Accurate relativistic energy-consistent pseudopotentials for the superheavy elements 111 to 118 including quantum electrodynamic effects

Energy-consistent two-component semi-local pseudopotentials for the superheavy elements with atomic numbers 111-118 have been adjusted to fully relativistic multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian, including perturbative corrections for the frequency-dependent Breit interaction in the Coulomb gauge and lowest-order quantum electrodynamic effects. The pseudopotential core includes 92 electrons corresponding to the configuration [Xe]4f(14)5d(10)5f(14). The parameters for the elements 111-118 were fitted by two-component multi-configuration Hartree-Fock calculations in the intermediate coupling scheme to the total energies of 267 up to 797 J levels arising from 31 up to 62 nonrelativistic configurations, including also anionic and highly ionized states, with mean absolute errors clearly below 0.02 eV for averages corresponding to nonrelativistic configurations. Primitive basis sets for one- and two-component pseudopotential calculations have been optimized for the ground and excited states and exhibit finite basis set errors with respect to the finite-difference Hartree-Fock limit below 0.01 and 0.02 eV, respectively. General contraction schemes have been applied to obtain valence basis sets of polarized valence double- to quadruple-zeta quality. Results of atomic test calculations in the intermediate coupling scheme at the Fock-space coupled-cluster level are in good agreement with those of corresponding fully relativistic all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian. The results demonstrate besides the well-known need of a relativistic treatment at the Dirac-Coulomb level also the necessity to include higher-order corrections for the superheavy elements.     

Improved valence basis sets for divalent lanthanide 4f-in-core pseudopotentials

Improved energy-optimized (6s5p4d) and (7s6p5d) primitive valence basis sets have been derived for energy-consistent scalar-relativistic 4f-in-core pseudopotentials of the Stuttgart-Cologne variety modeling divalent lanthanides with a $4\hbox{f}^{n+1}$ occupation (n = 0?C13 for La?CYb). Segmented contracted basis sets covering the range of polarized double-, triple-, and quadruple-zeta quality, augmented by 2f1g correlation sets, were created for use in molecular calculations. The basis sets contain smaller (4s4p3d) and (5s5p4d) primitive subsets, which are designed in particular for solid state calculations of crystals containing divalent lanthanide ions. Hartree?CFock, density functional theory and coupled cluster results obtained with the new basis sets for lanthanide atomic ionization potentials as well as of geometry optimizations of various test molecules, i.e. selected lanthanide mono- and dihydrides, mono- and difluorides, and monooxides, show a satisfactory agreement with experimental data as well as with corresponding scalar-relativistic all-electron results. Core-polarization potentials are found to improve the results, especially for the atomic first and second ionization potentials.     

Energy-consistent relativistic pseudopotentials for the 4d elements: atomic and molecular applications

Recently reported energy-consistent relativistic pseudopotentials have been used with series of matching correlation consistent basis sets in benchmark calculations of various atomic and molecular properties. The basis set convergence of the 4d metal electron affinities and 5s2-->5s0 excitation energies are reported at the CCSD(T) level of theory, and the effects of valence and 4s4p correlation are investigated. In addition the impact of correlating the low-lying 3d electrons was also studied in all-electron Douglas-Kroll-Hess (DKH) calculations, which also included the ionization potentials and 5s2-->5s1 excitation energies. For all four atomic properties, higher order coupled cluster calculations through CCSDTQ are reported. The final calculated values are generally all within 1 kcal/mol of experiment. A notable exception is the ionization potential of Tc, the currently accepted experimental value of which is suggested to be too high by about 3 kcal/mol. Molecular calculations are also reported for the low-lying electronic states of ZrO and RuF, as well as the ground electronic state of Pd2. The effects of spin-orbit coupling are investigated for these cases in pseudopotential calculations. Wherever possible, the pseudopotential results have been calibrated against DKH calculations with correlation consistent basis sets of triple-zeta quality. In all cases the calculated data for these species are in very good agreement with experiment. In particular, the correct electronic ground state for the RuF molecule (4Phi92) was obtained, which was made possible by utilizing systematic sequences of correlation consistent basis sets.     

Energy-adjusted pseudopotentials for the rare earth elements

Nonrelativistic and quasirelativistic energy-adjusted pseudopotentials and optimized (7s6p5d)/[5s4p3d]-GTO valence basis sets for use in molecular calculations for fixed f-subconfigurations of the rare earth elements, La through Lu, have been generated. Atomic excitation and ionization energies from numerical HF, as well as SCF pseudopotential calculations using the derived basis sets, differ by less than 0.1 eV from numerical HF all-electron results. Corresponding values obtained from CI(SD), CEPA-1, as well as density functional calculations using the quasirelativistic pseudopotentials, are in reasonable agreement with experimental data.     

Relativistic energy-consistent pseudopotentials--recent developments

The direct adjustment of two-component pseudopotentials (scalar-relativistic + spin-orbit potentials), to atomic total energy valence spectra derived from four-component multiconfiguration Dirac-Hartree-Fock all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian, has been made a routine tool for an efficient treatment of heavy main-group elements. Both large-core (nsp valence shell) and small-core ((n - 1)spd nsp valence shell) potentials have been generated for all the post-d elements of groups 13-17. At the example of lead and bismuth compounds (PbHal, BiH, BiO, BiHal (Hal = F, Cl, Br, I)), we show how small-core and large-core potentials can be combined in accurate, yet computationally economic, spin-free-state-shifted relativistic electronic structure calculations of molecular ground and excited states.     

Quasirelativistic energy-consistent 5f-in-core pseudopotentials for divalent and tetravalent actinide elements

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modeling divalent (5f ⁿ⁺¹ occupation with n = 5–13 for Pu–No) respectively tetravalent (5f ^n-1 occupation with n = 1–9 for Th–Cf) actinides together with corresponding core-polarization potentials have been generated. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets as well as 2f1g correlation functions have been derived and contracted to polarized double, triple, and quadruple zeta quality. Corresponding smaller (4s4p) and (5s5p) respectively (4s4p3d) and (5s5p4d) basis sets suitable for calculations on actinide(II) respectively actinide(IV) ions in crystalline solids form subsets of these basis sets designed for calculations on molecules. Results of Hartree–Fock test calculations for actinide di- and tetrafluorides show a satisfactory agreement with calculations using 5f-in-valence pseudopotentials. Electronic Supplementary Material The online version of this article doi: contains supplementary material, which is available to authorized users.     

On the use of common effective core potentials in density functional calculations. I. Test calculations on transition-metal carbonyls

The performance of effective core potentials adjusted at the Hartree-Fock level but applied in density functional calculations has been tested in a set of calculations using various basis sets and/or core potentials. Test molecules have been the first-row transition-metal carbonyls Cr(CO)₆, Fe(CO)₅, and Ni(CO)₄ and the second-row carbonyls Mo(CO)₆, Ru(CO)₅, and Pd(CO)₄. Only “small-core” potentials have been used, and these are able to reproduce molecular structures and bond energies from all-electron calculations. Relativistic effects have been estimated for the second-row carbonyls by using quasi-relativistic core potentials. © 1996 John Wiley & Sons, Inc.     

Energy-consistent pseudopotentials for quantum Monte Carlo calculations

The authors present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. They demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, they compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post-Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. They also show their pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for first and second rows, and n=D,T for third to fifth rows) optimized for our pseudopotentials are also presented.     

Quasirelativistic energy-consistent 4f-in-core pseudopotentials for tetravalent lanthanide elements

Quasirelativistic energy-consistent 4f-in-core pseudopotentials modeling tetravalent lanthanides (4f ^n?1 occupation with n = 1, 2, 3, 8, 9 for Ce, Pr, Nd, Tb, Dy) have been adjusted. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets contracted to polarized double- to quadruple-zeta quality as well as 2f1g correlation functions have been derived. Corresponding smaller (4s4p3d) and (5s5p4d) basis sets suitable for calculations on lanthanide(IV) ions in crystalline solids form subsets of these basis sets designed for calculations on neutral molecules. Calculations for lanthanide tetrafluorides using the 4f-in-core pseudopotentials at the Hartree–Fock level show satisfactory agreement with calculations using 4f-in-valence pseudopotentials. For cerium tetrafluoride the experimental bond length is well reproduced using the 4f-in-core pseudopotential at the coupled-cluster level with single and double excitation operators and a perturbative estimate of triple excitations. For cerium dioxide 4f-in-core and 4f-in-valence pseudopotential calculations agree quite well, if a proper f basis set instead of f polarization functions is applied.     

Energy-adjustedab initio pseudopotentials for the second and third row transition elements

Summary Nonrelativistic and quasirelativisticab initio pseudopotentials substituting the M^(Z–28)+-core orbitals of the second row transition elements and the M^(Z–60)+-core orbitals of the third row transition elements, respectively, and optimized (8s7p6d)/[6s5p3d]-GTO valence basis sets for use in molecular calculations have been generated. Additionally, corresponding spin-orbit operators have also been derived. Atomic excitation and ionization energies from numerical HF as well as from SCF pseudopotential calculations using the derived basis sets differ in most cases by less than 0.1 eV from corresponding numerical all-electron results. Spin-orbit splittings for lowlying states are in reasonable agreement with corresponding all-electron Dirac-Fock (DF) results.     

Quasirelativistic energy-consistent 5f-in-core pseudopotentials for trivalent actinide elements

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modelling trivalent actinides, corresponding to a near-integral 5f ⁿ occupation (n = 0–14 for Ac–Lr), have been generated. Energy-optimized (6s5p4d), (7s6p5d), and (8s7p6d) primitive valence basis sets contracted to polarized double to quadruple zeta quality as well as 2f1g correlation functions have been derived. Corresponding smaller basis sets (4s4p3d), (5s5p4d), and (6s6p5d) suitable for calculations on actinide(III) ions in crystalline solids form subsets of these basis sets designed for calculations on neutral molecules. Results of Hartree–Fock test calculations for actinide(III) monohydrates and actinide trifluorides show a satisfactory agreement with corresponding calculations using 5f-in-valence pseudopotentials. Even in the beginning of the actinide series, where the 5f shell is relatively diffuse, only quite acceptable small deviations occur as long as the 5f-shell does not participate significantly in covalent bonding. Electronic Supplementary Material Supplementary material is available in the online version of this article at and is accessible for authorized users.     

On the performance of two-component energy-consistent pseudopotentials in atomic Fock-space coupled cluster calculations

The four-component atomic intermediate-Hamiltonian Fock-space coupled cluster (IHFSCC) code of Landau et al. [J. Chem. Phys. 115, 6862 (2001)] has been adapted to two-component calculations with relativistic pseudopotentials of the energy-consistent variety. Recently adjusted energy-consistent pseudopotentials for group 11 and 12 transition elements as well as group 13 and 14 post-d main group elements, which were fitted to atomic valence spectra from four-component multiconfiguration Dirac-Hartree-Fock calculations, are tested in IHFSCC calculations for ionization potentials, electron affinities, and excitation energies of a variety of atoms and ions. Where comparison is possible, the deviations from experimental data are in good agreement with those found in previously published IHFSCC all-electron calculations: experimental data are usually reproduced within a few hundred wavenumbers.     

Development of new pseudopotential methods: improved model core potentials for the first-row transition metals

We have recently developed new nonrelativistic and scalar-relativistic pseudopotentials for the first-row transition metal and several main-group elements. These improved Model Core Potentials were tested on a variety of transition metal complexes to determine their accuracy in reproducing electronic structures, bond lengths, and harmonic vibrational frequencies with respect to both all-electron reference data as well as experimental data. The new potentials are also compared with the previous model core potentials available for the first-row transition metals. The new potentials do a superior job at reproducing atomic data, reproduce molecular data as well as the previous version, and in conjunction with new main-group pseudopotentials that have L-shell structure of the valence basis set, they are slightly faster.     

Quasirelativistic energy-consistent 5f-in-core pseudopotentials for pentavalent and hexavalent actinide elements

Quasirelativistic energy-consistent 5f-in-core pseudopotentials modeling pentavalent (5f ^n?2 occupation with n = 2–6 for Pa–Am) and hexavalent (5f ^n?3 occupation with n = 3–6 for U–Am) actinides have been adjusted. Energy-optimized (6s5p4d) and (7s6p5d) valence basis sets contracted to polarized double- to quadruple-zeta quality as well as 2f1g correlation functions have been derived. Corresponding smaller basis sets (4s4p3d) and (5s5p4d) suitable for calculations on actinide(V) and actinide(VI) ions in crystalline solids form subsets of these basis sets designed for calculations on neutral molecules. Calculations using the Hartree–Fock and the coupled-cluster method with single and double excitation operators and a perturbative estimate of triple excitations for actinide pentafluorides show satisfactory agreement with calculations using 5f-in-valence pseudopotentials and experimental data, respectively. However, in the hexavalent case the 5f-in-core approximation seems to reach its limitations except for hexavalent uranium (5f⁰), where results for both uranium hexafluoride and the uranyl ion deviate only slightly from the 5f-in-valence reference data.     

Relativistic small-core energy-consistent pseudopotentials for the alkaline-earth elements from Ca to Ra

Small-core ten-valence electron energy-consistent scalar- and two-component relativistic pseudopotentials for the alkaline-earth elements from Ca to Ra are presented. The accuracy and reliability of these pseudopotentials are discussed in terms of their ability to reproduce all-electron calculated and experimental dipole polarizabilities and ionization potentials.     

Molecular core-valence correlation effects involving the post-d elements Ga-Rn: benchmarks and new pseudopotential-based correlation consistent basis sets

Correlation consistent basis sets that are suitable for the correlation of the outer-core (n-1)spd electrons of the post-d elements Ga-Rn have been developed. These new sets, denoted by cc-pwCVXZ-PP (X=D,T,Q,5), are based on the previously reported cc-pVXZ-PP sets that were built in conjunction with accurate small-core relativistic pseudopotentials (PPs) and designed only for valence nsp correlation. These new basis sets have been utilized in benchmark coupled cluster calculations of the core-valence correlation effects on the dissociation energies and spectroscopic properties of several small molecules. As expected, the most important contribution is the correlation of the (n-1)d electrons. For example, in the case of the group 13 homonuclear diatomics (Ga(2),In(2),Tl(2)), this leads to a dissociation energy increase compared to a valence-only treatment from 1.5 to 3.2 kcal/mol, bond length shortenings from -0.076 to -0.125 A?, and harmonic frequency increases of 7-8?cm(-1). Even in the group 15 cases (As(2),Sb(2),Bi(2)), the analogous effects of (n-1)d electron correlation are certainly not insignificant, the largest values being +4.4?kcal/mol, -0.049 A?, and +9.6?cm(-1) for the effects on D(e), r(e), and ω(e), respectively. In general, the effects increase in magnitude down a group from 4p to 6p. Correlation of the outer-core (n-1)p electrons is about an order of magnitude less important than (n-1)d but larger than that of the (n-1)s. The effect of additional tight functions for Hartree-Fock and valence sp correlation was found to be surprisingly large, especially for the post-4d and post-5d elements. The pseudopotential results for the molecules containing post-3d elements are also compared to the analogous all-electron calculations employing the Douglas-Kroll-Hess Hamiltonian. The errors attributed to the PP approximation are found to be very small.     

Compact and efficient basis sets of s- and p-block elements for model core potential method

We propose compact and efficient valence-function sets for s- and p-block elements from Li to Rn to appropriately describe valence correlation in model core potential (MCP) calculations. The basis sets are generated by a combination of split MCP valence orbitals and correlating contracted Gaussian-type functions in a segmented form. We provide three types of basis sets. They are referred to as MCP-dzp, MCP-tzp, and MCP-qzp, since they have the quality comparable with all-electron correlation consistent basis sets, cc-pVDZ, cc-pVTZ, and cc-pVQZ, respectively, for lighter atoms. MCP calculations with the present basis sets give atomic correlation energies in good agreement with all-electron calculations. The present MCP basis sets systematically improve physical properties in atomic and molecular systems in a series of MCP-dzp, MCP-tzp, and MCP-qzp. Ionization potentials and electron affinities of halogen atoms as well as molecular spectroscopic constants calculated by the best MCP set are in good agreement with experimental values.     

Extended Gaussian-type valence basis sets for calculations involving nonempirical core pseudopotentials. II. PS-21 G Basis for Li to Ca and Ga to Kr atoms

Split valence basis sets adapted to Durand and Barthelat core pseudopotentials have been optimized for the three first rows of the main group elements. The reliability of these PS -21G basis functions has been checked by performing test calculations on about 30 small molecule and crystalline systems. These studies indicate a satisfactory overall agreement between the PS -21G results and those of all-electron calculations carried out with basis sets having similar valence-shell contractions.     

Ionic core effects on the mie resonance in lithium clusters

We investigate the effects of the atomic cores on the Mie resonance in lithium metal clusters, perturbing a jellium Hamiltonian with zero-range pseudopotentials. The resonance is red-shifted with respect to the classical formula by core effects, most important of which is the increased effective mass due to the core potentials. Much of the large shift seen in lithium clusters is thereby explained if the strength of the pseudopotentials is taken from band structure calculations. However, such pseudopotentials cause the resonance to be greatly broadened, contrary to observation.     

Valence basis sets for lanthanide 4f-in-core pseudopotentials adapted for crystal orbital ab initio calculations

Crystal orbital adapted Gaussian (4s4p3d), (5s5p4d) and (6s6p5d) valence primitive basis sets have been derived for calculating periodic bulk materials containing trivalent lanthanide ions modeled with relativistic energy-consistent 4f-in-core lanthanide pseudopotentials of the Stuttgart-Koeln variety. The calibration calculations of crystalline A-type Pm₂O₃ using different segmented contraction schemes (4s4p3d)/[2s2p2d], (4s4p3d)/[3s3p2d], (5s5p4d)/[2s2p2d], (5s5p4d)/[3s3p3d], (5s5p4d)/[4s4p3d], (6s6p5d)/[2s2p2d], (6s6p5d)/[3s3p3d] and (6s6p5d)/[4s4p4d] are discussed at both Hartree–Fock (HF) and density functional theory (DFT) levels for the investigation of basis set size effects. Applications to the geometry optimization of A-type Ln₂O₃ (Ln = La-Pm) show a satisfactory agreement with experimental data using the lanthanide valence basis sets (6s6p5d)/[4s4p4d] and the standard set 6-311G* for oxygen. The corresponding augmented sets (8s7p6d)/[6s5p5d] with additional diffuse functions for describing neutral lanthanide atoms were applied to calculate atomic energies of free lanthanide atoms for the evaluation of cohesive energies for A-Ln₂O₃ within both conventional Kohn-Sham DFT and the a posteriori-HF correlation DFT schemes.