The well-tempered model core potentials for the main-group elements Li–Rn

The well-tempered model core potential (wtMCP) parameters and valence basis sets, based on the well-tempered basis set expansion, were developed for the main-group elements Li–Rn. For the s–block elements, the valence space comprises the ns valence shell and the outermost core (n–1)p shell. For the p-block elements, the ns and np shells together with the (n–1)d shell make up the valence space. Nonrelativistic wtMCPs were developed for all atoms. Scalar-relativistic wtMCPs were prepared for all atoms heavier than Ar by using the relativistic elimination of small components to obtain the reference and core orbitals. The new potentials were tested at the restricted Hartree–Fock, second-order Mø øller–Plesset perturbation theory and density functional theory with Beckes three-parameter hybrid functional combined with Perdews 1991 gradient-corrected correlation functional levels for several diatomic molecules and the results were compared with those obtained from all-electron calculations and experimental values. Excellent agreement between the results was obtained.     

Theoretical calculation of the photodetachment spectra of XAuY- (X, Y = Cl, Br, and I)

The photodetachment spectra of the title molecules have been calculated, taking electron correlation and spin-orbit coupling into account and employing improved relativistic effective core potentials for gold and the halogen atoms. The calculated spectra have been compared with existing experimental spectra. The spin-orbit splitting of several degenerate electronic states has been calculated. The composition of the spin-orbit eigenstates are analyzed in terms of scalar relativistic electronic states. A comparison of the relative position of peaks in the calculated photodetachment spectra of the title molecules has been made.     

Electron localizability indicators from spinor wavefunctions

For the fully relativistic 4‐component many‐electron wavefunction six flavors of electron localizability indicators (ELI) have been proposed. Their counterparts, suitable for the application to the 2‐component wavefunctions, have been also derived. Six proposed indicators have been tested on Ar and Rn atoms and one of them, the ELI‐D for spatially antisymmetrized electron pairs, has been found to reveal atomic shell structures at quantitative level. Shell structures of all the atoms of periods 4–7 of the periodic table have been obtained using this indicator and compared with these obtained from the nonrelativistic limit calculations as well as from scalar‐relativistic (zero‐order regular approximation) calculations. © 2014 Wiley Periodicals, Inc.     

Relativistic multiconfiguration Hartree–SFock by means of direct perturbation theory

A relativistic direct perturbation theory approach has been implemented at the multiconfiguration Hartree–Fock level into the numerical program package LUCAS. The method has been applied to the closed-shell Be, Zn, Cd, and Hg atoms and to the rare gases Ne to Rn. The scalar relativistic valence-correlation correction to the rare gases is found to be very small, while for Zn, Cd, and Hg the first-order relativistic corrections to the valence-correlation energy are calculated to be −4.6, −6.3, and −17.4 mH, respectively. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 151–158, 1997     

Accurate ab initio pair potentials between helium and the heavier group 2 elements

Interactions between the heavier Group 2 metals (Ca, Sr, and Ba) and helium were studied using the well-tempered model core potential method. Accurate pair potentials, calculated at the coupled-cluster level of theory with very large basis sets, were used in bound state calculations. Three bound rovibrational states were found for each complex. The pair-potential parameters were used to predict how each of the metal atoms would be solvated by a helium nanodroplet. The Ca atom is not fully solvated by the droplet and the interaction between the helium and the metal decreases from Ca to Ba. This agrees with the experimental observation that the spectra of these atoms in a nanodroplet are intermediate between the spectra of the free atoms and the spectra in liquid helium.     

Microwave spectra of the Xe-N2 van der Waals complex: a comparison of experiment and theory

Rotational transitions for the Xe-N2 complex were measured in the frequency region from 4 to 18 GHz using a pulsed-nozzle Fourier-transform microwave spectrometer. Twelve (four) a-type transitions were recorded for the 132Xe-14N2 and 129Xe-14N2 (131Xe-15N)) isotopomers. In addition, the nuclear quadrupole hyperfine structures due to the presence of the 14N (nuclear-spin quantum number I=1) and 131Xe (I=32) nuclei were detected and analyzed. Two ab initio potential-energy surfaces were calculated at the coupled-cluster level of theory with single, double, and pertubatively included triple excitations. Dunning's augmented correlation-consistent polarized valence triple-zeta basis set was used for the nitrogen atoms. For the first surface, a well-tempered basis set with additional polarization functions was used for the Xe atom; for the second surface, a newly developed augmented correlation-consistent polarized valence quintuple-zeta basis set employing small-core relativistic pseudopotentials was used for the Xe atom. The basis sets were supplemented with bond functions for the van der Waals bond. The counterpoise correction was applied to reduce the basis-set superposition error. The resulting two surfaces both have a single minimum at a T-shaped geometry, with well depths of 122.4 and 119.3 cm(-1), respectively. Bound-state energies supported by the potential-energy surface were determined. The quality of the ab initio potential-energy surfaces was evaluated by comparison of the experimental transition frequencies and rotational and centrifugal distortion constants with those derived from the bound-state energies. A scaled potential-energy surface was obtained which has excellent agreement with the experimental data.     

Regulatory Mechanism of the Enantioselective Intramolecular Enone [2+2] Photocycloaddition Reaction Mediated by a Chiral Lewis Acid Catalyst Containing Heavy Atoms

The asymmetric catalysis of the intramolecular enone [2+2] photocycloaddition has been subject of extensive experimental studies, however theoretical insight to its regulatory mechanism is still sparse. Accurate quantum chemical calculations at the CASPT2//CASSCF level of theory associated with energy‐consistent relativistic pseudopotentials provide a basis for the first regulation theory that the enantioselective reaction is predominantly controlled by the presence of relativistic effects, that is, spin–orbit coupling resulting from heavy atoms in the chiral Lewis acid catalyst.     

Scalar relativistic calculations of hyperfine coupling tensors using the Douglas-Kroll-Hess method with a finite-size nucleus model

A scalar relativistic method to calculate hyperfine coupling tensors at the Douglas-Kroll-Hess level has been extended to incorporate a finite-size nucleus model using a Gaussian charge and magnetic moment distribution. Density functional calculations at gradient-corrected and hybrid functional levels have been carried out for the group 11 atoms and for a set of small group 12 molecules, comparing nonrelativistic as well as scalar relativistic results at second-order Douglas-Kroll-Hess level with and without finite-size nucleus. While nonrelativistic calculations underestimate isotropic hyperfine couplings increasingly with increasing nuclear charge, scalar relativistic calculations with point nucleus provide somewhat overestimated values. Inclusion of the finite-size nuclear model in the calculation of the wavefunction, and in the transformed hyperfine operators both decrease the magnitude of the hyperfine couplings. The effects, which are cumulative, improve agreement with experiment.     

Solvent effects on heavy atom nuclear spin-spin coupling constants: a theoretical study of Hg-C and Pt-P couplings.

The computation of indirect nuclear spin-spin coupling constants, based on the relativistic two-component zeroth order regular approximate Hamiltonian, has been recently implemented by us into the Amsterdam Density Functional program. Applications of the code for the calculation of one-bond metal-ligand couplings of coordinatively unsaturated compounds containing (195)Pt and (199)Hg, including spin-orbit coupling or coordination effects by solvent molecules, show that relativistic density functional calculations are able to reproduce the experimental findings with good accuracy for the systems under investigation. Spin-orbit effects are rather small for these cases, while coordination of the heavy atoms by solvent molecules has a great impact on the calculated couplings. Experimental trends for different solvents are reproduced. An orbital-based analysis of the solvent effect is presented. The scalar relativistic increase of the coupling constants is of the same order of magnitude as the nonrelativistically obtained values, making a relativistic treatment essential for obtaining quantitatively correct results. Solvent effects can be of similar importance.     

Long-range relativistic heavy atom effect on 1H NMR chemical shifts of selenium- and tellurium-containing compounds

Previously unknown manifestation of heavy atom effect on the NMR chemical shifts of β- and γ-protons initiated by the relativistic effects of the tellurium and selenium atoms has been investigated in the representative series of selenium- and tellurium-containing compounds. To approve the four-component density functional approach to be the appropriate tool for the investigation of the heavy atom on light atom effect (HALA), the benchmark calculations of the proton chemical shifts have been performed at the CCSD level using comprehensively chosen locally dense basis set with taking into account solvent, vibrational, and relativistic corrections. A good agreement with the experimental data was achieved. The magnitudes of the relativistic HALA corrections to β- and γ-proton chemical shifts were found to vary in a wide range, namely from −3.08 ppm for the γ-proton of methyltelluraldehyde to 14.51 ppm for β-proton in benzotelluraldehyde.     

A new method to generate spin-orbit coupled potential energy surfaces: effective relativistic coupling by asymptotic representation

A new method has been developed to generate fully coupled potential energy surfaces including derivative and spin-orbit coupling. The method is based on an asymptotic (atomic) representation of the molecular fine structure states and a corresponding diabatization. The effective relativistic coupling is described by a constant spin-orbit coupling matrix and the geometry dependence of the coupling is accounted for by the diabatization. This approach is very efficient, particularly for certain systems containing a very heavy atom, and yields consistent results throughout nuclear configuration space. A first application to a diatomic system is presented as proof of principle and is compared to accurate ab initio calculations. However, the method is widely applicable to general polyatomic systems in full dimensionality, containing several relativistic atoms and treating higher order relativistic couplings as well.     

Core–Valence correlations and relativistic effects in heavy atoms for molecular applications

Theoretical core effective potential methods are widely used in valence-only electron molecular calculations. These methods, which imply the frozen-core approximation, work well for the elements of the righthand side of the periodic table but are often unrealistic for metallic elements with highly polarizable cores. For these atoms one has to consider the polarization of the cores under the influence of the electric field created by the valence electrons. Moreover, relativistic corrections must be added for heavy atoms. Various theoretical approaches of core–valence interactions (polarization and core–valence correlations) will be reviewed, with a special emphasis on practical methods of calculation. The problem of handling the relativistic effects will mainly be discussed within the two-component Pauli formalism. It will be shown that the Foldy–Wouthuysen transformation is not the unique way for deriving relativistic corrections and that the second-order Dirac equation also provides a good starting point for obtaining relativistic corrections. Analytical exact results are given for the hydrogen atom. The accuracy of this approach is tested on many-electron atoms and molecules. It is finally shown that the problem of the core-valence separation is relevant to the general methodology of effective Hamiltonians that seems to provide the best promising way for filling the gap between the semiempirical and purely theoretical ab initio methods.     

First-order relativistic corrections to MP2 energy from standard gradient codes: Comparison with results from density functional theory

The evaluation of the first-order scalar relativistic corrections to MP2 energy based on either direct perturbation theory or the mass–velocity and Darwin terms is discussed. In a basis set of Lévy-Leblond spinors the one- and two-electron matrix elements of the relativistic Hamiltonian can be decomposed into a nonrelativistic part and a relativistic perturbation. Thus, a program capable of calculating nonrelativistic energy gradients can be used to calculate the cross-term between relativity and correlation. The method has been applied to selected closed-shell atoms (He, Be, Ne, and Ar) and molecules (CuH, AgH, and AuH). The calculated equilibrium distances and harmonic frequencies were compared with results from first-order relativistic density functional calculations. It was found that the cross-term is not the origin of the nonadditivity of relativistic and correlation effects. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1596–1603, 1998     

Near-Dirac–Hartree–Fock results for first-row atoms calculated with GTO basis sets

Relativistic basis sets for first-row atoms have been constructed by using the near-Hartree–Fock (nonrelativistic) eigenvectors calculated by Partridge. These bases generate results of near-Dirac–Hartree–Fock quality. Relativistic total and orbital energies, relativistic corrections to the total energy, and magnetic interaction energies for the first-row atoms have been presented. The smallest Gaussian expansions (13s8 p expansions) yield Dirac–Hartree–Fock total energies accurate through six significant digits, while the largest expansions (18s13p expansions) give these energies accurate through seven significant digits. These results are more accurate than some of the results reported earlier, particularly for the open-shell atoms, indicating that the basis employed is reasonably economical for relativistic calculations. © 1995 John Wiley & Sons, Inc.     

Relativistic all-electron molecular Hartree-Fock-Dirac-(Breit) calculations on CH4, SiH4, GeH4, SnH4, PbH4

Summary Results and details of molecular Fock-Dirac-(Breit) calculations on CH₄, SiH₄, GeH₄, SnH₄, and PbH₄ obtained with the MOLFDIR© program package are presented and compared with other calculations and experimental results. The relativistic ground state energies (including the Breit interaction) of the atoms C, Si, Ge, Sn, and Pb, necessary for reference purposes, have been calculated using a small relativistic CI. One of our findings is that for the heavier systems perturbation theory over-estimates the relativistic bond length contraction. The Breit interaction has only a small effect on the bond lengths.     

Molecular QTAIM Topology Is Sensitive to Relativistic Corrections

The topology of the molecular electron density of benzene dithiol gold cluster complex Au₄−S−C₆H₄−S′−Au′₄ changed when relativistic corrections were made and the structure was close to a minimum of the Born–Oppenheimer energy surface. Specifically, new bond paths between hydrogen atoms on the benzene ring and gold atoms appeared, indicating that there is a favorable interaction between these atoms at the relativistic level. This is consistent with the observation that gold becomes a better electron acceptor when relativistic corrections are applied. In addition to relativistic effects, here, we establish the sensitivity of molecular topology to basis sets and convergence thresholds for geometry optimization.     

Optimization of Gaussian basis sets for Dirac-Hartree-Fock calculations

We investigate the optimization of Gaussian basis sets for relativistic calculations within the framework of the restricted Dirac-Hartree-Fock (DHF) method for atoms. We compare results for Rn of nonrelativistic and relativistic basis set optimizations with a finite nuclear-size. Optimization of separate sets for each spin-orbit component shows that the basis set demands for the lower j component are greater than for the higher j component. In particular, the p _1/2 set requires almost as many functions as the s _1/2 set. This implies that for the development of basis sets for heavy atoms, the symmetry type for which a given number of functions is selected should be based on j, not on l, as has been the case in most molecular calculations performed to date.     

Relativistic free complement method for correctly solving the Dirac equation with the applications to hydrogen isoelectronic atoms

The accuracy of the relativistic free complement (FC) method, which was previously reported for solving the Dirac?CCoulomb equations of atoms and molecules, has been strictly examined with the applications to hydrogen isoelectronic atoms. The FC wave function grown up by the Hamiltonian automatically takes care of the correct relationship between large and small components, i.e., FC or ICI balance. Combining the FC method with the inverse Hamiltonian method can help to obtain correct solutions safely against to several obstacles characteristic to the Dirac?CCoulomb equation. To ensure the exactness of the obtained wave function, we examined the total square deviation from the exact wave function, local energy constancy, H-square error, and energy upper and lower bounds for hydrogen-like atoms.