Douglas Kroll Method Research Articles

An Intruder-Free Fock Space Coupled-Cluster Study of the Potential Energy Curves of LiMg+ within the (2,0) Sector.

The potential energy curves (PECs) and spectroscopic constants of the ground and excited states of a LiMg+ molecular cation were investigated. We obtained accurate results for the fifteen lowest-lying states of the LiMg+ cation using the Intermediate Hamiltonian Fock Space Multireference Coupled Cluster (IH-FS-CC) method applied to the (2,0) sector. Relativistic corrections were accounted for using the third-order Douglas-Kroll method. In each instance, smooth PECs were successfully computed across the entire range of interatomic distances from equilibrium to the dissociation limit. The results are in good accordance with previous studies of this molecular cation. Notably, this study marks the first application of IH-FS-CC in investigating a mixed alkali and alkaline earth molecular cation, proving its usability in determining accurate PECs of such diatomics and their spectroscopic constants.

Two-component transformation inclusive contraction scheme in the relativistic molecular orbital theory

The incorporation of a two-component transformation into the contraction coefficients of basis functions is suggested. Such a contraction, referred to as “two-component transformation inclusive contraction (TIC)”, effectively eliminates the computational cost associated with the two-component transformation and can be readily implemented into conventional quantum chemistry programs. TIC was verified through numerical calculations using the second- and third-order unitarized Douglas–Kroll method, which is newly derived for the validation of TIC, and the infinite-order two-component method. The numerical validation results suggested that TIC could sufficiently reproduce the results of primitive basis sets for both atoms and molecules.

Relativistic two-electron repulsion operator formulas for the Douglas–Kroll method

Two-electron repulsion operator formulas for the second- and third-order Douglas–Kroll (DK2 and DK3) method were derived and implemented. The formulas were applied to helium-like ions with Z = 10, 20, …, 130, and to rare gas atoms. The energies obtained for the DK2 and DK3 two-electron operators showed differences from the four-component method that were one order of magnitude smaller than those of the nonrelativistic and DK1 two-electron operators. Approximation formulas for the small component-type two-electron integrals were also derived; they were applied to the helium-like ions with Z = 80 and 130 and were numerically confirmed to reproduce unapproximated values very accurately.

Potential energy curves of the Na2+ molecular ion from all-electron ab initio relativistic calculations

ABSTRACTThe equation-of-motion coupled-cluster method for electron affinity calculations has been used to study potential energy curves (PECs) for the Na+2 molecular ion. Although the studied molecule represents the open shell system the applied approach employs the closed shell Na+ 22 ion as the reference. In addition the Na+ 22 system dissociates into the closed shell fragments; hence, the restricted Hartree–Fock scheme can be used within the whole range of interatomic distances, from 2 to 45 Å. We used large basis set engaging 268 basis functions with all 21 electrons correlated. The relativistic effects are included via second-order Douglas–Kroll method. The computed PECs, spectroscopic molecular constants and vibrational energy levels agree well with experimental values if the latter are available or with other theoretical data.

Model Core Potential and All-Electron Studies of Molecules Containing Rare Gas Atoms†

Molecules HRgF (Rg = Ar, Kr, Xe, Rn) were studied at levels of theory that included electron correlation, taking relativistic effects into account either by using the recently developed parametrization of the extended model core potentials and basis sets or by using the Douglas-Kroll method with all-electron basis sets. Charge distributions were calculated according to Mulliken, Lowdin, and natural bond orbital methods of population analysis and the results of these methods were compared, confirming that bonding in these molecules corresponds to interaction between the fluoride anion and the RgH(+) moiety. In contrast to previously reported results, the present calculations show that the radon compound, HRnF, is more stable than compounds of the lighter congeners. Trends in the first ionization energies, bond lengths, energies of formation and decomposition, and harmonic vibrational frequencies were discussed and found to be consistent with the periodic trends of the atomic properties of the rare gas atoms.

An arbitrary order Douglas–Kroll method with polynomial cost

A new Douglas-Kroll transformation scheme up to arbitrary order is presented to study the convergence behavior of the Douglas-Kroll series and the influence of different choices of parametrization for the unitary transformation. The standard approach for evaluating the Douglas-Kroll Hamiltonian suffers from computational difficulties due to the huge number of matrix multiplications, which increase exponentially with respect to the order of truncation. This makes it prohibitively expensive to obtain results for very high order Douglas-Kroll Hamiltonians. The highest order previously presented is 14th order, but it is not enough to obtain accurate results for systems containing heavy elements, where the Douglas-Kroll series converges very slowly. In contrast, our approach dramatically reduces the number of matrix multiplications, which only increase with a polynomial scaling. With the new method, orders greater than 100 and machine accuracy are possible. This fast method is achieved by employing a special transformation to all Douglas-Kroll operators and our algorithm is very simple. We demonstrate the performance of our implementation with calculations on one-electron systems and many-electron atoms. All results show very good convergence behavior of the Douglas-Kroll series. Very small differences are found between the different parametrizations, and therefore the exponential form, which is the simplest and fastest, is recommended.

Relativistic corrections to electrical first-order properties using direct perturbation theory

Direct perturbation theory (DPT) is applied to compute relativistic corrections to electrical properties such as dipole moment, quadrupole moment, and electric-field gradient. The corrections are obtained as second derivatives of the energy and are given via method-independent expressions that involve the first derivative of the density matrix with respect to the relativistic perturbation as well as property integrals with additional momentum operators. Computational results obtained using Hartree-Fock (HF), second-order Moller-Plesset (MP2) perturbation theory, and the coupled-cluster singles and doubles approach augmented by a perturbative treatment of triple excitations are presented for the hydrogen halides HX with X=F, Cl, Br, (I, At) and the magnitude of relativistic effects, their basis-set dependence, and the limitations of DPT are discussed. We compare our results to those obtained using the second-order Douglas-Kroll method and benchmark them using four-component HF (Dirac-HF) and MP2 calculations. Relativistic effects are shown to be already important for elements of the third row (Na-Ar) when aiming at a high-accuracy quantum-chemical treatment. DPT provides reliable results for compounds containing elements up to the fourth period (K-Kr) and only breaks down when applied in lowest order to heavier elements. As a first application of the present DPT treatment for electrical properties, we report calculations for bromofluoromethane (CH(2)FBr) which was investigated using rotational spectroscopy by Cazzoli et al. [Mol. Phys. 106, 1181 (2008)] and for which consideration of relativistic effects turns out to be essential for good agreement between theory and experiment in the case of the bromine quadrupole-coupling constant.

Sixth-order Douglas–Kroll: two-component reference data for one-electron ions from through

Abstract Quasirelativistic Douglas–Kroll calculations including spin–orbit interaction have been performed for hydrogen-like (one-electron) ions. The Douglas–Kroll operators have been implemented up to the sixth order, and a huge even-tempered basis set has been applied that gives results to microhartree accuracy for the energy levels 1 s 1 2 through 4 f 7 2 for all ions with nuclear charge from Z = 1 up to Z = 100. Besides providing reference data for other implementations, these results can be used to analyse the performance of the Douglas–Kroll method. Such an analysis is presented for the 1s energy levels and the spin–orbit splitting of the 2p shell. The leading order of the error of the Douglas–Kroll result is different for 2s and 2 p 1 2 although these levels are degenerate both at nonrelativistic and Dirac level.

Spin–orbit interactions and supersymmetry in two-component relativistic methods

The spin–orbit coupling effects are investigated for the Douglas–Kroll and infinite-order two-component relativistic hamiltonians. The infinite-order theory reproduce all important features of the four-component Dirac theory, including the supersymmetric pattern of energy level. The Douglas–Kroll method can not give the right values of the spin orbit splitting.

Correlated ab initio calculations of spectroscopic parameters of SnO within the framework of the higher-order generalized Douglas-Kroll transformation.

The first molecular calculations with the generalized Douglas-Kroll method up to fifth order in the external potential (DKH5) are presented. We study the spectroscopic parameters and electron affinity of the tin oxide molecule SnO and its anion SnO(-) applying nonrelativistic as well as relativistic calculations with higher orders of the DK approximation. In order to guarantee highly accurate results close to the basis set limit, an all-electron basis for Sn of at least quintuple-zeta quality has been constructed and optimized. All-electron CCSD(T) calculations of the potential energy curves of both SnO and SnO(-) reproduce the experimental values very well. Relative energies and valence properties are already well described with the established standard second-order approximation DKH2 and the higher-order corrections DKH3-DKH5 hardly affect these quantities. However, an accurate description of total energies and inner-shell properties requires superior relativistic schemes up to DKH5.

Electric moments of carbon dichalcogenides

The dipole and quadrupole moments and parallel dipole polarizabilities of the carbon dichalcogenides XCY (X, Y = O, S, Se and Te) are evaluated at the level of the CCSD(T) approximation with HyPolX basis sets. The relativistic effects are taken into account by using the scalar version of the two-component Douglas–Kroll method. The reported calculations provide a systematic study of the electron correlation and relativistic effects in the investigated series of carbon dichalcogenides and their results agree well with best-known experimental data. Particular attention has been given to the OCS molecule and the present calculations combined with recent studies by other authors may help to resolve the ambiguities concerning its quadrupole moment. Changes of dipole and quadrupole moments in the studied series of molecules are interpreted in terms of the electronegativities of the chalcogene atoms and their changes due to relativistic effects. The latter have been found to lead to significant changes in the calculated quadrupole moments of the heavy members of the series of molecules. Study of the basis set dependence of the calculated data shows that both the earlier PolX and the recently developed HyPolX basis sets are suitable for calculations of reliable values of the low-multipole moments and dipole polarizabilities of molecules containing heavy elements.

On the physical meaning of the ZORA Hamiltonian

By expanding the Foldy-Wouthuysen representation of the Dirac equation near the free-particle solution it is shown that the Hamiltonian of the zeroth-order regular approximation (ZORA) leads to an infinite summation of the leading relativistic corrections to the free-particle, non-relativistic energy. The analysis of the perturbation expansion of the ZORA Hamiltonian reveals that the ZORA Hamiltonian recovers all terms of the Breit-Pauli theory to second order. This result is general and applies not only to hydrogen-like atomic ions (as was demonstrated before) but also to a wide variety of physical problems. ZORA is analogous to the random phase approximation in many-body theory in the sense that both methods include an infinite-order summation of the asymptotically non-vanishing terms. This highlights the difference between ZORA and the Douglas-Kroll method, with the latter being analogous to finite-order many-body perturbation theory. On the basis of this analysis the performance of ZORA when calculating various molecular properties is discussed.

An ab initio study of excited states of U and UF

By using the third-order Douglas-Kroll method and spin–orbit multiconfiguration quasidegenerate perturbation theory we have been able to predict the spectrum of odd parity states of uranium atom within at worst 1000–2000 cm−1 accuracy for the 48 states presented. For UF we have given Morse potential parameters for the lowest 18 Ω states. The ground state was found to be I9/24. The importance of dynamic correlation effects including core correlation treated on the same footing with spin–orbit coupling is elucidated.

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.

A theoretical study of the low-lying states of the AuSi molecule: An assignment of the excited A and D states

The lowest 42 electronic states of diatomic gas phase gold silicide are calculated by complete active space self-consistent field and second-order multiconfigurational perturbation theory. The scalar relativistic effect is considered by the third-order Douglas-Kroll method and the spin–orbit coupling is treated perturbatively after the spin-free calculations. The ground state of AuSi is confirmed to be the Π21/2 state, and the spin–orbit splitting of Π21/2 to Π23/2 is estimated as 1527 cm−1. From the careful analysis of the potential energy curves and the spectroscopic constants, the experimentally observed A and D excited states are assigned to Ω=1/2(II) or Ω=1/2(III) and Ω=1/2(V) or Ω=1/2(VI), respectively.

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 Effects for Polarizabilities and Hyperpolarizabilities of Rare Gas Atoms

Abstract The relativistic effects for static polarizabilities α and hyperpolarizabilities γ of rare gas atoms (Ne through Rn) are investigated theoretically with the third-order Douglas–Kroll method. The results yield interesting trends of relativity in going from Ne to Rn: The relativistic effect decreases α values of Ne and Ar and increases the values of Kr, Xe, and Rn, and it decreases γ values of Rn slightly and increases the values of other rare gas atoms, while the electron correlation effect increases α and γ of all rare gas atoms.

The change of picture of the Hellmann–Feynman force operator in approximate relativistic methods

The calculation of energy gradients in one- and two-component approximate methods of relativistic quantum chemistry which follow from the block-diagonalisation of the Dirac hamiltonian, is considered. It is indicated that the transition to these approximate methods involves the so-called change of picture for all operators. In particular this applies to the Hellmann–Feynman force operator. To avoid explicit transformation of this operator to the new picture two schemes for its modelling in terms of the usual Coulomb attraction operators are proposed. One of them is based on the point charge nuclear dipole moment model. The other one, the so-called shifted nucleus model, involves a parametrised shift of the nucleus. Both these models lead to a semianalytical method for the evaluation of energy gradients in approximate relativistic calculations, in which the relaxation terms is obtained analytically whereas the matrix elements of the Hellmann–Feynman force are calculated by using the finite difference method. The accuracy and relative merits of the two models are analysed. The two models are tested within the one-component Douglas–Kroll method. A large change of picture contribution is found in calculations of intramolecular forces in the coinage metal hydrides. This shows that any approximate relativistic technique for calculations of the energy gradients and for the relativistic geometry optimisation must take into account the change of picture contribution to the evaluated Hellmann–Feynman force.

Spin-orbit effects in electric field gradients ofalkali metal atoms

Spin-orbit effects in electric field gradients (EFG) of alkaliatoms in the 2P3/2 state are investigated bycomparison of correlated (CCSD(T)) four-component Dirac-Fockresults with scalar relativistic Douglas-Kroll calculations.Since the Douglas-Kroll method is based on a unitarytransformation of the full four-component Dirac Hamiltonian toa two-component form the use of the untransformed EFG operatorleads to the well known picture change error. We therefore usethe point charge nuclear quadrupole moment method incombination with the Douglas-Kroll transformation which removespicture-change effects since only Coulomb-type operators are tobe transformed. We find that spin-orbit effects partiallycompensate scalar relativistic effects and are only importantfor the heavier elements Rb and Cs. A comparison to EFGs derivedfrom valence-only ⟨r-3⟩ values shows that Sternheimercorrections are roughly proportional to the dipolepolarizability of the positively charged atom and can becomevery large for the heaviest atom studied, Cs, amounting to 46%of the total EFG.

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.