Computational methods Computer programmes

Performance Analysis of a Family of Solid Propellants with Details of the Computer Programme and the Theoretical Method Used , Rept No RPE TN 225, Westcott (Engl) (1963) 6) S.A. Johnson et al, Research on Combustion of Solid Propellants , Rept No 641Q3, Contract DA-04-495-AMC-239(R), Lockheed Proplsn Co, Redlands(1964) 7) R.L.Coates, Research... 

This procedure demands a solution, based on numerical analysis and a rather long computer programme. Then, this method can be used only when detailed results are demanded, requiring the knowledge of the exact distribution of stresses and displacements between phases. 

From precise wavelength measurements of the fluorescence spectrum (which may be performed e. g. by interferometric methods accurate values for the molecular constants can be obtained since the wavelength differences of subsequent lines in the fluorescence progression yield the energy separation of adjacent vibrational and rotational levels as a function of v . From these spectroscopically deduced molecular constants, the internuclear distance can be calculated A special computer programm developed by Zare ) allows the potential curve to be constructed from the measured constants and, if the observed fluorescence progression... 

Based upon the method of calculation adopted, a complete computer programme consisting of three main parts can easily be written for support of such calculations. The three parts are as follows (a) the CNDO part or EHMO part with Madelung correction for calculation of the localized electron orbitals in the anionic group (b) the transition matrix element calculation part and (c) the second-order susceptibility part for the calculation of the microscopic susceptibility of the anionic group followed by the calculation of the macroscopic SHG coefficients of the crystal. 

Figure 36.6. Snapshot of the Method creator of the Immusoft computer programme serving for the establishment of assay protocols. The figure shows on the right-hand side the various steps of the protocol developed for the assay of alkaline phosphatase (ALP), in which the functionalisation of the microchannels (coating and blocking steps) is directly integrated in the assay progress.

NIRS involves the multidisciplinary approaches of the analytical chemist, statistician, and computer programmer. The word chemometrics refers to the application of mathematical or statistical methods to measurements made on chemical systems of varying complexity. Chemometrics is defined as the chemical discipline that uses mathematical, statistical, and other methods that apply formal logic to design or select optimal measurement procedures and experiments, and to provide maximum relevant chemical information by analyzing chemical data. 

The solution of a protein crystal structure can still be a lengthy process, even when crystals are available, because of the phase problem. In contrast, small molecule (< 100 atoms) structures can be solved routinely by direct methods. In the early fifties it was shown that certain mathematical relationships exist between the phases and the amplitudes of the structure factors if it is assumed that the electron density is positive and atoms are resolved [255]. These mathematical methods have been developed [256,257] so that it is possible to solve a small molecule structure directly from the intensity data [258]. For example, the crystal structure of gramicidin S [259] (a cyclic polypeptide of 10 amino acids, 92 atoms) has been solved using the computer programme MULTAN. Traditional direct methods are not applicable to protein structures, partly because the diffraction data seldom extend to atomic resolution. Recently, a new method derived from information theory and based on the maximum entropy (minimum information) principle has been developed. In the immediate future the application will require an approximate starting phase set. However, the method has the potential for an ab initio structure determination from the measured intensities and a very small sub-set of starting phases, once the formidable problems in providing numerical methods for the solution of the fundamental equations have been solved. 

Ab initio calculated geometrical parameters depend on the kind of applied basis sets (which is the main variable when using an ab initio computer programme like, for example, Pople s GAUSSIAN 90 ) and on the kind of calculational procedure The so-called Hartree-Fock limit is the theoretically best result obtainable with a single determinant MO basis. Because of the different weighting of inter-electron repulsion between electron pairs of like and unlike spin, Hartree-Fock calculations are in error. They may be improved by the use of configuration interaction methods (Cl) or by the use of perturbation theory, like the Moller-Plesset treatment of second, third or fourth order (MP2, MP3 or MP4). 

The simplifications of band-structure calculations which are now referred to as linear methods were introduced by Ole K. Andersen almost ten years ago. Since then these ideas have been taken up by several workers in the field and translated into computer programmes that generate the band structure of almost any material. As a result, running times on computers have been cut by orders of magnitude. 

At this stage it should be clear that the linear theory outlined in the preceding chapters may be applied at many levels of approximation. In the most favourable cases one may obtain energy levels simply by means of a pocket calculator, and this will often suffice for an overall picture of the band structure one wants to study. On the other hand, for most applications one must resort to calculations on a large-scale electronic computer, and to that end we now present a package of computer programmes, LMTOPACK, based on the linear muffin-tin orbital method. 

Historically the three parameters m, a, and b were the first to be used in band calculations using the LMTO method, and they also appear in various parts of the computer programmes presented in Chap.9. However, for the purpose of comparing different materials, they depend too strongly on E to be useful, and in Chap.4 we therefore introduce physically more significant parameters for comparisons between elements. 

The use of computers is imperative for the exact calculation of distillation columns run for the separation of multicomponent mixtures [264]. Possibilities of modelling a primary distillation of crude oil were discussed by Kirbach, Frenzel and Strankmiiller [265]. The computing programme as prepared for a ZRA 1 computer for ideal multicomponent mixtures was published by Leibnitz and Schuhler [253]. The calculating procedure for non-ideal multicomponent mixtures was dealt with by Nagel et al. [254]. Morozova and Platonov [266] developed a method for examin-... 

Modelling has become much more sophisticated since complex computer programmes can be handled easily. However, particularly for learning models, mathematical equations are preferred that are solvable by analytical methods. 

In cases where a sample contains several /3 "-emitters, the activity of the radionuclide of interest can be determined after chemical separation or by measurement of the annihilation radiation as a function of time followed by analysis of the decay curve. At the same time a best estimate for the half-life of the considered radionuclide may be obtained, which may serve as a qualitative criterion. Analysis of a decay curve may be carried out graphically. This method is however subjective and time-consuming, so that appropriate computer programmes are preferable. 

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