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Einstein coefficients for emission from high rotational states of the OH(X2∏) radical
Vol. 49, No. 3, pp.223-235, 1993 Printed in Great Britain. All rights reserved
J. Quonr. Spec~rosc. Radial. Transfer
0022~4073/93$6.00+ 0.00 Copyright 0 1993Pergamon Press Ltd
EINSTEIN COEFFICIENTS FOR EMISSION FROM HIGH ROTATIONAL STATES OF THE OH(X’l-I) RADICAL K. W. HOLTZCLAW,~
J. C. PERSON, and
B. D. GREEN
Physical Sciences Inc., 20 New England Business Center, Andover, MA 01810, U.S.A. (Received 20 May 1992)
Abstract-We report calculated Einstein coefficients for the pure-rotation and vibrationrotation transitions of 0H(X21T) with particular emphasis on those involving high-lying rotational levels. Coefficients are reported for rotational levels ranging from N = 42 (u = 0) to N = 9 (u = 9). The pure-rotation coefficients are the first to be reported in the literature. The effects of the centrifugal distortion are explicitly accounted for in these calculations and individual spin-sublevel-specific coefficients are reported. Centrifugal distortion in this work was accounted for assuming an expression for rotational kinetic energy appropriate in the Hund’s case (b) limit. Comparison with otherwise similar calculations by Nelson et al, which incorporate the effects of incomplete spin uncoupling on the rotational kinetic energy, shows that this approximation results in insignificant errors for N’ > 10. The implications of these coefficients for radiative decay of OH(X211) in the upper atmosphere are discussed.
INTRODUCTION Recent observations of long-wavelength radiances from the upper atmosphere and from the laboratory display emissions arising from high lying rotational states of OH(X*II). These observations include both vibration-rotation (Av > 0) as well as pure-rotation (Av = 0) transitions. For example, fluorescence from pure-rotation transitions has been observed in spectra from the CIRRZS IA mission from rotational levels with N up to 33 in the ground vibrational state, u = 0.’ Pure-rotation transitions from high N states in absorption have been observed in solar data.* In recent experiments using the LABCEDE facility at the Phillips Laboratory (Hanscom AFB, MA), the reaction of O’D with H, at low total pressures (0.02 torr) was used to produce OH in highly rotationally excited states. Pure-rotation transitions from levels as high as N = 29 (v = 0) were observed using a helium-cooled circular-variable-filter spectrometer.3 These data have been analyzed to extract relaxation rate coefficients of the v = 0, N = 29 level for several collision partners. Vibration-rotation spectra from the O’D + H, reaction have been observed via infrared chemiluminescence4,5 again including experiments performed in the LABCEDE facility. The latter have allowed observation of vibrational emission from levels as high as N = 26 (v = 1) using a Michelson interferometer and are presently being analyzed for collisional relaxation information. The radiative emission rates required to quantify the excited state densities giving rise to these emissions are not available in the literature. BACKGROUND
AND
COMPUTATIONAL
APPROACH
There are a number of compilations of vibration-rotation Einstein coefficients for OH now in the literature. Recent examples include those from Nelson et aL6 and Mies,’ and Langhoff et a1.8 These compilations were calculated in a similar manner and account for the effects of centrifugal distortion and the effects of incomplete electron spin uncoupling. Since these phenomena influence the relative intensities of the various transitions originating from a common upper rotational state, the corresponding transition moments are not easily separated into distinct vibrational and rotational factors. Consequently, the data for OH are voluminous and must be reported as coefficients for individual v’, J’, u”, J” transitions. The data of Refs. 6, 7 and 8 differ principally with respect to the electric dipole moment function (EDMF) employed. Only Nelson et al employ tTo whom all correspondence
should be addressed. 223
K. W. HOLTZCLAW et al ‘Table
I. Einstein A coefficients
for pure-rotation
transitions
in OH(X),
1‘ = C&2
an EDMF derived from experimental data. The dipole moment functions used by Mies and Langhoff et al were calculated ah initio. Nonetheless, a significant limitation of the compilations reported to date is that they extend to rotational states of no greater than J’ = 13.5-15.5. The vibration-rotation interaction, or centrifugal distortion, profoundly influences the relative intensities of individual vibration-rotation transitions in OH. The effect of centrifugal distortion on the radiative properties of diatomics was first discussed by Herman and Wallis’ who employed an approach based on perturbation theory. The interaction can be viewed as producing a mixing of the vibrational wavefunctions which diagonalise the Hamiltonian in the absence of centrifugal distortion: IY’ > =a’/ 0 > + b’J 1 > + C’I 2 > +. . . , IY”> It is common practice to expand internuclear distance as follows:
=a”lO>
+b”ll>
the dipole
moment
+c”l2>
+ ...,
as a Taylor
p(l)=&)+; g (r--l.)+; 2
() ‘C
(1
(r-r,y+
rr
series about
.’
the equilibrium
Einstein coefficients for the OH(XZfI) radical Table 2. Vibration-rotation
225
Einstein A coefficients for the 14 transition.
N’
Pl
Ql
R1
P2
f 3 4 :
10.13 7.80 11.48 12.51 14.16 13.38
9.16 3.70 1.92 1.15 0.51 0.74
0.00 3.36 3.86 3.63 2.60 3.16
12.50 12.73 12.88 13.43 14.65 14.04
1.18 5.30 0.58 0.37 0.20 0.26
3.96 0.00 4.12 3.74 2.62 3.20
7
14.84
0.37
2.03
15.22
0.16
2.04
t 10 11 12 11:
15.98 15.45 16.45 16.84 17.15 17.55 17.39
0.21 0.28 0.17 0.13 0.11 0.07 0.09
1.02 1.50 0.62 0.30 0.10 0.04 0.00
15.75 16.24 16.66 17.01 17.30 17.67 17.52
0.10 0.13 0.09 0.07 0.06 0.04 0.05
1.02 1.50 0.62 0.30 0.10 0.04 0.00
i:
17.65 17.67
0.06 0.05
0.52 0.21
17.76 17.75
0.04 0.03
0.52 0.21
11: 19 20 f?l
17.61 17.49 17.30 17.05 16.37 16.74
0.04 0.03 0.03 0.02
0.99 1.62 2.41 3.38 4.52 5.84
17.56 17.69 17.37 17.10 16.41 16.79
0.02 0.03 0.02 0.02 0.01 0.02
0.99 1.62 2.41 3.38 5.84 4.52
23 ::
15.94 14.96 15.48
0.02 0.01
7.35 10.91 9.04
15.99 15.52 15.00
0.01 0.01
7.34 10.91 9.03
26
14.41
0.01
12.97
14.44
0.01
12.97
91
13.21 13.83
0.01
17.65 15.21
13.24 13.86
0.01
15.21 17.64
3: 31 32
11.92 12.57 11.25 10.57
0.00 0.01 0.00 0.00
23.04 20.26 26.00 29.13
11.94 12.60 11.27 10.58
0.00 0.00 0.00
20.25 23.03 26.00 29.12
:2 35 36
9.88 9.20 8.52 7.84
0.00 0.00 0.00
32.43 35.88 39.48 43.23
9.89 9.21 8.53 7.85
0.00 0.00 0.00
32.42 35.87 39.47 43.23
iI 39
7.17 6.54 5.90
0.00 0.00
47.11 51.12 55.25
7.18 6.54 5.91
0.00 0.00
47.10 51.11 55.24
Q2
%
Calculation of the corresponding transition moment integral between two such mixed wavefunctions with the expanded dipole moment shows that the permanent dipole moment can now contribute to the transition moment where it would not in the absence of centrifugal distortion. If a harmonic oscillator basis set is assumed for the mixed wavefunctions, the surviving terms in the transition moment integral arelo (Y’l&)l
Y”) = a’a”~(r,)(olo>
+n’b”;g (
+ b’b”&,)(lll) (Olr-r,ll)+b’n”; ) 1,
+ *** $ 0
(llr-r,lO)+... ‘,
1 azp (Ol(r-r,)z12)+c’uM6 Y@ (2I(r--r,)*lo)+... (3) ( > ‘e ( > ‘C and it can be seen that these terms contain contributions from the permanent dipole moment. It should be noted that, since OH is decidedly anharmonic, the permanent dipole moment also contributes to some extent even in the absence of rotation. This contribution is independent of rotational level and is not responsible for the anomalous intensity variations observed in OH. However, as shown by Herman and Wallis, the permanent dipole moment contribution due to centrifugal distortion varies both in magnitude and sign, depending upon the rotational states involved in the transition as well as the change in rotational quantum number. fn’c”;
$
K. W. HOLTZCLAWetal
226
Table3.Vibration-rotation Einstein A coefficients forthe2-Itransition N’
1 2 3 4 5 6 7 : :: 12 13 14 15 16 17 18 19 20 21 22 :: 25 :: 28 29 30 32' 33 z 36 37
PI 10.73 14.12 16.21 17.85 19.27 20.54 21.68 22.70 23.60 24.39 25.06 25.60 26.02 26.32 26.50 26.55 26.48 26.32 26.02 25.59 25.09 24.52 23.84 23.07 22.21 21.32 20.36 19.39 18.34 17.29 16.21 15.10 14.02 12.92 11.83 10.76 9.73
Ql 12.17 4.93 2.57 1.53 0.99 0.68 0.49 0.36 0.28 0.22 0.17 0.14 0.11 0.09 0.08 0.06 0.05 0.04 0.04 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Rl 0.00 4.23 4.70 4.23 3.48 2.66 1.88 1.19 0.63 0.23 0.03 0.03 0.26 0.74 1.48 2.50 3.82 5.43 7.36 9.62 12.20 15.12 18.38 21.97 25.92 30.20 34.82 39.78 45.05 50.65 56.56 62.76 69.24 75.99 83.02 90.23 97.71
P2 17.56 17.47 18.22 19.19 20.24 21.27 22.25 23.17 23.99 24.71 25.34 25.84 26.23 26.50 26.66 26.69 26.61 26.43 26.12 25.68 25.17 24.59 23.91 23.13 22.27 21.37 20.41 19.43 18.37 17.32 16.24 15.13 14.04 12.94 11.85 10.78 9.74
Q2 7.02 1.56 0.76 0.48 0.34 0.26 0.20 0.16 0.13 0.11 0.09 0.08 0.06 0.05 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
t2 0.00 5.00 5.01 4.36 3.53 2.68 1.88 1.19 0.63 0.23 0.03 0.03 0.26 0.74 1.48 2.50 3.81 5.43 7.36 9.61 12.20 15.11 18.37 21.97 25.91 30.19 34.81 39.77 45.04 50.64 56.54 62.75 69.23 75.98 83.01 90.22 97.70
The vibrational wavefunctions of OH are particularly subject to substantial mixing via centrifugal distortion due to the large inertial constant of OH [see Eqs. (4) and (5) opposite]. This, combined with the large permanent dipole moment in OH (N 1.7 D), results in substantial contributions to the transition moment from the permanent dipole moment and thus pronounced “interference” effects between the contributions involving the permanent dipole moment and derivatives of the dipole moment. In molecules with smaller permanent dipole moments and inertia1 constants these effects are greatly diminished. For example, Billingsley has shown that these effects are much less pronounced in the vibration-rotation spectrum of NO(X*II) for J of 0.5-30.5.” The effects of centrifugal distortion are much less pronounced for pure-rotation transitions than for vibration-rotation transitions since the former depend primarily on the permanent dipole moment for oscillator strength rather than any derivative of the dipole moment. Consequently, Einstein coefficients can be easily estimated if a constant permanent dipole moment is assumed. However, more accurate coefficients require accounting for the variance of dipole moment with internuclear distance. This is most important at large N and at high v where the molecular wavefunctions have substantial amplitude at large internuclear distance. We have found no Einstein coefficients for the pure-rotation transitions of OH in the literature. In this paper, we present pure-rotation Einstein coefficients for N’ ranging to a maximum of 42 (v = 0) through 9 (v = 9). This includes all levels with term energies up to 26,895 cm-‘, the exothermicity of the reaction of H with 0, to form OH (v, J). Coefficients are also presented for the vibration-rotation transitions through the fourth overtone.
Einstein coefficients for the OH(X%) radical Table 4. Vibration-rotation
N'
*1 10.24 13.76 16.07 17.97 19.65 21.17 22.56 23.82 24.94 25.92 26.76 27.45 27.99 28.38 28.60 28.72 28.68 28.48 28.13 27.68 27.06 26.36 25.56 24.65 23.67 22.61 21.46 20.29 19.07 17.80 16.53 15.25 13.97 12.70
221
Einstein A coefficients for the 3-2 transition.
Ql 11.03 4.47 2.32 1.38 0.89 0.61 0.44 0.32 0.25 0.19 0.15 0.12 0.09 0.08 0.06 0.05 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00
3.52 3.68 3.07 2.26 1.48 0.81 0.31 0.04 0.03 0.30 0.88 1.80 3.08 4.74 6.80 9.28 12.19 15.54 19.34 23.61 28.35 33.54 39.21 45.36 51.96 59.02 66.46 74.40 82.75 91.47 100.65 110.07 119.84
32
02
16.80 17.07 18.10 19.36 20.67 21.95 23.18 24.32 25.36 26.28 27.07 27.72 28.23 28.59 28.78 28.89 28.82 28.61 28.24 27.79 27.15 26.45 25.63 24.72 23.73 22.66 21.51 20.34 19.11 17.84 16.57 15.28 13.99 12.72
6.35 1.39 0.67 0.42 0.30 0.22 0.17 0.14 0.11 0.09 0.08 0.06 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00
4.17 3.94 3.16 2.30 1.49 0.81 0.32 0.04 0.03 0.30 0.88 1.80 3.07 4.73 6.79 9.27 12.18 15.53 19.33 23.60 28.33 33.53 39.20 45.34 51.95 59.00 66.44 74.38 82.74 91.45 LOO.63 110.06 119.83
Rather than employ an approach based on perturbation theory as outlined by Herman and Wallis, we have taken a more exact approach in accounting for the effects of centrifugal distortion. We follow methods similar to those described by Oba et al’* in their calculations of transition probabilities of HF and of Refs. 6, 7, and 8 in their earlier calculations for OH. Numerical wavefunctions for the initial and final states were determined by solving the radial Schriidinger equation using the Numerov-Cooley method.13 A computer code using the algorithm described by Cashion14 and obtained from the Quantum Chemistry Program Exchange (Indiana University) was used for this purpose. In order to account for the strong distortion interaction in OH, a rotating molecule potential for both states involved in a given transition was computed. These potentials were computed by adding for the kinetic energy of rotation” to the potential for the non-rotating molecule to account for centrifugal distortion due to rotation, viz.
vR= vRKR +
&xation
.
The addition of the kinetic energy of rotation in this expression for the potential energy follows since the magnitudes of potential and kinetic energy are equivalent at the classical turning points of the motion that define the potential surface. The RKR potential given by Nelson et ali6 was used for the non-rotating-molecule potential. We have assumed the following expression appropriate in the Hund’s case (b) limit for the kinetic energy of rotation (see Ref. 15,p. 233):
(5) where B is the inertial constant, N the quantum number describing total angular momentum excluding electron spin, and n the projection of electron orbital angular momentum on the internuclear axis. This simplifying approximation becomes increasingly accurate in the limit of large J. QSRT
49/3-B
228
K. W. HOLTZCLAW et al Table 5. Vibration-rotation
Individual
Einstein
A coefficients
for the 4-3 transition
N’
*1
Ql
Rl
*2
Q2
%
: 3 4 5 6
10.70 7.70 12.85 14.71 16.40 17.96
3.07 7.57 1.59 0.94 0.61 0.41
2.07 0.00 1.91 1.33 0.73 0.27
13.31 12.68 14.51 15.87 17.27 18.65
0.95 4.35 0.45 0.28 0.20 0.15
2.45 0.00 2.04 1.37 0.74 0.27
ii 9 10 11 12 13 14 15 16 17 18
20.73 19.41 21.92 22.98 23.88 24.64 25.24 25.70 25.95 26.10 26.09 25.88
0.21 0.29 0.16 0.12 0.09 0.07 0.06 0.04 0.03 0.03 0.02 0.02
0.05 0.02 0.39 1.08 2.15 3.63 5.55 7.94 10.81 14.19 18.09 22.53
21.19 19.96 22.30 23.31 24.17 24.89 25.46 25.89 26.13 26.26 26.23 26.01
0.11 0.09 0.07 0.06 0.05 0.04 0.03 0.02 0.02 0.02 0.01 0.01
0.05 0.02 0.39 1.08 2.15 3.63 5.55 7.93 10.80 14.18 18.08 22.51
2': ff
25.09 25.57 23.78 24.49
0.01 0.00 0.01
27.52 33.08 45.88 39.20
25.20 25.69 23.86 24.58
0.01 0.00
27.51 33.07 45.87 39.19
23
22.94
0.00
53.15
23.01
0.00
53.14
it 26
21.02 22.01 19.93
0.00 0.00
69.39 60.97 78.23
22.08 21.08 19.99
0.00 0.00
60.95 69.37 78.21
:z
17.55 18.76
0.00
97.76 87.69
18.80 17.60
0.00
97.74 87.68
:z 31
15.07 16.32 13.77
0.00 0.00
119.02 108.08 130.37
16.36 15.10 13.80
0.00 0.00
119.01 108.06 130.36
vibration-rotation
transition
probabilities
were calculated
using the relation
where v is the transition frequency in cm-‘, R the transition moment integral, and S,.,.. the rotational linestrength factor. Since neither a Hund’s case (a) or (b) description is appropriate for most rotational levels in OH, linestrength formulae appropriate for angular momentum coupling intermediate between Hund’s cases (a) and (b) were employed. A subroutine” employing the expressions derived by Hill and Van Vleck18 was used. Transition frequencies were calculated using the term values of Coxon.” The transition moment integral was evaluated using a Simpson’s rule integration routine. The EDMF of Nelson et al6 was used in these integrals as given by U(T) = 1.6502 + 0.538(r - r,) - 0.796(r - r,)’ - 0.739(r - T~)~,
(7)
where the dipole moment is in Debye, r is the internuclear distance, and re the equilibrium internuclear distance for the non-rotating molecule (0.96966 A).2o The data giving rise to the EDMF of Nelson span the range u = G-9 and J < 9.5. Consequently the authors quote the range of internuclear distance over which Eq. (7) is valid as 0.7-I .76 A” where these limits are the classical turning points of the u = 9 vibration. Accurate evaluation of transition moment integrals involving states of sufficiently high v and N can require extrapolation of the EDMF outside the range of validity. This raises the possibility of significant errors in the EDMF and the resulting Einstein coefficients. We have attempted to determine whether all the transition probabilities reported here can be accurately calculated with the EDMF of Nelson et al. This was done by first integrating the transition moment integral over the range from 0.7 to 1.76 A then calculating the Einstein coefficients. The differences between these coefficients and those calculated with larger limits of integration give an indication of the importance of the range excluded to the
229
Einstein coefficients for the OH(X*fI) radical Table 6. Vibration-rotation
Einstein A coefficients for the 2-O transition.
N’
Pl
Ql
Rl
P2
Q2
%
i 3 4 5 6 ii
4.38 5.37 5.78 6.04 6.22 6.36 6.57 6.48
2.39 5.88 1.25 0.75 0.49 0.34 0.19 0.25
0.00 2.56 3.28 3.50 3.52 3.46 3.19 3.34
7.09 6.59 6.46 6.46 6.50 6.56 6.69 6.63
0.76 3.40 0.37 0.24 0.17 0.13 0.09 0.10
0.00 3.05 3.52 3.62 3.59 3.49 3.20 3.36
9 t:
6.66 6.79 6.72
0.15 0.10 0.12
3.02 2.65 2.84
6.75 6.85 6.80
0.07 0.05 0.06
3.02 2.65 2.84
ts 14 ix
6.86 6.83 6.88 6.90
0.07 0.08 0.06 0.04 0.05
2.27 2.46 2.07 1.70 1.88
6.88 6.91 6.92 6.93
0.04 0.03 0.03
2.46 2.27 2.07 1.70 1.88
17
6.89
0.04
1.51
6.91
0.02
1.51
t: 20 21 22 23
6.87 6.84 6.80 6.75 6.70 6.62
0.03 0.03 0.02 0.02 0.02
1.34 1.17 1.01 0.85 0.71 0.58
6.89 6.86 6.82 6.76 6.71 6.64
0.02 0.02 0.02 0.01 0.01
1.34 1.17 1.01 0.85 0.71 0.58
if
6.47 6.55
0.02
0.35 0.46
6.48 6.57
0.01
0.35 0.46
ff i8
6.29 6.39 6.08 6.19
0.01 0.01
0.26 0.18 0.06 0.11
6.30 6.40 6.09 6.20
0.01 0.01
0.18 0.26 0.06 0.11
30 3321
5.97 5.73 5.85
0.01 0.01
0.02 0.00
5.97 5.74 5.86
0.01 0.01
0.02 0.00
33 34
5.60 5.47
0.01 0.01
0.02 0.05
5.61 5.48
0.01 0.01
0.02 0.05
3: 37
5.20 5.33 5.05
0.01 0.01
0.10 0.17 0.27
5.20 5.34 5.06
0.01 0.00 0.00
0.10 0.17 0.27
entire transition moment integral and thus to the Einstein coefficient. It was found that integrating over the expanded range 0.6-l .9 8, gave coefficients which differed approx. 1% or less from those produced by integrating over the range OS-l.9 or 0X5-2.13 A for all coefficients reported here. Consequently, this modest extrapolation covers the range over which the wavefunctions involved in these calculations have significant amplitude. Errors in the coefficients due to inaccuracies in the EDMF between 0.6 to 0.7 and 1.76 to 1.9 8, are difficult to assess but are probably small for most transitions. A maximum for such an error can be estimated by comparing coefficients calculated over the range 0.6-1.76 or 0.7-1.9 A with those reported here. Except for those transitions involving the highest N, the discrepancies are typically less than l-2%. RESULTS
The Einstein coefficients for the pure-rotation transitions were calculated for u’ = O-9 and the vibration-rotation coefficients were calculated for u’ = 1-9, Au = 1-5. Due to the voluminous nature of the data, only the pure-rotation coefficients for u = O-2 and the vibration-rotation coefficients for u’ = 14, Au = l-2 are included as Tables l-8 in this paper. A complete set of coefficients may be obtained from the authors.? Einstein coefficients were calculated with the tA listing of all Einstein coefficients is available from the authors at Physical Sciences Inc., 20 New England Business Center, Andover, MA 01810, U.S.A.
230
K.
W. HOLTZCLAW et al
Table 7. Vibration-rotation
N'
I
f : 5 6 7 6
9 10 11 12 13 14 15 16 17 18 19 f! 22 23 24 25 26 27 28 29 i! 32 33 34
Pl 11.68 14.34 16.17 15.48
16.65 17.04 17.36 17.63 17.87 1 18.07 18.23 I 18.35 18.44 18.51 18.54 18.55 18.52 18.47 18.39 18.28 18.15 17.99 17.81 17.60 17.38 17.13 16.86 16.58 16.27 15.95 15.62 15.27 14.90 14.52
I
Einstein
Ql 15.72 6.40 3.36 2.02 1.33 0.93
0.68 0.51 0.40 0.32 0.26 0.22 0.18 0.15 0.13 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02
A coefficients
for the 3-l
(12
R1 0.00
6.80 8.71 9.28 9.33
9.14 8.80 8.38 7.93 7.43 6.92 6.39 5.87 5.35 4.84 4.33 3.83 3.36 2.91 2.48 2.07 1.70 1.36 1.05 0.77 0.54 0.34 0.19 0.08 0.02 0.00
0.04 0.13 0.27
transition.
18.96 17.63 17.31 17.32 17.43 17.60 17.78 17.96 18.13 18.29 18.41 18.50 18.57 18.63 18.64 18.63 18.60 18.54 18.45 18.34 18.20 18.04 17.85 17.64 17.41 17.16 16.89 16.61 16.30 15.98 15.64 15.29 14.92 14.54
9.06
0.00
2.01 0.98 0.62 0.45 0.34 0.27 0.22 0.19 0.16 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01
8.11 9.37 9.62
9.51 9.24
8.86 8.41 7.95 7.44 6.93 6.40 5.87 5.35 4.84 4.33 3.83 3.36 2.91 2.48 2.07 1.70 1.36 1.05 0.77 0.54 0.34 0.19 0.08 0.02 0.00
0.04 0.13 0.27
appropriate transition frequency for each of the four allowed vibration --rotation transitions associated with a particular P-, Q-, or R-branch band. The tabulated coefficients are the average of the two coefficients associated with a given spin sublevel. Figure 1 shows the pure-rotation coefficients for N’ through 42 for c = 0. The radiative rates exceed 500 set-’ for the highest levels considered and are quite large given the spectral region occupied by the corresponding transitions. This, of course, is due to the very large permanent dipole moment of OH. The coefficients are reported for the individual spin sublevels because, at low N. the frequencies of the F, and F, transitions are sufficiently different to result in substantial differences in the corresponding Einstein coefficients. These differences become negligible with increasing N. The prominent influence of centrifugal distortion on the radiative properties of OH is illustrated in Figs. 2 and 3 which show the Einstein coefficients for the 1-O and 4-2 vibration-rotation transitions respectively. The vibration-rotation coefficients tabulated here are best considered an extension of those presented by Nelson et al6 since. except for the case (b) kinetic energy approximation made here, the two sets were calculated in a similar manner using the same EDMF. Nelson et al, however, did not present results for the third and fourth overtones. Nonetheless, where there is overlap, comparison between the two calculations is useful. Figures 4 and 5 show the Einstein coefficients of the 1LO and 8-6 transitions respectively as calculated by Nelson et al6 and in this work. In general, the largest discrepancies occur for the lowest N. This is expected since the case (b) approximation made here is least accurate at low N. Nonetheless the discrepancies between P-branch and R-branch coefficients of Nelson et al and those of this work are ,<20% in all cases. Consequently, the case (b) approximation for kinetic energy used here does not
Einstein coefficients for the OH(X21T) radical Table 8. Vibration-rotation
231
Einstein A coefficients for the 4-2 transition,
N’
Pl
Ql
Rl
P2
Q2
%
;
25.33 20.61
27.76 11.33
11.94 0.00
31.21 33.55
15.96 3.52
14.28 0.00
3 4 5 6
27.37 28.61 29.50 30.21
5.96 3.59 2.36 1.64
15.25 16.23 16.29 15.90
30.67 30.68 30.91 31.23
1.71 1.08 0.77 0.59
16.45 16.85 16.62 16.08
i 9 10 11 12 13 14 15 16 17 18
30.81 31.32 31.74 32.10 32.41 32.64 32.82 32.94 33.00 33.02 32.96 32.87
0.91 1.20 0.71 0.57 0.46 0.38 0.32 0.27 0.23 0.20 0.18 0.15
15.28 14.53 13.68 12.79 11.86 10.90 9.96 9.02 8.09 7.19 6.31 5.47
31.58 31.91 32.22 32.50 32.74 32.92 33.06 33.15 33.18 33.18 33.10 33.00
0.47 0.39 0.32 0.27 0.23 0.20 0.18 0.16 0.14 0.12 0.11 0.10
15.39 14.60 13.72 12.81 11.87 10.91 9.96 9.02 8.09 7.19 6.31 5.47
21: 21 22 23 24 25 26
32.52 32.72 32.27 31.96 31.62 31.23 30.80 30.33
0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06
4.67 3.92 3.21 2.57 1.99 1.47 1.02 0.65
32.83 32.62 32.36 32.05 31.70 31.30 30.87 30.39
0.09 0.08 0.07 0.07 0.06 0.05 0.05 0.05
4.67 3.92 3.21 2.57 1.99 1.47 1.02 0.65
ii
29.27 29.82
0.05 0.06
0.15 0.36
29.33 29.88
0.04
0.15 0.36
ii 31
28.68 28.07 27.42
0.05 0.04 0.04
0.00 0.03 0.08
28.12 28.73 27.46
0.03 0.03
0.00 0.03 0.08
introduce large errors. For N greater than 10, convergence is essentially complete between the two sets of calculations and the case (b) approximation is thus quite adequate. This holds in comparison with the P- and R-branch coefficients of all vibrational transitions covered by Nelson et al. In contrast with the P- and R-branch data, there is essentially complete agreement at all N between the Q-branch coef?cients of Nelson et al and those presented here. In using the Hund’s case (b) limit expression for Eq. (5), we have assumed that the effects of incomplete electron spin uncoupling and of the incipient uncoupling of electron orbital angular momentum (L uncoupling) are negligible in these calculations. The comparison with the results of Nelson et al6 shows that, for the high N coefficients of greatest concern here, the assumption of complete uncoupling of electron spin is a good approximation. The effects due to neglect of L uncoupling, however, are more difficult to assess. The phenomenon is of concern since the result of L uncoupling is the addition of a small component of the A *Z+ state into the X*H wavefunction. The components can then contribute to the transition moment integral with the transition moments characteristic of the A-X system. Since the A-X transition dipole moment is large, any perturbations between the A and X states have the potential to significantly influence the overall radiative rate. We rely upon the results of Mies,’ who has used perturbation theory to estimate the importance of L uncoupling for the vibration-rotation Einstein coefficients. He concludes that, for P- and R-branch transitions with J < 20.5, the contribution to the transition moment would be roughly 5% or less of the total. Thus the maximum uncertainty in the corresponding Einstein coefficient would be on the order of 10%. For Q-branch transitions the effect is more pronounced for a given J since the contribution to the overall transition moment scales as .I*. However, these transitions become progressively weaker with increasing J, and thus they are less likely to be observed. The effects due to L uncoupling at higher N are more difficult to assess. Mies’ analysis suggests that, for the P- and R-branch transitions, the contribution to the overall transition moment from
K. W. HOLTZCLAW et al
232
‘O°C t
1
#
b 7
z
7
3
2
a
t
2
q
1.0
9 Ft
10
l
F2
7
tu I 0.1
1 0
-
10
20
30 N’
Fig. 1. The pure-rotation
Einstein
coefficients
20
40
60
N
Fig. 2. The vibration-rotation Einstein 1 0 transition.
50
40
0
coefficients
for the
for I‘ = 0
the A-X mixing scales roughly as J. Consequently, the contribution to the transition moment at N = 40 would be roughly twice that at N = 20 or less than about 10%. In turn this would indicate potential errors in the Einstein coefficients of roughly 20% or less. The errors at high N in the Q-branch coefficients would be much larger but again, these are very weak transitions and thu: the effect is still of little practical importance. Estimates of the importance of L-uncoupling for the pure-rotation Einstein coefficients may bc made from Mies’ analysis and a comparison of the vibration-rotation and pure-rotation transitior
r-d 0 PI
Fig. 3. The vibration-rotation
Einstein
coefficients
4-2 0 R
for the 4-2 transition
Einstein coefficients for the OH(XTI) radical
233
5-
4-
OR1
.Nelson&al.
3-
2-
l-
O0
,
Fig. 4. A comparison of the P- and R-branch Einstein coefficients of Ref. 6 with this work for the -0 transition.
moment integrals. The pure-rotation integrals are typically of the magnitude of the permanent integrals are always 6 0.2 D. dipole moment, i.e. - 1.7 D. The corresponding vibration-rotation As a result, the relative contribution to the overall transition moment due to the A 4’ component is much smaller for pure-rotation transitions. Consequently, we expect the effects of L-uncoupling on the pure-rotation coefficients to be negligible. The relative rates of the vibration-rotation and pure-rotation coefficients allow insight into the relaxation dynamics of the corresponding levels in the upper atmosphere. Although the radiative rates for the vibration-rotation transitions are substantial for many rotational levels, they are slower than the corresponding pure-rotation transitions for the higher rotational states. Consequently, Au = 0 fluorescence will be the dominant radiative decay process for molecules possessing greater than roughly 25 rotational quanta. Since these levels are less susceptible to collisionallyinduced rotational energy transfer due to large energy gaps between levels,’ Au = 0 decay can, at
rool---
0 Rt , Nelson et al.
Fig. 5. A comparison of the P- and R-branch Einstein coefficients of Ref. 6 with this work for the 8-6 transition.
234
sufficiently high altitudes, excited OH.
K. W. HOLTZCLAW et al
become
the dominant
overall
decay
channel
for highly
rotationally
SUMMARY The effects of the centrifugal distortion have been accounted for in the calculation of Einstein coefficients for transitions involving high lying rotational and vibrational levels of OH. Einstein A coefficients have been calculated for the Au = 0 through Av = 5 transitions of OH(X*H) for all rotation-vibration levels energetically accessible in the H + 0, +OH(v, J) + 0, reaction. The vibration-rotation coefficients for the Au = l-3 sequences can be considered an extension to higher N of those calculated by Nelson et al.6 The effects of incomplete spin uncoupling were ignored in accounting for the effects of centrifugal distortion in these calculations. However, the resulting coefficients are in reasonable agreement with those of Nelson et al at all n and are essentially identical (discrepancies 6 1%) for N > 13. The pure-rotation coefficients are the first to be reported in the literature. These coefficients will permit a better understanding of the kinetics and chemistry involving highly excited OH(X’lT) probed via infrared chemiluminescence. The pure-rotation radiative relaxation rates are extremely large. For the highest levels considered, these rates exceed 500 set’. They are significantly greater than the corresponding vibration-rotation radiative rates. Consequently, for vibrationally excited molecules possessing a large amount of rotational excitation, AC = 0 fluorescence will be the predominant radiative decay of the OH airglow layer, the rotational radiative rate is channel. At the -85 km altitudes comparable to the collisional relaxation rate if the collisional process proceeds at 1% of gas kinetic. For the large energy quanta contained in these high rotational states, relaxation cross-sections of this magnitude are reasonable. Previous observations of nonequilibrium rotational distributions for OH in the airglow layer’.“’ reflect the successful competition of radiative decay with collisional thermal equilibration. Future laboratory measurements of the collisional relaxation rate coefficients are needed. These measurements will compliment the radiative relaxation rates presented here and facilitate the modelling efforts required to predict the altitude dependence of the spectrum of chemiluminescent OH atmospheric emissions. Acknowledgements-This work was supported by Physical Contract F19628-88-C-0069 with the Geophysics Directorate
Sciences Inc. internal funding and in part by funding under of the Phillips Laboratory in support of the Air Force Office
of Scientific Research under Task 231064 and the Defense Nuclear Agency under Project SA. Task SA. The authors gratefully acknowledge the generous assistance of J. A. Coxon who provided OH(X%) term energies prior to publication. The authors would also like to acknowledge Physical Sciences Inc.
useful discussion
with L. G. Piper, W. T. Rawlins.
and W. J. Marinelli
of
REFERENCES I. D. R. Smith, W. A. M. Blumberg, R. M. Nadile, S. J. Lipson, E. R. Huppi. N. Wheeler, and J. A. Dodd, Geophys. Res. Lett. 19, 593 (1992). 2. A. J. Fauvel, Astrophys. J. 252, 330 (1984). 3. K. L. Carleton. W. J. Marinelli, K. W. Holtzclaw, and B. D. Green, submitted to J. Phys. Chem. (1993). 4. J. E. Butler, R. G. MacDonald, D. J. Donaldson, and J. J. Sloan, Chem. Phys. Lett. 95, 183 (1983). 5. W. J. Marinelli, K. L. Carleton, K. W. Holtzclaw, B. D. Green, S. J. Lipson, and W. A. M. Blumberg, in preparation (1993). 6. D. D. Nelson, A. Schiffman. D. J. Nesbitt, J. J. Orlando, and J. B. Burkholder. J. Chem. Phys. 93, 7003 (1990). 7. F. H. Mies, J. Molec. Spectrosc. 53, 150 (1974). 8. S. R. Langhoff, H. J. Werner, and P. Rosmus, J. Molec. Spectrosc. 118, 507 (1986). 9. R. Herman and R. F. Wallis, J. Chem. Phys. 23, 637 (1955). 10. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, Dover, New York, NY (1980). Appendix III. 11. F. P. Billingsley, J. Molec. Spectrosc. 61, 53 (1976). 12. D. Oba, B. S. Agrawalla, and D. W. Setser, JQSRT 34, 283 (1985). 13. J. W. Cooley, Muth. Computat. 15, 363 (1961). 14. J. K. Cashion, J. Chem. Phys. 39, 1872 (1963); ibid 41, 3988 (1963). 15. G. Herzberg, Spectra of Diatomic Molecules, pp. 425426, Van Nostrand-Reinhold. New York, NY (1950). 16. D. D. Nelson. A. Schiffman, D. J. Nesbitt, and D. J. Yaron, J. Chem. Phys. 90, 5443 (1989).
Einstein
coefficients
for the OH(X%)
radical
235
17. R. Engleman, P. E. Rouse, H. M. Peek, and V. D. Baiamonte, Scientific Report LA-4364, Appendix B, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM (1970). 18. E. Hill and J. H. Van Vleck, Phys. Reo. 38, 250 (1928). 19. J. A. Coxon, private communication (1991). 20. K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, p. 508, Van Nostrand-Reinhold, New York, NY (1979). 21. W. Pendelton, P. Espy, D. Baker, A. Steed, M. Fetow, and K. Henriksen, J. Geophys. Res. 94,505 (1989).
J. Quonr. Spec~rosc. Radial. Transfer
0022~4073/93$6.00+ 0.00 Copyright 0 1993Pergamon Press Ltd
EINSTEIN COEFFICIENTS FOR EMISSION FROM HIGH ROTATIONAL STATES OF THE OH(X’l-I) RADICAL K. W. HOLTZCLAW,~
J. C. PERSON, and
B. D. GREEN
Physical Sciences Inc., 20 New England Business Center, Andover, MA 01810, U.S.A. (Received 20 May 1992)
Abstract-We report calculated Einstein coefficients for the pure-rotation and vibrationrotation transitions of 0H(X21T) with particular emphasis on those involving high-lying rotational levels. Coefficients are reported for rotational levels ranging from N = 42 (u = 0) to N = 9 (u = 9). The pure-rotation coefficients are the first to be reported in the literature. The effects of the centrifugal distortion are explicitly accounted for in these calculations and individual spin-sublevel-specific coefficients are reported. Centrifugal distortion in this work was accounted for assuming an expression for rotational kinetic energy appropriate in the Hund’s case (b) limit. Comparison with otherwise similar calculations by Nelson et al, which incorporate the effects of incomplete spin uncoupling on the rotational kinetic energy, shows that this approximation results in insignificant errors for N’ > 10. The implications of these coefficients for radiative decay of OH(X211) in the upper atmosphere are discussed.
INTRODUCTION Recent observations of long-wavelength radiances from the upper atmosphere and from the laboratory display emissions arising from high lying rotational states of OH(X*II). These observations include both vibration-rotation (Av > 0) as well as pure-rotation (Av = 0) transitions. For example, fluorescence from pure-rotation transitions has been observed in spectra from the CIRRZS IA mission from rotational levels with N up to 33 in the ground vibrational state, u = 0.’ Pure-rotation transitions from high N states in absorption have been observed in solar data.* In recent experiments using the LABCEDE facility at the Phillips Laboratory (Hanscom AFB, MA), the reaction of O’D with H, at low total pressures (0.02 torr) was used to produce OH in highly rotationally excited states. Pure-rotation transitions from levels as high as N = 29 (v = 0) were observed using a helium-cooled circular-variable-filter spectrometer.3 These data have been analyzed to extract relaxation rate coefficients of the v = 0, N = 29 level for several collision partners. Vibration-rotation spectra from the O’D + H, reaction have been observed via infrared chemiluminescence4,5 again including experiments performed in the LABCEDE facility. The latter have allowed observation of vibrational emission from levels as high as N = 26 (v = 1) using a Michelson interferometer and are presently being analyzed for collisional relaxation information. The radiative emission rates required to quantify the excited state densities giving rise to these emissions are not available in the literature. BACKGROUND
AND
COMPUTATIONAL
APPROACH
There are a number of compilations of vibration-rotation Einstein coefficients for OH now in the literature. Recent examples include those from Nelson et aL6 and Mies,’ and Langhoff et a1.8 These compilations were calculated in a similar manner and account for the effects of centrifugal distortion and the effects of incomplete electron spin uncoupling. Since these phenomena influence the relative intensities of the various transitions originating from a common upper rotational state, the corresponding transition moments are not easily separated into distinct vibrational and rotational factors. Consequently, the data for OH are voluminous and must be reported as coefficients for individual v’, J’, u”, J” transitions. The data of Refs. 6, 7 and 8 differ principally with respect to the electric dipole moment function (EDMF) employed. Only Nelson et al employ tTo whom all correspondence
should be addressed. 223
K. W. HOLTZCLAW et al ‘Table
I. Einstein A coefficients
for pure-rotation
transitions
in OH(X),
1‘ = C&2
an EDMF derived from experimental data. The dipole moment functions used by Mies and Langhoff et al were calculated ah initio. Nonetheless, a significant limitation of the compilations reported to date is that they extend to rotational states of no greater than J’ = 13.5-15.5. The vibration-rotation interaction, or centrifugal distortion, profoundly influences the relative intensities of individual vibration-rotation transitions in OH. The effect of centrifugal distortion on the radiative properties of diatomics was first discussed by Herman and Wallis’ who employed an approach based on perturbation theory. The interaction can be viewed as producing a mixing of the vibrational wavefunctions which diagonalise the Hamiltonian in the absence of centrifugal distortion: IY’ > =a’/ 0 > + b’J 1 > + C’I 2 > +. . . , IY”> It is common practice to expand internuclear distance as follows:
=a”lO>
+b”ll>
the dipole
moment
+c”l2>
+ ...,
as a Taylor
p(l)=&)+; g (r--l.)+; 2
() ‘C
(1
(r-r,y+
rr
series about
.’
the equilibrium
Einstein coefficients for the OH(XZfI) radical Table 2. Vibration-rotation
225
Einstein A coefficients for the 14 transition.
N’
Pl
Ql
R1
P2
f 3 4 :
10.13 7.80 11.48 12.51 14.16 13.38
9.16 3.70 1.92 1.15 0.51 0.74
0.00 3.36 3.86 3.63 2.60 3.16
12.50 12.73 12.88 13.43 14.65 14.04
1.18 5.30 0.58 0.37 0.20 0.26
3.96 0.00 4.12 3.74 2.62 3.20
7
14.84
0.37
2.03
15.22
0.16
2.04
t 10 11 12 11:
15.98 15.45 16.45 16.84 17.15 17.55 17.39
0.21 0.28 0.17 0.13 0.11 0.07 0.09
1.02 1.50 0.62 0.30 0.10 0.04 0.00
15.75 16.24 16.66 17.01 17.30 17.67 17.52
0.10 0.13 0.09 0.07 0.06 0.04 0.05
1.02 1.50 0.62 0.30 0.10 0.04 0.00
i:
17.65 17.67
0.06 0.05
0.52 0.21
17.76 17.75
0.04 0.03
0.52 0.21
11: 19 20 f?l
17.61 17.49 17.30 17.05 16.37 16.74
0.04 0.03 0.03 0.02
0.99 1.62 2.41 3.38 4.52 5.84
17.56 17.69 17.37 17.10 16.41 16.79
0.02 0.03 0.02 0.02 0.01 0.02
0.99 1.62 2.41 3.38 5.84 4.52
23 ::
15.94 14.96 15.48
0.02 0.01
7.35 10.91 9.04
15.99 15.52 15.00
0.01 0.01
7.34 10.91 9.03
26
14.41
0.01
12.97
14.44
0.01
12.97
91
13.21 13.83
0.01
17.65 15.21
13.24 13.86
0.01
15.21 17.64
3: 31 32
11.92 12.57 11.25 10.57
0.00 0.01 0.00 0.00
23.04 20.26 26.00 29.13
11.94 12.60 11.27 10.58
0.00 0.00 0.00
20.25 23.03 26.00 29.12
:2 35 36
9.88 9.20 8.52 7.84
0.00 0.00 0.00
32.43 35.88 39.48 43.23
9.89 9.21 8.53 7.85
0.00 0.00 0.00
32.42 35.87 39.47 43.23
iI 39
7.17 6.54 5.90
0.00 0.00
47.11 51.12 55.25
7.18 6.54 5.91
0.00 0.00
47.10 51.11 55.24
Q2
%
Calculation of the corresponding transition moment integral between two such mixed wavefunctions with the expanded dipole moment shows that the permanent dipole moment can now contribute to the transition moment where it would not in the absence of centrifugal distortion. If a harmonic oscillator basis set is assumed for the mixed wavefunctions, the surviving terms in the transition moment integral arelo (Y’l&)l
Y”) = a’a”~(r,)(olo>
+n’b”;g (
+ b’b”&,)(lll) (Olr-r,ll)+b’n”; ) 1,
+ *** $ 0
(llr-r,lO)+... ‘,
1 azp (Ol(r-r,)z12)+c’uM6 Y@ (2I(r--r,)*lo)+... (3) ( > ‘e ( > ‘C and it can be seen that these terms contain contributions from the permanent dipole moment. It should be noted that, since OH is decidedly anharmonic, the permanent dipole moment also contributes to some extent even in the absence of rotation. This contribution is independent of rotational level and is not responsible for the anomalous intensity variations observed in OH. However, as shown by Herman and Wallis, the permanent dipole moment contribution due to centrifugal distortion varies both in magnitude and sign, depending upon the rotational states involved in the transition as well as the change in rotational quantum number. fn’c”;
$
K. W. HOLTZCLAWetal
226
Table3.Vibration-rotation Einstein A coefficients forthe2-Itransition N’
1 2 3 4 5 6 7 : :: 12 13 14 15 16 17 18 19 20 21 22 :: 25 :: 28 29 30 32' 33 z 36 37
PI 10.73 14.12 16.21 17.85 19.27 20.54 21.68 22.70 23.60 24.39 25.06 25.60 26.02 26.32 26.50 26.55 26.48 26.32 26.02 25.59 25.09 24.52 23.84 23.07 22.21 21.32 20.36 19.39 18.34 17.29 16.21 15.10 14.02 12.92 11.83 10.76 9.73
Ql 12.17 4.93 2.57 1.53 0.99 0.68 0.49 0.36 0.28 0.22 0.17 0.14 0.11 0.09 0.08 0.06 0.05 0.04 0.04 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Rl 0.00 4.23 4.70 4.23 3.48 2.66 1.88 1.19 0.63 0.23 0.03 0.03 0.26 0.74 1.48 2.50 3.82 5.43 7.36 9.62 12.20 15.12 18.38 21.97 25.92 30.20 34.82 39.78 45.05 50.65 56.56 62.76 69.24 75.99 83.02 90.23 97.71
P2 17.56 17.47 18.22 19.19 20.24 21.27 22.25 23.17 23.99 24.71 25.34 25.84 26.23 26.50 26.66 26.69 26.61 26.43 26.12 25.68 25.17 24.59 23.91 23.13 22.27 21.37 20.41 19.43 18.37 17.32 16.24 15.13 14.04 12.94 11.85 10.78 9.74
Q2 7.02 1.56 0.76 0.48 0.34 0.26 0.20 0.16 0.13 0.11 0.09 0.08 0.06 0.05 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
t2 0.00 5.00 5.01 4.36 3.53 2.68 1.88 1.19 0.63 0.23 0.03 0.03 0.26 0.74 1.48 2.50 3.81 5.43 7.36 9.61 12.20 15.11 18.37 21.97 25.91 30.19 34.81 39.77 45.04 50.64 56.54 62.75 69.23 75.98 83.01 90.22 97.70
The vibrational wavefunctions of OH are particularly subject to substantial mixing via centrifugal distortion due to the large inertial constant of OH [see Eqs. (4) and (5) opposite]. This, combined with the large permanent dipole moment in OH (N 1.7 D), results in substantial contributions to the transition moment from the permanent dipole moment and thus pronounced “interference” effects between the contributions involving the permanent dipole moment and derivatives of the dipole moment. In molecules with smaller permanent dipole moments and inertia1 constants these effects are greatly diminished. For example, Billingsley has shown that these effects are much less pronounced in the vibration-rotation spectrum of NO(X*II) for J of 0.5-30.5.” The effects of centrifugal distortion are much less pronounced for pure-rotation transitions than for vibration-rotation transitions since the former depend primarily on the permanent dipole moment for oscillator strength rather than any derivative of the dipole moment. Consequently, Einstein coefficients can be easily estimated if a constant permanent dipole moment is assumed. However, more accurate coefficients require accounting for the variance of dipole moment with internuclear distance. This is most important at large N and at high v where the molecular wavefunctions have substantial amplitude at large internuclear distance. We have found no Einstein coefficients for the pure-rotation transitions of OH in the literature. In this paper, we present pure-rotation Einstein coefficients for N’ ranging to a maximum of 42 (v = 0) through 9 (v = 9). This includes all levels with term energies up to 26,895 cm-‘, the exothermicity of the reaction of H with 0, to form OH (v, J). Coefficients are also presented for the vibration-rotation transitions through the fourth overtone.
Einstein coefficients for the OH(X%) radical Table 4. Vibration-rotation
N'
*1 10.24 13.76 16.07 17.97 19.65 21.17 22.56 23.82 24.94 25.92 26.76 27.45 27.99 28.38 28.60 28.72 28.68 28.48 28.13 27.68 27.06 26.36 25.56 24.65 23.67 22.61 21.46 20.29 19.07 17.80 16.53 15.25 13.97 12.70
221
Einstein A coefficients for the 3-2 transition.
Ql 11.03 4.47 2.32 1.38 0.89 0.61 0.44 0.32 0.25 0.19 0.15 0.12 0.09 0.08 0.06 0.05 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00
3.52 3.68 3.07 2.26 1.48 0.81 0.31 0.04 0.03 0.30 0.88 1.80 3.08 4.74 6.80 9.28 12.19 15.54 19.34 23.61 28.35 33.54 39.21 45.36 51.96 59.02 66.46 74.40 82.75 91.47 100.65 110.07 119.84
32
02
16.80 17.07 18.10 19.36 20.67 21.95 23.18 24.32 25.36 26.28 27.07 27.72 28.23 28.59 28.78 28.89 28.82 28.61 28.24 27.79 27.15 26.45 25.63 24.72 23.73 22.66 21.51 20.34 19.11 17.84 16.57 15.28 13.99 12.72
6.35 1.39 0.67 0.42 0.30 0.22 0.17 0.14 0.11 0.09 0.08 0.06 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00
4.17 3.94 3.16 2.30 1.49 0.81 0.32 0.04 0.03 0.30 0.88 1.80 3.07 4.73 6.79 9.27 12.18 15.53 19.33 23.60 28.33 33.53 39.20 45.34 51.95 59.00 66.44 74.38 82.74 91.45 LOO.63 110.06 119.83
Rather than employ an approach based on perturbation theory as outlined by Herman and Wallis, we have taken a more exact approach in accounting for the effects of centrifugal distortion. We follow methods similar to those described by Oba et al’* in their calculations of transition probabilities of HF and of Refs. 6, 7, and 8 in their earlier calculations for OH. Numerical wavefunctions for the initial and final states were determined by solving the radial Schriidinger equation using the Numerov-Cooley method.13 A computer code using the algorithm described by Cashion14 and obtained from the Quantum Chemistry Program Exchange (Indiana University) was used for this purpose. In order to account for the strong distortion interaction in OH, a rotating molecule potential for both states involved in a given transition was computed. These potentials were computed by adding for the kinetic energy of rotation” to the potential for the non-rotating molecule to account for centrifugal distortion due to rotation, viz.
vR= vRKR +
&xation
.
The addition of the kinetic energy of rotation in this expression for the potential energy follows since the magnitudes of potential and kinetic energy are equivalent at the classical turning points of the motion that define the potential surface. The RKR potential given by Nelson et ali6 was used for the non-rotating-molecule potential. We have assumed the following expression appropriate in the Hund’s case (b) limit for the kinetic energy of rotation (see Ref. 15,p. 233):
(5) where B is the inertial constant, N the quantum number describing total angular momentum excluding electron spin, and n the projection of electron orbital angular momentum on the internuclear axis. This simplifying approximation becomes increasingly accurate in the limit of large J. QSRT
49/3-B
228
K. W. HOLTZCLAW et al Table 5. Vibration-rotation
Individual
Einstein
A coefficients
for the 4-3 transition
N’
*1
Ql
Rl
*2
Q2
%
: 3 4 5 6
10.70 7.70 12.85 14.71 16.40 17.96
3.07 7.57 1.59 0.94 0.61 0.41
2.07 0.00 1.91 1.33 0.73 0.27
13.31 12.68 14.51 15.87 17.27 18.65
0.95 4.35 0.45 0.28 0.20 0.15
2.45 0.00 2.04 1.37 0.74 0.27
ii 9 10 11 12 13 14 15 16 17 18
20.73 19.41 21.92 22.98 23.88 24.64 25.24 25.70 25.95 26.10 26.09 25.88
0.21 0.29 0.16 0.12 0.09 0.07 0.06 0.04 0.03 0.03 0.02 0.02
0.05 0.02 0.39 1.08 2.15 3.63 5.55 7.94 10.81 14.19 18.09 22.53
21.19 19.96 22.30 23.31 24.17 24.89 25.46 25.89 26.13 26.26 26.23 26.01
0.11 0.09 0.07 0.06 0.05 0.04 0.03 0.02 0.02 0.02 0.01 0.01
0.05 0.02 0.39 1.08 2.15 3.63 5.55 7.93 10.80 14.18 18.08 22.51
2': ff
25.09 25.57 23.78 24.49
0.01 0.00 0.01
27.52 33.08 45.88 39.20
25.20 25.69 23.86 24.58
0.01 0.00
27.51 33.07 45.87 39.19
23
22.94
0.00
53.15
23.01
0.00
53.14
it 26
21.02 22.01 19.93
0.00 0.00
69.39 60.97 78.23
22.08 21.08 19.99
0.00 0.00
60.95 69.37 78.21
:z
17.55 18.76
0.00
97.76 87.69
18.80 17.60
0.00
97.74 87.68
:z 31
15.07 16.32 13.77
0.00 0.00
119.02 108.08 130.37
16.36 15.10 13.80
0.00 0.00
119.01 108.06 130.36
vibration-rotation
transition
probabilities
were calculated
using the relation
where v is the transition frequency in cm-‘, R the transition moment integral, and S,.,.. the rotational linestrength factor. Since neither a Hund’s case (a) or (b) description is appropriate for most rotational levels in OH, linestrength formulae appropriate for angular momentum coupling intermediate between Hund’s cases (a) and (b) were employed. A subroutine” employing the expressions derived by Hill and Van Vleck18 was used. Transition frequencies were calculated using the term values of Coxon.” The transition moment integral was evaluated using a Simpson’s rule integration routine. The EDMF of Nelson et al6 was used in these integrals as given by U(T) = 1.6502 + 0.538(r - r,) - 0.796(r - r,)’ - 0.739(r - T~)~,
(7)
where the dipole moment is in Debye, r is the internuclear distance, and re the equilibrium internuclear distance for the non-rotating molecule (0.96966 A).2o The data giving rise to the EDMF of Nelson span the range u = G-9 and J < 9.5. Consequently the authors quote the range of internuclear distance over which Eq. (7) is valid as 0.7-I .76 A” where these limits are the classical turning points of the u = 9 vibration. Accurate evaluation of transition moment integrals involving states of sufficiently high v and N can require extrapolation of the EDMF outside the range of validity. This raises the possibility of significant errors in the EDMF and the resulting Einstein coefficients. We have attempted to determine whether all the transition probabilities reported here can be accurately calculated with the EDMF of Nelson et al. This was done by first integrating the transition moment integral over the range from 0.7 to 1.76 A then calculating the Einstein coefficients. The differences between these coefficients and those calculated with larger limits of integration give an indication of the importance of the range excluded to the
229
Einstein coefficients for the OH(X*fI) radical Table 6. Vibration-rotation
Einstein A coefficients for the 2-O transition.
N’
Pl
Ql
Rl
P2
Q2
%
i 3 4 5 6 ii
4.38 5.37 5.78 6.04 6.22 6.36 6.57 6.48
2.39 5.88 1.25 0.75 0.49 0.34 0.19 0.25
0.00 2.56 3.28 3.50 3.52 3.46 3.19 3.34
7.09 6.59 6.46 6.46 6.50 6.56 6.69 6.63
0.76 3.40 0.37 0.24 0.17 0.13 0.09 0.10
0.00 3.05 3.52 3.62 3.59 3.49 3.20 3.36
9 t:
6.66 6.79 6.72
0.15 0.10 0.12
3.02 2.65 2.84
6.75 6.85 6.80
0.07 0.05 0.06
3.02 2.65 2.84
ts 14 ix
6.86 6.83 6.88 6.90
0.07 0.08 0.06 0.04 0.05
2.27 2.46 2.07 1.70 1.88
6.88 6.91 6.92 6.93
0.04 0.03 0.03
2.46 2.27 2.07 1.70 1.88
17
6.89
0.04
1.51
6.91
0.02
1.51
t: 20 21 22 23
6.87 6.84 6.80 6.75 6.70 6.62
0.03 0.03 0.02 0.02 0.02
1.34 1.17 1.01 0.85 0.71 0.58
6.89 6.86 6.82 6.76 6.71 6.64
0.02 0.02 0.02 0.01 0.01
1.34 1.17 1.01 0.85 0.71 0.58
if
6.47 6.55
0.02
0.35 0.46
6.48 6.57
0.01
0.35 0.46
ff i8
6.29 6.39 6.08 6.19
0.01 0.01
0.26 0.18 0.06 0.11
6.30 6.40 6.09 6.20
0.01 0.01
0.18 0.26 0.06 0.11
30 3321
5.97 5.73 5.85
0.01 0.01
0.02 0.00
5.97 5.74 5.86
0.01 0.01
0.02 0.00
33 34
5.60 5.47
0.01 0.01
0.02 0.05
5.61 5.48
0.01 0.01
0.02 0.05
3: 37
5.20 5.33 5.05
0.01 0.01
0.10 0.17 0.27
5.20 5.34 5.06
0.01 0.00 0.00
0.10 0.17 0.27
entire transition moment integral and thus to the Einstein coefficient. It was found that integrating over the expanded range 0.6-l .9 8, gave coefficients which differed approx. 1% or less from those produced by integrating over the range OS-l.9 or 0X5-2.13 A for all coefficients reported here. Consequently, this modest extrapolation covers the range over which the wavefunctions involved in these calculations have significant amplitude. Errors in the coefficients due to inaccuracies in the EDMF between 0.6 to 0.7 and 1.76 to 1.9 8, are difficult to assess but are probably small for most transitions. A maximum for such an error can be estimated by comparing coefficients calculated over the range 0.6-1.76 or 0.7-1.9 A with those reported here. Except for those transitions involving the highest N, the discrepancies are typically less than l-2%. RESULTS
The Einstein coefficients for the pure-rotation transitions were calculated for u’ = O-9 and the vibration-rotation coefficients were calculated for u’ = 1-9, Au = 1-5. Due to the voluminous nature of the data, only the pure-rotation coefficients for u = O-2 and the vibration-rotation coefficients for u’ = 14, Au = l-2 are included as Tables l-8 in this paper. A complete set of coefficients may be obtained from the authors.? Einstein coefficients were calculated with the tA listing of all Einstein coefficients is available from the authors at Physical Sciences Inc., 20 New England Business Center, Andover, MA 01810, U.S.A.
230
K.
W. HOLTZCLAW et al
Table 7. Vibration-rotation
N'
I
f : 5 6 7 6
9 10 11 12 13 14 15 16 17 18 19 f! 22 23 24 25 26 27 28 29 i! 32 33 34
Pl 11.68 14.34 16.17 15.48
16.65 17.04 17.36 17.63 17.87 1 18.07 18.23 I 18.35 18.44 18.51 18.54 18.55 18.52 18.47 18.39 18.28 18.15 17.99 17.81 17.60 17.38 17.13 16.86 16.58 16.27 15.95 15.62 15.27 14.90 14.52
I
Einstein
Ql 15.72 6.40 3.36 2.02 1.33 0.93
0.68 0.51 0.40 0.32 0.26 0.22 0.18 0.15 0.13 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02
A coefficients
for the 3-l
(12
R1 0.00
6.80 8.71 9.28 9.33
9.14 8.80 8.38 7.93 7.43 6.92 6.39 5.87 5.35 4.84 4.33 3.83 3.36 2.91 2.48 2.07 1.70 1.36 1.05 0.77 0.54 0.34 0.19 0.08 0.02 0.00
0.04 0.13 0.27
transition.
18.96 17.63 17.31 17.32 17.43 17.60 17.78 17.96 18.13 18.29 18.41 18.50 18.57 18.63 18.64 18.63 18.60 18.54 18.45 18.34 18.20 18.04 17.85 17.64 17.41 17.16 16.89 16.61 16.30 15.98 15.64 15.29 14.92 14.54
9.06
0.00
2.01 0.98 0.62 0.45 0.34 0.27 0.22 0.19 0.16 0.14 0.12 0.10 0.09 0.08 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.01
8.11 9.37 9.62
9.51 9.24
8.86 8.41 7.95 7.44 6.93 6.40 5.87 5.35 4.84 4.33 3.83 3.36 2.91 2.48 2.07 1.70 1.36 1.05 0.77 0.54 0.34 0.19 0.08 0.02 0.00
0.04 0.13 0.27
appropriate transition frequency for each of the four allowed vibration --rotation transitions associated with a particular P-, Q-, or R-branch band. The tabulated coefficients are the average of the two coefficients associated with a given spin sublevel. Figure 1 shows the pure-rotation coefficients for N’ through 42 for c = 0. The radiative rates exceed 500 set-’ for the highest levels considered and are quite large given the spectral region occupied by the corresponding transitions. This, of course, is due to the very large permanent dipole moment of OH. The coefficients are reported for the individual spin sublevels because, at low N. the frequencies of the F, and F, transitions are sufficiently different to result in substantial differences in the corresponding Einstein coefficients. These differences become negligible with increasing N. The prominent influence of centrifugal distortion on the radiative properties of OH is illustrated in Figs. 2 and 3 which show the Einstein coefficients for the 1-O and 4-2 vibration-rotation transitions respectively. The vibration-rotation coefficients tabulated here are best considered an extension of those presented by Nelson et al6 since. except for the case (b) kinetic energy approximation made here, the two sets were calculated in a similar manner using the same EDMF. Nelson et al, however, did not present results for the third and fourth overtones. Nonetheless, where there is overlap, comparison between the two calculations is useful. Figures 4 and 5 show the Einstein coefficients of the 1LO and 8-6 transitions respectively as calculated by Nelson et al6 and in this work. In general, the largest discrepancies occur for the lowest N. This is expected since the case (b) approximation made here is least accurate at low N. Nonetheless the discrepancies between P-branch and R-branch coefficients of Nelson et al and those of this work are ,<20% in all cases. Consequently, the case (b) approximation for kinetic energy used here does not
Einstein coefficients for the OH(X21T) radical Table 8. Vibration-rotation
231
Einstein A coefficients for the 4-2 transition,
N’
Pl
Ql
Rl
P2
Q2
%
;
25.33 20.61
27.76 11.33
11.94 0.00
31.21 33.55
15.96 3.52
14.28 0.00
3 4 5 6
27.37 28.61 29.50 30.21
5.96 3.59 2.36 1.64
15.25 16.23 16.29 15.90
30.67 30.68 30.91 31.23
1.71 1.08 0.77 0.59
16.45 16.85 16.62 16.08
i 9 10 11 12 13 14 15 16 17 18
30.81 31.32 31.74 32.10 32.41 32.64 32.82 32.94 33.00 33.02 32.96 32.87
0.91 1.20 0.71 0.57 0.46 0.38 0.32 0.27 0.23 0.20 0.18 0.15
15.28 14.53 13.68 12.79 11.86 10.90 9.96 9.02 8.09 7.19 6.31 5.47
31.58 31.91 32.22 32.50 32.74 32.92 33.06 33.15 33.18 33.18 33.10 33.00
0.47 0.39 0.32 0.27 0.23 0.20 0.18 0.16 0.14 0.12 0.11 0.10
15.39 14.60 13.72 12.81 11.87 10.91 9.96 9.02 8.09 7.19 6.31 5.47
21: 21 22 23 24 25 26
32.52 32.72 32.27 31.96 31.62 31.23 30.80 30.33
0.14 0.12 0.11 0.10 0.09 0.08 0.07 0.06
4.67 3.92 3.21 2.57 1.99 1.47 1.02 0.65
32.83 32.62 32.36 32.05 31.70 31.30 30.87 30.39
0.09 0.08 0.07 0.07 0.06 0.05 0.05 0.05
4.67 3.92 3.21 2.57 1.99 1.47 1.02 0.65
ii
29.27 29.82
0.05 0.06
0.15 0.36
29.33 29.88
0.04
0.15 0.36
ii 31
28.68 28.07 27.42
0.05 0.04 0.04
0.00 0.03 0.08
28.12 28.73 27.46
0.03 0.03
0.00 0.03 0.08
introduce large errors. For N greater than 10, convergence is essentially complete between the two sets of calculations and the case (b) approximation is thus quite adequate. This holds in comparison with the P- and R-branch coefficients of all vibrational transitions covered by Nelson et al. In contrast with the P- and R-branch data, there is essentially complete agreement at all N between the Q-branch coef?cients of Nelson et al and those presented here. In using the Hund’s case (b) limit expression for Eq. (5), we have assumed that the effects of incomplete electron spin uncoupling and of the incipient uncoupling of electron orbital angular momentum (L uncoupling) are negligible in these calculations. The comparison with the results of Nelson et al6 shows that, for the high N coefficients of greatest concern here, the assumption of complete uncoupling of electron spin is a good approximation. The effects due to neglect of L uncoupling, however, are more difficult to assess. The phenomenon is of concern since the result of L uncoupling is the addition of a small component of the A *Z+ state into the X*H wavefunction. The components can then contribute to the transition moment integral with the transition moments characteristic of the A-X system. Since the A-X transition dipole moment is large, any perturbations between the A and X states have the potential to significantly influence the overall radiative rate. We rely upon the results of Mies,’ who has used perturbation theory to estimate the importance of L uncoupling for the vibration-rotation Einstein coefficients. He concludes that, for P- and R-branch transitions with J < 20.5, the contribution to the transition moment would be roughly 5% or less of the total. Thus the maximum uncertainty in the corresponding Einstein coefficient would be on the order of 10%. For Q-branch transitions the effect is more pronounced for a given J since the contribution to the overall transition moment scales as .I*. However, these transitions become progressively weaker with increasing J, and thus they are less likely to be observed. The effects due to L uncoupling at higher N are more difficult to assess. Mies’ analysis suggests that, for the P- and R-branch transitions, the contribution to the overall transition moment from
K. W. HOLTZCLAW et al
232
‘O°C t
1
#
b 7
z
7
3
2
a
t
2
q
1.0
9 Ft
10
l
F2
7
tu I 0.1
1 0
-
10
20
30 N’
Fig. 1. The pure-rotation
Einstein
coefficients
20
40
60
N
Fig. 2. The vibration-rotation Einstein 1 0 transition.
50
40
0
coefficients
for the
for I‘ = 0
the A-X mixing scales roughly as J. Consequently, the contribution to the transition moment at N = 40 would be roughly twice that at N = 20 or less than about 10%. In turn this would indicate potential errors in the Einstein coefficients of roughly 20% or less. The errors at high N in the Q-branch coefficients would be much larger but again, these are very weak transitions and thu: the effect is still of little practical importance. Estimates of the importance of L-uncoupling for the pure-rotation Einstein coefficients may bc made from Mies’ analysis and a comparison of the vibration-rotation and pure-rotation transitior
r-d 0 PI
Fig. 3. The vibration-rotation
Einstein
coefficients
4-2 0 R
for the 4-2 transition
Einstein coefficients for the OH(XTI) radical
233
5-
4-
OR1
.Nelson&al.
3-
2-
l-
O0
,
Fig. 4. A comparison of the P- and R-branch Einstein coefficients of Ref. 6 with this work for the -0 transition.
moment integrals. The pure-rotation integrals are typically of the magnitude of the permanent integrals are always 6 0.2 D. dipole moment, i.e. - 1.7 D. The corresponding vibration-rotation As a result, the relative contribution to the overall transition moment due to the A 4’ component is much smaller for pure-rotation transitions. Consequently, we expect the effects of L-uncoupling on the pure-rotation coefficients to be negligible. The relative rates of the vibration-rotation and pure-rotation coefficients allow insight into the relaxation dynamics of the corresponding levels in the upper atmosphere. Although the radiative rates for the vibration-rotation transitions are substantial for many rotational levels, they are slower than the corresponding pure-rotation transitions for the higher rotational states. Consequently, Au = 0 fluorescence will be the dominant radiative decay process for molecules possessing greater than roughly 25 rotational quanta. Since these levels are less susceptible to collisionallyinduced rotational energy transfer due to large energy gaps between levels,’ Au = 0 decay can, at
rool---
0 Rt , Nelson et al.
Fig. 5. A comparison of the P- and R-branch Einstein coefficients of Ref. 6 with this work for the 8-6 transition.
234
sufficiently high altitudes, excited OH.
K. W. HOLTZCLAW et al
become
the dominant
overall
decay
channel
for highly
rotationally
SUMMARY The effects of the centrifugal distortion have been accounted for in the calculation of Einstein coefficients for transitions involving high lying rotational and vibrational levels of OH. Einstein A coefficients have been calculated for the Au = 0 through Av = 5 transitions of OH(X*H) for all rotation-vibration levels energetically accessible in the H + 0, +OH(v, J) + 0, reaction. The vibration-rotation coefficients for the Au = l-3 sequences can be considered an extension to higher N of those calculated by Nelson et al.6 The effects of incomplete spin uncoupling were ignored in accounting for the effects of centrifugal distortion in these calculations. However, the resulting coefficients are in reasonable agreement with those of Nelson et al at all n and are essentially identical (discrepancies 6 1%) for N > 13. The pure-rotation coefficients are the first to be reported in the literature. These coefficients will permit a better understanding of the kinetics and chemistry involving highly excited OH(X’lT) probed via infrared chemiluminescence. The pure-rotation radiative relaxation rates are extremely large. For the highest levels considered, these rates exceed 500 set’. They are significantly greater than the corresponding vibration-rotation radiative rates. Consequently, for vibrationally excited molecules possessing a large amount of rotational excitation, AC = 0 fluorescence will be the predominant radiative decay of the OH airglow layer, the rotational radiative rate is channel. At the -85 km altitudes comparable to the collisional relaxation rate if the collisional process proceeds at 1% of gas kinetic. For the large energy quanta contained in these high rotational states, relaxation cross-sections of this magnitude are reasonable. Previous observations of nonequilibrium rotational distributions for OH in the airglow layer’.“’ reflect the successful competition of radiative decay with collisional thermal equilibration. Future laboratory measurements of the collisional relaxation rate coefficients are needed. These measurements will compliment the radiative relaxation rates presented here and facilitate the modelling efforts required to predict the altitude dependence of the spectrum of chemiluminescent OH atmospheric emissions. Acknowledgements-This work was supported by Physical Contract F19628-88-C-0069 with the Geophysics Directorate
Sciences Inc. internal funding and in part by funding under of the Phillips Laboratory in support of the Air Force Office
of Scientific Research under Task 231064 and the Defense Nuclear Agency under Project SA. Task SA. The authors gratefully acknowledge the generous assistance of J. A. Coxon who provided OH(X%) term energies prior to publication. The authors would also like to acknowledge Physical Sciences Inc.
useful discussion
with L. G. Piper, W. T. Rawlins.
and W. J. Marinelli
of
REFERENCES I. D. R. Smith, W. A. M. Blumberg, R. M. Nadile, S. J. Lipson, E. R. Huppi. N. Wheeler, and J. A. Dodd, Geophys. Res. Lett. 19, 593 (1992). 2. A. J. Fauvel, Astrophys. J. 252, 330 (1984). 3. K. L. Carleton. W. J. Marinelli, K. W. Holtzclaw, and B. D. Green, submitted to J. Phys. Chem. (1993). 4. J. E. Butler, R. G. MacDonald, D. J. Donaldson, and J. J. Sloan, Chem. Phys. Lett. 95, 183 (1983). 5. W. J. Marinelli, K. L. Carleton, K. W. Holtzclaw, B. D. Green, S. J. Lipson, and W. A. M. Blumberg, in preparation (1993). 6. D. D. Nelson, A. Schiffman. D. J. Nesbitt, J. J. Orlando, and J. B. Burkholder. J. Chem. Phys. 93, 7003 (1990). 7. F. H. Mies, J. Molec. Spectrosc. 53, 150 (1974). 8. S. R. Langhoff, H. J. Werner, and P. Rosmus, J. Molec. Spectrosc. 118, 507 (1986). 9. R. Herman and R. F. Wallis, J. Chem. Phys. 23, 637 (1955). 10. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, Dover, New York, NY (1980). Appendix III. 11. F. P. Billingsley, J. Molec. Spectrosc. 61, 53 (1976). 12. D. Oba, B. S. Agrawalla, and D. W. Setser, JQSRT 34, 283 (1985). 13. J. W. Cooley, Muth. Computat. 15, 363 (1961). 14. J. K. Cashion, J. Chem. Phys. 39, 1872 (1963); ibid 41, 3988 (1963). 15. G. Herzberg, Spectra of Diatomic Molecules, pp. 425426, Van Nostrand-Reinhold. New York, NY (1950). 16. D. D. Nelson. A. Schiffman, D. J. Nesbitt, and D. J. Yaron, J. Chem. Phys. 90, 5443 (1989).
Einstein
coefficients
for the OH(X%)
radical
235
17. R. Engleman, P. E. Rouse, H. M. Peek, and V. D. Baiamonte, Scientific Report LA-4364, Appendix B, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM (1970). 18. E. Hill and J. H. Van Vleck, Phys. Reo. 38, 250 (1928). 19. J. A. Coxon, private communication (1991). 20. K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, p. 508, Van Nostrand-Reinhold, New York, NY (1979). 21. W. Pendelton, P. Espy, D. Baker, A. Steed, M. Fetow, and K. Henriksen, J. Geophys. Res. 94,505 (1989).