A generalized effective fitting Hamiltonian is tested against a model system of highly excited coupled Morse oscillators. At energies approaching dissociation, a very few resonance couplings in addition to the standard 1:1 and 2:2 couplings of the Darling-Dennison Hamiltonian suffice to fit the spectrum and match the large-scale features of the mixed regular and chaotic phase spaces, consisting of resonance zones organized around periodic orbits of low order that break the total polyad action.

We propose an analytical method for understanding the problem of multi-channel electron transfer reaction in solution, modeled by a particle undergoing diffusive motion under the influence of one donor and several acceptor potentials. The coupling between the donor potential and acceptor potentials are assumed to be represented by Dirac Delta functions. The diffusive motion in this paper is represented by the Smoluchowskii equation. Our solution requires the knowledge of the Laplace transform of the Green's function for the motion in all the uncoupled potentials.Understanding of electron transfer processes in condensed phase is very important in chemistry, physics, and biological sciences, for the experimentalists as well as theoreticians [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. A large amount of research in this area has been dedicated in the understanding of the behavior of electron transfer reactions exhibited by donor-acceptor pairs in solutions. Multi-channel electron transfer in condensed phase is one of the very interesting problem to study. In general the quantum "jumps" of a high frequency vibrational mode can open several new point reaction sinks for electron transfer [21], in contrast to the broadening effect by a low frequency mode [22]. The work of Jortner and Bixon [21] was based on a quantum mechanical treatment of the high frequency vibrational coordinate in place of classical one of Sumi and Marcus [15]. However, theoretical treatment of Jortner and Bixon [21] did not consider the dynamics of motion on the potential surfaces. In the following we propose a simple analytical method for understanding the problem of multi-channel electron transfer reaction in solution, modeled by a particle undergoing diffusive motion under the influence of one donor and several acceptor potentials explicitly. A molecule (donor -acceptors) immersed in a polar solvent can be put on an electronically excited potential (represents the free energy of the donor surface) by the absorption of radiation. The molecule executes a walk on that potential, which may be considered as random as it is immersed in the polar solvent. As the molecule moves it may undergo non-radiative decay from certain regions of that potential to several potentials (represents the free energy of acceptor potentials). So the problem is to calculate the probability that the molecule will still be on the electronically excited donor potential after a finite time t. We denote the probability that the molecule would survive on the donor potential by P d (x, t). We also use P (i) a (x, t) to denote the probability that the molecule would be found on the i-th acceptor potential. It is very usual to assume the motion on all the potentials to be one dimensional and diffusive, the relevent coordinate being denoted by x. It is also common to assume that the motion on all the potential energy surface is overdamped. Thus all the probabilitya (x, t)s may be found at x at the time t obeys a modified Smoluchowskii equation.∂P (1) a (...

We propose an exactly solvable model for the two state curve crossing problems. Our model assumes the coupling to be a delta function. It is used to calculate the effect of curve crossing on electronic absorption spectrum and resonance Raman excitation profile.

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We consider a suspended elastic rod under longitudinal compression. The compression can be used to adjust potential energy for transverse displacements from harmonic to double well regime. The two minima in potential energy curve describe two possible buckled states at a particular strain. Using transition state theory (TST) we have calculated the rate of conversion from one state to other. If the strain ε is between εc and 4εc, the saddle point is the straight rod. But for εc < 4εc, the saddle is S-shaped. At εc = 4εc the simple TST rate diverges. We suggest methods to correct this divergence, both for classical and quantum calculations. We also find that zero point energy contributions can be quite large (as large as 10 9 ) so that single mode calculations can lead to large errors in the rate.Considerable attention has recently been paid to two-state nano-mechanical systems [1,2,3,4,5,6,7] and the possibility of observing quantum effects in them. Roukes et al.[1] proposed to use an electrostatically flexed cantilever to explore the possibility of macroscopic quantum tunnelling in a nano-mechanical system. Carr et al. [5,8] suggested using the two buckled states of a nanorod and investigated the possibility of observing quantum effects. A suspended elastic rod of rectangular cross section under longitudinal compression is considered. As the compressional strain is increased to the buckling instability [8], the frequency of the fundamental vibrational mode drops continuously to zero. Beyond the instability, the system has a double well potential for the transverse motion (see Fig. 1). The two minima in the potential energy curve describe the two possible buckled states at that particular strain [8] and the system can change from one to the other, under thermal fluctuations or quantum tunneling. We use L, w and d (satisfying L >> w >> d) to denote the length, width and thickness of the rod [8,9,10, 11]. F is the linear modulus (energy per unit length) of the rod and is related to the elastic modulus Q of the material by F = Qwd. The bending moment κ is given by κ 2 = d 2 /12. We take the length of the uncompressed rod to be L 0 . We apply compression on the two ends, reducing the separation between the two to L. If y(x) denotes the displacement of the rod in the 'd' direction, then the length of the rod L total = L 0 dx 1 + (y ′ ) 2 ≈ L + 1/2 L 0 dx(y ′ ) 2 . The compression causes a contribution to the potential energy V elastic = F/(2L 0 )(L total − L 0 ) 2 . In addition, bending of the rod in the 'd' direction cause bending energy V b = F κ 2 /2 L 0 dx(y ′′ ) 2 . Thus the total potential energy is given byHere ε = (L − L 0 )/L 0 is the FIG. 1: Potential energy V as a function of the fundamental mode displacement Y . The shape of the potential energy is harmonic for ε > εc, quartic for ε = εc ≡ critical strain (εc < 0) and a double well for ε < εc.