FG matrix

A unitary transformation is then introduced whieh diagonalizes the FG matrix, yielding eigenvalues s, and eigenvectors q,. The kinetic energy operator is still diagonal in these eoordinates. 

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 ... 

The second rule for isotopomer harmonic frequencies is the so-called Sum Rule which follows from Equation 3.A1.8. Equation 3.A1.8 relates the sum of the squares of all the frequencies to the sum of the diagonal matrix elements of the (FG) matrix diagonalized to obtain the frequencies. When mass weighted Cartesian displacement coordinates are used to calculate the vibration frequencies, this means that the sum of the A s (A = 4n2v12)can be found as follows (Equation3.51)... 

Wavenumbers from the FG matrix analysis force constants from 4-31G calculations . ... 

When this procedure is carried out in Cartesian coordinates, there should be six (five for a linear molecule) eigenvalues of the FG matrix being exactly zero, corresponding to the translational and rotational modes. In real calculations, however, these values are not... 

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... 

In this case, analytic formulas can also be obtained for the second derivatives that govern the harmonic vibrations [10]. Table 2 lists the vibrational force constants and normal mode frequencies determined [10, 12] using Wilson s FG matrix treatment, the procedure standard for molecular vibrations [14]. The radial displacements from the minimum are combined in the usual way to form symmetry coordinates,... 

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... 

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