d2 and d8 Trigonal Energy Levels

1973 ◽  
Vol 28 (8) ◽  
pp. 1247-1257
Author(s):  
J. R. Perumareddi

The trigonal d2 and d8 symmetry adapted eigenvectors and energy determinants have been obtained in two different representations, with and without cubic orientation. Spin-orbit perturbation was applied last in both the representations. The complete theory developed here is necessary in a definitive and a meaningful study of the interpretation of the spectral and magnetic properties of trigonally distorted or substituted cubic systems and of trigonal systems in which the axial part of the ligand field is of such a magnitude that the cubic parentage has no meaning.

1974 ◽  
Vol 29 (1) ◽  
pp. 31-41 ◽  
Author(s):  
E. König ◽  
S. Kremer

The complete ligand field -Coulomb repulsion -spin orbit interaction matrices have been derived for the d4 and d6 electron configurations within octahedral (Oh) and tetrahedral (Td) symmetry. The calculations were perform ed in both the weak-field and strong-field coupling schemes and complete agreement of the results was achieved. The energy matrices are parametrically dependent on ligand field (Dq), Coulomb repulsion (B, C) and spin-orbit interaction (ζ). Correct energy diagrams are presentend which display the splittings by spin-orbit perturbation as well as the effect of configuration mixing. Applications to the interpretation of optical spectral data, to the detailed behavior at the crossover of ground terms, and to complete studies in magnetism are pointed out.


1974 ◽  
Vol 29 (3) ◽  
pp. 419-428 ◽  
Author(s):  
E. König ◽  
R. Schnakig ◽  
S. Kremer

The complete ligand-field, Coulomb interelectronic repulsion, and spin-orbit interaction matrices have been derived for the d5 electron configuration within octahedral (Oh) and tetrahedral (Td) symmetry. The calculations were performed in both the weak-field and strong-field coupling schemes and complete agreement of the results was achieved. The energy matrices are parametrically dependent on ligand field (Dq), Coulomb repulsion (B, C), and spin-orbit interaction (ζ). Correct energy diagrams are presented which display the splittings by spin-orbit perturbation as well as the effect of configuration mixing. Applications to the interpretation of electronic spectra, and to complete studies in magnetism are pointed out. The detailed behavior at the crossover of ground terms is considered


2021 ◽  
Author(s):  
◽  
Hani Hatami

<p>In the last few years, two dimensional crystals have become available for experimental studies. Good examples of such systems are monolayers and bilayers of graphene and monolayers of transition metal dichalcogenides such as MoS₂ and WSe₂. The availability of two dimensional crystals has encouraged physicists to study the electronic and magnetic properties of such systems. This thesis adds to the theoretical knowledge about electronic and magnetic properties of two dimensional crystals with the focus on graphene and MoS₂.  As a general theme in this thesis, we calculate how in general these systems interact with electric and magnetic fields and what their response is to such stimuli. In particular, we have studied the response of monolayer graphene to an in-plane electric field. We have also looked at spin-orbit coupling effects that arise from applying perpendicular or in-plane external electric fields, especially their consequences for transport properties of bilayer graphene. We investigated the electronic properties of charge carriers confined in a mesoscopic ring structure using a gate voltage in bilayer graphene. We also showed how spin-orbit coupling can affect the electrical properties of such rings. We found how spin-orbit coupling can affect the transport properties in bilayer graphene. We also investigated the RKKY or indirect exchange coupling between magnetic moments in monolayer MoS₂ through calculating wave vector dependent spin susceptibility.  We examined the electronic properties of electrons and holes confined electrostatically into a bilayer graphene ring. We presented an analytical solution for finding energy levels in the ring. We showed that the magnetic field dependence of the lowest energy level with fixed angular momentum in bilayer graphene rings, in contrast to usual semiconductor quantum rings, is not parabolic but displays an asymmetric “Mexican hat“. We found that introducing spin-orbit coupling in the ring can flatten this Mexican hat.  We studied the effect of an orbital Rashba type effect, induced by an in-plane electric field in monolayer graphene. Using perturbation theory, we showed that this term can affect the energy levels in a crossed electric and magnetic field such that the electron and hole levels repel each other. We calculated the AC transport of monolayer graphene in the linear-response regime and showed that taking the orbital Rashba term into account casts doubt on the universality of the minimum conductivity of monolayer graphene.  We studied the effect of spin-orbit coupling in transport properties of bilayer graphene systems by calculating tunnelling through npn and np junctions. We showed that at sufficiently large spin-orbit strength, normal transmission through a barrier which is forbidden in bilayer graphene becomes finite. We predict that in a weak Rashba spin-orbit regime, outgoing electrons show signals which are spin polarized. We also showed that considering spin-orbit coupling only in the barrier of an npn junction can invert the spin of the incoming electrons.  Finally, we obtained analytical expressions for the wave vector-dependent static spin susceptibility of monolayer transition metal dichalcogenides, considering both the electron-doped and hole-doped cases. These results are then applied to the calculations of physical observables of monolayer MoS₂. We claculated that the hole-mediated RKKY exchange interaction for in-plane impurity-spin components decays with a different power law from what is expected for a two-dimensional Fermi liquid. In contrast, we calculated that the out-of-plane spin response shows the conventional long-range behaviour.</p>


1972 ◽  
Vol 27 (12) ◽  
pp. 1820-1860 ◽  
Author(s):  
Jayarama Perumareddi

AbstractThe complete theory of Liehr and Ballhausen for d2 and d8 electronic configurations immersed in cubic fields has been extended to include noncubic ligand fields of quadrate, trigonal, and cylindrical symmetry. The complete set of symmetry adapted eigenvectors for the three symmetries have been derived in various coupling schemes in which the spin-orbit interaction, electron cor-relation, and ligand field in turn are varied from minor to dominant perturbations. The cor-responding energy matrices as a function of the parameters of the ligand field, electron correlation, and spin-orbit constant have been constructed in all the representations. Unitary transformations connecting different formalisms were obtained. The energy matrices have been solved for representative sets of parametric values and energy diagrams have been plotted in all the symmetries as well as in the square planar limit of the quadrate crystalline field. The secular determinants, the eigenfunctions, the energy diagrams, and the unitary transformations presented here are extremely useful in the study of the various aspects of spectroscopic, magnetic, and other properties of appropriate systems. The theory is applicable to quadrately distorted or substituted, trigonally distorted or substituted, octahedral and tetrahedral complexes and to compounds of cylindrical symmetry of d2 and d8 electronic configurations.


2021 ◽  
Author(s):  
Nicolaj Kofod ◽  
Patrick Nawrocki ◽  
Carlos Platas-Iglesias ◽  
Thomas Just Sørensen

The wide range of optical and magnetic properties of the lanthanide(III) ions is associated to their intricate electronic structures, which in contrast to lighter elements is characterized by strong relativistic effects and spin-orbit coupling. Nevertheless, computational methods are now capable of describing the ladder of electronic energy levels of the simpler trivalent lanthanide ions, as well as the lowest energy term of most of the series. The electronic energy levels result from electron configurations that are first split by spin-orbit coupling into groups of energy levels denoted by the corresponding Russel-Saunders terms. Each of these groups are then split by the ligand field into the actual electronic energy levels known as microstates or sometimes mJ levels. The ligand field splitting directly informs on coordination geometry, and is a valuable tool for determining structure and thus correlating the structure and properties of metal complexes in solution. The issue with lanthanide complexes is that the determination of complex structures from ligand field splitting remains a very challenging task. In this manuscript, the optical spectra – absorption, luminescence excitation and luminescence emission – of ytterbium(III) solvates were rec-orded in water, methanol, dimethyl sulfoxide and N,N-dimethylformamide. The electronic energy levels, that is the microstates, were resolved experimentally. Subsequently, density functional theory (DFT) calculations were used to model the structures of the solvates and ab initio relativistic complete active space self-consistent field (CASSCF) calculations were employed to obtain the microstates of the possible structures of each solvate. By comparing experimental and theoretical data, it was possible to determine both the coordination number and solution structure of each solvate. In water, methanol and N,N-dimethylformamide the solvates were found to be eight-coordinated and to have a square anti-prismatic coordination geometry. In DMSO the speciation was found to be more complicated. The robust methodology developed for comparing experimental spectra and computational results allows the solution structures of lanthanide complexes to be determined, paving the way for the design of complexes with predetermined properties. <br>


2019 ◽  
Vol 5 (1) ◽  
pp. 17 ◽  
Author(s):  
Hiroshi Sakiyama

The theoretical equations of Zeeman energy levels, including the zero-field energies and the first- and second-order Zeeman coefficients, have been obtained in closed form for nine states of the 3 T 1 ( g ) ground term, considering the axial ligand-field splitting and the spin-orbit coupling. The equations are expressed as the functions of three independent parameters, Δ , λ , and κ , where Δ is the axial ligand-field splitting parameter, λ is the spin-orbit coupling parameter, and κ is the effective orbital reduction factor, including the admixing. The equations are useful in simulating magnetic properties (magnetic susceptibility and magnetization) of the complexes with 3 T 1 ( g ) ground terms, e.g., octahedral vanadium(III), octahedral low-spin manganese(III), octahedral low-spin chromium(II), and tetrahedral nickel(II) complexes.


2021 ◽  
Author(s):  
◽  
Hani Hatami

<p>In the last few years, two dimensional crystals have become available for experimental studies. Good examples of such systems are monolayers and bilayers of graphene and monolayers of transition metal dichalcogenides such as MoS₂ and WSe₂. The availability of two dimensional crystals has encouraged physicists to study the electronic and magnetic properties of such systems. This thesis adds to the theoretical knowledge about electronic and magnetic properties of two dimensional crystals with the focus on graphene and MoS₂.  As a general theme in this thesis, we calculate how in general these systems interact with electric and magnetic fields and what their response is to such stimuli. In particular, we have studied the response of monolayer graphene to an in-plane electric field. We have also looked at spin-orbit coupling effects that arise from applying perpendicular or in-plane external electric fields, especially their consequences for transport properties of bilayer graphene. We investigated the electronic properties of charge carriers confined in a mesoscopic ring structure using a gate voltage in bilayer graphene. We also showed how spin-orbit coupling can affect the electrical properties of such rings. We found how spin-orbit coupling can affect the transport properties in bilayer graphene. We also investigated the RKKY or indirect exchange coupling between magnetic moments in monolayer MoS₂ through calculating wave vector dependent spin susceptibility.  We examined the electronic properties of electrons and holes confined electrostatically into a bilayer graphene ring. We presented an analytical solution for finding energy levels in the ring. We showed that the magnetic field dependence of the lowest energy level with fixed angular momentum in bilayer graphene rings, in contrast to usual semiconductor quantum rings, is not parabolic but displays an asymmetric “Mexican hat“. We found that introducing spin-orbit coupling in the ring can flatten this Mexican hat.  We studied the effect of an orbital Rashba type effect, induced by an in-plane electric field in monolayer graphene. Using perturbation theory, we showed that this term can affect the energy levels in a crossed electric and magnetic field such that the electron and hole levels repel each other. We calculated the AC transport of monolayer graphene in the linear-response regime and showed that taking the orbital Rashba term into account casts doubt on the universality of the minimum conductivity of monolayer graphene.  We studied the effect of spin-orbit coupling in transport properties of bilayer graphene systems by calculating tunnelling through npn and np junctions. We showed that at sufficiently large spin-orbit strength, normal transmission through a barrier which is forbidden in bilayer graphene becomes finite. We predict that in a weak Rashba spin-orbit regime, outgoing electrons show signals which are spin polarized. We also showed that considering spin-orbit coupling only in the barrier of an npn junction can invert the spin of the incoming electrons.  Finally, we obtained analytical expressions for the wave vector-dependent static spin susceptibility of monolayer transition metal dichalcogenides, considering both the electron-doped and hole-doped cases. These results are then applied to the calculations of physical observables of monolayer MoS₂. We claculated that the hole-mediated RKKY exchange interaction for in-plane impurity-spin components decays with a different power law from what is expected for a two-dimensional Fermi liquid. In contrast, we calculated that the out-of-plane spin response shows the conventional long-range behaviour.</p>


2022 ◽  
Author(s):  
Christoph van Wüllen ◽  
Eva M. V. Kessler

Building blocks containing $5d$ spin centres are promising for constructing single molecule magnets due to their large spin-orbit interaction, but experimental and computational results obtained so far indicate that this might not be the case for Re$^\textrm{IV}$ centres in an octahedral environment. Density functional results obtained in this work for [ReCl$_4$(CN)$_2$]$^{2-}$ and trinuclear complexes formed by attaching Mn$^\textrm{II}$ centres to the cyano ligands indicate that zero field splitting in such complexes exhibits large rhombicity (which leads to fast relaxation of the magnetisation) even if there are only small distortions from an ideal geometry with a four-fold symmetry axis. This is already apparent if second-order spin-orbit perturbation theory is applied but even more pronounced if higher-order spin-orbit effects are included as well, as demonstrated by wavefunction based calculations. Computational results are cast into a ligand field model and these simulations show that especially a distortion which is not along the $C_4/C_2$ axeshas a large effect on the rhombicity. Quantum simulations on these complexes are difficult because the zero field splitting strongly depends on the energetic position of the low-lying doublets from the $t_{2g}^3$ configuration.


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