J . Phys. Chem. 1989, 93, 8387-8387
Reply to Comments on “Electron-Phonon Coupling in Mlxed-Valence Compounds: Mode Mixing and Coupling Constants” Sir: This Comment is in response to the Comment1 by Girlando and Painelli (GP) with which we in large part disagree for the following reasons. The most important point made by G P can be summarized by T solution is intrinsically single mode, being quoting them: “...the F not able to account for the mode mixing introduced by e-mv coupling.”’ We show the incorrectness of this statement by here sketching out how mode mixing in fact is introduced into our model2 (which G P refer to as the PT solution). Interestingly, it is done in very close analogy to the way Painelli and Girlando (PG)3 introduce mode mixing in their discussion of VAM theory. To introduce mode mixing, we note that, as our treatment now stands,2 the normal coordinates (4,) are defined before electronic interaction is permitted between the monomer units. When the electronic coupling (Vo, V,’ in our notation2) is turned on, the three potential surfaces mix, as shown explicitly by our vibronic matrix (eq 14, ref 2). It is perfectly straightforward to find the lowest root of this matrix to lowest order in the vibronic coupling, and the result is
where W, (= Cal/2q,2hvu - ’/2(U - Vo’) - ‘I2R) and W , (= C,1/2qa.2hu, - V,l) are respectively the energies of the ground and first excited potential surfaces in the absence of vibronic coupling (all A, = 0), cI- is an explicit function (eq 13, ref 2) of V,, V,l and the interelectronic repulsion parameter, U, and R = [(U V O 2 8V2]’/2; A, and Y, are respectively the electron-phonon (vibronic) coupling parameter and oscillator frequency of the CY mode. Mode mixing is evident in eq 1 which in fact is the precise analogue in our model of the PG eq 24 obtained by an adiabatic Herzberg-Teller treatments3To find new normal coordinates (4,)‘ that will be linear combinations of q,, we diagonalize the force constant matrix obtained by taking second derivatives of (1) with respect to 4,. This is in exact analogy to PG eq 27 and 28 where the procedure is very clearly e ~ p l a i n e d . ~ Thus, our treatment with the extension just discussed includes mode mixing and provides a (perturbational) nonadiabatic solution (Le., solves the dynamic problem, (Hel T,) 9,= E,@.,. It thus should be superior to linear response (LR) theory (or the VAM model) in the multimode case for exactly the same reason it is superiofl in the one mode case. Specifically, our entire previous treatment2 continues to apply if it is simply understood that all the q, (and A, and v,) starting with eq 15 are actually the q‘, (and A,‘ and):v determined in the manner described above. Note that the discussion here has been for the two-site two-electron case, for example, (TCNQ-),. Mode mixing can equally well be introduced in the two-site one-electron case, for example, (TCNQ);, by proceeding in an exactly analogous manner starting with our vibronic matrix (39).2 In regard to the vibronic coupling constants (A,) which we deduced from matrix-isolated (TCNQ)< and (TCNQ-)*-which GP compare with the well-established values (g,) in their Table 11-we in no way stated or implied that ours were superior to those already in the literature. In fact, we state explicitly in the Abstract of our paper2 that they are based on “...some preliminary infrared data...”. The cases where sharp disagreements occur may well be due to inadequacies in our preliminary data, but it may also be the case that TCNQ dimers in an Ar matrix at 10 K have significant structural differences from those summarized in the G P Table I.’ (Note well: The numbers in their table in the column footnoted by ref c should be multiplied by the factor 2.0 because,
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(1) Girlando, A.; Painelli, A. J . Phys. Chem., preceding paper in this issue. (2) Prassides, K.; Schatz, P. N . J . Phys. Chem. 1989, 93, 83. (3) Painelli, A,; Girlando, A. J . Chem. Phys. 1986, 84, 5655. (4) Wong, K.Y. Inorg. Chem. 1984, 23, 1285.
0022-3654/89/2093-8387$01 .50/0
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in our formalism, g, = X,hv, for the one-electron case, (TCNQ)
