- Email: [email protected]
True potential energy curves and Franck-Condon factors of a few alkaline earth hydrides
Physica 113C (1982) 263-270 North-Holland Publishing Company
TRUE POTENTIAL ENERGY CURVES AND F R A N C K - C O N D O N FACTORS OF A FEW ALKALINE EARTH HYDRIDES
M.V. R A M A N A I A H and S.V.J. L A K S H M A N Spectroscopic Laboratories, Department of Physics, S.V. University, Tirupati-517502, India
Received 12 June 1981 The true potential energy curves for seven electronic states of Call, two states of SrH and seven states of BaH molecules have been constructed by the method of Lakshman and Rao. Franck-Condon factors for five band systems of Call, one band system of SrH and five band systems of BaH molecules have been evaluated by the method of Fraser and Jarmain. 1. Introduction
T h e construction of accurate potential energy curves is of considerable importance for the understanding of the physical problems arising in astrophysics, gas kinetics and molecular spectra. T h e C a l l molecule was first studied by Mulliken [1] and its astrophysical importance was discussed by Libarel and Weniger [2]. Fourteen electronic states including the ground state are reported for the C a l l molecule [3-17]. T h e latest studies on the A - X and B - X systems of C a l l were made by Berg and Klynning [16], whereas those of the C - X and D - X systems were made by G r u n d s t r 6 m [4, 6]. Watson and W e b e r [7] studied the E - X system. Aslam Khan [8] while studying the F - X system had shown that the u p p e r FzE state was predissociated since the bands a p p e a r e d only in absorption and not in emission at normal pressures. T h e ground and twelve electronically excited states have been reported [9, 18-28] for the SrH molecule. Watson and Fredrickson [18] observed only two bands in the A - X system whereas Watson et al. [20] analyzed the B - X and C - X systems rotationally. The latest rotational analyses of the C - X and D - X systems were made by M o r e and Cornell [21]. Since 1932, studies have been made on the B a H molecule reporting [9, 29-39] the ground state and seven excited electronic states. The
molecular p a r a m e t e r s of the A2rI1/z-X2E, A2II3/2X2E and B - X systems have been recently reevaluated by Veseth [39] whereas the reanalysis of the C - X and D - X systems was made by K o p p et al. [36]. Funke [32] observed and analyzed the E - X system whereas the F - X system was extended by Aslam and Khan [38].
2. True potential energy curves
The present p a p e r deals with the construction of potential energy curves by the R.K.R. method as modified by L a k s h m a n and R a o [40]. Chakraborty and Pan [41] in their review p a p e r pointed out that the method of L a k s h m a n and R a o involves little mathematical complications and gives very reliable results. The turning points are given in terms of f and g by r~, = [(f/g)+ fz]l/2_ f ,
f = [~£]-'/2
0378-4363/82/0000--0000/$02.75 O 1982 North-Holland
g =
rmax = [ ( f / g ) + fz]i/2 + f ;
(tocx¢)-'/2 In W~, (OAeXe)-3/2[Ole(4tOeXe U) 1/2
+ (2tO¢xeBe - %to¢) In Wi],
and W~ = (002 - 4toexeU)l/z/{to¢ - (4toex, U)l/z} .
264
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call, S r H and B a H potentials and F r a n c k - C o n d o n factors Table I Molecular constants of Call, SrH and BaH Molecule
State
Te (cm 1)
toe (cm 1)
toeXe (cm-t)
Be (cm-l)
ae (cm-1)
re (/~)
Call
X2E + A2IIr
0 14 413.0
1298.34 1333.00
19.10 20.00
0.0971) 0.1060
2.0025 1.9740
B 2'y+
15 762.0
1285.00
20.00
0.1021
1.9744
C2E +
28 276.0
1445.00
25.00
0.1181
1.9227
D22~+ E2H
22 602.0 20 418.0
1150.00 1248.60
33.00 21.80
0.0100 0.1124
F2E +
36 705.0
1487.00
28.00
4.2766 4.3477 a (4.3970) 4.3410 a (4.3921) 4.5800 a (4.6390) 2.5000 4.2840 a (4.3402) 4.6867 a (4.7517)
2.6200 2.001 ~ (1.9878) 1.9128 a (1.8997)
SrH
X2]£+ C2E +
0 26230.0
1206.20 1347.0[)
17.00 23.50
3.6751 4.0080
/).0814 0.1320
2.1456 2.0550
BaH
X2~ + A2H 1/2 A2H3/2 B2,~,+ C2~ + D2]~+ E2II3/2
0
1168.31 1110.55 1109.98 1088.90 1282.00 428.00 1228.60
14.50 15.29 13.59 15.40 15.00 4.50 16.90
3.3829 3.2789 3.3220 3.2660 3.5900 1.6200 3.5600
0.0660 0.0728 0.0820 0.0700 0.0640 0.0170 0.0750
2.2318 2.2490 2.2490 2.3080 2.1700 3.2200 2.1870
9457.45 9939.82 11 092.44 23 675.00 21 885.00 15 055,40
0.1300
a Values of B0 and r0. Values of Be and re are shown in parentheses, as calculated by the method described in the text.
Table II Potential energy curves of various electronic states of Call, SrH and BaH Molecule
Call
State
v
U
U + Te
rmin
rmax
(cm- l)
(cm- 1)
(/~)
(/~)
X2~ +
0 1 2 3 4 5 6 7 8 9 10
644.40 1 904.54 3 126.48 4 310.22 5 455.76 6 563.10 7 632.24 8 663.18 9 655.92 10 610.46 11 526.80
644.40 1 904.54 3 126.48 4 310.22 5 455.76 6 563.10 7 632.24 8 663.18 9 655.92 10 610.46 11 526.80
1.8532 1.7590 1.7006 1.6567 1.6211 1.5910 1.5650 1.5420 1.5214 1.5027 1.4856
2.1803 2.3327 2.4508 2.5563 2.6552 2.7505 2.8439 2.9364 3.0290 3.1223 3.2168
A21-1r
0 l 2 3 4 5 6 7
661.50 1 954.50 3 207.50 4 420.50 5 593.50 6 726.50 7 819.50 8 872.50
15 074.50 16 367.50 17 620.50 18 833.50 20 006.50 21 139.50 22 232.50 23 285.50
1.8275 1.7342 1.6763 1.6326 1.5972 1.5671 1.5410 1.5179
2.1503 2.3006 2.4172 2.5213 2.6190 2.7132 2.8055 2.8972
M. V. Ramanaiah and S. V.J. Lakshman / Call, SrH and B a H potentials and Franck-Condon factors Table II (contd) Molecule
State
U (cm -I)
U + Te (cm -1)
rmin (A)
rmax (/k)
8 9 10
9 885.50 10 858.50 11 791.50
24 298.50 25 271.50 26 204.50
1.4971 1.4781 1.4607
2.9889 3.0814 3.1752
B2~ +
0 1 2 3 4 5 6 7 8 9 10
637.50 1 882.50 3 087.50 4 252.50 5 377.50 6 462,50 7 507.50 8 512.50 9 477.50 10 402,50 11 287.50
16 399.50 17 644.50 18 849.50 20 014.50 21 139.50 22 224.50 23 269.50 24 274.50 25 239.50 26 164.50 27 049.50
1.8261 1.7315 1.6729 1.6287 1.5929 1.5626 1.5363 1.5129 1.4920 1.4729 1.4554
2.1549 2.3088 2.4284 2.5354 2.6361 2.7334 2.8289 2.9240 3.0193 3.1156 3.2136
C2~ ÷
0 1 2 3 4 5
716.25 2 111.25 3 456.25 4 751.25 5 996,25 7 191,25
28 992.25 30 387.25 31 732.25 33 027.25 34 272.25 35 467.25
1.7819 1.6939 1.6396 1.5989 1.5659 1.5382
2.0923 2.2395 2.3549 2.4587 2.5571 2.6527
D2~ +
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5
566,75 1 650.75 2 668.75 3 620,75 4 506.75 5 326.75 6 080,75 6 768,75 7 390,75 7 946,75 8 436.75 618.85 1 023.85 2 985.25 4 103.05 5 177.25 6 207.85 7 194.85 8 138.25 9 038.05 9 894.25 10 706.85 736.50 2 167.50 3 542.50 4 861.50 6 124.50 7 331.50
23 168.75 24 252.75 25 270.75 26 222.75 27 108.75 27 928.75 28 682.75 29 370.75 29 992.75 30 548.75 31 038.75 21 036.85 21 441.85 23 403.25 24 521.05 25 595.25 26 625.85 27 612.85 28 556.25 29 456.05 30 312.25 31 124.85 37 441.50 38 872.50 40 247.50 41 566.50 42 829.50 44 036.50
2.4518 2.3322 2.2483 2.1778 2.1140 2.0539 1.9955 1.9376 1.8790 1.8187 1.7680 1.8364 1.7974 1.6835 1.6397 1.6044 1.5746 1.5488 1.5259 1.5054 1.4867 1.4695 1.7615 1.6756 1.6229 1.5835 1.5517 1.5249
2.8015 2.9532 3.0716 3.1797 3.2847 3.3903 3.4992 3.6141 3.7375 3.8729 3.9936 2.1703 2.2303 2.4532 2.5653 2.6714 2.7747 2.8769 2.9792 3.0827 3.1882 3.2964 2.0677 2.2146 2.3305 2.4354 2.5353 2.6330
0 1 2 3 4
598.85 1 771.05 2 909.25 4 013.45 5 083.65
598.85 1 771.05 2 909.25 4 013.45 5 083.65
1.9917 1.8945 1.8342 1.7889 1.7523
2.3288 2.4852 2.6063 2.7142 2.8151
E2II
F2~ +
SrH
X2~ +
v
265
266
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call, S r H and B a H potentials and F r a n c k - C o n d o n factors Table II (contd) Molecule
State
v
5 6 7 8 9 10
BaH
U (cm ')
U + Te (cm -I)
rmin (]k)
rrnax (A)
6 119.85 7 122.05 80911.25 9 024.45 9 924.65 10 790.85
6 119.85 7 122.05 8090.25 9 024.45 9 924.65 10 790.85
1.7214 1.6946 1.6711 1.6500 1.63111 1.6136
2.9123 3.11073 3.11112 3.19511 3.2892 3.3844
C2.~,+
0 1 2 3 4 5
667.63 1%7.63 3 220.63 4 426.63 5 585.63 6 697.63
26897.63 28 197.63 29 450.63 30 656.63 31 815.63 32 927.63
1.9125 1.8274 1.7773 1.7416 1.7141 1.6923
2.2318 2.3888 2.5135 2.6266 2.7344 2.8399
X2~ +
0 1 2 3 4 5 6 7 8 9 10
580.53 1 719.84 2 830.15 3911.46 4 963.77 5 987.08 6 981.39 7 946.70 8 883.01 9 790.32 10 668.63
580.53 1 719.84 2 83(I. 15 3911.46 4 963.77 5 987.08 6 981.39 7 946.70 8 883.01 9 790.32 10 668.63
2.11748 1.9748 1.9123 1.8651 1.8267 1.7942 1.7659 1.7409 1.7185 1.6981 1.6795
2.4163 2.5724 2.6923 2.7983 2.8968 2.9911 3.0827 3.1727 3.2619 3.35111 3.441t4
A2HI/2
0 1 2 3 4 5 6 7 8 9 10
551.45 1 631.42 2 680.81 3 699.62 4 687.85 5 645.50 6 572.57 7 469.06 8 334.97 9 170.30 9 975.05
10 008.911 11 088.87 12 138.26 13 157.07 14 145.30 15 102.95 16 030.1)2 16 926.51 17 792.42 18 627.75 19 432.50
2.1069 2.0060 1.9436 1.8967 1.8589 1.8271 1.7996 1.7755 1.7540 1.7347 1.7171
2.4573 2.6201 2.7460 2.8580 2.9628 3.0635 3.1619 3.2592 3.3562 3.4535 3.5518
A2II3/2
0 1 2 3 4 5 6 7 8 9 10
551.59 1 634.39 2 690.01 3 718.45 4 719.71 5 693.79 6 640.69 7 560.41 8 452.95 9 318.31 10 156.49
10 491.41 11 574.21 12 629.83 13 658.27 14 659.53 15 633.61 16 580.51 17 500.23 18 392.77 19 258.13 20 096.31
2.0931 1.9945 1.9346 1.8905 1.8557 1.8270 1.8029 1.7824 1.7647 1.7495 1.7362
2.4434 2.6075 2.7345 2.8474 2.9528 3.0539 3.1524 3.2495 3.3460 3.4424 3.5394
0 1 2 3 4 5 6
540.60 1 598.70 2 626.00 3 622.50 4 588.20 5 523.10 6 427.20
11 633.04 12 691.14 13 718.44 14 714.94 15 680.64 16 615.54 17 519.64
2.1094 2.0067 1.9429 1.8947 1.8556 1.8225 1.7938
2.4633 2.6272 2.7539 2.8667 Z9722 3.0737 3.1729
B2y,+
M.V. Ramanaiah and S.V.J. Lakshman / Call, SrH and BaH potenfials and Franck-Condon factors
267
Table II (contd) Molecule
State
C2~+
D2X+
E2113/2
v
U (cm-1)
U + Te (cm-1)
main (/~)
rmax (/~)
7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
7 300.50 8 143.00 8 954.70 9 735.60 637.25 1 889.25 3 111.25 4 303.25 5 465.25 6 597.25 7 699.25 8 771.25 9 813.25 10 825.25 11 807.25 212.88 631.88 1 041.88 1 442.88 1 834.88 2 217.88 2 591.88 2 956.88 3 312.88 3 659.88 3 997.88 610.08 1 804.88 2 965.88 4 093.08 5 186.48 6 246.08 7 271.88 8 263.88 9 222.08 10 146.48 11 037.08
18 392.94 19 235.44 20 047.14 20 828.04 24 312.25 25 564.25 26786.25 27 978.25 29 140.25 30 272.25 31 374.25 32 446.25 33 488.25 34 500.25 35 482.25 22 097.88 22 516.88 22 926.88 23 327.88 23 719.88 24 102.88 24 476.88 24 841.88 25 197.88 25 544.88 25 882.88 15 665.48 16 860.28 18 021.28 19 148.48 20 241.88 21 301.48 22 327.28 23 319.28 24 227.48 25 201.88 26 092.48
1.7683 1.7454 1.7247 1.7056 2.0161 1.9195 1.8589 1.8128 1.7752 1.7432 1.7153 1.6905 1.6681 1.6477 1.6290 2.9611 2.7868 2.6747 2.5878 2.5154 2.4527 2.3970 2.3466 2.3003 2.2575 2.2174 2.0230 1.9262 1.8660 1.8206 1.7838 1.7526 1.7255 1.7016 1.6800 1.6606 1.6427
3.2710 3.3689 3.4673 3.5666 2.3419 2.4895 2.6023 2.7017 2.7938 2.8816 2.9667 3.0501 3.1325 3.2146 3.2966 3.5248 3.7718 3.9580 4.1204 4.2697 4.4110 4.5469 4.6793 4.8094 4.9381 5.0660 2.3561 2.5101 2.6289 2.7345 2.8332 2.9280 3.0206 3.1120 3.2031 3.2945 3.3858
T h e o t h e r s y m b o l s h a v e t h e i r usual significance.
T h e details of this m e t h o d h a v e b e e n e x p l a i n e d
The vibrational and rotational constants needed
by t h e a u t h o r s [43] in an e a r l i e r p u b l i c a t i o n . T h e
for t h e p r e s e n t w o r k a r e t a k e n f r o m H u b e r a n d H e r z b e r g [42] a n d are p r e s e n t e d in t a b l e I.
v a l u e of re is e s t i m a t e d using t h e r e l a t i o n re = (~/~)(1/~Be).
T h e v a l u e of Bo i n s t e a d of Be is a v a i l a b l e for the A2Hr, B2E ÷, C2~:÷, E2II a n d F2E ÷ states of t h e
T h e p o t e n t i a l e n e r g y c u r v e s ar e c o n s t r u c t e d for t h e X2~ +, A2Hr, B2~:+, C2~ +, D / E +, E2~ ÷ an d
C a l l m o l e c u l e . T h e v a l u e of Be is e s t i m a t e d using
F2~:+ states of C a l l , t h e X2~ + an d C2H + states of
t he e x p r e s s i o n Be = Bo + a d 2 . T h e v a l u e of ae
S r H , a n d t h e X~H +, A2Hln, A2H3n, B2~:+, C2~ ÷, D2~: + an d E2H3n states of t h e B a H m o l e c u l e s . T h e
is in turn tion
and
d e t e r m i n e d using t h e P e k e r i s relaNewton's
method
of a p p r o x i m a t i o n .
results are p r e s e n t e d in t ab l e II.
268
M. V. Ramanaiah and S. V.J. L a k s h m a n / Call, SrH and B a H potentials and Franck-Condon factors
3. F r a n c k - C o n d o n factors
P~,~,,=R~(~o,~,,)[f q,v,q,v,,dr 2
F r a n c k - C o n d o n factors have been evaluated by the m e t h o d of Fraser and Jarmain. The relative transition probability is written as
Table III Franck-Condon factors of various band systems of the Call molecule
e,~ 0
1
2
3
4
5
0.0005 0.0295 0.9220 0.0480 0.0000 0.0000
0.0000 0.0015 0.0430 0.8908 0.0646 0.0000
0.0000 0.0001 0.0030 0.0557 0.8596 0.0816
0.0000 0.0000 0.0003 0.0049 0.0675 0.8284
0.0011 0.0234 0.9357 0.0384 0.0014 0.0000
0.0001 0.0028 0.0311 0.9152 0.0484 0.0024
0.0000 0.0004 0.0050 0.0365 0.8974 0.11570
0.0000 0.0001 0.0008 0.0074 0.0400 0.8822
0.0118 0.1872 0.4692 0.3056 0.0258 0.0004
0.0013 0.0320 0.2344 0.3160 0.3720 0.0435
0.0002 0.0049 0.0570 0.2550 0.1957 0.421/0
0.0000 0.0007 0.0114 0.0833 0.2530 0.1066
0.0959 0.1320 0.1121 0.0726 0.1/374 0.0145
0.2512 0.1298 0.0270 0.0000 I).0105 0.0259
I},3436 0,0045 0.0373 11.0687 0.0526 0.0246
I}.2349 I).1371 11.1016 0.0138 I).0026 0.0215
0.0006 0.0046 0.9855 0.0062 11.0027 0.0002
0.0001 0.0017 0.0050 0.9823 0.0061 11.0043
0.0000 0.0002 0.0030 11.0045 0.9800 0.0053
0.00(/0 0.0000 0.0005 0.(X144 0.0035 0.9783
A2Ilr-x2~ + system 0 1 2 3 4 5
0.9843 0.0157 0.0000 0.0000 0.0000 0.0000
0.0152 0.9531 0.0317 0.0000 0.0000 0.0000
B2N-X2£ + system 0 1 2 3 4 5
0.9856 0.0142 0.0002 0.0000 0.0000 0.0000
0.0132 0.9591 0.0270 0.0007 0.0000 0.0000
C2£+-X2£ + system 0 1 2 3 4 5
0.8768 0.1191 0.0042 0.0000 0.0000 0.0000
0.1099 0.6561 0.2211 0.0128 0.0001 0.0000
= R 2~(r- ~,v,,)q~,~,,,, where Re is the electronic transition m o m e n t , ~0~, and qJ~,,are the vibrational wave functions of the molecule in the v' and v" states and q~,~,, denotes the F r a n c k - C o n d o n factor. The value of 6 a / a is found to be less than 5% for the A - X and B - X systems of Call, and for the AZlII/2-X2]~ +, AZH3/e--X2]~ +, B - X and C - X systems of B a H and hence the re-shift method of Jarmain and Fraser [44] has been used to evaluate the F - C factors for the above systems. For the C-X, D - X and E - X systems of Call, the C - X system of SrH and the D - X and E - X systems of BaH, the value of 6 a / a has been found to be greater than 5% and hence the method of Fraser and Jarmain [45] has been used in the evaluation of F - C factors for these systems. The results are presented in tables III, IV and V. In the A - X , B - X and E - X systems of Call, the Av = 0 sequence of bands seems to be very prominent as the factors of the other bands are very low as compared to those of the Av = 0 sequence of bands. In fact Watson and Weber [7] observed only the Av---0 sequence of bands in the E - X system of Call. In addition, in all these systems, the bands 3, 0 and beyond in the v'(v" = 0) progression and 0, 3 and beyond in the v"(v' = 0) progression may not be observed since their
D21£+-X2£ + system 0 1 2 3 4 5
0.0013 0.0045 0.01191 I).0142 0.0192 0.0234
0.(1182 0.0428 0.0621 0.0715 0.0718 0.0658
Table IV Franck-Condon factors of the C2£+-X2£ + band system of the SrH molecule v't,~"
0
I
2
3
4
5
0 1 2 3 4 5
0.8528 0.1410 0.0062 0.0001 0.0000 0.0000
0.1285 0.5979 0.2550 0.0184 0.0003 0.0000
0.0163 0.2096 0.3944 0.3425 0.0365 0.0007
0.(X)21 0.0424 0.2504 0.2389 0.4044 0.0603
1/.0003 0.0076 0.0724 0.2585 0.1271 0.4420
0.0000 0.0013 0.0169 0.1012 0.2418 0.0538
E2II-X2£ + system 0 1 2 3 4 5
0.9962 0.0033 0.0005 0.0000 0.0000 0.0000
0.0031 0.9900 0.0054 0.0014 0.0001 0.0000
M. V. Ramanaiah and S. V.J. Lakshman / Call, SrH and B a H potentials and Franck-Condon factors Table V Franck-Condon factors of various band systems of the BaH molecule U tt
1
2
3
4
5
0.0001 0.0123 0.%63 0.0212 0.0000 0.0001
0.00120 0.0003 0.0204 0.9479 0.0312 0.0001
0.0000 0.0000 0.0006 (I.0298 0.9264 (I.0427
0.0000 0.0000 0.0000 0.0010 0.0408 0.9016
0.0019 0.1755 0.5818 0.1951 0.0379 0.0065
0.0000 0.0056 0.2491 0.4507 0.2232 0.0563
0.0000 0.0000 0.0113 0.3133 0.3389 0.2368
0.0000 0.0000 0.0000 0.0190 0.3681 0.2455
0.0001 0.0104 0.1761 0.5451 0.2481 0.0198
0.0000 0.0006 0.0210 0.2173 0.4328 0.2970
0.0000 0.0000 0.0018 0.0349 0.2479 0.3315
0.0021 0.0099 0.0245 0.0424 0.0572 0.0639
0.0075 0.0283 0.0550 0.0721 0.0705 0.0532
0.0200 0.0581 0.0825 0.0733 0.0417 0.0124
0.0428 0.0899 0.0827 0.0380 0.0044 0.0030
0.0014 0.0664 0.8231 0.1079 0.0011 0.0000
0.0001 0.0043 0.0957 0.7540 0.1441 0.0019
0.0000 0.0003 0.0084 0.1220 0.6862 0.1801
0.0000 0.0000 0.0007 0.0138 0.1453 0.6199
A2IIt/r-X2~ ÷ system 0 1 2 3 4 5
0.9943 0.0057 0.0000 0.0000 0.0000 0.0000
0.0056 0.9817 0.0127 0.0000 0.0000 0.0000
B2X+-X2X+ system 0 1 2 3 4 5
0.9057 0.0857 0.0078 0.0008 0.0001 0.0000
0.0925 0.7332 0.1500 0.0211 0.0028 0.0004
C2X+-X2X+ system 0 1 2 3 4 5
0.9302 0.0680 0.0018 0.0000 0.0000 0.0000
0.0662 0.0034 0.7956 0.1254 0.1326 0.6665 0.0055 0.1930 0.0001 0.0115 0.0000 0.0002
D2X+-X2~+ system 0 1 2 3 4 5
0.0000 0.0002 0.0009 0.0022 0.0046 0.0079
0.0004 0.0022 0.0067 0.0145 0.0246 0.0352
E21-I3/2-"X2~+ system 0 1 2 3 4 5
0.9640 0.0358 0.0002 0.0000 0.0000 0.0000
0.0345 0.8932 0.0718 0.0005 0.0000 0.0000
F-C factors are practically zero. The Av = 0 sequence of the C - X system of C a l l is not so prominent as in the case of the other systems, and the band 0, 3 might be observed in addition to those of the above systems. The bands of the D - X system of the C a l l molecule are widely distributed as is revealed by their F-C factors.
269
Some more bands in the C - X system of SrH could be expected to be observed under improved experimental conditions than have been observed by More and Cornell [21]. Only the bands 3, 0 and beyond in the v'(v"= 0) progression and 0, 4 and beyond in the v"(v'= 0) progression might not be observed since their F-C factors are either zero or too small. In the A2II1/2-X2E +, A21I3/2---X2~ +, B-X, C-X, and E - X systems of the B a H molecule, the Av = 0 progression seems to be quite prominent whereas the D - X system would be very weak.
Acknowledgements One of the authors (MVR) is grateful to the U G C of India for a Teacher Fellowship.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R.S. MuUiken, Phys. Rev. 25 (1925) 509. G. Lib&ale and S. Weniger, Physica 41 (1969) 47. E. Hulthen, Phys. Rev. 29 (1927) 97. B. Grundstr6m, Z. Physik 69 (1931) 235. B. Grundstr6m, Z. Physik 75 (1932) 302. B. Grundstr6m, Z. Physik 95 (1935) 574. W.W. Watson and R. Weber, Phys. Rev. 48 (1935) 732. M. Aslam Khan, Proc. Phys. Soc. 80 (1%2) 593. G. Edvinson, I. Kopp, B. Lindgren and N. Aslund, Ark. Fys. 25 (1%3) 95. M. Aslam Khan, Proc. Phys. Soc. 87 (1966) 569. M. Aslam Khan and M.K. Afridi, J. Phys. B1 (1968) 260. M. Aslam Khan and S.S. Hussain, Nuovo Cimento: Soc. Ital. Fis. B18 (1973) 384. L.E. Berg and L. Klynning, Astron. Astrophy. Suppl. Ser. 13 (1974) 325. B. Kaving, B. Lindgren and D.A. Ramsay, Phys. Scr. 10 (1974) 73. B. Kaving and B. Lindgren, Phys. Scr. 10 (1974) 81. L.E. Berg and L. Klynning, Phys. Scr. 10 (1974) 331. L.E. Berg, L. Klynning and H. Martin, Opt. Commun. 17 (1976) 320. W.W. Watson and W.R. Fredrickson, Phys. Rev. 39 (1932) 765. W.R. Fredrickson, M.E. Hogan, Jr. and W.W. Watson, Phys. Rev. 48 (1935) 602. W.W. Watson, W.R. Fredrickson and M.E. Hogan, Phys. Rev. 49 (1936) 150. K.R. More and S.D. Cornell, Phys. Rev. 53 (1938) 806.
270 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call. S r H and B a H potentials and F r a n c k - C o n d o n factors
M. Aslam Khan, Proc. Phys. Soc. 81 (1963) 1047. M. Aslam Khan, Proc. Phys. Soc. 82 (1%3) 564. M. Aslam Khan, Proc. Phys. Soc. 89 (1966) 165. M. Aslam Khan and M.R. Butt, J. Phys. BI (1%8) 745. M. Aslam Khan, M. Raft and S.J.A. Hussainee, J. Phys. Bll (1976) 1953. M. Aslam Khan, M. Raft, Iqbal Khan and M. Aslam Baig, J. Phys. B9 (1976) 2313. I.A. Khan, M. Raft and M.A. Khan, Nuovo Cimento: Soc. Ital. Fis. B44B (1978) 394. A. Schaafsma, Z. Physik 74 (1932) 254. W.R. Fredrickson and W.W. Watson, Phys. Rev. 39 (1932) 753. W.W. Watson, Phys. Rev. 43 (1933) 9. G.W. Funke, Z. Physik 84 (1933) 610. P.G. Koontz and W.W. Watson, Phys. Rev. 48 (1925) 937. B. Grundstr6m, Z. Physik 99 (1936) 595. G.W. Funke and B. Grundstr6m, Z. Physik 100 (1936) 293.
[36] I. Kopp, N. Aslund, G. Edvinson and B. Lindgren, Ark. Fys. 30 (1965) 321. [37] I. Kopp, M. Kronekvist and A. Guntsch, Ark. Fys. 32 (1966) 371. [381 M. Aslam Khan, J. Phys. B1 (1968) 985. [39] L. Veseth, Mol. Phys. 25 (1973) 333. [40] S.V.J. Lakshman and T.V. Ramakrishna Rao, J. Phys. B4 (1971) 269. [41] B. Chakraborty and Y.K. Pan, Appl. Spectr. Rev. 7 (1973) 283. [42] K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. IV: Constants of Diatomic Molecules (Van Nostrand Reinhold Co., New York. 1979). [431 S.V.J. Lakshman and M. Venkataramanaiah, Curr. Sci. 49 (1980) 579. [44] W.R. Jarmain and P.A. Fraser, Proc. Phys. Soc. 66 (1953) 1153. [45] P.A. Fraser and W.R. Jarmain, Proc. Phys. Soc. 66 (1953) 1145.
TRUE POTENTIAL ENERGY CURVES AND F R A N C K - C O N D O N FACTORS OF A FEW ALKALINE EARTH HYDRIDES
M.V. R A M A N A I A H and S.V.J. L A K S H M A N Spectroscopic Laboratories, Department of Physics, S.V. University, Tirupati-517502, India
Received 12 June 1981 The true potential energy curves for seven electronic states of Call, two states of SrH and seven states of BaH molecules have been constructed by the method of Lakshman and Rao. Franck-Condon factors for five band systems of Call, one band system of SrH and five band systems of BaH molecules have been evaluated by the method of Fraser and Jarmain. 1. Introduction
T h e construction of accurate potential energy curves is of considerable importance for the understanding of the physical problems arising in astrophysics, gas kinetics and molecular spectra. T h e C a l l molecule was first studied by Mulliken [1] and its astrophysical importance was discussed by Libarel and Weniger [2]. Fourteen electronic states including the ground state are reported for the C a l l molecule [3-17]. T h e latest studies on the A - X and B - X systems of C a l l were made by Berg and Klynning [16], whereas those of the C - X and D - X systems were made by G r u n d s t r 6 m [4, 6]. Watson and W e b e r [7] studied the E - X system. Aslam Khan [8] while studying the F - X system had shown that the u p p e r FzE state was predissociated since the bands a p p e a r e d only in absorption and not in emission at normal pressures. T h e ground and twelve electronically excited states have been reported [9, 18-28] for the SrH molecule. Watson and Fredrickson [18] observed only two bands in the A - X system whereas Watson et al. [20] analyzed the B - X and C - X systems rotationally. The latest rotational analyses of the C - X and D - X systems were made by M o r e and Cornell [21]. Since 1932, studies have been made on the B a H molecule reporting [9, 29-39] the ground state and seven excited electronic states. The
molecular p a r a m e t e r s of the A2rI1/z-X2E, A2II3/2X2E and B - X systems have been recently reevaluated by Veseth [39] whereas the reanalysis of the C - X and D - X systems was made by K o p p et al. [36]. Funke [32] observed and analyzed the E - X system whereas the F - X system was extended by Aslam and Khan [38].
2. True potential energy curves
The present p a p e r deals with the construction of potential energy curves by the R.K.R. method as modified by L a k s h m a n and R a o [40]. Chakraborty and Pan [41] in their review p a p e r pointed out that the method of L a k s h m a n and R a o involves little mathematical complications and gives very reliable results. The turning points are given in terms of f and g by r~, = [(f/g)+ fz]l/2_ f ,
f = [~£]-'/2
0378-4363/82/0000--0000/$02.75 O 1982 North-Holland
g =
rmax = [ ( f / g ) + fz]i/2 + f ;
(tocx¢)-'/2 In W~, (OAeXe)-3/2[Ole(4tOeXe U) 1/2
+ (2tO¢xeBe - %to¢) In Wi],
and W~ = (002 - 4toexeU)l/z/{to¢ - (4toex, U)l/z} .
264
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call, S r H and B a H potentials and F r a n c k - C o n d o n factors Table I Molecular constants of Call, SrH and BaH Molecule
State
Te (cm 1)
toe (cm 1)
toeXe (cm-t)
Be (cm-l)
ae (cm-1)
re (/~)
Call
X2E + A2IIr
0 14 413.0
1298.34 1333.00
19.10 20.00
0.0971) 0.1060
2.0025 1.9740
B 2'y+
15 762.0
1285.00
20.00
0.1021
1.9744
C2E +
28 276.0
1445.00
25.00
0.1181
1.9227
D22~+ E2H
22 602.0 20 418.0
1150.00 1248.60
33.00 21.80
0.0100 0.1124
F2E +
36 705.0
1487.00
28.00
4.2766 4.3477 a (4.3970) 4.3410 a (4.3921) 4.5800 a (4.6390) 2.5000 4.2840 a (4.3402) 4.6867 a (4.7517)
2.6200 2.001 ~ (1.9878) 1.9128 a (1.8997)
SrH
X2]£+ C2E +
0 26230.0
1206.20 1347.0[)
17.00 23.50
3.6751 4.0080
/).0814 0.1320
2.1456 2.0550
BaH
X2~ + A2H 1/2 A2H3/2 B2,~,+ C2~ + D2]~+ E2II3/2
0
1168.31 1110.55 1109.98 1088.90 1282.00 428.00 1228.60
14.50 15.29 13.59 15.40 15.00 4.50 16.90
3.3829 3.2789 3.3220 3.2660 3.5900 1.6200 3.5600
0.0660 0.0728 0.0820 0.0700 0.0640 0.0170 0.0750
2.2318 2.2490 2.2490 2.3080 2.1700 3.2200 2.1870
9457.45 9939.82 11 092.44 23 675.00 21 885.00 15 055,40
0.1300
a Values of B0 and r0. Values of Be and re are shown in parentheses, as calculated by the method described in the text.
Table II Potential energy curves of various electronic states of Call, SrH and BaH Molecule
Call
State
v
U
U + Te
rmin
rmax
(cm- l)
(cm- 1)
(/~)
(/~)
X2~ +
0 1 2 3 4 5 6 7 8 9 10
644.40 1 904.54 3 126.48 4 310.22 5 455.76 6 563.10 7 632.24 8 663.18 9 655.92 10 610.46 11 526.80
644.40 1 904.54 3 126.48 4 310.22 5 455.76 6 563.10 7 632.24 8 663.18 9 655.92 10 610.46 11 526.80
1.8532 1.7590 1.7006 1.6567 1.6211 1.5910 1.5650 1.5420 1.5214 1.5027 1.4856
2.1803 2.3327 2.4508 2.5563 2.6552 2.7505 2.8439 2.9364 3.0290 3.1223 3.2168
A21-1r
0 l 2 3 4 5 6 7
661.50 1 954.50 3 207.50 4 420.50 5 593.50 6 726.50 7 819.50 8 872.50
15 074.50 16 367.50 17 620.50 18 833.50 20 006.50 21 139.50 22 232.50 23 285.50
1.8275 1.7342 1.6763 1.6326 1.5972 1.5671 1.5410 1.5179
2.1503 2.3006 2.4172 2.5213 2.6190 2.7132 2.8055 2.8972
M. V. Ramanaiah and S. V.J. Lakshman / Call, SrH and B a H potentials and Franck-Condon factors Table II (contd) Molecule
State
U (cm -I)
U + Te (cm -1)
rmin (A)
rmax (/k)
8 9 10
9 885.50 10 858.50 11 791.50
24 298.50 25 271.50 26 204.50
1.4971 1.4781 1.4607
2.9889 3.0814 3.1752
B2~ +
0 1 2 3 4 5 6 7 8 9 10
637.50 1 882.50 3 087.50 4 252.50 5 377.50 6 462,50 7 507.50 8 512.50 9 477.50 10 402,50 11 287.50
16 399.50 17 644.50 18 849.50 20 014.50 21 139.50 22 224.50 23 269.50 24 274.50 25 239.50 26 164.50 27 049.50
1.8261 1.7315 1.6729 1.6287 1.5929 1.5626 1.5363 1.5129 1.4920 1.4729 1.4554
2.1549 2.3088 2.4284 2.5354 2.6361 2.7334 2.8289 2.9240 3.0193 3.1156 3.2136
C2~ ÷
0 1 2 3 4 5
716.25 2 111.25 3 456.25 4 751.25 5 996,25 7 191,25
28 992.25 30 387.25 31 732.25 33 027.25 34 272.25 35 467.25
1.7819 1.6939 1.6396 1.5989 1.5659 1.5382
2.0923 2.2395 2.3549 2.4587 2.5571 2.6527
D2~ +
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5
566,75 1 650.75 2 668.75 3 620,75 4 506.75 5 326.75 6 080,75 6 768,75 7 390,75 7 946,75 8 436.75 618.85 1 023.85 2 985.25 4 103.05 5 177.25 6 207.85 7 194.85 8 138.25 9 038.05 9 894.25 10 706.85 736.50 2 167.50 3 542.50 4 861.50 6 124.50 7 331.50
23 168.75 24 252.75 25 270.75 26 222.75 27 108.75 27 928.75 28 682.75 29 370.75 29 992.75 30 548.75 31 038.75 21 036.85 21 441.85 23 403.25 24 521.05 25 595.25 26 625.85 27 612.85 28 556.25 29 456.05 30 312.25 31 124.85 37 441.50 38 872.50 40 247.50 41 566.50 42 829.50 44 036.50
2.4518 2.3322 2.2483 2.1778 2.1140 2.0539 1.9955 1.9376 1.8790 1.8187 1.7680 1.8364 1.7974 1.6835 1.6397 1.6044 1.5746 1.5488 1.5259 1.5054 1.4867 1.4695 1.7615 1.6756 1.6229 1.5835 1.5517 1.5249
2.8015 2.9532 3.0716 3.1797 3.2847 3.3903 3.4992 3.6141 3.7375 3.8729 3.9936 2.1703 2.2303 2.4532 2.5653 2.6714 2.7747 2.8769 2.9792 3.0827 3.1882 3.2964 2.0677 2.2146 2.3305 2.4354 2.5353 2.6330
0 1 2 3 4
598.85 1 771.05 2 909.25 4 013.45 5 083.65
598.85 1 771.05 2 909.25 4 013.45 5 083.65
1.9917 1.8945 1.8342 1.7889 1.7523
2.3288 2.4852 2.6063 2.7142 2.8151
E2II
F2~ +
SrH
X2~ +
v
265
266
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call, S r H and B a H potentials and F r a n c k - C o n d o n factors Table II (contd) Molecule
State
v
5 6 7 8 9 10
BaH
U (cm ')
U + Te (cm -I)
rmin (]k)
rrnax (A)
6 119.85 7 122.05 80911.25 9 024.45 9 924.65 10 790.85
6 119.85 7 122.05 8090.25 9 024.45 9 924.65 10 790.85
1.7214 1.6946 1.6711 1.6500 1.63111 1.6136
2.9123 3.11073 3.11112 3.19511 3.2892 3.3844
C2.~,+
0 1 2 3 4 5
667.63 1%7.63 3 220.63 4 426.63 5 585.63 6 697.63
26897.63 28 197.63 29 450.63 30 656.63 31 815.63 32 927.63
1.9125 1.8274 1.7773 1.7416 1.7141 1.6923
2.2318 2.3888 2.5135 2.6266 2.7344 2.8399
X2~ +
0 1 2 3 4 5 6 7 8 9 10
580.53 1 719.84 2 830.15 3911.46 4 963.77 5 987.08 6 981.39 7 946.70 8 883.01 9 790.32 10 668.63
580.53 1 719.84 2 83(I. 15 3911.46 4 963.77 5 987.08 6 981.39 7 946.70 8 883.01 9 790.32 10 668.63
2.11748 1.9748 1.9123 1.8651 1.8267 1.7942 1.7659 1.7409 1.7185 1.6981 1.6795
2.4163 2.5724 2.6923 2.7983 2.8968 2.9911 3.0827 3.1727 3.2619 3.35111 3.441t4
A2HI/2
0 1 2 3 4 5 6 7 8 9 10
551.45 1 631.42 2 680.81 3 699.62 4 687.85 5 645.50 6 572.57 7 469.06 8 334.97 9 170.30 9 975.05
10 008.911 11 088.87 12 138.26 13 157.07 14 145.30 15 102.95 16 030.1)2 16 926.51 17 792.42 18 627.75 19 432.50
2.1069 2.0060 1.9436 1.8967 1.8589 1.8271 1.7996 1.7755 1.7540 1.7347 1.7171
2.4573 2.6201 2.7460 2.8580 2.9628 3.0635 3.1619 3.2592 3.3562 3.4535 3.5518
A2II3/2
0 1 2 3 4 5 6 7 8 9 10
551.59 1 634.39 2 690.01 3 718.45 4 719.71 5 693.79 6 640.69 7 560.41 8 452.95 9 318.31 10 156.49
10 491.41 11 574.21 12 629.83 13 658.27 14 659.53 15 633.61 16 580.51 17 500.23 18 392.77 19 258.13 20 096.31
2.0931 1.9945 1.9346 1.8905 1.8557 1.8270 1.8029 1.7824 1.7647 1.7495 1.7362
2.4434 2.6075 2.7345 2.8474 2.9528 3.0539 3.1524 3.2495 3.3460 3.4424 3.5394
0 1 2 3 4 5 6
540.60 1 598.70 2 626.00 3 622.50 4 588.20 5 523.10 6 427.20
11 633.04 12 691.14 13 718.44 14 714.94 15 680.64 16 615.54 17 519.64
2.1094 2.0067 1.9429 1.8947 1.8556 1.8225 1.7938
2.4633 2.6272 2.7539 2.8667 Z9722 3.0737 3.1729
B2y,+
M.V. Ramanaiah and S.V.J. Lakshman / Call, SrH and BaH potenfials and Franck-Condon factors
267
Table II (contd) Molecule
State
C2~+
D2X+
E2113/2
v
U (cm-1)
U + Te (cm-1)
main (/~)
rmax (/~)
7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
7 300.50 8 143.00 8 954.70 9 735.60 637.25 1 889.25 3 111.25 4 303.25 5 465.25 6 597.25 7 699.25 8 771.25 9 813.25 10 825.25 11 807.25 212.88 631.88 1 041.88 1 442.88 1 834.88 2 217.88 2 591.88 2 956.88 3 312.88 3 659.88 3 997.88 610.08 1 804.88 2 965.88 4 093.08 5 186.48 6 246.08 7 271.88 8 263.88 9 222.08 10 146.48 11 037.08
18 392.94 19 235.44 20 047.14 20 828.04 24 312.25 25 564.25 26786.25 27 978.25 29 140.25 30 272.25 31 374.25 32 446.25 33 488.25 34 500.25 35 482.25 22 097.88 22 516.88 22 926.88 23 327.88 23 719.88 24 102.88 24 476.88 24 841.88 25 197.88 25 544.88 25 882.88 15 665.48 16 860.28 18 021.28 19 148.48 20 241.88 21 301.48 22 327.28 23 319.28 24 227.48 25 201.88 26 092.48
1.7683 1.7454 1.7247 1.7056 2.0161 1.9195 1.8589 1.8128 1.7752 1.7432 1.7153 1.6905 1.6681 1.6477 1.6290 2.9611 2.7868 2.6747 2.5878 2.5154 2.4527 2.3970 2.3466 2.3003 2.2575 2.2174 2.0230 1.9262 1.8660 1.8206 1.7838 1.7526 1.7255 1.7016 1.6800 1.6606 1.6427
3.2710 3.3689 3.4673 3.5666 2.3419 2.4895 2.6023 2.7017 2.7938 2.8816 2.9667 3.0501 3.1325 3.2146 3.2966 3.5248 3.7718 3.9580 4.1204 4.2697 4.4110 4.5469 4.6793 4.8094 4.9381 5.0660 2.3561 2.5101 2.6289 2.7345 2.8332 2.9280 3.0206 3.1120 3.2031 3.2945 3.3858
T h e o t h e r s y m b o l s h a v e t h e i r usual significance.
T h e details of this m e t h o d h a v e b e e n e x p l a i n e d
The vibrational and rotational constants needed
by t h e a u t h o r s [43] in an e a r l i e r p u b l i c a t i o n . T h e
for t h e p r e s e n t w o r k a r e t a k e n f r o m H u b e r a n d H e r z b e r g [42] a n d are p r e s e n t e d in t a b l e I.
v a l u e of re is e s t i m a t e d using t h e r e l a t i o n re = (~/~)(1/~Be).
T h e v a l u e of Bo i n s t e a d of Be is a v a i l a b l e for the A2Hr, B2E ÷, C2~:÷, E2II a n d F2E ÷ states of t h e
T h e p o t e n t i a l e n e r g y c u r v e s ar e c o n s t r u c t e d for t h e X2~ +, A2Hr, B2~:+, C2~ +, D / E +, E2~ ÷ an d
C a l l m o l e c u l e . T h e v a l u e of Be is e s t i m a t e d using
F2~:+ states of C a l l , t h e X2~ + an d C2H + states of
t he e x p r e s s i o n Be = Bo + a d 2 . T h e v a l u e of ae
S r H , a n d t h e X~H +, A2Hln, A2H3n, B2~:+, C2~ ÷, D2~: + an d E2H3n states of t h e B a H m o l e c u l e s . T h e
is in turn tion
and
d e t e r m i n e d using t h e P e k e r i s relaNewton's
method
of a p p r o x i m a t i o n .
results are p r e s e n t e d in t ab l e II.
268
M. V. Ramanaiah and S. V.J. L a k s h m a n / Call, SrH and B a H potentials and Franck-Condon factors
3. F r a n c k - C o n d o n factors
P~,~,,=R~(~o,~,,)[f q,v,q,v,,dr 2
F r a n c k - C o n d o n factors have been evaluated by the m e t h o d of Fraser and Jarmain. The relative transition probability is written as
Table III Franck-Condon factors of various band systems of the Call molecule
e,~ 0
1
2
3
4
5
0.0005 0.0295 0.9220 0.0480 0.0000 0.0000
0.0000 0.0015 0.0430 0.8908 0.0646 0.0000
0.0000 0.0001 0.0030 0.0557 0.8596 0.0816
0.0000 0.0000 0.0003 0.0049 0.0675 0.8284
0.0011 0.0234 0.9357 0.0384 0.0014 0.0000
0.0001 0.0028 0.0311 0.9152 0.0484 0.0024
0.0000 0.0004 0.0050 0.0365 0.8974 0.11570
0.0000 0.0001 0.0008 0.0074 0.0400 0.8822
0.0118 0.1872 0.4692 0.3056 0.0258 0.0004
0.0013 0.0320 0.2344 0.3160 0.3720 0.0435
0.0002 0.0049 0.0570 0.2550 0.1957 0.421/0
0.0000 0.0007 0.0114 0.0833 0.2530 0.1066
0.0959 0.1320 0.1121 0.0726 0.1/374 0.0145
0.2512 0.1298 0.0270 0.0000 I).0105 0.0259
I},3436 0,0045 0.0373 11.0687 0.0526 0.0246
I}.2349 I).1371 11.1016 0.0138 I).0026 0.0215
0.0006 0.0046 0.9855 0.0062 11.0027 0.0002
0.0001 0.0017 0.0050 0.9823 0.0061 11.0043
0.0000 0.0002 0.0030 11.0045 0.9800 0.0053
0.00(/0 0.0000 0.0005 0.(X144 0.0035 0.9783
A2Ilr-x2~ + system 0 1 2 3 4 5
0.9843 0.0157 0.0000 0.0000 0.0000 0.0000
0.0152 0.9531 0.0317 0.0000 0.0000 0.0000
B2N-X2£ + system 0 1 2 3 4 5
0.9856 0.0142 0.0002 0.0000 0.0000 0.0000
0.0132 0.9591 0.0270 0.0007 0.0000 0.0000
C2£+-X2£ + system 0 1 2 3 4 5
0.8768 0.1191 0.0042 0.0000 0.0000 0.0000
0.1099 0.6561 0.2211 0.0128 0.0001 0.0000
= R 2~(r- ~,v,,)q~,~,,,, where Re is the electronic transition m o m e n t , ~0~, and qJ~,,are the vibrational wave functions of the molecule in the v' and v" states and q~,~,, denotes the F r a n c k - C o n d o n factor. The value of 6 a / a is found to be less than 5% for the A - X and B - X systems of Call, and for the AZlII/2-X2]~ +, AZH3/e--X2]~ +, B - X and C - X systems of B a H and hence the re-shift method of Jarmain and Fraser [44] has been used to evaluate the F - C factors for the above systems. For the C-X, D - X and E - X systems of Call, the C - X system of SrH and the D - X and E - X systems of BaH, the value of 6 a / a has been found to be greater than 5% and hence the method of Fraser and Jarmain [45] has been used in the evaluation of F - C factors for these systems. The results are presented in tables III, IV and V. In the A - X , B - X and E - X systems of Call, the Av = 0 sequence of bands seems to be very prominent as the factors of the other bands are very low as compared to those of the Av = 0 sequence of bands. In fact Watson and Weber [7] observed only the Av---0 sequence of bands in the E - X system of Call. In addition, in all these systems, the bands 3, 0 and beyond in the v'(v" = 0) progression and 0, 3 and beyond in the v"(v' = 0) progression may not be observed since their
D21£+-X2£ + system 0 1 2 3 4 5
0.0013 0.0045 0.01191 I).0142 0.0192 0.0234
0.(1182 0.0428 0.0621 0.0715 0.0718 0.0658
Table IV Franck-Condon factors of the C2£+-X2£ + band system of the SrH molecule v't,~"
0
I
2
3
4
5
0 1 2 3 4 5
0.8528 0.1410 0.0062 0.0001 0.0000 0.0000
0.1285 0.5979 0.2550 0.0184 0.0003 0.0000
0.0163 0.2096 0.3944 0.3425 0.0365 0.0007
0.(X)21 0.0424 0.2504 0.2389 0.4044 0.0603
1/.0003 0.0076 0.0724 0.2585 0.1271 0.4420
0.0000 0.0013 0.0169 0.1012 0.2418 0.0538
E2II-X2£ + system 0 1 2 3 4 5
0.9962 0.0033 0.0005 0.0000 0.0000 0.0000
0.0031 0.9900 0.0054 0.0014 0.0001 0.0000
M. V. Ramanaiah and S. V.J. Lakshman / Call, SrH and B a H potentials and Franck-Condon factors Table V Franck-Condon factors of various band systems of the BaH molecule U tt
1
2
3
4
5
0.0001 0.0123 0.%63 0.0212 0.0000 0.0001
0.00120 0.0003 0.0204 0.9479 0.0312 0.0001
0.0000 0.0000 0.0006 (I.0298 0.9264 (I.0427
0.0000 0.0000 0.0000 0.0010 0.0408 0.9016
0.0019 0.1755 0.5818 0.1951 0.0379 0.0065
0.0000 0.0056 0.2491 0.4507 0.2232 0.0563
0.0000 0.0000 0.0113 0.3133 0.3389 0.2368
0.0000 0.0000 0.0000 0.0190 0.3681 0.2455
0.0001 0.0104 0.1761 0.5451 0.2481 0.0198
0.0000 0.0006 0.0210 0.2173 0.4328 0.2970
0.0000 0.0000 0.0018 0.0349 0.2479 0.3315
0.0021 0.0099 0.0245 0.0424 0.0572 0.0639
0.0075 0.0283 0.0550 0.0721 0.0705 0.0532
0.0200 0.0581 0.0825 0.0733 0.0417 0.0124
0.0428 0.0899 0.0827 0.0380 0.0044 0.0030
0.0014 0.0664 0.8231 0.1079 0.0011 0.0000
0.0001 0.0043 0.0957 0.7540 0.1441 0.0019
0.0000 0.0003 0.0084 0.1220 0.6862 0.1801
0.0000 0.0000 0.0007 0.0138 0.1453 0.6199
A2IIt/r-X2~ ÷ system 0 1 2 3 4 5
0.9943 0.0057 0.0000 0.0000 0.0000 0.0000
0.0056 0.9817 0.0127 0.0000 0.0000 0.0000
B2X+-X2X+ system 0 1 2 3 4 5
0.9057 0.0857 0.0078 0.0008 0.0001 0.0000
0.0925 0.7332 0.1500 0.0211 0.0028 0.0004
C2X+-X2X+ system 0 1 2 3 4 5
0.9302 0.0680 0.0018 0.0000 0.0000 0.0000
0.0662 0.0034 0.7956 0.1254 0.1326 0.6665 0.0055 0.1930 0.0001 0.0115 0.0000 0.0002
D2X+-X2~+ system 0 1 2 3 4 5
0.0000 0.0002 0.0009 0.0022 0.0046 0.0079
0.0004 0.0022 0.0067 0.0145 0.0246 0.0352
E21-I3/2-"X2~+ system 0 1 2 3 4 5
0.9640 0.0358 0.0002 0.0000 0.0000 0.0000
0.0345 0.8932 0.0718 0.0005 0.0000 0.0000
F-C factors are practically zero. The Av = 0 sequence of the C - X system of C a l l is not so prominent as in the case of the other systems, and the band 0, 3 might be observed in addition to those of the above systems. The bands of the D - X system of the C a l l molecule are widely distributed as is revealed by their F-C factors.
269
Some more bands in the C - X system of SrH could be expected to be observed under improved experimental conditions than have been observed by More and Cornell [21]. Only the bands 3, 0 and beyond in the v'(v"= 0) progression and 0, 4 and beyond in the v"(v'= 0) progression might not be observed since their F-C factors are either zero or too small. In the A2II1/2-X2E +, A21I3/2---X2~ +, B-X, C-X, and E - X systems of the B a H molecule, the Av = 0 progression seems to be quite prominent whereas the D - X system would be very weak.
Acknowledgements One of the authors (MVR) is grateful to the U G C of India for a Teacher Fellowship.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R.S. MuUiken, Phys. Rev. 25 (1925) 509. G. Lib&ale and S. Weniger, Physica 41 (1969) 47. E. Hulthen, Phys. Rev. 29 (1927) 97. B. Grundstr6m, Z. Physik 69 (1931) 235. B. Grundstr6m, Z. Physik 75 (1932) 302. B. Grundstr6m, Z. Physik 95 (1935) 574. W.W. Watson and R. Weber, Phys. Rev. 48 (1935) 732. M. Aslam Khan, Proc. Phys. Soc. 80 (1%2) 593. G. Edvinson, I. Kopp, B. Lindgren and N. Aslund, Ark. Fys. 25 (1%3) 95. M. Aslam Khan, Proc. Phys. Soc. 87 (1966) 569. M. Aslam Khan and M.K. Afridi, J. Phys. B1 (1968) 260. M. Aslam Khan and S.S. Hussain, Nuovo Cimento: Soc. Ital. Fis. B18 (1973) 384. L.E. Berg and L. Klynning, Astron. Astrophy. Suppl. Ser. 13 (1974) 325. B. Kaving, B. Lindgren and D.A. Ramsay, Phys. Scr. 10 (1974) 73. B. Kaving and B. Lindgren, Phys. Scr. 10 (1974) 81. L.E. Berg and L. Klynning, Phys. Scr. 10 (1974) 331. L.E. Berg, L. Klynning and H. Martin, Opt. Commun. 17 (1976) 320. W.W. Watson and W.R. Fredrickson, Phys. Rev. 39 (1932) 765. W.R. Fredrickson, M.E. Hogan, Jr. and W.W. Watson, Phys. Rev. 48 (1935) 602. W.W. Watson, W.R. Fredrickson and M.E. Hogan, Phys. Rev. 49 (1936) 150. K.R. More and S.D. Cornell, Phys. Rev. 53 (1938) 806.
270 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
M. V. R a m a n a i a h and S. V.J. L a k s h m a n / Call. S r H and B a H potentials and F r a n c k - C o n d o n factors
M. Aslam Khan, Proc. Phys. Soc. 81 (1963) 1047. M. Aslam Khan, Proc. Phys. Soc. 82 (1%3) 564. M. Aslam Khan, Proc. Phys. Soc. 89 (1966) 165. M. Aslam Khan and M.R. Butt, J. Phys. BI (1%8) 745. M. Aslam Khan, M. Raft and S.J.A. Hussainee, J. Phys. Bll (1976) 1953. M. Aslam Khan, M. Raft, Iqbal Khan and M. Aslam Baig, J. Phys. B9 (1976) 2313. I.A. Khan, M. Raft and M.A. Khan, Nuovo Cimento: Soc. Ital. Fis. B44B (1978) 394. A. Schaafsma, Z. Physik 74 (1932) 254. W.R. Fredrickson and W.W. Watson, Phys. Rev. 39 (1932) 753. W.W. Watson, Phys. Rev. 43 (1933) 9. G.W. Funke, Z. Physik 84 (1933) 610. P.G. Koontz and W.W. Watson, Phys. Rev. 48 (1925) 937. B. Grundstr6m, Z. Physik 99 (1936) 595. G.W. Funke and B. Grundstr6m, Z. Physik 100 (1936) 293.
[36] I. Kopp, N. Aslund, G. Edvinson and B. Lindgren, Ark. Fys. 30 (1965) 321. [37] I. Kopp, M. Kronekvist and A. Guntsch, Ark. Fys. 32 (1966) 371. [381 M. Aslam Khan, J. Phys. B1 (1968) 985. [39] L. Veseth, Mol. Phys. 25 (1973) 333. [40] S.V.J. Lakshman and T.V. Ramakrishna Rao, J. Phys. B4 (1971) 269. [41] B. Chakraborty and Y.K. Pan, Appl. Spectr. Rev. 7 (1973) 283. [42] K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, Vol. IV: Constants of Diatomic Molecules (Van Nostrand Reinhold Co., New York. 1979). [431 S.V.J. Lakshman and M. Venkataramanaiah, Curr. Sci. 49 (1980) 579. [44] W.R. Jarmain and P.A. Fraser, Proc. Phys. Soc. 66 (1953) 1153. [45] P.A. Fraser and W.R. Jarmain, Proc. Phys. Soc. 66 (1953) 1145.