Researchers are increasingly focused on accurately modelling the electronic structure of heavy elements, a challenge complicated by the simultaneous influence of relativistic effects and electron correlation. Mohamed Kahil, Fatima Fakih, and Nabil Joudieh, from the Department of Physics at Damascus University, along with Nidal Chamoun from the Department of Statistics, present a comparative study examining how these two phenomena impact atomic ionization energies across the gold to radon series. Their work demonstrates a significant non-linear relationship, revealing that relativistic and correlation effects are not simply additive, but instead interact in a complex, system-dependent manner. This finding is crucial because it highlights the necessity of simultaneously treating both relativity and correlation to achieve accurate predictions of ionization energies in heavy-element systems, moving beyond approximations that consider them as independent factors.
They performed two complementary analyses to achieve this. The first analysis compared relativistic corrections computed at both the Hartree-Fock (HF) and coupled cluster CCSD(T) levels to assess how electron correlation influences the magnitude of relativistic corrections. The second analysis involved a comparison of corresponding data.
Relativistic and correlation effects on atomic ionization energies are non-additive
Scientists computed relation corrections 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.
The accurate prediction of atomic properties, particularly ionization energies, requires careful treatment of quantum mechanical effects. Two of the most important contributions are electron correlation and relativistic effects. The Hartree-Fock (HF) method, which may incorporate relativistic corrections important in the study of heavy-element systems, provides a mean-field description of electron-electron interactions but neglects the instantaneous correlation between two electrons.
Coupled cluster (CC) theory, particularly the CCSD(T) method (“coupled cluster with singles, doubles and perturbative triples”), is a well-established approach for including electron correlation effects. The treatment of electron correlation and relativistic effects on equal footing is an active and important research area in computational electronic structure theory.
Previous studies have examined these effects separately or in combination for specific systems. For example, Schwerdtfeger and colleagues demonstrated that relativistic effects on ionization energies exceeded correlation effects for gold atoms. Similarly, other work showed that relativistic effects dominated correlation contributions for valence ionization energies in lead, although the reverse was true for the first ionization energy.
The first systematic study comparing both relativistic and non-relativistic effects at the correlated level appeared in 1990, followed by work in 1991 showing that relativistic effects increase the ionization potential of Au by 1.708 eV while correlation effects increase it by 1.269 eV. These studies established that both effects are important, but they did not systematically investigate how these effects interact with each other.
Despite these previous studies, to our knowledge, no systematic investigation has examined the mutual interplay between relativistic and correlation effects, that is, how one effect influences the magnitude and behavior of the other. This is the central question we address in this work. We investigate three key aspects: (1) whether relativistic corrections to ionization energies differ when computed with correlated versus uncorrelated electrons, (2) whether correlation corrections differ when computed within relativistic versus non-relativistic frameworks, and (3) we compared the separate relativistic and correlations effects in what regards their corrections contrasted to experiment.
We focus on the heaviest main-group elements (Au through Rn, with atomic numbers Z = 79, 86) because these elements exhibit chemical and physical properties distinctly different from their lighter counterparts, and relativistic effects are most pronounced in this region. Our analysis employs the Dirac-Coulomb (DC) Hamiltonian for relativistic calculations and the Schrödinger Hamiltonian for non-relativistic calculations, combined with the CCSD(T) method for correlation effects, while HF methods neglects them.
Our key finding is that relativistic and correlation effects are not independent: their combined effect is not additive, reflecting the inherent non-linearity of the quantum many-body problem. This conclusion aligns with earlier observations that these contributions are not independent, and our work provides a systematic quantitative demonstration of this non-additivity across a series of heavy atoms.
Not only the sum of the two types of corrections is not equal to the correction when combining the two effects, but the order in which one adds the separate effects also counts, in that the addition of the “pure” correlation effects to the relativistic HF result getting a “total” correction, is different in general from the other way round by adding the “pure” relativistic corrections to the non-relativistic CCSD(T) result giving another “total” correction. We acknowledge that earlier works have noted the non-independence of relativistic and correlation effects.
The novelty of our work lies in the systematic, quantitative exploration of this non-additivity across the complete 6th-row main-group elements (Au, Rn) using the CCSD(T) method within both Dirac, Coulomb and non-relativistic frameworks, providing a comprehensive dataset and analysis that were previously lacking, especially for Bi, Rn. All calculations were performed using the DIRAC 2025 program.
For relativistic calculations, we employed the Dirac-Coulomb (DC) Hamiltonian, while for non-relativistic calculations, we used the Schrödinger Hamiltonian (NR). Both sets of calculations utilized the dyall.4zp basis set and a nuclear Gaussian charge distribution. The electronic configurations of the studied atoms follow the pattern [Xe] 4f14 5d10 6sx 6py, where x and y vary depending on the element.
For Au and Hg, x = 1 and 2 respectively, with y = 0. For elements from Tl through Rn, x = 2 and y ranges from 1 to 6. In the non-relativistic treatment, electrons occupy the three p orbitals with single occupancy followed by pairing.
In the relativistic treatment, spin-orbit coupling causes the three p orbitals to split into one p1/2 and two p3/2 orbitals, with the p1/2 orbitals being more stable and filled before the p3/2 orbitals. The latter orbitals are occupied according to average-of-configuration (AOC) manner. For example the considered electron configuration for Bi atom in the relativistic and non-relativistic cases is as follows, where the highest electron in the relativistic case, to be put in the p3/2 four spinors, is allocated the spinor occupation value of 1⁄4 in DIRAC.
To specify the active set of spinors in the relativistic calculations, we retained the default DIRAC setup of ‘-10.0 20.0 1.0’, which selects all orbitals with energies between -10.0 and +20.Hartree (atomic units) with a minimum energy gap of 1.Hartree. This choice ensures inclusion of chemically relevant valence and sub-valence orbitals while excluding high-energy virtual orbitals that contribute negligibly.
For the Au atom, this resulted in 87 virtual orbitals and 33 correlated electrons, corresponding to the electrons beyond the closed [Xe] core: the 4f14 shell (14 electrons), the 5d10 shell (10 electrons), and the valence 6s1 electron, in addition to 8 electrons already in the [Xe] core: 5s25p6. The number of correlated electrons increases by one for each successive atom in the series.
For consistency, the non-relativistic calculations were configured to include the same number of correlated electrons and virtual orbitals as their relativistic counterparts. Test calculations with different number of correlated electrons changed the results by around 0.1%, confirming the robustness of our results.
Actually, electronic correlations become relatively pronounced in valence and neighbouring orbitals compared to core ones, due to many factors including:, Distance & Energy: Valence electrons are in the highest energy levels, farthest from the nucleus, meaning they have higher potential energy and are less attracted to the core., Shielding & Effective Nuclear Charge: Core electrons shield the valence electrons from the full nuclear charge, reducing the effective nuclear charge experienced by the valence electrons. We performed additional tests to assess the basis set convergence and found that the dyall.4zp basis set provides a good balance between accuracy and computational cost.
We see that including relativistic corrections at both HF and CCSD(T) levels improves the predictions for atoms ending in s and p1/2 orbitals (Au through Pb). However, for atoms ending in p3/2 orbitals (Bi through Rn), the situation is more complex. At the HF level, relativistic corrections worsen the agreement with experiment, while at the CCSD(T) level, they generally improve it (with the exception of At).
Notably, for Bi, Po, At and Rn, the non-relativistic HF values show surprisingly good agreement with experiment, better than when relativistic and/or correlation effects are included. This unexpected result may suggest that error cancellation occurs in these cases.
Relativistic and correlation contributions to ionisation energies in gold to radon
Investigations into the first ionization energies of heavy atoms, ranging from gold through radon with atomic numbers 79 to 86, reveal a non-linear relationship between relativistic and electron correlation effects. Analyses demonstrate that the combined impact of relativity and correlation on ionization energy is not simply the sum of their individual contributions, indicating a complex interplay between these phenomena.
Specifically, the research establishes that these effects are not independent, with their interaction varying depending on the atomic system under consideration. For certain atoms within the studied series, relativistic and correlation effects enhance one another, while in others, they partially offset each other.
Furthermore, the order in which these effects are applied significantly impacts the final result; adding relativistic corrections to a correlated outcome yields a different value than adding correlation corrections to a relativistic outcome. This finding underscores the non-additivity of relativistic and correlation effects, reflecting the inherent non-linearity of the quantum many-body problem.
Previous studies indicated relativistic effects on gold atoms exceeded correlation effects, and relativistic effects dominated correlation contributions for lead valence ionization energies, however, this work provides a systematic quantitative demonstration of non-additivity across a series of heavy atoms. The research employed the Dirac-Coulomb Hamiltonian for relativistic calculations and the Schrödinger Hamiltonian for non-relativistic calculations, both utilising the dyall.4zp basis set.
Calculations were performed using the coupled cluster CCSD(T) method to account for electron correlation, while Hartree-Fock methods were used to neglect correlation. This comprehensive dataset, spanning the 6th-row main-group elements, provides a detailed analysis previously lacking in the field.
Relativistic and correlation contributions to sixth-row main-group element ionization energies
Scientists have demonstrated a non-linear interplay between relativistic and electron correlation effects when calculating ionization energies for heavy elements ranging from gold to radon. These calculations reveal that the combined impact of relativistic and correlation effects on ionization energy is not simply the sum of their individual contributions, indicating a complex interaction dependent on the specific atomic system.
The magnitude of these effects, and how they combine, varies across the series of elements investigated. This research establishes that accurate predictions of ionization energies in heavy-element systems necessitate the simultaneous treatment of both relativistic and correlation effects, rather than approximating them as independent components.
The study systematically quantifies this non-additivity across the sixth-row main-group elements using a high-level coupled cluster method within both relativistic and non-relativistic frameworks, providing a comprehensive dataset previously unavailable. Furthermore, the order in which relativistic and correlation corrections are applied influences the final result, highlighting the inherent non-linearity of the quantum many-body problem.
The authors acknowledge prior observations of the non-independence of these effects, but emphasize the novelty of their systematic and quantitative analysis across a complete series of elements. Future research could explore the extent to which these non-additive effects influence other properties of heavy elements, such as excitation energies or chemical bonding. Understanding these interactions is crucial for refining theoretical models and improving the accuracy of calculations for systems containing heavy atoms.