SAP-X2C: Optimally-Simple Two-Component Relativistic Hamiltonian With Size-Intensive Picture Change

We present a simple relativistic exact 2-component (X2C) Hamiltonian that models two-electron picture-change effects using Lehtola’s superposition of atomic potentials (SAP) [S. Lehtola, J. Chem. Theory Comput. 15, 1593-1604 (2019)]. The SAP-X2C approach keeps the low-cost and technical simplicity of the popular 1-electron X2C (1eX2C) predecessor, but is significantly more accurate and has a well-defined thermodynamic limit, making it applicable to extended systems (such as large molecules and periodic crystals). The assessment of the SAP-X2C-based Hartree-Fock total and spinor energies, spin-orbit splittings, equilibrium bond distances, and harmonic vibrational frequencies suggests that SAP-X2C is similar to the more complex atomic mean-field (AMF) X2C counterparts in its ability to approximate the 4-component Dirac-Hartree-Fock reference.

💡 Research Summary

The paper introduces SAP‑X2C, a new variant of the exact two‑component (X2C) relativistic Hamiltonian that incorporates two‑electron picture‑change (2ePC) effects through Lehtola’s superposition of atomic potentials (SAP). Traditional one‑electron X2C (1eX2C) is attractive because it is cheap, easy to implement, and accurately captures scalar relativistic and spin‑orbit (SO) effects by block‑diagonalizing the one‑electron Dirac Hamiltonian. However, 1eX2C neglects 2ePC and suffers from a non‑size‑intensive picture‑change due to the divergent nuclear Coulomb potential, limiting its applicability to extended or periodic systems.

SAP‑X2C retains the same unitary transformation U used in 1eX2C but replaces the nuclear potential V and the small‑component matrix W with SAP‑based counterparts V_SAP and W_SAP. The SAP potential is constructed as a sum of atomic effective nuclear charges fitted to contracted s‑type Gaussian functions, plus a three‑center electron‑screening term. This yields V_SAP = V + V_e and W_SAP = W + W_e, where V_e and W_e are evaluated from three‑center two‑electron integrals that are readily available in modern integral libraries (e.g., Libint, Libcint). After the transformation, the screening contribution V_e is subtracted to avoid double‑counting. The only additional computational ingredient beyond 1eX2C is the evaluation of these three‑center integrals, making the implementation straightforward and inexpensive.

Theoretical advantages of SAP‑X2C are twofold. First, the SAP potential decays super‑polynomially with distance, guaranteeing absolute convergence for any dimensional lattice and thus providing a well‑defined thermodynamic limit. Second, the electron‑screening part implicitly accounts for the dominant intra‑atomic 2ePC contributions, improving both scalar relativistic and SO terms without the need for full four‑component (4C) mean‑field calculations.

Benchmark calculations were performed on (i) a diverse set of small molecules containing light to heavy elements, (ii) the heavy‑element dimer Og₂, (iii) core‑level SO splittings of Rn and Og, (iv) equilibrium bond lengths and harmonic vibrational frequencies, and (v) a size‑intensivity test using increasing fragments of an Xe crystal. Compared to 1eX2C, SAP‑X2C reduces total Hartree‑Fock (HF) energy errors from tens of millihartree for heavy atoms to a few millihartree across the board. For Og₂ spinor energies, SAP‑X2C achieves errors an order of magnitude smaller than 1eX2C and comparable to the more sophisticated atomic‑mean‑field (AMF) X2C schemes. Core‑level SO splittings are also markedly improved, with SAP‑X2C errors typically within a few hundred cm⁻¹, whereas 1eX2C errors can exceed several thousand cm⁻¹. Structural properties (bond lengths, vibrational frequencies) show similar trends: SAP‑X2C delivers corrections of ~0.01 Å and ~5 cm⁻¹ relative to 1eX2C, matching the performance of AMF‑X2C.

In terms of computational cost, AMF‑X2C and its extended variant eamfX2C are slightly more accurate but require separate atomic 4C Dirac‑Hartree‑Fock calculations to generate atomic densities and potentials, making them substantially more expensive and less black‑box. SAP‑X2C, by contrast, needs no prior atomic calculations and can be applied directly to large molecules or periodic cells with minimal code changes. The size‑intensivity test confirms that SAP‑X2C’s total energy error scales linearly with system size, demonstrating true size‑intensive behavior.

Overall, SAP‑X2C offers an “optimally simple” solution: it preserves the low cost and ease of 1eX2C while remedying its two major deficiencies—neglect of 2ePC and lack of size‑intensive picture‑change. Although AMF‑X2C variants retain a slight edge in raw accuracy, SAP‑X2C’s simplicity, absence of prerequisite atomic calculations, and robust performance for large‑scale and periodic systems make it a compelling choice for relativistic electronic‑structure work, especially when combined with post‑HF correlation methods or solid‑state applications.