Redundant symmetry coordinates

A. Allouche and J. Pourcin, Spectrchim. Acta, 49A, 571 (1993). Ab Initio Calculation of Vibrational Force Fields Determination of Non-Redundant Symmetry Coordinates by Least-Square Component Analysis. 

Finally, we should note that in cases where a redundant set of curvilinear co-ordinates Hi are defined, the transformation to curvilinear symmetry coordinates becomes more complicated. This difficulty is discussed briefly in ref. 12, but it will not be developed here. 

As a further illustration we will demonstrate here the use of this embedding to resolve a multiplicity case for the symmetry coordinates of a vibrating icosahedron [22]. An icosahedral cage with twelve atoms has 30 internal modes. Since the icosahedron is a deltahedron, the stretchings of the 30 edges form a non-redundant set of internal coordinates. The corresponding symmetry representations are given by ... 

The symmetry coordinate iR (y4,) in Eq. 12.21 represents a redundant coordinate (see Eq. 8.1). In such a case, a coordinate transformation reduce.s the order of the matrix by one, since all the G matrix elements related to this coordinate become zero. Conversely, this result provides a general method of finding redundant coordinates. Suppose that the elements of the G matrix are calculated in terms of internal coordinates such as those in Table 1-10. If a suitable combination of internal coordinates is made so that Gy =0 (where j refers to ail the equivalent internal coordinates), such a combination is a redundant coordinate. By using the U matrices in Eqs. 12.20 and 12.21, the problem of solving a tenth-order secular equation for a tetrahedral XY4 molecule is reduced to that of solving two first-order (y4, and E) and one quadratic (F2) equation. 

The use of redundant coordinates requires extensive modification of the lattice dynamical procedure. It is, however, often worth the additional complication to use redundancies if this facilitates the formulation of symmetry coordinates. When the Wigner projection operator (Wigner, 1931) is used to build such symmetry coordinates, it is necessary to first understand the results of the application of all symmetry operations of the applicable group to the displacement coordinates chosen. This is indeed relatively straightforward for the direction cosine displacement coordinates and therein lies their principal value. These coordinates transform like axial vectors in contrast to cartesian coordinates, which transform like polar vectors. 

In the present case (for q = 0) we use twelve displacement coordinates Aj(0A ) of which eight are independent, as already discussed, and the four relations of (2.53) represent the redundancy conditions R of (2.56). In this case the 4>j, for the four redundant coordinates will not be zero in the potential expansion (2.54). In fact, Walmsley and Pople (1964) construct symmetry coordinates... 

Symmetrically Complete Sets. It is easiest to construct each internal symmetry coordinate out of equivalent internal coordinates, i.e., internal coordinates which arc exchanged by the symmetry operations of the molecules, such as the four CII bond extensions in CH4. Then the construction of the symmetry coordinates for a given molecule breaks down into separate problems for the different equivalent sets. Further, it is very desirable to utilize symmetrically complete sets i.e., sets containing all the coordinates resulting from the application of the symmetry operations of the molecule to an arbitrarily chosen coordinate. Thus the six IICH bond angles in CH4 form such a set and should all bo used, even though only five are independent. This use of redundant coordinates will introduce zero roots into the secular equation, but they are most... 

If the molecule has any symmetry, these considerations can be applied separately to each species. The number of independent coordinates in each species can be obtained by reducing the representation formed by the cartesian coordinates and subtracting the translations and rotations appropriately. The number of internal symmetry coordinates is similarly obtained for each species, and any excess represents redundancy. Likewise the rank of G " should equal the number of independent coordinates of species y. 

Internal Coordinates for the In-plane Modes. The carbon-hydrogen (s) and the carbon-carbon t) bond stretchings yield twelve of the twenty-one required coordinates. There are twelve more coordinates 4>) in the set of hydrogen-carbon-carbon bendings, so that three redundancies are expected. The first step is the determination of the number of symmetry coordinates of each species which can be formed from the above internal coordinates. 

The fact that all elements involving a vanish shows that the redundancy predicted for this species in Sec. 10-2 is just the symmetry coordinate formed by taking the sum of the six a s. 

In the GVFF there is no inherent limit to the number of FyS that are included. There is only the restraint that the number of Fs should not exceed the number of observable frequencies in fact, it should generally be much smaller so as to overdetermine the force field. Since an independent set of coordinates can be chosen, e.g., local symmetry coordinates, and the redundancy conditions explicitly given, there is no need to include linear terms in F, and Eq. (5-1) is the most general representation of the force field. Our discussion will focus on applications of the GVFF. 

There are only three vibrations for CHCI3 and thus only three symmetry coordinates are necessary, indicating that one of the four is redundant and is either equal to zero or not independent of the others. This point will be discussed later in this section. The coordinates must first be shown to be orthogonal and transform properly. S i is orthogonal with 6 2 since... 

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

There are N(N-l)/2 distinct distances in a cluster of N atoms, disregarding symmetry-dictated equivalencies. This set of distances is of course redundant 3N-6 Cartesian coordinates are sufficient to determine molecular geometry, apart from the position of the center of mass and the orientation of the principle moments of inertia. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

As this point it is important to note that the B matrix is usually not square and therefore cannot be inverted. In fact, it transforms from Cartesian (dimension 3n) to internal coordinates (dimension 3n - 6). In simple cases, six dummy coordinates (Tx, Ty, Tz, Rx, Ry, Rz) may be added to the 3 - 6 internal ones in order to obtain in invertible square matrix. However, in some cases the symmetry of the problem makes it necessary to introduce redundant non-linearly independent coordinates (6 CCC angles for benzene or 6 HCH angles for CHq). Gussoni et al. (1975) has shown that it is possible to use the transposed matrix instead of the inverse one and that this choice is the only one which ensures invariance of the potential energy upon coordinate transformation. We can therefore write... 

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