Quality of contracted Gaussian-type function basis sets

The valence quality of contracted (C) Gaussian-type function (GTF) basis sets in molecular calculations is discussed for the first- through fourth-row atoms. The split-valence basis sets derived from minimal-type CGTF sets are compared with those derived from primitive (P) GTF sets. Using F, Cl, Br, and I atoms and their homonuclear diatomics as test species, we find that the split-valence CGTF sets have almost the same quality as PGTF sets with larger s and p expansion terms: for example, the (53/5), (533/53), (5333/533/5), and (53 333/5333/53) CGTF sets correspond approximately to the [9/5], [15/9], [19/15/5], and [22/18/7] PGTF sets for the first- to fourth-row atoms, respectively, where the slash separates the s, p, and d symmetries. For the main group atoms of the four rows, we recommend using the above-mentioned CGTFs or larger.     

Orbitals expanded in slater functions with single-exponent by shell and by subshell

Slater-type orbitals (STO s) with a single-exponent by shell or by subshell have been constructed to reduce the number of integrals evaluated in the electronic calculations. The expansion of orbitals in these new basis sets has been carried out in detail for the ground state of the Ne atom. We have carried out a study of STO basis sets with a different size for this atom that could help to propose empirical rules for the selection of these basis sets for other atoms. The usefulness of STO s with single-exponent by shell and subshell and the splitting of s and p functions are discussed. © 1994 John Wiley & Sons, Inc.     

New relativistic atomic natural orbital basis sets for lanthanide atoms with applications to the Ce diatom and LuF3

New basis sets of the atomic natural orbital (ANO) type have been developed for the lanthanide atoms La-Lu. The ANOs have been obtained from the average density matrix of the ground and lowest excited states of the atom, the positive ions, and the atom in an electric field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calculations of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.1 eV in most cases. Two molecular applications are included as illustration: the cerium diatom and the LuF3 molecule. In both cases it is shown that 4f orbitals are not involved in the chemical bond in contrast to an earlier claim for the latter molecule.     

Valence p functions for alkali and alkaline-earth atoms

Contracted Gaussian-type function (CGTF) basis sets are reported for valence p orbitals of the six alkali and alkaline-earth atoms Li, Be, Na, Mg, K, and Ca for molecular applications. These sets are constructed by Roothaan–Hartree–Fock calculations for the ns → np excited states of atoms, in which both linear and nonlinear parameters of CGTFs are variationally optimized. The present CGTF sets reproduce well the numerical Hartree–Fock ns → np excitation energies: the largest error is 0.0009 hartrees for Li. New CGTFs are tested with diatomic Li₂, Na₂, K₂, and MH molecules, where M = Li, Be, Na, Mg, K, and Ca, by self-consistent-field (SCF) and multiconfiguration SCF calculations. The resultant spectroscopic constants compare well with those of more elaborate calculations and are sufficiently close to experimental values, supporting the efficiency of the present set for the valence p orbitals. Received: 9 July 1998 / Accepted: 17 September 1998 / Published online: 1 February 1999     

Correlating basis sets for the H atom and the alkali-metal atoms from Li to Rb

 Contracted Gaussian-type function sets are developed for correlating p, d, and f functions for a valence electron of the hydrogen atom and alkali-metal atoms from Li to Rb. A segmented contraction scheme is used for its compactness and efficiency. Contraction coefficients and exponents are determined by minimizing the deviation from the K orbitals of the atoms. The present basis sets yield an accuracy comparable to the correlation-consistent basis set for the hydrogen atom and also give a similar high accuracy for the alkali-metal atoms. In the calculations of spectroscopic constants of alkali hydrides, the decontraction of the p function plays an important role, especially for LiH. The contributions of d and f functions are nontrivial for KH and RbH. Received: 6 September 2002 / Accepted: 13 November 2002 / Published online: 19 March 2003 Acknowledgements. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: T. Noro e-mail: tashi@sci.hokudai.ac.jp     

Quality of correlating functions generated from commonly used basis sets

Tests have been performed on the quality of correlating functions generated from commonly used Gaussian basis sets, such as the 4-31G and MIDI-4 sets. The atoms tested were carbon, nitrogen, and oxygen. Self-consistent field and configuration interaction (CI) calculations were performed for the ground and lower excited states of neutral atoms as well as for positive and negative ions, using the original sets. Next, after adding (1) one d, and (2) two d and one f primitive Gaussian-type functions (GTFs) to the original sets, the CI calculations were repeated. In order to investigate the quality of the correlating orbitals generated from the GTF sets, parallel calculations to those for the GTF sets were carried out with an extended set of Slater-type functions. It was found that the excitation energies change in a stepwise manner as the basis sets changed from the original sets to the original set + 1d and the original set +2d1f. The improvements in excitation energies and ionization energies were almost independent of the original sets and were found to be strongly dependent on the augmented correlation functions. © 1996 by John Wiley & Sons, Inc.     

4f fine‐structure levels as the dominant error in the electronic structures of binary lanthanide oxides

The ground‐state 4f fine‐structure levels in the intrinsic optical transition gaps between the 2p and 5d orbitals of lanthanide sesquioxides (Ln₂O₃, Ln = La…Lu) were calculated by a two‐way crossover search for the U parameters for DFT + U calculations. The original 4f‐shell potential perturbation in the linear response method were reformulated within the constraint volume of the given solids. The band structures were also calculated. This method yields nearly constant optical transition gaps between Ln‐5d and O‐2p orbitals, with magnitudes of 5.3 to 5.5 eV. This result verifies that the error in the band structure calculations for Ln₂O₃ is dominated by the inaccuracies in the predicted 4f levels in the 2p‐5d transition gaps, which strongly and non‐linearly depend on the on‐site Hubbard U. The relationship between the 4f occupancies and Hubbard U is non‐monotonic and is entirely different from that for materials with 3d or 4d orbitals, such as transition metal oxides. This new linear response DFT + U method can provide a simpler understanding of the electronic structure of Ln₂O₃ and enables a quick examination of the electronic structures of lanthanide solids before hybrid functional or GW calculations. © 2015 Wiley Periodicals, Inc.     

Relativistic correlating basis sets for the sixth-period d-block atoms from Lu to Hg

Contracted Gaussian-type function sets to describe valence correlation are developed for the sixth-period d-block atoms Lu through Hg. A segmented contraction scheme is employed for their compactness and efficiency. Contraction coefficients and exponents are determined by minimizing the deviation from accurate natural orbitals generated from configuration interaction calculations, in which relativistic effects are incorporated through the third-order Douglas-Kroll approximation. The present basis sets yield more than 99% of atomic correlation energies predicted by accurate natural orbital sets of the same size. Relativistic model core potential calculations with the present correlating sets give the spectroscopic constants of the AuH molecule in excellent agreement with experimental results.     

镧系元素4f轨道在成键中的作用的理论研究

用密度泛函方法在冻结或不冻结4f轨道的条件下计算一系列含镧系元素双原子 分子,对结果进行分析,得出以下结论：镧系元素4f轨道按传统的化学键理论观点 是不直接参与成键的,但对成键有一定作用：通过与匹配物的轨道混合使键长变短 ,键能增加,一般可达百分之几。随着镧系原子序数的增加4f轨道对成键的贡献减 少。电负性高或价态高的匹配物对4f轨道的作用较强,4f轨道对成键的影响比较大 。对于重镧系元素,匹配物不是F或O时,4f轨道对成键的贡献相当小,可以看成芯 轨道,但对于轻稀土,在比较精确的计算中则应作为价轨道处理。镧系元素与氟结 合时,只有对靠近Yb的重镧系元素才可以把4f当作芯轨道,而与氧结合时即使对于 YbO把Yb4f作为芯轨道仍会带来较大误差。     

Systematically convergent correlation consistent basis sets for molecular core-valence correlation effects: the third-row atoms gallium through krypton

The family of correlation consistent polarized valence basis sets has been extended in order to account for core-core and core-valence correlation effects within the third-row, main group atoms gallium through krypton. Construction of the basis sets is similar to that of the atoms boron through argon, where either the difference between core-correlated and valence-only correlation energies were calculated via configuration interaction (CISD) computations on the ground electronic states of the atoms (named cc-pCVnZ) or the sets were optimized with respect to the core-valence correlation energy and a small weight of core-core correlation energy (cc-pwCVnZ). Due to the correlation of 3d orbitals, added shells of higher angular momentum exponents compared to the valence sets are necessary to describe the core region. The pattern of added core-correlating functions is (1s1p1d1f) for double-zeta, (2s2p2d2f1g) for triple-zeta, (3s3p3d3f2g1h) for quadruple-zeta, and (4s4p4d4f3g2h1i) for quintuple-zeta. Atomic and molecular results show good convergence to the CBS limit, with the cc-pwCVnZ sets showing improved convergence compared to the cc-pCVnZ ones for molecular core-valence correlation effects. After testing the basis sets on the homonuclear diatomics Ga2-Kr2 with coupled cluster wave functions, it is concluded that a treatment of core-valence correlation effects is essential for high-accuracy ab initio investigations of third-row-containing molecules. Though the basis sets are optimal for 3s3p3d correlation, preliminary atomic and molecular results show the basis sets to be efficient with respect to 3d-only correlation, and these potentially could be used with 3d-only correlation for more qualitative studies on larger species.     

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.     

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.     

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.     

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.     

Contracted Gaussian-type basis functions revisited II. Atoms Na through Ar

We report five minimal-type contracted Gaussian-type function (CGTF) basis sets of the second-row atoms, Na – Ar, for molecular applications. Three of the present CGTF sets are revised versions of those given by Huzinaga and co-workers and the other two are newly developed for more accurate calculations. Practical utility and improved reliability of the present basis sets, augmented by polarization functions, are confirmed by test calculations on the P atom and P₂ molecule both at the self-consistent field (SCF) and configuration interaction (CI) levels. Received: 10 February 1997 / Accepted: 23 April 1997     

Basis set selections for relativistic self-consistent field calculations

Practical methods of generating reliable and economic basis sets for relativistic self-consistent fields (RSCF) calculations are developed. Large component basis sets are generated from constrained optimizations of exponents in the nonrelativistic atomic calculations for light atoms. For heavy atoms, large component basis sets for inner core orbitals are generated by fitting numerical atomic spinors of Dirac-Hartree-Fock calculations with appropriate number of Slater-type functions. Small component basis sets are obtained by using the kinetic balance condition and other computational criteria. With judicious selections of the basis sets, virtual orbitals in RSCF calculations become very similar to those in nonrelativistic calculations, implying that relativistic virtual orbitals can be used in electron correlation calculations in the same manner as the conventional nonrelativistic virtual orbitals. It is also evident that the Koopmans' theorem is also valid in RSCF results.