Symmetry Properties of iMolecular Collision Complexes
The Journal
(18) A. Abragam, "The Principles of Nuclear Magnetism", Clarendon Press, Oxford, 1961. (19) L. H. Lmle, "Infrared Spectra of Adsorbed Species", Academic Press, New York, 1966. (20) M. L. Hair, "Infrared Spectroscopy in Surface Chemistry", Marcel Dekker, New York, 1967. (21) H. R. Anderson, Jr., and J. D. Swaien, J. Adhes., 9, 197 (1978). (22) T. Ohnishi, A. Ishitani, H. Ishida, N. Yamamoto, and H. Tsubomira, J . Chem. Phys., 82!, 1989 (1978). 123) P. Chollet. J. Messier. and C. Rosilio. J. Chem. fhvs.. 64. 1042 (1976). (24j A. Hjortsberg, W. P. Chen, E. Burstein, and M. Pomerantz, Opt. Commun., 25, 65 (1978). (25) Opt. . . . . Y. Levy, C. Imbert, J. Cipriani, S. Racine, and R. Dupeyrat, Commun., 11, 66 (1974). (26) J. Cipriani, S.Racine, R. Dupeyrat, H. Hasmonay; M. Dupeyrat, Y. Levy, and C. Imbert, Opt. Commun., 11, 70 (1974). (27) J. Cipriiani, H. Hasmonay, Y. Levy, S. Racine, M. Dupeyrat, R. Dupeyrat, and C. Imbert, Jpn, J. Appl. Phys., 14, 93 (1975), Suppl. 14-1. (28) A. M. Yacynych, H. B. Mark, Jr., and C. H. Giles, J . fhys. Chem., ,80, 839 (1976). (29) R. Steiger, Helv. Chim. Acta, 54, 2645 (1971). (30) 0. S. Heavens, "Optical Properties of Thin Solid Films", Dover Publications, New York, 1965. (3 1) R. M. A. Azzam and N. M. Bashara, "Ellipsometry and Polarized Light", North Holland Publishing Co., Amsterdam, 1977. (32) B. G. Anex and W. 1'.Simpson, Rev. Mod. fhys., 32, 466 (1962). (33) M. R. Philpott and J. D. Swalen, J . Chem. Phys., 69, 2912 (1978). (34) M. Tacke and J. D. Swalen, to be published. (35) J. D. Swalen, M. Tacke, R. Santo, and J. Fischer, Opt. Commun., 18, 387 (1976). (36) P. K. Tien and R. Ulrich, J . Opt. SOC. Am., 60, 1325 (1970). (37) D. Marcuse, "Theory of Dielectric Optical Waveguides", Academic Press, New York, 1974. (38) A. Otto, Z . fhys., 216, 398 (1978). (39) E. Kretschmann and H. Raether, Z. Naturforsch. A , 23, 2135 (1968). (40) I.Pockrand, Surf. Sci., 72, 577 (1978). (41) H. Kuhn, D. Mobius,,and H. Bucher, "Spectroscopy of Monolayer Assemblies", in "Physical Methods of Chemistry", A. Weissberger and 8.W. Hossiter, Ed., Part IIIB, Wiley-Interscience, New York, 1972, Chapter VII.
of Physical Chemistry, Vol. 83, No. 11, 1979 1445
(42) G. L. Gaines, "Insoluble Monolayers at Liquid-Gas Interfaces", Interscience. New York. 1966. (43) J. D. Swalen,'R. Santo, M:Tacke, and J. Fischer, IBMJ. Res. Devel., 21, 168 (1977). (44) I. Pockrand and J. D. Swalen, J . Oot. SOC.Am., 68, 1147 (1978). i45j K. H. Drexhaae. "Interaction of Liaht " with MonomolecularDve , hvers". , . frog. Opt.,"l2, (1974). (46) D. Mobius, private communication. See also "Monolayer Assemblies", in "Topics in Surface Chemistry", IBM Research Symposium XII, P. Bagus and E. Kay, Ed., Plenum, New York, 1978. (47) D. R. Lide, Jr., J. Chem. fhys., 33, 1514 (1960). (48) J. D. Swalen, K. E. Rieckhoff, and M. Tacke, Opt. Commun., 24, 146 (1978). (49) C. W. Pitt and L. M. Walpita, Electron. Lett., 12, 479 (1976); 13, 210, 347 (1977). See also Hectrocomponent Sci. Technoi., 3, 191 (1977). (50) I.Pockrand, J. D. Swalen, J. G. Gordon, 11, and M. R. Philpott, Surf. Sci., 74, 237 (1978). (51) M. R. Philpott, J . Chem. fhys., 61, 5306 (1974). (52) G. Wegner, private communication. See also, B. Tieke, H.-J. Graf, G. Wegner, B. Naegele, H. Ringsdorf, A. Banerjie, D. Day, and J. B. Lando, ColloidPolym. Sci., 255, 521 (1977); B. Tieke, G. Wegner, D. Naegele, and H. Ringsdorf, Angew. Chem., Int. Edit. Engl., 15, 764 (1976). (53) D. Allara, private communication, measured the infrared spectrum with p polarized light at low angles of a film of 10 layers of cadmium arachidate on silver, made in our laboratory. (54) K. Siegbahn, 'C. Nordling, A. Fahlman, R. Nordberg, K. Hamrin, J. Hedman, G. Johansson, T. Bergmark, S. E. Karlson, I.Lindgren, and B. Lindberg, "ESCA-Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy", Almquist and Wiksells, Uppsala, 1967. (55) B. L. Henke, J . fhys., C4, 115 (1971). (56) D. T. Clark and H. R. Thomas, J . folvm. Sci.. Chem. Ed., 15, 2843 ( 1977). (57) P. Cadman, G. Gossedge, and J. D, Scott, J . Electron Specfrosc., 13, 1 (1978). (58) C. R. Brundle, H. Hopster and J. D. Swalen, J. Chem. Phys., in press. (59) J. Fischer, private communication (60) G. J. Sprokel, private communication. ~I
'
Symmetry Properties of Angular Correlations for Molecular Collision Complexes G. M. McClelland' and
D. R. Herschbach*+
Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 (Received November 27, 1978) fublicatlon costs assisted by the National Science Foundation
In reactive and inelastic scattering, forward-backward symmetry of the product angular distribution is often attributed to formation of an intermediate collision complex which persists for at least a few rotational periods. Here we show that this familiar criterion is often invalid and derive other symmetry properties for several potentially observable angular distributions. These include two-, three-, and four-vector correlations among the initial and final velocity and rotational angular momentum vectors. We consider a hierarchy of three complex types associated with increasing randomness: rotational, separable, and statistical complexes. Each type is characterized by distinct observable symmetries in the angular correlations and corresponding features of the collieion dynamics.
I. Introduction The classic treatments of molecular vibration and rotation by E. H. Wilson made exemplary use of symmetry analysis.' These elegant methods have found many applications in spectroscopy in identifying selection rules and classifying interactions or energy level patterns. This paper takes an analogous approach to the analysis of vector or directional properties associated with the formation and dissociation of transient molecular collision complexes. Our aim is to aid the interpretation of molecular scattering experiments by classifying the symmetries of angular distributions involving the initial and final relative velocity 3 Department of Chemistry, Stanford University, Stanford, Calif. 94305.
0022-3654/79/2083-1445$01.OO/O
and rotational angular momentum vectors. The distinction between direct and complex processes has a central role in the study of single-collisiondynamics.2 In a direct reaction, the products separate within a vibrational period or less (typically S s). A complex reaction persists for a time (typically 25 x s) long compared to vibrational and rotational periods, and thus tends to randomize internal energy and to average product vector properties. The most reliable experimental criterion for the presence of an intermediate long-lived collision complex has been forward-backward symmetry of the product angular di~tribution,~ which has been observed for maoy reaction^.^ Experimental advances now make accessible several other kinds of angular distributions. These involve the orientation of rotational angular momentum
0 1979 American
Chemical Society
1446
The Journal of Physical Chemistry, Vol. 83, No. 1 I, 1979
vectors j and j’ for reagent and product molecules in addition to the corresponding directions of relative velocity vectors k and k’. Reactive scattering experiments with oriented reagent molecules have been limited to symmetric tops,5 but new laserG8and electric focusinggmethods make such studies feasible for diatomic molecules. The orientation of product molecules has been measured in prototypical but primitive electric deflectionlOJ1 and chemiluminescence8J2 experiments, and there is good prospect for obtaining much more detailed data from laser-induced f l u o r e s ~ e n c e . ~These ~ - ~ ~developments invite analysis of symmetry properties for the new kinds of angular distributions now accessible. The collision properties considered here comprise angular correlations among two or more of the four observable vectors j, j’, k, k’ and represent averages over all unobserved or random vector quantities. Although the explicit analysis is limited to properties involving these four vectors, the approach is quite general. As yet the term “angular correlation” is unfamiliar in chemical physics, but it has found wide use in nuclear physics16 because many types of experiments have similar directional properties. For example, a measurement of the polarization direction for an emitted photon may be treated as analogous to a measurement of the flight direction of a product particle. From this viewpoint, all collision experiments in which only two vectors are observed belong to the same class, termed direction-direction correlations; all experiments in which three or four vectors are measured belong to higher classes, termed triple or quadruple correlations. Our analysis identifies all observable symmetries for the various angular correlations among j, j’, k, k’ and thereby provides criteria for a more definitive characterization of molecular collision complexes. The symmetries we treat are consequences of collision dynamics and are distinct from other properties governed by symmetries of reagent and product quantum states. Thus, familiar procedures for correlating electronic stated’ or molecular orbitals18offer a qualitative prediction of the activation energy or stereochemical features. Also, the permutation symmetry of identical nuclei results in selection rules for the product ro-vibronic states and spin statistics.lg In contrast to these procedures, our analysis of angular correlations will use classical mechanics. The derivations are then simplified but the results are valid in the quantum domain, which must exhibit the same symmetry properties. Other aspects of angular correlations among j, j’, k, k’ have been examined in several recent theoretical studies, likewise provoked by the emergence of experimental techniques for observing angular momentum vectors. Classical trajectory calculationsz0and a kindred impulsive modello have investigated the “steric effects” associated with preferred reagent and product orientation. Quantum state-to-state cross sections for specified magnetic sublevels have been computed from nearly exactz1and approximatez2 methods. A “mobile” model for angular momentum coupling in reactive collisions provides quantities analogous to vector-addition coefficient^.^^ An information theory analysis shows that the triple correlation among j’, k, k’ often yields a very large entropy deficiencyaZ4The most pertinent previous work formulates a general statistical phase space theory for angular correlations in which the observable properties are expressed as polynomial expansions, with coefficients determined by moments involving averages over unobserved vector^.^^^^^ In section I1 we review the notion of angular correlations and in section I11 define a hierarchy of three types of
G. M. McClelland and D. R. Herschbach
a. Linear and Angular Momentum Vectors
b. Any Colllsion
c. Rotational Complex
d SeparaMe Complex
e. Statlstlcal Praduct Channel
Figure 1. Dynamical vector symmetries for each type of collision complex. Figure !a, 1b shows the vectors of interest and the azimuthal symmetry about k present for all collisions. Figure ic-e indicate the additional symmetries for each successively more random variety of complex.
collision complexes that correspond to increasing randomness. In section IV the observable symmetries are derived for each type of complex and limiting cases imposed by kinematics or experimental conditions are noted. The commonly used association of forward-backward symmetric scattering distributions with long collision lifetimes is shown to be valid only for particular limiting cases. In section V we discuss features of the complex motion that are involved in the dynamical symmetries. The important attributes are complex lifetime, geometry, rotational energy, vibrational energy transfer within the complex, and orientation dependence of the reagent and product channels. Finally, in section VI we outline the diagnostic criteria for identifying the various complex types which result from symmetries of the angular correlations. These criteria are the analogues for other angular correlations of the forward-backward symmetry of the product angular distribution.
11. Angular Correlations We consider the prototype reaction of an atom with a molecule to produce an atom and a molecule. As illustrated in Figure l a , the observable vectors are the directions of the initial and final relative velocity k,-k’ and the initial and final rotational angular momenta j, j’. The carat diadem notation indicates a unit vector. The other vectors of interest are the reagent and product orbital angular momenta 1, l’, which are perpendicular to the relative velocities, and the total angular momentum J which is conserved j + 1 = J = j ’ + 1’ (1) Although most of our discussion pertains to reactive encounters, the conclusions are equally valid for nonreactive collisions, and can in many cases be extended to systems with two molecules as reagents or products. The most general angular correlation involving all four ofithe observable vector directions is denoted by W(k,j,k’,j’), If the magnitude of the vectors is held constant, this correlation is proportional to the differentia! cross section for formivg products characterized by k’,j’ from reagents with k, j, or
The Journal of Physical Chemistty, Vol. 83,No. 11, 1979
Symmetry Properties of Molecular Collision Complexes
11447
__ All Colllsions
The four-vector correlation is a function of five angles; these can be taken as any three of the “interbond” angles between the four vectors (drawn as if sprouting from a common origin) and two dihedral angles between pairs of the three planes containing the “interbond” angles. The angular correlation represents an average over the directions of 1, 1’ and any other unobservable vectors. Likewise, an experiment which does not measure one or more of the observable vectors effectively imposes a further average over such vectors; the three- and two-vector correlations hence are averages of the four-vector correlation over the unobserved vector_s: Thi; -yjeld; J o y three-ye_ctorcorrelations involving (k,j,k’),(k,j,j’),(j,k’,j’), and (k,k’,j’), respectively. These are each functions of three angles, which comprise two “interbond” angles and one dihedql aFg1e.- ‘There are sixAtwo;vector cor_re!ations involving (k, lk’), (k9j’),(j,k’), (j,j’), (k,j), and (k’,j’),respectively. Thiese are functions of one angle, that between the pair of unit vectors. For example, the three-vector correlation W(k,k’,j’)can be measured in a crosseg-beam electric deflection experiment.1° The vectors k and k’ are determinedby the geometry of the reagent beams and detector and j’ from analysis of trajectories of the products in an inhomogeneous electric field. Since j is unspecified, the measured correlation is an average over initial rotation A
W(k,k’,j’) = (W(k,j,k’,j’))]
I
A
(3)
In analyzing symmetry properties of the angular correlations, we treat all vectors as having fixed magnitudes; the symmetry transformations alter only vector directions. Of course, vector directions are often strongly influenced by their magnitudes because of conservation laws or dynamical effects. An experiment usually averages over many magnitudes, and thus probably measures smaller asymmetries than are present for particular magnitudes. However, this averaging will not create or destroy symmetries in the directional correlations.
111. Categories of Collision Complexes According to classical mechanics, the course of a molecular collision, including its lifetime and exact product trajectories, is fully determined by the initial conditions. However, if we average over the relative phases of rotation, vibration, and translation of the reagents, we can describe a “complex” in terms of its distribution of lifetimes and product attributes. The complex “decays” to form these distributions. It is in this average sense that we speak of a complex here. For any molecular collision with unpolarized reagents, each of the vectors j, 1, J, j’, l’, and k’ will have azimuthal symmetry about the initial relative velocity k, as shown in Figure l b . The additional dynamical symmetries for various complex types are illustrated in other parts of Figure 1. A. T h e Rotational Complex. During the time between formation and breakup of a complex, it will rotate about o,its angular velocity vector. If the lifetime distribution is long and broad compared to the rotational period, the average direction of w will be along the total angular momentum J and each of the product vectors will be uniformly distributed abpyt J. Thus, by definition the azimuths about J of the k, j, and 1 vgctps of _a rotational complex are not correlated with the k’, j’ and 1’ azimuths. There m a y be a cprelation bet_ween,lfor example, the azimuths of k and j, or between k’ and j’, due to dynamic effects in the reagent or product channels. Figure ICil-
3: j’-. -j‘
Figure 2. Venn diagram showing the logical relationship among the complex mechanisms. Each successiveiy more random type is a subset of a less restrictive set. The most useful characteristic observable symmetries are also indicated in an abbreviated notation. A number 1 through 4 indicates the number of vectors which must be defined simultaneously to observe the symmetries; this number is omitted when a symmetry holds for all correlations. A bracket indicates an invarinnce with respect to the transformation of two vectors simultaneously.
lustrates the azimuthal symmetcy of the k , k, pr 3/ vectors of a rotational complex about J for a given k. B. The Separable Complex. The directional properties of the reagent and product trajectories of a separable complex are uncorrelated except by the conservation of total angular momentum. In the nomenclature of information theory,*’ the “relevance” between the reagent and product properties is zero. As shown in detail in section IV and indicated in Figure Id, two separable complex product trajectories which differ only by inversion are equally probable from a given reagent trajectory. All separable cornplexes are also rotational complexes. C. Statistical Channels. The reagent or product channels of a complex are statistical if the correlation of the vectors of that channel is determined solely by angular momentum conservatipn, For any collision process, the relative directions-of J, l’, and jlare determined by the sum rule J = 1’ j’; in addition, k’ is perpendicular to 1’. If the products are statistical, this is the only restrictiop on k’, implying that k’ is azimuthally symmetric about 1’ as illustrated in Figure le. Any complex for which either the reagent or product channel is statistical is also separable. We note that the rotational and separable complexes are defined in terms of the relation between the reagent and product channels, whereas the statistical complex is specified by the dynamics of a single channel. The three complex definitions form a progression of increasing randomness. Their logical relationship is illustrated by the Venn diagram of Figure 2. D. Special Cases. For certain limiting cases imposed by special mass combinations or experimental conditions, the angular correlations acquire additional symmetries. Supersonic reagent beamsz8produce rotational cooling and very low j . If a rotational moment of inertia IT is small, then j = (2E,1J1I2must be small and J = 1. A simiilar limiting case may apply for the product channel, and also for the reduced mass and the orbital angular momentum 1 or 1’ of the collision partners when one of the product or reagent molecules is light. If the rotational velocity of a molecule is negligible compared to the orbital and vibrational motion of a col-
+
1448
The Journal of Physical Chemistry, Vol, 83, No. 11, 1979
lision, reversing the direction of rotation and inverting the rotational angular momentum will have no effect on the collision. Thus in the low j or j’ limit, we generally expect vector correlations to become invariant to inversion of j or j’, respectively. Jn this case the correlation need not be independent of j or j’, since orientation effects in the reagent or product channel are still possible.
IV. Observable Symmetry Properties We call a correlation symmetric if it is invariant to a symmetry operation; thus
W(Oi,Oz,...)= W(TO1, TO2,...)
(4)
The symmetry operation T which transforms the observable vectors vl, v2, ... into Tvl, Tvz, ... depends only on the observed vectors and on no unobseyvedlquantity. For example, for a separabke coTplex_ W(k,k’,j’) is symmetric to the operation Tk = k,_Tk’ = Tk’, Tj’ = j’. However, althougb the azimuth of k about J is randomly distributed, since J is unobserved this azimuthal symmetry is unobservable. In this section we outline a scheme for deducing the symmetries of angular correlations. Then we derive the symmetries for a single (reagent or product) channel and go on to treat two-, three-, and four-vector correlations involving both channels. A . Finding Observable Symmetries. We consider a particular set of reagent and product vectors for one trajectory and transform them by a sequence of dynamical symmetry operations which are allowed either generally by mechanics or by the complex definitions. These operations are as follows: (1)rotating the entire trajectory (all reactions); (2) inverting the entire trajectory (all reactions with optically inactive reagents and products); (3) rotating only the reagent or product trajectory about the total angular momentum (rotational complex); (4)inverting only the reagent or product trajectory (separable complex); ( 5 ) r o t a t h g k about 1 or k’ about 1’ (statistical channel). Although 1, l’, and J are never observed, we shall see that in some cases rotation about one of these vectors can be expressed without reference to its direction but only to directions of observable vectors. To search exhaustively for observable symmetries in the angular correlations, we want to apply the dynamical symmetry operations in all possible combinations and sequences. It is straightforward to show that all these operations commute. Thus any dynamical symmetry operation which is a combination of those listed above can be rearranged in an arbitrary order. We will apply the rotation operations 1, 3, and 5 first and the inversion operations 2 and 4 last. These inversion operations themselves always correspond to observable symmetries, inyerting k and or k’ (which are polar vectors) but not j or j’ (axial vectors). Using TR to represent one or more of the operations 1,3, and 5 and TI to represent one or both of operations 2 and 4,we can represent our search scheme as W(Cl,fp,...)
7
W(TR+l,TRe2,...)
7
W(TITRC~,TITR+~, ...) (5)
Because the inversion operations 2 and 4 are always observable, if applying T R yields an observable symmetry, T I T R will produce additional observable symmetries. If T R yields no observable symmetry, T I T R will also produce none. B. Single-Channel Correlations. For a separable complex, correlations of reagent with product vectors can be expressed in terms of two single-channel correlations,
G. M. McClelland and D. R. Herschbach
one involving the reagent channel, and one involving the product channel. Since for such a complex the reagentproduct correlations arise solely from conservation of total angular momentum, these correlations can be computed by multiplying single-channel factors and integrating over all directions of the total angular momentum W(O,,O,) = J d J W,(J,O,) Wp(J,Op)
(6)
Here W(J,OJ and W(J,O ) represent single-channel correlations of the total angurar momentum with one or more of the reagent or product vectors. Suppose the overall correlation is invariant to an operation T which transforms the vectors 8, TO,, Op TO , (In general, T will have different effects on 6, and O,(s Then
- -
JdJ
W,(J,TB,) Wp(J,TOp) = I d J Wr(J,Or) Wp(J,Op)
(7) These integrals will in geperal be equal only if the integrands as a function of J ?re equal or related by some one to one transformation J T,J; thus W,(TjJ,TO,) W,(TJJ,Ttp) = Wr(J,Or)Wp(J,tpp) (8)
-
Since the reagent and product functions are independent, the single-channel correlations must be equal W,(T,J,TO,) = w,(J,o,)
w,(T,J,To,)=
(94
wP(J,cp)
(9b)
Hence the overall symmetries in the observed vector properties arise from symmetries in the single-channel correlations. All single-channel correlations are rotationally invariant. Those involving two vectors are also invariant to inversion of both vectors or reflection of both vectors in a plane, since both of these operations are equivalent to a rotation. Further symmetries exist because mechanics isjnversion symmetric, and inveJting a-trajectory inverts k (a polar vector) but lefiyes J andl jl(axial vectors) unchanged. Therefore, W(J,k) and W(J,k,j) are invariant to inverting k. For a statistical channel, the linear and angular momentum vectors are illustrated in Figure le. This pertains to the product channel but the same situation holds for a statistical reagent channel with prim$s omitted. The vectors J , j’, and- 1’ are coplanar, and k’ is azimuthally symmetric about l’, so that the equally probable k’ vectors generate a disk perpendicular to the plane of the angular momenta. The situation is symmetric to reflection in this plane. Since any reflection is equivalent to inversion followed by ,a _rotation, y e cpnslude that for a statistical channel, W(J,k,j) = W(-J,-k,-j) and that all three v,ec$p may be reflectlei in any plane, without changing FV(J,k,j). Of course W(J,k,j) is also invariant to rotation of k about 1, but since 1 is unobservable, this operation yields no observable symmetries. We note also two limiting cases. As 1 0 , j J, and f is-not constrained by angular momentum conse;v_ation, so k is isotropic. In this case, for-eFample, W(J,k,j) ?ill 0 , l - J so have the same symmetries asW(J,k). As j k becomes perpendicular to J, and j is isotropic. Figure 3 provides a summary of these various symmetry properties of single-channel Correlations. C. Tal0 Vector Correlations. Any two-vector correlation depends only on the angle between the two vectors, which is rotationally invariant. We call the correlation even if it is unchanged by changing the- angle to its supplement. The correlations of k or k’ with j or j’ will always be even,
- -
-
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
Symmetry Properties ot Molecular Collision Complexes SiNGLE
CHANNEL
W ( Jk)
W( J , j )
1449
W(k,k) W(J,k,j
1
GENERAL
l=O
j=O
i.0
j’=O
T
ALL COLLISIONS
ALL COLLISIONS
A L L R=O
ROTATIONAL COMPLEX
A L L j =O
SEPARABLE COMPLEX
STATISTICAL
STATISTICAL REAGENTS
STATIST1CAL
STATISTICAL PRODUCTS
8=0 STATISTICAL
Flgure 3. Properties of single channel correlations for all collisions and for the special case of statistical channels. All vectors are unit vectors; the caret is omitted. A downward arrow means that a property indicated at the origin of the arrow also pertains to any box through which the arrow passes or terminates. The letter i followed by a vector indicates an invariance of the correlation to inversion of that vector; the letter o indicates an isotropic vector. When two or more inversions are listed together separated by commas, the correlation is symmetric to simultaneous inversion of these vectors.
because inverting the trajectory inverts a velocity vector but leaves an angular momentum unchangsd. Likewise, no dynamical syminetry ever requires W(j,j’)to be even. The only two-vector correlation for which the evenness depends on dynamics is the angular distribution W(k,k’). For a rotational complex,Ak’is azimuthally symmetric about J. However, unless J is perpendicular to k or k’ 0’ or j ’ = 0), this does not require the angular distribution to be even. For the separable complex, two product trajectprjes which are inverted pairs are equally probable, so W(k,k’) must be even and the angular distribution has backward-forward symmetry. Figure 4 ingisates the complex types and special cases for which W(k,k’) is even or isotropic. D. Three- and Four-Vector Correlations. Among the three-vector copelatips, jt is sufficient to consider only those relating k arid j to k’ or j’ since the others may be obtained from microscopic reversibility. 1. Noncomplex. For any reaction with optically inactive reagents and products we may invert a trajectory and leave the correlation unchanged. This yields
~ ( k , j , k ’=) ~ ( - k , j4’) , = w(k,c,(kx itl)j,k’) (loa) W(k,j,5/) = W(-k,j,j’) = W(k,-j,C,(k x 3)j’) (10b) W(k,j,k’,j’, = W(-k,j2-k,ji) = IY(k,Cz(kX k)f,k’,C2(kX k’);’) (lOc) The notations Cz(k&X3) and qz(kX k ) indicate rotation by 180’ about k X j and k X k’. The second equality on each line above is demonstrated by rotating the vectors of the inverted trajectcry by 180’ about the appropriate axis, returning k and k’ to the original directions. 2. Rotational Complex. No additional symmetries occur for a rotational complex except in the four special cases noted kelow. The product vectors are invariant to rotation about J but this symmetry is unobservable because J is not measurgd. 4s 1 0, j*- J. Then W(k,j,k’,j’)is invariant_tqd y(k,j,k’) but not in W(k,j,k’,j’). As j,’ 0,l’ J and k’ becomes perpe-ndicular ,t?J. Then k,c+ be inverted by rotation around J and W(k,j,k’) = W(k,j,-k’). 3. Separable Complex. As we have discussed, the separable complex differs from the rotational complex only in that the exit channel trajectory can be inverted sepaqately from- the-entrance channel. This operation inverts k’ leaving k, j, j’ unchanged. The-qdfiitional Fymmetries for !heseparable cpmp!ex_ are W(k,j,k’)= W(k,j,-k’) and W(k,j,k’,j’)= W(k,j,-k’,j’). 4. Statistical Reagent Channel. When the reagent channel is statistical the vector correlations have two azimuthal symmetries: rotation of k about l?nd relative rotation of the products and reagents about J. Together with the single-channel properties listed in Figure 3, these symmetries impose an important constraint on the three-vector c?r_reJations. For the reagent channel -WJ(J,j,k)1 vr(-J,j,k).,For any separable reaction Wp(J,j’) = Wp(-J,-j’) and Wp(J,k’)= W,(-J,k’). We conclude that
- -
W(k,-i,k’) = W(k,j,k) W(k,-j,-jO = W(k,j,j’,
(114 (1lb)
The gtatjstical reagent channel adds no symmetries to W(k,j,k’,j’)because no-dynamical symmetry of the product vectors which inverts J produces an observable symmetry of the product vectors. As 1 0, j J; hence and therefore k are unconstrained by angular momentum conservation and become isotropic.
- -
1450
The Journal of Physical Chemistry, Vol. 83, No. 11, 1979
G. M. McClelland and D. R. Herschbach W( k, j, k‘)
ALL COLLISIONS ROTATIONAL COMPLEX SEPARABLE COMPLEX
1
ALL COLLISIONS ROTATIONAL COMPLEX SEWRABLE COMPLEX
STATISTICAL REAGENTS
STATISTICAL REAGENTS
STATISTICAL PRODUCTS
STATISTICAL PRODUCTS
!
jl I
I
i
r
BOTH STATISTICAL
Obseryable symmetries of the correlation W(k,j,r). Symmetries of W(J,k’,r)can be deduced from this figure using the principle of microscopic reversibility. See Figure 4 for an explanation of notation.
Figure 5.
- -
As 1’ 0, i’ 3 and Wp(8,.?,k’) = Wp(-8,-j’,k), Combined with t,he-W, symmetry, this requires that W(k,j,k’,j’) W(k,-j,k’,-j’). As j 0, j becomes isotropic and k becomes perpendicular to J . _ As j’- 0, k’ becomes perpendicular to 3, but this does not influence the symmetries. 5 . Statistical Product Channel. Sinse as noted i n Figure 3, the two-vector correlations Wp(J,j’)and Wp(J,k’) are no more symmetric for a statistical channel than for a npptatistical cojljsion, no symmetries are added to W(k,j,k’) or W(k,j,j’), We also need not discuss the four-vector correlation because by microreversibility the symmetries are analogous to those of a statistical reagent reaction. As I’ 0 or j’ 0, the vectors k’ or 3/, respectively, become isotropic. 6. B o t h Reagents and Pro
