Reduction of the Pathways of Photochemical Gasphase Reactions Induced by Multiple Photon IR Laser Excitation of Molecules

Two different cases, (1) thermal initiation and (2) collisionless IR multiple photon 
(MP) initiation, of initiating the model reaction of the successive decay of polyatomic 
molecules ABC → AB


INTRODUCTION
It is well known that during thermal initiation of gas-phase radical chemical reactions of polyatomic molecules, a great number of final products are usually formed and the yield of the desirable product is often rather low.This rather common situation is conditioned by a great number of potential chemical reactions in multicomponent mixtures, when under thermodynamic equilibrium the initial, inter- mediate and final products have the same high temperature.
Of considerable interest in this respect are chemical reactions under nonequilibrium and nonstationary conditions, when the temperatures of the initial, intermediate and final products are quite different.
Photochemical reactions may fall in this category.As known, photo- chemical reactions taking place as the electronic states of molecules are excited, essentially differ from thermal reactions (see, for example, Ref. 1).
Recent years have seen one more possibility of performing photo- chemical reactions in gas-phase through multiphoton excitation of the high vibrational levels of a polyatomic molecule by a powerful pulse of IR laser radiation. 2-5In some works successful experiments have been performed on synthesis of molecular compounds under the conditions of a gas-phase radical reaction during IR multiphoton excitation of one of the initial molecular compounds.In Ref. 6, for example, effective synthesis of the (CF3)3CI molecule during IR photodissociation of (CF3)3CBr in the presence of I2 was observed.
In this experiment one could observe controllable direct synthesis of the (CF3)3CI molecule with a high yield (up to 60%), and the forma- tion of other products typical of thermal reactions was suppressed.
In qualitative respect this is quite understandable since in IR photo- chemical dissociation of original molecules (as well as in UV photo- chemistry) the resulting radical-products have comparatively low energy.
The purpose of this paper is to give a more general explanation to a high degree of directivity, that is, reduction of the number of path-ways of the photochemical reaction stimulated by the MP excita- tion of the molecule.Using a simple model we are going to show that, in the process of IR MP photodissociation, it is possible to obtain the only desirable reaction product with a yield close to the limiting one, i.e., 100%.

PHOTOCHEMICAL REACTION MODEL
Here we are going to restrict ourselves to considering the case of successive dissociation of a polyatomic molecule ABC according to the scheme We shall not discriminate between the cases when the products AB, C, A and B are molecules or radicals.In the last case it is assumed that the reaction takes place with a sufficient amount of radical scavenger, so that all the resultant radicals are bound into stable final products.
To demonstrate a high degree of directivity of laser-induced reac- tions let us consider the case when the thermal rate of reaction ( 2) is higher than the rate of reaction (1), i.e., k2 >>kl (3) and AB is the desirable product.It is clear that in this case it is impossible to obtain a high yield of AB at thermal initiation since, according to (3), AB dissociates even faster than it is formed.We are going to show that, if reactions (1) and (2) take place under the conditions of MP collisionless photodissociation, the yield of AB, despite (3), may be as high as 100%.The resulting AB molecules are cold, because their average energy is less, by at least the value of energy of the AB--C bond, than that of dissociating ABC molecules.Their further decay (2) is possible only after absorbing some energy comparable to the energy of breaking of the A--B bond DE.Therefore, if the rate of MP excitation is properly chosen, dissoci- ation (2) does not occur and the desirable AB product is produced.
But when the reaction proceeds under thermal equilibrium, the result- ing AB molecules get quickly heated in collisions with the hot initial substance ABC and so dissociate at a very high rate.

THERMAL INITIATION OF REACTION
Let us consider the case of reaction under thermal equilibrium.The kinetic equations for the reaction of successive dissociation (1) and (2) can be presented in the form: 7   dfl kl(1-/ dt (4) dAB k (1 k 2AB dt where/ and /An are respectively the primary yield of dissociation reaction (1) and the yield of the desirable AB product in the presence of reaction (2), i.e., [ABC] Let the solution of system (4) be presented as the dependence of the relative yield of the desirable AB product /3aB//3= lAB ]/([AB + [A ]) on the total yield of ABC dissociation.It is the value/3aB//3 that reflects the degree of directivity of the chemical reaction.With k2 k solution ( 4) is expressed by the formula Aj= 1 [ kz/kl-1 With kz/k >> 1 a high relative yield of the desirable product is possible only at low degrees of processing of the initial product/3 << 1.At the same time for considerable values of/3 the value of An/ is very small.Specifically, with/3 1 / 2 and kz/kx << 1 expression (6) Thus, at thermal initiation of reaction it is impossible to carry on effective synthesis of the desirable AB product.
It must be noted that, in order to apply expression (6), it is necessary and sufficient that the ratio k/k should be constant in the course of the reaction.The area of its applicability, however, is not limited by reactions performed in a thermostat.Even when the medium is heated by pulsed IR radiation, it turns out that the basic part of the reaction takes place under temperature stabilization when its vari- ation, as/ increases for example from 0.1-0.9, is very small.Besides, the variation of k/k can be neglected even though the temperature changes greatly but the activation energies of reactions (!) and (2) are close to each other.

IR MULTIPHOTON INITIATION OF REACTION
Here we consider the case when reactions (1) and (2) are initiated by IR MP absorption by ABC and AB molecules.In this case it is possible to realize effective formation of the desirable AB product with/3an//3 1 at high yields/3 1, even despite condition (3).To find the values/3 and Ban it is necessary that we should know the distribution over vibrational states formed during MP excitation (E is the vibrational energy of molecule).To derive the function '(E) one should solve the systems of appropriate kinetic equations describ- ing energy absorption from the laser field and dissociation (see, for example, Refs. 3 and 4).As a matter of fact, they cannot be solved in the general form and so numerical computation is usually used in correlation with experiment.In certain approximations, however, it is possible to obtain rather simple analytical solutions of these equations.For instance, in an approximation of an s-multiple degener- ate linear oscillator at the levels where dissociation does not play a significant role the distribution of vibrational energy of a polyatomic molecule turns out to be a Boltzmann function 9'1   f,, gn [exp (-nhw] where g, (n + s 1)!/n !(s 1)! is the multiplicity of degeneration of the n-th level, and z denotes the vibrational statistical sum.The temperature T depends on the value of stored energy E (or the number of stored IR quanta ri).E s (8) hw exp (hw/kT)-I where w is the vibrational quantum frequency.For distribution (7) we can easily obtain the value of dispersion, that is, the energy distribution value , (E) 6E hw6n (h +2/S)1/2 (9) With >> 1 and s >> 1 distribution (7) is narrow enough, that is, 6n/a 1/s+1/<< 1.It should be noted that the calculations 4 of the excitation dynamics for more realistic models result in narrower distributions.To estimate the values of dissociation yield and relative yield of desirable product Ban/B we are going to use relation (9) thus overestimating the distribution f(E) width.Even though f(E) inevitably causes the value of Ban/ to be underestimated, we shall see that the values of an/B obtained will be rather high.The whole ensemble of the ABC, AB and A molecules that absorb radiation will be described with the composite function FMp(E) depending on the total energy E which comprises not only the vibrational energy but also the energy of breaking of bonds D1 and DE FMp(E)=faSC(E)+fAa(E-D)+fA(E-DI-D2) (10) The functions fABC (E), faa (E) and FA(E) describe the vibrational energy distribution in ABC, AB and A respectively.Definition (10)   comprising the shifts of initial vibrational energy by D for AB and by DI+D2 for A is convenient because the function FMp(E) (10)   does not change during unimolecular dissociation (1) or (2).Indeed, the decrease of the ABC molecules with the vibrational energy E is exactly compensated for by the increase in the number of the AB molecules with the vibrational energy E-D.Thus, the members related to dissociation turn out to be excluded from the equations for FMp(E).So the function FMp(E) turns out to be rather smooth like the one discussed above (7), whereas each of the distribution functions for separate components fABC (E), lAB (E) and fA(E) may be distorted greatly by dissociation.
To find the yields/3 and [3AB using the function Fp(E) one can take advantage of the well-known fact that the unimolecular dissoci- ation rate is highly dependent on the molecular energy E. This makes it possible to introduce such boundary energies Eo and Eo that we can consider with a sufficient accuracy such that ABC molecules with their energy E < F. do not dissociate and all the molecules with E >Eo dissociate ito AB and C (similarly for the AB molecule with its energy E approximating E:).If we neglect completely the molecular collisiens at all the stages of activation and reaction, we can consider tha ED1 =Da and ED2--Dx +DE.If the energy loss is allowed for, say, in their collisions with a cold buffer, ED1 >D1.In this case Eo is determined by that critical overexcitation level of the molecule when its dissociation rate equals the rate of cooling.The difference (Eo-Di) increases as the number of atoms in a molecule increases.
Thus, the values of/3 and/A are determined as The exact definition of the composite function F(N) for an arbitrary molecule is, generally speaking, a dicult task.But in our case this is not necessary since for the problem concerned it is only the width of energy distribution function that is of principal import- ance.Below we shall consider that the distribution width is given by formula (9) for a s-multiple degenerate harmonic oscillator.The s value corresponds to the number of vibrational degrees of freedom of the molecule.It must be also stressed that the formula turns out to be valid for an arbitrary system in the following two limiting cases.
First, with g << s expression (9) transforms to the Poisson distribution dispersion 8n which, as known, is formed during MP excitation in the case when opposite induced downward transitions can be neglected.Secondly, with g >> s it follows from (8) that n/g 1/ which agrees with the Boltzmann distribution width calculated with the use of the quasi-classical expression for density of states 0(N) ( + E)s, where is the energy of zero-point vibrations.
In both cases the distribution function F(N) with >> 1 and s >> 1 turns out to be close to the Gaussian one.For deniteness we shall consider the distribution Fe(E) to be Gaussian in such an arbitrary case, too () (12) where E is the energy stored by a molecule, and the dispersion 3E is given by expression (9).Now it is possible to write explicit 6 expressions for and n through the error functions err x l[erf (]._2EDI. (E--ED2 The fact that the distribution FM,(E) is narrow, that is, the condition can be fulfilled, enables us to make use of the asymptotic expansion 6   of err x in (13a) and (13b).Let us give an expression for the value (BaB/8)I/2, that is, a relative yield of desirable product with/3 1 / 2 .
Such a value of/3 can be obtained with E ED1, and in this case From expression (16) under condition (15) it follows that AB/ 1.This means that in case of IR MP initiation of reaction effective synthesis of the desirable AB product is possible.
It should be also noted that expressions (13a) and (13b) only approximate the true values/3 and BAn, rather roughly in some cases.
First of all, the values/3 and BAn calculated with the use of expressions (11) from composite distribution function (10) are approximated.
Besides, the distribution Ftp(E) will be distorted in dissociation since the cross-sections of radiation absorption and quantum state density are different for the ABC, AB and A molecules, with the total energy being the same.All this, however, (and especially the use of expression (12)) does not result in any essential limitations on the conclusion on the possibility of effective synthesis of desirable product during MP excitation.Indeed, of principal importance is only the narrowness (condition (15)) of laser-induced energy distribution Ftp(E) which is a characteristic feature of MP excitation.It is because of this that the possibility of effective synthesis of desirable product turns out to be a universal feature of collisionless MP excitation of molecules.

COMPARISON BETWEEN THERMAL AND MP INITIATIONS OF REACTIONS
The simple model of successive dissociation of molecules considered above enables one to draw an important qualitative conclusion on reducing the number of path-ways of photochemical reaction when changing from thermal (equilibrium) to IR MP collisionless (non- equilibrium) initiation.
Let us consider now a specific case of dissociation of the (CF3)3CBr molecule 6 (ff 36) and assume Eo2-Eol 20 000 cm-1.With T 1000 K and equal preexponentials this results in kE/k 318. Figure l a presents the calculated dependences of the relative yield of desir- able product BAn/B on the total yield / for the cases of thermal initiation (expression (6)) and MP initiation (expression (13)).In case of MP initiation (curve 2) the value of BAn/B remains close to unity almost up to full conversion of initial substance, that is, B 1. At the same time in case of thermal initiation (curve 1) the value of An/B tends quickly to zero as/ increases.Now we are going to compare the values of ([3AB/)1/2 (expressions ( 6) and ( 16).At thermal initi- ation (An/)I/2 3" 10-3", at MP initiation (3A/)/2 0.998.The comparison of these two values vividly demonstrates the possibility of highly effective directed synthesis at collisionless MP initiation, whereas at thermal initiation such a process is impossible.
For the cause of such a high increase in efficiency of desirable product yield A/ when changing from thermal to MP initiation of reaction to be more vivid, let us consider the vibrational energy distributions for the ABC, AB and A molecules formed in these reactions (Figure 2).As it may be seen from Figures la and lb, the average energy of the AB molecules at MP initiation is much lower (byDl) than that at thermal initiation.Really, AB molecules are formed (at the instant of dissociation) cold, that is, with their energy FIGURE Dependence of the relative yield of desirable product (BAn/B) on the total yield of reaction at thermal (curves 1)land MP (curves 2) initiation of reaction.being EABc-D1.But in a thermal reaction they quickly gain in collisions some additional energy ---D1.This comes about in times shorter than the dissociation time, and so right after the ABC molecules dissociate the AB product is also "ready" for dissociation (Figure 2b, curve 2).In this case the vibrational temperatures T of all the components are equal, that is, TA TAB TAnc.At the same time in case of collisionless MP dissociation the AB molecules remain sufficiently cold, and there will be no dissociation of AB if MP excitation does not go on up to the value Eo2.If some small part of AB should dissociate (Figure 2a, curve fan (E)), the average energy of the A molecules is smaller than that of AB by -D2.Thus, at MP initiation the relation TA < TAn < TAnc is valid.
6. CONCLUSION Thus, it has been shown herewith that one of the characteristic features of collisionless IR MP dissociation of polyatomic molecules is the possibility of suppressing the successive stages of reaction which bring about considerable fragmentation at thermal dissociation.Reducing the number of reaction pathways, that is, increasing the degree of dissociation directivity, as we pass from thermal dissociation to MP dissociation, results from extreme nonequilibrium of MP initiated reactions and a small width of vibrational energy distribution.
It should be noted that a change from thermal to laser-induced reaction can be observed in experiments as the exciting laser pulse duration or the pressure are decreased.The following reactions of (1)-(2) type were investigated at laser excitation; (CF3)3fBr -(CF3)3C" -' (CF3)2C: 6 and cyclobutyl acetatecyclo- butene-1,3-butadiene. 12 Both papers point to the possibility of effective synthesis of intermediate product when irradiated by rather short pulses.In Ref. 6 the predicted increase of intermediate (desir- able) product was observed as the pulse duration was decreased, that is, when passing from collisional to collisionless initiation of dissoci- ation.
In conclusion, we should note the following' 1. Everywhere in the above we considered the case when k2 >> k that is less favourable for effective formation of the desirable AB product.The result does not change qualitatively with k2 k since the use of MP excitation instead of thermal initiation also increases the efficiency of formation of the desirable product.The results of calculation on the basis of Eq. ( 4) (curve 1) and expressions (13) (curve 2) are given in Figure lb.
2. A greater gain in changing from thermal to MP initiation of reaction can be attained on condition (3), when the dissociation product AB (a molecule or a radical) is thermally less stable than the original product ABC.In the reverse case when the desirable AB product is thermally much more stable than the initial ABC product, the use of IR MP initiation of reaction instead of thermal initiation must not cause an essential increase in efficiency of formation of the desirable product.

FIGURE 2
FIGURE 2 Vibrational energy distributions for the initial molecule ABC, the desir- able product AB and the secondary product A. (a) Thermal initiation of reaction.(b) IR MP initiation of reaction.