Renner Hamiltonian

The derivation of the relativistic Renner Hamiltonian of a 11 state has been described in detail in [39]. The result is... 

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 first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

Determination of the paiameters entering the model Hamiltonian for handling the R-T effect (quadratic force constant for the mean potential and the Renner paiameters) was carried out by fitting special forms of the functions [Eqs. (75) and (77)], as described above, and using not more than 10 electronic energies for each of the X H component states, computed at cis- and toans-planai geometries. This procedure led to the above mentioned six parameters... 

Hamiltonian equations, 627-628 perturbative handling, 641-646 II electronic states, 631-633 vibronic coupling, 630-631 ABC bond angle, Renner-Teller effect, triatomic molecules, 611-615 ABCD bond angle, Renner-Teller effect, tetraatomic molecules, 626-628 perturbative handling, 641-646 II electronic states, 634-640 vibronic coupling, 630-631 Abelian theory, molecular systems, Yang-Mills fields ... 

Floquet theory principles, 35—36 single-surface nuclear dynamics, vibronic multiplet ordering, 24—25 Barrow, Dixon, and Duxbury (BDD) method, Renner-Teller effect tetraatomic molecules, Hamiltonian equations, 626-628 triatomic molecules, 618-621 Basis functions ... 

EWW Hamiltonian, Renner-Teller effect, triatomic molecules, 610—615 Expanding potential, molecular systems,... 

We wish to introduce here a very reliable calculation procedure to determine the eigenstates of a general Renner-Teller problem. It is convenient to start writing the matrix representation of the Hamiltonian of a Renner-Teller system on two degenerate p-like electronic states fx, fy and with the interaction part in the form given in Section 2, neglecting at the moment the quartic or higher order terms ... 

After these transformations, the Renner-Teller Hamiltonian adopts the following simple and very convenient aspect... 

We consider now the complete basis set of vibronic functions I f m, n) obtained as the direct product of the electronic functions l/+) and 1/1) by the phonons wave functions Im, n), where the integer numbers m and n (positive or zero) label the occupation numbers of the boson operators b b+ and b b, respectively. The Renner-Teller Hamiltonian (14) when applied to any vibronic function I f m,n) couples it with three states (at most) in fact it holds that ... 

The above results indicate that the Renner-Teller Hamiltonian (14) can be mapped exactly into a set of independent double chains of interacting states, as shown in Fig. 1. 

When the orbital functions are chosen as d-like or/-like, the axial symmetry of the molecule again leads to factorize the Renner-Teller Hamiltonian in a double chain. In effect the total angular momentum Jz remains a constant of motion and can be put in a diagonal form by means of the same type of canonical transformation, so we have... 

In Fig. 3, the computed vibronic energy levels are shown for 0Renner-Teller Hamiltonian of the type/g4. Looking at Figs 2 and 3, we can see that for small g values (near g = 0) the behavior of the vibronic energy levels without and with quartic terms is similar. However, with increasing g, relevant... 

Renner-Teller Effect in Nonlinear Molecules Choice of Hamiltonian Minimal Models Pragmatic Models Benchmark Handling Effective Hamiltonians Beyond the Two-State Renner-Teller Effect... 

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