Special functions theory

It has become fashionable to talk about density functional theory, so we had better come to grips with the term linear functional. It is a highly specialized term in modern algebra with the following meaning (for the mathematicians amongst you, and paraphrased from Birkhoff and MacLane (1965 182)) ... 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

The precursor of such atomistic studies is a description of atomic interactions or, generally, knowledge of the dependence of the total energy of the system on the positions of the atoms. In principle, this is available in ab-initio total energy calculations based on the loc density functional theory (see, for example, Pettifor and Cottrell 1992). However, for extended defects, such as dislocations and interfaces, such calculations are only feasible when the number of atoms included into the calculation is well below one hundred. Hence, only very special cases can be treated in this framework and, indeed, the bulk of the dislocation and interfacial... 

During the last decade, density-functional theory (DFT)-based approaches [1, 2] have advanced to prominent first-principles quantum chemical methods. As computationally affordable tools apt to treat fairly extended systems at the correlated level, they are also of special interest for applications in medicinal chemistry (as demonstrated in the chapters by Rovira, Raber et al. and Cavalli et al. in this book). Several excellent text books [3-5] and reviews [6] are available as introduction to the basic theory and to the various flavors of its practical realization (in terms of different approximations for the exchange-correlation functional). The actual performance of these different approximations for diverse chemical [7] and biological systems [8] has been evaluated in a number of contributions. 

Compound 53 has an orifice just wide enough to allow the potential entry of small ions, atoms (He), or molecules (H2, D2, T2) inside its cavity as judged from high-level transition state calculations at the density functionally theory level (B3LYP/6-31G, Figure 19).1791 The low barrier of insertion for helium makes it a specially attractive target, but other spe-... 

Although one could consider the electron density as just one of the many quantum chemical descriptors available, it deserves special attention. In QSM, it is the only descriptor used for a number of reasons. The idea of using the electron density as the ultimate molecular descriptor is founded on the basic elements of quantum mechanics. First of all, it is the all-determinmg quantity in density functional theory (DFT) [11] and also holds a very close relation to the wave function. Convincing arguments were given by Handy and are attributed to Wilson [12], although initial ideas can also be traced back to Bom [13] and von... 

Chatteijee, A. 2002. Edited the Special Issue on Application of Density Functional Theory in Chemical Reactions , Int. J. Molec. Sci., Issue 4 (April) pp. 234 444. http // www.mdpi.org/ijms/list02.htmhttp //www.mdpi.org/ijms/index.htm... 

Miller, W., Jr.(1968), Lie Theory and Special Functions, Academic Press, New York. 

Vilenkin, N. I. (1968), Special Functions and the Theory of Group Representations, Transl. Math. Monogr., Am. Math. Soc. 22, Providence. 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

The electronic coupling of donor and acceptor sites, connected via a t-stack, can either be treated by carrying out a calculation on the complete system or by employing a divide-and-conquer (DC) strategy. With the Hartree-Fock (HF) method or a method based on density functional theory (DFT), full treatment of a d-a system is feasible for relatively small systems. Whereas such calculations can be performed for models consisting of up to about ten WCPs, they are essentially inaccessible even for dimers when one attempts to combine them with MD simulations. Semiempirical quantum chemical methods require considerably less effort than HF or DFT methods also, one can afford application to larger models. However, standard semiempirical methods, e.g., AMI or PM3, considerably underestimate the electronic couplings between r-stacked donor and acceptor sites and, therefore, a special parameterization has to be invoked (see below). 

Generating functions occur in other contexts than moments and, in particular, the theory of special functions is full of them. Their use in deriving the solution to a whole family of equations is well illustrated in the reprint M, p. 337, Chapter 10. A simple example would be the ideal residence time tracer the chain of identical C s governed by... 

A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd., New Delhi, 1978 H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes Academic Press, New York, 1982. 

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