  At high pressure, bismuth acquires a complex incommensurate host-guest
structure, only recently discovered. Characterizing the structure and
properties of this incommensurate phase from first principles is challenging
owing to its non-periodic nature. In this study we use large scale DFT
calculations to model commensurate approximants to the Bi-III phase, and in
particular to describe the atomic modulations with respect to their ideal
positions, shown here to strongly affect the electronic structure of the
lattice and its stability. The equation of state and range of stability of
Bi-III are reproduced in excellent agreement with experiment using a fully
relativistic model. We demonstrate the importance of employing large unit-cells
for the accurate description of the geometric and electronic configuration of
Bi-III. In contrast, accurate description of the equation of state of bismuth
is found to be primarily sensitive to the choice of pseudopotential and
exchange-correlation function, while almost completely insensitive to the
commensurate approximation.
