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Noble-gas broadening of vibration-rotation lines belonging to diatomic molecules—III. Theoretical calculations of CO linesshifts and widths
J. Quanr. Specmsc. Radiur. TrmsJer. Vol. 13, pp. 911-921. Pergamon Press 1973. Printed
in Great
Britain
NOBLE-GAS BROADENING OF VIBRATION-ROTATION LINES BELONGING TO DIATOMIC MOLECULES-III. THEORETICAL CALCULATIONS OF CO LINESHIFTS AND WIDTHS C. BOULET,
P. ISNARD and A. LEVY
Laboratoire d’Infrarouge, Equipe de Recherche AssociQ au C.N.R.S., UniversitC de Paris VI, Campus d’orsay, BLtiment 350 (91405) Orsay, France (Received 21 September 1972)
Abstract-Noble-gas-broadened CO vibration-rotation linewidths and shifts are calculated, within the impact approximation, after a critical analysis of available experimental data. Several molecular parameters related to local electronic distribution have been computed in order to express explicitly the dispersion energy. The R-* dispersioncontributionis shown to be of slight, but not of negligible, influence. The importance of angular dependence of the repulsive potential is emphasized, together with the effect of cross-terms in the interruption function. The calculated results agree well with experimental data for almost all of the lines.
THE VIBRATIONALdependence of shifts and widths of noble-gas-broadened HCl lines has been extensively described. (I) Introduction into Anderson’s theory of the vibrational states of the active molecule has recently been shown to account successfully for this effect.‘2’ It may be asked whether a similar phenomenon occurs with the same magnitude for molecules other than the hydrogen halides and, in particular, for CO. Thus, starting from a review of the available experimental results,‘3-g’ we first point out variation of the line widths and shifts with vibrational quantum number for the CO molecule. The next step consists of computing the numerical values of the parameters involved in the dispersionenergy expression. In the last part, we use the generalized Anderson-Tsao-Curnutte theory in order to compare the theoretical results with experimental data.
1. REVIEW
OF EXPERIMENTAL
RESULTS
1. Line-widths (a) Argon broadening. Table 1 shows that, for the fundamental band, DRAEGERT’S’~’ measurements are in good agreement with the recent results of BOUANICH.(‘) For the O-2 transition, the experiments of BOUANICH~‘)provide results which practically coincide with those of THIBAULT.‘~’ Rank’s values(3’ are much greater. As regards the O-3 transition, the values reported by Bouanich and Haeusler in their recent paper’g) are quite similar, for 3 5 Im( I 13, to the widths measured by these same authors in the fundamental band. 911
912
C. BOLJLET,P. ISNARD and A. LEVY
TABLE 1. CO-ARGON MIXTURES.EXPERIMENTALAND THEORETICALRESULTS(HALF-WIDTH, EXPRESSED IN MK/BAR) Iml
1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
61.4 59.6 57.9 54.3 52.3 48.9 48.5 46.6 46.8 46.1 45.8 44.2 44.5 43.5 43.4 42.1 43.4 40.6 39.7 39.7
57.1 55.7 55.7 52.4 51.8 46.9 45.5 44.1 43.2 43 42.5 40.5 40.5 39 37.9 36.4 36.3 35.2 31.6 30.4
3
70.4 66.5 61.4 56.9 53.4 50.8 48.5 47.2 46.3 45.6 44.6 44.4 43.8 43.5 43.1 42.3 41.4 40.7 40 -
4
65.5 50.5 40.5 35 41 38.5 40 39 -
5
6
7
8
69.9 62.5 57.8 53.7 50.6 48.8 47.5 46.4 45.7 45.1 44.6 44.1 43.5 42.8 41.9 41 40.3 39.5 38.8 38.2
69.5 63.5 57.5 53.5 50 48.5 45.5 42.5 42.5 42.5 42.5 41.5 41 42 38 42 42 42 37.5 38.5
84.5 81.5 76.5 69.5 63.5 63.5 55 59.5 56 55
80.7 78 74.5 70.3 66 62.2 58 54.2 51 48 45 42.5 40 38 36 34.2 32.5 31 30 28.5
51.5 49.5 -
9
87 82.3 78 73 68 63.5 59 55 51.6 48.5 45.8 43.3 41 39 31.2 35.6 34 33 31.7 - 30.6
Experimental results. Column 1: O-l BOUANICH et al.Is); 2 : C-l CRANE-R• 4: O-1 RANK(~); 5: O-2 B~UANICH(‘); 6: O-2 THIBAULT@); 7: G2 RANK(~).
10
11
12
75 71.5 68 64 60.5 54.5 53.2 50 47 44.2 41.5 39 36.7 34.7 33 31.4 30 29 28 21.3
71.3 67.6 63 52.8 46.8 42.9 40.8 39.7 39.2 38.9 38.7 38.4 38 37.5 36.9 36.3 35.6 34.9 34.1 33.4
70.8 66.1 57.6 50.8 46.2 43.3 41.4 40.2 39.4 38.8 38.4 38 37.6 37.2 36.8 36.3 35.8 35.7 35.5 35.2
BJNS~N'@ ; 3 : &l
DRAEGERT’~) ;
Theoretical results. The horizontal line indicates the limit of validity of Anderson’s cut-off. Column 8: R.G.G. formalism without the R-* disperson energy; 9: R.G.G. formalism including the Rm8 dispersion terms; 10: H.T. potential without the (ei . R,) repulsive term and with d,, = 0.21 A; 11 : H.T. complete potential; 12 : H.T. complete potential with velocity averaging.
It is then seen that no vibrational effect is suggested by the experiments for Ar-broadening. (b) Xenon broadening (Table 2). Experimental data are scarce in this case. By comparing Draegert’s values for the O-l band with those of Bouanich for the O-2 band, in view of the agreement of their measurements for Ar-broadening, no noticeable vibrational variation is apparent. 2. Frequency shifts (Table 3) We note disagreement between the values of RANK(~) and BOUANICH.(~) Since no other experimental data are presently available, the only information we can obtain on CO frequency shifts has to do with (a) their order of magnitude (the shifts are much smaller than the corresponding ones for HCl) and (b) their vibrational dependence. The values reported in Rank’s paper are the only ones allowing a comparison of two different transitions. By contrast with the linewidths, the shifts exhibit a noticeable variation with the vibrational number o. 2. INTERMOLECULAR I.
POTENTIAL
R.G.G.
FORMALISM”’
Molecular parameters
All of the molecular constants of carbon monoxide are not known, particularly A,, and A, (components of the hyperpolarizability tensor). But, it is not possible to ignore a priori,
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
913
TABLE 2. CO-XE: EXPERIMENTAL AND THEORETICALRESULTS(HALF WIDTH EXPRESSED IN MK/BAR). THE NUMBERING OF COLUMNS IS THE SAME AS IN TABLE 1
Iml
2
3
5
7
8
9
10
11
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
72.8 68.1 64.2 614 58.2 53.9 51.3 48.9 48.6 47.3 46.7 44.2 47.7 42.7 41.7 40.4 37.2 36.9 33.3 34
89.3 80 73 68 63.9 60 56.6 54.7 52.9 51.3 49.3 48.5 47.3 46.5 45.7 45.2 45 43.7 42.4 -
84 74.5 68.9 64.5 60.8 58.1 55.8 53.9 52.6 51.7 51.2 50.8 50.4 49.6 48.2 46.9 45.8 44.8 44 43.3
92 82.5 75.5 73.5 68.5 65 62 60 60.5 57.5 56.5 59.5 51 57 48.5 -
93 88.4 82 75.2 68.7 64 57.5 53 48.8 45.2 42 39.2 36.7 34.4 32.4 30.7 29.3 28.5 27.4 27
98.5 92 85.2 77.4 70.2 65 58.8 54.2 50.4 47 44.2 41.8 39.8 38 36.7 35.4 34.5 33.7 33 32.4
86 81 75.4 69 63.2 58 53 49 45.4 42 39.2 36.7 34.4 32.5 31 30 29.5 29.2 29 28.8
84.8 79.1 72.3 64 50.8 45 42.7 41.9 41.5 41 40.5 39.7 38.9 37.9 36.9 35.8 34.8 33.8 32.8 32
83.5 77.2 69.7 60.6 51.8 46.5 43.9 42.4 41.4 40.6 39.9 39.2 38.7 37.6 36.8 35.8 35.2 35 34.8 34.6
in the dispersion-energy expression, the R-’ contribution which involves precisely these components. Consequently, let us consider the dispersion energy formulation derived from the variational method, as was previously done in the case of HC1.“oV”’ Its major advantage is that it involves molecular constants that are more easily computed than A ,, or A,, since they require knowing only the electronic wave functions for the fundamental state. The definition of these constants (d\;‘; dy’; d\t’; dy’) can be found in Ref. (lo), and the detailed calculation method in Ref. (12). The various parameters related to the electronic orbitals of CO were calculated from Nesbet’s data. (13) The numerical values obtained in this way for CO-He, CO-Ne and CO-Ar interactions are given in Table 4. It is then possible to calculate the dispersion energy with terms up to RF8 included (see equation (2) of Ref. 10). As in the case of HCl, the d\:‘,, show very slight variations from one rare gas to another. Accordingly, an order of magnitude for A,, and A, (which do not depend on the perturbing atom) may be determined. For this purpose, the previously reported relations are used (relation (8) in Ref. 2). We then find A,, 2: 26.7 x lo-33 cm4; A, N 0.5 x 1O-33 cm’. The dll, (f), which appear in the R-’ energies, are much larger than for HCl. Consequently, we can no longer assume the corresponding contributions to be ineffective in line-broadening. 2. Repulsive potential We confine ourselves first to an isotropic form for the repulsive forces, as we previously did for HCl. 3. Vibrational dependence of CO molecular parameters Table 5 gives the different coefficients, for the expansion of the dipole moment and polarizability, in terms of normal coordinates. The v-diagonal matrix elements of the normal
O+l
1
1.9
1.9 ~ 1.1 1.5 2.3
~ 2.2 1.9
2.4 2.8 -
1
3 4 5 6 7
8 9 10 11
12 13 14 15 16 17 18
R
3.5 3.6 3.1 3.1 3.7 -
2.8 3 3.2 3.3
1.8 2 2.2 2.4 2.6
1.7
1.6
(1)
obs. RANK(~) talc.
2
Iml
3.7 3.8 3.8 3.8 3.8 -
3.7 3.7 3.7 3.7
3 3.7 3.7 3.1 3.1
2.6
2.4
(2)
talc.
2.8 3.3 3.2 3.1 3.5 3.8 3.8
2.4 2.1 3 2.3
-0.7 0.1 0.6 1.5 1.7
- 1
-0.8
R
0+2
4.1 4.4 4.2 4.7 4.8 5.2 5.6 6.4 1.3 -
0.5 1.6 2.1 2.1 3.1 3.6 2.9 3.6 3.4 3 3.6 3.5 3.2 3.4 3.4 3.6 3.5 3.4
R
7.2 7.8 6.7 6.6
5.9 6.2 -
5.5 6 ~ 5.5 ~
-
6.2
P
obs. RANKC3’
P
obs. BOUANICH(“)
Argon
6.9 7.1 7.3 7.4 7.4 1.2 7
5.5 5.9 6.3 6.6
3.1 4 4.3 4.7 5.1
3.4
3.2
(1)
talc.
2.4 2.6 2.9 3.2 3.6 4.1 4.5 5 5.4 5.7 6 6.3 6.4 6.5 6.5 6.4 6.2 6.2
4.1 5.1 5.9 7.5 7.4 7.4 7.4 1.4 7.4 7.4 1.4 1.4 1.5 1.5 7.6 7.6 7.7 7.7
(1)
talc.
(2)
talc.
0-l
6.6 6.6 6.7 6.8 6.8 6.8 6.8
6.3 6.4 :4
4 5 6.4 6.3 6.3
3.5
3.2
(2)
talc.
Xenon
1.4 3.3 3.8 4.4 6.4 6.6 6.1 8.1 8.4 7.8 8.2 8.1 7.9 8.6 7.8 8.1 -
0.5 0.8 2 3 4.3 5 6.2 6.2 6.1 6.6 I 6.5 1.5 7.3 7.9 1.7
P
R
12.5 13.4 13.5 -
11.3 12 -
11.7 12.1
11.4
10.7 10.1
-
8.5
P
9.6
10 9 -
5.8 6 8.2 -
5.2
6.9
R
obs. _‘3’
o-+2
7.9 9.8 12.7 12.6 12.6 12.6 12.1 12.8 12.9 13.1 13.3 13.4 13.5 13.6 -
5.2 5.1 6.4 7.2 8.1 9 9.9 10.7 11.4 12 12.5 12.9 13 13 12.8 -
6.4 7
4.7
(2)
talc. (1)
talc.
FORMALISMINCLUDING DISPERSIONTERMS IN R-';
obs. BOUANICH(~‘)
TABLE 3. SHIFTS TOWARDS SMALLERWAVE NUMBERS(EXPRESSEDIN MK/BAR). THEORETICAL VALUES: (1) R.G.G. (2) H.T. FORMALISM(WITH THE COMPLETEPOTENTIAL)
$y < <
?
g *
F 5 %
71
-3
E
0
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
915
TABLE 4. PARAMETERS IN THE DISPERSION ENERGY (VARIATIONALcALcuLATIoN)
d(,f’ A
dy’ A2
d\;’ A2
CO-He
0.0165
0.5449
0.1349
0.9357
CO-Ne
0.0169
0.5626
0.1325
0.9537
CO-Ar
0.0179
0.6846
0.1168
1.0735
HCl-Ar
0.038
0.164
0.01
0.031
coordinates, on the basis of the anharmonic oscillator wave functions, may be calculated from the work of HERMAN and SHULER. (t4) In computing the vibrational variation of the L-J parameters, c1 and or, we apply the previous evaluations of FRIEDMAN and KIMEL(~ 5, which gave satisfactory results for HCl (Table 6). 3. APPLICATION
OF R.G.G.
FORMALISM
The molecular parameters of CO are taken from Refs. (20) and (21). Those concerning the noble gases are reported in Ref. (18). In deriving the results, a straight path trajectory is assumed with constant velocity for the perturber molecules (with u = ij); furthermore, Anderson’s cutoff procedure is applied. Tables 1-3, together with Figs. 1 and 2, enable us to show the relative importance of the various mechanisms responsible for the broadening of rotation-vibration lines of CO. (1) The R-* dispersion energy is not included (column 8, curve 1) The calculated results do not compare well with experimental data for small or large Jm( values. This discrepancy cannot be attributed to the cutoff procedure since, for most of TABLE 5. p1 AND a, EXPANSIONSIN TERMSOF NORMAL COORDINATES:(a) J. P. BOUANICH et al., .I. Phys. 29, 641 (1968); (b) W. F. MURPHY et al., Appl. Spectrosc. 23,211 (1969) ;
see also Ref. (20)
Aal
.
01 a I0 = 1.95 A” a,, = 1.5A2
p,O = -0.1240 p , , = 3.084 D/A pi2 = -0.574 D/A=
TABLE 6. VARIATIONOF THEE, AND a, CONSTANTS OF CO AS A FIJNCTION OF u
Friedmann-Kimel
Present work
O-l
G2
Gl
(r2
0.009
0.018
0.006
0.012
0.0019
0.0038
0.0012
0.0024
C. BOULET, P. ISNARD and A. LEVY
916
1
I
I 15
I IO
5
I
20
)
ml FIG. 1. Argon broadened CO lines. Halfwidths in mK/bar. Experimental values: + Bouanich t&l, n Crane-RobinsonO-1, x Thibault C&2, 0 Bouanich O-2. Theoretical predictions: Curve(l) R.G.G. formalism, (2) H.T. formalism without the (e, I&) repulsive term, (3) H.T. formalism.
the lines, the critical value of the impact parameter is much larger than the closest approach distance. This is an essential difference from HCI, resulting from the fact that the rotational constant B, of CO is much smaller than that of HCl. It may then be concluded that the disagreement arises from the inadequate expression used for the CO-noble gas interaction energy.
t
40----~.
I
I
I
I
I
5
IO
I5
20
)
Iml FIG. 2. Xenon-broadened
CO lines-Halfwidths
in mK/bar.
For symbols
refer to Fig. 1.
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
917
A careful analysis of the various contributions to S,(b,) shows that the predominant term [which reaches about 75 per cent of S,(b,)] is k’S2(bo) with k, = 1, both for Ar and Xe. This emphasizes the importance of a preliminary determination of the distances d\,% characterizing the center of dispersion forces. Be that as it may, the calculated and observed values remain substantially different. It could then be asked whether this discrepancy should not be even greater when R-’ contributions (d((f),l constants) are introduced. (2) Injluence of the R- 8 term in the dispersion energy (column 9) A great number of additional terms arise in the S,(b) expression, particularly if non additivity effects’i6’ are taken into account. The set of these new contributions is listed in Ref. (22). Their introduction results, however, in a very slight increase in the calculated widths (- 1 mK for \m(2 5). Nevertheless, an analysis of the different parts of S,(b,) with respect to the order k, of spherical harmonics (Fig. 3) reveals that the influence of these new terms is far from insignificant. Firstly, the prevalence of k, = 1 contributions has slightly decreased. The new terms in k, = 4 and also the terms in k, = 2 resulting from the Re6 dispersion energy are practically negligible. Secondly, the terms related to the Re8 energy are nearly equal (-20 per cent) to the k, = 3 contributions. Finally, no important change is made when a complete expression is used for the dispersion energy. Nevertheless, we must point out the great diversity in the nature of collisions contributing to the deexcitation of CO energy levels. For smaller Im( values, inelastic collisions (with respect to rotation) are predominant since, for k, = 1, the selection rules arising from the Clebsch-Gordan coefficients do not allow angular frequencies ojj, equal to zero. For higher jrnl values, the situation is more complex. It is therefore necessary, to examine the influence of repulsive forces.
Iml
Iml
(0)
(b)
FIG. 3. S,(b,) splitting-R.G.G. formalism including the terms resulting from R-’ energy. (a) CO-Ar 61, (b) CO-Xe O-2, ‘S,(bo) negligible. p It1 1 : terms resulting from the R-’ dispersion energy. 1 [7 k: = 2 terms resulting from the Rw6 dispersion energy. 0 k, = 2 Sum of the additional terms appearing when the Ak,=4 > Re8 dispersion energy is taken into account.
dispersion
918
C. BOULET,P. ISNARD and A. LEVY
(3) Vibrational efict.
Inadequacy of an isotropic exchange potential
At this stage, a first positive result of our calculation is, that lines with the same Irnlhave nearly equal widths for both transitions (the difference never exceeds 0.5 mK). The reason is that the isotropic term OS,(b), the only one which is strongly u-dependent, plays here a negligible role, contrary to the situation observed for HCl. This results in a predominance of rotational relaxation mechanisms in CO-noble gas mixtures, instead of the “dilation effect” found for HCl.“” But the calculated shifts appear much greater than those measured by Rank, which are themselves greater than Bouanich’s measurements. In order to account at least for the order of magnitude of these shifts, a reexamination of the vibrational dependence of E~ and pi seems necessary. If we assume somewhat smaller vibrational variations than given by Friedman and Kimel (Table 6), it becomes possible to account both for the magnitude of Rank’s shifting values and for the observed vibrational effect (Table 3). At the same time, a decrease in ‘S,(b,) is obtained, but the corresponding influence on calculated widths is almost negligible, owing to the small value of this term. This necessity of modified values for &I and c1 variations could perhaps be due to an influence of the perturber on the vibrational dependence of the active molecule parameters. This effect has been suggested previously by Giraud(17’ for HCl-Ar interactions. Up to now, we have used an isotropic repulsive potential. The corresponding contributions to S,(b) are only in k, = 0. But the whole OS,(b) term is almost negligible. This procedure therefore fails to take account of the influence of repulsive forces. It is thus necessary to choose, for the exchange energy, a model involving some angular dependence. For this purpose, we consider the additional terms appearing in S,(b) when a repulsive contribution (labelled rep.) to the interaction energy is introduced and expanded in terms of spherical harmonics. If an attractive term (labelled att) of the same order k, exists, two new terms arise, namely, “S,:$ which is positive and klSz$‘P(which is a negative cross-term and involves negative powers of b, lower in absolute value than those involved in the pure repulsive term). If these new terms are added to S,(b), remembering that b, decreases as (ml increases, it may be reasonably expected that the negative one will prevail at low Iml-values (since b, is large); on the other hand, for higher [ml-values, the purely repulsive term, the variation of which is much more rapid, will compensate the cross-term effect. This should result in a decrease in width for lower Iml-values and an increase at higher Jml-values, leading to better agreement with experiment. But, since the actual form of the repulsive potential is unknown, the only possibility is to resort to a semi-empirical formulation. (4) Angular dependence of the repulsive potential The most simplified model we can assume is rather similar to that proposed by Artman and Gordon and used by GIRAUD et al.(l*) for their study of perturbation effects in the pure rotation spectrum of HCI-noble gas mixtures. The corresponding expression is written as V,,,(R) = (4ear2/Rr2)[l
-(r1/2)+
3y,(q
.
Ro)2/21.
Additional k, = 2 contributions are then to be added in S2(b).(22) The sum of the new terms never amounts to more than a few per cent of S,(b,), for any ho(m). The calculated widths, therefore, remain unchanged. Nevertheless, it cannot be concluded that the repulsive forces are ineffective in the broadening process. For CO, the dominant contributions to S,(b) are those which exhibit an odd angular dependence (especially the k, = 1 terms)
Noble-gas broadening.of vibration-rotation
lines belonging to diatomic molecules-III
919
and characterize the long-range force effects during inelastic collisions (see Fig. 3). It is reasonable to assume the short-range forces to behave in a similar manner. We must then introduce a k, = 1 term in the exchange potential. In spite of its angular dependence, the influence of this term should not be negligible, owing to the smallness of the B, constants of co. The difficulty of determining such a term within the framework of the present formalism is quite clear. We, therefore, use the approximate model of Herman and Tipping. This model, which assumes the uniqueness of the center of dispersion forces, leads to a more tractable formalism. 4.
FORMALISM
DERIVED
FROM
THE
TIPPING
AND
HERMAN
POTENTIAL
It has been seen that, in the case of HCl, (2) this potential is an approximate form of Buckingham’s as regards long-range forces. This is due to the particular electronic distribution in the hydrogen halides which allows us to identify d,, (H.T. formalism) with the arithmetic mean (df’+dy’)/2 in the R.G.G. formulation. This is no longer possible with CO. For this molecule, we find (d\:’ +dy’)/2 = 0.35 8, as compared with the experimental value d,, = 0.21.A.* Nevertheless, it must be borne in mind that both formalisms exhibit the same angular dependence, at least for long-range forces (see Table 1 of Ref. 2). The same holds true with respect to repulsive potentials provided that the (el . R,) term is taken from the H.T. formulation [compare equation (7) of Ref. (2) with equation (1) of the present paper]. It is, therefore, not surprising that the theoretical curves calculated from the R.G.G. treatment (using (d\f’ +dy’)/2 = 0.35) and from the H.T. “truncated formulation” (with d,, = 0.21) should exhibit the same disappointing behaviour (curves 1 and 2 of Figs. 1 and 2). Thus the two potentials are almost equivalent and we are now in a position, with the aid of Herman and Tipping’s model, to check our assumption of an’additional (er . R,) repulsive potential. In column 11 and curves 3 are shown the results obtained with the complete expression for this potential. The resulting improvement in the calculated values is evident. For lower Im)-values, the difference between calculated and measured values is smaller than the experimental errors. For (ml 2 5, the comparison is less satisfactory. For argon, the relative difference remains within the current limits of errors (5 per cent). For Xe, greater disagreement appears, which reaches about 10 mK. For both Ar and Xe, the calculated curves exhibit a shape similar to the mean experimental curve, whereas this was not the case previously. Furthermore, it will be noticed that no cutoff problem arises, even for the higher values of Irnl. Splitting of the S,(b,) function allows us to emphasize the influence of the different kinds of collisions in the case of CO-rare gas mixtures. We consider here the case of the &2 transition for CO-Ar, the remaining ones behaving similarly. Several characteristic features appear. Thus, there is a small influence of the isotropic contributions to the potential (as in the R.G.G. formalism). Another similarity is that, among the purely attractive terms, the k, = 1 terms are predominant. The k, = 2 terms are, here again, negligible, except perhaps for low Im)-values. The k, = 1 contributions are found to behave entirely as predicted. This reveals the prime importance of the cross-terms, as was already pointed out by * See: H. FRIEDMANN, Adu. them. Phys 4, 225 (1962).
920
C. BOULET, P. ISNARD and A. LEVY
.
GIRAUD et ~1.“~’ These non-additivity effects in the interruption function for the two different kinds (attractive vs. repulsive) of intermolecular energies are, hence, of the greatest interest. Finally, it may be said that deexcitation of CO energy levels by rare gases occurs primarily through rotationally inelastic collisions inducing variations of one rotation quantum (k, = 1) under the influence of the dispersion energy. For small [ml-values, longrange forces prevail, whereas for large (ml-values, repulsive forces become predominant (‘s,::;; lS,f&). Either for long or for short-range forces, the separation of centers plays an important role.
The calculated shifts. If the e, and c1 vibrational variations of Friedmann and Kimel are again used too large values are found for the shifts. By contrast, introducing the new variations reported in our R.G.G. formalism study, yields rather good agreement with experiment (Table 3).
5. INFLUENCE
OF
VELOCITY
AVERAGING”’
Column 12 shows that the explicit inclusion of the velocity distribution in the calculation does not substantially improve the results for large lm(-values. Although the functions g,(k) and g,(k) are rather different, no appreciable variation in the widths is obtained. Moreover, the isotropic term, which is the most sensitive to velocity averaging, is negligible in our case. Nonetheless, a positive result of the present calculations is clearly seen from an analysis of the various contributions to S,(b,). The influence of repulsive forces has been lessened and, accordingly, the importance of attractive forces increased. Let us consider, for instance, the line Iml = 16. In the assumption of u = 17,‘S,$: amounts to 15 per cent of S,(b,) and the sum ‘S,:zL+ ‘S,$ to 85 per cent. In the case of velocity averaging, the above contributions reach, respectively 50 and 48 per cent. This means that long-range forces remain dominant. Only for the larger Iml-values (when b, approaches d) do repulsive forces actually prevail, due to overlapping of electronic clouds. 6. CONCLUSION
The present study enabled us to arrive at a satisfactory interpretation of pressure effects in CO-noble gas mixtures. It was necessary for this purpose to use very detailed expressions for the potential describing the interaction of a linear molecule and a noble gas atom. This reflects the complexity of such interactions. This work and the previous study’*’ allowed us to point out the basic difference in the relaxation mechanisms, according to the nature of the active molecule. Further progress will require the solution of two kinds of problems : those related to the description of trajectories during the encounter (cutoff procedures, velocity variations . . . ) and those arising from the empirical models used for the repulsive potential.
Acknowledgements-We would like to thank Mr. J. P. B~UANICH for kindly making the results of his measurements available to us prior to publication. The authors wish also to express their thanks to Professors L. GALATRY and D. ROBERT for making appropriate comments and significant improvements in many parts of the present work.
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules-111
921
REFERENCES A. LEVY, E. PIOLLET-MARIELand C. B~ULET, JQSRT 13,673 (1973). C. BoULET, P. ISNARDand A. LEVY, JQSRT 13,897 (1973). D. H. RANK, D. P. EASTMAN,B. S. RAO and T. A. WIGGINS,J. Molec. Spectrosc. lo,34 (1963). D. A. DRAEGWTand D. E. WILLIAMS,J. opt. Sot. Am. 58, 1399 (1968). J. P. B~UANICHand R. FARRENQ,C.R. Acad. Sci. Paris 271, 507 (1970). C. CRANE-ROBINSON and H. W. THOMPSON,Proc. R. Sot. 272,453 (1963). J. P. BOUANICH,JQSRT 12, 1609 (1972). R. J. THIBALJLT,J. H. JAFFE and E. K. PLYLER, J. Res. Natn. Bur. Stand. 67, 113 (1963). J. P. BOUANICHand C. HAEUSLER,JQSRT 12,695 (1972). D. ROBERT,C. GIRARDETand L. GALATRY,Chem. phys. L.&t. 3,102 (1969). C. GIRARDET,D. ROBERTand L. GALATRY, C.R. Acad. Sci. Paris 270,798 (1970). C. GIRARDET,“Third Cycle” Thesis, Besancon (1970). R. K. NESBET,J. them. Phys. 40, 3719 (1964). R. C. HERMANand K. E. SHULER,J. them. Phys. 22,481 (1954). H. FRIEDMANand S. KIMEL, J. them. Phys. 43, 3925 (1965). D. ROBERT, M. GIRAUDand L. GALATRY, J. them. Phys. 51,2192 (1969). M. GIRAUD, Thesis, Marseille (1972). M. GIRAUD, D. ROBERTand L. GALATRY, J. them. Phys. 53, 352 (1970). C. BOULET.E. ARIE, J. P. BOUANICHand N. LACOME.Cart. J. Phvs. 50.2178 (1972). J. 0. HIRSCHFELDER, C. F. CuRTIssand R. B. BIRQ Molecular Theoryof&sesandLiqui& Wiley, N.Y. (1954). D. H. RANK, A. G. ST PIERREand T. A. WIGGINS,J. Molec. Spectrosc. 18,418 (1965); E. K. PLYLER, L. R. BLAINEand W. S. CONNOR,J. opt. Sot. Am. 45,102 (1955). 22. P. ISNARD,C. BOULETand A. LEVY, Internal Report (1972). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
in Great
Britain
NOBLE-GAS BROADENING OF VIBRATION-ROTATION LINES BELONGING TO DIATOMIC MOLECULES-III. THEORETICAL CALCULATIONS OF CO LINESHIFTS AND WIDTHS C. BOULET,
P. ISNARD and A. LEVY
Laboratoire d’Infrarouge, Equipe de Recherche AssociQ au C.N.R.S., UniversitC de Paris VI, Campus d’orsay, BLtiment 350 (91405) Orsay, France (Received 21 September 1972)
Abstract-Noble-gas-broadened CO vibration-rotation linewidths and shifts are calculated, within the impact approximation, after a critical analysis of available experimental data. Several molecular parameters related to local electronic distribution have been computed in order to express explicitly the dispersion energy. The R-* dispersioncontributionis shown to be of slight, but not of negligible, influence. The importance of angular dependence of the repulsive potential is emphasized, together with the effect of cross-terms in the interruption function. The calculated results agree well with experimental data for almost all of the lines.
THE VIBRATIONALdependence of shifts and widths of noble-gas-broadened HCl lines has been extensively described. (I) Introduction into Anderson’s theory of the vibrational states of the active molecule has recently been shown to account successfully for this effect.‘2’ It may be asked whether a similar phenomenon occurs with the same magnitude for molecules other than the hydrogen halides and, in particular, for CO. Thus, starting from a review of the available experimental results,‘3-g’ we first point out variation of the line widths and shifts with vibrational quantum number for the CO molecule. The next step consists of computing the numerical values of the parameters involved in the dispersionenergy expression. In the last part, we use the generalized Anderson-Tsao-Curnutte theory in order to compare the theoretical results with experimental data.
1. REVIEW
OF EXPERIMENTAL
RESULTS
1. Line-widths (a) Argon broadening. Table 1 shows that, for the fundamental band, DRAEGERT’S’~’ measurements are in good agreement with the recent results of BOUANICH.(‘) For the O-2 transition, the experiments of BOUANICH~‘)provide results which practically coincide with those of THIBAULT.‘~’ Rank’s values(3’ are much greater. As regards the O-3 transition, the values reported by Bouanich and Haeusler in their recent paper’g) are quite similar, for 3 5 Im( I 13, to the widths measured by these same authors in the fundamental band. 911
912
C. BOLJLET,P. ISNARD and A. LEVY
TABLE 1. CO-ARGON MIXTURES.EXPERIMENTALAND THEORETICALRESULTS(HALF-WIDTH, EXPRESSED IN MK/BAR) Iml
1
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
61.4 59.6 57.9 54.3 52.3 48.9 48.5 46.6 46.8 46.1 45.8 44.2 44.5 43.5 43.4 42.1 43.4 40.6 39.7 39.7
57.1 55.7 55.7 52.4 51.8 46.9 45.5 44.1 43.2 43 42.5 40.5 40.5 39 37.9 36.4 36.3 35.2 31.6 30.4
3
70.4 66.5 61.4 56.9 53.4 50.8 48.5 47.2 46.3 45.6 44.6 44.4 43.8 43.5 43.1 42.3 41.4 40.7 40 -
4
65.5 50.5 40.5 35 41 38.5 40 39 -
5
6
7
8
69.9 62.5 57.8 53.7 50.6 48.8 47.5 46.4 45.7 45.1 44.6 44.1 43.5 42.8 41.9 41 40.3 39.5 38.8 38.2
69.5 63.5 57.5 53.5 50 48.5 45.5 42.5 42.5 42.5 42.5 41.5 41 42 38 42 42 42 37.5 38.5
84.5 81.5 76.5 69.5 63.5 63.5 55 59.5 56 55
80.7 78 74.5 70.3 66 62.2 58 54.2 51 48 45 42.5 40 38 36 34.2 32.5 31 30 28.5
51.5 49.5 -
9
87 82.3 78 73 68 63.5 59 55 51.6 48.5 45.8 43.3 41 39 31.2 35.6 34 33 31.7 - 30.6
Experimental results. Column 1: O-l BOUANICH et al.Is); 2 : C-l CRANE-R• 4: O-1 RANK(~); 5: O-2 B~UANICH(‘); 6: O-2 THIBAULT@); 7: G2 RANK(~).
10
11
12
75 71.5 68 64 60.5 54.5 53.2 50 47 44.2 41.5 39 36.7 34.7 33 31.4 30 29 28 21.3
71.3 67.6 63 52.8 46.8 42.9 40.8 39.7 39.2 38.9 38.7 38.4 38 37.5 36.9 36.3 35.6 34.9 34.1 33.4
70.8 66.1 57.6 50.8 46.2 43.3 41.4 40.2 39.4 38.8 38.4 38 37.6 37.2 36.8 36.3 35.8 35.7 35.5 35.2
BJNS~N'@ ; 3 : &l
DRAEGERT’~) ;
Theoretical results. The horizontal line indicates the limit of validity of Anderson’s cut-off. Column 8: R.G.G. formalism without the R-* disperson energy; 9: R.G.G. formalism including the Rm8 dispersion terms; 10: H.T. potential without the (ei . R,) repulsive term and with d,, = 0.21 A; 11 : H.T. complete potential; 12 : H.T. complete potential with velocity averaging.
It is then seen that no vibrational effect is suggested by the experiments for Ar-broadening. (b) Xenon broadening (Table 2). Experimental data are scarce in this case. By comparing Draegert’s values for the O-l band with those of Bouanich for the O-2 band, in view of the agreement of their measurements for Ar-broadening, no noticeable vibrational variation is apparent. 2. Frequency shifts (Table 3) We note disagreement between the values of RANK(~) and BOUANICH.(~) Since no other experimental data are presently available, the only information we can obtain on CO frequency shifts has to do with (a) their order of magnitude (the shifts are much smaller than the corresponding ones for HCl) and (b) their vibrational dependence. The values reported in Rank’s paper are the only ones allowing a comparison of two different transitions. By contrast with the linewidths, the shifts exhibit a noticeable variation with the vibrational number o. 2. INTERMOLECULAR I.
POTENTIAL
R.G.G.
FORMALISM”’
Molecular parameters
All of the molecular constants of carbon monoxide are not known, particularly A,, and A, (components of the hyperpolarizability tensor). But, it is not possible to ignore a priori,
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
913
TABLE 2. CO-XE: EXPERIMENTAL AND THEORETICALRESULTS(HALF WIDTH EXPRESSED IN MK/BAR). THE NUMBERING OF COLUMNS IS THE SAME AS IN TABLE 1
Iml
2
3
5
7
8
9
10
11
12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
72.8 68.1 64.2 614 58.2 53.9 51.3 48.9 48.6 47.3 46.7 44.2 47.7 42.7 41.7 40.4 37.2 36.9 33.3 34
89.3 80 73 68 63.9 60 56.6 54.7 52.9 51.3 49.3 48.5 47.3 46.5 45.7 45.2 45 43.7 42.4 -
84 74.5 68.9 64.5 60.8 58.1 55.8 53.9 52.6 51.7 51.2 50.8 50.4 49.6 48.2 46.9 45.8 44.8 44 43.3
92 82.5 75.5 73.5 68.5 65 62 60 60.5 57.5 56.5 59.5 51 57 48.5 -
93 88.4 82 75.2 68.7 64 57.5 53 48.8 45.2 42 39.2 36.7 34.4 32.4 30.7 29.3 28.5 27.4 27
98.5 92 85.2 77.4 70.2 65 58.8 54.2 50.4 47 44.2 41.8 39.8 38 36.7 35.4 34.5 33.7 33 32.4
86 81 75.4 69 63.2 58 53 49 45.4 42 39.2 36.7 34.4 32.5 31 30 29.5 29.2 29 28.8
84.8 79.1 72.3 64 50.8 45 42.7 41.9 41.5 41 40.5 39.7 38.9 37.9 36.9 35.8 34.8 33.8 32.8 32
83.5 77.2 69.7 60.6 51.8 46.5 43.9 42.4 41.4 40.6 39.9 39.2 38.7 37.6 36.8 35.8 35.2 35 34.8 34.6
in the dispersion-energy expression, the R-’ contribution which involves precisely these components. Consequently, let us consider the dispersion energy formulation derived from the variational method, as was previously done in the case of HC1.“oV”’ Its major advantage is that it involves molecular constants that are more easily computed than A ,, or A,, since they require knowing only the electronic wave functions for the fundamental state. The definition of these constants (d\;‘; dy’; d\t’; dy’) can be found in Ref. (lo), and the detailed calculation method in Ref. (12). The various parameters related to the electronic orbitals of CO were calculated from Nesbet’s data. (13) The numerical values obtained in this way for CO-He, CO-Ne and CO-Ar interactions are given in Table 4. It is then possible to calculate the dispersion energy with terms up to RF8 included (see equation (2) of Ref. 10). As in the case of HCl, the d\:‘,, show very slight variations from one rare gas to another. Accordingly, an order of magnitude for A,, and A, (which do not depend on the perturbing atom) may be determined. For this purpose, the previously reported relations are used (relation (8) in Ref. 2). We then find A,, 2: 26.7 x lo-33 cm4; A, N 0.5 x 1O-33 cm’. The dll, (f), which appear in the R-’ energies, are much larger than for HCl. Consequently, we can no longer assume the corresponding contributions to be ineffective in line-broadening. 2. Repulsive potential We confine ourselves first to an isotropic form for the repulsive forces, as we previously did for HCl. 3. Vibrational dependence of CO molecular parameters Table 5 gives the different coefficients, for the expansion of the dipole moment and polarizability, in terms of normal coordinates. The v-diagonal matrix elements of the normal
O+l
1
1.9
1.9 ~ 1.1 1.5 2.3
~ 2.2 1.9
2.4 2.8 -
1
3 4 5 6 7
8 9 10 11
12 13 14 15 16 17 18
R
3.5 3.6 3.1 3.1 3.7 -
2.8 3 3.2 3.3
1.8 2 2.2 2.4 2.6
1.7
1.6
(1)
obs. RANK(~) talc.
2
Iml
3.7 3.8 3.8 3.8 3.8 -
3.7 3.7 3.7 3.7
3 3.7 3.7 3.1 3.1
2.6
2.4
(2)
talc.
2.8 3.3 3.2 3.1 3.5 3.8 3.8
2.4 2.1 3 2.3
-0.7 0.1 0.6 1.5 1.7
- 1
-0.8
R
0+2
4.1 4.4 4.2 4.7 4.8 5.2 5.6 6.4 1.3 -
0.5 1.6 2.1 2.1 3.1 3.6 2.9 3.6 3.4 3 3.6 3.5 3.2 3.4 3.4 3.6 3.5 3.4
R
7.2 7.8 6.7 6.6
5.9 6.2 -
5.5 6 ~ 5.5 ~
-
6.2
P
obs. RANKC3’
P
obs. BOUANICH(“)
Argon
6.9 7.1 7.3 7.4 7.4 1.2 7
5.5 5.9 6.3 6.6
3.1 4 4.3 4.7 5.1
3.4
3.2
(1)
talc.
2.4 2.6 2.9 3.2 3.6 4.1 4.5 5 5.4 5.7 6 6.3 6.4 6.5 6.5 6.4 6.2 6.2
4.1 5.1 5.9 7.5 7.4 7.4 7.4 1.4 7.4 7.4 1.4 1.4 1.5 1.5 7.6 7.6 7.7 7.7
(1)
talc.
(2)
talc.
0-l
6.6 6.6 6.7 6.8 6.8 6.8 6.8
6.3 6.4 :4
4 5 6.4 6.3 6.3
3.5
3.2
(2)
talc.
Xenon
1.4 3.3 3.8 4.4 6.4 6.6 6.1 8.1 8.4 7.8 8.2 8.1 7.9 8.6 7.8 8.1 -
0.5 0.8 2 3 4.3 5 6.2 6.2 6.1 6.6 I 6.5 1.5 7.3 7.9 1.7
P
R
12.5 13.4 13.5 -
11.3 12 -
11.7 12.1
11.4
10.7 10.1
-
8.5
P
9.6
10 9 -
5.8 6 8.2 -
5.2
6.9
R
obs. _‘3’
o-+2
7.9 9.8 12.7 12.6 12.6 12.6 12.1 12.8 12.9 13.1 13.3 13.4 13.5 13.6 -
5.2 5.1 6.4 7.2 8.1 9 9.9 10.7 11.4 12 12.5 12.9 13 13 12.8 -
6.4 7
4.7
(2)
talc. (1)
talc.
FORMALISMINCLUDING DISPERSIONTERMS IN R-';
obs. BOUANICH(~‘)
TABLE 3. SHIFTS TOWARDS SMALLERWAVE NUMBERS(EXPRESSEDIN MK/BAR). THEORETICAL VALUES: (1) R.G.G. (2) H.T. FORMALISM(WITH THE COMPLETEPOTENTIAL)
$y < <
?
g *
F 5 %
71
-3
E
0
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
915
TABLE 4. PARAMETERS IN THE DISPERSION ENERGY (VARIATIONALcALcuLATIoN)
d(,f’ A
dy’ A2
d\;’ A2
CO-He
0.0165
0.5449
0.1349
0.9357
CO-Ne
0.0169
0.5626
0.1325
0.9537
CO-Ar
0.0179
0.6846
0.1168
1.0735
HCl-Ar
0.038
0.164
0.01
0.031
coordinates, on the basis of the anharmonic oscillator wave functions, may be calculated from the work of HERMAN and SHULER. (t4) In computing the vibrational variation of the L-J parameters, c1 and or, we apply the previous evaluations of FRIEDMAN and KIMEL(~ 5, which gave satisfactory results for HCl (Table 6). 3. APPLICATION
OF R.G.G.
FORMALISM
The molecular parameters of CO are taken from Refs. (20) and (21). Those concerning the noble gases are reported in Ref. (18). In deriving the results, a straight path trajectory is assumed with constant velocity for the perturber molecules (with u = ij); furthermore, Anderson’s cutoff procedure is applied. Tables 1-3, together with Figs. 1 and 2, enable us to show the relative importance of the various mechanisms responsible for the broadening of rotation-vibration lines of CO. (1) The R-* dispersion energy is not included (column 8, curve 1) The calculated results do not compare well with experimental data for small or large Jm( values. This discrepancy cannot be attributed to the cutoff procedure since, for most of TABLE 5. p1 AND a, EXPANSIONSIN TERMSOF NORMAL COORDINATES:(a) J. P. BOUANICH et al., .I. Phys. 29, 641 (1968); (b) W. F. MURPHY et al., Appl. Spectrosc. 23,211 (1969) ;
see also Ref. (20)
Aal
.
01 a I0 = 1.95 A” a,, = 1.5A2
p,O = -0.1240 p , , = 3.084 D/A pi2 = -0.574 D/A=
TABLE 6. VARIATIONOF THEE, AND a, CONSTANTS OF CO AS A FIJNCTION OF u
Friedmann-Kimel
Present work
O-l
G2
Gl
(r2
0.009
0.018
0.006
0.012
0.0019
0.0038
0.0012
0.0024
C. BOULET, P. ISNARD and A. LEVY
916
1
I
I 15
I IO
5
I
20
)
ml FIG. 1. Argon broadened CO lines. Halfwidths in mK/bar. Experimental values: + Bouanich t&l, n Crane-RobinsonO-1, x Thibault C&2, 0 Bouanich O-2. Theoretical predictions: Curve(l) R.G.G. formalism, (2) H.T. formalism without the (e, I&) repulsive term, (3) H.T. formalism.
the lines, the critical value of the impact parameter is much larger than the closest approach distance. This is an essential difference from HCI, resulting from the fact that the rotational constant B, of CO is much smaller than that of HCl. It may then be concluded that the disagreement arises from the inadequate expression used for the CO-noble gas interaction energy.
t
40----~.
I
I
I
I
I
5
IO
I5
20
)
Iml FIG. 2. Xenon-broadened
CO lines-Halfwidths
in mK/bar.
For symbols
refer to Fig. 1.
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules--III
917
A careful analysis of the various contributions to S,(b,) shows that the predominant term [which reaches about 75 per cent of S,(b,)] is k’S2(bo) with k, = 1, both for Ar and Xe. This emphasizes the importance of a preliminary determination of the distances d\,% characterizing the center of dispersion forces. Be that as it may, the calculated and observed values remain substantially different. It could then be asked whether this discrepancy should not be even greater when R-’ contributions (d((f),l constants) are introduced. (2) Injluence of the R- 8 term in the dispersion energy (column 9) A great number of additional terms arise in the S,(b) expression, particularly if non additivity effects’i6’ are taken into account. The set of these new contributions is listed in Ref. (22). Their introduction results, however, in a very slight increase in the calculated widths (- 1 mK for \m(2 5). Nevertheless, an analysis of the different parts of S,(b,) with respect to the order k, of spherical harmonics (Fig. 3) reveals that the influence of these new terms is far from insignificant. Firstly, the prevalence of k, = 1 contributions has slightly decreased. The new terms in k, = 4 and also the terms in k, = 2 resulting from the Re6 dispersion energy are practically negligible. Secondly, the terms related to the Re8 energy are nearly equal (-20 per cent) to the k, = 3 contributions. Finally, no important change is made when a complete expression is used for the dispersion energy. Nevertheless, we must point out the great diversity in the nature of collisions contributing to the deexcitation of CO energy levels. For smaller Im( values, inelastic collisions (with respect to rotation) are predominant since, for k, = 1, the selection rules arising from the Clebsch-Gordan coefficients do not allow angular frequencies ojj, equal to zero. For higher jrnl values, the situation is more complex. It is therefore necessary, to examine the influence of repulsive forces.
Iml
Iml
(0)
(b)
FIG. 3. S,(b,) splitting-R.G.G. formalism including the terms resulting from R-’ energy. (a) CO-Ar 61, (b) CO-Xe O-2, ‘S,(bo) negligible. p It1 1 : terms resulting from the R-’ dispersion energy. 1 [7 k: = 2 terms resulting from the Rw6 dispersion energy. 0 k, = 2 Sum of the additional terms appearing when the Ak,=4 > Re8 dispersion energy is taken into account.
dispersion
918
C. BOULET,P. ISNARD and A. LEVY
(3) Vibrational efict.
Inadequacy of an isotropic exchange potential
At this stage, a first positive result of our calculation is, that lines with the same Irnlhave nearly equal widths for both transitions (the difference never exceeds 0.5 mK). The reason is that the isotropic term OS,(b), the only one which is strongly u-dependent, plays here a negligible role, contrary to the situation observed for HCl. This results in a predominance of rotational relaxation mechanisms in CO-noble gas mixtures, instead of the “dilation effect” found for HCl.“” But the calculated shifts appear much greater than those measured by Rank, which are themselves greater than Bouanich’s measurements. In order to account at least for the order of magnitude of these shifts, a reexamination of the vibrational dependence of E~ and pi seems necessary. If we assume somewhat smaller vibrational variations than given by Friedman and Kimel (Table 6), it becomes possible to account both for the magnitude of Rank’s shifting values and for the observed vibrational effect (Table 3). At the same time, a decrease in ‘S,(b,) is obtained, but the corresponding influence on calculated widths is almost negligible, owing to the small value of this term. This necessity of modified values for &I and c1 variations could perhaps be due to an influence of the perturber on the vibrational dependence of the active molecule parameters. This effect has been suggested previously by Giraud(17’ for HCl-Ar interactions. Up to now, we have used an isotropic repulsive potential. The corresponding contributions to S,(b) are only in k, = 0. But the whole OS,(b) term is almost negligible. This procedure therefore fails to take account of the influence of repulsive forces. It is thus necessary to choose, for the exchange energy, a model involving some angular dependence. For this purpose, we consider the additional terms appearing in S,(b) when a repulsive contribution (labelled rep.) to the interaction energy is introduced and expanded in terms of spherical harmonics. If an attractive term (labelled att) of the same order k, exists, two new terms arise, namely, “S,:$ which is positive and klSz$‘P(which is a negative cross-term and involves negative powers of b, lower in absolute value than those involved in the pure repulsive term). If these new terms are added to S,(b), remembering that b, decreases as (ml increases, it may be reasonably expected that the negative one will prevail at low Iml-values (since b, is large); on the other hand, for higher [ml-values, the purely repulsive term, the variation of which is much more rapid, will compensate the cross-term effect. This should result in a decrease in width for lower Iml-values and an increase at higher Jml-values, leading to better agreement with experiment. But, since the actual form of the repulsive potential is unknown, the only possibility is to resort to a semi-empirical formulation. (4) Angular dependence of the repulsive potential The most simplified model we can assume is rather similar to that proposed by Artman and Gordon and used by GIRAUD et al.(l*) for their study of perturbation effects in the pure rotation spectrum of HCI-noble gas mixtures. The corresponding expression is written as V,,,(R) = (4ear2/Rr2)[l
-(r1/2)+
3y,(q
.
Ro)2/21.
Additional k, = 2 contributions are then to be added in S2(b).(22) The sum of the new terms never amounts to more than a few per cent of S,(b,), for any ho(m). The calculated widths, therefore, remain unchanged. Nevertheless, it cannot be concluded that the repulsive forces are ineffective in the broadening process. For CO, the dominant contributions to S,(b) are those which exhibit an odd angular dependence (especially the k, = 1 terms)
Noble-gas broadening.of vibration-rotation
lines belonging to diatomic molecules-III
919
and characterize the long-range force effects during inelastic collisions (see Fig. 3). It is reasonable to assume the short-range forces to behave in a similar manner. We must then introduce a k, = 1 term in the exchange potential. In spite of its angular dependence, the influence of this term should not be negligible, owing to the smallness of the B, constants of co. The difficulty of determining such a term within the framework of the present formalism is quite clear. We, therefore, use the approximate model of Herman and Tipping. This model, which assumes the uniqueness of the center of dispersion forces, leads to a more tractable formalism. 4.
FORMALISM
DERIVED
FROM
THE
TIPPING
AND
HERMAN
POTENTIAL
It has been seen that, in the case of HCl, (2) this potential is an approximate form of Buckingham’s as regards long-range forces. This is due to the particular electronic distribution in the hydrogen halides which allows us to identify d,, (H.T. formalism) with the arithmetic mean (df’+dy’)/2 in the R.G.G. formulation. This is no longer possible with CO. For this molecule, we find (d\:’ +dy’)/2 = 0.35 8, as compared with the experimental value d,, = 0.21.A.* Nevertheless, it must be borne in mind that both formalisms exhibit the same angular dependence, at least for long-range forces (see Table 1 of Ref. 2). The same holds true with respect to repulsive potentials provided that the (el . R,) term is taken from the H.T. formulation [compare equation (7) of Ref. (2) with equation (1) of the present paper]. It is, therefore, not surprising that the theoretical curves calculated from the R.G.G. treatment (using (d\f’ +dy’)/2 = 0.35) and from the H.T. “truncated formulation” (with d,, = 0.21) should exhibit the same disappointing behaviour (curves 1 and 2 of Figs. 1 and 2). Thus the two potentials are almost equivalent and we are now in a position, with the aid of Herman and Tipping’s model, to check our assumption of an’additional (er . R,) repulsive potential. In column 11 and curves 3 are shown the results obtained with the complete expression for this potential. The resulting improvement in the calculated values is evident. For lower Im)-values, the difference between calculated and measured values is smaller than the experimental errors. For (ml 2 5, the comparison is less satisfactory. For argon, the relative difference remains within the current limits of errors (5 per cent). For Xe, greater disagreement appears, which reaches about 10 mK. For both Ar and Xe, the calculated curves exhibit a shape similar to the mean experimental curve, whereas this was not the case previously. Furthermore, it will be noticed that no cutoff problem arises, even for the higher values of Irnl. Splitting of the S,(b,) function allows us to emphasize the influence of the different kinds of collisions in the case of CO-rare gas mixtures. We consider here the case of the &2 transition for CO-Ar, the remaining ones behaving similarly. Several characteristic features appear. Thus, there is a small influence of the isotropic contributions to the potential (as in the R.G.G. formalism). Another similarity is that, among the purely attractive terms, the k, = 1 terms are predominant. The k, = 2 terms are, here again, negligible, except perhaps for low Im)-values. The k, = 1 contributions are found to behave entirely as predicted. This reveals the prime importance of the cross-terms, as was already pointed out by * See: H. FRIEDMANN, Adu. them. Phys 4, 225 (1962).
920
C. BOULET, P. ISNARD and A. LEVY
.
GIRAUD et ~1.“~’ These non-additivity effects in the interruption function for the two different kinds (attractive vs. repulsive) of intermolecular energies are, hence, of the greatest interest. Finally, it may be said that deexcitation of CO energy levels by rare gases occurs primarily through rotationally inelastic collisions inducing variations of one rotation quantum (k, = 1) under the influence of the dispersion energy. For small [ml-values, longrange forces prevail, whereas for large (ml-values, repulsive forces become predominant (‘s,::;; lS,f&). Either for long or for short-range forces, the separation of centers plays an important role.
The calculated shifts. If the e, and c1 vibrational variations of Friedmann and Kimel are again used too large values are found for the shifts. By contrast, introducing the new variations reported in our R.G.G. formalism study, yields rather good agreement with experiment (Table 3).
5. INFLUENCE
OF
VELOCITY
AVERAGING”’
Column 12 shows that the explicit inclusion of the velocity distribution in the calculation does not substantially improve the results for large lm(-values. Although the functions g,(k) and g,(k) are rather different, no appreciable variation in the widths is obtained. Moreover, the isotropic term, which is the most sensitive to velocity averaging, is negligible in our case. Nonetheless, a positive result of the present calculations is clearly seen from an analysis of the various contributions to S,(b,). The influence of repulsive forces has been lessened and, accordingly, the importance of attractive forces increased. Let us consider, for instance, the line Iml = 16. In the assumption of u = 17,‘S,$: amounts to 15 per cent of S,(b,) and the sum ‘S,:zL+ ‘S,$ to 85 per cent. In the case of velocity averaging, the above contributions reach, respectively 50 and 48 per cent. This means that long-range forces remain dominant. Only for the larger Iml-values (when b, approaches d) do repulsive forces actually prevail, due to overlapping of electronic clouds. 6. CONCLUSION
The present study enabled us to arrive at a satisfactory interpretation of pressure effects in CO-noble gas mixtures. It was necessary for this purpose to use very detailed expressions for the potential describing the interaction of a linear molecule and a noble gas atom. This reflects the complexity of such interactions. This work and the previous study’*’ allowed us to point out the basic difference in the relaxation mechanisms, according to the nature of the active molecule. Further progress will require the solution of two kinds of problems : those related to the description of trajectories during the encounter (cutoff procedures, velocity variations . . . ) and those arising from the empirical models used for the repulsive potential.
Acknowledgements-We would like to thank Mr. J. P. B~UANICH for kindly making the results of his measurements available to us prior to publication. The authors wish also to express their thanks to Professors L. GALATRY and D. ROBERT for making appropriate comments and significant improvements in many parts of the present work.
Noble-gas broadening of vibration-rotation
lines belonging to diatomic molecules-111
921
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