G and f matrix

In addition, G and F matrix elements have been tabulated (see Appendix VII in Nakamoto 1997) for many simple molecular structure types (including bent triatomic, pyramidal and planar tetratomic, tetrahedral and square-planar 5-atom, and octahedral 7-atom molecules) in block-diagonalized form. MUBFF G and F matrices for tetrahedral XY4 and octahedral XY molecules are reproduced in Table 1. Tabulated matrices greatly facilitate calculations, and can easily be applied to vibrational modeling of isotopically substituted molecules. Matrix elements change, however, if the symmetry of the substituted molecule is lowered by isotopic substitution, and the tabulated matrices will not work in these circumstances. For instance, C Cl4, and all share full XY4 tetrahedral symmetry (point group Tj), but... 

Example calculation <-> "Cr substitution in the chromate anion. An example calculation on the chromate anion [Cr04] makes use of tabulated G and F matrix elements. 

Once the B (or G) and F matrix elements are known for a given symmetry species, the solution of the secular equation leads to the desired eigenvalues and eigenvectors. It should again be noted that the dimensionality of the B or F matrices is reduced to that of an asymmetric unit in the unit cell. 

Evidently, the values of force constants depend on the force field initially assumed. Thus, a comparison of force constants between molecules should not be made unless they are obtained by using the same force field. The normal coordinate analysis developed in Secs. 1.12-1.14 has already been applied to a number of molecules of various structures. Appendix Vll lists the G and F matrix elements for typical molecules. 

Although the bond distance is involved in both the G and F matrices, it is canceled during multiphcation of the G and F matrix elements. Therefore, any unit can be used for the bond distance. 

The four normal modes of vibrations of a pyramidal XY3 molecule are shown in Fig. 2.8. All four vibrations are both infrared- and Raman-active. The G and F matrix elements of the pyramidal XY3 molecule are given in Appendix VII. Table 2.3a lists the fundamental frequencies of XH3-type molecules. Several bands marked by an asterisk are split into two by inversion doubling. As is shown in Fig. 2.9, two configurations of the XH3 molecule are equally probable. If the potential barrier between them is small, the molecule may resonate between the two structures. As a result, each vibrational level splits into two levels (positive and negative) [560]. Transitions between levels of different signs are allowed in the infrared spectrum, whereas those between levels of the same sign are allowed in the Raman spectrum. The transition between the two levels at u = 0 is also observed in the microwave region (v = 0.79 cm ). 

Figure 2.17 illustrates the four normal modes of vibration of a tetrahedral XY4 molecule. All four vibrations are Raman-active, whereas only V3 and V4 are infrared-active. Appendix VII lists the G and F matrix elements for such a molecule. 

The matrices [G] and [F] are column matrices with row numbers n and k, respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of G occur in complex conjugate pairs. In general, we may write... 

If we choose a proper U matrix from symmetry consideration, it is possible to factor the original G and F matrices into smaller ones. This, in turn, reduces the order of the secular equation to be solved, thus facilitating their solution. These new coordinates R are called symmetry coordinates. 

Once the G and F matrices are obtained, the next step is to solve the matrix secular equation ... 

We have here been very careful to refer to precise constructions and theorems in density matrix theory and resonance formation. It is clear, however, that the present development can be generalized to other more general processes in non-equilibrium quantum statistics, where the two natural basis partners g and f refer to density matrices and/or transition matrices [11],... 

Note that this matrix is block-diagonal, with the subblocks separate bases of symmetry coordinates transforming as one IR at a time. The eigenvalues in the last three modes can now be read directly from the 1 x 1 subblocks in the F G matrix. 

Once the frequencies have geen determined the next step is to define the form of the normal mode of vibration in terms of the internal coordinates. In order to do this the L matrix should be evaluated. As an example of the procedure let us imagine we have multiplied 3 x 3 G and F matrices to give an H matrix (GF = H) from which we can construct a secular determinant by subtracting X from each diagonal element. 

Here G and F are the mass and potential energy matrices on the internal coordinate basis R, A is the diagonal matrix of the squared vibrational frequencies, and L is the transformation matrix to normal coordinates, R = LQ. The choice of internal coordinates is the key to the success of the GF method. Internal coordinates are directly related to chemical descriptions of molecular motions in terms of bond stretches, bends, and torsions. The internal coordinates in Fig. 6.3, for example, describe in-plane vibrations of polyenes. 

In the previous. section we discussed the reference force field of /-PA (see Table 6.2) derived from the force field of butadiene. In the Og symmetry block, the high frequency C—H stretch is decoupled from the other modes and thus from tt electrons. We are left with three relevant Ug modes their reference and experimental frequencies are reported in Table 6.3 and, as discussed in Section II, fix the matrix and the x cd) curves in Fig. 6.4. The A matrix is written on the basis of the reference normal coordinates Q . It consequently depends on both the G and F matrices and varies with molecular or polymeric structure. The e-ph coupling constants g, thus vary even with isotopic substitution. To define coupling constants independent of mass, we use the symmetry coordinates to solve the GF problem for the reference state. In fact, diagonalization of GF gives both eigenvector matrix L in the S basis. The L matrix is used to transform Jin Eq. (12) back to the S basis ... 

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