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Angular momentum distributions in a direct photodissociation process of linear triatomic molecules
Volume 177,number 1
CHEMICALPHYSICS LETTERS
8 February 1991
Angular momentum distributions in a direct photodissociation process of linear triatomic molecules Kuo-mei Chen Department ofChemistry,NationalSun Yat-senUniversity,Kaohsiung, Taiwan,ROC Received 26 October 1990
With the aid of an addition theorem of spherical harmonics, the transformations between the angular basis functions of photofragments in the atom-diatom collisional coordinates and the rotational wavefunctions of parent molecules in Euler angles are found. These transformation relations are employed to treat the photofragmentation of linear triatomic molecules quantum mechanically. In a direct photodissociation process, the rotational and orbital angular momentum distributions of photofragments as a function of the rotational and the bending-vibrational quantum numbers of parent molecules are derived.
1. Introduction
Experimental and theoretical studies on photodissociation processes of polyatomic molecules have received much attention in recent years [ l-6 1. In particular, Freed and co-workers [ 7-9 ] have developed an exhaustive Franck-Condon (FC) framework to treat the simplest system, the photofragmentation of linear triatomic molecules. Following the theoretical scheme of Morse and Freed, Heather and Light [ 10,111 have extended the FC framework to three-dimensional scattering calculations. In a different approach, Shapiro and Balint-Kurti [ 5,12- 14] have employed the artificial channel method to calculate the photofragmentation cross sections. In the latter treatment, the initial state and the transition dipole operator should be expanded in the natural scattering coordinates. Conceptually, the FC framework of Freed and co-workers provides a clear physical insight into the photodissociation dynamics. The extension of the FC framework to the potential scatterings (halfcollisions) can be executed either by the method of driven equations [ 151 or the discrete variable theory [ 111. In this Letter, we will present the transformation relations between the angular basis functions of photofragments and parent molecules. The expansion coefficients of the transformation are shown to be independent of the bending normal coordinates in realistic systems. These characteristics of the basis functions facilitate the separation of angular variable integrals from Franck-Condon integrals over normal coordinates. The ultimate aim of this Letter is to derive analytic distribution functions of photofragments as a function of the rotational and the bending-vibrational quantum numbers of parent molecules. In section 2, we present the abovementioned transformation relations. The dependence of the expansion coefficients on the bending coordinate is examined in section 3. In section 4, angular momentum distributions of pdotofragments are derived in analytic form for a direct photodissociation process.
2. Transformation relations between angular basis functions For a direct photodissociation process, the transition amplitude Pf+i is given by the following expression [1,91, pf+i=Cl
(y~‘flt^‘rl?)
I* 3
0009-2614/9 1 /$03.500 I99 1 - Elsevier Science Publishers B.V. ( North-Holland )
(1)
73
Volume 177,number I
CHEMICAL PHYSICSLETTERS
8 February 1991
where 1P,) and 1Yf) are the initial bound state and the final repulsive state wavefunction, respectively; i is the polarization unit vector of the excitation photons and i-r is the transition dipole operator. For a linear triatomic molecule, 1Y,) assumes the form [9]
I~ui~l’~‘>iI~~~Q~~I~~~Q~~l~~~~~~IJ~~~~
(2)
where 1‘C+ ) 1is the ground state electronic wavefunction, 1n,, Q, ) and 1n2, Qz> are the harmonic oscillator wavefunctions with Q, as the totally symmetric stretching mode and Q, as the asymmetric stretching mode. In eq. (2), I u, k, S) is a two-dimensional isotropic harmonic oscillator wavefunction with vibrational angular momentum k along the figure axis and is given by [9] (Kdp exp(
-K*62/2)L’(~‘,k,),2(K282)
,
(3)
where
(
K= rf,lm,
4243 t t&/m,
+ (r,* + r23)2/m2
112
Obend h >
’
(4)
6 (measured in rad) is the angular deviation from the linear configuration (cf. fig. I ) and L$! ,k,j,2 is an associated Laguerre polynomial. In eq. (2), the rotational wavefunction IJkh4) is given by (5) where Dt is a Wigner rotation matrix [ 161. For a repulsive final state, ) Yf) assumes the form [ 91
Fig. I. The geometricconfiguration of a linear triatomic molecule with bond lengths r 12,rz3and masses m,, m2, m,. The atom with mass m, recedes in the photofragmentation. Other parameters are drawn on the graph. 14
Volume 177,number 1
CHEMICALPHYSICS LETTERS
8 February 1991
We have assumed that the excited repulsive state has a ‘C+ symmetry to avoid complications due to the Renner-Teller effect which splits a repulsive, dipole-allowed II electronic state. In eq. (6)) In, Q,2) describes the oscillatory motion of photofragment 12, IEl,, R ) dictates the interfragmental motion, and the angular basis function )$f,‘,y ( i12, 2 ) ) is given by
where p = (2J’+ 1 )“2 and ( : : : ) is a 3-j symbol [ 161; F12and I? denote the orientation of r12 and R in the space-fixed frame (SFF), respectively. From the transformation properties of spherical harmonics [ 161, we have Y~,m,(FL*)= c @&;(~PY)
Y/,m;(7l2) >
(8a)
mi
and L,(~)
= c z&&Br) 4
K*,;(ff’ ) ,
(8b)
where ii2 and l?’ denote the orientation of r,2and R in the molecule-fixed frame (MFF), respectively. Combining eqs. (7), (8a), (8b) and the summation identity of rotation matrices, (9) we have
From the addition theorem of spherical harmonics J;
( _
l)‘l-‘z+q
([I, 12,J’) ml
m2
_k’
[ 17 1, it can be proven that
Y,,m;(7l2) hpI;@
)=(4x)-“%(X)
Y,.,,(i)
,
(11)
where F/,‘,,(x) is a Schulten-Gordon coeffkient, x+,/r, (cf. fig. 1 ), and 1 is along the figure axis which has Euler angles (000) in the SFF. Mathematical properties and the algorithm for the numerical calculation of the Schulten-Gordon coefficients have been explored recently [ 18,191. From the transformation properties of symmetric top wavefunctions #‘, we can ascertain that
I~~~(F12,~))=F:;,*(X)IJ)k)MI).
(12)
Conversely, we have IJ’k’M’> = IF;;F&*(X) l~:;~(~l2,~0
(13)
from the orthogonality relation of the Schulten-Gordon coefftcients. The normalization condition of eq. ( 13) is properly maintained since C,,,~[F~,~(X)]~= 1 [ 191. Eq. ( 12) is the relation which correlates the angular basis functions of the initial and final states. It should be noted that F{h(x) depends parametrically on the #’ See for example eq. (3.124)of ref. [ 161.
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8 February1991
CHEMICAL PHYSICS LETTERS
Volume 177, number I
bending normal coordinate. This fact implies that both sides of eq. ( 12) have the same number of degrees of freedom as dictated by basic theory.
3. The dependenceof F&Jx) on 6 Before formally integrating the matrix element in eq. ( 1 ), it is appropriate to examine the dependence of the Schulten-Gordon coefficient F&(X) on the bending coordinate 6. Referring to fig. 1, the displacements y,, y2 and y3 are proportional to 6 according to the normal mode analysis of linear triatomic molecules [ 201 and are given by (14)
(15)
and y,=-A, r23m3
where A = RK’/o&~. sin ~=cos
A=
From fig. 1, one has
‘y
=A6
r23m2;f (‘I2+r23)m1 .
(17)
r12r23ml m2
From the law of cosine,
R2= (ST+&-2
(z)
cos(k-b)
(18)
For small amplitude bending oscillations, it can be easily proven that d=
(m, +m2)R m, +m2+m, =
ml +m2 m, +mz+m3
Since sin +y3/d
(
r12m1+r23ml +r23m2 ml +m2
and x=rJr,
rd23m1d2
2(r12ml +r2,m,
(19) +r23m2)
> ’
=sin 8/sin 4, one has the final result
(20) Thus, the ratio x has a second-order dependence on 6. After substituting structural parameters of real systems into the above equation, it is found that the second term makes negligible contribution to X. We conclude that F&,(x) is independent of 6 in every realistic system.
4. Angularmomentumdistributionfunctionsin a directphotodissociationprocess After deriving the transformation relation between the angular basis function and proving the weak dependence of x on 8, it is a straightforward routine to calculate the matrix element in eq. ( 1). For plane-polarized
76
Volume 177, number
1
CHEMICAL PHYSICS LElTERS
8 February 1991
excitation photons and a ‘C++-‘Z+ electronic transition, the transition dipole operator takes the form D&(c~fl[)~~,where rOis the dipole operator in the MFF. Under the Condon approximation, we have pf+i=
1 C(FO)2q
MM
(21) where the result of integration over the product of three rotation matrices and the orthogonality relations of 3-j symbols have been invoked. In the above equation, ( Fo)2= I ( IX+, fl P,,1‘C+, i) (’ is an electronic-transition-matrix element, qE(Z,n;n,n2= [ ( n, Q,2; El,, R 1n,, Q1; n2, Q2) ]* is a two-dimensional Franck-Condon factor which has been thoroughly studied by Freed and co-workers [ 1,9]. Zv,kin eq. (2 1) is given by mK 2[(v-lkl)/2]!
Iv,k=
[I (
[(v+
IQ )/A! )
(Kd)‘k’t?Xp(-K2d2/2)
2’~~+~[~((v+/kl),2+1)]‘[~(,kl,2+l)]’
1 2
1’2
Jt&k,),2(K2d2)
6dd
(22)
where the results of a standard integral have been invoked [ 2 1, p. 8441. f’ and 2Fl in the above equation are the gamma and the confluent hypergeometric function, respectively. In the Airy function approximation [ 11, qE12,nin,n2is independent of the orbital angular momentum quantum number 12.Secondly, (TOO> 2 is a constant parameter for transitions to the same absorption band. Finally, the k dependence of the rotational energy (B[J(Jt 1) - k2] ) of the initial state can be neglected for transitions to a repulsive state. Thus, it is feasible to define the rotational angular momentum distribution function of photofragment 12 as (23) where k= v, v-2, .... - v+2, -v and J’=J- 1, J, Jt 1. The summation over k accounts for the k-degeneracy of the vth bending-vibrational level of the ground state. Similarly, we can define the orbital angular momentum distribution function as (24) with identical confinements on k and J’. Numerical calculations by programming eqs. (23) and (24) and their implications will be published elsewhere.
5. Summary
To calculate efficiently the transition amplitude for the photofragmentation of linear triatomic molecules, the transformation relations between angular basis functions of photofragments and parent moleculesare sought. After examining the dependence of the expansion coefftcients of the transformation on the bending coordinate, we find a complete separation of the angular basis function from the bending-vibrational wavefunction. Thus, the transition amplitude is factored into products of a two-dimensional Franck-Condon factor qE/z,n;n,n2, a function I,, of the bending-vibrational quantum numbers v and k, a square of a 3-j symbol and a square of a Schulten-Gordon coefftcient F{,*(X), while the parameter x depends on the bond lengths and masses of the constituent atoms. Accordingly, the rotational and orbital angular momentum distributions of photofragments 77
Volume 177, number 1
CHEMICALPHYSICSLETTERS
as a function of the rotational and the bending-vibrational
8 February1991
quantum numbers of parent molecules are presented.
Acknowledgement This research was supported by the National Science Council of the Republic of China.
[ I ] K.F. Freedand Y.B.Band,Excitedstates,Vol.3, ed. E.C.Lim (AcademicPress,NewYork,1977)pp. 109-201G [2] M. Shapiroand R. Bersohn,Ann. Rev. Phys. Chem. 33 (1982) 409. [3] S.R. Leone, Advan. Chem. Phys. 50 (1982) 255. [4] P. Bmmer and M. Shapiro, Advan. Chem. Phys. 60 ( 1985) 37 I. [5] G.G. Balint-Kurti and M. Shapiro, Advan. Chem. Phys. 60 (1985) 403. [6] M.N.R. Ashfold and J.E. Baggott, eds., Molecular photodissociation dynamics (The Royal Society of Chemistry, London, 1987). [7] M.D. Morse, K.F. Freed and Y.B. Band, Chem. Phys. Letters 44 (1976) 125. [S] M.D. Morse, K.F. Freed and Y.B. Baud, J. Chem. Phys. 70 (1979) 3604. [9] M.D. Morse and K.F. Freed, J. Chem. Phys. 74 (1981) 4395. [lo] R.W. Heather and J.C. Light, J. Chem. Phys. 78 (1983) 5513. [ 1I ] R.W. Heather and J.C. Light, J. Chem. Phys. 79 (1983) 147. [12] M. Shapiro and G.G. Balint-Kurti, J. Chem. Phys. 71 (1979) 1461. [ 13] G.G. Balint-Kurti and M. Shapiro, Chem. Phys. 6 1 ( I98 1) 137. [ 141I.F. Kidd and G.G. Balint-Kurti, J. Chem. Phys. 82 (1985) 93. [ 151Y.B. Band, K.F. Freed and D.J. Kouri, J. Chem. Phys. 74 (1981) 4380. [ 161R.N. Zarc, Angular momentum (Wiley, New York, 1988). [ 171K. Chen and C. Pei, Chem. Phys. Letters 124 (1986) 365. [ 181K. Chen, Proc. Natl. Sci. Count. 6 ( 1982) 148. [ 19) K. Chen and C. Pei, Chem. Phys. 76 ( 1983) 219. [20] G. He&erg, Molecularspectra and molecular structure, Vol. 2. Infrared and Raman spectra of polyatomic molecules (Van Nostrand Reinhold, New York, 1945) p. 173. [2 1 ] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products (Academic Press, New York, 1965).
78
CHEMICALPHYSICS LETTERS
8 February 1991
Angular momentum distributions in a direct photodissociation process of linear triatomic molecules Kuo-mei Chen Department ofChemistry,NationalSun Yat-senUniversity,Kaohsiung, Taiwan,ROC Received 26 October 1990
With the aid of an addition theorem of spherical harmonics, the transformations between the angular basis functions of photofragments in the atom-diatom collisional coordinates and the rotational wavefunctions of parent molecules in Euler angles are found. These transformation relations are employed to treat the photofragmentation of linear triatomic molecules quantum mechanically. In a direct photodissociation process, the rotational and orbital angular momentum distributions of photofragments as a function of the rotational and the bending-vibrational quantum numbers of parent molecules are derived.
1. Introduction
Experimental and theoretical studies on photodissociation processes of polyatomic molecules have received much attention in recent years [ l-6 1. In particular, Freed and co-workers [ 7-9 ] have developed an exhaustive Franck-Condon (FC) framework to treat the simplest system, the photofragmentation of linear triatomic molecules. Following the theoretical scheme of Morse and Freed, Heather and Light [ 10,111 have extended the FC framework to three-dimensional scattering calculations. In a different approach, Shapiro and Balint-Kurti [ 5,12- 14] have employed the artificial channel method to calculate the photofragmentation cross sections. In the latter treatment, the initial state and the transition dipole operator should be expanded in the natural scattering coordinates. Conceptually, the FC framework of Freed and co-workers provides a clear physical insight into the photodissociation dynamics. The extension of the FC framework to the potential scatterings (halfcollisions) can be executed either by the method of driven equations [ 151 or the discrete variable theory [ 111. In this Letter, we will present the transformation relations between the angular basis functions of photofragments and parent molecules. The expansion coefficients of the transformation are shown to be independent of the bending normal coordinates in realistic systems. These characteristics of the basis functions facilitate the separation of angular variable integrals from Franck-Condon integrals over normal coordinates. The ultimate aim of this Letter is to derive analytic distribution functions of photofragments as a function of the rotational and the bending-vibrational quantum numbers of parent molecules. In section 2, we present the abovementioned transformation relations. The dependence of the expansion coefficients on the bending coordinate is examined in section 3. In section 4, angular momentum distributions of pdotofragments are derived in analytic form for a direct photodissociation process.
2. Transformation relations between angular basis functions For a direct photodissociation process, the transition amplitude Pf+i is given by the following expression [1,91, pf+i=Cl
(y~‘flt^‘rl?)
I* 3
0009-2614/9 1 /$03.500 I99 1 - Elsevier Science Publishers B.V. ( North-Holland )
(1)
73
Volume 177,number I
CHEMICAL PHYSICSLETTERS
8 February 1991
where 1P,) and 1Yf) are the initial bound state and the final repulsive state wavefunction, respectively; i is the polarization unit vector of the excitation photons and i-r is the transition dipole operator. For a linear triatomic molecule, 1Y,) assumes the form [9]
I~ui~l’~‘>iI~~~Q~~I~~~Q~~l~~~~~~IJ~~~~
(2)
where 1‘C+ ) 1is the ground state electronic wavefunction, 1n,, Q, ) and 1n2, Qz> are the harmonic oscillator wavefunctions with Q, as the totally symmetric stretching mode and Q, as the asymmetric stretching mode. In eq. (2), I u, k, S) is a two-dimensional isotropic harmonic oscillator wavefunction with vibrational angular momentum k along the figure axis and is given by [9] (Kdp exp(
-K*62/2)L’(~‘,k,),2(K282)
,
(3)
where
(
K= rf,lm,
4243 t t&/m,
+ (r,* + r23)2/m2
112
Obend h >
’
(4)
6 (measured in rad) is the angular deviation from the linear configuration (cf. fig. I ) and L$! ,k,j,2 is an associated Laguerre polynomial. In eq. (2), the rotational wavefunction IJkh4) is given by (5) where Dt is a Wigner rotation matrix [ 161. For a repulsive final state, ) Yf) assumes the form [ 91
Fig. I. The geometricconfiguration of a linear triatomic molecule with bond lengths r 12,rz3and masses m,, m2, m,. The atom with mass m, recedes in the photofragmentation. Other parameters are drawn on the graph. 14
Volume 177,number 1
CHEMICALPHYSICS LETTERS
8 February 1991
We have assumed that the excited repulsive state has a ‘C+ symmetry to avoid complications due to the Renner-Teller effect which splits a repulsive, dipole-allowed II electronic state. In eq. (6)) In, Q,2) describes the oscillatory motion of photofragment 12, IEl,, R ) dictates the interfragmental motion, and the angular basis function )$f,‘,y ( i12, 2 ) ) is given by
where p = (2J’+ 1 )“2 and ( : : : ) is a 3-j symbol [ 161; F12and I? denote the orientation of r12 and R in the space-fixed frame (SFF), respectively. From the transformation properties of spherical harmonics [ 161, we have Y~,m,(FL*)= c @&;(~PY)
Y/,m;(7l2) >
(8a)
mi
and L,(~)
= c z&&Br) 4
K*,;(ff’ ) ,
(8b)
where ii2 and l?’ denote the orientation of r,2and R in the molecule-fixed frame (MFF), respectively. Combining eqs. (7), (8a), (8b) and the summation identity of rotation matrices, (9) we have
From the addition theorem of spherical harmonics J;
( _
l)‘l-‘z+q
([I, 12,J’) ml
m2
_k’
[ 17 1, it can be proven that
Y,,m;(7l2) hpI;@
)=(4x)-“%(X)
Y,.,,(i)
,
(11)
where F/,‘,,(x) is a Schulten-Gordon coeffkient, x+,/r, (cf. fig. 1 ), and 1 is along the figure axis which has Euler angles (000) in the SFF. Mathematical properties and the algorithm for the numerical calculation of the Schulten-Gordon coefficients have been explored recently [ 18,191. From the transformation properties of symmetric top wavefunctions #‘, we can ascertain that
I~~~(F12,~))=F:;,*(X)IJ)k)MI).
(12)
Conversely, we have IJ’k’M’> = IF;;F&*(X) l~:;~(~l2,~0
(13)
from the orthogonality relation of the Schulten-Gordon coefftcients. The normalization condition of eq. ( 13) is properly maintained since C,,,~[F~,~(X)]~= 1 [ 191. Eq. ( 12) is the relation which correlates the angular basis functions of the initial and final states. It should be noted that F{h(x) depends parametrically on the #’ See for example eq. (3.124)of ref. [ 161.
75
8 February1991
CHEMICAL PHYSICS LETTERS
Volume 177, number I
bending normal coordinate. This fact implies that both sides of eq. ( 12) have the same number of degrees of freedom as dictated by basic theory.
3. The dependenceof F&Jx) on 6 Before formally integrating the matrix element in eq. ( 1 ), it is appropriate to examine the dependence of the Schulten-Gordon coefficient F&(X) on the bending coordinate 6. Referring to fig. 1, the displacements y,, y2 and y3 are proportional to 6 according to the normal mode analysis of linear triatomic molecules [ 201 and are given by (14)
(15)
and y,=-A, r23m3
where A = RK’/o&~. sin ~=cos
A=
From fig. 1, one has
‘y
=A6
r23m2;f (‘I2+r23)m1 .
(17)
r12r23ml m2
From the law of cosine,
R2= (ST+&-2
(z)
cos(k-b)
(18)
For small amplitude bending oscillations, it can be easily proven that d=
(m, +m2)R m, +m2+m, =
ml +m2 m, +mz+m3
Since sin +y3/d
(
r12m1+r23ml +r23m2 ml +m2
and x=rJr,
rd23m1d2
2(r12ml +r2,m,
(19) +r23m2)
> ’
=sin 8/sin 4, one has the final result
(20) Thus, the ratio x has a second-order dependence on 6. After substituting structural parameters of real systems into the above equation, it is found that the second term makes negligible contribution to X. We conclude that F&,(x) is independent of 6 in every realistic system.
4. Angularmomentumdistributionfunctionsin a directphotodissociationprocess After deriving the transformation relation between the angular basis function and proving the weak dependence of x on 8, it is a straightforward routine to calculate the matrix element in eq. ( 1). For plane-polarized
76
Volume 177, number
1
CHEMICAL PHYSICS LElTERS
8 February 1991
excitation photons and a ‘C++-‘Z+ electronic transition, the transition dipole operator takes the form D&(c~fl[)~~,where rOis the dipole operator in the MFF. Under the Condon approximation, we have pf+i=
1 C(FO)2q
MM
(21) where the result of integration over the product of three rotation matrices and the orthogonality relations of 3-j symbols have been invoked. In the above equation, ( Fo)2= I ( IX+, fl P,,1‘C+, i) (’ is an electronic-transition-matrix element, qE(Z,n;n,n2= [ ( n, Q,2; El,, R 1n,, Q1; n2, Q2) ]* is a two-dimensional Franck-Condon factor which has been thoroughly studied by Freed and co-workers [ 1,9]. Zv,kin eq. (2 1) is given by mK 2[(v-lkl)/2]!
Iv,k=
[I (
[(v+
IQ )/A! )
(Kd)‘k’t?Xp(-K2d2/2)
2’~~+~[~((v+/kl),2+1)]‘[~(,kl,2+l)]’
1 2
1’2
Jt&k,),2(K2d2)
6dd
(22)
where the results of a standard integral have been invoked [ 2 1, p. 8441. f’ and 2Fl in the above equation are the gamma and the confluent hypergeometric function, respectively. In the Airy function approximation [ 11, qE12,nin,n2is independent of the orbital angular momentum quantum number 12.Secondly, (TOO> 2 is a constant parameter for transitions to the same absorption band. Finally, the k dependence of the rotational energy (B[J(Jt 1) - k2] ) of the initial state can be neglected for transitions to a repulsive state. Thus, it is feasible to define the rotational angular momentum distribution function of photofragment 12 as (23) where k= v, v-2, .... - v+2, -v and J’=J- 1, J, Jt 1. The summation over k accounts for the k-degeneracy of the vth bending-vibrational level of the ground state. Similarly, we can define the orbital angular momentum distribution function as (24) with identical confinements on k and J’. Numerical calculations by programming eqs. (23) and (24) and their implications will be published elsewhere.
5. Summary
To calculate efficiently the transition amplitude for the photofragmentation of linear triatomic molecules, the transformation relations between angular basis functions of photofragments and parent moleculesare sought. After examining the dependence of the expansion coefftcients of the transformation on the bending coordinate, we find a complete separation of the angular basis function from the bending-vibrational wavefunction. Thus, the transition amplitude is factored into products of a two-dimensional Franck-Condon factor qE/z,n;n,n2, a function I,, of the bending-vibrational quantum numbers v and k, a square of a 3-j symbol and a square of a Schulten-Gordon coefftcient F{,*(X), while the parameter x depends on the bond lengths and masses of the constituent atoms. Accordingly, the rotational and orbital angular momentum distributions of photofragments 77
Volume 177, number 1
CHEMICALPHYSICSLETTERS
as a function of the rotational and the bending-vibrational
8 February1991
quantum numbers of parent molecules are presented.
Acknowledgement This research was supported by the National Science Council of the Republic of China.
[ I ] K.F. Freedand Y.B.Band,Excitedstates,Vol.3, ed. E.C.Lim (AcademicPress,NewYork,1977)pp. 109-201G [2] M. Shapiroand R. Bersohn,Ann. Rev. Phys. Chem. 33 (1982) 409. [3] S.R. Leone, Advan. Chem. Phys. 50 (1982) 255. [4] P. Bmmer and M. Shapiro, Advan. Chem. Phys. 60 ( 1985) 37 I. [5] G.G. Balint-Kurti and M. Shapiro, Advan. Chem. Phys. 60 (1985) 403. [6] M.N.R. Ashfold and J.E. Baggott, eds., Molecular photodissociation dynamics (The Royal Society of Chemistry, London, 1987). [7] M.D. Morse, K.F. Freed and Y.B. Band, Chem. Phys. Letters 44 (1976) 125. [S] M.D. Morse, K.F. Freed and Y.B. Baud, J. Chem. Phys. 70 (1979) 3604. [9] M.D. Morse and K.F. Freed, J. Chem. Phys. 74 (1981) 4395. [lo] R.W. Heather and J.C. Light, J. Chem. Phys. 78 (1983) 5513. [ 1I ] R.W. Heather and J.C. Light, J. Chem. Phys. 79 (1983) 147. [12] M. Shapiro and G.G. Balint-Kurti, J. Chem. Phys. 71 (1979) 1461. [ 13] G.G. Balint-Kurti and M. Shapiro, Chem. Phys. 6 1 ( I98 1) 137. [ 141I.F. Kidd and G.G. Balint-Kurti, J. Chem. Phys. 82 (1985) 93. [ 151Y.B. Band, K.F. Freed and D.J. Kouri, J. Chem. Phys. 74 (1981) 4380. [ 161R.N. Zarc, Angular momentum (Wiley, New York, 1988). [ 171K. Chen and C. Pei, Chem. Phys. Letters 124 (1986) 365. [ 181K. Chen, Proc. Natl. Sci. Count. 6 ( 1982) 148. [ 19) K. Chen and C. Pei, Chem. Phys. 76 ( 1983) 219. [20] G. He&erg, Molecularspectra and molecular structure, Vol. 2. Infrared and Raman spectra of polyatomic molecules (Van Nostrand Reinhold, New York, 1945) p. 173. [2 1 ] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products (Academic Press, New York, 1965).
78