Electron affinity of 7Li

Variationally optimized exponentially correlated Gaussian functions are employed to obtain nonrelativistic wave functions of the lithium atom and its negative ion. The energy levels are computed by means of the expansion in powers of the fine-structure constant alpha. The first term of this expansion corresponds to the nonrelativistic energy. The higher order terms represent the relativistic and radiative corrections and are determined by some effective Hamiltonians. Highly accurate expectation values of singular operators entering these Hamiltonians are computed using a set of expectation value identities. The resulting electron affinity of lithium atom 4984.96(18) cm(-1) agrees very well with 4984.90(17) cm(-1) of the latest measurements.     

Relativistic effect on 77Se NMR chemical shifts of various selenium species in the framework of zeroth-order regular approximation

The relativistic effects on absolute magnetic shielding tensors (σ(Se)) are explicitly evaluated for various selenium species (40 species) with the DFT(BLYP)-GIAO method. Calculations are performed under relativistic and nonrelativistic conditions with the Slater-type basis sets in ADF 2010 in the framework of ZORA, employing the optimized structures under nonrelativistic conditions at B3LYP of Gaussian 03. Quadruple zeta all electron with four polarization functions (QZ4Pae) are mainly applied to evaluate σ(Se). Ranges of the effect on diamagnetic (σ(d)(Se)), paramagnetic shielding tensors (σ(p)(Se)), and σ(d+p)(Se) (= σ(d)(Se) + σ(p)(Se)) are -24 to -20 ppm, -115 to -3 ppm, and -136 to -26 ppm, respectively. The spin-orbit terms (σ(so)(Se)) are evaluated to be 92-225 ppm with QZ4Pae, which clarifies the effect on total shielding tensors (σ(t)(Se) = σ(d+p)(Se) + σ(so)(Se)) to be -8 to 152 ppm, at the spin-orbit ZORA level. The calculated σ(t)(Se) values reproduced well the observed values.     

Relativistic quantum chemistry and rigorous variational analysis

A brief review of relativistic quantum chemistry is given here. Relativistic effects and their importance in chemistry are discussed. An outline of different theoretical aspects is presented. Aspects of variation techniques relevant to relativistic calculations are discussed in detail. These involve the derivation of min-max theorems for Dirac, Dirac-Hartree-Fock and Dirac-Coulomb calculations. The consequence of relativistic Hamiltonians being unbounded are also discussed for other lines of investigation. The upper bounds derived are physically interpreted. Sample Dirac-Hartree-Fock results for the Be atom, calculated using both STO and GTO bases for the nonrelativistic orbitals and the upper components of the relativistic orbitals, are given. The inadequacy of the so-called kinetically balanced basis set is discussed and illustrated with these results. The importance of the variational or dynamical balance and hence the merit of the LCAS-MS scheme is pointed out. The possibility of calculating quantum electrodynamical pair energy from relativistic configuration interaction calculations on a two-electron atom is discussed and exemplified. The present status of relativistic molecular calculations is briefly reviewed. Conclusions on the aspects of variational analysis and molecular calculations are enclosed.     

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.     

Direct perturbation theory in terms of energy derivatives: scalar-relativistic treatment up to sixth order

A formulation of sixth-order direct perturbation theory (DPT) to treat relativistic effects in quantum-chemical calculations is presented in the framework of derivative theory. Detailed expressions for DPT6 are given at the Hartree-Fock level in terms of the third derivative of the energy with respect to the relativistic perturbation parameter defined as λ(rel)=c(-2). They were implemented for the computation of scalar-relativistic energy corrections. The convergence of the scalar-relativistic DPT expansion is studied for energies and first-order properties such as dipole moment and electric-field gradient within the series of the hydrogen halides (HX, X = F, Cl, Br, I, and At). Comparison with spin-free Dirac-Coulomb calculations indicates that the DPT series exhibits a smooth and monotonic convergence. The rate of convergence, however, depends on the charge of the involved nuclei and significantly slows down for heavy-element compounds.     

Nonrelativistic molecular quantum mechanics without approximations: electron affinities of LiH and LiD

We took the complete nonrelativistic Hamiltonians for the LiH and LiH- systems, as well as their deuterated isotopomers, we separated the kinetic energy of the center of mass motion from the Hamiltonians, and with the use of the variational method we optimized the ground-state nonadiabatic wave functions for the systems expanding them in terms of n-particle explicitly correlated Gaussian functions. With 3600 functions in the expansions we obtained the lowest ever ground-state energies of LiH, LiD, LiH-, and LiD- and these values were used to determine LiH and LiD electrons affinities (EAs) yielding 0.330 30 and 0.327 13 eV, respectively. The present are the first high-accuracy ab initio quantum mechanical calculations of the LiH and LiD EAs that do not assume the Born-Oppenheimer approximation. The obtained EAs fall within the uncertainty brackets of the experimental results.     

Direct perturbation theory in terms of energy derivatives: fourth-order relativistic corrections at the Hartree-Fock level

In this work, the quantum-chemical treatment of relativistic effects by means of direct perturbation theory is extended from its lowest order, DPT2, to the next higher order, DPT4. The required theory is given in terms of energy derivatives with the DPT4 energy correction defined as the corresponding second derivative with respect to the relativistic perturbation parameter λ(rel) = c(2) and c as the speed of light. To facilitate the implementation in standard quantum-chemical program packages, a general formulation of DPT starting from a nonrelativistic Lagrangian is developed, thereby expanding both wave function and operators in terms of λ(rel). The corresponding expressions, which incorporate in an additive manner scalar-relativistic and spin-orbit contributions, are given at the Hartree-Fock level and have been implemented in the CFOUR program package using the available analytic second-derivative techniques. The accuracy of the DPT4 corrections at the HF level is investigated by comparison with rigorous four-component calculations. Scalar-relativistic and spin-orbit contributions are analyzed individually and the importance of the various terms to those corrections is discussed. Furthermore, the basis-set dependence of the computed DPT4 corrections is investigated.     

Ionization potentials and electron affinities of Cu,Ag, and Au: Electron correlation and relativistic effects

The electron correlation and relativistic effects on ionization potentials and electron affinities of Cu, Ag, and Au are investigated in the framework of the coupled cluster method and different 1-component approximations to the relativistic Dirac-Coulomb Hamiltonian. The first-order perturbation approach based on the massvelocity and Darwin terms is found to be sufficiently accurate for Cu and Ag while it fails for Au. The spin-averaged Douglas-Kroll no-pair method gives excellent results for the studied atomic properties. The ionization potentials obtained within this method and the coupled cluster scheme for the electron correlation effects are 7.733(7.735) eV for Cu, 7.461(7.575) eV for Ag, and 9.123(9.225) eV for Au (experimental values given in parentheses). The calculated (experimental) electron affinity results for Cu, Ag, and Au are 1.236(1.226), 1.254(1.303), and 2.229(2.309) eV, respectively. There is a marked relativistic effect on both the ionization potential and electron affinity of Ag which sharply increases for Au while Cu exhibits only a little relativistic character. A similar pattern of relativistic effects is also observed for electric dipole polarizabilities of the coinage metal atoms and their ions. The coupled cluster dipole polarizabilities of the coinage metal atoms calculated in this article in the Douglas-Kroll no-pair formalism (Cu: 46.50 au; Ag: 52.46 au; Au: 36.06 au) are compared with our earlier data for their singly positive and singly negative ions. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 557–565, 1997     

A combined experimental and quantum chemistry study of selenium chemical shift tensors

A comprehensive investigation of selenium chemical shift tensors is presented. Experimentally determined chemical shift tensors were obtained from solid-state 77Se NMR spectra for several organic, organometallic, or inorganic selenium-containing compounds. The first reported indirect spin-spin coupling between selenium and chlorine is observed for Ph(2)SeCl(2) where 1J(77Se,35Cl)iso is 110 Hz. Selenium magnetic shielding tensors were calculated for all of the molecules investigated using zeroth-order regular approximation density functional theory, ZORA DFT. The computations provide the orientations of the chemical shift tensors, as well as a test of the theory for calculating the magnetic shielding interaction for heavier elements. The ZORA DFT calculations were performed with nonrelativistic, scalar relativistic, and scalar with spin-orbit relativistic levels of theory. Relativistic contributions to the magnetic shielding tensor were found to be significant for (NH4)2WSe4 and of less importance for organoselenium, organophosphine selenide, and inorganic selenium compounds containing lighter elements.     

Accurate relativistic energy-consistent pseudopotentials for the superheavy elements 111 to 118 including quantum electrodynamic effects

Energy-consistent two-component semi-local pseudopotentials for the superheavy elements with atomic numbers 111-118 have been adjusted to fully relativistic multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian, including perturbative corrections for the frequency-dependent Breit interaction in the Coulomb gauge and lowest-order quantum electrodynamic effects. The pseudopotential core includes 92 electrons corresponding to the configuration [Xe]4f(14)5d(10)5f(14). The parameters for the elements 111-118 were fitted by two-component multi-configuration Hartree-Fock calculations in the intermediate coupling scheme to the total energies of 267 up to 797 J levels arising from 31 up to 62 nonrelativistic configurations, including also anionic and highly ionized states, with mean absolute errors clearly below 0.02 eV for averages corresponding to nonrelativistic configurations. Primitive basis sets for one- and two-component pseudopotential calculations have been optimized for the ground and excited states and exhibit finite basis set errors with respect to the finite-difference Hartree-Fock limit below 0.01 and 0.02 eV, respectively. General contraction schemes have been applied to obtain valence basis sets of polarized valence double- to quadruple-zeta quality. Results of atomic test calculations in the intermediate coupling scheme at the Fock-space coupled-cluster level are in good agreement with those of corresponding fully relativistic all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian. The results demonstrate besides the well-known need of a relativistic treatment at the Dirac-Coulomb level also the necessity to include higher-order corrections for the superheavy elements.     

The accuracy of hyperfine integrals in relativistic NMR computations based on the zeroth-order regular approximation

The accuracy of the hyperfine integrals obtained in relativistic NMR computations based on the zeroth–order regular approximation (ZORA) is investigated. The matrix elements of the Fermi contact operator and its relativistic analogs for s orbitals obtained from numerical nonrelativistic, ZORA, and four–component Hartree–Fock–Slater calculations on atoms are compared. It is found that the ZORA yields very accurate hyperfine integrals for the valence shells of heavy atoms, but performs rather poorly for the innermost core shells. Because the important observables of the NMR experiment—chemical shifts and spin–spin coupling constants—can be understood as valence properties it is concluded that ZORA computations represent a reliable tool for the investigations of these properties. On the other hand, absolute shieldings calculated with the ZORA might be substantially in error. Because applications to molecules have so far exclusively been based on basis set expansions of the molecular orbitals, ZORA hyperfine integrals obtained from atomic Slater-type basis set computations for mercury are compared with the accurate numerical values. It is demonstrated that the core part of the basis set requires functions with Slater exponents only up to 10⁴ in the case where errors in the hyperfine integrals of a few percent are acceptable.     

Time-dependent quasirelativistic density-functional theory based on the zeroth-order regular approximation

A time-dependent quasirelativistic density-functional theory for excitation energies of systems containing heavy elements is developed, which is based on the zeroth-order regular approximation (ZORA) for the relativistic Hamiltonian and a noncollinear form for the adiabatic exchange-correlation kernel. To avoid the gauge dependence of the ZORA Hamiltonian a model atomic potential, instead of the full molecular potential, is used to construct the ZORA kinetic operator in ground-state calculations. As such, the ZORA kinetic operator no longer responds to changes in the density in response calculations. In addition, it is shown that, for closed-shell ground states, time-reversal symmetry can be employed to simplify the eigenvalue equation into an approximate form that is similar to that of time-dependent nonrelativistic density-functional theory. This is achieved by invoking an independent-particle approximation for the induced density matrix. The resulting theory is applied to investigate the global potential-energy curves of low-lying LambdaS- and omega omega-coupled electronic states of the AuH molecule. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are in good agreement with those of time-dependent four-component relativistic density-functional theory and ab initio multireference second-order perturbation theory. Nonetheless, this two-component relativistic version of time-dependent density-functional theory is only moderately advantageous over the four-component one as far as computational efforts are concerned.     

Picture change error correction of radon atom electron density

The importance of the picture change error (PCE) correction in the quasirelativistic electron density of radon atom is presented. PCE correction is considered for the infinite order two-component (IOTC) and second order Douglas-Kroll-Hess (DKH2) wave functions. Implementation details of PCE correction of electron density are outlined. The result section presents the radial distributions of electron density ρ(r) and the 4πr(2)ρ(r) function of the radon atom in the nuclear region. The PCE corrected and contaminated DKH2, IOTC electron densities are presented and compared with the Dirac-Coulomb Hamiltonian and nonrelativistic electron densities. Besides, some additional effects in electron density and SCF energy are considered, such as spin-orbit coupling, the inclusion of the Gaunt term, and the finite nucleus model effects. The effects of p(2)Vp(2) and p(2)ρ(r)p(2) analytic integral classes within IOTC Hamiltonian and PCE correction of IOTC electron density are considered.     

A comparison of two-component and four-component approaches for calculations of spin-spin coupling constants and NMR shielding constants of transition metal cyanides

Relativistic density functional theory (DFT) calculations of nuclear spin-spin coupling constants and shielding constants have been performed for selected transition metal (11th and 12th group of periodic table) and thallium cyanides. The calculations have been carried out using zeroth-order regular approximation (ZORA) Hamiltonian and four-component Dirac-Kohn-Sham (DKS) theory with different nonrelativistic exchange-correlation functionals. Two recent approaches for representing the magnetic balance (MB) between the large and small components of four-component spinors, namely, mDKS-RMB and sMB, have been employed for shielding tensor calculations and their results have been compared. Relativistic effects have also been analysed in terms of scalar and spin-orbit contributions at the two-component level of theory, including discussion of heavy-atom-on-light-atom effects for (1)J(CN), σ(C), and σ(N). The results for molecules containing metals from 4th row of periodic table show that relativistic effects for them are small (especially for spin-spin coupling constants). The biggest effects are observed for the 6th row where nonrelativistic theory reproduces only about 50%-70% of the two-component ZORA results for (1)J(MeC) and about 75% for heavy metal shielding constants. It is important to employ a full Dirac picture for calculations of heavy metal shielding constants, since ZORA reproduces only 75%-90% of the DKS results. Smaller discrepancies between ZORA-DFT and DKS are observed for nuclear spin-spin coupling constants. No significant differences are observed between the results obtained using mDKS-RMB and sMB approaches for magnetic balance in four-component calculations of the shielding constants.     

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.     

Two-component relativistic methods for the heaviest elements

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.     

Accurate non-Born-Oppenheimer calculations of the complete pure vibrational spectrum of D2 with including relativistic corrections

In this work we report very accurate variational calculations of the complete pure vibrational spectrum of the D(2) molecule performed within the framework where the Born-Oppenheimer (BO) approximation is not assumed. After the elimination of the center-of-mass motion, D(2) becomes a three-particle problem in this framework. As the considered states correspond to the zero total angular momentum, their wave functions are expanded in terms of all-particle, one-center, spherically symmetric explicitly correlated Gaussian functions multiplied by even non-negative powers of the internuclear distance. The nonrelativistic energies of the states obtained in the non-BO calculations are corrected for the relativistic effects of the order of α(2) (where α = 1/c is the fine structure constant) calculated as expectation values of the operators representing these effects.     

Symmetric group approach to relativistic CI. I. General formalism

General formulas for matrix elements of spin-dependent operators in a basis of spin-adapted antisymmetrized products of orthonormal orbitals are derived. The resulting formalism may be applied to construction of the Hamiltonian matrices both for Pauli and for projected no-pair relativistic configuration interaction methods. From a formal point of view, it is a generalization of the symmetric group approach to the CI method for the case of spin-dependent Hamiltonians. © 1997 John Wiley & Sons, Inc.