Douglas-Kroll-Hess Research Articles

Relativistic Douglas–Kroll–Hess calculations of hyperfine interactions within first-principles multireference methods

A relativistic magnetic hyperfine interaction Hamiltonian based on the Douglas–Kroll–Hess (DKH) theory up to the second order is implemented within the ab initio multireference methods, including spin–orbit coupling in the Molcas/OpenMolcas package. This implementation is applied to calculate relativistic hyperfine coupling (HFC) parameters for atomic systems and diatomic radicals with valence s or d orbitals by systematically varying active space size in the restricted active space self-consistent field formalism with restricted active space state interaction for spin–orbit coupling. The DKH relativistic treatment of the hyperfine interaction reduces the Fermi contact contribution to the HFC due to the presence of kinetic factors that regularize the singularity of the Dirac delta function in the nonrelativistic Fermi contact operator. This effect is more prominent for heavier nuclei. As the active space size increases, the relativistic correction of the Fermi contact contribution converges well to the experimental data for light and moderately heavy nuclei. The relativistic correction, however, does not significantly affect the spin-dipole contribution to the hyperfine interaction. In addition to the atomic and molecular systems, the implementation is applied to calculate the relativistic HFC parameters for large trivalent and divalent Tb-based single-molecule magnets (SMMs), such as Tb(III)Pc2 and Tb(II)(CpiPr5)2 without ligand truncation using well-converged basis sets. In particular, for the divalent SMM, which has an unpaired valence 6s/5d hybrid orbital, the relativistic treatment of HFC is crucial for a proper description of the Fermi contact contribution. Even with the relativistic hyperfine Hamiltonian, the divalent SMM is shown to exhibit strong tunability of HFC via an external electric field (i.e., strong hyperfine Stark effect).

Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory.

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Ab initio relativistic potential energy surface with analytical long-range part of benzene-Rn complex and its application to intermolecular vibrations

We study the benzene-Rn (BzRn) complex in its electronic ground state using an ab initio scalar relativistic method in the framework of the coupled-cluster level of theory with single, double, and perturbative triple excitations [CCSD(T)]. Two different approaches to treat the scalar relativistic effects have been tested — the small-core pseudopotential (PP) and the Douglas–Kroll–Hess (DKH) Hamiltonian. These methods were used with the augmented Dunning’s double zeta basis set for C and H atoms, and the augmented triple PP basis set for Rn supplemented by midbond functions. Additionally, the effects of the basis extension for C and H atoms and the extension of the correlation space to include the internal orbitals of Rn and C have been tested. The comparison of the binding energy De and the equilibrium coordinate ze found with different methods has allowed to choose the optimal one based on PP for the construction of the analytical potential energy surface (PES). The long-range part of the potential has been fitted separately and the obtained long-range coefficients have been compared to their counterparts calculated from the first principles using the CCSD polarization propagator. The vibrational energy level pattern of BzRn has been studied, and the revealed polyad structure of the complex has been compared to the previously studied benzene-Kr and benzene-Xe complexes. A Python code that yields the interaction energy at any Rn position is provided.

Δ‐SCF calculations of core electron binding energies in first‐row transition metal atoms

AbstractCore electron binding energies (CEBE) of nickel and copper atoms have been calculated using single‐configuration energy differences, the so‐called Δ‐self‐consistent field (Δ‐SCF) method. Basis set convergence has been examined for calculated L‐shell and M‐shell core electron binding energies, and a wide array of density functionals have been evaluated. Scalar relativistic corrections have been estimated using the popular Douglas‐Kroll‐Hess (DKH) approximation. While basis set convergence and functional dependence mirror the behavior reported in the literature for main group elements, the simplest Δ‐SCF calculations with pure Hartree‐Fock (HF) exchange and no correlation surprisingly outperform all density functionals in reproducing free‐atom CEBE values for Ni and Cu.

Coupled cluster study of the partial charge of hydrogen in HnX molecules in s and p block: Relationship with electronegativity

Bader analysis of HnX was performed using densities generated by second-order one electron Douglas-Kroll-Hess (DKH) Hamiltonian RCCSD(T)//aug-cc-pVTZ level of theory for the 2p, 3p and 4p subshells, and by DKH Hamiltonian RCCSD(T)//aug-cc-pVTZ-PP for the 5p and 6p subshells. The partial charges of hydrogen atoms are closely related to the original definition of electronegativity. Direct computational results of hydrogen partial charges were tabulated in a table and compared to the Pauling electronegativity scale as well as other electronegativity scales. A strong correlation between these different scales and the computational results for hydrogen partial charges was found.

Low-lying electronic terms of diatomic molecules AB (A = Sc–Ni, B = Cu/Ag/Au)

Low-lying electronic terms of 24 heteronuclear diatomic molecules AB (A = Sc–Ni, B = Cu/Ag/Au) have been systematically studied. Scalar relativistic effects were included by use of the spin-free Douglas–Kroll–Hess (DKH) Hamiltonian. The complete active space self-consistent field (CASSCF) method, followed by multi-reference configuration interaction (MRCI), was used to construct full potential energy curves (PECs). Spectroscopic constants as well as dipole moments are also reported. Spin-orbit coupling was subsequently calculated perturbatively via the spin-orbit terms of the Breit–Pauli Hamiltonian based on the CASSCF wavefunctions. Full spin-orbit coupled PECs were thus constructed for the low-lying terms of NiCu and NiAg. In addition, Kramers-restricted configuration interaction (KRCI) with the exact 2-component (X2C) Hamiltonian was utilised for the atoms and for the Cu-containing diatomic molecules. The influence of relativistic effects, spin-orbit coupling, core correlation effects and basis set incompleteness was probed for some selected cases. It is shown that for 23 out of the 24 diatomic molecules considered here a direct correspondence exists between the ground state terms (GSTs) of the monocations and those of the diatomic molecules with and .

All-electron basis sets augmented with diffuse functions for He, Ca, Sr, Ba, and lanthanides: application in calculations of atomic and molecular properties.

Non-relativistic and Douglas-Kroll-Hess (DKH) basis sets augmented with diffuse functions for He, Ca, Sr, Ba, and lanthanides are generated. These sets are appropriated to describe electrons away from the nuclei. Using the DKH augmented sets along with the B3LYP functional, bond lengths, dissociation energies, harmonic vibrational frequencies, adiabatic ionization potentials, adiabaticelectron affinities, anddipole moments for CaH, SrH, and BaH are computed. These results agree well with the most recent experimental and benchmark theoretical data published in the literature. The DKH mean dipole polarizabilities reported in this work for some elements are close to the recommended values. Scalar relativistic effects are also estimated.

Ab initio relativistic potential energy surfaces of benzene-Xe complex with application to intermolecular vibrations.

The benzene-Xe (BXe) complex in its electronic ground state is studied using ab initio methods. Since this complex contains the heavy Xe atom, the relativistic effects cannot be neglected. We test two different approaches that describe the scalar relativistic effects in the framework of the coupled-cluster level of theory with single, double, and perturbative triple excitations, used for the interaction energy calculations. The first one is based on the small core pseudopotential (PP), and the second one is based on the explicit treatment of scalar relativistic effects using the Douglas-Kroll-Hess (DKH) Hamiltonian. A few basis sets are tested with the PP and DKH, and for each one, the analytical potential energy surface (PES) is constructed. It is shown that the difference between PESs determined with PP and DKH methods is small, if the orbitals of the 4d subshell in Xe are correlated. We select the most appropriate approach for the calculation of the potential energy surface of BXe, with respect to accuracy and computational cost. The optimal level of theory includes a small Dunning's basis set for the benzene monomer and a larger PP basis set for Xe supplemented by midbond functions. The PES obtained using such an approach provides a reasonable accuracy when compared to the empirical one derived from the microwave spectra of BXe. The empirical and the theoretical values of intermolecular vibrational energies agree within 0.5 cm-1 up to second overtones. The vibrational energy level pattern of BXe is characterized by a distinct polyad structure.

Real-space quasi-relativistic quantum chemistry

We present for the first time real-space, arbitrarily-accurate representations of the operators required for up to second-order Douglas-Kroll-Hess (DKH), a model for constructing quasi-relativistic electronic Hamiltonians. The approach can be extended to other operator-based quasi-relativistic models. The representations are in the form of sums of Gaussian functions with positive coefficients and thus enable efficient and numerically-accurate formulations using conventional Gaussian basis sets or other bases such as multiwavelets. The operators are demonstrated with application to hydrogen-like systems using the relativistic-kinematic and first-order DKH Hamiltonians.

Jahn-Teller effect versus spin-orbit coupling: The structure of the free molybdenum pentafluoride molecule

The complete active space self-consistent field (CASSCF) and multiconfigurational quasi-degenerate second-order perturbation (MCQDPT2) calculations have been used to study the ground and low-lying excited electronic states of MoF5. Scalar-relativistic effects and spin-orbit coupling (SOC) have been taken into account employing the third-order Douglas-Kroll-Hess (DKH) Hamiltonian and full Breit-Pauli operator, respectively. The optimization and the calculation of the Hessian matrix have been performed by non-routine procedure employing the symmetry coordinates for all considered (D3h, C2v and C4v) geometrical configurations. The trigonal bipyramidal D3h structure of molybdenum pentafluoride MoF5 possesses two-fold degenerate ground first excited electronic orbital state and undergoes Jahn-Teller distortion to the C2v configurations. Accounting of SOC quenches the JT effect. As a result the D3h structure of MoF5 is no longer the conical intersection of two adiabatic potential energy surfaces (APES) but two separated APES corresponding to ground (12E1/2) and nearest excited (12E3/2) spin-orbit states are appeared instead. The D3h trigonal bipyramidal structure in the ground spin-orbit state corresponds to the 2-nd order saddle point on APES and lies above the minima (C2v structure, 12E1/2 state) by 244 cm−1. The warping barrier (another C2v structure) in the trough of APES and the Berry pseudorotation barrier (C4v structure, ground spin-orbit state 2E3/2) are 19 and 310 cm−1, respectively. Therefore MoF5 molecule is characterized in the gas phase by a large-amplitude motion of fluorine atoms in the equatorial plane and non-rigid intramolecular rearrangement by Berry mechanism. Despite such non-rigid nuclei dynamic, the calculated harmonic frequencies of MoF5 molecule do not contradict to available experimental IR spectrum. The repeat refinement of MoF5 gas electron diffraction data has been carried out using static and pseudoconformer models based on the SO-MCQDPT2 results. Corresponding to only extreme points on APES the configurations have been included into the pseudoconformer model. The both models have been fitted to the experimental data by the same quality.

Jahn-Teller effect enhanced by spin-orbit coupling. From theory to Experiment:Example of niobium tetrachloride NbCl4

The ground and lower-lying excited electronic states of NbCl4 molecule were systematically studied by the complete active space self-consistent field (CASSCF) and multiconfigurational quasi-degenerate second-order perturbation (MCQDPT2) methods. Scalar-relativistic effects and spin-orbit coupling (SOC) have been taken into account employing the third-order Douglas-Kroll-Hess (DKH) Hamiltonian and full Breit-Pauli operator, respectively. The high-symmetrical Td structure with spin-orbit ground 2G3/2 state undergoes Jahn-Teller distortion along all non-totally symmetric vibrational coordinates. As a result the compressed (2E1/2 state) and elongated (2E3/2 state) D2d structures correspond to minimum and first order saddle point on the lower sheet of adiabatic potential energy surface (APES). The height of the warping barrier in the trough of APES and the Jahn-Teller stabilization energy are 179 and 438 cm−1, respectively. For two-mode 2E⊗e Jahn-Teller problem the vibronic Schrödinger equation has been solved in the diabatic basis. It was shown, the topology of the lower sheet determined mainly the energies and wave functions ground and few low-lying vibronic levels. The original methodology of the simulation of gas-phase electron diffraction (GED) data was developed. This methodology is based on the analytic description of some APES numerically obtained. The computed radial distribution function and thermal average parameters were compared with the experimental GED data. A good agreement between experimental and theoretical results supports the reliability of the theoretical data.

A simple model for scalar relativistic corrections to molecular total atomisation energies

ABSTRACTScalar relativistic corrections to atomisation energies of first- and second-row molecules can be rationalised in terms of a simple additive model, linear in changes in atomic s populations. In a sample of 200 first-and second-row molecules, such a model can account for over 98% of the variance (99% for the first-row subset). The remaining error can be halved again by adding a term involving the change in atomic p populations: those coefficients need not be fitted but can be fixed from atomic electron affinity calculations. This model allows a fairly accurate a priori estimate for the importance of scalar relativistic corrections on a reaction energy, at essentially zero computational cost. While this is not a substitute for explicit calculation of Douglas–Kroll–Hess (DKH) or exact two-component (X2C) relativistic corrections, the model offers an interpretative tool for the chemical analysis of scalar relativistic contributions to reaction energies.

All-electron triple zeta basis sets for the actinides

Abstract Non-relativistic and Douglas-Kroll-Hess (DKH) segmented all-electron basis sets of valence triple zeta quality plus polarization functions (TZP) for the actinides are developed. To describe accurately the properties that depend on a good description of the electrons far away from the nuclei, the corresponding augmented sets (ATZP and ATZP-DKH) are reported as well. For the actinide monoxides, bond distances, dissociation energies, natural charges and populations of the valence orbitals of the actinides, and bond indices are calculated with the B3LYP/TZP-DKH procedure. For Am and No, the B3LYP/ATZP-DKH static mean dipole polarizabilities are also computed. To assess the performance of these small size all-electron basis sets, comparison with theoretical and experimental data reported previously in the literature is done. These sets must be also helpful on calculations of properties involving simultaneously core and valence electrons.

Excited interatomic potential energy surfaces of Rb + He that correlate with Rb terms 52S through 72S.

The excited state interatomic potential energy surfaces for Rb + He are computed at the spin-orbit multi-reference configuration interaction level of theory using all-electron basis sets of triple and quadruple-zeta quality that have been contracted for Douglas-Kroll-Hess (DKH) Hamiltonian and includes core-valence correlation. Davidson-Silver corrections (MRCI+Q) are employed to ameliorate size consistency error. An extrapolation of CASSCF energies is performed using the procedure of Karton and Martin whereas extrapolation of correlation energy is performed using an expression involving the inverse powers of (lmax + 1/2), the highest angular momentum value present in the basis set. The spin-orbit energies in the limit of complete basis set are obtained by replacing the energy eigenvalues in the spin-orbit matrix by the relativistic-corrected MRCI+Q energies extrapolated to the complete basis set limit. MRCI diabatic potential energy surfaces for a few selected 2Σ states are calculated to study the general topology and avoided crossings and repulsive form of the 6s 2Σ+ state. Important features of the potential energy surfaces are discussed with implications for alkali laser spectroscopy.

Gaussian basis sets for use in correlated molecular calculations. XI. Pseudopotential-based and all-electron relativistic basis sets for alkali metal (K-Fr) and alkaline earth (Ca-Ra) elements.

New correlation consistent basis sets based on pseudopotential (PP) Hamiltonians have been developed from double- to quintuple-zeta quality for the late alkali (K-Fr) and alkaline earth (Ca-Ra) metals. These are accompanied by new all-electron basis sets of double- to quadruple-zeta quality that have been contracted for use with both Douglas-Kroll-Hess (DKH) and eXact 2-Component (X2C) scalar relativistic Hamiltonians. Sets for valence correlation (ms), cc-pVnZ-PP and cc-pVnZ-(DK,DK3/X2C), in addition to outer-core correlation [valence + (m-1)sp], cc-p(w)CVnZ-PP and cc-pwCVnZ-(DK,DK3/X2C), are reported. The -PP sets have been developed for use with small-core PPs [I. S. Lim et al., J. Chem. Phys. 122, 104103 (2005) and I. S. Lim et al., J. Chem. Phys. 124, 034107 (2006)], while the all-electron sets utilized second-order DKH Hamiltonians for 4s and 5s elements and third-order DKH for 6s and 7s. The accuracy of the basis sets is assessed through benchmark calculations at the coupled-cluster level of theory for both atomic and molecular properties. Not surprisingly, it is found that outer-core correlation is vital for accurate calculation of the thermodynamic and spectroscopic properties of diatomic molecules containing these elements.

Effects of All-Electron Basis Sets and the Scalar Relativistic Corrections in the Structure and Electronic Properties of Niobium Clusters.

In this paper, an augmented all-electron double-ζ basis set is used in calculations of the structure and electronic properties of small niobium clusters. The B3PW91 and M06 DFT functionals with and without second order Douglas-Kroll-Hess (DKH) scalar relativistic corrections are also utilized. Furthermore, an additional d Gaussian type function is introduced in the standard basis sets in order to improve the description of the clusters orbitals in the valence band. Our findings show that the extra d function is important to yield accurate results of electronic properties and, in addition, the DKH corrections can be relevant when the all-electron basis sets are used in the calculations. Our best results are obtained with the M06 functional together with the DKH second order corrections and with the extra d function added to the all-electron basis set.

Segmented all-electron Gaussian basis sets of double and triple zeta qualities for Fr, Ra, and Ac

Segmented all-electron basis sets of valence double and triple zeta qualities plus polarization functions for the elements Fr, Ra, and Ac are generated using non-relativistic and Douglas-Kroll-Hess (DKH) Hamiltonians. The sets are augmented with diffuse functions with the purpose to describe appropriately the electrons far from the nuclei. At the DKH-B3LYP level, first atomic ionization energies and bond lengths, dissociation energies, and polarizabilities of a sample of diatomics are calculated. Comparison with theoretical and experimental data available in the literature is carried out. It is verified that despite the small sizes of the basis sets, they are yet reliable.

Estimation of ΔR/R values by benchmark study of the Mössbauer Isomer shifts for Ru, Os complexes using relativistic DFT calculations

The present study applies all-electron relativistic DFT calculation with Douglas-Kroll-Hess (DKH) Hamiltonian to each ten sets of Ru and Os compounds. We perform the benchmark investigation of three density functionals (BP86, B3LYP and B2PLYP) using segmented all-electron relativistically contracted (SARC) basis set with the experimental Mössbauer isomer shifts for 99Ru and 189Os nuclides. Geometry optimizations at BP86 theory of level locate the structure in a local minimum. We calculate the contact density to the wavefunction obtained by a single point calculation. All functionals show the good linear correlation with experimental isomer shifts for both 99Ru and 189Os. Especially, B3LYP functional gives a stronger correlation compared to BP86 and B2PLYP functionals. The comparison of contact density between SARC and well-tempered basis set (WTBS) indicated that the numerical convergence of contact density cannot be obtained, but the reproducibility is less sensitive to the choice of basis set. We also estimate the values of ΔR/R, which is an important nuclear constant, for 99Ru and 189Os nuclides by using the benchmark results. The sign of the calculated ΔR/R values is consistent with the predicted data for 99Ru and 189Os. We obtain computationally the ΔR/R values of 99Ru and 189Os (36.2 keV) as 2.35×10−4 and −0.20×10−4, respectively, at B3LYP level for SARC basis set.

All-electron Gaussian basis sets of double zeta quality for the actinides.

For the actinides, two segmented all-electron basis sets of valence double zeta quality plus polarization functions (DZP) are developed. One of them must be used along with the non-relativistic Hamiltonian, whereas the other with the Douglas-Kroll-Hess (DKH) one. Adding diffuse functions of s, p, d, f, and g symmetries to the non-relativistic and relativistic sets, augmented basis sets are developed. These functions are essential to describe correctly electrons far away from the nuclei. For some compounds, geometric parameters, atomic charges and valence orbital populations of the actinides, and bond dissociation energies are calculated using the Becke 3-parameter (exchange) and the Lee, Yang, and Parr (correlation) functional in conjunction with the DZP-DKH basis set. For Am and No, the static electric mean dipole polarizabilities are also reported. Comparison with benchmark theoretical and experimental values found in the literature is carried out. It is verified that the performances of the relativistic compact size basis sets generated in this work are regular, efficient, and reliable. They will be extremely helpful in molecular property calculations that need explicitly to consider the core electrons.

Dipole (hyper)polarizabilities of neutral silver clusters**Project supported by CNPq, CAPES, and FAPES (Brazilian Agencies).

At the Douglas–Kroll–Hess (DKH) level, the B3PW91 functional along with the all-electron relativistic basis sets of valence triple and quadruple zeta qualities are used to determine the structure, stability, and electronic properties of the small silver clusters (Agn, n ⩽ 7). The results presented in this study are in good agreement with the experimental data and theoretical values obtained at a higher level of theory from the literature. Static polarizability and hyperpolarizability are also reported. It is verified that the mean dipole polarizability per atom exhibits an odd-even oscillation and that the polarizability anisotropy is directly related to the cluster shape. In this article, the first study of hyperpolarizabilities of small silver clusters is presented. Except for the monomer, the second hyperpolarizabilities of the silver clusters are significantly larger than those of the copper clusters.