Computational problems

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

More complicated reactions and heat capacity functions of the foiiii Cp = a + bT + cT + are treated in thermodynamics textbooks (e.g., Klotz and Rosenberg, 2000). Unfortunately, experimental values of heat capacities are not usually available over a wide temperature range and they present some computational problems as well [see Eq. (5-46)]. 

We have chosen to cover a large number of topics, with an emphasis on when and how to apply computational techniques rather than focusing on theory. Each chapter gives a clear description with just the amount of technical depth typically necessary to be able to apply the techniques to computational problems. When possible, the chapter ends with a list of steps to be taken for difficult cases. 

It is likely that there will always be a distinction between the way CAD/CAM is used in mechanical design and the way it is used in the chemical process industry. Most of the computations requited in mechanical design involve systems of linear or lineatizable equations, usually describing forces and positions. The calculations requited to model molecular motion or to describe the sequence of unit operations in a process flow sheet are often highly nonlinear and involve systems of mixed forms of equations. Since the natures of the computational problems are quite different, it is most likely that graphic techniques will continue to be used more to display results than to create them. 

Design and Operation of Azeotropie Distillation Columns Simulation and design of azeotropic distiUation columns is a difficult computational problem, but one tnat is readily handled, in most cases, by widely available commercial computer process simulation packages [Glasscock and Hale, Chem. Eng., 101(11), 82 (1994)]. Most simida-... 

It is worth pointing out that the wide range of coefficients may cause computational problems for the optimization software. This is commonly referred to as the scaling problem. One way of circumventing this problem is to define scaled flowrates of MSAs in units of 10 ° m /s and scaled residual loads in units of 10 ° kmol/s, i.e., let... 

The first step in reducing the computational problem is to consider only the valence electrons explicitly, the core electrons are accounted for by reducing the nuclear charge or introducing functions to model the combined repulsion due to the nuclei and core electrons. Furthermore, only a minimum basis set (the minimum number of functions necessary for accommodating the electrons in the neutral atom) is used for the valence electrons. Hydrogen thus has one basis function, and all atoms in the second and third rows of the periodic table have four basis functions (one s- and one set of p-orbitals, pj, , Pj, and Pj). The large majority of semi-empirical methods to date use only s- and p-functions, and the basis functions are taken to be Slater type orbitals (see Chapter 5), i.e. exponential functions. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

At that time I was handicapped by my remembering a misinterpretation that I had made of some results obtained in 1932 by one of my students, Ralph Hultgren (6). He had begun to make a thorough study of sets of equivalent spd hybrid bond orbitals, and soon found that he could not handle the computational problem in those precomputer days. I pointed out that the best hybrid orbitals have cylindrical symmetry about... 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

Two macromolecular computational problems are considered (i) the atomistic modeling of bulk condensed polymer phases and their inherent non-vectorizability, and (ii) the determination of the partition coefficient of polymer chains between bulk solution and cylindrical pores. In connection with the atomistic modeling problem, an algorithm is introduced and discussed (Modified Superbox Algorithm) for the efficient determination of significantly interacting atom pairs in systems with spatially periodic boundaries of the shape of a general parallelepiped (triclinic systems). 

The simulation of macromolecular systems involves, in principle, the same difficulties as that of compounds of low-molecular mass, but the polymeric nature of the molecules tends to aggravate the computational problems faced by investigators of small molecules. 

Presently, only the molecular dynamics approach suffers from a computational bottleneck [58-60]. This stems from the inclusion of thousands of solvent molecules in simulation. By using implicit solvation potentials, in which solvent degrees of freedom are averaged out, the computational problem is eliminated. It is presently an open question whether a potential without explicit solvent can approximate the true potential sufficiently well to qualify as a sound protein folding theory [61]. A toy model study claims that it cannot [62], but like many other negative results, it is of relatively little use as it is based on numerous assumptions, none of which are true in all-atom representations. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

Due to their spatial localization, it follows that the interaction eneigy of an occupied LMO with any distant virtual LMO will be zero, and so the computational problem becomes reduced to annihilating matrix elements connecting LMOs that are close in space. These LMOs can be easily identified from the molecular connectivity table given the requirement that any allowed LMO spans one or two atoms. The Fock matrix element, Fif takes the form ... 

Local-density potentials greatly simplify the computational problems associated with defect calculations. In practice, however, such calculations still are very computer-intensive, especially when repeated cycles for different atomic positions are treated. In most cases the cores are eliminated from the calculation by the use of pseudopotentials, and considerable effort has gone into the development of suitable pseudopotentials for atoms of interest (see Hamann et al., 1979). 

I am grateful to Paul Weymans for drawing several complicated figures and to Tiny Verhoeven for solving several computer problems. 

The additional number of differential equations and increased complexities of the equilibrium relationships may also be compounded by computational problems caused by widely differing magnitudes in the equibbrium constants for the various components. As discussed in Section 3.3.2, it is shown that this can lead to widely differing values in the equation time constants and hence to stiffness problems for the numerical solution. 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

If the values calculated in these two ways are identical, the fluorescence decay is indeed a single exponential. Otherwise, for a multi-component decay, t < tm. In this case, several series of images have to be acquired at different frequencies (at least 5 for a triple exponential decay because three lifetimes and two fractional amplitudes are to be determined), which is a challenging computational problem. 

The path computation problem is formulated in terms of logic programming (Goto et al., 1997). The binary relation of X and Tis a fact denoted by ... 

---