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Realistic Franck-Condon factors and related integrals for diatomic molecules—II. The O2 Herzberg I system
1. Quanr.
Spectrosc.
Rod&.
Transfer.
Vol.
12, pp. 603417.
Pergamon Press 1972. Printed in Great Britain
REALISTIC FRANCK-CONDON FACTORS AND RELATED INTEGRALS FOR DIATOMIC MOLECULES-II. THE 0, HERZBERG I SYSTEM W. R. JARMAIN Department of Physics, University of Western Ontario, London 72, Canada (Received 2 July 1971) Abstract-Tabulated Klein-Dunham potentials are given for the Herzberg I system of 0,. calculated by the method of paper I in this series. Accurate numerical treatment also includes orthogonality tests of wave functions; comparisons of observed and calculated eigenvalues and B, for both states ; Franck-Condon factors and r-centroids over ranges 0 I u’ I 11, 0 < u” I 24; Franck-Condon densities in the continua for 0 < u” I 24 and graphs of their variation for 0 I u” 5 4.
INTRODUCTION
system of oxygen, a forbidden transition, is an important contributor to the ultraviolet night airglow. Measurements by DITCHBURN and YOUNG(‘) support earlier suggestions that absorption in the Herzberg continuum is the main source of oxygen atoms below 80 km, mostly below 25 km. In spite of its relative weakness and strong continuum overlapping by the Schumann-Runge system, it is therefore of considerable interest. Extensive numerical results which make up the bulk of the paper are intended to provide accurate theoretical information on the Herzberg I system and to illustrate the method of JARMAIN. The present study is much more complete than that of JARMAIN and NICHOLLS(~) except that analysis of electronic transition moment and total oscillator strength in the continuum is not repeated here.
THE HERZBERG I (A3C:--X3X;)
MOLECULAR
DATA
For the upper state, vibrational assignments are those fixed by BROIDA and GAYDON and confirmed by DEGEN and NICHOLLS.(51 The basic constants used in the program appear in Table 1. Values of I, are slightly higher than previously because of currently accepted values of h, c and N:. Bi is taken a trifle smaller than in Ref. (3) from BABCOCK and HERZBERG.(@ POTENTIAL
ENERGY
FUNCTIONS
Table 2 shows a Klein-Dunham potential energy function for the upper state in terms of turning points r1,2 at intervals of 0.5 in vibrational quantum number. All energies are 603
604
W. R. JAKMAIN
TABLE I. OS HERZ~EKG I SYSTEM: MOLECULARCONSTANTS
T:, - G"(0) (cm D‘.
(cm
I) I)
34610.7,t 6649.,t
B,, (cm 420471
* h = 6.62554 x IO 27 erg sec., c = 2.997925 x 10” I- Data taken from Ref. (3). : Data taken from Ref. (6).
I)
r,. (A) cm:sec.
N,
0.9106t 1.5215*
I .4457: I .2075*
= 6.0250 x IO*’
TABLE 2. KLEIN-DUNHAM POTENTIAL ENERGY I-UNCTION U(r) FOR 0,A3C:
v=
v+;
L’ (cm
‘)
r1 (A)
rz (A)
0.5 I .o I.5 2.0 2.5
396.42 786.20 I 169.34 1545.33 1913.13
1.454 I.429 I.41 I 1.397 I.385
I.600 1.637 1.668 1.696 I.722
3.0 3.5 4.0 4.5 S.0
2272.72 2624. I2 2966.73 3299.51 3622.44
1.375 1.366 1.357 1.350 1.343
1.748 I.772 I.797 1.822 1.846
5.5 6.0 6.5 7.0 7.5
3935.54 4237.87 4527.82 4805.39 5070.57
1.337 1.332 1.326 1.322 I.317
LX72 1.897 I.924 I.953 I.982
8.0 8.5 9.0 9.5 10.0
5321.78 5556.08 5773.47 5973.95 6154.63
I.313 I.310 I.306 I.304 I .301
2.013 2.050 2.089 2.131 2.187
IO.5 I I.0 1I.5
6309.73 6439.26 6543.22
1.298 I .296,, 1.296,
2.245 7 7’3 -.. 7.423
referred to the minimum of the function where I = re. Basic data are from HERZBERG,“’ with appropriate adjustments for an increase of one unit in v’ as in Ref. (3). The Klein-Dunham function for the lower state is similarly tabulated in Table 3. Vibrational data are from Ref. (6) for 11”= 0 and 1 ; FEAST@) for D” = 7-12; LOCHTEHOLTGRFXN and DIEKE(9) for v“ = 13-21; HERMAN et al. (lo) for v” = 22-24. Energies are interpolated for v” = 2-6 from a least squares ‘best’ polynomial fit and all are adjusted for a shift of origin from v” = 0 to v” = -3, the minimum of the potential curve. B”‘s are from the same sources, as compiled by WALLACE” ‘) except that for the last three groups, ranges of v” are 7-11, 12-20 and 21-24. Rotational constants B, to B, are also interpolated.
Realistic
Franck-Condon
factors
TABLE 3. KLEIN-DUNHAM
v=v++
and related
integrals
for diatomic
molecules-11
POTENTIAL ENERGY FUNCTION U(r) FOR 0,X3
U (cm-‘)
r1 (A)
r2 (A)
1.5 2.0 2.5
787.5 1568.8 2343.9 3113.1 3876.4
1.159 1.141 1.127 1.1 16 1.107
1.263 1.288 1.308 1.325 1.341
3.0 3.5 4.0 4.5 5.0
4633.9 5385.6 6131.5 6871.7 7606.1
1.098 1.091 1.084 1.078 1.072
1.356 1.371 1.384 1.397 1.410
5.5 6.0 6.5 7.0 7.5
8334.8 9057.7 9775.1 10486.9 11193.2
1.067 1.062 1.057 1.053 1.049
1.422 1.434 1.446 1.457 1.469
8.0 8.5 9.0 9.5 10.0
11893.2 12587.6 13276.5 13959.9 14637.8
1.045 1.041 1.037 1.034 1.031
1.480 1.491 1.502 1.513 1.524
10.5 11.0 11.5 12.0 12.5
15309.7 15975.7 16635.8 17291.1 17940.6
1.028 1.024 1.021 I.019 1.016
1.535 1.545 1.556 1.566 1.577
13.0 13.5 14.0 14.5 15.0
18584.4 19222.5 19854.4 20480.5 21100.9
1.013 1.011 1.008 1.006 1.003
1.588 1.598 1.609 1.619 1.630
15.5 16.0 16.5 17.0 17.5
21715.5 22326.5 22930.5 23527.5 24117.5
1.001 0.999 0.997 0.995 0.993
1.640 1.651 1.661 1.672 1.683
18.0 18.5 19.0 19.5 20.0
24702.9 25282.5 25856.4 26424.5 26985.5
0.99 1 0.989 0.987 0.985, 0.984
1.693 I.704 1.714 I.725 1.736
20.5 21.0 21.5 22.0 22.5
27540.5 28089.5 28632.5 29169.1 29699.5
0.982 0.980, 0.979 0.977 0.975,
1.747 1.758 1.769 1.780 1.791
23.0 23.5 24.0 24.5
30223.6 30741.5 31250.9 31752.5
0.974 0.972 0.970, 0.969
1.802 1.814 1.825 1.837
0.5 1.0
Z;
605
W. R. JARMAIN
606 ORTHOGONALITY
OF
WAVE
FUNCTIONS
Wave functions generated by the method of Ref. (2) for both states of the Herzberg I system have been tested for orthogonality by means of the important ‘noise factor’, equation (13) of that paper. In the upper state this factor varies from about IO-” to lo- ’ 5, and in the lower from 10-i’ to lo-*’ . In addition, the integral of the square of any wave function has a maximum deviation from unity of about 5 x lo-‘, occurring in the last place carried by the computer. Most are within 1 x lo-‘. EIGENVALUES
AND
ROTA-i’IONAL
ENERGIES
A comparison of observed eigenvalues E, for the upper state with those calculated by the program is shown in Table 4. Similarly, observed and calculated rotational energies B, are compared. Absolute values of deviations in both instances average practically the same as in Ref. (3) : near 0.8 cm- ’ for E, and 5 x 10e4 cm- ’ for B,. These are considered to be entirely acceptable.
TABLE 4. CALCULATEDAND OBSERVED EIGENVALUES AND B, (cm- ‘) FOR 0,,
0’ 0
6 I 8 9 10 11
A ‘Z:
Eigenvalue (talc.)
Eigenvalue (obs.)
Deviation
B, (talc.)
(obs.)
395.74 1168.83 1912.78 2622.85 3297.93 3934.41
396.42 1169.34 1913.13 2624.12 3299.5 1 3935.54
-0.68 -0.51 -0.35 - 1.27 - 1.58 - 1.13
0.9032 0.8874 0.8692 0.8488 0.8273 0.8030
0.9033 0.888 I 0.8687 0.849 I 0.8273 0.8034
- 0.0007 + 0.0005 - 0.0003 0.0000 ~ 0.0004
4526.75 5069.88 5555.41 5973.32 6309.93 6542.54
4527.82 5070.57 5556.08 5973.95 6309.73 6543.22
- 1.07 - 0.69 -0.61 -0.63 +0.20 -0.68
0.7761 0.7442 0.7061 0.6574 0.5926 0.4964
0.7765 0.7447 0.7064 0.658 I 0.5936 0.4970
- 0.0004 - 0.0005 - 0.0003 - 0.0007 -0.0010 - 0.0006
4
Deviation -O.OOQl
The same format is used in Table 5 for comparison of vibrational and rotational energies in the lower state, X3X;. Here absolute deviations in eigenvalues at high energies appear large but the maximum percentage deviation is actually smaller than for A3C:. Percentage deviations in B,‘s are very similar to those for the upper state. FRANCK-CONDON
FACTORS
An extended array of Franck-Condon factors (0 I u’ I 11, 0 I v” < 24) appears in Table 6. These are believed to approach maximum accuracy obtainable, using experimental data in Tables 1,4 and 5 and the method of Ref. (2). The sum rule can be checked only in the u’ = 0 progression with maximum u” = 24. There the deviation from unity is +4 x 10e6. For v’ = 1, the sum is nearly complete to seven places, with a deviation estimated to be of the same order as in v’ = 0.
Realistic
Franck-Condon
factors
and related
integrals
for diatomic
TABLE 5. CALCULATED AND OBSERVEDEIGENVALUESAND B, (cm-‘)
0”
Eigenvalue (talc.)
0 1
Eigenvalue (obs.)
607
molecules--II FOR O,, X3X;
Deviation
B” (talc.)
B” (obs.)
Deviation
2 3 4 5
787.4 2343.9 3876.5 5385.2 6871.3 8335.3
787.5 2343.9 3876.4 5385.6 6871.7 8334.8
-0.1 0.0 0.1 -0.4 -0.4 0.5
1.438 1.423 1.407 1.392 1.377 1.361
1.438 1.423 1.407 1.392 1.377 1.361
0.000 0.000 0.000 0.000 0.000 0.000
6 7 8 9 10 11
9775.3 11192.3 12587.2 13959.8 15309.5 16635.5
9775.1 11193.2 12587.6 13959.9 15309.7 16635.8
0.2 -0.9 -0.4 -0.1 -0.2 -0.3
1.346 1.330 1.315 1.299 1.283 1.267
1.346 1.329 1.316 1.298 1.282 1.269
0.000 0.001 -0.001 0.001 0.001 - 0.002
12 13 14 15 16 17
17938.3 19220.4 20478.6 21713.9 22927.3 24115.0
17940.6 19222.5 20480.5 21715.5 22930.5 24117.5
-2.3 -2.1 -1.9 -1.6 -3.2 -2.5
1.251 1.236 1.219 1.204 1.187 1.171
1.250 1.235 1.221 1.202 1.186 1.171
0.001 0.001 -0.002 0.002 0.001 0.000
18 19 20 21 22 23 24
25278.7 26419.7 27536.4 28627.9 29693.2 30735.1 31749.6
25282.5 26424.5 27540.5 28632.5 29699.5 30741.5 31752.5
-3.8 -4.8 -4.1 -4.6 - 6.3 -6.4 -2.9
1.155 1.139 1.123 1.106 1.089 1.073 1.053
1.156 1.139 1.121 1.105 1.091 1.075 1.055
-0.001 0.000 0.002 0.001 -0.002 - 0.002 - 0.002
FRANCK-CONDON
DENSITIES
Franck-Condon densities for the continua adjoining all progressions of bands (0 s a” I 24) are given in Table 7. Eigenvalues (cm- ‘) range from 6.7 x lo3 to 1.00 x 10’ and densities are in erg-’ as produced by the program. Variations of density in continua corresponding to 0 I u” I 4 are portrayed by Figs. l-5 respectively. For higher u”, TABLE 6. FRANCK--CONWN
FACTORSFOR THE HERZBERG I SYSTEMOF 0,
1
2
1.81-06 1.49-05 6.30-05 1.84-04 4.20-04
3.34-05 2.38-04 8.76 - 04 2.22 - 03 4.44 - 03
2.92 - 04 1.76-03 5.49 - 03 1.18-02 1.99-02
1.60-03 8.01-03 2.04-02 3.55 -02 4.82 -02
6.25 - 03 2.48 - 02 4.92 - 02 6.52 - 02 6.47-02
5 6 7 8 9
7.97 -04 1.31-03 1.87-03 2.37-03 2.62-03
7.38 - 03 1.07-02 1.35-02 1.54-02 1.54-02
2.79 -02 3.41-02 3.68 - 02 3.57-02 3.10-02
5.37 - 02 5.13-02 4.28-02 3.17-02 2.09-02
5.00-02 3.04 - 02 1.39-02 4.15-03 4.59 -04
10 11
2.46 - 03 1.71-03
1.33-02 8.76 - 03
2.37 -02 1.43 -02
1.23-02 6.01-03
4.93 - 05 3.51-04
0”
0
3
4
608
W. R. JARMAIN
TABLE6. (continued)
,“’
5
6
7
8
9
ii 2 3 4
1.84-02 5.52-02 7.92 - 02 7.05 - 02 4.10-02
4.26 - 02 8.98-02 8.16-02 3.59-02 3.90-03
7.93 - 02 1.05-01 4.46 - 02 1.44-03 1.16-02
1.21 -01 8.30-02 4.49 - 03 1.70-02 4.59-02
1.55-01 3.50-02 1.05-02 5.48 - 02 3.56-02
5 6 I 8 9
1.34-02 8.58 -04 1.72-03 7.86-03 1.23-02
2.71-03
3.37 -02
1.67-02
3.61-02
2.67 - 02 2.64 - 02 1.95-02
2.22 - 02 8.09 - 03 1.18-03
X54-02 1.03-02 3.75 -05 3.34-03 1.07 - 02
3.17-03 4.85-03 2.01 -02 2.45 - 02 1.77-02
10
1.25-02 8.57-03
1.15-02 5.37 -03
2.01 -05 4.54-04
1.20-02 8.31 -03
9.05 - 03 3.43 - 03
I
II t”’
10 I1 ,>I’
10
II
I2
13
I4
1.65-01 1.51 -03 5.49-02 4.68 - 02 1.70-03
1.50-01 1.51-02 7.59-02 6.82 - 03 1.85-02
1.14-01 6.77 -02 4.34 - 02 9.76 - 03 5.16-02
7.40-02 1.19-01 3.53-03 5.48 - 02 2.97 - 02
4.12-02 1.36-01 1.38-02 6.57-02 2.93 - 05
1.41-02 3.36-02 2.52-02 8.21 -03 4.56-04
4.23 - 02 1.96-02 6.21 -04 4.75 - 03 1.31-02
1.83-02 6.87 - 04 1.83-02 2.55-02 1.61 -02
1.70-03 3.02-02 2.85 -02 7.53 -03 9.23 -08
3.69 - 02 3.08 - 02 1.77-03 6.10-03 1.70-02
6.29 - 04 1.68-03
1.39-02 8.75 -03
5.93 - 03 1.34-03
X85-03 4.20-03
1.58-02 8.42 ~ 03
IS
I6
17
18
I9 _
1.95 - 02 1.14-01 7.00-02 2.43 - 02 2.96 - 02
7.92 - 03 7.58 - 02 1.20-01 3.64 - 04 6.59-02
2.74 ~ 03 4.07 - 02 1.28-01 3.86-02 4.16-02
7.96-04 1.78-02 9.68 -02 1.01-01 1.93-03
1.98-04 6.46 ~ 03 5.65 - 02 1.28-01 2.12-02
4.24 - 02 7.71 -04 1.66-02 2.81-02 1.47-02
5.23 - 03 2.23 - 02 3.47 - 02 8.13-03 3.91 -04
1.40-02 4.58 - 02 5.83 -03 6.54-03 2.11-02
5.87-02 1.34-02 1.14-02 3.27-02 1.52-02
5.14-02 5.42 - 03 4.35-02 1.30-02 7.20-04
3.06 - 03 1.15-04
6.42 - 03 7.30-03
1.70-02 7.28 - 03
1.40-03 l-75-04
1.01 -02 1.03-02
20
21
22
4.31-05 1.97-03 2.65 - 02 1.09-01 8.47 ~ 02
7.85 -06 5.15-04 1.02-02 6.86-02 1.26-01
1.22-06 1.16-04 3.21-03 3.34-02 1.16-01
23
2.14-07 2.10-05 8.24 ~ 04 1.30-02 7.60-02
24
2.19-08 2.62-06 1.73-04 4.13-03 3.86 -02
Realistic
Franck-Condon
factors
and related integrals
for diatomic
609
moleculesPI
TABLE 6. (continued)
23
24
20
21
22
6 7 8 9
7.26 - 03 4.97 - 02 2.13-02 3.72-03 2.50-02
1.17-02 5.70 - 02 1.09-03 3.71-02 2.05 - 02
7.31-02 1.33-02 3.99 - 02 2.91 -02 3.66 - 05
1.23-01 6.11-03 5.93-02 1.36-04 2.52 -02
1.23-01 6.51 -02 1.98-02 2.82-02 3.47 - 02
10 11
1.92-02 6.64 - 03
1.20-03 7.14 - 04
1.21-02 1.30-02
2.44-02 7.84 - 03
3.51-03 6.66 - 04
V”
v’ 5
The negative
number
in each entry is the power of 10 by which it is multiplied.
densities are significant for energies greater than 1.00 x 105, hence the sums of FranckCondon factors and integrated densities fall increasingly short of unity. At u” = 0, 1,2,3,4, the sums are respectively 0.999, 0.998, 0.999, 0.998 and 0.999. r-CENTROIDS
For each band with a calculated Franck-Condon factor, there is an r-centroid presented in Table 8. These have been derived from the original definition of r-centroid as in equation (19) of Ref. (2). A number of them depart from regularity (meaning a smooth relationship between r-centroid and wave length), notably the value for the (9,13) band. Its correspondence to a very small Franck-Condon factor probably makes it meaningless. No attempt has been made to smooth or ‘correct’ the array, because such procedures are outside the terms of reference for this paper. 40 000
I 0
;
0
- 30 000 x ^a & 25000 ; 20 000 z z % 15 000 S zz
10 000
0” : : L
5 000 0 5 000
15 000
25-000 Eigenvalue,
FIG. 1. Variation
of Franck-Condon
35000
45000
55.000
65-000
cm-lx 10’ density
with wave numbers,
v” = 0.
,131 11 ,319 11 ,544 11 ,772 11 ,977 11 ,114 12 ,133 12 .132 12 .118 12 ,976 11
,177 12 ,180 12 ,177 12 ,169 12 ,158 12 ,145 12 ,115 12 .851 11 ,604 11 ,412 11 ,272 11 ,175 11 ,110 11
,135 12 ,113 12 ,931 11 ,762 11 ,618 11 ,499 11 ,316 11 .195 11 ,118 11 ,700 10 .408 10 ,235 10 ,133 10
3.0 3.2 3.4 3.6 3.8
4.0 4.2 4.4
.756 11 ,559 11 ,395 11
2.5 2.6 2.7 2.8 2.9
.103 12 .127 12 ,132 12 ,121 12 ,969 11
.I74 11 .405 11 ,142 11 ,136 10 .176 10
.604 11 ,926 11 ,123 12 ,147 12 ,166 12
,270 12 ,242 12 ,213 12 ,185 12 ,159 12
2.0 2.1 2.2 2.3 2.4
.879 11 ,824 11 ,481 11 .144 11 ,123 9
,484 11 ,108 11 ,301 9 ,156 11 ,434 11 .700 11 ,876 11 ,930 11
,555 10 ,340 11 ,691 11 ,962 11 ,109 12 .107 12 ,961 11 ,799 11
,798 10 .294 11 .532 11
.702 8 ,721 10 ,275 11 ,525 11 ,745 11
,928 11 ,106 12 ,105 12 ,925 11 ,718 11
,675 11 ,390 1I ,166 11 ,345 10 ,981 8
,120 12 .lll 12 .784 11 ,404 11 ,114 11
,138 11 .246 8 ,101 11 ,362 11 ,671 11
,421 11 .200 10 .120 11 ,549 11 ,993 11
,116 12 .148 12 ,139 12 ,998 11 ,510 11
,453 11 .653 10 ,236 10 ,267 11 .654 11
,150 12 ,177 12 ,177 12 .155 12 .119 12
,414 11 .lOO 11 ,454 6 .842 10 .305 11
,335 12 ,337 12 .330 12 ,315 12 ,295 12
1.5 1.6 1.7 1.8 1.9
,329 10 ,725 11 .154 12 .169 12 .115 12
,184 12 ,828 11 .107 11 ,682 10 .566 11
1.1 1.2 1.3 1.4
,105 12 .180 12 .206 12 ,175 12 ,110 12
,569 11 .545 10 ,668 10 ,466 11 ,102 12
,341 12 ,296 12 ,232 12 ,160 12 ,934 11
,179 12 ,223 12 .263 12 .298 12 ,322 12
1.0
.265 12 ,262 12 ,266 12 ,176 12 .393 11
.220 11 ,303 11 ,516 11 ,178 12 ,236 12
,149 12 ,128 12 ,965 11 ,468 10 .237 11
,414 12 ,396 12 .384 12 ,282 12 ,158 12
,271 12 ,270 12 ,284 12 ,337 12 ,358 12
,551 11 ,563 11 ,624 11 ,966 11 .137 12
0.67 x 10“ 0.68 0.70 0.80 0.90
5
4
2
1
eigenvalue
0
3
,144 11 ,484 9 ,500 10
.866 10 ,477 11 ,762 11 ,717 11 ,435 11
,856 11 ,564 11 ,263 11 ,580 10 ,128 9
,132 10 ,259 11 ,642 11 ,936 11 ,101 12
,108 12 ,139 12 .113 12 ,563 11 ,108 11
,196 12 ,148 12 ,459 11 ,623 7 .396 11
.125 12 ,102 12 ,658 11 ,598 10 .114 12
6
,617 11 ,371 11 ,124 11
,356 11 ,131 10 ,126 11 .471 11 ,683 11
,442 11 ,734 11 ,875 11 ,820 11 ,617 11
,102 12 ,571 11 ,156 11 ,140 6 .137 11
,351 11 ,705 7 .309 11 ,868 11 ,117 12
,249 11 ,143 11 ,106 12 ,160 12 ,116 12
,314 11 ,445 11 ,772 11 ,216 12 ,158 12
7
,118 12 ,124 12 .513 11 ,112 10 ,240 11 ,799 11 ,104 12 ,766 11 ,286 11 ,100 10
,259 11 ,355 10 .582 11 ,112 12 ,106 12 ,542 11 ,807 10 ,390 10 ,363 11 .743 11 .897 11 .750 11 ,427 11 ,129 11 ,110 9 ,769 10 .518 11 ,684 11 ,396 11 ,626 10
,314 11 ,778 11 ,101 12 .881 11 .514 11 ,158 11 .159 9 ,900 10 ,337 11 ,599 11 ,757 11 .613 11 ,183 11 ,628 8 ,184 11 ,480 11 ,619 11 ,524 11
,111 12 ,133 12 .839 11 ,206 11 ,686 9
,271 10 ,252 11 ,489 11
.535 11 ,169 12 ,112 12 ,769 10 .325 11
,973 11 ,229 9 ,677 11 ,151 12 .115 12
.132 12 ,176 12 ,819 11 ,254 10 .334 11
,204 11 ,506 9 ,867 10
,330 11 .991 6 ,270 11 ,603 11 ,531 11
.106 11 .434 11 ,726 11 ,791 11 ,618 11
.849 11 ,106 12 ,152 11 ,192 12 ,181 11
,640 11 .425 11 ,137 11 ,847 11 ,202 12
,253 12 ,245 12 ,234 12 ,621 11 ,110 11
10
9
8
TABLE 7. FRANCK-CONDON DENSITIES(erg-‘) FOR THE CONTINUAOF THE HERZBERGI SYSTEMOF 0, (0 5 u” < 24)
,561 11 ,338 11 ,683 10
.702 11 .488 11 .470 10 ,993 10 ,447 11
,471 11 .102 11 ,762 9 ,203 11 ,505 11
,935 10 ,712 10 ,516 11 ,893 11 ,849 11
,159 11 .152 11 ,893 11 ,118 12 ,679 11
,162 12 ,176 11 .392 11 ,144 12 .112 12
,244 12 ,221 12 ,177 12 ,841 8 .144 12
11
E
.691 11
,289 11 ,516 11 ,424 11
,699 10 ,560 11 ,569 11 .145 11 ,164 10
.I22 11 ,833 11 ,587 11 ,220 11 ,996 9
h
,209 11 ,733 10 ,108 12 .754 11 ,188 11 ,945 9 ,316 11
r P 2
,127 12 .107 12
,581 10 ,141 12 ,136 12 ,130 11 ,352 11
.112 11 ,207 10 .494 10 ,201 12 .107 12
12
9 8 8 8 7 1 6 6 6 5 5 4 3 3 2
,233 ,944 ,377 ,151 ,601 ,234 ,920 ,375 ,148 ,529 ,195 ,484 ,623 ,356 ,363
5.0 5.3 5.6 5.9 6.2
6.5 6.8 7.1 7.4 1.1
8.0 8.5 9.0 9.5 10.0
0 ,752 9 ,421 9
v”
4.6 4.8
eigenvalue (cm - I)
,445 ,828 .253 ,394 .125
6 5 5 4 4
,428 ,105 .222 ,651 .130
,283 ,126 .558 ,242 ,102 7 7 6 5 5
9 9 8 8 8
.288 10 ,136 10 ,628 9
,482 9 .207 9 .878 8 8 8 7 7 7
,593 10
,374 ,157 ,639 ,261 .lll
,117 11
,110 10
,271 11 ,181 11
2
,243 10
,675 10 .408 10
1
8 7 7 6 6
10 9 9 9 8
,138 ,659 .308 .142 .642 ,285 .716 .182 .452 ,112
11 11 11 10 10
,339 ,197 ,107 ,560 ,282
.627 11 ,471 11
3
.138 .375 ,102 ,264 ,697
,483 ,249 ,125 .610 ,293
,643 ,440 ,276 ,162 ,902
9 8 8 7 6
10 10 10 9 9
11 11 11 11 10
,887 11 ,780 11
4
11 10 10 10 10 9 9 8 8 7
,518 .156 .445 .126 ,338
11 11 11 11 11
,128 ,719 .390 .204 .104
,800 ,683 .509 ,345 ,216
,713 11 ,802 11
5
6
10 9 9 8 8
11 9 10 10 10
,263 ,164 ,970 .549 ,300 ,159 .521 ,161 .483 .I39
11 11 11 11 11
,598 ,716 .675 ,547 ,396
.220 11 ,428 11
TABLE 7. (continued)
,404 ,146 .494 ,157 ,483
,428 ,330 ,196 .121 .714
,184 ,448 ,615 ,639 .555
10 10 9 9 8
11 11 11 11 10
11 11 11 11 11
,477 9 ,401 10
7
,870 ,350 ,129 ,445 ,145
,544 ,442 ,326 ,222 ,143
,214 ,968 ,321 ,516 .584
10 10 10 9 9
11 11 11 11 11
9 10 11 11 11
,300 11 ,945 10
8
11 10 10 11 11 11 11 11 11 11 11 10 10 10 9
.230 ,167 .470 ,237 ,428 ,526 .519 ,443 ,340 ,241 ,160 ,726 ,295 ,110 ,386
,554 11 ,434 11
9
11 11 10 10 9
.254 ,131 ,593 ,242 ,914
,346 ,208 ,106 ,475 .194
11 11 11 10 10
11 11 11 11 11
,128 ,296 ,417 ,454 .420 11 11 11 11 11
,355 .468 ,488 ,435 ,347
,402 ,289 .168 ,837 ,373
,282 .977 ,251 ,372 ,421
11 11 11 10 10
9 10 11 11 11
10 11 11 11 10 11 11 11 10 3
,367 ,454 ,261 ,495 .823
,486 ,254 ,346 .209 ,172
,622 ,331 ,422 .259 ,591
,162 11 ,513 9
,967 9 ,165 11 ,317 11 ,486 11 11 11 10 10 11
12 11
10
,418 ,559 ,967 ,770 ,261 ,568 ,189 ,568 ,766 ,639 ,320 ,120 ,415 ,585 ,244 ,876 ,163 ,429 .456 ,250
2.0 2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9
3.0 3.2 3.4 3.6 3.8
4.0 4.2 4.4 4.6 4.8 8 11 11 1I II
11 10 II 11 11
8 11 11 11 11
10 11 11 11 11
IO 11 12 11 I1
.318 ,239 ,778 ,333 ,445
.684 .385 .924 ,295 .554
,747 .298 .102 ,115 ,447
11 10 10 11 1I
11 11 7 11 11
11 11 10 11 11
11 11 10 11 11
12 11 10 11 12
.135 ,824 ,217 .396 ,107
,703 ,358 ,115 ,980 ,221
1.5 1.6 1.7 1.8 1.9 ,872 ,199 ,320 ,478 ,871
10 12 12 11 11
.758 .I00 .154 ,184 ,402
,189 ,549 ,192 .139 ,109
1.0 1.1 1.2 1.3 1.4
12 11 II 12 12
12 12 11 11 12
.201 ,160 .898 .808 ,195
12 12 12 11 11
.174 ,195 .233 .750 ,478
f’
0.67 x lo4 0.68 0.70 0.80 0.90
\‘)
14
(cm
13
eigenvalue
IO 11 11 9 11 11 11 10 10 1I
.504 ,366 ,621 ,292 ,246
11 11 11 11 10
,438 ,798 ,706 ,308 ,211 ,748 ,609 ,419 ,958 ,206
1I 12 11 11 10
,464 ,101 ,756 ,160 ,337
11 10 11 II IO
,298 ,540 ,542 ,430 ,254 11 11 11 11 9
I1 10 11 11 11
,112 ,448 ,422 ,737 ,653
,143 ,449 .387 ,101 .642
11 10 I1 11 11
12 11 10 11 12
,140 ,533 ,381 ,845 ,108
10 1I 12 11 8
,115 ,640 ,125 ,571 .122 ,339 ,186 ,541 ,941 ,633
11 11 12 11 11
12 12 10 12 12
,587 ,575 ,168 ,236 ,479
,167 ,102 ,662 ,136 ,104
12 12 12 6 12
,263 ,270 ,265 ,193 ,206
,384 ,177 .696 .211 .831
10 11 11 12 7
16
15
,391 ,103 ,395 ,390 ,133
,689 .269 ,457 ,492 ,422
.865 ,502 .627 ,666 .425
.976 .850 ,153 ,815 ,619
.481 .952 ,113 ,158 .220
,108 ,149 ,580 .135 ,978
,135 ,852 ,184 ,224 ,504
17
10 I1 11 II 11
11 11 10 11 11
1I 11 10 10 11
11 11 11 10 11
9 1I 12 11 11
12 12 9 12 11
12 11 11 12 I1
1I 10 10 11 11
II 1I 11 10 11
,100 ,648 ,231 ,454 ,455 ,404 .478 ,772 .351 ,383
II 11 11 11 10
,185 ,685 ,770 ,371 ,241
12 1I 11 12 II
,141 ,247 ,295 ,117 ,675 9 11 11 11 10
12 1I 12 11 II
,137 ,222 ,177 ,280 ,587
,241 ,479 ,988 ,585 ,355
,332 ,314 ,245 ,822 ,172
11 11 12 11 12
,471 ,869 ,181 .948 .126
,428 ,381 .505 ,620 ,316
,247 ,145 ,607 ,188 ,510
.328 .338 ,314 ,716 ,651
.104 ,250 .770 ,710 ,872
,704 ,123 .791 ,183 ,753
,396 .188 ,994 ,137 ,899
19
18
TABLE 7. (continued)
11 11 10 IO 11
11 11 11 11 IO
II 8 11 11 11
12 1I 10 1I I1
10 12 11 9 1I
11 12 9 12 1I
12 12 12 11 12
,623 ,408 ,351 ,477 ,540
,704 ,139 ,200 ,562 ,140
,829 ,654 ,124 ,474 ,441
.269 .101 ,650 .203 ,303
.134 .455 ,723 .115 ,161
,211 .I88 .183 ,324 ,731
,888 ,367 ,428 ,267 ,222
20
10 11 11 10 10
11 11 11 11 11
I1 1I 11 10 11
11 12 11 IO 11
12 10 11 12 11
12 10 12 1I 11
11 11 9 12 1I
,945 ,798 ,392 ,318 ,408
.159 .642 ,560 .261 ,506
,430 ,558 ,793 .393 ,145
,525 ,187 ,701 .900 ,239
,231 ,140 ,355 ,255 .115
,774 ,210 ,781 ,141 ,800
10 10 11 11 10
11 11 10 11 11
10 11 11 11 10
11 10 11 1I 11
11 12 II 11 12
9 12 10 12 11
.107.12 ,171 12 .291 12 ,382 10 ,260 12
21
,441 ,538 ,103 ,380 ,281
.162 ,303 ,532 ,882 ,321
,528 ,143 .255 .713 ,615
,894 ,838 ,444 .342 ,902
,117 ,145 ,117 ,728 ,239
,238 ,486 ,184 ,352 ,922
,399 ,353 ,218 ,248 ,152
22
11 10 11 11 11
11 11 11 9 11
II 10 11 11 11
11 1I 10 1I 11
12 10 12 11 10
12 10 12 11 11
12 12 12 12 11
,372 .368 ,225 ,132 ,368
,700 ,242 ,434 ,393 ,251
.724 .734 ,155 .561 ,505
,248 ,560 ,982 .240 ,932
,509 ,136 ,366 .820 .996
,275 ,209 ,205 ,149 ,662
,821 ,212 ,170 ,188 ,216
23
II 11 10 1I 11
11 10 11 11 9
11 II 11 10 11
10 11 11 11 IO
11 12 10 11 11
11 12 11 12 1I
11 11 11 12 12
12 11 12 11 12
12 12 12 11 12
1I 11 11 11
,359 ,402 .283 ,380 ,163
,278 .525 ,855 .510 ,240
10 1I 11 9 11
11 1I 9 11 1I
,184 9 ,477 11 ,791 11 .350 11 ,264 9
.163 ,275 ,945 ,467
.185 11 .495 11 .109 12
,897 11 ,235 11 ,136 12
,194 ,357 ,181 ,362 ,116
,138 ,231 ,383 ,378 ,156
24
z
2
;cl
<
m L
01 ZS9' II CEI' II 8tZ' II ISE' II 06E'
0'01 5'6 0'6 5'8 O'S
aq1 II POI‘ II 161‘ II 862' 11 99E' II EOE'
aql 01 laqwnu II ZSI' II 6PZ' II ZEE' II PZE' II ZLI’
L’L P’L I’L 8'9 S'9
klUa q3ea JO II@ II II II II 01
II SEE' II 9IZ' 019Ll' 8 E8L' 019E9'
JaMod ayl sau?xpu!
II 9PZ' II EOE' II 592' II 911' 8 PLE’
II 061' 01 9P9' 8 OOZ' 01 OE9' II EEZ'
6q 01 JO
II ELZ’ II 6LZ’ 119LI' 0108Z' 01 SPE'
01 OLS’ L SP8' 0168s II SIC II 8EE'
q3y~
II 181‘
II ELZ’ II OZZ' 01 PZ8' 8 P6E' II 9ZI'
8 PZI' 01 ZZS' II P61' II LIE' II ZLZ‘
s! J!
II 811’
II EPZ' IllPI' 01 891' 0168E' II IIZ'
01 LEP' II ILI' II 882' II 99Z' II 901'
‘pagd!$lnw 01 IS-S’ 01 8E9‘ 6 EZI' 11911' II SEZ'
II 6PI' II E9Z' II Z9Z' II ZZI' 6 081'
01 PZI’ 01 LZI’ 01 LSE’ 11981' II t81' II 8fZ' I 19SZ' II ObI' 01001' 01785'
ZOZ' 16Z' ZZE' 622' LZS'
6 ZOI' 01 166' II OIZ' 01556' II LPZ‘ II LSI’ 01 PPZ' 01 SLZ’ II ZOZ'
019lZ' II LSI’ II SLI’ 01 LLI’ 11 01 6 II II
Z'9 63 9'S E'S O'S
ZLI’ ItiP' L8L' S-PI' LLZ’
01919' 8 9fI' 01 626' II PPZ' II 602'
II 6PZ' II E6E' II 9057 01819' 01 EPP
6 S6E' 0166P' II 661' II EEZ' 01 ZIL’
E9E' 68Z' L99' 685' ZIE'
01681' II SPI' II ZEZ' II SZI' 8 EPI'
II II 01 01 II
(I_ ~13)anpua%!a
11 6LZ' 01 191' 01 ISI' II 952' II 8OP'
I!
01 L88' 6 POS' II OOZ‘ II 98E‘ II SLI'
EI
8 LZZ. II 9PI' II ZSE' II EZZ' L OL9'
PI
01 P86' II 60E' II Z9Z' 01 OLI' II 9SI'
ST
8SZ' L8Z' 9LS S99' 89E'
IPZ' 16L' IIE' PSI' 80E'
91
II II 01 01 II
01 01 II II 01
LI
II 762' II III' 01 ZEI' II 082' II 882'
01 181‘ II 8fZ‘ II 9PZ' 6 6EE' II EPZ'
81
II S91' 8 659' II PLI’ II EEE' 6 PIS'
11 ObI' II 182' 01 LL9' 01 EE8' II 8% zz
61
OS
EZ
‘1 TIEV~
IZ
PZ
(pmqUOJ)
614
W. R. JARMAIN TABLE 8. PCENTROIDS(A) FOR THE HERZBERGSYSTEMOF O2
V’I
0
1
2
3
4
5
6
7
8
9
u’ 0
1.35
1.36
1.38
1.40
1.41
1.43
1.45
1.47
1.49
1 2 3 4
1.34 1.33 1.32 1.32
1.35 1.35 1.34 1.33
1.37 1.36 1.35 1.35
1.39 1.38 1.37 1.36
1.40 1.39 1.39 1.38
1.42 1.41 1.40 1.39
1.44 1.43 1.42 1.40
1.46 1.44 1.42 1.43
1.41 1.46 1.46 1.45
1.51 1.49 1.49 1.47 1.46
5 6 7 8 9
1.31 1.31 1.30 1.30 1.29
1.33 1.32 1.32 1.31 1.31
1.34 1.33 1.33 1.32 1.32
1.35 1.35 1.34 1.34 1.33
1.37 1.36 1.36 1.35 1.33
1.38 1.37 1.38 1.37 1.37
1.41 1.40 1.39 1.38 1.38
1.42 1.41 1.40 1.40 1.38
1.44 1.43 1.34 1.42 1.42
1.45 1.46 1.44 1.44 1.43
10 11
1.29 1.29
1.30 1.30
1.32 1.31
1.33 1.33
1.39 1.35
1.36 1.36
1.37 1.37
1.47 1.40
1.41 1.41
1.42 1.42
0’3
10
11
12
13
14
15
16
17
18
19
1 2 3 4
1.53 1.50 1.51 1.49 1.41
1.55 1.54 1.53 1.51 1.51
1.57 1.56 1.54 1.54 1.52
1.60 1.58 1.56 1.56 1.54
1.62 1.61 1.60 1.58 1.40
1.65 1.63 1.62 1.60 1.59
1.67 1.66 1.64 1.68 1.61
1.70 1.68 1.61 1.66 1.64
1.73 1.71 1.70 1.68 1.64
1.76 1.74 1.72 1.71 1.70
5 6 7 8 9
1.48 1.41 1.46 1.45 1.42
1.49 1.48 1.45 1.48 1.47
1.51 1.53 1.50 1.49 1.48
1.55 1.53 1.51 1.50 3.59
1.56 1.54 1.52 1.54 1.52
1.58 1.54 1.56 1.55 1.54
1.59 1.59 1.58 1.56 1.60
1.63 1.61 1.59 1.60 1.59
1.65 1.63 1.63 1.62 1.60
1.67 1.68 1.65 1.63 1.66
10 11
1.46 1.45
1.46 1.46
1.47 1.46
1.51 1.50
1.52 1.51
1.53 1.49
1.56 1.56
1.58 1.57
1.58 1.64
1.63 1.62
uII
20
21
22
23
24
i I 2 3 4
1.79 1.77 1.75 I .74 1.72
1.83 1.81 1.78 1.77 1.75
1.88 1.84 1.82 1.80 1.78
1.86 1.87 1.85 1.83 1.81
1.86 1.94 1.89 1.86 1.84
5 6 7 8 9
1.69 1.69 1.67 1.68 1.66
1.74 1.71 1.73 1.69 1.68
1.76 1.73 1.73 1.71 1.86
1.79 1.79 1.76 1.65 1.74
1.82 1.81 1.78 1.78 1.76
10 1’ .
1.65 1.64
I.65 1.69
1.70 1.69
1.72 1.71
1.73 1.77
c’ 0
CONCLUSION
Klein-Dunham potential energy functions in Tables 2 and 3 are the basis for all other numerical results. It seems desirable, therefore, to compare them with previously published ‘equivalent’ RKR potentials. For the A3C: state, turning points agree closely with those of VANDERSLICE et al.“*) Deviations range from 0.000 to 0.002 A with the exception of r2
Realistic FranckCondon
factors and related integrals for diatomic molecules-11
615
40~000
25.000
5.000
0 5000
15caY
25000
35occJ
E~genvolue. FIG. 2. Variation of Franck-Condon
45000
55-000
65ooO
cm-lx IO’
density with wave numbers, 0” = 1.
,
for a’ = 9 where it is 0.004 8. When there are variations, present values run consistently lower on both limbs, possibly because the energy levels are slightly lower or I, differs. For the ground state we find even better agreement with the curve of VANDERSLICE et a1.‘t3’ and deviations are not systematic. Finally, Franck-Condon factors in Table 6 clearly supersede the quite different limited set of JARMAIN et ~1.“~’ The method used is not only considerably more accurate but incorporates the revised apparently correct vibrational assignment of Ref. (4).
0 0 -
: El
0
48000
-
42000
-
I6 000
v I
‘:
12000
s t
6000
0 5000
15.000
25000
35.000
Elgenvalue, FIG. 3. Variation of Franck-Condon
45coo
55000
650X
cni’x 103
density with wave numbers, u” = 2.
W. R. JARMAIN
616 24.000
I
- 12 000 )I c E g 9 000 z z
6-000
0” i k IL’
3000 0 5000
15000
25000
35.ccc Etgenval ue,
FIG. 4. Variation
of FranckkCondon
Etgenvotue, FIG. 5. Variation
of FranckkCondon
45000 cm-lx
density
55 000
65 000
IO3
with wave numbers,
a” = 3.
cm-’ x IO’ density
with wave numbers,
U” = 4
Acknowledgements~This project has been supported by grants from the National Research Council whose assistance is gratefully acknowledged. Consultations with Dr. R. W. NICHOLLS and Mr. J. C. MCCALLUM have been most helpful.
of Canada,
Realistic
Franck-Condon
factors and related integrals for diatomic
617
molecules--II
REFERENCES 1. R. W. DITCHBURN and P. A. YOUNG, J. Amos. Terr. Phys. 24, 127 (1962) 2. W. R. JARMAIN, JQSRT 11,421 (1971). 3. W. R. JARMAIN and R. W. NICHOLLS, Proc. Phys. Sot. 90,545 (1967). 4. H. P. BROIDA and A. G. GAYDON. Proc. R. Sot. A222. 181 (1954). 5. V. DEGEN and R. W. NICHOLLS, J. phys. B, Proc. Phys. Sot. 11, I240 (1969). 6. H. D. BABCOCK and L. HERZBERG, Astrophys. J. 108, 167 (1948). 7. G. HERZBERG, Can. J. Phys. 30, 185 (1952). 8. M. W. FEAST, Proc. Phys. Sot. A63, 549 (1950). 9. W. LOCHTE-HOLTGREVENand G. H. DIEKE, Ann. phys. Leipzig Ser. 5,3,937 (1929). 10. L. HERMAN, R. HERMAN and D. RAKOTOARIJIMY, J. Phys. Rad. Ser. 8.22, 1 (1961). Il. L. WALLACE, Astrophys. J. Suppl. Ser. 7, 165 (1962). 12. J. T. VANDERSLICE.E. A. MASON, W. G. MAISCH and E. R. LIPPINCOTT, J. them. Phys. 33,614 13. J. T. VANDERSLICE,E. A. MACON and W. G. MAISCH, J. them. Phys. 32,5 15 (1960). 14. W. R. JARMAIN, P. A. FRASER and R. W. NICHOLLS, Astrophys. J. 122,55 (1955).
(1960).
Spectrosc.
Rod&.
Transfer.
Vol.
12, pp. 603417.
Pergamon Press 1972. Printed in Great Britain
REALISTIC FRANCK-CONDON FACTORS AND RELATED INTEGRALS FOR DIATOMIC MOLECULES-II. THE 0, HERZBERG I SYSTEM W. R. JARMAIN Department of Physics, University of Western Ontario, London 72, Canada (Received 2 July 1971) Abstract-Tabulated Klein-Dunham potentials are given for the Herzberg I system of 0,. calculated by the method of paper I in this series. Accurate numerical treatment also includes orthogonality tests of wave functions; comparisons of observed and calculated eigenvalues and B, for both states ; Franck-Condon factors and r-centroids over ranges 0 I u’ I 11, 0 < u” I 24; Franck-Condon densities in the continua for 0 < u” I 24 and graphs of their variation for 0 I u” 5 4.
INTRODUCTION
system of oxygen, a forbidden transition, is an important contributor to the ultraviolet night airglow. Measurements by DITCHBURN and YOUNG(‘) support earlier suggestions that absorption in the Herzberg continuum is the main source of oxygen atoms below 80 km, mostly below 25 km. In spite of its relative weakness and strong continuum overlapping by the Schumann-Runge system, it is therefore of considerable interest. Extensive numerical results which make up the bulk of the paper are intended to provide accurate theoretical information on the Herzberg I system and to illustrate the method of JARMAIN. The present study is much more complete than that of JARMAIN and NICHOLLS(~) except that analysis of electronic transition moment and total oscillator strength in the continuum is not repeated here.
THE HERZBERG I (A3C:--X3X;)
MOLECULAR
DATA
For the upper state, vibrational assignments are those fixed by BROIDA and GAYDON and confirmed by DEGEN and NICHOLLS.(51 The basic constants used in the program appear in Table 1. Values of I, are slightly higher than previously because of currently accepted values of h, c and N:. Bi is taken a trifle smaller than in Ref. (3) from BABCOCK and HERZBERG.(@ POTENTIAL
ENERGY
FUNCTIONS
Table 2 shows a Klein-Dunham potential energy function for the upper state in terms of turning points r1,2 at intervals of 0.5 in vibrational quantum number. All energies are 603
604
W. R. JAKMAIN
TABLE I. OS HERZ~EKG I SYSTEM: MOLECULARCONSTANTS
T:, - G"(0) (cm D‘.
(cm
I) I)
34610.7,t 6649.,t
B,, (cm 420471
* h = 6.62554 x IO 27 erg sec., c = 2.997925 x 10” I- Data taken from Ref. (3). : Data taken from Ref. (6).
I)
r,. (A) cm:sec.
N,
0.9106t 1.5215*
I .4457: I .2075*
= 6.0250 x IO*’
TABLE 2. KLEIN-DUNHAM POTENTIAL ENERGY I-UNCTION U(r) FOR 0,A3C:
v=
v+;
L’ (cm
‘)
r1 (A)
rz (A)
0.5 I .o I.5 2.0 2.5
396.42 786.20 I 169.34 1545.33 1913.13
1.454 I.429 I.41 I 1.397 I.385
I.600 1.637 1.668 1.696 I.722
3.0 3.5 4.0 4.5 S.0
2272.72 2624. I2 2966.73 3299.51 3622.44
1.375 1.366 1.357 1.350 1.343
1.748 I.772 I.797 1.822 1.846
5.5 6.0 6.5 7.0 7.5
3935.54 4237.87 4527.82 4805.39 5070.57
1.337 1.332 1.326 1.322 I.317
LX72 1.897 I.924 I.953 I.982
8.0 8.5 9.0 9.5 10.0
5321.78 5556.08 5773.47 5973.95 6154.63
I.313 I.310 I.306 I.304 I .301
2.013 2.050 2.089 2.131 2.187
IO.5 I I.0 1I.5
6309.73 6439.26 6543.22
1.298 I .296,, 1.296,
2.245 7 7’3 -.. 7.423
referred to the minimum of the function where I = re. Basic data are from HERZBERG,“’ with appropriate adjustments for an increase of one unit in v’ as in Ref. (3). The Klein-Dunham function for the lower state is similarly tabulated in Table 3. Vibrational data are from Ref. (6) for 11”= 0 and 1 ; FEAST@) for D” = 7-12; LOCHTEHOLTGRFXN and DIEKE(9) for v“ = 13-21; HERMAN et al. (lo) for v” = 22-24. Energies are interpolated for v” = 2-6 from a least squares ‘best’ polynomial fit and all are adjusted for a shift of origin from v” = 0 to v” = -3, the minimum of the potential curve. B”‘s are from the same sources, as compiled by WALLACE” ‘) except that for the last three groups, ranges of v” are 7-11, 12-20 and 21-24. Rotational constants B, to B, are also interpolated.
Realistic
Franck-Condon
factors
TABLE 3. KLEIN-DUNHAM
v=v++
and related
integrals
for diatomic
molecules-11
POTENTIAL ENERGY FUNCTION U(r) FOR 0,X3
U (cm-‘)
r1 (A)
r2 (A)
1.5 2.0 2.5
787.5 1568.8 2343.9 3113.1 3876.4
1.159 1.141 1.127 1.1 16 1.107
1.263 1.288 1.308 1.325 1.341
3.0 3.5 4.0 4.5 5.0
4633.9 5385.6 6131.5 6871.7 7606.1
1.098 1.091 1.084 1.078 1.072
1.356 1.371 1.384 1.397 1.410
5.5 6.0 6.5 7.0 7.5
8334.8 9057.7 9775.1 10486.9 11193.2
1.067 1.062 1.057 1.053 1.049
1.422 1.434 1.446 1.457 1.469
8.0 8.5 9.0 9.5 10.0
11893.2 12587.6 13276.5 13959.9 14637.8
1.045 1.041 1.037 1.034 1.031
1.480 1.491 1.502 1.513 1.524
10.5 11.0 11.5 12.0 12.5
15309.7 15975.7 16635.8 17291.1 17940.6
1.028 1.024 1.021 I.019 1.016
1.535 1.545 1.556 1.566 1.577
13.0 13.5 14.0 14.5 15.0
18584.4 19222.5 19854.4 20480.5 21100.9
1.013 1.011 1.008 1.006 1.003
1.588 1.598 1.609 1.619 1.630
15.5 16.0 16.5 17.0 17.5
21715.5 22326.5 22930.5 23527.5 24117.5
1.001 0.999 0.997 0.995 0.993
1.640 1.651 1.661 1.672 1.683
18.0 18.5 19.0 19.5 20.0
24702.9 25282.5 25856.4 26424.5 26985.5
0.99 1 0.989 0.987 0.985, 0.984
1.693 I.704 1.714 I.725 1.736
20.5 21.0 21.5 22.0 22.5
27540.5 28089.5 28632.5 29169.1 29699.5
0.982 0.980, 0.979 0.977 0.975,
1.747 1.758 1.769 1.780 1.791
23.0 23.5 24.0 24.5
30223.6 30741.5 31250.9 31752.5
0.974 0.972 0.970, 0.969
1.802 1.814 1.825 1.837
0.5 1.0
Z;
605
W. R. JARMAIN
606 ORTHOGONALITY
OF
WAVE
FUNCTIONS
Wave functions generated by the method of Ref. (2) for both states of the Herzberg I system have been tested for orthogonality by means of the important ‘noise factor’, equation (13) of that paper. In the upper state this factor varies from about IO-” to lo- ’ 5, and in the lower from 10-i’ to lo-*’ . In addition, the integral of the square of any wave function has a maximum deviation from unity of about 5 x lo-‘, occurring in the last place carried by the computer. Most are within 1 x lo-‘. EIGENVALUES
AND
ROTA-i’IONAL
ENERGIES
A comparison of observed eigenvalues E, for the upper state with those calculated by the program is shown in Table 4. Similarly, observed and calculated rotational energies B, are compared. Absolute values of deviations in both instances average practically the same as in Ref. (3) : near 0.8 cm- ’ for E, and 5 x 10e4 cm- ’ for B,. These are considered to be entirely acceptable.
TABLE 4. CALCULATEDAND OBSERVED EIGENVALUES AND B, (cm- ‘) FOR 0,,
0’ 0
6 I 8 9 10 11
A ‘Z:
Eigenvalue (talc.)
Eigenvalue (obs.)
Deviation
B, (talc.)
(obs.)
395.74 1168.83 1912.78 2622.85 3297.93 3934.41
396.42 1169.34 1913.13 2624.12 3299.5 1 3935.54
-0.68 -0.51 -0.35 - 1.27 - 1.58 - 1.13
0.9032 0.8874 0.8692 0.8488 0.8273 0.8030
0.9033 0.888 I 0.8687 0.849 I 0.8273 0.8034
- 0.0007 + 0.0005 - 0.0003 0.0000 ~ 0.0004
4526.75 5069.88 5555.41 5973.32 6309.93 6542.54
4527.82 5070.57 5556.08 5973.95 6309.73 6543.22
- 1.07 - 0.69 -0.61 -0.63 +0.20 -0.68
0.7761 0.7442 0.7061 0.6574 0.5926 0.4964
0.7765 0.7447 0.7064 0.658 I 0.5936 0.4970
- 0.0004 - 0.0005 - 0.0003 - 0.0007 -0.0010 - 0.0006
4
Deviation -O.OOQl
The same format is used in Table 5 for comparison of vibrational and rotational energies in the lower state, X3X;. Here absolute deviations in eigenvalues at high energies appear large but the maximum percentage deviation is actually smaller than for A3C:. Percentage deviations in B,‘s are very similar to those for the upper state. FRANCK-CONDON
FACTORS
An extended array of Franck-Condon factors (0 I u’ I 11, 0 I v” < 24) appears in Table 6. These are believed to approach maximum accuracy obtainable, using experimental data in Tables 1,4 and 5 and the method of Ref. (2). The sum rule can be checked only in the u’ = 0 progression with maximum u” = 24. There the deviation from unity is +4 x 10e6. For v’ = 1, the sum is nearly complete to seven places, with a deviation estimated to be of the same order as in v’ = 0.
Realistic
Franck-Condon
factors
and related
integrals
for diatomic
TABLE 5. CALCULATED AND OBSERVEDEIGENVALUESAND B, (cm-‘)
0”
Eigenvalue (talc.)
0 1
Eigenvalue (obs.)
607
molecules--II FOR O,, X3X;
Deviation
B” (talc.)
B” (obs.)
Deviation
2 3 4 5
787.4 2343.9 3876.5 5385.2 6871.3 8335.3
787.5 2343.9 3876.4 5385.6 6871.7 8334.8
-0.1 0.0 0.1 -0.4 -0.4 0.5
1.438 1.423 1.407 1.392 1.377 1.361
1.438 1.423 1.407 1.392 1.377 1.361
0.000 0.000 0.000 0.000 0.000 0.000
6 7 8 9 10 11
9775.3 11192.3 12587.2 13959.8 15309.5 16635.5
9775.1 11193.2 12587.6 13959.9 15309.7 16635.8
0.2 -0.9 -0.4 -0.1 -0.2 -0.3
1.346 1.330 1.315 1.299 1.283 1.267
1.346 1.329 1.316 1.298 1.282 1.269
0.000 0.001 -0.001 0.001 0.001 - 0.002
12 13 14 15 16 17
17938.3 19220.4 20478.6 21713.9 22927.3 24115.0
17940.6 19222.5 20480.5 21715.5 22930.5 24117.5
-2.3 -2.1 -1.9 -1.6 -3.2 -2.5
1.251 1.236 1.219 1.204 1.187 1.171
1.250 1.235 1.221 1.202 1.186 1.171
0.001 0.001 -0.002 0.002 0.001 0.000
18 19 20 21 22 23 24
25278.7 26419.7 27536.4 28627.9 29693.2 30735.1 31749.6
25282.5 26424.5 27540.5 28632.5 29699.5 30741.5 31752.5
-3.8 -4.8 -4.1 -4.6 - 6.3 -6.4 -2.9
1.155 1.139 1.123 1.106 1.089 1.073 1.053
1.156 1.139 1.121 1.105 1.091 1.075 1.055
-0.001 0.000 0.002 0.001 -0.002 - 0.002 - 0.002
FRANCK-CONDON
DENSITIES
Franck-Condon densities for the continua adjoining all progressions of bands (0 s a” I 24) are given in Table 7. Eigenvalues (cm- ‘) range from 6.7 x lo3 to 1.00 x 10’ and densities are in erg-’ as produced by the program. Variations of density in continua corresponding to 0 I u” I 4 are portrayed by Figs. l-5 respectively. For higher u”, TABLE 6. FRANCK--CONWN
FACTORSFOR THE HERZBERG I SYSTEMOF 0,
1
2
1.81-06 1.49-05 6.30-05 1.84-04 4.20-04
3.34-05 2.38-04 8.76 - 04 2.22 - 03 4.44 - 03
2.92 - 04 1.76-03 5.49 - 03 1.18-02 1.99-02
1.60-03 8.01-03 2.04-02 3.55 -02 4.82 -02
6.25 - 03 2.48 - 02 4.92 - 02 6.52 - 02 6.47-02
5 6 7 8 9
7.97 -04 1.31-03 1.87-03 2.37-03 2.62-03
7.38 - 03 1.07-02 1.35-02 1.54-02 1.54-02
2.79 -02 3.41-02 3.68 - 02 3.57-02 3.10-02
5.37 - 02 5.13-02 4.28-02 3.17-02 2.09-02
5.00-02 3.04 - 02 1.39-02 4.15-03 4.59 -04
10 11
2.46 - 03 1.71-03
1.33-02 8.76 - 03
2.37 -02 1.43 -02
1.23-02 6.01-03
4.93 - 05 3.51-04
0”
0
3
4
608
W. R. JARMAIN
TABLE6. (continued)
,“’
5
6
7
8
9
ii 2 3 4
1.84-02 5.52-02 7.92 - 02 7.05 - 02 4.10-02
4.26 - 02 8.98-02 8.16-02 3.59-02 3.90-03
7.93 - 02 1.05-01 4.46 - 02 1.44-03 1.16-02
1.21 -01 8.30-02 4.49 - 03 1.70-02 4.59-02
1.55-01 3.50-02 1.05-02 5.48 - 02 3.56-02
5 6 I 8 9
1.34-02 8.58 -04 1.72-03 7.86-03 1.23-02
2.71-03
3.37 -02
1.67-02
3.61-02
2.67 - 02 2.64 - 02 1.95-02
2.22 - 02 8.09 - 03 1.18-03
X54-02 1.03-02 3.75 -05 3.34-03 1.07 - 02
3.17-03 4.85-03 2.01 -02 2.45 - 02 1.77-02
10
1.25-02 8.57-03
1.15-02 5.37 -03
2.01 -05 4.54-04
1.20-02 8.31 -03
9.05 - 03 3.43 - 03
I
II t”’
10 I1 ,>I’
10
II
I2
13
I4
1.65-01 1.51 -03 5.49-02 4.68 - 02 1.70-03
1.50-01 1.51-02 7.59-02 6.82 - 03 1.85-02
1.14-01 6.77 -02 4.34 - 02 9.76 - 03 5.16-02
7.40-02 1.19-01 3.53-03 5.48 - 02 2.97 - 02
4.12-02 1.36-01 1.38-02 6.57-02 2.93 - 05
1.41-02 3.36-02 2.52-02 8.21 -03 4.56-04
4.23 - 02 1.96-02 6.21 -04 4.75 - 03 1.31-02
1.83-02 6.87 - 04 1.83-02 2.55-02 1.61 -02
1.70-03 3.02-02 2.85 -02 7.53 -03 9.23 -08
3.69 - 02 3.08 - 02 1.77-03 6.10-03 1.70-02
6.29 - 04 1.68-03
1.39-02 8.75 -03
5.93 - 03 1.34-03
X85-03 4.20-03
1.58-02 8.42 ~ 03
IS
I6
17
18
I9 _
1.95 - 02 1.14-01 7.00-02 2.43 - 02 2.96 - 02
7.92 - 03 7.58 - 02 1.20-01 3.64 - 04 6.59-02
2.74 ~ 03 4.07 - 02 1.28-01 3.86-02 4.16-02
7.96-04 1.78-02 9.68 -02 1.01-01 1.93-03
1.98-04 6.46 ~ 03 5.65 - 02 1.28-01 2.12-02
4.24 - 02 7.71 -04 1.66-02 2.81-02 1.47-02
5.23 - 03 2.23 - 02 3.47 - 02 8.13-03 3.91 -04
1.40-02 4.58 - 02 5.83 -03 6.54-03 2.11-02
5.87-02 1.34-02 1.14-02 3.27-02 1.52-02
5.14-02 5.42 - 03 4.35-02 1.30-02 7.20-04
3.06 - 03 1.15-04
6.42 - 03 7.30-03
1.70-02 7.28 - 03
1.40-03 l-75-04
1.01 -02 1.03-02
20
21
22
4.31-05 1.97-03 2.65 - 02 1.09-01 8.47 ~ 02
7.85 -06 5.15-04 1.02-02 6.86-02 1.26-01
1.22-06 1.16-04 3.21-03 3.34-02 1.16-01
23
2.14-07 2.10-05 8.24 ~ 04 1.30-02 7.60-02
24
2.19-08 2.62-06 1.73-04 4.13-03 3.86 -02
Realistic
Franck-Condon
factors
and related integrals
for diatomic
609
moleculesPI
TABLE 6. (continued)
23
24
20
21
22
6 7 8 9
7.26 - 03 4.97 - 02 2.13-02 3.72-03 2.50-02
1.17-02 5.70 - 02 1.09-03 3.71-02 2.05 - 02
7.31-02 1.33-02 3.99 - 02 2.91 -02 3.66 - 05
1.23-01 6.11-03 5.93-02 1.36-04 2.52 -02
1.23-01 6.51 -02 1.98-02 2.82-02 3.47 - 02
10 11
1.92-02 6.64 - 03
1.20-03 7.14 - 04
1.21-02 1.30-02
2.44-02 7.84 - 03
3.51-03 6.66 - 04
V”
v’ 5
The negative
number
in each entry is the power of 10 by which it is multiplied.
densities are significant for energies greater than 1.00 x 105, hence the sums of FranckCondon factors and integrated densities fall increasingly short of unity. At u” = 0, 1,2,3,4, the sums are respectively 0.999, 0.998, 0.999, 0.998 and 0.999. r-CENTROIDS
For each band with a calculated Franck-Condon factor, there is an r-centroid presented in Table 8. These have been derived from the original definition of r-centroid as in equation (19) of Ref. (2). A number of them depart from regularity (meaning a smooth relationship between r-centroid and wave length), notably the value for the (9,13) band. Its correspondence to a very small Franck-Condon factor probably makes it meaningless. No attempt has been made to smooth or ‘correct’ the array, because such procedures are outside the terms of reference for this paper. 40 000
I 0
;
0
- 30 000 x ^a & 25000 ; 20 000 z z % 15 000 S zz
10 000
0” : : L
5 000 0 5 000
15 000
25-000 Eigenvalue,
FIG. 1. Variation
of Franck-Condon
35000
45000
55.000
65-000
cm-lx 10’ density
with wave numbers,
v” = 0.
,131 11 ,319 11 ,544 11 ,772 11 ,977 11 ,114 12 ,133 12 .132 12 .118 12 ,976 11
,177 12 ,180 12 ,177 12 ,169 12 ,158 12 ,145 12 ,115 12 .851 11 ,604 11 ,412 11 ,272 11 ,175 11 ,110 11
,135 12 ,113 12 ,931 11 ,762 11 ,618 11 ,499 11 ,316 11 .195 11 ,118 11 ,700 10 .408 10 ,235 10 ,133 10
3.0 3.2 3.4 3.6 3.8
4.0 4.2 4.4
.756 11 ,559 11 ,395 11
2.5 2.6 2.7 2.8 2.9
.103 12 .127 12 ,132 12 ,121 12 ,969 11
.I74 11 .405 11 ,142 11 ,136 10 .176 10
.604 11 ,926 11 ,123 12 ,147 12 ,166 12
,270 12 ,242 12 ,213 12 ,185 12 ,159 12
2.0 2.1 2.2 2.3 2.4
.879 11 ,824 11 ,481 11 .144 11 ,123 9
,484 11 ,108 11 ,301 9 ,156 11 ,434 11 .700 11 ,876 11 ,930 11
,555 10 ,340 11 ,691 11 ,962 11 ,109 12 .107 12 ,961 11 ,799 11
,798 10 .294 11 .532 11
.702 8 ,721 10 ,275 11 ,525 11 ,745 11
,928 11 ,106 12 ,105 12 ,925 11 ,718 11
,675 11 ,390 1I ,166 11 ,345 10 ,981 8
,120 12 .lll 12 .784 11 ,404 11 ,114 11
,138 11 .246 8 ,101 11 ,362 11 ,671 11
,421 11 .200 10 .120 11 ,549 11 ,993 11
,116 12 .148 12 ,139 12 ,998 11 ,510 11
,453 11 .653 10 ,236 10 ,267 11 .654 11
,150 12 ,177 12 ,177 12 .155 12 .119 12
,414 11 .lOO 11 ,454 6 .842 10 .305 11
,335 12 ,337 12 .330 12 ,315 12 ,295 12
1.5 1.6 1.7 1.8 1.9
,329 10 ,725 11 .154 12 .169 12 .115 12
,184 12 ,828 11 .107 11 ,682 10 .566 11
1.1 1.2 1.3 1.4
,105 12 .180 12 .206 12 ,175 12 ,110 12
,569 11 .545 10 ,668 10 ,466 11 ,102 12
,341 12 ,296 12 ,232 12 ,160 12 ,934 11
,179 12 ,223 12 .263 12 .298 12 ,322 12
1.0
.265 12 ,262 12 ,266 12 ,176 12 .393 11
.220 11 ,303 11 ,516 11 ,178 12 ,236 12
,149 12 ,128 12 ,965 11 ,468 10 .237 11
,414 12 ,396 12 .384 12 ,282 12 ,158 12
,271 12 ,270 12 ,284 12 ,337 12 ,358 12
,551 11 ,563 11 ,624 11 ,966 11 .137 12
0.67 x 10“ 0.68 0.70 0.80 0.90
5
4
2
1
eigenvalue
0
3
,144 11 ,484 9 ,500 10
.866 10 ,477 11 ,762 11 ,717 11 ,435 11
,856 11 ,564 11 ,263 11 ,580 10 ,128 9
,132 10 ,259 11 ,642 11 ,936 11 ,101 12
,108 12 ,139 12 .113 12 ,563 11 ,108 11
,196 12 ,148 12 ,459 11 ,623 7 .396 11
.125 12 ,102 12 ,658 11 ,598 10 .114 12
6
,617 11 ,371 11 ,124 11
,356 11 ,131 10 ,126 11 .471 11 ,683 11
,442 11 ,734 11 ,875 11 ,820 11 ,617 11
,102 12 ,571 11 ,156 11 ,140 6 .137 11
,351 11 ,705 7 .309 11 ,868 11 ,117 12
,249 11 ,143 11 ,106 12 ,160 12 ,116 12
,314 11 ,445 11 ,772 11 ,216 12 ,158 12
7
,118 12 ,124 12 .513 11 ,112 10 ,240 11 ,799 11 ,104 12 ,766 11 ,286 11 ,100 10
,259 11 ,355 10 .582 11 ,112 12 ,106 12 ,542 11 ,807 10 ,390 10 ,363 11 .743 11 .897 11 .750 11 ,427 11 ,129 11 ,110 9 ,769 10 .518 11 ,684 11 ,396 11 ,626 10
,314 11 ,778 11 ,101 12 .881 11 .514 11 ,158 11 .159 9 ,900 10 ,337 11 ,599 11 ,757 11 .613 11 ,183 11 ,628 8 ,184 11 ,480 11 ,619 11 ,524 11
,111 12 ,133 12 .839 11 ,206 11 ,686 9
,271 10 ,252 11 ,489 11
.535 11 ,169 12 ,112 12 ,769 10 .325 11
,973 11 ,229 9 ,677 11 ,151 12 .115 12
.132 12 ,176 12 ,819 11 ,254 10 .334 11
,204 11 ,506 9 ,867 10
,330 11 .991 6 ,270 11 ,603 11 ,531 11
.106 11 .434 11 ,726 11 ,791 11 ,618 11
.849 11 ,106 12 ,152 11 ,192 12 ,181 11
,640 11 .425 11 ,137 11 ,847 11 ,202 12
,253 12 ,245 12 ,234 12 ,621 11 ,110 11
10
9
8
TABLE 7. FRANCK-CONDON DENSITIES(erg-‘) FOR THE CONTINUAOF THE HERZBERGI SYSTEMOF 0, (0 5 u” < 24)
,561 11 ,338 11 ,683 10
.702 11 .488 11 .470 10 ,993 10 ,447 11
,471 11 .102 11 ,762 9 ,203 11 ,505 11
,935 10 ,712 10 ,516 11 ,893 11 ,849 11
,159 11 .152 11 ,893 11 ,118 12 ,679 11
,162 12 ,176 11 .392 11 ,144 12 .112 12
,244 12 ,221 12 ,177 12 ,841 8 .144 12
11
E
.691 11
,289 11 ,516 11 ,424 11
,699 10 ,560 11 ,569 11 .145 11 ,164 10
.I22 11 ,833 11 ,587 11 ,220 11 ,996 9
h
,209 11 ,733 10 ,108 12 .754 11 ,188 11 ,945 9 ,316 11
r P 2
,127 12 .107 12
,581 10 ,141 12 ,136 12 ,130 11 ,352 11
.112 11 ,207 10 .494 10 ,201 12 .107 12
12
9 8 8 8 7 1 6 6 6 5 5 4 3 3 2
,233 ,944 ,377 ,151 ,601 ,234 ,920 ,375 ,148 ,529 ,195 ,484 ,623 ,356 ,363
5.0 5.3 5.6 5.9 6.2
6.5 6.8 7.1 7.4 1.1
8.0 8.5 9.0 9.5 10.0
0 ,752 9 ,421 9
v”
4.6 4.8
eigenvalue (cm - I)
,445 ,828 .253 ,394 .125
6 5 5 4 4
,428 ,105 .222 ,651 .130
,283 ,126 .558 ,242 ,102 7 7 6 5 5
9 9 8 8 8
.288 10 ,136 10 ,628 9
,482 9 .207 9 .878 8 8 8 7 7 7
,593 10
,374 ,157 ,639 ,261 .lll
,117 11
,110 10
,271 11 ,181 11
2
,243 10
,675 10 .408 10
1
8 7 7 6 6
10 9 9 9 8
,138 ,659 .308 .142 .642 ,285 .716 .182 .452 ,112
11 11 11 10 10
,339 ,197 ,107 ,560 ,282
.627 11 ,471 11
3
.138 .375 ,102 ,264 ,697
,483 ,249 ,125 .610 ,293
,643 ,440 ,276 ,162 ,902
9 8 8 7 6
10 10 10 9 9
11 11 11 11 10
,887 11 ,780 11
4
11 10 10 10 10 9 9 8 8 7
,518 .156 .445 .126 ,338
11 11 11 11 11
,128 ,719 .390 .204 .104
,800 ,683 .509 ,345 ,216
,713 11 ,802 11
5
6
10 9 9 8 8
11 9 10 10 10
,263 ,164 ,970 .549 ,300 ,159 .521 ,161 .483 .I39
11 11 11 11 11
,598 ,716 .675 ,547 ,396
.220 11 ,428 11
TABLE 7. (continued)
,404 ,146 .494 ,157 ,483
,428 ,330 ,196 .121 .714
,184 ,448 ,615 ,639 .555
10 10 9 9 8
11 11 11 11 10
11 11 11 11 11
,477 9 ,401 10
7
,870 ,350 ,129 ,445 ,145
,544 ,442 ,326 ,222 ,143
,214 ,968 ,321 ,516 .584
10 10 10 9 9
11 11 11 11 11
9 10 11 11 11
,300 11 ,945 10
8
11 10 10 11 11 11 11 11 11 11 11 10 10 10 9
.230 ,167 .470 ,237 ,428 ,526 .519 ,443 ,340 ,241 ,160 ,726 ,295 ,110 ,386
,554 11 ,434 11
9
11 11 10 10 9
.254 ,131 ,593 ,242 ,914
,346 ,208 ,106 ,475 .194
11 11 11 10 10
11 11 11 11 11
,128 ,296 ,417 ,454 .420 11 11 11 11 11
,355 .468 ,488 ,435 ,347
,402 ,289 .168 ,837 ,373
,282 .977 ,251 ,372 ,421
11 11 11 10 10
9 10 11 11 11
10 11 11 11 10 11 11 11 10 3
,367 ,454 ,261 ,495 .823
,486 ,254 ,346 .209 ,172
,622 ,331 ,422 .259 ,591
,162 11 ,513 9
,967 9 ,165 11 ,317 11 ,486 11 11 11 10 10 11
12 11
10
,418 ,559 ,967 ,770 ,261 ,568 ,189 ,568 ,766 ,639 ,320 ,120 ,415 ,585 ,244 ,876 ,163 ,429 .456 ,250
2.0 2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9
3.0 3.2 3.4 3.6 3.8
4.0 4.2 4.4 4.6 4.8 8 11 11 1I II
11 10 II 11 11
8 11 11 11 11
10 11 11 11 11
IO 11 12 11 I1
.318 ,239 ,778 ,333 ,445
.684 .385 .924 ,295 .554
,747 .298 .102 ,115 ,447
11 10 10 11 1I
11 11 7 11 11
11 11 10 11 11
11 11 10 11 11
12 11 10 11 12
.135 ,824 ,217 .396 ,107
,703 ,358 ,115 ,980 ,221
1.5 1.6 1.7 1.8 1.9 ,872 ,199 ,320 ,478 ,871
10 12 12 11 11
.758 .I00 .154 ,184 ,402
,189 ,549 ,192 .139 ,109
1.0 1.1 1.2 1.3 1.4
12 11 II 12 12
12 12 11 11 12
.201 ,160 .898 .808 ,195
12 12 12 11 11
.174 ,195 .233 .750 ,478
f’
0.67 x lo4 0.68 0.70 0.80 0.90
\‘)
14
(cm
13
eigenvalue
IO 11 11 9 11 11 11 10 10 1I
.504 ,366 ,621 ,292 ,246
11 11 11 11 10
,438 ,798 ,706 ,308 ,211 ,748 ,609 ,419 ,958 ,206
1I 12 11 11 10
,464 ,101 ,756 ,160 ,337
11 10 11 II IO
,298 ,540 ,542 ,430 ,254 11 11 11 11 9
I1 10 11 11 11
,112 ,448 ,422 ,737 ,653
,143 ,449 .387 ,101 .642
11 10 I1 11 11
12 11 10 11 12
,140 ,533 ,381 ,845 ,108
10 1I 12 11 8
,115 ,640 ,125 ,571 .122 ,339 ,186 ,541 ,941 ,633
11 11 12 11 11
12 12 10 12 12
,587 ,575 ,168 ,236 ,479
,167 ,102 ,662 ,136 ,104
12 12 12 6 12
,263 ,270 ,265 ,193 ,206
,384 ,177 .696 .211 .831
10 11 11 12 7
16
15
,391 ,103 ,395 ,390 ,133
,689 .269 ,457 ,492 ,422
.865 ,502 .627 ,666 .425
.976 .850 ,153 ,815 ,619
.481 .952 ,113 ,158 .220
,108 ,149 ,580 .135 ,978
,135 ,852 ,184 ,224 ,504
17
10 I1 11 II 11
11 11 10 11 11
1I 11 10 10 11
11 11 11 10 11
9 1I 12 11 11
12 12 9 12 11
12 11 11 12 I1
1I 10 10 11 11
II 1I 11 10 11
,100 ,648 ,231 ,454 ,455 ,404 .478 ,772 .351 ,383
II 11 11 11 10
,185 ,685 ,770 ,371 ,241
12 1I 11 12 II
,141 ,247 ,295 ,117 ,675 9 11 11 11 10
12 1I 12 11 II
,137 ,222 ,177 ,280 ,587
,241 ,479 ,988 ,585 ,355
,332 ,314 ,245 ,822 ,172
11 11 12 11 12
,471 ,869 ,181 .948 .126
,428 ,381 .505 ,620 ,316
,247 ,145 ,607 ,188 ,510
.328 .338 ,314 ,716 ,651
.104 ,250 .770 ,710 ,872
,704 ,123 .791 ,183 ,753
,396 .188 ,994 ,137 ,899
19
18
TABLE 7. (continued)
11 11 10 IO 11
11 11 11 11 IO
II 8 11 11 11
12 1I 10 1I I1
10 12 11 9 1I
11 12 9 12 1I
12 12 12 11 12
,623 ,408 ,351 ,477 ,540
,704 ,139 ,200 ,562 ,140
,829 ,654 ,124 ,474 ,441
.269 .101 ,650 .203 ,303
.134 .455 ,723 .115 ,161
,211 .I88 .183 ,324 ,731
,888 ,367 ,428 ,267 ,222
20
10 11 11 10 10
11 11 11 11 11
I1 1I 11 10 11
11 12 11 IO 11
12 10 11 12 11
12 10 12 1I 11
11 11 9 12 1I
,945 ,798 ,392 ,318 ,408
.159 .642 ,560 .261 ,506
,430 ,558 ,793 .393 ,145
,525 ,187 ,701 .900 ,239
,231 ,140 ,355 ,255 .115
,774 ,210 ,781 ,141 ,800
10 10 11 11 10
11 11 10 11 11
10 11 11 11 10
11 10 11 1I 11
11 12 II 11 12
9 12 10 12 11
.107.12 ,171 12 .291 12 ,382 10 ,260 12
21
,441 ,538 ,103 ,380 ,281
.162 ,303 ,532 ,882 ,321
,528 ,143 .255 .713 ,615
,894 ,838 ,444 .342 ,902
,117 ,145 ,117 ,728 ,239
,238 ,486 ,184 ,352 ,922
,399 ,353 ,218 ,248 ,152
22
11 10 11 11 11
11 11 11 9 11
II 10 11 11 11
11 1I 10 1I 11
12 10 12 11 10
12 10 12 11 11
12 12 12 12 11
,372 .368 ,225 ,132 ,368
,700 ,242 ,434 ,393 ,251
.724 .734 ,155 .561 ,505
,248 ,560 ,982 .240 ,932
,509 ,136 ,366 .820 .996
,275 ,209 ,205 ,149 ,662
,821 ,212 ,170 ,188 ,216
23
II 11 10 1I 11
11 10 11 11 9
11 II 11 10 11
10 11 11 11 IO
11 12 10 11 11
11 12 11 12 1I
11 11 11 12 12
12 11 12 11 12
12 12 12 11 12
1I 11 11 11
,359 ,402 .283 ,380 ,163
,278 .525 ,855 .510 ,240
10 1I 11 9 11
11 1I 9 11 1I
,184 9 ,477 11 ,791 11 .350 11 ,264 9
.163 ,275 ,945 ,467
.185 11 .495 11 .109 12
,897 11 ,235 11 ,136 12
,194 ,357 ,181 ,362 ,116
,138 ,231 ,383 ,378 ,156
24
z
2
;cl
<
m L
01 ZS9' II CEI' II 8tZ' II ISE' II 06E'
0'01 5'6 0'6 5'8 O'S
aq1 II POI‘ II 161‘ II 862' 11 99E' II EOE'
aql 01 laqwnu II ZSI' II 6PZ' II ZEE' II PZE' II ZLI’
L’L P’L I’L 8'9 S'9
klUa q3ea JO II@ II II II II 01
II SEE' II 9IZ' 019Ll' 8 E8L' 019E9'
JaMod ayl sau?xpu!
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II 061' 01 9P9' 8 OOZ' 01 OE9' II EEZ'
6q 01 JO
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II 181‘
II ELZ’ II OZZ' 01 PZ8' 8 P6E' II 9ZI'
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II 811’
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01 LEP' II ILI' II 882' II 99Z' II 901'
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II 6PI' II E9Z' II Z9Z' II ZZI' 6 081'
01 PZI’ 01 LZI’ 01 LSE’ 11981' II t81' II 8fZ' I 19SZ' II ObI' 01001' 01785'
ZOZ' 16Z' ZZE' 622' LZS'
6 ZOI' 01 166' II OIZ' 01556' II LPZ‘ II LSI’ 01 PPZ' 01 SLZ’ II ZOZ'
019lZ' II LSI’ II SLI’ 01 LLI’ 11 01 6 II II
Z'9 63 9'S E'S O'S
ZLI’ ItiP' L8L' S-PI' LLZ’
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II 6PZ' II E6E' II 9057 01819' 01 EPP
6 S6E' 0166P' II 661' II EEZ' 01 ZIL’
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01681' II SPI' II ZEZ' II SZI' 8 EPI'
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01 L88' 6 POS' II OOZ‘ II 98E‘ II SLI'
EI
8 LZZ. II 9PI' II ZSE' II EZZ' L OL9'
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ST
8SZ' L8Z' 9LS S99' 89E'
IPZ' 16L' IIE' PSI' 80E'
91
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01 01 II II 01
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01 181‘ II 8fZ‘ II 9PZ' 6 6EE' II EPZ'
81
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‘1 TIEV~
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(pmqUOJ)
614
W. R. JARMAIN TABLE 8. PCENTROIDS(A) FOR THE HERZBERGSYSTEMOF O2
V’I
0
1
2
3
4
5
6
7
8
9
u’ 0
1.35
1.36
1.38
1.40
1.41
1.43
1.45
1.47
1.49
1 2 3 4
1.34 1.33 1.32 1.32
1.35 1.35 1.34 1.33
1.37 1.36 1.35 1.35
1.39 1.38 1.37 1.36
1.40 1.39 1.39 1.38
1.42 1.41 1.40 1.39
1.44 1.43 1.42 1.40
1.46 1.44 1.42 1.43
1.41 1.46 1.46 1.45
1.51 1.49 1.49 1.47 1.46
5 6 7 8 9
1.31 1.31 1.30 1.30 1.29
1.33 1.32 1.32 1.31 1.31
1.34 1.33 1.33 1.32 1.32
1.35 1.35 1.34 1.34 1.33
1.37 1.36 1.36 1.35 1.33
1.38 1.37 1.38 1.37 1.37
1.41 1.40 1.39 1.38 1.38
1.42 1.41 1.40 1.40 1.38
1.44 1.43 1.34 1.42 1.42
1.45 1.46 1.44 1.44 1.43
10 11
1.29 1.29
1.30 1.30
1.32 1.31
1.33 1.33
1.39 1.35
1.36 1.36
1.37 1.37
1.47 1.40
1.41 1.41
1.42 1.42
0’3
10
11
12
13
14
15
16
17
18
19
1 2 3 4
1.53 1.50 1.51 1.49 1.41
1.55 1.54 1.53 1.51 1.51
1.57 1.56 1.54 1.54 1.52
1.60 1.58 1.56 1.56 1.54
1.62 1.61 1.60 1.58 1.40
1.65 1.63 1.62 1.60 1.59
1.67 1.66 1.64 1.68 1.61
1.70 1.68 1.61 1.66 1.64
1.73 1.71 1.70 1.68 1.64
1.76 1.74 1.72 1.71 1.70
5 6 7 8 9
1.48 1.41 1.46 1.45 1.42
1.49 1.48 1.45 1.48 1.47
1.51 1.53 1.50 1.49 1.48
1.55 1.53 1.51 1.50 3.59
1.56 1.54 1.52 1.54 1.52
1.58 1.54 1.56 1.55 1.54
1.59 1.59 1.58 1.56 1.60
1.63 1.61 1.59 1.60 1.59
1.65 1.63 1.63 1.62 1.60
1.67 1.68 1.65 1.63 1.66
10 11
1.46 1.45
1.46 1.46
1.47 1.46
1.51 1.50
1.52 1.51
1.53 1.49
1.56 1.56
1.58 1.57
1.58 1.64
1.63 1.62
uII
20
21
22
23
24
i I 2 3 4
1.79 1.77 1.75 I .74 1.72
1.83 1.81 1.78 1.77 1.75
1.88 1.84 1.82 1.80 1.78
1.86 1.87 1.85 1.83 1.81
1.86 1.94 1.89 1.86 1.84
5 6 7 8 9
1.69 1.69 1.67 1.68 1.66
1.74 1.71 1.73 1.69 1.68
1.76 1.73 1.73 1.71 1.86
1.79 1.79 1.76 1.65 1.74
1.82 1.81 1.78 1.78 1.76
10 1’ .
1.65 1.64
I.65 1.69
1.70 1.69
1.72 1.71
1.73 1.77
c’ 0
CONCLUSION
Klein-Dunham potential energy functions in Tables 2 and 3 are the basis for all other numerical results. It seems desirable, therefore, to compare them with previously published ‘equivalent’ RKR potentials. For the A3C: state, turning points agree closely with those of VANDERSLICE et al.“*) Deviations range from 0.000 to 0.002 A with the exception of r2
Realistic FranckCondon
factors and related integrals for diatomic molecules-11
615
40~000
25.000
5.000
0 5000
15caY
25000
35occJ
E~genvolue. FIG. 2. Variation of Franck-Condon
45000
55-000
65ooO
cm-lx IO’
density with wave numbers, 0” = 1.
,
for a’ = 9 where it is 0.004 8. When there are variations, present values run consistently lower on both limbs, possibly because the energy levels are slightly lower or I, differs. For the ground state we find even better agreement with the curve of VANDERSLICE et a1.‘t3’ and deviations are not systematic. Finally, Franck-Condon factors in Table 6 clearly supersede the quite different limited set of JARMAIN et ~1.“~’ The method used is not only considerably more accurate but incorporates the revised apparently correct vibrational assignment of Ref. (4).
0 0 -
: El
0
48000
-
42000
-
I6 000
v I
‘:
12000
s t
6000
0 5000
15.000
25000
35.000
Elgenvalue, FIG. 3. Variation of Franck-Condon
45coo
55000
650X
cni’x 103
density with wave numbers, u” = 2.
W. R. JARMAIN
616 24.000
I
- 12 000 )I c E g 9 000 z z
6-000
0” i k IL’
3000 0 5000
15000
25000
35.ccc Etgenval ue,
FIG. 4. Variation
of FranckkCondon
Etgenvotue, FIG. 5. Variation
of FranckkCondon
45000 cm-lx
density
55 000
65 000
IO3
with wave numbers,
a” = 3.
cm-’ x IO’ density
with wave numbers,
U” = 4
Acknowledgements~This project has been supported by grants from the National Research Council whose assistance is gratefully acknowledged. Consultations with Dr. R. W. NICHOLLS and Mr. J. C. MCCALLUM have been most helpful.
of Canada,
Realistic
Franck-Condon
factors and related integrals for diatomic
617
molecules--II
REFERENCES 1. R. W. DITCHBURN and P. A. YOUNG, J. Amos. Terr. Phys. 24, 127 (1962) 2. W. R. JARMAIN, JQSRT 11,421 (1971). 3. W. R. JARMAIN and R. W. NICHOLLS, Proc. Phys. Sot. 90,545 (1967). 4. H. P. BROIDA and A. G. GAYDON. Proc. R. Sot. A222. 181 (1954). 5. V. DEGEN and R. W. NICHOLLS, J. phys. B, Proc. Phys. Sot. 11, I240 (1969). 6. H. D. BABCOCK and L. HERZBERG, Astrophys. J. 108, 167 (1948). 7. G. HERZBERG, Can. J. Phys. 30, 185 (1952). 8. M. W. FEAST, Proc. Phys. Sot. A63, 549 (1950). 9. W. LOCHTE-HOLTGREVENand G. H. DIEKE, Ann. phys. Leipzig Ser. 5,3,937 (1929). 10. L. HERMAN, R. HERMAN and D. RAKOTOARIJIMY, J. Phys. Rad. Ser. 8.22, 1 (1961). Il. L. WALLACE, Astrophys. J. Suppl. Ser. 7, 165 (1962). 12. J. T. VANDERSLICE.E. A. MASON, W. G. MAISCH and E. R. LIPPINCOTT, J. them. Phys. 33,614 13. J. T. VANDERSLICE,E. A. MACON and W. G. MAISCH, J. them. Phys. 32,5 15 (1960). 14. W. R. JARMAIN, P. A. FRASER and R. W. NICHOLLS, Astrophys. J. 122,55 (1955).
(1960).