Fig. 1. (Left) Radial densities of the 5f, 6d and 7s valence spinors of U in the 5f³6d¹7s² ground state configuration in comparison to the radial densities of the U³²⁺, U¹⁴⁺ and U⁶⁺ cores from average-level multi-configuration Dirac–Hartree–Fock calculations [18]. (Right) Non-relativistic orbital energies (Hartree–Fock, HF) and relativistic spinor energies (Dirac–Hartree–Fock, DHF) for valence shells of U from average level calculations [18]. The horizontal dotted lines correspond to a “chemical” core-valence separation based on one-particle energies.

Fig. 2. Third (left bars) and fourth (right bars) ionization potentials of the actinides estimated by pseudopotential (PP) multi-reference averaged coupled-pair functional (MR-ACPF) calculations including spin–orbit corrections and extrapolation to the basis set limit (dotted bars) [19]. Relativistic contributions estimated from the difference of state-averaged multi-configuration Dirac–Hartree–Fock (Dirac–Coulomb–Breit Hamiltonian) and Hartree–Fock calculations (striped bars) [18] and electron correlation contribution estimated from PP MR-ACPF correlation energies extrapolated to the basis set limit (filled bars) [19]. Crosses with error bars denote the experimentally measured values for Th and the semiempirical estimates for U [17].

Fig. 3. Total relativistic contributions estimated from the difference of state-averaged Dirac–Hartree–Fock (Dirac–Coulomb–Breit Hamiltonian) and Hartree–Fock calculations (striped bars) [18] and spin–orbit contributions estimated from pseudopotential calculations with and without spin–orbit operator (filled bars) [19] to the third (left bars) and fourth (right bars) ionization potentials of the actinides.

Fig. 4. Schematic representation of the effective core potential (ECP) approximation in a coordinate system of three axes determining the quality of the many-electron Hamiltonian, the one- and the many-electron basis sets.

Fig. 5. Errors in spin–orbit corrected coupled-cluster singles, double and perturbative triples results for IP₄ of Ce and Th. The experimental values are 36.76 and 28.65 eV for Ce and Th, respectively. The errors of uncorrelated calculations are denoted by a circle and square for Ce and Th, respectively. Relativistic energy-consistent small-core ab initio pseudopotentials (32 valence electrons) and large uncontracted basis sets (16s15p12d10f8g8h8i for Ce and 14s13p10d8f6g6h6i for Th) have been applied. The errors for basis sets including functions up to angular quantum number l are linear in 1/l³ for l>3 and can be extrapolated to the basis set limit.

Electronic ground configurations and states of the actinides Anⁿ⁺ (n=0–4) [17]

f, d and s denote the 5f, 6d and 7s shells. Calculations find a Pa²⁺ ground state [19].

Relative Dirac–Hartree–Fock/Dirac–Coulomb (DHF/DC) energies (eV) of the 2J+ 1-weighted average of all J levels belonging to a non-relativistic configuration [18] with respect to the value for the U [Rn] 5f³ 6d¹ 7s² ground state configuration. Only subconfigurations outside the Rn core are listed (conf)

For each f occupation number only the energetically lowest configuration for each ionization level is listed. Contributions of the Breit interaction evaluated in first-order perturbation theory are also given (+B). The frozen-core errors (eV) in the relative energies are given for 32, 14 and 6 valence electron systems. The frozen core was taken from the neutral atom in the ground state configuration. Frozen cores: U Q=6: 1s–5d, 6s, 6p; Q=14: 1s–5d; Q=32: 1s–4d.

Bond lengths R_e (Å), vibrational constants ω_e (cm⁻¹) and binding energies D_e (eV) of ThO in the ground state from energy-consistent small-core pseudopotential calculations [3] in comparison to experimental data

The calculated values are given without/with Boys–Bernardi counter-poise correction of the basis set superposition error.

Overview over recent calibration studies and applications of energy-consistent relativistic small-core actinide pseudopotentials in chronological order

Comparison of the U–F bond length R_e (Å), the infrared frequencies ν_i (cm⁻¹) and the bond dissociation energy D_e (kcal/mol) of UF₆ from various pseudopotential density functional theory calculations to experimental data

The intensities of the main peaks are listed in parentheses. For explanation of the (standard) acronyms characterizing the computational method as well as references for experimental values see Refs. [49] and [63].

Bond length (Å) and binding energies (eV) of actinide(III) monohydrate complexes An(H₂ O)³⁺from 5f-in-core (LC) and 5f-in-valence (SC) pseudopotential (PP) Hartree–Fock [70] and all-electron (AE) Dirac–Hartree–Fock [71] calculations

The PP results are given for small/large basis sets. The 5f Mulliken population is also listed. Basis sets: AE An (30s25p19d13f2g), O (9s4p1d), H (4s1p) for large components; PP small basis sets: LC An (8s7p6d2f); PP SC An (14s13p10d8f)/[6s6p5d4f], O and H cc-pVDZ; PP large basis sets: LC An (8s7p6d2f1g); PP SC An (14s13p10d8f6g)/[6s6p5d4f3g], O and H aug-cc-pVQZ.