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.     

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.     

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.     

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-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.     

Interaction of Cucurbit[n = 6∼8]urils and Benzimidazole Derivatives

The structures and optical properties of host–guest complexes produced from cucurbit[n = 6–8]urils and some benzimidazole derivatives have been investigated by ¹H NMR spectroscopy, electronic absorption spectroscopy and fluorescence spectroscopy. The experimental results reveal that calculations of A∼N_Q[n]/N_guest and I_f∼N_Q[n]/N_guest for the same association complex both support a good fit to an identical binding model. In particular, the A∼N_Q[n]/N_guest, I_f∼N_Q[n]/N_guest calculations and the ¹H NMR determinations for three Q[6]–ge(1∼3) complexes and three Q[8]–ge(1∼3) complexes all support a binding model of 1:1 and 1:2 respectively.     

Valence Fluctuation of Uranium Ions in Uranium Sesquinitride Revealed by Dynamical Mean-field Theory Merged with Density Functional Theory

The electronic properties, in particular, the occupation number of 5f electrons and the valence state of U ions in uranium sesquinitride (U₂N₃) are studied by using density functional theory (DFT) calculations merged with dynamical mean-field theory (DMFT). The results demonstrate that j=5/2 and j=7/2 manifolds are in the weakly correlated metallic and weakly correlated insulating regimes, respectively. The quasi-particle weights indicate that LS coupling scheme is more feasible for 5f electrons, which are not in the orbital-selective localized state. The weighted summation of the occupation probabilities of 5fⁿ (n=0,1,2,3,4) atomic configurations suggests that 5f electrons have the inter-configuration fluctuation, or the mixed-valence state for U ions, together with an average occupation number of 5f electrons n_5f∼2.234, which is in good agreement with the electron localization function (ELF) and occupation analysis based on other DFT-based calculations. The 5fⁿ-mixing-driven inter-configuration fluctuation might originate from the dual nature of 5f electrons, and the flexible electronic configuration of U ions. Finally, the so-called quasiparticle band structure is also discussed.     

Scalar-relativistic 5f-in-core pseudopotentials and core-polarization potentials for trivalent actinides: calibration calculations for Ac<Superscript>3+</Superscript>, Cm<Superscript>3+</Superscript> and Lr<Superscript>3+</Superscript> complexes

The performance of recently proposed 5f-in-core pseudopotentials for the trivalent actinides was investigated in calculations for model complexes An³⁺L ⁿ⁻ for three selected actinides (An³⁺ = Ac³⁺, Cm³⁺, Lr³⁺) and eight simple ligands with atoms from the first three periods of the table of elements (L ⁿ⁻ = F⁻, Cl⁻, OH⁻, SH⁻, CO, NH₂⁻, H₂O, H₂S, NH₃). Results of Hartree-Fock and Coupled Cluster with singles, doubles and perturbative triples calculations using basis sets of quadruple-zeta quality are compared to corresponding reference data obtained with scalar-relativistic energy-adjusted 5f-in-valence small-core pseudopotentials. The inclusion of core-polarization potentials in the 5f-in-core pseudopotential calculations and corrections of the basis set superposition error by the counterpoise correction leads to very good agreement between the 5f-in-valence and 5f-in-core pseudopotential results for bond lengths, bond angles and binding energies. Results from 5f-in-core pseudopotential calculations using different density functionals also show reasonable agreement with the more rigorous Coupled Cluster results. It is argued that the An 5f rather than the An f population is a useful criterion for the applicability of a specific An 5f-in-core pseudopotential.     

On the Bond Length Change upon 4f 1 → 5d 1 Excitations in Eightfold Coordination: CaF2:Ce3+ Cubic Defects

The bond lengths of 4f ¹ and 5d ¹ electronic states of cubic (CeF₈)⁵⁻ defects in fluorite have been calculated using quantum mechanical embedding, spin-free relativistic Hamiltonian, and dynamic electron correlation through second-order perturbation theory. The results predict the bond length between Ce and the surrounding eight F to shorten upon 4f ¹ → 5d ¹ excitation. This result coincides with previous findings for lanthanide and actinide ions in sixfold octahedral complexes where the ligand field splitting of the 5d (6d) orbitals is inverted with respect to the eightfold cubic field splitting. Altogether, the results of sixfold and eightfold coordination indicate that the bond shrinkage experienced upon 4f ⁿ to 4f ^n-15d ¹ excitations (5f ⁿ to 5f ^n-16d ¹ in the actinides) seems to be a general result of f-element complexes. These theoretical results contradict a widespread assumption according to which the bond length increases upon f→ d excitation and, therefore, experimental measurements of the sign of the bond length distortion that either validate or refute the quantum chemical predictions are most desirable.     

Contracted Gaussian-type basis functions revisited. IV. Atoms Rb to Xe

We report minimal-type contracted Gaussian-type function (GTF) sets, #n=(n3333/n33/n3) with n=5 and 6, #7= (74333/743/74), and #8= (84333/843/75), for the fourth-row atoms from Rb to Xe. Test calculations are performed on the Ag₂ molecule. Spectroscopic constants given by split valence sets derived from #5 and #6 are a little contaminated by basis set superposition error. However, we find that the fully valence split #8 set, (8433111/84111/711111), yields essentially the same results as a large GTF set, (22s15p12d), with a general contraction, when p-, d-, and f-type polarization functions are augmented. The present #7 and #8 CGTF sets are recommended for ab initio molecular calculations including fourth-row atoms. Received: 15 January 2002 / Accepted: 16 April 2002 / Published online: 24 June 2002     

Reliability of atomic natural orbital basis sets in calculations involving pseudopotentials

The performance of Atomic Natural Orbital (ANO) basis sets for calculations involving nonempirical core pseudopotentials has been studied by comparing the results for atomic and molecular nitrogen obtained using contracted ANO basis sets with those obtained using both the primitive set and a segmented one. The primitive set has been optimized at the SCF level for atomic N treated as a five-electron pseudo-atom, and consists of 7s and 7p primitive GTOs supplemented by 2d and 1f GTOs optimized at the CI level. From this primitive set three contracted [3s 3p 2d 1f] sets have been obtained. The first one has been derived from the ANOs of the neutral atom, the second has been obtained from an averaged density matrix and the third one is a segmented set. For the atom, the segmented set gives a zero contraction error at the SCF level as it must be in valence-only calculations. The ANO basis sets show some small contraction error at the SCF level but perform better in CI calculations. However, for the diatomic N₂ molecule the ANO basis sets exhibit a rather large contraction error in the calculated SCF energy. A detailed analysis of the origin of this error is reported, which shows that the conventional strategy used to derive ANO basis sets does not work very well when pseudopotentials are involved.     

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.     

Determination of thermodynamic stability of CdMoO4 by knudsen effusion vapor pressure measurement method

Thermodynamic stability of CdMoO₄ was determined by measuring the vapor pressures of Cd and MoO₃ bearing gaseous species. Th vaporization reaction could be described as CdMoO₄(s)+MoO₂(s) =Cd(g)+2/n(MoO₃)_n (n=3, 4 and 5). The vapor pressures of the cadmium (p _Cd) and trimer (p _(MoO3)3) measured in the temperature range 987≤T/K≤1111 could be expressed, respectively, as ln (p _Cd/Pa) = –32643.9/T+29.46±0.08 and ln(p _(MoO3)3/Pa) = –32289.6/T+29.28±0.08. The standard molar Gibbs free energy of formation of CdMoO₄(s), derived from the vaporization results could be expressed by the equations: °_f G _{CdMoO4 (s)} ⁰= –1002.0+0.267T±14.5 kJ mol^–1 (987≤T/K≤1033) and °_f G _{CdMoO4 (s)} ⁰ = –1101.9+0.363T±14.4 kJ mol^–1 (1044≤T/K≤1111). The standard enthalpy of formation of CdMoO₄(s) was found to be –1015.4±14.5 kJ mol^–1 .     

Density matrix averaged atomic natural orbital (ANO) basis sets for correlated molecular wave functions

Generally contracted basis sets for the first row transition metal atoms Sc-Zn have been constructed using the atomic natural orbital (ANO) approach, with modifications for allowing symmetry breaking and state averaging. The ANOs are constructed by averaging over the three electronic configurationsd ⁿ ,d ^n–1 s, andd ^n–2 s ² for the neutral atom as well as the ground state for the cation and the ground state atom in an external electric field. The primitive sets are 21s15p10d6f4g. Contraction to 6s5p4d3f2g yields results that are virtually identical to those obtained with the corresponding uncontracted basis sets for the atomic properties, which they have been designed to reproduce. Slightly larger deviations are obtained with the 5s4p3d2f1g for the polarizability, while energetic properties still have only small errors. The design objective has been to describe the ionization potential, the polarizability and the valence spectrum as accurately as possible. The result is a set of well-balanced basis sets for molecular calculations, which can be used together with basis sets of the same quality for the first and second row atoms.     

Accurate analytical self-consistent field wave functions for Tm2+ and Tm3+

The analytical expansion self-consistent field method was employed to perform ab initio calculations for the ground states of the rare-earth ions, Tm²⁺, 4f¹³, ²F, and Tm³⁺, 4f¹², ³H, (Z = 69). In each case the total number of basis functions used in the analytical expansions was 29, distributed as follows: 10, 8, 5, and 6, for the symmetries s, p, d, and f, respectively. All of the orbital exponents of the basis functions were optimized repeatedly, to the extent of the single-precision computer representation. Values of 〈rⁿ〉 for the 4f orbital of both ions are also presented, for the convenience of experimentalists.     

Ab initio HF and DFT calculations on an organic non-linear optical material

The molecular geometric optimization, vibrational frequencies, and gauge-including atomic orbital (GIAO) ¹H and ¹³C chemical shift values of 3-[(1E)-N-ethylethanimidoyl]-4-hydroxy-6-methyl-2H-pyran-2-one have been investigated by using ab initio Hartree–Fock (HF) and density functional method (B3LYP: Becke-3-Lee–Yang–Parr) with 6–31G(d) and 6–31++G(d,p) basis sets. Also, the first hyperpolarizabilities have been calculated at the HF and B3LYP levels employing the corresponding basis sets. To understand this phenomenon in the context of molecular orbital picture, we examined the molecular HOMOs and molecular LUMOs generated via HF and B3LYP levels. The computed vibrational frequencies are used to determine the types of molecular motions associated with each of the experimental bands observed. Data of 3-[(1E)-N-ethylethanimidoyl]-4-hydroxy-6-methyl-2H-pyran-2-one display significant second-order molecular nonlinearity and provide the basis for design of efficient nonlinear optical materials.     

Photoelectron spectrum and quantum chemical analysis of the structure of bis(1,3,6-trimethyluracilyl-5)methane

Photoelectron spectra of bis(1,3,6-trimethyluracilyl-5)methane (I) and 1,3,6-trimethyluracil (II) were studied; AM1 optimization of geometric characteristics was carried out. The total energy minimum and the best agreement between the values of IP_m and -ɛ_m were obtained for conformations with nearly orthogonal location of uracilyl fragments. In such conformations, the highest occupied orbitals are pseudodegenerate. To interpret the photoelectron spectra, we employed ab initio calculations in STO-3G and 4-31G basis sets. For uracil and its derivatives, all methods give the π, π, n⁻, n⁺, π sequence of the highest orbitals. A. E. Arbuzov Institute of Organic and Physical Chemistry, Kazan Scientific Center, Russian Academy of Sciences. Translated fromZhurnal Struktumoi Khimii, Vol. 36, No. 1, pp. 102–107, January–February, 1995. Translated by L. Smolina