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Reactive resonances in the photodissociation of symmetric triatomic molecules: an interpretation of collinear CO2 in terms of polar coordinates
Volume
124, number 6
CHEMICAL PHYSICS LETTERS
14 March 1986
REACT WE RESONANCES IN THE PHOTODISSOCIATION OF SYMMETRIC TRIATOMIC MOLECULES: AN INTERPRETATION OF COLLINEAR CO, IN TERMS OF POLAR COORDINATES
Reinhard SCHINKE and Volker ENGEL Max-Planck-lnstitut ftir Striimungsforschung, D-3400 Giittingen, Federal Republic of Germany Received 19 November
1985; in final form 24 January 1986
We discuss the origin of reactive scattering resonances in absorption cross sections following the collinear photodissociation of symmetric triatomic molecules. The analysis of spectra for CO, dissociation, previously calculated by Kulander and Light, becomes very simple in terms of polar coordinates. Of particular interest is a symmetry effect which is not observable m ordinary scattering calculations.
Resonances in reactive scattering are fascinating features which have attracted much theoretical work over the last two decades. For a comprehensive overview which contains most of the relevant references see the article of Kuppermann [ 11. Of particular interest are those systems which have a purely repulsive potential energy surface as one goes along the reaction coordinate from the reactant (or product) channel up to the saddle point, i.e. the “transition state”. The prototype system is Hj for which reactive scattering resonances are known since the earliest exact quantal calculations [2-41. In scattering the resonances occur as more or less sharp structures of the various reaction or inelastic transition probabilities. A most illustrative interpretation of reactive resonances has been established in the last five years in terms of polar coordinates (‘J, a) where p is the polar radius and (Yis the polar angle, respectively *: The resonances are due to bound or quasi-bound states supported by minima of the adiabatic potential energy curves which are obtained by solving the onedimensional Schriidinger equation for the internal (or)motion for fured radius p [6-91. These minima occur in the vicinity of the potential barrier and thus the reactive resonances are a direct probe of the transition state.
Although there are some indirect indications for F t H2 [lo] it is still questionable whether reactive resonances can be observed directly in a scattering experiment. Due to the summation over, usually, many partial waves, each leading to slightly different resonance positions on the energy scale, it is most likely that the sharp resonance structures are washed out. This is different from the photodissociation of an initially bound molecule, which - in principle - can be prepared in a single total angular momentum (J) state, preferentially the J = 0 ground state, either by extreme cooling in a jet or by infrared excitation [ Ill. Due to the AJ = 0, -+I selection rule [12] at most three total angular momentum states have to be considered in the expansion of the total dissociation wavefunction on the repulsive excited-state potential energy surface. Therefore, the photon absorption spectrum could be the ideal candidate to detect reactive scattering resonances [ 131 provided the.excited-state potential surface has two accessible dissociation channels as is always the case for symmetric molecules like CO, and H,O, for example. Indeed, there are many examples in the literature [ 141 showing diffuse bands which might be interpreted in terms of such resonances. The absorption cross section for the process
* For a comprehensive
is in first-order perturbation theory (“golden rule”) given by the matrix element
overview of polar coordinates tive scattering calculations see ref. [S].
504
in reac-
ABC@)t Ao + A t BC(n
(1)
0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics publishing Division)
Volume 124, number 6
CHEMICAL PHYSICS LETTERS
where I& is the vibrational wavefunction for the ABC molecule initially in vibrational state i on the ground electronic surface, Jl-$-’ is the nuclear-motion wavefunction for dissociation on the excited electronic surface into state f, and the electronic transition moment has been assumed constant. Except for slight changes of the boundary condition [ 121 the wavefunctions describing dissociation and ordinary scattering on the same potential are identical. The scattering energy is E=Ei+hw. Since resonances are inherent to the total wavefunction, they are observable at identical energies in any measurable quantity derived from this wavefunction, for example, the reaction probability, the absorption spectrum, etc. The only question is how clearly pronounced are the resonance structures for the various observables. In a recent Letter we calculated absorption spectra for the collinear H + H2 + H; system, where Hz denotes some electronically excited state of H, which supports many bound states [ 1.51. Not surprisingly, they showed the same resonance pattern as the H t H, + H2 t H reaction probability. In the present Letter we interpret previously reported absorption spectra for collinear dissociation of CO,. They have been calculated exactly by Kulander and Light [ 161 using the R-matrix method. As for H, the interpretation of these spectra is extremely simple in terms of polar coordinates and leads readily to a symmetry effect not observable in ordinary scattering. In figs. la and 2a we show two spectra for the dissociation out of the (0,O) vibrational ground and the (OJ) vibrational excited state of CO2 taken directly from figs. 2 and 3 of ref. [ 161 for final CO vibrational channels n = 2 and n = 0, respectively. The energy is the total available energy on the excited-state surface. While the (OJ) spectrum is a very smooth function over the entire range of energies probed, the (0,O) spectrum shows a pronounced sequence of resonance lines. Kuiander and Light [ 161 interpreted the resonances “as due to the configuration interaction between discrete states corresponding to the bound symmetric stretch vibrations and the several continua associated with the different possible vibrational levels in the CO fragments”. If that is true we do not understand why the resonance structures are not present in the (0,l) spectrum, because the total dissociation wave-
14 March 1986
I
01
2.5
I
30
35
9 rA1 Fig. 1. (a) Partial absorption spectrum for COz + 0 + CO@ = 2) dissociation out of the (0,O) vibrational ground state. The energy is the photon energy above the dissociation threshold. Redrawn from Kulander and Light [ 161 with permission of the authors and AIP. (b) Adiabatic potential energy curves et(p) for the gerade symmetry as obtained from the model excited-state potential of ref. [ 16).
functions are identical in both cases. Kulander and Light [ 161 argued that due to the node along the asymmetric stretch coordinate the resonance structures are yashed out. We believe that a more transparent inter3 (a)
2
2 5
1
0'
I 2.5
I
30
I 35
p CA1 Fig. 2. (a) The same as in fii. la but for then = 0 final state of CO and the (OJ ) vibrational state of CO2. (b) The same as in f%. lb but for the ungerade symmetry. 505
pretation of the spectra in figs. 1 and 2 becomes readily apparent if polar coordinates are used for both the ground- and the excited-state wavefunctions. Incidentally we note that the partial spectra for different final CO vibrational states n are very similar for a given ground state of CO,. We especially note that the resonance structures (peaks, dips) occur at the same energies independent of n. The ground- and excited-state wavefunctions are both expanded in an orthonormal set of functions $r(culP) which are defined by the one-dimensional Schriidinger equation [S]
---21.( p2 l
l d2 + @Jr) do2
$J%P)
=
en(P) $&lP),
(3)
where V(p,cr) is either the ground- or the excitedstate potential energy surface. The wavefunctions $r and the corresponding energies en depend parametrically on the polar radius p. The potential energy surface for a collinear and symmetric system is symmetric with respect to an angle h, : h = 30” for H, and h = 27.6” for CO,, and consequently the expansion functions are either of gerade (g) or of ungerade (u) symmetry. Expanding the total wavefunctions according to \(l(P,4
= Jlg(P,4
+ $,(P,ar),
(4) i = is u,
(5)
leads to the usual set of coupled equations for the radial wavefunctions x&p). Because of symmetry the total set of equations completely de-couples into one set for-the gerade part, $,, and one set for the ungerade part, J/, . The bound-state wavefunctions have a definite symmetry, they are either gerade or ungerade. We calculated the tirst five bound states of collinear ground-state CO, using the simple potential of Erkog et al. [17] (eqs. (4) and (5)). The corresponding energies, symmetries and normal mode quantumnumbers are summarized in table 1. The (0,O) vibrational ground state is gerade and the (0,l)vibrationally excited state is ungerade. Loosely speaking, the symmetric stretch quantum number determines the nodes along p and the asymmetric quantum number determines the nodes along (Yand therefore the symmetry. The correct boundary condition for the excitedstate wavefunction requires an outgoing wave in only 506
14 March 1986
CHEMICAL PHYSICS LETTERS
Volume 124, number 6
Table 1 Energies and symmetries of the five lowest bound states of collinear CO2 in the electronic ground state a)
a)
i
(n. ml
Ei @VI
symmetry
1 2 3 4 5
(090) (LO) (0,l) GO) (1,l)
0.232 0.397 0.522 0.562 0.683
g g U
g U
Calculated with the potential energy surface given in eqs. (4) and (5) of ref. [ 171.
one of the reaction channels and therefore I&’ has no definite symmetry, i.e. both $, and $, in eq. (4)
are non-zero. Insertingsxpansion (4) for both wavefunctions into the general expression (2) for the absorption cross section it follows immediately that either the gerade or the ungerade parts of $&) are projected out depending on the symmetry of the ground state. Therefore, for a constant transition moment function, which has trivially gerade symmetry, gerade (ungerade) bound states correlate with the gerade (ungerade) part of the scattering wavefunction. This is different from ordinary reactive scattering, where a linear combination of the two symmetry contributions is required to construct the S matrix. For example, the reactive S matrix is given by Sr = f (Sg - S,), where Sg and,!& are the S matrices as obtained from the gerade and the ungerade sets of coupled equations, respectively [7]. It should be stressed that J/ and I& are calculated completely independently from each other. In fig. lb we show the adiabatic potential energy curves e(p) for the gerade symmetry obtained from the excited-state potential energy surface used by Kulander and Light [ 161. The potentials for quantum numbers n > 3 exhibit local minima which become gradually deeper and shift farther outward with increasing n. These minima support quasi-bound states which then give rise to resonance structures in the gerade part of the scattering wavefunction. These resonances are transparent in the cross sections for absorption out of gerade bound states, for example. They also occur in the gerade S matrix Sg, which however has no direct physical meaning. In order to approximately locate the resonances we performed simple one-dimensional scattering calculations for each adla-
Volume 124, number 6
CHEMICAL PHYSICS LETTERS
batic potential and analyzed the energy dependence of the phase of the elastic S matrix. The resonance energies are indicated as horizontal lines in fig. 1 and indeed good correlation with the structures in the absorption cross section is observed. Similar structures are also seen in the absorption spectrum for the (1,0) ground state (fig. 4 of ref. [ 161) which is also of gerade symmetry. Since the resonances are caused by the excited-state wavefunction they occur at the same energies independent of the particular ground state which is dissociated. Since the resonance energies are above the asymptotic limit of the respective adiabatic potential curve the resonances are of the shape-type. In fig. 2b we show the adiabatic potential curves for the ungerade symmetry. As known for Hg,for example, the ungerade curves start to exhibit potential wells at much higher quantum numbers, n 2 8 in the present case. The potential curves for the lower quantum numbers are monotonic with broad shoulders at around the transition state. Consequently, the cross sections for absorption out of ungerade states are smooth curves and do not exhibit resonance structures, at least not in the range of energies considered by Kulander and Light [ 161. Resonance structures might be expected for higher energies, E > 1.6 eV, which would be due to local minima in the potential curves for n > 9. Except for the resonances the (0,O) and (0,l) spectra are very similar because the p dependence of the corresponding bound state wavefunctions is roughly the same. The additional node for the (0,l) wavefunction is in the Q!direction. The true energy position and the shape of the resonances (peaks, dips or even more complicated profiles), particularly their width can only be inferred from exact calculations and not by the simple one-dimensional adiabatic calculations described above. The reactive resonances are sensitive to details of the potential energy surface, especially at the transition state. In test calculations we changed the Sato parameter of the LEPS surface [ 161 and obtained drastically different adiabatic potentials which probably would yield very different adsorption spectra. In this sense the experimental resolution of reactive scattering resonances in photodissociation processes would be true “transitionstate spectroscopy” [ 181. The experimental CO2 absorption spectrum for the band near 1300 A indeed shows a progression of distinct structures [14]. However, the experimental
14 March 1986
energy spacing is roughly half of that extracted from the calculations of Kulander and Light [ 161. As speculated by these authors either the model potential energy surface is inaccurate or a completely different mechanism is responsible for these distinct structures. Some time ago Pack [19] and later Heller [20] discussed vibrational structures in the total absorption spectrum for a symmetric triatomic molecule using a simple approximation to the total dissociation wavefunction on top of the barrier. Although the exact total cross section of Kulander and Light resembles that predicted by this simple model we believe that the structures have different origins in both cases. In the model of Pack the vibrational structure is due to the summation of very narrow but smooth (approximate) partial cross sections while in the exact calculation of Kulander and Light the structures are already present in the individual partial cross sections. The question whether the model of Pack and the resonance picture discussed in this Letter are equivalent should be investigated in future studies by analysing the exact two-dimensional dissociation wavefunction. The major difficulty in calculating photodissociation cross sections is the excited-state potential energy surface, which is not well known for most systems. The photodissociation of H20 in the first absorption band is a very fortunate case where we know the corresponding potential from ab initio calculations [21]. The agreement of approximate dynamical calculations with experimental data is encouraging [22-241. The ideas discussed in this Letter on the basis of model calculations will be further investigated for the dissociation of Hz0 but using the calculated potential energy surface. The influence of the rotational degree of freedom and the incoherent summation over several initial (primarily rotational) states, which can be considerable if the temperature of the parent gas is high, will be of particular interest to us. Recently, Clary and Henshaw [2.5] calculated three-dimensional absorption cross sections for H20 using a LEPS fit to some of the data of ref. [21] and indeed found resonance structures. Their absorption spectrum is, however, only in fair agreement with the experimental curve, which is very smooth in the wavelength region around 1600-1700 A.
507
Volume 124, number 6
CHEMICAL PHYSICS LETTERS
We are grateful to Dr. R.T Pack (Los Alamos) for critical comments on the manuscript and D.C. Clary (Cambridge) for drawing our attention to the work of Launay and le Dourneuf.
Note added in proof After this Letter was accepted for publication we became aware that similar ideas had been formulated by Launay and le Dourneuf [26].
References [ll A. Kuppermann, in: Potential energy surface and dynamical calculations, ed. D.G. Truhlar (Plenum Press, New York, 1981) p. 375. 121 D.J. Diestler and V. McKay, J. Chem. Phys. 48 (1968) 2951. 131 D.G. Truhlar and A. Kuppermann, J. Chem. Phys. 52 (1970) 3841. ]41 S.-F. Wu and R.D. Levine, Mol. Phys. 22 (1971) 881. ]51 J. Manz, Comm. At. Mol. Phys. (1985). 161 A. Kuppermamr, J.A. Kaye and J.P. Dwyer, Chem. Phys. Letters 74 (1980) 257. 171 G. Hauke, J. Manz and J. RBmelt, J. Chem. Phys. 73 (1980) 5040. ]81 J. RomeIt, Chem. Phys. 79 (1983) 197. 191 J.M. Launay and M. le Dourneuf, J. Phys. B15 (1982) L455.
508
14 March 1986
[lo] D.M. Neumark, A.M. Wodtke, G.N. Robinson, C.C. Hayden and Y.T. Lee, J. Chem. Phys. 82 (1985) 3045; D.M. Neumark, A.M. Wodtke, G.N. Robinson,C.C. Hayden, K. Shobatake, R.K. Sparks, T.P. Schafer and Y.T. Lee, J. Chem. Phys. 82 (1985) 3067. [ 1 l] P. Andresen, V. Beushausen, D. Htiusler, H.W. Liilf and E.W. Rothe, J.Chem. Phys. 83 (1985) 1429. [ 121 G.G. Balint-Kurtiand M. Shapiro, Chem. Phys. 61(1981) 137; 72 (1982) 456. [ 131 D.C. Clsry, J. Phys. Chem. 86 (1982) 2569. [ 141 H. Okabe, Photochemistry of small molecules (Wiley, New York, 1978). [15] V. Engeland R. Schinke,Chem.Phys. Letters 122 (1985) 103. [16] K.C. Kulander and J.C. Light, J. Chem. Phys. 73 (1980) 4337. [17] S. Erkog, J.N. Murrell and D.C. Clary, Chem. Phys. Letters 72 (1980) 264. [ 181 H.J. Foth, J.C. Polanyi and H.H. Telle, J. Phys. Chem. 86 (1982) 5027. [ 191 R.T Pack, J. Chem. Phys. 65 (1976) 4765. [20] E.J. Heller, J. Chem. Phys. 68 (1978) 3891. [21] V. Staemmler and A. Pahna, Chem. Phys. 93 (1985) 63. [ 221 R. Schinke, V. Engel and V. Staemmler, Chem. Phys. Letters 116 (1985) 165. [23] R. Schinke, V. Engel and V. Staemmler, J. Chem. Phys. 83 (1985) 4522. [24] R. Schinke, V. Engel, P. Andresen, D. Hiiusler and G.G. Balint-Kurti, Phys. Rev. Letters 55 (1985) 1180. [ 251 D.C. Ciary and J.P. Henshaw, in: Theory of chemical reaction dynamics, ed. D.C. Clary (Reidel, Dordrecht), to be published. [26] J.M. Launay and M. le Dourneuf, in: XII International Conference on the Physics of Electronic and Atomic Collisions, ed. S. Datz (North-Holland, Amsterdam, 1981) p. 1017.
124, number 6
CHEMICAL PHYSICS LETTERS
14 March 1986
REACT WE RESONANCES IN THE PHOTODISSOCIATION OF SYMMETRIC TRIATOMIC MOLECULES: AN INTERPRETATION OF COLLINEAR CO, IN TERMS OF POLAR COORDINATES
Reinhard SCHINKE and Volker ENGEL Max-Planck-lnstitut ftir Striimungsforschung, D-3400 Giittingen, Federal Republic of Germany Received 19 November
1985; in final form 24 January 1986
We discuss the origin of reactive scattering resonances in absorption cross sections following the collinear photodissociation of symmetric triatomic molecules. The analysis of spectra for CO, dissociation, previously calculated by Kulander and Light, becomes very simple in terms of polar coordinates. Of particular interest is a symmetry effect which is not observable m ordinary scattering calculations.
Resonances in reactive scattering are fascinating features which have attracted much theoretical work over the last two decades. For a comprehensive overview which contains most of the relevant references see the article of Kuppermann [ 11. Of particular interest are those systems which have a purely repulsive potential energy surface as one goes along the reaction coordinate from the reactant (or product) channel up to the saddle point, i.e. the “transition state”. The prototype system is Hj for which reactive scattering resonances are known since the earliest exact quantal calculations [2-41. In scattering the resonances occur as more or less sharp structures of the various reaction or inelastic transition probabilities. A most illustrative interpretation of reactive resonances has been established in the last five years in terms of polar coordinates (‘J, a) where p is the polar radius and (Yis the polar angle, respectively *: The resonances are due to bound or quasi-bound states supported by minima of the adiabatic potential energy curves which are obtained by solving the onedimensional Schriidinger equation for the internal (or)motion for fured radius p [6-91. These minima occur in the vicinity of the potential barrier and thus the reactive resonances are a direct probe of the transition state.
Although there are some indirect indications for F t H2 [lo] it is still questionable whether reactive resonances can be observed directly in a scattering experiment. Due to the summation over, usually, many partial waves, each leading to slightly different resonance positions on the energy scale, it is most likely that the sharp resonance structures are washed out. This is different from the photodissociation of an initially bound molecule, which - in principle - can be prepared in a single total angular momentum (J) state, preferentially the J = 0 ground state, either by extreme cooling in a jet or by infrared excitation [ Ill. Due to the AJ = 0, -+I selection rule [12] at most three total angular momentum states have to be considered in the expansion of the total dissociation wavefunction on the repulsive excited-state potential energy surface. Therefore, the photon absorption spectrum could be the ideal candidate to detect reactive scattering resonances [ 131 provided the.excited-state potential surface has two accessible dissociation channels as is always the case for symmetric molecules like CO, and H,O, for example. Indeed, there are many examples in the literature [ 141 showing diffuse bands which might be interpreted in terms of such resonances. The absorption cross section for the process
* For a comprehensive
is in first-order perturbation theory (“golden rule”) given by the matrix element
overview of polar coordinates tive scattering calculations see ref. [S].
504
in reac-
ABC@)t Ao + A t BC(n
(1)
0 009-2614/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics publishing Division)
Volume 124, number 6
CHEMICAL PHYSICS LETTERS
where I& is the vibrational wavefunction for the ABC molecule initially in vibrational state i on the ground electronic surface, Jl-$-’ is the nuclear-motion wavefunction for dissociation on the excited electronic surface into state f, and the electronic transition moment has been assumed constant. Except for slight changes of the boundary condition [ 121 the wavefunctions describing dissociation and ordinary scattering on the same potential are identical. The scattering energy is E=Ei+hw. Since resonances are inherent to the total wavefunction, they are observable at identical energies in any measurable quantity derived from this wavefunction, for example, the reaction probability, the absorption spectrum, etc. The only question is how clearly pronounced are the resonance structures for the various observables. In a recent Letter we calculated absorption spectra for the collinear H + H2 + H; system, where Hz denotes some electronically excited state of H, which supports many bound states [ 1.51. Not surprisingly, they showed the same resonance pattern as the H t H, + H2 t H reaction probability. In the present Letter we interpret previously reported absorption spectra for collinear dissociation of CO,. They have been calculated exactly by Kulander and Light [ 161 using the R-matrix method. As for H, the interpretation of these spectra is extremely simple in terms of polar coordinates and leads readily to a symmetry effect not observable in ordinary scattering. In figs. la and 2a we show two spectra for the dissociation out of the (0,O) vibrational ground and the (OJ) vibrational excited state of CO2 taken directly from figs. 2 and 3 of ref. [ 161 for final CO vibrational channels n = 2 and n = 0, respectively. The energy is the total available energy on the excited-state surface. While the (OJ) spectrum is a very smooth function over the entire range of energies probed, the (0,O) spectrum shows a pronounced sequence of resonance lines. Kuiander and Light [ 161 interpreted the resonances “as due to the configuration interaction between discrete states corresponding to the bound symmetric stretch vibrations and the several continua associated with the different possible vibrational levels in the CO fragments”. If that is true we do not understand why the resonance structures are not present in the (0,l) spectrum, because the total dissociation wave-
14 March 1986
I
01
2.5
I
30
35
9 rA1 Fig. 1. (a) Partial absorption spectrum for COz + 0 + CO@ = 2) dissociation out of the (0,O) vibrational ground state. The energy is the photon energy above the dissociation threshold. Redrawn from Kulander and Light [ 161 with permission of the authors and AIP. (b) Adiabatic potential energy curves et(p) for the gerade symmetry as obtained from the model excited-state potential of ref. [ 16).
functions are identical in both cases. Kulander and Light [ 161 argued that due to the node along the asymmetric stretch coordinate the resonance structures are yashed out. We believe that a more transparent inter3 (a)
2
2 5
1
0'
I 2.5
I
30
I 35
p CA1 Fig. 2. (a) The same as in fii. la but for then = 0 final state of CO and the (OJ ) vibrational state of CO2. (b) The same as in f%. lb but for the ungerade symmetry. 505
pretation of the spectra in figs. 1 and 2 becomes readily apparent if polar coordinates are used for both the ground- and the excited-state wavefunctions. Incidentally we note that the partial spectra for different final CO vibrational states n are very similar for a given ground state of CO,. We especially note that the resonance structures (peaks, dips) occur at the same energies independent of n. The ground- and excited-state wavefunctions are both expanded in an orthonormal set of functions $r(culP) which are defined by the one-dimensional Schriidinger equation [S]
---21.( p2 l
l d2 + @Jr) do2
$J%P)
=
en(P) $&lP),
(3)
where V(p,cr) is either the ground- or the excitedstate potential energy surface. The wavefunctions $r and the corresponding energies en depend parametrically on the polar radius p. The potential energy surface for a collinear and symmetric system is symmetric with respect to an angle h, : h = 30” for H, and h = 27.6” for CO,, and consequently the expansion functions are either of gerade (g) or of ungerade (u) symmetry. Expanding the total wavefunctions according to \(l(P,4
= Jlg(P,4
+ $,(P,ar),
(4) i = is u,
(5)
leads to the usual set of coupled equations for the radial wavefunctions x&p). Because of symmetry the total set of equations completely de-couples into one set for-the gerade part, $,, and one set for the ungerade part, J/, . The bound-state wavefunctions have a definite symmetry, they are either gerade or ungerade. We calculated the tirst five bound states of collinear ground-state CO, using the simple potential of Erkog et al. [17] (eqs. (4) and (5)). The corresponding energies, symmetries and normal mode quantumnumbers are summarized in table 1. The (0,O) vibrational ground state is gerade and the (0,l)vibrationally excited state is ungerade. Loosely speaking, the symmetric stretch quantum number determines the nodes along p and the asymmetric quantum number determines the nodes along (Yand therefore the symmetry. The correct boundary condition for the excitedstate wavefunction requires an outgoing wave in only 506
14 March 1986
CHEMICAL PHYSICS LETTERS
Volume 124, number 6
Table 1 Energies and symmetries of the five lowest bound states of collinear CO2 in the electronic ground state a)
a)
i
(n. ml
Ei @VI
symmetry
1 2 3 4 5
(090) (LO) (0,l) GO) (1,l)
0.232 0.397 0.522 0.562 0.683
g g U
g U
Calculated with the potential energy surface given in eqs. (4) and (5) of ref. [ 171.
one of the reaction channels and therefore I&’ has no definite symmetry, i.e. both $, and $, in eq. (4)
are non-zero. Insertingsxpansion (4) for both wavefunctions into the general expression (2) for the absorption cross section it follows immediately that either the gerade or the ungerade parts of $&) are projected out depending on the symmetry of the ground state. Therefore, for a constant transition moment function, which has trivially gerade symmetry, gerade (ungerade) bound states correlate with the gerade (ungerade) part of the scattering wavefunction. This is different from ordinary reactive scattering, where a linear combination of the two symmetry contributions is required to construct the S matrix. For example, the reactive S matrix is given by Sr = f (Sg - S,), where Sg and,!& are the S matrices as obtained from the gerade and the ungerade sets of coupled equations, respectively [7]. It should be stressed that J/ and I& are calculated completely independently from each other. In fig. lb we show the adiabatic potential energy curves e(p) for the gerade symmetry obtained from the excited-state potential energy surface used by Kulander and Light [ 161. The potentials for quantum numbers n > 3 exhibit local minima which become gradually deeper and shift farther outward with increasing n. These minima support quasi-bound states which then give rise to resonance structures in the gerade part of the scattering wavefunction. These resonances are transparent in the cross sections for absorption out of gerade bound states, for example. They also occur in the gerade S matrix Sg, which however has no direct physical meaning. In order to approximately locate the resonances we performed simple one-dimensional scattering calculations for each adla-
Volume 124, number 6
CHEMICAL PHYSICS LETTERS
batic potential and analyzed the energy dependence of the phase of the elastic S matrix. The resonance energies are indicated as horizontal lines in fig. 1 and indeed good correlation with the structures in the absorption cross section is observed. Similar structures are also seen in the absorption spectrum for the (1,0) ground state (fig. 4 of ref. [ 161) which is also of gerade symmetry. Since the resonances are caused by the excited-state wavefunction they occur at the same energies independent of the particular ground state which is dissociated. Since the resonance energies are above the asymptotic limit of the respective adiabatic potential curve the resonances are of the shape-type. In fig. 2b we show the adiabatic potential curves for the ungerade symmetry. As known for Hg,for example, the ungerade curves start to exhibit potential wells at much higher quantum numbers, n 2 8 in the present case. The potential curves for the lower quantum numbers are monotonic with broad shoulders at around the transition state. Consequently, the cross sections for absorption out of ungerade states are smooth curves and do not exhibit resonance structures, at least not in the range of energies considered by Kulander and Light [ 161. Resonance structures might be expected for higher energies, E > 1.6 eV, which would be due to local minima in the potential curves for n > 9. Except for the resonances the (0,O) and (0,l) spectra are very similar because the p dependence of the corresponding bound state wavefunctions is roughly the same. The additional node for the (0,l) wavefunction is in the Q!direction. The true energy position and the shape of the resonances (peaks, dips or even more complicated profiles), particularly their width can only be inferred from exact calculations and not by the simple one-dimensional adiabatic calculations described above. The reactive resonances are sensitive to details of the potential energy surface, especially at the transition state. In test calculations we changed the Sato parameter of the LEPS surface [ 161 and obtained drastically different adiabatic potentials which probably would yield very different adsorption spectra. In this sense the experimental resolution of reactive scattering resonances in photodissociation processes would be true “transitionstate spectroscopy” [ 181. The experimental CO2 absorption spectrum for the band near 1300 A indeed shows a progression of distinct structures [14]. However, the experimental
14 March 1986
energy spacing is roughly half of that extracted from the calculations of Kulander and Light [ 161. As speculated by these authors either the model potential energy surface is inaccurate or a completely different mechanism is responsible for these distinct structures. Some time ago Pack [19] and later Heller [20] discussed vibrational structures in the total absorption spectrum for a symmetric triatomic molecule using a simple approximation to the total dissociation wavefunction on top of the barrier. Although the exact total cross section of Kulander and Light resembles that predicted by this simple model we believe that the structures have different origins in both cases. In the model of Pack the vibrational structure is due to the summation of very narrow but smooth (approximate) partial cross sections while in the exact calculation of Kulander and Light the structures are already present in the individual partial cross sections. The question whether the model of Pack and the resonance picture discussed in this Letter are equivalent should be investigated in future studies by analysing the exact two-dimensional dissociation wavefunction. The major difficulty in calculating photodissociation cross sections is the excited-state potential energy surface, which is not well known for most systems. The photodissociation of H20 in the first absorption band is a very fortunate case where we know the corresponding potential from ab initio calculations [21]. The agreement of approximate dynamical calculations with experimental data is encouraging [22-241. The ideas discussed in this Letter on the basis of model calculations will be further investigated for the dissociation of Hz0 but using the calculated potential energy surface. The influence of the rotational degree of freedom and the incoherent summation over several initial (primarily rotational) states, which can be considerable if the temperature of the parent gas is high, will be of particular interest to us. Recently, Clary and Henshaw [2.5] calculated three-dimensional absorption cross sections for H20 using a LEPS fit to some of the data of ref. [21] and indeed found resonance structures. Their absorption spectrum is, however, only in fair agreement with the experimental curve, which is very smooth in the wavelength region around 1600-1700 A.
507
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CHEMICAL PHYSICS LETTERS
We are grateful to Dr. R.T Pack (Los Alamos) for critical comments on the manuscript and D.C. Clary (Cambridge) for drawing our attention to the work of Launay and le Dourneuf.
Note added in proof After this Letter was accepted for publication we became aware that similar ideas had been formulated by Launay and le Dourneuf [26].
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