jats:titleAbstract</jats:title> jats:pWe use first-principles density functional theory to conduct an extensive structure search using the jats:monospaceAIRSS</jats:monospace> package for elemental sulfur in the range 50–550 GPa. We then obtain the low-temperature phase diagram of sulfur in the same pressure range, including vibrational effects through the harmonic approximation. We do not find any structures lower in energy than those already reported in experiment, although the phase diagram below 100 GPa is found to be crowded with structures separated by only a few meV. We report the transition sequence jats:inline-formula jats:tex-math

</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:miI</mml:mi> mml:msub mml:mrow mml:mn4</mml:mn> </mml:mrow> mml:mrow mml:mn1</mml:mn> </mml:mrow> </mml:msub> mml:mrow <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi mathvariant="italic">acd</mml:mi> mml:mo→</mml:mo> mml:miP</mml:mi> <mml:mover accent="true"> mml:mrow mml:mn1</mml:mn> </mml:mrow> mml:mrow mml:mo¯</mml:mo> </mml:mrow> </mml:mover> mml:mo→</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="njpab6068ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> ICM jats:inline-formula jats:tex-math

</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">→</mml:mo> <mml:mspace width="0.25em" /> mml:miC</mml:mi> mml:mn2</mml:mn> mml:mrow mml:mo/</mml:mo> </mml:mrow> mml:mim</mml:mi> <mml:mo stretchy="false">→</mml:mo> mml:miR</mml:mi> <mml:mover accent="true"> mml:mrow mml:mn3</mml:mn> </mml:mrow> mml:mrow mml:mo¯</mml:mo> </mml:mrow> </mml:mover> mml:mim</mml:mi> <mml:mo stretchy="false">→</mml:mo> mml:mrow mml:miI</mml:mi> mml:mim</mml:mi> </mml:mrow> <mml:mover accent="true"> mml:mrow mml:mn3</mml:mn> </mml:mrow> mml:mrow mml:mo¯</mml:mo> </mml:mrow> </mml:mover> mml:mim</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="njpab6068ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> and obtain accurate pressures for each transition, although we find the second-order jats:inline-formula jats:tex-math

</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:miC</mml:mi> mml:mn2</mml:mn> mml:mrow <mml:mo stretchy="true">/</mml:mo> </mml:mrow> mml:mim</mml:mi> mml:mo→</mml:mo> mml:miR</mml:mi> <mml:mover accent="true"> mml:mrow mml:mn3</mml:mn> </mml:mrow> mml:mrow mml:mo¯</mml:mo> </mml:mrow> </mml:mover> mml:mim</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="njpab6068ieqn3.gif" xlink:type="simple" /> </jats:inline-formula> transition particularly difficult to define. Contrary to previous first-principles works (Pavel jats:italicet al</jats:italic>; Rudin and Liu 1999 jats:italicPhys. Rev. Lett.</jats:italic> jats:bold83</jats:bold> 3049–52), we do not reproduce a trigonal jats:inline-formula jats:tex-math

</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:mo→</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="njpab6068ieqn4.gif" xlink:type="simple" /> </jats:inline-formula> simple cubic transition at either the static lattice or harmonic level. We also undertake a detailed analysis of the incommensurately modulated (ICM) phase of sulfur phase using a commensurate approximant found in the structure search. We find that the modulation amplitude is zero above 96 GPa; some 40 GPa below the experimentally reported transition to the unmodulated phase. We find that the body-centred atoms in the relaxed ICM approximant are, in addition to the dominant transverse modulation (which is a frozen-in optical phonon mode), slightly displaced longitudinally in the jats:italicb</jats:italic>-direction. We subsequently discover that this (small) longitudinal modulation is coupled to the transverse mode, and hence report previously unnoticed weak-mode coupling between transverse and longitudinal optical phonons in the ICM phase.</jats:p>