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Linestrengths for the electric quadrupole fundamental band of the oxygen molecule with rotation-vibration interaction corrections
JOURNAL
OF MOLECULAR
SPECTROSCOPY
153,26-31 ( 1992)
Linestrengths for the Electric Quadrupole Fundamental Band of the Oxygen Molecule with Rotation-Vibration Interaction Corrections T. K. BALASUBRAMANIAN, V. P. BELLARY,AND ROMOLA D’CUNHA Spectroscopy Division, Bhabha Atomic Research Centre, Trombay, Bombay-400 085, India
AND K. NARAHARIRAO Department of Ph.vsics, Ohio State University,
I74 West 18th Avenue, Columbus, Ohio 43210
The effect of rotation-vibration interaction on the rovibrational wavefunctions belonging to a 32 electronic state is investigated in the harmonic approximation. The results are used to obtain the Herman-Wallis type corrections to the branch linestrengths of the quadrupole fundamental band of Oz. The usefulness of the refined linestrength formulas and their limiting forms for Hund’s case (b) are discussed. 0 1992 Academic press, [nc.
I. INTRODUCTION
The oxygen molecule is known to display a weak rotation-vibration spectrum attributable to an electric quadrupole transition ( 1, 2). In a recent work devoted to generalized electric quadrupole linestrengths for a 32( int)-3Z( int) transition, we (3) presented revised calculations of absorption intensities of all prominent lines in the quadrupole fundamental band of OZ. This represented an improvement over an earlier intensity listing by Rothman and Goldman ( 4) based on Hund’s case (b) treatment. In particular, our work demonstrated the need for an intermediate coupling approach while dealing with some of the low-N transitions. Concurrently with the detection of the quadrupole lines, there has been intense search for magnetic dipole rovibrational transitions in the infrared spectrum of 02a possibility raised by the unpaired electron spin (S = 1) in the ground state X32;. Indeed Dang-Nhu et al. (5) have now found clear evidence for such transitions in the fundamental band region. Based on these observations they have also reported detailed calculations of the magnetic dipole line intensities. Since some of the rovibronic selection rules, in particular the parity rules, are common to the two kinds of transitions, it is apparent that lines belonging to the P, Q, R branches may, in general, have intensity contributions from both types of transitions. The correct apportioning of the observed intensities between the two mechanisms would be necessarily incumbent on the accuracy with which the individual contributions may be theoretically ascertained. Realizing this, we have thought it worthwhile to refine our quadrupole linestrengths further so as to include the effect of rotation-vibration interaction, which unlike the intermediate coupling calculation, is expected to affect the high-N transitions. This step seemed all the more necessary since certain aspects of the magnetic dipole rotationvibration intensity theory dealt with in Ref. (5) appear to us to be incomplete (6). 0022-2852192 $5.00 Copyright
0 1992 by Academtc
All rights of reproduction
26 Press. Inc.
in any form reserved.
LINESTRENGTHS FOR O2 FUNDAMENTAL
27
2. THEORY
The essential ideas underlying the derivation of linestrength factors for a quadrupole transition between multiplet states have been expounded in our previous work (3), wherein a generalized linestrength expression in the form of a master formula is given. In order to apply it to a specific case, it is essential to start with the eigenfunctions of the initial and final states involved in the transition. In the problem on hand we are concerned with the states X3X; [ Fi (J); o 1, with i = 1,2, 3 and n = 0 or 1. Rotationvibration interaction tends to mix the J-manifolds of u = 0 and 1 in a complicated manner which we need to treat first. To this end, we start with the appropriate rovibrational Hamiltonian H = r/e(r) + T” + ($)X(r)(3S:
- S2) f B(r)(J
- S)‘.
(1)
Here I$( r) is the electronic energy and T, is the nuclear kinetic energy operator. X and B have their customary meanings ( 7, 8). We may for convenience, rewrite Eq. fl)as I-I = V,,(r) + IE, + 2X(r)S: + B(r)[J2
- 2J,S: - (J+S_ f J-S,)],
(2)
where I’,,( P) = Ye(r) + [ B( r) - ( 3 )X(Y)] S2 and S2 may be replaced by its eigenvalue S( S + 1) = 2, for a triplet spin state. In Eqs. ( 1) and (2) we have chosen to ignore the spin-rotation interaction parameter y(r) as it is relatively unimpo~ant in the mixing process. The rotation-vibration interaction is essentially caused by the dependence of B and X on the internucle~ separation T, which we propose to include through a perturbation calculation. To do this, let us first expand B and X as a series in the Dunham variable $j = (P - r,)/r, so that X(r)
= A, f
B(r)
= &(r,/ry
[i\e = re(6A/6r),][ = B,[l
-
+ O(t2)
2[ + O(,$‘)].
In view of the above, Eq. (2) may be recast as H = Ho + Ii’,
(3)
with HO = V,,(r) + T, + 2X& + &[J2 - 2J,& - (J+S_ + J-S+)]
(4)
H’ = 24[ieS$ - Be[J2 - 2&J, - (J+S_ + J-S,)]).
(5)
In Eq. (5) above only terms linear in 5 have been retained. The effect of H’ can now be included through standard perturbation theory using the zero-order rovibronic parity basis functions (8, 9) defined by 13Z;Jj; v> = 2-“2( j3&-J) rf j3XfJ)) 13Z,Je; v> = j3Z;Je; v).
Iv)
(ha) (6bl
Here 1u) is the eigensoiution of the equation (7) involving only the vibrational part. Since what we are seeking are small corrections to the linestrengths, a first-order treatment in the harmonic potential approximation should suffice. A straightforward calculation with the additional proviso that w, 9
BALASUBRAMANIAN
28
ET AL.
2 1X, 1, finally yields the following perturbed substate wave functions (written with subscript p) 32-, 2, = 1: [%;JJ;
2, = l& = l’Z;J;)
x (Jz, = 1) - 2p(z - 2- t)[ 10 = 0) - E/z z1= 2)]} + S(4, e)4p fi13320Je)[lv
= 0) - filv
= 2)]
(8a)
= 0) - fi\v
= 2)]
(8b)
132;;Je; 0 = l)P = 13Z;Je) x {IV = 1) - 2pz[lv = 0) - lJzlV = 2)]} + 4p fi13Z;Je)[lv 3X, 2) = 0: l?Z;J~; 2) = O), = 132;JT) x { 12,= 0) + 2p(z - 2 - t)[lv = l)} - S(T, e)4p\rz13Z;Je)lv 13Z6Je; 2, = 0), = I’Z;Je){
= 1)
(8~)
= 1).
(Sd)
Jz, = 0) + 2pzlv = 1)) - 4pG13Z:Je)lv
In the above equations p = ( BJw,) 3’2, E = [r,(SA/Sr),]/Be = x,/B,, and z = J(J+ 1). Note that the off-diagonal part 2[B,( J+S_ + J-S,) of H’ [ Eq. (5)] can mix adjacent vibrational states belonging to different Q substates. Also the Kronecker symbol S(T, e) = 6(e or f, e) in Eqs. (8a) and (8~) ensures that the levels with e and f parity are not mixed. In intermediate coupling the perturbed eigenfunctions of the three rotational term series of a 32-(~) vibronic level may be expressed in terms of the perturbed functions given by Eq. (8) as I Fi(J)e;
v)~ = (~.$il + C.$i3)j32;Je;
z))~ + (C_dil - SJ6i3) X 13Z;Jee; D)~
(9a)
with i = 1 or 3 and
IWJ)f;
u)p = 13z;Jf; +p.
(9b)
In Eq. (9a), 6, represents the Kronecker symbol and the coefficients cJ and sJ are defined in Ref. (3). Use of Eqs. (8 ) and (9) in the master formula described in Ref. (3)) after some lengthy algebra, leads to the desired quadrupole branch linestrength expressions listed in Table I. These incorporate both intermediate coupling and corrections due to rotation-vibration interaction. The latter involve the additional assumption that the variation of the quadrupole moment function Q2( r) with Yis linear ( no electrical anharmonicity ) . 3. DISCUSSION
In the linestrengths given in Table I, the rotation-vibration by the terms containing the parameter Q = (&l~,)~‘~(u
= olQ2lv
= O)/(
corrections are described
1 IQ2lo),
Q2 being the electric quadrupole moment operator. Upon setting u = 0, the present formulas reduce to the ones derived by us previously (3) without rotation-vibration correction. It is important to realize that these corrections are nothing but the quad-
LINESTRENGTHS
FOR 02 FUNDAMENTAL
29
TABLE I Electric Quadrnpole Linestrengths for the Fun~mental Rovibmtional Transitions in a ‘Z State with Rotation-Vibration Interaction Corrections in Intermediate Coupling
(lrl\termediate Llnc strcngthe'b'c
Branch
couplinK)
%,,(J)
%,(J) %,(J) %z,,CJ)
%,(Jl %,(J) %ltJf Oo**fJf sS,,CJ) “%,CJ) %x,(J)
*‘&z(J) oPs,(J) oP,,(J) %z
fJ 1
%3,1J) %,,(J) o%,(J) UQw(J) %m(Jl oPdJ) %CJl %,(J)
a All expressions should be multiplied by the transition moment factor b c = r<0~g~~>/lt5/o*~='2* =
(,{Jf
=
[J(J+~)]'~=,
"(Jf
=
[J(J+2fJ*‘=
and
W(J)
=
tJtJ+sff
I’Z
z*
.
rupole transition counterparts of the usual Herman-Wallis corrections encountered in the discussion of electric dipole rotation-vibration intensities. It turns out that in intermediate coupling these corrections are not expressible as simple multiplicative factors. It is interesting to estimate the relative changes in the intensity caused by these corrections in a typical case. Let us consider for instance, the Sz2(9) line in the O2 fundamental. From Ref. (4) we take Be = 1.446 cm-‘, w, = 1580.2 cm-‘, (2)= 01$&10) = 0.82 X 1O-26 esu, and (V = Il&/v = 0)) = 0.15 X 1O-26esu [the last value being from Ref. (Z)] which gives c = 1.5 I X 10e4.Substitution in the appropriate line strength expression (Table I) yields a ro~tion-vibmtion intensity correction amounting to -3% for this line. It is clear that these corrections become more significant only at higher J (or N) values. But in this situation the coupling condition in the X3Z, of 02 approaches Hund’s case (b) rather rapidly. It is then useful to know the forms these formulas take in this limit. The substitution of the case (b) valuessJ = [(J-l- 1)/(2Ji1)]*‘2, cJ = [ .I/( 2 J -I- 1)] 1’2in the expressions in Table I yields the desired formulas which are separately listed in Table II. Although these case (b) linestrengths have been expressed as functions of J,it is sometimes desirable
30
BALASUBRAMANIAN
ET AL.
TABLE II Hund’s Case (b) Limit of the Refined Linestrength Expressions in Table I Branch
“&al(J) s&~(J) k,(J) “G,(J) -&i(J) k(J) p%,(J) o4c(J) %.zt(J) “&tJJl %,2(J) %&r(J) +,,(J) %z(J) *C&(J) =%,(J) k,(J) %z,(J) %,(J) O%(J) %,(J) O%(J) %n(J)
Line strength”bCcese
(b) limitlc
0
35(5+1)(25+5)Cl-4a(2J+1)12/2(2J+l)(2J+3) 3E1-40.(2J+1)72/(2J+1) 9[1-&(2J+1)]2/(2J-1)(2J+1)(2J+3) (J+1)(2Jt3)(J-1)2/J(2J-1).(2J+1) 3(J+l)/J(2Jtl) 6/(2J-3)(2J-1)(2Jtl) 3(J-2)(J-1)(2J+l)C1+4~(2J-3)3~/2(2J-3)(2J-l) 3JCJ+3)[1-9(25+3)1'/2(25+3) 311-4n(25+3)1+/(2J+3) 3(5+2)/(5+1)(25+3) (2J+1)[J(J+1)-31p/J(J+l)~ZJ-l)~ZJ+3) 3(5-1)/5(25-l) 3[1+4(2J-1)12/(2J-1) 3(J-2)(J+1)[1+&(25-1)1'/2(2J-1) 3(5+2)(5+3)(2J+l)Cl-Q.(2J+5)12/2(2J+3)(2J+5) 6/(25+1)(25+3)(25+5) 35/(5+1)(25+1) J(2J-1)(J+2)2/(J+1)(2J+l~~2J+3) 9C1+~(2J+1)1=/(25-1)(25+1)(25+3) 3Cl+~(2J+1)1z/(2Jt1) 3J(J+1)(2J-3)C1+&(2J+l)lf/2C2J-l)(ZJi-1) 0
o All expressions should be multiplied by moment factor'. b
the
transition
0 = C<0(Q(0>/clle,I0>l(Bo~~*~='=.
c In the case (b) limit the Q(AN=0) form branches are effacted by rotation-vibration corrections.
not
to switch to the quantum number N in its place, which may be accomplished by the simple substitution J = N -t- 1, J = N, and J = N - 1 for transitions involving, respectively, the F’i, F$, and F; components. Note that in the case (b) limit the intensities of the Q-form (AN = 0) branches are entirely free from the rotationvibration correction-a fact that could have been readily anticipated. Also in this limit the corrections to the linestrengths of the other branches take on the more familiar form of suitable multiplicative “Herman-Wallis” factors. In the foregoing section we have treated the problem of corrections to the rovibrational intensities in the fundamental vibrational transition, for which the harmonic approximation should be adequate. If a more accurate calculation is desired, one would have to resort to a direct numerical diagonalization of the appropriate rovibrational matrix set up using the Hamiltonian defined by Eq. (2). This first step, of course, would call for a knowledge of the potential energy curve of the 32 state. The second stage of the calculation would involve the various vibrational matrix elements of the quadrupole moment operator Q2( r), which can be evaluated only if its rdependence is known a priori. Conversely, a Herman-Wallis analysis of the accurately
LINESTRENGTHS
FOR O2 FUNDAMENTAL
31
measured relative rovibrational intensities, preferably of the S( A J = 2) and 0 ( AJ = -2 ) branch transitions f which cannot have magnetic dipole intensity admixtures), can in principle, be used to derive the quadrupole moment function &(r) in the electronic state concerned. ACKNOWLEDGMENT We thank Dr. V. B. Kartha for his keen interest in the work. RECEIVED:
January 9, 1992 REFERENCES
I. A. GOLDMAN, J. REID, AND L. S. ROTHMAN, Geaphys. Res. L&t. 8,77-X? ( 198 I f. 2. J. REID, R. L. SINCLAIR,A. M. ROBINSON,AND A. R. W. MCKELLAR, Phys. Rev. A 24, 1944-1949
(1981). 3. T. K. BALASUBRAMANIAN,R. D’CUNHA, AND K. NARAHARI RAO, J. Mol. Spectmc. 144, 374-380 (1990). 4. L. S. ROTHMAN AND A. GOLDMAN,Appl. Opt. 20,2 182-2 I84 ( 198 1). 5. M. DANENHU, R. ZANDER, A. GOLDMAN, AND C. P. RINSLAND, J. Mol. Spectrosc. 144, 366-373 (1990). 6. V. P. BELLARY,private communi~tion ( 199 1). 7. G. HERZBERG,“Spectra of Diatomic Molecules,” Van Nostrand, New York, 1950. 8. J. T. HOUGEN,“The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules,” National Bureau of Standards (U.S.) Monograph 1IS, U.S. Government Printing Office, Washington, D.C., 1970. 9. H. LEFEBVRE-BRION AND R. W. RELD, “Perturbations in the Spectra of Diatomic Molecules,” Academic Press, New York, 1986.
OF MOLECULAR
SPECTROSCOPY
153,26-31 ( 1992)
Linestrengths for the Electric Quadrupole Fundamental Band of the Oxygen Molecule with Rotation-Vibration Interaction Corrections T. K. BALASUBRAMANIAN, V. P. BELLARY,AND ROMOLA D’CUNHA Spectroscopy Division, Bhabha Atomic Research Centre, Trombay, Bombay-400 085, India
AND K. NARAHARIRAO Department of Ph.vsics, Ohio State University,
I74 West 18th Avenue, Columbus, Ohio 43210
The effect of rotation-vibration interaction on the rovibrational wavefunctions belonging to a 32 electronic state is investigated in the harmonic approximation. The results are used to obtain the Herman-Wallis type corrections to the branch linestrengths of the quadrupole fundamental band of Oz. The usefulness of the refined linestrength formulas and their limiting forms for Hund’s case (b) are discussed. 0 1992 Academic press, [nc.
I. INTRODUCTION
The oxygen molecule is known to display a weak rotation-vibration spectrum attributable to an electric quadrupole transition ( 1, 2). In a recent work devoted to generalized electric quadrupole linestrengths for a 32( int)-3Z( int) transition, we (3) presented revised calculations of absorption intensities of all prominent lines in the quadrupole fundamental band of OZ. This represented an improvement over an earlier intensity listing by Rothman and Goldman ( 4) based on Hund’s case (b) treatment. In particular, our work demonstrated the need for an intermediate coupling approach while dealing with some of the low-N transitions. Concurrently with the detection of the quadrupole lines, there has been intense search for magnetic dipole rovibrational transitions in the infrared spectrum of 02a possibility raised by the unpaired electron spin (S = 1) in the ground state X32;. Indeed Dang-Nhu et al. (5) have now found clear evidence for such transitions in the fundamental band region. Based on these observations they have also reported detailed calculations of the magnetic dipole line intensities. Since some of the rovibronic selection rules, in particular the parity rules, are common to the two kinds of transitions, it is apparent that lines belonging to the P, Q, R branches may, in general, have intensity contributions from both types of transitions. The correct apportioning of the observed intensities between the two mechanisms would be necessarily incumbent on the accuracy with which the individual contributions may be theoretically ascertained. Realizing this, we have thought it worthwhile to refine our quadrupole linestrengths further so as to include the effect of rotation-vibration interaction, which unlike the intermediate coupling calculation, is expected to affect the high-N transitions. This step seemed all the more necessary since certain aspects of the magnetic dipole rotationvibration intensity theory dealt with in Ref. (5) appear to us to be incomplete (6). 0022-2852192 $5.00 Copyright
0 1992 by Academtc
All rights of reproduction
26 Press. Inc.
in any form reserved.
LINESTRENGTHS FOR O2 FUNDAMENTAL
27
2. THEORY
The essential ideas underlying the derivation of linestrength factors for a quadrupole transition between multiplet states have been expounded in our previous work (3), wherein a generalized linestrength expression in the form of a master formula is given. In order to apply it to a specific case, it is essential to start with the eigenfunctions of the initial and final states involved in the transition. In the problem on hand we are concerned with the states X3X; [ Fi (J); o 1, with i = 1,2, 3 and n = 0 or 1. Rotationvibration interaction tends to mix the J-manifolds of u = 0 and 1 in a complicated manner which we need to treat first. To this end, we start with the appropriate rovibrational Hamiltonian H = r/e(r) + T” + ($)X(r)(3S:
- S2) f B(r)(J
- S)‘.
(1)
Here I$( r) is the electronic energy and T, is the nuclear kinetic energy operator. X and B have their customary meanings ( 7, 8). We may for convenience, rewrite Eq. fl)as I-I = V,,(r) + IE, + 2X(r)S: + B(r)[J2
- 2J,S: - (J+S_ f J-S,)],
(2)
where I’,,( P) = Ye(r) + [ B( r) - ( 3 )X(Y)] S2 and S2 may be replaced by its eigenvalue S( S + 1) = 2, for a triplet spin state. In Eqs. ( 1) and (2) we have chosen to ignore the spin-rotation interaction parameter y(r) as it is relatively unimpo~ant in the mixing process. The rotation-vibration interaction is essentially caused by the dependence of B and X on the internucle~ separation T, which we propose to include through a perturbation calculation. To do this, let us first expand B and X as a series in the Dunham variable $j = (P - r,)/r, so that X(r)
= A, f
B(r)
= &(r,/ry
[i\e = re(6A/6r),][ = B,[l
-
+ O(t2)
2[ + O(,$‘)].
In view of the above, Eq. (2) may be recast as H = Ho + Ii’,
(3)
with HO = V,,(r) + T, + 2X& + &[J2 - 2J,& - (J+S_ + J-S+)]
(4)
H’ = 24[ieS$ - Be[J2 - 2&J, - (J+S_ + J-S,)]).
(5)
In Eq. (5) above only terms linear in 5 have been retained. The effect of H’ can now be included through standard perturbation theory using the zero-order rovibronic parity basis functions (8, 9) defined by 13Z;Jj; v> = 2-“2( j3&-J) rf j3XfJ)) 13Z,Je; v> = j3Z;Je; v).
Iv)
(ha) (6bl
Here 1u) is the eigensoiution of the equation (7) involving only the vibrational part. Since what we are seeking are small corrections to the linestrengths, a first-order treatment in the harmonic potential approximation should suffice. A straightforward calculation with the additional proviso that w, 9
BALASUBRAMANIAN
28
ET AL.
2 1X, 1, finally yields the following perturbed substate wave functions (written with subscript p) 32-, 2, = 1: [%;JJ;
2, = l& = l’Z;J;)
x (Jz, = 1) - 2p(z - 2- t)[ 10 = 0) - E/z z1= 2)]} + S(4, e)4p fi13320Je)[lv
= 0) - filv
= 2)]
(8a)
= 0) - fi\v
= 2)]
(8b)
132;;Je; 0 = l)P = 13Z;Je) x {IV = 1) - 2pz[lv = 0) - lJzlV = 2)]} + 4p fi13Z;Je)[lv 3X, 2) = 0: l?Z;J~; 2) = O), = 132;JT) x { 12,= 0) + 2p(z - 2 - t)[lv = l)} - S(T, e)4p\rz13Z;Je)lv 13Z6Je; 2, = 0), = I’Z;Je){
= 1)
(8~)
= 1).
(Sd)
Jz, = 0) + 2pzlv = 1)) - 4pG13Z:Je)lv
In the above equations p = ( BJw,) 3’2, E = [r,(SA/Sr),]/Be = x,/B,, and z = J(J+ 1). Note that the off-diagonal part 2[B,( J+S_ + J-S,) of H’ [ Eq. (5)] can mix adjacent vibrational states belonging to different Q substates. Also the Kronecker symbol S(T, e) = 6(e or f, e) in Eqs. (8a) and (8~) ensures that the levels with e and f parity are not mixed. In intermediate coupling the perturbed eigenfunctions of the three rotational term series of a 32-(~) vibronic level may be expressed in terms of the perturbed functions given by Eq. (8) as I Fi(J)e;
v)~ = (~.$il + C.$i3)j32;Je;
z))~ + (C_dil - SJ6i3) X 13Z;Jee; D)~
(9a)
with i = 1 or 3 and
IWJ)f;
u)p = 13z;Jf; +p.
(9b)
In Eq. (9a), 6, represents the Kronecker symbol and the coefficients cJ and sJ are defined in Ref. (3). Use of Eqs. (8 ) and (9) in the master formula described in Ref. (3)) after some lengthy algebra, leads to the desired quadrupole branch linestrength expressions listed in Table I. These incorporate both intermediate coupling and corrections due to rotation-vibration interaction. The latter involve the additional assumption that the variation of the quadrupole moment function Q2( r) with Yis linear ( no electrical anharmonicity ) . 3. DISCUSSION
In the linestrengths given in Table I, the rotation-vibration by the terms containing the parameter Q = (&l~,)~‘~(u
= olQ2lv
= O)/(
corrections are described
1 IQ2lo),
Q2 being the electric quadrupole moment operator. Upon setting u = 0, the present formulas reduce to the ones derived by us previously (3) without rotation-vibration correction. It is important to realize that these corrections are nothing but the quad-
LINESTRENGTHS
FOR 02 FUNDAMENTAL
29
TABLE I Electric Quadrnpole Linestrengths for the Fun~mental Rovibmtional Transitions in a ‘Z State with Rotation-Vibration Interaction Corrections in Intermediate Coupling
(lrl\termediate Llnc strcngthe'b'c
Branch
couplinK)
%,,(J)
%,(J) %,(J) %z,,CJ)
%,(Jl %,(J) %ltJf Oo**fJf sS,,CJ) “%,CJ) %x,(J)
*‘&z(J) oPs,(J) oP,,(J) %z
fJ 1
%3,1J) %,,(J) o%,(J) UQw(J) %m(Jl oPdJ) %CJl %,(J)
a All expressions should be multiplied by the transition moment factor b c = r<0~g~~>/
(,{Jf
=
[J(J+~)]'~=,
"(Jf
=
[J(J+2fJ*‘=
and
W(J)
=
tJtJ+sff
I’Z
.
rupole transition counterparts of the usual Herman-Wallis corrections encountered in the discussion of electric dipole rotation-vibration intensities. It turns out that in intermediate coupling these corrections are not expressible as simple multiplicative factors. It is interesting to estimate the relative changes in the intensity caused by these corrections in a typical case. Let us consider for instance, the Sz2(9) line in the O2 fundamental. From Ref. (4) we take Be = 1.446 cm-‘, w, = 1580.2 cm-‘, (2)= 01$&10) = 0.82 X 1O-26 esu, and (V = Il&/v = 0)) = 0.15 X 1O-26esu [the last value being from Ref. (Z)] which gives c = 1.5 I X 10e4.Substitution in the appropriate line strength expression (Table I) yields a ro~tion-vibmtion intensity correction amounting to -3% for this line. It is clear that these corrections become more significant only at higher J (or N) values. But in this situation the coupling condition in the X3Z, of 02 approaches Hund’s case (b) rather rapidly. It is then useful to know the forms these formulas take in this limit. The substitution of the case (b) valuessJ = [(J-l- 1)/(2Ji1)]*‘2, cJ = [ .I/( 2 J -I- 1)] 1’2in the expressions in Table I yields the desired formulas which are separately listed in Table II. Although these case (b) linestrengths have been expressed as functions of J,it is sometimes desirable
30
BALASUBRAMANIAN
ET AL.
TABLE II Hund’s Case (b) Limit of the Refined Linestrength Expressions in Table I Branch
“&al(J) s&~(J) k,(J) “G,(J) -&i(J) k(J) p%,(J) o4c(J) %.zt(J) “&tJJl %,2(J) %&r(J) +,,(J) %z(J) *C&(J) =%,(J) k,(J) %z,(J) %,(J) O%(J) %,(J) O%(J) %n(J)
Line strength”bCcese
(b) limitlc
0
35(5+1)(25+5)Cl-4a(2J+1)12/2(2J+l)(2J+3) 3E1-40.(2J+1)72/(2J+1) 9[1-&(2J+1)]2/(2J-1)(2J+1)(2J+3) (J+1)(2Jt3)(J-1)2/J(2J-1).(2J+1) 3(J+l)/J(2Jtl) 6/(2J-3)(2J-1)(2Jtl) 3(J-2)(J-1)(2J+l)C1+4~(2J-3)3~/2(2J-3)(2J-l) 3JCJ+3)[1-9(25+3)1'/2(25+3) 311-4n(25+3)1+/(2J+3) 3(5+2)/(5+1)(25+3) (2J+1)[J(J+1)-31p/J(J+l)~ZJ-l)~ZJ+3) 3(5-1)/5(25-l) 3[1+4(2J-1)12/(2J-1) 3(J-2)(J+1)[1+&(25-1)1'/2(2J-1) 3(5+2)(5+3)(2J+l)Cl-Q.(2J+5)12/2(2J+3)(2J+5) 6/(25+1)(25+3)(25+5) 35/(5+1)(25+1) J(2J-1)(J+2)2/(J+1)(2J+l~~2J+3) 9C1+~(2J+1)1=/(25-1)(25+1)(25+3) 3Cl+~(2J+1)1z/(2Jt1) 3J(J+1)(2J-3)C1+&(2J+l)lf/2C2J-l)(ZJi-1) 0
o All expressions should be multiplied by moment factor
the
transition
0 = C<0(Q(0>/clle,I0>l(Bo~~*~='=.
c In the case (b) limit the Q(AN=0) form branches are effacted by rotation-vibration corrections.
not
to switch to the quantum number N in its place, which may be accomplished by the simple substitution J = N -t- 1, J = N, and J = N - 1 for transitions involving, respectively, the F’i, F$, and F; components. Note that in the case (b) limit the intensities of the Q-form (AN = 0) branches are entirely free from the rotationvibration correction-a fact that could have been readily anticipated. Also in this limit the corrections to the linestrengths of the other branches take on the more familiar form of suitable multiplicative “Herman-Wallis” factors. In the foregoing section we have treated the problem of corrections to the rovibrational intensities in the fundamental vibrational transition, for which the harmonic approximation should be adequate. If a more accurate calculation is desired, one would have to resort to a direct numerical diagonalization of the appropriate rovibrational matrix set up using the Hamiltonian defined by Eq. (2). This first step, of course, would call for a knowledge of the potential energy curve of the 32 state. The second stage of the calculation would involve the various vibrational matrix elements of the quadrupole moment operator Q2( r), which can be evaluated only if its rdependence is known a priori. Conversely, a Herman-Wallis analysis of the accurately
LINESTRENGTHS
FOR O2 FUNDAMENTAL
31
measured relative rovibrational intensities, preferably of the S( A J = 2) and 0 ( AJ = -2 ) branch transitions f which cannot have magnetic dipole intensity admixtures), can in principle, be used to derive the quadrupole moment function &(r) in the electronic state concerned. ACKNOWLEDGMENT We thank Dr. V. B. Kartha for his keen interest in the work. RECEIVED:
January 9, 1992 REFERENCES
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(1981). 3. T. K. BALASUBRAMANIAN,R. D’CUNHA, AND K. NARAHARI RAO, J. Mol. Spectmc. 144, 374-380 (1990). 4. L. S. ROTHMAN AND A. GOLDMAN,Appl. Opt. 20,2 182-2 I84 ( 198 1). 5. M. DANENHU, R. ZANDER, A. GOLDMAN, AND C. P. RINSLAND, J. Mol. Spectrosc. 144, 366-373 (1990). 6. V. P. BELLARY,private communi~tion ( 199 1). 7. G. HERZBERG,“Spectra of Diatomic Molecules,” Van Nostrand, New York, 1950. 8. J. T. HOUGEN,“The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules,” National Bureau of Standards (U.S.) Monograph 1IS, U.S. Government Printing Office, Washington, D.C., 1970. 9. H. LEFEBVRE-BRION AND R. W. RELD, “Perturbations in the Spectra of Diatomic Molecules,” Academic Press, New York, 1986.