Vibrational dynamics Hamiltonian modes

Inspired by recent experimental results on multiphoton IR excitation of the C—O stretch [136], we are specifically interested in the anharmonicity of this mode as well as its coupling to the other modes. The vibrational dynamics is investigated within a 2D model with the Hamiltonian... 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

The total Hamiltonian H should include all the vibrational modes involved in the hydrogen bond dynamics, as well as the various couplings taking place between these modes. At last, H would include the relaxation mechanisms. 

The relaxation and coherence dynamics contributions to Eq. (128) within the BOA approximation are considered. For this purpose, g = (a, ), m — (b, a ), and n — (c, w ) are defined, where, for example, u represents the vibrational quantum numbers of the vibrational modes of the electronic ground state. For the electronic coupling between the two excited states, it is assumed that the interaction Hamiltonian is given by H — c)Jcb(Q)(b + h.c. where Q denotes the normal coordinate. The coupled GMEs, for example, for pbv,cw can be written as [67]... 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

Stretching mode 0 g. The original idea was that static JT distortions could produce the large observed anisotropy however, a simple static treatment showed comparable energies of the zero-point vibration (fiw = 350 cm or 225 cm" ), the JT stabilization (Ejt = 516 cm" ), the spin-orbit interaction = 435 cm" ) and the rhombic field due to the azide ion (Hrj, = 60 cm" ). Therefore all interactions had to be considered simultaneously including dynamic effects. The used Hamiltonian was... 

The parameters of the electronic Hamiltonian are calculated by performing extensive ab initio calculations [21]. Results of calculations of static aspects of the electronic PESs, viz, the equilibrium minimum of the states and energetic minimum of the seam of the CIs within a LVC mo l summarized in Table 1. It can be seen from the data that the minimum of the X—A CIs occurs 0.02 eV above the equilibrium minimum of the A state. The same for the B CIs occurs at 0.84 eV and 0.06eV above the equilibrium minimum of the A and B states, respectively. The minimum of B-C CIs occurs 1.10eV above the equilibrium minimum of the C state. The minimum of the X—B CIs occurs 2.5eV above the equilibrium minimum of B state. The X and C CIs occur at much higher energy and are not considered here. Analysis of the coupling parameters of all 36 vibrational modes revealed the importance of 9 ai (vi3 — V5), 9 2 (r 36 — r zv), 2 az (vie and 4 bi (vzo - V17) vibrational modes in the nuclear dynamics in the X—A—B—C electronic states of PA + [21]. 

This Hamiltonian is an approximation to the true one, obtained by setting the CN moment of inertia to infinity (the more rigorous treatment is discussed in Ref. 28). In Eq. (23), p, denotes the reduced mass of H and CN, and V is the potential surface. The CN vibration is ignored in the treatment, since its frequency is so high, and the full potential energy surface suggests that this mode is unlikely to play a significant role for the states considered here (and for the isomerization dynamics). 

The above apparent non-RRKM and intrinsic RRKM and non-RRKM dynamics are reflections of a molecule s phase space structure. Extensive calculations and study of the classical dynamics of vibrationally excited molecules have shown that they may have different types of motions, e.g. regular and irregular [63]. A trajectory is regular if its motion may be represented by a separable Hamiltonian, for which each degree of freedom is uncoupled and moves independent of the other degrees of freedom. All trajectories are regular for the normal mode Hamiltonian, i.e. 

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

By restricting our interest to pure vibrational modes, the dynamical symmetry Hamiltonian operator (4.23) (excluding the Majorana operator... 

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