Liquid theory for the instantaneous normal modes of a liquid

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Abstract
At short enough times, the dynamics of a liquid can be resolved rigorously into independent simple harmonic motions called instantaneous normal modes. The spectrum of such modes is easily accessible via computer simulation, but, despite the existence of theories for other kinds of liquid modes, it has been difficult to come up with analytical approaches of power sufficient to explain the universal appearance of instantaneous normal‐mode spectra—though Wu and Loring were recently able to arrive at a theory by exploiting the analogy between this problem and the master equation. In this paper we propose a hierarchy of liquid‐theoretical treatments that do show the analogy between instantaneous normal modes and other collective excitations in liquids, but are nonetheless capable of leading to accurate predictions of instantaneous normal‐mode spectra. The crucial ideas are that the theoretical treatment must respect the fact these modes conserve momentum and must also recognize the strongly local character of intermolecular force constants. We discuss two theories in detail—a renormalized mean‐field theory, which turns out to be identical to the Wu–Loring theory, and a higher‐order theory—and apply both to a simple atomic liquid. Both theories successfully predict the results of computer simulations, including the fact that the spectrum depends much more on density than on temperature in the normal liquid range. The higher‐order theory, though, gives a slightly more accurate prediction of the fraction of imaginary modes.