Self-consistent Fitting Approach

Hence, the set of incremental coefficients ACq is included in the fitting process. The authors used this formulation within a self-consistent fitting approach [81] to obtain very accurate amino acid-specific Karplus parameters for the six vicinal -couplings that probe the side-chain torsion angle x by making use of six additional AC parameters. In a subsequent study, the results were further improved by introducing an additional sine term to account for asymmetry effects. 

At equilibrium the rate of all elementary reaction steps in the forward and reverse directions are equal therefore, this condition provides a check point for studying reaction dynamics. Any postulated mechanism must both satisfy rate data and the overall equilibrium condition. Additionally, for the case of reactions occurring at charged interfaces, the appropriate model of the interface must be selected. A variety of surface complexation models have been used to successfully predict adsorption characteristics when certain assumptions are made and model input parameters selected to give the best model fit (12). One impetus for this work was to establish a self-consistent set of equilibrium and kinetic data in support of a given modeling approach. 

In specifying rate constants in a reaction mechanism, it is common to give the forward rate constants parameterized as in Eq. 9.83 for every reaction, and temperature-dependent fits to the thermochemical properties of each species in the mechanism. Reverse rate constants are not given explicitly but are calculated from the equilibrium constant, as outlined above. This approach has at least two advantages. First, if the forward and reverse rate constants for reaction i were both explicitly specified, their ratio (via the expressions above) would implicitly imply the net thermochemistry of the reaction. Care would need to be taken to ensure that the net thermochemistry implied by all reactions in a complicated mechanism were internally self-consistent, which is necessary but by no means ensured. Second, for large reaction sets it is more concise to specify the rate coefficients for only the forward reactions and the temperature-dependent thermodynamic properties of each species, rather than listing rate coefficients for both the forward and reverse reactions. Nonetheless, both approaches to describing the reverse-reaction kinetics are used by practitioners. 

We note that the linear and volumetric CTEs in Refs.60 and 61 are not self-consistent in that the latter do not follow from the former according to the formal expression (for a monoclinic cell) Xv=Xa+Xb+Xc+(cosp/sinp)Pxp.. The reason for this is that xv was determined from a direct fit to the temperature dependent unit cell volume rather than from the linear CTEs (M. Herrmann, private communication). Although the same approach was used here, we obtained self-consistency to the percent level. 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

Tight binding (TB) and linear combination of atomic orbitals (LCAO) methods represent the more chemical approach to the problem of surface state calculations. They are basically fitting techniques, but, given a reasonable choice of parameters, they can add considerable detail to the basis provided by self-consistent calculations. The method, as applied to surfaces, was initiated by Hirabayashi [73] and developed into a useable form by Pandey and Phillips [74, 75]. 

As in the simpler jellium model, we retain the simple description of independent electrons, each moving in a confining potential U(r). Here however, that potential is not an arbitrary, made-up model potential chosen to fit data or to make a calculation convenient this potential includes such effects as the exchange interaction with the other electrons. In this, the present approach is quite reminiscent of the Hartree-Fock self-consistent procedure, which will be described next. There is one essential difference. Unlike the Hartree-Fock procedure, here the exchange term is approximated as a local function, depending only on the one-electron density. This approximation yields fast convergence to a self-consistent density. As in the Hartree-Fock... 

A number of theoretical models use a single-chain approach to simulate topological constraints in real polymer networks. The basic idea is that one starts from the statistical mechanics of a single network chain which is subjected to a spatial domain of constraints. The constraining potential is introduced in a heuristic manner and cannot be calculated within the frame of the chosen model self-consistently. Hence, the strength of the topological interaction must be characterized by best-fit parameters of the model. 

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