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.     

Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian

We report the implementation of analytic energy gradients for the evaluation of first-order electrical properties and nuclear forces within the framework of the spin-free (SF) exact two-component (X2c) theory. In the scheme presented here, referred to in the following as SFX2c-1e, the decoupling of electronic and positronic solutions is performed for the one-electron Dirac Hamiltonian in its matrix representation using a single unitary transformation. The resulting two-component one-electron matrix Hamiltonian is combined with untransformed two-electron interactions for subsequent self-consistent-field and electron-correlated calculations. The "picture-change" effect in the calculation of properties is taken into account by considering the full derivative of the two-component Hamiltonian matrix with respect to the external perturbation. The applicability of the analytic-gradient scheme presented here is demonstrated in benchmark calculations. SFX2c-1e results for the dipole moments and electric-field gradients of the hydrogen halides are compared with those obtained from nonrelativistic, SF high-order Douglas-Kroll-Hess, and SF Dirac-Coulomb calculations. It is shown that the use of untransformed two-electron interactions introduces rather small errors for these properties. As a first application of the analytic geometrical gradient, we report the equilibrium geometry of methylcopper (CuCH(3)) determined at various levels of theory.     

A finite-nucleus model for relativistic electronic structure calculations using a Douglas-Kroll-transformed Hamiltonian

Summary We report the implementation of a Gaussian finite-nucleus model in the framework of the spin-free no-pair Hamiltonian obtained from the Douglas-Kroll transformation of the no-pair operator with external-field projectors. The finite nucleus regularizes the weak singularity of the wavefunction at the locations of the nuclei and provides a means for efficient exponent optimization using a spinaveraged relativistic one-component operator. We report and discuss basis sets for the gold atom obtained from various optimization procedures, making use of a point nucleus as well as employing various finite-nucleus models.Dedicated to Prof. Dr. Werner Kutzelnigg on occasion of his sixtieth birthday     

Spin densities in two-component relativistic density functional calculations: noncollinear versus collinear approach

With present day exchange-correlation functionals, accurate results in nonrelativistic open shell density functional calculations can only be obtained if one uses functionals that do not only depend on the electron density but also on the spin density. We consider the common case where such functionals are applied in relativistic density functional calculations. In scalar-relativistic calculations, the spin density can be defined conventionally, but if spin-orbit coupling is taken into account, spin is no longer a good quantum number and it is not clear what the "spin density" is. In many applications, a fixed quantization axis is used to define the spin density ("collinear approach"), but one can also use the length of the local spin magnetization vector without any reference to an external axis ("noncollinear approach"). These two possibilities are compared in this work both by formal analysis and numerical experiments. It is shown that the (nonrelativistic) exchange-correlation functional should be invariant with respect to rotations in spin space, and this only holds for the noncollinear approach. Total energies of open shell species are higher in the collinear approach because less exchange energy is assigned to a given Kohn-Sham reference function. More importantly, the collinear approach breaks rotational symmetry, that is, in molecular calculations one may find different energies for different orientations of the molecule. Data for the first ionization potentials of Tl, Pb, element 113, and element 114, and for the orientation dependence of the total energy of I+2 and PbF indicate that the error introduced by the collinear approximation is approximately 0.1 eV for valence ionization potentials, but can be much larger if highly ionized open shell states are considered. Rotational invariance is broken by the same amount. This clearly indicates that the collinear approach should not be used, as the full treatment is easily implemented and does not introduce much more computational effort.     

改进的相对论量子化学计算ZORA方法

提出一种改进的ZORA(Zeroth-OrderRegularApproximationtotheDiracEquation)方法,其单电子方程为：[σ·p(c^2/2c^2-V~0)σ·p+V(r)]ψ=εψ。式中V~0为空间限域的势能函数：V~0(r=~A∑V~0^A(r~A),r~A=|r-R~A|,V~0^A(r~A)=V^A(r~A){1+exp[α(r~A-r~0^A]}^-^1。其中A表示分子的某个组成原子,R~A为A原子的位置矢量,V^A(r~A)为自由A原子的势函数,α和r~0^A为参数。改进的ZORA方法具有原来方法的所有优点,避免了原有ZORA方法因不满足标度变换不变性带来的缺陷,而且计算过程简单。具体计算表明,通过适当选择参数α和r~0^A,用本研究提出来的方法,在计算分子几何结构和键合能时,基本上消除了ZORA方法由于标度变换依赖性产生的误差。     

General variation method for the relativistic calculations of atoms and molecules

Use of the general variation method of Weinstein and MacDonald for the relativistic calculation of atoms and molecules is proposed. It is shown from the numerical calculations for hydrogenlike atomic systems that this method is useful in judging an accuracy of energies and wave functions obtained with a relativistic Hamiltonian whose spectra are not bounded. It is also shown that this method can be used to find spurious solutions such as 1p_½ or 2d_3/2 appearing in atomic systems. Problems in extending the method to many-electron atoms and molecules are discussed.     

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.     

Fast method for quantum chemical calculations of large molecules with the approximation of the DFT Hamiltonian

The fundamentally new method NESE is proposed for quantum chemical calculations of large molecules, which employs the approximation of the Hamiltonian of the commonly used DFT method and is as fast as the AM1 and PM3 semi-empirical methods or the DFTB method. The parameters for the new method were chosen by the least-squares method based on the comparison of its matrix elements with the reference DFT/PBE Hamiltonian. The initial non-iterative version NESE-0 was computer-implemented and approved on many thousands of various molecules containing H, C, N, and O atoms. The NESE-0 method moderately outperforms the DFTB approach and is an order of magnitude better than the AM1, PM3, and PM6 levels in reproducing the one-electron energies calculated in terms of the DFT/PBE.     

Scaled one-electron hamiltonian model for open-shell LCAO–MO–SCF calculations: Comparison with the restricted open-shell method of roothaan

The performance of a recently proposed scaled one-electron Hamiltonian (SOEH ) model is tested against parallel sets of restricted open-shell calculations by the method of Roothaan. It is found that the energy calculated by SOEH model, in general, lies slightly higher than the energy computed by the restricted open-shell method of Roothaan lending credibility to the application of variational argument to the scaled pseudoenergy functional (E^av) for deriving the SOEH model. The numerical stability of the converged SOEH energy with respect to changes in trial vectors indicates the reliability of the method. The SOEH model is shown to perform well in the calculation of geometries of radicals and ions. The convergence behavior of the SOEH model is compared with that of the restricted open-shell method of Roothaan.     

Accurate ionization potentials for UO and UO2: a rigorous test of relativistic quantum chemistry calculations

Accurate ionization potential (IP) measurements provide essential thermodynamic information and benchmark data that can be used to evaluate the validity of electronic structure models. Calculations of the first IP of UO2 using relativistic methods consistently predict values that are approximately 0.7 eV higher than the accepted experimental value. The present measurements validate the theoretical calculations and show that the previous determinations corresponded to the ionization of thermally excited molecules. Similarly, new measurements of the IP for UO show that the currently accepted value is too low by 0.4 eV.     

A new semi-empirical method of correcting large-scale configuration interaction calculations for incomplete dynamic correlation of electrons

We show that large-basis-set configuration interaction calculations including all single and double excitations from a complete-active-space optimized by MCSCF methods recover an approximately constant fraction of the external correlation energy. Therefore, by scaling this contribution to the energy, more accurate potential energy curves, barrier heights to chemical reactions, and potential energy surfaces can be calculated.     

The scaled one-electron Hamiltonian model for open-shelllcao-mo-scf calculations: further corrections to thesoeh energy

The wavefunction generated by the scaled one-electron Hamiltonian (soeh) model has been modified further by including all single excitations from the space of thesoeh function. A perturbation-variation ansatz has been invoked for obtaining the corresponding energy correction (ΔE ₂). [E(soeh)+ΔE ₂] is shown to reproduceE(Roothaan) very closely. It has been demonstrated that by making ΔE ₂ stationary with respect to change in μ results. The correctedsoeh [E(soeh)+ΔE ₂] is shown to be quite useful for the calculation of geometrical parameters of open-shell systems even though it lacks the ‘upper-boundedness’ exhibited byE(soeh).     

Fully relativistic pseudopotentials for alkaline atoms: Dirac-Hartree-Fock and configuration interaction calculations of alkaline monohydrides

Fully relativistic four-component energy-adjusted pseudopotentials and corresponding valence basis sets have been derived for the alkaline atoms Li through Cs, treating them as one-valence electron systems. Core-valence correlation effects are accounted for by a core-polarization potential, deviations of the core-nucleus repulsion from a point charge model by a suitable correction. The results of Dirac-Hartree-Fock and configuration interaction calculations are presented for atomic properties not used in the pseudopotential adjustment, i.e. electron affinities and dipole polarizabilities, as well as for the spectroscopic constants of the ground states of the alkaline monohydrides. The analytic form of the cut-off function for the electric field in the core-polarization term and its effects on atomic and molecular properties is discussed.     

Pt on graphene monolayers supported on a Ni(111) substrate: relativistic density-functional calculations

The structural, energetic, and magnetic properties of Pt atoms and dimers adsorbed on a Ni-supported graphene layer have been investigated using density-functional calculations, including the influence of dispersion forces and of spin-orbit coupling. Dispersion forces are found to be essential to stabilize a chemisorbed graphene layer on the Ni(111) surface. The presence of the Ni-substrate leads not only to a stronger interaction of Pt atoms and dimers with graphene but also to a locally increased binding between graphene and the substrate and a complex reconstruction of the adlayer. The stronger binding of the dimer also stabilizes a flat adsorption geometry in contrast to the upright geometry on a free-standing graphene layer. These effects are further enhanced by dispersion corrections. Isolated Pt adatoms and flat dimers are found to be non-magnetic, while an upright Pt dimer has strongly anisotropic spin and orbital moments. For the clean C/Ni(111) system, we calculate an in-plane magnetic anisotropy, which is also conserved in the presence of isolated Pt adatoms. Surprisingly, upright Pt-dimers induce a re-orientation of the easy magnetic axis to a direction perpendicular to the surface, in analogy to Pt(2) on a free-standing graphene layer and to the axial anisotropy of a gas-phase Pt(2) dimer.     

Periodic trends in indirect nuclear spin-spin coupling tensors: relativistic density functional calculations for interhalogen diatomics

There have been significant advances in the calculation and interpretation of indirect nuclear spin-spin coupling (J) tensors during the past few years; however, much work remains to be done, especially for molecules containing heavy atoms where relativistic effects may play an important role. Many J tensors cannot be explained based solely on a nonrelativistic Fermi-contact mechanism. In the present work, the relativistic zeroth-order regular approximation density-functional (ZORA-DFT) implementation for the calculation of J has been applied to the complete series of homonuclear and heteronuclear diatomic halogen molecules: F(2), Cl(2), Br(2), I(2), At(2), ClF, BrF, IF, ClBr, ClI, and BrI. For all of these compounds, the reduced isotropic coupling constant (K(iso)) is positive and the reduced anisotropic coupling constant (DeltaK) is negative. With the exception of molecular fluorine, the magnitudes of K(iso) and DeltaK are shown to increase linearly with the product of the atomic numbers of the coupled nuclei. ZORA-DFT calculations of J for F(2) and ClF are in excellent agreement with the results obtained from multiconfigurational self-consistent-field calculations. The relative importance of the various coupling mechanisms is approximately constant for all of the compounds, with the paramagnetic spin-orbit term being the dominant contributor to K(iso), at approximately 70-80%. Available experimental stimulated resonant Raman spectroscopy data are exploited to extract the complete J((127)I,(127)I) tensor for iodine in two rotational states. The dependence of K(iso) and DeltaK on bond length and rovibrational state is investigated by using calculated results in combination with available experimental data. In addition to providing new insights into periodic trends for J coupling tensors, this work further demonstrates the utility of the ZORA-DFT method and emphasizes the necessity of spin-orbit relativistic corrections for J calculations involving heavy nuclei.     

Local energy calculations employing importance sampling: Positronium hydride wavefunctions as severe tests of the method

Ground-state positronium hydride wavefunctions are generated by least squares solutions of the Schrödinger equation and compared with variational results. Effects on wavefunction quality of configuration space sampling and of low-lying virtual excited states are detailed.     

Projector Monte Carlo method based on Slater determinants: a new sampling method for singlet state calculations

A new sampling method is proposed for projector Monte Carlo (PMC) calculations based on Slater determinants (SD) in singlet states. Using the symmetry of the ?? and ?? electron determinants, the number of configurations to be considered can be about one-half of the original sampling. We applied the new sampling to the PMC-SD calculations of the H₂O molecule in the ground state. The results were always improved by the new sampling method both for the equilibrium and for bond-stretched structures.