ir overtone spectrum of the vibrational soliton in crystalline acetanilide.

The self-trapping (soliton) theory which was recently developed to account for the anomalous amide-I band at 1650 cm/sup -1/ in crystalline acetanilide (a model system for protein) has been extended to predict the anharmonicity constant of the overtone spectrum. These infrared-active overtones which have been detected at 3250, 4803, and 6304 cm/sup -1/ yield an anharmonicity constant that is in good agreement with the theory.

Recently, we have proposed that an anomalous band at 1650 cm ' in crystalline acetanilide (ACN), a model system for protein, could be assigned to a self-trapped state' similar to a Davydov soliton. This assignment has been questioned by several researchers. 81anchet and Fincher have suggested that the existence of a band at 3250 cm ' with the same temperature dependence as the 1650-cm ' band is evidence for an assignment based upon a structural transition. 5 From a more conventional perspective, Johnston and Swanson have argued that the 1650-cm ' band can be assigned to Fermi resonance.
In the context-of this ongoing controversy, it is interesting to consider the predictive power of the self-trapping theory. Here we show that this theory is able to predict the anharmonicity constant for an overtone spectrum which is in good agreement with independent measurements.
The first overtone of this spectrum is at 3250 cm Consider a single, hydrogen bonded chain of crystalline ACN. The classical Hamiltonian can be written in terms of complex mode amplitudes (a") for amide-I vibrations at molecules (indexed by n) along the chain, and the normal coordinates (qk) for low-frequency optical-phonon modes (indexed by wave number k). Assuming the self-trapped state to be stationary implies jk = 0; thus the Hamiltonian is 0 = g i~I I a" I -I(a"+qa"'+ a""a"q) n, k + -, ' wk qk'xk"qk I a. I'l Here co~is the amide-I frequency ignoring phonon interactions, J is the nearest-neighbor dipole-dipole coupling energy, 8'k is an harmonic potential for the phonon mode at wave number k, and the Xk"are nonlinear coupling coefficients between the amide-I and phonon modes. Variation with respect to qk implies that qk --(xk"/ Wk ) I a" I', so the Hamiltonian reduces to 0 = X i~110 I J(&n+1&~+&n &n+t) 2 y l&n I where (assuming xk" is independent of n) g xk /W» k Detailed calculations' show that J = 3.96 cm ' and y = 44.7 + 1.5 cm (4) where the uncertainty arises because peak positions in Ref.
2 are recorded only to the nearest wave number. Since 21 (( y, the second term in the summation of (2) can be neglected. This corresponds to assuming that the selftrapped energy is concentrated at a single molecule, as is observed in detailed numerical calculations.
Thus the Hamiltonian reduces to a sum of noninteracting terms of the form  Using the value of y given in (4)  In order to experimentally detect the overtone bands predicted by this theory, we have been guided by the following criteria: (a) a soliton overtone must display the same temperature-dependent intensity as the fundamental band at 1650 cm ', (b) a soliton overtone must be shifted by N'5 substitution consistently with the isotopic shift displayed by the fundamental band, (c) a soliton overtone must disappear in deuterated ACN, and (d) a soliton overtone must have maximum absorption at the same polarization as the fundamental.
A Nicolet 7000 series FT-IR instrument was used to observe the fundamental and the first two overtones, and a Cary 14 spectrophotometer was used for the third overtone.
Our estimates of absolute precision are included in Table I.
The samples consisted of single crystals of ACN of appropriate thickness, or of pressed KBr pellets of microcrystalline ACN when the material available was insufficient to grow single crystals.
The theory predicts that the first overtone of the soliton band should be observed near 3255 cm '. The ir spectrum of ACN in this region displays several bands, most of which can be assigned to overtones enhanced by Fermi resonance with the NH stretching at 3300 cm, as is usual with hydrogen bonded amides.
This region of the spectrum of ACN has been studied by Abbott and Elliott using dichroism and isotopic dilution methods. However, these authors concluded that "the origin of the structure in the NH and ND stretching modes remains obscure. " A band at 3250 cm ' is observed (see Fig. 1) and assigned to the first soliton overtone from the following considerations: (i) The 3250-cm band displays similar temperature dependence as the 1650-cm ' band.
(ii) %'5 substitution moves the 3250 band to 3246 cm with a red shift nearly twice the 2.7 cm ' observed for the fundamental (reported in Fig. 3 of Ref. 1). This shift is seen only for the 3250-cm band in this region of the ir spectrum.
(iii) In several runs on partially deuterated samples, not reported here, the intensity of the 1650 band is reduced and the intensity of the 3250-cm ' band is also reduced.
(iv) The 3250-cm ' band has the same dichroism as the 1650-cm ' band. A complete assignment of the other bands in this region is difficult and beyond the scope of the present work. But we note that the relatively strong band at 3260 cm ' of unknown origin is essentially unaffected by N' substitution and is also temperature independent in the temperature range investigated.
In Table I we report the observed frequency and the relative intensity of the first overtone of the soliton band assigned at 3250 cm The second overtone is predicted by the theory to occur near 4816 cm '. In this region we found a band at 4803 cm ' which has the required temperature dependence (see Fig. 2), disappears in deuterated samples, and has the correct dichroism. The 4803-cm ' band was observed both in single crystals and in KBr measurements, and it is assigned to the second soliton overtone. The relative intensity of this band is reported in Table I. For the observation of the third overtone (see Fig. 3) a single crystal of approximately 5 mm in length was used in a Cary 14 spectrophotometer.
A band at 6304 cm ' with the required temperature behavior was observed. The relative intensity of this band, reported in Table I,  Observation of the fourth overtone was not attempted, since the expected intensity, based on extrapolation of the values given in Table I, would require a crystal several centimeters long. %'e note that a second overtone of the normal amide-1 band has been detected at 4962 cm ' and a third overtone at 6588 cm ', but with lower intensities with respect to the fundamental.
The observed full bandwidths at half maximum (FWHM) for the overtone lines are also reported in Table I. Due to difficulties in subtracting background absorption, only the orders of magnitude of the intensities and bandwidths should be considered significant.
To test Eq. (9) a least-squares fit was made to the measured overtone frequencies by adjusting the parameters duo and y. In this minimization, terms were weighted by the reciprocal of the squared measurement error. The "fitted" parameters were found parameters were found to be roof -- Since this is well within the accuracy to which the measured frequencies are determined, the data is in excellent agreement with (9).
Qur interpretation of the 3250-cm ' band as a two quanturn soliton state is of biological significance because it is almost exactly resonant with the 10 kcaltmole of free energy released by the hydrolysis of adenosine triphosphate.