J. Phys. Chem. 1994,98, 3307-3316
3307
Vector Correlations in the Photopredissociation of van der Waals Molecules Octavio Roncero, Pablo Villarreal, and Gerard0 Delgado-Barrio Institituo de Matembticas y F'isica Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain
Nadine Halberstadt Laboratorie de Photophysique Molgcularie,t Centre Universitaire, 91405 Orsay, France
J. A. Beswick' LURE,$ Centre Universitaire, 91 405 Orsay, France Received: November I , 1993'
Quantum calculations of spatial distributions and angular momentum polarization (orientation and alignment) of the fragments in the photon-induced vibrational predissociation of Ne...ICl and Ne..C12 van der Waals complexes are presented. The general theory of vector correlations in photodissociation is particularized to the case of slow photopredissociation where the lifetime is long compared with the rotational period of the molecule. The formalism is applied to Ne.-ICl and N e 4 1 2 using empirical potentials determined before. Photodissociation line shapes have been computed by 3-D converged integration of the close-coupled vibrational and rotational equations. Vibrational predissociation line widths, final rotational-state distributions (scalar properties), and spatial anisotropy parameter of the angular distribution of the fragments as well as the orientation and alignment of the rotational angular momentum of the diatomic fragment (vectorial properties) have been calculated. The results are discussed in terms of the symmetry properties of the transition dipole matrix elements and the ro-vibrational dynamics during dissociation. In particular, the role of the mixing of final channels with different helicity quantum numbers (the projection i l of the total angular momentum J on the dissociation coordinate) is stressed. It is found that vectorial properties are much more sensitive than the scalar quantities on the detail of final-state interactions and Coriolis couplings. A diabatic vibrational golden rule (DVGR) treatment and an infinite-order sudden approximation (IOSA) of the rotation in the final vibrational channel channels have also been implemented and checked against the converged full line shape calculations. It is shown that while DVGR is an excellent approximation for both scalar (lifetimes and final-state rotational distributions) and vectorial properties, IOSA is very poor for describing the latter.
I. Introduction The study of vector properties such as angular distributions, alignment, and orientation of the products in molecular collisions and photofragmentation has received an increasing interest over the past years' as a means to obtain the most detailed information on stereochemicalaspects of reaction dynamics. Photodissociation is a particularly fundamental and relatively simple process which should be amenable to a full theoretical understanding.2 The measurement of vector properties of photofragments has therefore evolved rapidly in the past years, in particular by the use of Doppler-resolvedlaser probe techniques.' These measurements have provided important information on the coupling between electronic and nuclear motions, especially when nonadiabatic electroniccouplings are in~olved.~ Photodissociation can also be used to produce oriented and aligned fragments for subsequent collisional experiment^.^ A fascinating class of photodissociation experiments involves the use of van der Waals complexes and clusters as precursors for studying orientationally restricted bimolecular reactions induced by photon absorption.6 Vector properties in molecular photofragmentation are associated with the correlations between the polarization e of the incident light and hence the transition dipole moment d, the recoil direction of the fragments k, and the electronic or rotational angular momentum j of the fragment^.^ The most obvious one is the angular distribution of photofragments in space, which is
* To whom correspondence should be addressed. t Laboratoire du CNRS. t Laboratoire du CNRS, CEA et MEN. * Abstract published in Aduonce ACS Abstracrs, March 15, 1994.
an (e,d,k) correlation.8 Thealignment andorientation, alsocalled angular momentum polarization, are (e,dj) correlations and correspond respectively to the odd and even moments of the probability distribution of the projections m of j onto a particular axis (usually chosen as the polarization vector e of the incident photon)? These correlations have been most often studied for direct photodissociationprocesses or in fast predissociationwhere the lifetime is shorter than the rotational period. Although the average over several rotational periods of the molecule is expected to substantially reduce any spatial anisotropy,1° it is still possible to observe significant vector correlations even in the case of very slow predissociation.7 van der Waals molecules provide ideal model systems for the study of photopredissociation processes in molecules.' 1 Usually, the electronic states involved are well known, and the potential energy surfaces, as well as the transitions dipole moments, are reasonably well characterized. These species exhibit vibrational predissociation in a time scale which is often very long as compared with the rotational periods and are therefore well suited for the study of vector properties in the case of slow photopredissociation. It is the purpose of this work to calculate angular distributions as well as alignment and orientation of the rotational angular momentum of the fragments in the photopredissociation of N w I C l and Ne...C12 van der Waals complexes. For these two systems, rotational populations have been determined experimentally,12J3 and quantum calculations using empirically determined potentials have been performed.14J5 The organization of the paper is as follows. In section I1 we present the general theory and its application to the case of slow
QQ22-3654/94/2Q98-33QI~Q4.50/00 1994 American Chemical Society
Roncero et al.
3308 The Journal of Physical Chemistry, Vol. 98, No. 13. 1994 predissociation. In section I11 the photopredissociation of a triatomic van der Waals system is discussed. Finally, section IV is devoted to the discussion of the results. 11. Theory
-
We consider a generic one-photon photodissociation process hw A B in which one of the fragments carries an angular momentum j with projections 52 on a body-fixed z axis. For cases where the two fragments have angular momentum different from zero, j below has to be understood as the sum j = jA jB, Then a final transformation is needed in order to relate the vector correlations of j with those of j, and jB.I6 For an initially randomly oriented system in a state with total angular momentum Ji, the angularly resolved photoexcitation matrix for such a process is given by17J8 AB
+
+
where the reduced matrix elements are defined by
+
where Mi is the projection of the initial total angular momentum Ji on the space-fixed axis. In eq 1 , k = (ek,f$k)denotes the fragments’ recoil direction in the space-fixed frame, d is the dipole moment operator, and e is the polarization vector of the incident photon of frequency w . The dissociative plane waves q j m can be expressed in terms of wave functions with well-defined total angular momentum18
where L&, are Wigner rotation matrices and M is the projection of the total angular momentum J on the space-fixed z axis. The Q;f(r,R) wave function in eq 2 can be in turn expanded in a body-fixed angular basis set according to
where pjn(r)describes the asymptotic states of the two fragments and OR,4~ are the polar angles of the vector R joining the centers of mass of the two fragments. With this choice, the body-fixed z axis is parallel to R. The initial wave function can be similarly expanded as
(@&f(R)(PIFI(r)ldn~Q‘(r,R)l~i~~(R) UywW ) (7) Substituting eqs 2 and 6 into eq 1, the differential photoexcitation matrix in the body-fixed frame takes the form
where
is the polarization function.’’ Since the s2 is the projection of j on the recoil direction of the fragments, the differential excitation matrix in the body-fixed frame, as provided by eq 8, gives direction information on the (j,k) correlation (intrinsic correlation). What we also need in order tocalculate the angular distribution and angular momentum polarizationof the fragments is the excitationmatrix in the spacefixed frame. This can be conveniently done by the use of the momentsof the Q distribution, i.e., by the state multipoles defined by%17
and which transform under rotations as tensors of rank K. Substituting eq 8 into eq 10, we obtain
Ti$(k) =
*w42K
zcc
+1 X
C
+
+
(-l)@p4+J’-Jid(2J 1)(2J’+ 1)(2P 1 ) X
Note that in eqs 3 and 4 indices running over degrees of freedom of the fragments other than their total angular momentum and its projection have been omitted in order to simplify the notation. They should of course be included in the summation, since the asymptotic states of the fragments are mixed in the molecule. The transition operator d.e* for electric dipole transitions can be written as
whered,are the tensorialcomponentsoftheelectricdipoleoperator of the system in the body-fixed frame and e, are those of the polarization vector of the exciting light in the space fixed frame. Using eqs 3 and 4, it is obtained
with
$::(a)
=J ( 2 J
+ 1)(2Jf+ l)(-lpcw
X
The state multipoles defined by eq 11 contain the same information on the photodissociation dynamics as the body-fixed differential photoexcitation matrix in eq 8. The corresponding space-fixed multipoles can be obtained using
Photopredissociationof van der Waals Molecules
Using eq 11, one obtains
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3309
where the integrated multipole moments QQ = JQQ(k)d(ws ek) d& are given by @Q
= 1-2Q+K-J+J’-J~-Cl+Wd ( 2 J + l)(W’+ 1) x
whereK= 0, ...,2j+ 1and Ys,Q-paretheusualsphericalharmonics with IK - 4 I S I K P. When the sum over Q,Q’is performed in eq 14, the odd values of S generally vanish due to symmetry properties of the reduced matrix elements in eq 12. The spacefixed multipoles given in eq 14 have been obtained before in the axial recoil approximation,lgi.e., when the dissociation is fast as compared to the rotational period of the molecule. Equation 14 provides the general result valid for fast as well as for very slow predissociation. From eq 14 all the relevant vector correlations in the spacefixed frame can be obtained by using the relationship between state multipoles and the differential excitation matrixg.1’
+
From the definition of the polarization function ,!?KQ(e) given in eq 9, it is clear that the integrated multipoles are different from zero only for K = 0, 1, and 2. Therefore, the orientation and the alignment defined by eqs 18 are enough to characterize the m distribution?J For the case of slow predissociation, where the rotational spacings are larger than the predissociationwidths, it is possible to excite only one particular Ji J transition. Therefore, in all equations above the sums over J reduce to only one term. In this limiting case the spatial anisotropy parameter /3f given by eq 17b takes the formZo
-
st’ = (3p2 - 2)[3C(C + 1) - 8J(J + l ) ] cx 4 ( 2 J + 3)(2J- l)cl(J,lld(lJjQ)lz n
In particular, from eq 15 one obtains
which is the well-known expression for the differential photodissociation cross section.* Using eq 14, the integrated partial photodissociation cross section u p and the anisotropy parameter in eq 16 have the explicit form
+
+
where C = Ji(Ji 1) - J(J 1) - 2. From eq 20 we note that the anisotropy parameter for the angular distribution of photofragments is essentially independent of j (except for the dependenceof the transition dipole matrix elements). Also, when J is very large and only a few Q contribute to the wave functions, the well-known semiclassicallimitingvalues of 2 (for J H- a)and -1 (for J >> Q) are recovered in the case of linearly polarized incident light.lOszO Substituting eqs 12 and 19 into eq 18a, the orientation of j can be written in the limit of slow predissociation as p[J,(J,
+ 1) - J(J + 1) - 21
T
X
r.
J ( J + )l*
JII
(J,lldllJ.Q
+ 1) (JilldllJjQ) * + cc
