- Email: [email protected]
Dependence of the collisional relaxation of highly vibrationally excited polyatomic molecules on the population distribution function
2 September 1994
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 227 ( 1994) 164-l 69
Dependence of the collisional relaxation of highly vibrationally excited polyatomic molecules on the population distribution function Eduardo A. Coronado, Carlos A. Rinaldi, Gustav0 F. Velardez, Juan C. Ferrer0 INFIQC-Departamentode Fisicoquimica,Fact&ad de CienciasQuimicos,UniversidadNationalde Cdrdoba, CC 61, Sue. 16.5016 C&rdoba,Argentina Received 14 May 1994
Abstract The influence of the population distribution function on the collisional relaxation of highly vibrationally excited polyatomic molecules is analyzed in a non-reactive system. Unimodal and bimodal energy distributions are considered. Calculations made with unimodal energy distributions showedthat the energy decay is almost independent of its initial shape and also of the
collisionaltransition probability models, provided they have the same dependenceon energy.When the initial distribution is bimodal,the rate of energy decay, for constant averageenergytransferred, depends on the fraction of moleculesexcited, but if the decay is exponential,the energy loss profile is almost independent of the fraction q of moleculesexcited. These results are discussedin relation to the use of IR multiphoton absorption as an experimentaltechnique for the study of energy transfer processes.
1. Intruduction The collisional relaxation of highly vibrationally excited molecules has been the subject of much research, mainly with the purpose of measuring the average amount of energy transferred per collision with bath gases of different complexity and also as a function of the internal energy of the parent molecules [ l31. From an experimental point of view, modem, direct studies aim to measure the temporal evolution of the internal energy of excited species [ 4,5 1. In an ideal experiment, an ensemble of molecules is produced in a strictly monoenergetic initial distribution, N( E, to), and its decay is monitored by UV absorption [ 5 ] or IR fluorescence [ 4 1, under experimental conditions so that the width of the distribution re-
mains narrow at all times measured, compared with the total energy content of the excited molecules. In this case, the bulk average energy transferred, (( AE)) , becomes almost equal to the microscopic value,
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00805-l
E.A. Coronado et al. / Chemical PhysicsLetters 227 (1994) 164-169
to that corresponding to the difference between the electronic levels involved. Another limitation of this technique is that it can be applied only to those molecules which undergo fast internal conversion without any serious side complications that could interfere with the measurements. An alternative potentially powerful procedure to prepare vibrationally excited molecules in their ground electronic state is IR multiphoton absorption using a TEA CO2 laser as irradiation source [ 7- 10 1. The advantage of this technique is its applicability to a large number of molecules of different complexity, since many species have at least a moderate absorption cross section at the emission wavelength of the CO2 laser. Also, the initial excitation of the absorbing molecules can be easily varied by changing the incident laser energy. However, even though there is ample evidence that the absorbed photons are distributed over a broad range of vibrational levels and that, in the case of small molecules, the population density is bimodal [ 8,10,11], the real energy distribution is unknown and it is only characterized by the average energy absorbed. This can be a serious complication for the determination of the dependence of (AE) on energy. Taking into account these considerations, it seems of utmost importance to determine the influence of the energy distribution function on the time evolution of its first moment, ((E)). In this respect, in 1957, Montroll and Shuler, in agreement with Bethe and Teller [ 121, demonstrated that the relaxation of the mean energy of a system of harmonic oscillators depends only on the internal energy and not on its distribution [ 13 1. More recently, the problem of the relaxation of the moments of a distribution has been studied by several authors [ 6,14,15 1. Analytical solutions for the master equation have been possible only for certain cases and under restricted conditions for the collisional transition probability models [ 15,161. One solution of interest, which results in equal values of the microscopic and macroscopic average energy transferred, concerns linear decays with exponential transition probabilities [ 171. Another important case corresponds to the exponential relaxation of energy. According to the well-known theorem of Shuler et al. [ 18 1, the necessary and sufficient conditions for such decay is a linear dependence of the average energy
165
transferredonE,i.e. (AE)=c+& [19]. In order to overcome the limitations of the analytical solutions, in this Letter we present extensive numerical integration of the master equation, analyzing the influence of the internal energy distribution on the rate of energy decay for an ensemble of polyatomic molecules, its effect on the IR fluorescence measurements and on the determination of (AE). In this respect we consider different initial energy distributions, including uni- and b&modal functions, and we also discuss the effect of different models of the collisional transition probabilities, P(E’, E), on the energy relaxation process. The results are presented in relation to the use of IR multiphoton absorption as an experimental technique to study energy transfer processes.
2. Calculations The time evolution of the population N(E,t), in the absence of any process except collisional relaxation, is given by the following master equation [ 15 ] : d&5
) ldt
= -oN(E’
) +o c P(E', E)N(E) .
(1)
Here, o is the collision frequency and P(E', E) is the collisional transition probability from energy E to E of the excited molecule. This equation was numerically integrated with a predictor corrector algorithm [ 201, using a constant time interval of 0.5 ns and with an energy grain size of 50 cm-‘. All of the calculations were made for an arbitrary, non-reactive species, with initial excitation energies of 9000 and 15000 cm-‘. The collision frequency was o= 1 x IO*s-‘. The initial energy level populations were selected to encompass a wide spectrum of possibilities, going from the strictly monoenergetic Dirac function to the broad Boltzmann distribution, passing through the symmetric Gauss and Poisson functions, all at the same initial average energy value. Some calculations were also made using bimodal initial distributions. In this case, the initial population density was assumed to be the sum of a fraction q of molecules characterized by a Gaussian function plus a fraction 1 -q with a Boltzmann distribution at T= 300K. Obviously, as q decreases the maximum of the Gaus-
166
E.A. Coronado et al. / Chemical Physics Letters 227(1994) 164-169
Sian component shifts to higher energies since all of the calculations were made at the same value of the total average energy. The parameters corresponding to the different distribution functions are shown in Tables 1 and 2. The transition probabilities for deactivating collisions were calculated using the ordinary step-ladder and exponential models [ 2 11, which depend on the value of the average energy removed per collision, ( AE) ,+ The probabilities for activating collisions were obtained from the detailed balance condition. The density of vibrational states was calculated by exact count, using the following vibrational frequencies and degeneracies: 2265 (l), 914 (2), 747 (2), 658 (l), 364 (l), 259 (2). Different series of calculations were made considering a constant ( AE) d for step-ladder and exponential models and also assuming a linear dependence with the internal energy, i.e. (AE)d=a+b(E) for the exponential and the biexponential models of P( E’ , E) . The values of (A@ d as well as parameters a and b are given in the captions to the figures. Resolution of the master equation provided the time evolution of the level populations Ni, from which the temporal average energy (( E( t ) )>, was calculated. The IR fluorescence intensity was obtained using the expression of Durana and McDonald [ 22 ]
(2) where N,, is the density of excited molecules, A i’,Ois the Einstein coefficient for the fundamental spontaneous emission of the observed vibrational mode and p,(E) is the density of vibrational states for s oscillators at energy E. In these calculations, the IR fluorescence was supposed to be measured at 1500 cm-‘.
3. Results and discussion The temporal evolution of IR fluorescence and internal energy calculated by numerical integration of Eqs. ( 1) and (2 ), for different initial distribution functions and excitation energies (9000 and 15000 cm- * ) are displayed in Figs. l-4. These results show 16000 14000 12000
5
loooo \
Table 1 Expressions for the distribution functions
Gj< \ ._
BoltzmannNi= (gi/gi_i)Ni_, exp[ (E,-E,_, )/kT] with T= 1950 K for ((E)) = 9000 cm-i
..
..__ <
\
I
Gaussian
Ni= (2raZ)-“2exp[ - (Ei-E,,)‘/~u*], with a=3250cm-’ and EmuI= 15000 cm-’ and -Lx2 =9000 cm-’
Dirac
N,=S( (E) -E;)Ni
0.5
N.. 1
I
I\,
Ni=((E)/i!)exp(-(E))
Table 2 Parameters for the bimodal energy distribution: Ni = qNpm + (1 -q)Np’-=’ with T=300 Kand a=4000 cm-r 0 Q
E,,
0.5
30000 19000 22750 13750
0.5 0.7 0.7
(Gauss)/cm-’
((0 15000 9000 15000 9000
500
1000 Time
1500
2000
I ns
Fig. 1. Energy and IRF decays for two different initial average excitation energies and different energy distributions calculated with a stepladder transition probability model with ( AE)d = 250 cm-‘. (-) Boltzmann, (---) Dirac, (...) Gauss and (---) Poisson.
167
E.A. Coronado et al. / Chemical Physics Letters 227 (1994) 164-169 16000
16000 16000 - \
i . I
14000 12000 -
‘\
10000 -
‘\
6000 6000 4000 2000 0 0.6 I\
i\, --
I
I
I
.
0.5 0.4 -
04 2
L.
\
0.1
1..
--_
i
0.3 0.2 -
.\..\
0.2 I\
0.0
2 -
“\
0.3
/
I
I
I
0
500
1000
1500
0.1 -
0.0:
\_ 2000
Time I ns
0
1
500
1000
1500
2000
Time I ns
Fig. 2. Energy and IBF decays for two different initial average excitation energies and different energy distributions calculated with an exponential transition probability model with (U)d=250cm-1. (-) Boltzmann, (---) Dirac and (...) Gauss. The Boltzmann and Gauss functions give indistinguishable results at the lowest energy.
Fig. 3. Energy and IBF decays for two different initial average excitation energies and different energy distributions calculated with an exponential transition probability model with (AI!C),=100+0.02(E) cm-‘. (-) Bohzmaml, (---) Dime and (. . ) Gauss. The Boltzmann and Gauss functions give indistinguishable results at both energies.
that, for the unimodal case, both decays are almost independent of the initial energy distribution WE, to). The present calculations also demonstrate that the shape of the energy decay is independent of the transition probability model, provided the energy dependence is the same. Thus, calculations with step-ladder and exponential transition probability models, with the same energy independent (AQd, yielded indistinguishable linear decays of (( LL!?)) (Figs. 1 and 2). Also, calculations with an average energy transferred linearly dependent on internal energy, ( AQd= a + bE, resulted in nearly exponential energy decays, when either exponential or biexponential transition probability models, with the same ( AQd value, were used (Figs. 3 and 4). The independence of the evolution of the excitation energy on the shape of the initial distribution of vibrational states was first demonstrated by Montroll and Shuler for an ensemble of diatomic molecules
[ 13 1. Our results show that this behavior is more general, as it is also followed by polyatomic molecules regardless of the particular model for transition probabilities and of the energy dependence of ( AE) + As a consequence, it is possible to equate the macroscopic to the microscopic averages, that is, <(&)=(A&. The above results are restricted to unimodal energy distributions. In the case of infrared multiphoton excitation, there is ample evidence that the initial distribution can consist of two different ensembles of molecules [ 8 1. This situation is simulated by the calculations with a bimodal distribution, in which a fraction q of molecules is excited to upper levels, following an arbitrary selected Gaussian distribution while the rest remains in the Boltzmann distribution at 300 IL The observable experimental energy is the bulk average energy absorbed by the sample which differs from the true excitation function ((E)) 4 by a
168
E.A. Coronado et al. /Chemical Physics Letters 227 (1994) 164-169
16000
k
14000
14000
12000
12000
10000
b
10000
6000
;
6000
6000
p
; IA ;
6000
4000
4000
2000
2000
C
0
/
14000
0.5
9
k
0.3 0.2
\‘.... \ .-.. \ -.....
10000
i
6000
y
6000
,
‘;;;.._
12000 0.4
I
4000 0.1
\
2000
0.0, -0
0 500
1000
1500
2000
0
I
I
I
500
1000
1500
Time I ns
Fig. 4. Energy and IRF decays an exponential transition proba) with (AE),=lOO+O.O2(E) cm-‘andfor bility model (a biexponential model (...): P(E’, E)=0.25 exp[ - (EE’)/(150+0.03E)]+0.75exp[-(E-E’)/(SO+O.OlE)].
factor q. Two different situations arise related to the dependence of ( AE) d on E. Considering first the calculations with a (AE)d linearly dependent on E, the evolution of the global average internal energy exhibits the typical exponential decay, as expected. The results (Fig. 5 ) show that the system relaxes according to the average energy of the whole distribution, irrespective of the value of the fraction q. On the other hand, when a constant ( AZQdis used, the rate of energy decay depends on the average energy of the fraction q, (E),, instead of the mean value of the whole ensemble. Since ((A,!?)) is calculated from ~0 = -d<-Q)ldt, and <
2000
Timefns
Fig. 5. (a) Energy decay for an exponential transition probability cm-’ calculated for differmodel with ( AE) d=100+0.02(E) ent bimodal distributions.: qzO.5 (-), q-O.7 (...) and q= 1 (- - -). (b) Energy decay for a step-ladder transition probability model with (Ai?)., = 250 cm-’ calculated for different bimodal distributions: (-) q=O.5, (...) q=0.7and (---) q=l.
the fraction of molecules excited. However, this is not the case for systems relaxing with a constant (A&, where a knowledge of q is required. This fact has particular importance when the preparation of highly vibrationally excited molecules is achieved by infrared multiphoton excitation.
Acknowledgement The authors thank CONICET and CONICOR for partial financial support.
References [I ] I. Oref and D.C. Tardy, Chem. Rev. 90 ( 1990) 1407. [ 21 J.R. Barker and B.M. Toselli, Intern. Rev. Phys. Chem. 12 (1993) 305.
[ 31 H. Hippler and J. Troe, Bimolecular collisions (The Royal Society of Chemistry, London, 1989).
E.A. Coronado et al. / Chemical Physics Letters 227 (1994) 164-169 [4] M.L. Yet-ram, J.D. Brenner, K.D. King and J.R. Barker, J. Phys. Chem. 94 ( 1990) 6341. [S] H. Hippler, J. Troe and H. Wendelken, J. Chem. Phys. 78 (1983) 6709. [6] W. Forst and J.R. Barker, J. Chem. Phys. 83 (1985) 124. [7] J.M. Zellweger, T.C. Brown and J.R. Barker, J. Chem. Phys. 83 (1985) 6261. [8] V.S. Letokhov, Laser spectroscopy of highly vibrationally excited molecules, USSR Academy of Sciences, Institute of Spectroscopy (Hilger, Bristol, 1989). [9] D.W. Lupo and M. Quack, Chem. Rev. 87 (1987) 181. [lo] J.C. Jam-Wren, D.W. Setser and J.C. Ferrero, J. Phys. Chem. 89 (1985) 414. [ 111 B.M. Toselli, J.C. Ferrer0 and E.H. Star&o, J. Phys. Chem. 89 (1985) 1192. [ 121 H. Bethe and E. Teller, Deviations from thermal equilibrium in shock waves, Ballistic Research Laboratory, report X 117 (1941).
169
[ 131 E.W. Montroll and E.K. Shuler, J. Chem. Phys. 26 (1957) 454. [ 141 G. Gidiotis and W. Forst, J. Phys. Chem. 90 (1986) 3496. [ 151 J. Tree, J. Chem. Phys. 77 (1982) 3485. [ 161 A.P. Penner and W. Forst, J. Chem. Phys. 67 (1977) 5296; W. Forst and A.P. Penner, J. Chem. Phys. 72 (1980) 1435; W. Forst, J. Chem. Phys. 80 (1984) 2504. [ 171 W. Forst and G.Y. Xu, J. Chem. Phys. 86 (1987) 1269. [ 18) K.E. Shuler, G.H. Weiss and K. Andersen, J. Math. Phys. 3 (1962) 550. (191 W.Forst,G.Y.XuandG.Gidiotis,Can. J.Chem.65 (1987) 1639. [20] W.H. Press, B.P. Flannery, S.A. Tenkolsky and W.T. Vetterling, Numerical recipes (Cambridge Univ. Press, Cambridge, 1989). [21] D.C. Tardy and B.S. Rabinovitch, Chem. Rev. 77 ( 1977) 369. [22] J.F. Durana and J.D. McDonald, J. Chem. Phys. 64 (1976) 2518.
ELSEVIER
CHEMICAL PHYSICS LETTERS
Chemical Physics Letters 227 ( 1994) 164-l 69
Dependence of the collisional relaxation of highly vibrationally excited polyatomic molecules on the population distribution function Eduardo A. Coronado, Carlos A. Rinaldi, Gustav0 F. Velardez, Juan C. Ferrer0 INFIQC-Departamentode Fisicoquimica,Fact&ad de CienciasQuimicos,UniversidadNationalde Cdrdoba, CC 61, Sue. 16.5016 C&rdoba,Argentina Received 14 May 1994
Abstract The influence of the population distribution function on the collisional relaxation of highly vibrationally excited polyatomic molecules is analyzed in a non-reactive system. Unimodal and bimodal energy distributions are considered. Calculations made with unimodal energy distributions showedthat the energy decay is almost independent of its initial shape and also of the
collisionaltransition probability models, provided they have the same dependenceon energy.When the initial distribution is bimodal,the rate of energy decay, for constant averageenergytransferred, depends on the fraction of moleculesexcited, but if the decay is exponential,the energy loss profile is almost independent of the fraction q of moleculesexcited. These results are discussedin relation to the use of IR multiphoton absorption as an experimentaltechnique for the study of energy transfer processes.
1. Intruduction The collisional relaxation of highly vibrationally excited molecules has been the subject of much research, mainly with the purpose of measuring the average amount of energy transferred per collision with bath gases of different complexity and also as a function of the internal energy of the parent molecules [ l31. From an experimental point of view, modem, direct studies aim to measure the temporal evolution of the internal energy of excited species [ 4,5 1. In an ideal experiment, an ensemble of molecules is produced in a strictly monoenergetic initial distribution, N( E, to), and its decay is monitored by UV absorption [ 5 ] or IR fluorescence [ 4 1, under experimental conditions so that the width of the distribution re-
mains narrow at all times measured, compared with the total energy content of the excited molecules. In this case, the bulk average energy transferred, (( AE)) , becomes almost equal to the microscopic value,
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00805-l
E.A. Coronado et al. / Chemical PhysicsLetters 227 (1994) 164-169
to that corresponding to the difference between the electronic levels involved. Another limitation of this technique is that it can be applied only to those molecules which undergo fast internal conversion without any serious side complications that could interfere with the measurements. An alternative potentially powerful procedure to prepare vibrationally excited molecules in their ground electronic state is IR multiphoton absorption using a TEA CO2 laser as irradiation source [ 7- 10 1. The advantage of this technique is its applicability to a large number of molecules of different complexity, since many species have at least a moderate absorption cross section at the emission wavelength of the CO2 laser. Also, the initial excitation of the absorbing molecules can be easily varied by changing the incident laser energy. However, even though there is ample evidence that the absorbed photons are distributed over a broad range of vibrational levels and that, in the case of small molecules, the population density is bimodal [ 8,10,11], the real energy distribution is unknown and it is only characterized by the average energy absorbed. This can be a serious complication for the determination of the dependence of (AE) on energy. Taking into account these considerations, it seems of utmost importance to determine the influence of the energy distribution function on the time evolution of its first moment, ((E)). In this respect, in 1957, Montroll and Shuler, in agreement with Bethe and Teller [ 121, demonstrated that the relaxation of the mean energy of a system of harmonic oscillators depends only on the internal energy and not on its distribution [ 13 1. More recently, the problem of the relaxation of the moments of a distribution has been studied by several authors [ 6,14,15 1. Analytical solutions for the master equation have been possible only for certain cases and under restricted conditions for the collisional transition probability models [ 15,161. One solution of interest, which results in equal values of the microscopic and macroscopic average energy transferred, concerns linear decays with exponential transition probabilities [ 171. Another important case corresponds to the exponential relaxation of energy. According to the well-known theorem of Shuler et al. [ 18 1, the necessary and sufficient conditions for such decay is a linear dependence of the average energy
165
transferredonE,i.e. (AE)=c+& [19]. In order to overcome the limitations of the analytical solutions, in this Letter we present extensive numerical integration of the master equation, analyzing the influence of the internal energy distribution on the rate of energy decay for an ensemble of polyatomic molecules, its effect on the IR fluorescence measurements and on the determination of (AE). In this respect we consider different initial energy distributions, including uni- and b&modal functions, and we also discuss the effect of different models of the collisional transition probabilities, P(E’, E), on the energy relaxation process. The results are presented in relation to the use of IR multiphoton absorption as an experimental technique to study energy transfer processes.
2. Calculations The time evolution of the population N(E,t), in the absence of any process except collisional relaxation, is given by the following master equation [ 15 ] : d&5
) ldt
= -oN(E’
) +o c P(E', E)N(E) .
(1)
Here, o is the collision frequency and P(E', E) is the collisional transition probability from energy E to E of the excited molecule. This equation was numerically integrated with a predictor corrector algorithm [ 201, using a constant time interval of 0.5 ns and with an energy grain size of 50 cm-‘. All of the calculations were made for an arbitrary, non-reactive species, with initial excitation energies of 9000 and 15000 cm-‘. The collision frequency was o= 1 x IO*s-‘. The initial energy level populations were selected to encompass a wide spectrum of possibilities, going from the strictly monoenergetic Dirac function to the broad Boltzmann distribution, passing through the symmetric Gauss and Poisson functions, all at the same initial average energy value. Some calculations were also made using bimodal initial distributions. In this case, the initial population density was assumed to be the sum of a fraction q of molecules characterized by a Gaussian function plus a fraction 1 -q with a Boltzmann distribution at T= 300K. Obviously, as q decreases the maximum of the Gaus-
166
E.A. Coronado et al. / Chemical Physics Letters 227(1994) 164-169
Sian component shifts to higher energies since all of the calculations were made at the same value of the total average energy. The parameters corresponding to the different distribution functions are shown in Tables 1 and 2. The transition probabilities for deactivating collisions were calculated using the ordinary step-ladder and exponential models [ 2 11, which depend on the value of the average energy removed per collision, ( AE) ,+ The probabilities for activating collisions were obtained from the detailed balance condition. The density of vibrational states was calculated by exact count, using the following vibrational frequencies and degeneracies: 2265 (l), 914 (2), 747 (2), 658 (l), 364 (l), 259 (2). Different series of calculations were made considering a constant ( AE) d for step-ladder and exponential models and also assuming a linear dependence with the internal energy, i.e. (AE)d=a+b(E) for the exponential and the biexponential models of P( E’ , E) . The values of (A@ d as well as parameters a and b are given in the captions to the figures. Resolution of the master equation provided the time evolution of the level populations Ni, from which the temporal average energy (( E( t ) )>, was calculated. The IR fluorescence intensity was obtained using the expression of Durana and McDonald [ 22 ]
(2) where N,, is the density of excited molecules, A i’,Ois the Einstein coefficient for the fundamental spontaneous emission of the observed vibrational mode and p,(E) is the density of vibrational states for s oscillators at energy E. In these calculations, the IR fluorescence was supposed to be measured at 1500 cm-‘.
3. Results and discussion The temporal evolution of IR fluorescence and internal energy calculated by numerical integration of Eqs. ( 1) and (2 ), for different initial distribution functions and excitation energies (9000 and 15000 cm- * ) are displayed in Figs. l-4. These results show 16000 14000 12000
5
loooo \
Table 1 Expressions for the distribution functions
Gj< \ ._
BoltzmannNi= (gi/gi_i)Ni_, exp[ (E,-E,_, )/kT] with T= 1950 K for ((E)) = 9000 cm-i
..
..__ <
\
I
Gaussian
Ni= (2raZ)-“2exp[ - (Ei-E,,)‘/~u*], with a=3250cm-’ and EmuI= 15000 cm-’ and -Lx2 =9000 cm-’
Dirac
N,=S( (E) -E;)Ni
0.5
N.. 1
I
I\,
Ni=((E)/i!)exp(-(E))
Table 2 Parameters for the bimodal energy distribution: Ni = qNpm + (1 -q)Np’-=’ with T=300 Kand a=4000 cm-r 0 Q
E,,
0.5
30000 19000 22750 13750
0.5 0.7 0.7
(Gauss)/cm-’
((0 15000 9000 15000 9000
500
1000 Time
1500
2000
I ns
Fig. 1. Energy and IRF decays for two different initial average excitation energies and different energy distributions calculated with a stepladder transition probability model with ( AE)d = 250 cm-‘. (-) Boltzmann, (---) Dirac, (...) Gauss and (---) Poisson.
167
E.A. Coronado et al. / Chemical Physics Letters 227 (1994) 164-169 16000
16000 16000 - \
i . I
14000 12000 -
‘\
10000 -
‘\
6000 6000 4000 2000 0 0.6 I\
i\, --
I
I
I
.
0.5 0.4 -
04 2
L.
\
0.1
1..
--_
i
0.3 0.2 -
.\..\
0.2 I\
0.0
2 -
“\
0.3
/
I
I
I
0
500
1000
1500
0.1 -
0.0:
\_ 2000
Time I ns
0
1
500
1000
1500
2000
Time I ns
Fig. 2. Energy and IBF decays for two different initial average excitation energies and different energy distributions calculated with an exponential transition probability model with (U)d=250cm-1. (-) Boltzmann, (---) Dirac and (...) Gauss. The Boltzmann and Gauss functions give indistinguishable results at the lowest energy.
Fig. 3. Energy and IBF decays for two different initial average excitation energies and different energy distributions calculated with an exponential transition probability model with (AI!C),=100+0.02(E) cm-‘. (-) Bohzmaml, (---) Dime and (. . ) Gauss. The Boltzmann and Gauss functions give indistinguishable results at both energies.
that, for the unimodal case, both decays are almost independent of the initial energy distribution WE, to). The present calculations also demonstrate that the shape of the energy decay is independent of the transition probability model, provided the energy dependence is the same. Thus, calculations with step-ladder and exponential transition probability models, with the same energy independent (AQd, yielded indistinguishable linear decays of (( LL!?)) (Figs. 1 and 2). Also, calculations with an average energy transferred linearly dependent on internal energy, ( AQd= a + bE, resulted in nearly exponential energy decays, when either exponential or biexponential transition probability models, with the same ( AQd value, were used (Figs. 3 and 4). The independence of the evolution of the excitation energy on the shape of the initial distribution of vibrational states was first demonstrated by Montroll and Shuler for an ensemble of diatomic molecules
[ 13 1. Our results show that this behavior is more general, as it is also followed by polyatomic molecules regardless of the particular model for transition probabilities and of the energy dependence of ( AE) + As a consequence, it is possible to equate the macroscopic to the microscopic averages, that is, <(&)=(A&. The above results are restricted to unimodal energy distributions. In the case of infrared multiphoton excitation, there is ample evidence that the initial distribution can consist of two different ensembles of molecules [ 8 1. This situation is simulated by the calculations with a bimodal distribution, in which a fraction q of molecules is excited to upper levels, following an arbitrary selected Gaussian distribution while the rest remains in the Boltzmann distribution at 300 IL The observable experimental energy is the bulk average energy absorbed by the sample which differs from the true excitation function ((E)) 4 by a
168
E.A. Coronado et al. /Chemical Physics Letters 227 (1994) 164-169
16000
k
14000
14000
12000
12000
10000
b
10000
6000
;
6000
6000
p
; IA ;
6000
4000
4000
2000
2000
C
0
/
14000
0.5
9
k
0.3 0.2
\‘.... \ .-.. \ -.....
10000
i
6000
y
6000
,
‘;;;.._
12000 0.4
I
4000 0.1
\
2000
0.0, -0
0 500
1000
1500
2000
0
I
I
I
500
1000
1500
Time I ns
Fig. 4. Energy and IRF decays an exponential transition proba) with (AE),=lOO+O.O2(E) cm-‘andfor bility model (a biexponential model (...): P(E’, E)=0.25 exp[ - (EE’)/(150+0.03E)]+0.75exp[-(E-E’)/(SO+O.OlE)].
factor q. Two different situations arise related to the dependence of ( AE) d on E. Considering first the calculations with a (AE)d linearly dependent on E, the evolution of the global average internal energy exhibits the typical exponential decay, as expected. The results (Fig. 5 ) show that the system relaxes according to the average energy of the whole distribution, irrespective of the value of the fraction q. On the other hand, when a constant ( AZQdis used, the rate of energy decay depends on the average energy of the fraction q, (E),, instead of the mean value of the whole ensemble. Since ((A,!?)) is calculated from ~0 = -d<-Q)ldt, and <
2000
Timefns
Fig. 5. (a) Energy decay for an exponential transition probability cm-’ calculated for differmodel with ( AE) d=100+0.02(E) ent bimodal distributions.: qzO.5 (-), q-O.7 (...) and q= 1 (- - -). (b) Energy decay for a step-ladder transition probability model with (Ai?)., = 250 cm-’ calculated for different bimodal distributions: (-) q=O.5, (...) q=0.7and (---) q=l.
the fraction of molecules excited. However, this is not the case for systems relaxing with a constant (A&, where a knowledge of q is required. This fact has particular importance when the preparation of highly vibrationally excited molecules is achieved by infrared multiphoton excitation.
Acknowledgement The authors thank CONICET and CONICOR for partial financial support.
References [I ] I. Oref and D.C. Tardy, Chem. Rev. 90 ( 1990) 1407. [ 21 J.R. Barker and B.M. Toselli, Intern. Rev. Phys. Chem. 12 (1993) 305.
[ 31 H. Hippler and J. Troe, Bimolecular collisions (The Royal Society of Chemistry, London, 1989).
E.A. Coronado et al. / Chemical Physics Letters 227 (1994) 164-169 [4] M.L. Yet-ram, J.D. Brenner, K.D. King and J.R. Barker, J. Phys. Chem. 94 ( 1990) 6341. [S] H. Hippler, J. Troe and H. Wendelken, J. Chem. Phys. 78 (1983) 6709. [6] W. Forst and J.R. Barker, J. Chem. Phys. 83 (1985) 124. [7] J.M. Zellweger, T.C. Brown and J.R. Barker, J. Chem. Phys. 83 (1985) 6261. [8] V.S. Letokhov, Laser spectroscopy of highly vibrationally excited molecules, USSR Academy of Sciences, Institute of Spectroscopy (Hilger, Bristol, 1989). [9] D.W. Lupo and M. Quack, Chem. Rev. 87 (1987) 181. [lo] J.C. Jam-Wren, D.W. Setser and J.C. Ferrero, J. Phys. Chem. 89 (1985) 414. [ 111 B.M. Toselli, J.C. Ferrer0 and E.H. Star&o, J. Phys. Chem. 89 (1985) 1192. [ 121 H. Bethe and E. Teller, Deviations from thermal equilibrium in shock waves, Ballistic Research Laboratory, report X 117 (1941).
169
[ 131 E.W. Montroll and E.K. Shuler, J. Chem. Phys. 26 (1957) 454. [ 141 G. Gidiotis and W. Forst, J. Phys. Chem. 90 (1986) 3496. [ 151 J. Tree, J. Chem. Phys. 77 (1982) 3485. [ 161 A.P. Penner and W. Forst, J. Chem. Phys. 67 (1977) 5296; W. Forst and A.P. Penner, J. Chem. Phys. 72 (1980) 1435; W. Forst, J. Chem. Phys. 80 (1984) 2504. [ 171 W. Forst and G.Y. Xu, J. Chem. Phys. 86 (1987) 1269. [ 18) K.E. Shuler, G.H. Weiss and K. Andersen, J. Math. Phys. 3 (1962) 550. (191 W.Forst,G.Y.XuandG.Gidiotis,Can. J.Chem.65 (1987) 1639. [20] W.H. Press, B.P. Flannery, S.A. Tenkolsky and W.T. Vetterling, Numerical recipes (Cambridge Univ. Press, Cambridge, 1989). [21] D.C. Tardy and B.S. Rabinovitch, Chem. Rev. 77 ( 1977) 369. [22] J.F. Durana and J.D. McDonald, J. Chem. Phys. 64 (1976) 2518.