Femtosecond quantum structure, equilibration and time reversal for the CH-chromophore dynamics in CHD2F

Femtosecond quantum structure, equilibration and time reversal for the CH-chromophore dynamics in CHD2F

Volume 2 12, number 5 CHEMICAL PHYSICS LETTERS 17 September 1993 Femtosecond quantum structure, equilibration and time reversal for the CH-chromoph...

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Volume 2 12, number 5

CHEMICAL PHYSICS LETTERS

17 September 1993

Femtosecond quantum structure, equilibration and time reversal for the CH-chromophore dynamics in CHDzF * David Luckhaus, Martin Quack and Jiirgen Stohner Laboratoruun fiir Physikalische Chemle, ETH Ziirich (Zenirum), CH-8092 Zurich, Switzerland Received 10 May 1993;in final form 5 July 1993

We present a new numerical scheme to compute femtoseeond quantum molecular dynamics involving several strongly coupled degrees of freedom. This method is based on a combination of contracted grid basis eigenstate calculations with time-dependent propagation under coherent infrared multiphoton excitation with transformation to the quasi-resonant basis. Using data from our previous high-resolution spectroscopic investigation of CHD*F, we present quantum wavepacket results for the CH-chromophore dynamics, including coherent time evolution of CH stretching excitation in terms of explicit probability densities for the corresponding normal coordinate, relaxation towards approximate, partial quasi-equilibrium on the 100 fs time scale and a discussion of symmetry under time reversal in the ps time range.

1, Introduction The time-dependent quantum dynamics of polyatomic molecules under coherent excitation is of considerable current interest [ 1,2], particularly in relation to recent efforts in real time “laser femtochemistry” [ 31 and collisional relaxation [4]. One way to approach this problem by a combined experimental-theoretical method is to investigate important dynamical functional groups by high-resolution spectroscopy and to derive a Hamiltonian and time-dependent properties via this route [ 51. The alkyl-CH-chromophore with its stretching and bending modes being strongly coupled by anharmonic interactions has been thoroughly studied by high-resolution spectroscopy over the last decade [ 6 1. In Cs,CHXs molecules with a C, symmetry axis one finds decoupling of one degree of freedom, corresponding to an approximately conserved vibrational angular momentum quantum number I (about the C, axis) on short time scales [ 7,8]. Thus, with coherent excitation at either the CH stretching or bending frequency one has essentially two-dimensional quan* Presented at the Latsis Symposium on Intramolecular Kinetics and Reaction Dynamics, Zurich (1992), and dedicated to Jack D. Dunitz on the occasionof his 70th birthday.

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turn wavepacket dynamics on short time scales, leading to interesting effects of delocalization and nonclassical randomization for just two coupled degrees of freedom on the subpicosecond time scale

19,101. When one reduces the symmetry at the CH-chromophore to C, or C, [ 51, it is now well established spectroscopically that the short time dynamics is dominated by anharmonic coupling between the three CH (stretching and two bending) modes, mostly of the Fermi and Darling-Dennison resonance type [ 6,11- 13 1. The time-dependent coherent wavepacket motion becomes correspondingly more complex and difficult to analyze. It is the aim of the present Letter to contribute to this question, We present novel contracted time-dependent numerical grid methods and calculate the reduced time-dependent observables, total intramolecular energy, structural parameters such as CH bond length (or normal coordinate extension), and time-dependent intramolecular nonequilibrium entropy including behaviour under time reversal. We use as starting point the experimentally derived CH-chromophore Hamiltonian (fit 3 from ref. [ 121) and the electric dipole hypersurface from ref. [ 131 (ab initio-MPZ). The latter gives a satisfactory representation of the experimental band inten-

0009-2614/93/$06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved

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CHEMICAL PHYSICS LETTERS

sities and therefore should be adequate to describe the electric dipole coupling to intense radiation fields within the approximations used below.

2. Time-dependent quantum dynamics in a contracted grid basis for several vibrational degrees of freedom We consider the problem of coherent quantum motion for the three strongly coupled CH vibrational modes in CHDIF and related molecules without and with near-resonant infrared excitation. With spectroscopically known couplings in the molecular Hamiltonian [ 11,121, this will be a reasonable approximation on short time scales. We shall disregard rotational fine structure. Rotational levels were included previously for longer time-scale calculations on the asymmetric top ozone [ 141 and could be introduced here similarly at considerable computational expense. For the questions addressed here this neglect is not too serious and we shall discuss the magnitude of some of its consequences in section 3. The general equation to be solved for time-dependent quantum dynamics is

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sentially the number of grid points) and become therefore numerically impractical. In order to combine the merits of the two different approaches, we use the eigenstate basis calculated on a grid as described by us recently, using a systematic contraction scheme [ 121. This contracted grid eigenstate basis $ is then diagonal in the molecular Hamiltonian to within the approximations used, giving the eigenfrequency matrix W = { Wkk} fLllo,@k= & [email protected]~

(2)

The time-dependent differential equations are solved with a z-polarized classical periodic laser field (angular frequency w) with slowly varying field amplitude E,( t), using a coupling in the electric dipole approximation with the three-dimensional electric dipole hypersurface (operator p) from [ 12,131: i$

= [W+Vcos(wt+q)]U,

(3)

U can be used to represent explicitly the wavefuno tion y (SI, YY(417

where U is the time evolution operator, in practice to be written as matrix representation in some basis. As the number of coupled degrees of freedom included in the calculation increases, the numerical convergence of the calculations with the size of the basis becomes a major concern. As has been argued before, by far the best choice for near-resonant coherent infrared rovibrational excitation is the rovibronic “spectroscopic” eigenstate basis with transformation to the quasi-resonant basis and judicious selection of the basis of eigenstates with respect to dipole coupling strength and resonance defects [ 1416 1. Disadvantages result from the lack of numerical flexibility in the eigenstate calculation itself and in the treatment of continuum states. Direct numerical wavepacket propagation on multidimensional grids circumvent these disadvantages [ 17-211. However, for intermediate and long time propagation in multidimensional problems these direct methods do not allow any easy basis set contraction (basis size is es-

17 September

Wt)=U(t,

42,439

q2,

q3,

-**, f)=

toMto)

-.,

0,

Cbk(t)~k(ql,42,43,.-.), k

(5)

(6)

and also any operator Q in the basis & or density matrix P using the integrated forms of the Heisenberg equation of motion and Liouville-von Neumann equations in the well known way [ 21. In practice, we have used the URIMIR package [ 221 and extensions thereof for the time-dependent problem, using the quasi-resonant or Floquet approximations with or without transformation to the quasi-resonant basis [ 2,15 1, The numerical calculations become then very efficient and the approximations have been checked explicitly by comparing quasi-resonant and Floquet results in selected test cases, as before [ 10,141. The slowly varying field amplitude E,,(t) is approximated by sequences of steps with constant E. during short time intervals. The combined technique for wavepacket propagation, which has not been used before, has the following advantages: (i) The wavefunction and prob435

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ability density are obtained on the multidimensional grid as in other grid techniques. (ii) The molecular Hamiltonian used can be checked against, or derived from, spectroscopic results as in other “eigenstate basis” calculations. (iii) The grid basis can easily be extended to include “continuum” states applying absorbing boundary conditions [ 18-2 1,23 1, although the excitations reported below remain in the bound state region. (iv) The method can treat rotational motion and include rovibrational coupling as in other spectroscopic basis set expansions. (v) Contraction of the basis can be carried out in two stages, first in the grid eigenstates calculation [ 121, second in the quasi-resonant selection of important levels for optical excitation [ 14- 161. Methods somewhat similar to the quasi-resonant selection of levels have been proposed using “artificial intelligence” schemes [ 241, subsequently and largely independently from the original work using simple “natural intelligence” [ 14-161. We conclude this section with a brief discussion of the grid basis, the magnitudes of the contraction and some definitions used in the evaluations below. The molecular Hamiltonian is represented on a three-dimensional, equidistant grid in the dimensionless CH stretching ( qs) and the two CH bending (“in C, plane and out of plane” qi, qO) normal coordinates of CHD2F with 25X31 x31 =24025 points. The discretization in the coordinate basis I q) = I qs, qi, qO) is equivalent to an expansion in trigonometric interpolation functions and has been developed as a one-dimensional scheme early on by Meyer, and used over the years for a variety of spectroscopic problems [25], including also some simple kinetic applications [ 26 1. A modified version has recently been applied by Colbert and Miller [ 23 ] to reactive scattering. The grid basis is then contracted [ 121 by first solving one-dimensional problems for each coordinate and then combining the 1D bending functions to a two-dimensional basis of bending eigenfunctions xb (with qs=O) by diagonalizing the appropriate Hamiltonian in that subspace (label b for these bending functions). These are then combined with the 1D stretching functions { to form a basis q,, %)s,b= Ik.?b>

.

(7)

This intermediate basis is characterized by a well defined stretching mode quantum number v, for every 436

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basis state. The 3D CH-chromophore Hamiltonian is diagonalized in this basis p, requiring about 2500 basis functions [ 121 for the lowest 280 eigenfuno tions 4, most relevant for the subsequent optical excitation calculation, eqs. (2)-(6). Thus one has a contraction of the original basis by a factor of about 10 in the intermediate step and another factor of 10 in the second step, giving overall savings of a factor of 100 with respect to direct grid propagation (i.e. calculation of U in the basis of ~24000 grid point functions 1qs, qiyqO) ). Even further savings are then possible with transformation to the quasi-resonant basis, which is meaningful only if one has first derived the basis @ [ 161. While with a basis of a few hundred coupled levels for the basis vector B in eq. (6) the numerical problem of optical excitation is quite easy to solve, these successive reductions will become necessary when one includes rotational levels and further vibrational modes. On the other hand, the grid basis in the background of all these calculations is useful because of its numerical ease, and even for practical evaluation of final data for graphical representation. For instance, we calculate the probability distribution in the stretching coordinate

f’(CIs,t)=

JJIIY(t,q,,qi,%)l’dqidq~

by efficient integration at the intermediate level of eq. (7).

(8) grid basis

3. Results and discussion 3.1. Time-dependent structure during and after IR multiphoton excitation The first observable quantity of interest is the intramolecular energy during and after multiphoton excitation. Fig. 1 shows the time-averaged absorbed energy ( (E&s) I, at times without dissociation) for molecules initially in the ground state

calculated in the quasi-resonant approximation with various laser wavenumbers &_at constant intensities of 2 TW cm -* and 20 TW cm-* (“multiphoton absorption spectrum”). The higher intensity corre-

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Fig. 1. (a) Long time average of absorbed energy as a function of laser wavenumber for multiphoton excitation of CHD2F at intensity 2 TW cme2 (see also text). (b) As {a) with 20 TW cm-2. (c) Time-dependent polyad population for N=O (----) and N= 1 (- - -), &=2865 cm-‘,1=20TWcm-‘. (d) As (~),with&=2935cm-‘,1=20TWcm-~.

sponds to an average (absolute) coupling matrix ele ment between quasi-resonant levels of about 30 cm-‘. About 99W of the time-averaged population resides in the lowest 100 states, below 11500 cm-‘. We have also chosen various types of shaped pulses of the general kind known from experiment [27 1,

but defer here discussion of these results. For the example, rotation is considered to be frozen and the molecule fixed, CH normal coordinate axis is taken in the z direction

of the laboratory frame as is also the polarization of the laser. In practice, this implies that we calculate the electric dipole coupling matrix elements using the projection of the dipole moment vector onto the z axis. This is a possible, physically reasonable assumption for frozen rotation. In order to check the effect of this neglect of rotation, we took as another extreme assumption all dipole coupling matrix elements to be equal to their respective total absolute value. The resulting multiphoton absorp-

tion spectrum hardly differs from the one shown in fig. 1, except for some of the weaker features. It is therefore not represented. This and other tests show that the conclusions to be drawn below do not sensitively depend upon the particular simplifying assumptions made about freezing rotation. Of course, the multiphoton absorption spectrum itself and the dynamics would, indeed, depend upon the inclusion of a large number of available rotational states, requiring, however, much more expensive calculations

[141. At the lower intensity of 2 TW cm-* in fig. la, one recognizes the major l-photon, 2-photon and 3-photon resonance structures, with maxima at increasing average absorbed energies as expected. Increasing the intensity by a factor of 10, rather close to the ionization threshold, the spectral structures become much richer and more complex (fig. 1b). The heights and appearance of narrow peaks depend upon the wave437

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number resolution of the calculation and, indeed, the plotting routine. In the lower part of fig. 1 ( lc and Id), we show the time-dependent polyad populafor the lowest two polyads tions PN=&Pj(polyadN) (N= 0 is the ground state ) and excitation at two selected wavenumbers (2865 and 2935 cm-t), corresponding to the 3-photon and 2-photon resonance respectively. One finds the expected, rapid decay of the ground state, which recurs approximately once or several times over a time of 2 ps, depending upon excitation wavenumber. Appreciable population ( x20%) is obtained in the polyad N=4 with excitation at 2865 cm-‘, but polyads with N>6 show little population under these conditions. Beyond about 1 ps the time dependence of the populations becomes sensitive to details of the spline fits to the dipole surface and to the shape of the grids used in the numerical calculations. A second observable is the time-dependent quantum mechanical “structure” of the molecule_ This corresponds to the probability density 1Y( qs, qi, qO,t ) I* for the present analysis. In contrast to the effectively 2D coupled oscillator problem in CHXs molecules [ 9,1 O], for the 3D coupled oscillator system a complete graphical representation is not easy, not even in snapshot pictures at selected times. We have chosen here to show as a reduced structural information the probability to find a certain CH stretching normal coordinate extension as defined by eq. (8) as a function of time. This corresponds to a 1D oscillator representation of the complete 30 dynamics in fig. 2. Because q, is approximately proportional to the CH bond length with an isolated CH oscillator, the results have an easy intuitive interpretation. An alternative way to measure such a quantity on the fs time scale, in principle, would be a femtosecond electron diffraction experiment [ 3 1. We note that our results can be considered as experimental to the extent that the parameters of the Hamiltonian are experimental data and that the approximations in the analysis are valid. Fig. 2a shows the short time coherent oscillation of the probability density with the characteristic period of about 11 fs. The coherent multiphoton excitation does not lead to good energy selection, in fact rather to a wide distribution over many energy levels. However, the phase relations lead to a good coherent spatial selection on this short time scale, which to some 438

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extent is more important for chemistry than energy selection, as chemical reactions such as bond breaking and bond formation are spatially local phenomena, but not necessarily energy selected. On the contrary, spatial localization as a rule requires some lack of energy selection. This situation is often overlooked. At longer times, with substantial excitation at 400 fs, the probability density broadly extends over a larger range without too pronounced maxima and oscillatory behaviour (fig. 2b ). Some oscillatory behaviour, but complicated, recurs at times around 1.4 ps. Fig. 2c shows the probability density in the inplane bending coordinate at long times, when the stretching coordinate is excited. This type of result provides a detailed picture of how stretching excitation flows into bending excitation. One recognizes some quasi-periodicity in the bending excitation with the period of 11 fs of the stretching vibration. The out-of-plane bending vibration gets much less excited. The reverse phenomenon is observed as well. When exciting in the range of bending wavenumbers between 13 18 and 1335 cm-’ one finds first coherent bending motion, then some spreading and stretching excitation. However, the physical applicability to CHD2F of these model results with bending excitation is not clear, because of the close, intense CF stretching and CD2 bending fundamentals (around 1000 cm-’ [ 281) and more serious effects arising from neglect of rotation. We thus do not show these results in detail. 3.2. Intramolecular without field

motion of the isolated molecule

In the examples mentioned so far, intramolecular energy is not a conserved quantity, due to coupling to the strong radiation field. It is also of interest to study the time evolution of certain initial states under isolated-molecule conditions, when energy is conserved (but not necessarily sharply defined). The selection of initial states may follow chemical intuition and the practical experimental preparation of such states may not be obvious. However, the superposition principle guarantees their existence. The time evolution is governed by eq. (3) with V=O (field amplitude E0 = 0). Fig. 3 shows the time evolution of an initial state corresponding to six quanta of stretch excitation and

Volume 2 12, number 5 CHD2F1021 50 - 150

CHEMICAL PHYSICS LETTERS is

Cexc.

2865

cm-11

CHDZROZNI

I7 September I993 400 - 450 fs (WC.

2865 cm-l)

(4

CHDZF(O2NI 950-1050 fs lexc

2865 cm-l) nsel=3

Fig. 2. (a) Time-dependent probability density P( q,, f ) , eq.( 8 ), under IR-multiphoton excitation with I= 20 W cm-‘, FL=2865 cm-‘. Abscissa shows a dimensionless normal coordinate range -4.S
no bending excitation in the basis of eq. (7 ), giving a typical fast decay of initial state population p( ZJ,= 6, h=O=Vi=Vo) in about 100 fs (fig. 3a). The result can be compared with similar decays calculated in the somewhat different effective Hamiltonian basis [ 11,12 1. The initial decay is similar; larger differences arise in the long time oscillations. We note that in fig. 3 of ref. [ 121 a scaling error has led to a compression of the time scale by a factor of 2.3 (i.e. all values on the time axis should be multiplied by 2.3 to obtain correct values). Figs. 3b and 3c show

the probability density in the stretching coordinate as a function of time with this initial state, which at t=O looks like a II,= 6 eigenstate for a Morse oscillator and then evolves in time to provide a broad probability distribution with modest oscillatory motion, which is compared in fig. 3d to the randomly selected state from a corresponding microcanonical probability distribution (i.e. equal weight for all levels with N= v,+ 0.5 (y+ v,) = 6 and random phases, in the basis of eq. (7)). This provides an intuitive picture of relaxation towards microcanonical equi439

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Volume 2 12. number 5

CHDZF

I.0

1

I

I

I

0

-

100

is

16.0,0>

I

.a .6 -;

L’

.4 .2

CHDZF

100

-

200

fs

16,O.O)

CHDZF

0

-

100

is

N = 6

(stat.1

Fig. 3. (a) Decay of initial state population after initial excitation of 6 quanta of CH stretching (0 quanta in bending) in basis of cq. (7). (b) Probability density P(q,,t) for initial state as (a), time step 0.5 fs, time range 0 to 100 fs, -4Scq,< +8.5. (c) As (b) 100 to 200 fs. (d) As (b) but with randomly selected initial state in polyad N=6.

lib&m, similar to the densities considered for the 2D case [ 9, lo]. This relaxation can also be studied by looking at the expectation values for the normal coordinate extension ( QS(t) > and its root-meansquare deviation ( )Q,(t) 12} ‘12. The former relaxes to small values, close to Qs=QS,==O; the latter to some larger value describing the spreading, both oscillating somewhat with time. 440

3.3. Evolution of entropy and time-reversal symmetry In order to describe intramolecular relaxation for the subset of strongly coupled modes, one may use as relaxation parameter a suitably defined microcanonical nonequilibrium entropy [ 5,111. There is some arbitrariness in the definition of such an en-

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tropy parameter. Using the fine-grained Pauli detinition [ 29 ] S,=-kCpjlnpj i

(10)

this still depends on the definition of basis states in which the populations pj are to be measured, which amounts to defining what time-dependent relaxation observables are to be measured. For the example S, in fig. 4 we have chosen the effective Hamiltonian basis with 6 quanta in the initial pure CH stretching excitation. In the effective Hamiltonian basis the interpolyad couplings have been removed by the approximate similarity transformation [ 71 and thus a microcanonical ensemble corresponding to the 28 A’ states of the N= 6 polyad can easily be defined. The microcanonical normalization factor S,, is then just kln 28. A somewhat more suitable “coarse-grained” observable would be, for instance, just the CH stretching quantum number, defining a level with statistical weight g( v,) =N+ 1 - v, (corresponding to the possible A’ states combining with a given v, for the integer polyad quantum number N). This defines an average population &US) (Pj(vs))=P(v,)lg(v,)=

E,

Pk(v,)lg(ys) 3

(11)

with a coarse-grained quantum entropy SQ=-k

5: (Pi) ln

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CHEMICAL PHYSICS LETTERS

(12)

shown in the dashed line of fig. 4. Eq. (11) concerns

Fig. 4. Time-dependent reduced intramolecular entropy for the subset of three coupled oscillators with 6 quanta of CH stretch initially excited (in the effective Hamiltonian basis). Time reversal at t= I ps (see discussion in the text). (-) Sr(r)/S_ fine grained; (- - -) S,(t) /S,, coarse grained.

nondegenerate levels j with a given v,, whereas in eq. ( 12) we sum over all levels j with (pj} being taken from eq. (11) for (pj( v,)}, with the values of v, as applicable for the j considered. As expected, So is somewhat larger than &, but the differences are not large. We refer to ref. [ 5 ] for some further discussion. Other definitions of entropy may give different answers. In the extreme, one may argue that the entropy for our example of a pure quantum state remains constant (zero) at all times [ 30 ] (but see also ref. [31]). The relaxation described in fig. 4 reaches a substantial fraction of the equilibrium entropy in about 100 fs (i.e. less than 10 vibrational periods) with some subsequent

fluctuations.

These and the devia-

tions from maximum entropy are, of course, expected for a small quantum subsystem of just three coupled vibrational modes. Fig. 4 also shows the effect of time reversal at t = 1 ps. At this time t’we take the time-reversed state vector b*(t’) from forward propagation as initial state (i.e. a state with all momenta reversed) and continue to propagate. As expected, we find mirror-image evolution corresponding to backwards motion with S= 0 at t= 2 ps as at 1=0 ps. Because of the Hermitian spectroscopic Hamiltonian used, which shows time-reversal symmetry as does U, this back integration amounts to nothing more than a simple numerical check on the calculation. However, at a more fundamental level it opens the possibility to look for time-reversal symmetry violating effects in intramolecular dynamics, for which from the spectroscopic experiments available to date there is no evidence. If there were de lege violation of time-reversal symmetry, entropy would not return to its value zero at 2 ps. Even more speculative possibilities concern CPT violation [ 51. It may be useful to comment on why dynamics in the complex spectra of polyatomic molecules may be an interesting place to look for such small symmetry violating effects, rather than in the simplest and more commonly investigated cases of elementary particle dynamics. The high complexity of polyatomic molecules gives rise to a high degree of “accidental” and also systematic degeneracies. In these types of system one may hope for amplification and selection of weak effects, such as suggested for the ubiquitous (but very weak) parity violation in molecular spectra [32]. Different from parity violation, whose magnitude we can estimate, (CP)T violation would 441

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rely at present on effects of unpredicted magnitude. Nevertheless the result in fig. 4 might be suggestive for future studies, as molecules, in contrast to macroscopic systems, which also have the merit of complexity, can be studied under approximate conditions of isolation [ 51, and thus de lege T violation (if any! ) could be identified with significance.

4. Conclusions (i ) The combination of a contracted grid basis stationary state vibrational variational calculation with time-dependent quantum molecular dynamics for IR multiphoton excitation using transformation to the quasi-resonant basis leads to an efficient technique for obtaining the short time evolution in molecules with more than two strongly coupled vibrations. (ii) Intramolecular quantum oscillator dynamics in CHD,F shows coherent “Gaussian” CH stretching oscillations for about 10 vibrational periods with subsequent mode-selective transfer of excitation to bending, the motion in the C, plane being preferred over out-of-plane motion in this example. (iii) Intramolecular relaxation of highly excited CH stretching states demonstrates fast delocalization and quasi-equilibration in about 200 fs and timereversal symmetry on the ps time scale. Our results on coherent quantum molecular dynamics derived from spectroscopy are an alternative to real time femtosecond studies [ 3,4] and can lead to further insights and applications in molecular reaction control and selective multiple laser spectroscopies [ 33-361. Conceptually it is of interest that we are able to derive a detailed picture of time-dependent molecular quantum structure [ 51, beyond the established analysis of stationary or average structures from high-resolution spectroscopy [ 37 ] or Xray crystal structure analysis [ 38 1. The relatively short time for coherent vibrational motion ( 10 periods) in the alkyl CH-chromophore, with fast redistribution of excitation to the two bending vibrations within about 100 fs may be compared with other cases. In particular the acetylenic CH-chromophore is a case in question. The parent compound acetylene has been studied extensively to high vibrational energies by stimulated emission pumping [ 391 but should be considered as a case by 442

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itself. For the monosubstituted acetylenes CX3C=C-H we have shown about a decade ago that the isolated =CH-chromophore vibration may last well into the ps time scale, for thousands of vibrational periods because of the adiabatic separation of the high-frequency CH stretching mode [ 5,401. This has recently been confirmed by molecular beam spectroscopy at much higher resolution [ 4 1,42 1. The fast ( 100 fs) alkyl group redistribution discussed here is thus group specific (and different from the behaviour of other functional groups such as =CH) and also mode specific, because the remaining six degrees of freedom in CHD*F do not participate on the very short time scale.

Acknowledgement Discussions with Roberto Marquardt are gratefully acknowledged. Our work is supported linancially by the Schweizerischer Schulrat and the Schweizerischer Nationalfonds.

References [ I] J. Broeckhove and L. Lathouwers, eds., NATG ASI Series B, Vol. 299. Time dependent quantum molecular dynamics (Plenum Press, New York, 1992). [2] M. Quack, Advan. Chem. Phys. 50 (1982) 295. [ 3 ] A.H. Bewailand R.B. Bernstein, in: The chemical bond, Real time laser femtochemistry, ed. A.H. Zewail (Academic Press, New York, 1992); L.R. Khundkhar and A.H. Zewail, Ann. Rev. Phys. Chem. 41 (1990) 15; A.H. Zewail, Latsis Symposium Abstracts, ETH Zurich (1992). [ 41 Th. Eltisser and W. Raiser, Ann. Rev. Phys. Chem. 42 (1991) 83; R.E. Weston and G.E. Flynn, Ann. Rev. Phys. Chem. 43 (1992) 559. [ 51 M. Quack, Jerus. Symp. 24 ( 1991) 47; J. Mol. Struct. 292 (1993) 171. (61 M. Quack, Ann. Rev. Phys. Chem. 41 (1990) 839. [ 71 M. Lewerenzand M. Quack, J. Chem. Phys. 88 (1988) 5408. [ 81D. Luckbaus and M. Quack, Chem. Phys. Letters 205 (1993) 277. [9] R. Marquardt, M. Quack, J. Stohner and E. Sutcliffe, J. Chem. Sot. Faraday Trans. II 82 ( 1986) I 173. [lo] R. Marquardt and M. Quack, J. Chem. Phys. 95 (1991) 4854.

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