2054
J. Phys. Chem. 1983,87,2054-2059
Vibrational Relaxation Pathways in C2H2and C2D2 Sighart F. Fischer" and A. Irgens-Defregger Physik-Department der Technischen Universitat, Muchen, 8046 Garching, West Germany (Received: January 14, 1983)
Collision-inducedtransitions between various vibrational states in the energy regime below 3500 cm-' we estimated for C2H2and for C2D2. The anharmonic mixing of the vibrational eigenstates is incorporated fully up to fourth order and partly to fifth order in the expansion of the potential in terms of normal modes. The collision model is based on an extended distorted wave approximation and it incorporates rotational energy transfer via an effective collision mass. For the transition out of the excited antisymmetric C-H stretching mode vg of C2H2 we determine a very rapid population of the CEC stretching mode vz (- lo3collisions). For the corresponding excitation of CzD2we predict a somewhat slower temperature-dependent population of the v2 mode. The estimated relaxation time out of this mode is extremely long (1.76 X lo4 collisions) for C2H2in qualitative agreement with experiments. For CzD2we predict an even longer lifetime of this mode (9 X lo4 collisions).
Introduction The problem of vibrational relaxation within polyatomic molecules is of interest from several points of view. For isolated molecules one would like to understand under what conditions the energy distributes equally over the available phase space. The corresponding classical motion is either regular or chaotic.' For medium-induced transitions the detailed pathway of the energy flow needs to be found. In the so-called small-molecule limit the quantum system may be described by discrete eigenstates. These states are, in general, not harmonic. States with the same symmetry will mix. The amount of anharmonic mixture is important to understand details of the vibronic spectra such as intensity borrowing. Strong anharmonic mixture indicates the onset of chaotic motion if a control parameter such as the energy is increased. If there are many anharmonically mixed states within the spectral width of the excitation pulse, the energy will distribute among all of them. If they are not overlapping,the induced transitions can be only induced by a medium. The rate of the induced vibrational energy transfer can still depend very strongly upon the anharmonic mixing of these states. Thus, by studying such transitions, one may also learn about the intramolecular potential as well as the intermolecular interaction. In this paper we want to analyze the energy flow in CzHz and CzDz after excitation of the antisymmetric C-H stretching mode. Experiments are available for C2H2in solution (CCl,)' and for several collision partners in the gas phase.3 We shall find that the vibrational relaxation between almost isoenergetic states may differ by several orders of magnitude, depending on the symmetry which may or may not allow for anharmonic mixing. We have chosen acetylene, first, because effects of selective intramolecular mixing show up so strongly, so that very short lifetimes of 1 ps and also very long ones of 240 ps are observed in the liquid.2 Secondly, we wanted to show that the mixed eigenstates of the isolated molecule (1) AIP Conf. Proc., No. 46,1-403 (1978); V. I. Arnold and A. Avez, 'Ergodic Problems of Classical Mechanics", W. A. Benjamin, New York, 1974. ~. ~
(2) C. Kolmeder, W. Zinth, and W. Kaiser in "Springer Series in Chemical Physics", Vol. 23, K. B. Eisenthal, R. M. Hochstrasser, W. Kaiser, and A. Laubereau, Eds., Springer-Verlag, West Berlin; W. Zinth, C. Kolmeder, B. Benra, A. Irgens-Defregger,S. F. Fischer, and W. Kaiser, J . Chem. Phys., 78,3916 (1983). (3) J. Hager, W. Kruger, T. Ruegg, and H. Walther, J . Chem. Phys., 72,4286 (1980).
can be entirely determined from the empirical anharmonic force field whkh is known up to fourth order for a~etylene.~ In this connection we analyze the origin of the anharmonic coupling and find that the harmonic force field for valence bonds, together with anharmonic corrections for the local stretching bonds, gives a very good approximation if the nonlinear character of the transformation between valence-bond and normal-mode coordinates is properly included. Finally, we develop a model for the intermolecular collisions, which takes into account the detailed structure of the normal-mode displacements during the collision and the translational as well as the rotational energy transfer. Still the model is sufficiently simple, so that similar studies could be done for larger molecules.
Anharmonic Force Field Starting from an empirical anharmonic force field which is given in terms of valence coordinates we obtain the force constants corresponding to normal-mode coordinates by using the nonlinear transformation between normal-mode and valence-bond coordinates. We use the seven valence coordinates as defined by Strey and Mills4 (Figure 1): three stretching coordinates ( R , = Ar, (CH), Rz = Ar2 (CH), R3 = Ar, (CC)) and four bending coordinates (R4x= sin O1 cos $', RgX= sin Oz cos &, R , = sin O1 sin $', R , = sin Oz sin &). dl and e2 are the bending angles at the carbon atoms C1 and C2; and & are the phase angles of the bending planes about the molecular axis. There are five modes: the symmetric C-H stretching mode vl( Zg+), the antisymmetric C-H stretching mode v3(Zu+), the symmetric C 4 stretching mode vz(Xg+),and the two doubly degenerate bending vibrations v4(n,)and u,(n,). We obtain the transformation between normalmode coordinates Q, and valence coordinates R, with the help of the L-tensor formalism as introduced by Hoy et aL5 We define r p = (xp,siP,zp)( p = 1, 2, 3) as the bond vector for the p-th stretching coordinate, and rpoas their equilibrium lengths. For each bending coordinate Rqx,R, (q = 4,5) we denote by ri and rj the bond vectors which enclose the bending angles. Thus, the L tensor up to fourth derivatives reads for p = 1, 2, 3 L,' = azp/aQ' (4)G.Strey and I. M. Mills, J. Mol. Spectrosc., 59, 103 (1976). (5)A. R.Hoy, I. M. Mills, and G. Strey, Mol. Phys., 24,1265 (1972).
0022-3654/83/2087-2054$01.50/00 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 72, 7983 2055
Vibrational Relaxation Pathways in C,H, and C,D,
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