Accurate atomic correlation and total energies for correlation consistent effective core potentials II: Rb-Xe elements

We employ correlation-consistent effective core potentials (ccECPs) to perform exact or nearly exact correlation and total energy calculations for the fifth-row elements (Rb-Xe). Total energies are calculated using various correlated methods: configuration interaction (CI), coupled-cluster (CC) up to perturbative quadruple excitations whenever feasible, and stochastic quantum Monte Carlo (QMC) approaches. In order to estimate the energy at the complete basis set (CBS) limit, the basis sets are constructed systematically through aug-cc-p(C)VnZ for each ccECP and further extrapolated to the CBS limit within the corresponding methods. Kinetic energies are evaluated at the FCI/CISD level to provide insights into the electron density and localization of the ccECPs. We also provide data sets for widely used diffusion Monte Carlo (DMC) to quantify fixed-node biases with single-reference trial wavefunctions. These comprehensive benchmarks validate the accuracy of ccECPs within the CC, CI, and QMC methodologies, thus providing accurate and tested valence-only Hamiltonians for many-body electronic structure calculations.

💡 Research Summary

This paper presents a comprehensive benchmark of correlation‑consistent effective core potentials (ccECPs) for the fifth‑row elements ranging from rubidium to xenon. The authors combine state‑of‑the‑art quantum chemistry methods (CISD, RCCSD(T), UCCSD(T), CCSDT(Q), and full configuration interaction where feasible) with stochastic quantum Monte Carlo (QMC) techniques to obtain near‑exact total and correlation energies for each atom, extrapolated to the complete‑basis‑set (CBS) limit.

The methodology begins with the construction of correlation‑consistent Gaussian basis sets (aug‑cc‑p(C)VnZ, n = D, T, Q, 5, 6) tailored to each ccECP. Hartree–Fock energies are extrapolated using an exponential form, while correlation energies employ a two‑term inverse‑power series in (n + 3/8). This dual‑extrapolation scheme is deliberately conservative to avoid over‑estimation, especially for transition metals where the (n + 3/8)⁻⁵ term can underestimate correlation contributions.

For electronic correlation, the authors perform CISD on all atoms, supplementing open‑shell cases with both restricted and unrestricted CCSD(T). When the valence electron count is six or fewer, full CI (FCI) calculations are carried out, providing exact reference values within the chosen basis. For heavier atoms where FCI is prohibitive, CCSDT(Q) is employed; if even CCSDT(Q) cannot be performed with the largest basis, the missing high‑cardinality correlation energy is estimated by scaling the UCCSD(T) result using the ratio of CCSDT(Q) to UCCSD(T) energies obtained in the next‑smaller basis. This ratio‑based extrapolation has been validated to stay within roughly 1 % of the true valence correlation energy.

Diffusion Monte Carlo (DMC) calculations are performed with single‑determinant Slater‑Jastrow trial wavefunctions. The T‑moves algorithm ensures a variational upper bound despite the non‑local nature of the ccECP’s averaged relativistic effective potential (AREP). Time‑step errors are removed by linear extrapolation from four time steps (0.02, 0.01, 0.005, 0.0025 Ha⁻¹). The study quantifies fixed‑node (FN) biases, revealing that the error spikes when a new angular‑momentum channel (p, d, …) becomes occupied for the first time. Notably, 4d transition metals (Y–Cd) exhibit smaller FN errors than their 3d counterparts because the 4d orbitals are pushed outward by the filled 3d core, leading to smoother nodal surfaces.

Kinetic energies are evaluated from CISD/FCI wavefunctions to probe electron‑density localization, since ECPs break the virial theorem. The authors extrapolate kinetic energies to the CBS limit using a simple two‑point scheme, and they also report QMC kinetic energies with and without three‑body Jastrow terms. The kinetic‑energy analysis shows that Jastrow optimization tends to lower kinetic energy, indicating a redistribution of electron density toward the valence region and a possible under‑representation of the core‑region density in variational QMC.

Overall, the paper delivers a rigorously validated dataset of atomic total, correlation, and kinetic energies for Rb–Xe using ccECPs, together with detailed assessments of DMC fixed‑node errors. These results provide a reliable reference for future many‑body calculations on molecules and solids that employ ccECPs, facilitating the development of high‑accuracy electronic‑structure methods for heavy elements where relativistic and core‑valence correlation effects are significant.