Computational tractability

Concluding, it is essential to represent complex, real-life flow situations by computationally tractable models that retain adequate details. As an example, a computational snapshot approach that simulates the flow in stirred reactors or other vessels for any arbitrary impeller has been developed [5]. This approach lets the engineer simulate the detailed fluid dynamics around the impeller blades with much less computations that would otherwise be required. Improvements in CFD technique are likely to encourage further work along these lines. 

To be computationally tractable, the CA lattice must be of finite size. Boundaries, therefore, exist at the edge of the lattice. 

Herein lie two fairly different approaches to this problem. One approach considers the computational tractability to be paramount, and in this case, one is willing to sacrifice accuracy for tractability. Protein-ligand docking is an example of this approach, and due to its tractability, has become a very prevalent means to quickly assay how well a ligand can bind to a protein. 

The physics problem that needs to be addressed is that of an isolated impurity in an infinite crystal. This problem is clearly too complex to treat exactly specific geometrical arrangements have to be chosen that closely represent the physical situation while being computationally tractable. 

For cases with only one internal coordinate, either approach can be implemented in a CFD code (but the computational cost for the same accuracy can be very different). However, for cases with more than one internal coordinate, only the quadrature methods are computationally tractable on current computers. Thus, in the examples below, we will describe only CFD models based on the... 

The MO theory differs greatly from the VB approach and the basic MO theory is an extension of the atomic structure theory to molecular regime. MOs are delocalized over the nuclear framework and have led to equations, which are computationally tractable. At the heart of the MO approach lies the linear combination of atomic orbitals (LCAO) formahsm... 

Water Potentials. The ST2 (23), MCY (24), and CF (2J5) potentials are computationally tractable and accurate models for two-body water-water interaction potentials. The ST2, MCY and CF models have five, four, and three interaction sites and have four, three and three charge centers, respectively. Neither the ST2 nor the MCY potentials allow OH or HH distances to vary, whereas bond lengths are flexible with the CF model. While both the ST2 and CF potentials are empirical models, the MCY potential is derived from ab initio configuration interaction molecular orbital methods (24) using many geometrical arrangements of water dimers. The MCY+CC+DC water-water potential (28) is a recent modification of the MCY potential which allows four body interactions to be evaluated. In comparison to the two-body potentials described above, the MCY+CC+DC potential requires a supercomputer or array processor in order to be computationally feasible. Therefore, the ST2, MCY and CF potentials are generally more economical to use than the MCY+CC+DC potential. 

There are several future research directions for this project. First, results from the FMO reaction check are not infallible due to the qualitative nature of this check. A more precise, yet computationally feasible model may be possible. Second, more work remains in the WLN rearranger a full system based on our concepts would require knowledge of the entire complement of WLN rules. It may also be desirable to adopt or develop another, more computationally tractable line notation for the purpose of synthetic analysis. Finally, we would like to extend our work to more reaction classes to examine its potential in more detail. 

In addition to phase change and pyrolysis, mixing between fuel and oxidizer by turbulent motion and molecular diffusion is required to sustain continuous combustion. Turbulence and chemistry interaction is a key issue in virtually all practical combustion processes. The modeling and computational issues involved in these aspects have been covered well in the literature [15, 20-22]. An important factor in the selection of sub-models is computational tractability, which means that the differential or other equations needed to describe a submodel should not be so computationally intensive as to preclude their practical application in three-dimensional Navier-Stokes calculations. In virtually all practical flow field calculations, engineering approximations are required to make the computation tractable. 

The adoption of hierarchical classification schemes for both chemical and biological entities makes the storage and analysis of annotated chemical libraries computationally tractable and reduced to managing binary annotation matrices. These binary matrices are often visually illustrated as heatmaps (Fig. 7), where red indicates that the molecule is annotated to a target (i.e. shows biological activity under a certain criteria), while green... 

From the last column of the table, we see that the ratio of the parallel-spin to the total correlation energy is remarkably independent of the size of the basis set. Contrary to expectation, the parallel-spin correlation contribution appears to be about as difficult to account for within a finite basis-set approach as the antiparallel-spin correlation. Our investigation does not provide a careful study of the basis-set saturation behavior in MP2 calculations, such as is given in Refs. [74,72,75,33]. However, our results show that, with small- and moderate-sized basis sets which are sufficiently flexible for most purposes and computationally tractable in calculations on larger systems, there is no evidence that the parallel-spin correlation contribution converges more rapidly than the antiparallel-spin contribution. A plausible explanation for this effect is that, for small interelectronic separations, the wavefunction becomes a function of the separation, which is difficult to represent in a finite basis-set approach for either spin channel. The cusp condition of Eq. (19) is a noticeable manifestation of this dependence, but does not imply that the antiparallel-spin channel is more difficult to describe with a moderate-sized basis set than the parallel channel. In fact, in the parallel correlation hole, there is a higher-order cusp condition, relating the second and third derivatives with respect to u [76]. 

With an appropriate /(r12) function, e.g., in the original linear form f(r-[2) — C12, the operator product r firu) is no longer singular. Such cancellation is not possible with Slater determinants alone and this is what allows explicitly correlated wave functions to achieve accurate correlation energies with relatively small basis sets. With the single explicitly correlated term, therefore, we effectively include a linear combination of an infinite set of Slater determinants, but without the need to solve an infinite set of equations to determine the corresponding amplitudes. The R12 method constructs wave functions that are more compact and computationally tractable than naive Slater-determinant-based counterparts. 

The right-hand side can be evaluated with mere 0(nWRl) arithmetic operations (nocc is the number of occupied spin orbitals) using just computationally tractable two-electron integrals. For atoms, nonzero contributions to this sum occur only from the RI basis functions with angular quantum numbers up to 3Locc, where Locc is the maximum angular quantum number of occupied spin orbitals. 

While many aspects of the theory of classical condensed phase rate processes are well developed, no computationally tractable scheme based on a theory with controlled approximations exists for quantum systems, or for mixed quantum-classical systems, where some degrees of freedom, typically those of the solvent, may be treated classically. Kapral s research in this area has... 

HF is the simplest of the ab initio methods, named after the fact that they provide approximate solutions to the electronic Schrodinger equation without the use of empirical parameters. More accurate, correlated, ab initio methods use an approximate form for the wavefunction that goes beyond the single Slater determinant used in HF theory, in that the wavefunction is approximated instead as a combination or mixture of several Slater determinants corresponding to different occupation patterns (or configurations) of the electrons in the molecular orbitals. When an optimum mixture of all possible configurations of the electrons is used, one obtains an exact solution to the electronic Schrodinger equation. This is, however, not computationally tractable. 

Quantum chemical studies of cyclizations of enediynes and enyneallenes have been reviewed.180 The intermediates are computationally tractable as a result of the unrestricted broken spin symmetry (UBS) approach using GGA functionals for the description of open-shell biradicals. The intermediacy of biradicals in Cope-type rearrangements, to which the Bergman and Myers-Saito reactions belong, are shown to be predictable using a very simple rule biradicals are likely to be intermediates if they are stabilized either by allyl resonance or by aromaticity. 

The exact functional of DFT is unknown, and, even if it was known, there is no guarantee that it would be significantly more computationally tractable to evaluate energies and densities from it than it is to obtain them from exact solutions of the Schrodinger equation within the wavefunction approach. However, many approximate functionals are known which give excellent results for a range of chemical problems, and it is with those that we are concerned. All of these functionals adopt the same basic energy expression ... 

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