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Vibrational energy exchange between an oxygen molecule and an adsorbed hydrogen atom
Volume 183, number
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23 August 1991
Vibrational energy exchange between an oxygen molecule and an adsorbed hydrogen atom H.K. Shin Department of Chemistry, UniversityofNevada. Rena, NV89557. USA Received 28 March 1991; in final form 3 June 1991
The dynamics of collision between an oxygen molecule and a hydrogen atom adsorbed on a metal surface has been studied. Time evolution of vibrational energies and bond vibrations has been studied in classical mechanics. The frequency difference between the colliding oscillators has been shown to vary with time during the collision, and the interaction system becomes temporarily a near-resonant case due to the collision-induced distortion. A semiclassical procedure has been developed to calculate VV energy exchange probabilities. We find that this procedure leads to a well-defined oscillatory probability representing an efficient VV energy exchange between the colliding oscillators. In addition, we find the formation of peroxyl radicals when there is a suffkiently strong attractive interaction between the collision partners.
1. Introduction
Adsorption and energy transfer are two key elementary steps in gas-surface interactions involving HZ as one of the collision partners. Here the dissociative chemisorption of Hz on surface sites takes place [ I], and metals that chemisorb Hz then catalyze the subsequent reaction such as H2+0,-+20H; another reaction of this type is Hz+Dz+2HD [ 21. This mechanism involves the collision of O,(g) or D*(g) with H(ads), where g indicates a molecule in the gas phase and ads means an atom adsorbed on the surface of the metal. Another type of adsorbate involved in gas-surface chemistry is the situation when the adatom is an impurity, in which case the effects of impurity atoms on gas-surface interactions can be of importance; e.g., measurements of thermal accommodation coefficients are sensitively affected by such impurities [ 3,4]. The O,(g)/H(ads) collision in the Hz+OZ system is a prototype model for the development of general concepts in gas-surface interactions involving adatoms. The fundamental problem in this system is to study the dynamics of intermolecular energy transfer between the adatom and O2 before the reaction takes place. Specifically, the problem of vibration-tovibration (VV) energy exchange between the collision partners is of major importance in successful interpretation of the Hz+ Or reaction on metal surfaces. The efficiency of a VV energy exchange process depends sensitively on the magnitude of frequency mismatch between the collision partners [ 51. During the collision, the interaction causes the distortion of vibrational motion of each molecule, and this collision-induced timedependent distortion alters the frequency mismatch [ 61. This distortion will have profound effects on the collision dynamics. The questions that we are interested in answering in the O2 (g) /H(ads) interaction are what is the effect of the time-varying frequency on the VV energy exchange probability between the adatom and the gas molecule and how does it depend on the collision energy? To answer these questions, we first solve the classical equations of motion to analyze the time evolution of energy transfer and frequency distortion, and then develop a semiclassical procedure to calculate VV energy exchange probabilities. In addition, we will comment on the formation of the HO2 intermediate. Tungsten metal is chosen as the surface.
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2. Interaction model We consider the interaction of the incident O2 molecules with the adsorbed H atom and represent each atomatom interaction potential by a Morse function. The distance between the center of mass of the incident molecule and the surface is R, the distance of the adatom from the surface is d, +x1, and the instantaneous bond distance of O2 is dz+x,, where d, is the equilibrium separation of bond i; see fig. 1. Then, the intramolecular potential for each oscillator expressed in terms of a Morse function is V,(~,)=~,[l-exP(-~~l~i)l’,
where Di and b; are potential constants to be determined. The potential energy between the adatom and O2 is represented by the sum of two H...O interactions. The distances between H and the atoms of the rotating O2 ( ~0~0~) are z,,b=([R-(d,+X,)]2+[~(d2+x~)]2T[R-(d,+X,)](d~+X~)cose}“2. As shown in section 3, d, = 1.7 A, d2 z 1.2 A and the distance of the closest approach between O2 and the adatom is z 4 A, so the interatomic distances are zaB,bZR-(d,+x,)~j(dZ+xz)~~~8. Thus, the interaction potential expressed as the sum of two atom-atom
interactions is
, V=D,{exp[ (GO- ~,~lal-~e~~~~~a~-~,~/~~l~+~~i~~~~~~~~-~~~lal-~~~~~~~~-~~~/~al~ where Ds and a are intermolecular potential constants. Since the equilibrium distances are z=,,~ z RO- d, T id2 cos 0, the interaction potential energy is
-2exp[(R,-R)/2a+x,/2n+x,cos8/4a]{l+exp[-(d~+~Z)cos~/2a])}. HerewesetD,=D,=D.Thetermexp[-(d2+X2)cos8/a] effect of the H...O,, interaction, where the interatomic ( d2+x2) cos 8. We express the total potential energy as
(1)
in{l+exp[-(dz+x,)cosB/a]}representsthe distance is farther than the H...O, distance by
Fig. 1. Interaction coordinates.
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3. Numerical procedures The equations of motion for the relative coordinate, surface-H (S-H) bond, and O2 bond are d(t)=-au/aR,
@a)
M,_igt)=-au/ax,,
(2b)
M*j1*(f)=-aU/aX2,
(2c)
where p is the reduced mass of the collision system. We determine the time dependence of the rotational angle from & (2EJZ) ‘I* as 8= (2EJZ) ‘12f+ ~9, [ 7-101. Here $ is the initial phase, ER is the rotational energy and Z is the moment of inertia of 02. Standard numerical routines [ 11,121 will be used to solve eq. (2) for the initial conditions (t= to) ,
R(f,)=~ln[D/E+(D/E)~]+2aln{cosh[(E/2p)’~~f~/a]-[D/(E+D)]‘~~} Xj(to,~Y,i.~,I~~i~~{[1+(~,,il~i)”2sin(~,fo+~i)l/(~-~E,,;l~i)}
,
WI WI
where E is the collision energy and Q, = [ 2 (Di -EY,i) /M,] “‘lb,. Here S,, E,,, and Ml are the initial vibrational phase, initial vibrational energy and reduced mass of bond i, respectively. R(t) and X,(I) depend on the initial phases 6, and S,, and the phase-dependent quantities, such as vibrational energy transfer, will be averaged over all trajectories sampled using a standard Monte Carlo routine [ IO, p. 11131. We sample 720 trajectories at each collision energy E. Trajectories are begun at a distance R( lo) of 15 8, and integrated until they reach the same distance after the collision. At each E, the integration step is adjusted such that it is about $,th of the O2 vibrational period; e.g., at E= I .O eV, At= 1 fs. The potential constants used in the calculation are determined as follows. The dissociation energy and equilibrium distance of H from the tungsten surface are known to be 45 kcal/mol and I .67 A, respectively [ 13,141. While we can use 45 kcal/mol for D,, no data are available for bl. We estimate the latter constant as an average of the Morse parameters for tungsten and H2. We use the available value of the Morse potential constant with dimensions of reciprocal distance 1.4 116 A-’ for the pairwise atomic interaction in the tungsten metal [ 15 1. For Hz, the reciprocal distance determined from (ll1~~/2D~~)“*c++~ is 1.943 A-‘, where we take D,,= Dt=4.481 eV+ f&, and the HZ fundamental 4401 cm-’ [ 171. Then, l/b, obtained as an average of these two reciprocal distances is 1.677 A-’ or b, =0.596 A. The vibrational frequency of the S-H bond determined from the equation o, = (2D,/m,)‘/‘/b, is 3.242~ lOI s-l or 1720 cm-‘. For Oz, Dg=5.1156 eV, d,= 1.207 A and w, = 1580 cm- * [ 16 1. (Note that the angular frequency w2 for O2 used throughout this paper is 27~. ) The parameter 6, = ( 2D2/M2)‘/z/02 is found to be 0.376 A. To estimate the parameters for the H...O interaction, we use Doz. .02= 100 K and l/ao2...02 =4.7A-’ [17].ThevalueofbforH2is0.514A;sincethesame Morse form is used for Hz and H...O, we take one-half this value (0.257 A) and evaluate a as 4 (0.213+0.257)=0.235 A. The potential well depth D for H...O is determined as (D,,D,,.,~)“2=0.202 eV.
4. Results and discussion We consider the collision of the ground-state O2 molecule (I?,,~ = f fiw,=0.098 eV) with the adatom, where the initial vibrational energy of the S-H bond corresponds to E,,, =$t20, =0.320 eV, in the collision energy range of Ez0.05 to 2.0 eV for three rotational energies E,=O.OOl, 0.01, and 0.1 eV. Throughout the energy range, the phase averaged VT energy transfer between the gas molecule and the S-H bond is very small, and the latter bond is always deexcited. At Ea60.01 eV, the results of vibrational energy transfer are not significantly different from those of the collinear case. At Ea=O.Ol eV, the phase-averaged S-H vibrational energy 137
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removed by translation is only 5.8 x 10m3eV at E=O.OS eV and rises to 2.2~ 10e2 eV at E=2.0eV; these values are only 1.8% and 6.9% of EV,,, respectively. Even at ER= 0.1eV, the corresponding VT energy transfers are 6.8x IO-’ eV (2.1%) and 2.5x 10e2 eV (7.8%). Note that for a homonuclear diatomic molecule such as 02, the latter rotational energy is unrealistically high. For example, the most probable rotational energy at room temperature is only about 1.8~ 1Ov4 eV, whereas the average kinetic energy is 0.039 eV. Therefore, the present study indicates that the principal pathway for the relaxation of the vibrationally excited S-H bond is VV energy transfer to the gas molecule. It should be mentioned that because of the heavy mass of metal atoms, energy flow into the bulk solid phase is not efficient. Such energy flow blockage by heavy-mass atoms has already been noted in earlier studies [ 181. At a given collision energy, the vibrational energy rapidly jumps between the S-H bond and the incident gas molecule many times before the collision is over. This situation is clearly seen in fig. 2a, where the time evolution of energy transfer to or from the oscillator A&, and that to the S-H bond AE, are plotted for a representative trajectory at E= 1.0 eV and E,=O.Ol eV. The vibrational energy of each bond is defined as the sum of the kinetic and potential energies, and is a function oft, E and the initial phases. The difference between this sum and the initial energy is the amount of energy transfer to the oscillator. Both energy transfers, especially that to the S-H bond, rapidly oscillate during the collision before leveling off to a limiting value. This oscil-
0
2 4 6 FREQUENCY,0 (c’)
Y-.---J 75 0
1
TIME &-i5 Fig. 2. Time evolution of vibrational energy transfer to the S-H bond (AE,) and the gas molecule (A&) for a representative trajectory at E= 1.0eV and &=O.Ol eV, see (a). Collisiontrajectory is shown in (b).
138
Fig. 3. Dynamics and spectra of a sample trajectory representing the S-H dissociative case at E= 1.0 eV and Ea=O.Ol eV. (a) Time dependence of S-H and H-O> bond distances. AIao shown is the collision trajectory R(I). Plots show the dissociation of SH bond and the formation of H-O? bond. (b) Power spectrum ofthe H-O, bond vibration; peaksappear at 2143 and 1527 cm-‘. (c) Power spectrum of the O-O bond vibration; the major peak appears at 1544 cm-‘.
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lation shows there is rapid energy flow between the bonds in the close range of interaction with excess energy being supplied or removed by the translational motion. Fig. 2b shows the collision trajectory R(t). The OScillation of these energy transfer curves starts near t= -0.16 ps and ends at +O. 16 ps, so the duration of collision is about 0.32 ps. The limiting value of energy transfer to the adsorbed atom AR, (co) is negative, while AE13(co) is positive, indicating VV energy transfer from the adatom to Oz. We note that H+02+M+HOz+M is closely related to the chain branching step H+O,-OH+0 in the H2+02 reaction [ 19,201. This three-body step contributes significantly to both the energy release and the O2 consumption rate of the oxidation reaction [ 19,211. In Oz(g)/H(ads), the surface can act as a third body, and HOz can form if the interaction potential is sufficiently strong. In fact, we find that at E= 1.0 eV and E,=O.Ol eV the reaction S-H+02-+surface+HOz occurs when the well depth D of the S-H+02 interaction energy given by eq. ( 1) is greater than 1.2 eV. Note that the H-O, bond energy is known to be 1.83 eV [ 191. We assume the dissociation of S-H to have occurred when the bond displacement x1 exceeds 3.5 A. We confirm the occurrence of dissociation for each trajectory by checking the amount of energy accumulated in the S-H coordinate, which must exceed D, -$iw, = 1.63 eV. Fig. 3a shows the variation of S-H and H-O2 bond displacements for a sample trajectory representing the dissociative case, where we take D= 1.83 eV. The H-O2 bond displacement rapidly oscillates around 1 A, indicating that the newly formed bond is in a highly excited state. The S-H bond dissociates and the H-O2 bond forms near t = +O. 18 ps. Since the collision begins at t= -0.16 ps, it takes about 0.34 ps for the bond to dissociate and the new one to form after the collision has started. The power spectra of this newly formed H-O2 bond and the weakened O-O bond in HOz are shown in figs. 3b and 3c. Although a complete analysis of the internal motion of HOz requires use of normal modes, a local mode description reveals interesting features of the vibration of individual bonds. The newly formed H-O2 bond oscillates at 2143 cm-’ and the O-O bond is redshifted to 1544 cm-’ from the O2 fundamental in line with the lengthening of the O-O bond. The spectrum of H-O2 vibration shows a strong secondary peak at 1527 cm- ‘, which is the resonant frequency caused by the adjacent O-O bond. A low intensity peak at 2 157 cm-’ in the O-O spectrum is the resonance frequency of H-O*, and such a low intensity indicates a much weaker effect of H-O2 on the vibration of the heavier diatomic mass unit O-O; see fig. 3c. During the collision the distortion of vibrational frequencies can become significant. Since the S-H bond has the initial energy E,,, = $ko, and this energy is removed by 02, the distortion of O2 frequency, which will be denoted by w; (t), can play an important role in the deexcitation of the S-H bond. Thus the time-varying effective frequency difference for the deexcitation process is Aw (t ) = w, - [w2 + o; ( t) ] = Ao- w; (t), where AU= 140 cm-‘. In fig. 4, we show the phase-averaged time dependence of this quantity for E=O. 1, 0.5, and 1.0 eV at En=O.Ol eV. Also shown in fig. 4 is the phase-averaged collision trajectoryR(t) at E= 1.0 eV. A@(t) takes a minimum value at the distance of closest approach, which occurs at t=O, and the appearance of this minimum indicates an increase in the vibrational frequency of O2 during the collision. At low collision energies, where the perturbation of the internal motion of O2 is weak, the time-dependent term w; (t) is not important. For example, at E = 0. I eV, when the collision partners are at the distance of closest approach, Aw = 140 cm- ’ has decreased to Ao(t) = 111 cm-‘. However, at E= 1.0 eV, &(I) is large enough to balance do; i.e. the collision becomes temporarily resonant. At still higher collision energies, where the perturbation is large, w; (t) may exceed Aw at the closest approach, thus the extent of nonresonance now increases during the collision. For the present collision system involving low-lying vibrational states, where the displacement of internal coordinates is not very large, the potential energy appropriate for the VV process obtained by truncating the Taylor expansion in vibrational coordinates of U( R, x,, x2, 13)after the bilinear term is i=
1.2
A similar expansion is applied to Vi(x,) to obtain a harmonic form. We use these equations with parameters computed from exact classical trajectories to derive analytic semiclassical VV transition probabilities based on the treatments of classical relative and rotational coordinates and quantum vibrational coordinates. Earlier 139
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~~/--_1 -.1
0
.I
.2
Fig. 4. The time dependence of Ao-w;(l) during the collision at E=O. I, 0.5 and 1.OeV, E,=O.O I eV. Also shown is the collision trajectory at E= 1.OeV.
.3
COLLISIONTIME (ps)
studies on VV energy exchange in collinear diatom-diatom collisions based on such trajectories have produced the results which agree well with the exact quantum values when the oscillators are near resonance [ 221. The VV deexcitation of S-H (E,,, =$hw, ) corresponds to VV energy exchange between the harmonic oscillators S-H( I )+Oz(0)+S-H(O)+O,( 1). For this process, where vibrational energy is transferred to 02, we therefore include the time-dependent distortion in the vibrational frequency of Oz. When ladder operators (af, 4, ) [23] are introduced, the collision-induced term 4x$( a’v/ax$), becomes 4 (fi/2M2w2)(a2V/ax&, x (a$’ +a: +a$az +a2af ). Since 43’ and a$ represent the two-quantum excitation and deexcitation of O2 through pure VT energy transfer, respectively, we include only (Zt/2M, w2) ( a2V/ax<), (ala, + 4 ) in the present W process. The operator portion Of the W coupling term (8~/aX&)&f~ is (aft4 + ala2 +aT&+ u, al ), where the latter two operators are responsible for W energy exchange. Therefore, the Hamiltonian which is appropriate for the present VV energy exchange process is ~(~)=Aw,(at~,+t)+~(~*+f~2~2~22)(at~,+f)+~(~)(uta*+u,af)
9
(5)
where v22= (avjax:), and F(+
(h/2(M,M,w,w,)“2](a2v/ax,
ax,),
.
The operators ata2, a,41 and atu, -ala2 a,af]=afu,-afa2,
solution of the time-dependent IYu(O>=c(t)
140
form a closed system with respect to commutation; i.e. [aTa2, and [a,al, [afa2, a,af]] =2a,af. Thus we look for the Schrildinger equation iii I !&t ) ) =H( t) I !P(t) ) in the form [6 1
[ala*, [ala,, alaS]]=-2ata2,
exp[_f(t)441
expk(Oata21
ew[h(t)bta,-afa2)l
I V--~)>,
(6)
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where c(t), f(t), g(t) and h(t) are complex-valued functions of time. When the initial state I Y( -co)) is specified, this timedependent wavefunction provides a complete description of the interaction dynamics throughout the collision. Introducing eq. (6) in the Schriidinger equation and setting the coefficients of the operators aTal, aiaf, utu, and a$an equal to zero, we obtain 9$(t)=-F(l)f(t)‘-fi[Aw-w;(t)If(t)+F(t),
0)
ifi&t)=2F(f)g(t)f(t)+fi[Ao-w;(t)]g(t)+F(t),
WI
ifih(t)=F(t)f(t)+fh[Aw--W;(t)],
(7e)
where Aw-c&(t) with w; (t) = fMZw2( a2 V/ax]), determines the frequency mismatch during the collision. We solve these equations for f( lo) =g(tO) =h( to) = 0 with the time dependence of F(t) determined from the classical trajectory, the solution of eq. (2a). Note that c(l)=exp[fi((w,+o,)i+
jw;(r’)dl’)]. (0
The phase-averaged limiting (t-too) values off(t), g(t) and h(t) will then be used to calculate the transition probability given below. For the IO-+01 VV energy exchange process, using eq. (6) with 1y/( -co)) = IlO> in the transition probability defined as . P lo_ol = lim I<01 I Y(t)> 12, t-m
we obtain
P~o_~~= ,“; I(01 It(t)
explf(Oo~afl ewIs(t)aTa~l exp[h(t)(aTa, -ah~)l110>12
= lim IS(t) exp[h(t)] lz.
(8)
,-CC
Note that it is necessary to solve eqs. (7a)-( 7c) for both the real and imaginary parts off(t) and g(r), but for only the real part of h(t). The magnitude and strong oscillatory behavior of the VV energy exchange probability P,o_O, shown in fig. 5 indicate an efficient energy exchange between the collision partners. Below ER = 0.0 1 eV, the calculated probabilities are not significantly different from the collinear case, and only the 0.01 eV case is shown in the figure. When ER is raised to 0.1 eV, there is a shifting of the probability toward a higher collision energy E, but the same oscillatory behavior persists. In the E range considered in fig. 5, the vibrational energy of the S-H bond
Fig. 5. Tbe 1.2
l.4
COLLISIONENERGYW
lb
1.)
26
semiclassicalVV energy exchange probability P,,,+
for surface-H(
l)+OZ(0)+surface-H(0) +O,( 1) as a function of the collision energy at Ea=O.Ol and 0. IO eV.
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efftciently transfers its energy to the incident molecule near 0.1, 1.O, and 1.8 eV. The energy is transferred back to the bond near 0.5 and 1.4 eV. That is, in this collision, where the effective frequency difference Am(t) is not very large during the collision, vibrational energy efficiently jumpsbetween the oscillators. Note that ACID=140 cm- I, but the quantity which actually determines the frequency mismatch during the collision is Aw(z) = Aw- w;( I ), which changes during the collision. Further calculations indicate that the probability continues to oscillate as the collision energy increases from 2.0 eV. Finally, we note that the oscillatory structure similar to the above probability is a result we often encounter in two-state resonant scattering problems [24,25]. For resonant VV energy exchange between harmonic oscillators, it is simple to formulate an approximate but analytic probability expression which explicitly shows oscillatory behavior. For this purpose, we introduce IY(f))=e-‘Ho”*~I(t)) in the Schrijdinger equation and obtain ifiI~(t))=l?(r)]@(t)), where &=fiw,(afa,
+~)+fiw,(a~a,+~)
and R(t)=F(t)
e’HO”*(alaz+a,aJ)
.
e-iHot’*.
The solution of this equation in closed form is (@(l))=Texp(-(i/fi)
/~(P)dll)lQ( Kl
-m))=T[~~o(-ilfi)‘(n!)-“(
jd.(l’)di’)‘], ,0
where T stands for the time-ordered product [ 23,261. In principle, we can proceed indefinitely with iteration to solve this equation, However, for the case of exact resonance we can leave out the time ordering and express ]@(r))=exp(-(ijfi)
jI?(r’)dP)]@(-m)),
which is characteristic of the sudden approximation. P IO-01= lim ](Ol(exp
I-cc
-(i/h)
[&(l’)dt’ 10
The probability is then
]10}]2. >
In this procedure, if we neglect the x,- and +-dependence as e[Ro-RWl/a= {(s+s2) I/z cosh[ (E/~P)“‘~ t/a]--S}-’ where s= D/E [ 271. For this trajectory, the integral is a sine function
in eq. (2a), the collision trajectory can be obtained
,
j:, I?‘(t’) dt’ takes standard
tabulated forms and the result
I
-
iD 4a2,/Ml Mzwl co1
2
[eI~o-R(r)I/~_teI%-R(~)l/*~](~f~~+~,ut)
dt’
(10) 142
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where
/J=,/pE/2a2M,M2w,o.+
[ I+mE
(x/Z+tan-ImE)]
.
Thus this probability expression oscillates between zero and one as the collision energy increases. The second part in the square bracket of /3 determines the effects of molecular attraction, so for the purely repulsive interaction the probability is a simple oscillatory function of E ‘I*, P,o_o, =sin’[ (pE/2a2M, M20,w2)‘r2]
.
5. Concluding comments We have shown that the contribution of the time-varying frequency term induced by the collision leads the O,( g)/H( ads) system to behave as if it is a near-resonant case. The deexcitation of a vibrationally excited surface-H bond takes place mainly through VV transfer to the gas molecule. Vibration-to-vibration energy exchange between the surface-H bond and the incident oxygen molecule is efficient, and the semiclassical probability describing this VV process rapidly oscillates as the collision energy increases. The oscillatory dependence is due to the collision-inducednear match of vibrational frequencies. We have shown that the effect of rotation on the vibrational deexcitation is not very important especially at low rotational energies. When the attraction between H(ads) and O,(g) is sufliciently strong, the surface-H bond dissociates and the intermediate HO2 forms.
Acknowledgement This research was supported by an NSF Advanced Computing Resources grant (CHE4390039P) at the Pittsburgh Supercomputing Center.
References [ I ] S.L. Bemasekand G.A. Somotjai,J. Chem. Phys. 62 ( 1975) 3 149. [ 2] K.J. Laidler, Chemical kinetics (Harper & Row, New York, 1987) ch. 7. [ 31 H.Y. Wachman, Am. Rocket Sot. J. 32 ( 1962) 2. [4] S.C. Saxena and R.K. Joshi, Thermal accommodation
and adsorption coefficients ofgases (Hemisphere, New York, 1989).
[ 51J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980) ch. 5. [6] H.K. Shin, J. Chem. Phys. 79 (1983) 4285. [ 71 G.D.B. Sorensen,J. Chem. Phys. 57 (1972) 5241. (81 L.H. Scntman, Chem. Phys. Letters 18 (1973) 493. [ 91 J. Stricker, J. Chem. Phys. 64 ( 1976) 126 1. [lo] J. Ree and H.K. Shin, J. Chem. Phys. 93 (1990) 6463. [ 111 C.W. Gear, Numerical initial value problems in ordinary differential equations (Prentice-Hall, Englewood Cliffs, 197 1). [ 121 MATH/LIBRARY (IMSL, Houston, 1989) p. 640. [ 131 J.K. Roberts, Proc. Roy. Sot. A 152 (1935) 445. [ 1411. Higuchi, T. Reeand H. Eyring, J. Am. Chem. Sot. 79 (1957) 1330. [ 151 L.A. Girifa1eoandV.G. Weizer, Phys. Rev. 114 (1959) 687. [ 161 K.P. Hubcr and G. Henberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [ 171 J.B.Ca1vertandR.C. Amme, J.Chem. Phys. 45 (1966) 4710. [ 181 H.K. Shin, J. Chem. Phys. 92 (1990) 5223, and references therein. [ 19lT.H. Dunning,S.P. WalchandM.M.Goodgame, J. Chem.Phys. 76 (1981) 3482.
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[20] T.H. Dunning, S.P. Walch and A.F. Wagner, in: Potential energy surfaces and dynamics calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) pp. 329-357. [2l]J.A.MillerandR.J.Kee,J.Phys.Chem.81 (1977)2534. [22] R.T. Skodje, W.R. Gentry and CF. Clayton, Chem. Phys. 74 ( 1983) 347, and references therein. [23] W.H. LouiselI, Quantum statistical properties of radiation (Wiley, New York, 1973) pp. 60-65. [24] N.F. Mott and H.S.W. Massey, The theory of atomic collisions (Oxford Univ. Press, Oxford, 1965) chps. XV and XIX. [25] D. Rapp, Quantum chemistry (Holt, Rinehart, Winston, New York, 1971) ch. 25. [26] F.J. Dyson, Phys. Rev. 75 (1949) 486. [27] H.K. Shin and P.L. Ahick, Chem. Phys. Letters 52 ( 1977) 580.
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Vibrational energy exchange between an oxygen molecule and an adsorbed hydrogen atom H.K. Shin Department of Chemistry, UniversityofNevada. Rena, NV89557. USA Received 28 March 1991; in final form 3 June 1991
The dynamics of collision between an oxygen molecule and a hydrogen atom adsorbed on a metal surface has been studied. Time evolution of vibrational energies and bond vibrations has been studied in classical mechanics. The frequency difference between the colliding oscillators has been shown to vary with time during the collision, and the interaction system becomes temporarily a near-resonant case due to the collision-induced distortion. A semiclassical procedure has been developed to calculate VV energy exchange probabilities. We find that this procedure leads to a well-defined oscillatory probability representing an efficient VV energy exchange between the colliding oscillators. In addition, we find the formation of peroxyl radicals when there is a suffkiently strong attractive interaction between the collision partners.
1. Introduction
Adsorption and energy transfer are two key elementary steps in gas-surface interactions involving HZ as one of the collision partners. Here the dissociative chemisorption of Hz on surface sites takes place [ I], and metals that chemisorb Hz then catalyze the subsequent reaction such as H2+0,-+20H; another reaction of this type is Hz+Dz+2HD [ 21. This mechanism involves the collision of O,(g) or D*(g) with H(ads), where g indicates a molecule in the gas phase and ads means an atom adsorbed on the surface of the metal. Another type of adsorbate involved in gas-surface chemistry is the situation when the adatom is an impurity, in which case the effects of impurity atoms on gas-surface interactions can be of importance; e.g., measurements of thermal accommodation coefficients are sensitively affected by such impurities [ 3,4]. The O,(g)/H(ads) collision in the Hz+OZ system is a prototype model for the development of general concepts in gas-surface interactions involving adatoms. The fundamental problem in this system is to study the dynamics of intermolecular energy transfer between the adatom and O2 before the reaction takes place. Specifically, the problem of vibration-tovibration (VV) energy exchange between the collision partners is of major importance in successful interpretation of the Hz+ Or reaction on metal surfaces. The efficiency of a VV energy exchange process depends sensitively on the magnitude of frequency mismatch between the collision partners [ 51. During the collision, the interaction causes the distortion of vibrational motion of each molecule, and this collision-induced timedependent distortion alters the frequency mismatch [ 61. This distortion will have profound effects on the collision dynamics. The questions that we are interested in answering in the O2 (g) /H(ads) interaction are what is the effect of the time-varying frequency on the VV energy exchange probability between the adatom and the gas molecule and how does it depend on the collision energy? To answer these questions, we first solve the classical equations of motion to analyze the time evolution of energy transfer and frequency distortion, and then develop a semiclassical procedure to calculate VV energy exchange probabilities. In addition, we will comment on the formation of the HO2 intermediate. Tungsten metal is chosen as the surface.
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2. Interaction model We consider the interaction of the incident O2 molecules with the adsorbed H atom and represent each atomatom interaction potential by a Morse function. The distance between the center of mass of the incident molecule and the surface is R, the distance of the adatom from the surface is d, +x1, and the instantaneous bond distance of O2 is dz+x,, where d, is the equilibrium separation of bond i; see fig. 1. Then, the intramolecular potential for each oscillator expressed in terms of a Morse function is V,(~,)=~,[l-exP(-~~l~i)l’,
where Di and b; are potential constants to be determined. The potential energy between the adatom and O2 is represented by the sum of two H...O interactions. The distances between H and the atoms of the rotating O2 ( ~0~0~) are z,,b=([R-(d,+X,)]2+[~(d2+x~)]2T[R-(d,+X,)](d~+X~)cose}“2. As shown in section 3, d, = 1.7 A, d2 z 1.2 A and the distance of the closest approach between O2 and the adatom is z 4 A, so the interatomic distances are zaB,bZR-(d,+x,)~j(dZ+xz)~~~8. Thus, the interaction potential expressed as the sum of two atom-atom
interactions is
, V=D,{exp[ (GO- ~,~lal-~e~~~~~a~-~,~/~~l~+~~i~~~~~~~~-~~~lal-~~~~~~~~-~~~/~al~ where Ds and a are intermolecular potential constants. Since the equilibrium distances are z=,,~ z RO- d, T id2 cos 0, the interaction potential energy is
-2exp[(R,-R)/2a+x,/2n+x,cos8/4a]{l+exp[-(d~+~Z)cos~/2a])}. HerewesetD,=D,=D.Thetermexp[-(d2+X2)cos8/a] effect of the H...O,, interaction, where the interatomic ( d2+x2) cos 8. We express the total potential energy as
(1)
in{l+exp[-(dz+x,)cosB/a]}representsthe distance is farther than the H...O, distance by
Fig. 1. Interaction coordinates.
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3. Numerical procedures The equations of motion for the relative coordinate, surface-H (S-H) bond, and O2 bond are d(t)=-au/aR,
@a)
M,_igt)=-au/ax,,
(2b)
M*j1*(f)=-aU/aX2,
(2c)
where p is the reduced mass of the collision system. We determine the time dependence of the rotational angle from & (2EJZ) ‘I* as 8= (2EJZ) ‘12f+ ~9, [ 7-101. Here $ is the initial phase, ER is the rotational energy and Z is the moment of inertia of 02. Standard numerical routines [ 11,121 will be used to solve eq. (2) for the initial conditions (t= to) ,
R(f,)=~ln[D/E+(D/E)~]+2aln{cosh[(E/2p)’~~f~/a]-[D/(E+D)]‘~~} Xj(to,~Y,i.~,I~~i~~{[1+(~,,il~i)”2sin(~,fo+~i)l/(~-~E,,;l~i)}
,
WI WI
where E is the collision energy and Q, = [ 2 (Di -EY,i) /M,] “‘lb,. Here S,, E,,, and Ml are the initial vibrational phase, initial vibrational energy and reduced mass of bond i, respectively. R(t) and X,(I) depend on the initial phases 6, and S,, and the phase-dependent quantities, such as vibrational energy transfer, will be averaged over all trajectories sampled using a standard Monte Carlo routine [ IO, p. 11131. We sample 720 trajectories at each collision energy E. Trajectories are begun at a distance R( lo) of 15 8, and integrated until they reach the same distance after the collision. At each E, the integration step is adjusted such that it is about $,th of the O2 vibrational period; e.g., at E= I .O eV, At= 1 fs. The potential constants used in the calculation are determined as follows. The dissociation energy and equilibrium distance of H from the tungsten surface are known to be 45 kcal/mol and I .67 A, respectively [ 13,141. While we can use 45 kcal/mol for D,, no data are available for bl. We estimate the latter constant as an average of the Morse parameters for tungsten and H2. We use the available value of the Morse potential constant with dimensions of reciprocal distance 1.4 116 A-’ for the pairwise atomic interaction in the tungsten metal [ 15 1. For Hz, the reciprocal distance determined from (ll1~~/2D~~)“*c++~ is 1.943 A-‘, where we take D,,= Dt=4.481 eV+ f&, and the HZ fundamental 4401 cm-’ [ 171. Then, l/b, obtained as an average of these two reciprocal distances is 1.677 A-’ or b, =0.596 A. The vibrational frequency of the S-H bond determined from the equation o, = (2D,/m,)‘/‘/b, is 3.242~ lOI s-l or 1720 cm-‘. For Oz, Dg=5.1156 eV, d,= 1.207 A and w, = 1580 cm- * [ 16 1. (Note that the angular frequency w2 for O2 used throughout this paper is 27~. ) The parameter 6, = ( 2D2/M2)‘/z/02 is found to be 0.376 A. To estimate the parameters for the H...O interaction, we use Doz. .02= 100 K and l/ao2...02 =4.7A-’ [17].ThevalueofbforH2is0.514A;sincethesame Morse form is used for Hz and H...O, we take one-half this value (0.257 A) and evaluate a as 4 (0.213+0.257)=0.235 A. The potential well depth D for H...O is determined as (D,,D,,.,~)“2=0.202 eV.
4. Results and discussion We consider the collision of the ground-state O2 molecule (I?,,~ = f fiw,=0.098 eV) with the adatom, where the initial vibrational energy of the S-H bond corresponds to E,,, =$t20, =0.320 eV, in the collision energy range of Ez0.05 to 2.0 eV for three rotational energies E,=O.OOl, 0.01, and 0.1 eV. Throughout the energy range, the phase averaged VT energy transfer between the gas molecule and the S-H bond is very small, and the latter bond is always deexcited. At Ea60.01 eV, the results of vibrational energy transfer are not significantly different from those of the collinear case. At Ea=O.Ol eV, the phase-averaged S-H vibrational energy 137
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removed by translation is only 5.8 x 10m3eV at E=O.OS eV and rises to 2.2~ 10e2 eV at E=2.0eV; these values are only 1.8% and 6.9% of EV,,, respectively. Even at ER= 0.1eV, the corresponding VT energy transfers are 6.8x IO-’ eV (2.1%) and 2.5x 10e2 eV (7.8%). Note that for a homonuclear diatomic molecule such as 02, the latter rotational energy is unrealistically high. For example, the most probable rotational energy at room temperature is only about 1.8~ 1Ov4 eV, whereas the average kinetic energy is 0.039 eV. Therefore, the present study indicates that the principal pathway for the relaxation of the vibrationally excited S-H bond is VV energy transfer to the gas molecule. It should be mentioned that because of the heavy mass of metal atoms, energy flow into the bulk solid phase is not efficient. Such energy flow blockage by heavy-mass atoms has already been noted in earlier studies [ 181. At a given collision energy, the vibrational energy rapidly jumps between the S-H bond and the incident gas molecule many times before the collision is over. This situation is clearly seen in fig. 2a, where the time evolution of energy transfer to or from the oscillator A&, and that to the S-H bond AE, are plotted for a representative trajectory at E= 1.0 eV and E,=O.Ol eV. The vibrational energy of each bond is defined as the sum of the kinetic and potential energies, and is a function oft, E and the initial phases. The difference between this sum and the initial energy is the amount of energy transfer to the oscillator. Both energy transfers, especially that to the S-H bond, rapidly oscillate during the collision before leveling off to a limiting value. This oscil-
0
2 4 6 FREQUENCY,0 (c’)
Y-.---J 75 0
1
TIME &-i5 Fig. 2. Time evolution of vibrational energy transfer to the S-H bond (AE,) and the gas molecule (A&) for a representative trajectory at E= 1.0eV and &=O.Ol eV, see (a). Collisiontrajectory is shown in (b).
138
Fig. 3. Dynamics and spectra of a sample trajectory representing the S-H dissociative case at E= 1.0 eV and Ea=O.Ol eV. (a) Time dependence of S-H and H-O> bond distances. AIao shown is the collision trajectory R(I). Plots show the dissociation of SH bond and the formation of H-O? bond. (b) Power spectrum ofthe H-O, bond vibration; peaksappear at 2143 and 1527 cm-‘. (c) Power spectrum of the O-O bond vibration; the major peak appears at 1544 cm-‘.
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lation shows there is rapid energy flow between the bonds in the close range of interaction with excess energy being supplied or removed by the translational motion. Fig. 2b shows the collision trajectory R(t). The OScillation of these energy transfer curves starts near t= -0.16 ps and ends at +O. 16 ps, so the duration of collision is about 0.32 ps. The limiting value of energy transfer to the adsorbed atom AR, (co) is negative, while AE13(co) is positive, indicating VV energy transfer from the adatom to Oz. We note that H+02+M+HOz+M is closely related to the chain branching step H+O,-OH+0 in the H2+02 reaction [ 19,201. This three-body step contributes significantly to both the energy release and the O2 consumption rate of the oxidation reaction [ 19,211. In Oz(g)/H(ads), the surface can act as a third body, and HOz can form if the interaction potential is sufficiently strong. In fact, we find that at E= 1.0 eV and E,=O.Ol eV the reaction S-H+02-+surface+HOz occurs when the well depth D of the S-H+02 interaction energy given by eq. ( 1) is greater than 1.2 eV. Note that the H-O, bond energy is known to be 1.83 eV [ 191. We assume the dissociation of S-H to have occurred when the bond displacement x1 exceeds 3.5 A. We confirm the occurrence of dissociation for each trajectory by checking the amount of energy accumulated in the S-H coordinate, which must exceed D, -$iw, = 1.63 eV. Fig. 3a shows the variation of S-H and H-O2 bond displacements for a sample trajectory representing the dissociative case, where we take D= 1.83 eV. The H-O2 bond displacement rapidly oscillates around 1 A, indicating that the newly formed bond is in a highly excited state. The S-H bond dissociates and the H-O2 bond forms near t = +O. 18 ps. Since the collision begins at t= -0.16 ps, it takes about 0.34 ps for the bond to dissociate and the new one to form after the collision has started. The power spectra of this newly formed H-O2 bond and the weakened O-O bond in HOz are shown in figs. 3b and 3c. Although a complete analysis of the internal motion of HOz requires use of normal modes, a local mode description reveals interesting features of the vibration of individual bonds. The newly formed H-O2 bond oscillates at 2143 cm-’ and the O-O bond is redshifted to 1544 cm-’ from the O2 fundamental in line with the lengthening of the O-O bond. The spectrum of H-O2 vibration shows a strong secondary peak at 1527 cm- ‘, which is the resonant frequency caused by the adjacent O-O bond. A low intensity peak at 2 157 cm-’ in the O-O spectrum is the resonance frequency of H-O*, and such a low intensity indicates a much weaker effect of H-O2 on the vibration of the heavier diatomic mass unit O-O; see fig. 3c. During the collision the distortion of vibrational frequencies can become significant. Since the S-H bond has the initial energy E,,, = $ko, and this energy is removed by 02, the distortion of O2 frequency, which will be denoted by w; (t), can play an important role in the deexcitation of the S-H bond. Thus the time-varying effective frequency difference for the deexcitation process is Aw (t ) = w, - [w2 + o; ( t) ] = Ao- w; (t), where AU= 140 cm-‘. In fig. 4, we show the phase-averaged time dependence of this quantity for E=O. 1, 0.5, and 1.0 eV at En=O.Ol eV. Also shown in fig. 4 is the phase-averaged collision trajectoryR(t) at E= 1.0 eV. A@(t) takes a minimum value at the distance of closest approach, which occurs at t=O, and the appearance of this minimum indicates an increase in the vibrational frequency of O2 during the collision. At low collision energies, where the perturbation of the internal motion of O2 is weak, the time-dependent term w; (t) is not important. For example, at E = 0. I eV, when the collision partners are at the distance of closest approach, Aw = 140 cm- ’ has decreased to Ao(t) = 111 cm-‘. However, at E= 1.0 eV, &(I) is large enough to balance do; i.e. the collision becomes temporarily resonant. At still higher collision energies, where the perturbation is large, w; (t) may exceed Aw at the closest approach, thus the extent of nonresonance now increases during the collision. For the present collision system involving low-lying vibrational states, where the displacement of internal coordinates is not very large, the potential energy appropriate for the VV process obtained by truncating the Taylor expansion in vibrational coordinates of U( R, x,, x2, 13)after the bilinear term is i=
1.2
A similar expansion is applied to Vi(x,) to obtain a harmonic form. We use these equations with parameters computed from exact classical trajectories to derive analytic semiclassical VV transition probabilities based on the treatments of classical relative and rotational coordinates and quantum vibrational coordinates. Earlier 139
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~~/--_1 -.1
0
.I
.2
Fig. 4. The time dependence of Ao-w;(l) during the collision at E=O. I, 0.5 and 1.OeV, E,=O.O I eV. Also shown is the collision trajectory at E= 1.OeV.
.3
COLLISIONTIME (ps)
studies on VV energy exchange in collinear diatom-diatom collisions based on such trajectories have produced the results which agree well with the exact quantum values when the oscillators are near resonance [ 221. The VV deexcitation of S-H (E,,, =$hw, ) corresponds to VV energy exchange between the harmonic oscillators S-H( I )+Oz(0)+S-H(O)+O,( 1). For this process, where vibrational energy is transferred to 02, we therefore include the time-dependent distortion in the vibrational frequency of Oz. When ladder operators (af, 4, ) [23] are introduced, the collision-induced term 4x$( a’v/ax$), becomes 4 (fi/2M2w2)(a2V/ax&, x (a$’ +a: +a$az +a2af ). Since 43’ and a$ represent the two-quantum excitation and deexcitation of O2 through pure VT energy transfer, respectively, we include only (Zt/2M, w2) ( a2V/ax<), (ala, + 4 ) in the present W process. The operator portion Of the W coupling term (8~/aX&)&f~ is (aft4 + ala2 +aT&+ u, al ), where the latter two operators are responsible for W energy exchange. Therefore, the Hamiltonian which is appropriate for the present VV energy exchange process is ~(~)=Aw,(at~,+t)+~(~*+f~2~2~22)(at~,+f)+~(~)(uta*+u,af)
9
(5)
where v22= (avjax:), and F(+
(h/2(M,M,w,w,)“2](a2v/ax,
ax,),
.
The operators ata2, a,41 and atu, -ala2 a,af]=afu,-afa2,
solution of the time-dependent IYu(O>=c(t)
140
form a closed system with respect to commutation; i.e. [aTa2, and [a,al, [afa2, a,af]] =2a,af. Thus we look for the Schrildinger equation iii I !&t ) ) =H( t) I !P(t) ) in the form [6 1
[ala*, [ala,, alaS]]=-2ata2,
exp[_f(t)441
expk(Oata21
ew[h(t)bta,-afa2)l
I V--~)>,
(6)
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where c(t), f(t), g(t) and h(t) are complex-valued functions of time. When the initial state I Y( -co)) is specified, this timedependent wavefunction provides a complete description of the interaction dynamics throughout the collision. Introducing eq. (6) in the Schriidinger equation and setting the coefficients of the operators aTal, aiaf, utu, and a$an equal to zero, we obtain 9$(t)=-F(l)f(t)‘-fi[Aw-w;(t)If(t)+F(t),
0)
ifi&t)=2F(f)g(t)f(t)+fi[Ao-w;(t)]g(t)+F(t),
WI
ifih(t)=F(t)f(t)+fh[Aw--W;(t)],
(7e)
where Aw-c&(t) with w; (t) = fMZw2( a2 V/ax]), determines the frequency mismatch during the collision. We solve these equations for f( lo) =g(tO) =h( to) = 0 with the time dependence of F(t) determined from the classical trajectory, the solution of eq. (2a). Note that c(l)=exp[fi((w,+o,)i+
jw;(r’)dl’)]. (0
The phase-averaged limiting (t-too) values off(t), g(t) and h(t) will then be used to calculate the transition probability given below. For the IO-+01 VV energy exchange process, using eq. (6) with 1y/( -co)) = IlO> in the transition probability defined as . P lo_ol = lim I<01 I Y(t)> 12, t-m
we obtain
P~o_~~= ,“; I(01 It(t)
explf(Oo~afl ewIs(t)aTa~l exp[h(t)(aTa, -ah~)l110>12
= lim IS(t) exp[h(t)] lz.
(8)
,-CC
Note that it is necessary to solve eqs. (7a)-( 7c) for both the real and imaginary parts off(t) and g(r), but for only the real part of h(t). The magnitude and strong oscillatory behavior of the VV energy exchange probability P,o_O, shown in fig. 5 indicate an efficient energy exchange between the collision partners. Below ER = 0.0 1 eV, the calculated probabilities are not significantly different from the collinear case, and only the 0.01 eV case is shown in the figure. When ER is raised to 0.1 eV, there is a shifting of the probability toward a higher collision energy E, but the same oscillatory behavior persists. In the E range considered in fig. 5, the vibrational energy of the S-H bond
Fig. 5. Tbe 1.2
l.4
COLLISIONENERGYW
lb
1.)
26
semiclassicalVV energy exchange probability P,,,+
for surface-H(
l)+OZ(0)+surface-H(0) +O,( 1) as a function of the collision energy at Ea=O.Ol and 0. IO eV.
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efftciently transfers its energy to the incident molecule near 0.1, 1.O, and 1.8 eV. The energy is transferred back to the bond near 0.5 and 1.4 eV. That is, in this collision, where the effective frequency difference Am(t) is not very large during the collision, vibrational energy efficiently jumpsbetween the oscillators. Note that ACID=140 cm- I, but the quantity which actually determines the frequency mismatch during the collision is Aw(z) = Aw- w;( I ), which changes during the collision. Further calculations indicate that the probability continues to oscillate as the collision energy increases from 2.0 eV. Finally, we note that the oscillatory structure similar to the above probability is a result we often encounter in two-state resonant scattering problems [24,25]. For resonant VV energy exchange between harmonic oscillators, it is simple to formulate an approximate but analytic probability expression which explicitly shows oscillatory behavior. For this purpose, we introduce IY(f))=e-‘Ho”*~I(t)) in the Schrijdinger equation and obtain ifiI~(t))=l?(r)]@(t)), where &=fiw,(afa,
+~)+fiw,(a~a,+~)
and R(t)=F(t)
e’HO”*(alaz+a,aJ)
.
e-iHot’*.
The solution of this equation in closed form is (@(l))=Texp(-(i/fi)
/~(P)dll)lQ( Kl
-m))=T[~~o(-ilfi)‘(n!)-“(
jd.(l’)di’)‘], ,0
where T stands for the time-ordered product [ 23,261. In principle, we can proceed indefinitely with iteration to solve this equation, However, for the case of exact resonance we can leave out the time ordering and express ]@(r))=exp(-(ijfi)
jI?(r’)dP)]@(-m)),
which is characteristic of the sudden approximation. P IO-01= lim ](Ol(exp
I-cc
-(i/h)
[&(l’)dt’ 10
The probability is then
]10}]2. >
In this procedure, if we neglect the x,- and +-dependence as e[Ro-RWl/a= {(s+s2) I/z cosh[ (E/~P)“‘~ t/a]--S}-’ where s= D/E [ 271. For this trajectory, the integral is a sine function
in eq. (2a), the collision trajectory can be obtained
,
j:, I?‘(t’) dt’ takes standard
tabulated forms and the result
I
-
iD 4a2,/Ml Mzwl co1
2
[eI~o-R(r)I/~_teI%-R(~)l/*~](~f~~+~,ut)
dt’
(10) 142
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where
/J=,/pE/2a2M,M2w,o.+
[ I+mE
(x/Z+tan-ImE)]
.
Thus this probability expression oscillates between zero and one as the collision energy increases. The second part in the square bracket of /3 determines the effects of molecular attraction, so for the purely repulsive interaction the probability is a simple oscillatory function of E ‘I*, P,o_o, =sin’[ (pE/2a2M, M20,w2)‘r2]
.
5. Concluding comments We have shown that the contribution of the time-varying frequency term induced by the collision leads the O,( g)/H( ads) system to behave as if it is a near-resonant case. The deexcitation of a vibrationally excited surface-H bond takes place mainly through VV transfer to the gas molecule. Vibration-to-vibration energy exchange between the surface-H bond and the incident oxygen molecule is efficient, and the semiclassical probability describing this VV process rapidly oscillates as the collision energy increases. The oscillatory dependence is due to the collision-inducednear match of vibrational frequencies. We have shown that the effect of rotation on the vibrational deexcitation is not very important especially at low rotational energies. When the attraction between H(ads) and O,(g) is sufliciently strong, the surface-H bond dissociates and the intermediate HO2 forms.
Acknowledgement This research was supported by an NSF Advanced Computing Resources grant (CHE4390039P) at the Pittsburgh Supercomputing Center.
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and adsorption coefficients ofgases (Hemisphere, New York, 1989).
[ 51J.T. Yardley, Introduction to molecular energy transfer (Academic Press, New York, 1980) ch. 5. [6] H.K. Shin, J. Chem. Phys. 79 (1983) 4285. [ 71 G.D.B. Sorensen,J. Chem. Phys. 57 (1972) 5241. (81 L.H. Scntman, Chem. Phys. Letters 18 (1973) 493. [ 91 J. Stricker, J. Chem. Phys. 64 ( 1976) 126 1. [lo] J. Ree and H.K. Shin, J. Chem. Phys. 93 (1990) 6463. [ 111 C.W. Gear, Numerical initial value problems in ordinary differential equations (Prentice-Hall, Englewood Cliffs, 197 1). [ 121 MATH/LIBRARY (IMSL, Houston, 1989) p. 640. [ 131 J.K. Roberts, Proc. Roy. Sot. A 152 (1935) 445. [ 1411. Higuchi, T. Reeand H. Eyring, J. Am. Chem. Sot. 79 (1957) 1330. [ 151 L.A. Girifa1eoandV.G. Weizer, Phys. Rev. 114 (1959) 687. [ 161 K.P. Hubcr and G. Henberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [ 171 J.B.Ca1vertandR.C. Amme, J.Chem. Phys. 45 (1966) 4710. [ 181 H.K. Shin, J. Chem. Phys. 92 (1990) 5223, and references therein. [ 19lT.H. Dunning,S.P. WalchandM.M.Goodgame, J. Chem.Phys. 76 (1981) 3482.
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[20] T.H. Dunning, S.P. Walch and A.F. Wagner, in: Potential energy surfaces and dynamics calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) pp. 329-357. [2l]J.A.MillerandR.J.Kee,J.Phys.Chem.81 (1977)2534. [22] R.T. Skodje, W.R. Gentry and CF. Clayton, Chem. Phys. 74 ( 1983) 347, and references therein. [23] W.H. LouiselI, Quantum statistical properties of radiation (Wiley, New York, 1973) pp. 60-65. [24] N.F. Mott and H.S.W. Massey, The theory of atomic collisions (Oxford Univ. Press, Oxford, 1965) chps. XV and XIX. [25] D. Rapp, Quantum chemistry (Holt, Rinehart, Winston, New York, 1971) ch. 25. [26] F.J. Dyson, Phys. Rev. 75 (1949) 486. [27] H.K. Shin and P.L. Ahick, Chem. Phys. Letters 52 ( 1977) 580.