Combined representation method for use in band-structure calculations: Application to highly compressed hydrogen

Abstract

A representation is described whose basis functions combine the important physical aspects of a finite set of plane waves with those of a set of Bloch tight-binding functions. The chosen combination has a particularly simple dependence on the wave vector $\overrightarrow{\mathrm{k}}$ within the Brillouin zone, and its use in reducing the standard oneelectron band-structure problem to the usual secular equation has the advantage that the lattice sums involved in the calculation of the matrix elements are actually independent of $\overrightarrow{\mathrm{k}}$. For systems with complicated crystal structures, for which the Korringa-Kohn-Rostoker, augmented-plane-wave, and orthogonalized-plane-wave methods are difficult to use, the present method leads to results with satisfactory accuracy and convergence. It is applied here to the case of compressed molecular hydrogen taken in a $\mathrm{Pa}3$ ($\alpha -\mathrm{nitrogen}$) structure for various densities but with mean interproton distance held fixed. The bands show a marked free-electron character above 5 to 6 times the normal density, and the overall energy gap is found to vanish at 9.15 times normal density. Within the approximations made, this represents an upper bound for the molecular density in the transition to the metallic state from an $\alpha -\mathrm{nitrogen}$ structure.