8184
J. Phys. Chem. 1991, 95, 8184-8193
chloride this might occur a t a field strength of approximately 0.1 V cm-I or smaller. This estimate has been made by assuming that WE is again 30 kHz, and it has been verified by an analysis of the probabilities of all possible transitions. If there is really a field dip, its minimum is expected to depend on the strength of the guiding field as well as the orienting field. Since the guiding field is lower for methyl fluoride (2.5 V cm-') than it is for methyl chloride (15 V cm-I), the deviation of the adiabatic curve from the experimental points is expected to start at higher orienting field for methyl fluoride than for methyl chloride. Although it is not reflected in Figures 1 and 3, this possibility cannot be excluded, because of the experimental uncertainties in the methyl chloride measurements. In addition to the somewhat speculative effect mentioned here, field gradients at the reaction site may play a role. Especially for orienting field strengths 51 V cm-' they may well be responsible for a further degradation of the orientati0n.I A more sensitive test of the adiabatic criterion for a deflection of the field, in the experiments discussed above, would be to keep the polarity of the guiding field the same when reversing the orientation field. As noted above, the measured steric asymmetry should then drop to zero, rather than to cos (57') times the expected adiabatic value, if the sudden approximation is valid. Janssen also measured the effect of varying the guiding field strength, keeping the orienting field strength constant, on the steric a~ymmetry.~ He found a decrease of the steric asymmetry if the guiding field strength was decreased below the value needed to obtain saturation. This cannot be explained by the adiabatic approximation, but it can be explained by invoking the sudden approximation in combination with the assumption of a guiding-field-dependent field dip. Without the latter assumption the sudden approximation also leads to a decrease of the steric asymmetry, but much less than observed, and the transition from the guiding field to the orienting field would have to be extremely fast for the adiabatic approximation to break down.
Conclusions For the 1111 ) state of methyl halides, the electric field dependence of the orientation as calculated by using the sudden approximation differs only little from the result of the adiabatic approximation, provided that the field does not change direction. This also holds for the field dependence of the alignment.
If the possibility of a sudden rotation is taken into account, as may be necessary for the description of the field dependence of the steric asymmetry of the reactions of methyl fluoride and chloride with Ca*, the approximations of adiabatic passage and sudden field rotation constitute an upper and lower limit for the saturation curve. The experimental saturation curves in the region of low field strengths, i.e. below 10 V cm-I, are partly within these limits, so that one might expect that they can be explained by a gradual transition from the adiabatic to the sudden (rotation) approximation. If the field changes monotonically on going from the guiding field to the orienting field, such an explanation must be excluded, since under this condition a breakdown of the adiabatic approximation is highly improbable, except at weak fields (definitely less than 1 V cm-I). If, however, a strong field dip, with a minimum of 0.1 V cm-I or lower, is present in the region between the guiding and orienting field, the adiabatic approximation may easily break down, which could explain the experimental results. At orienting fields lower than ca. 1 V cm-' field inhomogeneitiesat the reaction site may become important, leading to a further degradation of the orientation. The field dependence of the asymmetry of the polarized laser-induced photodissociation and ionization of methyl iodide,2 involving no change in field direction, is explained satisfactorily by the adiabatic as well as by the sudden approximation. At intermediate field strengths, the latter seems to give a somewhat better agreement, but at low field strengths it leads to quite large deviations. Over the entire experimental field range, however, the applicability of the sudden approximation requires an extremely fast transition from guiding to orienting field, which makes it very unlikely. Apart from these conclusions the analysis given here has provided useful guidelines for future experimental field-dependent orientation studies.
Acknowledgment. We thank Dr. R. D. Jarvis and J. van Leuken for helpful discussions and critical comments on the manuscript. We also thank Dr. G. Wlodarczak for sending us a listing of a computer program on the Stark spectroscopy of CHpX. In addition, we ackowledge ECC grant SCI-006-c and the NATO Advanced Research Program for support. Registry No. Ca, 7440-70-2; CH3F,593-53-3; CH,CI, 74-87-3; CHJ, 74-88-4.
Stereodirected States in Molecular Dynamics: A Discrete Basis Representation for the Quantum Mechanical Scattering Matrix V. Aquilanti, S. Cavalli, C. Crossi, Dipartimento di Chimica dell'Universit6, 06100 Perugia, Italy
and R. W . Anderson* Chemistry Department, University of California, Santa Cruz, California 95604 (Received: May 7, 1991) A recently developed discrete basis for the quantum mechanics of anisotropic interactions is shown to provide a representation for stereodirected states in molecular dynamics. The representation is labeled with a steric quantum number and an angular momentum projection on a Jacobi coordinate vector. Exact calculations for a standard model for rotational energy transfer are presented and discussed.
1. Introduction In the dynamics of elementary chemical processes, steric effects, such as the specific role of molecular anisotropy on scattering properties, are often interpreted within a classical mechanics framework. For example, the trajectory of an atom approaching a molecule from a given direction can be followed to compute probabilities for elastic, inelastic, or reactive scattering in other
directions. Recent progress in molecular collision experiments with oriented beams requires we establish their link with such classical pictures. Elementary chemical processes obey quantum mechanical laws, and therefore such sharp pictures must be amended: the full description of a collision is contained in quantities such as the elements of the scattering matrix S, and a well-known but complicated link connects S-matrix elements
0022-3654191 12095-8184%02.50/0 0 1991 American Chemical Society
Stereodirected States in Molecular Dynamics and cross sections for states of sharp polarization. In this paper, we propose a representation for the S matrix that gives an answer to the question whether it is possible to analyze a scattering event in terms of correlations for the relative angle between directions of the incident reactants and the departing products. Our representation provides a positive answer, although within the intrinsic limits of quantum mechanics, i.e. our states will not be sharply directed in space but will be so only in the classical (high angular momentum) regime. In the construction of such stereodirected representation we will exploit some results of previous analyses of angular momentum coupling and decoupling in atom molecule collision^.^-^ Such analysis lead to representations analogous to the so-called molecular Hund’s cases (a) and (b) for the rotational states of diatomics. These cases are also interesting for the quantum mechanical treatment of open-shell atom collision^.^ As is well-known, a useful classification for such states, reflecting the overall symmetry due to the relative orientation of orbitals, is provided by the magnitude of the projection of electronic angular momentum on the internuclear distance, providing the 2, II, A, ..., character: in the collisions of open-shell atoms, this classification gives insight on orbital alignment with respect to the interatomic distance vector. In our analysis of the atom-molecule system,’J we considered (a)and (8) coupling cases, where an artificial projection quantum number (denoted by u in the following) measures the relative directions of the approaching atom and the molecular axis: accordingly, u will be called steric quantum number. As discussed elsewhere,’ the procedure can also be extended to the description of reactive systems. The analysis we present in the following is limited to the simplest case of an atom colliding with a diatomic molecule considered as a rigid rotor. Our application is explicitly to a model for AI-N, first proposed by Pattengill et al.s and now a standard one for exact and approximate calculations.6 Although more accurate interactions for the Ar-N, system are now available,’ we continue to use this model on which we also based a previous study, in part related to the present one. Some variants of the model, introduced to exhibit particular effects, such as the sensitivity to different anisotropies, will also be considered. Previous work has now most clearly exhibited the main characteristics of the system especially from the viewpoint of possible approximations to a complete dynamical approach: decoupling schemes, such as coupled states or infinite order sudden approximations, were studied: we provided an adiabatic analysis.* In the present paper, we calculate the exact S matrix for the problem, for prescribed energy and total angular momentum, and then obtain an alternative representation. In the next section, we summarize the problem and the angular momentum coupling schemes. In section 3 the explicit construction of the stereodirected representations is discussed. Numerical techniques and characteristics of the model are presented in section 4. Results and discussion follow in section 5, and conclusions in section 6. An Appendix outlines the correspondence between the quantum number u and the classical angle between vectors. 2. Angular Momentum Coupling Schemes The general problem of defining directed states of molecules has been tackled by Kais and Levine? alignment of diatomics (1) Aquilanti, V.; Grossi, G. k r t . Nuouo Cimenro 1985, 42, 157.
(2) Aquilanti, V.; Beneventi, L.; Grossi, G.;Vecchiocattivi, F. J . Chem. Phys. 1988,89,75 1. ( 3 ) Aquilanti, V.; Cavalli, S.;Grossi, G. Theor. Chim. Acra 1991, 79,283. (4) (a) Aquilanti, V.; Grossi. G.J . Chem. Phys. 1980, 73, 1165. (b) Aquilanti, V.; Casavecchia, P.; Grcwsi, G.; Lagan&,A. J. Chem. Phys. 1980, 73, 1173. (c) Aquilanti, V.; Grosi, G.;LaganB, A. krr.Nuovo Cimenro 1981, 863. 7. ~. ( 5 ) Pattengill, M. D.; La Budde, R. A.; Bernstein, R.B.;Curtiss, C. F. J . Chem. Phys. 1971.55, 5517. (6) Connor, J. N. L.; Sun, H.; Hutson, J. M. J. Chem. Soc., Faraday Trans. 1990.86, 1649. (7) (a) Bowers, M. S.;Tang, K. T.; Toennies, J. p. J. Chem. phys. 1987, 88, 5465. (b) Beneventi, L.; Casavecchia, P.; Volpi, G.G.;Wong, C.C. K.; McCourt, F. R. W. XIII Inlernarional Symposium on Molecular Beams; El Escorial: Spain, 1991; Book of Abstracts, paper E-10. (8) Kais, S.; Levine, R. D. J . Phys. Chem. 1987, 91, 5462. ~
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8185 and orientation of spherical top molecules were considered with respect to axes directed in space. Here we focus our attention to the role that the specific relatiue directionality of molecular approach plays in inelastic and reactive collisions. These processes are formulated as quantum mechanical scattering on intermolecular potential energy surfaces, which depend on the mutual interparticle distances. Convenient representations of potential energy surfaces for scattering calculations are in terms of orthogonal vectors of the Jacobi type or variants thereof: the actual representations exploit different parametrizations of components, lengths, mutual orientations of such vector^.^ We restrict our attention to elastic and inelastic atom-diatomic molecule collisions: extensions to the reactive case,also when more than three particles are involved, is possible, for example through the hyperspherical formulation of the few-body p r ~ b l e m . ~In general, the problem is essentially equivalent to that of scattering from an anisotropic potential energy surface: it is this anisotropy, that occurs in the representation of potential energy, which is of interest here. 1. Atom-Diatom Scattering Theory. In the treatment of inelastic atom-diatom collisions, the natural Jacobi vectors are r, the internuclear distance for the diatom, and R, the distance of the approaching atom to the molecular center of the mass. The quantum mechanical scattering problem is solved when the solutions of Schrodinger equation with proper scattering boundary conditions (these solutions are denoted as F in eq 1 below) are analyzed at large distances to give the observables (cross sections) in terms of states with definite polarization, i.e., sharp angular (spin, electronic, rotational) momenta and their projections. Although these states may not be easy to be prepared in practice and real-life experiment typically will give some coherent or incoherent superposition, these cross sections are the finest theoretically obserbable quantum mechanical scattering properties. The usual theoretical link between the asymptotic properties of F and experimental observables is the scattering matrix S. The elements of the scattering matrix are conveniently labeled by some typical quantum numbers from the problem: for the example of the atom-diatom case, initial and final vibrational (neglected in the following), relative orbital ( 1 in the following), and molecular rotational 0’ in the following); alternatively, one can consider helicities, i.e., angular momentum projections in body-fixed directions (Q and A below). The alternative representations are obtained by similarity transformations of the S matrix, which preserve its properties (unitarity and symmetry). The representation introduced in this paper involves a label (formally an angular momentum projection u), which is connected with the angle 19 between directions of Jacobi vectors. This angle is a typical parameter in the representation of potential energy surfaces and is therefore a natural indicator of steric effects on dynamics. From here on we neglect molecular vibration. Setting r to be constant simplifies the formalism and the calculations, but it does not restrict the illustration of the general features, including the following important ones: (i) We will associate the quantum number, u, to an angle 9, (see eq 15 below). The finite number of values for u allows only a discrete representation for the continuous angular range of 9 ; this is a manifestation of the quantum nature of the problem. (ii) The representation needs to be completed by specifying a helicity quantum number (Qor A, see below) in addition to the u quantum numbers. This has to be compared with the designation of terms of diatomic molecules, which besides the 2, n, A, ..., character (associated to the electronic orbital symmetry) need also an additional labeling ( Q or K,in Hund’s coupling schemes (a) or (b), respectively). The interaction Dotential V(R.01 . , , between a structureless atom and a diatomic moka.de. when the vibrational deeree of freedom is neglected, depends on R, the distance betwee; the atom and the center_of mass of the djatom, and on the orientation angle COS 9 = (R-f), where f and R are the orientations of r and R. The (9) Aquilanti, V.; Cavalli, S.;Grossi, G.; Anderson, R. W. J. Chem. Soc., Faraday Trans. 1990, 86, 1681.
8186 The Journal of Physical Chemistry, Vol. 95, No. 21, 19'91
quantum mechanical theory of such a system has been extensively studied.I0 It is formulated as the scattering of a particle from a rigid rotor with a moment of inertia 1. Its angular momentum j is combined with the orbital angular momentum of the atomdiatom system I to give the total angular momentum J with a component M along an axis fixed in space. To describe the collision process, one has to solve, under scattering boundary conditions, a multichannel Schriidinger equation, for a fixed value of the total angular momentum J and total energy E:
where p is the reduced mass of the atom-diatom system. The effective potential energy matrix V'(R) is a representation of the sum of three operators:
Vrot + Vcentr + Vint The first term in eq 2 is the Hamiltonian of the rotor: Vrot = (h 2/20i2 V,,,,
(2) (3)
is the centrifugal operator:
Vwntr= (h2/2pR2)12
(4)
and Vinlis the scalar function V ( R , 6 )and 1 is the unit matrix. Alternative representations in terms of different coupling schemes correspond to the possible choices of the relative role of these terms. The following representation stresses the analogy with the angular momentum coupling schemes analysis of the interaction of open-shell atoms, analysis modeled on Hund's cases of the spectroscopy of rotating molecule^.^ As suggested previously,' we use Greek letters to denote the five coupling cases. 2. Space-Fixed Frame: Coupling Case (e). In the laboratory system of coordinates'O the eigenfunctions of the molecule and of the atom are, respectively, the spherical harmonics yj,(i)and VI,,,,& where mj and ml are space fixed components of/j and 1, respectively. Since the potential is not spherically symmetric, neither I nor j are conserved during the collision; however, an expansion of the total wave function in a set of functions where j and 1 are good quantum numbers is adequate in the asymptotic situation, where the potential goes to zero. The eigenfunctions IJMjl) (or more simply V I ) ) of the total angular momentum J and of its projection M a r e obtained by vector coupling, and from the asymptotic behavior of the radial functions F(R) in eq 1 a t large R values it is possible to compute the elements S i J 7 ( E )of the scattering matrix which is unitary and symmetric. All the observables can be computed from it. We designate the representation bl) as coupling case (e). In this representation, the matrices of the rotational and centrifugal operators (the first and second terms in eq 2) are diagonal with eigenvalues
(h2/21)j(j + 1)
(h2/2pR2)l(+ l 1)
respectively. The interaction potential matrix (third term in eq 2 ) is not diagonal, and its off-diagonal elements are the coupling terms in eq 1. Therefore, in the V I ) representation, the channel coupling in eq 1 is due exclusively to the interaction potential. 3. Body-Fixed (BF)Frames: Coupling Cases ( 7 )and (6). The coupling cases (y) and (6) are referred to representations V Q ) and IlA), respectively.' The quantum number Q is the component on the quantization axis R of J and j: along this body-fixed frame axis the projection of I is zero: JR=jR=Q
IR=O
(5)
If we choose r as quantization axis (case a), the j component along (IO) (a) Arthurs, A. M.; Dalgarno, A. Proc. R. Soc. London, Ser. A 1960, 256, 540. (b) Bernstein, R. B.; Dalgarno, A.; Massey, H.S. W.; Percival, 1. C. Proc. R. Soc. London, Ser. A 1963, A274,427. (c) Lester, W. A.; Bernstein. R. 9. J . Chem. Phvs. 1968.48.4896. (dl See: Atom Molecule Collision Theory: A Guide for ihe Experimentalis;; Bernstein, R. B., Ed.; Plenum: New York, 1979.
Aquilanti et al. this axis, jr,is zero, while the corresponding 1 and J projections, 1, and J,, are labeled by A:
J,=I,=A j,=O (6) We can therefore use two alternative base PO) and [/A) to expand the scattering total wave function. Both have been considered in the The PI), PQ),and IIA) representations are related one to each other by orthogonal transformations, similarly for the S matrix (see next section). In the PO) representation the matrix of the molecular rotational interaction remains diagonal, while couplings are introduced in the centrifugal interaction matrix, which is now tridiagonal for Q,Le., Q' = Q f 1, and in the interaction potential matrix which is diagonal in fl but couples states with j # j'. In the IIA) representation the angular momentum operator of the orbital motion is diagonal, the molecular rotational matrix is diagonal in I and couples states with A' = A f 1 (all the relevant matrix elements for the case (6) can be obtained from the ones for the case y interchanging both j and I, and O and A). The interaction potential matrix is now diagonal in A and couples states with I # 1'. Therefore, choosing different bases to expand the scattering wave function, one obtains sets of equations of the same dimensionality but with a different structure. 4. Stereodirected (SD) Representation: Coupling Cases (a) and @). Representations for which the potential matrix is diagonal (such as Hund's cases (a) and (b) in the open-shell atomatom problem) have been introduced only recently' for the present problem where the interaction is a continuous function of the angle t9 between Jacobi vectors, so the treatment requires a discretization ~r0cedure.l~We define a case (a)and a case (8) as the discrete representations corresponding respectively to the (y) and (6) coupling schemes. Our recipe' starts by establishing a connection between cutting the angular range (0 I 9 I r ) in N slices and introducing an artificial vector A whose length is A classically, [ A ( A 1)]'/2 quantally, and A + semiclassically." Putting A = N / 2 and interpreting the integer or half-integer u which counts the slices (-N/2 I v I N / 2 ) as a projection quantum number, we can exploit angular momentum algebra at its full power. The consequence is that in computing matrix elements for V'(R) all the integrals are replaced by summations. We will see later that in the present context N i s not arbitrary but uniquely related to the largest diatomic rotational quantum number allowed by energy conservation. The basic tools of the method are the discrete analogs of spherical harmonics Y,,,,,which are essentially Clebsch-Gordan coefficients (see below). For N tending to infinity, we have for I
