All-electron scalar relativistic basis sets for the elements Rb–Xe

Segmented all-electron relativistically contracted (SARC) basis sets are presented for the elements ₃₇Rb–₅₄Xe, for use with the second-order Douglas–Kroll–Hess approach and the zeroth-order regular approximation. The basis sets have a common set of exponents produced with established heuristic procedures, but have contractions optimized individually for each scalar relativistic Hamiltonian. Their compact size and loose segmented contraction, which is in line with the construction of SARC basis sets for heavier elements, makes them suitable for routine calculations on large systems and when core spectroscopic properties are of interest. The basis sets are of triple-zeta quality and come in singly or doubly polarized versions, which are appropriate for both density functional theory and correlated wave function theory calculations. The quality of the basis sets is assessed against large decontracted reference basis sets for a number of atomic and ionic properties, while their general applicability is demonstrated with selected molecular examples.     

The employment of relativistic adapted Gaussian basis sets in Douglas–Kroll–Hess scalar calculations with diatomic molecules

The prolapse-free relativistic adapted Gaussian basis sets (RAGBSs), developed by our research group on the basis of the four-component approach, are used for the first time in Douglas–Kroll–Hess 2nd order scalar relativistic calculations (DKH2) of simple diatomic molecules containing Hydrogen and the halogens from Fluorine up to Iodine: HX and X₂, where X = F, Cl, Br, and I. To this end, the RAGBSs were contracted with the general contraction scheme to triple-, quadruple-, and quintuple-zeta sets. Polarization functions were also added to the basis sets by optimization with the configuration interaction method including single and double excitations into the DKH2 environment, DKH2-CISD. The molecular properties were then calculated with the coupled cluster electronic correlation treatment and the DKH2 scalar relativistic method, DKH2-CCSD(T), and indicated that our RAGBSs should be contracted as quadruple-zeta basis sets. The results achieved with the DKH2-CCSD(T) calculations and the selected quadruple-zeta RAGBSs are able to reproduce the experimental data of equilibrium distances, dissociation energies, and harmonic vibrational frequencies with root-mean-square (rms) errors of 0.015 Å, 3.6 kcal mol⁻¹, and 21.7 cm⁻¹, respectively.     

The error due to neglecting the core spin–orbit splitting in valence ZORA calculations with frozen core approximation and its elimination

In valence zeroth-order regular approximation (ZORA) calculations with frozen core approximation, when the basis set optimized to the related scalar relativistic ZORA calculations is used, neglecting the core spin–orbit splitting may result in additional basis set truncation errors. It is found that the error is negligible for most elements except the 6p-block elements. When the basis set is extended by a p-type STO function put on the 6p element atoms with the ζ value proper to 5p_1/2 orbitals, the error can be reduced to be negligible. The calculated atomic properties related to valence orbitals can be improved greatly by use of this extended basis set. The frozen core approximation calculations of some molecules containing Tl, Pb and Bi with closed shells show that neglecting the core spin–orbit splitting only slightly affects the calculated bond lengths and bond energies, and the calculated molecular property can also be improved slightly by use of the extended basis sets.     

Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Hydration of gas-phase ytterbium ion complexes studied by experiment and theory

Hydration of ytterbium (III) halide/hydroxide ions produced by electrospray ionization was studied in a quadrupole ion trap mass spectrometer and by density functional theory (DFT). Gas-phase YbX₂ ⁺ and YbX(OH)⁺ (X?=?OH, Cl, Br, or I) were found to coordinate from one to four water molecules, depending on the ion residence time in the trap. From the time dependence of the hydration steps, relative reaction rates were obtained. It was determined that the second hydration was faster than both the first and third hydrations, and the fourth hydration was the slowest; this ordering reflects a combination of insufficient degrees of freedom for cooling the hot monohydrate ion and decreasing binding energies with increasing hydration number. Hydration energetics and hydrate structures were computed using two approaches of DFT. The relativistic scalar ZORA approach was used with the PBE functional and all-electron TZ2P basis sets; the B3LYP functional was used with the Stuttgart relativistic small-core ANO/ECP basis sets. The parallel experimental and computational results illuminate fundamental aspects of hydration of f-element ion complexes. The experimental observations??kinetics and extent of hydration??are discussed in relationship to the computed structures and energetics of the hydrates. The absence of pentahydrates is in accord with the DFT results, which indicate that the lowest energy structures have the fifth water molecule in the second shell.     

Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn

The introduction of the resolution-of-the-identity (RI) approximation for electron repulsion integrals in quantum chemical calculations requires in addition to the orbital basis so-called auxiliary or fitting basis sets. We report here such auxiliary basis sets optimized for second-order Møller–Plesset perturbation theory for the recently published (Weigend and Ahlrichs Phys Chem Chem Phys, 2005, 7, 3297–3305) segmented contracted Gaussian basis sets of split, triple-ζ and quadruple-ζ valence quality for the atoms Rb–Rn (except lanthanides). These basis sets are designed for use in connection with small-core effective core potentials including scalar relativistic corrections. Hereby accurate resolution-of-the-identity calculations with second-order Møller–Plesset perturbation theory (MP2) and related methods can now be performed for molecules containing elements from H to Rn. The error of the RI approximation has been evaluated for a test set of 385 small and medium sized molecules, which represent the common oxidation states of each element, and is compared with the one-electron basis set error, estimated based on highly accurate explicitly correlated MP2–R12 calculations. With the reported auxiliary basis sets the RI error for MP2 correlation energies is typically two orders of magnitude smaller than the one-electron basis set error, independent on the position of the atoms in the periodic table.     

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.     

Reduced-size polarized basis sets for calculations of molecular electric properties. IV. First-row transition metals

Recent studies of the perturbation-dependent basis sets have indicated the possibility of a significant reduction of the size of the usual CGTO sets without considerable loss of accuracy in calculations of molecular electric properties. The resulting (ZPolX) basis sets have been developed for several atoms of the first and second row of the Periodic Table. The same method of the ZPolX basis set generation is extended for the first-row transition metals and the corresponding contracted ZPolX basis sets of the size [6s5p3d1f] are determined for both nonrelativistic and scalar relativistic calculations. The performance of the ZPolX basis sets is verified in calculations on the first-row transition metal oxides at the level of the ROHF, ROHF/CASPT2, and ROHF/CCSD(T) approximations. Also the study of the dipole polarizability of TiCl₄ confirms the excellent features of these very compact basis sets. The ZPolX basis sets for nonrelativistic and relativistic calculations of molecular electric properties are available on the web page http://www.chem.uni.torun.pl/zchk/basis-sets.html.     

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.     

Heats of formation for Fe(CO)n(n=1–4)

The dissociation energies of Fe(CO)_n (n=2–4) are computed using correlation consistent basis sets and the CCSD(T) approach. The dissociation energies are extrapolated to the CBS limit and are corrected for core–valence (CV), scalar relativistic, spin–orbit, zero-point, and thermal effects. Our iron carbonyl bond strengths agree with experiment within the respective error bars. We use our dissociations energies at 298 K to obtain the heats of formation of Fe(CO)_n (n=1–4).     

An improved generator coordinate Hartree–Fock method applied to the choice of contracted Gaussian basis sets for first‐row diatomic molecules

Accurate Gaussian basis sets (18s for Li and Be and 20s11p for the atoms from B to Ne) for the first‐row atoms, generated with an improved generator coordinate Hartree–Fock method, were contracted and enriched with polarization functions. These basis sets were tested for B₂, C₂, BeO, CN⁻, LiF, N₂, CO, BF, NO⁺, O₂, and F₂. At the Hartree–Fock (HP), second‐order Møller–Plesset (MP2), fourth‐order Møller–Plesset (MP4), and density functional theory (DFT) levels, the dipole moments, bond lengths, and harmonic vibrational frequencies were studied, and at the MP2, MP4, and DFT levels, the dissociation energies were evaluated and compared with the corresponding experimental values and with values obtained using other contracted Gaussian basis sets and numerical HF calculations. For all diatomic molecules studied, the differences between our total energies, obtained with the largest contracted basis set [6s5p3d1f], and those calculated with the numerical HF methods were always less than 3.2 mhartree. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 15–23, 2000     

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.     

On the use of contracted basis functions in relativistic Hartree-Fock calculations

A procedure is suggested to build up contracted basis sets for relativistic atomic and molecular Hartree-Fock calculations when corresponding non-relativistic results are available or easy to obtain. Significant reductions of the size of relativistic Fock-Roothaan matrices are expected.     

Ab initio relativistic effective potentials with spin-orbit operators. VI. Fr through Pu

Ab initio averaged relativistic effective core potentials (AREP ), spin-orbit (SO ) operators, and valence basis sets are reported for the elements Fr through Pu in the form of expansions in Gaussian-type functions. Gaussian basis sets with expansion coefficients for the low-energy states of each atom are given. Atomic orbital energies calculated under the j–j coupling scheme within the self-consistent field approximation and employing the AREP 'S in their unaveraged form (REP 'S) agree to within 10% of orbital energies due to numerical all-electron Dirac–Fock calculations. The accuracy of the AREP 'S and so operators is also shown to be good through comparisons of calculated so splitting energies with all-electron Dirac–Fock results.     

Basis-set extensions for two-component spin-orbit treatments of heavy elements

The accuracy of standard basis sets of quadruple-zeta and lower quality for the use in two-component self-consistent field procedures including spin-orbit coupling is investigated for the elements In-I and Au-At. Spin-orbit coupling leads to energetic and spatial splittings of inner shells, which are not described accurately with standard basis sets optimized for scalar relativistic calculations. This results in large errors in total atomic energies and significant errors in atomization energies of compounds containing these atoms. We show how these errors can be corrected by adding just a few steep sets of basis functions and demonstrate the quality of the resulting extended basis sets.     

Comparison of ab initio and density functional methods for vibrational analysis of TeCl4

Vibrational analysis of tellurium tetrachloride, TeCl₄, was performed with Hartree–Fock (HF), MP2, and generalized gradient approximation density functional theory (DFT) methods supplemented with polarized double-zeta split valence (DZVP) basis sets and relativistic effective core potentials (RECP) of Hay and Wadt. The molecular geometry is best reproduced at the HF and MP2/RECP+DZVP [polarized Hay and Wadt RECP for Te and 6–31G(d) basis set for Cl] levels of theory. The DFT methods gave rise to poorer results, especially those using Becke's 1988 exchange functional. Generally, the vibrational frequencies calculated by the MP2 and B3-type DFT methods with the all electron and RECP+DZVP basis sets as well as at the HF/RECP level were in satisfactory accord with the experimental data. The agreement was good enough to assist the assignment of the measured vibrational spectra. The best agreement with the experimental vibrational frequencies was achieved with the scaled HF/RECP force field. Consistent results were obtained for the unobserved A₂ (ν₄) fundamental, where the results of the best methods were within 4 cm⁻¹. The best force fields were obtained with the following methods: Becke3–Lee–Yang–Parr and Becke3–Perdew/all electron basis, MP2 and Becke3-Perdew/RECP+DZVP, and HF/RECP. The methods using RECPs are advantageous for large-scale computations. The RECP basis set effectively compensates the errors of the HF method for TeCl₄; however, it provides poor results with correlated methods. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 308–318, 1998     

Relativistic all-electron Dirac-Fock-Breit calculations on xenon fluorides (XeFn,n = 1, 2, 4, 6)

The MOLFDIR package of programs is used to perform fully relativistic all-electron Dirac-Fock and Dirac-Fock-Breit calculations for the the XeF_n (n = 1, 2, 4, 6) molecules assuming experimental symmetries. The Xe-F bound length for XeF₂, XeF₄, and XeF₆ is optimized and the total ground-state energies are reported. The variation of the relativistic energy and the Breit correction with the internuclear distance is plotted. The role of relativistic corrections in the proper prediction of the Xe-F distance and the dissociation energy of the molecule is discussed. The problem of the reduction of the number of scalar two-electron integrals is studied. Our results illustrate the possibilities, difficulties, and limitations of the finite basis Dirac-Fock calculations for polyatomic molecules of different symmetries. © 1997 by John Wiley & Sons, Inc.