Effective Hamiltonian formalisms, review

In the effective Hamiltonian formalism just reviewed, the diabatic state energies are obtained as the diagonal matrix elements of the effective Hamiltonian, while the resonance interaction between product-like and reactant-like diabatic surfaces is obtained as the off-diagonal matrix elements. The adiabatic states are obtained as the eigenvalues. Thus, the effective Hamiltonian corresponds to the projection of the full Cl Hamiltonian onto the subspace of the (product-like and reactant-like) Heitler-London and no-bond configurations. We now wish to comment briefly on the physical interpretation of the effective Hamiltonian computed via Eq. (17). 

The development of the effective Hamiltonian has been due to many authors. In condensed phase electron spin magnetic resonance the so-called spin Hamiltonian [20,21] is an example of an effective Hamiltonian, as is the nuclear spin Hamiltonian [22] used in liquid phase nuclear magnetic resonance. In gas phase studies, the first investigation of a free radical by microwave spectroscopy [23] introduced the ideas of the effective Hamiltonian, as also did the first microwave magnetic resonance study [24], Miller [25] was one of the first to develop the more formal aspects of the subject, particularly so far as gas phase studies are concerned, and Carrington, Levy and Miller [26] have reviewed the theory of microwave magnetic resonance, and the use of the effective Hamiltonian. 

This formalism has been adopted, and modified, by Cramer and Truhlar (Cramer and Truhlar, 1991, 1992) in a procedure which belongs to the category of continuum effective Hamiltonian quantum methods (for reviews see Cramer and Truhlar, 1994, 1995a, 1995b). 

To summarize, in the preceding discussion we have presented two formalisms for the computation of diabatic wavefunctions using an effective Hamiltonian. In the Van VIeck method the diabatic wavefunctions are orthogonal In the non-orthogonal transformation method the diabatic wavefunctions are non-orthogonal. In practice, perturbation theory is normally used to solve the equation system (20) the review of Spiegelmann and... 

For 3 + lanthanide and actinide ions, almost all transitions within the f shell are electric dipole in nature. These transitions are formally parity (Laporte) forbidden. That such transitions are observable is attributed to non-centro-symmetric terms in the crystal-field Hamiltonian. Such terms have the effect of mixing higher-lying, opposite-pairty d and g states into the f shell. As Judd (1988) noted in a review of atomic theory and optical spectroscopy of rare earths No doubt that we shall eventually be able to calculate much of what we want with a high degree of accuracy. That day has not yet arrived. . 

For many atoms and molecules, especially small open-shell systems of high symmetry, it is necessary to include spin-orbit interaction to achieve even qualitative agreement with experiment. For large systems with low symmetry or closed shells, the effect is less important because spin—orbit interaction is quenched, and these systems can therefore usually be described with a one-component method. In some cases this can also be achieved in a perturbation formalism at little additional cost. Few computer program systems have been developed for treating spin-orbit interactions at the all-electron level with a transformed Hamiltonian. In a recent review,the method and results from such calculations were discussed. Calculations including spin-orbit interactions at the RECP level have been carried out for many years.We will not discuss results, but it is clear that this will be an important method for large systems. 

There has been much excitement in the relativistic quantum chemistry community regarding the possibility of constructing a formally exact two-component Hamiltonian for molecular calculations [50-56], as outlined in several review articles recently [56-59]. To be specific, an exact Hamiltonian can be constructed relatively straightforwardly at the one-electron level. Many-electron effects can be built into the approach in a pragmatic way with the help of model potentials [60-62]. For perspectives on a systematic incorporation of electron correlation into relativistic quantum chemical methods with many-electron wavefunctions, see Kutzelnigg [58] Liu [59] Saue [56] Saue and Visscher [63]. For a perspective on DFT, see van Wiillen [64]. 

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