Real normal coordinates

In this equation, m( ) is the displacement vector of the g -th atom and is given as a sum of terms associated with degenerate pairs (a and b) of real normal coordinates Qf and gj ,... 

By definition (2.53) the normal coordinates Q( ) are complex quantities. For many applications and, in particular, for effecting the transition to quantum mechanics, it is more convenient to re-express the complex normal coordinates in terms of r al normal coordinates [2.2,6]. The most obvious way of expressing Q(S) in terms of real normal coordinates is as follows ... 

From (2.62) it immediately follows that ,(" ) = ai( ) and ao( ) = - < p( )> SO that only half of the real normal coordinates a (p are independent and we have as many independent normal coordinates as there are degrees of freedom (2N for the longitudinal modes of the diatomic chain with N unit cells). The independent normal coordinates thus obtained by restricting... 

It is also possible to define real normal coordinates n(j) which describe running waves. In contrast to the real normal coordinates a ( ), the coordinates n( ) are not just linear combinations of Q(S) and Q ( ) but also of the momenta P(p and P (. They are coordinates which can be obtained by a canonical transformation, but we shall introduce them here in an elementary way as follows [2.6] ... 

In Sect.2.1.4 we have shown that tn terms of the real normal coordinates a (S) or n(j), the Hamiltonian functions assume the simple and completely diagonalized forms of (2.63) and (2.69), respectively. In terms of the coordinates n( ) and their conjugate momenta < ( ) we have obtained J j... 

As in the case of the diatomic chain, it is possible to introduce real normal coordinates a (j) (standing waves) or n(j) (running waves). They are defined by expressions analogous to (2.62,67), respectively. 

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

Symmetry coordinates can be generated from the internal coordinates by the use of the projection operator introduced in Chapter 4. Both the symmetry coordinates and the normal modes of vibration belong to an irreducible representation of the point group of the molecule. A symmetry coordinate is always associated with one or another type of internal coordinate—that is pure stretch, pure bend, etc.—whereas a normal mode can be a mixture of different internal coordinate changes of the same symmetry. In some cases, as in H20, the symmetry coordinates are good representations of the normal vibrations. In other cases they are not. An example for the latter is Au2C16 where the pure symmetry coordinate vibrations would be close in energy, so the real normal vibrations are mixtures of the different vibrations of the same symmetry type [7], The relationship between the symmetry coordinates and the normal vibrations can be... 

Figure 1. Normal coordinate and real space description of pseudorotating K,. Circles represent energy contours for the trimer potential surface (3).

Since only the yth normal coordinate and its time derivative appear in equation (3-59), direct integration yields the time dependence of the motion of this coordinate. Clearly, it is in general difficult to have any "feel" for what motion each normal coordinate represents since it has a complicated dependence on all of the real coordinates. Nevertheless, the sum of the motions of all of the normal coordinates is identically equal to the sum of the motions of all of the real coordinates because one is just a linear transform of the other [equation (3-56)]. 

It must be reemphasized that the exact nature of [( ] is not necessary to the physical solution of our problem. Because the normal-coordinate approach merely represents a linear transformation of the real coordinates, the motion of the polymer represented by all the qls will be identical to the motion of the polymer represented by all the jc/s. Our problem thus becomes the rather simple one of finding a diagonal representation of the (z + 1) x (z + 1) matrix [A]. This rather well known result (a similar form applies in the treatment of a vibrating string, among others) is derived in the appendix at the end of this chapter, and is merely stated here ... 

As mentioned above, however, the exact nature of the normal coordinate in terms of the real coordinates is not easily perceived. Likewise, the exact nature of the real perturbation is not easily visualized in terms of perturbation to the normal coordinates. Thus to carry out our calculation exactly, we would have to transform the perturbation into the normal-coordinate framework. This is exactly the technique used by Bueche.6 The perturbation, that is, the boundary condition, used to solve equation (3-67) was that every normal coordinate was instantaneously displaced to the position of qt 0) at time zero and then no additional forces were put on the system. This perturbation corresponds neither to creep nor to stress relaxation. Although boundary conditions corresponding to these real experiments are more complicated in terms of normal coordinates,6 it can be shown2 that the relaxation times that arise in a stress-relaxation experiment are just one-half as large as those calculated above. Thus, from here on rp will be used to denote a relaxation time and is given by... 

When proper account is taken of the relative masses of the moving nuclei, the symmetry coordinates are converted to 3A —6 non-interacting normal coordinates, each of which contributes a single term of the form AiQ, where Ai = 4x1/ . All of the A,s are positive and the frequencies (i/,) real. Except for possible accidental degeneracies, which are sufficiently rare that they can usually be ignored, the numerical values of A,- - and hence of i/, - will differ from one another. 

For the term related to the reaction coordinate F, only the real contribution has been included in (Xo hereafter), in view of the possible application for the calculation of reaction rate constants. Other terms in Eq. 10.34 are = 27tc ,-and a = tfi, K j. and are, respectively, the third and fourth derivatives of the potential energy (expressed here in wave numbers) with respect to the mass-weighted normal coordinates Q, and is the kinetic contribution to... 

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