Internal Coordinate Axes

Thus the transfonnation matrix for the gradient is the inverse transpose of that for the coordinates. In the case of transfonnation from Cartesian displacement coordmates (Ax) to internal coordinates (Aq), the transfonnation is singular becanse the internal coordinates do not specify the six translational and rotational degrees of freedom. One conld angment the internal coordinate set by the latter bnt a simpler approach is to rise the generalized inverse [58]... 

The first block, at the upper left, is for the two Ax modes, and the two remaining blocks, which are identical, are for the a and b components of the E modes. Thus, the factorization is into two second-order equations, as expected. If the internal coordinates and the symmetry coordinates (SALCs) had not been selected and arranged so that the Ar, and A0, sets matched correctly, as specified earlier, the factorization of the 4 x 4 E block into two identical 2x2 blocks would not have been accomplished. 

The kinetic and potential energies of a polyatomic molecule can be expressed in terms of Cartesian coordinates (Ax, Ay, Az) or internal coordinates such as increments of bond length (Ar) and bond angles (Aa). In the former case, 3N coordinates are required for an iV-atom molecule. Figure 1-44 shows the nine Cartesian coordinates of the H20 molecule. Since the number of normal vibrations is 3 (3 x 3 — 6), this set of Cartesian coordinates includes six extra coordinates. On the other hand, only three coordinates (A/-, Ar2 andAa) shown in Fig. 1-44 are necessary to express the energies in terms of internal... 

Working in Cartesian coordinates is easier since they are independent parameters whereas the equations of constraint for internal coordinates are difficult to handle. Therefore the Aq have to be transformed to AX (a = 1,2,3 represent x,y,z respectively i = 1,..., N). These transformations are made by regarding Ag as a small quantity and expanding in a power series e.g. for Ary one obtains ... 

The numerical procedure used in association with the p, ax coordinates is as follows (12). The internal configuration space is divided into a number of spherical hyperradial sectors. The two-dimensional LHSFs are then determined at the center p of each sector and used to obtain the coupling matrix V7111 (p p) over the entire sector. Equation (178) is transformed into the firstorder nonlinear Bessel-Ricatti matrix differential equation... 

Fig. 9.3 Occurrence of octahedral solid circles) and tetrahedral hollow circles) coordination in AX solids. (Reproduced with permission from [10] Copyright 1959 International Union of Crystallography)...

The expression d(u n[rp) /dx can be obtained by a process analogous to that carried out in equations (6.2.50b,c), here in a control volume of dimension Ax, Ay, Az consider a control volume of internal particle coordinate dimensions Ax, Ay, Az located in the internal particle coordinate space. 

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