J. Phys. Chem. 1983, 87, 3420-3424
3420
Intramolecular Energy Transfer in Isolated Molecules and the Chaotic Behavior of Nonlinear Systems Nimrod Moiseyevt Depeflmnt of Chemlshy, Technion-Israel Institute of Technology, tfalfa 32000, Israel (Received: November 5, 1982)
It has been proposed that chaos in the Henon-Heiles and related systems may be interpreted in terms of rapid decay of vibrational excitations. However, it is not clear whether this idea can be applied in reverse, i.e., that the mechanism of vibrational relaxation of an excited mode requires chaotic behavior of the classical system. Studies reported in this paper of the decay of vibrational excibtions in separable, integrable and nonintegrable, nonlinear, systems show that, by using physically reasonable initial wave functions, it is unnecessary, and sometimea unreasonable,to relate intramolecularenergy transfer to the question of "what is the quantum analogue of classical chaos".
1. Introduction
Intramolecular energy transfer in isolated molecules has been associated with the chaotic behavior of nonlinear systems.'P2 The molecular potential surface is described by harmonic oscillators which are coupled by nonlinear terms. As the energy of the systems is varied, quasiperiodic trajectories in the classical phase space disappear, and, when a critical energy (E,) is related, most of the regions in phase space become irregular (i.e., contain chaotic trajectories only).2 Nordholm and Rice2suggested that the RRKM theory is applicable to a molecule whose E, is less than its dissociation energy. Recently, it was suggested, on the basis of a study of the classical autocorrelation function for a mode energy, that chaotic behavior be associated with rapid vibrational relaxation of an initially excited mode.3 Rapid vibrational relaxation is obtained whenever the excited mode shares its energy very rapidly with all the other degrees of freedom, and, therefore, the RRKM theory should be applied in this case. More recently, Moiseyev and Certain4 extended this idea3by carrying out quantum mechanics calculations on the Henon-Heiles system. It was found that for an initial state of two harmonic oscillators (one of which is excited and the other is not) a sharp transition occurs from periodic oscillation of the energy in the excited mode to rapid vibrational relaxation; this occurs at a total energy close to the classical critical energy, E,. Even if it is granted that chaos in nonlinear systems can be related to the rapid decay of vibrational excitations, it does not necessarily follow that we can always apply this idea in reverse: namely, that the mechanism of vibrational relaxation of the excited mode is correlated with the existence of quasiperiodic and chaotic classical trajectories in the phase space. There is no doubt that the properties of an isolated molecule, even in the classical limit, should be studied within the framework of quantum mechanics. Very recently, I have shown,5for the Henon-Heiles system, that quantum-mechanical calculations can provide an estimate of the classical critical energy, E,, and the location of "large-size" irregular regions in the classical phase space. The object of the present investigation is to study the interrelation between rapid vibrational relaxation (which indicates the applicability of the RRKM theory) with the chaotic behavior of classical nonlinear systems. The energy in a given mode can be time dependent, even for one-dimensional systems which have regular spectra 'Yigal Allon Foundation Fellow.
only. Since there are no other modes with which the excited mode can share its energy, and since the anharmonic perturbational energy terms are small, the time-dependent mode energy will deviate only slightly from its initial value. In two-dimensional separable systems, composed of two coupled harmonic oscillators, the energy of the initial excited mode is changed in time much more strongly than in the one-dimensional case. However, for the separable systems described in the next section the energy in the excited mode returns to its initial value within a "short" period of time, and no rapid vibrational relaxation is observed. These results, together with the numerical results which had been obtained previously for the Henon-Heiles system," support the assumption that rapid vibrational relaxation-and therefore the applicability of the RRKM theorem-could be ascribed to the chaotic motion observed in the classical dynamics of systems which are both are both nonlinear and nonintegrable. However, the results presented in section 3 clearly indicate that rapid vibrational relaxation of an initially excited mode can occur even though the system is integrable and does not show any chaotic behavior in the classical trajectory calculations. Such a rapid vibrational relaxation can occur even with the absence of Fermi resonances. Therefore, it is unnecessary and perhaps even unreasonable, to connect the mechanism of intramolecular energy transfer with the question of "what is the quantum analogue of the classical chaos". In section 4,the mechanism of intramolecular energy transfer in nonintegrable systems is studied. Rapid vibrational relaxation can be related to the presence of Fermi resonances.6 Indeed, as we shall show here, in the case of a Henon-Heiles system of coupled nondegenerate harmonic oscillators, the rapid vibrational relaxation of the excited mode occurs at a "critical" energy at which Fermi resonances are present; when the latter were removed, relaxation did not take place. Calculations which were carried out for small changes in the frequency of one of the harmonic oscillators showed that the rate of vibrational (1)R. A. Marcus, Faraday Discuss.Chem. Soc., 55,34 (1973);W.P. Reinhardt, J.Phys. Chem., 86,2158 (1982),and references therein. (2) K. S.J. Nordholm and S. A. Rice, J.Chem. Phys., 61,768(1974). ( 3 ) M. L. Koszyrowski, D. W. Noid, M. Tabor, and R. A. Marcus, J. Chem. Phys., 74,2530 (1981). (4)N. Moiseyev and P.R. Certain, J. Phys. Chem., 86, 1149 (1982). (5)N. Moiseyev, to be submitted for publication. For a review of criteria for quantum chaos, see E. J. Heller and M. J. Davis, J. Phys. Chem., 86, 2118 (1982),and references therein. (6)E. J. Heller, E. B. Stechel, and M. J. Davis, J. Chem. Phys., 73, 4720 (1980);D.W.Noid, M. L. Koszykowski, and R. A. Marcus, ibid., 71, 2864 (1979);D.W.Noid and R. A. Marcus, ibid., 67, 559 (1977).
0 1983 Amerlcan Chemical Society
Intramolecular Energy Transfer
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983 3421
relaxation of the excited mode increases continuously12 with the energy of the system. This continuously, but very sharply, increasing rate of intramolecular energy transfer can be related to the increase with energy of the density of the vibrational states of the anharmonic y mode. In the case of a totally bound potential of two harmonic oscillators coupled by an x2y2potential term, the gap between the nondegenerate harmonic energy levels increases with the energy, rather than decreasing as in molecular systems. In this case, the rate of intramolecular energy transfer is reduced as the energy increases even though increasingly large irregular regions were obtained in the classical calculations.' The conclusion is that, for both integrable and nonintegrable systems investigated here, rapid decay of the excited mode is obtained whenever the density of the vibrational states increases with the energy, and the rate of decay is not directly related to chaotic trajectories in the classical system. Degeneracy or accidental degeneracy of the coupled harmonic oscillators can also have a large effect on the increasing of the rate of the vibrational relaxation of the excited mode. The initial wave packet that we used are two pure states of harmonic oscillators where one of which is excited and the other is not. It should be stressed that because of the initial width of the wave packet one can not relate the behavior of the time evolution wave packet to the classical result obtained for an initial condition of a single point in the phase space. In our quantum-mechanical calculations the initial wave packet has the uncertainty of Ax = [hw,(n, + 1/2)]1/2 hp,
= m1/2Ax
My= m1l2Ay
For example, in the quartic potential for large enough n, ( h w = 1) the area in the Poincar6 surface of section that is "covered" by the initial quantum wave packet contains both quasiperiodic and chaotic trajectories. Therefore, the quantum behavior of the quartic potential system cannot be simply compared with the quasiperiodic trajectory obtained for initial condition of x = 0, P, = 0 or y = 0; P = 0. Note that, as pointed out very recently by Senitzke+ then for initial state of coherent wave packet, unlike our initial states, the classical analogue is completely deterministic and it is a single point in the phase space even for h # 0, whereas the classical analogue to the energy states of the harmonic oscillator even for h 0 is an ensemble of points on the Poincar6 surface of section with a specific probability density that satisfies the three conditions which were given and discussed in ref 8.
-
2. Decay of Vibrational Excitations i n Separable
Systems Two separable systems are studied. One is defined by the separable Hamiltonian: A = uta btb X(atb bta) (1) where ut, bt, and a, b are the harmonic oscillator creation and annihilation operations ( h = w = 1). This Hamiltonian can be rewritten as a separable one A = (1 X ) C ~ C (1 - X)dtd (2) where
+
+
+
= (at
+ bt)/21/2
dt = (at - bt)/21/2
+
+
(7) R. A. Pullen and A. R. Edmonds, J . Phys. A, 14, L477 (1981). (8)I. R. Senitzky, Phys. Rev. Lett., 47,1503 (1981).
(3)
(4)
The time-dependent operator at#) is expressed by a t ( t ) = ct exp[-i(1
+ X ) t ] + dt exp[-i(1 - X ) t ]
(5)
A similar expression holds for bt(t). By substituting eq 3 and 4 into eq 5 one can get the following expression for the time-dependent individual mode energy operator at(t) ~ ( t=) [uta
+ btb + (uta - btb)
COS (2Xt) - (bta atb) sin (2Xt)]/2 (6)
Let us define the initial state in which one of the oscillators is excited to its l-th vibrational level and the other one to its m-th vibrational level; Le., the initial state 9(0)is defined by
@(O) = [ ( ~ t ) ~ / l ! ' / ~ ] [ ( b t ) " / m ! ' / ~ ] 1 0(7) ) The time-dependent individual mode energy can be analytical obtained and is given by
E N = ( W l a t a l d t ) ) = (dO)lat(t) a(t)ldO)) = (1 - m) cos2 ( A t )
+m
(8)
We see that for an initial state of two identically excited harmonic oscillators, 1 = m, E, is time independent, and for 1 # m the individual mode energy oscillates in time with the periodicity of r / h . In the second example, the decoupled Hamiltonian is obtained by reversing the sign of the x2y term in the Henon-Heiles potential. This "anti" Henon-Heiles Hamiltonian is defined therefore by
A = (Px2+ x2)/2 + (P; + y 2 ) / 2 + W y - y3/3
+ 1/2)]1/2
Ay = [hw,(n,
Ct
(9)
where X = -1 instead of X = 1 in the Henon-Heiles system. The fact that the Hamiltonian defined by eq 9 is separable can easily be seen by making the coordinates transformation: u = (x
+ y)/21/2
(10)
u = (x
- y)/21/2
(11)
By substituting eq 10 and 11 into eq 9 one obtains
A = (P;
+ u2 + 8 1 / 2 ~ 3 / 3 ) / +2 (P: + u2 - 8'l2u3/3)/2 (12)
For initial states in which one oscillator in eq 9 is excited and and the other one is not, the energy in the x mode shows (see Figure 1) periodic oscillations in time, and no rapid decay of E,@) is observed. For the two separable systems studied here, of course, all regions of the classical phase space are regular, and we find that there is never a rapid decay of vibrational excitation. The results are consistent with the assumption that rapid decay of energy in an excited mode (i.e., rapid vibrational relaxation) is related to the existence of regions in the classical phase space which are irregular (contain chaotic trajectories only). 3. Decay of Vibrational Excitations in Integrable
Systems Recently the dynamics of the second-generation Hamiltonian has been studied by Katriel and Moiseyev? One of the results obtained was that the average number of low-frequency photons decayed very rapidly in time if the (9) J. Katriel and N. Moiseyev, J. Chem. Phys., 78,876 (1983).
3422
Moiseyev
The Journal of Physical Chemistry, Voi. 87, No. 18, 7983
-
Flgure 1. Energy in the “ x mode” as function of time for the “anti” Henon-Heiles Hamiltonian. X is the strength coupling parameter and (n , m ) denotes the oscillator quantum numbers of the inltlal state as defined in the text.
initial conditions correspond to an experiment starting with a beam of low-frequency photons; Le., a rapid vibrational relaxation of the low-frequency excited mode is obtained even though the system is integrable and has a regular spectrum only. In this section we shall study the mechanism of the decay of vibrational excitations for a more general form of nonlinear model Hamiltonian. It will be shown that the rate of the decay of the energy in the excited mode increases if the perturbation potential terms couple isoenergetic levels of the harmonic oscillators, and if these coupling terms are very nonlinear. Two classes of nonlinear systems are studied. The first class of Hamiltonians is defined by
fil = uta + btb + [ ~ / 2 ~ / ~ ] [ ( +a ~btb’] )’b
(13)
where 1 = 2,3, .... These model Hamiltonians only couple states within finite-dimensional subspaces of the complete Hilbert space and therefore are exactly soluble. Note that H(1=2)is the integrable Hamiltonian obtained by applying the rotating wave approximation to the Henon-Heiles system with the perturbational strength parameter €.lo The second class of Hamiltonians studied is defined by
I?, = uta + lbtb
+ [ ~ / 2 ~ / ~ ] [ (+a bta’] ~)’b
(14)
where 1 = 2,3, .... In this case, the coupling terms commute with-the two harmonic oscillator Hamiltonians, de_noted by Ho, and therefore only isoenergetic levels of Ho are coupled by the nonlinear potential terms. The initial states are chosen to be the eigenfunctions of the unperturbed Hamiltonian Hoand denoted by (&,nb) where n, and nb stand for the na-th and nb-thvibrational levels of the two harmonic oscillators. For an initial state of (90,0), the time-dependent energies of the excited mode weze calculated for the two model Hamiltonians H1and Hzwith 1 = 2. The results are presented in Figure 2a. In the first case no decay of the energy in the excited mode was obtained. However, in the case where the nonlinear terms coupled isoenergetic levels of Ho a rapid decay of the energy in the excited mode was observed. After few oscillations the excited mode shared ita energy with the second mode and was stabilized at the energy which was about the average energy of the two modes at time zero. (10) J. A. Sanders, J. Chem. Phys., 74, 5733 (1981).
J lT 21T
00
t Flgure 2. Energy in the “a mode” as a !unction of time for the nonlinear, Integrable model Hamiltonians: H,,2 = a t a a,,*6Ib f [e/ 23’2][(a t)’6 +, bta’] where ai = 1, a2 = I , and therefore [H,,V] = 0. (a) I = 2, /+, Is the rotating wave approximationof the Henon-Heiles Hamllto_nian,H , is the second-harmonic generation Hamiltonian. (b) I = 3 , H 3 is the third-harmonic generatlon Hamiltonian.
+
When the value of 1 is increased, the Hamiltonians become more nonlinear. As a result, a rapid decay of the energy in the epciAtedmode was obtained also in the first case (where [Ho,v]# 0). For example, see the results presented in Figure 2b which were obtained for 1 = 3. The excited mode shares its energy very rapidly with the second mode even though V does not copple isoenergetic leyels of Ho.Still for the Hamiltonian H2(1=3),where [Ho,v]= 0, the energy deFy of the excited mode is much faster than in the case of H1(1=3). The results presented here show that an excited mode can share its energy with the other degrees of freedom (and therefore the RRKM theory could be applied) even though the system is integrable and has a regular spectrum only. 4. Decay of Vibrational Excitations in Nonintegrable Systems Two cases were studied: the Henon-Heiles system, which has been investigated1’ extensively, and a system of a totally bound potential.’ The surface of section pictures for the two systems are similar in the sense that most ~~
~
~~
(11) M. Henon and C. Heiles, Astron. J.,69,73 (1964). (12) This continuous transition can be related to the continuation of
certain sequences of eigenvalues from the quasiperiodic to the chaotic regime as discussed by D. W. Noid, M. L. Koszykowski, M. Tabor, and R. A. Marcus, J. Chem. Phys., 72,6169 (1980).
The Journal of Physical Chemktty, Vol. 87, No. 18, 1983 3423
Intramolecular Energy Transfer
of the regions in the classical phase space become irregular within a very narrow energy range.’ However, the quantal energy spectrum of the two Hamiltonians is quite different. In the Henon-Heiles system, a harmonic oscillator is coupled with an anharmonic oscillator by a cubic term and therefore, when the energy is increased, the density of states is increased. In the other case, two harmonic oscillators are coupled by a quartic term and the gap between the nondegenerate harmonic levels is increased with energy. Henon-Heiles System. The Henon-Heiles system is defined by H = H, H y Q (15)
n 16,
--I
wx=l
w;=w
a ‘ 0.I
-
1
c
X
W
0.05
+ +
where
fix= y.(px2 4- ux2x2) fiy = 1/2(p:
+ wyy2)
Q = x2y - y3/3
(16)
Ob
200
(17)
The Henon-Heiles potential surface models the coupling between the symmetric and the antisymmetric vibrational stretching of a linear triatomic molecule. We have pointed out previously4 that, for initial states in which one oscillator is excited and the other not, a transition is observed from quasiperiodic behavior of the energy in the excited mode to rapid decay as the energy increases. This transition was obtained at a total energy which is in good agreement with the critical energy obtained in classical mechanics for the transition from the case where most of the regions in the phase space are regular to the case where most of them are irregular. In this section we show that, for nondegenerate coupled harmonic oscillators, the rapid intramolecular energy transfer from one excited mode to the other one can be related to the “almost” accidental degeneracy (Fermi resonances) of the two harmonic oscillators. In the almost-degenerate case, where w wr, an abrupt, though continuous, increase in the rate of vibrational relaxation is obtained for a specific value of the energy, similar to the results obtained previously for the degenerate case. a. “degenerate Case. In this case, rapid vibrational relaxation in an individual mode can be observed if the system energy exceeds the value for which “almost” Fermi resonances are obtained. For example, when w, = 1, wy = (3/4)1/2there are “almost” Fermi resonances when n, = 4, ny = 5, and w,n, - u p y 0. Indeed, an abrupt change in the relaxation behavior was observed when the initial state was (0,5) or (4,0)! (See Figure 3 in ref 4.) Following this analysis, rapid vibrational relaxation is not expected, for example, when w, = 1 and wy = !N4.In this case there are no Fermi resonances and the lowest almost-degenerate levels are obtaineg for n, = 5 and ny = 6. However, the coupling term in V, defined in eq 17, does not couple the odd harmonic states with the even ones in the finite basis set constructed of zero-order eigenfunctions. Therefore, as is presented in Figure 3a, ,we do not observe any vibrational relaxation when the system energy is smaller than (21/2)ho,+ ( 1 / 2 ) h ~ ywhich , is associated with the initial state (10,O). b. Almost-Degenerate Case. In order to have a better understanding of the dynamic of the Henon-Heiles system when w, = wy = 1, we studied the case where w, = 1, wy = (1.05)’/’ and consequently some of the oscillations in the mode energies which are due to degeneracy are removed. From the results presented in Figure 3b one can see the following: (a) Rapid vibrational relaxation is obtained for E = 0.11 similar to the results that has been obtained before4 for exact degeneracy.
360
5b0
400
t 0.15
b
-c
X
W
“e
(6,O)-
(4,O)-
0.05
(2
(0,O)-
=
=
(0,O) IO0
‘0
100
200
300
400
500
t Flgure 3. Energy in the “ x mode” as a function of time for the H a non-Helles Hamlttonian, with h = 0.0125: (a) the unperturbed Ham ittonlan is not degenerated and (b) the unperkrbed Hamlttonian is almost degenerated.
(b) The transition from slow to rapid vibrational decay is continuous, though abrupt. Namely, even at low energies the energy in the x mode does not return completely to its initial value with time. The deviation of the local maxima of E,(t) from its initial value continuously (but not linearly) increases as the system energy does. Totally Bound System. Pullen and Edmonds’ compared the quantal energy spectrum with the classical motion for the totally bound system defined by the Hamiltonian =
Y2(px2 + p:
+ x 2 + y2) + ax2y2
(18)
For a = 0.05 most of the regions in the classical phase space are irregular at energies above 15, and the motion is almost wholly chaotic at energies above 50. The transition from regular to irregular behavior (quasiperiodic/ chaotic) is sharp. If it is ture that chaos in nonlinear systems should be related to rapid decay of vibrational excitations, then a rapid decay of the energy in excited modes should be observed at E z 15. The initial state is taken as two harmonic oscillator normal modes where one is excited to the m-th vibrational level, @(O) = Im,O), and m is an even integer. The wave function @(t)is the solution of the time-dependent Schrodinger equation @(t)= exp(ifit/h)lm,O)
(19)
Since m is an even number, only the eigenstates that have
3424 401
The Journal of Physical Chemistty, Vol. 87, No. 18, 1983 I
1
numerical results show that chaos in this totally bound system cannot be simply related to a rapid decay of the energy in the excited mode.
I
I
Moiseyev
30
Discussion In previous quantum-mechanical calculations the initial state of the system was chosen to be a coherent wave packet whose width is reduced to zero in the classical limit (h 0). However, in our study of the mechanism of intramolecular energy transfer in an isolated molecule we take the initial state to be an eigenfunction of the “zeroorder” Hamiltonian constructed of noncoupled potentials; e.g., harmonic oscillators in the normal or local mode approximation. In the case of the Henon-Heiles system a possible initial state is a product of two harmonic wave functions where one is excited and the other is not. In such a case a rapid decay of the vibrational energy in the excited mode was observed at the same energy for which most of the classical trajectories are chaotic. This result was obtained previously for degenerate coupled harmonic oscillators. In this work it is shown that in the nondegenerate case the rapid vibrational relaxation is at an energy at which Fermi resonances are present. When the Fermi resonances were removed, the rapid vibrational relaxation did not take place. These results raise the question of whether intramolecular energy-transfer mechanisms should be related to the chaotic behavior of classical systems. The study of several specific model Hamiltonians provide counterexamples to a close interconnection between intermolecular energy transfer and chaos. For the model of a totally bound potential, even though there is chaotic behavior in the classical systems, no rapid change in the rate of the energy transfer from the excited mode to the second one was observed, even at high energies. Therefore, we conclude that classical chaotic behavior is not sufficient conditions to have a rapid decay of the energy in the excited mode. Morever, the results obtained for nonseparable integrable systems indicate that the chaotic behavior is also not a necessary condition to have a rapid decay of the excited-mode energy. Even for systems which only have a regular spectra, rapid vibrational relaxation of the energy in the excited mode can be obtained. Increasing density of states with the energy and/or degeneracy or almost degeneracy of the coupled harmonic oscillator energy levels have a large effect on increasing of the rate of the energy decay in the excited mode. The presence or absence of “classical chaos” would seem to be irrelevant in the cases that we have studied. It should be stressed that the results and the conclusions depend on the choice of the initial states, however, and the question o,f what is the physically relevant initial state, or what is H,,is still open for discussion.
-
I
I
1000
2000
t
3000
4000
5000
Flow@4. Energy in the “xmode” as tcnction of time for a totally bound potential, V = ( x 2 y 9 / 2 0.05x2y2. The Initial energy Is Increased by exciting the harmonic oscillator in the x directlon.
+
+
the Al and B1symmetry should be calculated. IC;.(Al)and IC;.(BJ were obtained by the variation method +j(Al) = C C2:,ny(bx,ny)+ 1 n ~ , n , ) ) / 2 l / ~ (20) n,,n,
lkj(B1) = C @:,ny(bx,ny)- Iny,nx))/21/2 (21) nnny
where n, and ny are even quantum numbers. Therefore, the time-dependent energy of the x modes is given by
E,@)= (4J(t)l’/,(R,”+ X2)ldt))
where
Ej and kj in eq 23 and 24 are the eigenvalues of H which are associated with tj(Al) and lC;.(Bl), respectively. The resulta for the mitial states IlO,O), l20,0), and 130,O) are presented in Figure 4. The rate of energy transfer from the excited mode to the other one is much slower than the rate in the case of the Henon-Heiles system. Moreover, the rate of intramolecular energy transfer is larger for the lower excited initial state Il0,O) than for the higher excited initial states, 120,O) and 130,O). For the initial state of Il0,O) the period of the oscillations in the x-mode energy is smaller than the period obtained for the initial state of 120,0),and for 120,O) is smaller than in the (30,O)case. However, the classical system is more chaotic at higher energies than at lower ones. Therefore, these
Acknowledgment. It is a pleasure to thank Professor E. Amitai Halevi of the Technion and the members of the Theoretical Chemistry Institute in Madison for most helpful discussions and comments. This work received partial support from the National Science Foundation (U.S.) Grant CHE-8107628 given to Prof. P. R. Certain, to whom I am grateful for his kind hospitality.
