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Transition probabilities for the B1Σ+u-X1Σ+g band system of H2
J. Quont.
Specbosc.
Radiat.
Transfer.
Vol.
TRANSITION
9, pp.
1543-1551.
Pergamon Press 1969. Printed in Great Britain
PROBABILITIES FOR THE B’Z,+-X’C,+ BAND SYSTEM OF H2 A. C. ALLISONand A. DALGARNO
Harvard
College Observatory
and Smithsonian
Astrophysical
(Received
Observatory,
Cambridge,
Massachusetts
02138
14 March 1969)
Abstract-The electronic dipole moment functions of Browne are used in calculations of the individual radiative transition probabilities for all the bands of the Lyman system of molecular hydrogen. The resulting lifetimes of the vibrational levels of the upper electronic state are in harmony with the experimental data of Hesser, and an apparent discrepancy between two experiments is resolved.
INTRODUCTION BROWNE(‘) has recently calculated the electronic dipole moment of the X’C; -B’C: transition of molecular hydrogen as a function of the internuclear distance R. Taken in conjunction with the calculations of KOLOS and WOLNIEWICZ(‘)of the potential energy curves, T/,(R) of the X’C: state and V"(R) of the B’ZJ state, it becomes possible to predict accurately the transition probabilities of all the individual bands of the Lyman system. The calculations can be compared to experimental absorption’3V4) and electron scattering measurements(5) for the lowest vibrational level, u” = 0, of the ground state X’C: and to experimental measurements of the average lifetime of the u’ = 3-7 levels of the excited state B’Z+ (6) In ad&ion to their basic interest in molecular spectroscopy, the transition probability data are required for a quantitative analysis of the Stecher-Williams mechanism’7’ for the destruction of molecular hydrogen in interstellar HI regions.
THEORY
The band oscillator strength of the electric dipole transition from the v” vibrational level of the X’C: state of H, to the v’ vibrational level of the B’C: state is given by
L-v, = %G - -%,)I(xWW ,,WMWI
2,
(1)
where E,. -E,., is the transition energy in atomic units, x:,,(R) is the vibrational wavefunction of the initial state and x$(R) that of the final state, D,,(R) is the parallel component of the dipole moment in atomic units, and R is the internuclear separation, also measured in atomic units. The band emission transition probability for the transition from the u’ level of the upper state to the Y”level of the lower state can be derived from f,,,,, by the 1543
A. C. ALLISON and A. DALGARNO
1544
formula
A v,urr= 6.6702 x 10’5f,~&~~v~~,
(2)
where &,.. is the transition wavelength in A units. The total discrete emission from the u’ level is given by
A,, =
transition
probability
for
1 A,,,,.. v”
The
vibrational
solutions
wave functions
xpO@) and x:,(R)
are the well-behaved
normalized
of the equations (4)
&(W =0, where M/m is the ratio of the mass of the hydrogen
(5)
atom to the mass of the electron.
The definition (1) of the band oscillator strength assumes that the centrifugal repulsion terms in (4) and (5), arising from the nuclear rotational angular momentum, have little effect on the vibrational wave functions xgV,,(R)and x$(R). The study of Franck-Condon factors
by HALMANN and LAULICHT@) suggests that for J” < 5, the effect on the band oscillator strengths
for U” = 0 is less than 25 per cent, a conclusion
measurements Consistent calculated
of HESSER et ~1.‘~’ and with independent
with the definition (l), the band oscillator with J” = J’ = 0 in equations (4) and (5).
generally
calculations strengths
consistent
with the
we have carried
out.
that we shall present were
For the potential energy functions I/,(R) and Vu(R) in (4) and (5), we adopted the results of KOLOS and WOLNIEWICZ(~) continued to larger separations by the asymptotic formulae of KOLOS(~) and CHAN and DALGARNO. (lo) The equations were solved by numerical integration using the Numerov method. In previous calculations of the band oscillator strengths for bands originating in the u” = 0 level,“‘) we employed a semi-empirical electronic dipole moment function D,,(R), based upon calculations by SCHIFF and PEKERIS, (12) by ROTHENBERG and DAVIDSON, and upon
unpublished material of Browne. More precise calculations have now been reported by BROWNE,(‘) in which both the dipole length and dipole velocity formulations are employed. The three dipole moment functions are shown in Fig. 1. RESULTS
AND DISCUSSION
In order to illustrate the sensitivity to the dipole moment function D,,(R), the band oscillator strengths corresponding to absorption from the v” = 0 level are listed in Table 1 for each of the three dipole moment functions shown in Fig. 1. The small differences provide a measure of the uncertainty in the theoretical predictions.
Transition probabilities for the B’CJ-X’E:
1545
band system of H,
I
I
I
I
2
4
6
8
I IO
R(a,)
FIG. 1. Electronic dipole moment of the Lyman transition of molecular hydrogen in atomic units. length; (c) B~~~~~(‘)--dipole velocity. (a) DALGARNO and ALLIKJN (''I;(b) B~~~~~(‘)--dipole
The values shown in Table 1 are in harmony, within the experimental error, with the data of GEIGER and TOPSCHOWSKY~‘) and of HESSER et al.(4) but, with the exception of the CM level, not with the data of HADDAD et ~1.‘~’ TABLE ~.THEORETICALBANDOSCILLATOR DALGARNO
0’ 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
ALLISON 0.16 0.57 1.1 1.7 2.2 2.5 2.6 2.6 2.4 2.2 1.9 1.6 1.4 1.2 0.95 0.78 O-64 0.52 0.42
and
STRENGTHS&X
BROWNE(‘) length
BROWNE(‘) velocity
0.14 0.47 0.94 1.4 1.8 2.1 2.2 2.2 2.1 1.9 1.7 1.4 1.2 1.0 O-86 0.71 0.58 0.48 0.39
0.16 0.57 1.1 1.7 2.3 2.6 2.8 2.8 2.7 2.4 2.2 1.9 1.6 1.4 1.2 0.96 0.79 0.65 0.54
u’ 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36
~O’)FOR THE THREED,,(R)IN F1c.1 DALGARNOand ALLISON(’I) 0.34 027 @22 0.18 0.14 @12 @094 0076 0061 0.049 0.039 0.032 0.026 0.02 1 0016 0.011 00060 o+IO13
BROWNE”) length
BROWNE(‘) velocity
0.32 0.26 0.21 0.17 0.14 0.12 CO96 0.079 0.064 0052 0.043 0.036 0.030 0.024 0.018 0.012 0.0072 0.0016
0.44 0.36 0.30 0.24 0.20 0.17 0.14 0.11 0.092 0.075 O-062 0.052 0.043 0,034 0.027 0.019 0.010 ONI
1546
A. C. ALLISONand A. DALGARNO
Most theoretical predictions of band oscillator strengths assume that the electronic dipole moment function is constant, so that the calculation reduces to the evaluation of Franck-Condon factors (6). FranckkCondon factors qorpvfare useful in the analysis of experimental data, and we present in Table 2 a list of qvpgus for all the discrete Lyman bands. They should be more precise than earlier calculations,“4*15) which were based on less accurate and less extensive potential energy functions V,(R) and T/,(R). The additional theoretical values we shall present are calculated by taking the arithmetic mean of the band strengths corresponding to the dipole length and dipole velocity moment functions of BROWNE. (‘) A comparison of these mean values with the electron scattering data of GEIGER and TOPSCHOWSKY~‘) is presented in Fig. 2 in the form of relative
I 0
I
I
2
4
I 6
I 8
I IO
I I2
I
I
I4
I6
+ I8
+ 20
v’ FIG. 2. Relativeband oscillator strengthsf,,./f,,.
The points are experimental electron scattering datajs) Curve (a) is the present theoretical results, and curve (b) shows the Franck-Condon factors.
band oscillator strengthsf,,. normalized to a value of unity at U’= 6. The figure includes the relative band oscillator strengths obtained from the Franck-Condon factors of Table 2. The comparison demonstrates the importance of taking into account the variation of D,,(R) with R,as GEIGER and TOPSCHOWSKY(‘) and HESSER et ~1.‘~’have earlier emphasized. Radiative lifetimes z,,,,, have been measured by HESSEX(~) for various bands of the Lyman system. He presents his results in the form of a mean lifetime of (8 + 2) x lo- lo set or mean transition probability of (1.3 f 0.3) x lo9 set- 1 for some unspecified distribution of excited levels among u’ = 3, 4, 5, 6, and 7. It appears appropriate to compare these numbers with our calculations of total transition probabilities A,, , and this we do in Table 3. Within the experimental error, the theory and experiment are in harmony, with the possible exception of v’ = 7. Because of the possibility of radiative decay into continuum levels of the lower electronic state, the lifetime of a particular vibrational level is not given by the reciprocal of A,.. The contribution of the discrete-continuum emission to the total transition probability
,
“‘1.11
0 1 1
3 4
I
TABLE 2. FRANCK~ONDON b
7 8
9
ID
FACTORSFOR THE H2 LYMAN SYSTEM I:
I*
I,
i4
A. C. ALLISON and A. DALGARNO
1548
TABLE 3. TOTALDISCRETETRANSITION
c’ 0
1 2 3 4 5 6 1 8 9 10
11 12 13 14 15 16 17 18 The experimental
&(lO’
set- ‘)
162 151 141 133 125 117 110 86.0 74.0 59.1 51.2 55.2 426 42.2 37.7 34.7 33.9 29.0 27.8 value for U’ = 3-7
PROBABILITIESFORTHE B'YZT STATEOF H,
v’ 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36
.4,(10’
set- ‘)
25.0 24.0 22.9 21.7 20.7 19.1 17.9 16.3 14.9 13.5 12.4 11.6 10.6 9.4 8.0 6.1 3.6 0.8
is 125.@
can be estimated approximately from the difference between unity and the sum over u” of the Franck-Condon factors in Table 2. It appears that the lifetimes for levels u” I 6 are little affected by decay into the continuum and the lifetimes for levels 7 I II’ I 11 are reduced by a factor of about 2, and that for levels u’ 2 12 the discrete-continuum emission is the dominant mode of radiative decay. We present in Tables 4 and 5 the theoretical values of the band oscillator strengths and transition probabilities A,.,., for all the discrete bands of the Lyman system of f !J”V’ molecular hydrogen. By using Franck-Condon factors to predict band oscillator strengths for the other discrete bands HESSER(@ derived a total oscillator strength ~~~fO,~of 0.51, a value much in excess of the eiperimental value of @28 derived by GEIGER and TOPSCHOWSKY,(‘) which is in harmony with our theoretical value of 0.291. The agreement evidenced in Table 3 shows, as Hesser suggested might occur, that the discrepancy is removed by the use of an accurate dipole moment function that varies with nuclear separation.
8. 07
7.62
5. 66
4.64
3.80
3.11
2.55
2.09
1.72
1.41
18
19
20
21
22
23
24
3. 66 3.40
2. 68
2.31
0.47
0.10
0. 78
0. 64
0. 52
0.44
0. 36
0.29
0.22
0. lb
0. 09
0. 02
28
29
30
31
32
33
34
35
36
0. a2
1.14
1.43
1. 72
2.02
3.12
0.20
0.90
1.55
2.09
2.51
2.87
3. ia
3.91
3.61
27
4. la 4.11
4.12
1.17
0. 96
26
4.13
3.97
3.69
3.26
2. 69
2.00
1.26
0. 57
25
4. 65
5.23
-5.83
6.46
7.07
8.34
0.09
8. 36
17
0. 05
0.69
2.26
4.88
8. 37
12.17
15.13
15.74
12.a2
6.72
0.85
1.88
16.13
8.05
6. a9
7.33
16
10.09
14
6.14
4. 52
8. 35
12.08
2.67
0.95
0. 02
0.74
4.06
10. bl
20.11
30.55
38.06
15
14. 32
12
21.62
9
13
23.61
8
Lb.74
24.87
7
11
24.99
b
19.23
23.63
5
10
15.93
20.54
3
4
40.85
56.65
37.86
27.59
5.20
10.43
I
38.90
11.38
1.51
2
1
0
2
0
did
0.16
0. 68
1.12
1.42
1.57
1.61
1.57
1.43
1.27
1.06
0. a0
0.51
0.24
0.05
0.01
0.17
0.65
1.49
2.75
4.39
6.27
a. IO
9.44
9.73
a.52
5.79
2.35
0.07
1.54
8. 30
18.30
24.31
17.93
2. 98
6.21
50.30
79.48
3
TABLE
0. 02
0. 07
0. 09
0. oa
0. 04
0. 01
0.00
0. 05
0. 16
0. 39
0. 77
1.31
2.02
2. aa
3. a7
4. aa
5. 78
6. 39
6.49
5. 90
4. 59
2.74
0. 91
0. 00
1.08
4. 74
10.28
15.04
15.10
8. 70
0. aa
3. 65
22.33
34. 82
12.23
IO. 17
107. 52
4
4. BAND
7.44
0. 04
0.20
0.40
0. 64
0.93
1.28
1.70
2.13
2.65
3.23
0. 17
0.77
1.34
1.82
2.20
2.48
2.66
2.68
2.63
2.44
1.51 2.06
3.77
0. e9
cl. 33
0. 02
0.15
0.91
2.39
4.46
b. 65
a. 20
8.23
6.25
2.84
0. la
1. 32
7.61
15.39
lb.25
0. 13
0. 56
0. 88
1.04
1.03
0. 90
0.68
0. 42
0. 18
0. 02
0. 04
0. 33
0. 99
2.01
3. 33
4. 75
5. 90
6. 34
5. 72
3. 97
1. 66
0. 09
0.85
4.75
10.28
13.20
9. 55
1.88
1. 65
15.59
24. b2
0.24 6.57
7. 09
7. 65
47.53
16.59
33.24
4.01
102. 07
64.44 23.12
27.74
7
(f,,,,,
64.61
b
STRENGTHS
4.19
4.40
4.35
3.96
3. Zb
2.26
I. 15
0.26
0.03
0.91
3.13
6.44
9.77
11.38
9.69
4.86
0.38
2.15
12.61
22.91
17.16
0.95
15.24
49.31
7.79
103.25
5
OSCILLATOR
x
0. 01
0. 03
0. 03
0. 01
0. 00
0. 07
0.24
0. 55
I. 04
1.74
2. 61
3. 51
4.27
4. 70
4. 59
3. 82
2.48
0. 97
0. 04
0. 55
2.99
6. 77
9. 8b
9.56
5. 08
0. 34
2.65
13.38
19.10
7.24
1.87
29. 35
22.24
10.31
72.40
72.99
7. 19
8
0. 04
0.19
0.40
0. 69
1.05
1.50
2.01
2.49
2.96
3.33
3.44
3. lb
2.48
1.54
0.58
0. 02
0. 33
1.82
4.31
6. a7
8. 00
6.44
2.70
0. 03
2.71
10.61
15.14
7.79
0.04
14.59
27.10
i.9a
33.55
30.76
102.68
24.46
0. a9
9
43.29
2.77
0. 02
10
SYSTEM
0. 14
0. 61
1.09
1.53
1.88
2.15
2.27
2.18
1.95
1. 54
0. 96
0. 37
0. 02
0. lb
1.00
2. 53
4.41
5. 87
6. 03
4. 38
1. b3
0. 00
2.03
7. 65
11.83
8. 36
0.75
4. 54
19. a9
14.21
1. 06
38. 97
10.38
113.04
103)FOR THE LYMAN
52.63
0.13
0. 58
0. 94
1.16
1.21
1.11
0. a7
0. 56
0.25
0. 03
0. 05
0.44
1.28
2.46
3.71
4. 54
4.45
3.20
1.28
0.03
1.03
4.65
8.49
a.27
3.01
0.19
8. 32
lb.86
6. 19
0. Ob
0. 26
0. 38
0. 39
0.30
0. 16
0. 04
0. 00
0.12
0.45
1. 06
I.85
2.66
3.22
3.23
2.53
1. 30
0. 20
0. 20
1.97
4. a7
6. 63
4.97
1.02
0. 79
7.93
11.88
4.07
5. 07
16. 58
20.09
31.26 4.47
136. 36
0. 02
0. 06
0. 07
0. 04
0. 01
0. 01
0. 08
0.26
0. 56
0.99
1.49
1.91
2.10
1.91
1.33
0. 55
0. 03
0.29
1. 56
3.43
4. ia
3.31
0. 52
0.22
6. 09
5.12
a.49
7.14
0. 30
69.25
115.95
7. a2
2.08
0.65
41.73
0.66
0. 00
0. 00
7.07
121.49
13 0. 00
0. II
0. 01
3. 24
0. 01
12 0. 00
11 0. 00
14
0. 00
0. 00
0. 00
0. 00
0. 01
0. 06
0.12
0. 30
0. 36
0.76
0. 65
1.15
0. 56
1.00
0. 07
0. 31
0.49
0.01
3.49
0.14
6.40
0. 59
4. 06
12.75
2.00
31.49
17.13
12.59
60.49
9.21
11.61
6.18
0. 00
0. LO
0.00
0. 00
0. 00
195.55
198.42
59.91
*
3.29
2. b?
2.08
1.47
0. 83
0. 18
33
34
35
36
0.91
4. 05
7.08
9. 82
12.23
14.61
17.02
3.99
30
32
24. 93
19.29
4.71
29
31
26.44
22.16
1.66
7. 30
12.47
16.73
19.97
22.69
28.19
29.88
25.60
31.00
31.19
5. 67
33.10
29.33
6. 90
10.08
25
30.48
28.96
36. 94
41.00
27
12.09
24
28
14.51
23
26.54
45.21
23.15
8.57
3.82
0. 62
8. 38
17.47
22
49.40
59.45
0. 32
4.40
26
21.05
18.83
53. 32
36.81
21
13.82
56. 69
44.22
17
I8
30. 56
59.17
52. 98
16
25. 37
60.24
63.24
L5
19
56.28
75. I2
14
20
50.35
88.44
13
14.1,
102.94
12
41.42
50.17
17. 32
118. Ll
1,
29.81
71.38
29.96
86.82
0. 13
6. 07
146.69
133.14
9
10
88.27
70.20
4. 5;
156.83
24.23
161.65
7
8
4.45
35.91
146. 62
158.79
5
6
61.81
9. 55
169.
114.26
79. 51
205.17
94.00
124. 36
3
04
4
12
263.
140.44
29. 05
1
2
174.94
1
56.22
0
8.19
0
did”
TABLE
87
30
57.58
26. 14
34.03 38.04
27.24 17.35
8.51
3. 55
1.20
5.17
8.47
10.65
11.66
11.90
11.47
10.39
9. 12
7. 55
0. 13
0. 50
0. 65
0.55
0.29
0. 06
0.02
0.31
1.10
2. 62
5. 07
12.94
5. 62
18.24
1. 67
24. I2
30.01
35.02
0. 36
0. 04
1.15
4.20
38.07
25.99
38.22
9. 56
15.25
23.08
IO.46
1.85
0. I2 1.00
IS.
5. 17 5. 97 6. 08 5. 19 3. 29
15.10 13.47 11.26 8.31 4.81 1.09
8. 31 6.07
1.36 0. 29
2.64
4.22
0.77
3. 87
16. 04
10.89
13.56
2. 34
51
14.22
I. 79 0.20
11.87
8. 59
5. 20
17.08
0.10
4.99
22.20
23. 94
0. 79
4.45
0.05
0. 16
0. 16
0.04
0. 02
0. 35
1.28
2. 92
5.48
9. 04
0.21
1.03
2.12
3. 61
5.48
7. 74
10.24
12.54
14.16
16.40
16. 71
15.12
17.74 13.37
11.71
7.14
2.65
0. 10
1.48
7.91
18.35
28.68
32.60
25.73
10.55
0. 10
10.09
38.49
53.45
26. 74
0. 12
47.12
84.75
5. 99
97.90
86. 48
277.73
63. 51
2.21
LYMAN
21.29
18.18
11.59
30.93
0. 17
2.44
12.95
28.69
40.86
38.73
20.09
I. 33
9.94
48.88
67.80
24.97
6.24
94.86
69.48
29.26
16. 01
16. 68
20. 06
23.20
25.43
26.36
25.72
23.19
18.74
17
72
31.07
210.
203.
19.27
‘) FOR THE
4.82
set-
12.43
6. 39 12.77
18.63 27.38
33.33 22.76
1.42
7. b5
40.33
3. 74 0. 39
39.69
29. 52
0. 16
4.79
16.21
32. 63
4.96
55.47 48.49
20.53
48.49
0.01
56. 08 13.15
54. 36 43.39
24.29
5. 82
48.13
32. 03 0.83
32. 79
55.29
7. 36 38. 32
64. 63
6. 29
88.24
0. 92
66.41
24. 64
62. 15
25. 70
154.
12. 56
307.75
22. 54
1.73
9. 48
54.12
95.60
b9.57
3.72
lo6
80. 39
A,,.,...(
46.01
51.47
73.48
71.98
40.42
3.98
16.06
26.11
120.63
58.04
78. 38
27.71 181. 61
37 68
203. 210.
343.85
PROBABILITIES
5.64
12.73
0. 46
7. 96
41.81
89.91
116.24
95. 34
144.23
83.33
13.43
39.48
402.
49.07
-
TRANSITION
27. I9
113.33
326. 02
3
5. BAND
32
0. 69
3.10
5. 50
7.66
9. 35
10.55
11.03
IO. 50
9.29
7.23
4.46
1.71
0. 09
0. 73
4.37
10.85
18. 52
24.22
24. 38
17.33
6. 32
0. 01
7. 53
27.65
41.67
28.67
2. 50
14.70
62. 52
43.26
3. I2
110.81
28.46
298.
109. 76
6. 73
0. 05
BAND
0. 65
2.81
4. 57
5. 59
5. 75
5. 22
4.08
2. 57
1.13
0. 14
0.21
I. 93
5. 50
10.42
15.45
18.60
17.88
12. 62
4. 96
0. 10
3.80
16.81
30.00
20.50
10.10
0. 62
0. 30
1.23
1.79
1.82
I. 38
0. 75
0. 19
0.01
0. 51
1.98
4. 53
7. 82
11.06
13.16
12.98
9. 99
5. 03
0. 74
0. 75
7.18
17.34
23. 09
lb.90
3.36
2. 54
24.86
12.02 36. 18
51.94
14.50
45.07
53. 67
351.26
103.44
1. 57
0.25
0.02
0.00
26.42
18.47
12.93
87. 31
19.05
32 31
131. 315.
7. 75
0. 02
0. 00
SYSTEM
I. 50
0.22
0. 07
0.26
0. 30
0. 18
0. 03
0. 03
0. 36
1.12
2.41
4.21
6.23
7.88
8.49
7. 59
5.21
2.13
0.11
1.08
5. 64
12.15
14.45
11.20
1.71
0. 72
0.01
0. 02
0.01
0. 00
0. 04
0.27
0. 50
1. 30
1. 52
3. 16
2. 65
4. b?
2. 24
3. 90
0. 20
1.16
I. 80
0.05
12.43
0. 50
21.78
1.94
13.17
40.25
6. 16
93.98
15. 56 19.04
49. 61
35.34
164. 36
24. 17
29.40
15.09
0. 00
25. 07
20.43
0.83
185.44
299.64
19.48
4.99
0. 01
0. 00
0. 01
0. 00
0. 00
0. 00
Transition probabilities for the B’Z:-X’E: Acknowledgement-This
work has been partly
supported
band system of Hz
1551
by NASA grant NGR 22-007-136.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
J. C. BROWNE.Astrophys. J. 156, 397 (1969). W. KOLOS and L. WOLNIEWICZ,J. Chem. Phys. 43,2429 (1965), 45,509 (1966). G. N. HADDAD, K. H. LOKAN,A. J. D. FARMERand J. H. CARVER,JQSRTS, 1193 (1968). J. E. H~ssw, N. H. BRWKS and G. M. LAWRENCE,Astrophys. J. (Letters) 153, L65 (1968); J. Chem. Phys. 49,548s (1968). J. GEIGERand M. TOPSCHOWSKY, 2. Naturforsch. 21a, 626 (1966). J. E. HESSER,J. Chem. Phvs. 48. 2518 (1968). T. P. STECHER and D. A. WILLIAMS,Astropiys. J. (Letters) 149, L29 (1967). M. HALMANNand I. LAULICHT,JQSRT 8,935 (1968). W. KOLOS,Internat. J. &ant. Chem. 1. 169 (1967). Y. M. CHAN and A. DALGARNO,Mole&ar ihys.‘14,101 (1968). A. DALGARNOand A. C. ALLISON,Astrophys. J. (Letters) 154, L95 (1968) B. SCHIFFand C. L. PEKERIS,Phys. Reo.lG, A638 (1964). R. ROTHENBERG and E. R. DAVIDSON,J. Molec. Spectrosc. 22, 1 (1967). R. W. NICHOLLS,Astrophys. J. 141, 819 (1965). R. J. SPINDLER,JQSRT9, 597 (1969).
Specbosc.
Radiat.
Transfer.
Vol.
TRANSITION
9, pp.
1543-1551.
Pergamon Press 1969. Printed in Great Britain
PROBABILITIES FOR THE B’Z,+-X’C,+ BAND SYSTEM OF H2 A. C. ALLISONand A. DALGARNO
Harvard
College Observatory
and Smithsonian
Astrophysical
(Received
Observatory,
Cambridge,
Massachusetts
02138
14 March 1969)
Abstract-The electronic dipole moment functions of Browne are used in calculations of the individual radiative transition probabilities for all the bands of the Lyman system of molecular hydrogen. The resulting lifetimes of the vibrational levels of the upper electronic state are in harmony with the experimental data of Hesser, and an apparent discrepancy between two experiments is resolved.
INTRODUCTION BROWNE(‘) has recently calculated the electronic dipole moment of the X’C; -B’C: transition of molecular hydrogen as a function of the internuclear distance R. Taken in conjunction with the calculations of KOLOS and WOLNIEWICZ(‘)of the potential energy curves, T/,(R) of the X’C: state and V"(R) of the B’ZJ state, it becomes possible to predict accurately the transition probabilities of all the individual bands of the Lyman system. The calculations can be compared to experimental absorption’3V4) and electron scattering measurements(5) for the lowest vibrational level, u” = 0, of the ground state X’C: and to experimental measurements of the average lifetime of the u’ = 3-7 levels of the excited state B’Z+ (6) In ad&ion to their basic interest in molecular spectroscopy, the transition probability data are required for a quantitative analysis of the Stecher-Williams mechanism’7’ for the destruction of molecular hydrogen in interstellar HI regions.
THEORY
The band oscillator strength of the electric dipole transition from the v” vibrational level of the X’C: state of H, to the v’ vibrational level of the B’C: state is given by
L-v, = %G - -%,)I(xWW ,,WMWI
2,
(1)
where E,. -E,., is the transition energy in atomic units, x:,,(R) is the vibrational wavefunction of the initial state and x$(R) that of the final state, D,,(R) is the parallel component of the dipole moment in atomic units, and R is the internuclear separation, also measured in atomic units. The band emission transition probability for the transition from the u’ level of the upper state to the Y”level of the lower state can be derived from f,,,,, by the 1543
A. C. ALLISON and A. DALGARNO
1544
formula
A v,urr= 6.6702 x 10’5f,~&~~v~~,
(2)
where &,.. is the transition wavelength in A units. The total discrete emission from the u’ level is given by
A,, =
transition
probability
for
1 A,,,,.. v”
The
vibrational
solutions
wave functions
xpO@) and x:,(R)
are the well-behaved
normalized
of the equations (4)
&(W =0, where M/m is the ratio of the mass of the hydrogen
(5)
atom to the mass of the electron.
The definition (1) of the band oscillator strength assumes that the centrifugal repulsion terms in (4) and (5), arising from the nuclear rotational angular momentum, have little effect on the vibrational wave functions xgV,,(R)and x$(R). The study of Franck-Condon factors
by HALMANN and LAULICHT@) suggests that for J” < 5, the effect on the band oscillator strengths
for U” = 0 is less than 25 per cent, a conclusion
measurements Consistent calculated
of HESSER et ~1.‘~’ and with independent
with the definition (l), the band oscillator with J” = J’ = 0 in equations (4) and (5).
generally
calculations strengths
consistent
with the
we have carried
out.
that we shall present were
For the potential energy functions I/,(R) and Vu(R) in (4) and (5), we adopted the results of KOLOS and WOLNIEWICZ(~) continued to larger separations by the asymptotic formulae of KOLOS(~) and CHAN and DALGARNO. (lo) The equations were solved by numerical integration using the Numerov method. In previous calculations of the band oscillator strengths for bands originating in the u” = 0 level,“‘) we employed a semi-empirical electronic dipole moment function D,,(R), based upon calculations by SCHIFF and PEKERIS, (12) by ROTHENBERG and DAVIDSON, and upon
unpublished material of Browne. More precise calculations have now been reported by BROWNE,(‘) in which both the dipole length and dipole velocity formulations are employed. The three dipole moment functions are shown in Fig. 1. RESULTS
AND DISCUSSION
In order to illustrate the sensitivity to the dipole moment function D,,(R), the band oscillator strengths corresponding to absorption from the v” = 0 level are listed in Table 1 for each of the three dipole moment functions shown in Fig. 1. The small differences provide a measure of the uncertainty in the theoretical predictions.
Transition probabilities for the B’CJ-X’E:
1545
band system of H,
I
I
I
I
2
4
6
8
I IO
R(a,)
FIG. 1. Electronic dipole moment of the Lyman transition of molecular hydrogen in atomic units. length; (c) B~~~~~(‘)--dipole velocity. (a) DALGARNO and ALLIKJN (''I;(b) B~~~~~(‘)--dipole
The values shown in Table 1 are in harmony, within the experimental error, with the data of GEIGER and TOPSCHOWSKY~‘) and of HESSER et al.(4) but, with the exception of the CM level, not with the data of HADDAD et ~1.‘~’ TABLE ~.THEORETICALBANDOSCILLATOR DALGARNO
0’ 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
ALLISON 0.16 0.57 1.1 1.7 2.2 2.5 2.6 2.6 2.4 2.2 1.9 1.6 1.4 1.2 0.95 0.78 O-64 0.52 0.42
and
STRENGTHS&X
BROWNE(‘) length
BROWNE(‘) velocity
0.14 0.47 0.94 1.4 1.8 2.1 2.2 2.2 2.1 1.9 1.7 1.4 1.2 1.0 O-86 0.71 0.58 0.48 0.39
0.16 0.57 1.1 1.7 2.3 2.6 2.8 2.8 2.7 2.4 2.2 1.9 1.6 1.4 1.2 0.96 0.79 0.65 0.54
u’ 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36
~O’)FOR THE THREED,,(R)IN F1c.1 DALGARNOand ALLISON(’I) 0.34 027 @22 0.18 0.14 @12 @094 0076 0061 0.049 0.039 0.032 0.026 0.02 1 0016 0.011 00060 o+IO13
BROWNE”) length
BROWNE(‘) velocity
0.32 0.26 0.21 0.17 0.14 0.12 CO96 0.079 0.064 0052 0.043 0.036 0.030 0.024 0.018 0.012 0.0072 0.0016
0.44 0.36 0.30 0.24 0.20 0.17 0.14 0.11 0.092 0.075 O-062 0.052 0.043 0,034 0.027 0.019 0.010 ONI
1546
A. C. ALLISONand A. DALGARNO
Most theoretical predictions of band oscillator strengths assume that the electronic dipole moment function is constant, so that the calculation reduces to the evaluation of Franck-Condon factors (6). FranckkCondon factors qorpvfare useful in the analysis of experimental data, and we present in Table 2 a list of qvpgus for all the discrete Lyman bands. They should be more precise than earlier calculations,“4*15) which were based on less accurate and less extensive potential energy functions V,(R) and T/,(R). The additional theoretical values we shall present are calculated by taking the arithmetic mean of the band strengths corresponding to the dipole length and dipole velocity moment functions of BROWNE. (‘) A comparison of these mean values with the electron scattering data of GEIGER and TOPSCHOWSKY~‘) is presented in Fig. 2 in the form of relative
I 0
I
I
2
4
I 6
I 8
I IO
I I2
I
I
I4
I6
+ I8
+ 20
v’ FIG. 2. Relativeband oscillator strengthsf,,./f,,.
The points are experimental electron scattering datajs) Curve (a) is the present theoretical results, and curve (b) shows the Franck-Condon factors.
band oscillator strengthsf,,. normalized to a value of unity at U’= 6. The figure includes the relative band oscillator strengths obtained from the Franck-Condon factors of Table 2. The comparison demonstrates the importance of taking into account the variation of D,,(R) with R,as GEIGER and TOPSCHOWSKY(‘) and HESSER et ~1.‘~’have earlier emphasized. Radiative lifetimes z,,,,, have been measured by HESSEX(~) for various bands of the Lyman system. He presents his results in the form of a mean lifetime of (8 + 2) x lo- lo set or mean transition probability of (1.3 f 0.3) x lo9 set- 1 for some unspecified distribution of excited levels among u’ = 3, 4, 5, 6, and 7. It appears appropriate to compare these numbers with our calculations of total transition probabilities A,, , and this we do in Table 3. Within the experimental error, the theory and experiment are in harmony, with the possible exception of v’ = 7. Because of the possibility of radiative decay into continuum levels of the lower electronic state, the lifetime of a particular vibrational level is not given by the reciprocal of A,.. The contribution of the discrete-continuum emission to the total transition probability
,
“‘1.11
0 1 1
3 4
I
TABLE 2. FRANCK~ONDON b
7 8
9
ID
FACTORSFOR THE H2 LYMAN SYSTEM I:
I*
I,
i4
A. C. ALLISON and A. DALGARNO
1548
TABLE 3. TOTALDISCRETETRANSITION
c’ 0
1 2 3 4 5 6 1 8 9 10
11 12 13 14 15 16 17 18 The experimental
&(lO’
set- ‘)
162 151 141 133 125 117 110 86.0 74.0 59.1 51.2 55.2 426 42.2 37.7 34.7 33.9 29.0 27.8 value for U’ = 3-7
PROBABILITIESFORTHE B'YZT STATEOF H,
v’ 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36
.4,(10’
set- ‘)
25.0 24.0 22.9 21.7 20.7 19.1 17.9 16.3 14.9 13.5 12.4 11.6 10.6 9.4 8.0 6.1 3.6 0.8
is 125.@
can be estimated approximately from the difference between unity and the sum over u” of the Franck-Condon factors in Table 2. It appears that the lifetimes for levels u” I 6 are little affected by decay into the continuum and the lifetimes for levels 7 I II’ I 11 are reduced by a factor of about 2, and that for levels u’ 2 12 the discrete-continuum emission is the dominant mode of radiative decay. We present in Tables 4 and 5 the theoretical values of the band oscillator strengths and transition probabilities A,.,., for all the discrete bands of the Lyman system of f !J”V’ molecular hydrogen. By using Franck-Condon factors to predict band oscillator strengths for the other discrete bands HESSER(@ derived a total oscillator strength ~~~fO,~of 0.51, a value much in excess of the eiperimental value of @28 derived by GEIGER and TOPSCHOWSKY,(‘) which is in harmony with our theoretical value of 0.291. The agreement evidenced in Table 3 shows, as Hesser suggested might occur, that the discrepancy is removed by the use of an accurate dipole moment function that varies with nuclear separation.
8. 07
7.62
5. 66
4.64
3.80
3.11
2.55
2.09
1.72
1.41
18
19
20
21
22
23
24
3. 66 3.40
2. 68
2.31
0.47
0.10
0. 78
0. 64
0. 52
0.44
0. 36
0.29
0.22
0. lb
0. 09
0. 02
28
29
30
31
32
33
34
35
36
0. a2
1.14
1.43
1. 72
2.02
3.12
0.20
0.90
1.55
2.09
2.51
2.87
3. ia
3.91
3.61
27
4. la 4.11
4.12
1.17
0. 96
26
4.13
3.97
3.69
3.26
2. 69
2.00
1.26
0. 57
25
4. 65
5.23
-5.83
6.46
7.07
8.34
0.09
8. 36
17
0. 05
0.69
2.26
4.88
8. 37
12.17
15.13
15.74
12.a2
6.72
0.85
1.88
16.13
8.05
6. a9
7.33
16
10.09
14
6.14
4. 52
8. 35
12.08
2.67
0.95
0. 02
0.74
4.06
10. bl
20.11
30.55
38.06
15
14. 32
12
21.62
9
13
23.61
8
Lb.74
24.87
7
11
24.99
b
19.23
23.63
5
10
15.93
20.54
3
4
40.85
56.65
37.86
27.59
5.20
10.43
I
38.90
11.38
1.51
2
1
0
2
0
did
0.16
0. 68
1.12
1.42
1.57
1.61
1.57
1.43
1.27
1.06
0. a0
0.51
0.24
0.05
0.01
0.17
0.65
1.49
2.75
4.39
6.27
a. IO
9.44
9.73
a.52
5.79
2.35
0.07
1.54
8. 30
18.30
24.31
17.93
2. 98
6.21
50.30
79.48
3
TABLE
0. 02
0. 07
0. 09
0. oa
0. 04
0. 01
0.00
0. 05
0. 16
0. 39
0. 77
1.31
2.02
2. aa
3. a7
4. aa
5. 78
6. 39
6.49
5. 90
4. 59
2.74
0. 91
0. 00
1.08
4. 74
10.28
15.04
15.10
8. 70
0. aa
3. 65
22.33
34. 82
12.23
IO. 17
107. 52
4
4. BAND
7.44
0. 04
0.20
0.40
0. 64
0.93
1.28
1.70
2.13
2.65
3.23
0. 17
0.77
1.34
1.82
2.20
2.48
2.66
2.68
2.63
2.44
1.51 2.06
3.77
0. e9
cl. 33
0. 02
0.15
0.91
2.39
4.46
b. 65
a. 20
8.23
6.25
2.84
0. la
1. 32
7.61
15.39
lb.25
0. 13
0. 56
0. 88
1.04
1.03
0. 90
0.68
0. 42
0. 18
0. 02
0. 04
0. 33
0. 99
2.01
3. 33
4. 75
5. 90
6. 34
5. 72
3. 97
1. 66
0. 09
0.85
4.75
10.28
13.20
9. 55
1.88
1. 65
15.59
24. b2
0.24 6.57
7. 09
7. 65
47.53
16.59
33.24
4.01
102. 07
64.44 23.12
27.74
7
(f,,,,,
64.61
b
STRENGTHS
4.19
4.40
4.35
3.96
3. Zb
2.26
I. 15
0.26
0.03
0.91
3.13
6.44
9.77
11.38
9.69
4.86
0.38
2.15
12.61
22.91
17.16
0.95
15.24
49.31
7.79
103.25
5
OSCILLATOR
x
0. 01
0. 03
0. 03
0. 01
0. 00
0. 07
0.24
0. 55
I. 04
1.74
2. 61
3. 51
4.27
4. 70
4. 59
3. 82
2.48
0. 97
0. 04
0. 55
2.99
6. 77
9. 8b
9.56
5. 08
0. 34
2.65
13.38
19.10
7.24
1.87
29. 35
22.24
10.31
72.40
72.99
7. 19
8
0. 04
0.19
0.40
0. 69
1.05
1.50
2.01
2.49
2.96
3.33
3.44
3. lb
2.48
1.54
0.58
0. 02
0. 33
1.82
4.31
6. a7
8. 00
6.44
2.70
0. 03
2.71
10.61
15.14
7.79
0.04
14.59
27.10
i.9a
33.55
30.76
102.68
24.46
0. a9
9
43.29
2.77
0. 02
10
SYSTEM
0. 14
0. 61
1.09
1.53
1.88
2.15
2.27
2.18
1.95
1. 54
0. 96
0. 37
0. 02
0. lb
1.00
2. 53
4.41
5. 87
6. 03
4. 38
1. b3
0. 00
2.03
7. 65
11.83
8. 36
0.75
4. 54
19. a9
14.21
1. 06
38. 97
10.38
113.04
103)FOR THE LYMAN
52.63
0.13
0. 58
0. 94
1.16
1.21
1.11
0. a7
0. 56
0.25
0. 03
0. 05
0.44
1.28
2.46
3.71
4. 54
4.45
3.20
1.28
0.03
1.03
4.65
8.49
a.27
3.01
0.19
8. 32
lb.86
6. 19
0. Ob
0. 26
0. 38
0. 39
0.30
0. 16
0. 04
0. 00
0.12
0.45
1. 06
I.85
2.66
3.22
3.23
2.53
1. 30
0. 20
0. 20
1.97
4. a7
6. 63
4.97
1.02
0. 79
7.93
11.88
4.07
5. 07
16. 58
20.09
31.26 4.47
136. 36
0. 02
0. 06
0. 07
0. 04
0. 01
0. 01
0. 08
0.26
0. 56
0.99
1.49
1.91
2.10
1.91
1.33
0. 55
0. 03
0.29
1. 56
3.43
4. ia
3.31
0. 52
0.22
6. 09
5.12
a.49
7.14
0. 30
69.25
115.95
7. a2
2.08
0.65
41.73
0.66
0. 00
0. 00
7.07
121.49
13 0. 00
0. II
0. 01
3. 24
0. 01
12 0. 00
11 0. 00
14
0. 00
0. 00
0. 00
0. 00
0. 01
0. 06
0.12
0. 30
0. 36
0.76
0. 65
1.15
0. 56
1.00
0. 07
0. 31
0.49
0.01
3.49
0.14
6.40
0. 59
4. 06
12.75
2.00
31.49
17.13
12.59
60.49
9.21
11.61
6.18
0. 00
0. LO
0.00
0. 00
0. 00
195.55
198.42
59.91
*
3.29
2. b?
2.08
1.47
0. 83
0. 18
33
34
35
36
0.91
4. 05
7.08
9. 82
12.23
14.61
17.02
3.99
30
32
24. 93
19.29
4.71
29
31
26.44
22.16
1.66
7. 30
12.47
16.73
19.97
22.69
28.19
29.88
25.60
31.00
31.19
5. 67
33.10
29.33
6. 90
10.08
25
30.48
28.96
36. 94
41.00
27
12.09
24
28
14.51
23
26.54
45.21
23.15
8.57
3.82
0. 62
8. 38
17.47
22
49.40
59.45
0. 32
4.40
26
21.05
18.83
53. 32
36.81
21
13.82
56. 69
44.22
17
I8
30. 56
59.17
52. 98
16
25. 37
60.24
63.24
L5
19
56.28
75. I2
14
20
50.35
88.44
13
14.1,
102.94
12
41.42
50.17
17. 32
118. Ll
1,
29.81
71.38
29.96
86.82
0. 13
6. 07
146.69
133.14
9
10
88.27
70.20
4. 5;
156.83
24.23
161.65
7
8
4.45
35.91
146. 62
158.79
5
6
61.81
9. 55
169.
114.26
79. 51
205.17
94.00
124. 36
3
04
4
12
263.
140.44
29. 05
1
2
174.94
1
56.22
0
8.19
0
did”
TABLE
87
30
57.58
26. 14
34.03 38.04
27.24 17.35
8.51
3. 55
1.20
5.17
8.47
10.65
11.66
11.90
11.47
10.39
9. 12
7. 55
0. 13
0. 50
0. 65
0.55
0.29
0. 06
0.02
0.31
1.10
2. 62
5. 07
12.94
5. 62
18.24
1. 67
24. I2
30.01
35.02
0. 36
0. 04
1.15
4.20
38.07
25.99
38.22
9. 56
15.25
23.08
IO.46
1.85
0. I2 1.00
IS.
5. 17 5. 97 6. 08 5. 19 3. 29
15.10 13.47 11.26 8.31 4.81 1.09
8. 31 6.07
1.36 0. 29
2.64
4.22
0.77
3. 87
16. 04
10.89
13.56
2. 34
51
14.22
I. 79 0.20
11.87
8. 59
5. 20
17.08
0.10
4.99
22.20
23. 94
0. 79
4.45
0.05
0. 16
0. 16
0.04
0. 02
0. 35
1.28
2. 92
5.48
9. 04
0.21
1.03
2.12
3. 61
5.48
7. 74
10.24
12.54
14.16
16.40
16. 71
15.12
17.74 13.37
11.71
7.14
2.65
0. 10
1.48
7.91
18.35
28.68
32.60
25.73
10.55
0. 10
10.09
38.49
53.45
26. 74
0. 12
47.12
84.75
5. 99
97.90
86. 48
277.73
63. 51
2.21
LYMAN
21.29
18.18
11.59
30.93
0. 17
2.44
12.95
28.69
40.86
38.73
20.09
I. 33
9.94
48.88
67.80
24.97
6.24
94.86
69.48
29.26
16. 01
16. 68
20. 06
23.20
25.43
26.36
25.72
23.19
18.74
17
72
31.07
210.
203.
19.27
‘) FOR THE
4.82
set-
12.43
6. 39 12.77
18.63 27.38
33.33 22.76
1.42
7. b5
40.33
3. 74 0. 39
39.69
29. 52
0. 16
4.79
16.21
32. 63
4.96
55.47 48.49
20.53
48.49
0.01
56. 08 13.15
54. 36 43.39
24.29
5. 82
48.13
32. 03 0.83
32. 79
55.29
7. 36 38. 32
64. 63
6. 29
88.24
0. 92
66.41
24. 64
62. 15
25. 70
154.
12. 56
307.75
22. 54
1.73
9. 48
54.12
95.60
b9.57
3.72
lo6
80. 39
A,,.,...(
46.01
51.47
73.48
71.98
40.42
3.98
16.06
26.11
120.63
58.04
78. 38
27.71 181. 61
37 68
203. 210.
343.85
PROBABILITIES
5.64
12.73
0. 46
7. 96
41.81
89.91
116.24
95. 34
144.23
83.33
13.43
39.48
402.
49.07
-
TRANSITION
27. I9
113.33
326. 02
3
5. BAND
32
0. 69
3.10
5. 50
7.66
9. 35
10.55
11.03
IO. 50
9.29
7.23
4.46
1.71
0. 09
0. 73
4.37
10.85
18. 52
24.22
24. 38
17.33
6. 32
0. 01
7. 53
27.65
41.67
28.67
2. 50
14.70
62. 52
43.26
3. I2
110.81
28.46
298.
109. 76
6. 73
0. 05
BAND
0. 65
2.81
4. 57
5. 59
5. 75
5. 22
4.08
2. 57
1.13
0. 14
0.21
I. 93
5. 50
10.42
15.45
18.60
17.88
12. 62
4. 96
0. 10
3.80
16.81
30.00
20.50
10.10
0. 62
0. 30
1.23
1.79
1.82
I. 38
0. 75
0. 19
0.01
0. 51
1.98
4. 53
7. 82
11.06
13.16
12.98
9. 99
5. 03
0. 74
0. 75
7.18
17.34
23. 09
lb.90
3.36
2. 54
24.86
12.02 36. 18
51.94
14.50
45.07
53. 67
351.26
103.44
1. 57
0.25
0.02
0.00
26.42
18.47
12.93
87. 31
19.05
32 31
131. 315.
7. 75
0. 02
0. 00
SYSTEM
I. 50
0.22
0. 07
0.26
0. 30
0. 18
0. 03
0. 03
0. 36
1.12
2.41
4.21
6.23
7.88
8.49
7. 59
5.21
2.13
0.11
1.08
5. 64
12.15
14.45
11.20
1.71
0. 72
0.01
0. 02
0.01
0. 00
0. 04
0.27
0. 50
1. 30
1. 52
3. 16
2. 65
4. b?
2. 24
3. 90
0. 20
1.16
I. 80
0.05
12.43
0. 50
21.78
1.94
13.17
40.25
6. 16
93.98
15. 56 19.04
49. 61
35.34
164. 36
24. 17
29.40
15.09
0. 00
25. 07
20.43
0.83
185.44
299.64
19.48
4.99
0. 01
0. 00
0. 01
0. 00
0. 00
0. 00
Transition probabilities for the B’Z:-X’E: Acknowledgement-This
work has been partly
supported
band system of Hz
1551
by NASA grant NGR 22-007-136.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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