Classical chaos and quantum simplicity: Highly

Classical chaos and quantum simplicity: Highly excited vibrational states of HCNS ) Kevin K. Lehmann, George J. Scherer,b) and William Klemperer Department o/Chemistry, Harvard University, Cambridge. Massachusetts 02138

(Received 14 May 1982; accepted 2 June 1982) Direct overtone spectra of H 12C 14N, H l3C 14N, and H 12C 15N have been measured between 15000 and 18 500 em-I with a precision of 0.001 em-I. These were obtained using intracavity photoacoustic spectroscopy, with a fully automated laser system. The spectra are unperturbed. The transition energies and rotational constants are in good agreement with predictions of first order anharmonic constants. Classical trajectories for HCN have been computed on the best experimentally parameterized potential, and found to be stochastic 12990 em-I above the ground state. Quantal density of states were computed for HCN and show that if extensive vibrational coupling occurs, the observed states would be highly perturbed. The simplicity of the observed states is shown to be expected given a Franck-Condon type limitation on significantly perturbing states. The results show the inapplicability of classical dynamics for predicting the dynamics of molecular vibrations.

The detailed study of highly excited vibrational states of polyatomic molecules is likely to be relevant to such subjects as RRKM theory and its limitations, multiphoton dissociation, internal conversion and intersystem crossing, and the possible industrial use of lasers to drive chemical processes in a state specific way. To explore highly excited vibrational states, we chose to look at direct overtone transitions in hydrogen containing molecules. Since this approach was the only way to study vibrational absorption spectra before the midIR region became experimentally accessible, work in this field has an extensive early history. 1 After the opening of the mid IR, attention shifted away from the photographic IR. With the advent of tunable dye lasers, the field has attracted renewed attention. One result of the more recent work, which has almost all been done at low resolution, has been the rediscovery of Mecke's local mode description of molecular vibrations. 2 This has been justified by the linear Birge-Sponer plots of hydrogen stretching overtones, and the appearance of a few strong bands compared with the great number predicted by a normal mode analysis. s A second conclusion has been that for most polyatomics, these excited local modes decay irreversibly into a quasi continuum of vibrational levels, with a lifetime of only - 50 fs. 4 This has been justified by the very wide Lorentzian line shapes (-100 cm -1) observed for many polyatomics. These ideas are not obvious from the usual normal mode, slightly anharmonic oscillator picture of excited levels of polyatomic molecules. We looked at small polyatomics with high resolution so that individual rotational lines could be observed, and the complexity of inhomogeneous broadening reduced. To observe these highly forbidden transitions, we used the technique of intracavity photoacoustic spectroscopy.5 To make it practical to take high resolution spectra of large numbers of bands, we constructed a fUlly automated laser spectrometer. The success of the automation is a testimonial to the power of small

computers in a laboratory. The details of this system will be given below. We have studied several small polyatomics with this apparatus. This paper discusses the results of studies on H 1ZC 14N, H 13C 14N, H 12C 15N. 6 We measured the isotopically substituted species because isotope shifts are sensitive to the character of the vibrational wave function. We have extensive data, soon to be reported, on 12C2H2, 12C 1sCH2, 13CZ Hz, and 12C2HD. High resolution spectra have already been taken of HN s and CDsH. The high resolution study of molecules of increasing complexity, starting with HCN, the simplest polyatomic Vibrator, will provide valuable data by which different theories of highly excited molecules can be tested. This paper contains a careful comparison of our results for HCN with those predicted by classical dynamics. EXPERIMENTAL

Acoustic cell Our photoacoustic cell is shown schematically in Fig. 1. The acoustic detector is a Knowles 1834 electret microphone, which contains an internal FET amplifier. We epoxied the microphone to a cutout in the middle of

RESONANT

PHOTO ACOUSTIC CELL

KNOWLES MICROPHONE RECTANGULAR WAVEGUIDE I/S" x 1/4"x 7"

T

3"- -

1 1 - - - - - - - 12·7.::.."-------1

alWork supported by National Science Foundation. blNSF Predoctoral Fellow. J. Chern. Phys. 77(6). 15 Sept. 1982

FIG. 1. Longitudinal resonant photoacoustic cell. air, f r =900 Hz, Q = 10.

0021·9606/82/182853-09$02.10

For 1 atm

© 1982 American Institute of Physics

2853

Downloaded 03 Apr 2004 to 128.112.81.10. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Lehmann, Scherer, and Klemperer: Highly excited HeN

2854

Ar'

DYE LASER CAVITY

_ _~__ LASER

t-~

LASER

POWER

OUTPUT MIRROR PIEZO PHOTOACOUSTIC DETECTOR

THIN ETALON FSR - 900 GHZ TUNED BY GALVO SCANNER

AID FABRY PEROT

E TALON FSR- 75 GHZ PIEZO TUNED

MICROPHONE

FIG. 2. Schematic of modified Spectra Physics 580A dye laser.

a section of copper waveguide, which has cross section i. d. 2.2 mm x 5 mm, and is 18 cm long. This waveguide is held in a larger cell and acts as an open acoustic resonator. 7 The lowest frequency mode has a pressure node at each end and a maximum in the center of the waveguide. For one atmosphere of air, the cell has a resonant frequency of - 900 Hz., and a Q of -10. This design gives a very small active volume/unit length, ensuring high sensitivity since the signal is proportional to l/V. Background due to window absorption is greatly reduced, because of the small amplitude of the resonant mode outside the waveguide. Calibration with CH 4 gave a background equivalent absorption of -1. 6 x 10-8 cm-t, with a rms noise of 3 x 10-10 cm -1. In contrast to radial resonant cells 8 with Q- 200, the resonant frequency is easily calculated with sufficient accuracy, and ambient temperature changes do not detune the acoustic resonance. Dye laser

In our experiments we use a cw dye laser (Spectra Physics model 580A), pumped by an Ar+ ion laser. A schematic of the laser is shown in Fig. 2. We have extended the cavity apprOXimately 35 cm to give room for

PMT

o

1 FLUORESCENCE CELL 2

P HOTODIODE

SPECTRUM ANALYZER

POWER METER

S-P 580A CW DYE LASER MODIFIED

Ar+ LASER

FIG. 3. Schematic of optica used for control and frequency calibration of laser.

D/A THIN ETALON

PULSE BIREFRINGENT LINES I---~ FIL TER

FIG. 4. Schematic of measurement and control electronics.

the acoustic cell inside. We use a flat output mirror, but move the fold mirror closer to the dye jet to stay centered in the stability range of the cavity. 9 Three tuning elements are used to control the laser frequency. A homebuilt birefringent filter, 10 with specifications given by Preuss, 11 reduces the laser linewidth to -40 GHz; the filter is tuned by rotating it with a stepping motor 12 through a wormgear. A solid etalon, with a free spectral range of 900 GHz, reduces the laser linewidth to -18 GHz. It is mounted on a galvoscanner to allow electronic adjustment of its angle. An analog squareroot circuit is used to make the frequency shift linear with applied voltage. An air spaced etalon, with a 75 GHz free spectral range, forces the laser to oscillate on a single longitudinal cavity mode. It is scanned by applying a voltage to a PZT spacer. The output coupler is mounted on a PZT, which is used to shift the cavity modes. The cavity modes are spaced by - O. 2 GHz (0.006 cm -1). The remaining optics are shown in Fig. 3. An invarspaced, plane-parallel, 2 GHz etalon, sits in a hermetically sealed thermally insulated housing. Since completing our HCN work, the invar etalon has been replaced by a cervit etalon. Cervit has a thermal expansion coefficient 100 times smaller than invar. 13 The transmission of light through the etalon is monitored with a photodiode, providing frequency markers. Iodine fluorescence is detected with a red sensitive PMT (RCA # 4832) prOViding absolute frequency calibration. The laser power is monitored by a photodiode for power normalization and control of the laser. A scanning etalon is used for visual monitoring of the laser mode structure.

Data aquisition and control Figure 4 contains a schematic of the electronics of the experiment. A reference signal modulates the laser

J. Chern. Phys., Vol. 77, No.6, 15 September 1982



2855

frequency at the resonant frequency of the acoustic cell, by means of a voltage on the air spaced etalon. To average out the effect of the discrete caVity modes, the output coupler is moved over several modes at -100 Hz. The signals from the microphone, the photodiode monitoring the etalon transmission, and 12 fluorescence are all demodulated by lock-in detectors, normalized to the laser power by an analog divider, and read by A/D converters. The laser power is also read by an A/D. A Commodore Pet microcomputer controls the laser through a home-built interface. 14 The average laser frequency is scanned 100 GHz by applying a ramp voltage to the two etalons with a 12 Bit D/A. The data set is recorded in memory, and printed out in real time on an IDS 440 graphics printer. At the end of a scan, the computer stores the data on a minifloppy disk. Using a second D/A, the solid etalon is moved to the next order of the 75 GHz etalon, and its position is adjusted to maximize laser power. Repeating this procedure eventually moves the solid etalon more than one of its free spectral ranges (900 GHz). When this happens the computer jumps it back to its zeroth order. The high linearity of the galvoscanner control allows the 900 GHz etalon to be reset one whole order to an accuracy much better than needed to have its transmission peak centered on the correct order of the 75 GHz etalon. Finally, the birefringent filter is advanced to maximize power, and then advanced half a scan further. The computer is now ready to start the next scan. With a channel size of 0.05 GHz, it takes about 15 min for a 100 GHz scan, the rate being limited by the speed of our graphics printer. In this way, the computer has piecewise continuously tuned the laser over several hundred A, with no manual adjustments to the laser. The only human intervention reqUired to scan for several days is to replace the minifloppy disks once every twelve hours.

Data analysis Since we use frequency modulation, and lock-in detection, the lines have a derivative shape. This is advantageous for two reasons. The first reason is that it does not matter lf a line goes off scale; its zero crossing is still on scale. The second reason is that it is easy for the computer to fit a straight line near the zero crossing to find its position. Finding the peak of a line is more difficult. This is because peak search algorithms involve taking numerical derivatives, which is time consuming and amplifies the noise relative to the signal. We found that the 12 spectra gave much poorer fits when unmodulated spectra were taken. In addition, nonresonant absorption produces much smaller signals. In analyzing the spectra, the computer scans through the three data channels, and finds all zero crossings that occur after a signal has passed below a preset threshold. It then fits a straight line to the closest nine points to get a channel number for the crossing. Peaks of the 12 and acoustic spectra are then assigned etalon fringe numbers by linear interpolation between the nearest fringes. Closely spaced fringes are important as the PZT driven scans are qUite nonlinear. A pro-

FIG. 5. Bandhead of H!2d 4N (0,0,0)-(0,0,5) transition. Shown top to bottom: Etalon transmission peaks 0.05 cm-! apart, 12 fluorescence, acoustic spectrum. This figure has been enhanced by draftsman for clarity.

gram then looks at the 12 lines in the overlap between scans, and finds the relative numbering of the fringes that gives the most 12 lines of the same frequency in both si>ectra. The process is repeated to find the relative order numbers of the reference etalon peaks throughout the run, often more than a hundred scans. The observed I2 lines are compared with a computer file of the 12 atlas of Gerstenkorn and Luc. 15 Reported frequencies are corrected by - O. 0056 cm -1 as determined by a later recalibration of the atlas. 16 An 12 line on the first scan is identified from the atlas, and its frequency given to the computer. The computer now piecewise assigns all the lines in the atlas to the spectra, and does a least-squares fit, three spectra long, to get an absolute frequency for each scan. Absolute frequencies of the acoustic lines and their strength are then printed out to be labeled on the hard copy spectra. The computer also saves the line positions and intensities for use by automated assignment programs. The reproducibility of line positions, based on both repeated scans, and from combination differences, has a standard deviation of 0.001-0.002 cm-1. For comparision, the Doppler width of the observed lines is - O. 045 cm -1. The absolute frequency calibration of the 12 atlas has a residual uncertainty of 0.002 cm-1. An example of a spectrum taken with this system is shown in Fig. 5. Shown is the R branch of the transition to a state of five quanta of C-H stretch.

Samples Spectra of HCN were taken with 15 Torr in the cell. The l1atural isotope sample (Fumico) was used as obtained. The H 13C UN and H 12C 15N were made from isotopically substituted KCN (Merke and Sharp Co. ) by hydrolyzing with a mixture of H3 PO t and P 20 5• Residual H20 impurity lines were identified with the H20 lines listed in the Atmospheric Absorption Compendium. 17 EXPERIMENTAL RESULTS

The observed ~ states of the three isotopes are listed in Table I. The quantum numbers for the vibrations

J. Chem. Phys., Vol. 77, No.6, 15 September 1982



2856

TABLE I. Measured vibrational and rotational constants of

~

states of HCN and its isotopes.

HI2 C I4 N _ DJ3 cm-I >< 103

V o cm-I

005 303 502 204 105 006

Obs.

Calc.

Obs.

Calc.

15551. 9443(10) 15710.5326(7) 16640.3129(17) 16674,2107(6) 17550.3922(13) 18377.0063(14)

15253 15720 16649 16681 17547 18317

55,2184(45) 60,6573(79) 68,462(31) 63,4907(92) 64.103(13) 67.181(12)

52.1 61. 5 71.1 61. 8 62.1 62.5

B o =1.47822162 cm-I

Do =2. 9093(12) x 10-6 cm- I

AD cm- I x 106 Obs.

(J

- 0.1011(43)

- o. 046(16) + 0, 239(87) -0,327(25) -0.090(22) - 0.127(20)

x 103 cm- I 1.7 1.3 2,5 1.0 2.3 2.5

constrained in fit-microwave values

HI3 C 14 N 15455,1999(6) 15572. 7998(12) 16 544.77528(46) 17431,93503(65) 18257.2095(11)

005 303 204 105 006

51,1537(32) 57.456(22) 58,3883(48) 59.8674(64) 62.300(10)

15430 15581 16554 17428 18205

B o =1,439995 cm-I

48.2 58.2 58.1 58.0 57.9

Do =2.786(33) >< 10-6 em-I

- O. 1002(29)

- o. 046(71) -0.047(9) -0,099(12) -0.115(18)

1.3 1.8 0.8 1.2 1.9

constrained in fit-microwave values

HI2 C I5 N 15539.68327(50) 15615.63734(78) 15603.55778(55) 17509.87835(55) 18359.871 82(64)

005 303 204 105 006

53,0722(31) 58. 224l(84) 59.6113(47) 61. 7473(46) 64.5829(49)

15513 15625 16614 17507 18304

50. 1 59.0 59.4 59.8 60.1

- 0.1010(43) -0.035(16) - o. 0505(85) - o. 0739(86) - 0.1211(62)

1.1 1.4 1.1 1.1 1.3

Combining fitted B o, Do and constraining B o = 1.435248 gives Do = 2.766(13) x 10-6 cm- I • Errors in

2(J

from least-squares fit.

Calibration has systematic error of 0.002 em-I.

are: VI = C-N stretch, V2 = bend, and Vs = C-H stretch. 18 ApprOXimately 55 lines per band were fitted. 18 All uncertainties are two standard deviations in the last place, given by the least-squares fit. The standard deviations of the fits are listed in the last column, in units of 0.001 cm -1. For H 12C UN and H 13C UN, the lower state constants were constrained to the literature values. 19 For H 12.C 15 N, our determined Do was more precise than the literature value, so unconstrained fits were done. The observed Bo's and Do's were then used with their correlations to give a best Dl» with Bo constrained to its microwave value. 19 That fit gave Do =2.766(13)X10- 6 cm-I, with chi squared (4 degrees of TABLE II.

15452. 0928(23) 1,481606(91) 54.278(25) 2.76(15) -0,430(53) 7,460(68) 0,2888(80) 4.0

V o obs. DJ3 x 103 106 AD X 106 3 qOlO X 10 Aqx 103 (Jx103

DOlO X

V o cal DJ3 x 103

microwave

x 10 3 D OlO IRX10 6 qOlo

Errors in

The calculated Vo's and AB's are produced from the harmonic frequencies and the anharmonic constants XI/ s, a/ s listed by Nakagawa and Morino. 20 These

(0,1,5)-(0,1,0) HeN band (em-I),

BOlO

BOlO

freedom) equal to 2. 5. Table II contains the observed II-II sequence transitions. The quality of these fits is not as good due to partially resolved 1 doubling. It is important to emphasize that the very small standard deviations of the fits are due to measurement uncertainty, rather than to a failure of the Hamiltonian. This is demonstrated by the fact that combination differences for both upper and lower state give about the same standard deViation.

2(J

15428 52.1 1.481837 7.4877 2.9729

15358.6998(11) 1.443 155(38) 50.11 75(88) 2.858(41) - 0.099(13) 7.077(31) 0.3362(30) 2.3 15338 48.2 1. 440000 7.1662

-

15440.2616(14) 1.438 622(54) 51.928(11) 2.824(68) o. 065(19) 7.052(44) 0.3288(48) 2.9 15418 50.1 1. 435 249 7.06949

from least-squares fit.



Lehmann, Scherer, and Klemperer: Highly excited HCN TABLE m. Isotope shifts (em-I) of vibrational and rotational constants of HCN. H12C14N_H13CI4N State

Obs.

Cal.

-103 xAB Obs. Cal.

005 303 204 105 006 015

96 138 130 119 120 93

93 139 127 119 112 90

4.07 3.19 5.10 4.32 4.88 4.16

Vo

3.85 3.29 3.73 4.18 4.62 3.85

HI2et4N_HI2et~

005 303 204 105 006 015

12 95 71

40 17 12

10 95 67 40 13 10

2.15 2.44 3.88 2.36 2.60 2.35

1.99 2.43 2.42 2.40 2.39 1.99

were calculated from a quartic force field derived from a least-squares flt of perturbation theory predictions to observed infrared spectral data. The agreement of conventional anharmonic oscillator theory and experiment is quite good, especially since only the first order corrections are included. Table nI shows the predicted and observed isotopic shifts, and the agreement is even better. The only large discrepancy is for the (2, 0, 4) level of H 12C UN, which is in Fermi resonance with the (5, 0, 2) level. The mixture allows the (5, 0, 2) level to steal enough intensity to be seen, and causes the anomalous distortion constants of the interacting states. In all other cases, the intensity of the transitions qualitatively agrees with expectations with (0, 0, 5) » (0, 0, 6) > (1, 0, 5) > (2,0,4) > (3, 0, 3). The excited stretching states with more than six quanta of excitation are too weak to be observed, even though many are in the energy region studied. There is no evidence in the spectrum for any ~ states of excited bending character. With the exception of the Fermi resonance, the levels appear unperturbed. All levels, even the Fermi interacting pair, fit a distortable rotor Hamiltonian to within experimental error, which is only - 0. 001 em -I. The spectra are much simpler than most high resolution infrared spectra. CLASSICAL TRAJECTORY CALCULATIONS

In a recent communication, we described the results of classical trajectory calculations on HCN. 21 The purpose of the calculations was to find the energy at which the classical motion of HCN changed from quasiperiodic to stochastic. (See recent review articles. 22 • 23) For an N-dimensional coupled system, trajectories are quasiperiodic when their time evolution can be expanded in a convergent series of N fundamental frequencies and combination frequencies. These trajectories have the property that points initially infinitesimally close in phase space will diverge no faster than linearly. A quasiperiodic trajectory wUl stay confined to a compact N-dimensional subspace of 2N-dimensional phase space, an in-

2857

variant torus. A stochastic trajectory does not have a convergent Fourier series for its time evolution. Initially close stochastic trajectories diverge exponentially in time. A stochastic trajectory does not stay on a torus, but lies in a higher dimensional subspace. It is conjectured that for N;> 2 the trajectory is dense on the 2N-1 dimensional energy shell. This is referred to I as Arnold diffusion. The trajectory may even be ergodic, which means that the time average of any quantity is equal to the space average over the energy shell. For low excitation energies, almost all trajectories are quasiperiodic, but at some energy, a finite measure of the trajectories becomes stochastic. This energy is referred to as the stochastic transition energy. For an HCN potential, we used the best experimentally parameterized potential, given by Carter, MUls, and Murrell. 24 This potential includes the experimental information on the harmoniC force fields of HCN and HNC, the calculated barrier to rearrangement, 33 as well as the dissociation energies and fragment potentials. The anharmonic potential constants are in good agreement with force field calculations, and variational vibrational frequencies for the first few states are in close agreement With observations. In comparing results from this potential with experiment, the differences between it and the "true" Born-Oppenheimer potential must be addressed. Given the amount of experimental data incorporated, and the absence of any incorrect'potential minima or maxima, we believe it should do a good job of predicting the qualitative classical dynamics in different regions of phase space. If it cannot, then the use of computational techniques to predict the type of dynamics is probably useless for any real chemical system. We integrated classical trajectories with various amounts of energy initially in the C-H bond. The C-N bond and the bond angle were initially displaced to the limits of their zero point motion. We tested for the character of the motion by looking at the power spectra of the time evolution of the coordinates. 25 For quasiperiodic trajectories, the power spectra have three fundamental frequencies, and various overtones and combinations. For stochastic trajectories, the power spectra have a dense set of lines. Figure 6 shows the spectra at 12319 em -1 above the ground state. This trajectory is quasiperiodic. Figure 7 shows spectra of a trajectory 12990 em -I above the ground state. It is stochastic. Figures 8 and 9 show spectra of trajectories with excitation energies near the experimentally studied region. As resolution of the Fourier transformation increases a stochastic spectrum splits into more and more lines. All trajectories run above 12990 cm-1 were stochastic, even those with a linear geometry. All trajectories run below 12 319 em -I were quasiperiodic. It is possible that regions of quasiperiodic behavior persist above this energy, and/or that other regions behave stochastically at lower energy. Our search of phase space was limited by computational resources. Our choice of initial conditions reflects our assignment of the observed states to be predominently C-H stretching excitation.




2858

observed in this work, those quantum states must be far into the classical stochastic region, if the potential even qualitatively gives the correct dynamics. It is quite common for regions of quasiperiodic behavior to exist far above the stochastic transition energy. Such trajectories would reqUire changing the initial conditions, and would correspond to quantum states of different occupation numbers. We believe these are the first high resolution vibrational spectra of quantum mechanical states that appear to be in classically stochastic regions of phase space.

o. C-H E ; 15798(cm- l )

I

I

I

I

I

CLASSICAL VERSUS QUANTUM CHAOS

b. BOND ANGLE

The real HCN molecule does not obey classical dynamics, but quantum dynamics. It is commonly believed that there exists quantal chaos that is analogous to claSSical stochasticity.23 The exact definition of quantal chaos is not agreed upon; it depends upon which features of classical chaotic motion are believed to be most important. Kosloff and Rice 27 focus on the Kolmogorov entropy, which classically is zero for quasiperiodic trajectories and nonzero for stochastic ones.

>-

l-

i/) Z W I-

Z

c.

CoN

0

C-H

I

E: 16469(cm- 1)

1000

2000

3000

I

4000

FREQUENCY (cm- I )

I

.1

I

BOND ANGLE I

>-

We conclude that the classical dynamics of HCN becomes stochastic at energies less than 0.36 of the dissociation energy. The coupling responsible for the transition is primarily kinetic coupling between local modes. All coordinates become stochastic at the same energy, even the bending angle, whose degree of excitation does not appear to appreciably change the stochastic transition energy. Finally, it appears that the transition energy is relatively potential insensitive; as Farantose's model for O:! becomes stochastic at about the same fraction of dissociation. However, it is commonly believed that classical dynamics is critically dependent upon the potential. 35 Since the transition energy we calculate is much less than the energy of the states

t.

b.

FIG. 6. Power spectra of coordinates for classical trajectory of HCN 12319 cm-1 above ground state. HCN has 3479 cm-1 of zero point energy. Trajectories, run on the same potential, but with a very heavy carbon atom to eliminate kinetic coupling between local modes, were found to be quasiperiodic at 25000 cm- 1 of excitation energy. Farantose and Murrell26 found that O:! trajectories are stochastic above - 2900 em-I, about the same fraction of dissociation energy as we have found for HCN.

Al ~~

A.

I

I

l-

i/) Z W

I,

I-

Z

~l c.

..

CoN

-I

.

l

I I

1

I

I

I

i\

1000

.J. . j 2000 FREQUENCY (cm-

. 3000 I

,I 4000

)

FIG. 7. Power spectra of coordinates for classical trajectory of HCN 12990 cm-1 above the ground state.

J. Chem. Phys., Vol. 77, No.6, 15 September 1982



behavior. The quantal chaotic system should display hypersensitivity, having very large derivatives of its eigenvalues and eigenfunctions as a function of the parameters (masses, rotational quantum numbers) of the Hamiltonian. The experimental consequences of a vibrational level being quantally chaotic woold be a disruption of its rotational progressions, an erratic isotope shift, and a mixing of its transition intensity over many levels. Our results show that the observed states of HCN are not quantally chaotic, even far into regions of phase space predicted to be classically stochastic. These results are in conflict with much speculation that a quantal system will become chaotic at the classical stochastic transition. Earlier work on the Henon-Heiles potential suggested that the quantal solutions did indeed become chaotic near the classical transition energy when the well supports large numbers of quantum states, 29 but not when the number of bound states are low. so Kay Sl looked at vibrational energy transfer on a linear Hs, with H2 parameterized Morse bond potentials, and only kinetic energy coupling between the bonds. He concluded that the quantum and classical solutions gave almost identical results in both the quasi-

o. C-H E = 18376 (em-I)

IoU

.l

L ...

~LL ~ L..u

l

bBOND ANGLE

>t:: en

z

LLJ

I~

c.

2859

CoN

o. C-H E = 21639 (cm-I )

j

1000

~ 2000 3000 FREQUENCY (cm -1)

4000


Since the quantal time evolution operator for a discrete spectrum is quasiperiodic, the quantal Kolmogorov entropy they define is zero for any bound system. This definition does not differentiate between types of bound states. Heller S6 focuses upon extensive vibrational energy transfer. Classically stochastic trajectories do not necessarily have large amounts of vibrational energy transfer, but often stay close in phase space to "destroyed" torri. Therefore, we believe this definition of quantal chaos is too strong to identify with classical stochasticity, and might better be identified with classically ergodic motion. The predicted observable consequence of this type of quantal chaos is the complete breakdown of any zero-order selection rules, and a sharing of intensity between all nearly isoenergetic states. Noid e tal, 28 building on the work of Pomphrey, 29 define quantum chaos as the existence of many overlapping avoided crossings. This is analogous to classical chaos which is believed to arise from overlapping classical resonances. We adopt this definition as it is basis set independent, agrees with spectroscopic notions of complexity, and prOVides a criterion for differentiating between types of quantal

b. BOND ANGLE

>t:: en Z

LLJ

IZ

c.

CON

1000

2000 3000 FREQUENCY (cm- I )

4000




2860

Lehmann, Scherer, and Klemperer: Highly excited HCN

periodic and stochastic regions, except for a few states near the dissociation energy. These results are significant since he chose a realistic potential, and because one would expect the differences between quantal and classical results to be largest for an all hydrogen system. In contrast to these earlier numerical results, we find that stochastic classical dynamics has no experimental consequences for the quantum solutions of the HCN potential. Both Kay31 and Marcus 32 have pro~ posed that quantal systems may not become chaotic if the overlapping classical resonances that produce classical chaos are much smaller in width than 11. If the classical motion does not move further than 11 from a destroyed torus, then the quantal solutions will average out the stochastic motion. However, even then, semiclassical quantization is not possible. QUANTUM CONSIDERATIONS

In classical mechanics, the density of states does not eXist, but Noid et al's20 notion of quantal chaos reqUires a minimum denSity of states large 'enough to produce overlapping avoided crossings. A simple harmonic counting will not produce an accurate density of states for HCN at these energies, since the bending potential is quite anharmonic. In fact, the barrier to produce HNC is calculated to be only 13840 cm -1 above the ground state. 33 To estimate the density of states, we calculated the bending level spectrum with a model consisting of a proton at a fixed distance from the CN center of mass, which was allowed to rotate in a potential V(8) = - 2551 XP1(COS 8) - 9845 xP2 ( cos 8) cm -1. 34 The coefficients were chosen to fit the calculated barrier to rearrangement and the energy difference of HCN and HNC. This potential gives a bending vibrational fundamental of 710 cm-1, compared With an observed fundamental of 712 cm -1. The energy levels were obtained by diagonalizing the Hamiltonian in a free rotor basis set. We estimated the density of states by summing over the bending spectrum with stretching levels given from the anharmonic constants. For a nonrotating molecule, 1 (eqUivalent to K for a symmetric top) is a good quantum number. CorioUs interactions connect states with ~l = ± 1. states separated by I ~ II > 1 can be mixed by indire ct CorioUs coupling. For example, the states (0, 2°, 0) and (0, 22, 0) of HCN are strongly mixed by this type of interaction. The effective density of states, therefore, depends upon whether 1 is a good quantum number or not. The avoided crossings of levels with I ~ll > 1 are, in general, narrower than those with ~l = 0, ± 1 since there is no direct matrix element between the states, but only higher order coupling. These crossings are much more frequent, however, due to the greater number of states. It is possible for these indirect CorioUs interactions to make a system quantally chaotic. In this case,