8096
J. Phys. Chem. 1991, 95,8096-8103
Interference Effects in the Polarized Emission Spectrum of Methyl Iodlde at 248 nm: Scattering through Two Coupled Optlcally Bright Excited States Michael R. Wedlock, E. Jensen, L. J. Butler,* and Karl F. Freed The James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637 (Received: January 31, 1991)
This work describes measurements of the emission spectrum of dissociating CH,I excited at 248 nm. The experiments are designed to observe how the angular distribution of the emission is influenced by interference contributions to the two-photon transition probability. We present both the polarized emission spectrum of the first five C-I stretching features in the emission spectrum and the model quantum calculationsthat motivated this experiment. The present results are analyzed and compared to previous experimental results obtained by using 266-nm excitation. The C-I stretching progression generated by excitation at 248 nm shows a very early change in the angular distribution of the emitted photons. We attribute this result to interference terms that may contribute to the emission when two electronic states are resonantly excited. Absorption of a 266-nm photon excites molecules primarily to the 3 Qsurface ~ via a parallel transition, but excitation at 248 nm also permits significant perpendicular excitation to the IQI surface. We derive exact quantum expressions for the contributions to the measured intensities from CHJ molecules that absorb and emit via a single transition moment, molecules that absorb and emit via mutually perpendicular transition moments, and intensity due to interference terms in the two-photon transition amplitudes. Comparison of experimental and theoretical emission spectra, where the latter include or neglect interference terms, confirms our expectation that interference contributions may dramatically affect the spatial anisotropy of emitted photons and emphasizes the importance of comparing with full quantum calculations when excitation is possible to more than one excited-state potential energy surface.
I. Introduction from the Franck-cOndon absorption region. By observing emission into a series of overtones, we essentially watch the progress of the Analysis of the emission from dissociating molecules' is an dissociation, important probe of the early dynamics of subpicosecond dissoWhen the molecule is excited to only one excited electronic state ciation processes, a subject of increasing interest.* Because the polarized emission intensity arises from only two sources:6 the electronic potential curve crossings and conical intersections are first is from molecules that dissociate along a single electronic pervasive in dissociative-state manifolds, it is important to develop surface, while the second is from molecules that cross to a second ways to probe molecular dissociation on coupled electronic states. surface in the course of the dissociation. If the electronic transition Recent work in our laboratory3 extends beyond the study of dipoles of the two surfaces have different orientations in body-fuced molecular dynamics on a single Born-Oppenheimer potential coordinates, then molecules that cross to the second surface emit energy surface and demonstrates that resolving the polarization photons with a different spatial distribution that those that emit of the emitted photons can probe dissociation along two excited from the surface initially excited. This allows us to resolve the electronic states that are nonadiabactically coupled. Polarized two contributions to the intensity by measuring the intensity at emission spectroscopy is an extension of a technique introduced a fKed detector for two different polarizations of the incident laser. by Imre et al.' and has been applied by our group to nitromethane4 and to the dissociation of methyl iodide through a curve c r ~ ~ ~ i n gSince ~ ~ the emission spectrum is a weighted sum of the intensity due to single surface and crossing terms for the 90' geometry of after excitation at 266 nm. Both molecules are examples of our measurements, we can resolve the contribution to the measured systems in which initial excitation promotes the molecules to a intensity from each process for each feature in the emission surface that does not correlate diabatically to all the primary spectrum using only two polarizations of the incident light. dissociation products. In each of these cases photon absorption This paper applies polarized emission spectroscopy to a system promotes molecules almost exclusively to a single excited state in which the excitation simultaneously promotes molecules to two which is coupled nonadiabatically to another excited state in parts coupled excited electronic surfaces. Then the emission intensity of the potential surface the molecule traverses after leaving the is e x p t e d to include contributions not only from the incoherent Franck-Condon region for absorption. processes described above, which are important when the excitation We interpret the polarized emission spectrum in terms of exis resonant with a single surface, but also from interference cited-state dissociation dynamics' by relying on a semiclassical contributions that arise for excitation to two initial states. The correlationSbetween the scattering wave function on the excited existence of such interference is well understood when the two potential surface and the ground electronic state vibrational wave resonant excited states are not coupled. When the electronic states functions. Features in the emission spectrum appear when the are coupled, additional interference terms dependent on that scattering wave function moves through different regions of the coupling as well as the incoherent curve-crossing terms described excited-state potential energy surface and develops good above can also change the polarization of the scattered Franck-Condon overlap with the ground electronic state vibraWe investigate the different contributions to the polarized emission tional wave functions. In a pseudodiatomic model of methyl spectra through comparison of emission spectra taken a t waveiodide,S the higher C-I stretch overtones arise from molecules lengths where excitation to one versus both electronic states is emitting in regions of the excited potential surface more distant dominant and through detailed theoretical analyses, using model coupled electronic states, of the predicted contributions from each (1) Imre. D.; Kinsey, J. L.; Sinha, A.; Krenos, J. J. Phys. Chem. 1984,88, scattering process. Thus, in this work we measure the emission 3956. spectrum of methyl iodide after excitation at 248 nm, a wavelength (2) Zewail, A. H.; Bernstein, R. B. Chem. Eng. News 1988, 66(45), 24. (3) (a) Lao, K. Q.; Person, M. D.; Xayariboun, P.; Butler, L. J. J . Chem. Phys. 1990, 92, 823. (b) See ref 3a and extensive references therein. (4) Lao, K. Q.; Jensen, E.; Kash, P. W.; Butler, L. J. J. Chem. Phys. 1990. 93, 3958. ( 5 ) (a) Heller, E. J.; Sundberg, R. L.; Tannor, D. J . Phys. Chem. 1982, 86, 1822. (b) Lee, S.-Y.;Heller, E. J. J . Chem. Phys. 1979, 71, 4777 and
references therein.
0022-365419 1/2095-8096$02.50/0
(6) This discussion considers only the resonant terms in the Raman scattering. The non resonant contributions can also be important, particularly for scattering into fundamentals, as sem by A. B. Myers, preprint. (7) Singer, S.J.; Lee, S.:Freed,K. F.;Band, Y. B. J . Chem. Phys. 1987, 87, 4762. (8) Heather, R.; Metiu, H. J . Chem. Phys. 1989, 90, 6903.
0 1991 American Chemical Society
Interference Effects in Polarized Emission Spectra at which excitation to two excited electronic surfaces is possible. The experimentally obtained polarized emission spectrum is compared to the previously measured one3 for excitation at 266 nm and to calculated polarized emission spectra of methyl iodide which we generate for 248-nm excitation. The next section describes the experimental apparatus and computational method. Section 111 contains a brief description of the previous work on the emission spectrum at 266 nm, followed by a derivation of the contributions to the polarized emission spectrum intensities when only one excited electronic state is initially excited. Section IV follows the same format for methyl iodide excited at 248 nm and includes expressions for the Raman intensity when two coupled electronic states are initially excited. The final sections compare the experimental resonance Raman spectra of methyl iodide to those generated computationally and discuss the principal processes that contribute to the observed polarized emission spectrum. 11. Method A. Experimental Apparatus. Polarized emission spectroscopy has been described in detail elsewhere." Briefly, our apparatus for this 248-nm measurement consists of either a Questek 2640 excimer laser or a Lambda-Physik EMG-103 MSC laser, a flowing gas cell, a monochromator, and data acquisition electronics. We use the laser with KrF to excite the sample molecules in the first continuum band. The laser beam passes through a MgF2 Pellin Broca prism which separates the unpolarized light into two linearly polarized components. The more intense of the two beams passes through a 248-nm half-wave plate into the sample cell, which is at right angles to the detector axis. Gaseous methyl iodide slowly flows through the stainless steel cell, equipped with Suprasil windows, a t a partial pressure of about 1.5 Torr, while about 7 Torr of helium simultaneously flushes the side arms. The emitted photons are collimated and focused by two lenses, f number = 6.9, through a depolarizing wedge, into a I-m Jobin-Yvon monochromator, where they are dispersed in first order by a 2400 groove/" grating. The resolution is about 65 cm-l. The emitted photons are collected by an EMG 9635QB photomultiplier tube located at the exit of the monochromator. The signal is integrated over a 30-11s gate width, digitized, and stored in our computer. The spectra are typically taken with pulse energies of 3 mJ/pulse, with a photon flux of about 7 X lOI5 photons/cm2, exciting less than 0.3% of the molecules in the interaction region. We measure the angular distribution of the emitted photons by rotating the polarization of the 248-nm exciting light with a half-wave plate to lie either perpendicular to or along the detector axis, and then measure the relative intensity of the emission features. Each spectrum is the sum of 20 scans, signal averaged for 20 shots at each data point. Data points are taken at 0.5-A intervals. We subtract background scans in 8 Torr of flowing helium at each polarization from the data to remove fluorescence signal from background gaseous impurities from our data. The area of each feature is integrated over a range extending 85 cm-l above and below its midpoint. B. Computational Formalism. The theoretical polarized emission spectrum from resonant electronic excitation to coupled dissociative surfaces is computed by using the time-independent quantum formalism devised by Singer et al.' This formalism begins from the second-order perturbation theory expression for the transition amplitude an from initial state i to final state f
as induced by the two perturbations HI and H2. The Singer et al. formalism is general for processes in which an initial quantum state evolves to a final quantum state by weakly allowed transitions through a manifold of coupled intermediate levels. For Raman scattering li> and 1 0 represent the initial and final vibrational state vectors on the ground electronic potential surface respectively, and the b> are eigenvectors of the excited-state Hamiltonian. HI and H2 are the absorption and emission transition dipole operators
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8097 that couple the ground and excited electronic potential surfaces. E is the total energy of the system, including that of the exciting photon, and we assume integration over continuum eigenstates. Singer et al. note that the sum over intermediate levels is just the energy eigenstate representation of Green's operator for the excited-state Hamiltonian H. We thus rewrite the transition amplitude as a matrix element of Green's operator
so that the second-order transition moment becomes = (2.3) Introducing the transition dipole operators pa and pb converts (2.3) into an
afi =
(2.4)
where GI and are unit vectors in the direction of the incident and scattered laser light polarizations respectively, and pa and pp are transition dipole operators connecting the ground and excited electronic states. We can then factor the transition amplitude as afi = ~((PB'Ep)(F..GI) a&
(2.5)
where [Aai> E pa(R)li> and Pa and PB are unit vectors. Following Heather and Metius it is convenient to introduce two definitions: TBa (P&)(Pa*s,) (2.6a) (2.6b)
CBf;iE
Heather and Metiu employ a time-dependent formalism in which they evaluate the reduced amplitudes Cfid from time correlation functions. We use the simplified notation , C = CbP4(E) for the remainder of this paper. However, we emphasize here that the reduced amplitudes depend not only on the electronic excited states, but also on the initial and final ground electronic state vibrational wave functions and the excitation energy. Since the methyl iodide system has two excited electronic states that contribute to the absorption at higher excitation energies, the transition amplitude (2.4) becomes adad = TIICII + T I Z C+I ~T~ICZI+ T22C22 (2.7) The four terms in (2.7) represent, respectively, parallel excitation followed by parallel emission, perpendicular excitation followed by parallel emission, parallel excitation followed by perpendicular emission, and perpendicular excitation followed by perpendicular emission. These four processes are represented schematically in Figure 1. In addition to the sum over states representation of eq 2.1, Green's function matrix can be expressed in terms of two linearly independent matrix solutions to the Schrodinger equation with the excited-state Hamiltonian' [ E l - H(R)]s(R)= 0 [El - H(R)]h(R) = 0
(2.8) where R is the reaction coordinate (the CH3-I separation), 1 is the unit operator and H(R) is the excited-state Hamiltonian. We enforce physical boundary conditions for Green's operator G(+)(E) by applying appropriate boundary conditions to the two solutions s(R) and h(R). The regular solution s(R) is chosen so that it vanishes as the radial scattering coordinate R 0. The irregular solution h(R) is forced to behave at large R as a diagonal matrix of Riccati-Bessel functions normalized to unit flux. The transition amplitude may then be recursively propagated by using the renormalized Numerov methodg and the propagation scheme devised by Singer et ale7
-
~~~~~
(9)(a) Johnson, B. R.J. Comput. Phys. 1973,13,445. (b) Johnson, B. R. J . Chem. Phys. 1977,67,4086.
8098 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
Wedlock et al. derivative coupling becomes a simple potential coupling. After an adiabatic expansion of the wave function N
~ ( R J )= CXn(R) n=O
Qn(W
(2.9)
in terms of electronic Q,,(R,r)and nuclear xn(R) functions, the Schrijdinger equation for the CH3-I excited surface dynamics becomes
where and Cl
dl
Figure 1. Two possible transition moments for excitation and two possible ones for emission lead to four possible Raman scattering processes: (a) parallel excitation followed by parallel emission; (b) parallel excitation followed by perpendicular emission: (c) perpendicular excitation followed by parallel emission; (d) perpendicular excitation followed by perpendicular emission. The processes depicted in (b) and (c) require nonzero potential coupling between the diabatic electronic states.
We use potential curves adapted from the ab initio computations of Tadjeddine et al.IOand of Yabushita and Morokuma." Yabushita and Morokuma generate two-dimensional adiabatic potential energy curves for methyl iodide using an a b initio CI calculation in which the spin-orbit interaction is included through an effective one-electron spin-orbit Hamiltonian. They also calculate nonadiabatic radial derivative matrix elements for the coupling between the relevant adiabatic potential energy surfaces. The potential curves calculated by Yabushita and Morokuma show an avoided crossing in C, symmetry, but in C, geometry they are qualitatively similar to the one-dimensional a b initio potential curves of Tadjeddine et al., for which numerical values are published. We employ a quasi-diatomic model with the reaction coordinate R as our only coordinate. Consequently, we use the Tadjeddine potentials 2AI (3Q0) and 3E ('QI) for our dynamics calculations after first interchanging the potential curves for internuclear distances larger than the intersection so that they undergo an avoided crossing rather than an intersection and thereby mimic the adiabatic potential energy curves computed for C, geometry. We assume that the potential surfaces do not radically change for small distortions from C3, symmetry. For nonadiabatic radial derivative coupling to be nonzero, the nuclear wave function must sample geometries out of C3, symmetry, and this limits the applicability of a quasi-diatomic model. The nonadiabatic radial derivative coupling element between the A' surfaces is presented as a function of R(C-I) in Figure 4 of Yabushita and Morokuma for the case where the C-I bond is bent 5' away from the symmetry axis of the molecule. Their a m putation is approximated with a Lorentzian-shaped coupling matrix element which is centered on the crossing region of the Tadjeddine potentials. A future paper will present results computed by using empirical potentials calculated by Guo and SchatzI2 which are based on earlier model potentials by Shapiro,13but which they modify to improve agreement with more recent experimental data. The Smith transf~rmationl~ is used to convert from an adiabatic to a diabatic representation in which the nonadiabatic radial (IO) Tadjeddine, M.; Flament, J. P.;Teichteil, C. Chem. Phys. 1987, 118,
45. (1 1) Yabushita, S.; Morokuma.
K.Chem. Phys. Lett. 1988,153,517. For a more recent treatment. see also: Amatatsu, Y.;Morokuma, K.;Yabushita, S . J. Chem. Phys. 1991, 94, 4858. (12) Guo, H.; Schatz, G. C. J . Chem. Phys. 1990, 93, 393. (13) Shapiro, M. J . Phys. Chem. 1986, 90,3644. (14) (a) Smith, F.T. Phys. Rev. 1969,179.11 1. (b) Baer. M.; Drolshagen, G.; Toennies. J. P. J . Chem. Phys. 1980, 73, 1690.
U is a diagonal matrix of electronic potential energies and R is the translational coordinate. The Smith transformation introduces a new basis set through the transformation matrix A, so that Q = Aq, x = A{, and $ = q'A'A{. After this transformation the Schrijdinger equation is simplified to
where the transformed potential matrix AtUA is obtained as T(Ull+Ud +
T1( u * l - u d501 2c
-(ull-ud 1
sin 2c
2 1
1
T ( U l 1 + U d --(u]]-uu) 2
; z ( U I 1 - U& rin 2c
-2c
(2.13)
where the V, are elements of the adiabatic potential energy matrix, where c = - ~ t a n - l ( 2(R 2
- xo) ) + I 4
(2.14)
and where xo is the value of the dissociation coordinate where the excited potentials cross. r is the full width at half-maximum of the coupling function, which is 0.05 au.
111. Excitation at 266 nm A. Emission spectrum of CH31for Excitation at 266 m. The first methyl iodide absorption band has been studied in detail by several m e t h ~ d s . ~ . ~Even J ~ J ~though five distinct states may arise from the n u* excitation in the strong spin-orbit coupling limit, only transitions to three of these are optically allowed in a single photon t r a n ~ i t i o n . ~ Crossed ~ . ~ ~ laser-molecular beam experimentsI6 a t 266 nm measure an anisotropy parameter j3 of approximately 2, corresponding to absorption via a transition moment parallel to the dissociativ_e C-l bond and indicating that the oscillator strength for the A X transition at 266 nm is carried primarily by the 3Q0excited electronic surface. However, the crossed laser-molecular beam experiments detect two significant product channels. The 3Q0surface correlates diabatically to spin-orbit excited iodine atoms I(*PIl2),while the 'QIsurface, which is not accessed significantly by absorption at 266 nm, correlates diabatically to ground state I(2P3/2). It is clear that
-
-
(15) Gcdanken. A.; Rowe, M. D . Chem. Phys. Lett. 1975, 31, 39. (16) At 266 nm: (a) Riley, S.J.; Wilson, K. R. Faraday Discuss. 1972, 53, 132. (b) Dzvonick, M.; Yang, S.;Bersohn, R. J . Chem. Phys. 1974,61. 4408. At 248 nm: (a) Van Veen, G. N. A.; Baller, T.; de Vries, A. E.;van Vetn, N. J. A. Chem. Phys. 1984,87,405. (b) Barry, M. D.; Gorry, P.A. Mol. Phys. 1984.52.461. At 229.4 nm: Penn, S.M.; Hayden, C. C.;Carlson Muyskens, K. J.; Crim, F. F. J . Chem. Phys. 1988,89, 2909. (17) (a) Mulliken, R. S.Phys. Rev. 1935, 47, 413. (b) Mulliken, R. S. Phys. Rev. 1937, 51, 310.
Interference Effects in Polarized Emission Spectra X
The Journal of Physical Chemistry, Vol. 95, No. 21, 1991 8099 TABLE I: Geometric Factors for Averaging over Molecular Orientations" A
A
Figure 2. Schematic diagram of experimental geometry. Incident photons propagate along the Yaxis (k, = -Y),and we detact scattered light along the 2 axis. q and 0 define the directions of incident and scattered light polarizations with respect to the lab-fixed X axis, respectively.
the two surfaces are coupled at C-I internuclear distances larger than the ground-state equilibrium C-I separation. Ab initio electronic structure calculations9J0confirm that these two surfaces intersect a t a C-I distance of -2.35 A. Absorption to and emission from the 'QOsurface occurs via a transition dipole moment parallel to the C-I bond ("parallel absorption and emission"), but absorption to and emission from the 'Q1 surface occurs through a transition dipole moment perpendicular to the C-I bond ("perpendicular absorption and emission"). As molecules promoted to the excited state dissociate, those that evolve along the diabatic 3Q0 surface emit via a parallel transition moment. However, some of the excited molecules cross to the 'Q1surface during the dissociation and undergo perpendicular emission. These two processes generate different spatial distributions of the emitted photons as described in a previously reported study of the polarized emission spectrum of CH31excited at 266 11111.' In that work, the emission intensities for two incident laser polarization directions, perpendicular to or parallel to the detector axis, provide a measure of the spatial anisotropy of the emitted photons. The Appendix shows that the predicted ratio of the emission intensity for the two polarization directions is 2:1 if the molecules both absorb and emit via a parallel transition. This ratio would be observed for emission features whose intensity comes from amplitude that propagates exclusively along the 3Q0 surface. However, the intensity ratio is 3:4 for molecules whose absorption moment is perpendicular to the emission mbment, as is the case for intensity due to amplitude that is excited to the 'QOsurface but crosses to and emits from the 'QIsurface. The photofragment anisotropy of /3 = 2 measured in crossed laser-molecular beam experiments suggests that the oscillator strength is carried by the 'QO 2 transition at 266 nm, so only two contributions to the emission spectrum are possible at this wavelength: intensity due to amplitude that propagates along a single potential energy surface (in this case the 'QO), and intensity due to amplitude that crosses to the 'QI surface. We thus weight the contributions from these two dynamical processes to probe the number of photons emitted from molecules that have crossed to the 'QIsurface. The "earliest" (shortest wavelength) features in the emission spectrum of methyl iodide excited at 266 nm have one or two quanta in the C-I stretch and show the nearly 2:l intensity ratio characteristic of absorption to and emission from a single surface. However, the intensity ratio for "later" (longer wavelength) features in the emission spectrum decreases steadily with increasing number of quanta n, in the final state C-I stretching (v3) mode, reflecting contributions from molecules which undergo parallel absorption to the 'QOsurface but which emit via a perpendicular transition from the 'Q1surface. The growth in perpendicular emission for final states with larger n indicates that amplitude crosses to the 'QI surface at C-I internuclear distances where there is good Franck-Condon overlap for emission into higher C-I vibrational overtones of the ground state. The next section outlines the quantum formalism for describing emission upon excitation to two coupled electronic states where the photon energy is such that only one of the two states is initially excited. B. Raman Polarizability at 266 MI. This section derives the Raman intensity after resonant excitation of methyl iodide. The expected intensity ratios are evaluated for cases where the excitation laser polarization direction is either perpendicular to or
-
"This table shows the analytic expressions derived for the geometric coefficients (T,,T,..). Terms not shown here average to zero when the average over molecular orientations is made: the expressions for the ( T,,Tkr)include products of direction cosines, and only terms containing the following forms survive integration over Euler angles, resulting in the values in section B, where F # F'and g # g'. along the detection axis. The special case of initial excitation to only one excited electronic state is treated first. The derivation is generalized in the next section to describe the Raman scattering intensity for excitation to two coupled excited electronic states. Methyl iodide may have dipole allowed transitions to three excited electronic surfaces in the first absorption band: 'QO, 3Q1, and IQ1. Crossed laser-molecular beam experiments suggest that the excitation at 266 nm is almost exclusively (>99%) to the 'QO surface.3b MCD15 experiments suggest that the absorption at 266 nm is 5% perpendicular. This means that contributions to the emission from the Cf6.d terms corresponding to absorption to the state labeled a! = 2 ('QIsurface) are small. If we neglect absorption to the *Q1 surface, the expression (2.7) for the Raman polarizability simplifies to The intensity is obtained by absolute squaring (3.1) as
The angular brackets in (3.2) indicate an average over molecular orientations. However, using the geometry shown in Figure 2 and choosing pI perpendicular to p2 in body-fixed coordinates, the last term disappears because the geometric factor vanishes after averaging over molecular orientations (see Appendix). This leaves only two contributions to the intensity, as mentioned earlier: one due to parallel absorption followed by parallel emission, and the other due to parallel absorption followed by perpendicular emission. The polarized emission spectroscopy experiment relies on the anisotropic distribution of photons emitted from molecules that have been excited with polarized light. This anisotropic distribution of photons allows preferential detection of light emitted through either parallel or perpendicular transitions. Varying the incident light polarization is equivalent to moving the detector. Hence, by measuring the relative intensities of the emission features for excitation laser polarization perpendicular to or along the detector axis we are able to separate the contributions to (3.2) and thereby to study how the potential surface intersections affect the dissociation process. The intensity ratio for the two different incident polarizations I,,,0/I,,lr12 may be written in terms of Tsol and cf#,ai as
where r ) is the angle between the direction of the incident light polarization and the lab-fixed X axis. The geometric coefficients are derived in the Appendix and presented in Table I. Examining the ratios of the geometric coefficients in (A.6) and (A.7), it becomes apparent that when photodissociative flux crosses from the 'QOsurface to the lQ1surface, the polarized emission intensity ratio should decrease from a value near 2 to an intermediate value between 0.75 and 2 as the molecule emits into higher and higher ground state overtones. Experimentally, the intensity ratio is observed to drop from near 1.9 to near 1.4, as seen in Figure 5.'*
Wedlock et al.
8100 The Journal of Physical Chemistry, Vol. 95, No. 21, 1991
The transition of amplitude to the 'Q,surface at the curve crossing is reflected in an increased contribution from the C21;literms for emission into the higher C-I stretch final states. IV. Excitation at 248 nm A. Raman Polarizability at 248 nm. Excitation to the perpendicular surface in methyl iodide should be proportionately larger at 248 nm than at 266 nm, but the excitation is still predominantly parallel.14 Thus, we now consider the more general case of Raman scattering at an incident frequency o1that permits excitation to both excited-state surfaces. We can no longer discard the terms in eq 2.7 for absorption to the second surface. Thus, the absolute value squared polarizability of (2.7) produces I
p:
+ TI2Cl2+ T21C2,+ T22C22>2
a < C Y ~ ~ ( W ~= ) > I C 2 1 I 2+ 1C22I2
+ 2Re (CII*CI2) + 2 Re (Cl,*C2,)+ 2Re (Cll*C22)+ 2Re (CI2*C2J+ 2Re (CI2*C2,)+ 2Re (C2,*C2,) (A.2) It is useful to distinguish two types of terms in eq A.2, namely, the incoherent terms, such as ICI 112, and the interference terms, such as 2 Re (Cll*C12).When the average over molecular orientations is taken, several of the interference terms vanish. We analyze Iafi12 using the scattering coordinate system of Lao et aL3 for analyzing their polarized emission studies of methyl iodide at 266 nm. The coordinate system is presented in Figure 2. The space-fixed axis system (X,Y,Z)is defined so that photons are incident along the Y axis (k, = -Y), and the detector is centered on the space-fixed Z axis. The angles q and 8 define the direction of polarization of the incident and scattered light with respect to the space-fKed X axis, respectively. Molecule-fixed coordinates (x,y,z) are established so that the z axis lies along the molecular symmetry axis. The transition dipole vectors in the space-fixed coordinate system are p1 = z = &x2+ &yP + l#Jzz2 (A.3) $2 = x = qQ + &yP + lpxz2 The &j are direction cosine matrix elements.24 Because we treat
8103
J . Phys. Chem. 1991,95,8 103-8 110 the methyl iodide molecule as quasi-diatomic, the cytindrical symmetry of the problem allows us to place the perpendicular transition dipole vector anywhere in the x-y plane. Lao et al. define their perpendicular transition dipole as being in the (l,l,O) direction, but the algebra is somewhat less cumbersome if the dipole is chosen along either the x or y molecule-fixed axes. The electric vectors for the incident and scattered radiation are given in our coordinate system by 8, = 2 cos 7 + 2 sin 7
8, = 2cos 8
+ P sin 8
(A.4) The scalar products of transition dipoles with polarization vectors produce 81-& = cos 7 dZx sin 7 q5zz
+ 4+i2= cos 7 dXx+ sin 7 q5xz 8s.fil = cos 8 Ozx + sin 8 &y
4.F2 = cos 0 & .+ sin 8 &y
(A.5) The nonzero orientation averaged geometric factors are then found to be (24) Zare, R. N. Mol. Photochem. 1972,4, 1.
+ 1) 0 + sin2 0 + 1)
< T I I T I I >= = (1/15)(2 cos2 7 cos2 B =
