variational collapse.

The authors are explicit that this argument does not extend to the N-electron case. The coordinate transformation relies on the homogeneity of the Coulomb potential, whereas the effective many-electron operators J and K are not homogeneous, and K is additionally nonlocal. Substituting V = V_ne + J - K into the transformed Hamiltonian therefore does not yield an equivalent Fock-like operator with the same spectral properties.

Accuracy is tested across one-, two-, and four-electron systems

The numerical assessment adds Dirac-Hartree-Fock ingredients progressively. One-electron systems (H, Ne9+, Kr35+, Xe53+, Hg79+) isolate the Green's-function propagation and the reconstruction of the opposite chirality in a fixed external potential. Two-electron closed-shell systems introduce a self-consistent Coulomb mean field. Four-electron closed-shell systems add the nonlocal Hartree-Fock exchange operator, giving the first benchmark containing all ingredients of the present Dirac-Coulomb implementation.

All calculations used the VAMPyR Python library with an adaptive multiwavelet representation, at c = 137.03599913900001 in atomic units, and the electron-electron interaction was restricted to the Dirac-Coulomb Hamiltonian without Breit or other non-instantaneous corrections. SWORD and the reference four-component implementation shared the same SCF workflow, the same non-relativistic starting orbitals, and identical convergence criteria, so the comparison isolates the effect of replacing four-component propagation with propagation plus reconstruction.

Across the tested precision ranges (MW5-MW8 for one-electron, MW4-MW7 for two-electron), both formulations systematically approach the reference energy as the multiwavelet threshold is tightened, and SWORD stays within the accuracy expected from the imposed numerical threshold. Energies were compared both with the four-component multiwavelet implementation under matched settings and with independent GRASP reference calculations using the corresponding nuclear model; hydrogen additionally served as a point-Coulomb control case.

Propagation cost falls, but total SCF wall time does not follow

The one-electron calculations show the reduced propagation most clearly: the production SCF stage is approximately a factor of two faster, consistent with propagating half as many spinor components, and reconstruction of the opposite chirality contributes only a small fraction of the update cost. Including setup and the common warm-up procedure reduces the total wall-time gain, but it remains substantial.

The two-electron benchmarks mark a different regime. The reduction in propagation cost persists, but the overall gain shrinks substantially once the self-consistent potential must be rebuilt and applied, because operations outside the Green's-function propagation can become more expensive for the SWORD orbitals. In the four-electron calculations the complete SWORD update, including reconstruction, required on average approximately 73.3% of the corresponding four-component propagation time. That is a clear reduction, but it stays above the idealized 50% suggested by the number of explicitly propagated components.

The authors attribute the gap to the prototype design: much of the existing four-component machinery is reused after the full spinor is reconstructed, so potential evaluation and Fock-matrix construction do not yet exploit the reduced representation. They present the timings as the performance of this prototype rather than as the computational limit of the formulation.

Derivative-sensitive starting guesses need a low-precision warm-up

Differentiation emphasizes short-wavelength components of a numerical function, so small-scale artifacts that are negligible in the orbital itself can become visible after differentiation and trigger extra local refinement in an adaptive multiwavelet representation. This matters directly for SWORD because the opposite chirality is reconstructed through an expression containing the momentum operator acting on the propagated left-handed component, exposing numerical structure at every reconstruction.

The sensitivity grows with nuclear charge, since core orbitals become more strongly localized and carry larger momentum components, and the momentum-dependent correction in the kinetically balanced Weyl guess becomes increasingly important relative to the leading non-relativistic term. In the reported calculations a short, deliberately low-precision four-component warm-up was sufficient to suppress these fine-scale features before the orbitals entered either production algorithm, and the same smoothed orbitals initialized both the four-component and SWORD runs. The authors describe this as an implementation issue tied to starting a derivative-sensitive adaptive calculation directly at high precision, consistent with the dynamic-precision strategy common in multiwavelet SCF work, rather than as an intrinsic limitation of SWORD.

The result is a working idea, not an optimized alternative

Across all three benchmark levels SWORD reproduces the corresponding four-component Dirac-Coulomb Hartree-Fock energies within the requested numerical precision, and both formulations agree consistently with the independent GRASP references. The authors state plainly that these results do not establish SWORD as an optimized alternative to existing relativistic methods; they establish that the underlying computational idea works.

The most useful way to read the paper is therefore as a split between what is already demonstrated and what is left open. Demonstrated: an exact Weyl-basis coupling relation, a Helmholtz-form propagation for one chirality, and energy agreement with four-component Dirac-Coulomb Hartree-Fock for the small closed-shell systems tested. Open: whether the formal reduction in propagation cost survives for larger and chemically relevant systems, and how much of the remaining SCF overhead can be reformulated around the reduced representation. The authors point to a production multiwavelet environment such as MRChem, with dynamic precision control and optimized low-level operations, as the natural next step for answering that question.

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