Relativistic spin-orbit splitting method

Relativistic mean field (RMF) models have been applied successfully to describe properties of rinite nuclei. In general ground state energies, spin-orbit splittings, etc. can be described well in terms of a few parameters ref. [18]. Recently it has lead to the suggestion that the bulk SE is strongly correlated with the neutron skin [19, 20] (see below). In essence the method is based upon the use of energy-density functional (EDF) theory. 

The photoionisation spectrum of the OSO4 molecule is one example were we definitely see the relativistic influence. Fig. 5 compares the experimental results with our DVM-DFS method and extensive HF-CI calculations. The splitting of the 3tj levels (originating from the spin-orbit splitting of the 5d-wavefunctions of Os) are quite well-reproduced. 

Another recent development is the implementation of DK Hamiltonians which include spin-orbit interaction. An early implementation shared the restriction of the relativistic transformation to the kinetic energy and the nuclear potential with the efficient scalar relativistic variant electron-electron interaction terms were treated in nonrelativistic fashion. Further development of the DKH approach succeeded in including also the Hartree potential in the relativistic treatment. This resulted in considerable improvements for spin-orbit splitting, g tensors and molecular binding energies of small molecules of heavy main group and transition elements. Application of Hamiltonians which include spin-orbit interaction is still computationally demanding. On the other hand, the SNSO method is an approximation which seems to afford a satisfactory level of accuracy for a rather limited computational effort. 

Kotzian et al. (1991) and Kotzian and Rosch (1992) applied their INDO/1 and INDO/S-CI methods to hydrated cerium(III), i.e. model complexes [Ce(H20) ] (n=8,9), in order to rationalize the electronic structure and the electronic spectrum of these species. Besides the scalar relativistic effects spin-orbit coupling was also included in the INDO/S-CI studies. The spin-orbit splitting of the 4f F and 5d states of the free Ce " ion was calculated as 2175cm" and 2320cm in excellent agreement with the experimental values of 2253 cm and 2489 cm" , respectively. The calculated energy separation between the F and states of approximately 44000cm" (estimated fi-om fig. 5 in Kotzian and Rosch 1992) is somewhat lower than the experimental value of 49943 cm" (Martin et al. 1978). 

It is known from quasi-relativistic calculations that this bare-potential approximation provides reliable results for valence electron properties. This is particularly true for the scalar-relativistic variants discussed in the next section [see, e.g.. Refs. [656,752]], but may be different for the truly two-component method including spin-orbit splitting [655]. For core-shell properties and for the spin-orbit splitting of high-angular-momentum orbitals this approximation will not be sufficient [643,659]. 

These were calculated by the method of Barnes and Smith from spin-orbit splittings in atomic spectra without relativistic correction (open circles). Values calculated from relativistic Hartree-Fock-Slater atomic wave functions are included for comparison (filled circles). 

Comparison of Spin—Orbit Configuration Interaction Methods Employing Relativistic Effective Core Potentials for the Calculation of Zero-Field Splittings of Heavy Atoms with a 2P° Ground State. 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

The Amsterdam Density Functional (ADF) method [118,119] was used for calculations of some transactinide compounds. In a modem version of the method, the Hamiltonian contains relativistic corrections already in the zeroth order and is called the zero-order regular approximation (ZORA) [120]. Recently, the spin-orbit operator was included in the ZORA Fock operator [121]. The ZORA method uses analytical basis fimctions, and gives reliable geometries and bonding descriptions. For elements with a very large SO splitting, like 114, ZORA can deviate from the 4-component DFT results due to an improper description of the pi/2 spinors [117]. Another one-component quasirelativistic scheme [122] applied to the calculations of dimers of elements 111 and 114[116,117]isa modification of the ZORA method. 

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