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Rotational dependence of Franck-Condon factors for OH+, NH+, SiH, MgH+, SiH+, and NO+
.( Qaant. Spectrosc. Radiat. Trans/er Vol. 27, No. 4, pp 471-479, 1982
0022-4073/82/040471-09$03.00[0
Printed in Great Britain.
Pergamon Press Ltd.
ROTATIONAL DEPENDENCE OF FRANCK-CONDON FACTORS FOR OH +, NH ÷, Sill, MgH ÷, Sill +, AND NO+t P. D. SINGHand A. A. DE ALMEIDA Departamentode Astronamia,lnstitutoAstronOmicoe Geoffsico,Universidadede S~o Paulo, CaixaPostal 30627,01000-SapPaulo,Brasil
(Received 15 October1980) Abstract--Therotationaldependenceof Franck-Condonfactors has been calculatedfor OH+, Nil+, Sill, MgH+, Sill+, and NO+ usingthe rotatingMorseoscillatormodel.A rotationaldependenceis noticedin the electronicbands of each of these molecularspecies;this dependencemay makea significantcontributionin the determinationof rotationaltemperatures,abundancesand opacitydistributionfunctions.
INTRODUCTION Recent absorption measurements with the Copernicus have established the existence in interstellar clouds of H2 in various rotational levels of the ground vibrational level) -3 These studies show that the rotational populations at low levels of the rotational quantum number J are characterized by one temperature and the rotational populations at high values of J are characterized by a second higher temperature. Comparison of the R(I) and P(1) lines with R(0) in the CIE+-X~2 + transition 4 of CO has led to a rotational temperature of 5 K for CO toward ,~ Ori, ¢ Per and a Cam stars. Because of a slight dependence of Franck-Condon factors (FCFs) on rotation and their important contribution in the determination of rotational temperatures, abundances and opacity distribution functions, we have calculated the rotational dependence of FCFs for selected bands of OH +, NH +, Sill, MgH +, Sill +, and NO +. Possibilities of OH +, Sill, Sill +, and NO + in interstellar space have been discussed by Singh and de Almeida: de Almeida and Singh,6 and Singh and Maciel.7 NH + and MgH + ions are of astrophysical importance s'9 and attempt to detect these ions at the absorption wavelengths with the Copernicus has also been made 4 in the directions of stars with E(B-V) ~<0.3. Recently, we have also made a theoretical study ~° on the existence of the NH + ion in interstellar space and in comets. METHOD AND RESULTS In an electronic band of a diatomic molecule, the relative intensities of the rotational lines depend mainly on the population distribution over the rotational levels, the H6nl-London factors, and the squares of the overlap integrals (the Franck-Condon factors), which are given by
q(v', J', v", J") = [(~,(v', J')~b(v", J"))l ~.
(1)
The eigenfunctions ~b of the upper (primed) and lower (double-primed) levels involved in a transition are solutions of the Schr6dinger equation d2@(v'J)
d~--~r
2/z[
V(r)+
~
]
-E(v,J) 6 ( v , J ) = 0 ,
(2)
where E(v, J) is the energy eigenvalue, V(r) is the effective potential energy for vibration of the nuclei, and the other letters have their usual meanings) l Normally, the small term in J(J + 1) in Eq. (2) is neglected and the FCFs are treated as constant throughout a band. This approximation is valid in many astrophysical applications but diatomic molecular eigenfunctions contain centrifugal terms involving the rotational quantum numbers and hence the FCFs are not strictly constant within a band) 2-~4 The rotational dependence of Franck-Condon factors has also been examined in a number of molecular systems) 5 We have used the eigenfunctions of a fWork partiallysupportedby CNPq-BrasilunderContractNo. 1111.4076/77-FA. 471
472
P.D. SINcuand A. A. DEAtMEtDA
rotating Morse oscillator, 16 given in convenient form by Learner, 17 to calculate the rotational dependence of the FCFs for selected bands. The accuracy of computing FCFs for rotating Morse oscillator wavefunctions is expected to be sufficient for most astrophysical and laboratory purposes} 8 We have modified the program of Bell et aL ~2 to calculate the rotational dependence of the FCFs for these molecular species and values are listed in Tables 2-5. Table l lists the band systems, previously published rotationless FCFs, and sources of input data for these six molecular species considered. DISCUSSION We discuss the rotational dependence of the FCFs separately. OH ÷ Using the Klein-Dunham potential, rotationless FCFs for A 3 7 r i - X 3 • - bands have recently been calculated by de Almeida and Singh. 36 Rotational lines of the red-degraded electronic bands (1,0), (0,0), (1, 1), (0, 1), (1,2), and (2, 1), up to J ~25, have been observed in the laboratory by Merer et aL '9 Among these, the (1,0), (0, 0) and (1, 1) electronic bands with heads at 3331, 3566, and 3986 ~,, respectively, are strong. The rotational dependence of the FCFs up to J --- 31 for the (v', 0) and (v', 1) bands, where v' = 0 - 6, were calculated (Table 2). The rotational dependence is larger in the R and Q branches than in the P branch of each of the strong bands [(0,0) and (1,0) of the OH + (A3"Iri-X3E-) system]. The ratios of the FCFs for J = 31 and J = 1 of the P branch are 0.414, 0.970 and 0.258 for the (0, 0), (1, 0), and (1, 1) bands, respectively. NH ÷ The rotational lines for J ~<10 of the (0, 0), (l, 0), (2, 1), (2, 0), and (1, 1) red-degraded bands, which appear at 2885, 2725, 2825, 2614, and 2980 ~, of the C2Y~÷-X2~rr system of the NH + ion, have been observed in the laboratory.:°-n Among these, the bands (0, 0) and (1,0) are strong and the rotational dependence is found to be larger in the R and Q branches than in the P branch for the (0,0) band. The ratios of FCFs for J = 31 and J = 1 of the P branch for (0, 0) and (1,0) bands are 0.666 and 1.323, respectively. The rotational lines for J ~ 16 of the bands (0, 0), (1,0), and (0, 1) belonging to (A:Y~ X!~rr) system of NH ÷ ion have been analysed by Colin and Douglas} 1 The (0, 0) band is strong compared to the (1,0) and (0, 1) bands. A large effect of rotation is noticed in the FCFs in each branch of the (0, 0) band (Table 2). FCFs decrease with increase of J and the decrease is fast in the R branch compared to that in the P or Q branches of the (0, 0) band.
TaMe 1. Sourcesof molecularconstantsof the molecularbandsystemsconsideredand previouspublished Franck-Condonfactors. Molecule
OH+
Band
System
References
A 3 ~ i - X3Z -
18,19,36
C2~+
20,22
- X2~
NH + A 2 ~ - - X2~ r
21-
SiH
A 2A
- X2~ r
23-26
Mg H+
BI~
- XIZ +
30,31
SiH+
AIH
- X IZ+
32,25,26,33
NO+
AI~
- XIs +
34,35
<
0.26583] 0.21603 0.31919 0.012331 0.270961 0.220241 0.311901 0.01410 0.28148 0.23034 0.29302 0.01915
0.29520 0.24667J 0.25114 0.03113
0.27385 0.22610 0.31454 0.01530 0.28499 0.23918 0.29927 0.02068 0.29798 0.25711 0.26541[ 0.03157[
0.30496[ 0.27798 0.20008 0.05408
0.62292 0.71881 0.33753 0.98708 0.59760 0.70050 0.30195 0.98422 0.55178 0.66785 0.24454 0.97791
0.47741 0.61555 0.16780 0.96433
0.36102 0.53311 0.08106 0.93353 0.18829 0.39929 0.01260 0.85821
0.63247 0.72726 0.34711 0.98827
0.61740 0.71807 0.32085 0.98689
0.58335 0.69606 0.27176 0.98288
0.52304 0.65711 0.20080 0.97351
0.42282 0.59246 0.11227 0,95143
0.26137 0.48276 0.02825 0.89566
0.61140 0.70856 0.32615 0.98555
0.57561 0.68079 0.28170 0.98093
0.51784 0.63699 0.21700 0.97177
0.42970 0.57082 0.13684 0.95289
0.29934 0.47064 0.05503 0.91128
0.12424 0.31584 0.00418 0.81234
0.16035 0.27351 0.00417 0.19516 I
0.301791 0.268501 0.168751 0.05938 I 0.25830[ 0.287661 0.044631 0.12572 I
0.26631 0.21738 0.32036 0.01256
0.26706 0.21830 0.32000 0.01282
0.63191 0.72537 0.35087 0.98799
0.63191 0.72537 0.35087 0.98799
0.63002 0.72369 0.34895 0.98776
0.28136 0.29400 0.09587 0.I0172
P
R
(1,0)
P
(0,0)
0+21524 0.28870 0.01806 0.15983
0.29634 0.28440 0.13189 0.07895
0.30219 0.26334 0.22717 0.04155
0.29042 0.24384 0.28057 0.02477
0.27799 0.22945 0.30636 0.01707
0.26956 0.22065 0.31721 0.01364
0.26631 0.21738 0.32036 0.01256
Q
P
P C.07751 6.04751 C.18362 C.00007 0.07714 0.04685 0.18452 0.00009 0.08195 0.04939 0.19126 0.00007
0.09339 0.05586 0.2003]
R
0.0781; 0.0480] 0.1842] 0.00003 0.08451 0.05254 0.19055 0.0000~ 0.09684 0.06106 0.2001] O.O000C
0.03319 0.01563 0.12247 0.00004
0.02723] 0,012251 0.113581 0.00009L 0.03900l 0.017121 0.133831 0.000021 0.06469; 0.027881 0.129601 0.000531 0.ii0291 0.054771 0.01355{ 0.02051(
0.00001 0,11444 0.06811 0.20061 0.00020
0.14850 0.09013 0.16197 0.00331 0.17785 0.12909 0.03800 0.02777
0.11702 0.07516 0.2048~ 0.00025 0.1469E 0.09766 0.18479 0.00223 0.17928 0.13224 0.10326 0.01400 0.13701 0.17148 0.00014 0.07319
0.11565 0.07694 0.00000 0.03980
0.08229 0.03955 0.11360 0.00197
0.04956 0.02327 0.13967 0.00001
0.02507 0.011811 0.10330 0.00006
I
o.o2183 i 0.00997l 0.09671i 0.000081
0.021201 0.009951 0.090931 0.00006
0.01993 0.00933 0.08633 0.00005
0.01975 0.00913 0.08761 0.00006,
0.01993 0.00933 0.08633 0.00005
(3,0)
(2,0)
Table 2. Rotational dependence of Franck-Condon factors for the OH + (A3m-X32-), NH + (C2Z+-X2~r,), NH + (AZ2 - - X27r,), and Sill (AZA-XZ~r,) systems. The following ordering was used: first entry-OH + (A3cr+-X3~-); second entry-NH + (C22+-X2¢r,); third entry--NH + (AZ£--X2~,); fourth entry-Sill (AzA-x2~,)
5
o
g
n~
1
J
0.00000 0.03474 0.00000 0.00000
0.03907 0.01385 0.08456
O. 00297 i 9.07464 3.03887 3.00000 .0O000]
).00756 ).00256 ).05920
).0000t
'.01266 .00415 .08266 .00000
0,01322 0,00507 0.07652
0.00000
0.02473 3.00957 3.09914 3.00010
).05339 .02711 ).02161 0.00847 ).06397 0.09503 ).00833 0.00081
}.05134 0.06732 <05757 I0.02364 " 0 0 0 0 0 _ ~]0.00000 00 .00000
0.025O3
0.01789 0.00639 0.09214 0.00001
3.0055~ 3.0019C 3.0446~ ),O000C
0.0083( 0.0031~ 0.05556 0.0000C
0.01003 0.00362 0.06764 0.00001
O.O000C
0.00676 0.00243 0.0497~
0.00537 0.00190 0.04015 0.00000
0.0048! 0.00161 0.0378~ 0.0000(
Q R
0.00000
0.02815 0.00868
0.00961 0.0029~ 0.06706 0.00044
0 00844
0.00452
0.00000[
0.00000 I
( .00000
q 0.04134 1 0.03776 ] ( <00000 0.01066 1 0.02056 1 ( <02183 o.oooooj o.ooooo I .00000
0.00075 I 0.00168 0.39412 0.03674 I 0.00000 0.39269 0.19308 0.00000 I 0.00000 0.07547 0.0]948 I 0.00000 3.00504 [ 0.01151 0.29770 3.00000 i 0.00000 0.42497 3,0000~__0.00000 0.03224 0.15007
0.00054 0.00009 0.01502 0.00000
0.01834 t ).01462
0.065331 0.000001 0.00000 0.05343 ]1 I <00000
0.011221
0.30814 0.25358 0.37816 0.01262
0.00122 0.00022 0.03005 0.00000
0.00632 0.00173 0.05901 0.00009
0002561 0.004851 ).00344 ).00000 I
0.004011 0.000981 O-04Q7RI 0.000011
0.00034 0.00006 0.00711 0.00001
0.29703 0.24219 0.37746 0.01064
R
).00367 ).00087 ].04466 ).00000
%00002
0.000001 0.00000
0.00000
i
0.00000
0.002051 0.00297 3.00135 0.00051] 0.00081~ 3.00031 0.029881 0.03614 3.02429
0.00000
0.00000:
0.00029 0.00004 0.00639 0.00001
0.00030 0.00005 0.00607 0.00001
Q
0.00048 0.00009 0.32809 0.01062 0.27389 0.00000 0.37520 0.01669 0.00086 0.00017 0.35576 0.03921 0.30403 0.00000 0.35789 0.02469 0.00216 0.00045 0.38551 0.03764 0.34460 0.00000 0.30645 0.04093
0.00067 0.00014 0.01297 0.00000
0.00041 0.00007 0.00805 0.00000
0.0003( 0.0000~ 0.0060] 0.00001
P
16,0)
Fable 2(Contd)
0.00034 0.00005 0.00874 0.00000
0.00180 0.00048 0.02309
0.00135 0.00034 0.01700 0.00000
0,00427 0.00127 0.04326
0.00139 0.00034 0.01985
0.00118 0.00029 0.01565 0.00000
0.0012] 0.00121 0.00031 0.0003G 0.00030 0.0000~ 0.01506 0.01506 0.00621 0.00000 0.00000 0.00001
p
(5,03
0.00236 0.00067 0.02696
0.00157 0.00042 0:01870 0.00000
0.00125 0.00031 0.01532 0.00000
0,0060~ 0.0022( 0.0430] 0.0000(
I0.00493 0.00173 0.03686 0.00000
0.0049 0.0017 0.0368~ O.O0001
0.0050 0.0017 0.0373 O.O000q
R
P
(4,03
R
!
r
i
0.38723 0.39014 i0.13074 I !3.08574
0.38771 0.33325 0.28488 0.04079
0.35407 0.29138 0.35340 0.02226
0.32442 0.26348 0.37566 0.01440
10.30493 I0.24682 0.37934 0.01112
0.29565 0.23961 0.37803 0.01006
0.29589 0.24093 0.37738 0.01041
P
0,I)
0.35377 0.41686 0.07428 0.11556
0.39643 0.36557 0.24110 0.05644
0.37168 0.31839 0.33270 0.03062
0.34044 0.28342 0.36845 0.01901
0.31624 0.25979 0.37799 0.01362
0.30139 0.24597 0.37825 0.01119
0.29589 0.24093 0.37738 0.01041
Q
0.0330
0.00879 /
0.02969[ 0.1334
2 _ 476
0.0406
0.140391
0.55015
0.00344 0.12879 0.1464 0.6993
0.04936 0.2109~ 0.07457 0.81447
0.1020( 0.26874 0.0282C 0.8691~
0.13980 0.30391 0.00861 0.89466
0.15831 0.31887 0.00273 0.90466
0.90436 /
0.00217I
0.3151o /
0.11763
(i,I)
).09217 ).00190 ).08188 ).34909
).00155 ).06913 ).16473 ).62911
9.02235 ).15455 ).10098 ).77378
3.07256 3.22456 ).04363 ).84564
3.11736 3.27424 3.01488 3.88161
0.14654 0.30408 0.00451 0.89875
0.15763 0.31510 0.00217 0.90436
Q
D
11
~6
~i
.6
1
6
1
0.27932 0.28508 0.08980 0.08421
0.27933 0.28444 0.08631 0.08468
0.27202 0.28346 0.06412 0.09436
0.25280 0.28061 0.02878 0.11745
0.21006 0.27119 0.00080 0.16250
0.12293 0.24274 0.03230 0.23751
0.00599 0.16231 I0.09249 0.28871
0.27853 0.28517 0.08818 0.08510
0.26984 0.28486 0.06915 0.09477
0.24970 0.28122 0.03616 0.11542
0.20942 0.26893 0.00467 0.15264
0.13576 0.23704 0.01217 0.21334
0.03091 0.16467 0.09102 0.28325
0.04024 0.03597 0.00164 0.20338
P
(2,1)
0.00447 0.09625 0.04682 0.26150
0.07302 0.20911 0.06266 0.26426
0.17489 0.25806 0.00168 0.18772
0.23319 0.27685 0.01461 0.13426
0.26212 0.28320 0.04997 0.10422
0.27522 0.28484 0.07834 0.08921
).27932 0.28508 0.08980 0.08421
Q 0.16171 0.11284 0.16175 0.00229 0.16124 0.11154 0.15892 0.00230 0.16500 0.11475 0.14077 0.00347 0.17198 0.12287 0.10160 0.00762 0.17647 0.13626 0.03960 0.02200 0.15402 0.15274 0.00131 0.07111 0.03761 0.15060 0.04399 0.14269
).16742 ).12017 ).14628 ).00359 1.17467 ).13107 ).11309 ).00710 ).17877 ).14573 ).05809 ).01726 ).16326 ).16052 ).00370 ).04777 ).08675 ).15885 ).04383 ).12347 ).01214 ).07761 ).00000 ).02474
P
,.16231 ).11362 L16068 ).00239
R
(3,1)
0.00243 0.12469 0.00000 0.10331
0.12551 0.16122 0.01686 0.09726
0.17362 0.14998 0.01782 0.03325
0.17680 0.13451 0.08013 0.01171
0.17013 0.12267 0.12790 0.00501
0.16417 0.11549 0.1533~ 0.00285
0.16171 0.11284 0.16175 0.00229
Q
Table 2( Contd)
=o
C~
n~
o
476
P.D. SINGH and A. A. DE ALMEIDA
Sill The electronic band system A2A-X2~rrof the Sill molecule appears in the visible region 23'24 and rotationless FCF calculations 25'26show that bands belonging to Av = 0 sequence are strong. The rotational lines of the (0, 0) band with J <~20 have been observed in the solar disc and sunspot spectra 27'28and an oscillator strength of [0o = 0.0037 is indicated} 9 The effect of rotation on the FCFs of the (0,0) band is small but it is appreciable in the (1, 1) band of the Sill (A2A-X2zrr) system (see Table 2). In each case, the FCFs decrease with an increase of J. The decrease is slower in the P branch than in the R and Q branches. The ratios of FCFs for J = 31 to J = 1 in the P branch are 0.907 and 0.508 for the (0, 0) and (1, l) bands, respectively. MgH + The B 1~r-XlY~+ system belonging to 24MgH+ was studied in the laboratory by Balfour 3° and constants are listed in a recent compilation by Huber and Herzberg. 3~ We used Pekeris relation ~6 to estimate to,,xe for the B~r state of 24MgH+. The rotational dependence in the FCFs for the 24MgH+ (B 17r-XIE +) bands is small (Table 3). Sill + The rotational lines up to J ~< 14 of the (0, 0) (0, 1), (0, 2), (1,0) and (1, l) electronic bands belonging to the Sill + (AJ~r-X1E +) system have been observed in the laboratory by Douglas and Lutz. 32Of these, the (0, 0) and (0, l) bands whose heads appear at 3993 and 4356 A., respectively, are of comparable intensity; the (1, 0) and (l, l) bands have heads at 3932 and 4284 A, respectively, and are somewhat less intense. The rotational dependence is large in the strong electronic (0, 0) and (l, 0) bands of the Sill + ion. FCFs decrease with an increase of J for these bands. FCFs are more affected by an increase of J in the R and Q branches than in the P branch (Table 4).
Table 3. Rotational dependence of Franck-Condon factors for the MgH + (B ~7";'-XI~+) system. The first, second, : third, fourth, fifth, sixth, and seventh entries refer to v' = 0, 1, 2, 3, 4, 5 and 6, respectively. (v' ,0)
(v' ,i)
J
i
R
P
Q
R
P
0.00371 0.01080 0.01779 0.02164 0.02127 0.01704 0.01005
0.00375 0.01090 0.01795 0.02184 0.02146 0.01721 0.01018
0.00375 0.01090 0.01795 0.02184 0.02146 0.01721 0.01018
0.02188 0.04784 0.06017 0.05739 0.04579 0.03124 0.01666
0.0220 0.0481 0.06051 0.0576[ 0.0459( 0.03135 0.01676
0.02207 0.04817 0.06051 0.05765 0.04596 0.03135 0.01676
0.00356 0.01039 0.01713 0.02085 0.01629 0.00943
0.00375 0.01091 0.01795 0.02182 0.02142 0.01716 0.010]i
0.00367 0.01070 0.01762 0.02143 0.02103 0.01681 0.00984
0.02117 0.04651 0.05878 0.05629 0.04503 0.03068 0.01615
0.0220~ 0.0481~ 0.06049 0.0576( 0.04589 0.03127 0.01666
0.02172 0.04751 0.05981 0.05708 0.04555 0.03]04 0.01646
0.00332 0.00973 0.01606 0.01953 0.01907 0.01501 0.00835
0.00365 0.01064 0.01750 0.02125 0.02081 0.01656 0.00958
0.00350 0.01023 0.01685 0.02048 10.02003 0.01587 0.00903
0.02001
0.02163
0.05437 0.04363 0.02961 0.01512
0.05678 0.04527 0.03077 0.01677
0.02090 0.04597 0.05815 0.05572 0.04456 0.03027 0.01572
0.00301 0.00884 0.01461 0.01772 0.01718 0.01319
0.00347 0.01010 0.01661 0.02014 0.01961 0.01541 0.00856
0.00325 0.00950 0.01567 0.01900 0.01846 0.01437 0.00772
0.01845 0.04129 0.05307 0.05151 0.04144 0.02785 0.01335
0.02071 0.04553 0.05758 0.05514 0.04403 0.02977 0.01517
0.01964 0.04355 0.05550 0.05349 0.04287 0.02893 0.01438
0.00319 0.00931 0.01531 0.01849 !0.01783 0.01369
0.00292 0.00856 0.01410 0.01702 0.01633 0.01232 0.00585
0.U1652 0.03745 0.04866 0.04758 0.03826 0.02514 0.01033
0.01934 0.04286 0.05461 0.05259 0.04204 0.02812 0.01344
0.01798 0.04028 0.05180 0.05027 0.04033 0.02680 0.0121]
0.02045
0.00674 0.00263 0.00776 0.01281 0.01545 0.01471
0.01084 0.00455
0.00699
0.04432 I 0.04730 0.05643, 0.05952
]
Q
Rotational dependence of Franck-Condon factors
477
Table 3 (Contd). (V',0)
11
(v',l)
R
P
Q
0.00221 0.00653 0,01074 0,01280 0.01182 0.00800 0.00152
0.00284 0.00829 0.01361 0.01633 0.01549 0,01138 0.00478
0.00252 0.00742 0.01220 0.01459 0.01369 0.00973 9.00327
R; 0.01430 0.03286 0.04317 0.04243 0.03~2 0.02107 0.00445
p
Q
0.01755 0.03932 0.05055 0.04896 0.03908 0.02555' 0.01049
0.01595 0.03618 0.04702 0.04590 0.03667 0.02354 0.00807
Table 4. Rotational dependence of Franck-Condon factors for the Sill + (AJ~r-X'E +) system. The first, second, third, and fourth entries refer ~,, to r' 0, 1, 2 and 3. =
(v',0)
11
(v',l)
R
P
Q
R
P
Q
0.01108 0.01242 0.00000 0.04916
0.01131 0.01276 0.00089 0.04856
0.01131 0.01276 0.00089 0.04856
0.03242 0.02967 0.00000 0.00467
0,03296 0.03032 0.00212 0.00490
0.03296 0.03032 0.00212 0.00490
0.01006 0.01077 O.OOOO0 0.05207
0.01115 0.01248 0.00000 0.04921
0,01070 0.01179 0.00000 0,05038
0.02987 0.02651 0.00000 0.00351
0.03255 0.02977 0.00000 0.00461
0.03147 0.02847 0.00000 0.00416
0.00832 0.00796 0.00000 0.05676
0.01019 0.01090 0.00000 0.05220
0.00934 0.00957 0.00000 0.05431
0.02545 0.02074 0.00000 0.00168
0.03017 0.02672 0.00000 0.00337
0.02805 0.02407 0.00000 0.00257
0.00605 0.00425 0.00000 0.06213
0.00850 0.00811 0.00000 0.05690
0.00733 0.00625 0.00000 0.05950
0.01938 0.01228 0.000001 0.00012
0.02586 0.02103 0.00000 0.00150
0.02280 0.01698 0.00000 0.00070
0.00353 0.00039 0.00000 0.06656
0.00623 0.00437 0.00000 0.06221
0.00487 0.00230 0.00000 0.06458
0.01216 0,00146 0.00000 0.001041
0.01982 0.01254 0,00000 0.00006
0.01606 0.00721 0.00000 0,00011
0.00128 0.00000 0.00000 0.06787
0.00367 0.00041 0.00000 0.06641
0.00238 0.00000 0.00000 0.06759
0.00494 0.00000 0.00000 0.00819
0.01254 0.00151 0.00000 0.00138
0.00858 0.00000 0.00000 0.00387
Table 5. Rotational dependence of Franck-Condon factors for the NO + (A~Tr-X~E+)system. The first, second, third, fourth, fifth, sixth, and seventh entries refer to v' = 0, l, 2, 3, 4, 5 and 6, respectively. (v', i)
(v',0)
P
Q
0.10017 0.16725 0.13329 0.05780 0.00824 0.00210 0.02279
3.10020 9.16727 9.13327 0.05777 0.00823 0.00211 0.02282
0.10020 0.16727 0.13327 0.15777 0.00823 0,00211 0.02282
0.09776 0.16549 0.13428 0.06003 0.00946 0.00152 0.02099
0.09920 0.16652 0.13367 0.05870 0.00873 0.00185 0.02202
0.09850 0.16602 0.13397 0.05935 0.00909 0.00168 0.02152
Q J
1
" R 0.02326 0.06972 0.11618 0.14296 0.14551 0.13010 0.10595
0.P32712 0.06974 0.11621 0.14299 0.14552 0.13010 0.10594
0.02327 0.06974 0.11621 0.14299 0.14552 0.13010 0.10594
21
0.02250 0.06793 0.11400 0.14123 0.14468 0.13017 0.10665
0.02296 0.06899 I 0.11529 ( 0.14223 I 0.14514 [ 0.13010 |,0.i0621
0.02273 0.06847 0.11466 0.14174 0.14492 0.13014 0.10642
478
P.D. SINGHand A. A. DE ALMEIDA Table 5. Contd). (v',l)
(v',O)
P
Q
0.09193 0.16092 0.13640 0.06556 0.01284 0.00047 0.01667
0.09474 0.16311 0.13537 0.06289 0.01117 0.00089 0.01862
3.09335 0.16204 0.13589 0.06421 0.01199 0.00066 0.01765
0.01839 0.05787 0.10110 0.13023 0.13861 0.12946 0.11003
0.08217 0.15229 0.13895 0.07508 0.01987 0.00009 0.00995
0.086301 0.15600 i 0.13795 0.07106! 0.01678 0.00001 0,01250
0.08424 0.15417 0.13848 0.07307 0.01829 0.00001 0.01121
0.01523 0.04961 0.08970 0.11956 0.13164 0.12718 0.11181
0.01450 0.04769 0.08702 0.11702 !0.12994 0.12657 0.11215
0.06785 0.13724 0.13981 0.08907 0.03345 0.00360 0.00243
0.07311 0.14296 0.13981 0.08408 0.02824 0.00177 0.00453
0.07047 0.14014 0.13988 0.08660 0.03080 0.00260 0.00341
0.01040 0.03609 0.06948 0.09859 0.11555 0.11886 0.11129
0.00967 0.03398 ,0.06622 0.09504 0.i1264 0.11711 0.11081
0.04880 0.11199 0.13322 0.10521 0.05727 0.01855 0.00120
0.05465 0.12023 0.13616 0.10092 0.04980 0.01293 0.00009
P
Q
R
41
0.02069 0.06359 0.10857 0.13676 0.14238 0.13012 0.10824
0.02157 0.06568 0.11116 0.13886 0.14342 0.13008 0.10743
0.02113 0.06464 0.10988 0.13782 0.14291 0.13011 0.10784
61
0.01778 0.05635 0.09911 0.12849 0.13760 0.12928 0.11051
0.01901 0.05940 0.10307 0.13193 0.13956 0.12959 O. 10952
81
0.01379 0.04579 0.08434 0.11443 0.12816 0.12587 0.11242
i01
0.00898 0.03195 0.06300 0.09148 0.10963 0.11522 0.11017
I
I 0.05169 0.11614 0.13481 0.10318 0.05354 0.01563 0.00048
NO + A recent rotational analysis of the A ~ r - X ~ E ÷ (emission) red-degraded (1, 1), (1, 0), (2, 0), (3, 0), and (4, 0) bands for J ~<30 was made by Alberti and Douglas 34 and has resulted in accurate molecular constants for the At~r and X~E ÷ states of the NO + ion. These constants were used to calculate the rotational dependence of FCFs and they are listed in Table 5. The (1, 1) band appears to be strong and the rotational dependence is larger in the R and Q branches than in the P branch of this band. In each branch of the (1, 1) band, the rotational FCFs decrease slowly with an increase of 3".The ratio of FCFs for J = I01 to J = 1 of the P branch is 0.719 for the (1, 1) band of the NO + ion. Acknowledgements--We are grateful to Dr. R. A. Bell for providing a computer program and useful correspondence concerning its use. We also thank Miss Massae Sato, Mr. Masayoshi Tsuchida and Mr. Luis Arakaki for their excellent assistance in modifying the FCF program. One of us (PDS) is indebted to CNPq for partial financial support under Contract No. 1111.4076/77-FA.
REFERENCES I. L. Spitzer, J. F. Drake, E. B. Jenkins, D. C. Morton, J. B. Rogerson, and D. G. York, Astrophys. J. (Lett.) 181, Lll6
0973).
2. L. Spitzer and W. D. Cochron, Astrophys. J. (Lett.) 186, 23 (1973). 3. L. Spitzer, W. D. Cochron, and A. Hirshfeld, Astrophys. Suppl. 28, 373 (1974). 4. E. B. Jenkins, J. F. Drake, D. C. Morton, J. B. Rogerson, L. Spitzer, and D. G. York, Astrophys. J. (Lett.) 181, L122 (1973). 5. P. D. Singh and A. A. de Almeida, Astron. Astrophys. 84, 177 (1980). 6. A. A. de Almeida and P. D. Singh, Astrophys. Space Sci. 56, 415 (1978). 7. P. D. Singh and W. J. Macid, Astrophys. Spcae Sci. 65, 87 (1980). 8. R. W. Nicholls, Ann. Rer. Astron. Astrophys. 15, 197 (1977). 9. P. D. Singh and W. J. Maciel, SolarPhys. 49, 217 (1976). 10. A. A. de Almeida and P. D. Singh, Astron. Astrophys. 1982 in press. I I. Herzberg, Spectra o/Diatomic Molecules. Van Nostrand, Princeton, New Jersey (1950). 12. R. A. Bell, D. Branch and W. L. Upson II, JQSRT 16, 177 (1976).
Rotational dependence of Franck-Condon factors
479
13. P. H. Dwivedi, D. Branch, J. N. Huffaker, and R. A. Bell, Astrophys. J. Supp. 36, 573 (1978). 14. R. A. Bell and D. Branch, Astrophys. J. 212, 591 (1977). 15. J. D. Brown, G. Burns, and R. J. Le Roy, Can. J. Phys. 51, 1664 (1973) and other references cited therein. 16. C. L. Pekeris, Phys. Rev. 45, 98 (1934). 17. R. C. M. Learner, Proc. R. Soc. A269, 311 (1%2). 18. L. A. Kuznetsova, N. E. Kuz'menko, Yu. Ya. Kuzyakov, and Yu. A. Plastinin, Sot,. Phys.-Ups 17, 405 (1974). 19. A. J. Merer, D. N. Matin, R. W. Marlin, M. Horani, and J. Rostas, Can. J. Phys. 53, 251 (1975). 20. M. W. Feast, Astrophys. J. 114, 344 (1951). 21. R. Colin and A. E. Douglas, Can. J. Phys. 46, 61 (1968). 22. G. Krishnamurthy and M. Saraswathy, Pramand 6, 235 (1976i. 23. A. E. Douglas, Can. J. Phys. 35, 71 (1957), 24. L. Klynning and B. Lindgren, Ark. Phys. 33, 73 (1%7). 25. T. V. R. Rao and S. W. J. Lakshman, Physica 56, 322 (1971). 26. H. S. Liszt and W. H. Smith, JQSRT 12, 947 (1972). 27. A. J. Sauval, SolarPhys. 10, 319 (1%9). 28. D. L. Lambert and E. A. Mallia, Mon. Not. R. Astron. Soc. 148, 313 (1970). 29. W. H. Smith and H. S. Liszt, JQSRT 11, 45 (1971). 30, W. J. Balfour, Can. J. Phys. 50, 1082 (1972). 31, K. P. Huber and G. Huber and G. Herzberg, Constants of Diatomic Molecule IV. Van Nostrand Reinhold, New York (1979). 32. A. E. Douglas and B. L. Lutz, Can. J. Phys. 48, 247 (1970). 33. T. A. Carlson, J. Copley, N. Duri~, N. Elander, P. Erman, M. Larsson, and M. Lyyra, Astron. Astrophys. 83, 238 (1980). 34. F. Alberti and A. E. Douglas, Can. J. Phys. 53, 1179 (1975). 35. R. W. Nicholls, J. Phys. B. I, 1192 (1968). 36. A. A. de Almeida and P. D. Singh, Astron. Astrophys. 95, 383 (1981).
0022-4073/82/040471-09$03.00[0
Printed in Great Britain.
Pergamon Press Ltd.
ROTATIONAL DEPENDENCE OF FRANCK-CONDON FACTORS FOR OH +, NH ÷, Sill, MgH ÷, Sill +, AND NO+t P. D. SINGHand A. A. DE ALMEIDA Departamentode Astronamia,lnstitutoAstronOmicoe Geoffsico,Universidadede S~o Paulo, CaixaPostal 30627,01000-SapPaulo,Brasil
(Received 15 October1980) Abstract--Therotationaldependenceof Franck-Condonfactors has been calculatedfor OH+, Nil+, Sill, MgH+, Sill+, and NO+ usingthe rotatingMorseoscillatormodel.A rotationaldependenceis noticedin the electronicbands of each of these molecularspecies;this dependencemay makea significantcontributionin the determinationof rotationaltemperatures,abundancesand opacitydistributionfunctions.
INTRODUCTION Recent absorption measurements with the Copernicus have established the existence in interstellar clouds of H2 in various rotational levels of the ground vibrational level) -3 These studies show that the rotational populations at low levels of the rotational quantum number J are characterized by one temperature and the rotational populations at high values of J are characterized by a second higher temperature. Comparison of the R(I) and P(1) lines with R(0) in the CIE+-X~2 + transition 4 of CO has led to a rotational temperature of 5 K for CO toward ,~ Ori, ¢ Per and a Cam stars. Because of a slight dependence of Franck-Condon factors (FCFs) on rotation and their important contribution in the determination of rotational temperatures, abundances and opacity distribution functions, we have calculated the rotational dependence of FCFs for selected bands of OH +, NH +, Sill, MgH +, Sill +, and NO +. Possibilities of OH +, Sill, Sill +, and NO + in interstellar space have been discussed by Singh and de Almeida: de Almeida and Singh,6 and Singh and Maciel.7 NH + and MgH + ions are of astrophysical importance s'9 and attempt to detect these ions at the absorption wavelengths with the Copernicus has also been made 4 in the directions of stars with E(B-V) ~<0.3. Recently, we have also made a theoretical study ~° on the existence of the NH + ion in interstellar space and in comets. METHOD AND RESULTS In an electronic band of a diatomic molecule, the relative intensities of the rotational lines depend mainly on the population distribution over the rotational levels, the H6nl-London factors, and the squares of the overlap integrals (the Franck-Condon factors), which are given by
q(v', J', v", J") = [(~,(v', J')~b(v", J"))l ~.
(1)
The eigenfunctions ~b of the upper (primed) and lower (double-primed) levels involved in a transition are solutions of the Schr6dinger equation d2@(v'J)
d~--~r
2/z[
V(r)+
~
]
-E(v,J) 6 ( v , J ) = 0 ,
(2)
where E(v, J) is the energy eigenvalue, V(r) is the effective potential energy for vibration of the nuclei, and the other letters have their usual meanings) l Normally, the small term in J(J + 1) in Eq. (2) is neglected and the FCFs are treated as constant throughout a band. This approximation is valid in many astrophysical applications but diatomic molecular eigenfunctions contain centrifugal terms involving the rotational quantum numbers and hence the FCFs are not strictly constant within a band) 2-~4 The rotational dependence of Franck-Condon factors has also been examined in a number of molecular systems) 5 We have used the eigenfunctions of a fWork partiallysupportedby CNPq-BrasilunderContractNo. 1111.4076/77-FA. 471
472
P.D. SINcuand A. A. DEAtMEtDA
rotating Morse oscillator, 16 given in convenient form by Learner, 17 to calculate the rotational dependence of the FCFs for selected bands. The accuracy of computing FCFs for rotating Morse oscillator wavefunctions is expected to be sufficient for most astrophysical and laboratory purposes} 8 We have modified the program of Bell et aL ~2 to calculate the rotational dependence of the FCFs for these molecular species and values are listed in Tables 2-5. Table l lists the band systems, previously published rotationless FCFs, and sources of input data for these six molecular species considered. DISCUSSION We discuss the rotational dependence of the FCFs separately. OH ÷ Using the Klein-Dunham potential, rotationless FCFs for A 3 7 r i - X 3 • - bands have recently been calculated by de Almeida and Singh. 36 Rotational lines of the red-degraded electronic bands (1,0), (0,0), (1, 1), (0, 1), (1,2), and (2, 1), up to J ~25, have been observed in the laboratory by Merer et aL '9 Among these, the (1,0), (0, 0) and (1, 1) electronic bands with heads at 3331, 3566, and 3986 ~,, respectively, are strong. The rotational dependence of the FCFs up to J --- 31 for the (v', 0) and (v', 1) bands, where v' = 0 - 6, were calculated (Table 2). The rotational dependence is larger in the R and Q branches than in the P branch of each of the strong bands [(0,0) and (1,0) of the OH + (A3"Iri-X3E-) system]. The ratios of the FCFs for J = 31 and J = 1 of the P branch are 0.414, 0.970 and 0.258 for the (0, 0), (1, 0), and (1, 1) bands, respectively. NH ÷ The rotational lines for J ~<10 of the (0, 0), (l, 0), (2, 1), (2, 0), and (1, 1) red-degraded bands, which appear at 2885, 2725, 2825, 2614, and 2980 ~, of the C2Y~÷-X2~rr system of the NH + ion, have been observed in the laboratory.:°-n Among these, the bands (0, 0) and (1,0) are strong and the rotational dependence is found to be larger in the R and Q branches than in the P branch for the (0,0) band. The ratios of FCFs for J = 31 and J = 1 of the P branch for (0, 0) and (1,0) bands are 0.666 and 1.323, respectively. The rotational lines for J ~ 16 of the bands (0, 0), (1,0), and (0, 1) belonging to (A:Y~ X!~rr) system of NH ÷ ion have been analysed by Colin and Douglas} 1 The (0, 0) band is strong compared to the (1,0) and (0, 1) bands. A large effect of rotation is noticed in the FCFs in each branch of the (0, 0) band (Table 2). FCFs decrease with increase of J and the decrease is fast in the R branch compared to that in the P or Q branches of the (0, 0) band.
TaMe 1. Sourcesof molecularconstantsof the molecularbandsystemsconsideredand previouspublished Franck-Condonfactors. Molecule
OH+
Band
System
References
A 3 ~ i - X3Z -
18,19,36
C2~+
20,22
- X2~
NH + A 2 ~ - - X2~ r
21-
SiH
A 2A
- X2~ r
23-26
Mg H+
BI~
- XIZ +
30,31
SiH+
AIH
- X IZ+
32,25,26,33
NO+
AI~
- XIs +
34,35
<
0.26583] 0.21603 0.31919 0.012331 0.270961 0.220241 0.311901 0.01410 0.28148 0.23034 0.29302 0.01915
0.29520 0.24667J 0.25114 0.03113
0.27385 0.22610 0.31454 0.01530 0.28499 0.23918 0.29927 0.02068 0.29798 0.25711 0.26541[ 0.03157[
0.30496[ 0.27798 0.20008 0.05408
0.62292 0.71881 0.33753 0.98708 0.59760 0.70050 0.30195 0.98422 0.55178 0.66785 0.24454 0.97791
0.47741 0.61555 0.16780 0.96433
0.36102 0.53311 0.08106 0.93353 0.18829 0.39929 0.01260 0.85821
0.63247 0.72726 0.34711 0.98827
0.61740 0.71807 0.32085 0.98689
0.58335 0.69606 0.27176 0.98288
0.52304 0.65711 0.20080 0.97351
0.42282 0.59246 0.11227 0,95143
0.26137 0.48276 0.02825 0.89566
0.61140 0.70856 0.32615 0.98555
0.57561 0.68079 0.28170 0.98093
0.51784 0.63699 0.21700 0.97177
0.42970 0.57082 0.13684 0.95289
0.29934 0.47064 0.05503 0.91128
0.12424 0.31584 0.00418 0.81234
0.16035 0.27351 0.00417 0.19516 I
0.301791 0.268501 0.168751 0.05938 I 0.25830[ 0.287661 0.044631 0.12572 I
0.26631 0.21738 0.32036 0.01256
0.26706 0.21830 0.32000 0.01282
0.63191 0.72537 0.35087 0.98799
0.63191 0.72537 0.35087 0.98799
0.63002 0.72369 0.34895 0.98776
0.28136 0.29400 0.09587 0.I0172
P
R
(1,0)
P
(0,0)
0+21524 0.28870 0.01806 0.15983
0.29634 0.28440 0.13189 0.07895
0.30219 0.26334 0.22717 0.04155
0.29042 0.24384 0.28057 0.02477
0.27799 0.22945 0.30636 0.01707
0.26956 0.22065 0.31721 0.01364
0.26631 0.21738 0.32036 0.01256
Q
P
P C.07751 6.04751 C.18362 C.00007 0.07714 0.04685 0.18452 0.00009 0.08195 0.04939 0.19126 0.00007
0.09339 0.05586 0.2003]
R
0.0781; 0.0480] 0.1842] 0.00003 0.08451 0.05254 0.19055 0.0000~ 0.09684 0.06106 0.2001] O.O000C
0.03319 0.01563 0.12247 0.00004
0.02723] 0,012251 0.113581 0.00009L 0.03900l 0.017121 0.133831 0.000021 0.06469; 0.027881 0.129601 0.000531 0.ii0291 0.054771 0.01355{ 0.02051(
0.00001 0,11444 0.06811 0.20061 0.00020
0.14850 0.09013 0.16197 0.00331 0.17785 0.12909 0.03800 0.02777
0.11702 0.07516 0.2048~ 0.00025 0.1469E 0.09766 0.18479 0.00223 0.17928 0.13224 0.10326 0.01400 0.13701 0.17148 0.00014 0.07319
0.11565 0.07694 0.00000 0.03980
0.08229 0.03955 0.11360 0.00197
0.04956 0.02327 0.13967 0.00001
0.02507 0.011811 0.10330 0.00006
I
o.o2183 i 0.00997l 0.09671i 0.000081
0.021201 0.009951 0.090931 0.00006
0.01993 0.00933 0.08633 0.00005
0.01975 0.00913 0.08761 0.00006,
0.01993 0.00933 0.08633 0.00005
(3,0)
(2,0)
Table 2. Rotational dependence of Franck-Condon factors for the OH + (A3m-X32-), NH + (C2Z+-X2~r,), NH + (AZ2 - - X27r,), and Sill (AZA-XZ~r,) systems. The following ordering was used: first entry-OH + (A3cr+-X3~-); second entry-NH + (C22+-X2¢r,); third entry--NH + (AZ£--X2~,); fourth entry-Sill (AzA-x2~,)
5
o
g
n~
1
J
0.00000 0.03474 0.00000 0.00000
0.03907 0.01385 0.08456
O. 00297 i 9.07464 3.03887 3.00000 .0O000]
).00756 ).00256 ).05920
).0000t
'.01266 .00415 .08266 .00000
0,01322 0,00507 0.07652
0.00000
0.02473 3.00957 3.09914 3.00010
).05339 .02711 ).02161 0.00847 ).06397 0.09503 ).00833 0.00081
}.05134 0.06732 <05757 I0.02364 " 0 0 0 0 0 _ ~]0.00000 00 .00000
0.025O3
0.01789 0.00639 0.09214 0.00001
3.0055~ 3.0019C 3.0446~ ),O000C
0.0083( 0.0031~ 0.05556 0.0000C
0.01003 0.00362 0.06764 0.00001
O.O000C
0.00676 0.00243 0.0497~
0.00537 0.00190 0.04015 0.00000
0.0048! 0.00161 0.0378~ 0.0000(
Q R
0.00000
0.02815 0.00868
0.00961 0.0029~ 0.06706 0.00044
0 00844
0.00452
0.00000[
0.00000 I
( .00000
q 0.04134 1 0.03776 ] ( <00000 0.01066 1 0.02056 1 ( <02183 o.oooooj o.ooooo I .00000
0.00075 I 0.00168 0.39412 0.03674 I 0.00000 0.39269 0.19308 0.00000 I 0.00000 0.07547 0.0]948 I 0.00000 3.00504 [ 0.01151 0.29770 3.00000 i 0.00000 0.42497 3,0000~__0.00000 0.03224 0.15007
0.00054 0.00009 0.01502 0.00000
0.01834 t ).01462
0.065331 0.000001 0.00000 0.05343 ]1 I <00000
0.011221
0.30814 0.25358 0.37816 0.01262
0.00122 0.00022 0.03005 0.00000
0.00632 0.00173 0.05901 0.00009
0002561 0.004851 ).00344 ).00000 I
0.004011 0.000981 O-04Q7RI 0.000011
0.00034 0.00006 0.00711 0.00001
0.29703 0.24219 0.37746 0.01064
R
).00367 ).00087 ].04466 ).00000
%00002
0.000001 0.00000
0.00000
i
0.00000
0.002051 0.00297 3.00135 0.00051] 0.00081~ 3.00031 0.029881 0.03614 3.02429
0.00000
0.00000:
0.00029 0.00004 0.00639 0.00001
0.00030 0.00005 0.00607 0.00001
Q
0.00048 0.00009 0.32809 0.01062 0.27389 0.00000 0.37520 0.01669 0.00086 0.00017 0.35576 0.03921 0.30403 0.00000 0.35789 0.02469 0.00216 0.00045 0.38551 0.03764 0.34460 0.00000 0.30645 0.04093
0.00067 0.00014 0.01297 0.00000
0.00041 0.00007 0.00805 0.00000
0.0003( 0.0000~ 0.0060] 0.00001
P
16,0)
Fable 2(Contd)
0.00034 0.00005 0.00874 0.00000
0.00180 0.00048 0.02309
0.00135 0.00034 0.01700 0.00000
0,00427 0.00127 0.04326
0.00139 0.00034 0.01985
0.00118 0.00029 0.01565 0.00000
0.0012] 0.00121 0.00031 0.0003G 0.00030 0.0000~ 0.01506 0.01506 0.00621 0.00000 0.00000 0.00001
p
(5,03
0.00236 0.00067 0.02696
0.00157 0.00042 0:01870 0.00000
0.00125 0.00031 0.01532 0.00000
0,0060~ 0.0022( 0.0430] 0.0000(
I0.00493 0.00173 0.03686 0.00000
0.0049 0.0017 0.0368~ O.O0001
0.0050 0.0017 0.0373 O.O000q
R
P
(4,03
R
!
r
i
0.38723 0.39014 i0.13074 I !3.08574
0.38771 0.33325 0.28488 0.04079
0.35407 0.29138 0.35340 0.02226
0.32442 0.26348 0.37566 0.01440
10.30493 I0.24682 0.37934 0.01112
0.29565 0.23961 0.37803 0.01006
0.29589 0.24093 0.37738 0.01041
P
0,I)
0.35377 0.41686 0.07428 0.11556
0.39643 0.36557 0.24110 0.05644
0.37168 0.31839 0.33270 0.03062
0.34044 0.28342 0.36845 0.01901
0.31624 0.25979 0.37799 0.01362
0.30139 0.24597 0.37825 0.01119
0.29589 0.24093 0.37738 0.01041
Q
0.0330
0.00879 /
0.02969[ 0.1334
2 _ 476
0.0406
0.140391
0.55015
0.00344 0.12879 0.1464 0.6993
0.04936 0.2109~ 0.07457 0.81447
0.1020( 0.26874 0.0282C 0.8691~
0.13980 0.30391 0.00861 0.89466
0.15831 0.31887 0.00273 0.90466
0.90436 /
0.00217I
0.3151o /
0.11763
(i,I)
).09217 ).00190 ).08188 ).34909
).00155 ).06913 ).16473 ).62911
9.02235 ).15455 ).10098 ).77378
3.07256 3.22456 ).04363 ).84564
3.11736 3.27424 3.01488 3.88161
0.14654 0.30408 0.00451 0.89875
0.15763 0.31510 0.00217 0.90436
Q
D
11
~6
~i
.6
1
6
1
0.27932 0.28508 0.08980 0.08421
0.27933 0.28444 0.08631 0.08468
0.27202 0.28346 0.06412 0.09436
0.25280 0.28061 0.02878 0.11745
0.21006 0.27119 0.00080 0.16250
0.12293 0.24274 0.03230 0.23751
0.00599 0.16231 I0.09249 0.28871
0.27853 0.28517 0.08818 0.08510
0.26984 0.28486 0.06915 0.09477
0.24970 0.28122 0.03616 0.11542
0.20942 0.26893 0.00467 0.15264
0.13576 0.23704 0.01217 0.21334
0.03091 0.16467 0.09102 0.28325
0.04024 0.03597 0.00164 0.20338
P
(2,1)
0.00447 0.09625 0.04682 0.26150
0.07302 0.20911 0.06266 0.26426
0.17489 0.25806 0.00168 0.18772
0.23319 0.27685 0.01461 0.13426
0.26212 0.28320 0.04997 0.10422
0.27522 0.28484 0.07834 0.08921
).27932 0.28508 0.08980 0.08421
Q 0.16171 0.11284 0.16175 0.00229 0.16124 0.11154 0.15892 0.00230 0.16500 0.11475 0.14077 0.00347 0.17198 0.12287 0.10160 0.00762 0.17647 0.13626 0.03960 0.02200 0.15402 0.15274 0.00131 0.07111 0.03761 0.15060 0.04399 0.14269
).16742 ).12017 ).14628 ).00359 1.17467 ).13107 ).11309 ).00710 ).17877 ).14573 ).05809 ).01726 ).16326 ).16052 ).00370 ).04777 ).08675 ).15885 ).04383 ).12347 ).01214 ).07761 ).00000 ).02474
P
,.16231 ).11362 L16068 ).00239
R
(3,1)
0.00243 0.12469 0.00000 0.10331
0.12551 0.16122 0.01686 0.09726
0.17362 0.14998 0.01782 0.03325
0.17680 0.13451 0.08013 0.01171
0.17013 0.12267 0.12790 0.00501
0.16417 0.11549 0.1533~ 0.00285
0.16171 0.11284 0.16175 0.00229
Q
Table 2( Contd)
=o
C~
n~
o
476
P.D. SINGH and A. A. DE ALMEIDA
Sill The electronic band system A2A-X2~rrof the Sill molecule appears in the visible region 23'24 and rotationless FCF calculations 25'26show that bands belonging to Av = 0 sequence are strong. The rotational lines of the (0, 0) band with J <~20 have been observed in the solar disc and sunspot spectra 27'28and an oscillator strength of [0o = 0.0037 is indicated} 9 The effect of rotation on the FCFs of the (0,0) band is small but it is appreciable in the (1, 1) band of the Sill (A2A-X2zrr) system (see Table 2). In each case, the FCFs decrease with an increase of J. The decrease is slower in the P branch than in the R and Q branches. The ratios of FCFs for J = 31 to J = 1 in the P branch are 0.907 and 0.508 for the (0, 0) and (1, l) bands, respectively. MgH + The B 1~r-XlY~+ system belonging to 24MgH+ was studied in the laboratory by Balfour 3° and constants are listed in a recent compilation by Huber and Herzberg. 3~ We used Pekeris relation ~6 to estimate to,,xe for the B~r state of 24MgH+. The rotational dependence in the FCFs for the 24MgH+ (B 17r-XIE +) bands is small (Table 3). Sill + The rotational lines up to J ~< 14 of the (0, 0) (0, 1), (0, 2), (1,0) and (1, l) electronic bands belonging to the Sill + (AJ~r-X1E +) system have been observed in the laboratory by Douglas and Lutz. 32Of these, the (0, 0) and (0, l) bands whose heads appear at 3993 and 4356 A., respectively, are of comparable intensity; the (1, 0) and (l, l) bands have heads at 3932 and 4284 A, respectively, and are somewhat less intense. The rotational dependence is large in the strong electronic (0, 0) and (l, 0) bands of the Sill + ion. FCFs decrease with an increase of J for these bands. FCFs are more affected by an increase of J in the R and Q branches than in the P branch (Table 4).
Table 3. Rotational dependence of Franck-Condon factors for the MgH + (B ~7";'-XI~+) system. The first, second, : third, fourth, fifth, sixth, and seventh entries refer to v' = 0, 1, 2, 3, 4, 5 and 6, respectively. (v' ,0)
(v' ,i)
J
i
R
P
Q
R
P
0.00371 0.01080 0.01779 0.02164 0.02127 0.01704 0.01005
0.00375 0.01090 0.01795 0.02184 0.02146 0.01721 0.01018
0.00375 0.01090 0.01795 0.02184 0.02146 0.01721 0.01018
0.02188 0.04784 0.06017 0.05739 0.04579 0.03124 0.01666
0.0220 0.0481 0.06051 0.0576[ 0.0459( 0.03135 0.01676
0.02207 0.04817 0.06051 0.05765 0.04596 0.03135 0.01676
0.00356 0.01039 0.01713 0.02085 0.01629 0.00943
0.00375 0.01091 0.01795 0.02182 0.02142 0.01716 0.010]i
0.00367 0.01070 0.01762 0.02143 0.02103 0.01681 0.00984
0.02117 0.04651 0.05878 0.05629 0.04503 0.03068 0.01615
0.0220~ 0.0481~ 0.06049 0.0576( 0.04589 0.03127 0.01666
0.02172 0.04751 0.05981 0.05708 0.04555 0.03]04 0.01646
0.00332 0.00973 0.01606 0.01953 0.01907 0.01501 0.00835
0.00365 0.01064 0.01750 0.02125 0.02081 0.01656 0.00958
0.00350 0.01023 0.01685 0.02048 10.02003 0.01587 0.00903
0.02001
0.02163
0.05437 0.04363 0.02961 0.01512
0.05678 0.04527 0.03077 0.01677
0.02090 0.04597 0.05815 0.05572 0.04456 0.03027 0.01572
0.00301 0.00884 0.01461 0.01772 0.01718 0.01319
0.00347 0.01010 0.01661 0.02014 0.01961 0.01541 0.00856
0.00325 0.00950 0.01567 0.01900 0.01846 0.01437 0.00772
0.01845 0.04129 0.05307 0.05151 0.04144 0.02785 0.01335
0.02071 0.04553 0.05758 0.05514 0.04403 0.02977 0.01517
0.01964 0.04355 0.05550 0.05349 0.04287 0.02893 0.01438
0.00319 0.00931 0.01531 0.01849 !0.01783 0.01369
0.00292 0.00856 0.01410 0.01702 0.01633 0.01232 0.00585
0.U1652 0.03745 0.04866 0.04758 0.03826 0.02514 0.01033
0.01934 0.04286 0.05461 0.05259 0.04204 0.02812 0.01344
0.01798 0.04028 0.05180 0.05027 0.04033 0.02680 0.0121]
0.02045
0.00674 0.00263 0.00776 0.01281 0.01545 0.01471
0.01084 0.00455
0.00699
0.04432 I 0.04730 0.05643, 0.05952
]
Q
Rotational dependence of Franck-Condon factors
477
Table 3 (Contd). (V',0)
11
(v',l)
R
P
Q
0.00221 0.00653 0,01074 0,01280 0.01182 0.00800 0.00152
0.00284 0.00829 0.01361 0.01633 0.01549 0,01138 0.00478
0.00252 0.00742 0.01220 0.01459 0.01369 0.00973 9.00327
R; 0.01430 0.03286 0.04317 0.04243 0.03~2 0.02107 0.00445
p
Q
0.01755 0.03932 0.05055 0.04896 0.03908 0.02555' 0.01049
0.01595 0.03618 0.04702 0.04590 0.03667 0.02354 0.00807
Table 4. Rotational dependence of Franck-Condon factors for the Sill + (AJ~r-X'E +) system. The first, second, third, and fourth entries refer ~,, to r' 0, 1, 2 and 3. =
(v',0)
11
(v',l)
R
P
Q
R
P
Q
0.01108 0.01242 0.00000 0.04916
0.01131 0.01276 0.00089 0.04856
0.01131 0.01276 0.00089 0.04856
0.03242 0.02967 0.00000 0.00467
0,03296 0.03032 0.00212 0.00490
0.03296 0.03032 0.00212 0.00490
0.01006 0.01077 O.OOOO0 0.05207
0.01115 0.01248 0.00000 0.04921
0,01070 0.01179 0.00000 0,05038
0.02987 0.02651 0.00000 0.00351
0.03255 0.02977 0.00000 0.00461
0.03147 0.02847 0.00000 0.00416
0.00832 0.00796 0.00000 0.05676
0.01019 0.01090 0.00000 0.05220
0.00934 0.00957 0.00000 0.05431
0.02545 0.02074 0.00000 0.00168
0.03017 0.02672 0.00000 0.00337
0.02805 0.02407 0.00000 0.00257
0.00605 0.00425 0.00000 0.06213
0.00850 0.00811 0.00000 0.05690
0.00733 0.00625 0.00000 0.05950
0.01938 0.01228 0.000001 0.00012
0.02586 0.02103 0.00000 0.00150
0.02280 0.01698 0.00000 0.00070
0.00353 0.00039 0.00000 0.06656
0.00623 0.00437 0.00000 0.06221
0.00487 0.00230 0.00000 0.06458
0.01216 0,00146 0.00000 0.001041
0.01982 0.01254 0,00000 0.00006
0.01606 0.00721 0.00000 0,00011
0.00128 0.00000 0.00000 0.06787
0.00367 0.00041 0.00000 0.06641
0.00238 0.00000 0.00000 0.06759
0.00494 0.00000 0.00000 0.00819
0.01254 0.00151 0.00000 0.00138
0.00858 0.00000 0.00000 0.00387
Table 5. Rotational dependence of Franck-Condon factors for the NO + (A~Tr-X~E+)system. The first, second, third, fourth, fifth, sixth, and seventh entries refer to v' = 0, l, 2, 3, 4, 5 and 6, respectively. (v', i)
(v',0)
P
Q
0.10017 0.16725 0.13329 0.05780 0.00824 0.00210 0.02279
3.10020 9.16727 9.13327 0.05777 0.00823 0.00211 0.02282
0.10020 0.16727 0.13327 0.15777 0.00823 0,00211 0.02282
0.09776 0.16549 0.13428 0.06003 0.00946 0.00152 0.02099
0.09920 0.16652 0.13367 0.05870 0.00873 0.00185 0.02202
0.09850 0.16602 0.13397 0.05935 0.00909 0.00168 0.02152
Q J
1
" R 0.02326 0.06972 0.11618 0.14296 0.14551 0.13010 0.10595
0.P32712 0.06974 0.11621 0.14299 0.14552 0.13010 0.10594
0.02327 0.06974 0.11621 0.14299 0.14552 0.13010 0.10594
21
0.02250 0.06793 0.11400 0.14123 0.14468 0.13017 0.10665
0.02296 0.06899 I 0.11529 ( 0.14223 I 0.14514 [ 0.13010 |,0.i0621
0.02273 0.06847 0.11466 0.14174 0.14492 0.13014 0.10642
478
P.D. SINGHand A. A. DE ALMEIDA Table 5. Contd). (v',l)
(v',O)
P
Q
0.09193 0.16092 0.13640 0.06556 0.01284 0.00047 0.01667
0.09474 0.16311 0.13537 0.06289 0.01117 0.00089 0.01862
3.09335 0.16204 0.13589 0.06421 0.01199 0.00066 0.01765
0.01839 0.05787 0.10110 0.13023 0.13861 0.12946 0.11003
0.08217 0.15229 0.13895 0.07508 0.01987 0.00009 0.00995
0.086301 0.15600 i 0.13795 0.07106! 0.01678 0.00001 0,01250
0.08424 0.15417 0.13848 0.07307 0.01829 0.00001 0.01121
0.01523 0.04961 0.08970 0.11956 0.13164 0.12718 0.11181
0.01450 0.04769 0.08702 0.11702 !0.12994 0.12657 0.11215
0.06785 0.13724 0.13981 0.08907 0.03345 0.00360 0.00243
0.07311 0.14296 0.13981 0.08408 0.02824 0.00177 0.00453
0.07047 0.14014 0.13988 0.08660 0.03080 0.00260 0.00341
0.01040 0.03609 0.06948 0.09859 0.11555 0.11886 0.11129
0.00967 0.03398 ,0.06622 0.09504 0.i1264 0.11711 0.11081
0.04880 0.11199 0.13322 0.10521 0.05727 0.01855 0.00120
0.05465 0.12023 0.13616 0.10092 0.04980 0.01293 0.00009
P
Q
R
41
0.02069 0.06359 0.10857 0.13676 0.14238 0.13012 0.10824
0.02157 0.06568 0.11116 0.13886 0.14342 0.13008 0.10743
0.02113 0.06464 0.10988 0.13782 0.14291 0.13011 0.10784
61
0.01778 0.05635 0.09911 0.12849 0.13760 0.12928 0.11051
0.01901 0.05940 0.10307 0.13193 0.13956 0.12959 O. 10952
81
0.01379 0.04579 0.08434 0.11443 0.12816 0.12587 0.11242
i01
0.00898 0.03195 0.06300 0.09148 0.10963 0.11522 0.11017
I
I 0.05169 0.11614 0.13481 0.10318 0.05354 0.01563 0.00048
NO + A recent rotational analysis of the A ~ r - X ~ E ÷ (emission) red-degraded (1, 1), (1, 0), (2, 0), (3, 0), and (4, 0) bands for J ~<30 was made by Alberti and Douglas 34 and has resulted in accurate molecular constants for the At~r and X~E ÷ states of the NO + ion. These constants were used to calculate the rotational dependence of FCFs and they are listed in Table 5. The (1, 1) band appears to be strong and the rotational dependence is larger in the R and Q branches than in the P branch of this band. In each branch of the (1, 1) band, the rotational FCFs decrease slowly with an increase of 3".The ratio of FCFs for J = I01 to J = 1 of the P branch is 0.719 for the (1, 1) band of the NO + ion. Acknowledgements--We are grateful to Dr. R. A. Bell for providing a computer program and useful correspondence concerning its use. We also thank Miss Massae Sato, Mr. Masayoshi Tsuchida and Mr. Luis Arakaki for their excellent assistance in modifying the FCF program. One of us (PDS) is indebted to CNPq for partial financial support under Contract No. 1111.4076/77-FA.
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0973).
2. L. Spitzer and W. D. Cochron, Astrophys. J. (Lett.) 186, 23 (1973). 3. L. Spitzer, W. D. Cochron, and A. Hirshfeld, Astrophys. Suppl. 28, 373 (1974). 4. E. B. Jenkins, J. F. Drake, D. C. Morton, J. B. Rogerson, L. Spitzer, and D. G. York, Astrophys. J. (Lett.) 181, L122 (1973). 5. P. D. Singh and A. A. de Almeida, Astron. Astrophys. 84, 177 (1980). 6. A. A. de Almeida and P. D. Singh, Astrophys. Space Sci. 56, 415 (1978). 7. P. D. Singh and W. J. Macid, Astrophys. Spcae Sci. 65, 87 (1980). 8. R. W. Nicholls, Ann. Rer. Astron. Astrophys. 15, 197 (1977). 9. P. D. Singh and W. J. Maciel, SolarPhys. 49, 217 (1976). 10. A. A. de Almeida and P. D. Singh, Astron. Astrophys. 1982 in press. I I. Herzberg, Spectra o/Diatomic Molecules. Van Nostrand, Princeton, New Jersey (1950). 12. R. A. Bell, D. Branch and W. L. Upson II, JQSRT 16, 177 (1976).
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479
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