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Inclusion of frequency shifts with electronic excitation in the calculation of multiple‐order resonance Raman profile line shapes from optical absorption data
1.D. L. Tonks and J. B. Page, Chem. Phys. Lett. 66, 449 (1979).
2.D. C. Blazej and W. L. Petioolas, J. Chem. Phys. 72, 3134 (1980).
3.S. Hassing and O. S. Mortensen, J. Chem. Phys. 73, 1078 (1980).
4.P. M. Champion and A. C. Albrecht, Chem. Phys. Lett. 82, 410 (1981).
5.J. B. Page and D. L. Tonks, J. Chem. Phys. 75, 5694 (1981).
6.V. Hizhnyakov and I. Tehver, Phys. Status Solidi 21, 755 (1967).
7.Although the methods used here can be applied to sum and difference spectra involving different normal modes, we will restrict the discussion to n‐phonon Stokes scattering involving just a single normal mode; i.e., to harmonic spectra. The number of normal modes of the scattering system will be kept arbitrary, however.
8.It should be noted that while the separation into orders is discussed in Bef. 5 in the context of purely linear electron—phonon coupling, phonon many‐body techniques can be employed to obtain an exactly analogous separation in the more general case of linear plus quadratic electron‐phonon coupling, as we have discussed for first‐order scattering in Ref. 9. In this more general case, which we will use here when considering the effects of vibrational frequency shifts with electronic excitation, the remarks characterizing the separation into orders and the identity of nth order and n‐phonon profiles at K remain valid.
9.D. L. Tonks and J. B. Page, Chem. Phys. Lett. 79, 247 (1981).
9. [The key reference (10) of that paper appeared incompletely there. It is: D. L. Tonks, Phys. Rev. B 22, 6420 (1980).]
10.D. L. Tonks and J. B. Page, Bull. Am. Phys. Soc. 26, 485 (1981).
11.The Φ defined by Eq. (2) is the complex conjugate of that used in Ref. 1, the present definition agreeing with that used by us in Ref. 5. Either definition leads to the same expression for the profiles, as is evident from the forms of Eqs. (3) and (4) of the present paper.
12.D. L. Tonks and J. B. Page (to be published).
13.It should be noted that the renormalized linear electron‐phonon coupling coefficients appearing in the present paper are different than those appearing in Ref. 9—there the re‐normalization of the linear coupling coefficients involved just the mode mixing, while here it is only the frequency shifts which enter the renormalization. In the general case treated in Ref. 9, in which both frequency shifts and mode mixing were allowed, the renormalization used there is the more useful for expressing the exact theoretical results. But with only frequency shifts present, the renormalization used here emerges as a natural quantity in terms of which to express the results, as is seen in the Appendix of the present paper.
14.V. Hizhnyakov and I. Tehver, Opt. Commun. 32, 419 (1980).
15.We should also mention here the work of H. C. Chow [Can. J. Phys. 57, 1, 11 (1979)],
15.and very recent work of D. J. Tannor and E. J. Heller (preprint). Chow treated vibrational frequency shifts, atomic equilibrium position shifts, and anharmonicity in RR scattering using the time correlator approach. However, Chow’s formalism was not applied to calculating RR profiles from optical absorption data. Moreover, he assumed that the frequency shifts were due to a change in the vibrational Hamiltonian with electronic excitation. Within each mode, this changes the kinetic and potential energy operators by the same factor and is an approximation to the general frequency shift case we have treated, in which only the vibrational potential energy operator is different in the electronic excited state. The latter case includes terms involving and omitted by Chow, and it is more difficult to treat theoretically. In the work of Tannor and Heller, vibrational frequency shifts, mode mixing, and atomic equilibrium position shifts in RR scattering are treated within a K wave‐packet propagation method. However, that work does not deal with the subject of the present paper, namely the inclusion of frequency shifts in the calculation of RR profiles from optical absorption data.
16.C. K. Chan and J. B. Page (to be published).
17.R. J. H. Clark, D. G. Cobbold, and B. Stewart, Chem. Phys. Lett. 69, 488 (1980);
17.R. J. H. Clark and B. Stewart, J. Am. Chem. Soc. 103, 6593 (1981).
18.Clark et al.. 17 did not give a value of directly, but rather gave their fit value (their notation) for the equilibrium position shift of the breathing mode upon electronic excitation. With frequency shifts present, δ is related to our unrenormalized linear coupling coefficient through where is the oxygen mass. In deriving this expression, we have assumed that Clark et al.. used the normalized breathing mode normal coordinate where the sum goes over the radial displacements of the four oxygen atoms. We used the above expression to compute needed to compute
19.T. I. Maksimova and N. B. Reshetnyak, Sov. Phys. Solid State 20, 669 (1978).
20.Our digitized input optical absorption is directly plotted in the figures. The rather “peaky” absorption of results in correspondingly structured real and imaginary parts of so that when the difference operator is applied in Eq. (4), a fair amount of structure appears in our computed profiles, as is seen in the figures. This structure is “real” in the sense that it stems from our digitized input optical absorption rather than being introduced by our calculation of the principal‐value integral needed for Re which was done analytically, as discussed in Ref. 1.
21.L. Chinsky, A. Laigle, W. L. Peticolas, and P. Y. Turpin, J. Chem. Phys. 76, 1 (1982).
22.In addition, the possibility of a rising background in the optical absorption data of Clark et al.. (Ref. 17) was recently suggested, but not firmly established, by Samoc̀ et al.. [M. Samoc̀, W. Siebrand, D. F. Williams, E. G. Woolgar, and M. Z. Zgierski, J. Raman Spectrosc. 11, 369 (1981)].
22.Such a rising background could render our computed profile line shapes somewhat less reliable at the higher frequencies.
23.Note that we are not advocating that the procedure employed here be used as a substitute for model calculations. Indeed, we have found that computed profiles can be quite sensitive to changes in the optical absorption, a point also noted in Ref. 4, so that better fits to profiles may well be achievable at the expense of less than perfect fits to the optical absorption. Again, our purpose here is to illustrate Eq. (4) and the importance of including frequency shifts.
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