Orbital interaction effective Hamiltonians

The life is relativistic and of the same kind should be the quantum chemistry. The four-component calculations involving large number of dynamic correlation are extremely time-consuming. The important thing, however, is that one is now able to formulate the fully equivalent two-component algorithms for those calculations. Since the final philosophy of the two-component calculations is similar to non-relativistic theory and most of mathematics are simply enough to comprehend the routine molecular relativistic calculations are possible. The standard nonrelativistic codes can be used with simple modification of the core Hamiltonian. The future is still in the development of the true two-component codes which will be able to deal with the spin-orbit interaction effect not in a posterior way as it is done nowadays. 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso ... 

We now consider how to eliminate either all relativistic effects or exclusively the spin-orbit interaction from the relativistic Hamiltonian. We start from the Dirac equation in the molecular field... 

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

For pN shells the effective Hamiltonian Heff contains two parameters F2 and 4>i, as well as the constant of spin-orbit interaction. The term with k = 0 causes a general shift of all levels, which is usually taken from experimental data in semi-empirical calculations, and can therefore be neglected. The coefficient at 01 is proportional to L(L + 1). Therefore, to find the matrix elements of the effective Hamiltonian it is enough to add the term aL(L + 1) to the matrix elements of the energy of electrostatic and spin-orbit interactions. Here a stands for the extra semi-empirical parameter. 

In quantum wires formed in a two-dimensional electron gas (2DEG) by lateral confinement the Rashba spin-orbit interaction is not reduced to a pure ID Hamiltonian H[s = asopxaz. As was shown in Ref. [4] the presence of an inplane confinement potential qualitatively modifies the energy spectrum of the ID electrons so that a dispersion asymmetry appears. As a result the chiral symmetry is broken in quantum wires with Rashba coupling. Although the effect was shown [4] not to be numerically large, the breakdown of symmetry leads to qualitatively novel predictions. 

An effective Hamiltonian for a static cooperative Jahn-Teller effect acting in the space of intra-site active vibronic modes is derived on a microscopic basis, including the interaction with phonon and uniform strains. The developed approach allows for simple treatment of cooperative Jahn-Teller distortions and orbital ordering in crystals, especially with strong vibronic interaction on sites. It also allows to describe quantitatively the induced distortions of non-Jahn-Teller type. 

The proposed approach to static cooperative Jahn-Teller effect is based on the exact effective Hamiltonian (7), acting in the reduced space of active one-centre distortions only. It involves effective force constants, which are analytically related to the parameters of the full microscopic Hamiltonian. Direct electronic interactions between sites, such as orbital-dependent electrostatic and exchange interactions [28], can be added to the effective Hamiltonian without modifying it. This approach proves to be especially efficient in the case of strong Jahn-Teller distortions, when the effects of second-order Jahn-Teller coupling become important. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Although this spin-orbit interaction is essentially a relativistic effect it may be approached classically. For hydrogen-like atoms the spin-orbit hamiltonian is... 

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