Non-exponential range of localized electron states in liquid chains

The ranges of electron states near a band edge in liquid chains have been found to depend not on an exponential decay factor, as previously believed, but on the `beat length' R of a related Bloch function. This leads to a new qualitative description of the eigenstates near the band edge of a liquid chain and gives a smooth transition to the band states. A method for determining the range R is given, and shown to agree with Monte Carlo computer calculations. These findings enable a correct application of the `local density' approximation - for example, to the determination of state densities. Some discussion is given of the relevance of the findings to work on real liquids, and on the Anderson model of a regular lattice with random potential strengths.