Simplification quadratic approximation

The system of equations obtained from the stage steady-state conditions can be solved without simplifications (it is reduced to a quadratic equation). However, a simple approximate solution is sufficient, based on the rate of nitrous oxide formation, r(3), being much less than the sum (r(1) + r<2)) of the rates along the routes iV and jV<2). 

In RSM, it is assumed that in the neighborhood of an optimum point, the concavity (whether concave up or down, or even twisted) of an arbitrary surface makes it reasonable to use a quadratic function as an approximation of the surface. The assumption may still hold even if the region of concern is not near an optimum point, but is small enough for the approximation to be valid. Most of the time, this simplification is good enough for practical purposes. 

An important simplification of the theoretical investigation into dynamics has been suggested by Miller, Handy and Adams [104] who proposed taking into account only the most important part of the PES, i.e. the reaction channel. The PES is approximated by the reaction path and its quadratic environment. All motions of nuclei are divided into the motion along the reaction path and the 3N — 7) harmonic vibration motions transverse to it. In this case, the classical reaction path Hamiltonian is given by ... 

The important assumption is that the barriers are all parabolic. This is the origin of the quadratic term in Equation (8.27) (below), which (in the present treatment) is responsible for most of the characteristic predictions of Marcus theory. We shall see later that it is an over-simplification (see Section 8.2.7). The assumption that the parabolic barriers for all the reactions in the series are congruent (which entails that for the symmetrical cases the intrinsic barrier is constant) simplifies the mathematics, and is an acceptable approximation in a first-attempt description, but it cannot be accurate, since the shapes of the potential-energy curves of AH and BH are related to the force-constants, which will vary with AG°. 

It is possible to justify Eq. (4) as follows The maximum number of contacts between a macromolecule and solvent molecules is given by the first portion of the far side of Eq. (4) [(z - 2)x/i2]- The coordination about a single unit of the macromolecule is given by z, and the -2 accounts for the two directions along which the macromolecule continues. This total number of macromolecular sites is to be multiplied by the site fraction on the macromolecules which is actually occupied by the smaller molecules. The site fraction is approximated by the volume fraction of small molecules v. The total number of contacts between macromolecules and small molecules must still be multiplied by some interaction parameter, Ah>i2- Equation (4), indicates that some simplifications are possible. The first one is that the product xn 2y can be replaced by the simpler product iV2. This can easily be seen from the definitions of the volume fractions. Next, the two constants, the coordination number (z - 2) and the interaction parameter (Am/j 2) be combined and expressed in units of RT. This new interaction parameter is commonly denoted by the letter %. It represents the total interaction energy per macromolecular volume element in units of RT. Making these substitutions leads to Eqs. (5) and (6). Note that AG has a quadratic... 

The Gaussian approximation can therefore be seen to be equivalent to a quadratic potential or linear elastic restoring force. Deviations from the Gaussian distribution will correspondingly yield nonlinear force terms in the dynamics. The Gaussian approximation should therefore be an appropriate simplification for describing systems close to equilibrium or at most linearly displaced from the equilibrium state. 

The second most frequent simplification of the PB equation is linearization of the charge density with respect to the potential. The resulting Debye-Hiickel equation may also be applied independently of, though usually in conjunction with, the preceding approximation of Eq. [433]. If the dielectric coefficient of the environment is constant, analytical solutions under a wide variety of boundary conditions may be obtained for systems with sufficient symmetry, as shown above. The obvious condition under which linearization is valid is that where the quadratic term in the expansion of the exponential is small compared to the linear term, that is,... 

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