Non-Cartesian coordinates for instantaneous normal mode theory of atomic liquids

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Abstract
A hybrid coordinate system of spherical polar coordinates for the mutual nearest-neighbor pairs and Cartesian coordinates for the unpaired atoms is introduced for instantaneous normal mode (INM) analysis of atomic liquids. Densities of states (dos) calculated with the hybrid coordinates in a unit-density, supercooled Lennard-Jones liquid differ from those obtained with Cartesian coordinates, primarily at imaginary frequency. A brief discussion of coordinate dependence is presented, with an analytic treatment of the frequency moments, and it is argued that the hybrid dos are more physically meaningful. INM theory strives to relate Im ω modes to diffusion and barrier crossing, but spurious nondiffusive contributions must be removed. Hybrid coordinates yield substantially fewer Im ω indicating that some nondiffusive modes are simply Cartesian artifacts. Normalized hybrid and Cartesian Re ω dos are nearly identical, as are velocity correlation functions C(t) obtained by treating the Re ω INM as a complete set of harmonic modes. These C(t) are in fair agreement with simulation, but, notably, reach an insufficiently deep negative minimum value at too short a time. A harmonic approximation using the hybrid-translational Re ω dos, in which the hybrid modes are projected onto the center-of-mass translations of the mutual neighbor pairs plus the unpaired atoms, yields much better agreement.