Relativistic Basis Sets Research Articles

All-electron relativistic computations on the low-lying electronic states, bond length, and vibrational frequency of CeF diatomic molecule with spin-orbit coupling effects.

Ab initio all-electron computations have been carried out for Ce+ and CeF, including the electron correlation, scalar relativistic, and spin-orbit coupling effects in a quantitative manner. First, the n-electron valence state second-order multireference perturbation theory (NEVPT2) and spin-orbit configuration interaction (SOCI) based on the state-averaged restricted active space multiconfigurational self-consistent field (SA-RASSCF) and state-averaged complete active space multiconfigurational self-consistent field (SA-CASSCF) wavefunctions have been applied to evaluations of the low-lying energy levels of Ce+ with [Xe]4f1 5d1 6s1 and [Xe]4f1 5d2 configurations, to test the accuracy of several all-electron relativistic basis sets. It is shown that the mixing of quartet and doublet states is essential to reproduce the excitation energies. Then, SA-RASSCF(CASSCF)/NEVPT2 + SOCI computations with the Sapporo(-DKH3)-2012-QZP basis set were carried out to determine the energy levels of the low-lying electronic states of CeF. The calculated excitation energies, bond length, and vibrational frequency are shown to be in good agreement with the available experimental data. © 2018 Wiley Periodicals, Inc.

Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K-Fr) and alkaline earth (Ca-Ra) elements.

New correlation consistent basis sets based on pseudopotential (PP) Hamiltonians have been developed from double- to quintuple-zeta quality for the late alkali (K-Fr) and alkaline earth (Ca-Ra) metals. These are accompanied by new all-electron basis sets of double- to quadruple-zeta quality that have been contracted for use with both Douglas-Kroll-Hess (DKH) and eXact 2-Component (X2C) scalar relativistic Hamiltonians. Sets for valence correlation (ms), cc-pVnZ-PP and cc-pVnZ-(DK,DK3/X2C), in addition to outer-core correlation [valence + (m-1)sp], cc-p(w)CVnZ-PP and cc-pwCVnZ-(DK,DK3/X2C), are reported. The -PP sets have been developed for use with small-core PPs [I. S. Lim et al., J. Chem. Phys. 122, 104103 (2005) and I. S. Lim et al., J. Chem. Phys. 124, 034107 (2006)], while the all-electron sets utilized second-order DKH Hamiltonians for 4s and 5s elements and third-order DKH for 6s and 7s. The accuracy of the basis sets is assessed through benchmark calculations at the coupled-cluster level of theory for both atomic and molecular properties. Not surprisingly, it is found that outer-core correlation is vital for accurate calculation of the thermodynamic and spectroscopic properties of diatomic molecules containing these elements.

On the long-range relativistic effects in the 15 N NMR chemical shifts of halogenated azines.

Long-range β- and γ-relativistic effects of halogens in 15 N NMR chemical shifts of 20 halogenated azines (pyridines, pyrimidines, pyrazines, and 1,3,5-triazines) are shown to be unessential for fluoro-, chloro-, and bromo-derivatives (1-2ppm in average). However, for iodocontaining compounds, β- and γ-relativistic effects are important contributors to the accuracy of the 15 N calculation. Taking into account long-range relativistic effects slightly improves the agreement of calculation with experiment. Thus, mean average errors (MAE) of 15 N NMR chemical shifts of the title compounds calculated at the non-relativistic and full 4-component relativistic levels in gas phase are accordingly 7.8 and 5.5ppm for the range of about 150ppm. Taking into account solvent effects within the polarizable continuum model scheme marginally improves agreement of computational results with experiment decreasing MAEs from 7.8 to 7.4ppm and from 5.5 to 5.3ppm at the non-relativistic and relativistic levels, respectively. The best result (MAE: 5.3ppm) is achieved at the 4-component relativistic level using Keal and Tozer's KT3 functional used in combination with Dyall's relativistic basis set dyall.av3z with taking into account solvent effects within the polarizable continuum solvation model. The long-range relativistic effects play a major role (of up to dozen of parts per million) in 15 N NMR chemical shifts of halogenated nitrogen-containing heterocycles, which is especially crucial for iodine derivatives. This effect should apparently be taken into account for practical purposes.

Calculation of 125Te NMR Chemical Shifts at the Full Four-Component Relativistic Level with Taking into Account Solvent and Vibrational Corrections: A Gateway to Better Agreement with Experiment.

Four-component relativistic calculations of 125Te NMR chemical shifts were performed in the series of 13 organotellurium compounds, potential precursors of the biologically active species, at the density functional theory level under the nonrelativistic and four-component fully relativistic conditions using locally dense basis set scheme derived from relativistic Dyall's basis sets. The relativistic effects in tellurium chemical shifts were found to be of as much as 20-25% of the total calculated values. The vibrational and solvent corrections to 125Te NMR chemical shifts are about, accordingly, 6 and 8% of their total values. The PBE0 exchange-correlation functional turned out to give the best agreement of calculated tellurium shifts with their experimental values giving the mean absolute percentage error of 4% in the range of ∼1000 ppm, provided all corrections are taken into account.

Quantum-chemical calculations of NMR chemical shifts of organic molecules: XV. Relativistic calculations of 29Si NMR chemical shifts of silanes

Calculations of 29Si NMR chemical shifts of 68 silanes possessing various substituents, in particular, with heavy halogens attached to silicon atom, were carried out applying an efficient calculation scheme of locally dense basis set in the framework of the electron density functional theory utilizing the Keal–Tozer functional combined with relativistic Dyall basis sets on a four-component relativistic level. The main factors of calculation accuracy of silicon chemical shifts were analyzed including the relativistic effects, environmental impact, and vibrational corrections. The mean absolute calculation error for the studied compounds series accounting for all mentioned factors was 14.0 ppm for the nonrelativistic calculation and 6.7 ppm for the four-component relativistic calculation at the range of silicon chemical shifts variation of ~250 ppm.

Relativistic ab Initio Accurate Atomic Minimal Basis Sets: Quantitative LUMOs and Oriented Quasi-Atomic Orbitals for the Elements Li-Xe.

Valence virtual orbitals (VVOs) are a quantitative and basis set independent method for extracting chemically meaningful lowest unoccupied molecular orbitals (LUMOs). The VVOs are formed based on a singular value decomposition (SVD) with respect to precomputed and internally stored ab initio accurate atomic minimal basis sets (AAMBS) for the atoms. The occupied molecular orbitals and VVOs together form a minimal basis set that can be transformed into orthogonal oriented quasi-atomic orbitals (OQUAOs) that provide a quantitative description of the bonding in a molecular environment. In the present work, relativistic AAMBS are developed that span the full valence orbital space. The impact of using full valence AAMBS for the formation of the VVOs and OQUAOs and the resulting bonding analysis is demonstrated with applications to the cuprous chloride, scandium monofluoride, and nickel silicide diatomic molecules.

All-electron Gaussian basis sets of double zeta quality for the actinides.

For the actinides, two segmented all-electron basis sets of valence double zeta quality plus polarization functions (DZP) are developed. One of them must be used along with the non-relativistic Hamiltonian, whereas the other with the Douglas-Kroll-Hess (DKH) one. Adding diffuse functions of s, p, d, f, and g symmetries to the non-relativistic and relativistic sets, augmented basis sets are developed. These functions are essential to describe correctly electrons far away from the nuclei. For some compounds, geometric parameters, atomic charges and valence orbital populations of the actinides, and bond dissociation energies are calculated using the Becke 3-parameter (exchange) and the Lee, Yang, and Parr (correlation) functional in conjunction with the DZP-DKH basis set. For Am and No, the static electric mean dipole polarizabilities are also reported. Comparison with benchmark theoretical and experimental values found in the literature is carried out. It is verified that the performances of the relativistic compact size basis sets generated in this work are regular, efficient, and reliable. They will be extremely helpful in molecular property calculations that need explicitly to consider the core electrons.

Potential energy curves and spectroscopic parameters of the 24 Λ-S states and 54 Ω states of the F2 + cation including the spin-orbit coupling effect*

This work calculated the PECs of 24 Λ-S states and 54 Ω states of F2 + cation. The calculations were done with the CASSCF method, which was followed by the internally contracted MRCI approach. Core-valence correlation correction, scalar relativistic correction and basis set extrapolation were taken into account. Of these 24 Λ-S states, the 22Σg −, 22Σu −, 24Σg −, 14Δu, and 24Πg states were found to be repulsive. The X2Πg, A2Πu 14Δg, 14Πg and 24Πg states were found to be inverted with the spin-orbit coupling effect included. The 12Δg, 24Πu, 14Πg, 14Σu +, 22Πu, 14Σg −, 24Σu −, and 12Σg + states were found to be weakly bound. The 24Σu − state had double wells. The avoided crossings of PECs were observed between the A2Πu and 22Πu states, the X2Πg and 22Πg states, the 12Σu − and 22Σu − states, the 14Πu and 24Πu states, and the 14Σ− g and 24Σ− g states. Some spectroscopic parameters were determined and the vibrational properties of several weakly-bound states were predicted. The spin-orbit coupling effect on the spectroscopic parameters was evaluated. Comparison with available experimental data shows that the methodology used in this paper is highly accurate for this system.

Dipole (hyper)polarizabilities of neutral silver clusters**Project supported by CNPq, CAPES, and FAPES (Brazilian Agencies).

At the Douglas–Kroll–Hess (DKH) level, the B3PW91 functional along with the all-electron relativistic basis sets of valence triple and quadruple zeta qualities are used to determine the structure, stability, and electronic properties of the small silver clusters (Agn, n ⩽ 7). The results presented in this study are in good agreement with the experimental data and theoretical values obtained at a higher level of theory from the literature. Static polarizability and hyperpolarizability are also reported. It is verified that the mean dipole polarizability per atom exhibits an odd-even oscillation and that the polarizability anisotropy is directly related to the cluster shape. In this article, the first study of hyperpolarizabilities of small silver clusters is presented. Except for the monomer, the second hyperpolarizabilities of the silver clusters are significantly larger than those of the copper clusters.

Ab initio calculations of X-ray magnetic circular dichroism spectra within the projector augmented wave method: An implementation into the VASP code

X-ray absorption and X-ray magnetic circular dichroism (XMCD) are very powerful tools for probing the orbital and spin moments of each atomic species orbital of magnetic materials. In this work, we present the implementation of a module for computing the X-ray absorption and XMCD spectra into the VASP code. We provide a derivation of the absorption cross-section in the electric dipole approximation. The matrix elements, which make up the X-ray absorption cross-section for a given polarization of light, are then computed using either the momentum operator p or the position operator r, within the projector augmented wave method. The core electrons are described using the relativistic basis-set whereas for the valence electrons, the spin–orbit coupling is added perturbatively to the semi-relativistic Hamiltonian. We show that both the p and the r implementations lead to the same results. The results for the K-edge and L23-edges of bcc-iron are then computed and compared to experiment.

Normal halogen dependence of 13 C NMR chemical shifts of halogenomethanes revisited at the four-component relativistic level.

The 'Normal Halogen Dependence' of 13 C NMR chemical shifts in the series of halogenomethanes is revisited at the four-component relativistic level. Calculations of 13 C NMR chemical shifts of 70 halogenomethanes have been carried out at the density functional theory (DFT) and MP2 levels with taking into account relativistic effects using the four-component relativistic theory of Dirac-Coulomb within the different computational methods (4RPA, 4OPW91) and hybrid computational schemes (MP2 + 4RPA, MP2 + 4OPW91). The most efficient computational protocols are derived for practical purposes. Relativistic shielding effect reaches as much as several hundreds of ppm for heavy halogenomethanes, and to account for this effect in comparison with experiment at the qualitative level, relativistic Dyall's basis sets of triple-zeta quality or higher are to be used within the framework of the four-component relativistic theory taking into account solvent effects. Relativistic geometrical optimization (as compared with the non-relativistic level) is essential for the molecules containing at least two iodines at one carbon atom. Copyright © 2016 John Wiley & Sons, Ltd.

Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the light elements H–Ar

Relativistic basis sets of double-zeta, triple-zeta, and quadruple-zeta quality have been optimized at the self-consistent field level with a Gaussian nuclear charge distribution for the light elements H–Ar. Correlating functions were optimized for all shells in MR-CISD calculations on the ground configuration. Energies and properties evaluated with the basis sets are compared with those produced by several other basis sets. Prescriptions are given for constructing contracted basis sets. The basis sets are available as an internet archive.

Troubles in the Systematic Prediction of Transition Metal Thermochemistry with Contemporary Out-of-the-Box Methods.

The recently developed DLPNO-CCSD(T) method and seven popular DFT functionals (B3LYP, M06, M06L, PBE, PBE0, TPSS, and TPSSh) with and without an empirical dispersion term have been tested to reproduce 111 gas phase reaction enthalpies involving 11 different transition metals. Our calculations, corrected for both relativistic effects and basis set incompleteness, indicate that most of the methods applied with default settings perform with acceptable accuracy on average. Nevertheless, our calculations also evidenced unexpected and nonsystematic large deviations for specific cases. For group 12 metals (Zn, Cd, Hg), most of the methods provided mean unsigned errors (MUE) less than 5.0 kcal/mol, with DLPNO-CCSD(T) and PBE methods performing excellently (MUE lower 2.0 kcal/mol). Problems started with group 4 metals (Ti and Zr). The best performer for Zr complexes with MUE of 1.8 kcal/mol, PBE0-D3, provides MUE larger than 8 kcal/mol for Ti. DLPNO-CCSD(T) provides a reasonable MUE of 3.3 kcal/mol for Ti reactions but gives MUE a larger than 14.4 kcal/mol for Zr complexes, with all the larger deviations for reactions involving ZrF4. Large and nonsystematic errors have been obtained for group 6 metals (Mo and W), for eight reactions containing Fe, Cu, Nb, and Re complexes. Finally, for the whole set of 111 reactions, the DLPNO-CCSD(T), B3LYP-D3, and PBE0-D3 methods turned out to be the best performers, all providing MUE below 5.0 kcal/mol. Since DFT results cannot be systematically improved and large nonsystematic deviations of 20-30 kcal/mol were obtained even for best performers, our results indicate that current DFT methods are still unable to provide robust predictions in transition metal thermochemistry, at least for the functionals explored in this work. The same conclusion holds for both DLPNO-CCSD(T) and canonical CCSD(T) methods when used entirely as out-of-the-box. However, if careful investigation of core correlation is performed, relativistic effects are properly included and the quality of the reference wave function is properly checked, CCSD(T) methods can still provide good quality results that might even be used to validate DFT methods due to paucity of accurate thermodynamic data for realistic-sized transition metal complexes.

A Density Functional Theory and ab Initio Investigation of the Oxidation Reaction of CO by IO Radicals.

To get an insight into the possible reactivity between iodine oxides and CO, a first step was to study the thermochemical properties and kinetic parameters of the reaction between IO and CO using theoretical chemistry tools. All stationary points involved were optimized using the Becke's three-parameter hybrid exchange functional coupled with the Lee-Yang-Parr nonlocal correlation functional (B3LYP) and the Møller-Plesset second-order perturbation theory (MP2). Single-point energy calculations were performed using the coupled cluster theory with the iterative inclusion of singles and doubles and the perturbative estimation for triple excitations (CCSD(T)) and the aug-cc-pVnZ (n = T, Q, and 5) basis sets on geometries previously optimized at the aug-cc-pVTZ level. The energetics was then recalculated using the one-component DK-CCSD(T) approach with the relativistic ANO basis sets. The spin-orbit coupling for the iodine containing species was calculated a posteriori using the restricted active space state interaction method in conjunction with the multiconfigurational perturbation theory (CASPT2/RASSI) employing the complete active space (CASSCF) wave function as the reference. The CCSD(T) energies were also corrected for BSSE for molecular complexes and refined with the extrapolation to CBS limit while the DK-CCSD(T) values were refined with the extrapolation to FCI. The exploration of the potential energy surface revealed a two-steps mechanism with a trans and a cis pathway. The rate constants for the direct and complex mechanism were computed as a function of temperature (250-2500 K) using the canonical transition state theory. The three-parameter Arrhenius expressions obtained for the direct and indirect mechanism at the DK-CCSD(T)-cf level of theory is 1.49 × 10(-17) × T(1.77) exp(-47.4 (kJ mol(-1))/RT).

MP2 calculation of (77) Se NMR chemical shifts taking into account relativistic corrections.

The main factors affecting the accuracy and computational cost of the Second-order Möller-Plesset perturbation theory (MP2) calculation of (77) Se NMR chemical shifts (methods and basis sets, relativistic corrections, and solvent effects) are addressed with a special emphasis on relativistic effects. For the latter, paramagnetic contribution (390-466 ppm) dominates over diamagnetic term (192-198 ppm) resulting in a total shielding relativistic correction of about 230-260 ppm (some 15% of the total values of selenium absolute shielding constants). Diamagnetic term is practically constant, while paramagnetic contribution spans over 70-80 ppm. In the (77) Se NMR chemical shifts scale, relativistic corrections are about 20-30 ppm (some 5% of the total values of selenium chemical shifts). Solvent effects evaluated within the polarizable continuum solvation model are of the same order of magnitude as relativistic corrections (about 5%). For the practical calculations of (77) Se NMR chemical shifts of the medium-sized organoselenium compounds, the most efficient computational protocols employing relativistic Dyall's basis sets and taking into account relativistic and solvent corrections are suggested. The best result is characterized by a mean absolute error of 17 ppm for the span of (77) Se NMR chemical shifts reaching 2500 ppm resulting in a mean absolute percentage error of 0.7%.

Relativistic Environmental Effects in 29Si NMR Chemical Shifts of Halosilanes: Light Nucleus, Heavy Environment

Relativistic calculations of (29)Si NMR shielding constants (chemical shifts) in the series of halosilanes SiX(n)H(4-n) (X = F, Cl, Br and I) are performed within a full four-component relativistic Dirac's scheme using relativistic Dyall's basis sets. Three different theoretical levels are tested in the computation of (29)Si NMR chemical shifts in comparison with experiment: namely, four-component relativistic GIAO-DFT, four-component relativistic GIAO-RPA, and a hybrid scheme of a nonrelativistic GIAO-MP2 with taking into account relativistic corrections using the four-component relativistic GIAO-RPA. The DFT results give larger relativistic effects as compared to the RPA data which might be rationalized in terms of the manifestation of correlation effects taken into account at the DFT level and not accounted for at the uncorrelated RPA level. Taking into account solvent effects slightly improves agreement with experiment, however, being not a matter of principle. Generally, relativistic pure nonempirical wave function methods perform much better as compared to relativistic DFT methods when benchmarked to experiment.

Estimating the Impact of an All-Electron Basis Set and Scalar Relativistic Effects on the Structure, Stability, and Reactivity of Small Copper Clusters

Basis sets of valence double and quadruple zeta qualities and the Douglas-Kroll-Hess (DKH) approximation are used to estimate the impact of an all-electron basis set and scalar relativistic effects on the structure, stability, and electronic properties of small neutral copper clusters (Cun, n ≤ 8). At the Becke three-parameter for exchange and Perdew-Wang 91 for correlation (B3PW91) non-relativistic and relativistic levels of theory, the bond length, binding energy, ionization potential, electron affinity, chemical potential, chemical hardness, and electrophilicity index are calculated. The results show that the agreement with experiment improves significantly when the DKH Hamiltonian combined with an all-electron relativistic basis set is used. Polarizabilities and hyperpolarizability are also reported. At the B3PW91 level, all-electron basis sets are shown to be more reliable than effective core potential valence basis sets in the determination of the second hyperpolarizability of copper clusters.

High-resolution spectroscopy of the chiral metal complex [CpRe(CH₃)(CO)(NO)]: a potential candidate for probing parity violation.

Heavy-metal containing chiral compounds have been suggested as promising candidates for studying parity-violation effects. We report herein the broadband rotational spectroscopy study of the chiral complex [CpRe(CH3)(CO)(NO)] in the gas phase. The spectra obtained are very rich due to the two rhenium isotopologues ((185)Re and (187)Re), hyperfine structure arising from the rhenium and nitrogen nuclei, and the asymmetry of the chiral complex. Since rhenium is located very close to the molecular center of mass, the rotational constants for the two rhenium isotopologues are very similar. However they can be differentiated by their characteristic nuclear quadrupole hyperfine splitting patterns. Comparison with calculated nuclear quadrupole coupling constants shows that all-electron relativistic basis sets are necessary for a correct description of the rhenium atom in this type of complex. The present study is an important step towards future precision studies on chiral molecules.

The Remarkable [ReH9]2– Dianion: Molecular Structure and Vibrational Frequencies

The equilibrium geometries and vibrational frequencies of the extraordinary [ReH9](2-) dianion (D3h symmetry) are investigated using Hartree-Fock (HF) theory, coupled cluster theory with single and double excitations (CCSD), and coupled cluster theory with single, double, and perturbative triple excitations [CCSD(T)]. The new generation of energy-consistent relativistic pseudopotentials and correlation consistent basis sets [cc-pVXZ-PP (Re) and cc-pVXZ (H) (X = D, T, Q)] are used. Anharmonicity was considered using second-order vibrational perturbation theory. The predicted geometries and vibrational frequencies generally agree with experimental findings. In order to stabilize the [ReH9](2-) dianion, the M2ReH9 (M = Na, K) sandwich complexes (D3h symmetry) are studied at the CCSD(T)/VTZ (VTZ = cc-pVTZ-PP (Re) and cc-pVTZ (H, Na, K)) level of theory. Compared to the [ReH9](2-) dianion, the predicted vibrational frequencies involving Re-H stretching modes are improved, indicating the importance of considering counterions in electronically dense systems. The natural bond orbital analysis shows that each H atom only bonds with the Re center, and the 5d orbitals of Re and 1s orbitals of H are major factors for the covalent Re-H bonding.

Structural and electronic properties of the BnY (n=1-11) clusters

The geometric structures, electronic properties, average binding energies, second-order energy differences and energy gaps of BnY (n=1-11) clusters are systematically studied using the density functional theory (DFT) TPSSh method with 6-311+G(d) basis set for B atoms and Lanl2dz relativistic effective core potential basis set for Y atom. It is found that with the size increasing, the lowest energy structures of BnY (n=1-11) clusters gradually evolve from planar shape to cubic structure. With the atoms of B increasing, the average binding energies of the ground state of BnY (n=1-11) clusters increase. The second-order energy differences and the energy gaps of the ground states of BnY (n=1-11) clusters show that B3Y, B5Y and B7Y clusters possess relatively high stabilities. The polarization and the first static hyperpolarizability studied show that the plane structures of B5Y, B4Y, B3Y and B6Y clusters have larger nonlinear optical properties.