Molecular behavior in the quasi‐periodic and stochastic regimes
01 Dec 1980-Annals of the New York Academy of Sciences (New York Academy of Sciences)-Vol. 357, Iss: 1, pp 169-182
TL;DR: In this paper, nonlinear dynamics has been applied to chemistry in a variety of ways, for example, in the analysis of spatial and temporal oscillations in chemical reactions, and the reaction rate at various regions of space are nonlinear in such cases.
Abstract: Nonlinear dynamics has been applied to chemistry in a variety of ways, for example, in the analysis of spatial and temporal oscillations in chemical reactions. The
equations for the reaction rate at various regions of space are nonlinear in such cases. Several other papers in this volume touch upon this subject. Nonlinear dynamics has
also been used to treat collisions between molecules by solving Hamilton’s equations for their motion. Many recent experimental data that have become available on
collisions and reaction dynamics have been treated in this way.
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TL;DR: In this article, the authors studied the family of systems in which a particle moves freely within hard-walled triangles (vibrations of triangular membranes), with X and Y labelling two of the angles.
Abstract: ‘Accidental’ degeneracies between energy levels E j and E j +1 of a real Hamiltonian can occur generically in a family of Hamiltonians labelled by at least two parameters X , Y ,... Energy-level surfaces in E , X , Y space have (locally) a double-cone (diabolo) connection and we refer to the degeneracies themselves as ‘diabolical points’. We studied the family of systems in which a particle moves freely within hard-walled triangles (vibrations of triangular membranes), with X and Y labelling two of the angles. Using an efficient Green-function technique to compute the levels, we found several diabolical points for low-lying levels (as well as some symmetry degeneracies); the lowest diabolical point occurred for levels 5 and 6 of the triangle 130.57°, 30.73°, 18.70°. The conical structure was confirmed by noting that the normal derivative u of the wavefunction ψ at a boundary point changed sign during a small circuit of the diabolical point. The form of the variation of u around a circuit, and the changing pattern of nodal lines of ψ , agreed with theoretical expectations. An estimate of the total number of degeneracies N d ( j ) involving levels 1 through j , based on the energy-scaling of cone angles and the level spacing distribution, gave N d ( j ) ~ ( j + ½) 2.5 , and our limited data support this prediction. Analytical theory confirmed that for thin triangles (where our computational method is slow) there are no degeneracies in the energy range studied.
354 citations
TL;DR: In recent years there has been an increasing use of laser spectroscopic and other techniques to investigate unimolecular dissociations of molecules, both to initiate a dissociation and to measure the formation of the individual quantum states of the immediate reaction products.
Abstract: In recent years there has been an increasing use of laser spectroscopic and other techniques to investigate unimolecular dissociations of molecules, both to initiate a dissociation and to measure the formation of the individual
quantum states of the immediate reaction products. This chapter is concerned with a description of statistical theories used to calculate the rates of such dissociations, for example, of a molecule AB, AB → A + B, and to calculate the distribution of the quantum states of the fragments A and B.
212 citations
TL;DR: In this paper, a generic family of plane billiards has been discovered and the shape of the boundary is given by the quadratic conformal image of the unit circle, and is thus real analytic.
Abstract: A generic family of plane billiards has been discovered recently. The shape of the boundary is given by the quadratic conformal image of the unit circle, and is thus real analytic. For small deformations of the unit disc the billiard is a typical KAM system, but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point. The Kolmogorov entropy has been calculated, and it increases with the deformation of the boundary. The author studies aspects of the quantum chaos for this billiard. He solves numerically the eigenvalue problem for the Laplace operator with Dirichlet's boundary condition. He examines the spectrum, and inspects the avoided crossings at which mixing of nearby states occurs. The variation of the nodal structure and of the localisation properties of the eigenfunctions is studied. In analysing the level spacing distribution he finds a continuous transition from the Poisson distribution towards the Wigner distribution. The exponent in the level repulsion law varies continuously along with a generic perturbation. For small perturbations it seems to be proportional to the square root of the perturbation parameter.
148 citations
TL;DR: In this article, the authors show that the inapplicability of classical dynamics for predicting the dynamics of molecular vibrations can be demonstrated by using photoacoustic spectroscopy with a fully automated laser system, and show that if extensive vibrational coupling occurs, the observed states would be highly perturbed.
Abstract: Direct overtone spectra of H 12C 14N, H 13C 14N, and H 12C 15N have been measured between 15 000 and 18 500 cm−1 with a precision of 0.001 cm−1. These were obtained using intracavity photoacoustic spectroscopy, with a fully automated laser system. The spectra are unperturbed. The transition energies and rotational constants are in good agreement with predictions of first order anharmonic constants. Classical trajectories for HCN have been computed on the best experimentally parameterized potential, and found to be stochastic 12 990 cm−1 above the ground state. Quantal density of states were computed for HCN and show that if extensive vibrational coupling occurs, the observed states would be highly perturbed. The simplicity of the observed states is shown to be expected given a Franck–Condon type limitation on significantly perturbing states. The results show the inapplicability of classical dynamics for predicting the dynamics of molecular vibrations.
97 citations
TL;DR: In this article, a uniform semiclassical theory of avoided crossings is presented for the case where that behavior is generated by a classical resonance, and the parameters in the expression are evaluated from canonical invariants (phase integrals) obtained from classical trajectory data.
Abstract: Avoided crossings influence spectra and intramolecular redistribution of energy. A semiclassical theory of these avoided crossings shows that when primitive semiclassical eigenvalues are plotted vs a parameter in the Hamiltonian they cross instead of avoiding each other. The trajectories for each are connected by a classically forbidden path. To obtain the avoided crossing behavior, a uniform semiclassical theory of avoided crossings is presented in this article for the case where that behavior is generated by a classical resonance. A low order perturbation theory expression is used as the basis for a functional form for the treatment. The parameters in the expression are evaluated from canonical invariants (phase integrals) obtained from classical trajectory data. The results are compared with quantum mechanical results for the splitting, and reasonable agreement is obtained. Other advantages of the uniform method are described.
80 citations
References
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Book•
01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
11,008 citations
TL;DR: In this article, the authors demonstrate the mechanism for a universal instability, the Arnold diffusion, which occurs in the oscillating systems having more than two degrees of freedom, which results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations.
3,527 citations
TL;DR: In this paper, the existence of a third isolating integral of motion in an axisymmetric potential was investigated by numerical experiments and it was found that the third integral exists for only a limited rage of initial conditions.
Abstract: The problem of the existence of a third isolating integral of motion in an axisymmetric potential is investigated by numerical experiments. It is found that the third integral exists for only a limited rage of initial conditions.
1,728 citations
TL;DR: In this article, the relation between the solutions of the timeindependent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables.
Abstract: The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.
1,208 citations
TL;DR: For a separable or non-separable system an approximate solution of the Schrodinger equation is constructed of the form Ae i h −1S. From the singlevaluedness of the solution, assuming that A is single-valued, a condition on S is obtained from which follows A as mentioned in this paper, which yields a classical mechanical principle for determining the type of quantum number to be used in any particular instance.
527 citations