Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (II): The restricted open‐shell approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent two‐component relativistic calculations, termed the Kramers‐restricted open‐shell HF (KROHF) method, is developed. The present KROHF method is defined as a relativistic analogue of ROHF using time‐reversal symmetry and quaternion algebra, based on the Kramers‐unrestricted HF (KUHF) theory reported in our previous study (Int. J. Quantum Chem., doi: 10.1002/qua.25356 ). As seen in the nonrelativistic ROHF theory, the ambiguity of the KROHF Fock operator gives physically meaningless spinor energies. To avoid this problem, the canonical parametrization of KROHF to satisfy Koopmans' theorem is also discussed based on the procedure proposed by Plakhutin et al. (J. Chem. Phys. 2006 , 125, 204110). Numerical assessments confirmed that KROHF using Plakhutin's canonicalization procedure correctly gives physical spinor energies within the frozen‐orbital approximation under spin–orbit interactions.     

Spin‐orbit ZORA and four‐component Dirac–Coulomb estimation of relativistic corrections to isotropic nuclear shieldings and chemical shifts of noble gas dimers

Hartree–Fock and density functional theory with the hybrid B3LYP and general gradient KT2 exchange‐correlation functionals were used for nonrelativistic and relativistic nuclear magnetic shielding calculations of helium, neon, argon, krypton, and xenon dimers and free atoms. Relativistic corrections were calculated with the scalar and spin‐orbit zeroth‐order regular approximation Hamiltonian in combination with the large Slater‐type basis set QZ4P as well as with the four‐component Dirac–Coulomb Hamiltonian using Dyall's acv4z basis sets. The relativistic corrections to the nuclear magnetic shieldings and chemical shifts are combined with nonrelativistic coupled cluster singles and doubles with noniterative triple excitations [CCSD(T)] calculations using the very large polarization‐consistent basis sets aug‐pcSseg‐4 for He, Ne and Ar, aug‐pcSseg‐3 for Kr, and the AQZP basis set for Xe. For the dimers also, zero‐point vibrational (ZPV) corrections are obtained at the CCSD(T) level with the same basis sets were added. Best estimates of the dimer chemical shifts are generated from these nuclear magnetic shieldings and the relative importance of electron correlation, ZPV, and relativistic corrections for the shieldings and chemical shifts is analyzed. © 2015 Wiley Periodicals, Inc.     

Full four‐component relativistic calculations of the one‐bond 77Se–13C spin‐spin coupling constants in the series of selenium heterocycles and their parent open‐chain selenides

Four‐component relativistic calculations of ⁷⁷Se–¹³C spin–spin coupling constants have been performed in the series of selenium heterocycles and their parent open‐chain selenides. It has been found that relativistic effects play an essential role in the selenium–carbon coupling mechanism and could result in a contribution of as much as 15–25% of the total values of the one‐bond selenium–carbon spin‐spin coupling constants. In the overall contribution of the relativistic effects to the total values of ¹J(Se,C), the scalar relativistic corrections (negative in sign) by far dominate over the spin‐orbit ones (positive in sign), the latter being of less than 5%, as compared to the former (ca 20%). A combination of nonrelativistic second‐order polarization propagator approach (CC2) with the four‐component relativistic density functional theory scheme is recommended as a versatile tool for the calculation of ¹J(Se,C). Solvent effects in the values of ¹J(Se,C) calculated within the polarizable continuum model for the solvents with different dielectric constants (ε 2.2–78.4) are next to negligible decreasing negative ¹J(Se,C) in absolute value by only about 1 Hz. The use of the locally dense basis set approach applied herewith for the calculation of ⁷⁷Se–¹³C spin‐spin coupling constants is fully justified resulting in a dramatic decrease in computational cost with only 0.1–0.2‐Hz loss of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.     

Kramers' unrestricted Hartree–Fock and second-order Møller–Plesset perturbation methods using relativistic effective core potentials with spin–orbit operators: Test calculations for HI and CH3I

The Kramers' restricted Hartree–Fock (KRHF) and second-order Møller–Plesset perturbation (KRMP2) methods using relativistic effective core potentials (RECP) with spin–orbit operators and two-component spinors are extended to the unrestricted forms, KUHF and KUMP2. As in the conventional unrestricted methods, the KUHF and KUMP2 methods are capable of qualitatively describing the bond breaking for a single bond. As a result, it is possible to estimate spin–orbit effects along the dissociation curve at the HF and MP2 levels of theory as is demonstrated by the test calculations on the ground states of HI and CH₃I. Since the energy lowering due to spin–orbit interactions is larger for the I atom than for the closed-shell molecules, dissociation energies are reduced and bond lengths are slightly elongated by the inclusion of the spin–orbit interactions. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 91–98, 1998     

Relativistic and electron‐correlation effects on magnetizabilities investigated by the Douglas‐Kroll‐Hess method and the second‐order Møller‐Plesset perturbation theory

Isotropic and anisotropic magnetizabilities for noble gas atoms and a series of singlet and triplet molecules were calculated using the second‐order Douglas‐Kroll‐Hess (DKH2) Hamiltonian containing the vector potential A and in part using second‐order generalized unrestricted Møller‐Plesset (GUMP2) theory. The DKH2 Hamiltonian was resolved into three parts (spin‐free terms, spin‐dependent terms, and magnetic perturbation terms), and the magnetizabilities were decomposed into diamagnetic and paramagnetic terms to investigate the relativistic and electron‐correlation effects in detail. For Ne, Kr, and Xe, the calculated magnetizabilities approached the experimental values, once relativistic and electron‐correlation effects were included. For the IF molecule, the magnetizability was strongly affected by the spin‐orbit interaction, and the total relativistic contribution amounted to 22%. For group 17, 16, 15, and 14 hydrides, the calculated relativistic effects were small (less than 3%), and trends were observed in relativistic and electron‐correlation effects across groups and periods. The magnetizability anisotropies of triplet molecules were generally larger than those of similar singlet molecules. The so‐called relativistic‐correlation interference for the magnetizabilities computed using the relativistic GUMP2 method can be neglected for the molecules evaluated, with exception of triplet SbH. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Formulation of unrestricted and restricted Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) method, dealing with Bogoliubov orbitals, which consist of particle and hole part, can provide states with pair correlations associated with Cooper pairs. The dimension of HFB Fock matrices can be reduced by restrictions of spin states of Bogoliubov orbitals similarly to ordinary Hartree–Fock (HF) equations such as restricted HF (RHF), unrestricted HF (UHF), and generalized HF (GHF). However, there are few studies of moderate restricted HFB equations such as UHF‐based HFB equations. In this article, formulation and calculations of restricted HFB equations are described. The solutions of general and restricted HFB equations are compared. Pair correlations taking account of restricted and general HFB equations are discussed. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Static and Frequency‐Dependent Dipole–Dipole Polarizabilities of All Closed‐Shell Atoms up to Radium: A Four‐Component Relativistic DFT Study

We test the performance of four‐component relativistic density functional theory by calculating the static and frequency‐dependent electric dipole–dipole polarizabilities of all (ground‐state) closed‐shell atoms up to Ra. We consider 12 nonrelativistic functionals, including three asymptotically shape‐corrected functionals, by using two smooth interpolation schemes introduced by the Baerends group: the gradient‐regulated asymptotic connection (GRAC) procedure and the statistical averaging of (model) orbital potentials (SAOP). Basis sets of doubly augmented triple‐zeta quality are used. The results are compared to experimental data or to accurate ab initio results. The reference static electric dipole polarizability of palladium has been obtained by finite‐field calculations using the coupled‐cluster singles, doubles, and perturbative triples method within this work. The best overall performance is obtained using hybrid functionals and their GRAC shape‐corrected versions. The performance of SAOP is among the best for nonhybrid functionals for Group 18 atoms but its precision degrades when considering the full set of atoms. In general, we find that conclusions based on results obtained for the rare‐gas atoms are not necessarily representative of the complete set of atoms. GRAC cannot be used with effective core potentials since the asymptotic correction is switched on in the core region.     

GRECP/5e‐MRD‐CI calculation of the electronic structure of PbH

The correlation calculation of the electronic structure of PbH is carried out with the generalized relativistic effective core potential (GRECP) and multireference single‐ and double‐excitation configuration interaction (MRD‐CI) methods. The 22‐electron GRECP for Pb is used and the outer core 5s, 5p, and 5d pseudospinors are frozen using the level‐shift technique, so only five external electrons of PbH are correlated. A new configuration selection scheme with respect to the relativistic multireference states is employed in the framework of the MRD‐CI method. The [6, 4, 3, 2] correlation spin–orbit basis set is optimized in the coupled cluster calculations on the Pb atom using a recently proposed procedure, in which functions in the spin–orbital basis set are generated from calculations of different ionic states of the Pb atom and those functions are considered optimal that provide the stationary point for some energy functional. Spectroscopic constants for the two lowest‐lying electronic states of PbH (²Π_1/2, ²Π_3/2) are found to be in good agreement with the experimental data. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Ab initio quasirelativistic calculations on electronic transitions in ICl by the multireference many‐body perturbation theory

A quasirelativistic perturbative method of ab initio calculations on ground and excited molecular electronic states and transition properties within the relativistic effective core potential approximation is presented and discussed. The method is based on the construction of a state‐selective many‐electron effective Hamiltonian in the model space spanned by an appropriate set of Slater determinants by means of the second‐order many‐body multireference perturbation theory. The neglect of effective spin–orbit interactions outside of the model space allows the exploitation of relatively high nonrelativistic symmetry during the evaluation of perturbative corrections and therefore dramatic reduction of the cost of computations without any contraction of the model‐space functions. One‐electron transition properties are evaluated via the perturbative construction of spin‐free transition density matrices. Illustrative calculations on the X0⁺ ? A1, B0⁺, and (ii)1 transitions in the ICl molecule are reported. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

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.     

Ab initio investigation of ground‐state properties of group‐12 fluorides

The performance of wavefunction‐based correlation methods in theoretical solid‐state chemistry depends on reliable Hartree–Fock (HF) results for infinitly extended systems. Therefore, we optimized basis sets of valence‐triple‐ζ quality based on HF calculations for the periodic system of group‐12‐metal difluorides. Scalar‐relativistic effects were included in the case of the metal‐ions by applying small‐core pseudopotentials. To assess the quality of the proposed basis sets, the structural parameters, bulk moduli as well as cohesive and lattice energies of the systems were evaluated at the HF and the density functional theory levels. In addition to these two mean‐field approaches and to assess further employment of our basis sets to wavefunction‐based correlation methods we performed periodic local MP2 computations. Finally, the possibilities of pressure induced structural phase transitions occurring in the ZnF₂, CdF₂, and HgF₂ were investigated. © 2014 Wiley Periodicals, Inc.     

Halogen effect on structure and 13C NMR chemical shift of 3,6‐disubstituted‐N‐alkyl carbazoles

Structures of selected 3,6‐dihalogeno‐N‐alkyl carbazole derivatives were calculated at the B3LYP/6‐311++G(3df,2pd) level of theory, and their ¹³C nuclear magnetic resonance (NMR) isotropic shieldings were predicted using density functional theory (DFT). The model compounds contained 9H, N‐methyl and N‐ethyl derivatives. The relativistic effect of Br and I atoms on nuclear shieldings was modeled using the spin–orbit zeroth‐order regular approximation (ZORA) method. Significant heavy atom shielding effects for the carbon atom directly bonded with Br and I were observed (~?10 and ~?30 ppm while the other carbon shifts were practically unaffected). The decreasing electronegativity of the halogen substituent (F, Cl, Br, and I) was reflected in both nonrelativistic and relativistic NMR results as decreased values of chemical shifts of carbon atoms attached to halogen (C3 and C6) leading to a strong sensitivity to halogen atom type at 3 and 6 positions of the carbazole ring. The predicted NMR data correctly reproduce the available experimental data for unsubstituted N‐alkylcarbazoles. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical evaluation of pseudopotential matrix elements with Gaussian‐type solid harmonics of arbitrary angular momentum

An efficient formalism for evaluating pseudopotential matrix elements with Gaussian‐type solid harmonics of arbitrary angular momentum is presented. It is based on the tensor coupling technique, which is especially well suited for treating Gaussian‐type solid harmonics of arbitrary angular momentum. Closed analytical expressions are derived for the matrix elements as well as for their nuclear displacement derivatives. The efficiency of the implementation into our new parallel density functional program PARA GAUSS and the quality of the pseudopotential approach is tested for a set of representative molecules and cluster models. To this end the results of pseudopotential calculations are compared to those of nonrelativistic and scalar‐relativistic all‐electron calculations. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 209–221, 2000     

A gauge-independent zeroth-order regular approximation to the exact relativistic Hamiltonian--formulation and applications

A simple modification of the zeroth-order regular approximation (ZORA) in relativistic theory is suggested to suppress its erroneous gauge dependence to a high level of approximation. The method, coined gauge-independent ZORA (ZORA-GI), can be easily installed in any existing nonrelativistic quantum chemical package by programming simple one-electron matrix elements for the quasirelativistic Hamiltonian. Results of benchmark calculations obtained with ZORA-GI at the Hartree-Fock (HF) and second-order Moller-Plesset perturbation theory (MP2) level for dihalogens X(2) (X=F,Cl,Br,I,At) are in good agreement with the results of four-component relativistic calculations (HF level) and experimental data (MP2 level). ZORA-GI calculations based on MP2 or coupled-cluster theory with single and double perturbations and a perturbative inclusion of triple excitations [CCSD(T)] lead to accurate atomization energies and molecular geometries for the tetroxides of group VIII elements. With ZORA-GI/CCSD(T), an improved estimate for the atomization energy of hassium (Z=108) tetroxide is obtained.     

Quantum molecular dynamics

Electron or ion dynamics are treated using spin‐dependent quantum trajectories. These trajectories are inferred from the Dirac current, which contributes Schroedinger's current and additional spin‐dependent terms, all of which are of order c⁰ in the nonrelativistic regime of particle velocity, where c is the speed of light. The many‐body problem is treated precisely as in classical dynamics. Each electron or ion has its own equation of motion, which is the time‐dependent Dirac or the time‐dependent Schroedinger equation in the relativistic or nonrelativistic regime of particle velocity, respectively. As an example the theory is applied to the electronic structure of the helium atom, in which two electrons with opposite spin states are shown to correlate such that their quantum trajectories keep them on average on opposite sides of the nucleus. As the theory is time dependent, excited states are also generated. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011