Resonance Raman Lineshape Studies of Vibrational and Rotational Relaxation in Solution BY PAULA. MADDEN University Chemical Laboratory Lensfield Road Cambridge CB2 1EW Received 19th August 1976 The theory of resonance scattering of light is reviewed to show how the lineshape is determined by various relaxation processes. Possible applications of linewidth measurements are then discussed. The object of this paper is to clarify the relationship of the shape of a Raman line at resonance to the parameters which characterise the rotational and vibrational relaxation of molecules in their electronic ground states. The type of information which can be obtained will then be discussed. The reason for extending the study of Raman bands into the resonance region is that the scattering changes in three potentially useful ways (a) The scattering from very low concentrations of scatterers becomes detectable leading to the possibility that the vibrational relaxation in the neighbourhood of the chromophores in biological macromolecules may be investigated.(b) In addition to zero’th and second rank tensors which contribute to the ordinary Raman scattering a first rank tensor may also contribute giving the possibility of studying two moments of the reorientational distribution function in the same experimental conditions. (c) Long overtone progressions of totally symmetric vibrations of small molecules may be enhanced the relationship between their linewidths gives a way of probing the vibrational dephasing processes for small molecules and ions in solution.Before discussing further these possibilities it must be shown that these relaxation processes are indeed observable in the experiment. The Raman lineshape off-reson- ance is now well characterised and we shall use it as a reference point but at resonance the situation is confused because in addition to Raman scattering (RR) the resonance fluorescence(RF) will begin to contribute. RF is envisaged as a process of absorption followed by emission; the characteristic bandshape of the scattered light is thus expected to be that of an emission band and to reflect the relaxation behaviour of the excited electronic state. It will be seen that the theory of the next section includes both terms and that the predicted lineshape includes contributions from each; this means that ground state relaxation parameters are only recovered under certain limiting conditions.THEORY OF RESONANCE SCATTERING To show how the scattering process is related to resonance fluorescence and resonance Raman we present a simplified version of a theory which has been treated in detail e1sewhere.l We consider the time evolution of a density operator p which describes the occupation of states of the scattering molecule (a) and the radiation PAUL A. MADDEN field (x). In the absence of the field molecule interaction the density operator factorises po = tToxo. (1) p evolves subject to a Liouville operator where 9O A = [Hmol+ Hrad, A]-BA (3) where Hmolis the time independent part of the Hamiltonian for a molecule in a medium and Hradthe Hamiltonian for the radiation field.The time dependent parts of the molecule-medium interaction cause relaxation of the populations of molecular energy levels and this is described by g2.9’ is the part of the Liouville operator arising from the field-molecule interaction and for our purposes may be written as 9’A = (.9+ 9”A K [/mi A] (4) A] + [/lSES where pi and /is are the components of the molecular dipole operator along the polarisation vectors of the incident and scattered radiation and Eiand Es are the electric field operators for the modes of the radiation field which are populated by the incident and scattered beams. In the interaction representation the equation of motion of p is which is solved iteratively to obtain To understand the scattering process it is convenient to proceed via a description of absorption and emission.Experimentally in an absorption or emission experiment we observe the rate of change of the population of a final state of the molecule- radiation field system (called o,) when the system was prepared in an initial state Q. The molecular state in o is different from that in cc and the final radiation field state of o has one fewer photon in the mode i of the radiation field (for absorption) or one more in the mode s (for emission) than has the initial state cc.’ These processes arise from the second term in (8) Since we never explicitly observe the initial and final states of the radiation field only that one differs from the other by the presence of one photon we average over the radiation field (k= trXoA) dtl(ppl.@(r)&i(tl) + &s(t).@(tl)]aCC)02a (10) RESONANCE RAMAN LINESHAPE STUDIES the first term on the right represents the absorption process and the second the emission.We have omitted the cross terms L?i(t)@s(tl) etc. because the radiation field operators in the two modes are statistically independent (i.e. such products as Ei(t)Es(tavanish). In the scattering experiment the final molecular state differs from the inital molecular state (for a Raman process) and the number of photons in the final state is increased by one for the scattering mode and decreased by one for the incident mode from the occupancies of the initial state.The fourth order terms from (9) contribute to this process when by averaging the term over the radiation field we obtain where T1 = .@'(t)gs(t Yi(t2)2'(tJ + gi(t) Pi(tJ Ys( t2) L@ (tJ T2 = @(t)PS( (ti) =Pi( t2)Pi t3) + &( tl) LPs(t3)gi(t)Y'(t,) t3)Yi(ti)gi t2) gi(t)gi T3 = .@(t)gs( (t2) + .S@(rl)ps( (t3). By comparison with eqn (10) it can be seen that the term T1 corresponds to an absorp-tion process over the interval tz 3t3followed (after a delay note the time ordering of t3,tz tl and t)by emission from tl to t (and vice versa). It thus corresponds to the term normally called resonance fluore~cence.~ T2 and T3 on the other hand involve simultaneous " absorption " and " emission " processes and thus describe the Raman scattering process.This conclusion is borne out by the dispersion relationship' for the differential scattering cross section (a2C/~sZ8ws)(oi, us)for the two classes of terms. TI gives the classical RF dispersion relationship (see fig. 1) showing separate absorption and emission lineshapes; the T's are relaxation matrix elements thus rtlx Rqz ria (1 3) w+d 1 Ad FIG.1.-Explanation of the symbols used in eqn (12) and (IS). PAUL A. MADDEN is the linewidth for the a -+ q transition and is the inverse lifetime of the molecular level q. Away from resonance the T2 (and T3) cross sections reduce to the usual Raman scattering expressions where 6 is the shift from the peak of the Raman line and is the transition polarisability tensor. Note that eqn (15) identifies the parameter Tpa as the normal Raman width.Since all the terms in eqn (1 1) contribute to the scattering we must include them all in our expression for the lineshape. For the case of resonance with a single level we obtain' Frequently we expect to encounter the situation where the linewidth terms involving the intermediate state q are large compared to those involving only a and q because electronic relaxation and dephasing processes contribute only to the former. In this case and for Raman shifts not too far from the band centre (6 -r,,&the lineshape of the scattered light appears Lorentzian and the width at half height determines the desired parameter rpa. EXPERIMENTAL The experimental difficulties with these lineshape studies arises from sample heating by absorption of the laser light.This local heating tends to defocus the beam reducing the scattering intensity as well as making the temperature of the scattering medium an uncertain parameter. An additional diflticulty is that since we are dealing with solutions the solvent Raman bands will interfere with lineshape measurements. Rotating sample techniques have been developed by a number of workers and extensive reviews of this work have ap- ~eared.~*~ For lineshape studies the equipment must satisfy two basic requirements (i) the polarisation state of the incident and scattered radiation must be well defined (so that iso- tropic scattering may be separated). Consequently the polarisation vectors must always be tangent to (or avoid altogether) any curved surface through which the incident or scattered radiation passes.(ii) There must be some means of removing the solvent bands from the spectra. With these criteria in mind we have designed the system shown schematically in fig. 2. The cell consists of a piece of precision Suprasil tubing with windows fused to the ends so as to make a right angle at the corners (from which the scattering will be observed). The cell is divided into two halves one half contains the solution under study and the other pure solvent. The cell is rotated at -1800 r.p.m. by a hysteresis synchronous motor so that the position of the two halves is in phase with the sinusoidal voltage which drives the motor. A reference is taken from this voltage to control the gates of a two-channel photon counter (with a variable phase and duty cycle).We are thus able to measure the radiation scattered from the solution minus that from pure solvent. Complete elimination of the solvent bands RESONANCE RAMAN LINESHAPE STUDIES PC osc I ref I' S amp verticalT I \-pF-l FIG.2.-Schematic diagram of the spinning cell system; m is the hysteresis synchronous motor which is driven by a variable (0-100 c.P.s.) frequency oscillator (osc.) from which a reference signal (ref.) is taken to control the two channels of the photon counter (pc.) spec. is the spectrometer. is difficult because of noise and because the laser intensity in the (absorbing) solution is lower. The medium temperature in the laser focus may be determined by measuring the Stokes to anti-Stokes line intensities for the solvent bands.DISCUSSION OF APPLICATIONS The possible applications of the techniques labelled (a)and (b)in the Introduction have as yet received no attention and although the third is now fairly well character- ised more work is needed before significant results are obtained. (a)Many resonance Raman spectra of biological systems have been published particularly of the heme-pr~teins,~.~ but these studies have been concerned with the band positions and excitation profiles. The bands which are enhanced at resonance are those whose force field is appreciably altered by the electronic excitation. For example in the much studied cytochrome-c which consists of an iron atom bound to a porphyrin ring surrounded by protein the electronic excitation which is resonant is of the porphyrin and only ring modes are observed.An interesting systematic variation in the linewidths of these modes may be observed. The porphyrin has a distorted C, structure and we may label the modes according to their parentage in this group7 as Al A2 and B2 (there being several of each type). It is observed that the widths of these lines is such that r(AJ <r(A2)<r(B2).* Since these widths must arise from vibrational relaxation such a systematic variation suggests that specific information on the porphyrin-protein interaction could be learnt from a line- width study of these bands. (b)The possibilities posed by the presence of a first rank part in the scattering tensor for the study of reorientational motions may be more difficult to realise.McClain9 has published the group theoretical rules to be fulfilled by a band to give antisymmetric scattering. If a band were to give pure antisymmetric scattering (such as an A2 band of a C4 molecule) only a depolarised component would appear in the spectrum whose shape would be determined by both vibrational and rotational relaxation. We require instead a band which will induce both symmetric and anti- symmetric components in the polarisability tensor (e.g. the nominally A2 bands of cytochrome c) this can occur for the A2 vibrations of molecules of symmetry lower than C3,. We must then measure as well as the polarised and depolarised spectra the backscattering ~pectra,~*~~*~ in order to separate the contributions from the zero'th first and second rank polarisability tensors.It is even then unlikely that for the laser lines currently available and for the small molecules for which the re- PAUL A. MADDEN orientational motion is of interest antisymmetric scattering will be observed as it appears that the most stringent requirement is that the mode under study should vibronically mix two resonant excited states.11-13 It seems unlikely that there will be many molecules for which these criteria are met. (c) The overtone progressions of the totally symmetric vibrations of a variety of small molecules have been studied by a number of workers and the variation of half band width with the order of the overtone pubIished.l"-l6 The width of the isotropic part of these lines arises from vibrational relaxation and to understand the nature of the results we will briefly review a recent theory of this 1ine~idth.l~ The medium-molecule interaction is represented by a perturbation where the time dependence of Bl and B2arises from molecular motions in the medium and q is the oscillator coordinate.The theory of the relaxation rnatri~~*~' shows that rmn = ?/mn + rm/2 + rnl2 (19) where m and n label the vibrational levels and where ymnis the dephasing contribution to the linewidth and r,,,and the energy relaxation terms with rmand I'n involve the power spectrum of the correlation function of a medium operator at a vibrational frequency (of the order of say lo2 cm-l).Consider as an example the case where the term B,q arises from the vibrationally induced dipole- dipole interaction. in medium where Timis the dipole-dipole interaction tensor between a medium molecule j and the molecule under study. In this case which is a spectrum very similar to that of the dipole-induced dipole light scattering ZD~D(CO).'~ For small molecules this decays exponentially IDID(co> cc e-"/"O (25) with coo a frequency of the order of 10 cm-l. In this case we have Jl,(co = 100 cm-I) = e-I0 J,,(co) and we may neglect Trnand in eqn (19). Similar conclusions have been reached from a different point of view by 0thers.l' RESONANCE RAMAN LINESHAPE STUDIES If we assume that the oscillator is harmonic then qnn= 0 and q2, oc n.We then predict the width of the nth overtone Fnocc n2 (26) which is qualitatively what is observed (fig. 3) note that for this harmonic model I’ varies as n. We can then see by reference to eqn (20) that the power spectrum at zero frequency of the medium operators which cause the relaxation may be obtained from the experiments. FIG.3.The variation of overtone bandwith I‘, with 2.(a) is for the Add’ ion [ref. (1511 (b)is for Ti14in cyclohexane [ref. (16)] and (c) is for I2in CCl. [ref. (14)]. (d) shows the variation of (&)2 with n2 for a Morse oscillator basis the Morse potential is chosen to fit the observed frequencies for I2/CCI4. However as the figures show the experimental results tend to deviate from quadratic behaviour for high order overtones.To see if this is a defect of the har- homic model a Morse potential was determined from the measured anharmonicity constants its eigenvectors found and the matrix elements q, and q2, evaluated. For the Morse oscillator both qnnand q2, vary as n (q2, actually increases slightly more rapidly) and so the agreement with experiment is not improved. It should be noted that the experimental linewidths are subject to error for these cases as the bands are overlapping of low intensity and the solvent bands were not removed. An over-estimate of the linewidth would result if the observed spectra had a background contribution. The other possibility is that because rVcL is very large for these high overtones the approximate form of the lineshape from eqn (17) does not hold.This however seems unlikely as the resonant level for I2with the 5145 A laser line is t = 43 (in the gas phase)20 and the broadening of this level should be much higher than that of Y = 14 involved in the highest measured overtone. We have tried to give a survey of the possibilities for resonance Raman lineshape studies. It can be seen theoretically that the lineshape has too much information in the wings and that only the width is an important parameter (and then only under certain assumptions). Experimental studies of sufficiently high quality can now be accomplished and promise to yield very interesting information on the vibrational dephasing processes of bio-macromolecules and of small molecules.PAUL A. MADDEN P. A. Madden and H. Wennerstrom MoZ. Phys. 1976,31,1103. See e.g. C. P. Slichter Principles of Magnetic Resonance (Harper and Row N.Y. 1961). J. Behninger J. Raman Spectr. 1974,2,275. ‘W. Kiefer and H. J. Bernstein J. AppZ. Spectr. 1971,25 500. A very thorough review by W. Kiefer will appear in Adv. I.R. Raman Spectr. vol. 3 ed. R. J. H. Clark. See e.g. T. C. Strekas A. J. Packer and T. G. Spiro J. Raman Spectr. 1973,1 197. ’J. R. Nestor and T.G. Spiro J. Raman Spectr. 1973,1,539. a L. D. Barron and P. A. Madden unpublished. W. M. McClain J. Chem. Phys. 1971,55,2789. lo M. Pezolet L.A. Nafie and W. L. Peticolas J. Raman Spectr. 1973 1,455. l1 L. D. Barron MoZ. Phys. 1976,31,129. l2 0.Sonnich Mortensen Chem. Phys.Letters 1975 30,406. l3 J. Friedman and R. M. Hochstrasser Chem. Phys. Letters 1975 32,414. l4 W. Kiefer and H. J. Bernstein J. Raman Spectr. 1973 1,417. l5 Y.Bosworth and R. J. H. Clark J.C.S. Dalton 1975 381. l6 R. J. H. Clark and P. D. Mitchell J. Amer. Chem. Soc. 1973,95 8300. *’ P. A. Madden and R. M. Lynden-Bell Chem. Phys. Letters 1976,38,163. la T. I. Cox and P. A. Madden Chem. Phys. Letters 1976 41 188. l9 D. Oxtoby and S.A. Rice,preprint received. 2o R. F. Barrow and K.K. Yee,J.C.S. Faraday 11 1973,69,684.