Energy transfer in Ne and Xe collisions with CO2 at 1 eV

CHEMICAL

ENERGY

IN Ne AND Xe COLLISIONS

TRANSFER

7 June 1985

PHYSICS LEITERS

Volume 117, number 2

WITH

CO,

AT 1 eV

G.D. BILLING Department

I&caved

of Chemrstry,

12

Uniuersrty

of Copenhagen,

Panum

Itzrt~tute,

Copenhagen

2200

N,

Denmark

February1985; in final form 26 March 1985

A semiclassical colkion model has been used to calculate differential cross sections for rotational and vibrational excitation of Ct& colliding with Ne and Xe For We-CO, a previously determined semi-empirical potential surface was USA_ Reren~

electron-gas data were used to determinethe short-rangeXe-CO2 interaction

L Introduction

R2=R-dcosy+0.5d2sin2y/R

For heavy molecules hke CO or CO, it is not possible with present experimental techniques to obtain completely state-resolved differential cross sections. However, it turns out that the less resolved energy-loss spectra available from time-f-fhght measurements [ l] provide very sensitive probes of the anisotropy of the intermolecular potential [2]_ Thus it is possible by a trial and error approach to determine the intermolecular potential by comparing with spectra obtained from theoretical calculations. So far mainly calculations on rotational energy transfer have appeared_ It is the PUTpose of the present work to investigate rotational as well as vibratronal excitation in the Ne-CO2 and Xc-CO, systems.

2. The intemolemlar potentials In order to determine the intermolecular Xc-CO, potential, we used an approach previo-usly shown to be adequate for the Ar-CO2 system [3]. Thus we represent *e short-range interaction by a dumbbell _ potential of the form V&

=

C [exp(-cdl)

+ exp(-aR)]+B

exp(-trR)

and a=ao

+qR+a2R2.

(4)

Here R is the distance from the inert gas atom to the C atom and y the angIe between R and the O-C-O ,axis. When fitting the above expression to the HartreeFock part of the electron-gas data calculated by Dreyfus 143, we regarded the distanced as a parameter. Furthermore a Rae correction factor (C) [5] was used in the exchange energy, i e. VHF = VCod + v, where C = 0.6553

+

cv,

,

(44

for Xe-CO,,

The seven parameters in eqs. (l)-(4) were determined by a non-linear least-squares fit to 98 electron-gas points

in the range R = 4 5 (0.5) 8.0 bohr and y = 0, 8.888, 20.401,31.983,43.580,55.182,66.787,78343 and 90”. The mean-square deviation of the fit was about 4% and fig.1 shows the quality of the analytical expression at y = 0” and y = 90”. The long-range interaction was obtained from the calculations of Pack [6], i e.

(5)

,(l) Finally a switching function

where R1=R+dcosy+0.5d2sin2y/R,

(3)

02)

h(R)=1 h(R)

0 009-2614/85/s 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Divrsion)

(R>z_). = exp[--a@/R

- 1>2]

(R

dd) ,

(6) 145

CHEMICAL PHYSICS LElTERS

7 ke

Table 1 Intermolecular

potential parametersused for Xe-CO2 and

Ne-C02

Xe c

Ne

263 16

Qo

-0.1382 O-05939

=2

d

B

B

2.00

ff2

2 50

a

6.732

c% Ei

Fig. L Shortiange part of the Xe-CQ potential at 7 = 0’ and ‘I = 90”. Solid lines Iepresent an analytical fit to electrongas dais (.I.

where a =

a0 + a,P,(cos

161 _dJ)

a) 19 = 100 kJ/mol.

A*

o-0

IL-3

1.1615

A

3989x lo5

g

7.570

A*

3.557 2o.76b)

1115.6b)

Ci

0.0

0.0

37 _47b) 855.6b)

xi-1

180.9

162 gb)

2

pa)

4.180

A nEA6

4.785”)

; A6

98.480b)

2 A8

loo.lb) 18.89b)

z AS z Aa

b) From ref. [Cl.

3. calculations

(7)

-y) ,

was to give the total interaction V= VSR(l -h)+hVLR.

potential: (8)

For Ar-CO2 the parameters a and a in eq. (6) were determined from measurements of the second virial coeffkient [3 3.In the absence of such data for Xe-CO2, we use the magnitude and positions of the minima in the parallel and perpendicular geometry from ref. [7] to determine the parameters UO, a2 and a. The Ne-CO2 parameters used were those of surface III in ref. [S]. These parameters were obtained in a somewhat more approximate fashion, namely by modifying the 3TOG data from ref. [Y] as described in ref. [8]_ The only difference between the present Ne-CO2 potential and surface III m ref. [8] is that a different switching function (eq. (6)) is introduced. This has no influence in the energy region probed in the present calculations and only little effect at lower collision ene--es since both functions change rapidly from zero to unity at R = 3.6 A. The potential parameters used are listed in table l. 146

1.4347 19411 3.995

a0

Ullit

1438 0

2-178

a1

1985

The semiclassical method which has been used in the present calculations has been described previously [3, lo]. It treats the rotational and translational motion classically and the vibrational motion quantum mechanically_ It has been found that the cross sections for vibrational excitation of CO2 are well estimated within the harmonic oscillator (HO) approximation for the normal modes if the molecule is excited from its ground state [ 111. Since the HOmodel can be solved using the algebraic approach developed in ref. [ 123 it 1spossible to compute a sufficiently large number of trajectories to obtain converged angle-resolved energyloss distributions. It should be mentioned that the HO model does include “Fermi resonances” and “dynamical Conolis coupling” (see e.g. ref. [lo]). For propagating the trajectories we use the effective potential, i.e.

V= V&l

--h) +hVLK,

(9)

where V&=V&+Va_

(10)

Volume

CHEMICAL

117,number2

PHYSICS

The second term in eq. (10) arises from the coupling between the quantum and classical degrees of freedom. The introduction of an effective potential ensures that the Hamiltonian H=(P;

+ P; + P$/2/l+

+ V+E&

+ >1(9 -x2

(p; + p; + &)/2m --y2

-22)

(11)

conserves total energy. In eq. (11) Eht is the energy transferred to the vibrational degrees of freedom and the first two terms the kinetic energy of the translational and rotational motion of CO, respectively. The last term is a Lagrange multiplier term (see e-g_ ref. [lo]). A number of trajectories were run using the potential surfaces described in section 2. The initial rotational angular momentum of CO, was I1 = 0, the initial vibrational state (n1n2n3) = (000) and the kinetic energy I&

= 1 eV. The orbital

angular

momentum

2 was

selected randomly with 1502 trajectories in the I range 0 to 377s and 400 between 377 and 754E for Xe-C02. For Ne-CO,

we used a total of 2215 trajectories with

2077 in thel range 0 to 243R and 138 between 243 and 486A_ The corresponding impact parameter ranges are 0 to 3 and 3 to 6 A for both systems. The angular reso-

lutionwas set to AB = loo when analyzing the data. Tables 2 and 3 show that the average rotational exTable 2 Orbital~~momen~umrange.avPrageorbitalandrotational angularmo~lipt~asafimctionofscattering~eBfor Xe-CO2 .Nisthenumber oftrajectoriesintberangee *5' e 20 30 40 50 60 70

80 90 100 110 120 130 140 150 160 170

'lTlGl (11

'RGDC

309 361 340

434 403 360

541 517

314 264

342 321

261 215 210 162 150 109 71 42 9 1 3

297 267 242 225 194 172 146 119 89 68 41

(iti)

(i&j

34

28.5

1070

55

35.1

1700

472

78

38.8

2050

432 437

94 82

52.0 54.8

3640 3890

374 358 360 301 260 246 209 185 155 137 107

110 76 96 96 92 140 106 104 95 112 121

53s 48.2 39.2 47.0 40.0 34.3 31.9 31.1 25.1 24.4 17.0

3850 3170 2290 3030 2510 2000 1710 1790 1390 1570 805

N

7 June 1985

LFITERS

TabIe 3

orbital angular momentum range, average orbital and rotational angular momentum as a functmn of sattering angle0 ftx Ne-CO2

Nisth=numberofi~~~jectories titherange@f 5"

e

Iti

(2)

'max

N

(I&

(J&t>

20 30 40 SO 60 70 80 90 100 110 120 130 140 150 160 170

171 157 145 126 106 BB 82 54 42 14 3 0 0 1 0 1

201 193 182 167 154 138 123 112 97 80 70 54 40 37 24 27

265 243 251 240 221 208 196 188 167 156 147 135 108 116 101 99

153 142 130 114 111 120 125 126 93 81 128 120 94 113 82 68

14.6 24.1 33-E 39.6 45.0 550 52.0 55.8 51.8 59.1 56.0 52.6 49.7 50.4 45.2 38.5

269 726 1400 2030 2590 3720 3540 4030 3750 4660 4350 3960 3310 3510 2730 2720

citation ( jmt> peaks at 8 = 60’ for Xe and 8 = 100” for Ne+CO, and that the Ne atom is more efficient for rotational exc%tation of the CO, molecule_ Both observations may be explained by the larger size of the Xe atom which makes the collision less penetrating. Furthermore we calculated the intensity distribution I,.,,@, AE) where 0 is the scattering angle, A,!? the en-

ergy loss, n the irutial and n ’ the fmal state n = (TI~FZ~CJ~) Thus we have

where His a Gaussian distribution, Hnj;n’l’

=(,l/q3)-1

X exp[-_CAE,i;nvi. m

= Ef&d

+ AE)‘/B2]

_ E~id&id

,

(13) (14)

and AEni.nlr’=E,;n;n;l’-E ,

nIn2n,I.

(15)

The parameter B is determined by the experimental apparatus conditions and for Xe-CO2 at 1 eV, B==i32 meV [73. The total energy-loss spectra, i.e_

147

CHEMICAL

Volume 117, number 2

PHYSICS

7 June 1985

LlZITEXS

I

Yl 10

20

30

LO

50

60

70

Boy

Fig. 3. Theankotropy at 1 eV. i.e. a In v/a ms 7, as a function of 7 for the present potential (solid line) and that used in ref. [7] (dashed line)

trajectories, as in ref. [7]. albert in an effective (vrbrational distorted) potential the differences must be attributed to the potential. Fig. 3 compares the anisotropy at 1 eV of the potential used in ref. [7] and the one determined in the present work. We see

-10 -08 -06 -0; -02

ODAE

lV

-OK -0L -02

OII AE

Fg. 2. IntensitydistributionsforXe-CO, at 1 eVasafunction of sattering angle. Experimentally determined htributions (to the left) are from ref. [7].

I@,E) ~5

I,.@,

AE),

(16)

are shown in fig. 2 together wrth the experimentally determined distnbutions. The experimental distributions are relative and the maxima are therefore scaled to the calculated maxima. We notice that the agreement is reasonable for LaE I < O-4 eV but that the expenmental data show a second peak (rotational rainbow) at larger bE 1values for 0 = 1 OO”, 140° and 160”. According to the analysis in ref. [7] the second peak 1s due to scattering at small values of the y angle (7 2: 20”). It was demonstrated in ref. [7] that such trajectories led to large transfer of energy to the CO2 molecules whereas the broad peak at small IAEI values (see fig_ 2) was due to a “multiple collision rainbow” at larger y values. These fiidmgs were supported by two peaks in the rotational excitation function J(y) obtamed using an empirically determined potential energy surface [7]. We do not find this double peak behaviour - only the broad multiple-collision rainbow at small AE is present. Since we use 3D classical

that a In y/a cos y is slightly larger at y > 45O and smaller at 7 < 45O for the electron-gas potential as compared to the semi-empirical potential of ref. [7]. Since the second peak in ref. [7] was shown to be due to scattering from angles y < 45” the smaller anisotropy in the potential used here may explain the dtfferences. In ref. [7] 3D classical trajectory calculations were carned out in order to determine rhe intensity distribution theoretically. It was assumed that the vibrational excitation could be neglected. Table 4 shows that this is Justified for Xe-CO2 where the total cross section gets at most 10% contribution from vibrational excitation at 0 = 170° and less at smaller scattering angles. The Ne + CO, system is expected to show more vibrational excitation due to smaller reduced mass. Table 4 confirms this Fig. 4 shows the intensity drstribution obtained withB = 200 meV at 8 = 4.0”. 60”, 100” and 14-O” _The distributions are simiIar to the ones obtained for Xe-CO,, but somewhat broader at 100” and 1 40°. The contnbution from the excitation of the bending mode to 010 is also shown. We notice that the vibrational excitation does not change the shape of the distribution at level with the resolutron chosen here. Wrth smaller B values (B = 100 meV) the distributions at 100° and 140° show a pronounced bifurcation_ The Cl10 bending excitation is. not surprisingly, peaked sideways That the Same holds for the other vibrational transitions may be explained by Fermi coupling between the symmetric stretch and the bending mode, and Coriohs coupling between the 010 and 001 levels.

Volume 117, number 2

CHEMICAL

PHYSKS LFZI-ERS

7 June 1985

Table 4 Total ctoss sections (in A’) as a function of smttering angle for Xe-COa and Nk-COa atEkj,, = 1 eV. 7% initial rotational and vibrati~~~Istateisji=O and @ln2?13)= (000) e W%)

F&al vibrationalstate 000

010

loo+-020

110 +030

001

Xe-CO2

20 40 60 100 140 160 170

8.1 13 12 8-8 4.7 2.7 1.2

3.2(-3) 5.7(-2) 55(-2) 0.29 0.25 0.15 9 5(-2)

2.1(-5) P-7(-4) 3.8<4) 3-5(-21 4.91-2) 2-q-2) 2.3(-2)

l-1(-7) 2.1(-5) 2.9(-6) Ga(--3) l-9(-2) 6.0(-3) 5 8(-3)

l.l(-10) 7.2(-9) 3.0(-7) 2.5(-7) 1.11-S) l-4(-6) 7.5(-7)

Ne-CO2

20 40 60 100 140 160

17 11 9.5 3.5 2.1 0.72

0.14 0.69 1.4 0.90 0.48 3.6(-2)

1.4(-3) 3.4(-2) 0.18 0.26 0.12 7.9(-2)

1.0(-5) 1.3(--3) 1.9(-2) 6 St--2) 2.5(-2) lO(-2)

1.3(-S) 4.8(-S) i 2(--4) S-3(-5) 6.01~-5) 25(-a)

The procedure used to determine the potential energy surface for Xe-COz was similar to the one used previously for AI-CO, [3]. Smce good agreement between theory and experiment was obtained in ref. [3] for a number of buik properties-relaxation times, diffusion constant, rate constants-it is interesting to note that the agreement obtained here with the experimental energy-loss spectra is less satrsfactory. Thus it appears that the conclusion of the present work is similar to the one made rn ref. [2] for D, + CO, namely that the electron-gas potential is not accurate enough to predict Anne-resolved properties correctly.

Acknowledgement

Fig. 4. Intensity distibutionsfor Ne-COa at 1 eVasa function of starming angIe. The u311triiutior1from vibmticmalexcitation totbe olole~~~~o~~~~-

The Danish Natural Research Science Council is aclorowledged for granting CPU time for this research. Professor U. Buck, MI”I fii~ Str&mungsforschung, Gettingen, Federal Republic of Germany and Dr. C. Dreyfus, are acknowledged for sending me experimental data and electron gas data respectively prior to publication.

Volume

117. number 2

CHEMICAL

Referen-

[l ] U. Buclq F. Huisken, I. Schleusener and J. SchZfer, J. Chem. PhyS 72 (1980) 1512; J. Andre, LJ. Buck, F. Htiken, J. Schleusener and F. Toreno, J. Chem. Phys. 73 (1980) 5620; U_ Buck, F. Huisken, D. Otten and R Sclrinke. Chem. Phys Letters 101 (1983) 126. (21 G.D. BiWng and L.L Paulsen, Chem. Phys. Letters 99 (1983) 368. [3] G.D. BiUin& Chem Phys. 91 (1984) 327.

150

PHYSICS

LEI-I-ERS [4] [S] [6] [7] [8] [9] [lo] [ll] [12]

7 June 1985

C Dreyfus, private ozmununication ALM. Rae, Chem. Phys Letters 18 (1973) 574_ R T Pack, J. Chem Phys. 64 (1976) 1659. U. Buck. D. Otten, R. Schinke and D. Poppe. J_ &em. Phys. 82 (1985) 202. G-D. Billing, Chem. Phyr 49 (1980) 255. S. Odiot, S_ Fliszar and J.L. Candara, Chem. Phys. Letters 71(1980) 307. G-D. Bilhug, Computer Phys Rept. 1 (1984) 237. G-D_ FJdEng, Chem_ Phys. 60 (1981) 199. G D. Billing, Chem. Phys. 51 (1980) 417.