Hamiltonian orbital connections

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, E, P, Uj, It, Uc, and Ic-The terms of H involving the parameters aj, ac, and bo aie diagonal with respect to both the Ij and Ic quantum numbers, while the f>2 term connects with one another the basis functions with I j = Ij 2, 4- 2. The c terms... 

Values for the spin Hamiltonian are given in Table XIV. The 5D state of d6 has three orbital states for the ground state in octahedral symmetry. Since these three states are connected by the spin-orbit coupling, the spin-lattice-relaxation time is quite short and the zero-field splitting very large. In a distorted octahedral field the large zero-field distortion makes detection of ESR difficult. In the case of ZnF2 the forbidden AM = 4 transition was measured and fitted to Eq. (164). 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as... 

The operator [157] is a phenomenological spin-orbit operator. In addition to being useful for symmetry considerations, Eq. [157] can be utilized for setting up a connection between theoretically and experimentally determined fine-structure splittings via the so-called spin-orbit parameter Aso (see the later section on first-order spin-orbit splitting). In terms of its tensor components, the phenomenological spin-orbit Hamiltonian reads... 

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