Cartesian displacement coordinate

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

Defining mass-weighted Cartesian displacement coordinates r/ . 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

It is convenient to define a set of 3N mass-weighted Cartesian displacement coordinates q, q2,..., q N such that the first three q s are the components of Qi, the fourth, fifth and sixth q s are the components of Q2, and so on. The kinetic energy T can therefore be written... 

With the use of the Cartesian displacement coordinates defined in Fig. 3 a basis vector is of the form... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

Fig, 3 Cartesian displacement coordinates for the ammonia molecule, The Z(CO a is perpendicular to the plane of the paper (which is not a plane of symmetry)... 

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. 

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1,.., 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1,..., x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates... 

The matrix to be diagonalized for finding the vibrational frequencies is the matrix product of the above G matrix for Cartesian coordinates and the corresponding F matrix for Cartesian displacement coordinates. It is noted in passing that the GF matrix is generally not symmetric, i.e. 

The second rule for isotopomer harmonic frequencies is the so-called Sum Rule which follows from Equation 3.A1.8. Equation 3.A1.8 relates the sum of the squares of all the frequencies to the sum of the diagonal matrix elements of the (FG) matrix diagonalized to obtain the frequencies. When mass weighted Cartesian displacement coordinates are used to calculate the vibration frequencies, this means that the sum of the A s (A = 4n2v12)can be found as follows (Equation3.51)... 

The relationship between the force constant matrix in Cartesian displacement coordinates Fy, and the force constant matrix for mass weighted Cartesian coordinates F can be written as follows (only the first three rows and columns of the matrices are explicitly shown) ... 

In classical terms, if we use the mass-weighted Cartesian displacement coordinates, the kinetic energy of the moving nuclei isf... 

The point of changing from Cartesian displacement coordinates to normal coordinates is that it brings about a great simplification of the vibrational equation. Furthermore, we will see that the normal coordinates provide a basis for a representation of the point group to which molecule belongs. 

To simplify this equation, we define the mass-weighted Cartesian displacement coordinates qv...,q3N ... 

Thus if there are no degenerate vibrations, each normal coordinate is either unchanged or multiplied by — 1 upon application of a symmetry operation. Each Qk is a linear combination of the mass-weighted Cartesian displacement coordinates of the nuclei. If Qk is multiplied by — 1, each Cartesian displacement coordinate is multiplied by - 1, which reverses the directions of all the displacement vectors. If Qk is unchanged by a symmetry operation, then the symmetry operation sends the displacement vectors to a configuration indistinguishable from the original one. (The displacement vectors are defined relative to molecule-fixed axes, which in turn are defined relative to the nuclear positions. The effect of a symmetry... 

Cameron system of, 298 fundamental IR band of, table of, 174 rotational transitions of, table of, 168 Carbon-13 NMR, 356-357 Cartesian displacement coordinates, 236 Case (a) coupling, 188-189, 212 Case (b) coupling, 190-191,212 Cayley, A., 78, 387 Center of symmetry, 53 Central-force problem, 38 Centrifugal distortion in diatomics, 158,166-167 in polyatomics, 213, 216, 218 Chain rule, 20-21 Characters ... 

Figure 10.4 The matrix expressing the effect of the identity operation on the set of Cartesian displacement coordinates (Fig. 10.3) for C03"2.

In Figure 10.9 we reproduce the character table of the group Dy, and append to it the results just obtained for the characters of the operations in the reducible representation for which the 12 Cartesian displacement coordinates form a basis. This may be reduced by the methods of Section 4.3 with the following result ... 

Figure 10.9 The character table for the group with the characters for the representation generated from the 12 Cartesian displacement coordinates appended.

Symmetry Types of the Normal Modes. For this nonlinear four-atomic molecule there are 3(4) -6 = 6 genuine internal vibrations. Using a set of three Cartesian displacement coordinates on each atom, we obtain the following representation of the group C3l, ... 

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