Quantum Dynamics of Oblique Vibrational States in the Hénon–Heiles System

In this paper, we study the quantum time evolution of oblique nonstationary vibrational states in a Hénon–Heiles oscillator system with two dissociation channels, which models the stretching vibrational motions of triatomic molecules. The oblique nonstationary states we are interested in are the eigenfunctions of the anharmonic zero-order Hamiltonian operator resulting from the transformation to oblique coordinates, which are defined as those coming from nonorthogonal coordinate rotations that express the matrix representation of the second-order Hamiltonian in a block diagonal form characterized by the polyadic quantum number n = n1 + n2. The survival probabilities calculated show that the oblique nonstationary states evolve within their polyadic group with a high degree of coherence up to the dissociation limits on the short time scale. The degree of coherence is certainly much higher than that exhibited by the local nonstationary states extracted from the conventional orthogonal rotation of the original normal coordinates. We also show that energy exchange between the oblique vibrational modes occurs in a much more regular way than the exchange between the local modes.


INTRODUCTION
−10 From the theoretical point of view, the tracking of a nonstationary state is formally accomplished by solving the time-dependent Schrodinger equation, 4,11,12 with computational challenges arising from the dimensionality of the system and the numerical instability of the time propagation, which are gradually being addressed as algorithms improves and computational power increases. 4Nevertheless, it is also useful to develop complementary dynamical theories of internal vibrational energy redistribution, capable of qualitatively and quantitatively capturing the essence of such processes. 6,8,10he vibrational dynamics of nonstationary quantum states depends on the system in which the state evolves, as well as on the way in which the state is initially prepared. 9,13−24 It is clear that the possible coherent time evolution of nonstationary vibrational states is conditioned by their degree of energetic excitation, as well as by the couplings they have with other states, which usually end up randomizing the energy transfer between the vibrational modes in the medium and high energy regimes.The question that arises, then, is to what extent a proper design of nonstationary vibrational states can effectively circumvent such couplings so that their dynamics become more coherent and extend over longer periods of time.
Normal coordinates are, by definition, those that eliminate the second-order kinetic and potential couplings of the Hamiltonian operator of the system. 25The transformation from normal to local coordinates reintroduces therefore these couplings. 14,15Generally, this transformation is linear and usually orthogonal, meaning that local coordinates are expressed as an orthogonal combination of normal coordinates.−36 The resulting coordinates can be visualized as individual nonorthogonal rotations of the original normal coordinates, that is, oblique coordinates. 36or systems of two coupled harmonic oscillators, oblique coordinates are constructed by decoupling the second-order Hamiltonian matrix blocks characterized by the polyadic quantum number n = n 1 + n 2 .In a recent application 34 of these coordinates to Heńon−Heiles systems with two dissociation channels, useful for modeling the stretching vibrational modes of triatomic molecules, we have shown that oblique coordinates are much better than normal and local coordinates when calculating the energy levels and wave functions of the system using the eigenfunctions of the corresponding uncoupled Hamiltonian operator as basis functions.In oblique coordinates, this set of basis functions is formed by the eigenfunctions of each uncoupled polyadic block, including anharmonicities.Accordingly, by initially preparing the system on one of these oblique basis function states, it is expected for it to evolve over time with a very low probability of leaving the initial polyadic block, meaning that the oblique nonstationary state will maintain a high degree of coherence within its polyadic block.
In this article, we compare the quantum dynamics of oblique nonstationary vibrational states to the dynamics of the corresponding local nonstationary states in a Heńon−Heiles system with two dissociation channels.This work adds, therefore, to the numerous dynamic studies to which the already seminal coupled Heńon−Heiles oscillator system has been subjected 37−52 since its inception, 53 by opening up the exploration of the quantum dynamics of oblique nonstationary vibrational states and the energy flows taking place among the oblique vibrational modes in coupled oscillator systems and in more realistic stretching motions of polyatomic molecules.

Coordinate Systems and Hamiltonian Operators.
The Hamiltonian operator for the system of coupled Heńon− Heiles oscillators can be expressed as follows where x 1 and x 2 are the coordinates of the system, m 1 and m 2 are the masses of the oscillators, k 11 and k 22 are the harmonic force constants, and k 112 and k 222 are the cubic coupling and anharmonicity constants.The natural harmonic frequencies of the uncoupled oscillators are therefore given as ω 1 = (k 11 / m 1 ) The Hamiltonian operator thus becomes where m′ = (m 1 m 2 ) 1/2 is a reduced mass, and the redefined force constants are given by the following expressions The rotated coordinates r 1 and r 2 , which are equivalent to the local coordinates in the context of molecular vibrations, 14,15 are obtained by performing a 45°orthogonal rotation of coordinates y 1 and y 2 in the following form The Hamiltonian operator is then written as follows As discussed in our previous work on the Heńon−Heiles system, 34 general oblique coordinates, z 1 and z 2 , can be constructed by performing a nonorthogonal linear transformation of the rotated coordinates as follows z ab r ar 1 ( 1) The Journal of Physical Chemistry A where a and b are the transformation parameters.To preserve the symmetry of the energy potential function, the values of these parameters have to be equal (a = b), which simplifies the coordinate transformation as follows z a r ar 1 ( 1) z a ar r 1 ( 1) The Hamiltonian operator in oblique coordinates z 1 and z 2 becomes then with the reduced masses and the second-and third-order force constants given by  )   112 122 It is further convenient to redefine the parameter a in the angular form a = tan α, which allows us to write the nonorthogonal linear transformation in the following form z r r 1 cos 2 cos sin This transformation accounts for a symmetric aperture of the r 1 and r 2 axes in angles −α and α, respectively, as shown in Figure 3 of our previous work. 34he specific oblique coordinates of interest, referred to as simply oblique coordinates, are defined such that the matrix representation of the second-order Hamiltonian operator of the system can be written in a block diagonal form 33,34 characterized by the polyadic quantum number n = n 1 + n 2 , where n 1 and n 2 are the quantum numbers corresponding to coordinates z 1 and z 2 , respectively.For the Heńon−Heiles oscillator systems, we have previously demonstrated that the oblique rotation angle is given by 34 meaning that when the uncoupled oscillators are degenerate (ω 1 = ω 2 ), the oblique coordinates coincide with the rotated coordinates (α o = 0).In our previous work, 34 we have shown that the use of oblique coordinates for Heńon−Heiles systems with two dissociation channels, which mimic the behavior of the stretching modes of triatomic molecules, provides much faster variational convergence of the energy levels and eigenfunctions of the system than normal and rotated coordinates. 34ccordingly, to study the time evolution of nonstationary quantum states expressed in oblique coordinates, we have used the Heńon−Heiles system with masses m 1 = m 2 = 1, harmonic frequencies ω 1 = 1.3 and ω 2 = 0.7, and third-order force constants k 112 = −0.1 and k 222 = −0.01, in atomic units (ℏ = 1).This potential was introduced by Eastes and Marcus 54 to develop semiclassical methods of calculating bound states of coupled systems 54−58 and has also been used to study the propagation of Gaussian wave packets. 37,44,46,47,51The potential has a dissociation energy of E dis = 11.4601au and a total number of 83 bound states, which essentially have a regular, nonergodic structure. 37,58The values of the parameters for this Heńon−Heiles system in normal, rotated, and oblique coordinates are given in Table 1, and the contour plots of the potential energy function in the three coordinate systems are shown in Figure 1.

Stationary States.
The time-dependent quantum dynamics of the Heńon−Heiles system has been studied using the representation of the stationary states, which are given by the general expression where ψ i (z 1 , z 2 ) and E i are the eigenfunctions and eigenvalues of the Hamiltonian operator, respectively, i.e., the solutions of the time-independent Schrodinger equation.
This eigenvalue equation is solved by writing the Hamiltonian operator (eq 20) in the following form he Journal of Physical Chemistry A where H ̂(0) (z 1 , z 2 ) is the zero-order Hamiltonian of the system given by the sum of the individual anharmonic Hamiltonian of each coordinate, i.e. where and H ̂′(z 1 , z 2 ) is the operator containing the second-and thirdorder couplings.
The eigenvalue equation for the zero-order Hamiltonian operator is then given by where the eigenfunctions ϕ n (z 1 , z 2 ) are the products of onedimensional functions ϕ nd 1 (z 1 ) and ϕ nd 2 (z 2 ), that is, which are the solutions of the eigenvalue equations solved variationally by expressing the eigenfunctions ϕ nd i (z i ) as linear combinations of harmonic functions φ vd i (z i ) in the following form.
Further diagonalization of the resulting Hamiltonian matrix provides eigenvector coefficients c vd i (n i ) and eigenenergies ε nd i .The energy eigenvalues of the zero-order Hamiltonian E n (0) are then just the sums of the energies of the individual anharmonic oscillators, i.e.
The eigenfunctions of the zero-order system, ϕ nd 1 ,nd 2 (z 1 , z 2 ), are next used as basis functions to determine, also variationally, the energy levels and eigenfunctions of the complete system.Specifically, the eigenfunctions ψ i (z 1 , z 2 ) of the complete system are expressed as linear combinations of these basis functions as follows The Hamiltonian matrix for the complete system is then diagonalized, yielding the energy eigenvalues E i and coefficients C nd 1 ,nd 2 (i) of the eigenfunctions of the Heńon−Heiles system in the nonorthogonal linear coordinates z 1 and z 2 , which include the normal, rotated, and oblique coordinates as particular cases.
We have computed variationally the energy levels and eigenfunctions of the Heńon−Heiles potential in the three different coordinate systems.In all three cases, we selected the anharmonic basis functions as those belonging to successive polyadic blocks characterized by the quantum number n = n 1 + n 2 .To achieve convergence of all bound energy levels of the potential to the first five significant figures in both normal and rotated coordinates, it was necessary to use all of the basis functions up to the polyadic block n = 40, resulting in a total of 861 functions.However, for the more favorable oblique coordinates, 34 it was sufficient to use the basis functions comprised up to the polyadic block n = 23, which resulted in a total of solely 300 functions.
In Table 2, we include the variationally calculated energy levels for the Heńon−Heiles potential, sorted by polyadic groups, and their assignments made using the three coordinate systems.The calculated energy levels agree well with those obtained by Swimm and Delos 58 using a set of harmonic basis The Journal of Physical Chemistry A functions.For the normal coordinates, we give the dominant squared coefficients of the eigenfunctions, whose values decrease as the energy excitation increases until they are no longer reliable for assigning energy levels with normal quantum numbers.Levels misassigned in this way begin to appear in increasing numbers from the n = 6 polyadic block upward.However, this does not mean that these levels cannot be assigned well in normal coordinates.In fact, all eigenfunctions of the system maintain a regular structure practically up to the dissociation limit, as shown in Figure 2, where the eigenfunctions belonging to the n = 8 polyadic block are depicted.With the sole exception of the last bound state, 83, all states of this Heńon−Heiles potential can be assigned with no problems by visual inspection of their eigenfunctions, confirming the nonergodic character of the potential as revealed long ago by means of classical and semiclassical treatments. 37,58The limitations of normal coordinates for assigning bound states using the dominant coefficients of the eigenfunctions are due to the curvilinear profiles that the nodal lines of the eigenfunctions adopt as the energy excitation increases, which cannot be reproduced by the rectilinear normal coordinates used in this work.
When using rotated and oblique coordinates, direct assignments by the dominant coefficients of the wave functions are no longer feasible, since the basis functions do not mimic the exact eigenfunctions of the potential well. 17,32For these coordinates, it makes more sense to analyze to what extent the variational wave functions are composed of contributions from the basis functions belonging to the same polyad.These The Journal of Physical Chemistry A polyadic contributions are quantified using the polyadic probability P i (n), which is defined as follows Table 2 includes the polyadic probabilities of the vibrational states of the Heńon−Heiles potential calculated using rotated and oblique coordinates.As observed, the oblique polyadic probabilities are always larger than the rotated polyadic probabilities.In fact, the oblique polyadic probabilities allow us to assign the polyadic blocks of all of the bound states of the potential, while the rotated coordinates become less useful as energy increases and are no longer reliable from the polyadic block n = 7 upward, as shown in Figure 3, where the average values of the polyadic probabilities provided by both coordinate systems for each polyadic group n are plotted.

Time Evolution of Nonstationary States.
Let us now consider the study of the quantum dynamics of nonstationary states in the Heńon−Heiles oscillator.Suppose that the system is initially prepared, at time t = 0, in a given nonstationary state Ψ(z 1 , z 2 , t = 0) ≡ Ψ(z 1 , z 2 , 0).The wave function describing the system at any other instant of time can

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then be written as a linear combination of stationary states in the following form where the coefficients a i are given by a z z z z ( , ) ( , , 0) The probability for the system to stay in the initial nonstationary state, i.e., the survival probability, is then given by which, by using eq 43, can be developed as follows As we saw in the previous section, the use of oblique coordinates provides variational wave functions for the Heńon−Heiles system that can be represented, to a large extent, as linear combinations of the polyadic basis functions to which the variational eigenfunction belongs.From a dynamical point of view, this means that if the system is initially prepared in one of the nonstationary oblique basis functions, it is expected to evolve mostly within the polyadic block of the initial state, with a relatively low probability of leaving this block.
Let us consider then that the initial state is one of the eigenstates ϕ md 1 (z 1 )ϕ md 2 (z 2 ) of the oblique zero-order system, i.e.
This state contains m 1 quanta of vibrational energy in the oblique anharmonic mode z 1 and m 2 quanta in the oblique anharmonic mode z 2 .By substituting eqs 41 and 47 into eq 44, it is easy to verify that the coefficients a i of the time-dependent wave function (eq 43) are given by The initial wave function can be then expressed as follows and the time-dependent wave function and the survival probability become We can also compute the probability of the system undergoing a transition from the initial nonstationary state |m 1 , m 2 ⟩ to any other nonstationary state |m 1 ′, m 2 ′⟩.This transition probability is defined as ( , , ) and using again eqs 41 and 47, the transition probability becomes Furthermore, the probability that the initial nonstationary state |m 1 , m 2 ⟩ remains in its polyadic block m = m 1 + m 2 is equal to the sum of the survival probability and the transition probabilities to the remaining states of the polyadic block, i.e.  ,  ; ) This is the so-called polyadic survival probability of the nonstationary state |m 1 , m 2 ⟩.
As far as the energies of the nonstationary states of the system are concerned, since the system is conservative, that is, the Hamiltonian operator is time-independent, the total energy remains constant and is given by In contrast, the energies of the anharmonic vibrational modes z 1 and z 2 do vary with time.The Hamiltonian operators that account for these energies are h ̂1(z 1 ) and h ̂2(z 2 ), so we can quantify the energy contents of the vibrational modes, E 1 (t) and E 2 (t), as the expected values of the individual Hamiltonian operators, that is 20,40 E t z z t h z z t ( ) ( , , ) ( , , Using eq 50 for the time-dependent wave function, we obtain where Let us begin by analyzing the time evolution of the nonstationary states in the polyadic block m = 1.This block is composed of the basis functions ϕ 1 (z 1 ) ϕ 0 (z 2 ) ≡ |1,0⟩ and ϕ 0 (z 1 ) ϕ 1 (z 2 ) ≡ |0, 1⟩.When chosen as oblique initial states, these basis functions are given by the following expansions with energies E ∥1,0⟩ = E |0,1⟩ = 2.0736.As we can see, the dominant coefficients in both sets of coordinates correspond to the eigenfunctions ψ 2 and ψ 3 of the Heńon−Heiles system, which contribute the most to the m = 1 polyadic block (see Table 2).Moreover, the coefficients for the oblique states have absolute values larger than those for the local states.Therefore, the oblique initial nonstationary states include larger contributions from the polyadic variational functions ψ 2 and ψ 3 than those from their local initial nonstationary states.So, let us see what impact this has on the time evolution of the nonstationary states.
In Figure 4a, we plot the survival probabilities of the oblique and local |1, 0⟩ nonstationary states as a function of time.As observed, the oblique |1, 0⟩ nonstationary state evolves by displaying quasi-periodic oscillations with recurrent probability maxima that practically reach unity, while the local |1, 0⟩ state shows more irregular oscillations with lower and slightly outof-phase recurrence peaks.In Figure 4b, we further plot the polyadic survival probabilities of the two |1, 0⟩ states, which clearly show that the oblique state practically never leaves the polyad, while the local state occasionally leaks out of the polyad.The oblique |1, 0⟩ state evolves coherently within the polyadic block and does so with a recurrence period of 0.257 fs, which coincides technically with that obtained for an initial state constructed as a superposition of the polyadic stationary states ψ 2 and ψ 3 , given by τ rec.= 2.257/(E 3 − E 2 ) = 0.257 fs.The time evolution of the nonstationary state |0, 1⟩ is identical, for symmetry reasons, to that of the |1, 0⟩ state just described.
As for the energy content of the vibrational modes, the initial energies deposited in the vibrational mode of the oblique |1, 0⟩ nonstationary state are E 1 (0) = ε 1 = 1.4864 and E 2 (0) = ε 2 = 0.4979, and they evolve with time (eq 58; Figure 5a) in a perfectly regular and periodic way that accounts for a complete exchange of energy between the oblique modes in a period of time equal to the recurrence time of 0.257 fs.For the local |1, 0⟩nonstationary state, the initial energy contents of the vibrational modes are E 1 (0) = ε 1 = 1.5545 and E 2 (0) = ε 2 = 0.5202, and their evolution, although somewhat periodic as well, as observed in Figure 5b, is irregular and abrupt, with some extra energy losses and gains of the vibrational modes due to non-negligible couplings of the local nonstationary state |1, 0⟩ with local states belonging to polyadic groups other than m = 1.

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Let us consider next the time evolution of the nonstationary states of the polyadic m = 2 block.This block is formed by the basis functions |2, 0⟩, |1, 1⟩, and |0, 2⟩, which result to be expressed as the following linear combinations of the stationary states of the complete system in oblique coordinates   2).These coefficients are larger in absolute value for the oblique coordinates than for the local coordinates.
In Figures 6a,b, we plot the survival probabilities of the |2, 0⟩ and |1, 1⟩ nonstationary states, respectively, in both oblique and local coordinates.As observed, the oblique states evolve in a more regular oscillatory way than the local states and display noticeably higher recurrence peaks.The recurrence time of the oblique state |2, 0⟩ is 0.259 fs, which is similar to that of the oblique state |1, 0⟩, while the recurrence time of the oblique state |1, 1⟩ roughly halves to 0.130 fs.This value is in practical coincidence with the recurrence time of 2.130/(E 7 − E 4 ) = 0.130 fs of the nonstationary state formed by a linear combination exclusively of the third and fourth stationary states.
As for the energy content of the oscillators, the oblique |2, 0⟩ nonstationary state evolves again by nearly completing regular energy exchanges between the vibrational modes, as shown in Figure 7a, whereas the local |2, 0⟩ state exhibits more abrupt and irregular energy exchanges (Figure 7b).In contrast, the energy behavior of the nonstationary state |1, 1⟩ is completely different.In this case, the two vibrational modes are initially equally excited, each containing one vibrational quantum, and the transition probabilities to the other two nonstationary states within the same poliad, |2, 0⟩ and |0, 2⟩, are equal by symmetry.As a result, the mean values of the energy of each vibrational mode remain the same, with small irregular oscillations due to couplings with states of polyads other than m = 2, as shown in Figure 8 for both the oblique and local descriptions.Such irregular oscillations are more pronounced for the local state due to its stronger couplings with states of other polyadic blocks than the oblique state.
The dynamics of the oblique and local nonstationary states of higher polyadic blocks can, in principle, be qualitatively predicted based on what occurs in the polyadic blocks m = 1 and m = 2, except for any particular effect that may arise when progressively increasing the energy excitation of the vibrational modes.To explore this, we focused on the time evolution of the nonstationary states |m 1 , 0⟩, i.e., those initially excited exclusively in the z 1 mode.
In Figure 9a, we show the survival probabilities of the |5, 0⟩ oblique and local nonstationary states located in the mediumenergy regime of the potential, with energies 5.8065 and 6.0994, respectively, and in Figure 9b, we show the time evolution of the survival probabilities of the |10, 0⟩ oblique and local nonstationary states, with energies of 10.1980 and 10.8029, respectively, very close to the dissociation limit of 11.4601.As expected, the recurrence peaks decrease in height as time progresses but remain appreciably high for the oblique nonstationary states and certainly much higher than the recurrence peaks of the local nonstationary states, which, for the most excited |10.0⟩ state, fade rapidly.Furthermore, the polyadic survival probabilities remain higher for the oblique states than for the local states, as shown in Figure 10a,b.The medium-energy oblique state |5, 0⟩ continues to remain most of the time within its polyadic block, while the oblique state near the dissociation limit |10, 0⟩ clearly explores other polyads, albeit developing significant polyadic recurrences, in contrast to the corresponding local state that dissipates rapidly into other polyads.
In Table 3, we provide a comparison of the energies of all nonstationary |m 1 , 0⟩ oblique and local states, along with their first recurrence times and survival probabilities.As observed, the first recurrence times generally become longer as the The Journal of Physical Chemistry A excitation of the state increases.However, the first recurrence peaks of oblique states remain high, with values above 0.8, even in the most excited states, while the recurrences of local states decay rapidly with increasing excitation.
Finally, as far as the energy content of the vibrational modes is concerned, the energy exchange between the oblique modes remains quite regular and complete, even in the most excited nonstationary states such as |10, 0⟩, as shown in Figure 11a.In the local nonstationary state |10, 0⟩, the exchange is, as we already know, somewhat rough and, more importantly, less efficient as time goes on, as evidenced by the gradual decrease of the amplitudes of the energy oscillations seen in Figure 11b.

CONCLUSIONS
In this work, we have comparatively studied the quantum dynamics of oblique nonstationary states versus rotated, or local, nonstationary states in a system of coupled Heńon− Heiles oscillators with two dissociation channels that can be used to model the vibrational stretching motions of symmetric triatomic molecules.The local nonstationary states are the eigenstates of the zero-order Hamiltonian operator obtained by performing a 45°orthogonal rotation of the normal coordinates, while the oblique nonstationary states are the zero-order states obtained by performing a nonorthogonal coordinate rotation that allows us to write the matrix representation of the second-order Hamiltonian in a block diagonal form characterized by the polyadic quantum number n = n 1 + n 2 .
First, we performed variational calculations of all of the bound vibrational states of the Heńon−Heiles system up to the dissociation limit using the three coordinate systems, normal, local, and oblique, and confirmed that the vibrational spectrum   The Journal of Physical Chemistry A is practically regular over the entire range of bound energies.We have also shown that the oblique coordinates enable us to assign all states to their corresponding polyadic groups, as opposed to the local coordinates, whose polyadic assignments start to fail from the middle energy region upward.
As a consequence of the oblique coordinates' efficiency in organizing the stationary states of the Heńon−Heiles system into polyadic blocks with n = n 1 + n 2 , we have demonstrated that the time evolution of oblique nonstationary states is much more periodic and coherent than that of the corresponding local nonstationary states, as evidenced by both the survival probabilities and the energy exchange between vibrational modes.The quantum time dynamics of nonstationary oblique states occur mainly within the polyadic group to which the initial excited state belongs.Furthermore, the time dynamics of such states remain quite regular up to the dissociation limit on a short time scale, as demonstrated by the well-defined profiles of the recurrence peaks and the energy exchange between the oblique vibrational modes occurring in a regular, oscillating manner with little loss.
In view of our recent work on the usefulness of oblique coordinates in two-dimensional systems, both in the timeindependent 33−36 and time-dependent (present work) approaches, it is clear that the next step is to extend oblique coordinates to systems with a larger number of degrees of freedom such as the vibrational motions taking place in polyatomic molecules and in condensed phase systems.One of the advantages of oblique coordinates over normal coordinates is the greater number of parameters available in the former, which is useful to optimize them conveniently with greater flexibility.However, it is to be expected that as the dimensionality increases, the possibility of obtaining closed analytical expressions for the optimal values of the oblique parameters will quickly fade away, and numerical methods will

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have to be resorted to, as occurs with normal coordinates.All this opens, in our opinion, a wide range of possibilities for using oblique coordinates in systems with three or more degrees of freedom, both to construct highly separable oblique coordinate systems that computationally facilitate the variational determination of vibrational states and their physical interpretation, as well as to design multidimensional oblique nonstationary states that propagate with a high degree of coherence, facilitating the transfer of energy between vibrational modes in a more selective way.Logically, our future work on oblique coordinates points in all of these directions.

Figure 2 .
Figure 2. Variational wave functions of the n = 8 polyadic block in normal coordinates.

Figure 3 .
Figure 3. Average polyadic probabilities of the states belonging to the polyadic groups of the Heńon−Heiles potential in rotated (squares) and oblique (circles) coordinates.

Figure 5 .
Figure 5.Time evolution of the E 1 (t) (solid line) and E 2 (t) (dashed line) energies of the vibrational modes for the oblique (a) and local (b) |1, 0⟩ nonstationary states.

Figure 7 .
Figure 7. Time evolution of the E 1 (t) (solid line) and E 2 (t) (dashed line) energies of the vibrational modes for the oblique (a) and local (b) |2, 0⟩ nonstationary states.

Figure 8 .
Figure 8.Time evolution of the one-dimensional mode energies of the initial nonstationary state |1, 1,⟩ oblique (red) and local (blue).

Figure 11 .
Figure 11.Time evolution of the E 1 (t) (solid trace) and E 2 (t) (dashed trace) energies of the one-dimensional modes of the initial nonstationary states |10, 0⟩ oblique (a), and local (b).

Table 1 .
Parameters of the Heńon−Heiles System in Different Coordinate Systems

Table 2 .
Energy Levels and Polyaddition Probabilities of the Heńon−Heiles System a

Table 3 .
Energies and Recurrence Probabilities of the Nonstationary States of the Heńon−Heiles System