Relativistic electron densities in the four-component Dirac representation and in the two-component picture

Autschbach, Jochen; Schwarz, Winfried H.

doi:10.1007/s002149900108

Cited by 26 publications

(6 citation statements)

References 19 publications

(25 reference statements)

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“…In the following three subchapters, treatment of PCE in electron density, Laplacian of electron density, and the Fourier transform of electron density (structure factors) is briefly presented. In general, it has been found that PCE dominantly affects the core electron density and properties related to electron density in the vicinity and/or at the nucleus, such as hyperfine coupling, , quadrupole moments, contact densities ,, and effective densities. , Properties related to electron density in the valence region (e.g., dipole moments and/or polarizabilities) − are not affected by PCE and the same is found for the electron density itself. ,,, Nevertheless, a common exploration of PCE in electron density, its Laplacian or Fourier transform for compounds of heavy elements is being worthwhile. Foremostly, the direct comparison to the relativistic as well as electron correlation effects is to be addressed, including the radial significance with respect to the core region and the relevance on differences in structure factors with respect to sin(θ)/λ.…”

Section: Methodsmentioning

confidence: 99%

“…According to eq , diagonalization of the Dirac–Coulomb Hamiltonian simultaneously accounts for a change of the resulting quasirelativistic wave function Ψ̃. ,, This was recognized in the physics realm already with the introduction of the free particle Foldy–Wouthuysen transformation, and has been later explored also in the quantum chemistry community. , Why picture change affects a particular property at the quasirelativistic level of theory can be understood by the inspection of the formal mapping between the expectation value of a general property operator X̂ in the Dirac picture (Ψ) and the unitary transformed IOTC picture (Ψ̃): ,,, …”

Section: Methodsmentioning

confidence: 99%

“…89,90 Properties related to electron density in the valence region (e.g., dipole moments and/or polarizabilities) 91−95 are not affected by PCE and the same is found for the electron density itself. 39,58,80,88 Nevertheless, a common exploration of PCE in electron density, its Laplacian or Fourier transform for compounds of heavy elements is being worthwhile. Foremostly, the direct comparison to the relativistic as well as electron correlation effects is to be addressed, including the radial significance with respect to the core region and the relevance on differences in structure factors with respect to sin(θ)/λ.…”

Section: ∑ ∑mentioning

confidence: 99%

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Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth

2016

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This study investigates the possibility of detecting relativistic effects and electron correlation in single-crystal X-ray diffraction experiments using the examples of diphenyl mercury (HgPh2) and triphenyl bismuth (BiPh3). In detail, the importance of electron correlation (ECORR), relativistic effects (REL) [distinguishing between total, scalar and spin-orbit (SO) coupling relativistic effects] and picture change error (PCE) on the theoretical electron density, its topology and its Laplacian using infinite order two component (IOTC) wave functions is discussed. This is to develop an understanding of the order of magnitude and shape of these different effects as they manifest in the electron density. Subsequently, the same effects are considered for the theoretical structure factors. It becomes clear that SO and PCE are negligible, but ECORR and scalar REL are important in low- and medium-order reflections on absolute and relative scales-not in the high-order region. As a further step, Hirshfeld atom refinement (HAR) and subsequent X-ray constrained wavefunction (XCW) fitting have been performed for the compound HgPh2 with various relativistic and nonrelativistic wave functions against the experimental structure factors. IOTC calculations of theoretical structure factors and relativistic HAR as well as relativistic XCW fitting are presented for the first time, accounting for both scalar and spin-orbit relativistic effects.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: ∑ ∑mentioning

confidence: 99%

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Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth

2016

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“…The Atomic Mean Field spin-orbit operator (AMFI) [71,72,73] and/or the 2e spin-orbit interaction have not been accounted for in the calculations. The treatment of Picture Change Error (PCE) [74,68,75,76,77,78,79] has not been considered within this study. Nevertheless, it has been shown that PCE in the 1c spin density of the [OsCl 5 (Hpz)]…”

Section: Computational Detailsmentioning

confidence: 99%

Spin contamination analogy, Kramers pairs symmetry and spin density representations at the 2-component unrestricted Hartree–Fock level of theory

Bučinský

Malček

Biskupič

et al. 2015

Computational and Theoretical Chemistry

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Abstract"Kramers pairs symmetry breaking" is evaluated at the 2-component (2c) Kramers unrestricted and/or general complex Hartree-Fock (GCHF) level of theory, and its analogy with "spin contamination" at the 1-component (1c) unrestricted Hartree-Fock (UHF) level of theory is emphasized. The GCHF "Kramers pairs symmetry breaking" evaluation is using the square of overlaps between the set of occupied spinorbitals with the projected set of Kramers pairs. In the same fashion, overlaps between α and β orbitals are used in the evaluation of "spin contamination" at the UHF level of theory. In this manner, UHFŜ 2 expectation value is made formally extended to the GCHF case. The directly evaluated GCHF expectation value of theŜ 2 operator is considered for completeness. It is found that the 2c GCHF Kramers pairs symmetry breaking has a very similar extent in comparison to the 1c UHF spin contamination. Thus higher excited states contributions to the 1c so 2c unrestricted wave functions of open shell systems have almost the same extent and physical consequences. Moreover, it is formally shown that a single determinant wave function in the restricted open shell Kramers case has the expectation value ofK 2 operator equal to the negative number of open shell electrons, while the eigenvalue ofK 2 for the series of simple systems (H, He, He*-triplet, Li and Li*-quartet) are found to be equal to minus the square of the number of open shell electrons. The concept of unpaired electron density is extended to the GCHF regime and compared to UHF and restricted open shell Hartree-Fock spin density. The "collinear" and "noncollinear" analogs of spin density at the GCHF level of theory are considered as well. Spin contamination and/or Kramers pairs symmetry breaking, spin

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“…Therefore, it can lead to large differences in calculations of properties in a two-component framework. It is noted that the difference between ρ + and ρ 2c is significant mainly deep in the atomic cores and to some extent in the outer atomic core shells of heavy atoms [5,75,76]. In the valence regions of light and heavy atoms, the four-component and the two-component densities are very similar.…”

Section: Picture Changementioning

confidence: 95%

Relativistic calculations of magnetic resonance parameters: background and some recent developments

Autschbach

2014

Phil. Trans. R. Soc. A.

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This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.

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Relativistic electron densities in the four-component Dirac representation and in the two-component picture

Cited by 26 publications

References 19 publications

Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth

Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth

Spin contamination analogy, Kramers pairs symmetry and spin density representations at the 2-component unrestricted Hartree–Fock level of theory

Relativistic calculations of magnetic resonance parameters: background and some recent developments

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