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Line strengths in a 3Σ-3Σ quadrupole transition with intermediate coupling: Application to line intensities in the quadrupole fundamental band of the oxygen molecule
JOURNAL
OF MOLECULAR
SPECTROSCOPY
144, 374-380
(1990)
Line Strengths in a 3Z- 3E Quadrupole Transition with Intermediate Coupling: Application to Line Intensities in the Quadrupole Fundamental Band of the Oxygen Molecule T. K. BALASUBRAMANIAN AND ROMOLA D’CUNHA
AND
K. NARAHARI
RAO
Department yl’Physic:c Ohio State University 174 Hht 18th Avenue, Colwnh~ts. Ohio 43210
Closed form expressions are developed for the intensity factors of the 13 branches resulting from a 3Z’(int)-3Z’( int) electric quadrupole transition in a diatomic molecule. The rigorous intermediate coupling treatment which includes centrifugal distortion effects results in the occurrence of the L/-form and M-form branches which are forbidden in the limit of Hund’s case (b) in both the ‘Z states. The branch intensity factors are used to recalculate the absorption intensities of rovibrational transitions in the quadrupole fundamental band of 0:. It is shown that departure from the strict case (b) limit moderately affects the intensities of some of the low cel1990 Academic Press. Inc. J transitions.
I. INTRODUCTION
It has long been recognized that electric quadrupole rotation-vibration spectra. though intrinsically weak, may be important for the detection of homonuclear molecules in planetary atmospheres (1. 2). Following the recent identification of a few quadrupole rotation-vibration lines of I602 in the atmospheric window near 1603 cm -’ (3, 4). Rothman and Goldman (5) reported calculations of positions and intensities of prominent quadrupole lines in the fundamental band of ‘“O?. In the computation of line intensities these authors chose to treat the X3X.; state in the limit of Hund’s case (b). For transitions involving low rotational levels, it appears to be necessary to avoid this approximation. In this context it is useful to recall that in the ground state (X32; ) of 02, the spin splitting parameter A,, has the value 2.0 cm-’ (as against B0 = 1.4 cm-‘). Consequently the low N levels will exhibit incipient or partial spin uncoupling warranting a rigorous intermediate coupling description. In light of this we have developed closed form line strength formulas for the 23 branches that can arise for 3Z i-3X * quadrupole transitions with intermediate coupling. Using these we have recalculated the absorption line intensities in the quadrupole fundamental band of 02. The results are discussed in this paper. 0022.2852/90 Couynght
$3.00
(CJ1990 hy Academic Press, Inc
All rights of reprcduct~on in any form reserved.
374
LINE INTENSITIES
IN THE 02 FUNDAMENTAL
2. QUADRUPOLE (a)
LINE INTENSITY
375
FACTORS
General Line Strength Formda and Selection Rules
A lucid exposition of the basic theory underlying electric quadrupole transitions in atomic systems may be found in the book by Kemble (6). Chiu ( 7). after casting Kemble’s result in the more convenient spherical tensor notation, adapted it to the line strength problem in singlet-singlet electronic transitions in diatomic molecules. A straightforward generalization of his method to multiplet transitions yields the “master formula” S(jJ’,
iJ) = (2J + 1) I 2
urr2a,r2t x (n’h’S’
2’; u’lQ?h/nASZ:u)C(J2J’;~2Xa’)/’
(1)
which S(jJ’, iJ) is the line strength for the transition “‘+I A’[F;( J’)] - “+I h[F,( J)] , Fi( J) being an intermediate coupling term series designation; QZx is
in
a molecule-fixed spherical component of the electric quadrupole operator: I nhSS; Y) = / n A: U) I SS ) is a vibronic Hund’s case (a) substate, and C is a Clebsch-Gordon coefficient. a, is a typical element of the unitary matrix that connects the intermediate coupling rotational eigenfunctions 1F,(J)) and the case (a) basis functions. As usual singly primed quantities refer to the upper state. Since Q,, (A = -2. -1, . . , +2) in Eq. ( 1) is a purely orbital operator its matrix elements between case (a) substates vanish unless S’ = S and 8’ = S. Equation ( I ) i++ f,O++ 1)and )h’-A also implies the selection rules j J’ - JJ <2(withO+++O. = 9 ’ - 9 1 s 2. The additional selection rule + H +. - ++ - for the rovibronic parity molecules and the special rule g - g. II c-t II for the vibronic parity in charge-symmetric follow from the fact that all the components of the electric quadrupole operator are bilinear in the spatial coordinates.
(h) Line Strengths_f& the -‘S ’ (int) - 35 * (int) Transitiorl In order to apply Eq. (1) to a 35*-3X3’ transition we need the rotational eigenfunctions corresponding to the three term series F,(J) (i = I, 2, 3) of 32, valid for intermediate coupling. A rigorous calculation that includes centrifugal distortion effects yields ( 8)
where .yJ =
I F,(J))
= sJ13S,. 1J, +) + c,,I~~~, OJ),
IFAJ))
= 13E,. 1J. -)
/ Fj(J,)
= c.,(~P-,, 1 J, t)
j3Xi, lJ, -t) = 2”‘[13Z_,.
([F3(J> - Fz(J) + 4DJ(J+
-15)
(2) - .~,(~5~, OJ), 1
* 13X1, lJ)]
1)]/[F3(J)
- FI(J)])“‘, CJ = (1 - ,?)‘I?.
(3)
Note that, strictly, the case (a) basis functions I 35n, ClJ) should have an M-label (the eigenvalue of the space-fixed component Jz) which has been suppressed here. This is
376
BALASUBRAMANIAN.
D’CUNHA.
AND
RAO
because in Eq. ( 1) the summation over the M- M’-sublevels has already. For a 35’-32’ transition the selection rules stated above lead branches as shown in Fig. 1. Equation ( 1) requires that A’ - A = only nonvanishing matrix element of Qlh is ( IZ’~S; II’ ( &, 1n38; t‘)
N’
been carried
out
to a total of 23 A. Therefore, the = Q?, which can
“1;
J’
F* t F3 + F,
15
+
15 if
: I
,q_-L-------__---------_____-__-----------___-
14
f3--‘---------=r=======~=~~~~=~~~~=~=~~~~~~~~~~
,5__,_____----
I
I +
13 1
13
1: -
F2
j=====:" ,
12
12--Ill - -,13 - -,I 11-
11
1; r I lo--_I-
10
,; z
1,: I
4----c
9
8--I-IO ( l3--1-
8
---------
,---i_ q--,l----c
7
+=
A
I
I
/
I
I
I
I
I
1. 'd ILo I I
r!’ 3,
/
,”
I I
I
/
I
1
I
/
I
I I
_l
/
ccl = I
I
,
I
N
CL-
P
’ I
11
111 10 I 12 -
N" J"
/
I
I
I
I
1
I
/ I
i+
F, ;
F; F,
FIG 1. Energy level diagram showing the 23 branches in a ‘S;-“~ dx quadrupole transition. The CT-form (AN= +4) and M-form ( AN = -4) transitions (indicated by dashes) are forbidden in the limit of Hund’s case ( b) in both the states. Even N’ levels indicated by dashes do not occur for 1602.
377
LINE INTENSITIES IN THE O2 FUNDAMENTAL
be made real by a suitable choice of the phases of the basis functions. Consequently the branch intensities are governed by the single transition moment Q2. Explicit formulas for the Clebsch-Gordon coefficient appearing in Eq. ( 1) have been tabulated by Condon and Shortley ( 9). These in conjunction with relations ( 1) and (2) above finally yield the line strength expressions (the so-called Honl-London factors) given in Table I. For intermediate coupling even in one of the 32 states, all the 23 branches will have, in general, nonzero intensities. However, in the limit of pure case (b) in both the states the additional selection rule IN’ - N ] d 2 limits the number of branches to 2 1. Indeed, upon making the case (b) substitution sJ = s (25 + 1)I”? in the present formulas one may readily verify that the line strengths of the U-form (M = +4) and M-form (AN = -4) branches vanish, thus restoring the case (b) selection rule on N. The occurrence of the two extra branches is thus a qualitative feature of the intermediate coupling calculation presented here. ’ J=
[(J+
l)/(zJ+
l)]“‘,cJ=
TABLE I Line Strengths” in a ‘S-‘8 Electric Quadrupole Transition
dJ=
[J/
BALASUBRAMANIAN.
378
D’CUNHA.
AND
RAO
Table I includes, in the last column, limiting formulas for case (b). Equivalent expressions in terms of 3 - j and 6 - j symbols may be found in Ref. (5) which treats the intensity problem entirely in the case (b) limit. In passing it is useful to state a few “sum rules” obeyed by the line strengths. From the expressions in Table I together with the conditions sj + cj = 1, stemming from the normalization of the rotational eigenfunctions (Eq. (2)), it is easily verified that the sum of the strengths of all transitions to a common terminal level F,(J) is Qf( 25 + I)_ for each i. From the symmetrical manner in which the line strengths are defined, a similar result should also hold for all transitions sharing a fixed initial level. 3. APPLICATION
TO LINE
INTENSITIES
IN THE 0,
FUNDAMENTAL
BAND
The absorption intensity for the quadrupole rovibrational transition F, (t” = 1, J’) c F, (v = 0, J) within the X3S; state of 01 could be calculated from the relation, Jabs = CQSu3S(jJr, where u is the line position is the matrix element. Q2 = (X3X,v
iJ)exp[-F,(O. (in cm-‘),
J)hc/kT][l
- exp(-huc/kT)],
S is the line strength
= 1 IQ&Y3S,;,2)
(4)
listed in Table I, and Q?
= 0) = (v = 1 I&u
= 0).
The rotational term values for the u = 0 and 1 levels as well as the parameters (:, and sJ occurring in the line strength formulas were first generated by diagonalizing the X 39 Hamiltonian in case (a) basis (see Ref. (8)), employing suitable computational procldures. The molecular parameters needed for these computations were taken from Ref. (5). The line positions were obtained as the appropriate term differences. A temperature of 296 K was assumed in the intensity calculation and the product . Eq . (4) was fixed by requiring that the intensity for the &( 9) line, which is CQ, * m one of the strongest, agreed with the value reported in Ref. (5). As may be seen from Table I, the line strengths for the branches SZ1, QZ2, and OZZ are insensitive to the coupling condition. Hence. with the above normalization, the intensities for these three branches obtained from the present calculations agreed automatically with that reported in Ref. (5). The line wavenumbers, assignments, and intensities are listed in Table II. The line intensities given here in cm/molecule may be converted to the more familiar unit cm-* atm-’ using the appropriate conversion factor reported by Pugh and Rao (10). A stick spectrum generated using these data is displayed in Fig. 2. Note that in the X3X; state of 160Z even N levels cannot occur. 4. DISCUSSION
Since departure from Hund’s case (b) can be significant only at low rotational quantum numbers, it is natural to expect that intensities of transitions involving low J (or N) values will be particularly susceptible. This is borne out by a comparison of the present intermediate coupling data with the line intensities given in Ref. (5), based on case (b) assumption. For the QR,~ (J” = I ) line, for instance. the case (b) calculation overestimates the intensity by about 30%. We need scarcely emphasize that the present intensities based on a rigorous treatment of the coupling in the X3X; are more dependable. As more accurate and extensive experimental data on line intensities become available it becomes possible, with the help of the present line strength expressions, to go
379
LINE INTENSITIES IN THE O2 FUNDAMENTAL TABLE II Calculated Positions” and Absorption Intensities of Rovibrational Quadrupole Transitions in “‘0: vlrm
-1
)
Jnknslty
Rranch
N"
3"
v(cm
-1
1
Jntenslty
cm/molrc 1394.3154 1394.3970 1407.2888 1407.3106 1407.3726 1470.0951
2.01 7.07 3.99 4.11 4.34 7.41
E-30 F-30 F-30 F-30 F-30 F-30
8;; M;
5: 5:
D11 033
;:
:5
1420.1171 1470.1795
7.68 8.17
E-30 F-30
03;
;:
5:
1437.8153 1437.7930 1432.8279 1445.3811 1445.4038 1445.4167 1457.e5fJo
1.34 1.79 l.LJ 7.11 7.70 7.35 3.20
F-79 F-29 F-T9 F-79 F-79 F-79
::: 011 033 077 011 033
:1 71
s': 77
1: 1:
i8 ;I
1457.8945 1457.8817 1470.2770
3.63 3.37 4.53
F-79 F-79
011 "?? 033
1: 15
1'8 1;
1470.7461 1470.7599 1487.4716 1487.4969 1487 5115
4.80 5.71 5.91 6.34 6.91
F-29 F-29 F-79 F-73 F-79
077 011 013 077 011
1: 1: 13
57 74 7s
1s 1:
1497.5 1494.6846 65
7.01 1.33
F-79 E-30
51;
11
Ib
1494.6371 1494.6479
7.69 8.60
F-79
011
1:
i:
1504.6783 1496.5539 1506.6189 1506.6507 1506 66An 1508.5957 1518 5085 1516.5554
7.77 1.54 7.60 8.45 9 72 7.66 7.08 3.68
F-IO F-30 F-79 F-79 F 79 F-30 F-79 F-30
% 8::
'l z
'$ a
1518.5494 1518.5713 1570.5779 1578.3717 1530.2541 1530.3781 1530.3595 1537.3397 1540.0981 1540.0994 1541.9846 1547 0430 1541.1837 1544.0683
6.20 9.88 4 63 6.7? 5.15 6.54 8.72 8.64 I.18 1.16 3.04 6.04 2.30 7.07
F-79 F-79 F~30 F-30 F-73 F-79 E-29 F-30 F-73 F-79 F-79 F-79 F-30 F-71
1544.3 74 1544.1 3 64 1546.0647 1546.0660 1546.0700 1541.6774 1547.6793 1547 6333 1549.0609 1549 0677 1549.0667 1550.3657 1550.3670 1550.3703 1551.5407 1551.5475 1551.5464 1557.'376 ;;x; 3793
;:pj FI38 3.98 F-30
,551.zK: 1557 '1937 1557 9739 1553.07ll4 1553.4853 1553 5077 1553 5091 1551 5130 1553 5304 155? 3038 15s3 !l74? 1554 1170 1554.7463 1554.?667
3.67
F-30
:.::, &GO 6 a9 1.76
F-:8 r 7n F-30 F-79
i:1; 7.03 1.83 1.90 3.07 7 73 7.85 1.37 1.55 4.35 3.80
F-s; F-79 F-73 F-29 F-79 F-73 F 79 F Jfl F-30 F-79 F-79
;:;; :-;; ? TiFl r-10 3.49 r-30 5.77 F-79 4.31 F-29 Ci.lA F.79 'I.:!+ 5 51 1 40 9.73 I 51
I-% F 30 F-71 F-30 F-79
Rraoch
cmlmolec
[II P73 F";
i
z
I;:
:
7"
:
7
:
:
:
:
::: P73 013 P17 :::
:
6'
:
: 3
$;;
: 1
077 011 P73 Pl7 033
:
z ;
1554.7990
6.98
F-29
011
1554.3n49 1554.3011 1554.4755 1554.9636 1554.9659 1554.9695 1555.5017
6.19 5.87 1.97 7 R7 6.75 6.74 7.95
F-79 F-79 F-73 F-73 F-79 F-79
8:: R17 011 8;;
1555.5039 1555.5077 1555.9119 1555.9154 1555.9187 1556.1958 1556.7014
5.97 6.50 7.14 4.60 5.74 5.79 7.00
l--79 F-?Y F-79 F-79 F-79 F-79 F-79
"0:: nil $I;
1556.7766 1556.7077 1556.3573
7.73 1.5R 7.38
F-3ll F-79 F-79
2:: (111
1556.3663 1556.3607 1556.9150 1557.0093 1551.4807
1.40 8 37
F-30 F-JO
K
;-:: i:77
;::t: F-30
zi R73
3.55 5.64 9.47 7.77
F-30 F-3ll F-30
:;I pR;:
I-?? l.;, 7.37 2.54 6.R4 : f:
;I;; F-j9 F-71 F-30 F-79 F-7;
::I R77 ::;
1-i'; ;I;;
;:I ::: C?? 533 RPI
557. 731 557. 5 060 558.1577 557.9798 550.7374 550.7870 558.4366 56R.cY31 570.3699 570.4997 570.5788 577.4547 577.4556 579.7784 581.7483 581.7978 581.8677 58J.7471 590.099? 597.8417 591.8760 537.9108 594.8638 601.8774 603.7968 601.8766
16i c 614~
1614 6477 lGl4.6677 1616 6670 1673 4168 1675.734A 1675.J71fl 1675.3386 1677.3874 1635 8353 1535.86Ofi 1635.876R
Qll
011 033
P71
523
1 09
F-78
;.;g
;r;;
6.3: 7.q6 1 :z
K:oo ;I;;
9.81 :.A;
F-73 :::;
s:: <27 533 R71
1.49 1 ?Y
F-78 r 78
::: :??
1.46 : 57 1 4cI 1.30 70 ? :9
i I 6?
F-IO F-30 F--78 f-78 f-78 r-Jn F Jfl
I-:; i.li
;::i F-7a
9 17 9.36
F-79 F-73
; .;fl F-3;
I646 7141 1646.7589 1646.7747 1656.4837 185K.51J6 1656.5783 1666.59(1? 1666.67?3 1666.h367 1676.5589 1676.5879
:.g;
:r;;
: 3: 5 55 5 37 3 98 3.75 3 611 2.50 ? 17
F-z; r--is F-79 f-=79 r-79 F-71 F-79 fm?9
1676.5969 1686 ?GRR 1686.3377 1686 4065
7 ?; 1 io 16
I
1.5; F.79 F-?1
1696.0755 1696 0493 1696.0630 1705.5iilfl
$ 8: 7.58 3.9a
: :: F-30 r-30
N-
3"
BALASUBRAMANIAN.
D’CUNHA,
AND
RAO
I
/
.
FIG. 2. Computer generated “stick spectrum” showing rovibrational transitions in the quadrupole fundamental band of Oz. A temperature of 296 K was assumed in the computation of absorption intensities. Due to the near equality of the splitting parameters and the rapid spin uncoupling in the two states, each N-transition in the main branches 0, Q, and S occurs as a triplet whose components are too close to be seen separately in the figure.
just determining the parameter (33, and to sort out quantitatively the small contributions to the intensities of some of the branches due to the magnetic dipole mechanism which can concurrently, though weakly. occur along with the quadrupole mechanism ( 11). beyond
5. ACKNOWLEDGMENT One of us ( KNR) is gratified that part of this research was done during the tenure of a NASA-Ames State University consortium. RECEIVED
Ohio
June 5, 1990 REFERENCES
1. -7. 3. 4. 5. 6. 7. 8. Y.
10. 11.
G. G. A. J.
HERZBERG..h/rophys
J. 87.428-437 ( 1938). 163. 170 ( 1949).
HERZBERG. Nutuw (London)
GOLDMAN, J. REID, AND L. S. ROTHMAN, G‘ro.spl~~~RL:s. Ldf
8, 77-78
( 1981 ).
REID, R. L. SINCLAIR, A. M. ROBINSON, AND A. R. W. MCKELLAR, P/~Y
Rev. .-1 24, 1944-1949 (1981). L. S. ROTHMAN AND A. GOLDMAN, Appl. Opr. 20,2182-2184 (1981). E. C. KEMBLE, “The Fundamental Principles of Quantum Mechanics.” McGraw-Hill. New York. 1937. Y. N. CHILI, J. Chern. P~JX 42. 2671-2681 (1965). T. K. BALASUBRAMANIANAND V. P. BELLARY, .-lcfa PIzw. ffwzg. 63, 249-255 ( 1988). E. U. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra,” Cambridge Univ. Press. Cambridge, England, 1935. L. A. PUCH AND K. NARAHARI RAO, in “Molecular Spectroscopy: Modern Research” (K. Narahari Rao. Ed.), Vol. II. p. 170. Academic Press, New York. 1976. T. K. BALASUBRAMANIANAND V. P. BELLARY. “43rd Symposium on Molecular Spectroscopy, Columbus, Ohio,” Paper MF5.
OF MOLECULAR
SPECTROSCOPY
144, 374-380
(1990)
Line Strengths in a 3Z- 3E Quadrupole Transition with Intermediate Coupling: Application to Line Intensities in the Quadrupole Fundamental Band of the Oxygen Molecule T. K. BALASUBRAMANIAN AND ROMOLA D’CUNHA
AND
K. NARAHARI
RAO
Department yl’Physic:c Ohio State University 174 Hht 18th Avenue, Colwnh~ts. Ohio 43210
Closed form expressions are developed for the intensity factors of the 13 branches resulting from a 3Z’(int)-3Z’( int) electric quadrupole transition in a diatomic molecule. The rigorous intermediate coupling treatment which includes centrifugal distortion effects results in the occurrence of the L/-form and M-form branches which are forbidden in the limit of Hund’s case (b) in both the ‘Z states. The branch intensity factors are used to recalculate the absorption intensities of rovibrational transitions in the quadrupole fundamental band of 0:. It is shown that departure from the strict case (b) limit moderately affects the intensities of some of the low cel1990 Academic Press. Inc. J transitions.
I. INTRODUCTION
It has long been recognized that electric quadrupole rotation-vibration spectra. though intrinsically weak, may be important for the detection of homonuclear molecules in planetary atmospheres (1. 2). Following the recent identification of a few quadrupole rotation-vibration lines of I602 in the atmospheric window near 1603 cm -’ (3, 4). Rothman and Goldman (5) reported calculations of positions and intensities of prominent quadrupole lines in the fundamental band of ‘“O?. In the computation of line intensities these authors chose to treat the X3X.; state in the limit of Hund’s case (b). For transitions involving low rotational levels, it appears to be necessary to avoid this approximation. In this context it is useful to recall that in the ground state (X32; ) of 02, the spin splitting parameter A,, has the value 2.0 cm-’ (as against B0 = 1.4 cm-‘). Consequently the low N levels will exhibit incipient or partial spin uncoupling warranting a rigorous intermediate coupling description. In light of this we have developed closed form line strength formulas for the 23 branches that can arise for 3Z i-3X * quadrupole transitions with intermediate coupling. Using these we have recalculated the absorption line intensities in the quadrupole fundamental band of 02. The results are discussed in this paper. 0022.2852/90 Couynght
$3.00
(CJ1990 hy Academic Press, Inc
All rights of reprcduct~on in any form reserved.
374
LINE INTENSITIES
IN THE 02 FUNDAMENTAL
2. QUADRUPOLE (a)
LINE INTENSITY
375
FACTORS
General Line Strength Formda and Selection Rules
A lucid exposition of the basic theory underlying electric quadrupole transitions in atomic systems may be found in the book by Kemble (6). Chiu ( 7). after casting Kemble’s result in the more convenient spherical tensor notation, adapted it to the line strength problem in singlet-singlet electronic transitions in diatomic molecules. A straightforward generalization of his method to multiplet transitions yields the “master formula” S(jJ’,
iJ) = (2J + 1) I 2
urr2a,r2t x (n’h’S’
2’; u’lQ?h/nASZ:u)C(J2J’;~2Xa’)/’
(1)
which S(jJ’, iJ) is the line strength for the transition “‘+I A’[F;( J’)] - “+I h[F,( J)] , Fi( J) being an intermediate coupling term series designation; QZx is
in
a molecule-fixed spherical component of the electric quadrupole operator: I nhSS; Y) = / n A: U) I SS ) is a vibronic Hund’s case (a) substate, and C is a Clebsch-Gordon coefficient. a, is a typical element of the unitary matrix that connects the intermediate coupling rotational eigenfunctions 1F,(J)) and the case (a) basis functions. As usual singly primed quantities refer to the upper state. Since Q,, (A = -2. -1, . . , +2) in Eq. ( 1) is a purely orbital operator its matrix elements between case (a) substates vanish unless S’ = S and 8’ = S. Equation ( I ) i++ f,O++ 1)and )h’-A also implies the selection rules j J’ - JJ <2(withO+++O. = 9 ’ - 9 1 s 2. The additional selection rule + H +. - ++ - for the rovibronic parity molecules and the special rule g - g. II c-t II for the vibronic parity in charge-symmetric follow from the fact that all the components of the electric quadrupole operator are bilinear in the spatial coordinates.
(h) Line Strengths_f& the -‘S ’ (int) - 35 * (int) Transitiorl In order to apply Eq. (1) to a 35*-3X3’ transition we need the rotational eigenfunctions corresponding to the three term series F,(J) (i = I, 2, 3) of 32, valid for intermediate coupling. A rigorous calculation that includes centrifugal distortion effects yields ( 8)
where .yJ =
I F,(J))
= sJ13S,. 1J, +) + c,,I~~~, OJ),
IFAJ))
= 13E,. 1J. -)
/ Fj(J,)
= c.,(~P-,, 1 J, t)
j3Xi, lJ, -t) = 2”‘[13Z_,.
([F3(J> - Fz(J) + 4DJ(J+
-15)
(2) - .~,(~5~, OJ), 1
* 13X1, lJ)]
1)]/[F3(J)
- FI(J)])“‘, CJ = (1 - ,?)‘I?.
(3)
Note that, strictly, the case (a) basis functions I 35n, ClJ) should have an M-label (the eigenvalue of the space-fixed component Jz) which has been suppressed here. This is
376
BALASUBRAMANIAN.
D’CUNHA.
AND
RAO
because in Eq. ( 1) the summation over the M- M’-sublevels has already. For a 35’-32’ transition the selection rules stated above lead branches as shown in Fig. 1. Equation ( 1) requires that A’ - A = only nonvanishing matrix element of Qlh is ( IZ’~S; II’ ( &, 1n38; t‘)
N’
been carried
out
to a total of 23 A. Therefore, the = Q?, which can
“1;
J’
F* t F3 + F,
15
+
15 if
: I
,q_-L-------__---------_____-__-----------___-
14
f3--‘---------=r=======~=~~~~=~~~~=~=~~~~~~~~~~
,5__,_____----
I
I +
13 1
13
1: -
F2
j=====:" ,
12
12--Ill - -,13 - -,I 11-
11
1; r I lo--_I-
10
,; z
1,: I
4----c
9
8--I-IO ( l3--1-
8
---------
,---i_ q--,l----c
7
+=
A
I
I
/
I
I
I
I
I
1. 'd ILo I I
r!’ 3,
/
,”
I I
I
/
I
1
I
/
I
I I
_l
/
ccl = I
I
,
I
N
CL-
P
’ I
11
111 10 I 12 -
N" J"
/
I
I
I
I
1
I
/ I
i+
F, ;
F; F,
FIG 1. Energy level diagram showing the 23 branches in a ‘S;-“~ dx quadrupole transition. The CT-form (AN= +4) and M-form ( AN = -4) transitions (indicated by dashes) are forbidden in the limit of Hund’s case ( b) in both the states. Even N’ levels indicated by dashes do not occur for 1602.
377
LINE INTENSITIES IN THE O2 FUNDAMENTAL
be made real by a suitable choice of the phases of the basis functions. Consequently the branch intensities are governed by the single transition moment Q2. Explicit formulas for the Clebsch-Gordon coefficient appearing in Eq. ( 1) have been tabulated by Condon and Shortley ( 9). These in conjunction with relations ( 1) and (2) above finally yield the line strength expressions (the so-called Honl-London factors) given in Table I. For intermediate coupling even in one of the 32 states, all the 23 branches will have, in general, nonzero intensities. However, in the limit of pure case (b) in both the states the additional selection rule IN’ - N ] d 2 limits the number of branches to 2 1. Indeed, upon making the case (b) substitution sJ = s (25 + 1)I”? in the present formulas one may readily verify that the line strengths of the U-form (M = +4) and M-form (AN = -4) branches vanish, thus restoring the case (b) selection rule on N. The occurrence of the two extra branches is thus a qualitative feature of the intermediate coupling calculation presented here. ’ J=
[(J+
l)/(zJ+
l)]“‘,cJ=
TABLE I Line Strengths” in a ‘S-‘8 Electric Quadrupole Transition
dJ=
[J/
BALASUBRAMANIAN.
378
D’CUNHA.
AND
RAO
Table I includes, in the last column, limiting formulas for case (b). Equivalent expressions in terms of 3 - j and 6 - j symbols may be found in Ref. (5) which treats the intensity problem entirely in the case (b) limit. In passing it is useful to state a few “sum rules” obeyed by the line strengths. From the expressions in Table I together with the conditions sj + cj = 1, stemming from the normalization of the rotational eigenfunctions (Eq. (2)), it is easily verified that the sum of the strengths of all transitions to a common terminal level F,(J) is Qf( 25 + I)_ for each i. From the symmetrical manner in which the line strengths are defined, a similar result should also hold for all transitions sharing a fixed initial level. 3. APPLICATION
TO LINE
INTENSITIES
IN THE 0,
FUNDAMENTAL
BAND
The absorption intensity for the quadrupole rovibrational transition F, (t” = 1, J’) c F, (v = 0, J) within the X3S; state of 01 could be calculated from the relation, Jabs = CQSu3S(jJr, where u is the line position is the matrix element. Q2 = (X3X,v
iJ)exp[-F,(O. (in cm-‘),
J)hc/kT][l
- exp(-huc/kT)],
S is the line strength
= 1 IQ&Y3S,;,2)
(4)
listed in Table I, and Q?
= 0) = (v = 1 I&u
= 0).
The rotational term values for the u = 0 and 1 levels as well as the parameters (:, and sJ occurring in the line strength formulas were first generated by diagonalizing the X 39 Hamiltonian in case (a) basis (see Ref. (8)), employing suitable computational procldures. The molecular parameters needed for these computations were taken from Ref. (5). The line positions were obtained as the appropriate term differences. A temperature of 296 K was assumed in the intensity calculation and the product . Eq . (4) was fixed by requiring that the intensity for the &( 9) line, which is CQ, * m one of the strongest, agreed with the value reported in Ref. (5). As may be seen from Table I, the line strengths for the branches SZ1, QZ2, and OZZ are insensitive to the coupling condition. Hence. with the above normalization, the intensities for these three branches obtained from the present calculations agreed automatically with that reported in Ref. (5). The line wavenumbers, assignments, and intensities are listed in Table II. The line intensities given here in cm/molecule may be converted to the more familiar unit cm-* atm-’ using the appropriate conversion factor reported by Pugh and Rao (10). A stick spectrum generated using these data is displayed in Fig. 2. Note that in the X3X; state of 160Z even N levels cannot occur. 4. DISCUSSION
Since departure from Hund’s case (b) can be significant only at low rotational quantum numbers, it is natural to expect that intensities of transitions involving low J (or N) values will be particularly susceptible. This is borne out by a comparison of the present intermediate coupling data with the line intensities given in Ref. (5), based on case (b) assumption. For the QR,~ (J” = I ) line, for instance. the case (b) calculation overestimates the intensity by about 30%. We need scarcely emphasize that the present intensities based on a rigorous treatment of the coupling in the X3X; are more dependable. As more accurate and extensive experimental data on line intensities become available it becomes possible, with the help of the present line strength expressions, to go
379
LINE INTENSITIES IN THE O2 FUNDAMENTAL TABLE II Calculated Positions” and Absorption Intensities of Rovibrational Quadrupole Transitions in “‘0: vlrm
-1
)
Jnknslty
Rranch
N"
3"
v(cm
-1
1
Jntenslty
cm/molrc 1394.3154 1394.3970 1407.2888 1407.3106 1407.3726 1470.0951
2.01 7.07 3.99 4.11 4.34 7.41
E-30 F-30 F-30 F-30 F-30 F-30
8;; M;
5: 5:
D11 033
;:
:5
1420.1171 1470.1795
7.68 8.17
E-30 F-30
03;
;:
5:
1437.8153 1437.7930 1432.8279 1445.3811 1445.4038 1445.4167 1457.e5fJo
1.34 1.79 l.LJ 7.11 7.70 7.35 3.20
F-79 F-29 F-T9 F-79 F-79 F-79
::: 011 033 077 011 033
:1 71
s': 77
1: 1:
i8 ;I
1457.8945 1457.8817 1470.2770
3.63 3.37 4.53
F-79 F-79
011 "?? 033
1: 15
1'8 1;
1470.7461 1470.7599 1487.4716 1487.4969 1487 5115
4.80 5.71 5.91 6.34 6.91
F-29 F-29 F-79 F-73 F-79
077 011 013 077 011
1: 1: 13
57 74 7s
1s 1:
1497.5 1494.6846 65
7.01 1.33
F-79 E-30
51;
11
Ib
1494.6371 1494.6479
7.69 8.60
F-79
011
1:
i:
1504.6783 1496.5539 1506.6189 1506.6507 1506 66An 1508.5957 1518 5085 1516.5554
7.77 1.54 7.60 8.45 9 72 7.66 7.08 3.68
F-IO F-30 F-79 F-79 F 79 F-30 F-79 F-30
% 8::
'l z
'$ a
1518.5494 1518.5713 1570.5779 1578.3717 1530.2541 1530.3781 1530.3595 1537.3397 1540.0981 1540.0994 1541.9846 1547 0430 1541.1837 1544.0683
6.20 9.88 4 63 6.7? 5.15 6.54 8.72 8.64 I.18 1.16 3.04 6.04 2.30 7.07
F-79 F-79 F~30 F-30 F-73 F-79 E-29 F-30 F-73 F-79 F-79 F-79 F-30 F-71
1544.3 74 1544.1 3 64 1546.0647 1546.0660 1546.0700 1541.6774 1547.6793 1547 6333 1549.0609 1549 0677 1549.0667 1550.3657 1550.3670 1550.3703 1551.5407 1551.5475 1551.5464 1557.'376 ;;x; 3793
;:pj FI38 3.98 F-30
,551.zK: 1557 '1937 1557 9739 1553.07ll4 1553.4853 1553 5077 1553 5091 1551 5130 1553 5304 155? 3038 15s3 !l74? 1554 1170 1554.7463 1554.?667
3.67
F-30
:.::, &GO 6 a9 1.76
F-:8 r 7n F-30 F-79
i:1; 7.03 1.83 1.90 3.07 7 73 7.85 1.37 1.55 4.35 3.80
F-s; F-79 F-73 F-29 F-79 F-73 F 79 F Jfl F-30 F-79 F-79
;:;; :-;; ? TiFl r-10 3.49 r-30 5.77 F-79 4.31 F-29 Ci.lA F.79 'I.:!+ 5 51 1 40 9.73 I 51
I-% F 30 F-71 F-30 F-79
Rraoch
cmlmolec
[II P73 F";
i
z
I;:
:
7"
:
7
:
:
:
:
::: P73 013 P17 :::
:
6'
:
: 3
$;;
: 1
077 011 P73 Pl7 033
:
z ;
1554.7990
6.98
F-29
011
1554.3n49 1554.3011 1554.4755 1554.9636 1554.9659 1554.9695 1555.5017
6.19 5.87 1.97 7 R7 6.75 6.74 7.95
F-79 F-79 F-73 F-73 F-79 F-79
8:: R17 011 8;;
1555.5039 1555.5077 1555.9119 1555.9154 1555.9187 1556.1958 1556.7014
5.97 6.50 7.14 4.60 5.74 5.79 7.00
l--79 F-?Y F-79 F-79 F-79 F-79 F-79
"0:: nil $I;
1556.7766 1556.7077 1556.3573
7.73 1.5R 7.38
F-3ll F-79 F-79
2:: (111
1556.3663 1556.3607 1556.9150 1557.0093 1551.4807
1.40 8 37
F-30 F-JO
K
;-:: i:77
;::t: F-30
zi R73
3.55 5.64 9.47 7.77
F-30 F-3ll F-30
:;I pR;:
I-?? l.;, 7.37 2.54 6.R4 : f:
;I;; F-j9 F-71 F-30 F-79 F-7;
::I R77 ::;
1-i'; ;I;;
;:I ::: C?? 533 RPI
557. 731 557. 5 060 558.1577 557.9798 550.7374 550.7870 558.4366 56R.cY31 570.3699 570.4997 570.5788 577.4547 577.4556 579.7784 581.7483 581.7978 581.8677 58J.7471 590.099? 597.8417 591.8760 537.9108 594.8638 601.8774 603.7968 601.8766
16i c 614~
1614 6477 lGl4.6677 1616 6670 1673 4168 1675.734A 1675.J71fl 1675.3386 1677.3874 1635 8353 1535.86Ofi 1635.876R
Qll
011 033
P71
523
1 09
F-78
;.;g
;r;;
6.3: 7.q6 1 :z
K:oo ;I;;
9.81 :.A;
F-73 :::;
s:: <27 533 R71
1.49 1 ?Y
F-78 r 78
::: :??
1.46 : 57 1 4cI 1.30 70 ? :9
i I 6?
F-IO F-30 F--78 f-78 f-78 r-Jn F Jfl
I-:; i.li
;::i F-7a
9 17 9.36
F-79 F-73
; .;fl F-3;
I646 7141 1646.7589 1646.7747 1656.4837 185K.51J6 1656.5783 1666.59(1? 1666.67?3 1666.h367 1676.5589 1676.5879
:.g;
:r;;
: 3: 5 55 5 37 3 98 3.75 3 611 2.50 ? 17
F-z; r--is F-79 f-=79 r-79 F-71 F-79 fm?9
1676.5969 1686 ?GRR 1686.3377 1686 4065
7 ?; 1 io 16
I
1.5; F.79 F-?1
1696.0755 1696 0493 1696.0630 1705.5iilfl
$ 8: 7.58 3.9a
: :: F-30 r-30
N-
3"
BALASUBRAMANIAN.
D’CUNHA,
AND
RAO
I
/
.
FIG. 2. Computer generated “stick spectrum” showing rovibrational transitions in the quadrupole fundamental band of Oz. A temperature of 296 K was assumed in the computation of absorption intensities. Due to the near equality of the splitting parameters and the rapid spin uncoupling in the two states, each N-transition in the main branches 0, Q, and S occurs as a triplet whose components are too close to be seen separately in the figure.
just determining the parameter (33, and to sort out quantitatively the small contributions to the intensities of some of the branches due to the magnetic dipole mechanism which can concurrently, though weakly. occur along with the quadrupole mechanism ( 11). beyond
5. ACKNOWLEDGMENT One of us ( KNR) is gratified that part of this research was done during the tenure of a NASA-Ames State University consortium. RECEIVED
Ohio
June 5, 1990 REFERENCES
1. -7. 3. 4. 5. 6. 7. 8. Y.
10. 11.
G. G. A. J.
HERZBERG..h/rophys
J. 87.428-437 ( 1938). 163. 170 ( 1949).
HERZBERG. Nutuw (London)
GOLDMAN, J. REID, AND L. S. ROTHMAN, G‘ro.spl~~~RL:s. Ldf
8, 77-78
( 1981 ).
REID, R. L. SINCLAIR, A. M. ROBINSON, AND A. R. W. MCKELLAR, P/~Y
Rev. .-1 24, 1944-1949 (1981). L. S. ROTHMAN AND A. GOLDMAN, Appl. Opr. 20,2182-2184 (1981). E. C. KEMBLE, “The Fundamental Principles of Quantum Mechanics.” McGraw-Hill. New York. 1937. Y. N. CHILI, J. Chern. P~JX 42. 2671-2681 (1965). T. K. BALASUBRAMANIANAND V. P. BELLARY, .-lcfa PIzw. ffwzg. 63, 249-255 ( 1988). E. U. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra,” Cambridge Univ. Press. Cambridge, England, 1935. L. A. PUCH AND K. NARAHARI RAO, in “Molecular Spectroscopy: Modern Research” (K. Narahari Rao. Ed.), Vol. II. p. 170. Academic Press, New York. 1976. T. K. BALASUBRAMANIANAND V. P. BELLARY. “43rd Symposium on Molecular Spectroscopy, Columbus, Ohio,” Paper MF5.