J . Phys. Chem. 1988, 92, 4059-4062 carbon double bond with silicon orbitals can result in a modification of the force constants which yield a potential energy curve similar to that obtained experimentally.
Conclusions The quantitative determination of the ring-puckering potential energy function for silacyclopent-2-ene in this work demonstrates that this ring system is indeed unusually rigid. Molecular mechanics calculations, which are usually reliable in predicting the correct conformation and in estimating energy differences between conformers, fail for this molecule. This can be explained by a significant interaction between the silicon atom and the carboncarbon double bond. We are currently carrying out ab initio (GAUSSIAN 82) calculations on this molecule in order to gain an
4059
insight into the nature of the bonding. Whether silicon d orbitals or antibonding u orbitals are involved in the interaction is not clear. What is clear is that a type of bonding interaction which is relatively insignificant in the analogous oxygen and sulfur compounds is present for this cyclic silane. A similar interaction is apparently present in 2-phospholene,13 but the magnitude is much smaller.
Acknowledgment. We thank the National Science Foundation and the Robert A. Welch Foundation for financial support. Registry No. CH=CHSiH2CH2CH2, 6572-33-4; CH=CHSiD2CH2CH2, 110897-47-7.
-
(13) Harthcock, M. A.; Villarreal, J. R.; Richardson, L. W.; Laane, J. J. Phys. Chem. 1984,88, 1365.
Vibrational Relaxation Efficiency at Low and High Temperature Mark G . Sceats Department of Physical Chemistry, University of Sydney, N.S. W. 2006, Australia (Received: December 4, 1987)
A stochastic model of vibrational relaxation of a diatomic is developed and used to explain the deviation from Landau-Teller behavior at low temperatures. The model includes (i) attractive forces, associated with both encounter dynamics and acceleration effects, (ii) quantum effects associated with energy conservation and wavepacket spreading, and (iii) the anharmonicity of the diatomic. For a low-frequency diatomic vibration of 200 cm-I, a weak temperature dependence below 100 K is predicted for both light and heavy colliders.
Introduction Vibrational relaxation of diatomics in seeded supersonic expansions has been extensively studied'-5 and the primary observation, namely that the V-T relaxation becomes weakly temperature dependent at low temperatures, has been the subject of considerable These trends are also confirmed in bulb experiments,6.'@-I3 where deviations from classical Landau-Teller behavior become apparent at low temperature. In bulb experiments, predissociation of van der Waals complexes has been invoked,6 but the same behavior is observed in beams where selective excitation of the isolated molecule is possible. In this note a model is presented which explains the observed behavior over the entire temperature range, without change of mechanism, for the isolated molecule. Theoretical Development The quantity of interest is the collision efficiency for the transfer of vibrational energy from an oscillator AB with vibrational energy E . The efficiency Q is related to the relaxation rate kE by (1) Tusa, J.; Sulkes, M.; Rice, S. A. Proc. Narl. Acad. Sci. U.S.A. 1980, 77, 2367. (2) Sulkes, M.; Tusa, J.; Rice, S. A. J . Chem. Phys. 1980,72,5733. Tusa, J.; Sulkes, M.; Rice, S. A. J. Chem. Phys. 1979, 70, 3136. (3) Hall, G.; Liu, K.; McAuliffe, M. J.; Giese, C. F.; Gentry, W. R. J . Chem. Phys. 1984, 81, 5577. (4) McClelland, G. M.; Saenger, K. L.; Valentine, J. J.; Herschbach, D. R. J. Phys. Chem. 1979,83, 947. ( 5 ) Kable, S. H.; Knight, A. E. W. J . Chem. Phys. 1987, 86, 4709. (6) Ewing, G. Chem. Phys. 1978, 29, 253. (7) Gray, S. K.; Rice, S. A. J . Chem. Phys. 1985, 83, 2818. (8) Cerjan, C.; Rice, S. A. J. Chem. Phys. 1983, 78, 4929. Cerjan, C.; Lipkin, M.; Rice, S. A. J . Chem. Phys. 1983, 78, 4929. (9) Schwenke, D. W.; Truhlar, D. G. J . Chem. Phys. 1984, 81, 5586. (10) Lucht, R. A,; Cool, T. A. J . Chem. Phys. 1975, 63, 3962. (1 1) Zittel, P.; Moore, C. B. J . Chem. Phys. 1973, 59, 6636. (12) Billingsley, J.; Callear, A. B. Trans. Faraday SOC.1971, 67, 257. (13) Stephenson, J. C. J. Chem. Phys. 1974, 60, 4289.
where uAB,M is a reference hard-sphere diameter, p is the bath particle-diatom reduced mass, and pM is the bath density at temperature T. In the stochastic model used in this paper the energy relaxation rate is deduced from the frequency-dependent friction @(a) that the bath exerts on the oscillator. The interest in use of stochastic approaches is that they are readily extended to liquids where individual collision events may not be res01ved'~J~ and collective behavior of the bath molecules may play a role. If K ( t ) is the Fourier transform of @(o),then linear coupling theory gives16
kE = R X - K ( t ) Cvv(t) di with R = 1. Cvv(t) is the normalized velocity autocorrelation function of the unperturbed oscillator, and R is a correction factor introduced in an ad hoc manner to account for nonlinear coupling with heavy bath parti~1es.I~ For a Morse oscillator with harmonic frequency w and dissociation energy D
where 6 = E / D and a(€)= w(1 The semiclassical limits to this paper are found for a given quantum state u by using u
20 + -21 = [ l - (1 - 4 / 2 1 hw
(4)
(14) Sceats, M. G. Adu. Chem. Phys., 1988, 708, 358. (15) Nordholm, S.; Freasier, B. C.; Jolly, D. L. Chem. Phys. 1977, 23, 135. (16) Grote, R. F.; Hynes, J. T. J . Chem. Phys. 1982, 77, 3736.
0022-3654/88/2092-4059$01.50/0 0 1988 American Chemical Society
4060
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988
The coupling factor R is well described by the empirical relationship R = exp{-l.685(
))
+
mMmM2mB
(5)
for a homonuclear diatomic. This was deduced from collinear trajectory studies for an unexcited oscillator ( t = O)I7 but is also reasonable at higher energies.14 It is useful because it depends only on the masses and circumvents the extreme difficulties of dealing with nonlinear equations of motion. Further theoretical work is required to validate its use, and it is introduced here only to highlight the linear coupling assumption. At low bath densities collective motions of the bath are negligible and the friction arises from binary encounters. In the model of this work the bath-diatom friction is decomposed into bathatom terms. At short times the appropriate result is’*
Sceats assumes conservation of collisional energy and orbital angular momentum, which are not appropriate for inelastic phenomena. The use of only the attractive limb in determining RT and the clamping of RT at u at high temperature have been discussed elsewhere.20 The quantity ( Ih(=(w)l2) in (7) is the power spectrum of the force exerted on A in trajectories which enter the region R C R,. At low densities for an isolated atom, third-body effects are negligible. vAM(R)in the vicinity of the minimum is modelled by a Morse oscillator V(R) = t[exp(-2a[R-Re])
- 2 exp(-e[R-R,])]
(14)
where u = Re - In 2/a. The use of (14) in this region is introduced for two reasons. Firstly, the exponentially repulsive potential is a good representation for real systems.2’ Secondly, analytical results can be obtained.22 The force power spectrum for incoming energy E is I~~A(W)I’
= I(E
+ hWlP(R)IE)1*
(15)
where the force operator P(R) is -dV/dR. The results areI4J9
For a homonuclear diatomic AA, @(a) = PA(w). The form of (6) assumes that the stochastic forces on atoms A and B are uncorrelated and unscreened. The task is to model the atom frictions PB(w). In this case of interest h w > k T and quantum effects cannot be neglected. In recent work it was shown that at low densities the collisional friction arising from binary encounters is’4319
JAis the flux of bath particles across the transition state of the effective potential v A M ( R ) = VAM(R)- 2 k T In R
The transition state is found by considering the attractive limb of the atom-atom potential which is well modelled by the dispersion forces VAM(R) = - 4 t ( ~ / R ) ~
a = -1
(8)
2
2
n=O
and S(y) = sinh y/y and the wave vectors are defined by
(9)
k , = (2wE)’f2/h
where t is the well depth of the cor_responding Lennard-Jones potential. The transition state of VAM(R)is found from the maximum of (8) to be RT = ~ ( 1 2 / T * ] ~ / ~for T* = u
for T*
< 12
> 12
(18)
kf = (2/.~(E+h~))’/~/h
(19)
k, = ( 2 / . ~ t ) ~ / ~ / h
(20)
When (14) and (10) are used in ( 6 ) , the result is (10)
where T* = kT/c. The flux is
where by20
Q*IN
is a “collision integral” for inelastic processes given
= exp(4/T*)
for T* > 12
(13)
This definition is equivalent to that of canonical variational transition-state theory and while the value of R*IN is larger than the collision integrals of momentum transfer for elastic scattering, . ~ ” elastic scattering the temperature dependence is ~ i m i l a r . ’ ~ The (17) Kelly, J . D.; Wolfsberg, M . J . Chem. Phys. 1966, 44, 324. (18) Rodger, P. M.; Sceats, M. G . J . Chem. Phys. 1985, 83, 3358. (19) Sceats. M . G.:Millar, D. P.; Fell, G . E. Chem. Phys., submitted for
publication.
where B ( 6 ) accounts for adiabatic effects associated with the finite time scale of collision. When w = 0, B(w) = 1 by definition (see below) SO that PA(0) e PA is the friction
which is related to the diffusion coefficient of atom A by PA = kT/mADA. The model differs from exact theory of diffusion through the use of instead of R(1,1)*.20 The form of B(w) is B(w) =
im e-’[ + E ) ] l i 2 P ( x ) cosh’ dx
x( x
6(x)
where (20) Sceats, M . G., to be submitted for publication. (21) Devonshire, A. F. Proc. R. SOC.London A 1937, 158, 269. (22) Keck, J.; Carrier, G. J . Chem. Phys. 1965, 43, 2284.
(23)
Vibrational Relaxation of a Diatomic
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988 4061
P(x\ =
{.( ?(
y)) I
[
+
+
a rapidly decreasing function of w and only the leading term in A ( € )plays a role. In this caseIg the transition from impulsive to adiabatic collisions is modelled by the frequency dependence of B(w). In this regime of temperature the result is modeled by
x
(I):’
[
.
B(w) = 2 3 - exp --
exp(
--)
(33)
derived by Keck and Carrier.22 As the temperature is lowered further, w >> wo and B(w) is reducedI9 to the result obtained from Landau-Teller theory23
r
hw kT
_.
1
(25) The collision bandwidth wo, de Broglie frequency wq, and van der Waals frequency w, are defined by
In this regime the vibrational relaxation rate drops rapidly with lowering of temperature in agreement with experiment, and the ) exponent of (34) gives the well-known e ~ p ( - a T l / ~dependence. As the temperature approaches the regime where k T becomes comparable to the well depth t, there is an acceleration effect and B(w) from (22) reduces toI9
WAM
Before some limits of B(w) are explored, it is worthwhile collecting the terms in ( 2 2 ) and ( 3 ) to give, for a homonuclear diatomic,
tanh
(s)
B ( n w ( t ) ) (30)
When w = 0, A(c) = 1. The term ( u A M / u A A , M ) ’ is the ratio of surface areas and is given by (1 Re/2uAM)-I. Equations 29 and 30 represent the primary results of this work. The c dependence of (30) includes anharmonic effects of the oscillator and the w dependence of B(w) incorporates adiabatic effects arising from the finite time scale of collision. Quantum effects associated with energy conservation and wavepacket spreading on the repulsive limb are also incorporated in B(w). The role of the attractive forces is included not only in the collision integral but also in the phase shift 6 ( x ) in the formula for B(w) through the magnitude of w,. Consider first the extreme high-temperature limit in which h w
