On the spectra of highly vibrationally excited polyatomic molecules

Chemical Physics33 (1978) 35-43

0 North-HollandPublishing Company

ON THE SPkTRA

OF HIGHLY VIBRATIONALLY EXCITED POLYATOMIC MOLECULES *

Klaus Dieter HANSEL

.mr

Projektgtuppe jiir Laserforschzuzgder Mar-Plaack-Gesellschaft Fiirdenuzgder Wissenschaftene. V.. D-8046 Garchingnear Munich, Gertnany Received 8 March 1978

The spectra of highly vibrationally excited molecules are computed from the dipole-dipole correlation function using classical mechanics. The trajectories show a transition from quasiperiodic to stochastic behaviour; in the stochastic redon they are characterised by a stochasticity parameter. This parameter may be correlated with the broadening of the spectral lies. The implications of our calculations on the theoretical description of highly v&rationally excited molecules, particularly on m&photon dissociation, are discussed.

1. Introduction

difference in the dynamics between regions (1) and (2). [Region (3) belongs to the realm of classical scattering

Recent experimental developments have provided practical means of studying the spectral behaviour of highly excited molecules. These studies range from spectroscopic studies [l] to multiphoton dissociation experiments [2] to practical schemes of laser isotope

theory, and is not considered here.] In region (1) the motion of the system is qunsiperiodic. The classical

separation (see e.g. ref. [3]). The interpretation of these experimental results is often controversial, particularly in the case of multiphoton dissociation experiments [4-71. The first step to understanding these results is having a plausible level scheme, and information on the spectral behaviour of polyatomic molecules. Whereas quantum calculations for the highly vibrationally excited states are difficult, to say the least [8,9] , this

problem should lend itself to a classical treatment because of the high level density involved. This way, ambiguities arising from poor basis sets may be avoided. A few generalities about the dynamics of classical systems may be in order before proceeding to the details of the problem at hand [lo-121 - Basically $ree

energy regions may be distinguished in molecular systems [l l] : (1) The low energy regime where the energy levels are sparse; (2) a region of high level density, and (3) the dissociative continuum. There may be also a * This work was supported by the Bundesministerium fiir Forschung und Techaoiogie and Euratom.

mechanics may be described by low order perturbation theory [I 1,13,14]. In the high energy region the motion may be stoclzastic, i.e. for sufficiently large time intervals At the trajectories (Q(r), P(t)), (Q,P(t + At)) are uncorrelated. It should be emphasized that the distinction between the quasiperiodic and ergodic regime is not sharp: At low energies microscopically small regions of instability exist. Conversely, at high energies a small fraction of phase space is still occupied by quasiperiodic trajectories (for a detailed discussion see refs. [lo, 111). Nevertheless the transition from predominantly quasiperiodic to predominantlv stochastic behaviour can be defined [ZS], and therefore the distinction between a quasiperiodic and a stochastic energy range makes sense. The transition to stochastic motion is assumed to be responsible for the almost unlimited ability of polyatomic molecules of absorbing IR photons, i.e. the dissociation process is thought to be brought about by stochastic heating [ 151. Numerical evidence has recently been furnished by Noid et al. [16]. In this paper the main concern is the spectral behaviour in the high energy region. In the next section a model hamiltonian will be given, and a parameter describing the “stochasticity” of the molecule in a sense

K.D. HZns&pectra

36

of highly vibrationaUy excited mo~ecu!es

which will be made precise presently. Section 3 will contain some numerical results; these will be discussed in relation to present and future experiments and to other theoretical work.

w;2 = [c&(1

+72) + 2c&o&(72

- 1) + o&]‘~2,

the following normai frequencies ~1, w2 and normal modes are obtained

2. The stochastic hehaviour of polyatomic molecules For highly excited molecules the linear (i.e. harmonic oscillator) approximation is inadequate. Consequently, turning to the next simple model a molecule is considered to be a collection of anharmonic coupled oscillators. Rotational-vibrational coupling is still neglected here; its role presumably is to enhance the stochastic instabilities to be described below (cf. ref. [17]). Hence in.terms of normal coordinates the model hamiltonian is given by

= (7”&

t A2)-1/2

(

9

where A=$ [(l -7’)w$,-w;D-w~2].

where the V,, describes the anharmonic part of the potentialCommon approximations to the molecular potential are the retention of the cubic and quartic terms [ 18,191. When the high energy part of the potential is of interest Morse potential or some combination of Morse.potentials is commonly assumed [lo] _ In the model calculations described below the potential was chosen to be

Recently, Casartelli et al. performed numerical investigations on the stability of classical trajectories [22] _They introduced a stochastic parameter given by W-‘,QhW-‘,Q);~l

=

lim r~-,A~o

li;

In

IfI i=l

Wi)jl lid, (ri_l)

(3)

II ’

n-m

llA&)i

= llA,(t,)ll=

_.. = IIAo(tn_,)ll

=d,

A0(to) = A P, Ql.

t higher order terms,

(3

= Wor2, D is the dissociation energy, and LIis a range parameter_ 7 is a scale factor related to the masses of the atoms taking part in the vibration [21];it is considered as a free parameter of the model here_ This potential models a triatomic linear molecule with one bond of dissociation energy D and a second bond which is much longer. The following set of molecular parameters was assumed: wru =4_401 X IO-3 aeu,wzo = 8 X 10m3 aeu, D = 8.8 X 10M2 aeu, 7 = OS [I atomic energy unit (aeu) = 27.21 eV] _q1 and q2 are not the normal coordinates corresponding to the harmonic part of the_model eq. (2) Defining w&

The meaning of the stochastic parameter is illustrated in fig. 1. At time r. two trajectories are initiated at the point (P, Q), and at a near by point (P + AP, Q + AQ) at a distance d = IlAP, AQ II and the trajectories are propagated over a time interval At to time tl at which they are separated by a distance llA(t,)ll = llA(P(r,), Q(t,))u. Then initial conditions P(tl) + AP(tl) d/Wtl)ll, Q(r,)

+ AQ(t,)d/llh,)ll,

are selected for the second trajectory and the whole process is repeated all over again. For sufficiently small time intervals and separations‘ d the stochasticity parameter may be shown to converge to a number independent of A0 (to) and t. More-

37

K.D. HZnselfSpectra of highIy vibrationaIly excited molecules

bit31

/

I(w) = j

dt exp(iwt)(p(t)t)(O))

,

where is the dipole-dipole correlation function. The average I...) has to be taken over an appropriate ensemble (a microcanonical ensemble in the case of ergodic motion [ 161). To lowest order in the molecular coordinates the dipole-dipole correlation functiqn may be replaced by a sum over coordinatecoordinate correlation functions: Writing

one

Fig. 1. Defmition of the stochasticity parameter k. At times tl ._.f,r_l the initial conditions of the second trajectory are readjusted. At times tt. fs. _.. the divergence is observed.

over, -if the trajectory is in the stochastic energy range k depends on the total energy only. The parameter k is essentially a globai measure of local exponential instability. This is strictly true only if the dynamical system under consideration is ergodic (for a rigorous statement of the mathematics cf. refs. [22,31]). For the model eq. (2) it is not known whether at stifficiently high energies, the assumption of ergodicity is strictly true. The results of section 3, however, suggest it is. Proceeding under this assumption k is an invadant of the system, i.e. it is independent on the choice of coordinates, for example. In physical terms km1 has the dimension time, i.e. it is a characteristic time of the system. More precisely, k is the average rate with which two initially inftitesimally close trajectories separate. In other words it is characteristic of the memory of the system. I/k is to be compared with the vibrational periods Ti = 277/c+_If the two terms are of the same order of magnitude the phase of the vibration is subject to a random modulation; this will give rise to a broadening of the spectral lines [23,24]_ The spectral lineshapes may be directly extracted from the classical trajectories. According to linear response theory [ 1 l] the spectral density I(w) iS given by

has

and thus the electrical spectra may be expressed in terms of the mechanical spectra to lowest order. In this paper only the mechanical spectra are calculated.

3. Numerical computations

and results

Using the potential eq. (2) Hamilton’s equations were integrated numerically on an IBM 360/9 1 computer using the double precision version of the IMSL subroutine DREBS [29]. The relative error was required to be < 10-l”. The initial conditions were specified as follows: To zeroth order the system was assumed to consist of two independent harmonic oscillators_ Then the total energy E and the fraction of the energy in mode 1 and 2, E, , E2, was specified. The initial phases of the oscillators were selected by a random number generator. Then the total energy using the fi.111 hamiltonian was computed and the coordinates and momenta were scaled so that interacting system had the total energy specified initially (for details cf. ref. [30]). For every set of fixed initial conditions - total energy and fraction of energy in the different niodes about 12 trajectoiies were calculated. Each trajectory was propagated over a time inter&l At = 3.4 X IO5 (aeu)-l = 53 ps corresponding to 220 vibrational periods of the slower (Q1) mode. Such long integration

38

K-D- HrSnseljSpectra ofhighlyvibrationdlyexcitedmolecules

times were necessary to insure convergence of the spectra to about 10%. Concurrently the stochasticity parameter of eq. (3) was computed. The system under consideration having two degrees of freedom an addition characterisation of quasiperiodic versus stochastiG behaviour was possible: For every trajectory surfaces of section (e.g. the intersection of a trajectory with the plain PI = 0) were com-

puted (for a fuller definition cf. refs. [IO, 11,221). For quasiperiodic traject&ies a curve is traced out by the trajectory in the plain PI = 0.This reflects the fact that there are two constants of motion, i.e. the system is similar to a system of harmonic oscillators, and quasiperiodic motion obtains (fig. 2a). When the energy is increased a single trajectory may fill the whole energy surface; in this case the shotgun pattern of fig. 2b results.. The central point of the paper is *e application of the stochastic&! parameter to $e classical theory of vibrational spectra. It was computed as described in ref. [22] and in the preceding section. The following method was used to generate an initial separation

A(@. The direction of A was obtained by selecting random points on the edges of a four-dimensional hypercube centred around (P(O), Q(O)). The length of A(&-,) was chosen as a fraction of the length of the vector (P(O), Q(O)). d= tlA(@

-20

.

I -f

!

‘--

-WS..~~,~~..,~~~‘,“‘.“...L’~“’ -lJ.IS -0.05 0.05

0.15

Pl 120

7 J

.-.

*.=

100 -

eo 60 -

40 -

=fll(P(O),

Q(O)ll _

Convergence to about two significant figures was obtained by letting O-01 >f> 10d4_ The dependence on the step size of the integration routine was also checked. The choice A = 0.1 X 2drnin(ol, w2) proved to be satisfactory. Having verified the validity of our procedure production runs were performed. k was calculated for an energy range 6.2 X 10e3 .GE < 8.8 X 10m2 aeu. As

stated in the preceding paragraph trajectories were run for several initial phases at each energy/energy partitioning. At low energies k depends very strongly on the initial conditions: Typically, at E = 6.2 X 10e3 aeu k was found to vary between 4.54 X 10m5 and 1.20 X lOA aeu. At high energies the dependence on the initial conditions seems to disappear. At E = X.54 X 10e2 aeu one finds 6-l X lOA
0' 20 1 0' “ -20

A summary of the energy dependence of the stochasticity parameter is given in fig. 3. A few words about the method of presentation may be in order. The full circles represent average values of k according to

-a0 I

Fig_2. Surfaces of section for the model potential eq. (2) (a) in the quasiperiodicenergy region (E = 6.2 x lop3 aeu); (b) in the stochastic region (E = 4.5 x IO-* aeu).

where (P(i) (0), Q(i) (0)) denotes the ith set of initial conditions belonging to a particular batch of trajectories. E(E, {Ei}) is a microcanonical phase space aver-

K.D. HCmel/Spectraof highly vibrationally excited wolecu~es lO’klE1

+[a.~.3

Fig. 3. Energy dependence of the stochastic parameterk(E). age, it may be interpreted as the characteristic rate of decay of the correlations in that part of phase sampled by the trajectories belonging to a particular set (E, {Ei}i= I,&- The error bars given in fig. 2 are the RMS deviations of the stochasticity numbers E_If the motion were truly ergodic the stochasticity numbers wouId depend on the total energy ordy: k((P(‘)(O), Q(‘)(O)); A; f) = k(E). This seems to hold true in the high energy region, here the individual values of k agree to about 10%. Examination of fig. 3 shows that two regions are clearly discernible. At energies below E -2.5 X 10m2 aeu E(J?) is almost constant and fairly small (5 1 X 10m4 aeu). At E Z=3 X 10M2 aeu Estarts to increase linearly. This behavi0ur persists up to the dissociation limit. Thus there is evidence for some kind of transition, possibly from quasiperiodic to ergodic behaviour. In fact k = 0 is expected for quasipedodic trajectories; however E takes on a small but nonzero value indicating that the motion still contains an ergodic component_ Fig. 2 suggests a possible definition of the transition from predominantly quasiperiodic to predominantly ergodic behaviour of the dynamics. Assuming the linearly increasing part of the curve belongs to the ergo& regime, the onset of predominantly ergodic behaviour might be defined as the intersection of that straight line with the abscissa. This yields a critical energy ofEO x 1.7 X 10e2 aeu. The (mechanical) spectra of the system were obtained as the Fourier transform of the classical trajectories as discussed in detail by Noid et al. [16]_ The spectra at different energies are displayed in fig. 4. It

39

is seen that the linewidth increases as the total energ increases. Fig. 4a corresponds to a molecule with zero point energy: here the spectrum has the structure fam&r from vibrational spectroscopy, two strong fundamentals, and a number of weak combination frequencies and overtones. All the spectral lines are sharp. At E=2..5 X IO-’ aeu (fig. 4b) line broadening is apparent. There is still a lot of structure in the spectra, however. At E = 6.0 X 10W7aeu the Iines have become very broad. To a first approximation the spectra may be considered to consist of two broadened lines although intensity variations in the wings are indicative of some additional~structure. Finally at E = 8.0 X lo-’ a broad continuum absorption extending from w = 0 to w = 9 X IO-3 aeu is observed. Variations in the intensity still occur and peaks belonging to the different normal modes are clearly distinguishable (fig. 4d). For the mode associated with Q1 a substantial red shift accompanies the increase of the width; this is to be expected for a Morse potential. Thus, fig. 4 demonstrates that there is a intramolecular line broadening mechanism due to energy exchange between different modes. Comparison of figs. 3 and 4 shows that the linewidth of the spectra is correlated to the stochasticity parameter I;. This is to be expected; Eis a measure of how long the system remembers its initial conditions on the average. Similarly, the linewidth measures the rate of decay of aparticular state. It is intuitively clear that the two quantities should be correlated. An analysis of the relationship between zand the linewidth may be performed on the basis of the classical Liouvillc equation [23,28] and is in progress [25]. Nevertheless it may be stated that E is a measure of the linewidth. This is the main result of the paper. Returning to the model considered in this paper it is interesting to consider the time scales involved: First there is the memory of the system. It may be characterized by the time l/is= 1.5 X 10~~ (aen)-l_ This is the time scale on which details of the dynamics are important. If one envisages a statistical mechanical description of a system (e.g. in terms of a master equation) another time is important, namely the recurrence time, T,. It is given by the density of states, or = p(E). This is the time the system needs in order to return to a particular set of initial conditions. For the present model

40

K.D. &insel/Specrraof h&hly vibrationallyexcited molecules

160

25-r

1

160

20-

140 120

IS-

i

I

100

G Y

G

60

IO

4

40 20

3!_-_

3

la1

1

60

5

1

1 ,...‘.““I”“1

0.015

305 eoa

-?I.005

0.005

0.015

12

7oa 101 MO 30

500

360 V 300 40 200 20 LOO 0 -0 300

9:

-.

-005

0.015

0 -0 7c

60

50

_

200

40 ‘s

IS0

3 v 0-a30

v l-4

(cl

20 50

0

-0

IO

1,+.. , . .oos

.. 0.015

-"o,

‘

0.015

41

K.D. HFnselfSpecttaof highly vibtationaliy excited molecules

1

0.015 0

[a.e.u.l

o 1a.e.u.J

Fig. 4. Absorptionspectrafor the modelpotential,eq. (2). for mode 1 (left) and mode 2 (right) at several energies: (a) E = 6.6 X lo-* aeu: (bl E = 3.5 X lo-* aeu; (c)E = 6.0 X lo-” aeu; (d) E.= 8.0 X lop2 aeu. (Molecularparameters:D = 8.8 x 10W2aeu, W, = 4_4bi x 10e3 aeu, w1 = 8 X 10m3aeu,r, = 0.5.)

eq. (7) is a rough estimate yielding 7r x 5 X lo3 (aeu)-’ at E=8.4 X 10m2(cf. refs. [8,27]).Thus, the present system just falls short of fulfiiing the condition for the applicability of statistical mechanics l/E< f err

[27]. However, even for a system with four atoms this inequality may be expected to hold assuining potential parameters and energies of the same order of magnitude as in this paper.

4. Discussion’

Two main conclusions may be drawn from our computer study of the stochastic behaviour of small molecules. (1) The existence of an intramolecular line broadening mechanism due to energy exchange between the normal modes has been demonstrated. It has been shown to be related to the transition from quasiperiodic to stochastic motion. (2) The characteristic times extracted from the model studied in this paper suggest that molecules consisting.of four or more atoms may well be described in terms of statistical mechanics in the high energy region. The first conclusion is of some experimental consequence whereas the second point bears on the theoretical description of molecular rate processes.

Let us discuss point (1) first. Computer studies performed by Noid et al. [16] and in this laboratory suggest that there is an intramolecular line broadening mechanism which should be observable in highly vibrationally excited molecules; The question whether one is dealing with line broadening rather than splitting of spectral lines is partly a question of resolution just as it is a question of “numerical resolution” in the computer study. To our knowledge the internal line broadening mechanism discussed in this paper has never been identified in any experiments though its existence has been pointed out before by Pomphrey and Percival [9]. This is probably due to the considerable experimental difficulties which are anticipated for an experimental study of this line broadening mechanism. First there is the question of generating highly vibrationally excited molecules; this may be done with high power lasers. Secondly there is the question of detecting and identifying the intramolecular stochastic line broadening; or broadening mechanisms, e.g. predissociation will compete with it; inhomogeneous line broadening will be an additional complication. Three observations may be he1pti.d: (1) there is a threshold for the onset of stochasticity (and hence line broadening). (2) In the stochastic &&me the spectra should be independent of the way in which the molecule has been excited. (3) For sufficiently energetic molecules the spectral lines

42

K.D. H&net/Spectra

of bigMy vibrationally excited molecules

may become very broad (several 1O-3 aeu in the case considered here). This suggests looking for an increase in absorptivity in a spectral region in the IRwhere the molecule

does not absorb radiation in the ground state.

From a theoretical

point of view we have been able to go beyond a numerical demonstration of line broadening. The system (i.e. the molecule) was characterized by its total energy E and by a stochasticity number k which describes the memory of the system. This number is essentially the so-called K-entropy, an invariant of an ergodic system [7_2]_Qualitativeiy the linewidtli increases as the stochasticity parameter increases. More quantitative conclusions may be drawn on the basis of the stochastic theory of lineshapes [23] : Work along these lines is in progress_ The inverse of the stochastic@ parameter kQ is a lower limit to the time scale on which the system can be described in terms of a statistIca theory the upper limit being given by the recurrence time. It was found that l/k(E) in the ergodie region is approximately equal to a vibrational period of the system. This fact is particularly significant with respect to muftiphoton, dissociation. Introducing an extra degree of freedom for the electromagnetic field the formulations of this paper may be applied to the excitation process (cf. ref. [El). Assuming a stochastic transition takes place again two regimes will ensue: a quasiperiodic energy exchange between the molecule and the fietd (this corresponds to the Rabi oscillations), and a regime where stochastic heating takes place. Again this type of behaviour was observed in computer studies of multiphoton dissociation [26] _

References

[ 11 P.F. Moulton, D-M. Larsen. J.N. Walpole and A. Mooradian, Opt. Letters I (1977) 51; W. Fuss. J. Hartmann and W-E. Schmid. preprint. 1977. [2] F. Brunner. T.P. Cotter; K.L. Kompa and D. Proch, I. Chem. Phys. 67 (1977) 1547; R.V. Ambartzumian and VS. Letokhov, Accounts Chem. Res. 10 (1977) 61; M-J. Coggiola, P-A. Schulz, Y.T. Lee and Y.R. Shen, Phys. Rev. Letters 38 (1977) 17; P. Kolodner, C. Winterfeld and E. Yablonovitch, Opt. Commun. 20 (1977) 119. [3] J.P. Aldridge, J.H. Birely, CD. Cantrell and D.C. Cartwright,

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[4] N- Bloembergen,

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K. D. HEnseIfSpectra of highly vibrationally

[30] C.S. Sloane and W.L. Hase, J. Chem. whys. 66 (1977)

[27] G.L. Hofacker, J. Chem. Phys. 43 (196.5) S208; G.L. Hofacker, in: Chemische Elementarprozesse, H. Hartmann (Springer, Berlin, 1968). [28] K.C. MO. Physica 57 (1972) 445. [29] R. Bulirsch, J. Stoer, Num. Math. 8 (1966) 1.

43

excited molecules

ed.

1523. [31] M. Smorodinsky, Ergodic theory, entropy, Math., Vol. 214 (Springer, Berlin, 1971).

Lect. Not.