Functional parametric

Although the geometry can be expressed entirely mathematically in terms of formulas and relations, to do so may be awkward for those proficient in mathematics and quite incomprehensible for those who are not. It is often the case that curves and surfaces must be specified functionally, parametrically, or as piecewise sections, all of which add burden and potential error in specification. 

In rigorous quantum mechanics, something like an electronic base function parametrically dependent on nuclear configuration space cannot be. Such dependence would imply that the electronic quantum number of the base function depends upon the particular selected region of nuclear configuration space. 

FIG. 13 Illustration of the buffer capacity measurement by means of an applied sinusoidal perturbation of the titrant. Basically, the method can be described by an electrochemo-electrical transfer function, parametrically dependent on the buffer capacity in the chemical domain. 

According to Eq. (11.93), the decoupled Hamiltonian within the Foldy-Wouthuysen framework is formally given as a series of even terms of well-defined order in 1/c. In most presentations of the Foldy-Wouthuysen transformation the exponential function parametrization Hjj] = exp(W[j]) is applied for each transformation step. However, in the light of the discussion in section 11.4 the specific choice of this parametrization does not matter at all, since one necessarily has to expand Ui into a power series in order to evaluate the Hamiltonian. Consequently, in order to guarantee a most general analysis, the most general parametrization for the Foldy-Wouthuysen transformation should be employed [610]. Thus, li is parametrized as a power series expansion in an odd and antihermitean operator W, , which is of (2/+l)-th order in 1/c, (cf. section 11.4). After n transformation steps, the intermediate, partially transformed Hamiltonian f has the following structure. 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

Parametrized representations of individual damping dispersion functions were first obtained [127] by fitting ab initio damping functions [74] for Ft-Fl interactions. The one-parameter dampmg fiinctions of Douketis et al are [127] ... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

The separation of nuclear and electronic motion may be accomplished by expanding the total wave function in functions of the election coordinates, r, parametrically dependent on the nuclear coordinates... 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

As usually indicated by the semicolon, both the wave functions and eigenvalues [ (R)] depend parametrically on the internal nuclear coordinates. 

The first point to remark is that methods that are to be incorporated in MD, and thus require frequent updates, must be both accurate and efficient. It is likely that only semi-empirical and density functional (DFT) methods are suitable for embedding. Semi-empirical methods include MO (molecular orbital) [90] and valence-bond methods [89], both being dependent on suitable parametrizations that can be validated by high-level ab initio QM. The quality of DFT has improved recently by refinements of the exchange density functional to such an extent that its accuracy rivals that of the best ab initio calculations [91]. DFT is quite suitable for embedding into a classical environment [92]. Therefore DFT is expected to have the best potential for future incorporation in embedded QM/MD. 

The pore size distribution function (a) appears parametrically in the flux relations of Feng and Stewart, so their models certainly cannot be completely predictive in nature unless this distribution is known. It is... 

The most general method i.s a form of parametric mapping in which the transformation functions, and in Equation (2.26), are polynomials... 

Fig. 9. Parametric simulation using the MSD model of water flux vs feed concentration as a function of organic sorption coefficient, where for A,...

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

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