Unperturbed models

This simply means that one must be willing to identify one of the unperturbed states as the best approximation to the state being sought. This, of eourse, implies that one must therefore strive to find an unperturbed model problem, eharaeterized by H that represents the true system as aeeurately as possible, so that one of the Ok will be as elose as possible to /k. 

Again, as in the case of Eq. (3.16) the sum of the zero- and first-order energy contributions is equal to the expectation value of the full Hamiltonian in the unperturbed model state, i.e. 

If we look at the form of the perturbed T4, we see that it has been altered by mixing in a small contribution of the p orbital of Cj. This change means that the electron pair that in the unperturbed model was confined to the CH3 group... 

For the unperturbed model system, the Schrodinger equation may be written... 

To a first approximation, one can treat polymer chains on the basis of the highly simplified unperturbed model without intramolecular excluded volume interactions. This allows us to considerably simplify the problem by calculating the Gaussian function exp(- c [ -j[/6) which describes the distribution of segments i and j belonging to the species A and B inside an N-unit polymer. 

Hd) is called the perturbing Hamiltonian or the first-order Hamiltonian. The role of X is somewhat abstract. There may be no physical way of smoothly changing a system from some unperturbed model system to the system of interest however, the parameter X is a device that accomplishes this in a mathematical sense. When X is set to zero. Equation 8.70 gives the Hamiltonian for the zero-order or unperturbed situation, whereas when X is set to 1.0, the Hamiltonian of interest results. With this embedded parameter, the Schrodinger equation with H is really a family of equations covering all the choices of X values. Perturbation theory is a means of dealing with the entire family of equations, even though in the end, one may be interested only in the case where X = 1. 

We have only recently understood the phenomena that control rainwater pH in the natural, unpolluted environment. As pointed out in Section 16.2, these appear to be mainly the cycles of sulfur and nitrogen compounds. A model of the unperturbed system is necessary in order to understand and predict the changes that occur when strong sulfur- and nitrogen-acids are... 

In the theory of radiationless transitions as covered in this paper (6,4g), the two non BO terms are treated as perturbations (not externally plied, but arising as imperfections within this model of molecular structure) that can induce transitions between unperturbed states each of which is taken to be a specific Bom-Oppenheimer product state ... 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

In the RIS model for PE described by Abe et al. [143], the conformational partition function, Z, of the unperturbed chain with n bonds is formulated using a well-known statistical weight matrix for bond i, Uj, that is usually written in a 3 x 3 form. 

Isolated unperturbed polyoxyethylene chains have been simulated on the 2nnd lattice [154], The literature contains RIS models for a large number of polyethers [124], and it is likely that most of these chains could be mapped onto the 2nnd lattice with little difficulty. It is also likely that the work on PP [156,158] can be extended to other vinyl polymers, such as poly(vinyl chloride). This capability should permit the construction and complete equilibration of amorphous poly(vinyl chloride) cells larger than those described to date. They may be large enough to address issues arising from the weak crystallization reported for these systems [174]. 

In network BI in which chain expansion was the greatest, the measured results show more chain swelling than a network with f=3 but less than a network with f=4. Chain swelling was less than that of the phantom network model for the other two networks, and in one case, the chains coiled to a size slightly less than that of the unperturbed molecule. 

Figure 2 Model for the Bronsted sites in the supercage of a dealuminated HY, depending on the nature and location of the extraframework aluminic phase (the drawing of the extraframework phase is only schematic). In the center is the unperturbed Bronsted site.

The classic case distinguishes between an atomic core, which is essentially unperturbed by bonding, and a valence shell whose content may be accessible to bond formation. Since we suppose this simplifying assumption to be maintained in the MO treatment, an atomic orbital belonging to the valence shell will be termed a valence atomic orbital (VAO). For the construction of MOs, we utilize the following general results of the MO/LCAO model ... 

Our conclusions are that a) theta conditions are not necessary to obtain experimental mK values unperturbed by excluded volume interactions, b) the RIS and BAA approximations are applicable and c) mK is a very sensitive to the details of the RIS model, the tacticity and the composition of a copolymer. 

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