J . Phys. Chem. 1989, 93, 8 128-8 138
8128
FEATURE ARTICLE Theory of Dissociative Charge Exchange in Ion-Molecule Collisions V . Sidis Laboratoire des Collisons Atomiques et MolZkulaires (URA du CNRS No. 281). Universitd de Paris-Sud, B6t. 351, 91405 Orsay Cedex, France (Received: February 3, 1989; In Final Form: May 24, 1989) Dissociative charge exchange is one of the many processes resulting from vibronic transitions in ion-molecule collisions. Its particular interest lies in the molecular breakup that accompanies in specific conditions an electron-transfer process. The available theoretical descriptions of such reactions are surveyed. The salient findings of (i) the classical trajectory surface hopping model, (ii) the high-energy sudden approximation, (iii) the complex local potential, and (iv) the complex local interaction matrix approaches, as well as (v) an unprecedented vexact" coupled wavepackets description, are put to the fore. Attention is paid to the fundamental concepts conveyed by each treatment and emphasis is put on the relative importance of the various dynamical parameters of the considered reaction.
1. Introduction The microscopic phenomena of energy deposition, conversion, and flow in gases and the related conditions which determine both the Occurrence and the outcome of chemical reactions are largely governed by molecular collision processes. Therefore, the comprehensive understanding of binary collision processes of simple molecules is a fundamental prerequisite to physical chemistry studies of complicated media. Detailed knowledge of a given molecular encounter is provided by the complete characterization of the collision products as a function of scattering angle (with respect to the incident direction), collision energy, and initial internal state of the reactants; the importance of a given molecular collision process is measured by the size of its cross section.' The latter then enables one to derive reaction rate constants.2 The laws stemming from such studies thereby help building models which are hopefully transferable from the simplest to the most complicated systems. The present contribution is to be placed in such a context. Among the various processes which may occur in molecular collisions, dissociative charge exchange (DCE) has recently given rise to a succession of theoretical developments. These have been judged to be of sufficient general interest to be gathered in the present Feature Article. For simplicity, the molecular collisions treated here will merely involve an atom (A) and a diatomic molecule (BC). DCE is defined as the ion-neutral collision process
A+
+ BC(v,J)
-
A
+ (B+ + C), + Q(t)
(la)
(B
+ C), + A+ + Q(t)
(lb)
or
BC+(u,J)
+ A-
whereby an electron transfer between the collision partners induces a molecular breakup. In the specified reactions the molecular partner is prepared in the vibration rotation state (v,J) and the dissociation products separate with a relative kinetic energy t along an axis oriented in the direction e,@ with respect to the incident direction. The energy loss or gain (endo- or exoergicity) of the reaction is noted Q(c). Experimental Information. DCE reactions have been identified experimentally in a broad variety of conditions. For instance, it has long been known that charge exchange collisions of He+ ions ( I ) See e&: Johnson, R. E. Encyclopedia of Physical Science and Technology; Academic Press: New York, 1987; Vol. 2, p 224. (2) See e&.: Henchman, M. In Ion-Molecule Reactions; Franklin, E. J., Ed.; Plenum: New York, 1972; Vol. 1 , p 101.
0022-3654/89/2093-8 128$01.50/0
with diatomic) and triatomic4 molecules produce excited electronic states of the resulting molecular ions; a number of these states are either dissociative or predissociated and thereby entail the fragmentation of the molecular ion. Collisions of rare-gas molecule ions X2+with rare-gas atoms' and molecules6 give rise to charge exchange into the X2 electronic ground state; as the latter is essentially dissociative, the X2 molecule breaks up spontaneously. Similarly, charge exchange collisions between H2+ions with atoms whose ionization potential lies around 8 eV, as is the case of Mg,' populate with great efficiency the antibonding b3&+ state of H2 which dissociates into 2H(ls) atoms. (A similar process also occurs in HeH+ Mg encounters.8) Collisions of H2+,' HeH+,9 and O2+,I0molecular ions with alkali-metal atoms give rise to charge exchange into low-lying quasi-bound Rydberg states of the corresponding neutral molecules; a succession of studies" have shown that these quasi-bound states decay via predissociation by and/or radiative dissociation to repulsive electronic states. Interestingly, similar phenomena are expected for charge exchange collisions of molecular ions with metal surfaces.I2 Another noteworthy remark is that DCE also occurs in collisions of larger molecular ions (as, e.g., H3+ (ref 13a), CH3CO+ and CH3Cl+ (ref 13b)) with atoms; such a process may thereby be used to generate
+
(3) (a) Jones, E. G.; Wu, R. L. C.; Hughes, M.; Tiernan, T. 0.; Hopper, D. G. J . Chem. Phys. 1980,73,5631. (b) Dowek, D.; Dhuicq, D.; Pommier, J.; Tuan, V. N.; Sidis, V.; Barat, M. Phys. Reu. A 1981, 24, 2445. (c) Browning, R.; Latimer, C. J.;Gilbody, H. B. J . Phys. B 1969, 2, 533. (d) Bischof, G.;Linder, F. 2.Phys. D 1986, I , 303. (4) (a) McMillan, M. R.; Coplan, M. A. J . Chem. Phys. 1979, 71, 3063. (b) Anicich, V. G.; Laudenslager, J. B.; Huntress, W. T.; Futrell, J. H. J . Chem. Phys. 1979,67,4330. (c) Anicich, V. G.; Huntress, W. J. Survey of bimolecular ion-molecule reactions for use in modelling the chemistry of planetary atmospheres, cometary comae and interstellar clouds. JET Propulsion Laboratory, 1986; update of Ap. J. Suppl. 1977, 33, 495. (5) Maier, 11, W. B. J . Chem. Phys. 1975, 62, 4615. (6) Leventhal, J . J.; Earl, J. D.; Harris, H. H. Phys. Reu. Lett. 1975, 35, 719. (7) De Bruijn, D. P.; Neuteboom, J.; Sidis, V.; Los, J. Chem. Phys. 1984, 85, 233. (8) Van der Zande. W. J. Private communication. (9j Van der Zande,'W. J.; Koot, W.; De Bruijn, D. P.; Kubach, C. Phys. Reo. Letf. 1986. 55. 1219. ( l o ) Van der Zande, W. J.; Koot, W.; Peterson, J. R.; Los, J. Chem. Phys. Lett. 1987, 140, 175. (1 1) (a) Kubach, C.; Sidis, V.; Fussen, D.; Van der Zande, W. J. Chem. Phys. 1987, 117, 439. (b) Van der Zande, W. J.; Los, J.; Peterson, J. R.; Kubach, C. Chem. Phys. Left. 1988,149,14. (c) Van der Zande, W. J.; Koot, W.; Los, J.; Peterson, J. R. J . Chem. Phys. 1988, 89, 6758. (d) Van der Zande, W. J.; Koot, W.; Peterson, J. R.; Los, J. Chem. Phys. 1988, 126, 169. (12) (a) Eckstein, W.; Verbek, H.; Datz, S. Appl. Phys. Lett. 1975, 27, 527. (b) Heiland, W.; Beitat, U.;Taglauer, E. Phys. Rev. B 1977, 19, 1677. (c) Willerding, B.; Heiland, W.; Snowdon, K. J. Phys. Reu. Lett. 1984, 53, 203 1. ~~
~~~1
~~
0 1989 American Chemical Society
Feature Article and characterize molecules in neuralization-reionization experiments'3c*dwhich actually constitute a new trend in mass spectrometry. DCE has also been observed in collisions of cluster ions with atoms,14 particularly amazing is the recent finding that DCE collisions of Nan+ (n I21) cluster ions with Cs atoms lead to the evaporation of a monomer or at most a dimer.14 Finally, let us mention that collisions of multiply charged atomic ions with molecules also give rise to DCE;lS in particular when Ar2+ (ref 16), N7+or 07+ (ref 15) ions capture two electrons from H2 the resulting (H2)2+ion breaks up into two protons (Coulomb explosion). Until quite recently, most of the cited experiments were concerned with either of the following: identification of the products, elucidation of the charge-transfer mechanism, measurement of total cross sections (or reaction rate constants) for charge transfer, and determination of the relative importance of DCE. Only a few experiments have investigated the evolution of the energy spectrum of the DCE products and their angular distribution as functions of collision To date no experiment exists that also provides information on the DCE fragmentation spectra as functions of scattering angle of the collision partners. As is well-known, such an information is related (albeit sometimes in a complicated way) to the impact parameter dependence of the collision process and therefore it probes to some extent characteristic distances of closest approach of the colliding partners.' Theoretical Background. The theoretical treatment of a slow heavy particle (atoms, molecules) collision has been made possible by viewing it as a process of temporary formation and then breakup of a sort of supersystem consisting of all nuclei and electrons of the collision partners Thenceforth by making use of the Born and Oppenheimer separation of electronic and nuclear motions a collision problem splits into two parts:17J8 (i) determination of electronic wave functions and energies for fixed nuclei, and (ii) treatment of the nuclear motions in the average potentials thus generated by the electrons. It may be considered to date that both steps are well mastered for atomic collision problems. The knowledge acquired in that field forms the backbone of present theories of molecular collisions involving electron rearrangement processes. This is particularly the case of ion-molecule charge exchange which plays a central role in the considered DCE process. The above step (i) is in principle handled by methods of quantum chemistry more or less readapted to take into account some of the specifics of the collision problem, namely18 collision velocity, spatial extension of the nuclear motions, electronic shells involved, electronic transition mechanisms coming into play, characteristic zones where such transitions may take place, and the nature of the interactions inducing them. Though the related methodological aspects are well established, the treatment of this part of a collisional problem already constitutes such a tremendous computational task that it has been largely responsible for the belated developments of the theory of molecular collision processes. Still, certain charge exchange problems present special characteristics that may be exploited to palliate such difficulties. Indeed, it is well-known that the charge exchange processes which occur the most favorably and provide the largest cross sections (possibly reaching several tens of angstrom squared) actually involve small resonance energy defects, defined as the difference between the recombination energy of the ion and the ionization potential of ( 1 3) (a) Gellene, I.; Porter, R. F. J. Chem. Phys. 1983,79,5975. (b) Los, J.; Beijersbergen, J. H. M.; Kistemaker, J. In?.J. Mass. Spectrom. fon Phys., submitted for publication. (c) Terlouw, J. K.; Schwarz, H. Angew. Chem., fnt. Ed. Engl. 1987,26,805. (d) Wesdemiotis, C.; Mc Lafferty, F. W. Chem. Reo. 1987, 87, 485. (14) Brtchignac, C.; Cahuzac, Ph.; Leynier, J.; Pflaum, R.; Weiner, J. Phys. Rev. Lett. 1988, 61, 314. (1 5) Barat, M.; Gaboriaud, M. N.; Guilemot, L.; Roncin, P.; Laurent, H.; Andriamonje, S. J. Phys. B 1987, 20, 577. (16) Martin, S.; Heckman, V.; Stevens, J.; Pollack, E. In Electronic and Atomic Collisions; Abstracts of contributed papers of XIV ICPEAC; Coggiola, M., Huestis, D. L., Saxon, R. P., Eds.; Palo Alto, CA, 1985; p 581. (17) Tully, J. C. In Dynamics of Molecular Collisions;Miller, W. H., Ed.; Plenum: New York, 1976; p 217. ( 1 8) Sidis, V. In Collision Theory for Atoms and Molecules; Gianturco, F. A., Ed.; NATO-AS1 Series; Plenum: New York, 1989.
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8129
B
I
C
Figure 1. Set of internal coordinates R, r, and y for an A sional system.
+ BC colli-
Figure 2. Schematic view illustrating the origin of the y dependence of the exchange interaction. That interaction depends on the shape of the orbitals between which the active electron is being exchanged and, roughly speaking, it behaves as the overlap of these orbitals. It is seen from the figure that the exchange interaction between an s orbital of the atom and a a,,orbital of a homonuclear diatomic molecule should involve odd powers of cos y. The figure displays the particular situation of a molecule lying parallel to the incident direction in the case of a straight line classical trajectory with impact parameter p and velocity Y.
the neutral. Such nearly resonant charge exchange (NRCE) processes usually occur with small momentum transfer and frequently involve the sharing of a single (active) electron between two distant (inert) ion cores.I9 These features have actually been used to simplify the calculations implied in the above electronic step (i)20or to circumvent them by designing practical models.21*22 In order to maintain the length of the present subject within reasonable bounds, it will be assumed here that the related methods are actually familiar to the reader. Only some of the most general and salient aspects of this basic background will be recalled. Basic Characteristics of NRCE Problems. Briefly, at the large intermolecular distances where NRCE is likely to occur, the appropriate (clamped nuclei) description of the electronic wave functions is that steming from valence bond tbeory23v24where each constituent essentially keeps its individual electron cloud. This description provides so-called diabatic electronic wave functions in the sense that they display a smooth variation with nuclear g e ~ m e t r y l ~and J ~ *entail ~ ~ a nondiagonal matrix representation of the electronic Hamiltonian He, (as opposed to adiabatic wave functions which diagonalize He' but may display important changes with nuclear geometry, particularly near so-called pseudo crossing points17918). Diagonal matrix elements of Hel represent average electronic energies and constitute the potential energy surfaces (PES) which govern the nuclear motions. In the above-mentioned context, these PES are mere prolongations of the separated partner energy levels at finite (albeit large) intermolecular distances. To some extent these levels may be distorted by charge-quadrupole and charge-(permanent or induced) dipole perturbations. Offdiagonal matrix elements of He, involving two charge exchange states (A+ BC and A + BC+) constitute the so-called exchange interaction.'J9 The main characteristics of this interaction is its exponential behavior at large intermolecular distance (R).'*19*21922
+
(19) Smirnov, B. M. Asymptotic Methods in the Theory of Atomic Collisions; Atomizdat: Moscow, 1973 [in Russian]. (20) (a) Archirel, P. A.; Levy, B. Chem. Phys. 1986,106,51. (b) Grimbert, D.; Lassier-Covers, B.; Sidis, V. Chem. Phys. 1988, 124, 187. (21) (a) Bylkin, V. I.; Pakina, L. A.; Smirnov, B. M. Sou. Phys. JETP 1971,32,540. (b) Evseev, A. V.; Radtsig, A. A.; Smirnov, B. M. Sou. Phys. JETP 1980, 50, 283. (c) Evseev, A. V.; Radtsig, A. A.; Smirnov, B. M. J. Phys. B 1982, 15,4437. (22) Sidis, V.; De Bruijn, D. P. Chem. Phys. 1984, 85, 201. (23) Coulson, C. A. Valence; Oxford University Press: London, 1961. (24) Kubach, C.; Sidis, V. Phys. Reu. A 1976, 14, 153.
Sidis
8130 The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 I
1210
-
'
~
l,,LH;
2 8..-
H'+H ( M a )
r 08
04
1.2
2.
1.6
2.4
2.8
r(A
3.2
1
Figure 3. Electronic energy levels H,l(r) ( i = 1, 2) of the separated collision partners for the prototype H2+ + Mg system. Various quantities relevant to the discussion of direct DCE are indicated. The relative position of the H2+and H2* curves is determined by the difference between the vertical ionization potential of H2* (b3Z,) and that of Mg(3sZ IS). r*(c) is the classical turning point associated with the H-H dissociation with energy c in the potential HZ2(r).Qualitatively,this picture may be used to imagine how slices of the diabaric potential energy surfeces for fixed R and y would look like.
As a rule of thumb it may be expressed as A ( r ) exp[-X(r)Rlf(y). In this expression r-is the BC bond distance and y is the angle between the 7 and R vectors (Figure 1). The anisotropy factor fly) arises from the angular shapes of the atomic and molecular orbitals between which the active electron is being exchanged (Figure 2). The exponent X(r) involves the binding energy of the active electron in the molecule IBC(r)and in the atom IA as A = ( ( 2 I A ) I i 2 + (21Bc)1/2/2). More elaborate expressions of the exchange interaction may be found in the literature2'-22but their use here would unnecessarily complicate the matter. Direct and Indirect Dissociative Charge Exchange. Two types of dissociative charge exchange (DCE) processes may be distinguished. These will be illustrated in the case of the H2+(X2Zg+,v) Mg(3s2 IS) collisional system which will often serve us here as a reference prototype system. In direct DCE, charge exchange populates an antibonding state of the molecular partner which thereby dissociates spontaneously:
+
H2+ + Mg
-
[H2*(b3Z,+)]
+ Mg+(3s 2S)
-
2H(ls)
+ Mg+ (2a)
In indirect DCE, charge exchange first populates a bound state of the molecular partner which decays later on either via radiation of a photon toward a repulsive state, e.g. H ;
+ Mg
-
H2*(a3q', v )
lh H2*(b3C,')
+
Mg'(3s ' S )
-
(2b)
2H(ls)
or via predissociation, e.g. H2'
+
Mg
-t
H ~ * ( C ~ ~+ I ~Mg'(3~ . V ) 'S)
4
(2c)
mbcuhr rotaion
mation, one may consider that these curves represent cuts of the relevant diabatic potential energy hypersurfaces at certain fixed values of R and y for large values of R . The considered process is expected to occur the most favorably near the crossing of the attractive and repulsive curves showed in Figure 3 since, as discussed above, this situation exactly corresponds to resonant charge exchange. Figure 3 conveys the same image as that encountered in standard molecular predissociation problems. Still, in the present case the two intersecting diabatic states are only coupled at finite distances of approach (R) of the collision partners via the exchange interaction discussed above. Owing to these features direct dissociative near-resonant charge exchange (DD-NRCE) is seen to belong to the broader class of collision-induced predissociation (CIP) problems.2ss26 Having specified the main aspects characterizing DD-NRCE reactions and presented an overview of the general context in which the theoretical study is to be placed we are now ready to view some methods of treatment of the investigated process. 11. Trajectory Surface Hopping (TSH) Studies of DD-NRCE The oldest and probably best known method for treating molecular collision dynamics problems is the so-called quasi-classical trajectory method in which, owing to their important masses, all nuclei of the collisional system are treated according to classical mechanics.17 Briefly, given a potential energy surface (determined in the Born and Oppenheimer approximation as an average electronic energy) and a set of initial conditions for the nuclear positions and momenta, the classical equations of motions (e.g., Hamilton's equations) determine the outcome of an elemental collision event. Proper averaging over suitably chosen sets of initial conditions representing quantal distributions thereby provide the sought cross sections as well as the energy partitioning among the various nuclear degrees of freedom. In this form the method only applies to problems involving a single potential energy surface. Tully and Preston27have proposed an extension of +is approach to cases implying more than a single PES in order to enable one to describe molecular collision events involving electron transition processes. The method known as the trajectory surface hopping (TSH) approach proceeds exactly as in the single PES case until the current nuclear trajectory reaches a so-called transition seam where the system may hop from one PES to another. This transition seam often consists of an intersection between two PES (as e.g. the locus of the crossing point of Figure 3 when the pair R,y varies). As the hop from one PES to another is a quantal phenomenon associated with an electron transition, extra information is required to determine the probability of its occurrence. For the present purpose it will just suffice us to mention that many sound recipes17~26~28~B have been proposed to determine the hopping probability. When a hop actually occurs, the trajectory proceeds along the new surface after some possible readjustments of energy and momenta imposed by conservation laws. It keeps evolving until it encounters the transition seam anew or the collision ends. This method has been applied in some detail by Kuntz et a1.26 in a study of the competition between the following processes:
collision-induced predissociation (CIP or DD-NRCE)
H2*(b3CU+)--c 2H(ls)
Indirect DCE processes are thus seen to constitute ordinary cases of ion-molecule charge exchange. On the other hand, as a member of general problems involving a breakup process, direct DCE constitutes a still more complex issue. This is the reason why, from this point on, only direct DCE will be discussed. Moreover, for reasons already mentioned above, we will specifically consider direct DCE associated with near-resonant charge exchange (NRCE) processes. With these specifications, it is now important to have in mind the typical energy level disposition of the reactants and products. This is indicated in Figure 3 where one sees that the relevant electronic energy levels of the separated partners are merely the potential energy curves of the neutral and ionized molecular partner placed in such a way that their energy separation represents the vertical electronic resonance energy defect as a function of the molecule bond distance r. As a rough approxi-
-
He2+ + Ne
+ Ne+
(3a)
+ He + Ne
(3b)
2He
collision-induced dissociation (CID) He2+ + Ne ion transfer (IR)
He2+ + Ne
He+
HeNe'
+ He
(3c)
(25) Preston, R. K.; Thompson, D. L.; McLaughlin, D. R. J . Chem. Phys. 1978, 68, 13.
(26) Kuntz, P. J.; Kendrick, J.; Whitton, W. N. Chem. Phys. 1979, 38, 147. (27) Tully, J. C.; Preston, R. K. J . Chem. Phys. 1971, 55, 562. (28) Stine, J. R.; Muckerman, J. T.J . Phys. Chem. 1987, 91, 459 (29) Eaker, C. W. Chem. Phys. 1987,87, 4532.
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8131
Feature Article
V=OZeV
1.0
1.5 R l H e -He1
2.0
/=17eV
lmpocl Porameler
25
11.1
L:OB&
Figure 5. Theoretical impact parameter distributions from TSH calculations [ref 261 on the He2++ Ne collision. Results for dissociative charge exchange and collision-induceddissociation are shown by solid and dashed lines, respectively. The values of the corresponding cross sections (in A2) are indicated in each case. Reprinted with permission from ref 26. Copyright 1979 Elsevier Science Publishers. @ , b \'=0.8eV 0,a V = 1 . 7 e V
--30
--- Experimen!
'5
.-50C 20 Ln In
2
10
0
1
0 0 - 0 1.0
1.5 2.0 RlHe - HellAl
2.5
+
Figure 4. (a) Same as Figure 3 for the He2+ Ne collisional system. The vibrational energies V = 0.2, 0.8, and 1.7 eV are indicated with respect to the minimum of the He2+ground-statecurve. (b) Slice of the adiabatic (NeHe2)+potential energy hypersurface for collinear geometry (y = 0) at an Ne-He (closest neighbor) distance of 2.5 A. Vibrational
energies are referred to the same origin as in Figure 4a. It is noteworthy that, for V = 0.2 eV, the classical He-He vibrational motion may lead to dissociation by overcoming the barrier seen on the lowest adiabatic curve only for RNtHs 5 2.5 A. Reprinted with permission from ref 26. Copyright 1979 Elsevier Science Publishers. at relative collision energies in the range 2 eV IE I10 eV. IT was found to be negligible, and C I P largely dominated CID. Figures 4-7 borrowed from that work illustrate some salient aspects of the TSH dynamics of CIP. The main message conveyed by these data is the importance of the molecule vibrational degree of freedom in inducing CIP in cases when the transition seam runs parallel to the entrance channel of the surface.26 Figure 5 indeed shows that when the internal vibrational energy (V) of the molecular partner is such that the corresponding classical turning points encompass the diabatic crossing point of the relevant PES (Figure 4) the CIP process switches on at intermolecular distances as large as 4-5 A and produce cross sections in the range 10-30 A2 (Figure 6). On the contrary, CID requires more oiolent encounters (Figure 5 ) and thereby entails much smaller cross sections (Figure 6). If the molecule vibrational energy Vis insufficient to reach the diabatic curve crossing then the collision partners have to approach at smaller intermolecular distances (Figure 5 ) . This is understood by inspection of Figures 4 and 7: the smaller the R the larger the exchange interaction and thus the stronger splitting between the adiabatic PES at the diabatic curve crossing. Hence, with the decrease of intermolecular distance, the top of the barrier on the lowest adiabatic PES lowers down and may be overcome during the vibration of the molecule. Finally, Figure 7 shows a few trajectories which illustrate the simultaneous H e H e
>.e/;H , e;
He.,
P u;cr:
_------- _ _ _ _
, 1
2 L 6 8 10 Relative Collision Energy E (eV1
Figure 6. Theoretical total cross sections from TSH calculations [ref 261 for DCE (circles) and CID (triangles) in the Hez++ Ne collision. Results obtained for two internal vibrational energies of the molecular ion are shown. The corresponding experimental data of ref 5 are represented by the dashed line. (In that experiment the He2+beam contained a broad vibrational distribution.) The Langevin cross section (see e.g. ref 1) often referred to in ion-molecule reaction studies (full line) is seen to be inappropriate for DCE in the considered energy range: the reason is that the actual average impact parameter for DCE (see Figure 5) is much larger than the Langevin orbiting radius. Reprinted with permission from ref 26. Copyright 1979 Elsevier Science Publishers.
and Ne-He motions during the collision. Several cases are sampled: (i) He-He vibration in the diabatic He2+-like attractive curve, (ii) He-He vibration in the upper adiabatic potential curve resulting from the avoidance of the diabatic curve crossing, (iii) He-He breakup by overcoming the barrier on the lowest adiabatic potential curve, and (iv) He-He breakup by moving along the diabatic He2-like repulsive curve. The TSH study of Kuntz et al. thus provides the basic classical concepts that enable one to understand how DD-NRCE might proceed and which specific features determine its dynamics at low collision energies. 111. Dissociative Charge Exchange at High Energies: The Sudden Approximation In the early 1980s a few attempts flourished which aimed at incorporating an increasing amount of quantal ingredients in the theory of nonadiabatic molecular collisions30(Le., involving rovibronic transitions). Among these, the common trajectory approach30 describes only the relative motion classically whereas all of the internal degrees of freedom (Le., electronic + vibration + (30) Sidis, V. In Advances in Atomic, Molecular and Optical Physics; Bederson, B.; Bates, D. R., Eds.; Academic Press: New York, 1989; Vol. 26, in press.
Sidis
8132 The Journal of Physical Chemistry, Vol. 93, No. 25, 1989
f
1 1.4 -
3
a. 1.0
la.
c
1.0
,
!
:
!
!
!
.
I
a description conveys is that, contrary to the low-energy view of section 11, the dissociative charge exchange process (and more generally nonadiabatic transitions) at high energies are primarily induced by the relative motion since in energetic collisions the molecular partner may hardly vibrate or rotate. In order to get some deeper insight into what determines the characteristics of a DD-NRCE fragmentation spectrum at kiloelectronvolt energies one may represent in eq 6 the continuum vibration function F(t,r)describing the dissociation in the final electronic state by resorting to a 6 function approximation22 F(t,r) = 6(r-r*(€))ldH22/drlr.(t)-1/2
I
r * ( t ) is the classical turning point of the molecule motion at the energy t in the repulsive potential H22(r)(i.e. H22(r*)= t, Figure 3). One then gets the probability per unit energy for the dissociation of the molecule with the energy t while the fragments are ejected at an angle 0,+ relative to the incident direction asz2
1 0
1
2 Time
3
1
5
6
(7)
7
I programme iinitsl
Figure 7. Samples of time dependences of the He-He and Ne-He distances in He2' + Ne collisions. A few TSH trajectories with total (Le., translational + vibrational) energy of 6 eV are shown. The broken line (referred to the left-hand scale) locates the pseudocrossing seam (see Figure 4a.b). Open circles indicate that the system remains on the same adiabatic surface; closed circles indicate that it hops to the other adinbatic surface. For example, in the top view the system hops twice while oscillating in the diabatic He2'-like potential of Figure 4a and, then it remains in the upper adiabatic curve of Figure 4b for two passages across the seam, and finally it oscillates again in the diabatic He2+-likepotential. Clearly that trajectory does not lead to dissociation. In the next view, the initial He2+vibrational energy is insufficient to overcome the barrier in the lowest adiabatic curve of Figure 4b for Rp+Hs > 2.5 A; the system then oscillates in the attractive part of that curve. The increase of the exchange interaction with the approach of the Ne atom lowers that barrier and thereby enables (for RNtHc I 2.5 A) the adiabatic dissociation along the repulsive part of the lowest curve of Figure 4b. The other two views may be analyzed likewise. Reprinted with permission from ref 26. Copyright 1979 Elsevier Science Publishers.
P(t,0,+) = lG"l(r*(4)fll(0,+). 4 2 1 ( r * ( ~ ) ~ ~ ~ ~ ~ 1 ~ 1 d ~ 2 2 / d r(8) lr.(t)-l
G o l ( r )is the initial bound vibration function of the molecule in the initial electronic state 1 and R1the associated initial rotation function (for molecular partners initially oriented at random, 2,(0,+)= ( ~ T ) - I / ~ ) One . sees that the profile of a DD-NRCE spectrum at high energies depends on three functions, two of which, Gul(r)and Hzz(r),characterize the isolated molecule in the initial is deterand final states while the third one (1.421(r*(e),0,+)12) mined by the NRCE collision dynamics. If anisotropy (e,+ dependences) may be neglected then an estimate of 1A2112 may be obtained from the Demkov-Nikitin3I model for NRCE problems as
(-)
l.421(r)12 = sech2 nAH(r) sin2 l : A E ( r , R ( t ) ) d t (9) "2
rotation) are treated quantally. The problem then amounts to where solving a time-dependent (-like) Schriidinger e q u a t i ~ n l ~ ~ ' ~ ~ ~ ~ (4) driven by the common classical trajectory f i ( t ) . P is the (internal) = HeI rovibronic wave function of the collisinal system and Hi,, Ti is the internal Hamiltonian comprising the electronic Hamiltonian ( H e ] ) and the rovibration kinetic energy of the molecular partner Ti. ij designates the ele_ctronic coordinates collectively, r' the molecule bond vector, and R the intermolecular vector. Usually, P is expanded in terms of rovibronic wave functions.
is the (vertical) electronic resonance energy defect
+
'
= x 4 ] ( z ; r > R ? ys x)u , ( 7 ) a J U , ( f ) J
(5)
"J
H I 2 ( r , R )= H Z 1 ( r , R= ) A ( r ) exp[-X(r)R]
is the exchange interaction and y R the radial collision velocity at the critical transition point R, where
-
where .A2, is the probability amplitude associated with the electronic transition 1- 2 holding both the molecule vibration and rotation frozen. Clearly, the important message that such
(12)
2H12(rrRc)= W r )
For a straight line trajectory i ( t ) = p'
4j represents an electronic state determined for clamped nuclei
(it thus depends parametrically on r, R , and y,Figure 1); xu,is a rovibration wave function of the molecule appropriate for channel j . The inner summation is a discrete one for bound vibrational states or a continuous one for dissociative states (see section IV). Historically, application of this theoretical framework to dissociative charge exchange was first undertaken with a view to study the three reactions 2a,b,c at kiloelectronvolt energies.22 In this ~ s) are much shorter energy range typical collision times T , ~(I than both characteristic vibration ( T z~ ~ s) and rotation (T,, z IO-" s) times. A collisional electronic transition process may then be described as if the nuclei of the molecule are held fixed in space. This situation greatly simplifies the treatment of general nonadiabatic t r a n s i t i o n ~ . ~Indeed ~ , ~ ~ it* may ~ ~ be shown that the probability of a transition from the initial rovibronic state lu, to the final state 2u2 is given by'8922,30
(1 lb)
+k
(134
where p designates the impact parameter and v the collision velocity. One sees that the above expression of 1.A2,I2is a bellshaped function whose maximum occurs for AH(r) = 0 and whose width at half-maximum evolves as Xv,. This bell-shaped function is distorted in favor of small values of e in the DD-NRCE spectrum owing to the factor ldH22/drlr*(cylin eq 8. The above sudden approximation (eq 6) has been applied to the study of dissociative charge exchange in the H2+ Mg collision (eq 2a,b,c) at kiloelectronvolt energies by Sidis and de Bruijn.22 Realistic expressions of the exchange interaction were obtained by making reference to the so-called "asymptotic theory of charge
+
(31) (a) Demkov, YU. N. Sou. Phys. JETP 1964, 18, 138. (b) Nikitin,
E,E,in Aduances in euantum chemistry: Academic: N~~ Y&, 5 , p 135.
1970; vel,
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8133
Feature Article 1.5
,
1
v=o
1 A
0 2 4 6 Released Kinetic Energy
0
2
6
6 Released Kinetic Energy U e V )
4
IO
8 E
(eV)
Figure 9. Same as Figure 8a except that the H2+ beam is prepared in specific vibrational states (u = 0 or 1). The experimental spectra of ref 7 are shown as full lines (for the bottom view the data contain some admixture of u = 2). Results of the sudden approximation of section 111 are shown by the dashed line. The dotted line shows the result obtained by disregarding the r dependence of the vertical charge exchange probabilities (1Aj1I2).The importance of the three factors that determine the shape of the spectrum in the sudden approximation of section 111 are clearly seen. Reprinted with permission from ref 7. Copyright 1984 Elsevier Science Publishers. E=5keV
c
3
0
2 4 6 8 Released Kinelic Energy E(cV)
Figure 8. (a) Energy spectrum of H atoms produced by DCE in the H2+ + Mg collision at a laboratory energy of 5 keV. The two H atoms are scattered in the direction 0 = 80" with respect to the incident direction. The spectrum results from both integration over impact parameter and averaging over the azimuthal angle a. The experimental data of ref 7 are represented by the dashed curve. Theoretical results (ref 22) obtained within the sudden approximation framework of section I11 are shown as full and dashed-dotted lines. The former are obtained with an Airyfunction approximation of the continuum wave function describing the H-H dissociation whereas the latter involve a &function approximation. The vibrational distribution in the H2+ ion beam is assumed to be that suggested by Von Bush and Dunn (ref 32). The continuous part of the spectrum (e 5 7 eV) is seen to result from two contributions: (a) radiative dissociation arising from charge exchange into H2*(a3Z,+), eq 2b in text, and (b) direct dissociation arising from charge exchange into H2*(b32,'), eq 2a in text. The peaks seen for t > 7 eV are due to dissociation subsequent to neutralisation into H2*(c3nU),eq 2c in text. (This figure is taken from ref 22.) (b) Typical shape of the energy spectrum of H atoms one would obtain for process 2a in text with a charge exchange probability 1Aj,I2independent of bond distance r. (The vibrational population of the H2+ beam is the same as in Figure 8a.) Comparison of curve b in Figure 8a and the present figure clearly demonstrates the importance of the r dependence of the vertical charge exchange probabilities in the sudden approximation of section 111. Reprinted with permission from ref 7. Copyright 1984 Elsevier Science Publishers.
e ~ c h a n g e " and ~ ~ ~the * ~close ~ coupling equations determining the amplitudes Ajl(F)= C,(i,t++m), with
Cj(r,t+-)
= bjl
were solved numerically. Figures 8-1 0 illustrate sample results from that work. The contributions of direct and indirect dissociative charge exchange are shown in Figure Sa for a Von Bush and Dunn3* initial vibrational distribution of the H2+ions. (32) Von Bush, F.; Dum, G.H.Phys. Rev. A 1972, 5, 1726.
3
$
E .6eV
a-
1
E = 8eL
U
4-
:&
Na
0 O'
1
-0
30' 60' 90'
LO' 30' 60' 90'
Figure 10. Measured (full lines) anisotropy in the scattering angle 8 for H,' colliding with different targets at Elsb= 5 keV for different H-H dissociation energies 6. The theoretical results of ref 22 for the Mg target are shown as closed triangles. Reprinted with permission from ref 7. Copyright 1984 Elsevier Science Publishers.
Comparison of the direct contribution in this figure with the curve shown in Figure Sb reveals the importance of the dynamical factor .Az1in the above formulas. Figure 9 shows a dissociative charge exchange spectrum obtained with state selected H2+ions: the roles of the lFvl(r*(t))12, l.A211z,and ldH22/drlr.(c)-1form factors are clearly seen. Finally, Figure 10 shows some angular distributions of the fragments for certain dissociation energies. IV. The Local Complex Potential Approximation for DD-NRCE at Low Energy Let us come back to the 1-100-eV energy range already considered in section 11. Contrary to the point of view presented there, we wish in this section to treat the internal rovibronic motion quantally and the relative motion c l a s ~ i c a l l yas , ~ already ~ explained in the beginning of section 111. However, at the low energies considered here the typical collision time T~~~ is comparable with or even longer than characteristic vibration times svib of the molecular partner (the molecular rotation periods remain Hence, contrary to the much longer than both sail and sVib). conditions described in section 111, the molecule may vibrate and thence dissociate during the very collision event (see e.g. Figure 7) though it can barely rotate. In this context, the minimal quantal
8134
Sidis
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989
Figure 11. Energy diagram relevant to the discussion of DD-NRCE in the Hif + Mg collision at low energy ( E 5 100 eV). The process involves an initially bound vibrational state E , , (e.g., u = 0) embedded in a continuum t (shaded area). At finite intermolecular distance R the bound state 1 1 , ~ )interacts with the continuum 12,t) and thereby acquires a local predissociation width rv(R). Reprinted with permission from ref 33. Copyright 1987 Elsevier Science Publishers.
description of a DD-NRCE process requires the treatment of the interaction between an initially bound vibrational state u associated with an attractive potential energy curve of the molecule (Hi,) and the dissociation continuum t associated with the repulsive molecular electronic state (Ifz2)that results from the electrontransfer process (Figure 11). In this picture the electronic potential energy curves as such disappear and only a bound vibrational state embedded in a continuum remains. At finite intermolecular distances R the discrete vibronic state I l p ) interacts with the continuum 1 2 , ~ )and thereby acquires a local decay lifetime r;I( R,y) (and possibly a level ~ h i f t ) ;the ~ ~dynamical .~~ aspect of the DD-NRCE process put forward here is then the requirement that the time spent by the system near the point R,y matches the corresponding decay lifetime. The mathematical formulation of the dynamics of DD-NRCE is based on eq 4 and 5. The latter may be reexpressed as
Qzjdc
r
result is obtained within the local complex potential approach of section 1V (ref 34). The exchange interaction is assumed to be isotropic and the energy differences IE,, - €1 are assumed to be independent of both R and y. Reprinted with permission from ref 33. Copyright 1987 Elsevier Science Publishers.
the discrete state by adding to El, the imaginary energy -ir,,/2, With33,34 r,(R,y) = 2alh,(t=El,,R,y)12 (18) The survival probability amplitude is thus given by the simple decay law
which, when inserted back into eq 16, enables one to calculate the DD-NRCE spectrum. Again, to get deeper insight into what determines the characteristics of such a spectrum one can make use of the &function approximation (eq 7) for the continuum function F(c,r) to obtain the expression of the bound-continuum interaction appearing in eq 17
b(c,t) exp(-iS‘c -_ dt’)]72,(0,@) (15)
Most labels have the same meaning as in section 111: are the relevant electronic states, R 1is the initial rotation wave function of the molecule (its factorization out of the expansion is made legitimate by the slowness of molecular rotation compared to all other motions), and G, and F a r e respectively the initial bound vibration wave function and the continuum wave function representing the molecule dissociation at energy t; a, is the probability amplitude for the survival of the system in state l l p ) whereas b(t,t) is the probability amplitude that the system dissociates with energy t . Ib(t,t++m)12 thus provides the energy spectrum of the dissociation products. Insertion of eq 15 in the time-dependent Schrodinger equation (eq 4) yields an infinite continuous ser of coupled equations that may hardly be solved exactly (at least in this form, see section VI). A first-order solution may however be ~ b t a i n e d by ~ ~first . ~ ~solving formally for b(c,t), Le. ib(t,t) = Jldt’h,(c,R,y)
16.5
Figure 12. Perspective view of a DD-NRCE spectrum Ib(t,p,f-*m)12 for an (H2+ + Mg)-like prototype collision at Elab= 10 eV (p is the impact parameter and c the relative kinetic energy of the H fragments). This
a,(t) e x p [ - i ~ ~ ( E l u - cd) ~ ] (16)
h , ( t A y ) = (~(c,r)lHzi(r,R,y)IG,(r))
(17)
and then making the approximation that the major contribution t o this integral arises from the range t’= t . The result of this approximation is to replace the whole effect ofthe continuum on (33) Sidis, V.; Courbin-Gaussorgues, C. Chem. Phys. 1987, 1 1 1 , 287. (34) Sidis, V.; Grimbert, D.; Courbin-Gaussorgues, C. In Electronic and Atomic Collisions; Gilbody, H. B., Newell, W. R., Read, F. H., Smith, A. C. H., Eds.; Elsevier: New York, 1988; p 485.
Similarly to results of section 111, one finds that the DD-NRCE spectrum writes as the product of the form factor G;(r*(t)). IdH22/drlr.(t)-1 times a function characteristic of the collision dynamics. Sidis and Courbin-Ga~ssorgues~~ have undertaken a systematic analysis, in the above framework, of typical dependences of the DD-NRCE spectra on impact parameter, collision energy, and molecular orientation. Examples were also provided which showed the effect of both the bound-continuum energy differences A,(€) = It - E,,I and coupling strength on the general appearance of these spectra. Samples taken from that work are shown in Figures 12-15. For A,(€) independent of both R and y and for an isotropic exchange interaction the general appearance of the DD-NRCE spectrum is as shown in Figure 12. The spectrum is sharply peaked at resonance: c = t, = E l , for large impact parameters. Both the decrease of impact parameter and the increase of the collision velocity broaden the spectrum (Figures 12 and 13). The origin of these trends could be understood in the framework of weak and strong decay models.33 For example, for weak decay probabilities (a, z 1) eq 16 may be solved in closed analytical form when H 2 1in eq 20 has an R dependence of the form R* (exp(-XR). In such cases, except for the “static” form factors G,2(r*) and IdHz2/drlr.-l,the “dynamical” part of the spectrum can be expressed in terms of Bessel functions of the variable:
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8135
Feature Article
7
v
8.75
Figure 13. Slices of DD-NRCE spectra similar to that of Figure 12 at a large impact parameter (p = 15 ao)for Elab = 1 eV (full line), 10 eV
10,
7 3.1 hr
15
(dashed line), and 100 eV (dotted line). This figure illustrates the general trend of the spectra for weak decay probabilities. Note the decrease of the peak height as E-' and the broadening of the spectrum with Y. Reprinted with permission from ref 33. Copyright 1987 Elsevier Science Publishers. 44 a.u.
16
2 3.1
Figure 15. Same as Figure 14 except that the energy differences A = c - E,, are assumed to depend on R as A(R) = A(-) - 4R exp(-0.5R)/30 and the exchange interaction is divided by IO. Reprinted with permission from ref 33. Copyright 1987 Elsevier Science Publishers.
4.11
52 a.u.
change interaction involves the anisotropy factor f(y) = cos y. A trench is now seen on the rising front of the spectrum for that class of molecules having their axis lying parallel to the incident direction. To understand this new feature it is worth recalling that, for molecules having their axis fixed in space and oriented along the direction €),@-with respect to the incident direction, the relative angle y = (?,I?) (Figure 2) evolves as
2
+
cos y ( t ) = R'[psin 0 cos (@ - @J
4.11
Figure 14. Same as Figure 12 except that the exchange interaction is anisotropic; its y dependence is given by fly) = cos y. Two views are shown: 0 (noted 0, in the figure) = 0" and 90". (The results are averaged over the azimuthal angle @.) For 0 = 0" (Le., H fragments resulting from H2+molecules lying parallel to the incident direction, see Figure 2) a trench is seen at the resonance energy e, on the onset range of the spectrum. This feature is characteristicof weak decay probabilities when the exchange interaction depends on odd powers of cos y. Reprinted with permission from ref 33. Copyright 1987 Elsevier Science
Publishers. For large impact parameters p , the width at half-maximum (w) of such a DD-NRCE spectrum was found to behave as33 w
2(ln 2)(X/p)'12v
(22)
and the height of the spectrum at resonance to behave as a function of energy as E-I. This is clearly seen in Figure 13. As an additional feature one may mention that the multiplication of the exchange interaction by a reduction factor has the effect of pushing the whole spectrum toward small impact parameters. Moreover, the effect of varying the initial vibrational state is to move the peak at the new position of the resonance and to change the form factor G?(r*(t)). It should be pointed out, however, that the ventral and nodal structures of this function are only revealed when the spectrum is broad enough so that the relation r * ( ~is) completely reflected in the spectrum. Figure 14 shows how the spectrum is modified when the ex-
VI
cos 01
(23)
for a straight line trajectory (eq 13a) having an impact parameter vector p' = (p cos @p, p sin @p, 0 ) . As a consequence for 0 = 0, V@,cos y is an odd function of time. Equation 16 shows that, at large impact parameters where the weak decay model applies (a, s l), b(6,t-a) vanishes precisely at A"(€) = 0 when the interaction H,,is an odd function of time. (In ref 33 another argument is proposed, namely that in its rising front, the 0 = 0 spectrum is proportional to the derivative with respect to A,(€) of the 0 = 90° spectrum.) In the last place, Figure 15 shows the trends of DD-NRCE spectra when A,(t) is a function of intermolecular distance (e.g., AJc) = 6 - E,,(R) = BRc exp(-AR)). A few features are noteworthy. Firstly, the onset range of the spectra directly reflects the function EI,(R=p). Secondly, the oscillations appearing behind the rising front of the spectra are due to interference between two waves: one corresponds to the breakup of the molecule with energy t as the collision partners are approaching each other and the other its breakup with the same energy as the collision partners recede. Clearly, this interference phenomenon can only exist when the decay probability is not too strong for the system may survive until the receding stage of the collision. Finally, the appearance of the spectrum depends upon molecule orientation: the wavelet seen at the foot of the rising front of the spectrum and the rapid decrease of the oscillation amplitudes at small impact parameter for 8 = 0' and 90' respectively are characteristic of these orientation~.~~
Sidis
8136 The Journal of Physical Chemistry, Vol. 93, No. 25, 1989
1
%3
h,I+
V v.= .0,1,2,3 . initial =
0
(3
1 .bo
3.00
Figure 16. Survival 1 0 ~ 1and ~ vibrational laJ2 ( D = I , 2, 3) probabilities for DD-NRCE as determined from the complex interaction matrix approach of section V. The considered prototype system is the same as that of Figure 12 except that the collision energy is Elab= 100 eV and the exchange interaction is multiplied by 0.75. Reprinted with permission from ref 34. Copyright 1988 Elsevier.
V. The Local Complex Interaction Matrix: Vibrational Excitation via a Dissociation Continuum in DD-NRCE Processes Since the model of section IV predicts a broadening of the spectra with both the decrease of impact parameter and the increase of collision velocity, the energy range of the populated portion of the dissociation continuum (e) should overlap a few vibrational levels 11,~’) in the neighborhood of the incident state 11,u). This should result in vibrational excitation (Il,v) 11,u‘)) and in a concomitant modification of the DD-NRCEspectrum profile. The description of these additional effects can still be attempted along the same lines as in section IV except that the discrete component of the total wave function Q in eq 15 now involves a summation over all states 11 ,u’) that are likely to be populated.34 The use of the first-order approximation of section IV has the effect of adding an imaginary term not only to the energy of each discrete state as in eq 18 but also to each of their coupling matrix elements.34 Thus, even if the discrete vibrational = O), states had no direct mutual coupling (Le., if (GIdlHIIIGldJ) an interaction does appear between them via the coupling of each discrete state with the continuum. This is immediately seen by inspection of the coupled equations that govern the evolution of the discrete probability amplitudes.
-
dad z C-irhd(E,uft,R,y) hUtt(E,,Jt,y)ad’ e x p [ - i l l ( E d , dt u~~
4.5
Figure 17. Same as Figure 16 for the DD-NRCE spectrum.
limited range of applicability. Firstly, it is not clear how these approaches can at all be made to match the high-energy regime described in section 111. Secondly, because they are designed to describe the bound continuum interactions to first order only, multiple bound-continuum transitions, which may occur in strong coupling cases, are not taken into account. Strong electronic coupling cases may be suspected to constitute the “Achilles heel” of the mentioned descriptions. Indeed, these are built on the assumption that the underlying electronic diabatic energy curves shown in Figure 11 are sufficiently meaningful so that one may select associated vibrational wave functions. This is generally true at large intermolecular distances of approach where the exchange interaction H21is exponentially small. However, at small distances of approach H,, may become so large that neither the electronic diabatic basis nor the corresponding vibration expansion basis is particularly well suited anymore; as a well-known consequence, the system will spread over a large number of basis states. If the only way of treating the problem were a vibronic close-coupling formulation based on expansions of the form given in eq 5, the quantum mechanical formulation of DD-NRCE would be stuck at this point owing to computational hindrances. A way of overcoming such difficulties has been lately proposed by Gauyacq and S i d i ~ .It~ simply ~ consists, after the factorization of the initial as done in eq 15, of replacing the rotation wave function %,(e,@) inner sums in eq 5 by vibration wuuepackets Pj(r,t) that are each treated as a whole, i.e.
i-
E,) d r ] (24) The continuum amplitudes b(c,t) are given by eq 16 modified by summation sign in front of the integral. inserting a It is worth mentioning that when only two discrete states are considered, eq 24 may be solved in closed analytical form.35 Solution of this problem actually stands as an extension of the Demkov-Nikitin two state exponential model3I to imaginary couplings. Samples taken from the work of Sidis et al.34 (Figures 16 and 17) show the effect of taking more than a single discrete s t a t e into consideration. Two main new features are observed at small impact parameters: (i) vibrational excitation via the dissociation continuum has a sizeable probability (Figure 16), (ii) the dissociation spectrum slightly moves towards smaller energies than that of the initial discrete state and structure appears near the other resonances (Figure 17). VI. “Exact” Novel Treatment of DD-NRCE: The Coupled Wavepackets Description However wealthy in varied new information and physical insight the approaches of sections IV and V may be, they still have a (35) Sidis, V.;Grimbert, D.; Courbin-Gaussorgues, C. J . Phys. B 1988,
21, 2879.
so that with eq 4
I.L is the reduced mass of the molecular partner and the initial condition is
Pk(rrt=tinitiai) = 61/$u(r) exP(+Elu~initiaJ with u being the selected incident vibrational state of the molecule. The set of coupled partial differential equations (26) may be solved numerically by using finite difference method^.^'*^^ The wavepackets are determined at the nodes of a two-dimensional grid tK,rL(1 IK 5 N,, 1 I L IN,). At each time step (tK) the second-order derivative is calculated at each point r, of the grid by using a five-point formula39 and the propagation is achieved (36) (a) Gauyacq, J. P.; Sidis, V. Presented at 3rd European Conference on Atomic and Molecular Physics, Bordeaux, France, 1989. (b) Gauyacq, J. P.; Sidis, V. Europhys. Lett. 1989, 10, 225. (37) (a) Gauyacq, J. P. J . Phys. B 1980,13,4417. (b) Gauyacq, J . P. J . Phys. B 1980, 13, L501. (38) See e.g.: Press, W. H.; Flannery, B. P.; Teukolsky, S. A,; Wetterling, W. T. Numerical Recipes: Cambridge University Press: London, 1986.
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8137
Feature Article
-1 0 r,a, 1 Figure 18. Potential energy curves Ei = Ifii(?) ( i = 1, 2) representing slices of the diabatic potential energy surfaces used in the coupled wavepackets calculations of section VI. The considered two cases correspond to the repulsive curves I and 11. Reprinted with permission from ref 36. Copyright I989 Les Editions de Physique. -P
o.8
t
Figure 20. DD-NRCE spectrum for the same prototype system as that of Figure 19 except that the repulsive energy curve is curve I1 of Figure 18. Note that for large impact parameters the spectrum is peaked at the resonance energy (c, = Elo= 3.25 eV) as in Figures 12 and 13 whereas an important displacement of the main peak occurs at small impact parameters-see also Figure 21. Reprinted with permission from ref 36. Copyright 1989 Les Editions de Physique.
A 1
2
3
P,a,
Figure 19. Survival 1oOl2and total vibrational excitation Culaulz probabilities determined by coupled wavepackets calculations (ref 36) at ElPb = 20 eV for an (H2+ Mg)-like prototype system with energy curves corresponding to case I of Figure 18. The calculations assume an isotropic exchange interaction of the form 0.5 exp(-R). The survival probability determined using the local complex potential (LCP) approximation is shown for comparison. Reprinted with permission from ref 36. Copyright 1989 Les Editions de Physique.
+
by using eq 26 in a two-term Taylor expansion. Instabilities of the algorithm are avoided by carefully selecting the step sizes At and Ar (e&, At = fbA$,f< 1)38and preventing escaping wavepackets to reflect from the boundaries of the grid. Sample results of such “exact” calculations36are shown in Figures 18-22. The relative disposition of the curves Hjj(j= 1,2) is shown in Figure 18. In these calculations Hzl = H l z = 0.5 exp(-R); this choice enables one to probe weak and strong coupling conditions by varying the impact parameter. Figure 19 shows the survival probability in the initial (u = 0) discrete state and the vibrational excitation probability ( u > 0) for a collision energy Elab= 20 eV and the repulsive curve I in Figure 18. The behavior of the probability curves with the decrease of impact parameter is clearly unpredictable whhin the LCP approach of section IV. (Though for curve I1 in Figure 18 the ”exact” and LCP curves are found to lie closer to each other, Figure 19 demonstrates the limitation of the LCP approach.) The result of Figure 19 for small impact parameters may be understood qualitatively by referring to the corresponding adiabatic energy curves. As the exchange interaction grows up, the crossing in Figure 18 gets more and more avoided in the adiabatic picture and, roughly speaking, the wavepacket splits into two parts: one “follows” the lowest (unbound) adiabatic energy curve whereas the other follows the upper (bound) curve. For case I, an important part of the initial wavepacket follows the upper state and is thereby preserved against dissociation; vibrational excitation is readily understood in terms of the different shapes of the initial H I ,potential curve and the upper adiabatic curve. For case I1 an important part of the initial wavepacket follows the lowest adiabatic curve thereby leading to an important dissociation probability. Figure 20 shows an “exact” DD-NRCE spectrum obtained for case I1 at Elab= 20 eV. The onset range of the spectrum 3 a. Ip I6.4 a, behaves as predicted in section IV. Yet at similar impact parameters a new trend is observed, namely, an important displacement of the main peak ~
(39) See e&: Abramowitz, 1.; Stegun, A. Handbook of Mathematical Functions: Dover, New York, 1972.
e.ee
1.00
2-08
3.00
E,&
Figure 21. Comparison of a slice of the perspective view of Figure 20 for p = 0.2 a. (dotted curve) with the corresponding spectrum obtained by letting the initial wavepacket P I of section VI evolve in the lowest adiabatic potential surface resulting from the diagonalization of the Hi,(r,R) matrix (full curve).
Figure 22. Same as Figure 20 except that the collision energy is El.b = 500 eV. This figure reminds us of curve b in Figure 8a. The structure seen as a function of p is related to the secalled Stueckelberg interference pattern (see e.g. ref 1) associated with two electronic potential energy surfaces.
toward small dissociation energies. That the above interpretation in terms of adiahatic energy curves is appropriate at small impact parameters is attested by Figure 21. This figure compares a slice of the perspective view of Figure 20 at p = 0.2 no and the result of a single adiabatic channel calculaton which determines for p = 0.2 a. the evolution of the part of the initial wavepacket (say P l ( r , t ) )that follows the lowest adiabatic curve only, Le.:
is the lowest electronic adiabatic eigenvector). Finally, Figure 22 shows a result obtained at Elab= 500 eV. The spectrum displays all of the characteristics expected from the discussion
J. Phys. Chem. 1989, 93, 8138-8142
8138
of section 111. A particular feature which is absent in the LCP approach of section IV is the oscillatory behavior seen at small impact parameter. This behavior is related to the so-called Stueckelberg interference pattern arising in the two-state Demkov model)’ and is manifested by the sin2 factor in eq 9. This last result shows the ability of the coupled-wavepackets treatment to provide a unified description of both low- and high-energy regimes.
VII. Conclusion Several facets of the DD-NRCE (or CIP) phenomenon which have progressively emerged during the 1979-89 decade have been surveyed. Particular emphasis has been placed on the steps which have led to the quantum mechanical treatment of the internal motion. Within the common classical trajectory framework the presented theoretical tools allow both for an “exact” description of the evolution of a system undergoing DD-NRCE and a detailed understanding of the conditions governing its dynamics. The respective roles of vibrational and translational energies are clarified. The dependence upon impact parameter and its relation with coupling strength have been uncovered and spectacular effects
of molecular orientation have been revealed. The interest of the studies reviewed herein is that they tackle simultaneously two difficult problems: the dynamics of vibronic transitions and that of bound continuum interactions. The sustained endeavor to work out a consistent description of both phenomena at once has lately provided a computationally manageable approach based on coupled vibronic wavepackets. As a byproduct, such an approach naturally lends itself to the study of (bound-bound) vibronic transitions and collision-induced dissociation as well. The next stage in the development of DD-NRCE theory should cope with the problem of how to free the present description from the as yet constraining common classical trajectory condition. In the long run, a more ambitious task should be the extension of the theory to larger systems. There is actually no conceptual difficulty to extend the above theoretical methods to tetraatomic collisional systems. On the other hand, the understanding of the DCE dynamics in still larger systems may only arise from the combination of modeling and statistical approaches with concepts stemming from adaptations of the presented theory.
ARTICLES Ab Initio Calculations of the Transition-State Geometry and Vibrational Frequencies of the SN2 Reaction of CI- with CH,CI Susan C. Tucker and Donald G.Truhlar* Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 (Received: February 27, 1989; In Final Form: June 16, 1989)
-
We present ab initio geometries, energies, and frequencies at the MP2/6-3 1G** level for three distinct stationary points on the CI- + CH,CI’ CICH, + C1’- gas-phase reaction path. We compare the saddle point geometries and frequencies for several basis sets, with and without correlation energy, and we compare geometries, charge distributions, and energy differences for the two most reliable method/basis set combinations. Finally, we use the MP2/6-31G** ab initio properties to evaluate the conventional transition-state theory rate constant at 300 K. Comparison with a classical calculation is used to test the validity of classical simulation techniques for this reaction. Comparison with the experimental rate constant yields a semiempirical estimate of the barrier height of 3 kcal/mol, in reasonable agreement with our best ab initio value of 4.5 kcal/mol.
1. Introduction Bimolecular nucleophilic displacement (SN2)reactions have long played a fundamental role in organic chemistry. Reactions of this type involving halides and methyl halides have become prototypes of the class, and they have recently come under both theoretical and experimental scrutiny in studies using new techniques for gas-phase thermochemistry and kinetics of ions and for simulating reaction rates in solution. These studies show that the aqueous-phase rate coefficients are many orders of magnitude slower than the corresponding gas-phase ones for these reactions; thus they provide an interesting challenge for our ability to model solvent effects on reaction processes. In the present paper we study the chloride exchange reaction’-14 CI-
+ CH3CI’
+
CICH)
+ C1’-
(R1)
( I ) Chandrasekhar, J.; Smith, S. F.; Jorgensen, W. L. J . Am. Chem. SOC. 1985, 107. 154.
0022-3654/89/2093-8 138$01.50/0
which has been modeled in aqueous solution, for example, by Monte Carlo simulation of the free energy of activation,’ by (2) Jorgensen, W. L.; Buckner, J. K. J . Phys. Chem. 1986. 90, 4651. (3) Bergsma, J. P.; Gertner, B. J.; Wilson, K. R.;Hynes, J. T. J . Chem. Phys. 1987,86, 1356. (4) Huston, S.E.; Rossky, P. J.; Zichi, D. A. J . Am. Chem. Soc., in press. ( 5 ) Hwang, J.-K.; King, G.; Creighton, S.;Warshel, A. J . Am. Chem. SOC. 1988, 110, 5291. (6) Kozaki, T.; Morihashi, K.; Kikuchi, 0. J . Am. Chem. SOC.,in press. (7) Tucker, S. C.; Truhlar, D. G. Chem. Phys. Lett. 1989, 157, 164. (8) Luke, B. T.; Loew, G.H.; McLean, A. D. Int. J . Quantum Chem. 1986, 29, 883. (9) Mitchell, D. J.; Schlegel, H. B.; Shaik, S.S.;Wolfe, S. Can. J . Chem. 1985,63, 1642. (10) Carrion. F.: Dewar. M. J. S.J . Am. Chem. Soc. 1986. 106. 3531. (1 1) Bash, P.’A.fField, M. J.; Karplus, M. J . Am. Chem. S&. 1987, 109, 8092. (12) Barlow, S. E.; Van Doren, J. M.; Bierbaum, V. M. J . Am. Chem. Soc. 1988, 106,7240. (13) Dodd, J. A.; Brauman, J. I. J . Phys. Chem. 1986, 90, 3559 and references therein. Note footnote 14.
0 1989 American Chemical Society
