A Comparative Study of Correlation and Relativistic Effects on Atomic Ionization Energy

This study investigates the interplay between relativistic effects and electron correlation effects on the first ionization energies of heavy atoms (Au through Rn, Z = 79-86). We perform two complementary analyses: (1) comparing relativistic corrections computed at both the Hartree-Fock (HF) and coupled cluster CCSD(T) levels to assess how correlation influences the magnitude of relativistic corrections, and (2) comparing correlation corrections computed within both non-relativistic and relativistic frameworks to determine how relativity influences the magnitude of correlation corrections. Our results reveal a striking non-linear relationship between these two effects. Specifically, the combined effect of relativity and correlation on ionization energy does not equal the sum of their individual contributions. This non-additivity indicates that relativistic and correlation effects are not independent; they interact in complex ways that depend on the atomic system. We find that for some atoms, the two effects enhance each other, while for others they partially cancel. Moreover, the order in which one may add “separate” effects also counts, in that adding “pure” relativistic effects to the remaining outcome (including correlation) would give a different result than when adding “pure” correlation effects to the remaining outcome (including relativity). These findings demonstrate that relativistic and correlation effects are inherently non-additive, reflecting the non-linearity of the quantum many-body problem. Accurate computational predictions of ionization energies in heavy-element systems thus require simultaneous treatment of both effects rather than treating them as independent contributions.

💡 Research Summary

This paper presents a systematic investigation of how relativistic effects and electron‑correlation effects interact in determining the first ionization energies of the heavy main‑group elements from gold (Z = 79) to radon (Z = 86). The authors perform four sets of calculations for each atom: non‑relativistic Hartree‑Fock (HF‑NR), relativistic Hartree‑Fock (HF‑DC), non‑relativistic coupled‑cluster with singles, doubles and perturbative triples (CCSD(T)‑NR), and relativistic coupled‑cluster (CCSD(T)‑DC). Relativistic calculations employ the Dirac‑Coulomb Hamiltonian, while non‑relativistic ones use the Schrödinger Hamiltonian. All computations are carried out with the DIRAC 2025 program, the dyall.4zp basis set, and a Gaussian nuclear charge model. The same set of correlated electrons and virtual orbitals is retained for the relativistic and non‑relativistic runs to ensure a fair comparison; for Au this corresponds to 33 correlated electrons and 87 virtual orbitals, with one additional correlated electron added for each successive element.

The results are summarized in two tables. Table 1 shows the “pure” relativistic correction (HF‑DC – HF‑NR) and the total relativistic‑correlation result (CC‑DC). For Au, Hg, Tl and Pb the relativistic correction raises the ionization energy, reflecting scalar relativistic contraction of the valence s and p₁/₂ orbitals. For Bi, Po, At and Rn the correction lowers the ionization energy because spin‑orbit splitting destabilizes the p₃/₂ valence orbitals. Table 2 presents the “pure” correlation correction (CC‑NR – HF‑NR) and the total relativistic‑correlation result (CC‑DC). Correlation generally increases the ionization energy, but for Pb and Po the sign flips depending on whether relativity is included, underscoring the non‑uniform nature of correlation across the period.

A key observation is that the combined CC‑DC values are not equal to the sum of the separate relativistic and correlation contributions. Moreover, the order in which the “pure” corrections are added matters: adding the relativistic correction to the correlated non‑relativistic result yields a different total than adding the correlation correction to the relativistic Hartree‑Fock result. This non‑additivity demonstrates that relativistic effects modify the electronic wavefunction in a way that changes the magnitude and even the sign of correlation contributions, and vice‑versa. The authors quantify this by reporting Δ% values that range from +30 % (relativistic effect for Au) to –18 % (correlation effect for Pb) and show that the interaction can be either synergistic (Au, Hg, Tl) or partially canceling (Pb).

The paper also includes convergence tests: basis‑set enlargement from dyall.3zp to dyall.4zp changes ionization energies by ≤0.01 eV, and increasing the virtual‑orbital space from 82 to 143 orbitals alters the result by less than the second significant digit, confirming that the chosen computational protocol is near the basis‑set limit. Sensitivity to the number of correlated electrons is similarly small (≈0.1 % change for Hg when the correlated space is enlarged from 33 to 52 electrons).

In the discussion, the authors compare their findings with earlier work that treated relativistic and correlation effects separately, noting that those studies could not capture the observed non‑linear interplay. They argue that for heavy elements, especially those with valence p₃/₂ electrons, error cancellation can make a non‑relativistic HF calculation appear deceptively accurate, but this is accidental and not predictive. Consequently, reliable predictions of ionization energies, electron affinities, or spectroscopic constants for heavy atoms must be performed with a fully relativistic correlated method such as Dirac‑Coulomb CCSD(T).

The conclusions emphasize that (i) relativistic and correlation effects are intrinsically non‑additive, (ii) the magnitude and even the sign of each effect depend on the electronic configuration of the specific element, and (iii) future high‑accuracy modeling of heavy‑element chemistry and materials must treat both effects simultaneously. The study provides a comprehensive dataset for Au through Rn that can serve as a benchmark for method development and for assessing approximate relativistic or correlation schemes.