- Email: [email protected]
A quantum-mechanical collinear model study of the collision N2 + O2
ChemicalPhysics20 (1977) 285-298 Q North-HollandPublishingCompany
A QUANTUM-MECHANICAL COLLINEAR MODEL STUDYOF
THECOLLIS10NN2+02 Xavier CHAPUISAT, Guy BERGERON Loboratoire de Chimie Thiorique’, Universitc? de Paris-Sud, Centre d’Orsay - 91405 Onay, France
Jean-Michel LAUNAY Dipartement d’Astrophysique Fondame,ztaie. Observatoire de Paris. 92190 Meudon, France
Received10 August 1976
A model close-coupling study of the translation-vibration-vibration (TVV) energy transfer in the collinear collision of two non-identical harmonic diitomic molecules is presented. The numerical applications are for Nt + 02 driven by an cxponential repulsion. Numerous transition probabilities are given, and the various dynamical processes are classified according to their importance. The classification is true whatever the collision energy. A reasonable interpretation of the dynamics based on (i) our knowledge of the simpler atom-diatom collision, (ii) the results of auxiliary calculations using various truncated expansions of the interaction potential, and (iii) classical trajectory calculations for the same system, is finally presented. The main results arc: (i) The TV transition probabilities of 02 are greater than that of N2 by several orders of magnitude; this is due to 02 being the more excitable molecule and being acted on by the more exciting field. (ii) The secondary processes are interfering sequences of the two principal processes which arc, respectively. the TV one-quantum jump in 02 and the W one-quantum exchange between N2 and 02. (iii) In the case of the TV one-quantum jump in N2 the mechanism is not unique according as 02 is vibrationally excited - or not - prior to collision.
1. Introduction The model collinear collision of an atom and a diatomic molecule has been the object of numerous publications in the recent past (for a review, see refs. [ 121). A number of exact quantum-mechanical calculations using mode1 potentials have been published, the aim of which was twofold: (i) to reveal some basic features of the collision dynamics which hopefully are also exhibited within more realistic models, and (ii) to allow a comparison with the results of approximate methods at various levels of approximation. Only a few exact calculations have appeared concerning the more compIex problem of the collision of two diatomic molecules. Most of them were for two identical molecules *The Laboratoire de Chirnie Theorique is associated with the C.N.R.S. (ERA no 549).
[3-S]. The case of two non-identical diatomics was most rarely treated [6,7]. The purpose of the present communication is (i) to present such exact quantum-mechanical results for a case modeling the collision N2 t 0, (the molecules are harmonic and the interaction potential is an exponential repulsion between the nearest end atoms), and (ii) to propose a detailed physical interpretation of the results. This interpretation is based on our knowledge of the atom-diatom collision dynamics and on a comparison with results obtained for the same system either classically with the same potential or quantum-mechanically with various truncated expansions of the potential. The model is described in section 2 together with the relevant close coupling equations and the details of the calculations. The numerical results are presented‘ in section 3. Complementary calculations using the
X.
286
Chapuisat et aL/QM collinear model study
of N2 + 02 colksion
truncated potentials and some classical trajectories are presented in section 4 and used to elucidate the dy namics as far as possible.
2. Theory and calculations The model diatom-diatom collision system is pictured in fig. 1, along with the notation used below. Taking each diatomic molecule harmonic and using an exponential repulsion between the nearest end atoms, the total hamiltonian describing the system is: SC= (-fi2/2@/aS
+ exp(-ar)
where x is the centers of mass distance, yi (i = 1,2) the intramolecular displacement from equilibrium, II characterizes the steepness of the exponentially repulsive interaction and r =X - ylyl - 7*y2, where 7i = miz/ (m, + mi2) is the distance between the inner atoms to within an additive constant. As is usual in atom-diatom collisions [8], the problem is reduced to dimensionless coordinates and parameters. Here the left-hand molecule is totally reduced. Thus the reduced hamiltonian becomes: H = (-a2/2i14) a2/aX2
r=
hlm22(mll
m22P2
+ mdlq2ql(m21+
M, w and (Yare, respectively, the reduced mass, the reduced frequency and the reduced steepness parameter. The quantity 7 is clearly a dimensionless orientation parameter which, for given molecules, determines which atoms are the inner ones. The usual expansion JI = x,, &r &3 unI VI) u,,(Y2) of the total wave function solution of& = E*, where ~hr and unZ are harmonic oscillator wave functions whose eigenvalues are (nl + l/2) and w(n2 + l/2), respectively, results in the following set of coupled differential equations: {d2/d.X2 + (uMIo;L)[e-(nl -Mz+
+ l/2)
1/2)1)f,~.;(x)=(2Ml~2)ex~(-x)
+ exp[-X t cr(Y1 f 7Y2)] whereU,;.I~=(u,;(Yl)lexp(xY1)lu,,(Y1))isa
+$(-a2/ayf
+ Y:) +$.-a*/aYi
t w2~z),
where
molecule 1
.
reduced mass p, =!!h!?h rw+ml~ force constant k,
I
frequency o1 = (k, /bl I”’
-
coupling matrix element for which a closed form expression is available [IO].
(intermolecular reduced mass) molecule 2 reduced moss ~+EzI!?& m21+m22
force mnstant kl I frequency o2 = (k2 /pJ”’
Fii 1. Definition of the coordinates and notation used to describe the colh-tea~collisionof two diatonic molecules.
X. Chap&at ef al./QMcollinear modeistady ofN2 + 02 colfision
287
The coupledequationsare solvednurne~~~y by meansof a very reliableintegrator basedon the Fox method [ 11,121.In the calculationswhoseresultsare reported in section 3, all channelswhoseenergyis smallerthan the total energypIus one quantum ofN, are retained,i.e., all the open channelsand the Iowest closedchannels.Underthis condition, the largestcalculation for N2 + 0, is at a total energyequal to 6.25 (in units of hwN2);it includes45 channelsof which 8 are closed.
3.ResuIts
.
The resultsreported here are for the collisionsystem N, + 0, (M= 0.533, [Y= 0.114, u = 0.670,~ = 0.936)*. The notation rzIn2- rz;rz; (AE) denotes a globaltransitionin whichthe partial transitionin N2isnlttn;andthatin02isn2t-,n;.AEisthe energy exchangedin the transition.AIIthe energies are expressedin units of the ~bration~ quantum of N,. Some illustrativebasictransitionprobabilitiesare pictured in fig. 2. Clearly,over the whole rangeof total energy studied (erot is variedfrom 0.75 to 6.25), three different groups of processescan be distinguished: (i) two dominant processes, namely the pure TV jump of one quantumin O,, denoted by {TV,01 (0.67)) where0.67 is AE, and the non-resonantex. changeof one quantumbetweenN, and O,, denoted by {TVV,1I (0.33)}where T recallsthat the transition involvessome energy exchangebetween the vibrational and translationalmotion. In fig. 2 these processesare illustratedby the transitions00 - 01 and 01 tf IO,respectiveIy_ (ii) four seco&ap processes, for which the transition probabiiitiesare betweenfifty and one hundred times smaUerthan the transitionprobabilitiesof the dominantprocesses,whateverthe total energy. These processesare the pure TV jump of one quantumin N, {TV, IO (I .O>I,the pure TVjump of two quanta in Oz {TV,02 (1.34)), the non-resonantexchangeof one
* We use the syncconvention as&crestand Johnson[8J to evaluate w and a, i.e., a = S A-’ is used in exp(-ar) and the molecular constants are that given by Her&erg [V].Mand y are simple mass ratios.
-d-- 1f Fig.2. Variation withEtot (inunit5of the vi&rational quantum of N2) of some basic transition probabitities. This illustrates aiI the various processes occurring in thecollision NZf 02. The main (solid lines) and secondary (dashed lines) processes are represented, along with the nearly resonant process 20 4-b 03 (dotted lime).The processes not mentioned here are less important by orders of magnitude whatever atot. The ordinate scaleis logarithmic. quantumin N2 for two quantain O2{TVV,12 (0.34)) and, a bit lesslikely, the exchangeof two quanta between N, and 0, {TW, 22 (0.66)). (iii) all the other more intricate processesare much lessimportant, includingthe accidentallyresonantexchangeof two quanta in N, for three quanta in 0, {W, 23 (0.01)). It appearsthat both the valuesof the potential couplingmatrix elementsand the amount of energyexchangedwith translationare important factors in determiningthe probabilityof a giventransition.Finally, the probabilityof a pure TV processincreasesmore rapidlywith emI than the probabilityof a TW proc-
Table 1 ~u~er~#~ values at various coifision energiesof tra~~t~oflprobab~ties fw whichboth Nr and 0s are initial& in their t&etexcited vibrational state. The numbers in parentheses are the amountS of energy exchanged with tm&atioa Ail energies are in tits of the v&rational quantum of Nz. Theo v&w, denoted by a question mark, were not kgiile on the Wings. The notation 0.417 -2 is ‘used for 0.417X 10”
“co1 0.08 0.58
l&8 1.58 2.08
2.58 3.08 3.58 4.58
*f-r.of
Pggj
1 l-*01
0.328 -14 0.683 -9 0.159 -6 0.31f -5 a.247 -4 O*llT-3 0.384 -3 a.999 -3 0.417 -2
0.432-8 0.576 -4 0.881 -3 0.256 -2 0.611 -2 Ml@ -1 0.174 -1. &244 -1 0.377 -I
0.117 -I 1
0.512 -6 0.281 -4 0.336 -3 0.152 -2 0.464 -2 0.107 -1 0.209 -1 QSS6-1
O.fUQ-5
QlM -3 0.11s -2 CL362-2 0.758 -2 0.132 -1 &201-l 0.365 -I
a.130 -7 ? 0.107 -4 a.372 -4 WI? -3 0.344 -3 0.893 -3 0.438 -2
0.272 -3 &I92 -2 0.638 -2 0.159 -1 i3.548-1
? 0.X36-g 0.325 -6 U.618-5 0.485 -4 0.230 -3 0.194 -2
ess. For instance~~~~~~becomes greater than PbF::b at etol = 3.0. Thus at low etor the resonant P$$$i and the TW P(os34)_ 10_07 are favoured in relative value.
Moreover~~~3~~ CI becomes greater than P&-$0 and Plpo;3”dat emi = 4.5 and 5.5, respectively.
Wenow turn to interpret the va~~on with ear of probabilities starting from a given initial state. A numerical i~ustra~on is given in table 1 where N2 and 02 are both ~iti~~y in the first excited vibrational state. Al Iow ecol the absobtte value of the
various transition
energy exchangedwith translation(A.@controls mainIy the ~robab~~ti~of the transitions,and for a same IAEI the exoergie transition (AE CO) is more probable than the endaergic one. Thus ar ecol = 1S8: p\;O$’ -c > Pg$’
>$++;;I
>f&U$ -r+
> P;&$j
>p;o-;;’ L)
> Pp;;-+
-
%k dy Plr_,03 is not where it should be a~cor~mg to AE. indeed the associatedprocess{TW, I2j is secondary and non~~ct so that it is u~ke~y aloud AE is small.This order of the proba~~~es h&Is up to fcor Q X5. For greater EBBSthe probabilitiesobey strictly the classificationin dominant processes,secondary processes,etc.. Moreoverthe exoergic or endoergiccharacter of a transition no longer influencesits proba~i~ in a sigrdfrcentmanner. Thus at ewl = 4.58:
Fig. 3 showsseveraltransition probabilitiesof purely ly TV tr~sitions versuseaP Clearly at highest co~sion energiesthe strong coupling regimeis app~a~hed. indeed the dominant process transition probab~ties .PQQ_,Q~, Po~_,o~and I&,03 approach their maxhnal vahres,On the basisof the results reported in fig. 3, six points are to be mentioned: (i) The ex~ita~on (or deex~i~~nn) probab~ti~ of 0, by one quantum are about one hundred times greater than that of N2 at givenemI; this is due ta the combined effect of the A,!%and the couplingmatrix demerits. (ii) The excitation (or deex~~t~on) of 0, by two
quanta is, at givenecot,about as probable as the excitation (or ~x~~~on~ of Nz by one quantum. Due to the difference between the AR’sthe former is shgbtly less probable up to eeeI* 5.0; then it becomes more probable. (hi) Whenthe ~bra~~~ state of the collision partner is banged after collision,for both Nz and 02 the more excited the orations state prior to co& sion the larger the probabity of a pure TV transition, As in an atom-diatom collision this is due to the increase of the couplingmatrix element, (iv) The ~robab~ty of a one quantum TV transition of the type On+ In dependss~~~g~~on n, i.e., the ~bration~ state of 02 (n) is a crucial factor when N2 undergoesa pure TV transition. This is remarkabk
289
X. Chapuisatet al./QMcollinearmodel study of N2 + 02 collision
Fig. 3. Logarithmicplot of some purely TV transition probabilities versuscc01(ii units of the vibrational quantum of Nz). since it demonstrates that the dynamical role of O2 in the one quantum TV jump in N3 can by no means be reduced even approximately to that of a united atom. The more excited o2 the more excitable N2 is. For instance the ratioP~~~~~:Po~~ll:Pg2_1~ = I.O:4.25 : 10.0 is almost constant whatever ecoL.The main purpose of the next section will be to elucidate this point.
(v) On the contrary the variable vibrational state of N2 does not influence at all the values of the transition probabilities when 0, undergoes a pure TV transition. For instance Poo_,oL, P~J_,,, and P20_,2l on the one hand, andPOt,+02 and PLO_,12on the other-hand are almost identical whatever E,,.,~Numerical values of these probabilities are given in table 2. Clearly N2 acts as a united atom with respect to TV transitions in 0,. (vi) All TV processes other than {TV, 10) and {TV, 02) - such as {TV, 1l} or {TV, 20) - are less probable by orders of magnitude. It should be emphasized that if the vibration of N2 were mu& faster than that of O,, the results above could be interpreted as follows: 02 feels that N2 is a united atom because it has no time to see N2 vibrate, and thus perceives it on the average as a rigid moIecule. On the contrary N, sees 0, vibrating slowly, and classically, the more excited the vibration of 02 the faster the displacements of the oxygen atoms and the more efficient the collision with N,. Obviously this interpretation must be refined quantum mechanically to understand why vibrational frequencies whose ratio is 0.67 allow such well-marked results. Some TW transition probabilities are reported in fig. 4. There is no simple connection between the probabilities and the AE’s. Thus P~~~~~: Pl”d_?“d2 is equal to 75.0 and 6.0 at eCol= 2.75 and 5.25, respectively. On the average the TW curves are less steep than the pure TV ones because of their smaller AE’s. Here the dominant process is (TVV, 11) and the more excited the vibrational states of the two collision partners prior to collision, the greater the transition probability. Moreover this increase is rigorously the same regardless whether 0, or N, is excited. Thus Prr_,zu %Po2_,:r z I .7 X Pol_lo whatever ecol. The same is true for the secondary process {TW, 221. For instance Pr2-,3o * Po3_21 % 2.9 X P02_,20 whatever ecol. A rather differ-
Table 2 The probability of a given pure TV transition undergoneby 02 isquite insensitive to the vibrationalstate of N2.0.581 as 0.581 x 10-l %ol
POO-rOl
~lo+ll
1.25 2.25 3.25 4.25 5.25
0.514 0.333 0.465 0.213 0.581
0.512 0.336 0.464 0.209 0.556
-6 -3 -2 -1 -1
~zo-P21
-6
0.510 -6
-3 -2 -1 -1
0.338 -3 0.462 -2 0.205 -1
P10-12
~ao-m2
0.178 0.227 0.898 0.933
-8 -5 -4 -3
0.178 -8 0.231 -5 0.896 -4 0.901-3
t -1 is the Same
X. Chapuisatet aL@f c&tear model study@iv, J- 02 collision
Fig. 5. The secondary transitions(thin arrows) GUIbe viewed es sequences of dominant transitions(thick arrows).The vibration& levels of N2 arc on the left, those of 02 on the right and the mixed ones in the middle, e is in units of the viirational quantum of Nz. Fig. 4. Logarithmicplot of some TW transitionprobab~~tie~ versus‘co1 (in units of the vibrationalquantum cf N2).
ent situation is observed for the last secondary process {TVV, IZ}. Here the transition probabilitiesare stilI enhanced by 0, as welI as N, being initially excited pirt to a greater extent for 0, than for Nz. Thus Pll_.03 x 1.4 XPa_.12 * 2.8 XP,,_,,, whatever e&_ Therefore the rule is that in any case an initial vibrationaIexcitation of the collision partners favours a ‘WV process and the greater the number of quanta involved for a molecule in the process, the more favoured the processby the Initial excitation of this molecule.
4. Complementary calculationsand interpretation All the four secondary processescan be viewed as interfering sequences of the two dominant processes, according to the following schematic equations:
{TV,
02) = {TV, 0132,
{TVv,223={Tw,
11)2,
This is iIIustratedin fig. 5. For instance, 00 + 10 can be replaced by the sequence 00 + 0 13 10 and 10 + 02 by either 10 -* 013 02 or 10-t 11+ 02, etc.. In this section we attempt (i) to test to what extent this is true and (ii) to interpret as far as possible the various dynamical trends exhibited by the preceding results. For these purposeswe have found it useful to perform two types of complementary calculations.In the first calculationswe use a modified variablepotentiaI to elucidate the details of the couplingsin the co&sion. The second caIcuIatlonsare classicaltrajectories using the true potential. Although they do not provide quantitative information, they help to visuakze the adiabatic& effects due to the recoil bf the moIecuIesin the course of the co&ion. For quantitative comparison
291
X. Chapuisat et al.lQM collinear model study of N2 + 02 collision
purposes, classical transition probabilities from a histogram type analysis should be used or semiclassical corrections should be introduced [13]. Indeed average energy transfers represent a considerably larger degree of averaging. Instead of the “exact” repulsive interaction potential:
v(X, Yl, Y2)=exp[-X+a(Y1
+rY,)J
,
we now use a parametrized interaction potential of the form:
JTX,Yl, Y2)= exp(-x)(1+ Cl0aYl+ C01qY2 +&&Y~
+ C,,&Y,
Y2 ++ Co2cK272Y;).
This new potential withall the c’s equal to 1 results in the “exact” potential truncated to second order. Then the coupling matrix elements and the overall transition probabilities are very similar to those given by the “exact” potential (for a comparison, see table 3). In particular when an “exact” matrix element is less than 10S3, its counterpart in the truncated case is zero. However even then the associate transition probability is well reproduced. This is indicative of a non-direct transition mechanism. In addition the new potential is more flexible. Indeed it can be truncated in a physitally significant way by taking some of the C’s equal to zero (or to some other value). For instance taking Cl1 = 0 amounts to a suppression of the term produc-
Table 3 Comparison of the coupling matrix elements (a), and of the overall transition probabilities at atot = 3.5 (b), in case the “exact” repukive potential is used (lower numbers) or is truncated to second order (upper numbers). 0.1007 +I is used for 0.1007 X 10”. AU the matrix elements of the “exact” potential greater than 10d3 have nonzero and satisfactory counterparts when the potential is truncated and even for a zero matrix element the associate transition probability is satisfactory
00
01
0.1007 +1 0.1008 +l
0.9218 0.9287 0.1016 0.1016
10 -1 -1 +l +l
0.8061 0.8122 0.7431 0.7486 0.1014 0.1014
02 -1 -1 -2 -2 +1 +I
0.6008 0.6053 0.1304 0.1319 0.0 0.4880 0.1024 0.1025
-2 -2
-3 +l +1
20
03
0.7431 -2 0.7486 -2 0.8061 -1 0.8091-l 0.9218 -1 0.9347 -1 0.1051-1 0.1063 -1 0.1022 +1 0.1023 +l
0.4595 -2 0.4629 -2 0.0 0.4267 -3 0.1140 0.1152 0.0 0.2781 -4 0.1051 -1 0.1062 -1 0.1020 +1 0.1021+1
0.3221 0.1041 O.lOSl 0.0 0.2597 0.1597 0.1622 0.0 0.8476 0.0 0.1480
11
20
03
0.112 -7 0.143 -7 0.104 -4 0.968 -5 0.718 -3 0.772 -3 0.488 -2 0.408‘-2
0.28 -11 0.38 -11 0.417 -8 0.422 -8 0.7 14 -6 0.838 -6 0.313 -5 0.222 -5 0.255 -2 0.215 -2
0.432 0.549 0.576 0.664 0.676 0.193 0.663 0.705 0.807 0.236 0.809 0.309
_
21
12
11
0.0
0.0
0.0 -3 -1 -1 -4
-3 -5
0.4880 0.1051 0.1063 0.6008 0.6093 0.806i 0.8260 0.1304 0.1328 0.0 0.6923
-3 -1 -1 -2 -2 -1 -1
0.4267 -3
00
0.0 0.6060 -3 0.1162 0.1140
O2 11
-3
@I) Etot = 3.50 01
10
0.665 -2 0.729 -2
0.158 0.188 0.641 0.516
02 -4 -4 -2 -2
0.580 0.689 0.344 0.372 0.208 0.365
-5 -5 -2 -2 -4 -4
-9 -9 -6 -6 -8 -7 -3 -3 -5 -5 -8 -7
12
21
0.27 -12 0.37 -12 0.472 -9 0.494 -9 0.180 -7 0.206 -7 0.736 -6 0.677 -6 0.988 -4 0.105 -3 0.161 -5 0.579 -5
(0.14 -16) (0.19 -16)
o.
0.73 -10 0.65 -10 0.338 -7 0.321 -7
o2 11
+ Cl1
neglected
Cl0
CIO + c20 neglected
Cl1 neglected
neglcctcd
co2
neglected
CO1
neglected
c20
neglected
Ct0
* cm
Transition probability:
Pr0C43:
+ cm
0.202 -5
0.510 -8 ObO
0.0
0.120 -3
0.0
0.0
0.0
0.0
0.323 -5 (0,405 $1
0.819 -8
0.195 -3 (0.247 -3)
0.374 -7 (0.0)
0.157 -5
0.720 -10
0.253 -5 We
0.407 -8
0.225 -5
0.568 -8
0.136 -3
0.923 -4
0.160 -5 (0.211 -5)
0.4 13 -8
0.942 -4 (0.128 -3)
0.680 -15
0.176 -14
0.609 -10
0.157 -9 (0.250 -9)
0.220 -13 10.134 -24)
0.365 -10
0.767 -10
0.317 -10 (0.674 -10)
poo+o2
p01-42
Poo-a1 p10-01
TV02
TV01
0.281 -7
0.0
0.160 -7
0.140 -7
0,280 -7
0.117 -11
0,o
0.705 -12
0,389 -11
0.120 -11
0.314 -11
0.29% -11
0,129 -7 (0.0) 0,185 -7 (0.589 -7)
0,237 -11
0,935 -8 (0.282 -7)
f
PO14
poo+10
TWO
0.0
0,221 -3
Q.153 -5
0.211 -3
0.215 -3
5.310 -3
0.218 -3
0,x3 -3
POb+IO
WI1
0.0
0.225 -5
0.873 -%
0.202 -5
0.216 -5
0.291 -5
0.145 -9
0.0
0,490 -9
0.467 -7
0.%79 -11
0.367 -7
0.319 -7
0.361 -7
0.240 - S 0.215 -5
PI o-02 po2-+11
w12
Tablo 4 Transi!ion probabiljties obtained in neglecting certain terms in the potential at eIot = 1.96. “Exact” denotes hero probabilities obtained for the potential developed to second order. The numbers in parentheses correspondin% lo TV transitions arc obtained by blocking the collision partner vibration ----
X. Chapuisat
et al./QM collinear model study of Na f 02 collision
ing direct W exchanges. Thus the new potential is well designed for studying the couplings. In the first series of calculations we neglect successivelyC10,C20,C0~,C02,C~~,C~0+C20+C01+C02 (i.e., we keep only CD) and finally Cl0 + Cl1 (i.e., the potential terms which are responsible for the direct dominant processes). The results are given in table 4 at ttot = 1.96. These results allow us to clarify the dynamics considerably. For this we use the following facts known from numerous studies of the atom-diatom collinear collision: (a) The use of an interaction potential truncated to first order with respect to the internal coordinate of the diatomic leads to a large increase in the TV transition probabilities. This is true for close-coupling (CC) as well as distorted wave (DW)calculationsand
is frequently referred to as Mies’“anharmonicity factor” effect [ 141. (b) The atom-diatom collisionsare usually very adiabatic.This explainswhy the perturbed stationary state (PSS) treatment givesresults in much better agreement with CC results than the DW treatment [ 151. In the diatom-diatom case, we must take account of the compression of both molecules during the collision, which makes the collision probably more adiabatic than in the atom-diatom case. (c) In the atom-diatom collisionsome processes are essentially direct (Le., they occur mainly via the
matrix element connecting the initial to the fmal state). Other processes result from a sequence of transitions during the same collision; for instance the overall transition 0 + 3 occurs via the sequence 0 + 1 + 2 + 3. The same behaviour is to be expected in the diatomdiatom collisions. Hence the results in table 4 can be rationalized. For instance the behaviour of Pu,-,_ul (column 1) is reasonably interpreted if the transition 00+ 01 is mainly direct. Indeed, neglecting Cl0 amounts to preventing the compression of N2 during the collision, thus decreasing the adiabatic@ and increasing the probability from 0.942 -4 to 0.136 -3 (0.128 -3 when N2 is rigid). Neglecting($2 doubles the probability as ex-
pected from the disappearanceof the Miesfactor. NeglectingCo1 suppressesthe main cause of the transition and the probabihty drops to 0.253 -5. This analysis is true whatever the transition within {‘IV,01) (column 2 and 3) and the probabilities depend very little on the vibrational state of N2 (cf. table 2). Sii-
293
larly the behaviour ofPu,-,_o2 (column 4) is easily explained by the transition 00 + 02, being essentially a sequential two quanta transition in 0, depending on N2 through adiabatic&y effects only. Thus it is more affected by suppressing Co1 than Co2. In fact {TV, 01) + {TV, 02) constitute a closed group for which suppressing Cl1 is almost negligible*. Pul_, 10 and Puz, 11 (columns 7 and 8) behave in agreement with {TW, 1 l}, being essentially a direct one quantum exchange process governed by the Cl, term in the potential. Indeed suppressing Cl1 results in a sharp decrease of the probabilities (from 0253 -3 to 0.153 -5 forPO1+lo and from 0.240 -5 to 0.873 -8 for Po2_,D) whereas all other changes of the potential lead only to tiny modifications. {TW, 221, for which no probability appears in table 4 because of a too small erot (channel 20 is not opened) is also definitely a sequential process because (i) there is no matrix element connecting the initial and final states when the potential is truncated and Po2+20 is nevertheless correctly accounted for (0.3 13 -5 instead of 0.222 -5 at emI = 3.5 in table 3) and (ii) suppressing C, I modifies Po2_,20 drastically (see below, table 5). Similarly {TVV, 12) is clearly an interfering process whose composents are {TV, 01) and {TVV, 11) because P1o_,o2 is correctly accounted for with the truncated potentiaI(O.208 -4 instead of 0.365 -4 at etot = 3.5 in table 3) although the associate matrix element is zero and suppressing either Co1 (P10+02 decreases from 0.361 -7 to 0.879 -11 at etot = 1.96) or Cr, (0.490 -9) results in the probability breaking down. All other changes of the potential are about negligible in this respect. Last of all (TV, lo}, the one quantum jump in N2, is not as simply analysed as the former secondary processes. Indeed PoO_,lo and Pal, 11 (columns 5 and 6) behave quite differently. They correspond to the same excitation of N,, O2 being either in state 0 or 1. The only way we found to clarify the situation consists of assuming these transitions to occur via two different routes, namely: (a) a direct one quantum TV transition in N2, for instance 00+ 10; (b) a complex route involvinga W exchange, for * SuppressingCl1 forbids the W processesto occur as a primary effect. However, simultaneously the adiabaticityof me collision decreasesand the anharmonicityfactor issomewhat modified.
: : i
X. Chapthat et aL/QMcoNinear mode! study ofN2 + 02 collision
tribution of {W, 11) when O2 vibrates than when 02 does not vibrate. This is true whatever etot_ Moreover the greater the vibrational excitation of 0, the more important the influence of Ctl. Indeed, at etot = 3.5, Po~_~~ changes from 0.287 -5 at Cl1 =O to 0.182 -4 at C,, = 1.5 (multiplied by 6.3) and Paz, 11 from 0.852 -7 to 0.135 -5 (multiplied by 15.8). On the contrary increasing Cl1 slightly decreases Poo_,o~ and all the probabilities within {TV, 01} as well asPoO+o2This is due to a slight increase of the collision adiabaticity. Fiially {TVV, 11)) {TVV, 12) and {TW, 22) are all strongly enhanced when C,, increases. Next we report in fig. 6 and table 6 the results of trajectory calculations for the same system based on the true exponentialinteractionpotential. The total classical lagrangian is:
lcl =
1
&dx/dt)
2
295
- exp(-ar)
2 +z
IQ~i(d~ildt)~
-$&$I >
where the notation is that of fig. 1 and t is the time. ‘The lagrangian is reduced by introducing dimensionless time and coordinates: T =cqt, X=ar, Y =QYlYlP
L,, = @Z(dX/dl-)2 - exp(-X +f’ f TZ) +; [(dy/dT)l -v2]
f $ [(dz/dr)2 - w7_z2],
Fig. 6. The basic classicaltrajectory at ecol= 3.5. Both N2 and O2 are initially motionless (n2cl = PI:*= -0.5). The true coordinates X, y and L (solid lines), the displaced equilibrium positions (dotted lines) and the N2 displacement for rigid 02 (dashed line) ate plotted versus time. AU coordinates and time are reduced.
Table 6 I[lustrarion of theadiibaticity effect, (a) &antum mec~~i~ pore TV transition probabilities at various e&s. Classical energy ex~~~esaf Nz andO [b) at variuus ec& N2 and 02 being inifialiy claskdiy motionless, and cc) at given ~~1, varying the init&dvibrational states of Nz.and 02. The upper numbers all are obtained for the true diatom-diatom collision. The lower numbers in pa~&~eses represent the same quantities as the upper ones but are obtained within a model in which the vibration of the co&i&n pzhx is artificially bfockd. In (c) the lower numbers which are not in parentheses (for a;1 = 0 and t@ = 0 or 1.0) are quanNm mechanical energy tr@nsfers G0
.
fcol
0,935 -6 . (0.121 -5)
0.605-12 (0,254 -11)
1.96
(0.132 0.100 -3) -3
(0.289 iJ.110 -7 -7)
f0.713 0.420 -1O -101
3.50
0.129 -2 (0.983 -2)
0.188 -4 to.rlgti -4)
0.689 -5 CO.124-4)
1.30
0.39 -f3 (0.131-5)
0.124 -4 (0.176 -4)
196
0.406 -6 (0.295 -51
0.275 -3 (0.368 -3)
3SO
0.686 -4 @.I77 -3f
0.832 -2 Kl.113 -1)
1.30
0385 -11 10.265 -10)
4549 -9 (O.t29 -8)
(cl Et01= 3.50 *d1 -O.Z
42
2.0
0.0
42
1.0
4,
4,
42
42
0.684 -4 (0.177 -3)
C.832 -2 [0.113 -11
-0.558 -2 (-O.liZ-4)
u.121 -I
-8168 -1
0.197 -1
0.565 -2
0.524 -3 (0.581-2)
-0.725 -3 0.188 -4
0.482 -2 0.489 -2
-0.234 -1
B.f 33 -1
0.168 -1
-0.151 -1
0.899 -2 a.913
-u-982 -0.969 -2
-MS!
-2
0.564 -3
0.279 -I
-0.306 -1
0.187 -1
-0.245 -1
0.428 -3
-0.123 -1
where the parameters M, w and 7 are the same as above. Hence the eq~tio~s of motion are: Md2X/dr2 J exp(-X t y t YZ),
d2y{dr2=I-exp(-X +y f yz), d2z/d$ = -J.&Z - T exp(-x’ + y C 72) _
Ali reduced chical energies are measured ia units of kl(mll +m 2)2~tzzm~2.C!learlyy = @Y, and z =aY.p hG@ whereEctis a reduced classical enerThusE, = OL gy and csm the corresponding reduced quantum me-
chanfcalenergy, is., the true energy dividedby Fiol. In fig, 6 we plot the &placed reduced equilibrium
297
X. Chapuisnr et aL/QM collinear modeistudyof N2 + 02 colbion
positions of both N, and O2 during the collision.They are solutions 0f avpy= av/az =0,wheretr= $JJ + ~tJ2~2fex~-X+y+yz),andwrite:~=d~(1+~2/w2) and5=($w2)y, where A is solution ofd +(l f y2jw2) e-X+A = 0. Six initial conditions must be specified. For each
reduced oscillator a “classical”quantum number ni* an! a vibrational phas t are introduced so that ~1= n + f and ~2 = w(n 2 -!-$). Then at T = 0,y =ymax sin 52 and dz/dr= st 51, dy/dr =ymax cos &, z = it bax zmaxcos .$, wherey,,, = (u(2nl + 1)t12and zmsx = cYr(z?I;i + 1)/o] 112.Finally at sufficiently large X we specify the collisionenergy ecol and dX/dT= -[(2/M) (012ecoi- e- *+Y+rz)] *lz. Here the E’Sare in units jttil as above. The classicaltrajectories illustrate the great adiabaticity of the collision.In fig. 6 we plot the displacements during the collision of the internal coordinates of N2 and 02 as functions of 7. The trajectory is for ecol = 3.5 and $’ = ng’= -0.5, i.e., the moleculesare initially motionless.We observedthat the maximal compressionsof both moleculesare much greater than the finalvibrationalamplitudesand are mainly accounted for by the displacedequilibriumpositions at the heart of the collision.This strong adiabaticity effect is more pronounced for N2 than for Cl2due to the fact that 0, is a softer molecule than N2. Moreoverthe plotted N2 displacementfor frozen Q, has a sunstantiallylarger amplitudethan the true N2 displacementwhereasthe 0, displacementfor frozen N, is about indistinguishabIe from the true 0, displacement. Other quantum mechanicalcaIcuIations(a) and classical trajectories (b and c) are summarizedin table 6. In (a) we compare pure TV transition probabilitiestesuitingfrom the diatom-diatom encounter with transition probabilitiesobtained in blockingthe collision partner vibration. The transition probabilitiesof N2 are more affected than that of O,, Indeed at em1= 3.5, POO_Olis multiplied by 1.35,PotiOZ by 1.8 tid Poo+10 by 2.6 when the partner vibration is frozen. Similarlyin (b) we compare the classicalenergy transfers in N2 and O2 resultingfrom the trne collision (when both of them are initially motionless)with energy transfers at blocked collisionpartner. At cool= 3.5, c’ by 1.36 only. AE$ is multipliedby 2.6 and AEo2 Fir&y in (c), at eCol=3.5, we carry on the same comparisonwith ~‘1’= 0 and 1 and nit = Cl,1and 2”. In addition we report some more complex classicalener-
gy transfersand we present two elementsofcompar;on to test the validity of the classical calculation_ For n1 = n$ =O, A@ is extremely good (0.482 -2) compared with Aflorn (O-489-2),** whereasAI?? is bad (-0.725 -3 insteadof0.188 4) becaus%f a threshold effect. Indeed the vibrationalquantumof N2is largeand anegativeclassicaltransfer obtains. For nfl=0 and R; = 1 the thresholdeffects is absentand the classicaland quantum-mechanicaltransfers are in good agreement.If. transition probabilitieswere to be reproduced, the quasiclassicalresults would be meaningfulonly for the 11.+ 20,02 -+ 11 and 01 -+ 10 probabilitiesat high collisionenergy @co1> 4.0, see fig. 4), as shown by Cohen and Alexander [16]. These authors found that for collinearD, + D, collisions,quasiclassicalW transitions probabilitiesare significantlyin error except when the probabilitiesexceed 0.1. The study above showsthat 02 - which is more excitable than N2 because of its smallervibrational quantum - is plunged in a more exciting field during the collisionbecause the recoil of N2 is smaller.This explains why {TV, 011 is associated with probabilities greater than that of (TV, 10) by two orders of magni-
tude.
5. Conclusion
I Y
In this paper we have presented a detailed study df * HereA$ = jnE”(Ei)dt * (l/A’) $1 b@(k), and N is the tatal number of trajectories.It is equal to 20 when
the othermolecule is either initially motionless or rigidand to 20 X 20 = 4 0 when the two molecules vibrate initially.
c! = $ = 0), 50 X 50 = 2500 trajectories In one case (n1 were run to test the precision of the ink@ involved. The results are the same for 400 or 2500 trajectories, up to 3 ** s&j&ant di&s. In the collision N&zl) + 02(n2):
298
X. Chapuisatet aI.r.lQM collinearmodel study of N2 + 02
the vibrationalexcitation of the two moleculesin the collisionN2 t 02. Although our model is collinearand the potential used is just a crude one, the main trends
of the collisionare so pronounced as to be hopefully true, at least qualitatively,in more realisticmodels. Since the rotational degreesof freedom are not included, our mode1cannot describereaIisticaIlyall aspects of the dynamics.Whether the grossfeatures of W energy tiansfer are independent of rotation could only be detern&id by comparingthe collinearstudy with a 3dimensionaltreatment. Our conclusionsapply to a wide range of thermal conditions. However,we do not assert that our results are directly applicableto vibrational relaxation experiments. An interesting extension of the present work wiIIbe to go down to lower temperaturesin order to study the increasinginfluence of the resonant and nearly resonant processes [ 171. Then a technical difficuIty will be to obtain convergedvaluesof very small transition probabilities. Our fmt result is that the TV transition probabilities of 02 are greater than those of Nz by two orders of magnitudebecause 0, is the more excitable molecule (since its vibrational quantum is the smallerone) and is acted on by the more exciting field (N2 recoils less than 02). The second result is that the four secondary processes(m, lo}, {TV, 021, ITW, 123 and {TW, 22) are interfering sequencesof the two dominant processes,namely {TV, 011, the TV one-quantum jump in 02, and {TVV, 111, the W one-quantumexchangebetween Nz and 0,. The tbird result is that in the case cf {TV, IO}, the one-quantumjump in Nz,
collisian
the mechanismis not unique but depends on the fact whether 0, is vibrationaIIyexcited - or not - prior to collision. In the former case {W, 11) controls {TV, 10). In the latter case {TV, 10) is rather a direct process. The adiabatic@ is important in the excitation mechanism.AI1these resuItsare true whatever ecol over the range studied.
References [II D. Secrest, AM. Rev. Phys Chem A 24 (1973) 379. [2J J.N.L. Connor, Ann. Rep. Prog. Cbem. 70 (1973) 5. [3j M.E.Riky and A. Kuppermann, Chem. Phys. Lett. 1 (1968) 537. [4] V.P. Gutschick, V. McKay and D.J. Diestier, J. Chem. Phys. 52 (1970) 4807. [S] M.H. Alexander, I. Chem. Phys. 59 (1973) 6254. [6] D.J. Wilson, J. Chem. Phys. 53 (1970) 2075. [7] M.H. Alexander, J. Chem. Phys. 60 (1974) 4274. [8] D. Secrest and B.R. Johnson, J. Chem. whys. 45 (1966) 4556. (91 C. Herzbeg, Spectra of Diatomic Molecules, 2nd Ed. (van Nostrand, 1950) pp. 553 and 560. [IO] D. Rapp and T. Sharp, J. Chem. Phys. 38 (1963) 2641. [ 111 L. Fox, The Numerical Solution of Two-Point Boundary Value Problems in Ordinary Differential Equations (Oxford Univ. Press, London, 1957). [12] D.W. Norcross and M.J. Seaton, J. Phys B6 (1973) 614. [13] J.W. Duffand D.G. Troblar, Chem. Phys. 9 (1975) 243. [14] F.H. Mics, J. Chem. Pbys. 40 (1964) 523. [15J E. Thiele and R. Katz, J. Chem. Phys. 55 (1971) 3195, [16] SC. Cohen and M.H. Alexander, J. Chem. Phys. 61 (1974) 3961. [171 W.F. Calaway and G.E. Ewing, J. Chem. Phys. 63 (1975) 2842.
A QUANTUM-MECHANICAL COLLINEAR MODEL STUDYOF
THECOLLIS10NN2+02 Xavier CHAPUISAT, Guy BERGERON Loboratoire de Chimie Thiorique’, Universitc? de Paris-Sud, Centre d’Orsay - 91405 Onay, France
Jean-Michel LAUNAY Dipartement d’Astrophysique Fondame,ztaie. Observatoire de Paris. 92190 Meudon, France
Received10 August 1976
A model close-coupling study of the translation-vibration-vibration (TVV) energy transfer in the collinear collision of two non-identical harmonic diitomic molecules is presented. The numerical applications are for Nt + 02 driven by an cxponential repulsion. Numerous transition probabilities are given, and the various dynamical processes are classified according to their importance. The classification is true whatever the collision energy. A reasonable interpretation of the dynamics based on (i) our knowledge of the simpler atom-diatom collision, (ii) the results of auxiliary calculations using various truncated expansions of the interaction potential, and (iii) classical trajectory calculations for the same system, is finally presented. The main results arc: (i) The TV transition probabilities of 02 are greater than that of N2 by several orders of magnitude; this is due to 02 being the more excitable molecule and being acted on by the more exciting field. (ii) The secondary processes are interfering sequences of the two principal processes which arc, respectively. the TV one-quantum jump in 02 and the W one-quantum exchange between N2 and 02. (iii) In the case of the TV one-quantum jump in N2 the mechanism is not unique according as 02 is vibrationally excited - or not - prior to collision.
1. Introduction The model collinear collision of an atom and a diatomic molecule has been the object of numerous publications in the recent past (for a review, see refs. [ 121). A number of exact quantum-mechanical calculations using mode1 potentials have been published, the aim of which was twofold: (i) to reveal some basic features of the collision dynamics which hopefully are also exhibited within more realistic models, and (ii) to allow a comparison with the results of approximate methods at various levels of approximation. Only a few exact calculations have appeared concerning the more compIex problem of the collision of two diatomic molecules. Most of them were for two identical molecules *The Laboratoire de Chirnie Theorique is associated with the C.N.R.S. (ERA no 549).
[3-S]. The case of two non-identical diatomics was most rarely treated [6,7]. The purpose of the present communication is (i) to present such exact quantum-mechanical results for a case modeling the collision N2 t 0, (the molecules are harmonic and the interaction potential is an exponential repulsion between the nearest end atoms), and (ii) to propose a detailed physical interpretation of the results. This interpretation is based on our knowledge of the atom-diatom collision dynamics and on a comparison with results obtained for the same system either classically with the same potential or quantum-mechanically with various truncated expansions of the potential. The model is described in section 2 together with the relevant close coupling equations and the details of the calculations. The numerical results are presented‘ in section 3. Complementary calculations using the
X.
286
Chapuisat et aL/QM collinear model study
of N2 + 02 colksion
truncated potentials and some classical trajectories are presented in section 4 and used to elucidate the dy namics as far as possible.
2. Theory and calculations The model diatom-diatom collision system is pictured in fig. 1, along with the notation used below. Taking each diatomic molecule harmonic and using an exponential repulsion between the nearest end atoms, the total hamiltonian describing the system is: SC= (-fi2/2@/aS
+ exp(-ar)
where x is the centers of mass distance, yi (i = 1,2) the intramolecular displacement from equilibrium, II characterizes the steepness of the exponentially repulsive interaction and r =X - ylyl - 7*y2, where 7i = miz/ (m, + mi2) is the distance between the inner atoms to within an additive constant. As is usual in atom-diatom collisions [8], the problem is reduced to dimensionless coordinates and parameters. Here the left-hand molecule is totally reduced. Thus the reduced hamiltonian becomes: H = (-a2/2i14) a2/aX2
r=
hlm22(mll
m22P2
+ mdlq2ql(m21+
M, w and (Yare, respectively, the reduced mass, the reduced frequency and the reduced steepness parameter. The quantity 7 is clearly a dimensionless orientation parameter which, for given molecules, determines which atoms are the inner ones. The usual expansion JI = x,, &r &3 unI VI) u,,(Y2) of the total wave function solution of& = E*, where ~hr and unZ are harmonic oscillator wave functions whose eigenvalues are (nl + l/2) and w(n2 + l/2), respectively, results in the following set of coupled differential equations: {d2/d.X2 + (uMIo;L)[e-(nl -Mz+
+ l/2)
1/2)1)f,~.;(x)=(2Ml~2)ex~(-x)
+ exp[-X t cr(Y1 f 7Y2)] whereU,;.I~=(u,;(Yl)lexp(xY1)lu,,(Y1))isa
+$(-a2/ayf
+ Y:) +$.-a*/aYi
t w2~z),
where
molecule 1
.
reduced mass p, =!!h!?h rw+ml~ force constant k,
I
frequency o1 = (k, /bl I”’
-
coupling matrix element for which a closed form expression is available [IO].
(intermolecular reduced mass) molecule 2 reduced moss ~+EzI!?& m21+m22
force mnstant kl I frequency o2 = (k2 /pJ”’
Fii 1. Definition of the coordinates and notation used to describe the colh-tea~collisionof two diatonic molecules.
X. Chap&at ef al./QMcollinear modeistady ofN2 + 02 colfision
287
The coupledequationsare solvednurne~~~y by meansof a very reliableintegrator basedon the Fox method [ 11,121.In the calculationswhoseresultsare reported in section 3, all channelswhoseenergyis smallerthan the total energypIus one quantum ofN, are retained,i.e., all the open channelsand the Iowest closedchannels.Underthis condition, the largestcalculation for N2 + 0, is at a total energyequal to 6.25 (in units of hwN2);it includes45 channelsof which 8 are closed.
3.ResuIts
.
The resultsreported here are for the collisionsystem N, + 0, (M= 0.533, [Y= 0.114, u = 0.670,~ = 0.936)*. The notation rzIn2- rz;rz; (AE) denotes a globaltransitionin whichthe partial transitionin N2isnlttn;andthatin02isn2t-,n;.AEisthe energy exchangedin the transition.AIIthe energies are expressedin units of the ~bration~ quantum of N,. Some illustrativebasictransitionprobabilitiesare pictured in fig. 2. Clearly,over the whole rangeof total energy studied (erot is variedfrom 0.75 to 6.25), three different groups of processescan be distinguished: (i) two dominant processes, namely the pure TV jump of one quantumin O,, denoted by {TV,01 (0.67)) where0.67 is AE, and the non-resonantex. changeof one quantumbetweenN, and O,, denoted by {TVV,1I (0.33)}where T recallsthat the transition involvessome energy exchangebetween the vibrational and translationalmotion. In fig. 2 these processesare illustratedby the transitions00 - 01 and 01 tf IO,respectiveIy_ (ii) four seco&ap processes, for which the transition probabiiitiesare betweenfifty and one hundred times smaUerthan the transitionprobabilitiesof the dominantprocesses,whateverthe total energy. These processesare the pure TV jump of one quantumin N, {TV, IO (I .O>I,the pure TVjump of two quanta in Oz {TV,02 (1.34)), the non-resonantexchangeof one
* We use the syncconvention as&crestand Johnson[8J to evaluate w and a, i.e., a = S A-’ is used in exp(-ar) and the molecular constants are that given by Her&erg [V].Mand y are simple mass ratios.
-d-- 1f Fig.2. Variation withEtot (inunit5of the vi&rational quantum of N2) of some basic transition probabitities. This illustrates aiI the various processes occurring in thecollision NZf 02. The main (solid lines) and secondary (dashed lines) processes are represented, along with the nearly resonant process 20 4-b 03 (dotted lime).The processes not mentioned here are less important by orders of magnitude whatever atot. The ordinate scaleis logarithmic. quantumin N2 for two quantain O2{TVV,12 (0.34)) and, a bit lesslikely, the exchangeof two quanta between N, and 0, {TW, 22 (0.66)). (iii) all the other more intricate processesare much lessimportant, includingthe accidentallyresonantexchangeof two quanta in N, for three quanta in 0, {W, 23 (0.01)). It appearsthat both the valuesof the potential couplingmatrix elementsand the amount of energyexchangedwith translationare important factors in determiningthe probabilityof a giventransition.Finally, the probabilityof a pure TV processincreasesmore rapidlywith emI than the probabilityof a TW proc-
Table 1 ~u~er~#~ values at various coifision energiesof tra~~t~oflprobab~ties fw whichboth Nr and 0s are initial& in their t&etexcited vibrational state. The numbers in parentheses are the amountS of energy exchanged with tm&atioa Ail energies are in tits of the v&rational quantum of Nz. Theo v&w, denoted by a question mark, were not kgiile on the Wings. The notation 0.417 -2 is ‘used for 0.417X 10”
“co1 0.08 0.58
l&8 1.58 2.08
2.58 3.08 3.58 4.58
*f-r.of
Pggj
1 l-*01
0.328 -14 0.683 -9 0.159 -6 0.31f -5 a.247 -4 O*llT-3 0.384 -3 a.999 -3 0.417 -2
0.432-8 0.576 -4 0.881 -3 0.256 -2 0.611 -2 Ml@ -1 0.174 -1. &244 -1 0.377 -I
0.117 -I 1
0.512 -6 0.281 -4 0.336 -3 0.152 -2 0.464 -2 0.107 -1 0.209 -1 QSS6-1
O.fUQ-5
QlM -3 0.11s -2 CL362-2 0.758 -2 0.132 -1 &201-l 0.365 -I
a.130 -7 ? 0.107 -4 a.372 -4 WI? -3 0.344 -3 0.893 -3 0.438 -2
0.272 -3 &I92 -2 0.638 -2 0.159 -1 i3.548-1
? 0.X36-g 0.325 -6 U.618-5 0.485 -4 0.230 -3 0.194 -2
ess. For instance~~~~~~becomes greater than PbF::b at etol = 3.0. Thus at low etor the resonant P$$$i and the TW P(os34)_ 10_07 are favoured in relative value.
Moreover~~~3~~ CI becomes greater than P&-$0 and Plpo;3”dat emi = 4.5 and 5.5, respectively.
Wenow turn to interpret the va~~on with ear of probabilities starting from a given initial state. A numerical i~ustra~on is given in table 1 where N2 and 02 are both ~iti~~y in the first excited vibrational state. Al Iow ecol the absobtte value of the
various transition
energy exchangedwith translation(A.@controls mainIy the ~robab~~ti~of the transitions,and for a same IAEI the exoergie transition (AE CO) is more probable than the endaergic one. Thus ar ecol = 1S8: p\;O$’ -c > Pg$’
>$++;;I
>f&U$ -r+
> P;&$j
>p;o-;;’ L)
> Pp;;-+
-
%k dy Plr_,03 is not where it should be a~cor~mg to AE. indeed the associatedprocess{TW, I2j is secondary and non~~ct so that it is u~ke~y aloud AE is small.This order of the proba~~~es h&Is up to fcor Q X5. For greater EBBSthe probabilitiesobey strictly the classificationin dominant processes,secondary processes,etc.. Moreoverthe exoergic or endoergiccharacter of a transition no longer influencesits proba~i~ in a sigrdfrcentmanner. Thus at ewl = 4.58:
Fig. 3 showsseveraltransition probabilitiesof purely ly TV tr~sitions versuseaP Clearly at highest co~sion energiesthe strong coupling regimeis app~a~hed. indeed the dominant process transition probab~ties .PQQ_,Q~, Po~_,o~and I&,03 approach their maxhnal vahres,On the basisof the results reported in fig. 3, six points are to be mentioned: (i) The ex~ita~on (or deex~i~~nn) probab~ti~ of 0, by one quantum are about one hundred times greater than that of N2 at givenemI; this is due ta the combined effect of the A,!%and the couplingmatrix demerits. (ii) The excitation (or deex~~t~on) of 0, by two
quanta is, at givenecot,about as probable as the excitation (or ~x~~~on~ of Nz by one quantum. Due to the difference between the AR’sthe former is shgbtly less probable up to eeeI* 5.0; then it becomes more probable. (hi) Whenthe ~bra~~~ state of the collision partner is banged after collision,for both Nz and 02 the more excited the orations state prior to co& sion the larger the probabity of a pure TV transition, As in an atom-diatom collision this is due to the increase of the couplingmatrix element, (iv) The ~robab~ty of a one quantum TV transition of the type On+ In dependss~~~g~~on n, i.e., the ~bration~ state of 02 (n) is a crucial factor when N2 undergoesa pure TV transition. This is remarkabk
289
X. Chapuisatet al./QMcollinearmodel study of N2 + 02 collision
Fig. 3. Logarithmicplot of some purely TV transition probabilities versuscc01(ii units of the vibrational quantum of Nz). since it demonstrates that the dynamical role of O2 in the one quantum TV jump in N3 can by no means be reduced even approximately to that of a united atom. The more excited o2 the more excitable N2 is. For instance the ratioP~~~~~:Po~~ll:Pg2_1~ = I.O:4.25 : 10.0 is almost constant whatever ecoL.The main purpose of the next section will be to elucidate this point.
(v) On the contrary the variable vibrational state of N2 does not influence at all the values of the transition probabilities when 0, undergoes a pure TV transition. For instance Poo_,oL, P~J_,,, and P20_,2l on the one hand, andPOt,+02 and PLO_,12on the other-hand are almost identical whatever E,,.,~Numerical values of these probabilities are given in table 2. Clearly N2 acts as a united atom with respect to TV transitions in 0,. (vi) All TV processes other than {TV, 10) and {TV, 02) - such as {TV, 1l} or {TV, 20) - are less probable by orders of magnitude. It should be emphasized that if the vibration of N2 were mu& faster than that of O,, the results above could be interpreted as follows: 02 feels that N2 is a united atom because it has no time to see N2 vibrate, and thus perceives it on the average as a rigid moIecule. On the contrary N, sees 0, vibrating slowly, and classically, the more excited the vibration of 02 the faster the displacements of the oxygen atoms and the more efficient the collision with N,. Obviously this interpretation must be refined quantum mechanically to understand why vibrational frequencies whose ratio is 0.67 allow such well-marked results. Some TW transition probabilities are reported in fig. 4. There is no simple connection between the probabilities and the AE’s. Thus P~~~~~: Pl”d_?“d2 is equal to 75.0 and 6.0 at eCol= 2.75 and 5.25, respectively. On the average the TW curves are less steep than the pure TV ones because of their smaller AE’s. Here the dominant process is (TVV, 11) and the more excited the vibrational states of the two collision partners prior to collision, the greater the transition probability. Moreover this increase is rigorously the same regardless whether 0, or N, is excited. Thus Prr_,zu %Po2_,:r z I .7 X Pol_lo whatever ecol. The same is true for the secondary process {TW, 221. For instance Pr2-,3o * Po3_21 % 2.9 X P02_,20 whatever ecol. A rather differ-
Table 2 The probability of a given pure TV transition undergoneby 02 isquite insensitive to the vibrationalstate of N2.0.581 as 0.581 x 10-l %ol
POO-rOl
~lo+ll
1.25 2.25 3.25 4.25 5.25
0.514 0.333 0.465 0.213 0.581
0.512 0.336 0.464 0.209 0.556
-6 -3 -2 -1 -1
~zo-P21
-6
0.510 -6
-3 -2 -1 -1
0.338 -3 0.462 -2 0.205 -1
P10-12
~ao-m2
0.178 0.227 0.898 0.933
-8 -5 -4 -3
0.178 -8 0.231 -5 0.896 -4 0.901-3
t -1 is the Same
X. Chapuisatet aL@f c&tear model study@iv, J- 02 collision
Fig. 5. The secondary transitions(thin arrows) GUIbe viewed es sequences of dominant transitions(thick arrows).The vibration& levels of N2 arc on the left, those of 02 on the right and the mixed ones in the middle, e is in units of the viirational quantum of Nz. Fig. 4. Logarithmicplot of some TW transitionprobab~~tie~ versus‘co1 (in units of the vibrationalquantum cf N2).
ent situation is observed for the last secondary process {TVV, IZ}. Here the transition probabilitiesare stilI enhanced by 0, as welI as N, being initially excited pirt to a greater extent for 0, than for Nz. Thus Pll_.03 x 1.4 XPa_.12 * 2.8 XP,,_,,, whatever e&_ Therefore the rule is that in any case an initial vibrationaIexcitation of the collision partners favours a ‘WV process and the greater the number of quanta involved for a molecule in the process, the more favoured the processby the Initial excitation of this molecule.
4. Complementary calculationsand interpretation All the four secondary processescan be viewed as interfering sequences of the two dominant processes, according to the following schematic equations:
{TV,
02) = {TV, 0132,
{TVv,223={Tw,
11)2,
This is iIIustratedin fig. 5. For instance, 00 + 10 can be replaced by the sequence 00 + 0 13 10 and 10 + 02 by either 10 -* 013 02 or 10-t 11+ 02, etc.. In this section we attempt (i) to test to what extent this is true and (ii) to interpret as far as possible the various dynamical trends exhibited by the preceding results. For these purposeswe have found it useful to perform two types of complementary calculations.In the first calculationswe use a modified variablepotentiaI to elucidate the details of the couplingsin the co&sion. The second caIcuIatlonsare classicaltrajectories using the true potential. Although they do not provide quantitative information, they help to visuakze the adiabatic& effects due to the recoil bf the moIecuIesin the course of the co&ion. For quantitative comparison
291
X. Chapuisat et al.lQM collinear model study of N2 + 02 collision
purposes, classical transition probabilities from a histogram type analysis should be used or semiclassical corrections should be introduced [13]. Indeed average energy transfers represent a considerably larger degree of averaging. Instead of the “exact” repulsive interaction potential:
v(X, Yl, Y2)=exp[-X+a(Y1
+rY,)J
,
we now use a parametrized interaction potential of the form:
JTX,Yl, Y2)= exp(-x)(1+ Cl0aYl+ C01qY2 +&&Y~
+ C,,&Y,
Y2 ++ Co2cK272Y;).
This new potential withall the c’s equal to 1 results in the “exact” potential truncated to second order. Then the coupling matrix elements and the overall transition probabilities are very similar to those given by the “exact” potential (for a comparison, see table 3). In particular when an “exact” matrix element is less than 10S3, its counterpart in the truncated case is zero. However even then the associate transition probability is well reproduced. This is indicative of a non-direct transition mechanism. In addition the new potential is more flexible. Indeed it can be truncated in a physitally significant way by taking some of the C’s equal to zero (or to some other value). For instance taking Cl1 = 0 amounts to a suppression of the term produc-
Table 3 Comparison of the coupling matrix elements (a), and of the overall transition probabilities at atot = 3.5 (b), in case the “exact” repukive potential is used (lower numbers) or is truncated to second order (upper numbers). 0.1007 +I is used for 0.1007 X 10”. AU the matrix elements of the “exact” potential greater than 10d3 have nonzero and satisfactory counterparts when the potential is truncated and even for a zero matrix element the associate transition probability is satisfactory
00
01
0.1007 +1 0.1008 +l
0.9218 0.9287 0.1016 0.1016
10 -1 -1 +l +l
0.8061 0.8122 0.7431 0.7486 0.1014 0.1014
02 -1 -1 -2 -2 +1 +I
0.6008 0.6053 0.1304 0.1319 0.0 0.4880 0.1024 0.1025
-2 -2
-3 +l +1
20
03
0.7431 -2 0.7486 -2 0.8061 -1 0.8091-l 0.9218 -1 0.9347 -1 0.1051-1 0.1063 -1 0.1022 +1 0.1023 +l
0.4595 -2 0.4629 -2 0.0 0.4267 -3 0.1140 0.1152 0.0 0.2781 -4 0.1051 -1 0.1062 -1 0.1020 +1 0.1021+1
0.3221 0.1041 O.lOSl 0.0 0.2597 0.1597 0.1622 0.0 0.8476 0.0 0.1480
11
20
03
0.112 -7 0.143 -7 0.104 -4 0.968 -5 0.718 -3 0.772 -3 0.488 -2 0.408‘-2
0.28 -11 0.38 -11 0.417 -8 0.422 -8 0.7 14 -6 0.838 -6 0.313 -5 0.222 -5 0.255 -2 0.215 -2
0.432 0.549 0.576 0.664 0.676 0.193 0.663 0.705 0.807 0.236 0.809 0.309
_
21
12
11
0.0
0.0
0.0 -3 -1 -1 -4
-3 -5
0.4880 0.1051 0.1063 0.6008 0.6093 0.806i 0.8260 0.1304 0.1328 0.0 0.6923
-3 -1 -1 -2 -2 -1 -1
0.4267 -3
00
0.0 0.6060 -3 0.1162 0.1140
O2 11
-3
@I) Etot = 3.50 01
10
0.665 -2 0.729 -2
0.158 0.188 0.641 0.516
02 -4 -4 -2 -2
0.580 0.689 0.344 0.372 0.208 0.365
-5 -5 -2 -2 -4 -4
-9 -9 -6 -6 -8 -7 -3 -3 -5 -5 -8 -7
12
21
0.27 -12 0.37 -12 0.472 -9 0.494 -9 0.180 -7 0.206 -7 0.736 -6 0.677 -6 0.988 -4 0.105 -3 0.161 -5 0.579 -5
(0.14 -16) (0.19 -16)
o.
0.73 -10 0.65 -10 0.338 -7 0.321 -7
o2 11
+ Cl1
neglected
Cl0
CIO + c20 neglected
Cl1 neglected
neglcctcd
co2
neglected
CO1
neglected
c20
neglected
Ct0
* cm
Transition probability:
Pr0C43:
+ cm
0.202 -5
0.510 -8 ObO
0.0
0.120 -3
0.0
0.0
0.0
0.0
0.323 -5 (0,405 $1
0.819 -8
0.195 -3 (0.247 -3)
0.374 -7 (0.0)
0.157 -5
0.720 -10
0.253 -5 We
0.407 -8
0.225 -5
0.568 -8
0.136 -3
0.923 -4
0.160 -5 (0.211 -5)
0.4 13 -8
0.942 -4 (0.128 -3)
0.680 -15
0.176 -14
0.609 -10
0.157 -9 (0.250 -9)
0.220 -13 10.134 -24)
0.365 -10
0.767 -10
0.317 -10 (0.674 -10)
poo+o2
p01-42
Poo-a1 p10-01
TV02
TV01
0.281 -7
0.0
0.160 -7
0.140 -7
0,280 -7
0.117 -11
0,o
0.705 -12
0,389 -11
0.120 -11
0.314 -11
0.29% -11
0,129 -7 (0.0) 0,185 -7 (0.589 -7)
0,237 -11
0,935 -8 (0.282 -7)
f
PO14
poo+10
TWO
0.0
0,221 -3
Q.153 -5
0.211 -3
0.215 -3
5.310 -3
0.218 -3
0,x3 -3
POb+IO
WI1
0.0
0.225 -5
0.873 -%
0.202 -5
0.216 -5
0.291 -5
0.145 -9
0.0
0,490 -9
0.467 -7
0.%79 -11
0.367 -7
0.319 -7
0.361 -7
0.240 - S 0.215 -5
PI o-02 po2-+11
w12
Tablo 4 Transi!ion probabiljties obtained in neglecting certain terms in the potential at eIot = 1.96. “Exact” denotes hero probabilities obtained for the potential developed to second order. The numbers in parentheses correspondin% lo TV transitions arc obtained by blocking the collision partner vibration ----
X. Chapuisat
et al./QM collinear model study of Na f 02 collision
ing direct W exchanges. Thus the new potential is well designed for studying the couplings. In the first series of calculations we neglect successivelyC10,C20,C0~,C02,C~~,C~0+C20+C01+C02 (i.e., we keep only CD) and finally Cl0 + Cl1 (i.e., the potential terms which are responsible for the direct dominant processes). The results are given in table 4 at ttot = 1.96. These results allow us to clarify the dynamics considerably. For this we use the following facts known from numerous studies of the atom-diatom collinear collision: (a) The use of an interaction potential truncated to first order with respect to the internal coordinate of the diatomic leads to a large increase in the TV transition probabilities. This is true for close-coupling (CC) as well as distorted wave (DW)calculationsand
is frequently referred to as Mies’“anharmonicity factor” effect [ 141. (b) The atom-diatom collisionsare usually very adiabatic.This explainswhy the perturbed stationary state (PSS) treatment givesresults in much better agreement with CC results than the DW treatment [ 151. In the diatom-diatom case, we must take account of the compression of both molecules during the collision, which makes the collision probably more adiabatic than in the atom-diatom case. (c) In the atom-diatom collisionsome processes are essentially direct (Le., they occur mainly via the
matrix element connecting the initial to the fmal state). Other processes result from a sequence of transitions during the same collision; for instance the overall transition 0 + 3 occurs via the sequence 0 + 1 + 2 + 3. The same behaviour is to be expected in the diatomdiatom collisions. Hence the results in table 4 can be rationalized. For instance the behaviour of Pu,-,_ul (column 1) is reasonably interpreted if the transition 00+ 01 is mainly direct. Indeed, neglecting Cl0 amounts to preventing the compression of N2 during the collision, thus decreasing the adiabatic@ and increasing the probability from 0.942 -4 to 0.136 -3 (0.128 -3 when N2 is rigid). Neglecting($2 doubles the probability as ex-
pected from the disappearanceof the Miesfactor. NeglectingCo1 suppressesthe main cause of the transition and the probabihty drops to 0.253 -5. This analysis is true whatever the transition within {‘IV,01) (column 2 and 3) and the probabilities depend very little on the vibrational state of N2 (cf. table 2). Sii-
293
larly the behaviour ofPu,-,_o2 (column 4) is easily explained by the transition 00 + 02, being essentially a sequential two quanta transition in 0, depending on N2 through adiabatic&y effects only. Thus it is more affected by suppressing Co1 than Co2. In fact {TV, 01) + {TV, 02) constitute a closed group for which suppressing Cl1 is almost negligible*. Pul_, 10 and Puz, 11 (columns 7 and 8) behave in agreement with {TW, 1 l}, being essentially a direct one quantum exchange process governed by the Cl, term in the potential. Indeed suppressing Cl1 results in a sharp decrease of the probabilities (from 0253 -3 to 0.153 -5 forPO1+lo and from 0.240 -5 to 0.873 -8 for Po2_,D) whereas all other changes of the potential lead only to tiny modifications. {TW, 221, for which no probability appears in table 4 because of a too small erot (channel 20 is not opened) is also definitely a sequential process because (i) there is no matrix element connecting the initial and final states when the potential is truncated and Po2+20 is nevertheless correctly accounted for (0.3 13 -5 instead of 0.222 -5 at emI = 3.5 in table 3) and (ii) suppressing C, I modifies Po2_,20 drastically (see below, table 5). Similarly {TVV, 12) is clearly an interfering process whose composents are {TV, 01) and {TVV, 11) because P1o_,o2 is correctly accounted for with the truncated potentiaI(O.208 -4 instead of 0.365 -4 at etot = 3.5 in table 3) although the associate matrix element is zero and suppressing either Co1 (P10+02 decreases from 0.361 -7 to 0.879 -11 at etot = 1.96) or Cr, (0.490 -9) results in the probability breaking down. All other changes of the potential are about negligible in this respect. Last of all (TV, lo}, the one quantum jump in N2, is not as simply analysed as the former secondary processes. Indeed PoO_,lo and Pal, 11 (columns 5 and 6) behave quite differently. They correspond to the same excitation of N,, O2 being either in state 0 or 1. The only way we found to clarify the situation consists of assuming these transitions to occur via two different routes, namely: (a) a direct one quantum TV transition in N2, for instance 00+ 10; (b) a complex route involvinga W exchange, for * SuppressingCl1 forbids the W processesto occur as a primary effect. However, simultaneously the adiabaticityof me collision decreasesand the anharmonicityfactor issomewhat modified.
: : i
X. Chapthat et aL/QMcoNinear mode! study ofN2 + 02 collision
tribution of {W, 11) when O2 vibrates than when 02 does not vibrate. This is true whatever etot_ Moreover the greater the vibrational excitation of 0, the more important the influence of Ctl. Indeed, at etot = 3.5, Po~_~~ changes from 0.287 -5 at Cl1 =O to 0.182 -4 at C,, = 1.5 (multiplied by 6.3) and Paz, 11 from 0.852 -7 to 0.135 -5 (multiplied by 15.8). On the contrary increasing Cl1 slightly decreases Poo_,o~ and all the probabilities within {TV, 01} as well asPoO+o2This is due to a slight increase of the collision adiabaticity. Fiially {TVV, 11)) {TVV, 12) and {TW, 22) are all strongly enhanced when C,, increases. Next we report in fig. 6 and table 6 the results of trajectory calculations for the same system based on the true exponentialinteractionpotential. The total classical lagrangian is:
lcl =
1
&dx/dt)
2
295
- exp(-ar)
2 +z
IQ~i(d~ildt)~
-$&$I >
where the notation is that of fig. 1 and t is the time. ‘The lagrangian is reduced by introducing dimensionless time and coordinates: T =cqt, X=ar, Y =QYlYlP
L,, = @Z(dX/dl-)2 - exp(-X +f’ f TZ) +; [(dy/dT)l -v2]
f $ [(dz/dr)2 - w7_z2],
Fig. 6. The basic classicaltrajectory at ecol= 3.5. Both N2 and O2 are initially motionless (n2cl = PI:*= -0.5). The true coordinates X, y and L (solid lines), the displaced equilibrium positions (dotted lines) and the N2 displacement for rigid 02 (dashed line) ate plotted versus time. AU coordinates and time are reduced.
Table 6 I[lustrarion of theadiibaticity effect, (a) &antum mec~~i~ pore TV transition probabilities at various e&s. Classical energy ex~~~esaf Nz andO [b) at variuus ec& N2 and 02 being inifialiy claskdiy motionless, and cc) at given ~~1, varying the init&dvibrational states of Nz.and 02. The upper numbers all are obtained for the true diatom-diatom collision. The lower numbers in pa~&~eses represent the same quantities as the upper ones but are obtained within a model in which the vibration of the co&i&n pzhx is artificially bfockd. In (c) the lower numbers which are not in parentheses (for a;1 = 0 and t@ = 0 or 1.0) are quanNm mechanical energy tr@nsfers G0
.
fcol
0,935 -6 . (0.121 -5)
0.605-12 (0,254 -11)
1.96
(0.132 0.100 -3) -3
(0.289 iJ.110 -7 -7)
f0.713 0.420 -1O -101
3.50
0.129 -2 (0.983 -2)
0.188 -4 to.rlgti -4)
0.689 -5 CO.124-4)
1.30
0.39 -f3 (0.131-5)
0.124 -4 (0.176 -4)
196
0.406 -6 (0.295 -51
0.275 -3 (0.368 -3)
3SO
0.686 -4 @.I77 -3f
0.832 -2 Kl.113 -1)
1.30
0385 -11 10.265 -10)
4549 -9 (O.t29 -8)
(cl Et01= 3.50 *d1 -O.Z
42
2.0
0.0
42
1.0
4,
4,
42
42
0.684 -4 (0.177 -3)
C.832 -2 [0.113 -11
-0.558 -2 (-O.liZ-4)
u.121 -I
-8168 -1
0.197 -1
0.565 -2
0.524 -3 (0.581-2)
-0.725 -3 0.188 -4
0.482 -2 0.489 -2
-0.234 -1
B.f 33 -1
0.168 -1
-0.151 -1
0.899 -2 a.913
-u-982 -0.969 -2
-MS!
-2
0.564 -3
0.279 -I
-0.306 -1
0.187 -1
-0.245 -1
0.428 -3
-0.123 -1
where the parameters M, w and 7 are the same as above. Hence the eq~tio~s of motion are: Md2X/dr2 J exp(-X t y t YZ),
d2y{dr2=I-exp(-X +y f yz), d2z/d$ = -J.&Z - T exp(-x’ + y C 72) _
Ali reduced chical energies are measured ia units of kl(mll +m 2)2~tzzm~2.C!learlyy = @Y, and z =aY.p hG@ whereEctis a reduced classical enerThusE, = OL gy and csm the corresponding reduced quantum me-
chanfcalenergy, is., the true energy dividedby Fiol. In fig, 6 we plot the &placed reduced equilibrium
297
X. Chapuisnr et aL/QM collinear modeistudyof N2 + 02 colbion
positions of both N, and O2 during the collision.They are solutions 0f avpy= av/az =0,wheretr= $JJ + ~tJ2~2fex~-X+y+yz),andwrite:~=d~(1+~2/w2) and5=($w2)y, where A is solution ofd +(l f y2jw2) e-X+A = 0. Six initial conditions must be specified. For each
reduced oscillator a “classical”quantum number ni* an! a vibrational phas t are introduced so that ~1= n + f and ~2 = w(n 2 -!-$). Then at T = 0,y =ymax sin 52 and dz/dr= st 51, dy/dr =ymax cos &, z = it bax zmaxcos .$, wherey,,, = (u(2nl + 1)t12and zmsx = cYr(z?I;i + 1)/o] 112.Finally at sufficiently large X we specify the collisionenergy ecol and dX/dT= -[(2/M) (012ecoi- e- *+Y+rz)] *lz. Here the E’Sare in units jttil as above. The classicaltrajectories illustrate the great adiabaticity of the collision.In fig. 6 we plot the displacements during the collision of the internal coordinates of N2 and 02 as functions of 7. The trajectory is for ecol = 3.5 and $’ = ng’= -0.5, i.e., the moleculesare initially motionless.We observedthat the maximal compressionsof both moleculesare much greater than the finalvibrationalamplitudesand are mainly accounted for by the displacedequilibriumpositions at the heart of the collision.This strong adiabaticity effect is more pronounced for N2 than for Cl2due to the fact that 0, is a softer molecule than N2. Moreoverthe plotted N2 displacementfor frozen Q, has a sunstantiallylarger amplitudethan the true N2 displacementwhereasthe 0, displacementfor frozen N, is about indistinguishabIe from the true 0, displacement. Other quantum mechanicalcaIcuIations(a) and classical trajectories (b and c) are summarizedin table 6. In (a) we compare pure TV transition probabilitiestesuitingfrom the diatom-diatom encounter with transition probabilitiesobtained in blockingthe collision partner vibration. The transition probabilitiesof N2 are more affected than that of O,, Indeed at em1= 3.5, POO_Olis multiplied by 1.35,PotiOZ by 1.8 tid Poo+10 by 2.6 when the partner vibration is frozen. Similarlyin (b) we compare the classicalenergy transfers in N2 and O2 resultingfrom the trne collision (when both of them are initially motionless)with energy transfers at blocked collisionpartner. At cool= 3.5, c’ by 1.36 only. AE$ is multipliedby 2.6 and AEo2 Fir&y in (c), at eCol=3.5, we carry on the same comparisonwith ~‘1’= 0 and 1 and nit = Cl,1and 2”. In addition we report some more complex classicalener-
gy transfersand we present two elementsofcompar;on to test the validity of the classical calculation_ For n1 = n$ =O, A@ is extremely good (0.482 -2) compared with Aflorn (O-489-2),** whereasAI?? is bad (-0.725 -3 insteadof0.188 4) becaus%f a threshold effect. Indeed the vibrationalquantumof N2is largeand anegativeclassicaltransfer obtains. For nfl=0 and R; = 1 the thresholdeffects is absentand the classicaland quantum-mechanicaltransfers are in good agreement.If. transition probabilitieswere to be reproduced, the quasiclassicalresults would be meaningfulonly for the 11.+ 20,02 -+ 11 and 01 -+ 10 probabilitiesat high collisionenergy @co1> 4.0, see fig. 4), as shown by Cohen and Alexander [16]. These authors found that for collinearD, + D, collisions,quasiclassicalW transitions probabilitiesare significantlyin error except when the probabilitiesexceed 0.1. The study above showsthat 02 - which is more excitable than N2 because of its smallervibrational quantum - is plunged in a more exciting field during the collisionbecause the recoil of N2 is smaller.This explains why {TV, 011 is associated with probabilities greater than that of (TV, 10) by two orders of magni-
tude.
5. Conclusion
I Y
In this paper we have presented a detailed study df * HereA$ = jnE”(Ei)dt * (l/A’) $1 b@(k), and N is the tatal number of trajectories.It is equal to 20 when
the othermolecule is either initially motionless or rigidand to 20 X 20 = 4 0 when the two molecules vibrate initially.
c! = $ = 0), 50 X 50 = 2500 trajectories In one case (n1 were run to test the precision of the ink@ involved. The results are the same for 400 or 2500 trajectories, up to 3 ** s&j&ant di&s. In the collision N&zl) + 02(n2):
298
X. Chapuisatet aI.r.lQM collinearmodel study of N2 + 02
the vibrationalexcitation of the two moleculesin the collisionN2 t 02. Although our model is collinearand the potential used is just a crude one, the main trends
of the collisionare so pronounced as to be hopefully true, at least qualitatively,in more realisticmodels. Since the rotational degreesof freedom are not included, our mode1cannot describereaIisticaIlyall aspects of the dynamics.Whether the grossfeatures of W energy tiansfer are independent of rotation could only be detern&id by comparingthe collinearstudy with a 3dimensionaltreatment. Our conclusionsapply to a wide range of thermal conditions. However,we do not assert that our results are directly applicableto vibrational relaxation experiments. An interesting extension of the present work wiIIbe to go down to lower temperaturesin order to study the increasinginfluence of the resonant and nearly resonant processes [ 171. Then a technical difficuIty will be to obtain convergedvaluesof very small transition probabilities. Our fmt result is that the TV transition probabilities of 02 are greater than those of Nz by two orders of magnitudebecause 0, is the more excitable molecule (since its vibrational quantum is the smallerone) and is acted on by the more exciting field (N2 recoils less than 02). The second result is that the four secondary processes(m, lo}, {TV, 021, ITW, 123 and {TW, 22) are interfering sequencesof the two dominant processes,namely {TV, 011, the TV one-quantum jump in 02, and {TVV, 111, the W one-quantumexchangebetween Nz and 0,. The tbird result is that in the case cf {TV, IO}, the one-quantumjump in Nz,
collisian
the mechanismis not unique but depends on the fact whether 0, is vibrationaIIyexcited - or not - prior to collision. In the former case {W, 11) controls {TV, 10). In the latter case {TV, 10) is rather a direct process. The adiabatic@ is important in the excitation mechanism.AI1these resuItsare true whatever ecol over the range studied.
References [II D. Secrest, AM. Rev. Phys Chem A 24 (1973) 379. [2J J.N.L. Connor, Ann. Rep. Prog. Cbem. 70 (1973) 5. [3j M.E.Riky and A. Kuppermann, Chem. Phys. Lett. 1 (1968) 537. [4] V.P. Gutschick, V. McKay and D.J. Diestier, J. Chem. Phys. 52 (1970) 4807. [S] M.H. Alexander, I. Chem. Phys. 59 (1973) 6254. [6] D.J. Wilson, J. Chem. Phys. 53 (1970) 2075. [7] M.H. Alexander, J. Chem. Phys. 60 (1974) 4274. [8] D. Secrest and B.R. Johnson, J. Chem. whys. 45 (1966) 4556. (91 C. Herzbeg, Spectra of Diatomic Molecules, 2nd Ed. (van Nostrand, 1950) pp. 553 and 560. [IO] D. Rapp and T. Sharp, J. Chem. Phys. 38 (1963) 2641. [ 111 L. Fox, The Numerical Solution of Two-Point Boundary Value Problems in Ordinary Differential Equations (Oxford Univ. Press, London, 1957). [12] D.W. Norcross and M.J. Seaton, J. Phys B6 (1973) 614. [13] J.W. Duffand D.G. Troblar, Chem. Phys. 9 (1975) 243. [14] F.H. Mics, J. Chem. Pbys. 40 (1964) 523. [15J E. Thiele and R. Katz, J. Chem. Phys. 55 (1971) 3195, [16] SC. Cohen and M.H. Alexander, J. Chem. Phys. 61 (1974) 3961. [171 W.F. Calaway and G.E. Ewing, J. Chem. Phys. 63 (1975) 2842.