Hamiltonian mapping

Such Hamiltonian mappings are generated by a Poincare surface of section transverse to the orbits of the flow. Thus, v(q) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. 

R. E. GiUilan, G.S. Ezra, Transport and turnstUes in multidimensional Hamiltonian mappings for unimolecular fragmentation Application to van der Waals predissociation,... 

This study of Hamiltonian mapping models of fragmentation displays the differences and similarities between several systems. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near to the identity symplectic mappings with applications to symplectic integration algorithms. J. Stat. Phys. 74 (1994)... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

We will introduce the following notation to describe the flow map of a Hamiltonian system with Hamiltonian H ... 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

Two-dimensional Area-Preserving Maps Consider a Hamiltonian of the form... 

Assume that there exists a unitary operator U(it) which maps the Heisenberg operator Q(t) at time t into the operator (—<). Assume further that this mapping has the property of leaving the hamiltonian invariant, i.e., that U(it)SU(it)" 1 = H. Consider then the equation satisfied by the transformed operator... 

The action of the Hamiltonian, H, can be expressed as a superoperator mapping the Hilbert space 2 (iK j into itself by... 

Figure 1. Eigenvalues of the scaled champagne bottle Hamiltonian (Eq. (2)) for p = 0.00625, in the energy, , and angular momentum, k map. The eigenvalues, represented by points, are joined (a) by lines of constant bent vibrational quantum number, vt, and (b) by lines of constant linear quantum number, v = 2vt + k. ...

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

In order to establish conditions for the isolation of the image of point D under the EM map, the projection is performed by first taking a section through the surface ct = 0 at fixed J, an example of which is shown in the right-hand panel of Fig. 15, for the critical value = S — N. The shaded area of the (K, plane defines the classically allowed range for the specified value. The lines indicate energy contours for y = 0.5. Those that touch the section correspond to relative equilibria of the Hamiltonian, whose values... 

It was shown in the earlier sections that the existence or nonexistence of quantum monodromy in two-dimensional maps depends on the relative dispositions of the critical points and relative equilibria of the Hamiltonian, which involves a search for the stationary points of with respect to K and Qk- For L = 0 there is a root at the critical point K = J, and other possible roots given by... 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

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