Harmonics method

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

One of the advantages of the simulation of infrared and Raman spectra is the ability to separate the contribution that different conformers make to the overall spectrum. Mathieu and Grand [54] used a quasi-harmonic method using ab initio... 

The use of the impedance technique in the study of polymer coated steel, has been thoroughly described elsewhere. The present paper compares this technique with that of harmonic analysis, originally proposed by Meszaros ). The authors have presented preliminary data using the latter technique(3) wherein the early stages of polymer breakdown have been studied. The current paper extends this work to polymers which have been immersed for a considerable period of time. The harmonic method gives information not available from the impedance technique in the Tafel slopes and the corrosion current are directly measurable. A brief summary of the harmonic method and the equations used are given below. 

It can be seen that for severely degraded specimens, both the harmonic analysis and Impedance techniques are capable of detecting the presence of gross corrosion. The harmonics method provides a reasonable estimation of the corrosion rate when the Impedance data exhibits Warburg type behaviour. For less severely degraded specimens, especially those exhibiting blister attack, the Impedance method Is not as successful as the harmonic analysis technique. 

Levy, R.M., Srinivasan, A.R., Olson, W.K., Mccammon, J.A. Quasi-harmonic method for studying very low-frequency modes in proteins. Biopolymers 1984, 23, 1099-112. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

FDA, USP, and ICH U.S. Food and Drug Administration, U.S. Pharmacopeia, International Conferences on Harmonization Method validation 55,72-74... 

An analytical method must be properly validated if it is to be used at the national level for enforcement purposes. Validated methods stand on their own merits in contested legal actions in national courts. This situation has been recognized by all nations and international organizations. Initiatives to harmonize method validation criteria and approaches are currently being considered. 

Bogdanov, S., Martin, P., and Lullmann, C. (1997). Harmonized methods of the European Honey Commission. 

The project enables risk assessments (or components thereof) to be performed using internationally accepted methods, and these assessments can then be shared to avoid duplication and optimize use of valuable resources for risk management. It also promotes sound science as a basis for risk management decisions, promotes transparency in risk assessment and reduces unnecessary testing of chemicals. Advances in scientific knowledge can be translated into new harmonized methods. 

Of a different nature is the PI approximation (Modest, 2003), also known as the diffusion approximation (Ishimaru, 1997). Here the number of propagation directions is not restricted, but instead it is assumed that energy distributes quite uniformly over all these directions, as will be described in the next section. This approximation is the lowest order of the spherical harmonics method (also known as the Pn approximation). It is more versatile than two-and four-flux models, because it lends itself more easily to different geometries. 

Most discussions of harmonization revolve around excipients or general tests and assays. But the performance of even a harmonized method using different reference standards is not an optimal situation. In fact. 

Chemicals Programme on the development, updating, and expansion of scientifically valid harmonized methods for testing and assessment helps OECD countries (and, increasingly, non-OECD countries that use these methods) to prevent new chemicals that might present unacceptable risks from entering the market and to assess the safety of chemicals already in use. 

The intention of this Council Act is to ensure that data generated in the safety testing of chemicals are of high quality and are based on internationally harmonized methods. Such data can then be used to assess chemical hazards and to make decisions on appropriate activities to prevent or reduce risk to human health or the environment. Where the need for duplicative testing is minimized through MAD, testing costs can... 

The theoretical models which have been used to describe the bonding in cluster compounds of the main group and transition metal elements are reviewed. The historical development of these models is outlined and special emphasis is placed on those studies which have led to the elucidation of structure-electron count correlations. Theoretical treatments of cluster bonding are based on localised, delocalised (molecular orbital) or free electron methods derived from the solution of the Schrodinger equation for a particle on a sphere. A detailed analysis of the Tensor Surface Harmonic method, as an example of a free electron model, is presented. Group theoretical consequences of the model are also presented. 

Quasi-Harmonic Method for Studying Very Low Frequency Modes in Proteins. 

There are many experimental techniques for studying interfacial relaxations of soluble adsorption layers. Except for the wave damping techniques, these methods are developed and used only by individual research groups. Up to now, no commercial set-up exists and therefore, relaxation experiments are not so wide spread. New developments in this field will probably increase the number of investigators studying the dynamic and mechanical properties of adsorption layers, since instruments are easy to construct and data handling is relatively simple. In this section, wave damping and other harmonic methods as well as transient relaxation techniques will be described. 

Figure 5. Thermal - (radians) scatterplots for terminally blocked valine at T = 0.2 kcal/ mol (a, d), 0.6 kcal/mol (b, e), and 2.2 kcal/mol (c,f) and the corresponding descriptions produced by the angular linear quasi-harmonic (a-c) and multivariate wrapped Gaussian (d-f) models. The shaded regions are bounded by the contours where ( , )/ ( 0, °) = e 2. In contrast to the quasi-harmonic method, the MWG method accurately models the data set even at high temperatures. (Adapted from Church and Shalloway [38].)...

Harmonized method. The method to be used for monitoring in supply zones is the method of random sampling during the day (see Note 1, below). Monitoring must be carried out at the supply tap and at frequencies provided for the audit monitoring. 

The idea of the Green s function/principal component analysis is closely related to the essential dynamics approach recently introduced into biomolecular simulations. Other similar works include those by Garcia, Ichiye and Karplus, Go and coworkers,and developers of the quasi-harmonic method. " The basic idea of the essential dynamics approach is to diagonalize a covariance matrix a whose elements are given by the formula... 

Spherical Harmonics Method. Wigner recognized that the diffusion theory approximation in all of these calculations was deficient. However, he was reluctant to invoke more complicated transport theory because the neutrons were not monoenergetic, and any transport correction to the diffusion theory would be obscured by the error caused by the assumption that the thermal neutrons were monoenergetic. He and Breit had used the spherical harmonics method in their original calculations of the size of a bomb, and he supervised the later work at Chicago on the application of the spherical harmonics expansion to calculate the thermal utilization. 

A mathematical method for texture analysis has therefore to define the way of solving this integral equation in view of determining the unknown, i.e. f g). In 1965 Bunge [8] and Roe [9] independently proposed to use series expansions to solve this. Since then many improvements and variants have been proposed for this method which is referred to as the harmonic method in the literature. Other... 

In the following, we will restrict our explanations to some basic ideas in the framework of the harmonic method. 

The texture of a given material is then characterized with the set of the coefficients they are the unknowns in the texture analysis by using the harmonic method. Similarly the pole densities can be expanded into series. 

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