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Model calculations for the predissociative excitation of CS2
Volume 140. number I
CHEMICAL PHYSICS LETTERS
18 September 1987
MODEL CALCULATIONS FOR THE PREDISSOCIATIVE EXCITATION OF CS2 Konrad T. WU Department of Chemistry, State University of New York, College at Old Westbury, Old Westbury, NY 11568, USA Received 28 April 1987; in final form 7 July 1987
A collinear model for predissociation of linear triatomic molecules is presented. A very steep repulsive potential for CS (A) + S is obtained by fitting the experimental results from the dissociation of CS2 at various excitation energies. Qualitative agreement has also been obtained for the rotational energy distribution of the product fragment. The role of the interfragment effect in determining the CS (A) vibrational distributions is discussed.
1. Introduction Internal energy disposals in CS (A ‘l-I) have recently been investigated by the photon-excitation [ 1] and the metastable-argon-impact excitation [ 21 of C!$, in which the detailed dynamics of the dissociation process have been interpreted. These studies suggest that the Franck-Condon approach for near collinear dissociation via a repulsive potential is the basic model for the excited-state fragmentation of this molecule and that the predissociation occurs in a short-range region of the final repulsive potential that falls steeply as the fragments separate. Since a considerable amount of experimental data for the dissociative excitation of CS2 is available, this system seems to be an excellent candidate on which the dissociation theory of polyatomic molecules can be tested. Among various theoretical treatments of the polyatomic fragmentation dynamics, the collinear model is frequently used for triatomic molecules that have a linear equilibrium configuration in all electronic states involved, as in the case of CS2. The task, however, is to determine which potential-energy surfaces are responsible for controlling the fragment dynamics. A direct comparison between the experimental and theoretical final-state distributions of the molecular fragment provides valuable information regarding the repulsive potential surfaces of the dissociating molecule. It is highly desirable, from an experimentalist’s point of view, to obtain such infor-
mation analytically without laborious theoretical computations. In this paper we apply a simplified collinear model calculation [ 3-71 to the dissociation of CSI leading to the CS (A) formation in order to derive accurate repulsive potential for CS (A) + S and to demonstrate the feasibility of this simplified collinear model for predicting the fragment vibrationalstate distributions of CS2 at various excitation energies. It is hoped that the present model calculations may be used as a reference for future work on the predissociation of other linear triatomic molecules.
2. Collinear model for predissociation Band and Freed [ 3-71 have shown that the transition amplitude for predissociation of a triatomic molecule from the initial bound state to a final repulsive state can be reduced to a sum of one-dimensional bound-continuum Franck-Condon factors in the reaction coordinate for an effective oscillator by using harmonic-oscillator vibrational wavefunctions and the approximation of collinear dissociation. Contributions from initial bending vibration and overall rotation to the transition probability may be incorporated independently and evaluated analytically. The final Franck-Condon factor for the predissociation process is then simplified into a product of a stretching-translation factor and a rotationbending factor. The constraints of energy conserva-
0 009-26 14/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
31
Volume 140, number
1
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tion are contained in the former, while the latter contains the constraints of angular momentum conservation. These two factors can be treated separately.
PHYSICS
LETTERS
18 September
1987
The effective harmonic-oscillator potential representing the initial bound state has a standard form of
that the repulsive potential may be distorted during the recoil mechanism, which may then alter the Franck-Condon overlap integrals. This process is the interfragment effect. Based on the collinear model of Band and Freed [ 31, the inter-fragment effect can be incorporated into the repulsive potential (2) by adding a new potential that is capable of changing the vibrational excitation through Taylor’s expansion to the first order at the equilibrium position of CS (A). Starting with
VQ;)=fk(Q;--cd2,
VQL Qi>=Co ew(-D&.-d
2.1. Stretching-translation factor
(1)
while the repulsive potential describing the recoiling fragments is in the form of
(2) where Q; is the reaction coordinate, k is the normalmode force constant of the triatomic in the intermediate excited state, and r. is the equilibrium distance between the atom and the center of mass of the diatom in the intermediate state. V. and D are adjustable parameters for reproducing the experimental vibrational-state distribution of the product fragment. Since the CS (A) fragment is formed from predissociation of the Rydberg states converging into the ground state of the CS2 ion which retains its original linear geometry [ 21, the CS1 molecule must be initially prepared in its ground vibrational state upon electronic excitation. The Franck-Condon calculations for transition are much simplified for this mechanism because V= 0 is the only initial state that needs to be considered. Subsequent electronic relaxation takes place in the linear configuration via curve crossing, which may promote excitation of the C---S separating bond on a repulsive surface, while the CS (A) oscillator remains bound but may be excited due to a change in the bond length. It is clear that the Franck-Condon overlap integral obtained from the potential functions (1) and (2) is primarily determined by the geometric rearrangement intrafragment effect during the transformation from bound to unbound states. Depending on the role the recoiling S atom plays during separation from the diatomic fragment on the repulsive surface, the CS (A)oscillator can be either excited further or relaxed or vibra(i.e. either translational-+vibrational tionaldtranslational energy transfer). This implies 32
(3)
and then transforming it into the reaction coordinate, results in the final repulsive potential [ 31
VQL Q;> = Vo ew( -De;) x
(
1+
A!fc%Lfs
De;) ’
(4)
where Q; is the normal coordinate of CS (A) in the effective potential. The second term of eq. (4) contributes to distortion of the repulsive potential due to the interfragment effect which may modify the final vibrational distribution of CS (A). The value of Qi can be estimated from the equilibrium position of the effective oscillator at a particular transition energy by a simple relation [ 41
Q;=(RbCS-R~S)+C(Q~-ro>,
(5)
where R!& and R& are the C-S internuclear separations for the initial bound state and the CS (A) fragment, respectively, C is a function of normalmode force constants and atomic masses for CS2 and CS (A). And Q; is the transition point from the bound to the repulsive potential in the reaction coordinate at an available energy E, for which V( Q; = Qi, Q; = 0) = E on the repulsive potential [ 41. Eqs. (4) and (5) show that the interfragment effect may sensitively depend on the position of the transition point Q;;. However, this effect may only be valid at the point at which the bound and the continuum intersect at ~0. We need to extend our application to other instances when the two potentials cross at different regions, Fig. 1 displays different kinds of situations where electronic predissociation may take place from the initial ground vibrational level and demonstrates how we
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CHEMICAL PHYSICS LETTERS
embedded in the dissociation continuum. The interfragment repulsion never acts on the separating fragments and consequently Q;=O. The only contribution to the Franck-Condon factors in this case (IV) is the overlap integrals of the vibrational wavefunctions between the two potential functions (1) and (2).
c
E
18 September 1987
’
w’”
\
2.2. Rotation-bending factor
?lI u\ III
?
II
-r
v=o~--
I
-1
+l--
d:
>
Q; Fig. 1. Representative cases I-IV for the application of the collinear model for predissociation to a linear molecule in the reaction coordinate Q;. Upward arrows indicate the Qk positions for different cases. See text for details.
can use the current model to calculate the Franck-Condon factors for transition. When the excitation energy is sufficiently low as in case I the initial intermediate state does not cross the repulsive potential directly at v=O. The predissociation occurs in the classically forbidden region due to the tunneling effect [ 51. Qb is determined purely on energetic grounds, the same procedure as used in direct dissociation [4]. As the energy increases, the two curves have direct overlap (case II), so that the transition point Qh can be easily defined. After the transition takes place from the bound state to the continuum, the molecule flies apart immediately. The relative energies of the effective oscillator at Qi and various points on the repulsive curve corresponding to the different vibrational levels of CS (A) have no real physical significance. The Franck-Condon factors for transition are calculated from the overlap of vibrational wavefunctions between the bound and the modified repulsive potentials. The transition from bound to continuum takes place at the equilibrium position of the effective oscillator in case III, and Q; is thus equal to R& -R&. For case IV, the effective oscillator is
The results of the collinear model in the dissociation of triatomic molecules, calculated from the stretching-translation factor alone, may sometimes be significantly modified by the effects of parentmolecule bending and overall rotation of the triatomic system [ 5-91. Analytical formulas for evaluating these effects in the collinear model are readily available [ 5-71 and are used directly for the present calculations. As a good approximation, the effect of dynamic axis switching [7] is ignored; analytical expressions can then be simplified. Since lower levels of the bending mode have favorable populations at thermal temperature, only the 0’ and 1’ states need to be considered in the actual calculation. The thermal bending populations for CS: are 0’ = 8 1.4% and 1’ = 15.2%. Due to the close spacing of the rotational energy levels, the rotational distribution may be strongly influenced by final-state interactions (rotational energy transfer). The calculation of such an effect is extremely difficult to perform because so many final states are available. Fortunately, the final rotational distribution of CS( A) produced for photodissociation of CS2 at 130.4 nm [ 1 ] was found not to be significantly affected by final-state interactions but to be of Boltzmann character to accommodate the constraints of energy and total angular momentum conservations. Therefore, the interfragment effect due to rotation is not considered in the present model calculations.
3. Calculations and results The intermediate bound state is previously defined as a Rydberg state converging to the ground state of cs,+ ; its spectroscopic constants are therefore assumed to be identical to those of CS,’ (x). They 33
CHEMICAL PHYSICS LETTERS
Volume 140, number 1
Table 1 Transition point between potential curves, Q;, and the normal coordinate of CS (A), Q;, for the predissociation of CS2 at various available energies _??
18 September 1987
NJ’)
:m A B C D E
.I?(cm-‘)
Qi(h
lo2Qi(h
1900 2425 6100 7430 18750
2.686 2.680 2.658 a) a)
0.5973 0.4478 -0.1004 0 0
” The effective potential is embedded continuum.
t/
in the dissociation
are obtained from refs. [ 10,111 to calculate the parameters for characterizing the effective potential: L5.8~ lo5 dyn/cm and r0=2.7014 A. In order to find the best parameters V0and D for the repulsive potential, we first ignore both the interfragment effect of vibration and the rotation-bending factor. V. and D are varied initially until the Franck-Condon transition amplitudes due to the rotationless stretchingtranslation factor alone, calculated from the ground vibrational level (v= 0, J= 0) of the potential (1) to various energy levels corresponding to CS (A, v’, S ~0) on the repulsive potential (2), tit well to the experimental results [ 1,2] at different excitation energies. Values of the V0and D parameters are then adjusted until the absorption amplitudes calculated for a transition between the effective potential of CS2 (x) and the repulsive potential function (2), based on the Franck-Condon principle, also fit well with the absorption spectrum or the fluorescence excitation spectrum [ 121. A good fit is not always obtainable due to the diffusive nature of the absorption spectrum or the fluorescence excitation spectrum involving predissociation; however, it usually provides guidance for the selection of better parameters for V,andD [13]. After these two potential functions are defined, contributions from the interfragment effect due to vibration can be included in the repulsive potential curve. Table 1 lists values of Q; as a function of Q;, calculated from eq. (5)) for the predissociative excitation of CSI at various available energies. The spectroscopic constants required for such calculations are from refs. [ IO,1 1,141. The values of Qi change gradually from positive to negative as Qi 34
Ql
OO_
J’ Fig. 2. Relative rotational distribution for CS2 photodissociation from the rotation-bending contribution alone. (A) J=O and 30 for 0” and 1’ bending vibrations in the initial bound state. (B) After thermal averaging over J.
decreases. This suggests that there is a crossing point in between (Q&2.662 A) at which the C-S internuclear distance in the effective oscillator is equivalent to the equilibrium distance of CS (A). Thus when the dissociation occurs, the transformation from bound to continuum and then to separating fragments becomes such a natural process that there involves no other energy conversion mechanism than the one arising from the intrafragment effect. This result indicates that the vibrational inter-fragment effect is a consequence of the change in spectroscopic properties of the dissociating molecule. The rotation-bending contribution to the Franck-Condon transition amplitudes is then evaluated by using the analytical formulas presented in refs. [ 5-71. The fragment rotation distributions, arising from the rotation-bending effect for the 0’ and 1’ bending modes in the initial bound state, are shown in fig. 2 for the initial rotationless state (J= 0) and the most probable state at 300 K (J= 30) in the photodissociation of CS, at 130.4 nm. Information
18 September 1987
CHEMICAL PHYSICS LETTERS
Volume 140, number 1
CS,-CS(A’n)+S
CS,-+CS(A,v’)+S
0 V’
0
I
I
I
10
20
30
V’
2
4
6
V’
Fig. 4. Observed ( * ) and calculated (0 ) vibrational distributions as a function of vibrational quantum number v’ of CS (A, v’) at various available energies: (A) 1900 cm-‘; (B) 2425 cm-‘; (C) 6100 cm-i;(D) 743Ocm-‘; (E) 18750 cm-‘. Distributions are normalized to unity. I
40
50
J’ Fig. 3. Dependence of the stretching-translation factor ( II’,,,) on J’ for the photodissociation of CS2 at 130.4 nm. Data are obtained from J= 30.
on J= 0 is particularly useful for it provides a measure of the bending contribution from the parent molecule. The final rotation-bending factor is obtained by thermal averaging over the total angular momentum J in the initial bound state. The resulting distribution is also shown in fig. 2; the rotational distribution is peaked at J’ = 30, as would be predicted by the angular momentum conservation. In order to calculate the accurate Franck-Condon transition amplitude for the predissociation of CS2 the dependence of the stretching-translation factor on J and J’ has to be known. Fig. 3 shows the variation of the stretching-translation factor ( H’i+f) as a function J’ for the photodissociation of CS2 (J= 30) at 130.4 nm. This dependence clearly demonstrates the constraints of energy conservation on Franck-Condon transition amplitudes. The structure displayed in fig. 3 is due to the presence of nodes in the vibrational wavefunction on the repulsive surface of CS (A) + S. The same kind of structure is obtained for various initial J states. Their magnitudes change by less than 2W from J= 0 to J= 40. This indicates that the dependence of the rotational distribution on the initial J state of the molecule is so slight that detailed calculations for the stretching-
translation factor by thermal averaging over J is not necessary. The stretching-translation factors are therefore calculated from J=30 for various excitation energies in the present study. The final values of V, and D are chosen when the Franck-Condon transition amplitudes calculated from potential functions (1) and (4)) including the rotation-bending contribution, fit well with the experimental results at all energies. Only minor adjustments for V, and D are needed at this point. The best parameters thus obtained are V, = 3.2 x 105’ cm- ’ and D = 42.2 A- ‘. These parameters represent a very steep potential function. When a series of different sets of potential parameters for a shallow potential were used, the broad experimental vibrational distributions could never be reproduced. This indicates that the repulsive potential responsible for the dissociation of CS2 is indeed very steep, in agreement with our previous findings [ 21. The relative vibrational distributions calculated from the collinear model at various energies are compared with the experimental results as shown in fig. 4. It is interesting to note that these vibrational distributions are more sensitive to the bending contribution of the parent molecule and that inclusion of rotation has less effect. Similar observation has also been reported by Pattengill [ 81 in his classical trajectory study of collinear dissociation of ICN. Calculated vibrational distributions at i?= 7430 and 18750 cm-’ (figs. 4D and 4E, respectively) are obtained without considering the interfragment 35
Volume 140, number 1
CHEMICAL PHYSICS LETTERS
18 September 1987
4. Conclusion
7
p(J’)
t’ /’
0
I
I
I
I
I
I
I
10
20
30
40
so
60
70
J
.,
Fig. 5. Observed (-) and calculated (-) rotational distributions of CS (A, v’ =O) produced from photodissociation of CS2 at 130.4 nm. The non-smooth portion of the calculated function is due to the nodal structure present in the vibrational wavefunction of the repulsive potential curve.
effect. As demonstrated in tig. 1, the vibrational distributions remain unchanged with further increase in energy because the effective harmonic oscillator is embedded in the continuum at high energy. This prediction is consistent with the experimental observations that the CS (A+X) fluorescence persists with similar vibrational distributions when the incident photon energy applying to CS2 is further increased [ 2,151. The close agreement between theory and experiment suggests that the collinear model is suitable for interpreting the dissociation dynamics of cs*. It is also interesting to compare the final rotational distribution of CS (A), using the above potential parameters, with the experimental result obtained from the photodissociation of CS2 at 130.4 nm [ 11. As can be seen in fig. 5, they are in qualitative agreement. Since J’ x 50 is the highest rotational level that is allowed by the energy requirement, the broader rotational distribution observed experimentally may arise from the effect of final-state interactions due to J-d transfer. It is believed that a better fit could be obtained by smoothing out the variation of the stretching-translation factor with J’ by thermal averaging over J in the initial bound state and more importantly by taking into consideration the Iinalstate interaction due to rotation. This result (fig. 5) provides additional support for the shape of the repulsive potential curve obtained from the present study. 36
This study clearly demonstrates that the present collinear model for predissociation is useful for reproducing the experimental vibrational distributions of the product fragment for a linear triatomic molecule. The repulsive potential curve that is responsible for the dissociative excitation of CS2 has been proposed. This is an important step towards a complete understanding of the excited-state fragmentation of C!$. It is hoped that the present results will provide useful information for future theoretical studies in order to reconstruct the accurate potentialenergy surface for elucidating the predissociative excitation mechanism of CS2. We have also successfully applied the present model to the interpretation of the dynamics of the dissociative excitation of N20 leading to the formation of N2 (B) [ 131. This indicates that the present model may have a wide range of applicability in molecular systems that have a collinear dissociation configuration. Future work will involve a systematic approach to the linear triatomic molecular family in order to gain further insights into the dissociation dynamics. References [l] M.N.R. Ashfold, A.M. Quinton and J.P. Simons, J. Chem. Sot. Faraday Trans. II 76 (1980) 905. [2] K.T. Wu, J.Phys. Chem. 89 (1985) 4617. [ 31 Y.B. Band and K.F. Freed, J. Chem. Phys. 63 (1975) 3382. [4] Y.B. Band and K.F. Freed, J. Chem. Phys. 67 (1977) 1462. [ 51M.D. Morse, K.F. Freed and Y.B. Band, J. Chem. Phys. 70 (1979) 3604,362O. [ 61 M.D. Morse, K.F. Freed and Y.B. Band, Chem. Phys. Letters44 (1976) 125. [7] M.D. M0rseandK.F. Freed, J. Chem. Phys. 74 (1981) 4395. [ 8 ] M.D. Pattengill, Chem. Phys. 78 (1983) 229. [9] M.D. Pattengill, Chem. Phys. 87 (1984) 419. [ 10] G. Hersberg, Molecular spectra and molecular structure, Vol. 3. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand Reinhold, New York, 1966). [ 111 L.A. Pugh and K.N. Rao, in: Molecular spectroscopy: modern research, Vol. II, ed. K.N. Rao (Academic Press, New York, 1976) ch. 4. [ 121 H.J. Okabe, J. Chem. Phys. 56 (1972) 4381. [ 131 K.T. Wu, submitted for publication. [ 141 K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [ 151 L.C. Lee and D.L. Judge, J. Chem. Phys. 63 (1975) 2782.
CHEMICAL PHYSICS LETTERS
18 September 1987
MODEL CALCULATIONS FOR THE PREDISSOCIATIVE EXCITATION OF CS2 Konrad T. WU Department of Chemistry, State University of New York, College at Old Westbury, Old Westbury, NY 11568, USA Received 28 April 1987; in final form 7 July 1987
A collinear model for predissociation of linear triatomic molecules is presented. A very steep repulsive potential for CS (A) + S is obtained by fitting the experimental results from the dissociation of CS2 at various excitation energies. Qualitative agreement has also been obtained for the rotational energy distribution of the product fragment. The role of the interfragment effect in determining the CS (A) vibrational distributions is discussed.
1. Introduction Internal energy disposals in CS (A ‘l-I) have recently been investigated by the photon-excitation [ 1] and the metastable-argon-impact excitation [ 21 of C!$, in which the detailed dynamics of the dissociation process have been interpreted. These studies suggest that the Franck-Condon approach for near collinear dissociation via a repulsive potential is the basic model for the excited-state fragmentation of this molecule and that the predissociation occurs in a short-range region of the final repulsive potential that falls steeply as the fragments separate. Since a considerable amount of experimental data for the dissociative excitation of CS2 is available, this system seems to be an excellent candidate on which the dissociation theory of polyatomic molecules can be tested. Among various theoretical treatments of the polyatomic fragmentation dynamics, the collinear model is frequently used for triatomic molecules that have a linear equilibrium configuration in all electronic states involved, as in the case of CS2. The task, however, is to determine which potential-energy surfaces are responsible for controlling the fragment dynamics. A direct comparison between the experimental and theoretical final-state distributions of the molecular fragment provides valuable information regarding the repulsive potential surfaces of the dissociating molecule. It is highly desirable, from an experimentalist’s point of view, to obtain such infor-
mation analytically without laborious theoretical computations. In this paper we apply a simplified collinear model calculation [ 3-71 to the dissociation of CSI leading to the CS (A) formation in order to derive accurate repulsive potential for CS (A) + S and to demonstrate the feasibility of this simplified collinear model for predicting the fragment vibrationalstate distributions of CS2 at various excitation energies. It is hoped that the present model calculations may be used as a reference for future work on the predissociation of other linear triatomic molecules.
2. Collinear model for predissociation Band and Freed [ 3-71 have shown that the transition amplitude for predissociation of a triatomic molecule from the initial bound state to a final repulsive state can be reduced to a sum of one-dimensional bound-continuum Franck-Condon factors in the reaction coordinate for an effective oscillator by using harmonic-oscillator vibrational wavefunctions and the approximation of collinear dissociation. Contributions from initial bending vibration and overall rotation to the transition probability may be incorporated independently and evaluated analytically. The final Franck-Condon factor for the predissociation process is then simplified into a product of a stretching-translation factor and a rotationbending factor. The constraints of energy conserva-
0 009-26 14/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
31
Volume 140, number
1
CHEMICAL
tion are contained in the former, while the latter contains the constraints of angular momentum conservation. These two factors can be treated separately.
PHYSICS
LETTERS
18 September
1987
The effective harmonic-oscillator potential representing the initial bound state has a standard form of
that the repulsive potential may be distorted during the recoil mechanism, which may then alter the Franck-Condon overlap integrals. This process is the interfragment effect. Based on the collinear model of Band and Freed [ 31, the inter-fragment effect can be incorporated into the repulsive potential (2) by adding a new potential that is capable of changing the vibrational excitation through Taylor’s expansion to the first order at the equilibrium position of CS (A). Starting with
VQ;)=fk(Q;--cd2,
VQL Qi>=Co ew(-D&.-d
2.1. Stretching-translation factor
(1)
while the repulsive potential describing the recoiling fragments is in the form of
(2) where Q; is the reaction coordinate, k is the normalmode force constant of the triatomic in the intermediate excited state, and r. is the equilibrium distance between the atom and the center of mass of the diatom in the intermediate state. V. and D are adjustable parameters for reproducing the experimental vibrational-state distribution of the product fragment. Since the CS (A) fragment is formed from predissociation of the Rydberg states converging into the ground state of the CS2 ion which retains its original linear geometry [ 21, the CS1 molecule must be initially prepared in its ground vibrational state upon electronic excitation. The Franck-Condon calculations for transition are much simplified for this mechanism because V= 0 is the only initial state that needs to be considered. Subsequent electronic relaxation takes place in the linear configuration via curve crossing, which may promote excitation of the C---S separating bond on a repulsive surface, while the CS (A) oscillator remains bound but may be excited due to a change in the bond length. It is clear that the Franck-Condon overlap integral obtained from the potential functions (1) and (2) is primarily determined by the geometric rearrangement intrafragment effect during the transformation from bound to unbound states. Depending on the role the recoiling S atom plays during separation from the diatomic fragment on the repulsive surface, the CS (A)oscillator can be either excited further or relaxed or vibra(i.e. either translational-+vibrational tionaldtranslational energy transfer). This implies 32
(3)
and then transforming it into the reaction coordinate, results in the final repulsive potential [ 31
VQL Q;> = Vo ew( -De;) x
(
1+
A!fc%Lfs
De;) ’
(4)
where Q; is the normal coordinate of CS (A) in the effective potential. The second term of eq. (4) contributes to distortion of the repulsive potential due to the interfragment effect which may modify the final vibrational distribution of CS (A). The value of Qi can be estimated from the equilibrium position of the effective oscillator at a particular transition energy by a simple relation [ 41
Q;=(RbCS-R~S)+C(Q~-ro>,
(5)
where R!& and R& are the C-S internuclear separations for the initial bound state and the CS (A) fragment, respectively, C is a function of normalmode force constants and atomic masses for CS2 and CS (A). And Q; is the transition point from the bound to the repulsive potential in the reaction coordinate at an available energy E, for which V( Q; = Qi, Q; = 0) = E on the repulsive potential [ 41. Eqs. (4) and (5) show that the interfragment effect may sensitively depend on the position of the transition point Q;;. However, this effect may only be valid at the point at which the bound and the continuum intersect at ~0. We need to extend our application to other instances when the two potentials cross at different regions, Fig. 1 displays different kinds of situations where electronic predissociation may take place from the initial ground vibrational level and demonstrates how we
140, number 1
CHEMICAL PHYSICS LETTERS
embedded in the dissociation continuum. The interfragment repulsion never acts on the separating fragments and consequently Q;=O. The only contribution to the Franck-Condon factors in this case (IV) is the overlap integrals of the vibrational wavefunctions between the two potential functions (1) and (2).
c
E
18 September 1987
’
w’”
\
2.2. Rotation-bending factor
?lI u\ III
?
II
-r
v=o~--
I
-1
+l--
d:
>
Q; Fig. 1. Representative cases I-IV for the application of the collinear model for predissociation to a linear molecule in the reaction coordinate Q;. Upward arrows indicate the Qk positions for different cases. See text for details.
can use the current model to calculate the Franck-Condon factors for transition. When the excitation energy is sufficiently low as in case I the initial intermediate state does not cross the repulsive potential directly at v=O. The predissociation occurs in the classically forbidden region due to the tunneling effect [ 51. Qb is determined purely on energetic grounds, the same procedure as used in direct dissociation [4]. As the energy increases, the two curves have direct overlap (case II), so that the transition point Qh can be easily defined. After the transition takes place from the bound state to the continuum, the molecule flies apart immediately. The relative energies of the effective oscillator at Qi and various points on the repulsive curve corresponding to the different vibrational levels of CS (A) have no real physical significance. The Franck-Condon factors for transition are calculated from the overlap of vibrational wavefunctions between the bound and the modified repulsive potentials. The transition from bound to continuum takes place at the equilibrium position of the effective oscillator in case III, and Q; is thus equal to R& -R&. For case IV, the effective oscillator is
The results of the collinear model in the dissociation of triatomic molecules, calculated from the stretching-translation factor alone, may sometimes be significantly modified by the effects of parentmolecule bending and overall rotation of the triatomic system [ 5-91. Analytical formulas for evaluating these effects in the collinear model are readily available [ 5-71 and are used directly for the present calculations. As a good approximation, the effect of dynamic axis switching [7] is ignored; analytical expressions can then be simplified. Since lower levels of the bending mode have favorable populations at thermal temperature, only the 0’ and 1’ states need to be considered in the actual calculation. The thermal bending populations for CS: are 0’ = 8 1.4% and 1’ = 15.2%. Due to the close spacing of the rotational energy levels, the rotational distribution may be strongly influenced by final-state interactions (rotational energy transfer). The calculation of such an effect is extremely difficult to perform because so many final states are available. Fortunately, the final rotational distribution of CS( A) produced for photodissociation of CS2 at 130.4 nm [ 1 ] was found not to be significantly affected by final-state interactions but to be of Boltzmann character to accommodate the constraints of energy and total angular momentum conservations. Therefore, the interfragment effect due to rotation is not considered in the present model calculations.
3. Calculations and results The intermediate bound state is previously defined as a Rydberg state converging to the ground state of cs,+ ; its spectroscopic constants are therefore assumed to be identical to those of CS,’ (x). They 33
CHEMICAL PHYSICS LETTERS
Volume 140, number 1
Table 1 Transition point between potential curves, Q;, and the normal coordinate of CS (A), Q;, for the predissociation of CS2 at various available energies _??
18 September 1987
NJ’)
:m A B C D E
.I?(cm-‘)
Qi(h
lo2Qi(h
1900 2425 6100 7430 18750
2.686 2.680 2.658 a) a)
0.5973 0.4478 -0.1004 0 0
” The effective potential is embedded continuum.
t/
in the dissociation
are obtained from refs. [ 10,111 to calculate the parameters for characterizing the effective potential: L5.8~ lo5 dyn/cm and r0=2.7014 A. In order to find the best parameters V0and D for the repulsive potential, we first ignore both the interfragment effect of vibration and the rotation-bending factor. V. and D are varied initially until the Franck-Condon transition amplitudes due to the rotationless stretchingtranslation factor alone, calculated from the ground vibrational level (v= 0, J= 0) of the potential (1) to various energy levels corresponding to CS (A, v’, S ~0) on the repulsive potential (2), tit well to the experimental results [ 1,2] at different excitation energies. Values of the V0and D parameters are then adjusted until the absorption amplitudes calculated for a transition between the effective potential of CS2 (x) and the repulsive potential function (2), based on the Franck-Condon principle, also fit well with the absorption spectrum or the fluorescence excitation spectrum [ 121. A good fit is not always obtainable due to the diffusive nature of the absorption spectrum or the fluorescence excitation spectrum involving predissociation; however, it usually provides guidance for the selection of better parameters for V,andD [13]. After these two potential functions are defined, contributions from the interfragment effect due to vibration can be included in the repulsive potential curve. Table 1 lists values of Q; as a function of Q;, calculated from eq. (5)) for the predissociative excitation of CSI at various available energies. The spectroscopic constants required for such calculations are from refs. [ IO,1 1,141. The values of Qi change gradually from positive to negative as Qi 34
Ql
OO_
J’ Fig. 2. Relative rotational distribution for CS2 photodissociation from the rotation-bending contribution alone. (A) J=O and 30 for 0” and 1’ bending vibrations in the initial bound state. (B) After thermal averaging over J.
decreases. This suggests that there is a crossing point in between (Q&2.662 A) at which the C-S internuclear distance in the effective oscillator is equivalent to the equilibrium distance of CS (A). Thus when the dissociation occurs, the transformation from bound to continuum and then to separating fragments becomes such a natural process that there involves no other energy conversion mechanism than the one arising from the intrafragment effect. This result indicates that the vibrational inter-fragment effect is a consequence of the change in spectroscopic properties of the dissociating molecule. The rotation-bending contribution to the Franck-Condon transition amplitudes is then evaluated by using the analytical formulas presented in refs. [ 5-71. The fragment rotation distributions, arising from the rotation-bending effect for the 0’ and 1’ bending modes in the initial bound state, are shown in fig. 2 for the initial rotationless state (J= 0) and the most probable state at 300 K (J= 30) in the photodissociation of CS, at 130.4 nm. Information
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CS,-CS(A’n)+S
CS,-+CS(A,v’)+S
0 V’
0
I
I
I
10
20
30
V’
2
4
6
V’
Fig. 4. Observed ( * ) and calculated (0 ) vibrational distributions as a function of vibrational quantum number v’ of CS (A, v’) at various available energies: (A) 1900 cm-‘; (B) 2425 cm-‘; (C) 6100 cm-i;(D) 743Ocm-‘; (E) 18750 cm-‘. Distributions are normalized to unity. I
40
50
J’ Fig. 3. Dependence of the stretching-translation factor ( II’,,,) on J’ for the photodissociation of CS2 at 130.4 nm. Data are obtained from J= 30.
on J= 0 is particularly useful for it provides a measure of the bending contribution from the parent molecule. The final rotation-bending factor is obtained by thermal averaging over the total angular momentum J in the initial bound state. The resulting distribution is also shown in fig. 2; the rotational distribution is peaked at J’ = 30, as would be predicted by the angular momentum conservation. In order to calculate the accurate Franck-Condon transition amplitude for the predissociation of CS2 the dependence of the stretching-translation factor on J and J’ has to be known. Fig. 3 shows the variation of the stretching-translation factor ( H’i+f) as a function J’ for the photodissociation of CS2 (J= 30) at 130.4 nm. This dependence clearly demonstrates the constraints of energy conservation on Franck-Condon transition amplitudes. The structure displayed in fig. 3 is due to the presence of nodes in the vibrational wavefunction on the repulsive surface of CS (A) + S. The same kind of structure is obtained for various initial J states. Their magnitudes change by less than 2W from J= 0 to J= 40. This indicates that the dependence of the rotational distribution on the initial J state of the molecule is so slight that detailed calculations for the stretching-
translation factor by thermal averaging over J is not necessary. The stretching-translation factors are therefore calculated from J=30 for various excitation energies in the present study. The final values of V, and D are chosen when the Franck-Condon transition amplitudes calculated from potential functions (1) and (4)) including the rotation-bending contribution, fit well with the experimental results at all energies. Only minor adjustments for V, and D are needed at this point. The best parameters thus obtained are V, = 3.2 x 105’ cm- ’ and D = 42.2 A- ‘. These parameters represent a very steep potential function. When a series of different sets of potential parameters for a shallow potential were used, the broad experimental vibrational distributions could never be reproduced. This indicates that the repulsive potential responsible for the dissociation of CS2 is indeed very steep, in agreement with our previous findings [ 21. The relative vibrational distributions calculated from the collinear model at various energies are compared with the experimental results as shown in fig. 4. It is interesting to note that these vibrational distributions are more sensitive to the bending contribution of the parent molecule and that inclusion of rotation has less effect. Similar observation has also been reported by Pattengill [ 81 in his classical trajectory study of collinear dissociation of ICN. Calculated vibrational distributions at i?= 7430 and 18750 cm-’ (figs. 4D and 4E, respectively) are obtained without considering the interfragment 35
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4. Conclusion
7
p(J’)
t’ /’
0
I
I
I
I
I
I
I
10
20
30
40
so
60
70
J
.,
Fig. 5. Observed (-) and calculated (-) rotational distributions of CS (A, v’ =O) produced from photodissociation of CS2 at 130.4 nm. The non-smooth portion of the calculated function is due to the nodal structure present in the vibrational wavefunction of the repulsive potential curve.
effect. As demonstrated in tig. 1, the vibrational distributions remain unchanged with further increase in energy because the effective harmonic oscillator is embedded in the continuum at high energy. This prediction is consistent with the experimental observations that the CS (A+X) fluorescence persists with similar vibrational distributions when the incident photon energy applying to CS2 is further increased [ 2,151. The close agreement between theory and experiment suggests that the collinear model is suitable for interpreting the dissociation dynamics of cs*. It is also interesting to compare the final rotational distribution of CS (A), using the above potential parameters, with the experimental result obtained from the photodissociation of CS2 at 130.4 nm [ 11. As can be seen in fig. 5, they are in qualitative agreement. Since J’ x 50 is the highest rotational level that is allowed by the energy requirement, the broader rotational distribution observed experimentally may arise from the effect of final-state interactions due to J-d transfer. It is believed that a better fit could be obtained by smoothing out the variation of the stretching-translation factor with J’ by thermal averaging over J in the initial bound state and more importantly by taking into consideration the Iinalstate interaction due to rotation. This result (fig. 5) provides additional support for the shape of the repulsive potential curve obtained from the present study. 36
This study clearly demonstrates that the present collinear model for predissociation is useful for reproducing the experimental vibrational distributions of the product fragment for a linear triatomic molecule. The repulsive potential curve that is responsible for the dissociative excitation of CS2 has been proposed. This is an important step towards a complete understanding of the excited-state fragmentation of C!$. It is hoped that the present results will provide useful information for future theoretical studies in order to reconstruct the accurate potentialenergy surface for elucidating the predissociative excitation mechanism of CS2. We have also successfully applied the present model to the interpretation of the dynamics of the dissociative excitation of N20 leading to the formation of N2 (B) [ 131. This indicates that the present model may have a wide range of applicability in molecular systems that have a collinear dissociation configuration. Future work will involve a systematic approach to the linear triatomic molecular family in order to gain further insights into the dissociation dynamics. References [l] M.N.R. Ashfold, A.M. Quinton and J.P. Simons, J. Chem. Sot. Faraday Trans. II 76 (1980) 905. [2] K.T. Wu, J.Phys. Chem. 89 (1985) 4617. [ 31 Y.B. Band and K.F. Freed, J. Chem. Phys. 63 (1975) 3382. [4] Y.B. Band and K.F. Freed, J. Chem. Phys. 67 (1977) 1462. [ 51M.D. Morse, K.F. Freed and Y.B. Band, J. Chem. Phys. 70 (1979) 3604,362O. [ 61 M.D. Morse, K.F. Freed and Y.B. Band, Chem. Phys. Letters44 (1976) 125. [7] M.D. M0rseandK.F. Freed, J. Chem. Phys. 74 (1981) 4395. [ 8 ] M.D. Pattengill, Chem. Phys. 78 (1983) 229. [9] M.D. Pattengill, Chem. Phys. 87 (1984) 419. [ 10] G. Hersberg, Molecular spectra and molecular structure, Vol. 3. Electronic spectra and electronic structure of polyatomic molecules (Van Nostrand Reinhold, New York, 1966). [ 111 L.A. Pugh and K.N. Rao, in: Molecular spectroscopy: modern research, Vol. II, ed. K.N. Rao (Academic Press, New York, 1976) ch. 4. [ 121 H.J. Okabe, J. Chem. Phys. 56 (1972) 4381. [ 131 K.T. Wu, submitted for publication. [ 141 K.P. Huber and G. Herzberg, Molecular spectra and molecular structure, Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). [ 151 L.C. Lee and D.L. Judge, J. Chem. Phys. 63 (1975) 2782.