F-G matrix method

In Chapter 10, it is shown that by using symmetry considerations alone we may predict the number of vibrational fundamentals, their activities in the infrared and Raman spectra, and the way in which the various bonds and interbond angles contribute to them for any molecule possessing some symmetry. The actual magnitudes of the frequencies depend on the interatomic forces in the molecule, and these cannot be predicted from symmetry properties. However, the technique of using symmetry restrictions to set up the equations required in calculations in their most amenable form (the F-G matrix method) is presented in detail. 

In order to use the F-G matrix method to maximum advantage, it is first necessary to set up linear combinations of internal coordinates which provide the proper number of functions, transforming according to each of the irreducible representations spanned by the 3N - 6 normal modes of genuine... 

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

Because chemists seem to have become increasingly interested in employing vibration spectra quantitatively—or at least semiquantitatively—to obtain information on bond strengths, it seemed mandatory to augment the previous treatment of molecular vibrations with a description of the efficient F and G matrix method for conducting vibrational analyses. The fact that the convenient projection operator method for setting up symmetry coordinates has also been introduced makes inclusion of this material particularly feasible and desirable. 

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 time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

Fig. 14. Automatic correction of the thermograms by the state-function theory and the time-domain matrix methods the input f(t) the thermogram g(t) the corrected thermogram fc(t) T is the sampling period (20 sec). Reprinted from (62) with permission.

Finally, new material and more rigorous methods have been introduced in several places. The major examples are (1) the explicit presentation of projection operators, and (2) an outline of the F and G matrix treatment of molecular vibrations. Although projection operators may seem a trifle forbidding at the outset, their potency and convenience and the nearly universal relevance of the symmetry-adapted linear combinations (SALC s) of basis functions which they generate justify the effort of learning about them. The student who does so frees himself forever from the tyranny and uncertainty of intuitive and seat-of-the-pants approaches. A new chapter which develops and illustrates projection operators has therefore been added, and many changes in the subsequent exposition have necessarily been made. 

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 second reason to introduce the derivation (6 -9) is to note that all that is required to evaluate the absorption and emission probability F A (t, r) of (9) are matrix elements of the evolution operator exp(-i//r/h). (These matrix elements are the conventional probability amplitudes When considering a situation in which many different kinds of decay processes are involved, e.g. radiative and nonradiative decay, it is not always convenient to deal directly with the matrix elements of exp(-itfr/h), the af(t). Rather, it is simpler to introduce (imaginary) Laplace transforms 16) in the same manner that electrical engineers use them to solve ac circuit equations 33L Thus, if E is the transform variable conjugate to t, the transforms of af(t) are gf(E). The quantities gf (E) can also be labeled by the initial state k and are denoded by Gjk(E). It is customary in quantum mechanics to collect all these Gjk(E) into a matrix G(E). Since matrix methods in quantum mechanics imply some choice of basis set and all physical observables are independent of the chosen basis set, it is convenient to employ operator formulations. If G (E) is the operator whose matrix elements are Gjk(E), then it is well known that G(E) is the Green s function i6.3o.34) or resolvent operator... 

The F matrix can be written by assuming a suitable set of force constants. If the G matrix is constmcted by the following method, the vibrational frequencies are obtained by solving Eq. 1.116. The G matrix is defined as... 

The procedure for calculating harmonic vibrational frequencies and force constants by GF matrix method has been described in Sec. 1.12. In this method, both G (kinetic energy) and F (potential energy) matrices are expressed in terms of internal coordinates (R) such as increments of bond distances and bond angles. Then, the kinetic (7) and potential (V) energies are written as ... 

We use the effective mass approximation. The structure potential is approximated b y a c onsequence o f rectangular quantum barriers a nd wells. Their widths and potentials are randomly varied with the uniform distribution. The sequence of the parameters is calculated by a random number generator. Other parameters, e.g. effective masses of the carriers, are assumed equal in different layers. To simplify the transmission coefficient calculations, we approximate the electric field potential by a step function. The transmission coefficient is calculated using the transfer matrix method [9]. The 1-V curves of the MQW stmctures along the x-axis (growth direction) a re derived from the calculated transmission spectra... 

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