Redundant coordinate

Resorting to the definition of the Fourier transform, Eq. (2.9), we notice that for the redundant coordinates the harmonic kernel degenerates and becomes exp (0) = 1. Thus for the redundant coordinates the Fourier transform turns into a simple integration with respect to the respective reciprocal coordinate20 - a projection . 

This problem of spurious or, as they are conventionally called, redundant coordinates always arises when there are sets of angles that form a closed group, as in the case just considered, in planar cyclic molecules and also in three dimensions (e.g., tetrahedral and octahedral molecules). Several of the examples discussed in Section 10.7 will illustrate the point further. Redundant coordinates can usually be recognized without much difficulty, though troublesome cases sometimes arise. 

The state of the system consisting of n electrons and two nuclei is now specified in terms of 3n + 6 spatial coordinates referred to the centre of mass plus three centre-of-mass coordinates. Three of the spatial coordinates are redundant and can therefore be eliminated. Various choices of redundant coordinates can be made we choose to... 

We require only three nuclear coordinates to define the nuclear motion and we choose these to be R, the internuclear distance, and 0 the third Euler angle x is a redundant coordinate. In fact, because there are no nuclei lying off-axis in a diatomic molecule, X is undefineable it is, however, expedient to retain it because of simplification in the final form of the rotational Hamiltonian. We shall examine this point in more detail in... 

These equations do not involve the redundant coordinate / and can thus be inverted to... 

We now investigate the relationship between Pr and the original nuclear momenta Pa in more detail. We have introduced a redundant coordinate x in the transformation matrix 911(2.40). We therefore need to define our original (3n + 6) coordinates in terms of the (3 n + 7) final coordinates. To do this, we note that the position of the molecular centre of mass in the (X", Y", Z") coordinate system is... 

We have included the double-primed subscripts in (2.68) to emphasise that the differentiation is performed with space-fixed nuclear centre-of-mass electronic coordinates held constant. Equation (2.68) can be appreciated when we realise that the total Hamiltonian is independent of / so that we can take the eigenfunctions PYlK to be independent of x also. This relationship provides a crucial restriction on the redundant coordinates its form is such that we could, if we wished, write down the inverses of equations (2.59), (2.60) and (2.61). 

Along with the introduction of the redundant coordinate x, we define the normalisation condition as... 

Two different kinds of force fields have appeared in the hterature. In the first type a set of internal coordinates, sufficient to describe the molecular geometry, is choosen. The total strain energy is then calculated from the deviation of these internal coordinates (plus nonbonded interactions) from their strain free values. The redundant coordinates are given zero force constants and do not contribute to the potential energy. 

K. Kuczera,/. Mol. Struct., 160, 159 (1987). Uniquely Defined Harmonic Force Constants in Redundant Coordinates. 

Various methods have been suggested for dealing with redundant coordinates. The normal coordinate optimization method [52] can use a force field defined in redundant coordinates, but is restricted to rectilinear coordinates. A general force field, expressed in redundant coordinates, can be transformed to a non-redundant set [13]. The current method of choice is to carry out the optimization directly in the redundant coordinate space [ ] using the concept of a generalized inverse. If the total number of internal coordinates, including... 

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. 

Figure 13. Two distinct geometries of a five-atom molecule, which have the same values for nine bond lengths. A redundant coordinate is the length of the tenth bond, denoted by the dashed line (see Ref. 200).

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