Standards, convergence

Calculations using the semiempirical PM3 method with standard convergence criteria of 0.0003 aii on the maximum component of the gradient vector and either an energy change from the previous cycle of < 10 hartree or a maximum predicted displacement for the next step of < 0.0003 au. 

The cleaning of all the matrices by making equal to zero all quantities which in absolute value are smaller than 10-17 (in double precision) must be rendered more systematic. In this way we expect the accumulation of errors to diminish noticeably. The introduction of the Newton method as well as Ait.ken s extrapolation and other standard techniques to speed up convergency may be suitable in future application. However, before having recourse to these standard convergence techniques, we hope to find other more fundamental basic conditions - such as the spin equation which smoothed the oscillations away — in order to attain a complete control of the process. The most important question which remains open is the way in which the renormalizations of the p-RDM s is performed. Another possible improvement is to extend the spin-adaptation to the renormalization of the 4-RDM in order to make a thorough use of the partial traces of the different symmetries. 

The bubble and dew-point temperature calculations have been implemented by the FORTRAN IV subroutine BUDET and the pressure calculations by subroutine BUDEP, which are described and listed in Appendix F. These subroutines calculate the unknown temperature or pressure, given feed composition and the fixed pressure or temperature. They provide for input of initial estimates of the temperature or pressure sought, but converge quickly from any estimates within the range of validity of the thermodynamic framework. Standard initial estimates are provided by the subroutines. 

Convergence is usually accomplished in 2 to 4 iterations. For example, an average of 2.6 iterations was required for 9 bubble-point-temperature calculations over the complete composition range for the azeotropic system ehtanol-ethyl acetate. Standard initial estimates were used. Figure 1 shows results for the incipient vapor-phase compositions together with the experimental data of Murti and van Winkle (1958). For this case, calculated bubble-point temperatures were never more than 0.4 K from observed values. 

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

Figure 1 The course of energy minimization of a DNA duplex with different choices of coordinates. The rate of convergence is monitored by the decrease of the RMSD from the final local minimum structure, which was very similar in all three cases, with the number of gradient calls. The RMSD was normalized by its initial value. CC, IC, and SG stand for Cartesian coordinates, 3N internal coordinates, and standard geometry, respectively.

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

In attempt to handle uncertainties and unknowns, Tlieodore has proposed a modified version of the standard Delplii panel approach that the autliors have modestly defined as the FLTA (an acronym for tlie Flynn-Theodore Approach)." In order to generate better estimates, several knowlcdgctible individuals within and perhaps outside tlie organiziition are asked to independently provide estimates, with cxplanatoiy details on tliese estimates. Each individuiil in the panel is then allowed to independently review all responses. The cycle is tlien repeated until the group s responses approach convergence. 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

It can be seen from Fig. 7 that V is a linear function of the qf This qV relation was pointed out and discussed at some length in the papers in ref. 6. It is not simple electrostatics in that it would not exist for an arbitrary set of charges on the sites, even if the potentials are calculated exactly. The charges must be the result of a self-consistent LDA calculation. The linearity of the relation and fie closeness of the points to the line is demonstrated by doing a least squares fit to the points. The sums that define the potentials V do not converge at all rapidly, as can be seen by calculating the Coulomb potential from the standard formula for one nn-shell after another. The qV relation leads to a special form for the interatomic Coulomb energy of the alloy... 

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