J. Phys.: Condens. Matter 8 (1996) 9569–9573. Printed in the UK Localization and the glass transition Brian B Laird† and Scott D Bembenek Department of Chemistry, University of Kansas, Lawrence, KS 66045, USA Received 15 July 1996 Abstract. We explore the role of disorder-induced localization in the dynamics of glasses and supercooled liquids using instantaneous normal mode (INM) analysis. This study is motivated by the fact that such localized excitations (tunnelling states and soft harmonic vibrations) are believed to be important in the thermodynamics and dynamics of amorphous systems at very low temperatures. The results are presented for two simple model systems that show the existence of a temperature below which all unstable INMs become localized. The relationship of this temperature to the glass transition is discussed. 1. Introduction Amorphous materials at very low temperatures possess a variety of properties that are quite distinct from those of crystals. For example, the heat capacities of amorphous materials below about 1 K increase linearly with increasing T and are significantly larger than their crystalline counterparts (which vary generically as T 3 ) [1]. Similar anomalous behaviour is found for a variety of other properties, such as thermal conductivity. Although many issues remain unresolved, these anomalies are, at present, best explained by the presence of disorder-induced localized excitations that coexist with and dominate the sound waves at low frequencies. At very low temperatures these states are primarily tunnelling modes (two-level systems (TLSs)) [2, 3]. At higher temperatures there is evidence [4, 5] that the dominant excitations here are low-frequency quasi-localized (resonant) harmonic vibrations. Recent experiments indicate a correlation between the nature of the glass transition and the relative concentration of TLSs and the quasi-localized harmonic modes [6, 7]. Given this correlation and the dominance of localized modes at very low temperatures, it is natural to speculate as to the role of localization at higher temperatures in the vicinity of T g . In this work, we explore this question in two model systems using the technique of instantaneous normal mode (INM) analysis. 2. Instantaneous normal modes Like standard normal modes, INMs [8, 9] are defined by expanding the potential energy of an N -particle system about a chosen configuration and diagonalizing the second-derivative (force-constant) matrix; the resulting eigenvectors and eigenvalues are the modes and the squared mode frequencies, respectively. The configurations used to determine the INM spectrum are not potential minima, as in normal mode analysis, but are taken directly from the trajectory at a given temperature. The force-constant matrix thus produced † Author to whom correspondence should be addressed. 0953-8984/96/479569+05$19.50 c  1996 IOP Publishing Ltd 9569