Local mode state

Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983 Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-... 

Figure 9.16 shows the evolution of the local mode polyad phase spheres for H2O as I increases from 1 (N = vs + va = 1) to 3 (N = vs + va = 5). As I increases the lowest energy levels sequentially pass through the unstable fixed point B, depart from the normal mode resonance region, and become local mode states. At I = 1 (part (a)) there are only 2 levels and both are on the normal mode side of the separatrix. At I = 3 (part (e)) there are 6 levels and the lowest 4 of these have departed the resonance region and are local mode states. 

In local-mode sampling an individual local mode such as a CH bond in benzene is excited [28]. This type of trajectory calculation has been performed to determine the population of a local-mode state a) versus time, from which the absorption linewidth of the overtone state may be determined [29]. Good agreement has been found with both experimental and quantum mechanically calculated overtone linewidths for benzene [30] and linear alkanes [31]. 

Table 4.1. Energy Splitting between Symmetric and Antisymmetric H2O Eigenstates Corresponding to the (0,m) Local-Mode Stated...

When classical trajectories are calculated for the H2O model with two O—H stretches discussed in section 4.3.2 (p. 76), local-mode type motion is found for which energy is trapped in individual O—H bonds (Lawton and Child, 1979, 1981). The trajectories are quasiperiodic and application of the EBK semiclassical quantization condition [Eq. (2.72)] results in pairs of local-mode states in which there are n quanta in one bond and m in the other or vice versa. The pair of local-mode states (n,m) and (m,n) have symmetry-related trajectories which have the same energy. The local-mode trajectory for the (5,0) state is depicted in Figure 4.6d. 

As discussed above (section 4.3.2) these local-mode states are not the eigenstates for the system, but superposition states. However, since the classical motion is quasiperiodic for these local mode states (i.e., the state is a torus in the phase space), the system is trapped in the initially excited local-mode state and the quantum periodic oscillation between the n,m) and (m,n) local mode states is not observed classically. Thus, classical mechanics severely underestimates the rate of energy transfer. 

Mode-specific unimolecular decomposition has been observed in much of the theoretical work listed in table 8.1. In some cases more than one zero-order Hamiltonian is necessary to assign resonance states for an excited molecule. For the Henon-Heiles Hamiltonian two zero-order Hamiltonians are used to identify assignable resonances as either restricted processors (Q ) or quasiperiodic liberators (Q ) (Bai et al., 1983 Hose and Taylor, 1982). Similarly, Manz and co-workers (Bisseling et al., 1987 Joseph et al., 1988) assigned many stretching resonances of ABA molecules as either hyperspherical mode or local mode states. At the same energy, the 2" states for the Henon-Heiles Hamiltonian decay about an order of magnitude faster than the g states. 

VJ+2V3, and 3 3 (in ascending order), because this implies normal mode wavefunctions. An alternative and more appropriate labelling of the levels is illustrated in the Figure 5. Even this notation becomes somewhat clumsey when extended to more complicated molecules like NH3, CH. and C H a tidy and appropriate notation for local mode states has still to be found. 

Comprehensive data and assignments of nonfundamental modes were reported [7, 37]. Overtones of the v(CH) modes have been measured on liquid and gaseous Ge(CH3)4 up to 6 v (15818 cm ) and have been assigned on the basis of the local-mode model. This gave localmode CH stretching frequencies of 2990.2 + 0.9 cm for the vapor and 2988 + 4 cm for the liquid. The data are compared for M(CH3)4 compounds (M = C, Si, Ge, Sn) and bandwidths are related to theories of vibrational redistribution from highly excited local-mode states [106]. 

---