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Higher-order vibration intensities of polyatomic molecules
Higher-Order
Vibration
Application
intensities
of Polyatomic
Molecules
to Diatomic and Bent Xl’, Molecules
.I treatment of the intensities of vil)ration rotaCon IJantls for ~~ol~~al-tomicmolecules is given using a method involving contact transformations. The general results are applietl to the cases of &atomic and hent SF2 molecules, leading 111explicit espresssions relating the observed intensities to the anharmonic coefficients of t be expansions of the potential ~ncrgy and the dipole moment. ‘The case of the sulfur dioxide molecule is computctl.
According to the quantum theor!. of radiation, the problem of calculating the intensit! of a vibration---rotation band of a pol\.atomic ~nolc’c~~le, associated with the transition 2,’--t f”‘, requires the evaluation of the squares of the matrix elements
the $,, being the eigenfunctions
of the Schriidinger (IJ -
I:,$
rqu:ltion
= 0,
and M being the operator associated with the electric tiilx’le moment uf the molecule. It should be noted that we consider molecules whose electronic angular momentum and spin are zero. Also, we do not take account of the rotational contribution to the total absorption. It has been the usual procedure to evaluate the transition matris elements [Eq. (1 I] with the function $( replaced b\- the zero-order wavefunction (harmonic oscillator eigenfunction). A better result can be obtained by some method of successive appro\-imation, e.g., the perturbation theory. This method was applied to diatomic molecules; in particular by Crawford and Dinsmore (I), and in a recent paper (2, we computed the case of bent SIy2 molecules. An alternate method has been suggested b>- Nielsen e/ (I/. (3) based on contact transformations of the type used b!- Shaffer, Nielsen, and Thomas (4, 51, for calculation of energies of polyntomic molecules. Most of the calculations in current USC’ Ixrtain only, 10 lint intensities in vibration rotation bands (ci,, and are often applied to diatomic molecules. The present interest in I
2
SECKOUN,
Ih~KBE, Ah-D JOUVI’
contact transformations has led us to think that the time is ripe for an evaluation of the intensities of the vibration-rotation bands of polyatomic molecules, to higher-order approximations. GENERAL
THEORY
The contact-transformations method consists of transforming H into a diagonal matrix H’ in first-order approximation (7) :
the initial Hamiltonian
II’ = THT-‘, with T = The transformed
wavefunctions
eiXA',
are given by: $v’ = T$,,
and we may write the transition
matrix
{v’IMjv”)=j
element
[Eq. (l)] in the form (3) :
#UlT-‘M’T+dv.
(2)
It can be seen, by taking the transformed operator M’ = TMT-1 in place of the operator M, that we can compute the transition matrix elements by means of zero-order wavefunctions. Transformed
Electric Dipole Moment
We espand
the M operator
in orders of magnitude
as
M=Mo+XMl+X2Mz$X3M3f...,
(4
According to the customary notation, X is a parameter of smallness equal to unity, which denotes the order of magnitude of the different terms in the expansion. If T1 is a first contact transformation, making the first-order Hamiltonian diagonal in the vibrational quantum numbers zl, we write, in the usual manner. T1 = e+ihS~ = 1 + iXSl - :
Sl* - i i
S13 +
. . ..
In the calculation of band intensities, we only take account of the vibrational part of the Hamiltonian (and so for wavefunctions), the rotational contribution to the total intensity being negligible when no resonance due to an accidental degeneracy occurs in the molecule (2). The transformed electric dipole moment is 2 X”
M’ = TIMTl-l
=
1 + iXSl - ;
S12 ?”
x (MO+ AM,+
x~M~+ h3Mi+
..
+ ... > 1 - ix&y, - y-&z+
&,3
+ 0
.
) >
\~lBKA'I'IOhIh-‘L’I:I\-SI’I’I145
.$
and one may expand M’ in orders of magnitude: M’ = MO’ + AMI + XZM?’ + X”M:,’ + . . . . One finds for the different orders of the once transformed the notation of Amat et al. (7, $1,
electric dipole moment,
using
MO’ = M,,, M,’ = MI + I[SI, Mo],
with (&‘“,
M,)
= [&, [&, Mn]]
= SI[&, Mn] - [S1, MJSI.
By means of a second contact transformation T2 = e- lhzS~,one obtains of approximation with a twice transformed electric dipole moment.
a higher order
Mt = T2M’T2-’ = TzTIMT1-‘T2-’ with the results: M,’ = Mu’. M,+ = Ml’, I.5
Mz+ = M2’ + i[&, Mu’], M::t = Ma + i[Sr. M,‘]. T‘WLE
I
COEFFICIENTS APPEARING
T1
W"ab
(1)
M
pm'amb hzE mwabm( l+5am+6bm1 ab+"
ob
(11 Mobc
m
P +T& klm abc
(kO1
(I)
M
ab c
IN .I/'
‘2
i2$
)Icm ’
1 F,“SkY 2 n
U+~,,)+
abm (1+iTcmHl+5am+~h
)1”S ObPp
; $
/J” s;,
1-t
(l+5ap+abp)(l+scp)
knp sl,p(l+~mp)(l+~~pJ(l+Skn+~“p)
-
skp (1+5,,
1
One sees that by means of the different T; matrices, diagonalizing the Hamiltonian to the i-th-order of approximation, it is possible to compute the transformed electric dipole moment MC”) and so the transition matrix element (~‘1M ci) / ZJ”) with zero-order wavefunctions. Application of normal coordinates on harmonic oscillator eigenfunctions is known (9), and one must expand the M operator with respect to dimensionless normal coordinates qa: Jfll = PO,
M2
=
c
Pabqaqb,
(a2
b)
MS=
c
Pabcqoqbqc.
abc ((1 5 b 5
cl
OXCE-TRANSFORMED
COMPUTATION
The S1 function, which has been calculated reduced by Amat et al. (11):
(0 5
In this expression,
abc b 5
by Herman
(a 5
cl
and Shaffer (IO), has been
b)
the coefficients .yC”C’ and Cy,ibrare those given in Table VI of Ref.
(10. TABLE
11
TRANSITION MATRIX ELEMEXTS I
Band
da
Matrix
I
[ (“a+1
)/2]‘IR
element
)(a
11 c l/2
29,
$
3’a
F
HIM,,
ha+lHva+2
[’
(Va+l)(Vb+l
$,+db+dc
2 3,+3,
(1)~~~
ha+2)(v,+lj11/2[cl~
HIMY
_RZ (11 Mab
~~Va+,~~Vb+,~~V~+,~EIMa~-~2
$[(Va+l
I
I,)M,,_%’
1[
1
(11 Maa
l/z I[
v,+l)(v,+2)(va+3) 1/2
_A2
1
I
(,)MaSb-h2(l)M~-h2(r)MOd]
Maab
_t2
(I)
MF_h2
1,) Mtb]
TRANSITION MATRIX ELEMENTSFOR A DIATOXIC MOLECULE
>
>
(v+l) (vc2 )(v+3)
(
1111 + )I
2k
t&
fl,
I
Vsing formulas given 11~.Imat PI LZI.(I-2), one computes each commutator of Eq. (4) : in the results obtained, one neglects terms in the form X”p,,, which onI!- add a third-o&l c.orrcction to the fundamental hand. [S,, n/l,,] = 0,
-
$12”
c
c
abed in _< b)
/4fS~b~:Cm”(l
+
F,,“,
+
s,Aip,,phy,
+
y&&)(1
+ 6,,,,).
))I
may write the once-transformed electric dipole moment in orders of magnitude with respect to the dimensionless coordinates y,, and the conjugate momenta p,,: N’e
M’ = PO + x c Pcay, + A2 c
[c1,Mobqaq6
+
il$f@bpapttl
” (a<
+
A”[ ‘a<
c obc b<
b)
Ud’fabcqaqbqc cl
+
,l#cab+(fk$bqc c abc (a< b!
+
&p&b)].
(7)
SECROUN,
BARBE, TABLE
ASD
JOUVE
IV
TRANSITIONMATRIS ELE~~ENPSFORA BENT XY2 MOLECULI.:
Band
Transition
matrix
element
+
dl + WI kill
4
1 -iiF 1
~2(h':-W;)
NV, +2)(v2+3~[~222
ht-
k211
2 4w$AJ
+21z2
!?$
2
+ pq
u)l ‘lz2 4u.5 “f
+
‘TABLE
1’
OBSERVED II~TENS~TIE~ op SO,
Wove
number
(cm-’
Intensity
)
( dork )
1151.38
2
500
517.69
2
520
1361.76
18
900
2295.88
25 a
2715.46
1
1665.07
0.5b
2499.55
96
1875.55
26
* The 2~: band is not observable on account of its very small intensit>-. * The “I + ~2 band intensity has been determined from relative intensities and Fletcher (/il.
given by iYielsen, Shelton,
The coeflicients appearing in Eq. (7) are given in Table I. In these expressions, *CA ,,,, indicates a summation over all distinct permutations of the indexes u, b, c. The transition matrix element expression for each absorption band is given in ‘I’ablr I I. APPLICATION
TO DIATOMIC
~MOI,ECPl,I
71’aking the IKJtt!ntkl energy and the electric dipole moment function in forms similar to those for l)ol>xtomic molecules, it is possible to apply the results of the last section to the. case of :I diatomic molecule. The transition matrix elements obtained are given in ‘I’able III. It
In T&It: IV, are transition niIll:ts u,, is defined I)\-
matri\
vlcmc*nts for (‘:! trixtomic ,\,, = W,-‘Lw,,L’.
molecules.
Tn these for-
These results differ from those obtained by a first-order perturbation method (2). Present results introduce some progress: they enable one to compute dipole moment coefficients from the knowledge of potential constants and band intensities. As an example, one applies these relations to the case of the sulfur dioxide molecule. Placing the observed intensities (13) (given in Table V) and known potential constants TABLE VI DUJOLEMOMENT COEFFIC~EKTSOF SO,
Coefficient
r r
Sign of the squore root
1
2
This
research
-
0.13
-
0.19
+ 0.33
r3
+
+
+
+
+
3.95
1o-3
-
1 .45
1o-2
e
7.04
7o-3
- 2.0
1o-3
- 5.4
1o-3
_
1.06
1O-3
- 5.40
1o-3
e
5.57
1o-2
+
0.67
1O-2
-2
-4. 1 .22
10
_
1O-2
1 .72
(Debye
)
(14) in thvsc relations enables LIS to ~xlcul;ttc the F’S, The relative sign of t tw lintxr of the f’ or (I ~Jr~lnch c!uc dipole moment coefficients is fixed accordin g to enhancement to the interaction eilect. Some ambiguity remains in quadratic coefficients; WY give results corresponding to each sign in calculating the square roots (Table VI 1, Some remarks follow these results. E‘irst, electrical anharmonicit>. seems to be thr Ixedominant factor accounting for the observed intensities of the 2~1 and V$+ v3 bands. This statement leads IIS to think that the neglect of electrical anharrnonicit!in our last paper (.?I is not alwa~x justified. However, mechanical anharmonicity has ;I ver)- important contribution to the intvnsity of the Q + LJ:{ band. Thus, it seems that the coefficient ~13 has a positive value; thih woul~l mean that stretching of both bonds gives rise to a greater change in dipole momt’nt for the antis!.mmctric vibration and is just the opposite of the situation with the pote~ t ial constants, where kI3i is negative (asymmetric stretch causes the kIi3 term to be ;I larger negative contribution). It might be argued that this is in agreement with the foul principal resonance structures for SO, (16) (Fig. 11, where simultaneous stretching of both bonds during the antisymmetric vibration wrdd favor structures 1 and 2 and g:i\.c riw to ;I posilive value of pIzi.
1. fi. 1,. ~KA\\.FOKIl
ANO
H.
I,.
L)INSMOKE, J. 6k??f. l'/lgS. 18, 983 !1')5()1.
871 (19711. .i. H. HANSON,H. H. NIELSEN, \V. H. SHAFFEK, 8x1)J. U'acc;oxeK, .I.C‘/W~I~. /'/r_~s. 27, 10 j 103ir. 4. H. H. SIELSEN. Keit.Al& Pllys.23, 90 (1951). .T. !\‘. H. SHAEFEK, H. H. SIELSEN, AND L. H. TIII)M.U.I'//J.Y. licz.56. IOil (10301. 0. 1:. LEGAL, Cd. Plrys. 12, 416 (19X$. 7. M. GOLDSMITH, G. AMAT, AND H. H. NIELSEN, J. C'irern. 1'ilys. 24, 1178 (1956). of Imlyatomi~. I~oIc~~uIcs.” A’. G. ~\MAT, H. H. XIELSES, AND G. '~.IKK.Aco.“Rotation-villration 11. Dekkcr, Inc., Kew \-cd, 1971. 2'.(‘.SECXOUN
ff). If. 12. 13.
ANT) I'.JIJWE, J. P/I~.v. 32,
K. C. HERMAX AND \V. H. SIIAPFFA, J. Clwr?z.I'irys. 16, 453 (lY48). G. AMAT, M. GOLDWITH, AND H. H. NIELSEN, J. Clm~. Ylrys. 27, 838 (1957). G. AMAT, hl. GOLDSMITII, AND H. H. SIELSEX, J. Phys. Radium 16, 854 (1955). C. SECH~CS AND P. JOWE, C'.R. dmd. Sri. Puris, Ser. B 270, 1610 (19701.
III. A. RAKDE,. SHELTON, AND W. H. J.‘LETCIIER, J. Ploys.K&urn 15, 604 (lY51); I<. I). SAELTON, A. H. NIELSEN, AND LV. H. FLETCHER, J. C‘hem. P//p. 21, 2178 (19531.
if). Lv. CIOFFITT, /',',)r. I\'O,'. S0r. .4 200. 409 ~10.50~.
Vibration
Application
intensities
of Polyatomic
Molecules
to Diatomic and Bent Xl’, Molecules
.I treatment of the intensities of vil)ration rotaCon IJantls for ~~ol~~al-tomicmolecules is given using a method involving contact transformations. The general results are applietl to the cases of &atomic and hent SF2 molecules, leading 111explicit espresssions relating the observed intensities to the anharmonic coefficients of t be expansions of the potential ~ncrgy and the dipole moment. ‘The case of the sulfur dioxide molecule is computctl.
According to the quantum theor!. of radiation, the problem of calculating the intensit! of a vibration---rotation band of a pol\.atomic ~nolc’c~~le, associated with the transition 2,’--t f”‘, requires the evaluation of the squares of the matrix elements
the $,, being the eigenfunctions
of the Schriidinger (IJ -
I:,$
rqu:ltion
= 0,
and M being the operator associated with the electric tiilx’le moment uf the molecule. It should be noted that we consider molecules whose electronic angular momentum and spin are zero. Also, we do not take account of the rotational contribution to the total absorption. It has been the usual procedure to evaluate the transition matris elements [Eq. (1 I] with the function $( replaced b\- the zero-order wavefunction (harmonic oscillator eigenfunction). A better result can be obtained by some method of successive appro\-imation, e.g., the perturbation theory. This method was applied to diatomic molecules; in particular by Crawford and Dinsmore (I), and in a recent paper (2, we computed the case of bent SIy2 molecules. An alternate method has been suggested b>- Nielsen e/ (I/. (3) based on contact transformations of the type used b!- Shaffer, Nielsen, and Thomas (4, 51, for calculation of energies of polyntomic molecules. Most of the calculations in current USC’ Ixrtain only, 10 lint intensities in vibration rotation bands (ci,, and are often applied to diatomic molecules. The present interest in I
2
SECKOUN,
Ih~KBE, Ah-D JOUVI’
contact transformations has led us to think that the time is ripe for an evaluation of the intensities of the vibration-rotation bands of polyatomic molecules, to higher-order approximations. GENERAL
THEORY
The contact-transformations method consists of transforming H into a diagonal matrix H’ in first-order approximation (7) :
the initial Hamiltonian
II’ = THT-‘, with T = The transformed
wavefunctions
eiXA',
are given by: $v’ = T$,,
and we may write the transition
matrix
{v’IMjv”)=j
element
[Eq. (l)] in the form (3) :
#UlT-‘M’T+dv.
(2)
It can be seen, by taking the transformed operator M’ = TMT-1 in place of the operator M, that we can compute the transition matrix elements by means of zero-order wavefunctions. Transformed
Electric Dipole Moment
We espand
the M operator
in orders of magnitude
as
M=Mo+XMl+X2Mz$X3M3f...,
(4
According to the customary notation, X is a parameter of smallness equal to unity, which denotes the order of magnitude of the different terms in the expansion. If T1 is a first contact transformation, making the first-order Hamiltonian diagonal in the vibrational quantum numbers zl, we write, in the usual manner. T1 = e+ihS~ = 1 + iXSl - :
Sl* - i i
S13 +
. . ..
In the calculation of band intensities, we only take account of the vibrational part of the Hamiltonian (and so for wavefunctions), the rotational contribution to the total intensity being negligible when no resonance due to an accidental degeneracy occurs in the molecule (2). The transformed electric dipole moment is 2 X”
M’ = TIMTl-l
=
1 + iXSl - ;
S12 ?”
x (MO+ AM,+
x~M~+ h3Mi+
..
+ ... > 1 - ix&y, - y-&z+
&,3
+ 0
.
) >
\~lBKA'I'IOhIh-‘L’I:I\-SI’I’I145
.$
and one may expand M’ in orders of magnitude: M’ = MO’ + AMI + XZM?’ + X”M:,’ + . . . . One finds for the different orders of the once transformed the notation of Amat et al. (7, $1,
electric dipole moment,
using
MO’ = M,,, M,’ = MI + I[SI, Mo],
with (&‘“,
M,)
= [&, [&, Mn]]
= SI[&, Mn] - [S1, MJSI.
By means of a second contact transformation T2 = e- lhzS~,one obtains of approximation with a twice transformed electric dipole moment.
a higher order
Mt = T2M’T2-’ = TzTIMT1-‘T2-’ with the results: M,’ = Mu’. M,+ = Ml’, I.5
Mz+ = M2’ + i[&, Mu’], M::t = Ma + i[Sr. M,‘]. T‘WLE
I
COEFFICIENTS APPEARING
T1
W"ab
(1)
M
pm'amb hzE mwabm( l+5am+6bm1 ab+"
ob
(11 Mobc
m
P +T& klm abc
(kO1
(I)
M
ab c
IN .I/'
‘2
i2$
)Icm ’
1 F,“SkY 2 n
U+~,,)+
abm (1+iTcmHl+5am+~h
)1”S ObPp
; $
/J” s;,
1-t
(l+5ap+abp)(l+scp)
knp sl,p(l+~mp)(l+~~pJ(l+Skn+~“p)
-
skp (1+5,,
1
One sees that by means of the different T; matrices, diagonalizing the Hamiltonian to the i-th-order of approximation, it is possible to compute the transformed electric dipole moment MC”) and so the transition matrix element (~‘1M ci) / ZJ”) with zero-order wavefunctions. Application of normal coordinates on harmonic oscillator eigenfunctions is known (9), and one must expand the M operator with respect to dimensionless normal coordinates qa: Jfll = PO,
M2
=
c
Pabqaqb,
(a2
b)
MS=
c
Pabcqoqbqc.
abc ((1 5 b 5
cl
OXCE-TRANSFORMED
COMPUTATION
The S1 function, which has been calculated reduced by Amat et al. (11):
(0 5
In this expression,
abc b 5
by Herman
(a 5
cl
and Shaffer (IO), has been
b)
the coefficients .yC”C’ and Cy,ibrare those given in Table VI of Ref.
(10. TABLE
11
TRANSITION MATRIX ELEMEXTS I
Band
da
Matrix
I
[ (“a+1
)/2]‘IR
element
)(a
11 c l/2
29,
$
3’a
F
HIM,,
ha+lHva+2
[’
(Va+l)(Vb+l
$,+db+dc
2 3,+3,
(1)~~~
ha+2)(v,+lj11/2[cl~
HIMY
_RZ (11 Mab
~~Va+,~~Vb+,~~V~+,~EIMa~-~2
$[(Va+l
I
I,)M,,_%’
1[
1
(11 Maa
l/z I[
v,+l)(v,+2)(va+3) 1/2
_A2
1
I
(,)MaSb-h2(l)M~-h2(r)MOd]
Maab
_t2
(I)
MF_h2
1,) Mtb]
TRANSITION MATRIX ELEMENTSFOR A DIATOXIC MOLECULE
>
>
(v+l) (vc2 )(v+3)
(
1111 + )I
2k
t&
fl,
I
Vsing formulas given 11~.Imat PI LZI.(I-2), one computes each commutator of Eq. (4) : in the results obtained, one neglects terms in the form X”p,,, which onI!- add a third-o&l c.orrcction to the fundamental hand. [S,, n/l,,] = 0,
-
$12”
c
c
abed in _< b)
/4fS~b~:Cm”(l
+
F,,“,
+
s,Aip,,phy,
+
y&&)(1
+ 6,,,,).
))I
may write the once-transformed electric dipole moment in orders of magnitude with respect to the dimensionless coordinates y,, and the conjugate momenta p,,: N’e
M’ = PO + x c Pcay, + A2 c
[c1,Mobqaq6
+
il$f@bpapttl
” (a<
+
A”[ ‘a<
c obc b<
b)
Ud’fabcqaqbqc cl
+
,l#cab+(fk$bqc c abc (a< b!
+
&p&b)].
(7)
SECROUN,
BARBE, TABLE
ASD
JOUVE
IV
TRANSITIONMATRIS ELE~~ENPSFORA BENT XY2 MOLECULI.:
Band
Transition
matrix
element
+
dl + WI kill
4
1 -iiF 1
~2(h':-W;)
NV, +2)(v2+3~[~222
ht-
k211
2 4w$AJ
+21z2
!?$
2
+ pq
u)l ‘lz2 4u.5 “f
+
‘TABLE
1’
OBSERVED II~TENS~TIE~ op SO,
Wove
number
(cm-’
Intensity
)
( dork )
1151.38
2
500
517.69
2
520
1361.76
18
900
2295.88
25 a
2715.46
1
1665.07
0.5b
2499.55
96
1875.55
26
* The 2~: band is not observable on account of its very small intensit>-. * The “I + ~2 band intensity has been determined from relative intensities and Fletcher (/il.
given by iYielsen, Shelton,
The coeflicients appearing in Eq. (7) are given in Table I. In these expressions, *CA ,,,, indicates a summation over all distinct permutations of the indexes u, b, c. The transition matrix element expression for each absorption band is given in ‘I’ablr I I. APPLICATION
TO DIATOMIC
~MOI,ECPl,I
71’aking the IKJtt!ntkl energy and the electric dipole moment function in forms similar to those for l)ol>xtomic molecules, it is possible to apply the results of the last section to the. case of :I diatomic molecule. The transition matrix elements obtained are given in ‘I’able III. It
In T&It: IV, are transition niIll:ts u,, is defined I)\-
matri\
vlcmc*nts for (‘:! trixtomic ,\,, = W,-‘Lw,,L’.
molecules.
Tn these for-
These results differ from those obtained by a first-order perturbation method (2). Present results introduce some progress: they enable one to compute dipole moment coefficients from the knowledge of potential constants and band intensities. As an example, one applies these relations to the case of the sulfur dioxide molecule. Placing the observed intensities (13) (given in Table V) and known potential constants TABLE VI DUJOLEMOMENT COEFFIC~EKTSOF SO,
Coefficient
r r
Sign of the squore root
1
2
This
research
-
0.13
-
0.19
+ 0.33
r3
+
+
+
+
+
3.95
1o-3
-
1 .45
1o-2
e
7.04
7o-3
- 2.0
1o-3
- 5.4
1o-3
_
1.06
1O-3
- 5.40
1o-3
e
5.57
1o-2
+
0.67
1O-2
-2
-4. 1 .22
10
_
1O-2
1 .72
(Debye
)
(14) in thvsc relations enables LIS to ~xlcul;ttc the F’S, The relative sign of t tw lintxr of the f’ or (I ~Jr~lnch c!uc dipole moment coefficients is fixed accordin g to enhancement to the interaction eilect. Some ambiguity remains in quadratic coefficients; WY give results corresponding to each sign in calculating the square roots (Table VI 1, Some remarks follow these results. E‘irst, electrical anharmonicit>. seems to be thr Ixedominant factor accounting for the observed intensities of the 2~1 and V$+ v3 bands. This statement leads IIS to think that the neglect of electrical anharrnonicit!in our last paper (.?I is not alwa~x justified. However, mechanical anharmonicity has ;I ver)- important contribution to the intvnsity of the Q + LJ:{ band. Thus, it seems that the coefficient ~13 has a positive value; thih woul~l mean that stretching of both bonds gives rise to a greater change in dipole momt’nt for the antis!.mmctric vibration and is just the opposite of the situation with the pote~ t ial constants, where kI3i is negative (asymmetric stretch causes the kIi3 term to be ;I larger negative contribution). It might be argued that this is in agreement with the foul principal resonance structures for SO, (16) (Fig. 11, where simultaneous stretching of both bonds during the antisymmetric vibration wrdd favor structures 1 and 2 and g:i\.c riw to ;I posilive value of pIzi.
1. fi. 1,. ~KA\\.FOKIl
ANO
H.
I,.
L)INSMOKE, J. 6k??f. l'/lgS. 18, 983 !1')5()1.
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