Computational expense

The choice of parameterization and the design of a discretization method are not independent Some choices of parameters will facilitate symplec-tic/reversible discretization while others may make this task very difficult or render the resulting scheme practically useless because of the computational expense involved. 

For example, a 10 GPa (total strain = 0.06) shock wave in copper has a maximum total strain rate 10 s [21] the risetime would thus be (eje) 0.6 ns. For uniaxial-strain compression, y averaged over the entire shock front. The resolution of the shock wave in a large-scale, multidimensional finite-difference code would be computationally expensive, but necessary to get the correct strength f behind the shock. An estimate of the error made in not resolving the shock wave can be obtained by calculating dt/dy)o with y 10 s (the actual plastic strain rate) and y 10 s (the plastic strain rate within the computed shock wave due to a time step of 0.06 qs). From (7.41) with y = 10 s (actual shock wave) and y = 10 s (computation) ... 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Essentially, the RISM and extended RISM theories can provide infonnation equivalent to that obtained from simulation techniques, namely, thermodynamic properties, microscopic liquid structure, and so on. But it is noteworthy that the computational cost is dramatically reduced by this analytical treatment, which can be combined with the computationally expensive ab initio MO theory. Another aspect of such treatment is the transparent logic that enables phenomena to be understood in terms of statistical mechanics. Many applications have been based on the RISM and extended RISM theories [10,11]. 

EMGRESP is a source-term and dispersion emergency response screening tool for calculating downwind contours with a minimum of user input and computational expense in the event of a release of a hazardous chemical. The program provides hazardous contaminant information, calculates toxic concentrations at various distances downwind of a release, and c" the... 

Flynn et al." applied a finite element based numerical model to solve the problem of a push-pull flow with cross-drafts and demonstrate that the results show good agreement with experimental data. They note, however, that the numerical method is time consuming and therefore computationally expensive. 

The gradient of the energy is an off-diagonal element of the molecular Fock matrix, which is easily calculated from the atomic Fock matrix. The second derivative, however, involves two-electron integrals which require an AO to MO transformation (see Section 4.2.1), and is therefore computationally expensive. 

A significant advantage is that the constrained optimization can usually be carried out using only the first derivative of tlie energy. This avoids an explicit, and computationally expensive, calculation of the second derivative matrix, as is nomially required by Newton-Raphson techniques. 

Force fields split naturally into two main classes all-atom force fields and united atom force fields. In the former, each atom in the system is represented explicitly by potential functions. In the latter, hydrogens attached to heavy atoms (such as carbon) are removed. In their place single united (or extended) atom potentials are used. In this type of force field a CH2 group would appear as a single spherical atom. United atom sites have the advantage of greatly reducing the number of interaction sites in the molecule, but in certain cases can seriously limit the accuracy of the force field. United atom force fields are most usually required for the most computationally expensive tasks, such as the simulation of bulk liquid crystal phases via molecular dynamics or Monte Carlo methods (see Sect. 5.1). 

Generalizations derived from a few problem-solving instances. Solving branch-and-bound problems is computationally expensive. Thus we would like to be able to achieve improvements in problem solving as rapidly as possible. 

However, the direct dynamics calculations are computationally expensive, and cannot employ particularly high levels of electron correlation or large basis sets. If certain regions of the potential cannot be treated to within the required accuracy using a computationally affordable level of theory, the results may have unacceptably large errors. Nevertheless, direct dynamics calculations have played and will play a critical role in the discovery and analysis of competing pathways in chemical reactions. 

While the general features of halogen bonding are now well known, it has proven challenging to develop models with sufficient accuracy to predict spectroscopic features and bond energies. This is particularly problematic with iodine, where high quality basis sets are not readily available and are computationally expensive. There have been numerous approaches taken to address this issue during the past decade, many of which are discussed below. 

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