FG matrix method

IR absorption spectra of oxypentafluoroniobates are discussed in several publications [115, 157, 167, 185, 186], but only Surandra et al. [187] performed a complete assignment of the spectra. Force constants were defined in the modified Urey-Bradley field using Wilson s FG matrix method. Based on data by Gorbunova et al. [188], the point group of the NbOF52 ion was defined as C4V. Fifteen normal modes are identified for this group, as follows ... 

Solutions of the vibrational secular equation for [XeF] [IrF5] were carried out using the Wilson FG-Matrix Method for several sets of force constants to test the consistency of the observed spectrum with the postulated structure and to seek an explanation for the doublet at ca. 600 cm. attributed to Xe F stretching. 

The force constants Ki were calculated using the FG matrix method for an octahedral MXe molecule [33] ... 

If, however, the ab-initio calculation of a specific molecule jdelds the complete potential energy hypersurface (as discussed above) the FG-matrix method (7) can be used. This was done by, e.g. AUavena (52), Hartmann and Gliemann (55) and Menna, Mocda and Randaccio (44). Huff and Ellison (144) present an easier and simpler method which is very favourable for computational purposes it also depends on truncating the potential after the harmonic term the 3 N mass-weighted (mi) Cartesian displacement co-ordinates Axi are called qi, and are given by... 

Given the molecular geometry and a set of force constants for a polyatomic molecule, it is a routine matter to calculate the normal coordinates, a procedure known as normal coordinate analysis. Suites of computer programs are readily available that will calculate vibrational frequencies and the internal coordinate composition of each normal vibration. Most of the early calculation of vibration frequencies were made by Wilson s FG-matrix method, which is briefly summarized below. Today, a number of alternative techniques based on semiempirical methods, molecular mechanics, or density functional theory are also available, in convenient commercial software packages. 

In the Wilson FG-matrix method, the problem is framed in internal coordinates rather than in... 

The identification of the IRs according to which the normal coordinates transform can greatly reduce the computational labor associated with implementing the FG matrix method. It is frequently easy to set up symmetry coordinates, as linear combinations of internal coordinates, which transform according to IRs of the point group G. For CIF3, one choice of symmetry coordinates would be... 

In the present book we have used the cogredient expansion form (2.14), where, as distinct from the standard form, an additional normalizing factor has been introduced, namely (—l)< v/(2K + l)/4n. Our expansion of the classical probability density p(0, differs from the standard one in exactly the same way as the expansion of the quantum mechanical density matrix p over 2Tq differs from the expansion over lTg. In Section 5.3 we present a comparison between the physical meaning of the classical polarization moments pg, as used in the present book, and the quantum mechanical polarization moments fg, as determined by the cogredient method using normalization (D.ll). 

Depending on the X-ray source and the spectral modification devices, the LD are in the pg range for 2—3 kW X-ray tubes and in the fg range with excitation by means of synchrotron radiation. Figure 11.15 shows a typical TXRF spectrum the absolute detection limit values of typical TXRF instruments are shown in Fig. 11.10. Thus, TXRF permits to simultaneously determine trace elements in samples of small volume. Additional advantages are insensitivity to matrix effects, easy cahbration, fast analysis times and low cost. In practice, the method is in particular apphed for multi-element determinations in water samples of various nature and for the routine analysis of Si-wafer surfaces employed in the microelectronics industry. 

A normal coordinate analysis (NCA) for hexazacyclophane Cu(II) complex was performed by using the Wilson s force field geometrical matrix (FG) method . An INDO/1 optimized molecular geometry was used to build the G matrix . The CuN distance is 1.84 A for the copper atom in the macrocycle plane this bond length is relatively short in comparison to that reported for the non-macrocycle tris-(l,10-phenantroline)Cu(II) (2.1 A) the planar macrocycles copper porphin (2.031 and tetraazacyclotetradecane Cu(II) derivatives (2.08... 

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