Construction of the Foldy–Wouthuysen transformation and solution of the Dirac equation using large components only

Lenthe, Erik van; Baerends, E. J.; Snijders, J.G.

doi:10.1063/1.472104

Cited by 30 publications

(21 citation statements)

References 28 publications

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“…The order of magnitude of the ratio between small and large components is v/c where v is the velocity of the particle. 78 Up to order c −1 , the small component writes: 78,79 …”

Section: Acknowledgementmentioning

confidence: 99%

X-ray magnetic and natural circular dichroism from first principles: Calculation of K - and L1 -edge spectra

et al. 2017

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An efficient first principles approach to calculate X-ray magnetic circular dichroism (XMCD) and X-ray natural circular dichroism (XNCD) is developed and applied in the near edge region at the K -and L1 -edges in solids. Computation of circular dichroism requires precise calculations of X-ray absorption spectra (XAS) for circularly polarized light. For the derivation of the XAS cross section, we used a relativistic description of the photon-electron interaction that results in an additional term in the cross-section that couples the electric dipole operator with an operator σ · ( × r) that we name spin-position. The numerical method relies on pseudopotentials, on the gauge including projected augmented wave method and on a collinear spin relativistic description of the electronic structure. We apply the method to the calculations of K -edge XMCD spectra of ferromagnetic iron, cobalt and nickel and of I L1 -edge XNCD spectra of α-LiIO3, a compound with broken inversion symmetry. For XMCD spectra we find that, even if the electric dipole term is the dominant one, the electric quadrupole term is not negligible (8% in amplitude in the case of iron). The term coupling the electric dipole operator with the spin-position operator is significant (28% in amplitude in the case of iron). We obtain a sum-rule relating this new term to the spin magnetic moment of the p-states. In α-LiIO3 we recover the expected angular dependence of the XNCD spectra.

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“…The order of magnitude of the ratio between small and large components is v/c where v is the velocity of the particle. 78 Up to order c −1 , the small component writes: 78,79 …”

Section: Acknowledgementmentioning

confidence: 99%

X-ray magnetic and natural circular dichroism from first principles: Calculation of K - and L1 -edge spectra

et al. 2017

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“…The change of picture effect on the calculation of expectation values of the position operator in Ž . the zeroth-order regular approximation ZORA w x scheme 13 has been more recently discussed by w x van Lenthe et al 24 . The differences between the expectation values of the electron᎐nucleus distance r calculated in the two pictures have been found to be, with exception of the deep core states w x of heavy ions, rather small 22, 24 .…”

Section: Introduction He Progress In Applications Of Relativisticmentioning

confidence: 99%

Picture change and calculations of expectation values in approximate relativistic theories

Kellö

Sadlej

1998

Int. J. Quant. Chem.

140

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ABSTRACT:The effect of the so-called picture change on expectation values of Ž . one-electron operators in approximate two one -component relativistic theories is discussed. This effect is expected to be particularly large for operators which assume large values in the vicinity of heavy nuclei. The numerical results illustrating the picture change effect on electric field gradients at nuclei have been obtained in the spin-free Pauli and Douglas᎐Kroll approximations. It has been found that the picture change effect lowers the electric field gradient at I in HI by about 1 a.u. Very large picture change Ž . effect y8 a.u. has been calculated for HAt. It is concluded that in accurate calculations of expectation values of operators involving high inverse powers of the electron᎐nucleus distance the picture change, which accompanies the transformation of the Dirac Ž . Ž . Dirac᎐Coulomb equation to approximate two one -component relativistic Hamiltonians, must be taken into account.

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“…We note that the higher-order eects (the dierence between second order and the full contribution for Z 92) are rather similar for the Dirac and the Newton±Wigner and the ZORA values, except concerning the 1s state. The exceptional sensitivity of 1s to relativistic eects has already been mentioned [16].…”

Section: Resultsmentioning

confidence: 91%

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

Autschbach

Schwarz

2000

Theoretical Chemistry Accounts: Theory, Computation, and Modeli

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Analytic formulae of the relativistic radial functions of hydrogen-like atoms in the four-component standard Dirac picture and in two approximations, (Pauli and ZORA), to the two-component (so-called SchroÈ dinger or Newton±Wigner) picture and graphs of the respective relativistic changes of densities are presented and discussed. The two dierent pictures of the Dirac density of charge position and of the Newton± Wigner density of mass position are remarkably dierent in strongly inhomogeneous ®elds and result in respective dierences in position-dependent expectation values, hr m i. The fractional magnitudes of D rel hr m i, of D chargeamass hr m i, and of the gauge dependence of ZORA (which for small n states is comparable to the dierence of the two kinds of position observables) are all of order Z 2 a 2 .

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Construction of the Foldy–Wouthuysen transformation and solution of the Dirac equation using large components only

Cited by 30 publications

References 28 publications

X-ray magnetic and natural circular dichroism from first principles: Calculation of K - and L1 -edge spectra

X-ray magnetic and natural circular dichroism from first principles: Calculation of K - and L1 -edge spectra

Picture change and calculations of expectation values in approximate relativistic theories

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

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