Artículo 5 El derecho a la integridad personal

The potential energy surface is an area in which this model could be improved.

Following the competition of this work, the in-plane force field for benzene has been

re-examined by two groups: one using a higher level of

ab initio theory than PFB22 and

the other generating a force field23 from experimental frequencies for a large number of

isotopically labelled benzenes4. However, there is substantive quantitative agreement

between these new force fields and the one used here so that detailed dynamical studies

would be required to clarify the effect of the small differences between surfaces. Even for such a heavily studied system, it is surprising that there are still some uncertainties in the ground state spectroscopy24 and hence the force field for benzene.

Although we have a potential surface that reproduces the anharmonicity of a CH stretch very well and also the ring mode frequencies, it is mainly quadratic and hence it cannot be expected to be accurate when the amplitudes o f many modes become large. In particular, we have seen the in-plane wag receive very large amounts of energy and it is easily conceivable that correction to the high energy behaviour of this motion could have an observable effect on the dynamics. In order to investigate this possibility quartic terms,

ß ■

and r 2/? 2, were calculated for us25, at a similar level of

ab initio theory. (It

should be remembered that the cubic terms in the surface of PFB cannot be used alone for large amplitude vibrations where they require quartic or higher even order terms to ensure bounded motion). The quartic force constants appeared to be much too small to have any significant effect and it is not clear that such low level

ab initio theory is adequate for the

description of possibly important anharmonic terms. There has also been no information on relevant anharmonic potential couplings forthcoming from the recent investigations of the force field.

During the course of this work a number of other classical studies on the CH stretch overtones of C 6H 6 have appeared in the literature26,27,28. These include a number of classical studies that were involved, at least in part, in looking at the effect o f the potential energy surface on the dynamics. Unfortunately even the best of these potential surfaces does not describe all the bending modes accurately. Hence, these calculations cannot reproduce completely the dynamics of benzene which we have seen here, for they cannot exactly reproduce the important beating motions which rely on the

difference

between the bending mode frequencies. However, a number of features of these studies are of interest here. First, Bintz, Thompson, and Brady26,27 looking only at v = 6 and 9 , reported energy transfer out of the initially excited stretch and some energy transfer to the para, and to a lesser extent, ortho and meta CH oscillators on approximately the same timescale as here. Furthermore, they also noted the importance of the modes involving

48

CCH wag and found that the out-of-plane modes have negligible effect on the dynamics. In another study, Lu, Hase, and W olf28 looked at the effect of lowering the CCH wag potential coefficient upon excitation of the stretch. Briefly, they found that the decay times o f the CH overtone increase as the wag frequency is lowered by excitation o f the stretch.

The most important of the studies o f benzene appearing after the completion of this work is experimental work of Page et al.29 Using supersonic expansion and state selective multilevel saturation spectroscopy they obtained high resolution spectra o f the fundamental CH stretch and first and second overtone transitions of rotationally cooled (= 5K) benzene. The greatly reduced linewidths observed suggest that there are significant inhomogeneous contributions to the room temperature spectra of Berry et a l} '2 and the low temperature matrix spectrum of Perry and Zewail3. For v = 1 and 2 Page et al. found linewidths less than 1 c m '1 and for v = 3 they observed less than 10 cm '1 (a decay time more than 500 fs!). This contrasts strongly with the room temperature bandwidths of 43, 43 and 30 c m '1 for v = 1,2, 3. Owing to technical limitations, they were prevented from investigating above v = 3 so the effect o f inhomogeneous broadening to the higher overtones is not clear.

The size of the discrepancy between classical models and experiment may not be as large as the work of Page et al. may at first suggest. First, it is interesting that, prior to the new experiment, all the classical trajectory6,18,26'28,30, (as well as sem iclassical13, and quantum mechanical7,10,12) studies have reproduced linewidths (or decay times) that are in agreement with Berry et al. In particular, this has been seen here in the trends in the overtone decay rate with the v and with deuteration. This suggests that while the room tem perature bands for the

higher overtones

may have some inhom ogeneous components they still retain some significant homogenous bandwidth. Secondly, it should be noted that classical mechanics is least accurate at low energies, in particular, the inaccuracy involved with the classical description of the ground state becomes relatively large.

decay time is based on a number of assumptions, principally, that there is a large enough number of states coupled to the pure CH overtone state to lead to exponential decay. It is important to stress that the lifetime is related to the frequency width of a manifold of states each of which has some CH stretch character7. Clearly, the v = 3 spectrum of Page et al. does not contain a single Lorentzian manifold (corresponding to exponential decay) but in fact there are three other peaks around the central peak at 8827 cm '1. It thus appears that there are at least five states over a range o f about 70 c m '1, including probably two unresolved vibrational levels o f the central band, that have appreciable CH oscillator strength. It would not be expected that the width of this central peak, or a level resolved from it, would be in any way related to a single exponential of the type which we often observe. Rather, a nonexponential decay, possibly with some components much faster than 500 fs, would be expected from the spectroscopy. Their observations support two general points: that v = 3 is a poor local mode and that classical mechanics overestimates the decay rate. Furthermore, as there is significant variation o f the decay rates with overtone level, as well as some qualitative variation in the character o f the decay, including strongly nonexponential decay, it is difficult to make judgem ents about the correspondence of theory and experiment based solely on the v = 3 results. Clearly experimental results for higher overtones are needed.

Lu and H ase30' 32, following up their earlier work, have attempted to reconcile the classical trajectory results with the experiment of Page et al. In doing so they recognise the major problem is associated with the use o f classical mechanics as opposed to the potential energy surface; which they note would need physically unrealistic changes to produce a linewidth <10 c m '1 for v = 3. Specifically, they investigated means of correcting the improper classical treatment o f zero point motion through models with reduced dimensionality31 and with reduced zero point energy32. W hile both these approaches successfully decreased the overtone decay rate the reduced dimensionality models (CH3 and C 3H 3) appeared to follow a different relaxation pathway as the relative variation of overtone linewidth with v is different to the reduced zero point energy and standard trajectory values. It is pleasing to see that a reduced zero point motion model

50

reproduces the trends of the standard ground state model confirming our assumption that our own reduced zero point 'cold' trajectories provide an insight into the standard dynamics.

The use of classical mechanics can briefly be assessed. The decay rates out of CH stretching overtones are clearly overestimated but there is evidence'-to support the position that the trends are reasonably reproduced. This suggests that in classical mechanics the same relaxation pathways operate as do in quantum mechanics, at least in a case of relatively strong coupling. The errors in energy transfer rates therefore appear to be systematic and hence it should be possible to apply corrections to classical simulations to provide an accurate representation.

Some of these questions regarding the classical trajectory method are addressed further in the following chapters where the Fermi resonance mechanism is applied to energy transfer in hydrogen bonded amides.

In document LOS DERECHOS ECONÓMICOS, SOCIALES Y CULTURALES EN EL SISTEMA INTERAMERICANO: EL ARTÍCULO 26 DE LA CONVENCIÓN AMERICANA SOBRE DERECHOS HUMANOS (página 39-41)