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Algebraic expressions for vibration-rotation line intensities of near prolate asymmetric molecules: Application to SO2 molecule
J. Quonr. Specmsc. Rodior. 7imsfeer. Vol. 13, pp. 1325-l 332. krgamon Press 1973. Printed in Great Britain
ALGEBRAIC EXPRESSIONS FOR VIBRATION-ROTATION LINE INTENSITIES OF NEAR PROLATE ASYMMETRIC MOLECULES : APPLICATION TO SO, MOLECULE C. SECROUN,A. BARBEand P. JOUVE Laboratoire de Physique Mokculaire, Fact&e des Sciences, Reims, France (Received 9 April 1973)
Abstract-A treatment of the intensities of vibration-rotation lines of near prolate asymmetric molecules is given, leading to explicit expressions relating the observed intensities to the appropriate coefficients in the expansions of the potential energy and the dipole moment. The theoretical results are compared with experimental detetminations of line intensities in the vi-band of sulfur dioxide.
1. INTRODUCTION THE CALCULATION of the theoretical intensity of absorption for a molecule has been investigated by several authors. GORA(l) has given an interesting method for the computation of line strengths for low rotational quantum numbers. BRALAWSKYand BEN-ARYEH”) have calculated F factors for an asymmetric molecule like H,O, by a perturbation method. LEGAY’~) has proposed a contact transformation method to calculate intensities of vibrationrotation transitions. But these results were not derived for an asymmetric molecule. However, this method has been applied to linear triatomic moleculesC4) and to symmetric molecules of the type AB . . . X YZ, .(‘) Recently, a treatment of the intensities of vibration bands for polyatomic molecules has been derived, using a contact transformation method.@) In this paper, our purpose is to carry out calculations of line intensities using a similar method and to apply the results to near-prolate asymmetric molecules.
2. GENERAL
PRINCIPLES
The theoretical intensity of absorption for a molecule making a transition from a state z to a state r’ may be expressed by the relation (v’,r’JZ]v,r) = (87r3Nv/3hcZ)[exp(-E,/kT)][1-exp(-hv/kT)](~)2
(1)
where N is the molecular concentration (molecule-‘), v the frequency of the transition (cm-‘), Z the rotational state sum, E, the ground rotational energy (cm-‘), and (p) the dipole moment matrix element (Debye). Intensity is thus expressed in cm molecule- r. 1325
1326
C. SECROUN,A. BARBEand P. JOUVE
The rotational state sum for an asymmetric rotor has been given by GORDON(') as 2 = (nl'2/2)(kT/hc)3'2(ABC)_ 1'2,
(2)
where A, B and C are rotational constants (cm- ‘). The problem is the evaluation of the matrix elements (p). We need two steps: (i) a contact transformation method to take account of vibration-rotation interaction ; (ii) a perturbation method for the asymmetry. 3. THEORY
In order to use the wave function of the zero-order Hamiltonian in the evaluation of (p), two contact transformations are applied to the dipole moment. In Ref. (3), the transformed dipole moment & has been put in a symbolic symmetrized form and is II:=
P,Pp. ..u,+u,. c 2 a/?...a
. . PpP, $,
.#...aA;;Pc#b
. . .qd%?;
qeqd
’ . . Pbpa,
(3)
de... osb
dse
where q. and p. are, respectively, the dimensionless normal coordinate and its conjugate momentum, the a, are the direction cosines, and the P, are components of angular momentum. The coefficients JZ have been expressed with the coefficients of the contact transformation and of the non-transformed dipole moment.‘6) They are given in Table 1 for a first-order perturbation. In this table, the coefficients S are those of AMAT et al.@’ TABLE1. COEFFICIENTS OFTHETRANSFORMED DIPOLE MOMENT
The transformed dipole moment is a sum of terms, products of vibrational and rotational operators. The zero-order wave functions are separable into the coordinates of vibration and rotation. Therefore, the dipole moment matrix elements consist of rotational elements and vibrational elements. 1. Vibrational matrix elements The vibrational matrix elements are obtained from well-known, non-vanishing matrix elements in the linear harmonic oscillator representation.‘@ 2. Rotational matrix elements In calculating the transformed between rotation and vibration.
dipole moment, account is taken of the interaction
Algebraic expressions for vibration-rotation
line intensities
1327
In the symmetric rotator basis of D, symmetry, the matrix elements of the rotational operators may be written as (J, K, MJORIJ’, IS, M) = (J, K(O#‘,
K’)(J,
M(O’;(J’, M)
(4)
with o, =
pap@.. . y”“..
1
. p&J,*
@...a
The matrix elements involved in equation (3) are given in Table 2, where we choose the wave functions to conform with the phase conventions adopted by VAN VLECK('~)and NIELSEN. Thus, PzlJ, K, M) =
WJ, K W,
h PSJ, K, M> = Z~[(J-K)(J+K+l)]IJ.K+1,M)+~~[(J+K)(J-K+l)]~J,K-1,M),
PylJ, K, M) = $[(J-K)(J+K+
l)]IJ, K+ 1, M)
+[(J+K)(J-K+l)IJ,K-1,M).
(5)
A perturbation technique will be now developed, which is convenient for taking account of asymmetry in slightly asymmetric molecules (1~12 0.90). Using the Wang functions, where the JM, notation has been suppressed, the asymmetric rotor wave functions are given as follows :(‘) eK=
l_qE
(
I
SK+QSK-2+&+*9
where
B =
EK-&+~
’
and H is the reduced energy obtained in the Wang transformation. and fl for the various possible cases are given in Table 3, with ‘= f(J,4 f(J,
2(B - C) 2A-B-C’
= $J---n)(~---n+
~)(J+FI)(J+TI+
1) = &J(J- l)(J+ l)(J+2).
11,
Explicit formulas for a
1328
C. SECROUN, A. BARBE and P. JOUVE
I329
Algebraic expressions for vibration-rotation line intensities TABLE 3. COFSFICIENISIN THE EXPANSION OF ASYMMETRIC-ROTOR WAVE FUNCTIONS(Y REFERS TO THE PARITY OF J+K_,+K,)
K-1
Y
a
0 I
0 0
0 0
1 2
1 0 1 0 1
0
2
3 3 >3
W)f(J, v
B -(&/16)f(J, 2)“‘[1 -(s/32)5(5+ -(~/16)_f(J, 2)“‘[1+(~/32)J(J+
- (~/24)f(J, 3)“’
:(e/32)J(J+ (~/16)f(J,2)~“[1 l)]-’ (~/16)f(J, 2)“‘[1+(&/32)J(J + l)] - ’ [&/8(K- l)]f(J, K - 1)“’
If a and p refer to the asymmetric-rotor
l)]-’ 1)1-l
- (~/%)f(& 3)“2 -(&/32)f(J, 4)“’ - (~/32)f(J, 4)l’* - [E/~(K + l)]f(J, K + 1)“’
level J, K and Q’,/I?’to the level J’, K’, we find
that (II/&&,)
=
(1-@*+~*y”+fi’*I
(J, K(O,(J’, K) + cru’(J, K - 210&I’, K - 2)
+ PB’(J, K + 2(0,13’, K + 2), (+KIORl~~+l)
=
(7)
(J,KIO,IJ’,K+l)+a’(J,
KIO,@',K-1)
+ j?(J, K + 210,#‘, K + 1) +cta’(J, K - 210,15’, K - 1) + BB’(J, K + 210,&I’, K + 3), (q/KpRIg,_l)
=
1_a’+~‘:OL’*+~*
(8)
(J,K(O,(J',K1)+&J,
+ jY(J, K(O,(J', K + 1) + aa’(J,
K - 210,&J’,K - 3)
+ PD’(.J, K + 2(ORlJ’, K + 1). It is now possible to calculate matrix elements such as (IC/&#k) molecule (Table 4). 3. Calculation
K- 210&‘, K- 1)
(9) for an asymmetric
of intensities
The intensity of absorption is proportional to the line strength (p)*. In the absence of an external field, the space is isotropic and the squares of the matrix elements are M= +.I (P)* = 3 ,;_, I(+(u, JKWIP:IW+, J’K’W>l*.
(10)
From Tables 14, it is then possible to write the intensity for any transition in the case of a slightly asymmetric molecule. As an example, the expressions are now developed for C,, triatomic molecules (e.g. . .) for fundamental bands. SO,,O,,NO,.
C. SECROUN, A. BAREIEand P. JOUVE
1330
TABLE 4. DIRECTIONCOSINEMATRIXELBMLNTS FOR A NEAR PROLATE~sy~~rmuc lK
J
*;
K
yP.(*thMc.)
2’)l,
l-
aZ+~Z+a’2+~2
J-l
a2+/?2+a’2+~2
Kkl
&JTK)(JTK-I)I
2
1
+;&/[(JTK-Z)(JTK-3)]+$,/[JTK+Z)(JTK+l)]
K
2’/l(,
I
I- a*“‘~a”+8”
crpori.p3
K+l
1
,/(J2-K2)+aa’,/[J2-(K-2)2]+&?‘J[J2-(K+2)2]
2
[!
J
ROTOR
I
K+aa’(K_2)+fi,fI’(K+2)
1_a2+~2;a’2+B12
1
,/[JTK)(J+K+l)]+&‘[(JTK+2)(J+K-l)]
I
I
+$‘J[(JrK-2)(J+K+3)1
+(“a+i’~~)[“,:J[(J~K)(JrK+l)l I
K
2’/.lc, lI
a*‘p1+~‘2+p12
IJ
[(J+K+l)(J-K+l)]+aa’,/[(J+K-l)(J-K++)]
1
+/3B’,/(J+K+3)(J-K-l)
_I
Jfl
.,‘[(J+K+WJ+K+l)l
K&l
* K stands for
K- ,
Putting qJv = (8~~Nv/3hcZ) [exp( - E&T)] [ I- exp( - hv/kT)], we obtain the results listed below. Fundamental band v, of symmetry A, (perpendicular transition) (0, JWl,
J’K z!z1) = %VXCI;TMJ’4K){1 -w/w
x [q;K)
(y;;~“+qK);;,~lj
x (1 - (a2 + p2 + a’2 + /T2) + [26/6$$ +8~,~:i2K,+8~~;~t:~:]+
[“sL2Z/i’K, +,BS$
[l/~,K,][~::~,-“,+~:E;K~]}.
(11)
Fundamental band v, of symmetry B, (parallel transition) (0, JKIIIl, J’K) = ~~,‘~,24M,~~~,,[l -(fI2p&Y,,,(;:A”]
+ [2/X$
[aa’N:i2,,
+ /?/?‘N#}.
X (1 -(a2+p2+a’2
+/?“) (12)
Algebraic expressions for vibration-rotation
The different coefficients introduced different values of J’.
1331
line intensities
in these expressions are given in Table 5 for
TABLE 5. CQWFICIENTSOF LINE INTENS~XS
J-l J
J+l
J-l J
J+l
4K’
M.l
J
1
(JTK)(JTK-1)
G 2J+l
39W’
9m
8 (K’
2J+2K+l
+_2K+l
(JfK+2)(J+K+l)
8J(J + 1)
(JTK)(J+K+l)
+_2K+l
+2K+l
(J+_K+Z)(JTK-1)
1 8(J+ 1)
(J+K+Z)(JkK+l)
-2J+2K-1
+2K+l
(JTK)(JTK-1)
Jz-K2
(JfK-2)(JTK-3) (JTK-2)(J+_K+3) (JkK+4)(JfK+3)
25 0 2(J + 1)
(J-K+l:;J+K+l)
4.
APPLICABILITY
OF
THE
J2-(K+2)2 (K+2)* (J+K+3)(J-K-l)
METHOD
The applicability of relations (11) and (12) is limited by the validity of the perturbation methods used. When no resonance due to an accidental degeneracy occurs in the molecule, the results concerning vibration-rotation’ interaction are good approximations. But the
TABLE 6. CONSTANTS FOR SO2 USEDIN THE CALCULATION OF LINE INTENSITIES
Quantity
Symbol
Value
Reference
Permanent dipole moment
- 1.59 D
C.
Dipole moment derivatives
-0.13 D -0.19 D 0.33 D
C. SECROUN et al."' C. SECROUNet al.@) C. %XROUN et al.‘@
Vibrational wave numbers
Rotational constants
1167.6Ocm526.27 cm- ’ 1380.91 cm-’ 2.02736 cm- ’ 0.34417 cm-’
0.29353 cm- ’ Coriolis interaction constants
0.272 - 0.962
Centrifugal interaction constants in units 1O-2o g1j2 cm
17.9178 - 2.04505 18.7898 5.32138 7.38215 -6.86156 0.84256
SECROUN(")
A. H. N~SEN et al.‘12) A. H. NIELSENet a1.‘12’ A. H. NIELSENet aI.“2’ D. KWEUON’ls’ D. KM?LSON(15’ D. KIVEL.WN(‘~) A. BARBEand P. JouVE”~ A. BARBEand P. JOLWE”~)
C. SECROUN, A. BAREIEand P. JOUVE
1332
study of asymmetry implies LX, /3 << 1. Table 3 shows that this is not always true, specially when K/J is small. However, experimental determinations of intensities are not precise. Hence, theoretical errors would be lower than experimental ones and it is possible to compute line intensities, in good approximation, from very simple formulas. 5. APPLICATION
TO
THE
v, BAND
OF
SO,
Recently, HINKLEY et Al. measured line intensities in the v1 fundamental band. It is now possible to calculate intensities for the same transitions and compare them with experimental results. This comparison is given in Table 7, with different approximations, using the data of Table 6. We find good agreement between theoretical and experimental results. There is a discrepancy when K/J is small, which is a bad limit for applicability of the perturbation method (e.g., for the transition 14,,,, t 14,,,,). Without taking account of these particular cases, the average deviation is 0.15 x 10w2’ cm-molecule- ‘, TABLE 7. COMPARIXINOF THE THEORETICAL AND EXPERIMENTALLINEINTENSITIES,FOR THE V,+LJNDAMENTAL OF
SO, IN UNITSlO-‘l
Transition 30 10.20 + 23 5 19 + 186’12 + 13;,, + 22X,3 + 8 + 6,:, cl 8 + 2 11+
30,,,*, 24,,,8 19713 14& 2210.1, 81.7 61s
&,* 5’5 +- 40.4
14 *.;a + 140.1,
cm-mol-’
Symmetric rigid rotor
Asymmetric rigid rotor
Asymmetric molecule
Experimental results”‘)
0.79 1.78 2.12 2.34 1.24 2.58 2.05 1.70 1.99 7.22
0.88 1.63 2.02 2.30 1.30 2.19 1.90 1.66 2.00 1.45
0.84 1.46 1.95 2.11 1.24 2.19 1.89 1.66 1.97 1.45
0.66 1.44 1.67 1.96 1.04 2.96 2.74 1.43 1.86 2.51
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. K. Gout, J. Molec. Spectrosc. 2,259 (1958). J. BRALAWSLYand Y. BEN-ARYEH,J. them. Phys. 51,2233 (1969). F. LEGAY, Cal. Phys. 12,416 (1958). N. LEGAY-SOMUAIRE and F. LEGAY, J. Molec. Spectrosc. 8, 1 (1962). H. M. HANSON and H. H. NIELSEN,J. Molec. Spectrosc. 4,468 (1960). C. SECROUN,A. BARBEand P. Jouv~, J. Molec. Spectrosc. 45, 1 (1973). A. R. GORDON, J. them. Phys. 2,65 (1934). G. AUAT, H. H. NIELSEN and G. TARRAGO,Rotation-Vibration of Polyatomic Molecules. Dekker, New York (1971); G. AMAT, M. GOLDSMITHand H. H. NIELSEN,J. them. Phys. 27,838 (1957). D. R. LIDE, JR., J. them. Phys. 20, 1761 (1952). E. D. HINKLEY,A. R. CALAWA, P. L. KELLEYand S. A. CLOUGH,J. Appl. Phys. 43,3222 (1972). C. SECROUNand P. JOUVE,C.R. Acad. Sci. Paris 270, 1610 (1970). A. H. NIELSEN,R. D. SHELTONand W. H. FLETCHER,J. Phys. Rad. 15,604 (1964). J. H. VAN VLECK,Rev. Mod. Phys. 23,213 (1951). H. H. NIELSEN,Hundbuch der Physik, Vol. 37/l. Springer, Berlin (1959). D. KIVELSON,J. them. Phys. 22,904 (1954). A. BARBEand P. Jouv~, J. Molec. Spectrosc. 38,273 (1971).
ALGEBRAIC EXPRESSIONS FOR VIBRATION-ROTATION LINE INTENSITIES OF NEAR PROLATE ASYMMETRIC MOLECULES : APPLICATION TO SO, MOLECULE C. SECROUN,A. BARBEand P. JOUVE Laboratoire de Physique Mokculaire, Fact&e des Sciences, Reims, France (Received 9 April 1973)
Abstract-A treatment of the intensities of vibration-rotation lines of near prolate asymmetric molecules is given, leading to explicit expressions relating the observed intensities to the appropriate coefficients in the expansions of the potential energy and the dipole moment. The theoretical results are compared with experimental detetminations of line intensities in the vi-band of sulfur dioxide.
1. INTRODUCTION THE CALCULATION of the theoretical intensity of absorption for a molecule has been investigated by several authors. GORA(l) has given an interesting method for the computation of line strengths for low rotational quantum numbers. BRALAWSKYand BEN-ARYEH”) have calculated F factors for an asymmetric molecule like H,O, by a perturbation method. LEGAY’~) has proposed a contact transformation method to calculate intensities of vibrationrotation transitions. But these results were not derived for an asymmetric molecule. However, this method has been applied to linear triatomic moleculesC4) and to symmetric molecules of the type AB . . . X YZ, .(‘) Recently, a treatment of the intensities of vibration bands for polyatomic molecules has been derived, using a contact transformation method.@) In this paper, our purpose is to carry out calculations of line intensities using a similar method and to apply the results to near-prolate asymmetric molecules.
2. GENERAL
PRINCIPLES
The theoretical intensity of absorption for a molecule making a transition from a state z to a state r’ may be expressed by the relation (v’,r’JZ]v,r) = (87r3Nv/3hcZ)[exp(-E,/kT)][1-exp(-hv/kT)](~)2
(1)
where N is the molecular concentration (molecule-‘), v the frequency of the transition (cm-‘), Z the rotational state sum, E, the ground rotational energy (cm-‘), and (p) the dipole moment matrix element (Debye). Intensity is thus expressed in cm molecule- r. 1325
1326
C. SECROUN,A. BARBEand P. JOUVE
The rotational state sum for an asymmetric rotor has been given by GORDON(') as 2 = (nl'2/2)(kT/hc)3'2(ABC)_ 1'2,
(2)
where A, B and C are rotational constants (cm- ‘). The problem is the evaluation of the matrix elements (p). We need two steps: (i) a contact transformation method to take account of vibration-rotation interaction ; (ii) a perturbation method for the asymmetry. 3. THEORY
In order to use the wave function of the zero-order Hamiltonian in the evaluation of (p), two contact transformations are applied to the dipole moment. In Ref. (3), the transformed dipole moment & has been put in a symbolic symmetrized form and is II:=
P,Pp. ..u,+u,. c 2 a/?...a
. . PpP, $,
.#...aA;;Pc#b
. . .qd%?;
qeqd
’ . . Pbpa,
(3)
de... osb
dse
where q. and p. are, respectively, the dimensionless normal coordinate and its conjugate momentum, the a, are the direction cosines, and the P, are components of angular momentum. The coefficients JZ have been expressed with the coefficients of the contact transformation and of the non-transformed dipole moment.‘6) They are given in Table 1 for a first-order perturbation. In this table, the coefficients S are those of AMAT et al.@’ TABLE1. COEFFICIENTS OFTHETRANSFORMED DIPOLE MOMENT
The transformed dipole moment is a sum of terms, products of vibrational and rotational operators. The zero-order wave functions are separable into the coordinates of vibration and rotation. Therefore, the dipole moment matrix elements consist of rotational elements and vibrational elements. 1. Vibrational matrix elements The vibrational matrix elements are obtained from well-known, non-vanishing matrix elements in the linear harmonic oscillator representation.‘@ 2. Rotational matrix elements In calculating the transformed between rotation and vibration.
dipole moment, account is taken of the interaction
Algebraic expressions for vibration-rotation
line intensities
1327
In the symmetric rotator basis of D, symmetry, the matrix elements of the rotational operators may be written as (J, K, MJORIJ’, IS, M) = (J, K(O#‘,
K’)(J,
M(O’;(J’, M)
(4)
with o, =
pap@.. . y”“..
1
. p&J,*
@...a
The matrix elements involved in equation (3) are given in Table 2, where we choose the wave functions to conform with the phase conventions adopted by VAN VLECK('~)and NIELSEN. Thus, PzlJ, K, M) =
WJ, K W,
h PSJ, K, M> = Z~[(J-K)(J+K+l)]IJ.K+1,M)+~~[(J+K)(J-K+l)]~J,K-1,M),
PylJ, K, M) = $[(J-K)(J+K+
l)]IJ, K+ 1, M)
+[(J+K)(J-K+l)IJ,K-1,M).
(5)
A perturbation technique will be now developed, which is convenient for taking account of asymmetry in slightly asymmetric molecules (1~12 0.90). Using the Wang functions, where the JM, notation has been suppressed, the asymmetric rotor wave functions are given as follows :(‘) eK=
l_qE
(
I
SK+QSK-2+&+*9
where
B =
EK-&+~
’
and H is the reduced energy obtained in the Wang transformation. and fl for the various possible cases are given in Table 3, with ‘= f(J,4 f(J,
2(B - C) 2A-B-C’
= $J---n)(~---n+
~)(J+FI)(J+TI+
1) = &J(J- l)(J+ l)(J+2).
11,
Explicit formulas for a
1328
C. SECROUN, A. BARBE and P. JOUVE
I329
Algebraic expressions for vibration-rotation line intensities TABLE 3. COFSFICIENISIN THE EXPANSION OF ASYMMETRIC-ROTOR WAVE FUNCTIONS(Y REFERS TO THE PARITY OF J+K_,+K,)
K-1
Y
a
0 I
0 0
0 0
1 2
1 0 1 0 1
0
2
3 3 >3
W)f(J, v
B -(&/16)f(J, 2)“‘[1 -(s/32)5(5+ -(~/16)_f(J, 2)“‘[1+(~/32)J(J+
- (~/24)f(J, 3)“’
:(e/32)J(J+ (~/16)f(J,2)~“[1 l)]-’ (~/16)f(J, 2)“‘[1+(&/32)J(J + l)] - ’ [&/8(K- l)]f(J, K - 1)“’
If a and p refer to the asymmetric-rotor
l)]-’ 1)1-l
- (~/%)f(& 3)“2 -(&/32)f(J, 4)“’ - (~/32)f(J, 4)l’* - [E/~(K + l)]f(J, K + 1)“’
level J, K and Q’,/I?’to the level J’, K’, we find
that (II/&&,)
=
(1-@*+~*y”+fi’*I
(J, K(O,(J’, K) + cru’(J, K - 210&I’, K - 2)
+ PB’(J, K + 2(0,13’, K + 2), (+KIORl~~+l)
=
(7)
(J,KIO,IJ’,K+l)+a’(J,
KIO,@',K-1)
+ j?(J, K + 210,#‘, K + 1) +cta’(J, K - 210,15’, K - 1) + BB’(J, K + 210,&I’, K + 3), (q/KpRIg,_l)
=
1_a’+~‘:OL’*+~*
(8)
(J,K(O,(J',K1)+&J,
+ jY(J, K(O,(J', K + 1) + aa’(J,
K - 210,&J’,K - 3)
+ PD’(.J, K + 2(ORlJ’, K + 1). It is now possible to calculate matrix elements such as (IC/&#k) molecule (Table 4). 3. Calculation
K- 210&‘, K- 1)
(9) for an asymmetric
of intensities
The intensity of absorption is proportional to the line strength (p)*. In the absence of an external field, the space is isotropic and the squares of the matrix elements are M= +.I (P)* = 3 ,;_, I(+(u, JKWIP:IW+, J’K’W>l*.
(10)
From Tables 14, it is then possible to write the intensity for any transition in the case of a slightly asymmetric molecule. As an example, the expressions are now developed for C,, triatomic molecules (e.g. . .) for fundamental bands. SO,,O,,NO,.
C. SECROUN, A. BAREIEand P. JOUVE
1330
TABLE 4. DIRECTIONCOSINEMATRIXELBMLNTS FOR A NEAR PROLATE~sy~~rmuc lK
J
*;
K
yP.(*thMc.)
2’)l,
l-
aZ+~Z+a’2+~2
J-l
a2+/?2+a’2+~2
Kkl
&JTK)(JTK-I)I
2
1
+;&/[(JTK-Z)(JTK-3)]+$,/[JTK+Z)(JTK+l)]
K
2’/l(,
I
I- a*“‘~a”+8”
crpori.p3
K+l
1
,/(J2-K2)+aa’,/[J2-(K-2)2]+&?‘J[J2-(K+2)2]
2
[!
J
ROTOR
I
K+aa’(K_2)+fi,fI’(K+2)
1_a2+~2;a’2+B12
1
,/[JTK)(J+K+l)]+&‘[(JTK+2)(J+K-l)]
I
I
+$‘J[(JrK-2)(J+K+3)1
+(“a+i’~~)[“,:J[(J~K)(JrK+l)l I
K
2’/.lc, lI
a*‘p1+~‘2+p12
IJ
[(J+K+l)(J-K+l)]+aa’,/[(J+K-l)(J-K++)]
1
+/3B’,/(J+K+3)(J-K-l)
_I
Jfl
.,‘[(J+K+WJ+K+l)l
K&l
* K stands for
K- ,
Putting qJv = (8~~Nv/3hcZ) [exp( - E&T)] [ I- exp( - hv/kT)], we obtain the results listed below. Fundamental band v, of symmetry A, (perpendicular transition) (0, JWl,
J’K z!z1) = %VXCI;TMJ’4K){1 -w/w
x [q;K)
(y;;~“+qK);;,~lj
x (1 - (a2 + p2 + a’2 + /T2) + [26/6$$ +8~,~:i2K,+8~~;~t:~:]+
[“sL2Z/i’K, +,BS$
[l/~,K,][~::~,-“,+~:E;K~]}.
(11)
Fundamental band v, of symmetry B, (parallel transition) (0, JKIIIl, J’K) = ~~,‘~,24M,~~~,,[l -(fI2p&Y,,,(;:A”]
+ [2/X$
[aa’N:i2,,
+ /?/?‘N#}.
X (1 -(a2+p2+a’2
+/?“) (12)
Algebraic expressions for vibration-rotation
The different coefficients introduced different values of J’.
1331
line intensities
in these expressions are given in Table 5 for
TABLE 5. CQWFICIENTSOF LINE INTENS~XS
J-l J
J+l
J-l J
J+l
4K’
M.l
J
1
(JTK)(JTK-1)
G 2J+l
39W’
9m
8 (K’
2J+2K+l
+_2K+l
(JfK+2)(J+K+l)
8J(J + 1)
(JTK)(J+K+l)
+_2K+l
+2K+l
(J+_K+Z)(JTK-1)
1 8(J+ 1)
(J+K+Z)(JkK+l)
-2J+2K-1
+2K+l
(JTK)(JTK-1)
Jz-K2
(JfK-2)(JTK-3) (JTK-2)(J+_K+3) (JkK+4)(JfK+3)
25 0 2(J + 1)
(J-K+l:;J+K+l)
4.
APPLICABILITY
OF
THE
J2-(K+2)2 (K+2)* (J+K+3)(J-K-l)
METHOD
The applicability of relations (11) and (12) is limited by the validity of the perturbation methods used. When no resonance due to an accidental degeneracy occurs in the molecule, the results concerning vibration-rotation’ interaction are good approximations. But the
TABLE 6. CONSTANTS FOR SO2 USEDIN THE CALCULATION OF LINE INTENSITIES
Quantity
Symbol
Value
Reference
Permanent dipole moment
- 1.59 D
C.
Dipole moment derivatives
-0.13 D -0.19 D 0.33 D
C. SECROUN et al."' C. SECROUNet al.@) C. %XROUN et al.‘@
Vibrational wave numbers
Rotational constants
1167.6Ocm526.27 cm- ’ 1380.91 cm-’ 2.02736 cm- ’ 0.34417 cm-’
0.29353 cm- ’ Coriolis interaction constants
0.272 - 0.962
Centrifugal interaction constants in units 1O-2o g1j2 cm
17.9178 - 2.04505 18.7898 5.32138 7.38215 -6.86156 0.84256
SECROUN(")
A. H. N~SEN et al.‘12) A. H. NIELSENet a1.‘12’ A. H. NIELSENet aI.“2’ D. KWEUON’ls’ D. KM?LSON(15’ D. KIVEL.WN(‘~) A. BARBEand P. JouVE”~ A. BARBEand P. JOLWE”~)
C. SECROUN, A. BAREIEand P. JOUVE
1332
study of asymmetry implies LX, /3 << 1. Table 3 shows that this is not always true, specially when K/J is small. However, experimental determinations of intensities are not precise. Hence, theoretical errors would be lower than experimental ones and it is possible to compute line intensities, in good approximation, from very simple formulas. 5. APPLICATION
TO
THE
v, BAND
OF
SO,
Recently, HINKLEY et Al. measured line intensities in the v1 fundamental band. It is now possible to calculate intensities for the same transitions and compare them with experimental results. This comparison is given in Table 7, with different approximations, using the data of Table 6. We find good agreement between theoretical and experimental results. There is a discrepancy when K/J is small, which is a bad limit for applicability of the perturbation method (e.g., for the transition 14,,,, t 14,,,,). Without taking account of these particular cases, the average deviation is 0.15 x 10w2’ cm-molecule- ‘, TABLE 7. COMPARIXINOF THE THEORETICAL AND EXPERIMENTALLINEINTENSITIES,FOR THE V,+LJNDAMENTAL OF
SO, IN UNITSlO-‘l
Transition 30 10.20 + 23 5 19 + 186’12 + 13;,, + 22X,3 + 8 + 6,:, cl 8 + 2 11+
30,,,*, 24,,,8 19713 14& 2210.1, 81.7 61s
&,* 5’5 +- 40.4
14 *.;a + 140.1,
cm-mol-’
Symmetric rigid rotor
Asymmetric rigid rotor
Asymmetric molecule
Experimental results”‘)
0.79 1.78 2.12 2.34 1.24 2.58 2.05 1.70 1.99 7.22
0.88 1.63 2.02 2.30 1.30 2.19 1.90 1.66 2.00 1.45
0.84 1.46 1.95 2.11 1.24 2.19 1.89 1.66 1.97 1.45
0.66 1.44 1.67 1.96 1.04 2.96 2.74 1.43 1.86 2.51
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. K. Gout, J. Molec. Spectrosc. 2,259 (1958). J. BRALAWSLYand Y. BEN-ARYEH,J. them. Phys. 51,2233 (1969). F. LEGAY, Cal. Phys. 12,416 (1958). N. LEGAY-SOMUAIRE and F. LEGAY, J. Molec. Spectrosc. 8, 1 (1962). H. M. HANSON and H. H. NIELSEN,J. Molec. Spectrosc. 4,468 (1960). C. SECROUN,A. BARBEand P. Jouv~, J. Molec. Spectrosc. 45, 1 (1973). A. R. GORDON, J. them. Phys. 2,65 (1934). G. AUAT, H. H. NIELSEN and G. TARRAGO,Rotation-Vibration of Polyatomic Molecules. Dekker, New York (1971); G. AMAT, M. GOLDSMITHand H. H. NIELSEN,J. them. Phys. 27,838 (1957). D. R. LIDE, JR., J. them. Phys. 20, 1761 (1952). E. D. HINKLEY,A. R. CALAWA, P. L. KELLEYand S. A. CLOUGH,J. Appl. Phys. 43,3222 (1972). C. SECROUNand P. JOUVE,C.R. Acad. Sci. Paris 270, 1610 (1970). A. H. NIELSEN,R. D. SHELTONand W. H. FLETCHER,J. Phys. Rad. 15,604 (1964). J. H. VAN VLECK,Rev. Mod. Phys. 23,213 (1951). H. H. NIELSEN,Hundbuch der Physik, Vol. 37/l. Springer, Berlin (1959). D. KIVELSON,J. them. Phys. 22,904 (1954). A. BARBEand P. Jouv~, J. Molec. Spectrosc. 38,273 (1971).