J. Phys. Chem. 1987, 91, 5451-5455
= mD1/mHDgH2/2m~ +~
H / ~ H D # D ~ ’ /~ 8rpfD2/mD2 ~ D ,
= mD1/mHDlEH + m H / m H D ~ -
@ H I ~ I ~ I / ~ cos H D ‘%,D2 ~
(A6)
The thermal kinetic energy of the D2 molecule appears in the second term on the right-hand side of the equation. Its average is 3/2kTwith a mean square fluctuation of the same amount. cos 6 H D l is the cosine of the angle between the momenta of H and D,. Because the scattering cross section is a function of the relative energy this cosine does not take on all values uniformly between -1 and + l . However, the energy dependence is weak and we will assume this merely to estimate the uncertainty in ET. For lpD21
5451
we take an average (8kTmD2/r)‘/2. The energy ET in eq A6 is the relative kinetic energy of an H atom and a D2 molecule. To obtain the relative kinetic energy for an H and H D collision, mDl is replaced everywhere by mHD in eq A6. The values of EREL, EH, and ET calculated from equations in this appendix are given in Tables I1 and I. The outstanding conclusion is that the relative incident kinetic energy in these experiments is uncertain to the extent of +10 to 15%. There would be some improvement if the dissociation energy of H2S were more precisely known but the major problem is that the thermal velocity makes an appreciable and varying contribution to the relative kinetic energy. Registry No. H, 12385-13-6; D,, 7782-39-0; HD, ‘ 1983-20-5; HIS, 7783-06-4.
Dynamical Stereochemistry in Nonreactive Collisions A. J. McCaffery School of Chemistry and Molecular Sciences, University of Sussex, Brighton BNI 9QJ, U.K. (Received: March 6, 1987)
Information on stereochemical forces in molecular collision dynamics may be obtained by using high-resolution polarized fluorescence in molecules in thermal cells. Particular attention is paid to diatomics in Z states where laboratory-frame experiments in conjunction with theoretical models, rigorous or simple, allow a collision-frame determination of a preferred collision geometry. In the case of II-state diatomics and triatomics, the presence of optically accessible molecule-frame vectors permits direct determination of stereochemical preferences from simple thermal- or flow-cell experiments.
Experiments that are sensitive to the angle dependence of the intermolecular potential in reactive and nonreactive collisions are generally both difficult and expensive, and a perusal of contributions from Fifth Fritz Haber Symposium rapidly will convince the reader of this. Molecular-beam techniques in conjunction with inhomogeneous electric and magnetic orienting fields are often essential in any attempt to obtain stereochemical data at the single-molecule level. However, this level of sophistication is not always necessary, and this contribution demonstrates that useful stereochemical information may be obtained from precise polarization measurements in conjunction with very high spectroscopic resolution on molecules in simple thermal cells. The use of narrow-line, tunable lasers enables the experimenter to select a wide range of input channels, and high-resolution detection using either a second laser or emission spectroscopy allows many exit channels to be sampled. Velocity dependence in the collision frame may be obtained through laser scanning of line profiles. Furthermore, although it is intuitively clear that direct interpretation of laboratory-frame spectroscopic experiments in the molecular frame is most readily achieved when the electric vector of the light wave can be “tied” to or correlated with a vector that is fixed in the molecule frame, here we demonstrate techniques for interpreting laboratory-frame observations in atom-diatom collisions in terms of molecule-frame events.
and NaK,3 and Se, and Te; demonstrated that the phenomenon is widespread. Double-resonance work on Ba05 and a combined laser-molecular-beam investigation of Na? have confirmed that the m quantum number is remarkably slow-changing following collisional interactions. Theoretical interest in this problem revolved around the evolution of criteria for evaluating approximations to the full quantal treatment of the atom-diatom scattering problem. For example, exact close-coupled calculations on a variety of atom-molecule systems7-” indicated a strong propensity to conserve m in space and collision frames for elastic collisions, but no propensity was observed in calculations on inelastic processes until Khare et chose either the kinematic or the geometric apse as the quantization axis. An interesting feature of these calculations was that sets of u(jm -j’mq for other quantization axes could be obtained by rotating the dominant u(jp -J’p’)Am = 0 cross section from the geometric or the kinematic apse into the appropriate frame. This conservation of angular momentum projection for the case of strongly repulsive interactions may be demonstrated by using classical a r g ~ m e n t s that ’ ~ show that there can be no change of rotor orientation along the momentum transfer vector in the “sudden” limit. However, when long-range anisotropic interactions become important, the situation
1. Polarization in Elastic and Inelastic Collisions With the availability of stable single-mode dye lasers for studies of energy transfer and the consequent ability to prepare single quantum levels in the input channel, fully resolved polarized fluorescence investigations revealed a surprising result. This was that the magnetic quantum number m in the laboratory frame is changed only very slowly as a result of collisions. This result NaLi was shown first for the case of IZ.l Later studies on
(3) McCormack, J.; McCaffery, A. J. Chem. Phys. 1980, 51, 405. (4) Ibbs, K. G.; McCaffery, A. J. J . Chem. SOC.,Faraday Trans. 2 1981, 77, 637. ( 5 ) Silvers, S.; Gottscho, R.; Field, R. J . Chem. Phys. 1981, 74, 6000. (6) Mattheus, A.; Fisher, A.; Ziegler, G.;Gottwald, E.; Bergmann, K. Phys. Rev.Lett. 1986, 56, 712. (7) Kouri, D. J.; Shimoni, Y . ;Kumar, A. Chem. Phys. Lett. 1977, 52, 299. (8) Monchick, L. J. Chem. Phys. 1977, 67, 4626. (9) Alexander, M. H. J. Chem. Phys. 1977, 66, 59. (10) DePristo, A. E.; Alexander, M. H. J . Chem. Phys. 1977, 66, 1334. (11) Schaefer, J.; Meyer, W. J . Chem. Phys. 1979, 70, 344. (12) (a) Khare, V.; Kouri, D. J.; Hoffman, D. K. J. Chem. Phys. 1981, 74,2275. (b) Khare, V.; Kouri, D. J.; Hoffman, D. X. J . Chem. Phys. 1982,
(1) Kato, H.; Jeyes, S . R.; Rowe, M. D.; McCaffery, A. J. Chem. Phys.
-.
Left. 1976. 39. 573.
76. 4493.
(2) (a) Rowe, M. D.; McCaffery, A. J. Chem. Phys. 1978, 34, 81. (b) Rowe, M. D.; McCaffery, A. J.; Chem. Phys. 1979, 43, 3 5 .
(13)McCaffery, A. J.; Proctor, M. J.; Whitakel, B. J. Annu. Reo. Phys. Chem. 1986, 37, 223.
0022-365418712091-5451$01.SO10 0 1987 American Chemical Society
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The Journal of Physical Chemistry, Vol. 91, No. 21, I987
is changed, theoretically at least, since experimental tests are lacking at present, and m conservation is not predicted in any frame. Very recently Orlikowski and AlexanderI4 have presented an analysis that predicts the class of system and scattering conditions under which the m quantum number will be preserved and those under which it will not. The significance of the distinction is that in the second of these situations the measurement of m-resolved cross sections will yield unique information on the anisotropy and in the former case it cannot. In the above three approaches to the problem of collision-induced magnetic transitions (or reorientation), namely, experiment, exact quantum theory, and related approximations and considerations of classical angular momentum, each is suggestive of a favored stereochemistry in atom-diatom nonreactive encounters. They indicate that a coplanar collision geometry is favored in which rotor angular momentum and orbital angular momentum are parallel. This correlation of vectors occurring in the collision frame may readily be inferred from observations in the laboratory frame through use of a formulation of the atom-rigid rotor scattering problem developed recently by Alexander and cow o r k e r ~ , ’ ~which . ’ ~ uses the “translational-internal” coupling scheme of Hunter, Curtiss, and SniderI6 and Grawert.” This approach, which emphasizes the importance of the transferred angular momentum, is a valuable aid to visualization since it allows vector correlations to be made in cell experiments on molecules that do not possess molecule-fixed internal vectors that are accessible to the light beam. It will be described in more detail below after a brief word on how cross sections are determined in laser-cell experiments. I.i. Determination of Cross Sections. The methods that are used to determine unpolarized cross sections or rates of stateresolved energy transfer using optical techniques are well documented.I8 Less well-known are the techniques for obtaining cross sections for reorientation by elastic collision2 and the transfer of orientation (or alignment) by inelastic c ~ l l i s i o n .The ~ most appropriate formalism for describing polarized arrays of molecules is the tensor density matrix or state multipole formalism, and the reader is referred to the pioneering work of Case, McClelland, and Herschbach19 for a full description. It is the state multipoles of population ( K = 0), orientation ( K = l ) , and alignment ( K = 2 ) that are created upon selection of a single rotational level with laser radiation.I9 These state multipoles evolve independently in a spherically symmetric collision environment,2 and thus each may have its characteristic rate for transfer upon collision. These quantities may be obtained from the pressure dependence of polarization and intensity signals in emission, and the ratio of the two, the circular or linear polarization ratio, is often very useful since many difficult-to-calculate and difficult-to-compensate-for quantities cancel. Tabulations of quantities that are a good approximation to population transfer cross sections (ao)are widely available.’8 A few values of u1 are tabulated for destruction of orientation by elastic collision and for transfer of orientation by inelastic collision. As anticipated by the discussion at the beginning of this section, the former values are very small, fractions of A2, while the latter are comparable to population cross sections.I8 I.ii. Vector Correlationsfrom Laboratory-Frame Experiments. Following Alexander and Davis,15 the scattering amplitude may be expanded in terms of irreducible tensor components to obtain expressions for the m-resolved cross sections
(14) Orlikowski, T.;Alexander, M. H. J . Chem. Phys. 1984, 80, 4133. (15) Alexander, M. H.; Davis, S. L. J . Chem. Phys. 1983, 79, 227. (1.6) (a) Curtiss, C. F. J . Chem. Phys. 1968,49, 1952. (b) Hunter, L. W.; Curtiss, C. F. J . Chem. Phys. 1973, 58, 3884. (e) Hunter, L. W.; Snider, R. F. J . Chem. Phys. 1974, 61, 1151. (17) Grawert, G. 2.Phys. 1969, 225, 283. (18) See, for example, Ennen, G.; Ottinger, C. Chem. Phys. 1974,31,404 or Brunner, T.A.; Pritchard, D. E. Adu. Chem. Phys. 1979, 50, 586. (19) Case, D. A.; McClelland, G. M.; Herschbach, D. R. Mol. Phys. 1978, 35, 541; Mol. Phys. 1975, 30, 1537; J . Chem. Phys. 1976, 64, 4212.
McCaffery
R
Figure 1. Relationship between initial rotational state j , , final rotational state j,, and transferred angular momentum k .
ej,
where is the tensor opacity function and k represents the transferred angular momentum, as illustrated in Figure 1 . The tensor opacity may be written as
where the reduced T matrix element is given by
Linear combinations of the d j m - j’m? may be taken to obtain expressions for the multipolar cross sections:
In explicit form for the three state multipoles of single-photon polarized fluorescence jj’# = ( 2 j + 1)-1/2(2j’+ 1)-1/2 pi, k
The polarization observables such as the circular or linear polarization ratio may readily be expressed in terms of these cross sections multiplied by angular momentum factors2 The sensitivity of population, orientation and alignment transfer to a particular k channel is contained in the angular momentum factors, and we can define a relative multipole sensitivity S ( K ) for each transferred angular momentum channel k , in which the population response is set to unity: S(K) =
(-Ilk[
;, j, i“][j, ;,
q-’
A typical multipole sensitivity plot20 is shown in Figure 2 for the Aj = -1 transition from the initial s t a t e j = 5, and the value of such plots is that the quantities involved are defined in the collision frame. They may be related to the laboratory frame through the relative sensitivities. To give a concrete example, in the case of “sudden” RET experiments, in which we know that there is very efficient transfer of the orientation ( K = 1 ) state multipole, the plot in Figure 2 tells us that it is the very lowest k channels that contribute most strongly to the transfer of angular momentum. This simple correlation of initial and final angular momenta through the transferred angular momentum k confirms the coplanar stereochemistry of the rotationally inelastic process that is a feature of quantal calculations. In other processes K = 1 may be less efficiently transferred, but if carefully measured, particularly if K = 2 transfer can also be determined, the dominant k (20) Bain, A. J.; McCaffery, A. J., submitted to J . Chem. Soc., Faraday Trans. 2.
Nonreactive Collisions: Dynamical Stereochemistry
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5453
1.0-
0 No
0.5-
Figure 3. Proposed structure of the long-lived alkali diatomic ('II)-rare gas complex.
S(K)
A-
'b
(
k -0
k -2
-05-
10.
1.0-
,
\
\
Figure 2. Plot of multipole response S ( K ) as a function of allowed k channels in the Aj = -1 transition for an initial j of 5 .
channels can be ascertained and thus information on the mechanism may be derived. I.iii. Diatomics in II States. The method described above is useful in the case of diatomics in 2 states, but when a II state is excited, an additional factor may be employed to relate vectors in the molecular frame, namely, the A-doublet structure. In any rotationally resolved transition from a 2 state, only one A-doublet component of a II state may be excited and the geometry of the excited electron cloud relative to the rotational angular momentum vector is known. Collision-induced changes in each of these vectors may be determined spectroscopically and related to the molecular frame. An example of the way in which this procedure may be used to infer mechanism is in the recent discovery of a long-lived complex that is formed between excited II-state alkali diatomics and certain polarizable rare gases.21 The characteristic feature of complex formation is the observation of a change in the relative rates of rotational transfer A j = 1/-I. An increase in the rate of rotational excitation is observed as the pressure of certain rare gases (notably Xe, Kr, and on occasion Ar) increases. This increase is marked by an inverse temperature dependence, the slope of the temperature plot showing a direct relation to intermolecular potential and furthermore exhibiting a marked dependence on initial rotational level. Perhaps the most striking feature, however, is that these unusual dynamical properties are observed only when the e A-doublet level is excited, thefcomponent being quite normal in its pressure and temperature behavior. The interpretation of these results in terms of complex formation and mode-specific dissociation has been given in recent publications.21 In this case the relative geometries of rotational angular momentum and electron cloud in the molecule frame tell us that the successful trajectory for complex formation is perpendicular to the plane of rotation, forming a complex having the geometry shown in Figure 3. The dissociation process to form a rotationally excited diatomic with A-doublet surface crossing requires an overall increase in rotational angular momentum and a rotation of the electron cloud about the bond axis since the transfer involves a transition between A-doublet components. Thus a combined rotational and vibrational motion is required, an effect that would be facilitated by
ortho
Figure 4. Energy level diagram for the ortho state of NH2 displaying the rotational K stacks within the (0, 9,O) vibrational state of the 'Al excited state. Arrows depict laser excitation and collisional-transfer routes.
202
+
(21) (a) Al-Imarah, F. J. M.; Bain, A. J.; Mehde, M. S.; McCaffery, A. J. J . Chem. Phys. 1985, 82, 1298. (b) Fell, C. P.;Brunt, J.; Harkin, C. G.; McCaffery, A. J. Chem. Phys. Lett. 1986, 128, 87.
I
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