Wilson G matrix

Equation [85] is equivalent to Eq. [16] of the TB method, " with a difference in sign in the definitions of the Lagrange multipliers. A definition analogous to that of the Wilson G matrix of vibrational dynamics... 

The contravariant metric tensor gjk is known in the theory of small vibrations as Wilson s G matrix (kinematic matrix). 

From these frequencies and with the help of the corresponding G-Matrix elements (Wilson et al., 1955), the symmetry-adapted force constants (F) can be calculated directly. In the vibrations discussed here, F is a linear combination of stretching and interaction force constants / and f,r . 

According to the Born-Oppenheimer approximation, the potential function of a molecule is not influenced by isotopic substitution. Frequency shifts caused by isotopic substitution therefore provide experimental data in addition to the fundamentals which can yield information about the structure of a species. However, the half-widths of absorptions are too large to be resolved by the experimental techniques which are normally used, which is why these methods cannot reveal small isotopic shifts (some cm ). The half-widths of the bands are reduced drastically by applying the matrix-isolation technique (c.f. Sec. 4.4). The absorptions of many matrix-isolated species can therefore be characterized with the help of isotopic substitution, i.e., the molecular fragment which is involved in the vibration can be identified. The large - Si/" Si shift of the most intense IR absorption of matrix-isolated S=Si=S from 918 cm to 907 cm, for instance, demonstrates that silicon participates considerably in this vibration (Schnoeckel and Koeppe, 1989). The same vibration is shifted by 4 cm if only one atom is substituted by a atom. The band at 918 cm must be assigned to the antisymmetric stretching vibration, since the central A atom in an AB2 molecule with Doo/rsymmetry counts twice as much as the B atoms in the G-matrix (c.f. Wilson et al., 1955). 

The mathematical procedures for the calculation of force constants are described in Section 5.2 of this book. For calculations involving small (triatomic) species, the book by Fadini and Schnepel (1985) is recommended. A review covering force con.stant calculations has been presented by Becher (1968). In this chapter only relevant aspects are discussed. Force constant calculations mainly rely on correct assignments of the observed frequencies. These are compared with calculated frequencies, which can be obtained, for example, by the Wilson method (Wilson et al., 1955). First, the G-matrix based on the known or an assumed geometry is calculated. The eigenvalues (frequencies) of the vibrations for different sets of force constants are then calculated. Their values are derived mostly from the force constants of similar species. Since the measured frequencies can be reproduced with an infinite number of sets of force constants, additional experimentally... 

In Eq. (10) q are the internal coordinates, x are the Cartesian coordinates, g is the gradient, H is the Hessian, and the Wilson B matrix is given by B. Throughout this chapter, a superscript t denotes transpose. Finite displacements in redundant internal coordinates require that the back transformation of the positions to Cartesian coordinates be solved iteratively using Eq. (10) and... 

The generalization of a force constant, k, and reduced mass, fi, from a one dimensional harmonic oscillator to the normal mode oscillators of a polyatomic molecule is accomplished by the F, G matrix methods of Wilson, et al., (1955). For the present discussion it is sufficient to know that a force constant and a reduced mass may be uniquely defined for each of 3N — 6 linearly independent sets of internal coordinate displacements in a polyatomic molecule (see also Section 9.4.12).] The harmonic oscillator Hamiltonian... 

The G (mass and geometry dependent) and F (mass and geometry independent) matrices are discussed by Wilson et al, (1955). For two bond stretches coupled by a shared atom, the relevant F and G matrix elements are defined by... 

The kinetic energy contribution is expressed in terms of the internal coordinates, R and their conjugate momenta, P, = —ih d/dR,. Here t takes on any value between 1 and 3n - 6. If the internal coordinates are the usual stretch-bend extensions, then the G-matrix elements, which are functions of R are those of Wilson et al. (24). The terms that depend solely on the coordinates are the Born-Oppenheimer potential V and a mass dependent contribution V, this latter term resulting from the noncommunitivity of P, and R,. 

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... 

As already mentioned, Shimanouchi, Tsuboi, and Miyazawa (1961) developed a matrix treatment of lattice dynamics for molecular solids which is analogous to the molecular vibration F-G matrix formulation of Wilson, Decius, and Cross (1955). This method was applied to the q = 0 lattice motions of solid benzene by Harada und Shimanouchi (1966, 1967) and was subsequently used by many other authors (e.g., Bernstein, 1970). 

From the definition of the Gr matrix (Eq. (3-15)) or, more directly, from the analytical expressions of the Gr matrix elements provided by Wilson et al. [3], it transpires that the element (GR)y connecting internal coordinates i and j may be different from zero when they possess at least one atom in common. It follows that atoms away from the fimctional group under study certainly will not affect the group frequency Eq. (3-35) holds, i.e., the G matrix favours localization. 

A unit change in the cartesian coordinate causes a change in the internal coordinate Sj. Wilson has shown that the G matrix elements are given by... 

Wilson introduced a convenient vectorial method for obtaining matrix elements which does not use cartesian coordinates. For a complete treatment of this type, see Wilson, Decius, and Cross, Chapter 4, and also Meister and Cleveland. This method will not be discussed here since general formulas have been tabulated for all the common G matrix elements. ... 

Fig. 14.4. Schematic representations of the G matrix elements. (Redrawn from Molecular Vibrations by Wilson, Decius, and Cross. Copyright 1955, McGraw-Hill Book Company, Inc. Used by permission.)...

The geometrical factors needed to set up Wilson s G-matrix have already been prescribed. With F- as found above and the bending and stretching force constants derived above, the F-matrix is also complete. The straightforward solution of FG —EA = 0 gives the imaginary frequency v as well as the real frequencies v. From these frequencies the terms F in Eq. (4) can easily be evaluated. 

The Wilson F-G matrix method for calculating force constants is discussed in most texts on theoretical spectroscopy. 

G is the inverse of the G matrix as defined by Wilson, Decius, and Cross. 

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. 

E.R. Davidson. Matrix Eigenvector Methods in Methods in Computational Molecular Physics (ed. G.H.F. Dierksen and S. Wilson) Reidel, Dordrecht, 1983. 

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