Temperature dependence of the quasiclassical reactivity for vibrationally excited hydrogen molecules colliding with hydrogen atoms

Temperature Dependence of the Quasiclassical Reactivity for Vibrationally Excited Hydrogen Molecules Colliding with

Hydrogen Atoms

A. LAGANA Dipartimento di Chimica dell' Uniuersid, Via Elce di sotto, 8-Perugia Italy

Abstract

A discussion of reactants vibrational energy and temperature dependence of reactive rate constants for the hydrogen atom hydrogen molecule reaction is presented for a matrix of values calculated at 0 < u < 10 and temperatures in the range from 300 K to 4000 K. A parametrization of the results is attempted. A comparison with rate constant values obtained from an approximate quantum treatment is also reported.

Introduction

The hydrogen-atom hydrogen-molecule reaction has been investigated using a variety of theoretical methods [l-41. Converged close coupling (cc) three-dimensional (3D) quantum calculations of the H + H2 (u = 0) reaction probability have been performed up to an energy of 0.7 eV. The main goal of these calculations was to investigate the low-energy microscopic properties of this reaction and to provide a benchmark for approximate treatments that have to be used at larger energies. In fact, when total energy is large, the computational effort required for dealing with the basis set of the internal states wave- functions is very demanding. Such a difficulty can be overcome, to some extent, by introducing constraints into the dynamical treatment and by decoupling, accordingly, the corresponding set of quantum differ- ential equation [5-71.

When the goal of the calculation is the evaluation of rate constants, the amount of computer work required is even larger. In this case, reactive probabilities have to be estimated using a fine grid of collision energies for a sufficiently extended interval. This goal is more easily accomplished by using classical trajectory techniques for which estab- lished statistical treatments reach fairly converged results using random sampling of the initial conditions. Mayne and Toennies (MT) [BI have already calculated reactive rate constants for the reaction

(1) H + Hz (u) - Hz (U ') + H

International Journal of Chemical Kinetics, Vol. 18, 1003-1021 (1986) 0 1986 John Wiley & Sons, Inc. CCC 0538-8066/86/091009-13$04.00

1010 LAGANA

and its isotopic variants at u = 0, 1, and 2 by regularly scanning the translational energy variable and randomly selecting remaining initial parameters. However, although such an approach has the advantage of allowing an analysis of the evolution of the reactive probability with the collision energy, its demand for computer time is high when rate constants at several initial vibrational states have to be computed. As a consequence, this approach is not competitive with other treatments, making a more extensive use of random sampling of initial conditions.

Quasiclassical rate constant calculations have also been performed by Smith 191 on the potential energy surface of Yates and Lester [lo] at u 5 4. This surface, partially based upon ab initio calculations, is less accurate than the one fitted by Truhlar and Horowitz (LSTH) 1111 to the full set of ab initio values [12]. As already mentioned, on the LSTH surface quasiclassical calculations have been performed by Mayne and Toennies [SI. Using the same surface, extensive classical trajectory calculations have been performed also by Truhlar and coworkers [ 131. Their goal was the investigation of the relative importance of com- peting events occurring in hydrogen atom hydrogen molecule collisions and to sample the effect of vib-rotational excitation of reactants on the reactivity of the system.

In order to obtain a clearer picture of the effect of the vibrational excitation of the hydrogen molecule in reaction (11, we have recently calculated an extended matrix of state (u ) to state ( u ’ ) rate constant (K(u, u ’; 2’). The temperature dependence will be dropped when unnec- essary.) values using a classical trajectory technique. A practical reason for carrying out this investigation was the need to assess the efficiency of deactivation mechanisms for vibrationally excited hydrogen molecules colliding with hydrogen atoms in a H, discharge 1141. In fact, under certain circumstances, hydrogen atom collisions have been shown to be more efficient than impact onto the wall in deactivating hydrogen molecules.

In the present article, we report state-to-state rate constants evaluated at two extreme temperature situations (a preliminary presenta- tion of these results was given in refs. 1151 and [I611 and discuss the dependence of reactivity upon initial vibration. A parametrization of the dependence of rate constant values from the temperature is also performed.

Calculations and Results

Classical trajectory calculations were carried out on the Cray X-MP 12 at CINECA (Bologna-1) using a vectorized version of the program of Muckerman [171, which is based on a selection of initial conditions of the importance sampling type [MI. Modifications were

TEMPERATURE DEPENDENCE OF QUASICLASSICAL REACTIVITY 1011

introduced in order to improve the vector performance of the program (in particular, of the potential routine) on the Cray and to calculate detailed reactive and nonreactive rate constants. Calculations were carried out starting from reactants thermalized at a given temperature. At both 300 and 4000 K all reactant vibrational numbers from 1 to 9 were used.

Additional calculations were carried out at some intermediate temperatures in order to investigate with higher accuracy the temperature dependence of the detailed rate constants. For these additional temperatures, initial vibrational states as high as u = 3 were considered. Each run was based on the integration of a batch of 4000 trajectories. As a reasonable compromise for the convergence of calculated reactive rate constants (elastic rate constants are b,, dependent) and the number of trajectories to be integrated, we adopted a maximum impact parameter (b-) of 2 A.

Accuracy of the integration of the classical equations of motion was tested, as usual, against the constance of the total energy, total angular momentum, and collision time. Random back integrations were also performed. A step-size of 0.1 fs was found to be satisfactory.

Rate constants for the adiabatic and deactivation reactive processes are reported in Figure 1 as a function of the vibrational excitation of

8 0 0 7

V

Figure 1. Deactivation ( ) and vibrationally adiabatic ( H) reactive rate constants reported as a function of the initial vibrational number. Upper panel: values calculated at T = 4000 K, Lower panel: values calcuiated at T = 300 K. Connecting lines Save been drawn for aim of clarity.

1012 LAGANA

the reactant molecule. At both temperatures, in contrast with nonreactive collisions [15,161, deactivation is far more efficient than adiabatic mechanisms.

We have also compared our quasiclassical results with values derived from a BCRLM approach 1191. BCRLM calculations were performed up to a total energy above 4 eV in order to make it possible to evaluate rate constants at quite large u values [20]. In addition, in order to follow more accurately the trend of quasiclassical results within the investigated interval of temperatures, we have run classical trajectories also at T = 500 and 1000 K. In Figure 2 quasiclassical and BCRLM results for the u = 1 to u ‘ = all reactive transitions are reported as a function of the temperature. For comparison, results of ref. [8] are also reported. Mayne and Toennies results differ from ours mainly at high temperature. This may be due either to the different sampling of the collision energy or, as is more likely, to an insufficient extension of the collision energy range in MT calculations. According to our calculations, divergence with temperature of BCRLM results from quasiclassical values is less dramatic than the one suggested by extrapolation of conclusions drawn from a previous comparison U91. In the logarithmic scale used in Figure 2, the difference between the two results looks roughly constant, indicating that it may originate from a factor little or not dependent from the temperature.

e+ J B

1 2 3 4 T / lo3 K .

Figure 2. Logarithm of rate constant values for the u = 1 to all products’ states. Results of ref. [8 ] (+); quasiclassical values (m); BCRLM values (8). Connecting lines have been drawn for aim of clarity.


Values of state-to-state reactive rate constants estimated at T = 300 and 4000 K, respectively, are reported in Tables I and 11. Usually, vibrational enhancements of reactivity are analyzed in terms of devia- tion from a “prior” rate constant. Results of several authors [21] suggest that a relationship of the type

(2) In k(u, u ’; T)/ln ko(u, u ’; T) = A. + AI[E(u) - E(u ’)]/kT

(where E(u) and E(u ‘) are vibrational energies associated with u and u ’ and A. and A l are adjustable parameters) provides, in general, an acceptable fit to both experimental and computational data once k o (u , u ’; T ) is properly defined [221. This approach has been followed in ref. [91 for the rationalization of the isotopic effect for reaction (1) at low (u < 5) vibrational states. At the temperatures reported in Tables I and 11, the u dependence of reactive detailed rate Constants varies according to the region of initial vibrational number considered. This is particularly true for the highest temperature. This fact, as already pointed out in ref. 1151, is probably related to the more important role played by trajectories trapped inside the strong interaction region C231 when u increases and may make inadequate a compaction of calculated values using eq. (2).

A way of scaling calculated state-to-state rate constants is to report the value of M u , u ’ = u -n) as a function of n (the jump between initial and final vibrational numbers) at fixed u values [15]. In this case, in fact, curves calculated at different fixed u values show relevant simi- larities and are partially overlapping. Their behavior is characterized by a first region of increasing derivative followed by a decreasing one. As a consequence, all curves go first through a maximum and, after- word, decrease drastically (see Fig. 3). This representation of the vibrational energy dependence of detailed rate constants has been shown in refs. [151 and 1161 to also be more efficient than usual plots against u or u ’ .

The temperature dependence of some of the rate constants calculated at T = 300 K and T = 4000 K, was parametrized using an Arrhenius relationship. Parameters (E, and A) of the Arrhenius fit for the whole matrix of state-to-state rate constants obtained from our calculations are reported in Table 111. Obviously trends of the amplitude and of the activation energy parameters reflect the similar vibrational energy dependence of the corresponding rate constants. Accordingly, the value of A is smaller for transitions concerned with large n values and activation energy is maximum for low u transitions. Differences of the estimated parameters between u ’ = u and u ‘ < u processes are not large, due to the generalized property of quasiclassical reactive rate constants of not showing a strong bias towards vibrationally adiabatic processes.

TABL

E I.

Rea

ctiv

e st

ate

to s

tate

rat

e co

nsta

nts A

for

H +

Hz at

a te

mpe

ratu

re o

f T =

300 K

.

5 6

7 R

i V, v

0

n 1

3

1 .7

22

.14

(-1

3)

,592

.17 (

-1 3)

2 .9

05

.14

( - I

?)

.I lL.Ol( -

1 1

) .I

JC.O

?( -

11

)

3 .2

2_

+.0

?( -11 )

,34+.02( -11

)

.711.03( -11

) .57+.02(-11)

4 .5

7+.0

7(-1

1)

.56~

.08(

-ll)

.97-.08(-11)

.15$.01(-10)

. (-10)

5 .19+. 04 ( -

1 1

) .8

1r.

23( -11

) .l

ot. 01 (-10) .162.01(-10)

.303.01(-10)

.29+ .Ol

(-lO

)

6 .5

3~.1

3(-1

1) .612.09(-11)

.724

.09(

-11)

.16$.02(-10)

.21$.02(-10)

.38+.02(-10)

.402

.04(

-10)

7 .89+.43(-11)

.46+.07(-11)

.67+.10(-11)

.132.03(-10)

.19=.02(-10)

.28&.02(-10)

.49+.03(-10)

.44~

.03(

-10)

8 .2

9_+.

06(-

11)

.411.09(-11)

.61+.12(-11)

.73~

.10(

-11)

.14+.02(-10)

.19+.02(-10)

.32+.03(-10)

.53+

.03(

-10)

.6

12.0

5(-10)

9 .1

2~.0

4(-1

1)

.37~

.07(

-ll)

.4

01.0

7(-1

1)

.54+.10(-11)

.85+.13(-11)

.14+.02(-10)

.19~

.02(

-10)

.39>.04(-10)

.61+

.03(

-10)

.67~

.04(

-10)

All

entr

ies

in c

m3/

(mol

ecul

e x s

).

TABL

E 11.

Rea

ctiv

e st

ate

to s

tate

rat

e co

nsta

nts f

or H

+ Hz at

a te

mpe

ratu

re o

f 400

0 K

.

0 1

2 3

4 5

6 7

8 9

10

11

12

13

1 .78>04(-10)

.89+.04(-10)

.39+.03(-10)

.70+.10(-11)

.16+.04(-11) .31+.14(-12)

.50+

.20(

-12)

2 .65+.04(-10)

.12+.01(-

9) .ll+.Ol(- 9)

.41+.03(-10)

.12+.02(-10) .34+.06(-11)

.63+.22(-12)

.16+.10(-12)

3 .63+.05(-10) .ll;.Ol(-

9) .15+.0l(-

9) .13+.01(- 9)

.58+.05(-10)

,12+.02(-10)

.33+

.06(

-11)

.12+.03(-11) .70+.29(-12)

.29+.14(-12)

4 .45+.04(-10)

.93+.06(-10) .14+.01(-

9) .16>01(- 9)

.15+.01(- 9)

.55+.05(-10)

.24+.03(-10)

.51+.06(-11) .24+.05(-11)

.97+.29(-12)

.31+.19(-12)

5 .37+.03(-10)

.87+.06(-10)

.ll+.Ol(-

9)

.18+

.02(

- 9)

.172.02(- 9)

.1

62.0

1(-

9) .77+.07(-10)

.22+

.02(

-10)

.80+.11(-11)

.24+.06(-11)

.10+.03(-11)

.19+.10(-12)

6 .31>03(-10)

.66>

05(-1

0)

.95+.07(-10)

.13+.02(-

9) ,17+.02(-

9) .19+.02(-

9)

.15+

.02(

- 9) .96+.14(-10)

.38+

.07(

-10)

.90+.13(-11)

.442.08(-11)

.12+.04(-11)

.19+.15(-12)

7 .3

5+.0

5(-1

0)

.67+.07(-10)

.72+.06(-10)

.97+.08(-10)

.13+.02(-

9) .13+.01(-

9) .15+.02(-

9) .17+.02(-

9) .74+.07(-10)

.28+.03(-10)

.17+.03(-10)

.43;.14(-111

.22+.07(-11)

.35+.23(-12)

8 .18+.02(-10)

.45+.04(-10)

.64+.05(-10)

.83+.06(-10)

.93+.07(-10)

.16+

.02(

- 9) .12+.01(-

9) .13+.01(-

9) .14+.01(-

9) .61+.08(-10)

.34+.04(-10)

.12+.02(-10)

.53+.13(-11)

.14+.04(-11)

9 .17+.03(-10)

.31+.03(-10)

.60>06(-10)

.67+.07(-10)

.96+.15(-10)

.94+.07(-10)

.13+.01(-

9) .12+.01(-

9) .12+.01(-

9) .12+.01(-

9) .98+.20(-10)

.34+.04(-10)

.12+.03(-10) .33+.09(-11)

All e

ntri

es in

cm

3/(m

olec

ule x

s).

1016 LAGANA

8 4 0 n

Figure 3. State-to-state rate constants reported as a function of the gap ( n ) between initial and final vibrational state. Upper panel: values calculated a t T = 4000 K; Lower panel: values calculated at T = 300. * ( v = 91, Dfu = €9, A(u = 7), and o(u = 6). Connecting lines have been drawn for aim of clarity.

In order to have a quantitative indication of the accuracy of Ar- rhenius parameters evaluated in this way, we have recalculated them by including in the interpolation values estimated at T = 500 K and 1000 K. Parameters obtained in this way at u = 1, 2, and 3 are reported in the first two columns of Table IV. A comparison with values of Table I11 shows tha t while preexponential factors of a two- temperature Arrhenius fit may deviate up to 50% from those obtained by a four temperatures treatment, the two different sets of data lead to an almost identical value for the activation energy. This fact confirms that the assumption of a negligible temperature dependence of the slope of the plot obtained by reporting the logarithm of k versus 1/T (as implied in Arrhenius interpolation based only on T = 300 K and T = 4000 K results) is reasonable. In Table IV values of parameters estimated by using as an interpolator of the detailed rate constants a functional form having an explicit temperature factor in the preexponential term are also reported. Sometimes, this preexponential temperature factor has been given the form of a second power in T [24]. In order to gain additional flexibility, we allowed the T exponent to vary freely for the equation


TABLE 111. Arrhenius parameters.

v - v '

1 0

1 1

2 0

2 1

2 2

3 0

3 1

3 2

3 3

4 0

4 1

4 2

4 3

4 4

5 0

5 1

5 2

5 3

5 4

5 5

6 0

6 1

6 2

6 3

6 4

6 5

6 6

E:

.450( 1)

.472( 1 )

.276( 1)

.302( 1 )

.286( 1)

.216( 1 )

.224( 1 )

.197( 1 )

.202( 1 )

.133( 1)

.181( 1 )

.172( 1 )

.153( 1 )

.140( 1)

.191( 1 )

.153( 1 )

.155( 1 )

.156( 1)

.112( 1 )

.110( 1 )

.114( 1 )

.153( 1)

.166( 1 )

.135( 1)

.135( 1)

.104( 1)

.852( 0 )

A&

.147(- 9)

.161(- 9 )

.920(-10)

.176(- 9 )

.158(- 9)

.827(-10)

.146(- 9 )

.192(- 9 )

.168(- 9 )

.532(-10)

.117(- 9 )

.174(- 9)

.194(- 9 )

.179( - .9)

.471(-10)

.105(- 9)

.131(- 9 )

.219(- 9)

.196(- 9)

.184(- 9 )

.358( -1 0)

.801(-10)

.117(- 9)

.154(- 9 )

.201(- 9)

.216(- 9)

.167(- 9 )

7 0

7 1

7 2

7 3

7 4

7 5

7 6

7 7

8 0

8 1

8 2

8 3

8 4

8 5

8 6

8 7

8 8

9 0

9 1

9 2

9 3

9 4

9 5

9 6

9 7

9 8

9 9

EaB v - v '

.882( 0)

.173( 1)

.153( 1)

.130( 1)

.124( 1)

.989( 0)

.721( 0 )

.871( 0)

.118( 1)

.154( 1)

.151( 1 )

.157( 1)

.122( 1 )

.137( 1)

.852( 0)

.578( 0 )

.535( 0)

.171( 1)

.137( 1 )

.175( 1)

.162( 1 )

.156( 1)

.123( 1)

.124( 1 )

.724( 0)

.436( 0 )

.376( 0 )

A&

.391(-10)

.833( - 10)

.873( -10)

.114(- 9)

.152(- 9)

.147(- 9 )

.164(- 9)

.190(- 9)

.209(-10)

.546 ( -1 0 )

.774( -1 0 )

.101(- 9)

.108(- 9 )

.190(- 9)

.134(- 9 )

.140(- 9 )

.150(- 9)

.211(-10)

.368(-10)

.747 ( - 10)

.822( - 10)

.117(- 9 )

.110(- 9 )

.152(- 9 )

.131(- 9 )

.127(- 9)

.126(- 9)

'in kcal mol-'. 'in cm3/(molecule x s).

1018 LAGANA

Table IV. Parameters from interpolation over four temperatures.

v - v '

1 0

1 1

1 a l l

2 0

2 1

2 2

2 <2

2 a l l

3 0

3 1

3 2

3 3

3 (3

3 a l l

.436(1) .768(-10)

.464(1) .129(- 9)

.465(1) .246(- 9)

.263(1) .576(-10)

.291(1) .117(- 9 )

.282(1) .117(- 9 )

.280(1) .173(- 9)

.290(1) .330(- 9 )

.202(1) .564(-10)

.214(1) .918(-10)

.189(1) .138(- 9 )

.197(1) .132(- 9 )

.199(1) .287(- 9)

.207(1) .469(- 9)

E:

.227(1)

.394(1)

.302( 1)

.102( I )

.152(1)

.156(1)

.131(1)

.129(1)

.939( 0)

.386(0)

.630( 0 )

.102(1)

.618(0)

.642( 0)

C*

.162(-14)

.329(-11)

.551(-13)

.143( -1 3)

.909( -1 3)

.179( -1 2)

.821(-13)

.815(-13)

.218(-12)

.110(-13)

.205( -1 2)

.103(-11)

.245( -1 2 )

:305( -12)

q

.133(1)

.455(0)

.104(1)

.103(1)

.886(0)

.803(0)

.947(0)

.103(1)

.688( 0)

.112(1)

.806( 0)

.601(0)

.875(0)

.908(0)

'in kcal mol-'.

*in cm3/(mnlecule x s x T'). &' in cm3/(moIecuIes x s).

Values of the E,, C, and q parameters estimated in this way (see columns 3, 4, and 5 of Table IV) suggest that the choice of q = 2 is inappropriate when an extended range of temperatiires has t o he cov- ered. In fact, Q values reported in Table IV rarely exceed unity.

Another way of deriving the value of the activation energy from state-to-state rate constants is to use the equation

(4) E,(u, u ' ) = RT2(d In k ( u , v'; T ) / d T ) When rate constants are estimated from reactive probabilities calculated at a fine grid of collision energies, the activation energy can be evaluated accurately at each value of the temperature as the average energy of the reactive molecules 1251. In our case, however, the computer program directly works out the thermal rate constant without going through the determination of the reactive probability at a given grid of energy values. Therefore, a way of estimating the value of the activa-


5t

Figure 4. Activation energies obtained from a numeri 1 determination re- - - ported as a function of the reactants’ temperature. Upper panel: u = 1; Inter- mediate panel: u = 2; Lower panel: u = 3. @(vibrationally adiabatic), .(n = 1 deactivation), A (global deactivation). Connecting lines have been drawn for aim of clarity.

tion energy, without imposing a global functional dependence of the rate constant from the temperature, consists of approximating the derivative with its numerical evaluation. Values obtained following this procedure are illustrated in Figure 4 for the vibrationally adiabatic, n = 1 and global deactivation processes at u values of 1, 2, and 3 (for which the calculation has been carried out at the four temperatures). Figure 4 shows that for all these rate constants there is a similar increase of the activation energy with the temperature. In addition, the estimate of E , obtained from the Arrhenius equation is in fairly good agreement with the average value of E , derived from Figure 4.

Conclusions

Rate constant values obtained from extensive quasiclassical calculations performed at T = 300 K and T = 4000 K at values of the initial vibrational state covering the interval 1-9, have shown a dependence from the vibrational excitation of reactants varying as a function of the particular interval of u examined. A useful compaction of the results has been obtained by reporting detailed rate constants

1020 LAGANA

as a function of the jump between reactants’ and products’ vibrational states.

Results obtained a t four temperatures selected within the 300-4000 K interval, have shown that quasiclassical results deviate from approximate quantum BCRLM values less than expected and that such a difference can be assumed to be almost temperature independent.

Calculations at four temperatures have also allowed a more accurate evaluation of the temperature dependence of the rate constants and, consequently, a more refined interpolation of their values jointly with a not averaged determination of the activation energy.

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Received January 20, 1986 Accepted April 25, 1986

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Temperature dependence of the quasiclassical reactivity for vibrationally excited hydrogen molecules colliding with hydrogen atoms