Relativistic Basis Research Articles

Ab initio and DFT analysis of the low-lying electronic states of metal dihalides: quantum chemical calculations on the neutral BrMCl (M = Cu, Ag, Au)

The electronic configuration of the electronic ground and low-lying doublet excited states of neutral metal dihalides BrMCl (M = Cu, Ag, Au) has been investigated on the basis of CASSCF/CASPT2 methods taking into account scalar relativistic effects. A preliminary study of the electronic problem in BrAgCl, based on DFT and CASSCF/CASPT2 approaches and using various basis sets, namely relativistic all-electron basis sets, effective core potentials and ab initio model potentials (AIMP), as well as non-relativistic AIMP is discussed. It is shown that single-determinant methods are not flexible enough to describe the bonding of the neutral species in the electronic ground state regardless of the basis set. The failure to allocate the single electron of BrAgCl correlates with a wrong charge distribution within the complex, which is more accentuated when using pseudopotential basis sets. The inclusion of static and dynamic correlation effects by means of CASSCF/CASPT2 methods using large relativistic all-electron basis sets provides a correct qualitative picture of the electronic structure of the BrMCl series (M = Cu, Ag, Au). The spin unrestricted KS-DFT approach leads to a reasonable description of the degenerate electronic ground state ((2)Σ/(2)Π) bonding in these complexes with negligible spin contamination providing comparative spin densities in the series of molecules under investigation.

All-electron scalar relativistic basis sets for the 6p elements

New segmented all-electron relativistically contracted (SARC) basis sets have been developed for the elements 81Tl–86Rn, thus extending the SARC family of all-electron basis sets to include the 6p block. The SARC basis sets are separately contracted for the second-order Douglas–Kroll–Hess and the zeroth-order regular approximation scalar relativistic Hamiltonians. Their compact size and segmented construction are best suited to the requirements of routine density functional theory (DFT) applications. Evaluation of the basis sets is performed in terms of incompleteness and contraction errors, orbital properties, ionization energies, electron affinities, and atomic polarizabilities. From these atomic metrics and from computed basis set superposition errors for a series of homonuclear dimers, it is shown that the SARC basis sets achieve a good balance between accuracy and size for efficient all-electron scalar relativistic DFT applications.

Unified treatment of basis sets of relativistic exponential-type spinor orbitals for particles with arbitrary half-integral spin in position, momentum and four-dimensional spaces

New formulae for the 2(2s + 1)-component relativistic basis spinors useful for the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where s = 1/2, 3/2, 5/2, …. These spinors are reduced to independent sets of two-component spinors. The relations presented in this study can be useful in the linear combination of atomic orbital approximation for the solution of different problems arising in relativistic quantum mechanics when orthonormal basis sets of relativistic exponential-type spinor wave functions and Slater-type spinor orbitals in position, momentum and four-dimensional spaces are employed.

Relativistic segmented contraction basis sets with core-valence correlation effects for atoms 57La through 71Lu: Sapporo-DK-nZP sets (n = D, T, Q)

For the 15 lanthanide atoms 57La through 71Lu, we report Sapporo-DK-nZP sets (n = D, T, Q), which are natural extensions of the Sapporo-(DK)-nZP sets for lighter atoms and efficiently incorporate the correlation among electrons in the N through P shells as well as the relativistic effect. The present sets well describe the correlation among the 4s and 4p electrons, which are important in the excitation of 4f electrons. Atomic test calculations of 57La, 58Ce, 59Pr, and 60Nd at configuration interaction with the Davidson correction level of theory confirm high performance of the present basis sets. Molecular test calculations are carried out for 57LaF and 70YbF diatomics at the coupled-cluster level of theory. The calculated spectroscopic constants approach smoothly to the experimental values as the quality of the basis set increases.

Global spectroscopy of the water monomer

Given the large energy required for its electronic excitation, the most important properties of the water molecule are governed by its ground potential energy surface (PES). Novel experiments are now able to probe this surface over a very extended energy range, requiring new theoretical procedures for their interpretation. As part of this study, a new, accurate, global spectroscopic-quality PES and a new, accurate, global dipole moment surface are developed. They are used for the computation of the high-resolution spectrum of water up to the first dissociation limit and beyond as well as for the determination of Stark coefficients for high-lying states. The water PES has been determined by combined ab initio and semi-empirical studies. As a first step, a very accurate, global, ab initio PES was determined using the all-electron, internally contracted multi-reference configuration interaction technique together with a large Gaussian basis set. Scalar relativistic energy corrections are also determined in order to move the energy determinations close to the relativistic complete basis set full configuration interaction limit. The electronic energies were computed for a set of about 2500 geometries, covering carefully selected configurations from equilibrium up to dissociation. Nuclear motion computations using this PES reproduce the observed energy levels up to 39 000 cm(-1) with an accuracy of better than 10 cm(-1). Line positions and widths of resonant states above dissociation show an agreement with experiment of about 50 cm(-1). An improved semi-empirical PES is produced by fitting the ab initio PES to accurate experimental data, resulting in greatly improved accuracy, with a maximum deviation of about 1 cm(-1) for all vibrational band origins. Theoretical results based on this semi-empirical surface are compared with experimental data for energies starting at 27 000 cm(-1), going all the way up to dissociation at about 41 000 cm(-1) and a few hundred wavenumbers beyond it.

Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the 7p elements, with atomic and molecular applications

Relativistic basis sets of double-zeta, triple-zeta, and quadruple-zeta quality have been optimized for the 7p elements at the Dirac-Coulomb self-consistent field level of theory with a Gaussian nuclear charge distribution. For all of these sets, valence and outer-core correlating functions have been optimized in multireference CI calculations on the valence pn states. Diffuse functions are also provided. Prescriptions are given for constructing contracted basis sets, based on MRCI calculations for correlation of the atoms. The basis sets are applied to a range of atomic and molecular properties, to provide information on how to use the basis sets. Tests of the basis sets with an explicit representation of the 8s showed that the 8s is not needed. The basis sets are available as an internet archive and from the Dirac program web site, http://dirac.chem.sdu.dk.

New determination of the adiabatic ionization potential of the BaOH radical from laser photoionization-molecular beam experiments and ab initio calculations

The adiabatic ionization potential of the BaOH radical, as generated in a laser vaporization-supersonic expansion source has been determined by laser photoionization experiments to be (4.55 ± 0.03) eV. This value supports the three lowest out of seven previous experimental estimates, the former ranging from 4.35 to 4.62 eV. The present result is compared to ab initio calculations, as performed using both quantum chemistry at different levels of theory and density functional theory, and trying several effective core potentials and their accompanying basis sets for Ba. The most satisfactory agreement is obtained for either the adiabatic or vertical ionization potentials that derive from post-Hartree-Fock [MP2 and CCSD(T)] treatments of electron correlation, along with consideration of relativistic effects and extensive basis sets for Ba, in both BaOH and BaOH(+). Such conclusions extend to the results of related calculations on the Ba-OH dissociation energies of BaOH and BaOH(+), which were performed to help in calibrating the present computational study. Bonding in BaOH/BaOH(+), as well as possible sources of discrepancy with previous experimental determinations of the BaOH adiabatic ionization potential are discussed.

Theoretical characterization of single-electron iodine-bond weak interactions in CH3…I-Y(Y = BH2, H, CH3, C2H3, C2H, CN, NC) systems

Iodine-involved single-electron halogen bonds (SEXBs) weak interactions in the systems of CH3…I-Y(Y = BH2, H, CH3, CH=CH2, C≡CH, CN, NC) were investigated for the first time using B3LYP/6-311++G(d,p) and MP2/aug-cc-pVTZ computational levels (the relativistic effective core potential basis set of Lanl2dz was used on iodine atom). The interaction energies between two moieties with basis set super-position error corrections for the seven complexes are −0.57, −1.36, −3.80, −2.17, −4.49, −6.33 and −8.64 kJ mol−1 (MP2/aug-cc-pVTZ), respectively, which shows that SEXBs interactions are all weak. Natural bond orbital theory analysis revealed that charges flow from CH3 to the I-Y moiety. The total amount of natural bond orbital charge transfer (Δ NC) from the CH3 radical to I-Y increases in the order CH3…IBH2 < CH3…IH ≈ CH3…ICH3 ≈ CH3…IC2H3 < CH3…ICCH < CH3…ICN < CH3…INC. Atoms-in-molecules theory was used to investigate the topological properties of the bond critical points in the seven SEXB structures.

O(3P) + CO2 Collisions at Hyperthermal Energies: Dynamics of Nonreactive Scattering, Oxygen Isotope Exchange, and Oxygen-Atom Abstraction

The dynamics of O((3)P) + CO(2) collisions at hyperthermal energies were investigated experimentally and theoretically. Crossed-molecular-beams experiments at <E(coll)> = 98.8 kcal mol(-1) were performed with isotopically labeled (12)C(18)O(2) to distinguish products of nonreactive scattering from those of reactive scattering. The following product channels were observed: elastic and inelastic scattering ((16)O((3)P) + (12)C(18)O(2)), isotope exchange ((18)O + (16)O(12)C(18)O), and oxygen-atom abstraction ((18)O(16)O + (12)C(18)O). Stationary points on the two lowest triplet potential energy surfaces of the O((3)P) + CO(2) system were characterized at the CCSD(T)/aug-cc-pVTZ level of theory and by means of W4 theory, which represents an approximation to the relativistic basis set limit, full-configuration-interaction (FCI) energy. The calculations predict a planar CO(3)(C(2v), (3)A'') intermediate that lies 16.3 kcal mol(-1) (W4 FCI excluding zero point energy) above reactants and is approached by a C(2v) transition state with energy 24.08 kcal mol(-1). Quasi-classical trajectory (QCT) calculations with collision energies in the range 23-150 kcal mol(-1) were performed at the B3LYP/6-311G(d) and BMK/6-311G(d) levels. Both reactive channels observed in the experiment were predicted by these calculations. In the isotope exchange reaction, the experimental center-of-mass (c.m.) angular distribution, T(θ(c.m.)), of the (16)O(12)C(18)O products peaked along the initial CO(2) direction (backward relative to the direction of the reagent O atoms), with a smaller isotropic component. The product translational energy distribution, P(E(T)), had a relatively low average of <E(T)> = 35 kcal mol(-1), indicating that the (16)O(12)C(18)O products were formed with substantial internal energy. The QCT calculations give c.m. P(E(T)) and T(θ(c.m.)) distributions and a relative product yield that agree qualitatively with the experimental results, and the trajectories indicate that exchange occurs through a short-lived CO(3)* intermediate. A low yield for the abstraction reaction was seen in both the experiment and the theory. Experimentally, a fast and weak (16)O(18)O product signal from an abstraction reaction was observed, which could only be detected in the forward direction. A small number of QCT trajectories leading to abstraction were observed to occur primarily via a transient CO(3) intermediate, albeit only at high collision energies (149 kcal mol(-1)). The oxygen isotope exchange mechanism for CO(2) in collisions with ground state O atoms is a newly discovered pathway through which oxygen isotopes may be cycled in the upper atmosphere, where O((3)P) atoms with hyperthermal translational energies can be generated by photodissociation of O(3) and O(2).

Performance of Effective Core Potentials for Density Functional Calculations on 3d Transition Metals.

The performance of popular Hartree-Fock-based effective core potentials in Hartree-Fock and density functional calculations of 3d transition metals has been evaluated by basis-set convergence studies for ten cases: the equilibrium bond dissociation energy (De) for dissociation of ground-state Ti2 to ground and excited atoms, the ground-state dissociation energies of FeO, Cu2, ScH, TiH, Sc2, Fe2, and TiV(+), and the first excitation energy (Ex) of Ti atom. Each case is studied with 11 or 13 density functionals. For comparison, the accuracy of the all-electron def2-TZVP basis set is tested with both relativistic and nonrelativistic treatments. Convergence and accuracy are assessed by comparing to relativistic all-electron calculations with a nearly complete relativistic basis set (NCBS-DK, which denotes the cc-pV5Z-DK basis set for 3d metals and hydrogen and the ma-cc-pV5Z-DK basis set for oxygen) and to nonrelativistic all-electron calculations with a nearly complete nonrelativistic basis set (NCBS-NR, which denotes the cc-pV5Z basis set for 3d metals and hydrogen and the ma-cc-pV5Z basis set for oxygen). As compared to NCBS-DK results, all ECP calculations perform worse than def2-TZVP all-electron relativistic calculations when averaged over all 130 data (13 functionals and ten test cases). The compact effective potential (CEP) relativistic effective core potential (RECP) combined with a valence basis set developed for the many-electron Dirac-Fock (MDF10) RECP performs best in effective core potential calculations and has an average basis-set incompleteness error of 3.7 kcal/mol, which is much larger than that (0.9 kcal/mol) of def2-TZVP relativistic all-electron results. Hence, the def2-TZVP relativistic all-electron calculations are recommended for accurate DFT calculations on 3d transition metals. In addition to our general findings, we observed that all kinds of density functionals do not show the same trends. For example, when ECPs are used with hybrid functionals, which sometimes are not recommended for calculations of transition metal systems, they are found to perform better at achieving the basis-set limit than when used with local functionals and meta-GGA functionals. The most successful combination of RECP and basis set has a basis-set incompleteness error of 1.7-2.4 kcal/mol for hybrid generalized gradient approximations, which is smaller than that of nonrelativistic NCBS calculations (whose average basis-set incompleteness error for hybrid functionals is 2.7-2.9 kcal/mol). The average basis-set incompleteness error in Hartree-Fock calculations is 1.0-4.4 kcal/mol for five of the ECP basis sets but is 5.8-10.8 kcal/mol for six others.

AB INITIO STUDY OF ELECTRONIC AND MAGNETIC STRUCTURE OF INTERMETALLIC RE5Ge3 COMPOUNDS

A series of binary rare-earth metal germanides RE 5 Ge 3 ( RE = La , Ce , Pr and Nd ) adopting the Mn 5 Si 3-type hexagonal structure is studied. These intermetallic phases show a complex magnetic behavior. Using a modification of the local density approximation (LSDA + U) the magnetic and electronic properties of these compounds are calculated. The spin-orbit coupling (SOC) was included using a full relativistic basis. Besides the structural parameters, bonding characters, total and partial densities of state and band structures are analyzed and compared with the experimental findings.

Aluminum Foils: The Contrasting Characters of Hyperconjugation and Steric Repulsion in Aluminum Dimetallocenes

The novel sandwich complex Cp2*Al2I2, which was recently synthesized by Minasian and Arnold, has been characterized using ab initio and density functional methods. A large family of related compounds was also investigated. Although a few Al(II)–Al(II) bonds are known, this is the first such bond to be supported by Cp-type ligands. In addition, in the remarkable Cp4*Al4 synthesis by Roesky, Cp2*Al2I2 is the Al(II) intermediate; Cp4*Al4 is important as a precursor to novel organoaluminum species. Halogen and ligand effects on the Al–Al bond in Cp2*Al2I2 were systematically explored by studying a series of 20 Cp2*Al2I2 derivatives using density functional theory with relativistic basis sets for the halogens. Comparison was made with the focal point treatment, which uses extrapolation to estimate the full configuration interaction and complete basis set limit energy. Torsional potential energy curves, natural population analyses, and enthalpies of hydrogenation were computed. Using the focal point approach, torsional barriers were computed with 0.05 kcal mol(–1) uncertainty. The interplay of steric and electronic effects on the torsional potential energy curves, enthalpies of dehydrogenation reactions, and geometries is discussed. In species with small ligands (R = H, Me), hyperconjugative effects determine the torsional landscape, whereas steric repulsions dominate in species with Cp* alkyl ligands. Species with Cp ligands represent an intermediate case, thus providing insight into how ligands modulate the structures and properties of small metal clusters.

A question of balance: Kinetic balance for electrons and positrons

The kinetic balance criterion used in current relativistic basis set codes is satisfied by the electron solutions of the Dirac equation, but not the positron solutions. A proposal for applying kinetic balance to both sets of solutions is presented. The method is applied along with “normal” kinetic balance to one-electron systems, to investigate its possible relation to prolapse, and to the positron affinity of F−, to investigate the kinetic energy deficiency for positron solutions. The new method reduces but does not eliminate prolapse for energy-optimized basis sets, and provides faster and smoother convergence with basis set size for the positron affinity.

Assessment of higher-order spin–orbit effects on electronic g-tensors of d 1 transition-metal complexes by relativistic two- and four-component methods

The electronic g-tensors of a series of V, Cr, Mo, W, Tc, and Re d1 transition-metal complexes have been studied systematically by density functional theory (DFT) methods. The comparison between one-component second-order perturbation theory calculations with two- and four-component first-order perturbation calculations has allowed an assessment of the importance of higher-order spin-orbit contributions. Using an efficient matrix Dirac–Kohn–Sham implementation with relativistic kinetic balance basis sets, it has been possible for the first time to apply four-component DFT also to g-tensors of larger models for biological vanadium, molybdenum, and tungsten metal sites. Higher-order spin–orbit effects are generally crucial for an accurate determination of the g-tensors in such complexes, in many cases more important than the choice of non-hybrid or hybrid density functional. A systematic scaling analysis of the spin–orbit integrals shows that second-order spin–orbit effects may be of the same size as the leading first-order effects and thus alter the computed g-tensors fundamentally, in particular for the 5d species. In the latter case, even third-order effects may be non-negligible.

QM and QM/MM Studies of Uranyl Fluorides in the Gas and Aqueous Phases and in the Hydrophobic Cavities of Tetrabrachion

The structural properties and electronic structures of pentacoordinated uranyl complexes belonging to the [UO(2)F(n)(H(2)O)(5-n)](2-n) series have been studied in the gas and aqueous phases using density functionals with relativistic pseudopotentials and all-electron basis sets in the gas-phase calculations in combination with COSMO in the aqueous phase. In addition, the conformational orientation and structural and electronic properties of [UO(2)F(5)](3-) in the hydrophobic cavities of the right-handed coiled-coil (RHCC) protein of tetrabrachion have been determined using the hybrid QM/MM method. Although there is good agreement between the available experimental geometrical parameters and the values obtained in the aqueous phase using pseudopotentials or all-electron basis sets, variation of the uranyl U═O bond with the number of fluoride ligands is only truly captured after the inclusion of five water molecules in the second coordination sphere around the molecules. The docking procedure used in this work shows that there are only two possible orientations of the uranyl group of [UO(2)F(5)](3-) embedded in the hydrophobic cavities of the RHCC protein. The two orientations are exclusively along the axes perpendicular to the protein axial channel with no possible orientation of the uranyl group along the axial channel because of both steric effects and interaction with the alkyl chain of the isoleucine residues pointing into the axial channel. In addition, the embedded complex is always positioned nearer to the isoleucine residues at the N-terminal ends of the hydrophobic cavities. Energy analysis, however, reveals that both conformations can only be observed in cavity 2, the largest hydrophobic cavity. The structural and electronic properties of the ligand embedded in this cavity are very similar to those of the gas-phase structure. A comparable study of [Pt(CN)(6)](2-) and the anticancer drug cisplatin, [PtCl(2)(NH(3))(2)], in cavity 2, revealed the existence of just two orientations for the former, similar to the uranyl complex, and multiple orientations for the latter.

A quantum chemical study of the structure and energy of β-diketonates. XVII. Internal rotation of radical substituents in β-diketonate molecules

A quantum chemical study of the internal rotation of CX3 (X = H, F, CH3) radical substituents in scandium and calcium β-diketonate complexes is performed. Calculations are carried out using the GAUSSIAN-03 program at the HF and DFT/B3LYP levels with relativistic effective core pseudo-potentials and valence double zeta basis sets. Analytical expressions V(φ) are obtained that describe the potential energy variation during the rotation of CX3 groups. The rotation of CH3 and CF3 groups is shown to be described by a simple potential of the form V = V0/2 + V3cos(3φ), while for the description of the rotation of tert-butyl groups it is required to use a more complex function V = V0/2 + V3cos(3φ) + V6cos(6φ). Based on the obtained expressions for V(φ), the effective torsion angles φef of the rotation of CX3 groups are calculated for different temperatures. It is shown that φef obtained based on the quantum chemical calculations are close to the values obtained in electron diffraction experiments.

Ab initio study of Au nIr ( n = 1–8) clusters

The density functional method PW91P86 with relativistic effective core potentials (RECP) and LANL2DZ basis set have been used to investigate the possible stable geometrical configurations and relative stabilities of the lowest-lying isomers of iridium-doped gold clusters, Au n Ir ( n = 1–8). Several low-lying isomers were determined. The results indicate that the ground-state Au n Ir, n = 1–8, clusters have planar structures except for n = 7. The stability trend of the Au n Ir clusters shows that even-numbered Au n Ir clusters are relatively more stable than the neighbouring odd-numbered ones, and indicating the Au 2Ir clusters are magic cluster with high chemical stability.

High Resolution Millimeter-Wave Spectroscopy of Vinyltellurol

The millimeter-wave rotational spectrum of vinyltellurol has been recorded and assigned for the first time. To support the spectrum assignment, high level ab initio calculations have been carried out. Geometries, total electronic energies, and harmonic vibrational frequencies have been determined at the MP2 level. A small-core relativistic pseudopotential basis set (cc-pVTZ-PP) was employed to describe the tellurium atom. Two stable conformers, synperiplanar (sp) and anticlinal (ac), have been identified. The sp conformer is planar with a small negative inertia defect of -0.025 u Å(2). The ac conformer was found to be nonplanar with a C-C-Te-H dihedral angle of about 140° from sp. This conformer exhibits a large amplitude motion associated with the torsion about the C-Te bond. The barrier to internal rotation is about 1 kJ/mol, according to the theoretical calculations. For the ac conformation, a torsional potential function consisting of quartic and quadratic terms of the torsional angle has been partially determined from the observed rotational constants.

Density functional study of Au nRh ( n=1–8) clusters

The density functional method PW91P86 with relativistic effective core potentials (RECPs) and LANL2DZ basis set has been used to investigate the possible stable geometrical configurations and relative stabilities of the lowest-lying isomers of rhodium-doped gold clusters, Au n Rh ( n=1–8). Several low-lying isomers were determined. The results indicate that the ground-state Au n Rh, n=1–8, clusters have planar structures except for n=5 and 8. The stability trend of the Au n Rh clusters shows that the clusters with even numbers of atoms are more stable than the clusters with odd numbers of atoms, indicating that the Au 5Rh cluster is a magic cluster with high chemical stability.

All-electron double zeta basis sets for platinum: Estimating scalar relativistic effects on platinum(II) anticancer drugs

Segmented all-electron contracted double zeta valence plus polarization function (DZP) basis sets for the element Pt were constructed for use in conjunction with the non-relativistic and Douglas–Kroll–Hess (DKH) Hamiltonians. The DZP–DKH set is loosely contracted and thus offer computational advantages compared to the generally contracted relativistic basis sets, while their sufficiently small size allows it to be used in place of effective core potentials (ECP) for routine studies of molecules. Using the one-parameter hybrid functional mPW1PW, the performance of the basis sets is assessed for predicting the molecular structures and atomic charges of platinum(II) antitumor drugs, cisplatin and carboplatin. These results can be used as reference values to calibrate further ECP calculations. Despite their compact size, the DZP sets demonstrate consistent, efficient, and reliable performance and will be especially useful in calculations of molecular properties that require explicit treatment of the core electrons.