Volume 38, number ‘3 : CXEMICAL E?HYSIg J_EmERS 15 March-1976. ‘. -. - :

THEORETICAL STUDY- OF THE LOW-TEMP;EtiTURE YfBRATfONAL RELAXATION OF o-& IN COLLISIONS WITH ‘He*

MiIIard H. ALEXANDER Departmerlt of Chemistry. University af Maryland. College Park, Marylarld 20742, USA

Received 14 November 1975

Cross secfiws and rate constants for the vibrat$onal Maxation of H&J = I,i= I) in collisions with 4He were determined using the coupled states method with a fulIy-converged channel basis. The interaction potential was taken to be that of Gordon and Secrest with the elastic matrix elements modified to include the spherically-symmetric component of the semi-empirical Shafer-Gordon potential. First-order forbidden transitions play a sig’nificznt role in the overall relaxation process. The cross sections for the de-excitation of the u= i,j= I level are slightly larger than those for the u= 1, j= 13 level. Ihe ratio of the corresponding rate constants for vibrational relaxation varies smoothly from a value of 1.25 at 500 K to 1.63 at 60 K.

1, Introduction

Vjbration~ly inelastic collisions between He and Hz have long served as the model for V 4 T and V --) R,T energy transfer. Recently there have appear- ed a number of quanturr;mechanical [l-6] and semi-

classical [7-91 three-dimensional calculations of cross sections and rate constants for this system. Further impetus to these studies has been provided by the experimental determination [IO] of rate constants for the vibrational relaxation of n-H, dilute in He over the temperature range 60-477 K. These experi- ments measure only an overall relaxation rate, which refers to transitions to all possible final rotational states, and are therefore unable to resolve the relative importance of individual vibration-rotation transi- tions. This is unfortunate, since recent theoretical work [6] has shown that these “propensity rules” are sensitive functions of the assumed potential sur- face. By varying the concentrations of ortho and para hydrogen it is possib!e to extract rate constants for the vibrational deactivation of each of these species, which are characterized by distinct rotational mani-

* Research supported by the Computer Science Center of the University of ?&Wand and by the Office of Naval Research, contract N~OOl4~7-0239-02.

folds. This would provide additional bidirect experi- mental information on the coupling between rotation- al and vibrational inelasticity- Similar experiments have been carried out 11 I] for the seif-relaxation of molecular hydrogen and are presently underway [ 121 for the He-H2 system.

To aid in the interpretation of these future experi- ments as well as to discover whether our present

knowledge OF t&e He-H, interaction is sufficient to predict correctly the variation of relaxation rate with rotational quantum number, we have determined fully-converged coupled states cross sections for 4He-H, vibrationa! relaxation with the hydrogen molecule initially in the u = 1, i= 1 level. The Gordon- Secrest interaction potential [l3] was used with some modifications to the.elastic matrix elements. By integrating over an equilibrium distribution of colli- sion velocities, we then obtained thermal rate con- stants. Comparison with the rates for the de-excita- tion of H, in the u= l,j= 0 level, computed previously f6], simulates results of ortho-para reIaxation expzri- ments at temperatures below =2C!O R, where only the I=0 and 1 levels are significantly populated.

2. Cross s&ion caiculatioas

The He-HZ collision dynamics are treated using

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Volume 38, ntrmk’3 CHEMICAL PHYSICS LETTERS I.5 hfarch 1976 ..-

thecoupled-states (CS) approximation method of McCuir& and Ko&-i [14,15j. Considerable study [IS] has shown that th$s method yields a high degree of

-accuracy whenever the collision is dominated by short-range repulsive interactions, which is the case for the He-Z-I2 sySte%, Qnc must solve a set of coupled differential equations, indexed by the total angular momenkm J and tl-& helicity index p, namely

Shafer-Gordon potential [ l8]. The quantity %jp is‘the body-fixed angular coupling matrix element [ 141, which can be computed from the relations

pp _ iti+ I)--33ct2 I.1 -(2j+3)(2i- I)’ i=is

112 ii+Jf+2~V+~+~)

(2i+ 1)(2j+3) ]

o’--P+mf-ct+~)

(2j + 3) (2j + 5) I

II2

) p=j+z,

where x’is the distance between the He atom and the mid-point of the Hz molecule, m is the collision re- duced mass, and kvj is the channef wavevectot. In the calculations reported here the wavevectors refer to the exact H2 vibration-rotation energies as deter- mined by Schaefer and Lester [16]. From the asymp- totic behavior of the soWions to eq. (1) one can extract partial S-matrices and, subsequentty, total cross sections [6J. For transitions into (or out of) thej= 1 leyei, one must solve these coupled equations once for fr = 0 and once for p= 1 [17]. The total cross section is given in terms of the integral cross sections correspcinding to each helicity index 11 S, 171, namely

where j (andjor i) = I. In our previous study of the vibrational de-excita-

tion of the u = 1 .j = 0 level we found that quantitative agreement with experimental rate constants was ob- tained when the spherically-symmetric component of the elastic matrk elements of the repulsive Gordon- Secrest surface [13] was repkced by the sphericaliy- syrru%etric terin in the semi-empirical He-H, poten- tial of Shafer and Gordon’[lS]. We shall utilize the same “elastically corrected” surface here. The poten- tial matrix elements in eq. (1) are defmed by

?j v’l” . = sjl. (1 -suu.)(Uj[VD(X,R)IVj“)

+ sijs;,.v,(x)+{~iiIr2(X,R)lvj”) P;;?, (3) 9

where R is the internuclear distance of the I!I2 mole- cule, Yo and V2 are given by Gordon and Secrest f 131, and. V&(X) is the spherica%y~syritmetric part of the

= 0, j’#j,j-i. 2 _ (4)

As before [3,4,6], rotating-Morse functions are used to describe ihe vibrational motion of the H2 molecuIe.

Inelastic cross sections were determined at six dif- ferent total energies ranging from 0.52 1 eV to 0.830 eV, where for o-Hz the zero of energy is taken to be the u = 0, i = 1 level. The coupled equations (1) were solved using a version of the Gordon algorithm [ 191, which had been specifically modified to ensure stabil- ity in calculated S-matrix elements of very small mag- nitude [20]. As has been previously observed [I ,3,6 J , to obtain convergence in the ro-vibrationally inelastic S-matrix elements of interest it was necessary to in- clude a considerable number of both rotationally afld vibrationally cfosed channels. Specifically, we used a

Table 1 Comparison of ro-vibrationat de-excitation cross sections

.* I J iJ EcOu = 0.024 .V+ a.1 14 =va)

0 0 0 2.09 - 8 1 1 0 2.24 - 8 1 1 1 1.63 - 8 0 2 0 3.35 - 8 1 3 0 1.79 - 8 1 3 1 1.17 - 7 0 4 0 6.45 - 8 I 5 0 5.16 - 8 1 5 1 1.10 - 7 0 6 0 l.iS - 8 1 7 0 7.09 - 9 1 7 1 8.74 - g 0 8 0 2.07 - 11

1.01 - 6 1.34 - 6 7.87 - 7 L.20 - 6 7.88 - 7 4.64 - 6 3.55 - 6 2.76 - 6 4.43 - 6

8.16 - 7 4.68 - 7 3.96 - 7 1.07 - 8

a) E&l is the collition energy in the u = 1,~’ = 1 chann&

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channel basis consisting of the rotational levels j=l&5,7foru=0,1,2andj= 1 foru=3,resulting in a total of 13 channels.

Table 2

15 March 1976

Table 1 displays “helicity-indexed” cross sections for the ro-vibrational relaxation of the u = 1, j = 1 level [eq- (2)] and compares them with the u = 1 ,j =O cross sections already reported [6]. One sees that rotational transitions which are first-order forbidden (Aj > 2) contribute significantly to vibrational relaxa- tion. This has been previously observed in studies of the vibrational relaxation of the u = 1, i = 0 level [ 1,3-6 1. Secondly the y = 1 cross sections are in most cases larger than the JI = 0 values, in spite of the fact that the angular coupling potential given by eqs. (4) is everywhere larger In magnitude when p = 0. This clearly illustrates that dynamical effects can in- fluence the size of ro-vibrationally inelastic cross sec- tions.

The third observation concerns the relative mag- nitudes of the 10 + Oj and 11 + Oj + 1 transitions. For p = 0 the inelastic angular coupling matrix ele- ments are nearly equal for transitions between succes- sive even or successive odd rotational levels and the elastic matrix elements are larger in magnitude for the even levels (except forj =j’ = 0 where the coupling vanishes). This would imply that the even cross sec- tions (10 + Oj) should exceed the odd cross sections. However, simple energetic considerations would predict larger cross sections for the odd transitions (11 -+Oji I), h’ h w tc are characterized by smaller ener- a defects. For example, AELL _+ o5 = 2534 cm-’ whereas AElO_W = 2993 cm-l. In fact the 1_1= 0 11 --f Oj f 1 cross sections are smaller than the 10 -+ Ojvalues (except for the 11 + 01 transition), while the p= 1 11 + Oj + 1 cross sections, characterized by a smaller coupling potentia1, are kzrger than the p= 0 10 + Oj values. Clearly, it is not possible to isolate a single factor which is solely responsible for the ob- served variation between the ortho (j = I) and para fj = 0) vibrational relaxation cross sections. Their magnitude probably reflects, in a subtle way, coupling and resonance effects within the rotational manifolds of the upper and lower vibrational levels.

3. Rate constants

Table 2 presents the results of the present series

Cross sections for ‘He-H2 [u = 1, i = 1) vibrational relaxztion --

ECOII

(eV)b)

ql+O;WP)

j=l 3 5 7 totalc)

5.710-3 4.89-g 2.32-8 2.55-g 2.35-g S-59-8 l-171-2 1.76-8 3.48-8 3.84-8 3.52-g 9-43-8 2.398-2 1.83-S 8.42-8 9.02-8 8.19-g 2.01-7 5.471 -2 1.08-7. 4.37--i 4.61-7 4.20-S 1.05-6 1.140- 1 9.72-7 3.36-6 3.74-6 4.20-7 8.48-6 3.147-l 3.91-S 8.43-S 1.40-4 2.48-5 2.88-4

a) Eq. (2) of text. b),&l is the collision energy in the u = 1, j = 1 channel. c) E&J_ I.51 af test.

of calculations, tabulated as a function of the collision energy in the u = 1, j = 1 channel. Also listed are the total relaxation cross sections

all-rO /. = =Ql_Oj 3 (5)

where the sum ranges fromj = 1 to j = 7. The u = 0, j = 9 channel is closed for EoolL CO.09 eV and con- tributes insignificantly to the total cross sections at 0.114 eV and C.3 147 eV (last two lines in table 2). The total relaxation cross section displays a power- law dependence on the collision energy at high ener- gies with an increasing degree of positive curvature as the energy decreases, which is similar to what was observed for the relaxation of the u = 1 ,j = 0 level [6].

By numerical integration over an assumed Maxwell- Boltzmann distribution of cohision velocities one can convert the total relaxation cross sections into rate constants [2-6]. To do so the results in table 2 were

spline-fitted (in a log-log representation) over the

range 5.710X low3 eV<E,,l<3.147>( 10-L eV with linear and quadratic extrapolations at high and low energies, respectively.-Previous work [6] indicates that the rates so determined will be accurate from 60 K to = 500 K. The computed rates for the relaxa- tion of the u =l , j = 1 level are plotted in fig. 1 as a function of the temperature. Also shown are the u = 1, j = 0 rates [6] as well as n-H2 rates, defined by aver- aging the ortho (u = 1 ,j = 1) and para (u = 1, j = 0) rates over the normal ratio of these rotational isomers

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&!ume ~S,~numb& 3 . CHEhlICAL.PHYSlCS LETTERS ” 15 hfarch 1976

Y

330 500 200 100 50

T (“K)

Fig. 1. Calsulated rate constants for the vibrational relasation ofp-Hz(u=l,j=O) [6],o-Hz(u=l.j=l),and,z-Hz. The n-Hz rate is defined by k, = (.Q f 3X-&14. Tbc open circles are the experimental z~-H2 rates of Audibcrt et al. [lo]. Also shown is the ratio of the ortho 0’ = 1) to the para 0’ = 0) rates as a function of temperature.

These are compared with the experimental n-HZ rates .pf Audibert et al. [IO].

The u = 1, j = 1 level is seen to relax slightly faster than the u = 1, j = 0 level, which reflects the magnitude of the j = 1 cross sections with helicity index ,U = 1 (table 1). The ratio of thej = 1 to j = 0 vibrational relaxation rates is shtiwn on the bottom of fig. 1. This ratio increases smoothly from a value of 1.25 at 500 K to 1.65 at 60 K. At temperatures below ~200 K only the j = 0 andj = 1 levels of molecular hydrogen are

significantly populated. Consequently the thedreticai rates displayed in fig. 1 should accurately represent the low-temperature relaxation of ortho, para, and normal hydrogen. At higher temperatures the experi- .mentaI rates will include transitions from rotational levels-with j Z 2.

The temperature dependence and magnitude of our calculated ortho/para ratio is in semi-quantitative- igreement with the results of the’ experimental study -of self-relaxation in mixtures of +H2’and p-H, [ 111. Fqi example, at 80 K the measured ratio is 1.9 which should be compared to oui value 0~~1.55. In addition,

preliminary experimental data on the He-H, System [12] indicate a comparable ratio of the low-tempera- ture ‘relaxation rates. This is most satisfying, especially in view of the imperfectipns-in _ou; assumed potential surface f6] _ Whither this surface wilI prove capable of quantitative predictions of the dependence of the relazzation rate on the initial rotational level over a wide temperature range wili have to await further experimental results.

Acknowledgement

The-author is most grateful to Drs. M. Audibert and J. Lukasik as well as to Professor J. Ducuing for having suggested this study.

References

[I ] H. Rabitz and G. Zarur, J. Chem. Phys. 61 (1974) 5076. H. Rabitz and G. Zarur, J. Cbem. Phys. 62 (1975) 1425. hi-H. Alexander, J. Chem. Phys. 61 (1974) 5167. Jf.H. Alexander, Chem. Phys. 8 (1975) 86. P. i%fcGuire and J.P. Toennics, J. Chem. Phys. 62 (1975)

4623. h1.H. Alexander and P. McGuire, Cross Sections and Rate Constants for Low-Temperature 4He-H2 Vibra- tional Relaxation, J. Chcm. Phys. (15 January, 19761, to be published. H.K. Shin, J. Phys. Chem. 75 (1971)4001;Chem. Phys. Letters 37 (1976) 143. W.H. Miller and A.W. Raczkowski, Discussions Faraday sot. 55 (1973) 4s. G.B. Sorensen, J. Chem. Phys. 57 (1972) 5241; 61 (1974) 3340;

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Prograa 187;,Qu&tum Chemistry Pro&m Exchange,

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