Low-frequency vibrational modes and static vibrational hyperolarizabilities of long-chain molecules: polyenes and polyacetylene M. Tommasini, C. Castiglioni, G. Zerbi * Dipartimento di Chimica Industriale e Ingeneria Chimica del Politecnico di Milano; Piazza Leonardo Da Vinci 32, 20133 Milan, Italy Received 17 January 2000; accepted 9 February 2000 Abstract A model for calculating vibrational frequencies of Transverse Acoustical Modes (TAM) of all-trans polyenes of any length is presented. Based on the results obtained, the relevance of TAM modes in determining static vibrational polarizability and second vibrational hyperpolarizability of very long polyenes (polyacetylene) in real materials is discussed. The major role of the Effective Conjugation Coordinate (ECC) in determining the second vibrational hyperpolarizability is also presented and compared to that of the TAM modes. The results obtained point out the limitations of calculations on isolated molecules when the behaviour of real materials in the low-frequency domain is under investigation.  2000 Elsevier Science B.V. All rights reserved. Keywords: Vibrational modes; Vibrational hyperpolarizabilities; Polyenes; Polyacetylene 1. Introduction Large second-order molecular hyperpolarizabilities (g ) are required for obtaining molecular materials suita- ble for applications in the field of photonics [1–4]. Third harmonic generation (THG), degenerate four-wave mixing (DFWM) and Optical Kerr effect are only a few of the optical phenomena that have been observed in presence of large optical non-linearities and that can be used for the treatment of optical signals. Many g -active molecules have been detected, following a precise strategy, i.e. the search of compounds containing delocalized p electrons. For this reason organic conducting polymers and their oligomers are promising candidates for non-linear optical (NLO) applications. From the point of view of the fundamental research aimed to state firm structure/property relationships an important point is to study and rationalize the evolu- tion of NLO behaviour with chain length in series of compounds. A super-linear increase of the molecular second- order hyperpolarizability (g ) with the number of repeating units in the chain has indeed been observed in the case of relatively short oligomers of the same series [5,6]; saturation behaviour has sometimes been detected [7] in the longer terms of the series. On the other hand, the prediction of NLO behaviour in polymeric compounds is relevant for technology, since polymeric materials show peculiar macroscopic characteristics: for instance, methods developed for processing of polymeric materials (e.g. for obtaining Journal of Molecular Structure (Theochem) 500 (2000) 323–338 0166-1280/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S0166-1280(00)00433-4 www.elsevier.nl/locate/theochem * Corresponding author. Tel.: + 390-2-2399-3235; fax: + 390- 2-2399-3231. E-mail address: darstellung@icil64.cilea.it (G. Zerbi).