NMR shielding constants in hydrogen molecule isotopomers

We analyze the NMR shielding constants in three isotopomers of the hydrogen molecule: H₂, HD and D₂. The results obtained within the Born?COppenheimer approximation using the coupled-cluster singles-and-doubles model are very close to the previous theoretical values. In particular, the isotope shifts computed using significantly larger basis sets agree with the earlier literature results, confirming the disagreement of these calculations with the available experimental data. To examine the accuracy of the computed isotope shifts, we analyze in addition the relativistic corrections and estimate the role of the adiabatic and nonadiabatic effects. The relativistic corrections appear to be negligible; on the other hand, the changes in the shielding constants due to the adiabatic and nonadiabatic effects may account for the discrepancies between the computed and experimental isotope shifts.     

Towards Accurate Predictions of Proton NMR Spectroscopic Parameters in Molecular Solids

The factors contributing to the accuracy of quantum-chemical calculations for the prediction of proton NMR chemical shifts in molecular solids are systematically investigated. Proton chemical shifts of six solid amino acids with hydrogen atoms in various bonding environments (CH, CH₂, CH₃, OH, SH and NH₃) were determined experimentally using ultra-fast magic-angle spinning and proton-detected 2D NMR experiments. The standard DFT method commonly used for the calculations of NMR parameters of solids is shown to provide chemical shifts that deviate from experiment by up to 1.5 ppm. The effects of the computational level (hybrid DFT functional, coupled-cluster calculation, inclusion of relativistic spin-orbit coupling) are thoroughly discussed. The effect of molecular dynamics and nuclear quantum effects are investigated using path-integral molecular dynamics (PIMD) simulations. It is demonstrated that the accuracy of the calculated proton chemical shifts is significantly better when these effects are included in the calculations.     

Accurate potential energy curves for the group 12 dimers Zn2, Cd2, and Hg2

Potential energy curves of the electronic ground states of the group 12 dimers Zn₂ and Cd₂ were computed at the CCSD(T) level of theory, including full triple corrections $\Updelta$ T in the coupled-cluster procedure, and spin-orbit (SO) contributions from four-component coupled-cluster calculations, extrapolated to the complete basis set (CBS) limit. For Hg₂, the potential energy curve published recently (Pahl et al. in J Chem Phys 132:114301, 2010] is complemented in this work by non-relativistic calculations to quantify and discuss relativistic effects. We obtain very accurate fits of our CBS/CCSD(T) and CBS/CCSD(T)+ $\Updelta$ T data points to an analytically simple and computationally efficient extended Lennard Jones form. For the CBS/CCSD(T)+ $\Updelta$ T+SO curves, we obtain dissociation energies of D _e?=?226?cm^?1 and D _e?=?319?cm^?1 for Zn₂ and Cd₂ respectively, in very good agreement with recent theoretical calculations and experimental data. We also present equilibrium distances and rotational and vibrational spectroscopic constants to compare with available theoretical and experimental data. The results obtained for non-relativistically treated Hg₂ continue nicely the trends with increasing atom number preset by Zn₂ and Cd₂, confirming that indeed, relativistic effects account for the known peculiarities for the mercury dimer.     

M?ssbauer spectroscopy for heavy elements: a relativistic benchmark study of mercury

The electrostatic contribution to the M?ssbauer isomer shift of mercury for the series HgF _n (n?=?1,?2,?4) with respect to the neutral atom has been investigated in the framework of four- and two-component relativistic theory. Replacing the integration of the electron density over the nuclear volume by the contact density (that is, the electron density at the nucleus) leads to a 10% overestimation of the isomer shift. The systematic nature of this error suggests that it can be incorporated into a correction factor, thus justifying the use of the contact density for the calculation of the M?ssbauer isomer shift. The performance of a large selection of density functionals for the calculation of contact densities has been assessed by comparing with finite-field four-component relativistic coupled-cluster with single and double and perturbative triple excitations [CCSD(T)] calculations. For the absolute contact density of the mercury atom, the Density Functional Theory (DFT) calculations are in error by about 0.5%, a result that must be judged against the observation that the change in contact density along the series HgF _n (n?=?1,?2,?4), relevant for the isomer shift, is on the order of 50?ppm with respect to absolute densities. Contrary to previous studies of the ⁵⁷Fe isomer shift (F Neese, Inorg Chim Acta 332:181, 2002), for mercury, DFT is not able to reproduce the trends in the isomer shift provided by reference data, in our case CCSD(T) calculations, notably the non-monotonous decrease in the contact density along the series HgF _n (n?=?1,?2,?4). Projection analysis shows the expected reduction of the 6s _1/2 population at the mercury center with an increasing number of ligands, but also brings into light an opposing effect, namely the increasing polarization of the 6s _1/2 orbital due to increasing effective charge of the mercury atom, which explains the non-monotonous behavior of the contact density along the series. The same analysis shows increasing covalent contributions to bonding along the series with the effective charge of the mercury atom reaching a maximum of around +2 for HgF₄ at the DFT level, far from the formal charge +4 suggested by the oxidation state of this recently observed species. Whereas the geometries for the linear HgF₂ and square-planar HgF₄ molecules were taken from previous computational studies, we optimized the equilibrium distance of HgF at the four-component Fock-space CCSD/aug-cc-pVQZ level, giving spectroscopic constants r _e = 2.007 ? and ?? _e = 513.5?cm^?1.     

Ab initio calculations of relativistic and electron correlation effects in polyatomics using the universal Gaussian basis set: XeF2

Ab initio accurate all-electron relativistic molecular orbital Dirac–Fock self-consistent field calculations are reported for the linear symmetric XeF₂ molecule at various internuclear distances with our recently developed relativistic universal Gaussian basis set. The nonrelativistic limit Hartree–Fock calculations were also performed for XeF₂ at various internuclear distances. The relativistic correction to the electronic energy of XeF₂ was calculated as ～ ?215 hartrees (?5850 eV) by using the Dirac–Fock method. The dominant magnetic part of the Breit interaction correction to the nonrelativistic interelectron Coulomb repulsion was included in our calculations by both the Dirac–Fock–Breit self-consistent field and perturbation methods. The calculated Breit correction is ～6.5 hartrees (177 eV) for XeF₂. The relativistic Dirac–Fock as well as the nonrelativistic HF wave functions predict XeF₂ to be unbound, due to neglect of electron correlation effects. These effects were incorporated for XeF₂ by using various ab initio post Hartree–Fock methods. The calculated dissociation energy obtained using the MP 2(full) method with our extensive basis set of 313 primitive Gaussians that included d and f polarization functions on Xe and F is 2.77 eV, whereas the experimental dissociation energy is 2.78 eV. The calculated correlation energy is ～ ?2 hartrees (?54 eV) at the predicted internuclear distance of 1.986 Å, which is in excellent agreement with the experimental Xe—F distance of 1.979 Å in XeF₂. In summary, electron correlation effects must be included in accurate ab initio calculations since it has been shown here that their inclusion is crucial for obtaining theoretical dissociation energy (D_e) close to experimental value for XeF₂. Furthermore, relativistic effects have been shown to make an extremely significant contribution to the total energy and orbital binding energies of XeF₂. © 1995 John Wiley & Sons, Inc.     

Pseudopotential study of the ground and excited states of Yb2

The ground state of the van der Waals-type lanthanide dimer Yb₂ has been studied by means of relativistic energy-consistent ab initio pseudopotentials using three different core definitions. Electron correlation was treated by coupled-cluster theory, whereby core-valence correlation effects have been accounted for either explicitly by correlating the energetically highest coreorbitals or implicitly by means of an effective core-polarization potential. Results for the first and second atomic ionization potentials, the atomic dipole polarizability, and the spectroscopic constants of the molecular ground state are reported. Low-lying excited states have been investigated with spin-orbit configuration interaction calculations. It is also demonstrated for the whole lanthanide series that correlation effects due to the atomic-like, possibly open 4f-shell in lanthanides can be modeled effectively by adding a core-polarization potential to pseudopotentials attributing the 4f-shell to the core. Received: 3 April 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Electron correlation and relativistic effects in xenon tetrafluoride

Ab initio all-electron fully relativistic Dirac–Fock self-consistent field and Dirac–Fock–Breit calculations are reported for the XeF₄ molecule at various internuclear distances assuming the experimental D_4h geometry with our recently developed relativistic universal Gaussian basis set. The nonrelativistic limit Hartree–Fock calculations were also performed for XeF₄ at various internuclear distances. The calculated relativistic correction to the total energy of molecule at the Dirac–Fock level is ～ ?5856 eV, whereas the magnetic part of the Breit correction to the electron-electron interaction is calculated as ～ 177 eV. The electron correlation effects were included in the nonrelativistic Hartree–Fock calculations using the second-order Møller-Plesset (MP 2) theory, and the calculated correlation energy for XeF₄ is ?71 eV. The basis-set superposition error (BSSE ) was estimated by using the counterpoise method for Xe and F. The inclusion of both the relativistic and electron correlation effects in the calculated total energies of F, Xe, and XeF₄ predicts the Xe—F bond length and dissociation energy of XeF₄ as 1.952 Å and 5.59 eV, respectively, which are in excellent agreement with the experimental values of 1.953 Å and 5.69 eV, respectively, for XeF₄. The contribution of the electron correlation and relativistic effects to the dissociation energy of XeF₄ is 8.11 and 0.05 eV, respectively. The Breit interaction, however, contributes only 0.02 eV to the dissociation energy of XeF₄. Electron correlation is most significant for the prediction of an accurate value of dissociation energy, whereas relativistic effects are very important for the prediction of spin-orbital splitting as well as the energies of the orbitals, especially the inner orbitals of XeF₄. © 1995 John Wiley & Sons, Inc.     

Calculations of nuclear quadrupole coupling in noble gas-noble metal fluorides: interplay of relativistic and electron correlation effects

The nuclear quadrupole coupling constants (NQCCs) of noble gas and noble metal nuclei in the recently found noble gas-noble metal fluorides (NgMF, where Ng=Ar,Kr,Xe and M=Cu,Ag,Au) are obtained theoretically by high-level ab initio calculations, where both relativistic and electron correlation effects are included, and compared to experimental results. Fully relativistic four-component Dirac-Hartree-Fock (DHF) calculations are carried out at the basis set limit for electric field gradient that couples with the electric quadrupole moment of the nucleus, and uncorrelated relativistic effects are extracted by comparing DHF results to nonrelativistic (NR) HF calculations. Electron correlation effects are investigated both at fully relativistic second-order Moller-Plesset (DMP2) and at NR MP2 levels of theory, as well as at the NR coupled-cluster singles and doubles with perturbational triples [CCSD(T)] level. The validity of the approximation where relativistic effects, on the one hand, and nonrelativistically obtained correlation effects, on the other hand, are evaluated separately and assumed to be additive, is investigated by comparison with the DMP2 results. Inclusion of relativistic effects is shown to be necessary for obtaining the correct NQCC trends as the nucleus of interest and/or its neighbors become heavier. Electron correlation treatment is needed for approaching quantitative agreement with the experimental NQCCs. The assumption of additive electron correlation and relativistic effects, corresponding to the NR correlation treatment added on top of relativistic DHF data, gives qualitatively correct noble gas NQCCs. For noble metal NQCCs, correlation treatment at the relativistic level of theory is mandatory for reaching agreement with experimental results. Current work also confirms the experimental trends of NQCCs, which have been taken as an indication of nearly covalent interaction between noble gas and noble metal in the heaviest present systems, especially in XeAuF.     

A challenge for density functional theory: the XONO and XNO2 (X=F, Cl, and Br) molecules

The equilibrium geometries, harmonic frequencies, dipole moments, infrared intensities, and relative energies of the cis-XONO, trans-XONO, and XNO₂ (X=F, Cl, and Br) have been investigated using four functionals in common use in Kohn-Sham density functional theory (DFT) calculations. Two of the functionals include non-local or gradient correction terms, while the other two also incorporate some exact Hartree-Fock exchange and are labeled hybrid functionals. The quality of the results obtained from the functionals is determined by comparison to previously published high-level coupled-cluster calculations. The hybrid functionals perform better for prediction of the equilibrium geometries, where the two gradient corrected functionals yield qualitatively incorrect molecular structures for cis-FONO and cis-ClONO. None of the functionals perform well in predicting all six harmonic frequencies, showing that the correlation between equilibrium geometries and harmonic frequencies is not as strong for these DFT methods as it is for conventional wavefunction ab initio methods, such as coupled-cluster theory. Results from the various functionals generally come into better agreement with each other and also with the coupled-cluster results moving down the periodic table. Received: 12 February 1997 / Accepted: 25 March 1997     

Thirty years of relativistic self-consistent field theory for molecules: relativistic and electron correlation effects for atomic and molecular systems of transactinide superheavy elements up to ekaplutonium E126 with g-atomic spinors in the ground state configuration

Relativistic Hartree–Fock–Roothaan (RHFR) self-consistent field theory for molecules developed by Malli and Oreg (J Chem Phys 63, 830, 1975) is reviewed. Ab initio all-electron fully relativistic Dirac–Fock and the corresponding nonrelativistic Hartree–Fock calculations for a number of molecular systems of heavy and superheavy elements are discussed in order to asecrtain relativistic effects. It is pointed out for the first time that there are dramatic antibinding effects of relativity for diatomics of the superheavy elements ekagold and ekaastatine. These are first results of antibinding effects of relativity in relativistic quantum chemistry. Moreover, in order to take into account the relativistic and electron correlation effects simultaneously for these systems, relativistic Moeller Plesset second order (RMP2), coupled-cluster singles doubles (RCCSD) and RCCSD with inclusion of triple corrections perturbationally (RCCSD(T)) calculations performed by the author for a number of atomic and molecular systems of superheavy elements (SHE) including the primordial SHE ekaplutonium E126 (Z = 126) (with g atomic spinors occupied in the ground state atomic configuration) are reported. Such calculations and results have not been reported before for systems of superheavy elements. Contribution to the Serafin Fraga Memorial Issue. An erratum to this article can be found at     

Relativistic calculations of AuSi+ and AuSi−

Theoretical calculations of the electronic structure of the ground state and a series of excited states of the AuSi⁺ and AuSi⁻ molecules are presented. The calculations were carried out with the spin-free relativistic infinite-order two-component (IOTC) method and high-level complete active space self-consistent field/complete active space perturbation theory correlated methods. The spin-orbit (SO) coupling was introduced via the restricted active space state interaction method with the use of the atomic mean-field SO integrals. The work presents the spectroscopic parameters of calculated states and full potential energy curves of the ionic AuSi⁺ and AuSi⁻ structures for the first time. Electrostatic potential maps projected on the electron density surface illustrate the significant relativistic effects on going from nonrelativistic to scalar relativistic treatments.     

On the application of the incremental scheme to ionic solids: test of different embeddings

Within the application of the incremental scheme to cerium dioxide high-level quantum chemical calculations, using the coupled-cluster approach, have been performed for (Ce⁴⁺)_m(O²⁻)_n clusters. Two different approaches were considered. In the first one all increments were obtained from the nearly neutral Ce₄O₇-cluster. In the second approach different clusters were used for the evaluation of increments. Several embeddings were tested: from purely point charges up to pseudopotential-surrounding of oxygens to imitate the Ce-ions. The advantages and disadvantages of applied embedding schemes were discussed.     

Quantum chemical rovibrational analysis of aminoborane and its isotopologues

Aminoborane, H₂NBH₂ and its isotopologues, H₂N¹⁰BH₂, D₂NBD₂, and D₂N¹⁰BD₂, have been studied by high-level ab initio methods. All calculations rely on multidimensional potential energy surfaces and dipole moment surfaces including high-order mode coupling terms, which have been obtained from electronic structure calculations at the level of explicitly correlated coupled-cluster theory, CCSD(T)-F12, or the distinguishable cluster approximation, DCSD. Subsequent vibrational structure calculations based on second-order vibrational perturbation theory, VPT2, and vibrational configuration interaction theory, VCI, were used to determine rotational constants, centrifugal distortion constants, vibrationally averaged geometrical parameters and (an)harmonic vibrational frequencies. The impact of core-correlation effects is discussed in detail. Rovibrational VCI calculations were used to simulate the gas phase spectra of these species and an in-depth analysis of the ν₇ band of aminoborane is provided. Color-coding is used to reveal the identity of the individual progressions of the rovibrational transitions for this particular mode.     

The ground-state potential energy function of a beryllium dimer determined using the single-reference coupled-cluster approach

The accurate ground-state potential energy function of the beryllium dimer, Be(2), has been determined from large-scale ab initio calculations using the single-reference coupled-cluster approach in conjunction with the correlation-consistent core-valence basis sets up to septuple-zeta quality. Results obtained with the conventional and explicitly-correlated coupled-cluster methods were compared. The scalar relativistic and adiabatic (the diagonal correction) effects were also discussed. The vibration-rotation energy levels of Be(2) were predicted and found to be as accurate as those determined from the empirical potential energy function [J. M. Merritt et al., Science, 2009, 324, 1548]. The potential energy function of Be(2) was determined in this study to have a minimum at 2.444 ? and the well depth of 935 cm(-1).     

The Optical Spectrum of Au2+

The electronic structure of the Au₂⁺ cation is essential for understanding its catalytic activity. We present the optical spectrum of mass-selected Au₂⁺ measured via photodissociation spectroscopy. Two vibrationally resolved band systems are observed in the 290–450 nm range (at ca. 440 and ca. 325 nm), which both exhibit rather irregular structure indicative of strong vibronic and spin-orbit coupling. The experimental spectra are compared to high-level quantum-chemical calculations at the CASSCF-MRCI level including spin-orbit coupling. The results demonstrate that the understanding of the electronic structure of this simple, seemingly H₂⁺-like diatomic molecular ion strictly requires multireference and relativistic treatment including spin-orbit effects. The calculations reveal that multiple electronic states contribute to each respective band system. It is shown that popular DFT methods completely fail to describe the complex vibronic pattern of this fundamental diatomic cation.     

Calculations of electronic structure of the UF<Subscript>6</Subscript> molecule and the UO<Subscript>2</Subscript> crystal with a relativistic pseudopotential

The influence of relativistic effects on the properties of uranium hexafluoride was considered. Detailed comparison of the spectrum of one-electron energies obtained in the nonrelativistic (by the Hartree-Fock method), relativistic (by the Dirac-Fock method), and scalar-relativistic (using a relativistic potential of the uranium atom core) calculations was carried out. The methods of optimization of atomic basis in the LCAO calculations of molecules and crystals are discussed which make it possible to consider distortion of atomic orbitals upon the formation chemical bonds. The influence of the atomic basis optimization on the results of scalar-relativistic calculations of the molecule UF₆ properties is analyzed. Calculations of the electronic structure and properties of UO₂ crystals with relativistic and nonrelativistic pseudopotentials are fulfilled.     

Perturbative treatment of scalar-relativistic effects in coupled-cluster calculations of equilibrium geometries and harmonic vibrational frequencies using analytic second-derivative techniques

An analytic scheme for the computation of scalar-relativistic corrections to nuclear forces is presented. Relativistic corrections are included via a perturbative treatment involving the mass-velocity and the one-electron and two-electron Darwin terms. Such a scheme requires mixed second derivatives of the nonrelativistic energy with respect to the relativistic perturbation and the nuclear coordinates and can be implemented using available second-derivative techniques. Our implementation for Hartree-Fock self-consistent field, second-order Moller-Plesset perturbation theory, as well as the coupled-cluster level is used to investigate the relativistic effects on the geometrical parameters and harmonic vibrational frequencies for a set of molecules containing light elements (HX, X=F, Cl, Br; H2X, X=O, S; HXY, X=O, S and Y=F, Cl, Br). The focus of our calculations is the basis-set dependence of the corresponding relativistic effects, additivity of electron correlation and relativistic effects, and the importance of core correlation on relativistic effects.