J . Phys. Chem. 1993, 97, 12153-12157
12153
Raman Excitation Profiles for Titanium Tetraiodide: A Consistent Analysis Based on the Transform Method M. C. C. Ribeiro, W. J. Barreto, M. L. A. Temperini, and P. S. Santos' Laboratbrio de Espectroscopia Molecular, Instituto de Qujmica da Universidade de Sao Paulo, Sao Paulo, CEP 01 498, CP 20780, Brazil
Received: March 22, 1993; In Final Form: August 30, 1993'
The transform method is employed to reproduce the Raman excitation profiles of the v1, 2u1, and 3vl Raman bands, as well as those of the anti-Stokes components -VI, and -2v1 of titanium tetraiodide in cyclohexane solution. The theoretical fit of the profiles was achieved with a unique set of physical parameters; in particular, the line-shape fitting was based exclusively on the linear non-Condon to linear vibronic coupling ratio, m/(,that optimizes a t 0.055 f 0.005. The use of a different vibrational frequency in the excited electronic state does not improve the fit a t any extent. From the fit of the relative intensities of the profiles, an absolute value of = 3.7 f 0.1 was obtained, leading to a Ti-I bond-length variation in the excited electronic state of 0.105 A. The expressions of the transform method corresponding to an anti-Stokes component and Stokes overtones beyond the first are here presented explicitly with the simultaneous inclusion of linear vibronic and non-Condon couplings.
Introduction Our present knowledge of the nature of the resonance Raman effect has been derived, to a great extent, from comprehensive investigations of the resonance Raman spectrum of small, highly symmetric molecules. Titanium tetraiodide is an archetypical example of such a molecule, and its resonance Raman spectrum is particularly noteworthy due to the presence of a very long progression in the totally symmetric stretching mode, vI. On the other hand, its electric dipole allowed electronic transition is very conveniently located in the neighborhood of 500 nm, making possible the use of several Ar+, Kr+, and dye laser lines to generate excitation profiles of remarkable quality for VI and its overtones. In addition, since the Raman shift of V I is only 161 cm-1, the corresponding anti-Stokes components can be observed with appreciable intensities. Notwithstanding these favorable features, a comprehensive investigation of the Raman excitation profiles (REPs) of Ti14 involving V I , its overtones, and anti-Stokes components has not yet been undertaken. In fact, its resonance Raman spectrum in cyclohexane solution was first reported by Clark and Mitchel1,l whose analysis indicated a slight anharmonicity for V I in the ground electronic state (see also refs 2 and 3). More recently, a theoretical analysis was performed, although theoretical profiles were not calculated.4 One of the most serious difficulties in the calculation of REPs is the so-called multimode problem, intrinsic to the theory of the resonance Raman effect of polyatomic molecules. In this respect, the transform method allows one to circumvent this difficulty by transfering the multimode information contained in the optical spectrum directly to the REPs.5 Provided the REPs of the overtones have been determined, the transform method also permits, in principle, the determination of the absolute magnitude of the shift of the potential minimum in the excited electronic In practice, the limiting factor is the reliability of the theoretical fitting of the profiles. In this regard, Ti14 is an ideal system since high-quality profiles can be obtained. In addition, reliable profiles can also be obtained for the anti-Stokes components, to which the transform method can be applied using equations explicitly developed. Another consequence of the relatively low energy of the V I vibrational quantum in Ti14 is a significant population of excited vibrational levels of that mode .Abstract published in Advance ACS Absrracrs, October 15, 1993.
0022-365419312097-12153$04.00/0
in the ground electronic state at ambient temperatures. This, in turn, permits one to test the effect of temperature predicted by equations of the method, a subject that has been of recent interest.*,9 In the present paper, the transform method is applied in a convenient fashion to obtain the REPs for a fundamental (vI), for its overtones (2vl and 3Vl) and for the anti-Stokes components ( - V I and -2vl) using a unique set of physical parameters. In addition, the equations of the method corresponding to the antiStokes fundamental band, with simultaneous inclusion of linear electron-phonon and linear non-Condon couplings, plus a frequency shift in theexcited electronic state, as well as theequations for the Stokes overtones higher than 2vl with simultaneous inclusion of linear electron-phonon and non-Condon couplings are now made explicit.
Theory
In this section, a brief review of the equations used in the calculations is presented, with particular emphasis on those not explicitly available in the literature. The great advantage of the transform method is that, despite the fact that the Raman excitation profile is a multimodedependent function, only parameters associated with the particular vibrational mode under consideration are used. The complex interference between different vibrational modes is incorporated in the optical absorption spectrum, CY(WL), through the normalized line-shape function I(wL) = [Jdw (Y(O)/O]-~CY(~L)/OL that appears as the imaginary part of the complex function c$(oL), that can be expressed
where OL is a particular laser excitation frequency. The first term of the equation is the Hilbert transform of the imaginary part of ~ ( W L ) , and P denotes the principal value of the integral. The characteristic, long progression in the totally symmetric mode v1 in the resonance Raman spectrum of Ti14 suggests that the conventional Franck-Condon mechanism should be suitable for calculating the profiles. This straightforward model normally involves the so called standard assumptions, in which the REP, I",~coL), of a totally symmetry mode nvf (where n = 1 stands for the fundamental) with frequency wf is obtained by the application 0 1993 American Chemical Society
Ribeiro et al.
12154 The Journal of Physical Chemistry, Vol. 97,No. 47, 1993
where &(wr) ~ ( o L )= [ 1 - & w ~ ) ] ~ ( w=L~) ( w L -) ~ ( W L w).The thermal distribution factor (nr) is given by (nr) = [exp(hwf/kT) - l]-l, Cis a frequency-independent constant which will beignored in the following (the dependence on the fourth power of the scattered light will be compensated for in the experimental data), and [f is the magnitude of the linear electron-phonon coupling parameter for the normal mode Qf. The dimensionless parameter [f is related to the displacement of the minimum of the potential curve along the direction of the mode Qf in the excited electronic state. Equation 2 can be considered to be formally correct only in the limit of T = 0 K. If there is significant population of excited vibrational levels of the ground electronic state at room temperature, as is the case for TiI4, higher order terms can become important. For instance, for a fundamental band just one additional term requires that eq 2 be rewritten
' / 2 ~ f r ) ( t p ) ~ ( 1 +(np) )(( n r ) ) I a 3 ( ~ ~ r + + )+(wL)Iz (3) where the operator &3(wf,wp,-wp)denotes successiye applications of &(of)on the function ~ ( o L ) Le., , A(w) A(w) A(+r) ~ ( w L ) . The second term of eq 3 contains simultaneous contributions from the parameters of all normal modes. However, in the case of Ti14,the summation over other normal modes can be ignored. Only the mode V I is retained, since this is the only one for which the linear vibronic coupling parameter is expected to have a significant magnitude, as indicated by the extensive progression in the RRS and its considerable thermal population at room temperature ((n161) = 0.848 at room temperature). The significant value of (nf) in the present case makes the equation of the transform method more complex when the model is implemented beyond the standard assumptions. If the vibrational frequency in the excited electronic state (we,f) is different from that in the electronic ground state (w,,f = 161 cm-I), and the factor (nf) is negligible, the equation reduces to the form of eq 2, with &of)replaced by A(we,f),Le., the frequency in the excited state becoming an adjustable parameter in fitting the experimental profiles. In the case of TiI4, however, it is necessary to test the complete expression for the first order term. Thus, for a fundamental band6J0 I,,f(WL)
RR 2
= (1 + (nf))(tf 1 IA(%,f)U + 4 - a W c , f ) - (n,)[l G(0e.f-
+
~ g , f ) ~ (nf)Sf[&we,f) - &4g,f)lM(wL)I2
(4)
where [PRis a renormalization of [r: [ P R= [r 2(wa,f)2/[wc,dwe,r w,,r)] and 6f is a parameter related to the shift in thevibrational frequency: bf 55 (we,f- ug,f)/(we,r+ o,,f). In cases in which &and (nf) can be ignored, eq 4 becomes very similar in form to eq 2. In the case of TiI4, it is mandatory to proceed beyond the Condon approximation, i.e., the linear non-Condon coupling parameter mfmust be taken intoaccount; this parameter is related to the dependence of the electronic transition moment on the normal coordinate Qf. In this case, eq 2 for a fundamental becomes'
+
Il,f(wL) = (1 + (nf))ktPR[4(%) - 4(%
- %,f)l +
mPR[4(WL) + 4 @ L - %,f)l12 ( 5 ) where mPR p mr2w,,f/(w,,t + wg,f). In eq 5 , provision has been made for a frequency shift in the excited state, provided that the shift parameter, 61, and the thermal excitation, (nf), are small enough to allow one to neglect the additional corrections in eq 4.
The equation for the anti-Stokes component corresponding to eq 5 has been reported in the literature for the particular case in which there is no frequency shift.I2 In this limit, the exprasion for an anti-Stokes fundamental is obtained from eq 5 by changing ~ ( W L - wc,f)to ~ ( O + L wf), the factor (1 + (nf)) to (nr), and the sign of the first term -[f to [f, When non-Condon coupling is included, a formal consequence is the quadratic vibronic coupling, i.e., the shift in the vibrational frequency and/or a Duschinsky rotation." To test the contribution of a frequency shift to the fundamental (Stokes and anti-Stokes) when non-Condon coupling is considered, it was necessary to develop the analog of eq 5 for the anti-Stokes component. Using the time-correlator formalism,14 one obtains (see Appendix) L&L)
= (n,)I-tPRW,
+ (Jg,f) - 4 b L - [%,f - wg,rlN+
m p R w L+ Ug,f) + 4(% - [%,f - ~ g , f l ) l 1 2 ( 6 ) In the limit we,f = wg,f = wf, eq 6 reduces to eq 11 of ref 12; for the anti-Stokes fundamental, however, the result of the introduction of frequency shift is more complicated than simply changing wrtow,,r, as was thecase for thestokes fundamental."J5 In the case of overtones, the literature expression for the first overtone includes both linear vibronic and linear non-Condon couplings, but excludes frequency shift+
+ (nf))21Ef2[4(4-2 4 b L - of)+ 4@L - 20,) 1- 2tfmf[4(wL) - 4 b L - 2@f)112(7)
ZZ,f(WL) = I/2(1
This equation can be obtained from the time-correlatorformalism for the resonance Raman effectI4J7or from eq 5, without frequency shift if the operation on the function ~ ( w L )is applied twice and terms of order m? are ignored, which is justifiable in view of the relative smallness of the non-Condon coupling parameter Jthis is analogous to the successive application of the operator A(wr) in the case of the standard assumptions as in eq 2: or wen when the frequency shift is considered).6J0 Following this procedure, as also suggested in ref 16, three operations on ~ ( w L )as in eq 5 or two operations as in eq 6, both without frequency shift, lead respectively to the second Stokes overtone and the first antiStokes overtone, with the consideration of linear non-Condon coupling:
+ (nf>)3k - 34(WL-Wf) + - 2wf) - 4(mL - 3 q ) I + 3(tf)2mf[4(wL)~ ( W L- wf) - ~ ( W L- 20,) + ~ ( O L - 3wJIv L2,fbL) = '/2(nf>21(tf)2r4(wL>- 2 4 b L + 4+ 4 b L + 2Wf)l - 2Epf[4(wL + 20,) - 4(WL)112 z3,f(WL)
'/6(l
(8)
(9)
Summing up, in this section eqs 6, 8, and 9 of the transform method, heretofore not explicitly available in the literature, have been presented since they are essential for reproducing more satisfactorily the experimental profiles of the VI band and its overtone bands in TiI4. Experimental Setup
Titanium tetraiodide was purchased from Sigma Chemical Co. All manipulations were performed in a vacuum line since the compound is extremely moisture sensitive. Ti14 solutions (ca. I C 3 M) in cyclohexane (spectroscopic grade) were sealed in rotatory cells for liquids and used for obtaining the Raman spectra on a Jarrel-Ash 300 Raman spectrometer fitted with a photon counting detection system. The spectra were excited by several lines of Ar+ and Kr+ lasers, with the power being maintained at ca. 20 mW to avoid decomposition. The intensities, measured as the areas under the Raman bands, were measured relative to the intensity of the Raman band at ca. 806 cm-1 of cyclohexane as an internal standard. Since there is
Raman Excitation Profiles for Ti14
200
300
400 500 600 700 800 Wave1 e n g t h/ n m Figure 1. Absorption spectrum of Ti14 in cyclohexane solution. The dotted line is a Lorentzian curve which extrapolates the wings to zero. Theverticalbars indicate the laser lines used for excitation of the Raman spectra.
a considerable wavenumber difference between the Raman band chosen as internal standard and those of the sample, corrections have to be applied to the measured intensities for the dependence of the Raman intensity on the fourth power of the scattered light frequency, the dependence of the photomultiplier sensitivity on the scattered light frequency, and theeffective spectral slit width. Reabsorption of the scattered radiation was minimized by positioning the laser beam as close as possible to the exit window of the cell. Using the same cell, the vis-UV spectrum was determined with a HP spectrophotometer coupled to a PC, and the digitized absorption spectrum used directly as input in programs for the calculation of profiles by the transform method.
Results The vis-UV spectrum of Ti14 in cyclohexane solution is shown in Figure 1, the position of the exciting lines being indicated by vertical bars. Also shown in Figure 1 (dotted line) is a Lorentzian curve that smoothly extrapolates the wings of the absorption band to zero, a standard procedure in the use of the transform method, and that reveals the theoretical profiles to be practically insensitive to the details of the extrapolation9 (in fact, the Lorentzian used simulated very well the whole absorption band). Table I shows the relative intensities obtained in the present work for the Stokes ( V I , 2vl, and 3ul) and anti-Stokes components (-VI and -2vl) for several excitation lines. There is very satisfactory agreement with the results reported in ref 1 for several of the excitation lines. Figure 2 shows the experimental Raman excitation profiles and the best fit obtained by the transform method. All profiles were fit using a linear non-Condon coupling to linear vibronic coupling ratio of m r / b = 0.055 (*0.005)and wc,f = wg,r = wr = 161 cm-1 (the equations used were 5-9). All the profiles are shown on the same scale, and by comparison of the relative intensities, the absolute [r value was found to be & = 3.7 f 0.1 (hence mr = 0.203).
Discussion The first step in the use of the transform method is the calculation of the profile within the standard assumptions (eq 2), the only adjustable parameter being the magnitude of Q a scale factor which does not change the line shape of the profile. The significant progression in the u1 mode in the resonance Raman spectrum of TiI, strongly suggested that a conventional Condon mechanism would be sufficient to reproduce the experimental profiles. However, as shown for the fundamental Stokes band in Figure 3, there is a systematic deviation between the calculated and experimental profiles. The profiles for the overtones and
The Journal of Physical Chemistry, Vol. 97,No. 47, 1993 12155 anti-Stokes components (not shown in Figure 3 ) also exhibit similar deviations when calculated within the standard assumptions. Within the framework of the standard assumptions, the consideration of temperature effects alone fails to improve the theoretical profiles. In fact, the calculation of the fundamental Stokes profile from eq 3, which contains the third-order term,8*9 does not lead to any improvement in the theoretical profiles. Chan and Page9 have shown that there are two conditions under which the temperature corrections are negligible: a weak electronphonon coupling and a small mode frequency (compared to the smallest resolvable width in the absorption band). Despite the significant thermal populations of vibrational levels in the electronic ground state and the expected large [r value, the structureless absorption band of Ti14 in cyclohexane solution results in negligible higher order terms, thus allowing the firstorder expressions of the transform method to be used. The next possibility for improving the profiles is to proceed beyond the standard assumptions and use different vibrational frequencies for the ground and excited electronic states. If the frequency shift is small, such that 6r
