Cross sections and rate constants for the vibrational relaxation of D2 (υ = 1,j = 0) in collisions with 4He

CROSS SECTfGNS AND RATE CONSTANTS FOR THE ~B~~~ON~.~L~A~ON D,(u=~,~=O)INCOLLLSIONSWI~~~~~

:

OF

.

.

Millard H; ALEXANDER Departmtkt of Chemistry. University of Maryland, College Park, fifuryiund 20742. USA

Received 4 February 1976

vibrational reliaxationwere determined using the surface. First-order forbidden rotational transitions play a si@fkant role, comparable to that observed previously for the He-Hi system. At 60 R the u = 1, j = 0 level of D2 is predicted to relax ~4 times slower than the corresponding level of Hz. This difference decrexes with increasing temperature.

Fully converged quantum cross sections for 4He-D2 (u= l,i=O) coupled-states method and a modikied version of the Gordon-Secrcst

1.

Introduction

The vibrational relaxation of H, in collisions with He has been the object of considerable recent interest, both experimental [l-3] as welI as theoretical [4--91. At present, one can lneasure only global vibrational relaxation rates, which refer to transitions to all pos-

sible foal rotational levels. It is thus not yet possible to probe directly the coupling between vibrational . and rotational inelasticity, which would be revealed

and at 300 K [I 1,121. The present article reports the determination of low-energy cross sections for the vibrational relaxation of D2 (u = L,i = 0) in coltisions with 4H~.

2. Collision dynamics

The collision dynamics are treated within the coupled-states (CS) approximation of h&zGuire and Kouri [13]. Previous work [ 13,141 has confirmed the

only by the determination of the relative magnitude of rates for individual vibration-rotation processes. As recent theoretical work [S] has demonstrated, these “propensity rubs” are sensitive functions of the assumed potential surface of interaction. hzdirecr experimental information on the degree of V+R coupling can, however, be furnished by altering the spectrum of mofecular rotational levels. A candidate for future study is the He-D, system, which, relative to the He-Hz system, is characterized by a larger reduced collision mass but a smaller ratio of rotational to vibrational spacing. Experimentahy, this system has -been studied only under shock tube conditions [lo]

lowing our earlier studies [S,9] of the relaxation of ortho and para H,, we used the potential surface of Gordon and Secrest [IS], with the sphericalfy-symmetric component of the diagonal matrix elements replaced by the semi-empirical potential of Shafer and Gordon [16!_ The potential matrix elements which appear in the CS scattering equations are [S,9 ] I

* Research stipported in Fart by the Computer Science Center,

where g .is the body-fuced helicity index [ 131, X is the

Univer&ty of hkyland, and by the Office of Naval Research, Contract NOO~l4~7-02~~2.

accuracy of this method when applied to rotationally and ro-vibrationally inelastic He-H, collisions. FoZ-

v$f

&$

=

~~.(l-s~~)rui’rv,(x,~)l~~

~‘~j~Sw,~sc(~CcUi.‘lV,(X 0) distance between the He atom and the mid-point of the D2 molecule, R is the D2 internuciear distance, 101

.

angular coupiing mat@x element Et 31. ‘: -,thg bbdj-fii:d .‘-: &*k+forq, f759j, tE?e-ro-vibratiol~at‘mction of the D, ,Imbldcule is detiribed a&a rotating Morse oscillator. , To determine tie &&eve&ors of the vibration_‘. rot&n ch&mkls, we have tal& the experimektal Hz .; Dtinhant coefficients f X73,

(2) and ckverted- them to v&s a~~~r~pr~~e to D2 by the use of the Usual reiation [18] -

whers AT& and A& denote, respectively , the reduced masses of the I-$ ,and D, oscillators. Table 1 displays the calcuiatkd D, Dunham constants. Eq. (3) is, of course, only’ an approximation [ 181. The corrections will be, however, small [18], so that the resulting shifts in the D2 energy levels will have a negligible effect on the dakulzted vibrationagy-inelzstic cross sections. The everr rotational levels of Dz are associated wi’h the even nuclear spin states and are given the “ortho”’ ‘designation [19]. Fig. 1 displays the low-lying vibration-rotation levels of o-D2 and compares them with iho& of p-%Iz :20]. Both the rotational and vibrational spacing are fess in the case of Dz. Due to the mass factors in eq. (3) the ratio of the rotational to vibrational spacing also decreases by a factor of *21/11. The coupled scattering-equations were solved using the Cordon piogram [Zl], modified to permit the

Table 1

Non-zera Dunham coefficients used in determining D2 vibration-rotation

.’

-

‘;:

1500-

u

L.

i

-. 0

-

=

v=o O-D2

__~.

3 .P-Hz

Fig. 1. Low-1yinL:vibrati&otation levels of o& and pH2. The D2 values were computed using the Dunham co&ficients of tabte I; the I32 valuesare from ref. [Xl]. *

stable determination-of S-matrix elements of very small magnitude {22f. Total cross sections for the deexcitation of the u = f , i = 0 level of l$ were determined at six values of the total energy, relative to the u = O,i = 0 level, ranging from 0.376 eV to 0.685 eV. As in the case of the He-H, system I&9,23,24], it ivas necessary to add both rotationally ad vibrationally closed channels to obtain adequate convergence in the calculated Sr+trix. For E < O-48 eV, the channel basis included the even ieveisj = O-8 for u = 0, andi = O-6 for u = 1 and 2. This 8,6,6 basis resultrnd in a total of 13 channels. Above 0.48 eV we found it necessary to add the 0,lO and 1,8 ievels. Interestingly enough, the 0,lO level, which is resonant with the 1,2 level to within x 17 cm-l, was of only minor importance in desciibing the seiaxation of the I ,0 level, even at the higher cotlision energies.

3. Cross sections

The ~om?uted de-excitation cross sections are given in table 2 along with the total de-excitation cross sections,

energies Y,, tern-‘) a)

i 0 I2 ‘3

Cl=0

3123.48 -60.643 0.2872

I

2

30.414 -1.0820 0.0154

-0.0116

al I&‘: the Iack of an entry indicates that the correspond61%cdeff!cient is ne$igible..

for D2 and Ii2 [8l. The relative cross sections for three different collision energies are displayed in fig- 2, and compared with the corresponding He-HZ cross sections ES] _One can draw the foflowing major conclusions: (1) The important relsation pathways involve the i = 0,2, and 4 levels of u = 0. Although the i = 4 level & not directly,coupled to the 1,O state dy the interac. .

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CHEFWALPYYSICS

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_Table 2 Cross~section in A2 for the vibntional ~eiax&on of the u = 1, j 7 0 level of DZ in

n1o-+oj

j=

_._

c&is&s wiih.4ke

.&&W)

0 2 4 : lob)

:

total-D> c, total-Hz

1.5May 1976

’

LETTERS

d)

-

4.977-3

.1.098-2

2.398-2

2.74-9 2.61-9 2.58-9 l-92--10 3.0.5-l 1

4.70-9 4.61-9 4.75-9 3.78- 10 6.52-11

1.64-S 1.56-S 1.80-8 1.66-9 3.44- 10

8.14-9 3.27-S

1.45-S 4.95-S

5.i9812

-_

1.31-7

3.140-I

1.13-7

1.2%6

3.14-S

$68-8

7.07-7

2.454

1.57-7 2:05-8 4.91-9

2.14-6 3.80-7 1.03-8 1.18-10

l-08-4 4.9 l-5 4.75-6 2.70-7

4.36-6 6.60-6

2.18-4 2.j3-4

3.83-7 : 7.37-7

5.204

1.140-I

-I

’

a)

E coli is the collision energy in the u = 1, j = 0 channel;E,o = 2991.12 cm-‘- ,8,6,6 basis for E,,I~ < 0.06 eV; 10,8,6 basis at higher energies. b, The 0,lO channel opens up at E,,ll = 0.01933 eV; the l&O IO cross sections are negligibly small at 2.398-2 and S-398-2 eV. ‘) Total de-excitation cross section [see eq. (4) of text]. d) Total de-excitation cross section for HZ (v = 1, j = 0) from ref. [ 8).

tion potential [SS], the 10+04 transition plays a significant role at all energies. (2) At the higher collision energies the final state rotational distribution closely mimics that obtained for the He-H2 system [8]. As the energy drops, the flux

j Fig. 2. Relative cross sections for the de-excitation by 4He of the u = l,j= 0 levels of Hz (fiied circles, dotted line) and D2
becomes distributed more or less evenly among the j = 0,2, and 4 levels, which does represent a marked contrast with the Hz results, where the lo+06 transi-. tion is clearly dominant. A possibIe explanation involves simple resonance considerations. Transitions to j = 0,2, and 4 are aI strongly allowed dynamically. In the case of- Hz, the wider rotational spacing (relative to the vibrational spacing) implies that the j = 4 transition will be favored more strongly, since the energy gap is least. (3) On the other hand the cross sections for de-excitation to the j = 6 and 8 levels of u = 0 are small, contrary to what would be predicted by resonance considerations. Additionally, the lO+O 10 transition does not appear to be important, in spite of the 12-O 10 near resonance. Also plotted in fig. 2 are statistical or “prior” cross sections 1251, which are determined by assuming Lhat at a given collision energy the detailed transition rates are proportional to the volume of phase space associated with the various final states. Using the known expression for the density of final states [25], one obtains u IO-Oj

=f%f2(2j f I)

(1 t AE/E_,P,

(5)

where M is the collision reduced mass and AE = El0 - Eq is the amounf of internal energy transferred into translation. It is clear that for both Ha 103

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40,ntimlnx.1

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CHl#ICAL PHY‘SICS LE’lTERS

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May 1976

ind Dz the actual cr&se@ons are unrelated to the sratisticai predict~ons~ particuiary in-the case of D, st the lower collisibn energies. Most noticeably, the statistical theory attributes tbo large. a weight to the transitions with &j = 6 and 8. DynamicaUy, these transitions are unfavored for two r&sons: First, the interaction potential couples direcfiy only levels with s - O,%?. Consequently, 10-46 and it3+08 represent

higher-order proceses, involving a series of virtual transitions with ai = 2. Secondly,‘& low collision encrgies, only a smaIl number of total angular momentum values (par&d waves) contribute to The lotal cross section [24,26]. The 10406 and l-08 proccsf,cs become possible only for J > 6 and 8, respecrivrly, which, for Ec,u below ==0.02 eV, lie beyond the maxima in the partial cross section curves and arc ;&&ore of less importance 13.261. The reader should ziso note that purely statistical considerations predict that the vibrational de-excitation cross sections should i~rc~ilse with decreasing collision energy [eq. (511, which is in direct disagreement with the energy dependence displayed by the results in table 2. The total cross sections (table 2) are nearly equal :o the Hz values at Ecofi = 0.3 eV but become smaller in rekttivc magnitude as the eneru decreases. The ratio of O& to U& ranges from 1.1 at Ecofi = 0.31 eV to 4.0 at 5.0 X IO-3eV. Recen;ly, Shin IS] has arescnted an approximate dynamical treatment of the fie-ii2 system which can be easily adapted to the Gordon-Secrest surface [27]. Application of the resuiting expression for the de-excitation cross section to the f&-D2 system yields a cross section ratio which ranges from 0.5 at E = 0.3 1 eV to 2.6 at E = 5 X 10-S eV. In spite of the many dynamical approximations made by Shin 1271, the isotope effect is weI1 Fredicted. In contrast, Bergeroa and Chapuisat f28], using a purely classical treatment, concluded that Hz and D2 would display identical vibrationally inelastic behavior at all energies. Clearly, one must be extremely cautious in applying the results of a quasi-classical calculation to an energy regime where vibrational energy transfer is classically forbidden [3,29].

From the to&l de-excitaticn cross section displayed in table 2 one can calculate thermal rate con-

f “K Fig. 3- Rate constants for the relaxation of HZ and I&. initially in the state u = 1.j = 0, in collisions with “Hc, computed usins the data given in table 2. Also shown is the ratio of the relaxation rates as a function of temperature.

stants for vibrational relaxation by integrating over a Boltzmann velocity distribution 16-91. The resulting rates appear in fig. 3 and are compared with the comparable He-H, values [S] . Also shown is the ratio of the relaxation rates, kHz/kDt, as a function of temperature. Below = 100 K, o-D:! is primarily in the ground rotational level 0 = 0). Consequently, the rates displayed in fig. 3 could be directly compared with future iow-temperature experimental values. At temperatures greater than * 100 K higher rotational levels will start contributing to the overall relaxation process. To judge from earlier work [7,9], these higher levels will relax to the u = 0 manifold at a faster rate. The temperature dependence of the rate constant ratio implies that Dz might actually relax faster than Hz at higher temperatures, particularly if kvii increases with increasing initial rotational angular momentum. This behavior is expected on the basis of shock tube studies J&l O] , which attribute a larger relaxation rate to D2 for T> 1000 K. At 300 IS our calculated relaxation rate is 2.48 X lo-17 ems/s. This might be compared to the experimental values for tt-D* (67% u-Dz, 33% p1D2) of 6.98 X 1O-x8 cm3/s reported by Lukasik and Ducuing [12f and 1.84 X lo-l7 cm3/s reported by Hopkins and Chen [i l] . The calculated value is therefore =3.6 times larger than the experimental value of Lukasik and

V&utie 40, number 1

CHEMICALPHYSLCS L!ZlTERS

Ducuing; whereas the Hz (u = 1, j = 0) value at 300 K computed using the same potential surface [S] exceeds the experimental /r-H2 value [3] by a factor of only -1.8. At 300K the ratio of the theoretical rates for the relaxation of the u = l,j = 0 levels of Hz and D, is 1.8; which should be compared to the ex-

perimental ratio of k(rr-Ha) to k(rr-Dz) of 2.6 (using the rr-Da result of Lukasik and Ducuing [i 21) and 0.97 (using the result of tiopkins and Chen [ 111).

Acknowledgement

Part of this work was accomplished during a visit to the I_aboratoire d’Optique Quantique at the Ecole Polytechnique in Paiaiseau, France. The author wishes to express his gratitude to ?rofessor J. Ducuing for his hospitality and to Drs. M. Audibert and J. Lukasik for helpful and informative discussions.

References [ 1] MM. Audibert, C. Joffrin and J. Ducuin5, Chem. Phys. Letters 19 (1973) 26; J. Chem. Phys. 61 (1974) 4357. [2] J.E. Dove and H. Teitelbaum, Chem. Phys. 6 (1974) 431. [3] MM. Audibert, R. Vilaseca, J. Lukasik and J. Ducuing, Chem. Phys. Letters 37 (1976) 408. [4] G.B. Sorenson, J. Chem. Phys. 61 (1974) 3340; G.D. Billing, Chem. Phys. 9 (1975) 359. [5] H.K. Shin, J. Phys. Chem. 75 (1971) 4001:Chem. Phys. Letters 37 (1976) 143. [6] H. Rabitz and G. Zarur, J. Chem. Phys. 62 (1975) 1425. [7] M.H. Alexander, Chem. Phys. 8 (1975) 86.

1.5 May 19%

[S] M.H. Alexander and P. McCuire,J. Chem. Phys. 64 (1976)452. [91 NH. Alexander, Chem. &s. Letters 38 (1976) 417. [lo] P.F. Bird and W.D. Breshears, Chem. Phys. Letters 13 (1972) 529. [ll] B. Hopkinsand H.-L. Chen, J. Chem. Phys. 58 (1973) 1277. [ 121 J. Lukasik and J. Du&in_e, J. Chem. Phys. 60.(1974) 331_ 1131 P. h¶cGuire, Chem. Phys. Letters 23 (1973) 575: P. hfcGuire and D. Kouri, I. Chem. Phys. 60 (1974) 2448. [ 141 P. McGuire, 5. Chem. Phys. 62 (1975) 525; Chem. Phys. 8 (1975) 231. [IS] M.D. Gordon and D. Secrest, J. Chem. Phys. 52 (1970) 120. [I61 R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422. [ 171 J.V. Foltz, D.H. Rank and T.A. Wiggins, J. hid. Spcctry. 21 (1966) 203. [ 181 J.L. Dunham, Phys. Rev. 41 (1932) 721. [ 191 G. Hemberg, Spectra of diatomic moIecules (Van Nostrand, Princeton, 19.50) pp. 133-140. 1201 J. Schaefer and W.A. Lester Jr., Chem. Phys. Letters 20 (1973) 575. [21] R.G. Gordon, Program 187. Quantum Chemistry Program Exchange. [22] h1.H. Alexander, J. Comput. Phys. 20 (1976) 248. [ 231 H. Rabitz and G. Zarur, J. Chem. Phys. 6 I (1974) 5076; M-H. Alexander and P. McGuire,Chem. Phys. i 2 (1976) 1241 %I. Alexander, J. Chem. Phys. 61 (1974) 5167. 1251 I. Procaccia and R.D. Levine, J. Chem. Phys. 63 (1975) 4261. [26] J. Lukasik, Thesis, Univ. of ParirSud (1975). 1371 X1-H. Alexander. Chem. Phys. Letters 39 (1976) 485. t28) G. Bergeron and X. Chapuimt. J. Chem. Phys. 57 (1972) 3436. [ 291 WM. Miller and A.W. Raczkowski, Discussions Faraday sot. 55 (1973) 45.