Henon-Heiles potential

Figure 18. Trajectories in the Henon-Heiles potential for the initial condition xo ya = —0.37. The marks ( ) indicate the location of caustics. Taken from Ref. [29].

A Three-Dimensional Version of Henon-Heiles Potential... 

A Simple 3-DOF Hamiltonian. An example that illustrates in a nontrivial way some of the statements above is the three-dimensional Henon-Heiles potential... 

Figure 14. Potential energy surfaces, in two and three DOFs, for the Henon-Heiles potential, with (D = e = 1.

This change of shapes also happens in other contexts. The three-dimensional Henon-Heiles potential changes with nonzero J (called A in Wiesenfeld and Wiggins [56] see Section IV.B.l). Many nontrivial instances may be found in Cushman [63],... 

In 35 the numerical solution of the two-dimensional time-independent Schrodinger equation is studied using the method of partial discretization. The discretized problem is treated as a problem of the numerical solution of a system of ordinary differential equations and Numerov type methods are used to solve it. More specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe el al. and the minimum phase-lag method of Rao et al. are applied to this problem. The methods are applied for the calculation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heils potential. The results are compared with the results produced by full discretization. Conclusions are presented. 

Initial studies on the relationship between relaxation and chaos diagnostics focused on two degree of freedom systems37 wherein computations are simplified and where the K entropy, for systems which are chaotic over the entire energy hypersurface, equals the maximum Lyapunov exponent [see Eq. (29)]. We review the results for two flow cases, the Henon-Heiles potential,38 which is of general interest in nonlinear mechanics and is given by... 

Figure 20. Adiabatic (top) versus exact (bottom) eigenvalues for the m = 6400 a.u. even parity states of the Henon-Heiles potential. The numbers on top and bottom are eigenvalue indices and their horizontal position matches that of the eigenvalue, so as to discern between eigenvalues that are almost degenerate. (From Ref. 69.)...

As an example of doorway channel calculations consider the dissociation of the Henon-Heiles system. The potential used in this model69 is a modification of the Henon-Heiles potential. Specifically, in order to eliminate the competing effects of tunneling (which, as previously noted, tend to mask intramolecular effects), Eq. (4.25) was modified as follows ... 

Figure 2. For the Henon-Heiles potential with X=80 (equation (2) upper broken curve, ridge profile (6=0, 2tt/3 4tt/3) lower broken curve, valley bottom profile (6=Tr/3, ir, 5tt/3)), adiabatic potential curves (p) (equation 12) and corresponding nonadiabatic coupling matrix elements Pj j (p) (equation (13)) as a function of radial coordinate p for and A2 symmetry. Positions of levels indicated by continuous segments for those identified as quasiperiodic [3l] and by dotted segments for those not identified as quasiperiodic.

Figure 3. Adiabatic curves j (p), equation (12 , for the E symmetry of Henon-Heiles potential (equation (10)) for X close to 80" =0.1118. Slight changes in X affect mainly the large p region for example, the curves labelled as a, b and c show how the 2/3 state varies for x=0.110, 0.112, 0.144. The corresponding v=7 level varies as in inset, and thus would cross the v=2 level of the 20/3 state, practically unaffected by a change in x [33] (dashed curves) actually, the crossing is avoided and the levels behave as the continuous curves 1 and 2.

The idea of using B-spline basis sets for the representation of vibrational molecular wave functions emerged rapidly. For a Morse potential and a two-dimensional Henon-Heiles potential, we have assessed the efficiency of the B-splines over the conventional DVR (discrete variable representation) with a sine or a Laguerre basis sets [50]. In addition, the discretization of the vibrational continuum of energy when using the Galerkin method allows the calculation of photodissociation cross-sections in a time-independent approach. 

---