- Email: [email protected]
Collisional pumping of a-doublet transitions in CH and OH by H2 and He
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
15 November 1979
COLLISIONAL PUMPING OF A-DOUBLET TRANSITIONS IN CH AND OH BY H 2 AND He D.R. FLOWER Department o f Physics. University o f Durham, Durham, DHI 3LE, UK Received 1 August 1979
The mechanism of inversion of the rotational ground states of the CH and OH radicals, proposed by Bertojo, Cheung and Townes, is reexamined. From considerations of the symmetry and form of the interaction with the spherically symmetric perturbers para-H2 and He, we show that present knowledge of the potential energy sttrfaces enables the efficiency of the coUisional pumping mechanism to be determined only to within a rather wide range. We specify the minimum additional information on the potential which is required in order to solve the problem and attempt to infer what the answer may be.
1. Introduction The mechanism which gives rise to inversion of the populations of the ground rotational levels of the OH and CH molecules has been the subject of considerable speculation and study. Of the collisional pumping mechanisms which have been proposed, the most generally accepted is that advanced by Bertojo et al. [1]. On the basis of calculations of the relevant potential energy curves, these authors suggested that selective excitation of the A-doublet components of excited rotational levels, followed by radiative decay, could induce departures from LTE in the ground state A-doublets. In the case of CH, collisions with all three neutral perturbers considered (H, H2, He) were found to result in population inversion, whereas, in the case of OH, only H produced inversion, H 2 and He giving rise to the opposite effect. In more recent and detailed studies, Kaplan and Shapiro [2] confirm that H pumps the OH radical effectively, whilst Dixon and Field [3] also conclude that collisions with H 2 and He lead to inversion of the ground (J = 3/2) doublet. In the present paper, we reexamine the problem of collisional pumping of CH and OH by the closed shell perturbers H 2 and He, in the light of current knowledge of the relevant potential energy surfaces. We show that the treatment of Bertojo et al. may be improved in a number of respects, and these developments are described in section 2. Although qualitative statements as to the likely efficiency of collisional pumping may be made (section 3), quantitative results must await more complete calculations of the potential energy surfaces.
2. Theory Bertojo et al. [1 ] computed the potential energy curves for a perpendicular approach of the spherically symmetric perturber (H 2 X l y~, o = J = 0; He ls 2 1 S) with respect to the CH(OH) internuclear axis. There are two such curves, according to the orientation of the electron lobes of the perturbed molecule, parallel or perpendicular to the direction o f incidence of the perturber; Bertoj o e t al. refer to these cases as 1 and 2 respectively (see their fig.l). According to these authors, case 1 collisions favour rotational excitation about the axis of higher momentum of inertia, corresponding, in general, to the lower energy component of an excited A-doublet. They estimate the ratio of cross sections for the two cases from the radii at which the interaction potential becomes equal to the rotational excitation energy. 475
Volume 67, number 2,3
CHEMICAL PHYSICSLETTERS
15 November 1979
The analysis of Bertojo et al. seems unsatisfactory in at least two respects: (i) They assume that the whole of the interaction potential can give rise to rotational excitation, whereas only the anisotropic part will contribute to inelastic collision processes. (ii) They assume that cases 1 and 2 collisions may be considered separately, i.e. that the perturber follows a given potential energy curve during the collision. However, as Green and Zare [4] have shown, the perturber does not follow a single adiabatic curve but a diabatic combination of the two. Let us, therefore, consider the problem in more detail.
(a) In terac tion po ten tial The potential energy of interaction between a spherically symmetric perturber and the open shell molecule may be written V(p, R) = 4 , ~ OqQu(p) Yqu(i~)YQ_u(k), qQ,u
(1)
where p = (p, 0, ¢) is the intermolecular vector, in a coordinate system aligned with the principal axes of the target molecule, and k = (O, ~) are the polar coordinates of the unpaired electron. The adiabatic potential energy surfaces, V(p), are obtained by taking the expectation value of (1) with respect to symmetrised electronic eigenfunctions. Subject to the convention that the y-axis lies in the plane of symmetry, defined by the internuclear axis of the target molecule (z-axis) and the intermolecular vector p, we obtain:
V(p, O) = 4rr ~ VqQu(p)Yqu(O , 7r/2) (YI 1(O, dp) + YI-I (O, ~)[YQ_u(O, (b)l Yll(Q, ~) + YI-1 (O, ~ ) ) / 2 . qQ/a
(2) The + signs in (2) refer to symmetric (A') and the - signs to asymmetric (A") eigenstates, respectively. Retaining terms in the expansion up to q = Q = 2, (2) becomes V(p, 0) = VOO+ 31/2 VlO cos 0 + 51/2 V20(3 cos20 - 1)/2 + ~ 0222 sin20 ,
(3)
where Vqo = OqO0 - 0q20/5112 and the + signs refer to the symmetries of the adiabatic potential energy surfaces, defined above. For a perpendicular approach of the perturber relative to the axis of the target molecule (0 = 1r/2),
V(p, rr/2) = V00 - 51/21/20/2 + 23-0222 ,
(4)
whereas, for the collinear approaches (0 = 0 and 7r), V(p, 0) = V00 +31/2 V10 + 51[ 2 V20 ,
(5)
V(p, rr) = V00 - 31/2V10 + 51/2V20 .
(6)
and
Bertojo et al. computed the two adiabatic potential energy curves (their cases 1 and 2) for a perpendicular approach of the perturber. Referring to eq. (2), we see that [V(case 1) + V(case 2)]/2 = V00 - 51/2 V20/2
(7)
and 3
[V(case 1) - V(case 2)]/2 = + ~ o222 ,
(8)
where the + sign applies to the case of CH and the - sign to OH. Thus the difference between case 1 and case 2 energies yields the coefficient 0222 of the potential energy expansion (1), whilst the average energy comprises isotropic (V00) and anisotropic (V20) terms. In deriving (3), we have considered the two electronic states (symmetries) associated with the A-doubling in the open shell molecule. Following Green and Zare [4] we may equivalently discuss the interaction in terms of a single 476
Volume 67, number 2,3
CHEMICAL PHYSICSLETTERS
15 November 1979
electronic state (symmetry) with two rotational modes, in which case the potential becomes effectively q)-dependent For the state of A" symmetry, for example, we obtain V(,o, 0, qS)= V00 + 3t/2VloCOSO + 51/2 V20(3 cos20 - 1)/2 + 23-0222 sin20 (e 2iq~+ e-2i~)/2
(9)
and, when 0 = 7r]2,
V(p, 7r/2, (3) = VO0 - 51/2V20/2 + ~ O2z2(eZi¢ + e-2i~)/2 .
(10)
With the aid of eqs. (7) and (8), eq. (10) may be written in the form V(p, rr]2, 40 = IV(case 1) + V(case 2)]/2 -+([V(case 1) - V(case 2)]/2} (e 2i~ + e-2i~)/2 ,
(11)
where the + ( - ) sign refers to CH(OH). We note that (11) has the form of the effective 4~-dependent potential derived by Green and Zare. It is the second term in eq. (11) which gives rise to selective excitation of a given component of a A-doublet; the first term contributes to elastic and rotationally inelastic scattering but does not discriminate between the two components of the excited state. In order to determine the relative importance of selective to indiscriminate rotational excitation, the contribution of the anisotropic term, V20, to the average interaction energy (7) must be determined. Combining eqs. (5)-(7), we obtain VO0 =(V(p,O) + V(p,n) + 2 [V(case 1) + V(case 2)]}/6,
(12)
V20 ={V(p,0) + V(p, 70 - IV(case t) + V(case 2)1}/3X/5 •
(13)
and
In the limit of V(0, 0) + V(,o, 7r) >> V(case 1) + V(case 2), i.e. when the collinear approaches are much more strongly repulsive than the perpendicular approach, eqs. (12) and (13) show that the anisotropy parameter, b 2 - V20/Voo 2/51/2, whereas, in the opposite limit, V(p,O) + V(,o, 70 ,~ V(case 1) + V(case 2), b 2 -+ 1/51/2 . In general, the anisotropy parameter, b 2, is determined by the relative magnitude of the interaction energies for collinear and perpendicular approaches of the perturber and is a function of intermolecular separation, p. As the collinear potentials have not been calculated, we shall, in what follows, leave b 2 as a free parameter and subsequently consider which of the limits is likely to be approached.
(b) Collision cross sections In view of the uncertainties and the gaps in our knowledge of the relevant potential energy surfaces, we shall make only a semi-quantitative analysis of the collision process, derived from perturbation theory. Introducing the anisotropy parameter, b2, eq. (10) may be written in the form V(p, 7r/2, ~b)= VO0 (1 - b 2 [51/2/2 - ~ (0222/V20)(e 2i~ + e-2i~)/2] } .
(14)
We take the square of the matrix element of the anisotropic part of the potential as a measure of the rotational excitation cross section, Q, i.e.
Qo~l(ilb2Voo[51/2/2
_
~3 (0222/V20) (e2i0 + e-2i¢)/2] if)[2
where i and f a r e the initial and final states of the total system of radical and perturber. Using eqs. (7) and (8), we obtain
Q o~ 1(i1(51/2/2 ) [b2/(1 - 51/2b2/2)] ([V(case 1) + V(case 2)]/2} (1 ~/3)lf)l 2 , where _
2 V(case 1 ) - V(case 2) 1 - 51/2b2/2 e2i~ + e-2i¢ 51/2 V(case 1) + V(case 2) b2 2
(15)
477
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
15 November 1979
determines the relative magnitude of the cross sections for excitation of a given A-doublet component and is the quantity of interest to us here. In order that rotational excitation be probable, the anisotropic part of the interaction potential must be comparable with the rotational energy separation, i.e. rotational transitions will occur on the repulsive part of the potential energy curves. As both case 1 and case 2 energies have a similar exponential variation with p (cf. fig. 3 of Bertojo et al.), the ratio of the difference to the sum of these energies is not very sensitive to/9.
3. Discussion First consider the limiting values of the anisotropy parameter derived in section 2. (i) b 2 ~ 2/51/2. In this limit, fl -~ 0 and V20 -+ oo [cf. eq. (7)]. Clearly, this limit will not be attained in practice and we shall always have Ifll > 0. The relative importance of selective rotational excitation may be expected to become small as this limit is approached. (ii) b 2 ~ - 1 / 5 1 / 2 . We then have V(case 1) - V(case 2) e 2i~ + e -2i~ f l ~ - 3 V(case 1 ) ¥ V(case 2) 2 A further limit which it is useful to consider is that of b 2 ~ 0% which is equivalent to the assumption of Bertojo et al. that only the anisotropic term contributes to the interaction energy (7); then fl ~ _ V(case 1 ) - V(case2) e 2i~ + e -2i~ V(case 1) + V(case 2) 2 As may be seen from these expressions, the available potential energy calculations enable the values of fl to be determined only to within rather wide limits. Indeed, strictly speaking, fl is not bounded at all: eq. (15) shows that fl ~ oo as b 2 ~ 0, i.e. as V20 ~ 0, a possible if improbable limit. The only unequivocal statement which may be made regarding the efficiency of the collisional pumping mechanism is that, for a given potential anisotropy, the relative importance of selective rotational excitation will increase with the difference between case 1 and 2 interaction energies. The calculated potentials (Bertojo et al., fig. 3) then suggest that departures of level populations from their values in thermodynamic equilibrium, induced by collisions with H 2 and He, will be greater in the case of CH than in the case of OH. It is instructive to consider which of the above limits is likely to be approached in practice. The electron charge distribution of a diatomic molecule may be expected to have greater extent along the internuclear axis than perpendicular to this axis. Consequently, for a given centre of mass separation, the interaction should become more rapidly repulsive for a collinear than for a perpendicular approach of an inert perturber, i.e. V(p, 0) + V(p, 70 > V(case 1) + V(case 2) on the repulsive part of the potential energy surface. This simple reasoning suggests, therefore, that the limit b 2 = 2/51/2 is likely to be approached over that part of the potential which contributes significantly to rotational excitation. As mentioned above, fl -+ 0 as this limit is approached, and the efficiency of the collisional pumping mechanism will be low. More definite conclusions must await additional calculations of the relevant potential energy surfaces, in particular, for collinear approaches of the perturber; this is perhaps the most pressing requirement for the study of these interstellar radicals.
References [1] [2] [3] [4] 478
M. Bertojo, A.C. Cheung and C.H. Townes, Astrophys. J 208 (1976) 914. H. Kaplan and M. Shapiro, Astrophys. J. 229 (1979) L91. R.N. Dixon and D. Field, Proc. Roy. Soc. A (1979), to be published. S. Green and R.N. Zare, Chem. Phys. 7 (1975) 62.
CHEMICAL PHYSICS LETTERS
15 November 1979
COLLISIONAL PUMPING OF A-DOUBLET TRANSITIONS IN CH AND OH BY H 2 AND He D.R. FLOWER Department o f Physics. University o f Durham, Durham, DHI 3LE, UK Received 1 August 1979
The mechanism of inversion of the rotational ground states of the CH and OH radicals, proposed by Bertojo, Cheung and Townes, is reexamined. From considerations of the symmetry and form of the interaction with the spherically symmetric perturbers para-H2 and He, we show that present knowledge of the potential energy sttrfaces enables the efficiency of the coUisional pumping mechanism to be determined only to within a rather wide range. We specify the minimum additional information on the potential which is required in order to solve the problem and attempt to infer what the answer may be.
1. Introduction The mechanism which gives rise to inversion of the populations of the ground rotational levels of the OH and CH molecules has been the subject of considerable speculation and study. Of the collisional pumping mechanisms which have been proposed, the most generally accepted is that advanced by Bertojo et al. [1]. On the basis of calculations of the relevant potential energy curves, these authors suggested that selective excitation of the A-doublet components of excited rotational levels, followed by radiative decay, could induce departures from LTE in the ground state A-doublets. In the case of CH, collisions with all three neutral perturbers considered (H, H2, He) were found to result in population inversion, whereas, in the case of OH, only H produced inversion, H 2 and He giving rise to the opposite effect. In more recent and detailed studies, Kaplan and Shapiro [2] confirm that H pumps the OH radical effectively, whilst Dixon and Field [3] also conclude that collisions with H 2 and He lead to inversion of the ground (J = 3/2) doublet. In the present paper, we reexamine the problem of collisional pumping of CH and OH by the closed shell perturbers H 2 and He, in the light of current knowledge of the relevant potential energy surfaces. We show that the treatment of Bertojo et al. may be improved in a number of respects, and these developments are described in section 2. Although qualitative statements as to the likely efficiency of collisional pumping may be made (section 3), quantitative results must await more complete calculations of the potential energy surfaces.
2. Theory Bertojo et al. [1 ] computed the potential energy curves for a perpendicular approach of the spherically symmetric perturber (H 2 X l y~, o = J = 0; He ls 2 1 S) with respect to the CH(OH) internuclear axis. There are two such curves, according to the orientation of the electron lobes of the perturbed molecule, parallel or perpendicular to the direction o f incidence of the perturber; Bertoj o e t al. refer to these cases as 1 and 2 respectively (see their fig.l). According to these authors, case 1 collisions favour rotational excitation about the axis of higher momentum of inertia, corresponding, in general, to the lower energy component of an excited A-doublet. They estimate the ratio of cross sections for the two cases from the radii at which the interaction potential becomes equal to the rotational excitation energy. 475
Volume 67, number 2,3
CHEMICAL PHYSICSLETTERS
15 November 1979
The analysis of Bertojo et al. seems unsatisfactory in at least two respects: (i) They assume that the whole of the interaction potential can give rise to rotational excitation, whereas only the anisotropic part will contribute to inelastic collision processes. (ii) They assume that cases 1 and 2 collisions may be considered separately, i.e. that the perturber follows a given potential energy curve during the collision. However, as Green and Zare [4] have shown, the perturber does not follow a single adiabatic curve but a diabatic combination of the two. Let us, therefore, consider the problem in more detail.
(a) In terac tion po ten tial The potential energy of interaction between a spherically symmetric perturber and the open shell molecule may be written V(p, R) = 4 , ~ OqQu(p) Yqu(i~)YQ_u(k), qQ,u
(1)
where p = (p, 0, ¢) is the intermolecular vector, in a coordinate system aligned with the principal axes of the target molecule, and k = (O, ~) are the polar coordinates of the unpaired electron. The adiabatic potential energy surfaces, V(p), are obtained by taking the expectation value of (1) with respect to symmetrised electronic eigenfunctions. Subject to the convention that the y-axis lies in the plane of symmetry, defined by the internuclear axis of the target molecule (z-axis) and the intermolecular vector p, we obtain:
V(p, O) = 4rr ~ VqQu(p)Yqu(O , 7r/2) (YI 1(O, dp) + YI-I (O, ~)[YQ_u(O, (b)l Yll(Q, ~) + YI-1 (O, ~ ) ) / 2 . qQ/a
(2) The + signs in (2) refer to symmetric (A') and the - signs to asymmetric (A") eigenstates, respectively. Retaining terms in the expansion up to q = Q = 2, (2) becomes V(p, 0) = VOO+ 31/2 VlO cos 0 + 51/2 V20(3 cos20 - 1)/2 + ~ 0222 sin20 ,
(3)
where Vqo = OqO0 - 0q20/5112 and the + signs refer to the symmetries of the adiabatic potential energy surfaces, defined above. For a perpendicular approach of the perturber relative to the axis of the target molecule (0 = 1r/2),
V(p, rr/2) = V00 - 51/21/20/2 + 23-0222 ,
(4)
whereas, for the collinear approaches (0 = 0 and 7r), V(p, 0) = V00 +31/2 V10 + 51[ 2 V20 ,
(5)
V(p, rr) = V00 - 31/2V10 + 51/2V20 .
(6)
and
Bertojo et al. computed the two adiabatic potential energy curves (their cases 1 and 2) for a perpendicular approach of the perturber. Referring to eq. (2), we see that [V(case 1) + V(case 2)]/2 = V00 - 51/2 V20/2
(7)
and 3
[V(case 1) - V(case 2)]/2 = + ~ o222 ,
(8)
where the + sign applies to the case of CH and the - sign to OH. Thus the difference between case 1 and case 2 energies yields the coefficient 0222 of the potential energy expansion (1), whilst the average energy comprises isotropic (V00) and anisotropic (V20) terms. In deriving (3), we have considered the two electronic states (symmetries) associated with the A-doubling in the open shell molecule. Following Green and Zare [4] we may equivalently discuss the interaction in terms of a single 476
Volume 67, number 2,3
CHEMICAL PHYSICSLETTERS
15 November 1979
electronic state (symmetry) with two rotational modes, in which case the potential becomes effectively q)-dependent For the state of A" symmetry, for example, we obtain V(,o, 0, qS)= V00 + 3t/2VloCOSO + 51/2 V20(3 cos20 - 1)/2 + 23-0222 sin20 (e 2iq~+ e-2i~)/2
(9)
and, when 0 = 7r]2,
V(p, 7r/2, (3) = VO0 - 51/2V20/2 + ~ O2z2(eZi¢ + e-2i~)/2 .
(10)
With the aid of eqs. (7) and (8), eq. (10) may be written in the form V(p, rr]2, 40 = IV(case 1) + V(case 2)]/2 -+([V(case 1) - V(case 2)]/2} (e 2i~ + e-2i~)/2 ,
(11)
where the + ( - ) sign refers to CH(OH). We note that (11) has the form of the effective 4~-dependent potential derived by Green and Zare. It is the second term in eq. (11) which gives rise to selective excitation of a given component of a A-doublet; the first term contributes to elastic and rotationally inelastic scattering but does not discriminate between the two components of the excited state. In order to determine the relative importance of selective to indiscriminate rotational excitation, the contribution of the anisotropic term, V20, to the average interaction energy (7) must be determined. Combining eqs. (5)-(7), we obtain VO0 =(V(p,O) + V(p,n) + 2 [V(case 1) + V(case 2)]}/6,
(12)
V20 ={V(p,0) + V(p, 70 - IV(case t) + V(case 2)1}/3X/5 •
(13)
and
In the limit of V(0, 0) + V(,o, 7r) >> V(case 1) + V(case 2), i.e. when the collinear approaches are much more strongly repulsive than the perpendicular approach, eqs. (12) and (13) show that the anisotropy parameter, b 2 - V20/Voo 2/51/2, whereas, in the opposite limit, V(p,O) + V(,o, 70 ,~ V(case 1) + V(case 2), b 2 -+ 1/51/2 . In general, the anisotropy parameter, b 2, is determined by the relative magnitude of the interaction energies for collinear and perpendicular approaches of the perturber and is a function of intermolecular separation, p. As the collinear potentials have not been calculated, we shall, in what follows, leave b 2 as a free parameter and subsequently consider which of the limits is likely to be approached.
(b) Collision cross sections In view of the uncertainties and the gaps in our knowledge of the relevant potential energy surfaces, we shall make only a semi-quantitative analysis of the collision process, derived from perturbation theory. Introducing the anisotropy parameter, b2, eq. (10) may be written in the form V(p, 7r/2, ~b)= VO0 (1 - b 2 [51/2/2 - ~ (0222/V20)(e 2i~ + e-2i~)/2] } .
(14)
We take the square of the matrix element of the anisotropic part of the potential as a measure of the rotational excitation cross section, Q, i.e.
Qo~l(ilb2Voo[51/2/2
_
~3 (0222/V20) (e2i0 + e-2i¢)/2] if)[2
where i and f a r e the initial and final states of the total system of radical and perturber. Using eqs. (7) and (8), we obtain
Q o~ 1(i1(51/2/2 ) [b2/(1 - 51/2b2/2)] ([V(case 1) + V(case 2)]/2} (1 ~/3)lf)l 2 , where _
2 V(case 1 ) - V(case 2) 1 - 51/2b2/2 e2i~ + e-2i¢ 51/2 V(case 1) + V(case 2) b2 2
(15)
477
Volume 67, number 2,3
CHEMICAL PHYSICS LETTERS
15 November 1979
determines the relative magnitude of the cross sections for excitation of a given A-doublet component and is the quantity of interest to us here. In order that rotational excitation be probable, the anisotropic part of the interaction potential must be comparable with the rotational energy separation, i.e. rotational transitions will occur on the repulsive part of the potential energy curves. As both case 1 and case 2 energies have a similar exponential variation with p (cf. fig. 3 of Bertojo et al.), the ratio of the difference to the sum of these energies is not very sensitive to/9.
3. Discussion First consider the limiting values of the anisotropy parameter derived in section 2. (i) b 2 ~ 2/51/2. In this limit, fl -~ 0 and V20 -+ oo [cf. eq. (7)]. Clearly, this limit will not be attained in practice and we shall always have Ifll > 0. The relative importance of selective rotational excitation may be expected to become small as this limit is approached. (ii) b 2 ~ - 1 / 5 1 / 2 . We then have V(case 1) - V(case 2) e 2i~ + e -2i~ f l ~ - 3 V(case 1 ) ¥ V(case 2) 2 A further limit which it is useful to consider is that of b 2 ~ 0% which is equivalent to the assumption of Bertojo et al. that only the anisotropic term contributes to the interaction energy (7); then fl ~ _ V(case 1 ) - V(case2) e 2i~ + e -2i~ V(case 1) + V(case 2) 2 As may be seen from these expressions, the available potential energy calculations enable the values of fl to be determined only to within rather wide limits. Indeed, strictly speaking, fl is not bounded at all: eq. (15) shows that fl ~ oo as b 2 ~ 0, i.e. as V20 ~ 0, a possible if improbable limit. The only unequivocal statement which may be made regarding the efficiency of the collisional pumping mechanism is that, for a given potential anisotropy, the relative importance of selective rotational excitation will increase with the difference between case 1 and 2 interaction energies. The calculated potentials (Bertojo et al., fig. 3) then suggest that departures of level populations from their values in thermodynamic equilibrium, induced by collisions with H 2 and He, will be greater in the case of CH than in the case of OH. It is instructive to consider which of the above limits is likely to be approached in practice. The electron charge distribution of a diatomic molecule may be expected to have greater extent along the internuclear axis than perpendicular to this axis. Consequently, for a given centre of mass separation, the interaction should become more rapidly repulsive for a collinear than for a perpendicular approach of an inert perturber, i.e. V(p, 0) + V(p, 70 > V(case 1) + V(case 2) on the repulsive part of the potential energy surface. This simple reasoning suggests, therefore, that the limit b 2 = 2/51/2 is likely to be approached over that part of the potential which contributes significantly to rotational excitation. As mentioned above, fl -+ 0 as this limit is approached, and the efficiency of the collisional pumping mechanism will be low. More definite conclusions must await additional calculations of the relevant potential energy surfaces, in particular, for collinear approaches of the perturber; this is perhaps the most pressing requirement for the study of these interstellar radicals.
References [1] [2] [3] [4] 478
M. Bertojo, A.C. Cheung and C.H. Townes, Astrophys. J 208 (1976) 914. H. Kaplan and M. Shapiro, Astrophys. J. 229 (1979) L91. R.N. Dixon and D. Field, Proc. Roy. Soc. A (1979), to be published. S. Green and R.N. Zare, Chem. Phys. 7 (1975) 62.