J. Phys. Chem. 1994,98, 502-507
502
Temperature Dependence of N
+ Nz Rate Coefficients
Antonio L a g a d Dipartimento di Chimica, Uniuersitit di Perugia, Perugia, Italy
Emesto Garcia Departamento de Quimica Fisica, Uniuersidad del Pais Vasco, Bilbao, Spain Received: June 14, 1993; In Final Form: August 13, 19930
After the validity of two model potential energy surfaces was tested, quasiclassical calculations of vibrational deexcitation rate coefficients for the nitrogen-atom nitrogen-molecule reaction were performed for a range of reactant vibrational numbers extending up to u = 45 and a set of translational and rotational temperatures ranging from 500 to 4000 K. The different effect of varying either the rotational or the translational temperature on the relative efficiency of reactive and nonreactive vibrational transitions is discussed in detail.
Introduction We have recently started an investigation of bimolecular reactions of interest for the modeling of the gas-phase processes taking place round reentering spacecrafts. Among these, due to the particular abundance of N atoms at the shock wave temperature,' nitrogen-atom nitrogen-moleculereactions play a primary role. In particular, we have focused our attention on the
N('S,)
+ N2('2:,
u)
-
N('S,)
+ N2('2:,
u')
, DU+'Z: 2
(1)
process and on the effect that a variation of the temperature ( r ) of the system has on the efficiency of the Nz vibrational deexcitationprocess2 becauseof the key role played by vibrational deexcitation in determining the kinetics of cold p l a ~ m a . ~ The study of reaction 1 is also important from a theoretical point of view. N + N2 is, in fact, a suitable prototype of a three identical heavy atom system and, as a consequence, an interesting casestudy for rationalizingbimolecularreactivity. In particular, its investigation is useful to assess the role played by internal and translational motions in promotingvibrational deexcitationwhen the reactive system is not as light as the H + Hz onea4 In this paper we report the most extended calculation of the rate coefficients ever performed for reaction 1 and discuss the importanceof independently increasinginternal and translational temperatures.
Calculations Little theoretical work has been performed on reaction 1 up to this point. The main reason for this is the difficulty of carrying out an accurate ab initiocalculationof the potentialenergy surface (PES) of the N + N2 system as well as the scarcity of available experimental information.s An ab initio investigation of the electronicstructure of N3 was recently carried out by Petrongolo.6 Unfortunately, the calculated collinear transition state has an energy too high to explain the measured rea~tivity.~ Corrective actions can be taken either by assuming a highly efficient electronic surface transition during the collision or by empiricallyadjusting the minimum energy path to reaction. This can be understood with the help of Figure 1 where a schematic correlation diagram of the N + Nz system is shown. By assuming a complete nonadiabatic transition between the (%, + I&+) and the (2Dy + I&+) surface at fairly long range, one can describethe collision as occurring on a single surface having little or no barrier and supportinga rather stable intermediatecomplex. On the contrary, when assuming an inefficient surface crossing,one can play with *Abstract published in
\
Aduance ACS Absrracrs, December 1, 1993.
0022-3654/94/2098-0502~04.50/0
'
I
\
Figure 1. Correlationenergy diagram for the doublet and quartet state? of the nitrogen-atom nitrogen-molecule reaction.
the minimum energy path by lowering, for example, the height of the transition state with respect to its ab initio estimate. To reproduce the two limiting situations, we constructed two different LEPS surfaces (PES A and PES B). The parameters of both surfaces were chosen to reproduce the spectroscopic properties of the diatomic fragments.' These parameters are the equilibrium distance re = 1.0977 A, the dissociation energy De = 228.4 kcaljmol, and the exponentialparameter /3 = 2.689 A-1. The remaining S (Sato) parameter of PES A was set equal to 4.023. The resulting surface shows a barrier to reaction of 36 kcal/mol and has already been used for previous preliminary trajectory The Sat0 parameter of PES B was, instead, set equal to 0.58 to lead to a fairly deep intermediate well (36 kcaljmol). Although, thanks to the availability of vector and parallel computers: progress is being made toward accurate quantum calculations of the cross sections and rate coefficients of atomdiatom reactions,'Sl3 still their systematic evaluation can be carried out mainly by using quasiclassicalmeans14even for light systems. In fact, classical trajectory programs are more straightforward to implement on parallel computers, and elapsed times for their runs on concurrent processors can beorders of magnitude smaller than those obtained on sequential machines.10J5J6 On the other hand, especially for fairly heavy systems such as N + N2, to make quantum calculations converge, a large expansion basis set of bound functionsis needed. This not only makes scalar 0 1994 American Chemical Society
Temperature Dependence of N
+ N2 Rate Coefficients
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 503
TABLE 1: PES B Statetestate Rate Coefficients at 300 I ( ' ~~~~
ut
=0
VI
=1
01 =
v=5
v=9
u = 13
.42(-10) .39(-10) .12(-10) .14(-10) .18(-13) .49(-14)
.83(-11) .12(-10) .11(-10) .98(-11) .82(-11) .78(-11) -12(-10) .83(-11) .14(-10) .56(-11) .58(-11) .26(-11) .73(-14)
.36(-11) .69(-11) .45(-11) .58(-11) .82(-11) .63(-11) .73(-11) .55(-11) .99(-11) .41(-11) .50(-11) .38(-11) .61(-11) .44(-11) .75(-11) .29(-11) .71(-11) .27(-11) .26(-11) .13(-11) .60(-14)
.24(-11) .40(-11) .27(-11) .46(-11) .49(-11) S5(-11) .42(-11) .43(-11) .33(-11) .33(-11) .39(-11) .28(-11) .45(-11) .27(-11) .72(-11) .24(-11) .34(-11) .31(-11) .34(-11) .29(-11) .45(-11) .30(-11) .49(-11) .28(-11) .73(-11) .24(-11) .40(-11) .84(-12) .19(-14) .22(-14)
=3
0 '
01 5 ut
2
u=l
4
=5
'v = 6 ut
=7
ut = 8 0 '
=9
ut
= 10
ut=
11
ut=
12
ut = 13
ut = 14
a Nonreactive (upper rows) and reactive (lower rows) rate coefficients given in cm3 molecule-1 s-l (a(-x) stands for a X
calculations computationally very demanding but it also makes TABLE 2 ut
=0
ut=
1
ut
=2
ut
=3
u' = 4 ut
=5
ut 5 6 ut
=7
ut
=8
ut
=9
ut
= 10
ut=
0'
11
= 12
ut =
13
ut
14
0 '
= 15
the distribution of the calculation on concurrent processors inefficient to the end of gaining speedup. Detailed quasiclassical state (0) to state (u? rate coefficients (k,(T)) calculated on PES A and PES Bat a fixed temperature Tare given in Tables 1-3 (upper rows nonreactive, lower rows reactive). In Table 1 rate coefficients calculated on PES B are given for T = 300 K and u ranging from 1 to 13. Deexcitation rate coefficients calculated on PES B are larger than those of the correspondingvibrationally adiabatic processes, indicating that this surface makes the redistribution of the system energy over all available product states very efficient for both reactive and nonreactive processes. As shown by Table 2, a similar behavior was found for rate coefficients calculated at T = IO00 K. On the contrary, as shown by Table 3, rate coefficients calculated on PES A at T = 1000 K (at T = 300 K all collisions are vibrationally elastic on this surface), though nonnegligible for u' # u, show a preference for vibrational adiabaticity. Vibrationaldeexcitationstate-to-state rate coefficients calculated on PES A decrease steadily when plotted as a function of the difference n (n = u t - u ) between the reactant u and the product u'vibrational state. In spite of the largely empirical nature of the PES used, the thermal ( T = 3400 K) rate coefficient calculated on PES A (7 X cm3 molecule-I s-l) is in excellent agreement with the measuredvalue" ( 5 X l613cm3 molecule-1s-l). On the contrary, the same quantity calculated on PES B turns out to be 3 orders of magnitude larger (3.4X cm3 molecule-I s-l). For this reason, extended vibrational deexcitation rate coefficient calculations were performed only on PES A. To carry out the calculation, a proper limitingvalue (b,) had to be selected for the impact parameter b. The value of 6 , was determined by plotting the opacity function (P,(b)) at several values of the initial vibrational number and T = 4000 K. As
PES B Statetestate Rate Coefficients at 1000 K8 Ut1
0-3
.65(-1 0) .57(-10) .34(-10) .29(-10) .31(-11) .23(-11) .14(-12) .74(-13)
.30(-10) .37(-10) .30(-10) .23(-10) .24(-10) .17(-10) .17(-10) .11(-10) .19(-11) .12(-11) .12(-12) .39(-13) .42(-14)
0
15
.27(-10) .19(-10) .19(-10) .17(-10) .23(-10) .14(-10) 18(-10) .13(-10) .14(-10) .97(-11) .l O(-10) .49(-11) .43(-11) SO(-12) .30(-13) .27(-13) .65(-14)
.
u=7
u=9
.17(-10) .14(-10) -14(-10) .14(-10) .15(-10) * lo(-10) .13(-1 0) .17(-10) .13(-10) .99(-11) .15(-10) .88(-11) .11(-10) .61(-11) .78(-11) .29(-11) .11(-11) .60(-12) .51(-13) .33(-13) .70(-14)
.go(-1 1) *12(-10) .1O(-10) .99(-11) * 1l (-10) .lo(-10) lo(-10) .11(-10) .97(-11) .85(-11) .12(-10) .77(-11) .14(-10) .89(-11) .13(-10) .65(-1 1) .95(-11) .65(-11) .75(-11) .30(-11) .95(-12) .41(-12) .61(-13) .60(-13) .67(-14) .30(-14)
'0
11
.87(-11) .79(-11) .93(-11) .91(-11) .87(-11) .80(-11) .13(-10) .78(-11) .70(-11) .68(-11) .70(-11) .lo(-10) .90(-11) .66(-11) .74(-11) .65(-1 I ) .79(-11) .76(-11) .76(-11) .57(-11) .14(-10) .49(-11) .58(-11) .30(-11) .84(-12) SO(-1 2) .go(-13) .75(-13)
.
.27(-14)
u = 13
u = 15
.80(-11) .76(-11) .70(-11) .60(-11) .65(-11) .72(-11) .51(-11) .54(-11) .72(-11) .11(-10) .79(-11) .77(-11) .65(-1 1) .63(-11) .as(-1 1) .58(-11) .71(-11) .74(-11) .69(-11) .61(-11) .64(-11) .63(-11) .74(-11) .55(-11) .11(-10) .42(-11) .64(-11) .28(-11) .94(-12) .7 1(-12) .79(-13) .41(-13)
.43(-11) .65(-11) .52(-11) SO(-1 1) .53(-11) .47(-11) .50(-11) .80(-11) .50(-11) .50(-11) .61(-11) .53(-11) .47(-11) S9(-11) .45(-11) .68(-11) .52(-11) .60(-11) .54(-11) .61(-11) .52(-11) .61(-11) .92(-11) .86(-11) .74(-11) .72(-11) .89(-11) .58(-11) .74(-11) .42(-11) .73(-11) .25(-11) .lo(-1 1) .40(-12) .33(-13) .54(-13)
ut = 16
.80(-14) ut = 17
Nonreactive (upper rows) and reactive (lower rows) rate coefficients given in cm3 molecule-'
s-1
(a(-x) stands for a X 10-x).
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
504
LaganA and Garcia
TABLE 3: PES A State-to-State Rate Coefficientsat 1000 Ka U = l ut
v' = 1 ut
v=3
v=5
v=7
.19(-09)
.50(-14)
=2 .33(-14)
u' = 3
.19(-09)
u' = 4 0'
=5
.19(-09) .29(-14)
.50(-14)
uf = 6 ut
=7
ut
=8
VI
=9
.29(-14) .19(-09) .81(-14)
.35(-14) .28(-14) .55 (-1 4) .13(-13) .50(-14) .13(-13) .62(-14) .61(-14) .43(-1 4) .91(-14)
.12(-13) .19(-09) .l 1(-13)
v' = 10 v'= 11 0 '
u=9
u = 11
u = 13
=0 .43(-1 4) .37(-1 4) .47(-14) .13(-13) .13(-13) .55(-14) .91(-14) .54(-14) .83(-14) .70(-14) .22(-13) .82(-14) .15(-13)
.lo(-13) .la(-1 3) .93(-14) .13(-13) .19(-13) .25(-13) .42(-14) .19(-13) .15(-13) .29(-13) .27(-13) .13(-1 3) .22(-13) .lo(-1 3) .91(-14) .44(-14) .56(-13)
.14(-13) .68(-14) .77(-14) .27(-14) .17(-13) .73(-14) .59(-13) .19(-09) .48(-13)
.40(-13)
.11(-1 3) .42(-13) .34(-13) .68(-13) .14(-12)
= 12 .42(-14)
vf = 13 ut = 14
.8 5 (-1 4) ut=
15
ut = 16
u = 15
,591-1 3) .3o(-i 3j .55(-13) .52(-13) .20(-13) .27(-13) .55(-13) .36(-13) .21(-13) .3 1(-13) .32(-13) .31(-13) .64(-14) .34(-13) .20(-13) .62(-13) .20(-13) .44(-13) .24(-13) .56(-13) .48(-13) .70(-13) .34(-13) .15(-12) .41(-13) .15(-12) .14(-12) .26(-13) .25(-12) .19(-09) .17(-12) .18(-13)
u t = 17
.37(-14) a
Nonreactive (upper rows) and reactive (lower rows) rate coefficients given in cm3molecule-1 s-* (a(+ stands for a X 10-x).
TABLE 4 u
b,(A) M
Parameters of the Calculation
5. 2.00
10 15 20 25 30 35 40 45 2.25 2.25 2.50 2.75 3.00 3.25 3.50 3.75 12000 12500 12500 13400 14700 16200 18000 19800 21900
shown by Table 4, b, almost linearly increases with u. However, it must be stressed here that while such a choice of b,, is sufficient to ensure the convergence of the reactive rate coefficients, it may not be so for nonreactive ones. In fact, even when the impact parameter is so large that further reaction cannot occur (as is well-known, the convergence of calculated cross sections with the total angular momentum-to which b, is related-is faster for reactive than for nonreactive collisions), nonreactive encounters may still be effective in slightly perturbing the vibration of the Nz molecule. However, this negligibly alters the quasiclassical estimate of deexcitation rate coefficients because the half-integer boxing method used for discretizing classical internal energies is in any case insensitive to small perturbations of the vibrational state." As shown in Table 4, to not vary the density in b of calculated trajectories, their total number (M) was increased when b,,, was increased. Analysis of Calculated Rate Coefficients
After defining the parameters of the calculation, we started a massive computation of reactive rate coefficients for reaction 1 by varying independently the initial vibrational number as well as the rotational ( Tmf)and translational ( Ttr)temperatures of the system. A first goal of the investigation was to evaluate the effect on the overall reactivity of increasing the initial vibrational energy of the reactant nitrogen molecule up to u = 45 at different values of the gas temperature T ranging from 500 to 4000 K. Results are shown in Figure 2 where fixed u values of reactive rate coefficients k,( T ) (defined as k,( T ) = E,k,.iT)) are plotted as
V
Figure 2. Reactive rate coefficients &(, T ) for T = 500 K (dashed-dotted line), 1000 K (dotted line), 2000 K (dashed line), and 4000 K (solid line) plotted as a function of the initial vibrational number u.
a function of the initial vibrational number. As shown by the figure, at all temperatures rate coefficients become important beginning only at rather high reactant vibrational states, implying a less active role of vibration in promoting reactivity. On the contrary, even a modest increase of energy due to a rise of the system temperature is quite effective in enhancing reactivity. In the above mentioned calculations, a temperature increase affects both rotational and collision energies. To determine which type of energy is more effective in promoting reactivity, Tmtand T,, were varied independently from 500 to 4000 K. Calculated reactive rate coefficients k,( Ttr,Trot) are illustrated in Figures 3 and 4 where they are plotted a t constant Ttr as a function of Tml and at constant T,, as a function of Tfr. As Figures 3 and 4 clearly show, the effect of varying the rotational temperature is small, especially at high translational temperature. In fact, while at low translational temperatures (lower left panel) the effect of increasing the rotational temperature from 500 to 4000 K leads to an increment of k,( TU,Tmt)
Temperature Dependence of N
+ Nz Rate Coefficients
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 505
y -----
r-•
,... , ......................
;=3,
,...I.. .... ...............I .- .......... ......... -.
.
-.+
\
0
tzx-
~-,;.-,:,
-*---.--------
0
2000
4ooo 0
*-------. . . . . . ;. . . . . .
2000
I
4ooo
TtUtIK Figure 3. Reactive rate coefficients ko(Tu,Tmt)for TU = 500 K (lower left panel), TU = loo0 K (lower right panel), Tu = 2000 K (upper left panel), and Tu = 4OOO K (upper right panel) plotted as a function of the rotational temperature Trotat u = 45 (solid line), u = 35 (dashed line), u = 25 (dotted line), and u = 15 (dashed-dotted line).
60,
c s
..........
........
0
1
I
2000
v
Figure 5. Reactive rate coefficients for Tu = 500 K and Trot= 500 K (lower left panel), 1000K (lower right panel), 2000 K (upper left panel), and 4000 K (upper right panel) plotted as a function of the initial vibrational state u. Reactive and nonreactive vibrational decxcitation rate coefficients k,d(Tu,Tmt) are given as solid and dashed lines, respectively. For comparison,reactivevibrationalexcitationkDe(Tu,Tm) (dashed-dottedline) andvibrationallyadiabatic k,( Tu,Tm()(dotted line) rate coefficients are also plotted.
....... 4Ooo 0
2000
.-
4Ooo
ZrIK
0
0
20
4 0 0
20
40
Figure 4. As in Figure 3 for Trot= 500 K (lower left panel), Tml= 1000 K (lower right panel), Tm = 2000 K (upper left panel), and Trot= 4000 K (upper right panel) as a function of the translational temperature.
Figure 6. As in Figure 5 for Ttr= 1000 K.
ranging from 30% to 50% (in terms of percent, this effect is higher for lower reactant vibrational states), at Ttr = 4000 K the increment never exceeds 30%, while often being much smaller. On the contrary, the effect on kv(Ttr,Tml)of increasing the translational temperature of the system is definitely much larger. Increases of the reactive rate coefficients at fixed Trotvary from 300% to more than 10008. These results suggest that, though much smaller in absolute value, N + N2reactive rate coefficients have the same dependence on rotational and translational energies as those of the H + H2 reaction.' A comment is also deserved on more detailed quantities (such as the state-testate rate coefficients k,,(TIr,Trot)or their partial summationssuch as the vibrational deexcitation rate coefficients k,d( Tu,TmI) ~ u ~ < u k , , ( T ~ ,or Tm the t )vibrational excitation rate coefficients k t ( T ~ , T , t ) ~u&,dTtr,Tmt). Some of these quantities are plotted as a function of u for Ttr= 500,1000,2000, and 4000K,respectively, in Figures 5-8. Within each figure the four rotational temperatures Tm = 500, 1O00, 2000,and 4000 K are considered in the lower left, lower right, upper left, and
upper right side panels, respectively. Reactive and nonreactive vibrational deexcitations (solid and dashed curves of each panel) are'always the dominant processes (with the former invariably more efficient than the latter). An increase of the rotational temperature does not alter significantly the situation because it affects to the same extent both reactive and nonreactive k$( Ttr,Tmt) values. Processes mostly affected by an increase of the rotational temperature are the reactive vibrational excitations kVe(Ttr,Trot)(dashed-dotted curves) and, to a less extent, reactive adiabatic processes (dotted curves). However, although theeffect is large for both of them, at high u values reactive excitation becomes so efficient that the two curves tend to cross (the crossing point depending on thevalueof Tml).At high reactant vibrational states and rotational temperatures, reactive excitation increases to the point that it begins tocomparewithnonreactivedeexcitation. Reactive and nonreactive detailed state-to-state k,.(T) are plotted in Figures 9 and 10, respectively, as a function of n. Again, only curves related to u = 45, 35, 25, and 15 (solid, dashed, dotted, and dashed-dotted lines, respectively)have been considered to assess whether the regular shapes already found for H + H2
506 The Journal of Physical Chemistry, Vol. 98, No. 2, 1994
Laganii and Garcia
-
0
20
4 0 0
20
40
20
40
2
L -20
0 .-- ---0 -10
0
-20
-10
n Figure 9. Detailed nonreactive state-to-state rate coefficients &,{T) plotted as a function of n = u'- u for u = 45 (solid line), u = 35 (dashed line), u = 25 (dotted line), and u = 15 (dashed-dotted line) and T = 500 K (lower left panel), IO00 K (lower right panel), 2000 K (upp er left panel), and 4000 K (upper right panel).
V
Figure 7. As in Figure 5 for Ttr = 2000 K.
I
s
-
\
h?
-
G
2s
v
i
0 0
20
4 0 0 V
Figure 8. As in Figure 5 for Tll = 4000 K.
10
(see ref 4) and the lowest N + N2 (see ref 2) vibrational numbers could be extrapolated to larger vibrational numbers. The figures show that for both reactive and nonreactive collisions the u u' = u - 1 transition is the most efficient deexcitation process.
-
Conclusions
To investigatetheeffect on thedetailed N + N2 ratecoefficients of increasing the temperature of the system on an extended range of vibrational states, we have performed a massive trajectory calculation. After test calculations aimed at selecting the most appropriate potential energy surface between the two possible limiting situations (the most appropriate PES was found to be the one having a barrier to reaction of 36 kcal mol-', a value much smaller than that given by ab initio calculations), extended calculations were carried out by raising independently u to 45 and T,, and Trotto 4000 K. The effect of increasing the two temperatures was found to be similar to that of the lighter collinearly dominated H + H2 reaction. Namely, an increase of the initial translational temperature was found to be the most effective way of promoting reactivity; reactive vibrational deexcitation was found to be more efficient than the nonreactive one (and by far the dominant process for N + N2 collisions) and the vibrationalproduct distributions were found to have the same dependence on the vibrational state jump.
0
-10
-20 10
0
-10
-20
n
Figure 10. As in Figure 9 for reactive state-to-state rate coefficients.
Acknowledgment. The authors wish to thank the Italian Space Agency (ASI), the Progetti Finalizzati of CNR (Italy), and the Spanish DGICYT (Grant PS-89-0160) for financial support and computer time allocation. Financial support has also been given by the Integrated Spain-Italy Action (Grant HI-92-012). References and Notes (1) Giordano, D.; Maraffa. L. Proceedings of the AGARDCP-514 Symposium on Theoretical and Experimental Methods in HypersonicFlows
1992, 26-1. (2) Lagan&,A.; Garcia, E.; Ciccarelli, L. J. Phys. Chem. 1987,91,312. (3) Nonequilibrium Vibrational Kinetics; Capitelli, M., Ed.; Springer-Verlag: Berlin, 1986. Nonequilibrium PranssesinParriallyIotdzed Gases; Capitelli, M., Barsdley, J. N., Eds.;Plenum: New York, 1990. Loureiro, J. Chem.Phys. 1991,157,157. Armenise,I.;Capitelli,M.;Garcia,E.;Gorse, C.; Lagan&,A.; Longo, S. Chem. Phys. Lett. 1992,200, 597. (4) LaganA, A.; Garcia, E.; Mateos, J. Chem. Phys. Lett. 1991,176,273. ( 5 ) (a) Back, R. A.; Mui, J. Y . P. J . Phys. Chem. 1%2,66, 1362. (b) Bauer, S. H.; Tsang, S. C. Phys. Fluids 1963, 6, 182. (c) Bar Nan, E.; Lifshitz, A. J. Chem. Phys. 1969, 47, 2878. (6) Petrongolo,C.J. Mol.Struct. Theochem. 1988,175,215. Petrongolo, C. J . Mol. Struct. Theochem. 1989, 202, 135. (7) Hubcr, K. P.; Herzbcrg, G. Constants ofDiatomic Molecules; Van Nostrand: Toronto, 1979. (8) Frost, R. J.; Smith, I. W. M.Chem. Phys. Lett. 1987,140,499. (9) Laforenza, D. Theor. Chim. Acta 1991, 79, 155.
Temperature Dependence of N
+ N2 Rate Coefficients
(10) Lagan&,A. Comput. Phys. Commun. 1992, 70, 223. (1 1) Parker, G. A.; Pack, R.T.; Lagan&,A.; Archer, B. J.; Krese, J. D.; BaEic, 2.In Supercomputer Algorfrlmrfor Reactivity, DynomicsandKinetics of Small Molecules; Lagan&,A., Ed.; Kluwer: Dordrecht, The Netherlands, 1989. Parker, G. A.; Pack, R.T.;Lagan& A. Chem. Phys. L t t . 1993, 202, “e IJ.
(12) Lagan&,A.; Garcia, E.;Gervasi, 0.J. Chem. Phys. 1988,88,7238. (13) Lagan& A.; Garcia, E.;Gervasi, 0.;Baraglia, R.;Laforcnza, D.; Pereno. R. Theor. Chim. Acta 1991. 79. 323. (r4)Bunker, D. L. Methods Cor&&. Phys. 1971,lO. 287. Truhlar, D. G.; Muckeman, J. T. In Atom-Molecule Collision Theory: A Guide to rhe
The Journal of Physical Chemistry, Vol. 98, No. 2, 1994 507 Experimenlalist; Bernstein, R.B., Ed.; Plenum: New York, 1979. Clementi, E. J. Phys. Chem. 1985,89,4426. (15) Alvarino, J. M.;Garcia, E.;Lagan&,A.InSupercompurerAlgortthms for Reactiuity, Dymmics and Kinctics ofSmall Molecules; Lagan&,A.,Ed.; Kluwer: DordrWht, The NetherlanL, 1989, (16) Baraglia, R.; Ferrini, R.;Laforcnza, D.; Perego, R.; Gervasi, 0.; A. In High Performame Computing, Durand, M., El Dabaghi, F., Ms.; North Holland Amsterdam, 1991. (17) Connor, J. N.L.; Lagan&,A. Comput. Phys. Commun. 1979, 17, 145.
