Zero-order approximation, natural

An important problem in the application of QED methods to many-electron atoms is the choice of the zero-order approximation (actually the choice of the basis set of the one-electron relativistic wave functions in Eq(86). One natural choice is the approximation of noninteracting electrons when the potential V in Eq(2) is the Coulomb potential of the nucleus (28). This approximation is convenient for highly charged, few-electron ions. For a many-electron neutral atom a better choice is the Dirac-Hartree-Fock (DHF) approximation. 

It seems natural to suppose that the tetragonal distortion of the tri-anion results from the Jahn-Teller effect. In order to study the problem more thoroughly we undertook recently the DFT calculations of this cluster as well as of several other hexanuclear rhenium chalcohalide clusters. The technical details of these calculations can be found in the original publication [8]. Here we only want to note that the introduction of relativistic corrections for Re atoms is crucial for the correct reproduction of the geometry of clusters. In our calculations, this was done by the zero order regular approximation (ZORA) Hamiltonian [9] within ADF 2000.02 package [10]. 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

A natural way to calculate a CH eigenvector nvj) consists in applying the wave operator to the initial zero-order state /iv )°. In order to illustrate this point, we have performed four iterations of the wave operator scheme on the zero-order states j v,)° (n = 2, 3, 4, 5 and 6). Consequently, we have obtained a first approximation to the nv,) eigenvectors (n = 2, 3, 4, 5 and 6). The results obtained this way are compared to the exact ones in Table 1. 

The ability to assign a group of vibrational/rotational energy levels implies that the complete Hamiltonian for these states is well approximated by a zero-order Hamiltonian which has eigenfunctions /,( i)- The are product functions of a zero-order orthogonal basis for the molecule, or, more precisely, product functions in a natural basis representation of the molecular states, and the quantity m represents the quantum numbers defining tj>,. The wave functions are given by... 

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