MODELING THE VIBRATIONAL RELAXATION OF POLYATOMIC MOLECULES . THE METHYLFLUORIDE CASE STUDY

We present in this paper a theoretical analysis of the vibrational translational (V-T) relaxation process in CH3F, carried out by using a numerical model based on rate equations. In particular, we have analysed the dependence of the V-T relaxation time on the average vibrational energy absorbed per molecule. We have also investigated the influence of the dependence of the rate constants used in the model, on the gas translational temperature. The results of the model clearly outline the strongly nonlinear character of the V-T relaxation process in CH3F, a situation commonly observed in other important polyatomic molecules of intermediate size each as SF6, freons, and related methylhalides.


INTRODUCTION
In the last period, a number of papers has dealt with the gas phase collisional energy transfer in the electronic ground state of highly excited polyatomic molecules.The interest lies both in large polyatomics such as azulene, toluene, benzene, and smaller ones such as SF6, fluorochloromethanes, SiF4 (see for a review).In most of these experiments the key quantity measured is the average vibrational energy transfered per collision, (AE) and its dependence on the average vibrational energy initially stored by some means in one molecule.Often the results are a source of controversy and the dynamical factors governing this dependence are still under investigation, both experimentally and theoretically.
4][5][6] For example recent Monte-Carlo computer simu- lations for highly excited CS2 in Ar 3 were able to predict for (AE) values which are close to the experimental ones. 6The model yielded a linear dependence of (AE) on (E) for the whole investigated energy range.However, for CS2, as for many polyatomic molecules, the (AE) dependence on (E) shows a linear dependence -On leave from the Institute of Isotopic and Molecular Technology, Cluj-Napoca, Romania.147 in the low (E) range (up to 6000 cm-), followed by a stronger than linear dependence (where the V-T relaxation time is observed to decrease), and, finally, again a linear dependence in the high energy region (above 16,000 cm-).The same behaviour has been evidenced for SF6 4 and CF2HC15, although the above mentioned energy thresholds were found around 1000 and 4000 cm-1, respectively.
The CH3F molecule has been studied intensively for its interesting properties concerning the V-V relaxation, but also for its prospective use in the passive mode locking of the CO2 laser, 7 FIR radiation generation 8 and its possible applications concerning chemical and isotopic selectivity. 9e v3 C-F stretching mode has the fundamental (1081 cm-) resonant to the P(20) CO2 laser line, and, in the same time, is the lowest vibrational frequency which is believed to be the "doorway" for the V-T relaxation.The slow V-T relaxation caused by the high v3 value makes it possible to store the vibrational energy in the various vibrational modes for a relatively long period.The laser induced fluore- scence (LIF) techniques make thus it possible to study in great detail the intra-and inter-mode V-V relaxation and to measure the appropriate rate constants (see [1][2] and references cited therein).The V-T relaxation rate was also measured by LIF, double resonance, and thermal lens techniques.High resolution spectroscopy 3 revealed the existence of Fermi resonances and Coriolis interaction between the high vibrational levels 3v3 k'4 and (2V2, 2v5) 3v3.As a result the main rate constants for the vibrational energy exchange and transformation are rather well established.
The most natural way for a theoretical investigation of the relaxation phenomena in CH3F was thus to use these rate constants in kinetic models based on a given structure of the vibrational levels and on coupled rate equations for their population evolution.Indeed, such kinetic models 11'12 were able to reproduce the LIF results and also to predict 12 V-V rates for higher vibrational levels.
In the present paper we shall present the results obtained by adapting the model of Nakane and Tsuchiya, 2 hereafter called NT model, to the study of vibrational- translational relaxation processes.The NT model was originally developed for studying V-V intra-and inter-molecular relaxation in CH3F.In particular, we have paid great attention to the V-T relaxation time (TVT) dependence on (E), and we have also analyzed the dependence of (AE) on (E).By using this model we have, thus, been able to observe also in the case of CH3F the same three distinct regions characterizing the dependence of vr on (E) in SF6 and CF2HC1.'4'5 However, we have also observed some differences which will be discussed upon further on.Finally, the results obtained with the above model will be discussed by making reference to the abundant experimental findings on the beh.aviour of VT in medium size polyatomic molecules that we have gathered in the last few years. 5'4-7 2. THE KINETIC MODEL According to NT scheme, we have considered the vibrational levels up to 4000 cm-(i.e. up to 4v3) and divided them into 14 distinct levels, including the ground state.The energies E, degeneracies, and numeration of these levels are given in Table 1.energy levels of CHaF taken into account in the kinetic model.Here v25 (v2, vs) and v12 (vl, 2v2, 2v5,  Values for intramode and intermode rate of energy exchange are properly ascribed for these levels as shown in Table 2.The largest part of the rate values are deduced from experimentally measured quantities, another part is calculated from SSH theory, some of them were assigned with reasonable values12.As revealed by LIF experiments, the V-T relaxation occurs primarily through the v3 and v6 modes which present the two lowest energy levels of the scheme.The ratio of the v3 to v6 V-T rate is assumed to be about 0.4 as calculated from the SSH theory. The rate equations of the form dN/ -intra(V-V)terms + inter(V-V)terms + (V-T)terms (1)   have been written for the populations of all the Ni levels.Since the rates for the intra and intermode exchange, as well as the rates for the V-T relaxation satisfy the principle of detailed balancing, only 13 ordinary differential equations are needed, the 14th being redundant.
For the explicit form of the equations of system (1) the interested reader is refered to Eq. (17-35) of the NT paper.We have to mention however, some differences with respect to the NT model.Firstly, the excitation energy was not computed by assuming an absorption cross-section and a rotational relaxation rate, but by simply assuming that when the relaxation starts, the absorbed energy is found in the v3 ladder.This does not affect the relaxation modeling since we can neglect the population redistribution during the laser pulse.The fastest rate constant which characterizes the energy transfer from the v3 mode to other modes is estimated to be about 5 x 105 sec-torr-1, while for a standard CO2 laser pulse with a time duration of about 100 nsec this value would be reached only at p 20 torr.Therefore, at gas pressures of few torrs the assumption is justified.Such treatment gave us the possibility to study the dependence of the relaxation process not only on the average Table 2 Energy transfer processes and rate constants used in the kinetic model.The rate constants for the backward reactions are calculated assuming the detailed balancing rule.The values of the rate constants are taken from Ref. 12 where the interested reader can find reference to the original works (AE) Rate constant x 10 -5 Process (cm-1) (sec -1 x torr-) Intermode V-V energy transfer + 1049 ka 0.0037 x n + 1182 kt6 0.0087 x n energy absorbed but also on the initial vibrational energy distribution for a fixed value of Secondly, since the aim of this paper is not to investigate intra-and inter-mode dynamics, the rate equations for all the levels were written and numerically inte- grated over a relatively long time interval (up to 2 msec) with an integration step of about 0.1 /zs, in order to obtain the V-T characteristics.To this end, at each integration step we used the level populations Ni (t) to mpute the new vibrational energy 13 E(t) Ni (t) E (2) i=I and, consequently, the new translational-rotational energy and translational tem- perature.We have also avoided to assign a common vibrational temperature to all the levels during the integration (see discussion below).At each step, appropriate corrections of the rate constants values were made in order to take into consideration their dependence on the translational temperature.Thirdly and finally, we have rewritten Eq. ( 20) of ref. 12, which describes the dynamics of the lv3 (Ni) level as follows" dN1 (4kt3N14N-4ktb3N11N) + (3ktb3N4N-3kt3N11N) dt + (kd333N14No ku333N11N1) + (ku33N4N1 kd33NllNo) q-(ka,333)N,oN k333,4NllN ) with the notations used in Table 2 for the rate constants, N being the total number of molecules in the unit volume.
The system of coupled differential equations was integrated numerically by a standard Runge-Kutta-Gill method.The time dependent vibrational energy E(t) was used to compute both the instantaneous relaxation time rvrl(t) d{ln[E() E(t)/E(OO)dt E(0)]} (3) and the so-called "effective" relaxation time "'eff, 18 defined as the time in which the vibrational energy decays to the value" E(ree) [E(0) E(oo)]/e (4) In Eq. ( 3) and ( 4), E(0) and E(oo) are the vibrational energies at time 0 (immediately after the laser pulse) and when the V-T/R equilibrium has been reached, respectively.Since we have not used a common vibrational temperature, all the quantities which characterize the equilibrium situation were computed before starting the integration, by using an iterative numerical method to balance the total energy between the degrees of freedom of the system.At the end of the integration we checked that the final value of each quantity is nearly equal to the corresponding equilibrium value.The effective relaxation time yielded from each integration process has been used to compute the energy released per collision (AE), according to the following expression < AE> <E----> (5) preeZ where p and Z are the gas pressure and the rate of elastic collisions, respectively.

RESULTS AND DISCUSSIONS
All the results reported below refer to a pressure of 1 torr of pure CH3F gas.The system (1) has been integrated for initial average vibrational energies (per molecule) ranging from 10 cm -1 to 3000 cm-1.The result can be seen in Figure 1, where refe is plotted as a function of <E>.As one can see, after an initial plateau, the effective relaxation time strongly decreases by increasing the initial excitation energy, then reaches another plateau for energy values greater than about 1000 cm-1.The same behaviour was observed experimentally for many small and intermediate size molecules: SF6, 4 CF2HCI, 5 CS2, 6 and CF2C12.14<E> (era-) Figure 1 The dependence of the effective relaxation time (as defined by Eq. 4) on the energy absorbed per molecule.One can clearly observe the two plateaux corresponding to extremal excitation conditions.
We have also analysed the influence of the rate constants dependence on transla- tional temperature, upon the relaxation time.We have firstly solved the equations of system (1) by using rate constants independent of temperature.Then, we have obtained a new solution by using the same initial conditions but recalculating at each integration step new rate constants, assuming a T 1/2 power law dependence.The two sets of results were almost identical in the low energy region (which is the expected result since, in this case, the gas does not heat up too much) and were only slightly different in the high energy range.Therefore, one may conclude that it is not the heating of the gas which causes the strong decrease of vee, but the intrinsic nonlinearity of the relaxation process which is characteristic for high excited vibrational levels.
To illustrate this point we have plotted in Figure 2 the time dependence of the population of some vibrational levels, as governed by the rate equations for (E) 500 cm -1.Since we have used a semilogarithmic scale, the slope of each curve is proportional to the instantaneous V-T relaxation time of the corresponding level.It can be observed that during the relaxation the population of each level decays in a nonexponential manner, except perhaps the low levels v3 and v4 for which the effect is less pronounced.The fast rates observed in the first stage of the relaxation is due to the fact that a considerable amount of vibrational energy is transformed into kinetic energy during the V-V exchange processes.This effect has been observed experimen- Fibre 2 Time evolution of level populations for an initial excitation energy <E> 500 cm-.The following levels are included: va: ,3va: A A, va + v2: A, va + v4: , 4v3: v25: tally for molecules like CH3C1, CH3Br 19 and CF2HC1, which have the same number of atoms and similar spectroscopic structure.In addition one can see that the population ratio of certain levels is not constant.This means that the levels are not in equilibrium during the relaxation and, thus, one cannot define a common vibrational temperature to work with during the relaxation.The nonlinear behaviour was also observed by Nakane and Tsukiya 12 and was attributed to the existence of the 3v3 <---> v4 V-V energy transfer which generates the so-called "catastrophic cyclic path" mechanism proposed by Mandich and Flynn 2 for explaining the nonlinear behaviour of the OCS vibrational relaxation.Briefly, the vibrational energy absorbed initially in the v3 ladder by single-or multi-photon processes has the tendency to accumulate into high lying levels by means of up-the-ladder collisions.The existence of the 3v3 +---> v4 V-V relaxation channel allows the vibrational energy to relax into lower vibrational levels of the other normal modes, a process which causes their repopulation, closes the cycle but also produces kinetic energy.In our opinion, the existence of the 31;3 <---> v4 coupling (which is questionable, as shown in12) is not essential for starting the catastrophc cyclic path described above.This is because in the high levels region the vibrational energy will always find a channel to transfer to other modes, thus starting the cycle which will produce the nonlinear V-T relaxation.5O0 time (#see) 1000 Figure 3 Same as in Figure 2 in the case of low excitation ((E) 50 cm-1), with the following levels included 3v3, v3 + v12s, v3 + v4, 4v3, and with the same graphics as in Figure 2. .01 .oot .0001 500 time (#see) Figure 4 Same as in Figure 2 but with <E) 2500 cm-.
On the contrary, in the case of weak excitation we have observed that the population distribution in all the levels is in equilibrium at any time during the relaxation, and that the single vibrational temperature decays exponentially.This is well evidenced in Figure 3 where the same level populations as in Figure 2 are represented for low initial excitation energy ((E) 50 cmper molecule).
Finally, in order to get a better understanding of the dynamics of the level populations also in the case of intense molecular excitation, we have obtained a solution of system (1) for an energy value belonging to the region where a second plateau is observed in Figure 1, namely, (E) 2500 cm-.We recall again that a relaxation time independent of the excitation energy corresponds to a regime of exponential decay of the vibrational energy.The results are stored in Figure 4, where the time evolution of the population of the same vibrational levels as in Figure 2 is reported.
Because of the initial strong non-equilibrium conditions, in the first stages of the vibrational relaxation the population of some of the levels decays in a non- exponential manner.However, at later times (t -> 200/zs) all.thepopulation reach a quasi-equilibrium condition.This is contrary to what observed in Figure 2 ((E) 500 cm-1), where the nonlinear behaviour is evident at any time during the relaxation.Therefore, our results suggest that the exponential decay observed in CH3F, as well as in other polyatomics, in the case of high excitation can be a <E> (era-) Figure $ Log-Log plot of (AE) as a function of (E).The calculated slopes of the curve are unity for extremal energy conditions and 2 for the intermediate region.
consequence of the fast initial redistribution of the energy among the highly excited levels, followed by a V-T relaxation with a common rate.
The energy released by collision (AE) is represented in Figure 5 as a function of the energy absorbed per molecule (E) in a log-log plot.From this data we have calculated the power law dependences for the three distinct regions evidenced in Figure 1.We obtained a linear dependence AE : (E), (6)   for the two plateaus, and a quadratic dependence for the middle energy region.In fact one can verify from Figure 1 that fete depends on (E)in this last case.The same power laws were observed experimentally for CS. 6 For CFzHC1 we observed linear dependences for extremal energy conditions (very high and very low excitations) and stronger than linear dependence for intermediate energy.5 The power index in this last case was found however to depend on the laser frequency used for the excitation.As mentioned in the Section II we have also studied the influence of the initial energy distribution on the relaxation time.For the case of CH3F we have found very small differences in the instantaneous and effective relaxation times, when using the same initial amount of energy but stored in different levels of the v3 mode.This fact could be explained by the fact that, as seen in Table 2, the intra-mode V-V redistribution for the v3 mode is the fastest process in this molecule.In particular we observed that after few microseconds the level popula- tions reach about the same values for equal excitation energies, independently on the initial distribution of this energy.
The fast v3 intermode V-V relaxation would explain also the unusually low threshold for which the decreasing of ten occurs.As pointed out above, the re decreasing is produced when high lying levels begins to populate.Even in low fluence excitations, i.e. in the absence of multiphoton processes, these levels populate by the up-the-ladder type collisions, therefore compensating, at least partially, for the lack of multiphoton excitation.

CONCLUSIONS
We have carried out a theoretical analysis of the V-T relaxation process in CH3F molecules.To this end we have used a modified version of a model originally developed by Nakane and Tsuchiya for studying V-V intra-and inter-molecular ralaxation processes.Our study was mainly aimed to see whether a number of features characterizing the V-T relaxation in some important, intermediate size polyatomic molecules, such as SF6, freons and related methylhalides, were still present in CH3F.To this end, we have paid great attention on the V-T relaxation time dependence on (E).Our results have clearly evidenced (see Figure 1) that also aimed to testing the validity of our theoretical conclusions, and the final results will be the subject of a forthcoming publication.

Table
Vibrational