The normal mode representation

Many discussions in this text start by separating an overall system to a system of interest, referred to as the system and the rest of the world, related to the bath. The properties assigned to the bath, for example, assuming that it remains in thennal equilibrium throughout the process studied, constitute part of the model assumptions used to deal with a particular problem. 

This statement holds at zero temperature. At finite T we assign to the bath an additional property— 

Consider now the barrier crossing problem in the barrier controlled regime discussed in Section 14.4.3. The result, the rate expressions (14.73) and (14.74), as well as its non-Markovian generalization in which cor is replaced by k ofEq. (14.90), has the structure of a corrected TST rate. TST is exact, and the correction factor becomes 1, if all traj ectories that traverse the barrier top along the reaction coordinate (x ofEq. (14.39)) proceed to a well-defined product state without recrossing back. Crossing back is easily visualized as caused by collisions with solvent atoms, for example, by solvent-induced friction. 

Consider now the motion along this reaction coordinate. This is a motion that (1) connects between the reactant and the product basins of attraction, and (2) proceeds at the top of the barrier, that is, through the saddle point, with no coupling to other modes therefore no interactions or collisions that may cause reflection. This implies, given the original assumption that thermal equilibrium prevails in the reactant well, that TST must hold exactly. In other words, by choosing the correct reaction coordinate, the Kramers model in the barrier-controlled regime can be cast in terms of TST. 

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Substituting Eq. (25) into Eq. (23) we find the expression for the propagator of free diffusion in the normal mode representation ... 

The overbar specifies actions and angles in the normal mode representation, which differs from the local mode representation only by a rotation of tt/2 about the y axis). Thus,... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Although the normal mode representation is very useful, the physically observable coordinate is usually the system coordinate q. In the Langevin representation of the dynamics, the trajectory q(t) is characterized by a random force which affects its motion. In the equivalent Fokker-Planck representation of Langevin dynamics, all dynamical information lies in the joint probability distribution function that the particle at time t has position and velocity q, v, given that at time r = 0 its position and velocity were q, v. ... 

The same approach has been used in Ref. 68 to derive an eigenfunction of the Fokker-Planck operator as well as the stochastic separatrix in phase space (71-73). The dynamics in the normal mode representation was also used in conjunction with the re-... 

Fig. 5.15 Schematic representation of the normal modes of the Fe(ni)-azide complex with the largest iron composition factors. The individual displacements of the Fe nucleus are depicted by a blue arrow. All vibrations except for V4 are characterized by a significant involvement of bond stretching and bending coordinates (red arrows and archlines), hi such a case, the length of the arrows and archlines roughly indicate the relative amplitude of bond stretching and bending, respectively. Internal coordinates vibrating in antiphase are denoted by inward and outward arrows respectively (taken from [63])...

Fig. 6. Schematic representation of the normal modes of an adsorbed diatomic molecule neglecting the surface structure, after Richardson and Bradshaw . In parentheses the experimentally measured values for CO in the ontop position on Pt(lll). (a) A frustrated translation (60 cm (b) A frustrated rotation (not yet detected), (c) The metal-molecule stretch (460cm ) . (d) The intramolecular stretch model (2100cm" ) . ...

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

It is further shown in Chapter 10 that, when each of the normal modes is in its ground state, each of the y/j is totally symmetric and hence y/v is totally symmetric. If one of the normal modes is excited by one quantum number, the corresponding it may then belong to one of the irreducible representations other than the totally symmetric one, say T, and thus the entire vibrational wave function f/Y will belong to the representation T,. Simple methods for finding the representations to which the first excited states of the normal modes belong are explained in Chapter 10. In this section we will quote without proof results obtained by these methods. 

Thus, if there are any normal vibrations whose first excited states belong to any of these representations, there will be nonvanishing intensity integrals. By the methods of Chapter 10 it is easily found that the symmetries of the normal modes of an octahedral AB6 molecule are... 

Each of the normal modes forms a basis for or belongs to an irreducible representation of the molecule. 

The two characteristic features of normal modes of vibration that have been stated and discussed above lead directly to a simple and straightforward method of determining how many of the normal modes of vibration of any molecule will belong to each of the irreducible representations of the point group of the molecule. This information may be obtained entirely from knowledge of the molecular symmetry and does not require any knowledge, or by itself provide any information, concerning the frequencies or detailed forms of the normal modes. 

Figure 10.3 The set of 3n = 12 Cartesian displacement vectors used in determining the reducible representation spanning the irreducible representation of the normal modes of COj".

As shown in Section 5.1, the wave functions must form bases for irreducible representations of the symmetry group of the molecule, and the same holds, of course, for all kinds of wave functions, vibrational, rotational, electronic, and so on. Let us now see what representations are generated by the vibrational wave functions of the normal modes. Inserting Hn(Va4,) into 10.6-1, we obtain... 

A fundamental will be infrared active (/. e., will give rise to an absorption band) if the normal mode which is excited belongs to the same representation as any one or several of the Cartesian coordinates. 

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, ... 

The nature of these six vibrations may be further specified in terms of the contribution made to each of them by the various internal coordinates. We first note that Ag and Bu vibrations must involve only motions within the molecular plane, since the characters of the representations Ag and Bu with respect to ah are positive. The Au vibration will, however, involve out-of-plane deformation, since the character of Au with respect to oh is negative. Thus we may describe the normal mode of Au symmetry as the out-of-plane deformation. In order to treat the remaining five in-plane vibrations we need a set of five internal coordinates so chosen that changes in them may occur entirely within the molecular plane. A suitable set, related to the bonding in... 

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