Accuracy avoiding numerical problems

Choosing a standard GTO basis set means that the wave function is being described by a finite number of functions. This introduces an approximation into the calculation since an infinite number of GTO functions would be needed to describe the wave function exactly. Dilferences in results due to the quality of one basis set versus another are referred to as basis set effects. In order to avoid the problem of basis set effects, some high-accuracy work is done with numeric basis sets. These basis sets describe the electron distribution without using functions with a predefined shape. A typical example of such a basis set might... 

In order to reveal the effects of the JT vibronic interaction [74]-[76] one can employ the adiabatic approximation that was proved to provide a quite good accuracy in the description of the magnetic properties of MV clusters [77] and allowed to avoid numerical solutions of the dynamic problem. According to the adiabatic approach the magnetization can be obtained by averaging the derivatives —dUi(p, H)/dHa over the vibrational coordinates. In the case of an arbitrary p 7 0 the gap between... 

Provided that the large basis sets, which are obviously going to be required to perform very accurate electronic structure calculations, can be designed so as to avoid the numerical problems mentioned in the preceding section, it is likely that, with the use of increasingly parallel architectures in modem computers, we are poised to obtain a significant improvement in the accuracy of atomic and, particularly, molecular electronic structure calculations. 

The form of the response function to be fitted depends on the goal of modeling, and the amount of available theoretical and experimental information. If we simply want to avoid interpolation in extensive tables or to store and use less numerical data, the model may be a convenient class of functions such as polynomials. In many applications, however, the model is based an theoretical relationships that govern the system, and its parameters have some well defined physical meaning. A model coming from the underlying theory is, however, not necessarily the best response function in parameter estimation, since the limited amount of data may be insufficient to find the parameters with any reasonable accuracy. In such cases simplified models may be preferable, and with the problem of simplifying a nonlinear model we leave the relatively safe waters of mathematical statistics at once. 

Nevertheless, the second order closure models are avoided and usually not employed in industrial reactor flow simulations due to their complexity, negligible gain in accuracy and predictivity, and because of additional numerical convergence- and stability problems. Rather, the present trend in reactor modeling is to explore the capabilities of the large eddy simulation (LES) model. 

Inserting the same quadrature approximation to the source term integrals provides an approximate numerical type of closure avoiding the higher order moments closure problem on the cost of model accuracy [131]. This numerical approximation actually neglects the physical effects of the higher order moments. No reports applying this procedure to bubbly flows have been found so far. 

The very first systematic chiroptical analyses have been obtained with ORD measurements in the 1950s and 1960s. The dominance of the ORD analyses has been broken by commercial CD instruments in the 1960s when CD and ORD were measurable with an equivalent accuracy. Since that time ORD is just of interest for selected problems. It should be mentioned here that a new development based on the invention of new numerical quantum mechanical methods, which allows calculating [ajS with a sufficient accuracy, may lead to a situation in which even CD, VCD, and ORD measurements can be avoided. Here, in order to avoid experimental artifacts by association, etc., [a]5 has to be extrapolated with c- 0 section on Quantum me-... 

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