Valence-only spin-orbit Hamiltonian

Summarising, the valence-only spin-orbit CGWB-AIMP embedded cluster Hamiltonian, which results from choosing the CGWB-AIMP for the isolated cluster, reads ... 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

Obviously, the sfss technique is not bounded to be applied only in AIMP calculations or in other valence-only calculations, but it can be used with any relativistic Hamiltonian which can be separated in spin-free and spin-dependent parts [48]. Being a very simple procedure, it is an effective means for the inclusion of dynamic correlation and size consistency in spin-orbit Cl calculations with any choice of Cl basis, such as determinants, double-group adapted configuration state functions, or spin-free Cl functions. In the latter case [46], the technique reduces to changing the diagonal elements of the spin-orbit Cl matrix. 

There is no need to explicitly include terms for direct relativistic effects, such as the dependence of mass on velocity, which are important only in the core region, in the valence-electron Hamiltonian. These terms are included as a consequence of using the Diiac-Fock wave functions. Thus, the Hamiltonian for the valence electrons is composed of the nonrelativistic Hamiltoman for the valence electrons plus the RECPs, which include the effects of the core electrons as well as the relativistic effects on the valence electrons in the core region [37]. The RECPs thns represent, for the valence electrons, the dynamical effects of relativity from the core region, the repulsion of the core electrons, the spin-orbit interaction with the nucleus, the spin-orbit interaction with the core electrons, and an approximation to the spin-orbit interaction between the valence electrons [38], which has usually been found to be quite stnaU, especially for heavier element systems [39-41]. The REP operators can be written as a summation of spin-independent potential and the spin-orbit operator, as written below, and the readers are referred to reference [39] for details. 

If the Hartree-Fock equations associated with the valence pseudo-Hamiltonian (167) are solved with extended basis sets, then all the above F are almost basis-set-independent. At the present time, and for practical reasons, most of the ab initio valence-only molecular calculations use coreless pseudo-orbitals. The reliability of this approach is still a matter of discussion. Obviously the nodal structure is important for computing observable quantities such as the diamagnetic susceptibility which implies an operator proportional to 1/r. From the computational point of view, it is always easy to recover the nodal structure of coreless valence pseudo-orbitals by orthogonalizing the valence molecular orbitals to the core orbitals. This procedure has led to very accurate results for several internal observables in comparison with all-electron results. The problem of the shape of the pseudoorbitals in the core region is also important in relativity. For heavy atoms, the valence electrons possess high instantaneous velocities near the nuclei. Schwarz has recently investigated the compatibility between the internal structure of valence orbitals and the representations of operators such as the spin-orbit which vary as 1/r near the nucleus. ... 

As for the valence band, the picture is relatively complicated. Considering a single hole near the top of the Fg valence band (for zinc blende crystal structure) and the magnetic ion with five d orbitals occupied by N electrons, the Hamiltonian that is applicable would have ionic, crystal, and hybridization components. The p-d hybridization mediated kinetic exchange depends on the filling of only the orbitals, not all the one-electron d-orbitals of the magnetic ion. The spin-dependent part of the exchange Hamiltonian for interaction between the Fg valence band p-like electrons and all the three d-orbitals occupied by one electron can be described as... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

Both of these methods essentially use ab initio wavefunctions to deduce the composition of the MOs. An alternative approach is to use semi-empirical functions, obtained from the MINDO or CNDO methods, to calculate the constants directly from the hamiltonian. A major problem in such work is that in many cases only valence shell electrons are included and consequently spin-other orbit effects cannot be calculated directly. Hinkley, Walker, and Richards122 have shown from ab initio calculations that this shielding for a given atom often bears a constant ratio to theZ/r3 terms. Such a ratio could be... 

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