Quasiperiodic motion in collinear HeH2+

Volume 81, number 3

CHEMICAL PHYSICS LETTERS

1 August 1981

QUASIPERIODIC MOTION IN COLLINEAR HeH~ Howard R. MAYNE and Ralph J. WOLF

Lash Miller ChemicalLaboratory, University of Toronto, Toronto, Ontario, CanadaMSS 1A1 Received 30 March 1981; in final form 24 April 1981

The classical motion of the highly vibrationally excited HeH~ collinear complex has been investigated. An estimate of the fraction of phase space containing quasiperiodic trajectories versus total energy is reported. At some energies it is found that dissociative and quasiperiodic trajectories occupy disjoint regions of phase space.

1. Introduction

2. Computational methods

There is considerable interest in the intramolecular vibrational relaxation (IVR) in polyatomic systems [ 1 - 3 ] . Above the unimolecular reaction threshold, rapid IVR is a necessary assumption underlying statistical reaction rate theories [ 4 - 6 ] (e.g. RRKM, phase space). The assumption requires that all points in the molecular phase space are accessible and will lead ultimately to reaction. Numerical studies using model hamiltonians below the unimolecular reaction threshold show typically regular [7] (quasiperiodic) motion below a critical energy, E c [ 7 - 9 ] . Quasiperiodic trajectories are restricted to a subset of the available phase space, leading to infinitely slow IVR. A relatively abrupt transition from regular to irregular ("chaotic") motion occurs at energies above E e [8,9]. Above E c IVR will be rapid. In this letter we present collinear calculations on a realistic DIM (diatomics-in-molecules) surface for HeHH + [10,11]. The potential was developed for scattering calculations, but may also be applicable for studying the dynamics of van der Waals complexes. We have looked in detail for the critical energy, Ec, showing the onset of chaotic motion, using the method of Poincar6 surfaces of section [8,12].

Z 1. Hamiltonian and trajectory calculations We used the following hamiltonian:

H(x,y,px,Py)=p2x/21a +p2y/2m + V(x,y)=E,

(1)

where y is the HH distance, x is the H e - H H centre of mass distance and Px and py are their conjugate momenta, m is the reduced mass of HH and 12 is the reduced mass of He relative to HH. The potential is the DIM surface for HeHH + proposed by Kuntz [10] with the parameters of Chapman and Hayes [11 ]. This potential has a shallow well of - 0 . 0 1 7 eV (relative to the bottom of the He + H~ entry value) at x = 3.12 a0, y = 2.06 a 0 and a barrier of 0.023 eV at x = 4.5 a0, y = 2.0 a 0 . We thus have a bound HeH~ potential with a dissociation energy D - 0.040 eV in the region of this minimum. We shall measure total energy from the minimum of this basin. Trajectories are started in this basin and Hamilton's equations are integrated by a Hamming predictor-correcter routine with a step size of 1 X 10 -16 s. This conserves energy to better than one part in 105 . Trajectories are run for 5000 steps.

2. 2. Collinear sampling Sampling of the initial phase space was performed by two different methods. Method I is an approximate random selection of points proposed by Bunker [ 13]. 508

0 0 0 9 - 2 6 1 4 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

Volume 81, number 3

CHEMICAL PHYSICS LETTERS

1.0I--.-0.¢-~

The method is an extended rejection technique, generating points which are weighted by their phase space density. Method II involves setting the initial coordinates at the values for the minimum for the basin and partitioning the total energy randomly between the kinetic energy of the oscillators. The momenta are given by

Px = (2/1~'E)1/2'

Py = [2m(1 - ~')E] 1/2,

(2)

where ~"E [0,1] is a random number, and E is the total energy.

2.3. Poincarksurfacesof section Poincar6 surfaces of section (SOS) plots are produced by recording the value of (X,Px) as a trajectory passes through a given plane in the phase space (here Y =Y0 = 2.06 a0). If the m o m e n t u m py is greater than zero we print an X at the current value of (X,Px);if py is less than zero we print an o. The motion exhibited by a trajectory is then classified according to the appearance of the surface of section. If this shows smooth curves, the trajectory is quasiperiodic (regular) [8,12]. If a random "shotgun" pattern is obtained, the trajectory is non-quasiperiodic (irregular). If the pattern is obviously structured but not sufficiently smooth to be designated quasiperiodic, we label it as "uncertain". Fifty trajectories were run for each energy.

3. Results and discussion In fig. 1 we plot the fraction of the initial phase space which gives rise to quasiperiodic motion as a function of total energy. The two sampling methods give essentially the same results. We note that the regular region persists even above the unimolecular dissociation threshold D. There is thus no energy Ec, where a sudden transition from regular to irregular motion occurs; rather, a large range of energies from ~0.5 D to 2 - 3 D where the fraction of quasiperiodic trajectories remains non-zero. Quasiperiodic trajectories above the dissociation energy have been seen in a 3-D trajectory study on the HCC system [14]. In fig. 2 we present a selection of Poincar~ surfaces of section at several energies. These are intended to be representative for the cases (a) quasiperiodic, (b) nonquasiperiodic, and (c) dissociative behaviour.

1 August 1981

(a)

0.8 0.6 0.4 z 0.2~ o

c~ LL

(b) 0.8 0.6 0.~ 0.2 0.0

0

1

2

3

EID

Fig. 1. Fraction of trajectories displaying quasiperiodic (V)and quasiperiodic and uncertain (a) motion as a function of the total energy E.(a) Method I, (b) method II sampling of the initim phase space points. We observe several types of trajectory from the surfaces of section in fig. 2. Firstly, quasiperiodic trajectories occupy relatively little Of the (X,Px) plane energetically accessible to them. Of interest is the trajectory shown for E ---2D. Several of the quasiperiodic trajectories at high energies were of this type, having a highly structured SOS, typical of classical resonances [7,15]. Secondly, we note that the non-quasiperiodic trajectories (fig. 2b) are by no means "random" or "ergodic ~ since they fail to fill the (X,px) plane. Particularly, they fail to penetrate that region of the (X,Px) plane occupied by the quasiperiodic trajectories. Finally the dissociative trajectories (fig. 2c) also avoid the quasiperiodic region at low energies, but at high energies they seem to be fairly efficient at filling the energetically accessible space. (By making the distinction between non-quasiperiodic and dissociative we do not intend to imply that non-quasiperiodic trajectories, if run for a longer time, will never dissociate, although that possibility does exist.) The above observations are summarised in fig. 3, 509

Volume 81, number 3

CHEMICAL PHYSICS LETTERS

(c0

1 August 1981

(c)

(b) &T

E/D 0.5 7

se

a'.ee

~'.se

,¢.ee

s',eo

'a'.so

X3',Be

q'.ee

xS'.se

qLee

,,h

0.875 ",4

->1

';~' Be

a'.ee

xS'.se

~.',ee ~

&-

1.0625 7"

- - , . 5$

~ gO

,.

e

',

--

p

x

2.0

*

Ii

i

11

°

% *, ~

.

.

.

.

.se

°, t x

•

x,~,

,®

°

.se

a ~e

a eo

x a so

ee

~.s~

x

Fig. 2. Typical Poincar~ surfaces of section for various energies E. The bounding contour is the energetic limit. (a) Quasiperiodic, (b) non-quasiperiodic, (c) dissociation motion. × denotespy > 0; o denotespy < 0.

where we give a composite picture of the surfaces of section at various energies by/superimposing all the plots produced at that energy. At E = 0.5D we see two tori, one manifesting itself at the centre of the plane, the other near the energy limit. Around both we have trajectories which are non-quasiperiodic, but remain 510

fairly localised around the torus. By E = 0.875D the latter torus has disappeared, except for this residual structure, which is also appearing around the central torus. Non-quasiperiodic trajectories populate the area between. As the energy is raised above D, there remains no vestige of this torus, the energy limit having opened up to admit dissociation.

E/D

0.5

0.875

1.0625

2.0 •

o',,''~

4-" ,,,

%,, ~,'

-"------

~

+" +. • -t" ~

eo~ ,

"- ~ ,~

.

'~

e',

,;':

6~4,', ~., .' ~,°:;.*.

+ t

~.

+ 4-

.~o.L.,, h, 2"..,',,~'÷ ~ ..... ,.:+ ~+ i~.o+

o ,

~,

,.....~.

Fig. 3. Composite surfaces of section for various energies E. S m o o t h curves denote quasiperiodic m o t i o n , o denotes nonquasiperiodic, b u t highly localised motion. A dot denotes non-quasiperiodie, non-localised m o t i o n . + denotes dissociative motion.

fn~ ,,,n~

.2" X~ i

?.

3.8e

3.58

q.e8

q.58

S.ee

6.00

5.50

x

(b) o;-

BR

o 6 H

X~ CL~ i

~xox~@ Oo~xx x~°xxo~~ ~ x,~ ~ 0~ ~0x ~xx

3 ~. 0~

k

3.58

qt. eO

×

t4~.5e

5 ~. 0~

7

5 . bO

0~

I

6.00

Fig. 4. Composite surfaces of section for scattering trajectories at E = 2D. (a) Vibrational energy = 0.725D, translational energy = 0 . 8 5 0 D ; (b) vibrational energy = 1.275D, translational energy = 0.300D. Each plot is composed of surfaces of section for 41 trajectories with systematic variation of the phase angle of the H~ oscillator from 0 to 2~T. Symbols are as for fig. 2.

Volume 81, number 3

CHEMICAL PHYSICS LETTERS

For E > D , and indeed for all we calculated, the nonquasiperiodic trajectories avoid the area occupied by the torus. The dissociative trajectories, however, fill the space increasingly densely as the energy is raised, finally filling the whole plane by E = 2D. For energies below this, however, even the dissociative trajectories are not ergodic. Since only these trajectories can connect with asymptotic scattering states, this would seem to imply (by time-reversal) that there are areas of (x, Px) exclusively occupied by quasiperiodic motion inaccessible to scattering trajectories. This was also proposed recently by Hase and Wolf [16]. To test this hypothesis we have run several batches of scattering trajectories with constant energy but varying its vibrational and translational components. For energies of the order of those used in the bound-state calculations (specifically E = 2D), we see from fig. 4 that indeed there are areas of (X,Px) inaccessible from the scattering region.

4. Conclusions In this work we have examined the classical behaviour of a weakly bound complex, HeH~2. The results contrast with previous calculations using simple model hamiltonians in that there is no evidence of an abrupt change from regular to irregular behaviour: the change occurs gradually (and smoothly) up to energies three times the dissociation energy. The onset of chaos in model systems has been investigated using critical-point analysis of the potential energy surface [9,17]. In this regard, the surface used here is different in one major feature. The disparity in the HH and Hell force constants and bond energies makes the potential extremely asymmetric. This asymmetry should show up in a critical-point analysis, which we do not attempt here. Moreover, this behaviour is a sensitive function of the global form of the potential. Calculations on a different HeH~ potential showed the more usual abrupt regular-chaotic transition [ 18]. It is noted that the non-quasiperiodic, and even the

512

1 August 1981

dissociative trajectories at low energies are decidedly non-ergodic. By probing the (X,Px)plane from the asymptotic scattering region, we confirmed that, for sufficiently low energy, the quasi-periodic and scattering-accessible regions of the (X,px) plane are disjoint. It will be interesting to see whether such behaviour carries over to a 3-D treatment of a weakly bound complex. Work is in progress on this [19]. If such behaviour does occur in 3-D, we should expect the system to display non-RRKM behaviour.

References [1] S.A. Rice, in: Excited states, Vol. 2, ed. E.C. Lim (Academic Press, New York, 1975) p. 111. [2] M.J. Berry, Ann. Rev. Phys. Chem. 26 (1975) 259. [31 J.D. McDonald, Ann. Rev. Phys. Chem. 30 (1979) 29. [4] P.J. Robinson and K.A. Holbrook, Unimolecular reactions (Wiley, New York, 1972). [5] W.L. Hase, in: Dynamics of molecular collisions, Part B, ed. W.H. Miller (Plenum Press, New York, 1976) p. 121. [6] W.J. Chesnavich and M.T. Bowers, in: Gas phase ion chemistry, Vol. 1, ed. M.T. Bowers (Academic Press, New York, 1979) p. 119. [7] I.C. Percival, Advan. Chem. Phys. 36 (1977) 1. [8] M. Henon and C. Heiles, Astron. J. 69 (1964) 73. [9] P. Brumer, Intramolecular Energy Transfer: Theories for the Onset of Statistical Behaviour, Advan. Chem. Phys. (1981), to be published, and references therein. [10] P.J. Kuntz, Chem. Phys. Letters 16 (1972) 581. [11] F.M. Chapman and E.F. Hayes, J. Chem. Phys. 62 (1975) 4400. [12] D.W. Noid and R.A. Marcus, J. Chem. Phys. 67 (1977) 559. [13] D.L. Bunker, J. Chem. Phys. 37 (1962) 393. [14] R.J. Wolf and W.L. Hase, J. Chem. Phys. 73 (1980) 3779. [151 D.W. Noid and R.A. Marcus, J. Chem. Phys. 67 (1977) 559. [16] W.L. Hase and R.J. Wolf, in: Potential energy surfaces and dynamics calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 37. [17] C. Cerjan and W.P. Reinhardt, J. Chem. Phys. 71 (1979) 1819. [18] H.R. Mayne, Ph.D. Thesis, University of Manchester (1977). [19] R.J. Wolf and H.R. Mayne, in preparation.