Eigenfunctions model

Fig. 6. Outline of the successive eigenfunction model. Chemical patterns grow in domains when no flux boundary conditions are satisfied and > 0. For more detail, see Appendix.

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

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... 

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

It is found empirically and of course is predictable theoretically that, when using a model for molecular electronic structure, the set of eigenfunction equations associated with the operators commuting with H are constraints on the action of the variation principle if Et is computed from R subject to symmetry constraints and E2 is computed in the same model with no such constraints then (2)... 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

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