propensity rules in rotationally inelastic collisions of co2

Volume 98. number 4 CHEMICAL PHYSICS LETTERS 1 July 1983

PROPENSITY RULES IN ROTATIONALLY INELASTIC COLLISIONS OF CO2

Millard H. ALEXANDER Department of Chemistry. University of Maryland, ColIege Park. hfaryand 20742. USA

and

DC. CLARY Department of Chemistry, University of Manches:er Institute of Science and Technology, hfanciiester M60 IQD, UK

Received 30 March 1983

Propensity rules for the rotational quantum number dependence of cross sections for CO2 collisions with atoms are pre- dieted to have subtle characteristics for transitions involving levels in excited vibrational angular momentum states.

1. Introduction

Although there have been several recent theoretical

studies of rotational relaxation of linear polyatomics

in collisions with structureless atoms [l--6], most have

not dealt with the doubling of the rotational levels, denoted I-doubling [7,8], which occurs because of the

coupling between the nuclear rotational motion and

the degenerate bending vibrations_ By extending work

done earlier on rotationally inelasticcollissions of open-

shell diatomics 191, Alexander [5] was able to show

that for large values of the rotational angular momentum certain transitions among these 1 doublets would be more favored, in good agreement with the conclu-

sions of Cohen and Wilson [lo] from a microwave

double-resonance study of collisions of various rare

gases with HCN in the 01 1 0 vibrational manifold. Also

Clary, in a recent paper [6], used his vibrationalclose- coupling infinite-order-sudden (WC 10s) method [ 111 to calculate rotationally inelastic cross sections in collisions of CO, with He, both for transitions with-

in the 01 1 0 vibrational manifold and for transitions

between the OO”O and 01 1 0 manifolds. These calculations revealed a marked sawtooth structure in the varia- tion of the cross sections as a function of the final ro-

tational quantum number/, where transitions to either

odd or even rotational levels were strongly preferred.

We will show in the present article that this struc-

ture is a consequence of the same type of symmetry

selection rule that Alexander had previously demon-

strated [5,9,12], principally for collisions of open-shell diatom&. These selection rules, which are valid within

the infinite-order-sudden (10s) [13,14] limit, were

derived by consideration of the asymptotic limit of

certain 3j symbols [ 151 which appear in the 10s ex-

pressions for the degeneracy averaged cross sections.

We shall carry out the same type of analysis here,

using Clary’s previous formulation [6] of the IOS ap-

proximation for collisions of vibrating linear triatomics with structureless atoms, which is reviewed briefly in section 2. In sections 3 and 4 we derive the propensity

rules appropriate to rotationally inelastic collisions of

CO,, both for transitions in which the vibrational an-

gular momentum does not change (section 3) and for

transitions in which the vibrational angular momentum

changes from an initial value of 0 to a fmal value of 1 or 2 (section 4) ln the latter case we make use of a new semiclassical expression for certain 3j symbols [ 121.

The propensity rules and their limits of validity are

demonstrated both by Clarly’s previous calculations [6] and by new VCC IOS calculations on the CO,-He system reported here. A brief conclusion follows.

0 009-2614/83/0000-0000/S 03.00 0 1983 North-Holland 319

Volume 96. number 4 CHEhllCAL PHYSICS LETTERS 1 July 1983

2. Collision dynamics

The rotation-vibration wavefunctions of a linear polyatomic molecule are written. in Clary’s notation

161 . as

ItJjX??ZE) = CJ ItJjXnz> + ElUj--Xm)). (1)

where u is a collective indes for the vibrational quan-

tum numbers of the three normal modes (u = ut u2”u3 in spectroscopic notation); h is the vibrational angular momentum;j is the total angular momentum with space-fixed projection HZ: and (E = 21. cE = Y1/‘) for h > O.and (E = 0. cE = 1) for h = 0. Since in COz the nuclei all have spin 0. the nuclear wavefunction must be symmetric with respect to inversion of the nuclei. Only even-j states are allowed for X = 0. while. for h = 0. the allowed levels have a symmetry index given

by if51

e=(--I)] (2)

Wtthin the VCC 1OSapprosimation [6] the integral c10ss section for a transition from state ujXe to uY’X’E’, averaged over IH and summed over DI’. is given by

where k, is the wdvevector and.in Clary’s notation [6],

Here the -4:;; coefficients are integrals [6] over the products of the 10s orientatron-dependent S functions and the associated Legendre polynominak Ppmh”.

Since for CO2 the constraints of nuclear spin sym-

metry permit at most only one value of e for eachj,

the E index cdn be suppressed. Furthermore, the A coefficients c~tn be shown 161 to vanish unless L + h + h’ is even. ior u = OO”O. X’ > 0 or X > 0, u’ = OOOO, or unless L IS even for u = u’ and h = X’ > 0. The follow- ing analysis refers specifically to bending mode transitions in CO,.

The new VCC 10s calculations reported here were carried out with identical procedures to those described in ref. [6] _ A potential energy surface [ 161 constructed from SCF data was used in the computations_ We note that the previous VCC IOS rate constants for both the

vibrationally inelastic 01’0 + 0000 transition [ 161 and rotationally inelastic transitions, summed of over final j’ states, within the 01 1 0 manifold [6] agreed well with experiment [ 171 when this potenial was used.

3. Propensity rules: transitions elastic in 1

In the case of transitions in which the vibrational angular momentum h does not change, but in which other vibrational quantum numbers-may or may not changes, eq. (3) becomes

CJ _ UhJ-W’Aj’

If j and j’ are large compared to L. then the squared 3j symbol can be replaced by the known asymptotic limit [5,15]

= (2j’ + 1)--l z -l!,‘~~: [PC - i(h/j’)] 2 - + (6)

At large j’ the argument goes to zero, in which limit the associated Legendre polynomial vanishes unless L -j’ + j is odd_ This, coupled with the constraint that L be even, implies that for large j and j’ the uq +

u’Xj’ cross sections will become vanishingly small

u&ss(-Iv= (-l)j’. Thiiimplies, from eq. (2)apro- pensity toward conservation of the symmetry index E,

which we expect on the basis of Alexander’s previous discussion of rotationally inelastic collision of linear

polyatomics with vibrational angular momentum [S] _ The presence of the Xquantum number in the argument of the associated Legendre polynomial in eq. (6) indi- cates [5,9] that the propensity toward conservation of E will become apparent at lower values of j and j’ the smaller the value of X. Additionally, for a given value of h, the propensity will be most obvious at low jifAj=Jj’-jlissmall [5].

Since in the case of CO, only one E value is allowed for each j, and, further, the value of E alternates with j [eq. (2)], the propensity toward conservation of E implies that a plot of uhj + u’Aj’ cross sections as a

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Volume 98, number 4 CHEMICAL PHYSICS LETERS 1 July 1983

ij ;fi;

1 5 9 13 17 21 25 29 33 L

Fig. 1. (a) Plot of B& [eq. (5)] against L for He--C02 colliiions at a translational energy of 0.048 eV with u = u’ = 01’0. (b)PlotofBL UO &eq_ (7)) againstl at a translational energyof0.131eV&hv=00°0,u’=0110andh’=1.

function ofj’ will show marked sawtooth behavior.

Our derivation suggests that this alternation will be ap-

parent provided that j and j’ be greater than the values

of L which make a significant contribution to the sum

in eq. (5). In fig. la we plot the elPsl coefficients

as a function of L for transtions within the u = u’ = 01 1 0

vibrational state of CO,, in collisions with He, corre-

sponding to a VCC 10s calculation at an initial translational energy of 0.048 eV. We see that the coefficients

are large only for L 5 8, so that we would expect the

alternation to occur only ifj andj’ are both somewhat

greater than this value.

Fig. 2 show VCC IOS He--CO2 cross sections for

the 01 lO,j + 01 ‘Oj’ transitions as a function of j’,

for j = 2,16 and 30 and an initial translational energy

of 0.048 eV. These results have the structure predicted

from the propensity rules discussed above_ For j = 2 there is little oscillation in the cross-section distribu-

tion since the conditionjj’ %- Lcannot hold. However,

as j is increased, it is seen that the sawtooth structure becomes more pronounced, especially for large j’, with

the cross sections for transitions conserving the sym-

metry index e with even j’ having much larger magni-

tudes than those for transitions which change parity.

For j = 16, the oscillations disappear for j’ < 8 as, in

this case,j’ is not much larger than the significant values of L_ The structure in the OllOj+ OllO,j’

cross sections is also found in VCC 10s calculations

for higher translational energies and odd values ofj

163.

YO- 01’o-B 01’0 j=30

0., 20-

I’

Fig. 2. Plot of o0~10j _, 0110,~ againstj’ for He + CO2 colli- sionswithj= 2,16 and 30, and a translational energy of 0.048 eV.

4. Propensity rules: X = 0 + X’ = 1.2 transitions

For transitions from a vibrational state wivith X = 0

to a vibrational state with h’ # 3 eq. (3) becomes

where L takes only odd values when X’ is odd, and

only even values when h’ is even, for u = OO”O. In the

semiclassical limit, the squared symbol can be well ap-

proximated for X’ = 1 ,Z by the expressions recently

derived by Alexander and Dagdigian [13-l. We have,

for G E j’ + L + j even,

cos2@‘4 (8) dJ'[sin2w i- (JA'/J'@] 112

and, for G odd,

sin’@‘a) nZJ'[sir$ o -I- (Jk'/J'1)2] 112’ (9)

whereI=L+~,J=j+~,J’=j’+$.Herewistheangle

between J’ and I in the semiclassical angular momen-

tum triangle with sides J, J’ and I, so thar

cos w = [(J’)* + l2 -J*] 125’1. (10)

For 0 + A’ transitions, L is even when A’ is even and

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L is odd when A’ is odd, for IJ = OOOO. Thus propensity rules for these transitions will depend on whether K = j’ + X’ +j is even or odd and wetherj’ is much smaller or larger than j. For example, when X’ = 1 and j’ %j, COS~@‘W) * 1 and sinz(X’u) = 0. In this case transitions with j’ odd have K even, and as a result of eqs. (8) and (9) these transitions will have larger cross sections than those with j’ even. However for X’ = 1 and j’ <j. s&(h’w) > cos2 (h’w). so that now transitions witbj’ even will have large cross sections than those withj’ odd. These particular even-odd propensities for odd X’ will be reversed for X’ = 2. In terms of the symmetry index E. if X’ = I and j’ Z+j the strongest transitions will be those where E changes. and for j’ 4 j, those where E is conserved_ This propensity will be reversed for X’ = 2.

These propensity rules are seen to hold in the VCC 10s calculations of UooOo,j ..__ol ru,j’ for He + CO2 presented in fig. 3 for j = 2, 16 and 30 at an initial twnsiational energy of 0.131 eV. The oscillations are most severe for j = 2 since in this case the relationship j’ %-j holds best. Furthermore the subtfe propensity rule that transitions into odd j’ are favored when j’ B-j while transitions into even j’ are favored for j’ <j is seen to hold. particularly for j = 30. Note that for the transition (OOOO -+ 01 1 0). the factor& v,h, can be . reldtivcly large for quite large values of L and shows a

0 5 10 15 20 25 30 35 &O 45 50

I’

Fig. 3. Plot of retatixe cross xctionsP~~o~j+ 01104’ against j’ for He + COp with j = 2, 16 and 30, at a translational energy of 0.131 eV. These cross sections are normalized to the j’ = j value. Absoiuts ‘TOSS sccrions can be obtained by multipliu-

tion by 7-6 x IO*, 1.1 x iOe3, 1.1 X iOm3 uf forj= 2, 16, and 30, respectively.

small hump with a local minimum at L = 11 (see fig. lb). This is why the manifold of odd-j’ cross sections for the OOuOj = 2 + 01 ‘Oj transition shows a minor local minimum near j’ = i 3.

In the special case of transitions out ot the j = 0 level, the 3j symbol in eq. (7) vanishes unlessj’ = L which implies that j’ must be odd for a non-zero cross section. Tn this case the propensity toward odd j’ becomes a rigorous selection rule, at least in the IOS limit.

5. Discussion

We have shown how analysis of the semiclassical limit of certain 3j symbols which appear in the IOS expression for degeneracy-averaged integral cross sections for collisions of CO2 with atoms, results in the prediction of propensities for either conservation or change in the parity of the rotational quantum num- ber_ These propensities, and their limits of validity, were demonstrated by ex~inatiou of VCC IOS cross sections for the CO,-He system-The propensity rules were seen to have quite subtle characteristics that depend not only on the change in vibrational angular momentum but also on the initial rotational state j. For example, when j is small there are very pronounced oscillations in the cross section distributions for the OO”Oj -+ 0110,j’ transitions, which diminish for high j. Ho~vever,~~ontr~t,for~e~aseofOllO~jOllO~’ transitions, the oscillations are most significant for large values of j and are hardly present for low j.

Recently there has been interest in energy transfer involving the 10°O, 0220,0200, and 01’0 vibrational manifolds [ 181, since these processes are important in the 14and 16pm CO2 IasersTomodel accurately the relaxation processes in the lower laser level (01’0) or between the upper laser level (lO”O) and the near res- onant (0220) level it may well be necessary to take into account the rotational selection rules derivedhere. Also, it may become possible to use these lasers, or tunable diode lasers, to probe directly rotational relaxation in one of the low-lying vibrational manifolds of co,.

The research reported here was partially supported

322


by the National Science Foundation, grant CHE81- 08464. One of the authors (MHA) wishes to thank Tom Manuccia for helpful discussions about the experimen- ta] study of rotational relaxation in CO,.

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propensity rules in rotationally inelastic collisions of co2

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