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Influence of initial state bend—stretch couplings on product rotational distributions in photodissociation of bent triatomic molecules
Volume 182, number 3,4
CHEMICAL PHYSICS LETTERS
2 August 1991
Influence of initial state bend-stretch couplings on product rotational distributions in photodissociation of bent triatomic molecules Horatio
Grinberg ‘, Karl F. Freed
James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL 60637, US4
and Carl J. Williams Department of Chemistry, University ofNotre Dame, Notre Dame, IN 46556, VSA Received 30 March 1991; in final form 8 May 1991
Even when a simple valence force field is used to represent the bound state normal modes, the kinetic energy operator couples bending and stretching motions in bent triatomic molecules. These bend-stretch couplings have been ignored in previous theoretical descriptions of photodissociation and are shown here to significantly alter predicted rotationalenergydistributions. Examples consider model HCN and NOCl photodissociations from bent initial states.
1. Introduction The influence of the initially prepared parent states, the dissociative potential energy surfaces, and the excitation wavelength on product energy and angular distributions are crucial to understanding of photodissociation dynamics. Advances in laser technology enable the determination of final state distributions for a variety of species and photodissociation processes. Many experiments measure the product rotational energy distributions that are predicted theoretically to display a strong sensitivity to photodissociation dynamics and initial state selection [ l-31. However, despite these experimental advances, detailed theoretical understanding of product rotational distributions, even in triatomic photodissociations, is still rather limited. Two general theoretical approaches have been used for describing photodissociation dynamics. The first ’ Permanent address: Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina.
is based on advances in quantum inelastic scattering methods (both time dependent and independent) using ab initio or empirical potential surfaces and perhaps various tested approximations to the full collision dynamics [ 4-8 1. Second, alternative simple analytically tractable models exist in order to elucidate general qualitative trends, to produce computationally economical models for testing the adequacy of various dissociative potential energy surfaces, and to enable tests of assumptions inherent in full three-dimensional scattering calculations for photodissociation dynamics. Although the non-expert may assume that full three-dimensional scattering calculations are rigorous in describing photodissociation dynamics, most calculations contain a number of assumptions, including the neglect of electronic and spin angular momenta, the neglect of coupling to additional nearby electronic surfaces, and the simplified treatment of both the bound and dissociative state wavefunctions in the same single set of coordinate and rotational axis systems. It is the important testing of some of these assumptions that we consider here using our analytical three-dimen-
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sional rotationally infinite-order sudden (10s) treatment of photodissociation energy distribution [9,101. While complete three-dimensional coupled-state scattering calculations are possible for triatomic photodissociations, the expense of these calculations, the inaccuracy of even the best ab initio potential surfaces, and the availability of limited experimental data have led to the introduction of standard approximations, such as coupled states and rotational 10s methods [ 11,121, in order to simplify the otherwise enormous computational labor. Nevertheless, until recently, the majority of approximate scattering treatments of photodissociation dynamics have also been restricted to rotationless (J=O) initial states and therefore do not address the interesting questions of the partitioning of initial parent state excitation (J, Kf 0) into product rotational and angular distributions that is easily described using our analytical theory [9,10]. As mentioned above, full-quantum close-coupled calculations introduce a series of somewhat less obvious approximations some of which are analyzed here using our analytical quantum theory. The computational approaches generally employ the same Jacobi scattering coordinates for both the bound and dissociative potential surfaces. These familiar Jacobi coordinates are the fragment diatomic internuclear separation r, the distance R of the fragment atom from the diatomic center of mass, and the angle 0 between the vectors I and R. Thus, the initial bound state vibrational functions are expressed as basis set expansions that are separable in the Jacobi coordinates. Such an approximation can introduce severe errors and additionally becomes increasingly unwieldy as the size of the molecule grows further (tetra-atomics requiring six-dimensional expansions). In order to simplify the evaluation of threedimensional bound-continuum Franck-Condon factors the computations further introduce an approximation, which we term the neglect of bendstretch coupling, in which the bound state stretching modes are assumed to be functions only of r and Rj while the bend is a pure function of 8. This assumption is in stark contrast to standard normal mode analyses of bent triatomic molecules where the bends and stretches are functions of all three coordinates (hence the term bend-stretch coupling). When 298
2 August1991
bend-stretch couplings are ignored in spectroscopic applications, the bound state bending is more appropriately represented in terms ofthe triatomic bond angle 6, which may differ significantly from the Jacobi scattering coordinate 0 used to describe the bound state bending in many photodissociation models. This use of a common coordinate system for both initial and final potential energy surfaces implies several other implicit assumptions, such as the neglect of axis switching (the difference between the natural rotating frames for the molecule on the bound and dissociative surfaces). Our analytical 10s photodissociation theory is used in this paper to study the errors associated with introducing the bend-stretch approximation into the theory of bent triatomic molecule photodissociation. Model potential surfaces are used for NOCl T, ( 1 3A” ) [ 131 and HCN (c ‘A’) state [ 5 ] direct dissociations, and comparisons are made between computations that include or ignore the bend-stretch couplings. These couplings may be treated by the analytical quantum theory because this method uses two different coordinate systems, one appropriate to the bound and the second appropriate to the dissociative surface. This improvement, however, leads, in general, to completely nonseparable three-dimensional scattering amplitudes that are resolved in our theory by using the IOS approximation and an Airy function for the continuum wavefunction. The scattering amplitudes are thereby transformed into one-dimensional integrals that are represented analytically using efficient quadrature methods. The analytical nature of the theory enables the rapid computation of product distributions for given initial vibrational and rotational states of the parent molecule. We should note that Schinke’s well known and useful rotational reflection principle [ 14- 161 emerges as an approximation to the analytical quantum theory upon the introduction of the following series of approximations: ( I) Bend-stretch couplings are ignored and the initial bound state vibrations are assumed to be separable in r, R, and 0. (2) Axis switching and the nonlinear transformation between the natural coordinates for the bound and repulsive surfaces are ignored. (3 ) Classical models are frequently used for the
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CHEMICAL PHYSICS LETTERS
2 August 199 I
rotational scattering, although a quantum analog is also available [ 171. (4) The initial state is taken to be rotationless. A simple one-dimensional form of the rotational reflection principle has been shown by Schinke [ 161 to exhibit the initial bending state dependence of the rotational energy distribution that is predicted by Morse and Freed [ 18 1.The earlier version of the analytical theory [ 181 ignores the bend-stretch couplings and assumes an isotropic dissociative potential surface, limitations which are lifted in the current approach [lo]. Some of the calculations below also apply Schinke’s one-dimensional model rotational reflection principle in order to exhibit further some limitations of the above mentioned approximations. To keep within the confines of a Letter, the computations are restricted to rotationless initial states, but calculations displaying the interesting influence of initial rotational excitation will be presented elsewhere. These calculations will also consider Boltzmann averaged fragment rotational energy distributions for a range of temperatures. This thermal averaging may readily be studied with our model due to the computational speed afforded by the 10s analytical quantum theory. Here we consider only the adequacy of the bend-stretch approximation in describing the influence of initial state bending and strelching excitation on the fragment rotational energy distributions. The latter have a rather minor effect (through energy conservation and the energy dependence of translational contribution to FranckCondon factors) when bend-stretch couplings are ignored. However, when the couplings are retained, altered initial state angular motion and therefore fragment rotational distributions may emerge upon excitation of the initial state stretches.
simplest spectroscopic model for this wavefunction. The continuum wavefunction is represented in the 10s approximation as the product of a scattering wavefunction (in R), a diatomic fragment vibrational function (in r), and rotational functions for diatomic rotation and orbital motion of the fragment atom with respect to the diatomic. The scattering wavefunction is further approximated using Airy functions to enable the analytical evaluation of integrals over R. Complexity enters i.nto the analytical theory by virtue of the nonlinear transformation between the ground state normal coordinates Q and the dissociative surface,Jacabi coordinates Q’= (r, R, f?), the axis switching transformation between the different rotating coordinates appropriate to the two surfaces [ 18,191, and the subsequent theoretical analysis necessary to evaluate the resultant integrals [ lo]. The analytical quantum theory is similar to Light’s AiryDVR method [6] apart from different models for the centrifugal potential and the fact that the DVR method performs the Yand R integrals by numerical quadrature, while in our approach they are evaluated analytically. (Previous DVR computations also ignore bend-stretch couplings as do Morse and Freed. ) The potential energy for the initial bound state is taken to be a simple valence force field that is diagonal in the two bond displacements Y,- P,,~~and r2-r2,eq and the bond angle displacement S-S,. The normal modes Q are generated using standard FGmatrix methods [ 201 in which the kinetic energy operator [ 201 provides the bend-stretch couplings. (Additional ones would arise from using a more extensive force field.) Thus, the normal coordinates emerge in the standard from
2. Theory and computations
(~;j=M(:;l$
Despite the apparent complexity in evaluating the nonseparable three-dimensional transition amplitudes, the analytical quantum theory begins from a rather simple model for the bound and dissociative surface wavefunctions [ IO]. The initial bound state wavefunction is taken as a product of a rigid rotor wavefunction and harmonic oscillator functions for the stretching and bending normal modes, i.e. the
Introducing the bend-stretch approximation corrematrix elements the setting sponds to M,,=M,, =Mz3= M,,=O. The three force constants are chosen to reproduce the experimental vibrational frequencies from ref. [ 2 1] for NOCI and ref. [lo] for HCN. The nonlinear transformation between Jacobi and ground state displacement coor-
(1)
299
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dinates is given in ref. [ 10 ] and need not be reproduced here. The dissociative HCN surface is taken as potential 7 from ref. [5], while that for the T, state of NOCI is a spline fit of the form
to the ab initio potential surface of Segal and coworkers [131,where R,, is the value of R at the ground state equilibrium geometry. Details of the fitted NOCI potential will be presented elsewhere along with a more extensive study of the photodissociation on this surface. Suffice it to say that there are insufficient ab initio points to describe the quasibound excited surface bending states that are inferred from the experiment [ 221. Thus the potential is used merely as a model system for investigating how the kinetic energy induced bend-stretch couplings affect final state energy distributions. Previous computations for HCN using potential 7 from ref. [ 5 ] show that the analytical quantum theory is in good agreement with close-coupled computations [IO] when both calculations ignore bond-stretch coupling. All computations below take J= K= 0 in the initial bound state (rotational excitations will be treated elsewhere). For simplicity, the photodissociation is taken to proceed by scalar coupling (a constant dipole coupling matrix element) as the differences introduced by dipole coupling do not affect the qualitative results. The calculations consider a range of initial vibrational states and compare the rotational distributions with and without the bend-stretch couplings. When the initial state involves excitation only in the bending vibration, calculations are also provided using Schinke’s one-dimensional rotational reflection model in which the fragment rotational distribution P(j) for J=O is obtained as
P(j)=
II IJ
2 August1991
CHEMICAL PHYSICSLETTERS
3,4
2
sin 6 $(B, 0) w,,(0) d6 ,
0
3. Results Figs. l-3 present computations of the NO fragment rotational distributions P(j) for the n,=O-2 initial bending states of NOCl, respectively. The rapid oscillations will be shown elsewhere to be largely quenched upon thermal averaging over an initial rotational distribution corresponding to temperatures of a few kelvin. In addition, as found for Hz0 photodissociation calculations [ 231, the averaging over fragment atom and diatom fine structure states probably contributes to removing some of these rapid oscillations. Because our interest here lies solely with examining the accuracy of the bend-stretch approximation, the unaveraged distributions presented here suffice. (Although the experimental implications of this averaging are the interesting observation [ 221 of a predicted [ 181 mapping of the bending wavefunction onto P(j). ) The bend-stretch approximation in figs. l-3 is seen to produce a narrower P(j) than the full theory. Surprisingly, the one-dimensional rotational reflection principle model is some-
(a)
(4
0.2 0.0 (c)
::L 0
5
10
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Dlatomlc Rotational Angular Momentum
where 0 is the Jacobi coordinate, Yjo(0,O) is a spherical harmonic, and v,,(6) is the initial bending wavefunction.
300
:!5
j
Fig. 1. NO(ZP,,,) rotational distributions for model photodissociation of NOCl on the T, ( I ‘A”)surface. Initial state has J=O and {n,) = 0, and the maximum fragment asymptotic energy is E~0.3823 eV. The figures are: (a) one-dimensional rotational reflection model; (b) no bend-stretch coupling; (c) bend-stretch coupling included.
Volume 182,number 3,4
3 ._ 1 I 1 e n ‘0 ._ $
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CHEMICALPHYSICSLETTERS
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0.0 (W
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.o d
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I&_
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Fig. 2. Same as fig 1but for n3= I and E= 0.4324eV
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Fig. 4. Same as fig. 1 but for HCN tion to H+CN with E= 1.OOeV.
(C IA’)
state model dissocia-
e
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f
B r 2 b 2
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2!
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.
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40
j
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Diatomlc Rotational Angular Momentum
Fig. 3. Same as fig. 1 but for ng=2 and E=0.4849 eV.
what better; the model has P(j) too small at low j and too large at highj. Figs. 4-6 display similar computations for the model photodissociation of HCN. The initial ground state rotational distributions are quite similar for the three computations in fig. 4, but
25
j
Fig. 5. Same as fig. 4 but for n3= I
differences grow as the bending vibration becomes excited to n3 = 1 and 2 in figs. 5 and 6, respectively. The bend-stretch approximation shifts population to higher j for rz3= 1 and to lower j for n3=2. The one-dimensional rotational reflection principle ap301
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CHEMlCAL PHYSICS LETTERS
Volume 182, number 3,4
0.2 12 0.1
5 z s g E n 6
2 g ; 0 w 5 E B
0.0
0.2 0.1 0.0
I.
0.2
o.oo
0
0.1 0.0
5
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0
5
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Fig. 8. Same as fig. 7 but for Q=
I and E=0.4324 eV.
i
Diatomic Rotational Angular Momentum Fit. 6. Same as fig. 4 but for II, = 2.
(a)
(b)
excitation of the initial state stretching vibrations affects the bending motion. Thus, it is possible that stretching excitations likewise alter rotational energy distributions. Figs. 7 and 8 display computations in which the stretches have n, = 1 and n2= 1, respectively, for the model NOCl photodissociation. The bend-stretch approximation incurs a small error (a shift of population to higher j) in fig. 7, but larger discrepancies appear in fig. 8. Thus, it is clear that bend-stretch coupling must also be included to understand the influence of stretching excitations on product energy distributions.
Acknowledgement 5
10
15
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Diatomic Rotational Angular Momentum
i :!5 j
Fig. 7. Same as fig. 1but for n, = I and E=0.4324 eV. The figures are: (a) no bend-stretch coupling; (b) bend-stretch coupling included.
proximation is no better than that obtained by neglecting the bend-stretch approximation, but it is interesting to note that the one-dimensional calculation has additional oscillations in figs. 5 and 6 that are quenched by the averaging implicit in the full threedimensional analytical quantum theory. Because of the bend-stretch coupling in eq. ( I), 302
This research is supported, in part, by NSF Grant CHE89-13 123. HG is grateful to the Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and Consejo National de lnvestigaciones Cientiticas y Tecnicas, Republica Argentina for a leave of absence.
References [ 1] S.R. Leone, Advan. Chem. Phys. 50 (1982) 255. [2] J.P. Simons,J. Phys. Chem. 88 (1984) 1287. [3] R. Bersohn, J. Phys. Chem. 88 ( 1984) 5415.
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(41 E. Segev and M. Shapiro. J. Chem. Phys. 73 (1980) 2001; 77 (1982) 5604; 78 ( 1983) 4969. [ 51R.W. Heather and J.C. Light: J. Chem. Phys. 78 (1983) 5513.
[61 R.W. Heather and J.C. Light, J. Chem. Phys. 79 ( 1983) 147. [ 710. Atabek, J.A. Beswick and G. Delgado Barrio, J. Chem. Phys. 83 ( 1985) 2954. [ 81 K. Weide and R. Schinke, J. Chem. Phys. 90 (1989)7 150; V. Staemmler and A. Palma, Chem. Phys. 93 11985)63. [9] H. Grinberg, K.F. Freed and C.J. Williams, J. Chem. Phys. 86 (1987) 5456. [ IO] H. Grinberg, K.F. Freed and C.J. Williams, J. Chem. Phys. 92 (1990) 7283. [ 111 Y.B. Band, K.F. Freed and D.J. Kouri, J. Chem. Phys. 74 (1981) 4380.
[ 121 K.C. Kulander and J.C. Light, J. Chem. Phyr. 85 (1986) 1938.
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[ 131 Y.Y. Bai, A. Ogai, C.X.W. Quian, L. Iwata, G.A. Segal and H. Reisler, J. Chem. Phys. 90 (1989) 3903. [ 141 R. Schinke and V. Engel, Faraday Discussions Chem. Sot. 82 (1986) Ill. [ 1S] R. Schinke, in: Collision theory for atoms and molecules, ed. F.A. Gianturco (Plenum Press, New York, 1989). [ 161 R. Schinke, R.L. Vander Wal, J.L. Scott and F.F. Grim, J. Chem. Phys. 94 (1991) 283. [ 171 R. Schinke, J. Chem. Phys. 92 (1990) 2397. [ 181 M.D. Morse andK.F. Freed, J. Chem. Phys. 78 (1983) 6045. [ 191 M.D. Morse andK.F. Freed, J. Chem. Phys. 74 ( 1981) 4395. (201 E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations (Dover. New York, 1955). [21] J.K. McDonald, J.A. Merritt, V.F. Kalasinsky, H.L. Hensel and J.R. During, J. Mol. Spectry. I17 ( 1986) 69. [22] C.X.W. Quian, 4. Ogai. L. Iwata and H. Reisler, J. Chem. Phys. 92 ( 1990) 4296. [23] G.G. Balint-Kurti, J. Chem. Phys. 84 (1986) 443.
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Influence of initial state bend-stretch couplings on product rotational distributions in photodissociation of bent triatomic molecules Horatio
Grinberg ‘, Karl F. Freed
James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL 60637, US4
and Carl J. Williams Department of Chemistry, University ofNotre Dame, Notre Dame, IN 46556, VSA Received 30 March 1991; in final form 8 May 1991
Even when a simple valence force field is used to represent the bound state normal modes, the kinetic energy operator couples bending and stretching motions in bent triatomic molecules. These bend-stretch couplings have been ignored in previous theoretical descriptions of photodissociation and are shown here to significantly alter predicted rotationalenergydistributions. Examples consider model HCN and NOCl photodissociations from bent initial states.
1. Introduction The influence of the initially prepared parent states, the dissociative potential energy surfaces, and the excitation wavelength on product energy and angular distributions are crucial to understanding of photodissociation dynamics. Advances in laser technology enable the determination of final state distributions for a variety of species and photodissociation processes. Many experiments measure the product rotational energy distributions that are predicted theoretically to display a strong sensitivity to photodissociation dynamics and initial state selection [ l-31. However, despite these experimental advances, detailed theoretical understanding of product rotational distributions, even in triatomic photodissociations, is still rather limited. Two general theoretical approaches have been used for describing photodissociation dynamics. The first ’ Permanent address: Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina.
is based on advances in quantum inelastic scattering methods (both time dependent and independent) using ab initio or empirical potential surfaces and perhaps various tested approximations to the full collision dynamics [ 4-8 1. Second, alternative simple analytically tractable models exist in order to elucidate general qualitative trends, to produce computationally economical models for testing the adequacy of various dissociative potential energy surfaces, and to enable tests of assumptions inherent in full three-dimensional scattering calculations for photodissociation dynamics. Although the non-expert may assume that full three-dimensional scattering calculations are rigorous in describing photodissociation dynamics, most calculations contain a number of assumptions, including the neglect of electronic and spin angular momenta, the neglect of coupling to additional nearby electronic surfaces, and the simplified treatment of both the bound and dissociative state wavefunctions in the same single set of coordinate and rotational axis systems. It is the important testing of some of these assumptions that we consider here using our analytical three-dimen-
0009-26 14/91/S 03.50 0 199 1 - Elsevier Science Publishers B.V. (North-Holland)
297
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CHEMICALPHYSlCSLETTERS
sional rotationally infinite-order sudden (10s) treatment of photodissociation energy distribution [9,101. While complete three-dimensional coupled-state scattering calculations are possible for triatomic photodissociations, the expense of these calculations, the inaccuracy of even the best ab initio potential surfaces, and the availability of limited experimental data have led to the introduction of standard approximations, such as coupled states and rotational 10s methods [ 11,121, in order to simplify the otherwise enormous computational labor. Nevertheless, until recently, the majority of approximate scattering treatments of photodissociation dynamics have also been restricted to rotationless (J=O) initial states and therefore do not address the interesting questions of the partitioning of initial parent state excitation (J, Kf 0) into product rotational and angular distributions that is easily described using our analytical theory [9,10]. As mentioned above, full-quantum close-coupled calculations introduce a series of somewhat less obvious approximations some of which are analyzed here using our analytical quantum theory. The computational approaches generally employ the same Jacobi scattering coordinates for both the bound and dissociative potential surfaces. These familiar Jacobi coordinates are the fragment diatomic internuclear separation r, the distance R of the fragment atom from the diatomic center of mass, and the angle 0 between the vectors I and R. Thus, the initial bound state vibrational functions are expressed as basis set expansions that are separable in the Jacobi coordinates. Such an approximation can introduce severe errors and additionally becomes increasingly unwieldy as the size of the molecule grows further (tetra-atomics requiring six-dimensional expansions). In order to simplify the evaluation of threedimensional bound-continuum Franck-Condon factors the computations further introduce an approximation, which we term the neglect of bendstretch coupling, in which the bound state stretching modes are assumed to be functions only of r and Rj while the bend is a pure function of 8. This assumption is in stark contrast to standard normal mode analyses of bent triatomic molecules where the bends and stretches are functions of all three coordinates (hence the term bend-stretch coupling). When 298
2 August1991
bend-stretch couplings are ignored in spectroscopic applications, the bound state bending is more appropriately represented in terms ofthe triatomic bond angle 6, which may differ significantly from the Jacobi scattering coordinate 0 used to describe the bound state bending in many photodissociation models. This use of a common coordinate system for both initial and final potential energy surfaces implies several other implicit assumptions, such as the neglect of axis switching (the difference between the natural rotating frames for the molecule on the bound and dissociative surfaces). Our analytical 10s photodissociation theory is used in this paper to study the errors associated with introducing the bend-stretch approximation into the theory of bent triatomic molecule photodissociation. Model potential surfaces are used for NOCl T, ( 1 3A” ) [ 131 and HCN (c ‘A’) state [ 5 ] direct dissociations, and comparisons are made between computations that include or ignore the bend-stretch couplings. These couplings may be treated by the analytical quantum theory because this method uses two different coordinate systems, one appropriate to the bound and the second appropriate to the dissociative surface. This improvement, however, leads, in general, to completely nonseparable three-dimensional scattering amplitudes that are resolved in our theory by using the IOS approximation and an Airy function for the continuum wavefunction. The scattering amplitudes are thereby transformed into one-dimensional integrals that are represented analytically using efficient quadrature methods. The analytical nature of the theory enables the rapid computation of product distributions for given initial vibrational and rotational states of the parent molecule. We should note that Schinke’s well known and useful rotational reflection principle [ 14- 161 emerges as an approximation to the analytical quantum theory upon the introduction of the following series of approximations: ( I) Bend-stretch couplings are ignored and the initial bound state vibrations are assumed to be separable in r, R, and 0. (2) Axis switching and the nonlinear transformation between the natural coordinates for the bound and repulsive surfaces are ignored. (3 ) Classical models are frequently used for the
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rotational scattering, although a quantum analog is also available [ 171. (4) The initial state is taken to be rotationless. A simple one-dimensional form of the rotational reflection principle has been shown by Schinke [ 161 to exhibit the initial bending state dependence of the rotational energy distribution that is predicted by Morse and Freed [ 18 1.The earlier version of the analytical theory [ 181 ignores the bend-stretch couplings and assumes an isotropic dissociative potential surface, limitations which are lifted in the current approach [lo]. Some of the calculations below also apply Schinke’s one-dimensional model rotational reflection principle in order to exhibit further some limitations of the above mentioned approximations. To keep within the confines of a Letter, the computations are restricted to rotationless initial states, but calculations displaying the interesting influence of initial rotational excitation will be presented elsewhere. These calculations will also consider Boltzmann averaged fragment rotational energy distributions for a range of temperatures. This thermal averaging may readily be studied with our model due to the computational speed afforded by the 10s analytical quantum theory. Here we consider only the adequacy of the bend-stretch approximation in describing the influence of initial state bending and strelching excitation on the fragment rotational energy distributions. The latter have a rather minor effect (through energy conservation and the energy dependence of translational contribution to FranckCondon factors) when bend-stretch couplings are ignored. However, when the couplings are retained, altered initial state angular motion and therefore fragment rotational distributions may emerge upon excitation of the initial state stretches.
simplest spectroscopic model for this wavefunction. The continuum wavefunction is represented in the 10s approximation as the product of a scattering wavefunction (in R), a diatomic fragment vibrational function (in r), and rotational functions for diatomic rotation and orbital motion of the fragment atom with respect to the diatomic. The scattering wavefunction is further approximated using Airy functions to enable the analytical evaluation of integrals over R. Complexity enters i.nto the analytical theory by virtue of the nonlinear transformation between the ground state normal coordinates Q and the dissociative surface,Jacabi coordinates Q’= (r, R, f?), the axis switching transformation between the different rotating coordinates appropriate to the two surfaces [ 18,191, and the subsequent theoretical analysis necessary to evaluate the resultant integrals [ lo]. The analytical quantum theory is similar to Light’s AiryDVR method [6] apart from different models for the centrifugal potential and the fact that the DVR method performs the Yand R integrals by numerical quadrature, while in our approach they are evaluated analytically. (Previous DVR computations also ignore bend-stretch couplings as do Morse and Freed. ) The potential energy for the initial bound state is taken to be a simple valence force field that is diagonal in the two bond displacements Y,- P,,~~and r2-r2,eq and the bond angle displacement S-S,. The normal modes Q are generated using standard FGmatrix methods [ 201 in which the kinetic energy operator [ 201 provides the bend-stretch couplings. (Additional ones would arise from using a more extensive force field.) Thus, the normal coordinates emerge in the standard from
2. Theory and computations
(~;j=M(:;l$
Despite the apparent complexity in evaluating the nonseparable three-dimensional transition amplitudes, the analytical quantum theory begins from a rather simple model for the bound and dissociative surface wavefunctions [ IO]. The initial bound state wavefunction is taken as a product of a rigid rotor wavefunction and harmonic oscillator functions for the stretching and bending normal modes, i.e. the
Introducing the bend-stretch approximation corrematrix elements the setting sponds to M,,=M,, =Mz3= M,,=O. The three force constants are chosen to reproduce the experimental vibrational frequencies from ref. [ 2 1] for NOCI and ref. [lo] for HCN. The nonlinear transformation between Jacobi and ground state displacement coor-
(1)
299
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dinates is given in ref. [ 10 ] and need not be reproduced here. The dissociative HCN surface is taken as potential 7 from ref. [5], while that for the T, state of NOCI is a spline fit of the form
to the ab initio potential surface of Segal and coworkers [131,where R,, is the value of R at the ground state equilibrium geometry. Details of the fitted NOCI potential will be presented elsewhere along with a more extensive study of the photodissociation on this surface. Suffice it to say that there are insufficient ab initio points to describe the quasibound excited surface bending states that are inferred from the experiment [ 221. Thus the potential is used merely as a model system for investigating how the kinetic energy induced bend-stretch couplings affect final state energy distributions. Previous computations for HCN using potential 7 from ref. [ 5 ] show that the analytical quantum theory is in good agreement with close-coupled computations [IO] when both calculations ignore bond-stretch coupling. All computations below take J= K= 0 in the initial bound state (rotational excitations will be treated elsewhere). For simplicity, the photodissociation is taken to proceed by scalar coupling (a constant dipole coupling matrix element) as the differences introduced by dipole coupling do not affect the qualitative results. The calculations consider a range of initial vibrational states and compare the rotational distributions with and without the bend-stretch couplings. When the initial state involves excitation only in the bending vibration, calculations are also provided using Schinke’s one-dimensional rotational reflection model in which the fragment rotational distribution P(j) for J=O is obtained as
P(j)=
II IJ
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3,4
2
sin 6 $(B, 0) w,,(0) d6 ,
0
3. Results Figs. l-3 present computations of the NO fragment rotational distributions P(j) for the n,=O-2 initial bending states of NOCl, respectively. The rapid oscillations will be shown elsewhere to be largely quenched upon thermal averaging over an initial rotational distribution corresponding to temperatures of a few kelvin. In addition, as found for Hz0 photodissociation calculations [ 231, the averaging over fragment atom and diatom fine structure states probably contributes to removing some of these rapid oscillations. Because our interest here lies solely with examining the accuracy of the bend-stretch approximation, the unaveraged distributions presented here suffice. (Although the experimental implications of this averaging are the interesting observation [ 221 of a predicted [ 181 mapping of the bending wavefunction onto P(j). ) The bend-stretch approximation in figs. l-3 is seen to produce a narrower P(j) than the full theory. Surprisingly, the one-dimensional rotational reflection principle model is some-
(a)
(4
0.2 0.0 (c)
::L 0
5
10
15
20
Dlatomlc Rotational Angular Momentum
where 0 is the Jacobi coordinate, Yjo(0,O) is a spherical harmonic, and v,,(6) is the initial bending wavefunction.
300
:!5
j
Fig. 1. NO(ZP,,,) rotational distributions for model photodissociation of NOCl on the T, ( I ‘A”)surface. Initial state has J=O and {n,) = 0, and the maximum fragment asymptotic energy is E~0.3823 eV. The figures are: (a) one-dimensional rotational reflection model; (b) no bend-stretch coupling; (c) bend-stretch coupling included.
Volume 182,number 3,4
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0.0 (W
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.o d
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Fig. 2. Same as fig 1but for n3= I and E= 0.4324eV
5 Diatomic
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Fig. 4. Same as fig. 1 but for HCN tion to H+CN with E= 1.OOeV.
(C IA’)
state model dissocia-
e
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f
B r 2 b 2
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.
0.0. 0.2.
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j
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Diatomlc Rotational Angular Momentum
Fig. 3. Same as fig. 1 but for ng=2 and E=0.4849 eV.
what better; the model has P(j) too small at low j and too large at highj. Figs. 4-6 display similar computations for the model photodissociation of HCN. The initial ground state rotational distributions are quite similar for the three computations in fig. 4, but
25
j
Fig. 5. Same as fig. 4 but for n3= I
differences grow as the bending vibration becomes excited to n3 = 1 and 2 in figs. 5 and 6, respectively. The bend-stretch approximation shifts population to higher j for rz3= 1 and to lower j for n3=2. The one-dimensional rotational reflection principle ap301
2 August 199 1
CHEMlCAL PHYSICS LETTERS
Volume 182, number 3,4
0.2 12 0.1
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Fig. 8. Same as fig. 7 but for Q=
I and E=0.4324 eV.
i
Diatomic Rotational Angular Momentum Fit. 6. Same as fig. 4 but for II, = 2.
(a)
(b)
excitation of the initial state stretching vibrations affects the bending motion. Thus, it is possible that stretching excitations likewise alter rotational energy distributions. Figs. 7 and 8 display computations in which the stretches have n, = 1 and n2= 1, respectively, for the model NOCl photodissociation. The bend-stretch approximation incurs a small error (a shift of population to higher j) in fig. 7, but larger discrepancies appear in fig. 8. Thus, it is clear that bend-stretch coupling must also be included to understand the influence of stretching excitations on product energy distributions.
Acknowledgement 5
10
15
20
Diatomic Rotational Angular Momentum
i :!5 j
Fig. 7. Same as fig. 1but for n, = I and E=0.4324 eV. The figures are: (a) no bend-stretch coupling; (b) bend-stretch coupling included.
proximation is no better than that obtained by neglecting the bend-stretch approximation, but it is interesting to note that the one-dimensional calculation has additional oscillations in figs. 5 and 6 that are quenched by the averaging implicit in the full threedimensional analytical quantum theory. Because of the bend-stretch coupling in eq. ( I), 302
This research is supported, in part, by NSF Grant CHE89-13 123. HG is grateful to the Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and Consejo National de lnvestigaciones Cientiticas y Tecnicas, Republica Argentina for a leave of absence.
References [ 1] S.R. Leone, Advan. Chem. Phys. 50 (1982) 255. [2] J.P. Simons,J. Phys. Chem. 88 (1984) 1287. [3] R. Bersohn, J. Phys. Chem. 88 ( 1984) 5415.
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(41 E. Segev and M. Shapiro. J. Chem. Phys. 73 (1980) 2001; 77 (1982) 5604; 78 ( 1983) 4969. [ 51R.W. Heather and J.C. Light: J. Chem. Phys. 78 (1983) 5513.
[61 R.W. Heather and J.C. Light, J. Chem. Phys. 79 ( 1983) 147. [ 710. Atabek, J.A. Beswick and G. Delgado Barrio, J. Chem. Phys. 83 ( 1985) 2954. [ 81 K. Weide and R. Schinke, J. Chem. Phys. 90 (1989)7 150; V. Staemmler and A. Palma, Chem. Phys. 93 11985)63. [9] H. Grinberg, K.F. Freed and C.J. Williams, J. Chem. Phys. 86 (1987) 5456. [ IO] H. Grinberg, K.F. Freed and C.J. Williams, J. Chem. Phys. 92 (1990) 7283. [ 111 Y.B. Band, K.F. Freed and D.J. Kouri, J. Chem. Phys. 74 (1981) 4380.
[ 121 K.C. Kulander and J.C. Light, J. Chem. Phyr. 85 (1986) 1938.
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[ 131 Y.Y. Bai, A. Ogai, C.X.W. Quian, L. Iwata, G.A. Segal and H. Reisler, J. Chem. Phys. 90 (1989) 3903. [ 141 R. Schinke and V. Engel, Faraday Discussions Chem. Sot. 82 (1986) Ill. [ 1S] R. Schinke, in: Collision theory for atoms and molecules, ed. F.A. Gianturco (Plenum Press, New York, 1989). [ 161 R. Schinke, R.L. Vander Wal, J.L. Scott and F.F. Grim, J. Chem. Phys. 94 (1991) 283. [ 171 R. Schinke, J. Chem. Phys. 92 (1990) 2397. [ 181 M.D. Morse andK.F. Freed, J. Chem. Phys. 78 (1983) 6045. [ 191 M.D. Morse andK.F. Freed, J. Chem. Phys. 74 ( 1981) 4395. (201 E.B. Wilson, J.C. Decius and P.C. Cross, Molecular vibrations (Dover. New York, 1955). [21] J.K. McDonald, J.A. Merritt, V.F. Kalasinsky, H.L. Hensel and J.R. During, J. Mol. Spectry. I17 ( 1986) 69. [22] C.X.W. Quian, 4. Ogai. L. Iwata and H. Reisler, J. Chem. Phys. 92 ( 1990) 4296. [23] G.G. Balint-Kurti, J. Chem. Phys. 84 (1986) 443.
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