General numerical solution to the time-dependent master equation for non-steady-state reactions: Application to the photoinitiated isomerization of cycloheptatriene to toluene

Chemical Physics I57 ( 199 1) 3 15-325 North-Holland

General numerical solution to the time-dependent master equation for non-steady-state reactions: application to the photoinitiated isomerization of cycloheptatriene to toluene Robert L. Jackson IBM Research Division, Almaden Research Center, San Jose, CA 95120-6099,

USA

Received 6 May 199 1

An efficient numerical algorithm has been developed to solve the time-dependent master equation. The method may be used to calculate the probability of reaction as a function of pressure for a vibrationally hot reactant produced under non-steady-state reaction conditions. To test the numerical algorithm, calculations were performed on the probability of reaction versus buffer gas pressure for the 260 nm photoinitiated isomerization of cycloheptatriene to toluene, which occurs on the ground state potential surface following internal conversion. The average energy transferred per collision, (A&,,,), from cycloheptatriene to 2 5 buffer gases was determined via master equation fits to the Stem-Volmer data of Troe and Wieters [J. Chem. Phys. 7 1 ( 1979) 393 11. The calculated values of (A&,,,,) agree very well with those measured directly by Troe and co-workers.

1. Introduction Statistical rate theory has been used successfully for several decades to calculate the rates of gas-phase unimolecular reactions [ 1-4 1. Chemists have been able to apply theories like RRKM as a general tool for some time now, due to the ready availability of computer programs that perform the required calculations in a straightforward manner [ 5 1. RRKM theory is commonly used to calculate rate constants for thermal unimolecular dissociation or isomerization reactions [ 6,7 1. Rate constants at a given temperature are not calculated directly, however. RRKM theory yields a set of microcanonical rate constants, k(E), each defined at a specific vibrational energy, E #I. One must average the set of k(E)3 over the reactant’s vibrational energy distribution, which is a complex function of the rates of collisional energy transfer between the reactant population and a bath gas. Rigorously, unimolecular rate constants I’ Although we identify the energy variable of k(E) with the reactant’s vibrational energy, all non-fixed energy of the reactant is available to overcome the reaction barrier, including energy in vibrations, internal rotations, and, for symmetric tops, the external rotation associated with the top axis [ 6).

can be calculated only by solving the full steady-state master equation for reaction and energy transfer [ 8 1. A general analytical solution to the steady-state master equation is not available #‘. To circumvent this problem, the strong-collision approximation has been commonly employed in statistical rate theory calculations to describe the process of collisional energy transfer between the reactant and a bath gas [ 6,7]. In the strong-collision approximation, collisional deactivation of all reactant molecules with energy in excess of the reaction threshold is assumed to occur as a single elementary step in the overall reaction mechanism. The rate of this step is taken to be the rate of collisions between the reactant and the bath gas. Although this greatly simplifies the calculation of thermal unimolecular reaction rate constants and provides qualitatively correct results, it is now generally recognized that the strong-collision approximation leads to unacceptable errors in calculated unimolecular rate constants within the falloff pressure regime [ 8 1. Recent experiments [ lo-141 that have directly measured the amount of vibrational energy transferred per collision for several species with a variety of bath gases (12Special cases may be solved analytically. See ref. [ 91.

0301-O104/9 l/$03.50 0 199 1 Elsevier Science Publishers B.V. All rights reserved.

316

R.L. Jackson /Time-dependent

master equation for non-steady-state reactions

show that it is quite unreasonable to expect all bath gases to deactivate a reactant in a single step, independent of the reactant’s vibrational energy content. The accuracy of the strong-collision approximation may be improved by redefining the deactivation rate as the product of the collision rate and a “collision efficiency”, varying from 0 to 1. Even with this modification, however, the process of deactivation is still treated as a single elementary reaction step. Hence, no real physical improvement is achieved over the original strong-collision approximation. Fortunately, accurate numerical solutions to the steady-state master equation for thermal unimolecular reactions have been published [ 15-181. A computer program is now available, combining RRKM theory with a full master equation treatment to calculate rates for this class of reactions [ 5 1. Thus, the calculations of thermal unimolecular reaction rate constants may now be performed with the same simplicity that was previously possible only by assuming strong collisions, but with the accuracy afforded by a full master equation treatment. While major progress has been made in the implementation of statistical rate theory for steady-state reactions, less progress has been made for unimolecular reactions where the steady-state approximation is not applicable. Such reactions are quite common. They include: ( 1) photoactivation, where a molecule excited electronically undergoes rapid internal conversion and subsequent reaction from the electronic ground state [ 19,201, (2) overtone activation, where a molecule is excited above its reaction threshold by pumping a high overtone of a C-H stretching vibration [21-241 and (3) photochemical activation, where a primary photodissociation product retains enough of the initial excitation energy to undergo a secondary, non-photochemical reaction [ 25,261. In all of these cases, the reactant’s vibrational energy distribution evolves in time as reaction and collisional energy transfer take place. While the strongcollision approximation has been applied to statistical rate theory calculations for these types of processes, there is no reason to expect it to be any more accurate for non-steady-state processes than it is for thermal processes. Rigorously, rate constants for nonsteady-state reactions may be calculated via statistical rate theory only by solving the time-dependent form of the master equation. Numerical solutions to

the time-dependent master equation have been described in the literature for specific reactions [2729] but no simple, general solution has been published to date. In this paper, we describe a simple, numerical solution to the time-dependent master equation that may be applied to all of the above reaction classes. A computer program to implement this solution is available [ 301. An accurate solution to the time-dependent master equation is obtained using a simple algorithm that requires minimal computational power. To verify the accuracy of our solution, we compare it to solutions obtained by previously used numerical methods that require a very much larger expenditure of computer time [ 27 1. In making these comparisons, we have examined the photoinitiated isomerization of cycloheptatriene at 260 nm in the presence of a bath gas, as studied by Troe and coworkers [ 19,311. The probability of reaction as a function of bath-gas pressure has been extremely wellstudied in this system, making it an ideal test for our time-dependent master equation algorithm. We obtain excellent agreement between calculated and experimental values of the average energy transferred per collision for the 25 bath gases studied by Troe and Wieters [ 311. We also discuss the nonlinear Stem-Volmer behavior expected for quenching of a non-steady-state reaction at low pressure.

2. Time-dependent master equation We begin by writing a general mechanism for a nonsteady-state unimolecular reaction of the type described in section 1. Consider a reactant X that is photoexcited in some manner to yield a species A that undergoes unimolecular reaction from the electronic ground state, forming a set of products P. A general reaction mechanism for this process is: kp X(t) -

k(E.E) A(E’, 2) A(& l) t k(E*E’)1k(F ) kc.0 1

P

(1)

P

Here, /c, represents the photoexcitation rate coefftcient for production of the species A at time t with a specific vibrational energy E.The conversion of X to A could represent many processes. For a photoacti-

R.L. Jackson /Time-dependent

master equation for non-steady-state reactions

vation process or an overtone activation process (see section 1)) X and A will be chemically identical, but A will be vibrationally, and possibly rotationally, excited as a result of photon absorption. For a photochemical activation process, X dissociates to A, but A retains enough of the initial excitation energy to dissociate further to the set of products P. In all cases, A( E, t) and A( E' , t) are chemically identical but have different vibrational energies. The interconversion between A( E, t) and A(E’ , 2) results from collisional energy transfer to an inert bath gas, which is assumed to be present in large excess. The process leading from A to P is a unimolecular reaction; we assume that the rate constant for this reaction as a function of energy is describable by RRKM theory. Based on the mechanism outlined in eq. ( 1), the timedependent master equation may be written as

317

light source wavelength, I( t ) is the light source intensity as a function of time, and @(E) defines the probability per unit energy that photoexcitation of X results in the formation of A with energy between E and E+ dE. Integrating g(E) over the energy yields the total quantum yield for conversion of X to A. To define the rate coefficients for collisional energy transfer, k(E’, E) and k(E, E’ ), we make the usual separation of the rate coefficient into a collision term and a probability term [ 16 ] : k(E’, E) =wP(E’,

E) ,

(4)

where w is the collision rate of A with the bath gas, and thus depends linearly on the bath-gas pressure, and P( E’ , E) is the probability that one collision of A(E) with a bath-gas molecule yields A(E’). P(E’, E) is normalized such that OD s

0

P(E’, E) dE’ = 1 .

(5)

0

-

s

k(E’, E)A(E,

t) dE’-k(E)A(E,

t) .

(2)

0

As is typically done in writing the steady-state master equation [ 15 18 1, we take the vibrational energy E to be a continuous variable for convenience in writing our equations. Vibrational state densities are typically very high for molecules with more than a few atoms at reactive energies, so it is not unreasonable to treat the vibrational energy as a continuous variable. We note, however, that the vibrational energy is quantized and that numerical solutions to the master equation are obtained by grouping vibrational levels into finite energy grains, where all levels falling within a given energy grain are considered to be degenerate. The rate coefficient kp depends on both energy and time. The energy dependence arises because the conversion of X to A produces a range of vibrational energies in A. The time dependence arises because the light source intensity may vary strongly with time, such as for a pulsed laser source. Assuming that the reactant population of X is optically thin, we can write: k,=ol(O@(E)

dE,

where u is the absorption

(3) cross section of X at the

P(E’, E) is related to P(E, E’ ) by the principle detailed balance. Thus we may write [ 161 P(E, E’)p(E’) =P(E’,

of

exp( -E’/kT)

E)p(E)

exp( -E/kT)

,

(6)

where p(E) is the vibrational state density of A at energy E, k is Boltzmann’s constant, and T is the bathgas temperature. Current understanding of collisional vibrational energy transfer is not yet sufficiently advanced to supply a general functional form for P( E’ , E). Consequently, reasonable mathematical models for P( E’ , E) have been developed. An exponential model is commonly used in steady-state master equation calculations:

P(E’, E) = --!--exp(-F), N(E)

E>E’,

(7)

where N(E) is a normalization factor chosen to fultill the condition of eq. (5); CYis a parameter with units of energy (see below). An elegant method for computing N(E) was given in ref. [ 161, and we follow the same procedure in our work. P(E’, E) for E< E’ is defined from P( E, E’ ), using eq. (6) in conjunction with eq. (7). Another mathematical model for P( E' , E) that has been used in steady-state

R.L. Jackson /Time-dependent master equation for non-steady-state reactions

318

master equation function

calculations

is based on the Poisson

’ E-E’

--,,,(-y), N(E)

P(E’,E)=

E>E’.

a

(8) Finally, a very simple model for P(E’, E) that has also been used is the stepladder model, based on a delta function: P(E’, E)=cS(E-El-a)

,

E>,E’ .

(9)

The stepladder model, as it is typically used, does not obey the principle of detailed balance. This is not necessary, however, if eq. ( 9 ) is used only for E 2 E’ , obeying the condition of eq. (6) to obtain P( E’, E) for E-cE’. Substituting for k,,, k(E’, E), and k( E, E’ ) in eq. (2) and taking advantage of the normalization of P(E’, E), the time-dependent master equation then becomes

U(E, t)

~

dt

=aZ(t)@(E)

dEX(t)

m

P(E, E’)A(E’,

+w

t) dE’-oA(E,

t)

0

-k(E)A(E,

t) .

(10)

To reach the final form of equation, we normalize starting concentration of A(E,t)=A(E,t)/X(O) giving:

U(E, t)

~

dt

=oZ(t)@(E)

the time-dependent master all concentrations to the the reactant X at t = 0, i.e. and X(t)=X(t)/X(O),

dZLY(t)

Go

+w

I

P(E, E’)A(E’,

t) dE’-oA(E,

t)

0

-k(E)A(E,

t) .

(11)

In solving the time-dependent master equation ( 11 ), we wish to obtain the total fraction of reactant remaining at long times after the light source has been turned off and as the population of A approaches thermal equilibrium. Thus, we seek JA (E, t) dE at a sufficiently long time that unimolecular reaction ofA

no longer occurs at a significant rate. A general analytical solution of this kind to eq. ( 11) does not exist. A straightforward numerical solution to the timedependent master equation for any given time may be obtained by setting dt to a very short, finite time interval, integrating piecewise in time until the desired time is reached. This approach was used by Luu and Troe for a limited number of time-dependent master equation calculations on the photoinitiated isomerization of cycloheptatriene to toluene [ 191. The method is laborious, since the time increment must be small enough that A (E) does not change significantly during a given time increment. Luu and Troe used a time increment of 0.0 1o - ’ . A large commitment of computer resources is required to determine A as it approaches thermal equilibrium, since the computation may cover 2 lo6 time intervals. While the time increment can often be increased, large time increments can result in large accumulated numerical errors in A. If k( E) is significantly larger than o for any of the energy grains to be considered, a time increment of 0.0 lw - ’ may be too large. Our goal is to find an accurate, general numerical solution that does not require such a large commitment of computer resources. To that end, we have developed an algorithm to integrate eq. ( 11) with a time interval of one collision lifetime, i.e. 100 times longer than the method described above. The algorithm is described in detail in the next section.

3. Numerical solution to the time-dependent master equation We begin the integration of eq. ( 11) by considering a single time interval from tl to tl. We then separate eq. ( 11) into terms representing flux into and out of a given energy grain of A. The first two terms on the right-hand side of eq. ( 11) represent flux into A(E), due to photoexcitation of Xand collisional energy transfer from all other energy levels of A, while the last two terms represent flux out of A (E), due to reaction of A (E) and collisional energy transfer from A(E) to all other energy levels of A. We allow flux into A(E) only during an infinitesimally short time at the end of each time increment. Flux out of A(E) is taken to occur continuously over each time incre-

R.L. Jackson /Time-dependent

ment. Thus, for the time increment write t)-k(E)A(E,

dt

Integrating eq. ( 12) over straightforward, giving: A(E, tz)lA(E,

t, to tz, we may m

+

M(E, t> = -wA(E,

time

t) .

(12)

from

t, to t2 is

(13)

whereAt=tz-t,. Eq. ( 13) gives the probability that A(E) does not undergo either reaction or collision during the time increment t, to t2. The total probability of reaction plus collision of A (E) is thus simply unity minus the exponential function on the right-hand side of eq. ( 13 ) . We assume that a molecule undergoing a collision is transferred to another energy level and does not react during the time period t, to t2. We can then write the probability of reaction of A(E) and collision of A(E) individually during the time increment t, to tz. The probability of reaction, R (E, t2, t, ), is given by

R(E,

tz,t,)=

5

dZ?’ P(E, E’)A(E’,

k(E)

tl)

0

w w+k(E’)

x(1-exp(-[k(E’)+w]At)} +A(E,

t,)=exp{-]k(E)+wlAt],

319

master equation for non-steady-state reactions

t,) exp{-

[k(E)+w]At}

.

(16)

The first term in eq. ( 16) represents the flux into A(E) due to photoexcitation of X, the second term represents the flux into A(E) due to energy transfer from all other energy levels of A, while the last term represents the fraction of A (E, t, ) remaining after losses due to reaction and collisional energy transfer from A(E). To calculate A (E, t) at a given time, eq. ( 16) is used in steps from t=O to the desired time, taking A (E, tz) determined at the end of each time increment as A (E, t, ) for the next time increment. As a check on eq. ( 16 ), we show that it transforms back to the differential form, eq. ( 11), as At approaches zero. This transformation is readily accomplished by taking the Taylor series expansion of the exponentials in eq. ( 16). For very small time increments, we may keep only the .linear terms of the expansion. Eq. ( 16) then becomes

w+k(E)

x{l-exp(-[k(E)+wlAt)}, while the probability

(14)

of collision, C( E, tl, t L), is given

A(-& tz)=o
s

fZ))At@(E)

GX(t)

dZ? P(E, E’)A(E’,

t,)

0

by

C(E,

tz,t,)=

x { 1 -exp(

w

+A(&

w+k(E)

- [k(E)+w]At)}

t,)-A(E,

t,)[k(E)+wlAt

which may be rearranged .

(15)

We then consider the flux into A(E) at the end of the time increment tl to t2. The flux into,4 (E) due to photoexcitation of X is obtained from the first term of eq. ( 11) by simply averaging Z(t) over time for the time increment t, to tz. The flux into,4 (E) due to energy transfer from A (E’ ) for all E’ is obtained from the second term of eq. ( 11) by folding in the probability that A (E’ ) has undergone a collision during the time increment t, to t2, as obtained from eq. ( 15). We can then approximately solve eq. ( 11) over At, giving

,

(17)

to give

A(E’t2);tA(E’ tl) =a(Z(t,,

t2))@(E)

dZTX(t)

00

+w

I

dE’ P(E, E’)A(E’,

t,)

0

-k(E)A(E,

t,)-wA(E,

t,) .

(18)

For sufficiently small At, eq. ( 18) and eq. ( 11) are identical. In the next section, we demonstrate the utility of our time-dependent master equation solution by fitting the extensive buffer-gas quenching data of Troe and Wieters for the photoisomerization of cycloheptatriene [ 3 11. We selected this system because the key

320

R.L. Jackson /Time-dependent master equation for non-steady-state reaclions

parameters, including k(E), o, and P( E’ , E) , have been very well characterized. The quality of our master equation solution may thus be judged by direct comparison of the calculated results to experimental data.

4. Application: photoinitiated isomerization of cycloheptatriene The UV photoinitiated isomerization of cycloheptatriene to toluene has been studied by several groups. Srinivasan [ 32 ] showed that the photoisomerization process occurs by rapid internal conversion of electronically excited cycloheptatriene to the ground state, followed by rearrangement on the ground-state potential surface. He also showed that the isomerization reaction could be quenched by addition of a buffer gas, due to competition between reaction and collisional deactivation of vibrationally hot cycloheptatriene following the initemal conversion process. The UV photoisomerization of cycloheptatriene has been studied in great detail by Thrush and coworkers [33,34] and by Troe and co-workers [ 19,27,3 11. In 1973, the Troe group published an exhausitive study of the quenching process, measuring the probability of toluene formation as a function of pressure for number of buffer gases [ 19 1. They analyzed their results using a simple Stern-Volmer treatment: (19) where Q(P) represents the quantum yield of the isomerization process at a buffer gas pressure P,ycrepresents the ratio of the collisional deactivation rate to the collision rate, and k(E,) is the isomerization rate constant for cycloheptatriene at the light source wavelength ( EP is actually taken to be the photon energy plus the average thermal energy of a cycloheptatriene molecule at room temperature). The LennardJones collision rate at the buffer gas pressure P was used to calculate W, while RRKM theory was used to calculate k(E). The parameter y_ termed the collision efficiency, was then adjusted to fit the slope of the Stern-Volmer plot. This parameter is identical to the collision efficiency discussed in section 1 in the context of the strong-collision approximation. Thus,

analysis of the quenching probability as a function of pressure using eq. ( 19) is based on the strong collision approximation. In 1979, the Troe group extended their quenching study of the cycloheptatriene photoisomerization to include several different wavelengths, although this study employed only helium and cycloheptatriene (self-quenching) as buffer gases [ 3 11. At the same time, similar He- and selfquenching studies were also performed at several different wavelengths for the photoisomerization of 7methylcycloheptatriene, 7-ethylcycloheptatriene, and 7-isopropylcycloheptatriene. In order to obtain the most accurate possible k(E)‘s, Troe and co-workers also measured rate constants for the thermal isomerization of cycloheptatriene and each of the substituted cycloheptatrienes in a shock-wave at high temperature and pressure [ 35 1. The k( E)‘s were calculated via RRKM theory by adjusting the vibrational frequencies assigned to the transition state in order to fit the high-pressure Afactor determined from the shock wave data. The k(E)‘s calculated in this way have a rather large uncertainty, since correction factors must be applied to obtain the high-pressure A-factor and activation energy from the data. Consequently, direct measurements of k(E,)were also performed for 7-ethylcycloheptatriene and 7-isopropylcycloheptatriene, by following the isomerization process spectroscopically in real time at several different wavelengths [ 361. (Direct measurements of this kind were not possible for cycloheptatriene and 7-methylcycloheptatriene, since isomerization occurred faster than the nanosecond resolution of the experiments.) Comparison of the calculated and measured k(E)% revealed an error of approximately a factor of two in the calculated k(E)‘s, which was attributed to errors in the high-pressure activation energy and A-factor determined from the shock-wave data [ 361. Once this error was corrected, however, the calculated and measured rate constants agreed, giving a high degree of confidence in the accuracy of the k(E)% calculated for cycloheptatriene and each of the substituted cycloheptatrienes. More recently, Troe and co-workers [ lo-141 together with Barker and co-workers [ 12,14 ] have pioneered techniques that allow “direct” measurement of the average amount of vibrational energy removed per collision, (A&,,), for a vibrationally hot molecule upon collision with a buffer gas. Both the Troe

R.L. Jackson / Time-dependent master equation for non-steady-state reactions

and Barker groups used photoactivation (see the definition in section 1) to produce vibrationally hot molecules with a fairly narrow spread of vibrational energies and followed the loss of vibrational energy using optical techniques: calibrated UV absorption in the case of the Troe group and IR emission in the case of the Barker group. These experiments are extremely important, since they were the first to yield ( AEcoll) directly for large molecules. The molecules studied include azulene [ 121, toluene [ lo], hexafluorobenzene [ 131 and substituted cycloheptatrienes [ 111. These measurements of (AE,,,,) provide insight in the collisional energy transfer probability function, P(E’, E). ( AE,,,) at a given vibrational energy E is given by [ 37 ] (A&,,,)=

4 (E-E’)P(E’,E)

dE’ .

(20)

0

While measurements of ( AEO,,) were not performed for cycloheptatriene directly, it is reasonable to assume that ( AEcO,,) for cycloheptatriene and toluene will be nearly identical for a given buffer gas. This assumption is confirmed by the similarity of the ( AE_,,) measurements for toluene, 7-ethylcycloheptatriene, and 7-isopropylcycloheptatriene [ 111. The extensive data obtained by Troe and co-workers for cycloheptatriene thus provide information on all of the parameters required to perform a time-dependent master equation calculation on the quantum efficiency of cycloheptatriene photoisomerization as a function of buffer gas pressure. The Lennard-Jones collision rates and the k(E)‘s determined by Troe and co-workers may be used directly in eq. ( 16 ). We can then treat (Yas an adjustable parameter in order to fit the quenching data for the 25 buffer gases studied by Troe and Wieters [ 3 11, using the three mathematical modelsforP(E’,E)definedineqs. (7)-(9);omay then be used to calculate ( AE,,,) from eq. (20). The quality of our master equation calculations may then be judged by comparing the calculated and experimental values of ( AE,,,,) . In order to Iit the experimental data, we calculated the quantum efficiency G(P) for reaction of cycloheptatriene, which is given by

(21)

321

To obtain Q(P), we solved eq. ( 16) for the cycloheptatriene system using the computer program of ref. [ 301, with a time increment of e_-’ and an energy grain size of 100 cm-‘. The computation was stopped when A (E, f2 ) and A (E, t, ) differed by less than one part in 10’. We took the parameters required to calculate the Lennard-Jones collision rates from the appendix of ref. [ 35 ] for cycloheptatriene, and from ref. [ lo] for the 25 buffer gases. The RRKM program of ref. [ 51 was used to calculate k(E) for the isomerization of cycloheptatriene to toluene, employing the vibrational frequencies and E. given in the appendix of ref. [ 351 (model B). State densities were computed for the RRKM calculation by direct count within the harmonic oscillator approximation. We multiplied the k( E)‘s by a correction factor of 0.5, as suggested by Troe and co-workers in ref. [ 361. (See below for master equation calculations using alternate k(E)'s. ] The initial vibrational energy distribution function @(E) was taken to be the thermal distribution of cycloheptatriene at 295 K shifted by the energy of a 260 nm photon. Since we are calculating the quantum efficiency for a first-order photoreaction uncomplicated by side reactions, we may choose arbitrary values for aand Z(t) in eq. ( 16). We chose I(t) to be a delta function at t= 0. This is equivalent to defining the time integral of the first term in eq. ( 16) as A (E, 0), which we arbitrarily set at 0.1. Note that A (E, 0) defined in this way is equivalent to the denominator in eq. (2 1). For each buffer gas and each of the three forms of P(E' , E), master equation calculations were performed at six pressures, corresponding to 0.5, 1, 1.5, 2,2.5, and 3 times the inverse slope of the Stern-Volmer plot reported in ref. [ 3 11. This pressure range spans the range of 1.3 to 3.2 for @(O)/@(p), which is approximately the same range covered by the data in ref. [ 3 11. Calculations for the six pressures were repeated, adjusting cr until the least-squares StemVolmer slope through the six calculated points was within 1% of the experimental Stern-Volmer slope. (AE,,,,) was then determined from (Yvia eqs. (7), ( 8 ), ( 9 ) , and (20). The calculated results are given in table 1, together with the experimental results of Troe and co-workers [ lo]. Fig. 1 shows the calculated Stern-Volmer plot for quenching of the cycloheptatriene photoisomerization by CO*. The Stern-Volmer plot is clearly non-

R.L. Jackson /Time-dependent master equation for non-steady-state reactions

322

Table I Comparison of calculated and measured (ref. [ 10 ] ) ( LL!&,) for collisional vibrational energy transfer from cycloheptatriene. Calculations were performed as described in the text using the exponential, Poisson, and normalized stepladder models for P(E’,

Buffer gas

(hE,,,)

(cm-‘)

exponential He Ne Ar Kr Xe H2 DZ NZ 02 co CO2 ND CH, CF, SF6 C,H, CZFS GHs GFs C,H,o CSHU &HI., GHM GHs CsH,s

55

97 111 159 180 78 91 91 136 144 280 206 247 345 270 430 514 470 689 668 790 1104 1249 1281 1881

Poisson

stepladder

measured

55 95 110 156 176 78 90 90 134 141 269 200 239 330 259 405 472 436 622 595 699 945 1048 1074 1486

54 94 107 152 170 76 89 89 131 138 257 193 229 314 249 382 443 408 569 557 644 842 936 941 1265

75 84 130 130 140 92 100 130 160 160 280 230 260 320 400 380 370 520 480 640 740 840 930 770 1150

linear at low pressure. Nonlinearity of this kind is not unexpected for quenching of a vibrationally hot reactant and was in fact discussed more than 30 years ago [ 381. A thorough discussion was recently presented by Chandler and Miller [ 28 1. Nonlinearity occurs because the Stem-Volmer model of eq. ( 19) does not account for the time-dependence of the reactant’s vibrational energy distribution as collisional energy transfer and reaction take place. Eq. ( 19 ) is actually derived from the strong-collision approximation, with deactivation taken to be a single elementary reaction step occurring at the rate WY,. Even a simple modification of the Stem-Volmer model, allowing deactivation to occur as a sequence of steps, reproduces the type of nonlinearity shown in fig. 1 [ 381. Still, fig. 1 is quite linear above a pressure of about 12 Torr.

Fig. I. Calculated Stem-Volmer plot for quenching of the 260 nm cycloheptatriene photoisomerization reaction by COz. The exponential model of eq. (7) was used for P(E’, E). The line represents a least-squares fit to the six points (0 ) used to calculate (AE,,,,) in table 1.

Similar behavior was observed for all buffer gases examined in this work. Our fitting procedure was thus restricted to pressures in the linear region of the calculated Stem-Volmer plot, as defined by the solid points in fig. 1. The experimental data do not clearly show a deviation from linearity at low pressure, but the data do not typically cover the pressure range where nonlinear behavior is expected. It is rarely possible to study quenching at very low buffer gas pressures, since the buffer gas must be maintained in large excess relative to the reactant. We must still account for the nonlinearity in our data-fitting procedure, however, since the experimental data were tit to eq. ( 19) [ 3 11, which demands an intercept of unity. The vulues of@(O)/

Q(p) determined from the master equation must therefore be renormalized so that the linear portion of the calculated Stern- Volmer plot has the same intercept (unity) as the experimental data. Without this renormalization, (A&,,,) determined from the master equation fits would be too low by about 30°h. The agreement between the calculated and experimental values of (A&,,,,) is excellent. Differences are ~20% for all buffer gases except He, Nz, and SF6. One interesting result is that (A&,,,) varies significantly for the larger colliders depending on the mathematical model chosen for P(E’, E). It has been shown previously [ 391 that rate constants calculated

R. L. Jackson /Time-dependent master equation for non-steady-state reactions

for thermal unimolecular reactions via the steadystate master equation do not depend on the functional form of P( E’ , E), provided ( AEm,,) is identical for each model. Extending that idea to our work, we did not expect the value of (L&,,,) required to fit the quenching data to depend on the functional form of P( E’ , E). The agreement between the calculated and measured (AE_,,,) is significantly better, however, for the Poisson and normalized stepladder models than for the exponential model when (AE,,,) is large. Related observations have been reported by Chandler and Miller [ 281. The exponential model differs from the Poisson and stepladder models in one very important respect: P( E’ , E) is largest for E = E’ in the exponential mode, while P( E’, E) is largest for (E-E’ ) =a for both the Poisson and stepladder models. Thus, the exponential model emphasizes elastic collisions, while the Poisson and stepladder models emphasize inelastic collisions. It is reasonable to expect collisions between molecules that have a large number of vibrational degrees of freedom to be predominantly inelastic, although direct energy transfer measurements for large molecules are too limited at present to draw a firm conclusion [ 141. Our master equation calculations on the cycloheptatriene system use k(E)‘s derived by Troe and coworkers from their shock-wave data for the thermal isomerization of cycloheptatriene [ 35 1. A correction factor of 0.5 was applied to the k(E)‘s, as recommended by Troe and co-workers, based on a comparison of similarly derived k(E)‘s to measured rate constants for photoisomerization of substituted cycloheptrienes at several wavelengths [ 361. We also performed master equation calculations for 7 of the 25 buffer gases using an alternate set of k( E)‘s. These k(E)‘s were derived by adjusting the vibrational frequencies assigned to the transition state in order to fit the high-pressure A-factor determined by Klump and Chesick [40] for the thermal isomerization of cycloheptatriene. No correction factor was applied to the alternate set of k(E)‘s. The values of (AE_,,,) derived using the data-fitting procedure described above with the alternate set of k(E)‘s are given in table 2. The values of (AE,,l,) in table 2 are only slightly higher than those in table 1. Consequently, we conclude that the high-pressure Arrhenius parameters determined by Klump and Chesick (logA=13.54, E,=51.1 kcal/mol) are to be pre-

323

Table 2 Comparison of calculated and measured (ref. [ 10) ) (A.&) for collisional vibrational energy transfer from cycloheptatriene using an alternate set of k(E)‘s derived from the thermal isomerization data of ref. [ 401 Buffer gas

He Ar co CO2 CzH6 CsH,z CaH1s

(AEm,,) (cm-‘) exponential

Poisson

stepladder

measured

59 118 154 299 462 856 2082

59 117 151 287 433 762 1619

59 114 147 274 407 690 1369

15 130 160 280 380 140 1150

ferred, at least at temperatures in the vicinity of their measurements (632-68 1 K). Similar Arrhenius parameters have also been obtained by other workers [41,42]. The agreement between ( AE,,,,) determined from our calculations and the experiments of Troe and coworkers for the cycloheptatriene system supports the accuracy of our solution to the time-dependent master equation. The accuracy is verified by comparing the results obtained using our algorithm to the results obtained by piecewise direct time integration of eq. ( 11) using a small At. If At is sufficiently small, one can be certain that A (E, t) calculated by direct time integration of eq. ( 11) is accurate. To make a direct comparison, the quantum efficiency of the cycloheptatriene photoisomerization process was calculated at six different pressures for the buffer gases He, Ar, CO, COZ, C2H6, CSH ,2, and CBHL8using our algorithm and using direct time integration of eq. ( 11). For our algorithm, we used a time increment of w-‘, while for direct time integration of eq. ( 11 ), we used a time increment ofO.Olo-‘. The results are identical within O.Ol”~ for each calculation performed, which is approximately the numerical accuracy of either method. Our algorithm thus clearly provides an accurate numerical solution to eq. ( 11). Most importantly, a typical calculation performed using our algorithm for the cycloheptatriene system requires approximately a factor of 60 less computer time than the same calculation performed by direct time integration of eq. ( 11) with a time increment of O.Olw-‘. Of course, it is possible in many cases to use a time

324

R.L. Jackson /Time-dependent master equation for non-steady-state reactions

increment much larger than O.Olo- ’ in the directtime-integration method. For time increments approaching w- ‘, direct time integration of eq. ( 11) will actually require somewhat less computer time than our algorithm. We found that neither method converged to a stable solution when the time increment was larger than w-r. The main advantage of our algorithm, however, is that it always works for a time increment of c.-‘. One need not worry that the solution will vary with the choice of time increment, and one need not optimize the time increment for each calculation in order to minimize the use of computer time. Our time-dependent master equation algorithm may be extended to calculations involving non-steadystate reaction systems of other types. Examples include chemical activation systems, although a nonsteady-state master equation solution applicable to these systems has been published recently [ 43 1. Hightemperature thermal reaction systems may also be treated. The steady-state master equation may not be appropriate at high temperatures, since reaction may occur at such a high rate that a steady-state reactant vibrational energy distribution is never completely established. The primary difference between systems of this kind and the photochemical systems we have considered in this paper involves the rate coefficient k, in eqs. ( 1) and (3). This rate coefficient must be defined in a manner appropriate to the system of interest, including a definition of the initial reactant vibrational energy distribution function, corresponding to #(E) in eq. (3).

5. Conclusions An efficient numerical algorithm requiring minimal computational resources has been developed to solve the time-dependent master equation. Our method may be used to calculate the probability of reaction as a function of pressure for a vibrationally hot reactant produced by photoactivation, overtone activation, or a photodissociation process. Calculations were performed on the probability of quenching the 260 nm photoisomerization of cycloheptatriene as a .function of buffer gas pressure. Fits to the Stern-Volmer data of Troe and Wieters [ 3 1 ] provide the average energy transferred per collision,

(A&,,,), from cycloheptatriene to 25 buffer gases. Values of (A&.,,,) calculated in this manner agree very well with those measured directly by Troe and co-workers [ IO]. Three different models for the probability of energy transfer per collision were used in the calculations: the exponential, Poisson, and normalized stepladder models. For large colliders, the results obtained from the exponential model show significantly poorer agreement with the data than the results obtained from the Poisson and normalized stepladder models.

Acknowledgement The author gratefully acknowledges Dr. Rong Zhang for critically reading the manuscript and for numerous helpful discussions.

References [ 1) C.N. Hinshelwood, Proc. Roy. Sot. A 1 I3 ( 1927) 230. [2] L.S. Kassel, J. Phys. Chem. 32 (1928) 25, 1065. [ 31 O.K. Rice and H.C. Ramsperger, J. Am. Chem. Sot. 49 (1927) 1617;50(1928)617. [4] R.A. Marcus, J. Chem. Phys. 20 (1952) 359. [ 51R.G. Gilbert and SC. Smith, Quantum Chemistry Program Exchange, Indiana University, Bloomington Indiana, Program No. 460,1988. [6] W. Forst, Theory of Unimolecular Reactions (Academic Press, New York, 1973). [ 71 P.J. Robinson and K.A. Holbrook, Unimolecular Reactions ( Wiley-Interscience, New York, 1972). [8] D.C. Tardy and B.S. Rabinovitch, Chem. Rev. 77 (1977) 369. [9] J. Troe, J. Chem. Phys. 66 ( 1977) 4745. [IO] H. Hippler, J. Troe and H.J. Wendelken, J. Chem. Phys. 78 (1983) 6709. [ 111 H. Hippler, J. Troe and H.J. Wendelken, J. Chem. Phys. 78 (1983) 6718. [ 121 J.R. Barker, J. Phys. Chem. 88 (1984) 11; J. Shi and J.R. Barker, J. Chem. Phys. 88 (1988) 6219, and references cited therein; H. Hippler, L. Lindemann and J. Troe, J. Chem. Phys. 83 (1985) 3906; H. Hippier, B. Otto and J. Troe, Ber. Bunsenges. Physik. Chem. 93 (1989) 428. [ 131 M. Damm, H. Hippler, H.A. Olschewski, J. Troe and J. Willner, 2. Physik. Chem. NF 166 ( 1990) 129; T. Ichimura, Y. Mori, N. Nakashima and Y.J. Yoshihara, Chem. Phys. 83 (1985) 117;

R.L. Jackson / Time-dependent master equation for non-steady-state reactions T. Ichimura, M. Takahashi and Y. Mori, Chem. Phys. I14 (1987) 111. [ 141 I. Orefand D.C. Tardy, Chem. Rev. 90 (1990) 1407. [ IS] B.J. Gaynor, R.G. Gilbert and K.D. King, Chem. Phys. Letters 55 (1978) 40. [ 161 R.G. Gilbert, K. Luther and J. Troe, Ber. Bunsenges. Physik. Chem. 87 (1983) 169. [ 171 H.W. Schranz and S. Nordholm, Chem. Phys. 74 (1983) 365. [ 181 S.C. Smith and R.G. Gilbert, Intern. J. Chem. Kinetics 20 (1988) 307. [ 191 S.H. Luu and J. Troe, Ber. Bunsenges. Physik. Chem. 77 (1973) 325. [20] M. Damm, H. Hippler and J. Troe, J. Chem. Phys. 88 (1988) 3564. [2 I ] K.V. Reddy and M.J. Berry, Chem. Phys. Letters 52 ( 1977) 111. [ 22 ] D.W. Chandler, W.E. Fameth and R.N. Zare, J. Chem. Phys. 77 (1982) 4447. [23] J.M. Jasinski, J.K. Frisoli and C.B. Moore, J. Chem. Phys. 79 (1983) 1312. [24] [ 251 [26] [27]

F.F. Grim, Ann. Rev. Phys. Chem. 35 ( 1984) 657. R.L. Jackson, J. Chem. Phys. 92 ( 1990) 807. E. Weitz, J. Phys. Chem. 91 (1987) 3945. S.H. Luu and J. Troe, Ber. Bunsenges. Physik. Chem. (1974) 766.

78

325

[ 281 D.W. Chandler and J.A. Miller, J. Chem. Phys. 8 1 ( 1984) 455. [29] J.E. Baggott and D.W. Law, J. Chem. Phys. 85 (1986) 6475. [ 301 R.L. Jackson, Quantum Chemistry Program Exchange, Indiana University, Bloomington Indiana, to be published. [31] J. Troe and W. Wieters, J. Chem. Phys. 71 (1979) 3931. [32] R. Srinivasan, J. Am. Chem. Sot. 84 ( 1962) 3432. [ 331 R. Atkinson and B.A. Thrush, Proc. Roy. Sot. A 3 16 ( 1970) 123, 131, 143. [34] SW. Orchard and B.A. Thrush, Proc. Roy. Sot. A 329 (1970) 233. [35] D. Astholz, J. Troe and W. Wieters, J. Chem. Phys. 70 (1979) 5107. [ 361 H. Hippler, K. Luther, J. Troe and H.J. Wendelken, J. Chem. Phys. 79 (1983) 239. [37] R.G. Gilbert, Chem. Phys. Letters 96 (1983) 259. [38] G.B. Porter and B.T. Connelly, J. Chem. Phys. 33 (1960) 81. [39] J. Troe, J. Chem. Phys. 66 (1977) 4745. [40] K.N. Khimp and J.P. Chesick, J. Am. Chem. Sot. 85 ( 1963) 130. [41] B.J. Gaynor, R.G. Gilbert, K.D. King and J.C. Mackie, Intern. J. Chem. Kinetics 8 (1976) 695. [42] S.H. Luu, K. Glanzer and J. Troe, Ber. Bunsenges. Physik. Chem. 79 (1975) 855. [43] SC. Smith, M.J. McEwan and R.G. Gilbert, J. Chem. Phys. 90 (1989) 4265.