Wilson vibrational analysis

The energy levels of the vibrational modes can be predicted with a reasonable accuracy on the basis of the standard Wilson vibrational analysis (241,244) (called GF analysis). The vibrational motion of atoms in the polyatomic system is approximated by harmonic oscillations in a quadratic force field. Computations of the force constants are the subject of quantum chemistry. 

In the Wilson matrix analysis, the normal vibrational modes in methyl radical were assumed to be the same as those in both methyl ions. This assumption is rather crude however, it is believed to influence the results very little. The following values for the normal vibrational modes were obtained (in cm" ) 3100 3100 2915 1620 1620 1030. 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/6-31G and MP2/6-31G (0.25) constants (see next section), are evaluated widiin the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HO-IG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,... 

This basic approach, elaborated brilliantly in the classic book by Wilson et al. [1], was the only method used to perform vibrational analysis, both theoretically and experimentally during much of the 20th century. The approach is still used today in most of the popular electronic structure codes [2,3] and also has become a valuable tool in biomolecular chemistry [4]. 

Harmonic vibrational frequencies of base pairs were evaluated using the standard Wilson FG analysis. ... 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

Since compounds 10 and 30 are rather complicated, the authors could not attempt to determine the normal frequencies of the molecule, but had to restrict the analysis of spectra to the determination of characteristic frequencies. Thus, six frequencies have been found that can be used to identify 2-benzopyrylium cations. Band I appears between 1650-1610 cm-, and the pyrylium ring is responsible for this vibration [8a band according to Wilson s notation (34MI1)]. The position of this band and its intensity are dependent on the nature and position of substituents in the cation, and these changes are similar to data of monocyclic pyrylium salts [82AHC(Suppl)]. 

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. 

A critical pre-requisite to using Raman and resonance Raman spectroscopy to examine the excited-state structural dynamics of nucleic acids and their components, is the determination of the normal modes of vibration for the molecule of interest. The most definitive method for determining the normal modes is exhaustive isotopic substitution, subsequent measurement of the IR and Raman spectra, and computational analysis with the FG method of Wilson, Decius, and Cross [77], Such an analysis is rarely performed presently because of the improvements in accuracy of ab initio and semi-empirical calculations. Ab initio computations have been applied to most of the nucleobases, which will be described in more detail below, resulting in relatively consistent descriptions of the normal modes for the nucleobases. 

Normal Coordinate Analysis. In order to obtain quantitative information about the strength of the coordinate bond, it is necessary to carry out normal coordinate analysis using Wilson s GF method (55, 36). Such attempts have been made recently on ammine (57, 55), halogeno-ammine (57, 55), halogeno (15), nitro (51), and cyano (20) complexes of Co(III) and have given the following Urey-Bradley force constants for the coordinate bond stretching vibration ... 

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

A detailed analysis of the vibrational progressions in the X, A, B and C states is reported in [21], Such a discussion is out of the scope and we therefore highlight the main findings here. Extended progressions due to tire ai vibrational mode V13 (6a in Wilson s notation) have been observed in the X, A and B states. Peak spacings of... 

The interactions of electromagnetic radiation with the vibrations of a molecule, either by absorption in the infrared region or by the inelastic scattering of visible light (Raman effect), occur with the classical normal vibrations of the system (Pauling and Wilson, 1935). The goal of our spectroscopic analysis is to show how the frequencies of these normal modes depend upon the three-dimensional structure of the molecule. We will therefore review briefly in this section the nature of the normalmode calculation more detailed treatments can be found in a number of references (Herzberg, 1945 Wilson etal., 1955 Woodward, 1972 Cali-fano, 1976). We will then discuss the component parts that go into such calculations. 

This analysis highlights the need for information from all three t3T)es of vibrational spectroscopy and the crucial role that the INS intensities play in the assignments. The assignments can be substantiated by generating the INS spectrum by the Wilson GF method and the result is compared to the experimental spectrum in Fig. 6.25. The agreement for the internal modes is good but the analysis has omitted the strongest feature in the spectrum, the librational mode at 460 cm. ... 

The BrF radical has been identified from its e.s.r. spectrum as a product of the y-irradiation at 77 K of a mixture of BrF, and SFg. Analysis of the coupling-constant data indicates that the 4s spin density is 0.54, as compared with 0.46 for the 3 s orbital on Cl in CIF radical. Christe and Wilson have described the preparation of BrFg salts and have also obtained complete data for the vibrational spectra, from which they have calculated force constants for this cation the Br— F stretching force constant, 4.9 mdyn is the highest yet reported for any BrF bond. 

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. 

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