Wilson GF-matrix method

To model quantitatively an IINS spectrum, it is only necessary to obtain the amplitudes of motion of the atoms in the vibrational modes. These can be calculated by a variety of methods, such as the balls-and-springs approach of the Wilson GF matrix method, ab-initio calculations, and molecular dynamics this point expresses what is undoubtedly the greatest strength of IINS spectroscopy. Examples are presented below. 

Fig. 9. Comparison of the IINS spectrum (TFXA, ISIS) of the deactivated palladium catalyst (solid line) and the results from the Wilson GF matrix method analysis of the spectrum (dashed line). The model is shown in the top right-hand corner.

In the early 1950s Morino and his co-workers in Japan established a systematic treatment for the calculated of amplitudes of vibration from harmonic force fields, based on the widely used Wilson GF matrix method... 

Thus modes at 360 and 585 involve the wag, those at 386, 425 and 485 involve the rock and that at 457 cm" the torsion. A Wilson GF matrix method study of the cyclohexyl complex [42] came to similar conclusions, but considered that the modes were coupled to to the W-C stretches, which occur in the same region as the linear bends ( 11.2.6). This was done by introducing an interaction force constant between the... 

The experimental INS spectra of the Os and Co complexes and the spectrum of the Os complex modelled with the Wilson GF matrix method are shown in Fig. 7.12. Assignments for C symmetry are shown in Table 7.7. The vibrational modes of the osmium complex are illustrated in Fig. 7.14 (p. 311). 

The method adopted to obtain the foree constants from the vibrational frequencies treated the molecule as a system of point masses connected by springs that obeyed Hooke s law, so the system was purely harmonic. The approach was codified in a classic book [3] and is known as the Wilson GF matrix method. The basis of the method is described in 4.2.2 and in more detail in [4,5]. The key equation is ... 

Fig. 8.2 INS spectra of maleic anhydride [2] (a) experimental, (b) calculated using the Wilson GF matrix method, (c) from a DFT calculation (using the B3LYP functional with the 6-31G(d,p) basis set) of the isolated molecule and (d) from a periodic DFT calculation of the complete unit cell.

It is possible, albeit complex, to extract the force constants from an ab initio calculation and so is not usually done. Table 8.1 compares the force constants for maleic anhydride obtained from the Wilson GF matrix method with those obtained from the ab initio calculation. 

More polycyclic alkanes have been treated by INS spectroscopy. The spectrum of of adamantane (tricyclo[3.3.1.1 ]decane, CioHie) is shown in Fig. 4.6. Norbomane and some of its mono- and dimethyl derivatives (see Fig. 8.5 for the structures) were extensively studied and force fields developed using the Wilson GF matrix method [17—19]. Given the low symmetry and relatively large number of atoms present, this was a tour de force of spectroscopy. 

In this section we will consider polydimethylsiloxane (PDMS) as an example of the type of work that is possible with amorphous polymers. The structure and INS spectrum of PDMS are shown in Fig. 10.21a [40]. The repeat unit shown in Fig. 10.21b was used to model the spectrum using the Wilson GF matrix method [41]. The major features are reproduced skeletal bending modes below 100 cm", the methyl torsion and its overtone at 180 and 360 cm respectively, the coupled methyl rocking modes and Si-0 and Si-C stretches at 700-1000 cm and the unresolved methyl deformation modes 1250-1500 cm. The last are not clearly seen because the intensity of the methyl torsion results in a large Debye-Waller factor, so above 1000 em or so, most of the intensity occurs in the phonon wings. 

Fig. 11.3 INS spectrum of (a) K2[PtCl6] and (b) K2[PdCl4]. The dashed lines are a fit to the internal modes using the Wilson GF matrix method.

Morino and his co-workers established a systematic treatment for the calculation of mean amplitudes of vibration from harmonic force fields, based on the widely used Wilson GF matrix method, which is still the foundation for the majority of such calculations. The theory of such calculations has been developed considerably since, notably by Cyvin and coworkers. With the advent of modern computers, the calculation of mean amplitudes using these methods has become a routine procedure. The resultant extensive literature has been reviewed by Cyvin. ... 

Now it is possible to evaluate the vibrational frequencies using the Wilson GF matrix method,... 

Srivastava et al. carried out a complete study of the vibrational spectrum of acetylglycine, and evaluated its relation to the spectrum of acetylcholine [13]. A normal coordinate analysis of acetylglycine was carried out using the Wilson s GF matrix method. Vibrational frequencies were assigned, and the infrared spectra of acetylglycine and acetylcholine compared. Conformation-sensitive modes of acetylcholine were identified, and a transferable Urey-Bradley force field was also obtained. 

This is recognized as a secular equation which gives the solution to the problem in internal coordinates. This is the original formulation proposed by Wilson, leading to the term GF matrix method which is often encountered in the literature. The relation between internal and normal coordinates is given by... 

The well-known GF matrix technique of E. B. Wilson and his colleagues for calculating the harmonic frequencies of polyatomic molecules is based on the use of valence coordinates, also referred to as internal coordinates. What is presented here is merely a sketch of the method a fuller discussion would require extensive use of matrix algebra, which is beyond the scope of this book. The appendix on matrices in this chapter serves only as a very short introduction to such methods. For details reference should be made to the classical work of E. B. Wilson, J. C. Decius and P. C. Cross (WDC) in the reading list. 

The established method for calculating the vibrational frequencies of molecules is the Wilson GF method.27 In this method, the potential energy of a molecule is defined in terms of the force constants by a matrix F, and the kinetic energy, which depends on the geometry of the molecule, is defined by a matrix G. Using the methods of classical mechanics, the following equation may be derived. 

Conceptually the GF method of Wilson has been applied routinely to polymers, k-dependent F and G matrices similar to those defined in Eqs. (I) and (3) can be defined, thus bringing to the solution of the k-dependent dynamical matrix. 

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