Spin-orbit interaction mean-field

Low-spin iron(III) ions have an electron hole in the t2g orbitals. Therefore, these centers have S = 1/2 and spin-orbit interaction contribntes considerably to the magnetic hyperfine field. Low-spin iron(III) componnds in solution always show a rather complicated magnetic Mossbauer pattern at temperatures around 4.2 K and low external fields, which means that the relaxation rate of these centers is lower than the nnclear precession rate of 10 s. Sometimes a magnetic sphtting is observed even at 77 K. Therefore, in order to pin down 8 and A g, it is advisory to measure between 100 and... 

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

Again, it is assumed that the reader is familiar with the usual content of an undergraduate course in atomic physics, viz. the principles of quantum mechanics, the hydrogen atom, elementary treatments of angular momentum, of spin-orbit interaction, the Pauli principle, static mean fields, the central field model, the building-up principle, spectroscopic notation and the working of dipole selection rules. 

This means that F2 is not split, hut F4 and T s are each spht into four states. Note now that spin quantum number is no longer used to define the states, (b) When spin-orbit interaction is considered first, for F, = 4Vi, 3 Vi, 2Vi, and 1 Vi. When these states are subjected to the effects of an 0-field, from (i), all, except7 = IVi, are further split ... 

Mean-Field Approximations for Spin-Orbit Interaction... 

The spin-orbit operators for the model potential and pseudopotential approximations are one-electron operators. These operators include the effect of the two-electron spin-orbit interaction used in the mean-field approximation to derive the model potential or pseudopotential. Molecular calculations with these potentials therefore include, at least at the atomic level, the two-electron spin-orbit terms. This is just the kind of approximation we are looking for. 

Here another source of a conceptual problem of the seeond order-approach appears. The standard formulation of the J-O theory, even if extended by the dynamic coupling model, is based on the single configuration approximation. This means that in such a description all the eleetron correlation effects are neglected and it is well known that the transition amplitude strongly depends on them. At this point also the spin-orbit interactions should be taken into consideration as possibly important in the description of the spectroscopic patterns of the lanthanides. In the case of all of these possibly important physical mechanisms there is a demand for an extension of the standard Judd-Ofelt formulation. The transition amplitude in equation (10.17) has to be modified by the third-order contributions that originate from various perturbing operators introduced in addition to the crystal field potential that plays a... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

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