Douglas-Kroll Research Articles

Douglas-Kroll and infinite order two-component transformations of Dirac-Fock operator.

We extended the conventional Douglas-Kroll (DK) and infinite order two-component (IOTC) methods to a technique applicable to Fock matrices, called extended DK (EDK) and extended IOTC (EIOTC), respectively. First, we defined a strategy to divide the Dirac-Fock operator into zero- and first-order terms. We then demonstrated that the first-order extended DK transformation, which is the Foldy-Wouthuysen transformation for the zero-order term, as well as the second- and third-order EDK and EIOTC, could be well defined. The EDK- and EIOTC-transformed Fock matrix, kinetic energy operator, nuclear attraction operator, and density matrix were derived. These equations were numerically evaluated, and it was found that these methods were accurate. In particular, EIOTC was consistent with the four-component approach. Four-component and extended two-component calculations are more expensive than non-relativistic calculations due to small-component-type two-electron integrals. We developed a new approximation formula, RIS-V, for small-component-type two-electron integrals, including the spin-orbit interaction between electrons. These results suggest that the RIS-V formula effectively accelerates the four-component and extended two-component methods.

First Full-Dimensional Potential Energy and Dipole Moment Surfaces of SF6

A 15-dimensional analytical form for the potential energy and dipole moment surfaces of the SF6 molecule in the ground electronic state is obtained using ab initio methods. In order to calculate the equilibrium S-F distance, we applied the coupled cluster CCSD(T) method and several versions of the correlation-consistent basis sets from valence triple-zeta (VTZ) and augmented valence triple-zeta (AVTZ) to core-valence quadruple-zeta (CVQZ) with Douglas-Kroll (DK) relativistic corrections that provided good agreement with an empirical equilibrium value. Ab initio electronic energies on 15D grids of nuclear geometries are computed using the CCSD(T) method with VTZ and CVQZ-DK basis sets. The analytical representation of the potential energy surface is determined through an expansion in symmetry-adapted products of nonlinear coordinates up to the 5th order. The influence of additional redundant coordinates on the quality of the fit was investigated. Parameters of full-dimensional dipole moment surfaces are determined up to the 4th order expansion in normal mode coordinates. For validation of ab initio results, the fundamental vibration frequencies and absorption cross sections of the principal sulfur hexafluoride isotopologue are calculated, giving good agreement with cold (180 K) and room temperature (296 K) experimental spectra. Absorption cross sections calculated from our preliminary line list agree much better with these observations than the simulations using SF6 line-by-line lists constructed from effective models included in currently available spectroscopic databases.

The electronic structure of vanadium monochloride cation (VCl(+)): tackling the complexities of transition metal species.

Six electronic states (X (4)Σ(-), A (4)Π, B (4)Δ, (2)Φ, (2)Δ, (2)Σ(+)) of the vanadium monochloride cation (VCl(+)) are described using large basis set coupled cluster theory. For the two lowest quartet states (X (4)Σ(-) and A (4)Π), a focal point analysis (FPA) approach was used that conjoined a correlation-consistent family of basis sets up to aug-cc-pwCV5Z-DK with high-order coupled cluster theory through pentuple (CCSDTQP) excitations. FPA adiabatic excitation energies (T0) and spectroscopic constants (re, r0, Be, B0, D¯e, He, ωe, v0, αe, ωexe) were extrapolated to the valence complete basis set Douglas-Kroll (DK) aug-cc-pV∞Z-DK CCSDT level of theory, and additional treatments accounted for higher-order valence electron correlation, core correlation, and spin-orbit coupling. Due to the delicate interplay between dynamical and static electronic correlation, single reference coupled cluster theory is able to provide the correct ground electronic state (X (4)Σ(-)), while multireference configuration interaction theory cannot. Perturbations from the first- and second-order spin orbit coupling of low-lying states with quartet spin multiplicity reveal an immensely complex rotational spectrum relative to the isovalent species VO, VS, and TiCl. Computational data on the doublet manifold suggest that the lowest-lying doublet state ((2)Γ) has a Te of ∼11 200 cm(-1). Overall, this study shows that laboratory and theoretical rotational spectroscopists must work more closely in tandem to better understand the bonding and structure of molecules containing transition metals.

Large‐scale two‐component relativistic quantum‐chemical theory: Combination of the infinite‐order douglas–kroll–hess method with the local unitary transformation scheme and the divide‐and‐conquer method

Large‐scale two‐component (2c) relativistic quantum‐chemical (RQC) theory is reviewed. We briefly discuss the theories, advantages, and extensibilities of an overall linear‐scaling scheme in 2c relativistic theory. The theory is based on the infinite‐order Douglas–Kroll–Hess method, with the local unitary transformation scheme to produce the 2c relativistic Hamiltonian, and the divide‐and‐conquer method to achieve linear‐scaling of Hartree–Fock and electron correlation methods. Furthermore, perspectives for large‐scale RQC are explained to bring the practical usage and treatment of light and heavy elements in 2c relativistic calculations close to those in non‐relativistic methods. © 2014 Wiley Periodicals, Inc.

Taming the low-lying electronic states of FeH

The low-lying electronic states (X (4)Δ, A (4)Π, a (6)Δ, b (6)Π) of the iron monohydride radical, which are especially troublesome for electronic structure theory, have been successfully described using a focal point analysis (FPA) approach that conjoined a correlation-consistent family of basis sets up to aug-cc-pwCV5Z-DK with high-order coupled cluster theory through hextuple (CCSDTQPH) excitations. Adiabatic excitation energies (T(0)) and spectroscopic constants (r(e), r(0), B(e), B(0), D(e), ω(e), v(0), α(e), ω(e)x(e)) were extrapolated to the valence complete basis set Douglas-Kroll (DK) aug-cc-pwCV∞Z-DK CCSDT level of theory, and additional treatments accounted for higher-order valence electron correlation, core correlation, spin-orbit coupling, and the diagonal Born-Oppenheimer correction. The purely ab initio FPA approach yields the following T(0) results (in eV) for the lowest spin-orbit components of each electronic state: 0 (X (4)Δ) < 0.132 (A (4)Π) < 0.190 (a (6)Δ) < 0.444 (b (6)Π). The computed anharmonic fundamental vibrational frequencies (v(0)) for the (4,6)Δ electronic states are within 3 cm(-1) of experiment and provide reliable predictions for the (4,6)Π states. With the cc-pVDZ basis set, even CCSDTQPH energies give an incorrect ground state of FeH, highlighting the importance of combining high-order electron correlation treatments with robust basis sets when studying transition-metal radicals. The FPA computations provide D(0) = 1.86 eV (42.9 kcal mol(-1)) for the 0 K dissociation energy of FeH and Δ(f)H(298) (∘) [FeH((g))] = 107.7 kcal mol(-1) for the enthalpy of formation at room temperature. Despite sizable multireference character in the quartet states, high-order single-reference coupled cluster computations improve the spectroscopic parameters over previous multireference theoretical studies; for example, the X (4)Δ → A (4)Π and a (6)Δ → b (6)Π transition energies are reproduced to 0.012 and 0.002 eV, respectively, while the error for the problematic X (4)Δ → a (6)Δ intercombination excitation is reduced from at least 0.17 eV to about 0.04 eV.

Coupled cluster investigation on the low-lying electronic states of CuCN and CuNC and the ground state barrier to isomerization

The observation of several metal cyanides and isocyanides in interstellar space has raised much interest these molecules. Optimum molecular structures, harmonic vibrational frequencies, and dipole moments of the ground electronic states (X1Sigma+), triplet excited states, and open shell singlet excited states of CuCN and CuNC were determined using different levels of nonrelativistic and scalar relativistic (Douglas-Kroll) [Ann. Phys. 82, 89 (1979)] coupled cluster theory in conjunction with atomic natural orbital basis sets and correlation consistent basis sets. For the relativistic computations the specially contracted correlation consistent Douglas-Kroll (DK) basis sets were used. Moreover, barriers to isomerization from CuCN to CuNC were computed. The predicted structures of the X1Sigma+ state for CuCN are re(Cu-C)=1.826 A and re(C-N)=1.167 A, at the most sophisticated level of theory, the scalar relativistic DK-CCSD(T)/cc-pVQZ(DK) method. These results are in excellent agreement with the experimentally determined Cu-C bond length of 1.829 A and C-N bond distance of 1.162 A. At the same level of theory, the zero-point corrected barrier to isomerization from CuCN to CuNC is estimated to be 14.7 kcal mol(-1), and the cyanide is more stable than the isocyanide by 11.5 kcal mol(-1). For both CuCN and CuNC the 3Sigma+ state is the lowest lying excited electronic state. At the DK-CCSD/cc-pVQZ(DK) level of theory, the energetic ordering of excited states of CuCN and CuNC is X1Sigma+<a3Sigma+<b3Pi<2(1)Sigma+ approximately 3Delta<1Pi<1Delta. The variations of CN bond lengths in the optimized structures for the different electronic states and the CN stretching frequencies of the ground state and the excited states suggest that metal dpi to ligand pi charge transfer is insignificant, in contrast to previous results for isoelectronic NiCO.

Recent Development of Relativistic Molecular Theory

Today it is common knowledge that relativistic effects are important in the heavy-element chemistry. The continuing development of the relativistic molecular theory is opening up rows of the periodic table that are impossible to treat with the non-relativistic approach. The most straightforward way to treat relativistic effects on heavy-element systems is to use the four-component Dirac-Hartree-Fock approach and its electron-correlation methods based on the Dirac-Coulomb(-Breit) Hamiltonian. The Dirac-Hartree-Fock (DHF) or Dirac-Kohn-Sham (DKS) equation with the four-component spinors composed of the large- and small-components demands severe computational efforts to solve, and its applications to molecules including heavy elements have been limited to small- to medium-size systems. Recently, we have developed a very efficient algorithm for the four-component DHF and DKS approaches. As an alternative approach, several quasi-relativistic approximations have also been proposed instead of explicitly solving the four-component relativistic equation. We have developed the relativistic elimination of small components (RESC) and higher-order Douglas-Kroll (DK) Hamiltonians within the framework of the two-component quasi-relativistic approach. The developing four-component relativistic and approximate quasi-relativistic methods have been implemented into a program suite named REL4D. In this article, we will introduce the efficient relativistic molecular theories to treat heavy-atomic molecular systems accurately via the four-component relativistic and the two-component quasi-relativistic approaches. We will also show several chemical applications including heavy-element systems with our relativistic molecular approaches.

Extended Douglas–Kroll transformations applied to the relativistic many-electron Hamiltonian

A new generalized Douglas–Kroll (DK) approach is proposed for the relativistic many-electron Hamiltonian including the electron–electron interaction. In order to consider the higher-order DK transformation to the two-electron interaction, the present approach adopts the effective one-electron potential in the Dirac–Hartree–Fock/Dirac–Kohn–Sham operator as an expansion parameter in the DK transformation. Its numerical performance is tested for the atomic Hg and molecular HAt and At2 systems. The third-order DK transformation to both one-electron and two-electron Hamiltonians, which is the highest level of theory treated in this study, gives excellent agreement with the four-component relativistic approach. The first-order DK correction to the two-electron interaction is shown to be satisfactory for both atomic and molecular systems.

Relativistic Molecular Theory

This brief review contains surveys of both four-component and two-component relativistic molecular theories. First the four-component relativistic approach is reviewed. Emphasis is placed on efficient computational schemes for the four-component Dirac-Hartree-Fock and Dirac-Kohn-Sham methods. Next, in the twocomponent relativistic framework, two relativistic Hamiltonians, RESC and higher-order Douglas-Kroll (DK), are introduced. An illustrative application is shown for the relativistic study on valence photoelectron spectrum of OsO₄. The developing four-component relativistic and approximate quasi-relativistic methods have been packed in a program suite named REL4D.

The generalized Douglas–Kroll transformation

We derive the most general parametrization of the unitary matrices in the Douglas–Kroll (DK) transformation sequence for relativistic electronic structure calculations. It is applied for a detailed analysis of the generalized DK transformation up to fifth order in the external potential. While DKH2–DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy–Wouthuysen transformation. The freedom in the choice of this expansion coefficient vanishes consistently if the optimum unitary transformation is sought for. Since the standard protocol of the DK method is the application of unitary transformations to the one-electron Dirac operator, we analyze the DKH procedure up to fifth order for hydrogenlike atoms. We find remarkable accuracy of the higher-order DK corrections as compared to the exact Dirac ground state energy. In the case of many-electron atomic systems, we investigate the order of magnitude of the higher-order corrections in the light of the neglect of the DK transformation of the two-electron terms of the many-particle Hamiltonian. A careful analysis of the silver and gold atoms demonstrates that both the fourth- and fifth-order one-electron DK transformation yield a smaller contribution to the total electronic energy than the DK transformation of the two-electron terms. In order to improve significantly on the third-order correction DKH3, it is thus mandatory to include the DK transformation of the two-electron terms as well as the spin-dependent terms before proceeding to higher orders in the transformation of the one-electron terms. However, an analysis of the ionization energies of these atoms indicates that already DKH3 yields a highly accurate treatment of the scalar-relativistic effects on properties.

Relativistic calculations of electric field gradients using the Douglas–Kroll method

In this work, the analytical evaluation of Douglas–Kroll (DK) transformed electric field gradient (EFG) integrals was implemented. This allowed us for the first time to perform fully analytical and picture-change consistent calculations of nuclear quadrupole coupling constants in the framework of the DK method. Only scalar relativistic effects were taken into consideration in the present work. The method was applied to the calculation on a series of molecules at both Hartree–Fock and density functional levels of theory. The results were compared to the so-called point charge nuclear quadrupole model (PCNQM) numerical method. The results showed that, for the set of compounds investigated, the second-order DK transformation of the EFG integrals plays only a minor role in comparison with the first-order terms.

Relativistic electronic structure theory.

The theoretical and technical foundations are presented for the efficient relativistic electronic structure theories to treat heavy-atomic molecular systems. This review contains two surveys of four-component and two-component quasi-relativistic approaches. First, we review our highly efficient computational scheme for four-component relativistic ab initio molecular orbital (MO) methods over generally contracted spherical harmonic Gaussian-type spinors (GTSs). Illustrative calculations, which are performed with a new four-component relativistic ab initio molecular orbital program package REL4D, clearly show the efficiency of our computational scheme by the Dirac-Hartree-Fock (DHF) and Dirac-Hartree-Fock (DKS) methods. Next, in the two-component quasi-relativistic framework, two relativistic Hamiltonians, RESC and higher order Douglas-Kroll (DK) Hamiltonians, are introduced, and several illustrative calculations are shown. Numerical results for several systems show that good accuracy can be obtained with our third-order DK (DK3) Hamiltonian.

Nuclear quadrupole moments of Kr and Xe from molecular data

Nuclear quadrupole coupling constants for KrH+, XeH+ and XeD+ are combined with Douglas–Kroll (DK) CCSD(T) calculations to obtain the nuclear quadrupole moments of +259(1) and −114(1) millibarn (1 mb=10−31 m2 for 83Kr and 131Xe, respectively. For the radioactive isotopes 81Kr and 85Kr, the available atomic isotopic ratios give a Q of +644(4) and +443(3) mb, respectively. For the 83Kr 9.4 keV I=7/2 Mössbauer state, Q(83Kr∗) of 507(3) mb is derived from our data. For the 129Xe 40 keV I=3/2 Mössbauer state, Q(129Xe∗) is −393(10) mb.

Quasi-Relativistic Study of 199Hg Nuclear Magnetic Shielding Constants of Dimethylmercury, Disilylmercury and Digermylmercury

Quasi-relativistic ab initio calculations were performed for 199Hg nuclear magnetic shielding constants and chemical shifts in a series of Hg(XH3)2 (X = C, Si, and Ge) compounds. The relativistic terms included were the spin-free relativistic (SFR) term, the one- and two-electron spin−orbit (SO) terms, and the relativistic magnetic interaction (RMI) term. The second-order Douglas−Kroll (D-K) form was adopted for the one-electron SO, SFR, and RMI terms, and the Breit−Pauli form for the two-electron SO term. The calculated results show that the SFR, SO, and RMI terms are all important for calculating the 199Hg shielding constants and chemical shifts of the compounds in question. The SFR and SO terms strongly couple with each other, and the RMI term also strongly affects the paramagnetic and Fermi-contact (FC) terms. The calculated 199Hg chemical shifts are in reasonable agreement with experimental data only if the SFR, SO, and RMI terms are included and tight s basis functions of mercury are used. We found that the total trend of the chemical shifts in Hg(XH3)2 (X = C, Si, and Ge) originates from the sum of the FC and paramagnetic terms, which are the effects of the relativity and the electronegativity of the ligand, respectively.

The higher-order Douglas–Kroll transformation

The higher-order Douglas–Kroll (DK) Hamiltonians in an external potential are explicitly derived. Application of an exponential-type unitary operator with the 2n+1 rule significantly simplifies the formulations of the high-order DK Hamiltonians. The third-order DK method has been implemented practically. Numerical results for one- and many-electron systems show that good accuracy can be obtained even with the DK Hamiltonian correct to third order in the external potential.

The Scalar Relativistic Contribution to the Atomization Energies of CF, CF4, and SiF4

The one-electron Douglas Kroll (DK) approach and perturbation theory, accounting for the mass-velocity and Darwin (MVD) terms, are used to compute the scalar relativistic contribution to the atomization energies of CF, CF4, and SiF4. The difference between these two approaches is studied as a function of basis set and level of correlation treatment.

Ionization potentials of Zn, Cd, Hg and dipole polarizabilities of Zn+, Cd+, Hg+: correlation and relativistic effects

Abstract The ionization potentials (IPs) of the group IIB metals and the electric dipole polarizabilities of their positively charged ions were calculated using the relativistic one-component spin averaged Douglas–Kroll (DK) no pair approximation combined with the CCSD(T) treatment of the electron correlation. Ionization potentials 9.393 (9.394) eV for Zn and 8.941 (8.993) eV for Cd agree with experimental values (in parentheses) excellently and the IP for Hg, 10.362 (10.437) eV agrees with experiment reasonably well. The three factors that can affect the accuracy, i.e. basis set effects, electron correlation and relativistic effects, were carefully examined.

The scalar relativistic contribution to gallium halide bond energies

The one-electron Douglas–Kroll (DK) and perturbation theory (+R) approaches are used to compute the scalar relativistic contribution to the atomization energies of GaF n . These results are compared with previous GaCl n results. While the +R and DK results agree well for the GaCl n atomization energies, they differ for GaF n . The present work suggests that the DK approach is more accurate than the +R approach. In addition, the DK approach is less sensitive to the choice of basis set. The computed atomization energies of GaF2 and GaF3 are smaller than the atomization energies from the somewhat uncertain experiments. It is suggested that additional calibration calculations for the scalar relativistic effects in GaF2 and GaF3 would be valuable.

A two-centre implementation of the Douglas–Kroll transformation in relativistic calculations

An implementation of the Douglas–Kroll (DK) transformation is described within a new relativistic quantum chemistry code, MAGIC, which performs calculations on systems containing heavy atoms. This method reduces the computational cost in terms of memory requirements that are associated with completeness identities in the DK implementation by factorizing the one-electron matrices into smaller ones that depend only on two atoms at a time. Examples are presented.

Relativistic and correlation effects in CuH, AgH, and AuH: Comparison of various relativistic methods

The effects of relativity on the bond lengths, dissociation energies, and harmonic vibrational frequencies of the 1Σ+ electronic ground states of the group IB hydrides CuH, AgH, and AuH have been evaluated with a variety of ab initio methods. These properties were investigated with moderately-sized basis sets at the self-consistent field Hartree–Fock (SCF-HF) level and with second-order Mo/ller–Plesset (MP2) perturbation theory for electron correlation. Comparisons were made between all-electron results using the nonrelativistic Hamiltonian, perturbation theory (PT-MVD) at first-order with only the one-electron nonfine-structure terms of the Breit–Pauli Hamiltonian, the spin-free Douglas–Kroll (DK) transformed Dirac Hamiltonian and the untransformed Dirac Hamiltonian, and results using two sets of relativistic effective core potentials (RECPs). The expected trends of bond length decrease, dissociation energy increase, and harmonic vibrational frequency increase with both relativity and correlation are found. Both sets of RECPs are shown to give good results, if accompanied by a reasonable basis set. The DK method is demonstrated to be an inexpensive, reliable approximation to the DHF method.