Function-based

Grubmiiller described a method to induce conformational transitions in proteins and derived rate constants for these ([Grubmiiller 1994]). The method employs subsequent modifications of the original potential function based on a principal component analysis of a short MD simulation. It is discussed in more detail in the chapter of Eichinger et al. in this volume. 

Lonally, the templates were chosen by trial and error or exhaustive enumeration. A itafional method named ZEBEDDE (ZEolites By Evolutionary De novo DEsign) en developed to try to introduce some rationale into the selection of templates et al. 1996 Willock et al. 1997]. The templates are grown within the zeolite by an iterative inside-out approach, starting from a seed molecule. At each jn an action is randomly selected from a list that includes the addition of new (from a library of fragments), random translation or rotation, random bond rota-ing formation or energy minimisation of the template. A cost function based on erlap of van der Waals spheres is used to control the growth of the template ale ... 

The larger the value of n, the more uniform is the size distribution. Other types of distribution functions can be found in Reference 1. Distribution functions based on two parameters sometimes do not accurately match the actual distributions. In these cases a high order polynomial fit, using multiple parameters, must be considered to obtain a better representation of the raw data. 

Distribution models are curvefits of empirical RTDs. The Gaussian distribution is a one-parameter function based on the statistical rule with that name. The Erlang and gamma models are based on the concept of the multistage CSTR. RTD curves often can be well fitted by ratios of polynomials of the time. 

To date the majority of QM-MM applications have employed density functional methods ab initio or semiempirical methods in the quantum region. The energy tenns evaluated in these methods are generally similar, but there are specific differences. The relevant equations for the density functional based methods are described first, and this is followed by a description of the specific differences associated with the other methods. 

For density functional based QM-MM methods the electronic energy terms depend explicitly on the electron density, p(r), of the atoms in the quantum region [46]. The... 

Hydrosilation silicones or addition cure systems utilize a hydride functional crosslinker with a vinyl functional base polymer and a noble metal catalyst. While the cure can be initiated with UV [48,49], thermal cure versions dominate the commercial market [23,50]. In thermal cure systems, inhibitors are necessary for processing and anchorage additives are common. 

X is an acidity function based on the first-order approximation, Eq. (8-92). Values of X have been assigned by an iterative procedure. The data consist of values of Cb/cbh+ as functions of Ch+ for a large number of indicators. For each indicator an initial estimate of pXbh+ and m is made and X is calculated with Eq. (8-94). This yields a large body of X values, which are fitted to a polynomial in acid concentration. From this fitted curve smoothed X values are obtained, and Eq. (8-94), a linear function in X. allows refined values of pXbh + and m to be obtained. This procedure continues until the parameters undergo no further change. Table 8-20 gives X values for sulfuric and perchloric acid solutions. ... 

Becke formulated the following gradient-corrected exchange functional based on the LDA exchange functional in 1988, which is now in wide use ... 

Partitioning the wave function based on the wave function itself. 

The purpose of our study was to model the steady-state (capillary) flow behavior of TP-TLCP blends by a generalized mathematical function based on some of the shear-induced morphological features. Our attention was primarily confined to incompatible systems. 

This equation, with a = 0.25, was based on the observed electric dipole moments of HC1, HBr, and HI. Since then the value of the dipole moment of HF has been determined it is 1.98 D, which corresponds to 47% ionic character, whereas Equation 5 with a = 0.25 gives 59%. It seems justified to formulate an empirical function, based on the values 5, 11, 17, and 47% for the hydrogen halides HI, HBr, HC1, and HF, as calculated from their... 

Jorgensen et al. has developed a series of united atom intermolecular potential functions based on multiple Monte Carlo simulations of small molecules [10-23]. Careful optimisation of these functions has been possible by fitting to the thermodynamic properties of the materials studied. Combining these OPLS functions (Optimised Potentials for Liquid Simulation) with the AMBER intramolecular force field provides a powerful united-atom force field [24] which has been used in bulk simulations of liquid crystals [25-27],... 

Eh Linear energy functional based on the Hamiltonian H it acts on the space of A-particle Trace Class operators. 

Et Linear energy functional based on the Kinetic Energy operator. 

Constant Function Based Scheme If m(x) in Eq (29) is taken as a constant function on the element surrounding point Xy, the influence coefficients are given by performing the following integration... 

Funk OF, Kettmann V, Drimal J, Langer T. Chemical function based pharmacophore generation of endothelin-A selective receptor antagonists. J Med Ghent 2004 47 2750-60. 

Koskinen, P., Hakkinen, H., Seifert, G Sanna, S., Frauenheim, Th. and Moseler, M. (2006) Density-functional based tight-binding study of small gold clusters. New Journal of Physics, 8, 9—11. 

This is based on the example of Matko, Korba and Zupancic (1), who describe methods of simulating process transform functions, based on partitioned and nested forms of solution. Here the process transfer function is given by... 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

A number of problems formulated with data from the literature are given next as exercises. In addition, to the objective function given by Equation 15.11 the reader who is familiar with thermodynamic computations may explore the use of implicit objective functions based on fugacity calculations. 

A much improved description of PE melts is obtained with a more elaborate potential energy function, based on a spherically symmetric potential energy... 

It is a truism that in the past decade density functional theory has made its way from a peripheral position in quantum chemistry to center stage. Of course the often excellent accuracy of the DFT based methods has provided the primary driving force of this development. When one adds to this the computational economy of the calculations, the choice for DFT appears natural and practical. So DFT has conquered the rational minds of the quantum chemists and computational chemists, but has it also won their hearts To many, the success of DFT appeared somewhat miraculous, and maybe even unjust and unjustified. Unjust in view of the easy achievement of accuracy that was so hard to come by in the wave function based methods. And unjustified it appeared to those who doubted the soundness of the theoretical foundations. There has been misunderstanding concerning the status of the one-determinantal approach of Kohn and Sham, which superficially appeared to preclude the incorporation of correlation effects. There has been uneasiness about the molecular orbitals of the Kohn-Sham model, which chemists used qualitatively as they always have used orbitals but which in the physics literature were sometimes denoted as mathematical constructs devoid of physical (let alone chemical) meaning. 

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