text:
\section{\label{intro}Introduction}

Sulfur (S) is an element renowned for its structural diversity, possessing a considerable array of metastable allotropes at ambient pressure alone \cite{sulfur_allotropes,Meyer_sulfur}. This complexity persists at high pressures, where sulfur adopts elaborate catenated spiral structures above $1.5$ GPa \cite{chain_structures_sulfur} and an incommensurately modulated structure at $83$ GPa \cite{Degtyareva_and_Hemley_PRB,Luo_et_al}, which is driven by a charge density wave (CDW) instability.
\par
Self-hosting incommensurate host-guest (HG) structures, which comprise two distinct elemental sublattices embedded within one another in such a way that the ratio of their lattice lengths along a certain crystal axis is an irrational number, are well-known within the literature \cite{HG_Barium,HG_alkali_metals,Rb-IV,HG_Aluminium,HG_group_V,HG_Bismuth_SC,Haussermann_2002,HG_Strontium,HG_scandium,HG_Antimony,HG_Calcium}. The reasoning behind the stability of such host-guest phases is often explained through so-called Fermi surface effects, sometimes referred to as the Jones mechanism \cite{Jones_mechanism}, in which lattice distortions introduce Fourier components into the potential that lower the energy of Fermi level states.
\par
The physical properties of these elemental host-guest structures are of great interest and vary significantly between elements. Several examples are superconducting, and often exhibit high critical temperatures owing to favourable low-frequency \textit{quasi}-acoustic phonon modes \cite{HG_Bismuth_SC}, with a record $T_c$ of $29$ K among all elements (excluding hydrogen) being calcium in just such a phase (Ca-VII at $210$ GPa \cite{SC_elements_review,Calcium_experimental,chain_ordered_calcium}). Conversely, other realisations of elemental HG structures feature large pseudogaps in their electronic densities of states at the Fermi level, and form poorly-conducting electrides with significant amounts of electronic charge localised in interstitial regions \cite{HG_alkali_metals,HG_Aluminium}.
\par
X-ray diffraction studies on S up to $165$ GPa \cite{Degtyareva_and_Hemley_PRB,Luo_et_al} and $250$ GPa \cite{highest_experimental_pressure_sulfur,drozdov_hydrogen_sulfide_experiment} have determined that S transforms from the incommensurately-modulated CDW phase into the trigonal $\beta$-Po structure of $R\bar{3}m$ symmetry at ${\sim} 150$ GPa. Several first-principles structure searching investigations \cite{USPEX,Whaley_Baldwin,sulfur_no_SC} have predicted that the $R\bar{3}m$ structure remains stable up to ${\sim} 500$ GPa, where S then adopts a body-centered cubic (bcc) structure.
\par
The correct determination of the crystal structure of S at high pressures is of interest to experimentalists, in order to identify its presence as a decomposition product in high pressure hydrogen sulfide experiments on H$_2$S and H$_3$S and to resolve the resulting diffraction pattern \cite{hydrogen_sulfide_Goncharov,novel_sulfur_hydrides}. Knowledge of the true ground state is also important from a structure searching standpoint, so that the enthalpy of candidate H$_2$S / H$_3$S structures can be measured against decomposition into H and S \cite{hydrogen_sulfide_Cui,dissociation_H2S}.
\par
The high-pressure superconducting properties of pure S have long attracted attention, and at one time it held the highest experimentally verified $T_c$ in an element \cite{record_Tc_sulfur}. Further, the fact that the incommensurately modulated phase of S is superconducting with such a high $T_c$ is unusual, as the Fermi-level gaps opened up by CDW formation often work to suppress superconductivity \cite{CDW_sulfur}. As well as H$_2$S and H$_3$S, sulfur is a key constituent of several other high-$T_c$ compounds, including a recently claimed room-temperature superconductor \cite{RT_supercond}.
\par
In this work, we investigate the phase diagram of S up to $700$ GPa and report that under sufficient compression, S adopts a superconducting incommensurate HG structure similar to that of Barium in the IVa phase \cite{HG_Barium}, in addition to a previously unreported orthorhombic phase. We show that the guest structure in S undergoes several changes with increasing pressure, and link this to modulations of the host and guest atoms. We further study the electronic structure and charge density distribution of the HG phase, and compare the latter to other high-pressure HG structures.
\newpage

\section{\label{comp_details}Computational Details}

We used the \textit{ab-initio} Random Structure Searching (AIRSS) package \cite{airss_PRL,airss_JPCM} for our structure searching, with the plane-wave DFT code \verb|CASTEP| \cite{castep} performing the underlying electronic structure calculations. We generated a sulfur pseudopotential specifically designed for high-pressure work using the OTFG code in \verb|CASTEP| (see Supplemental Material \cite{supp}). Due to small enthalpy differences in certain parts of the phase diagram, we used an $850$ eV plane-wave cutoff and a \textbf{k}-point spacing of $2\pi \times 0.01$ \AA$^{-1}$, sufficient for absolute enthalpy convergence to $\pm 0.2$ meV. Relative enthalpies between structures were converged to an even tighter tolerance. Geometry optimisations were carried out using the L-BFGS algorithm and structures were relaxed until the forces on each atom were $5 \times 10^{-4}$ eV / \AA$^{-1}$ or smaller, and the residual stresses $1 \times 10^{-2}$ GPa or smaller.
\par
We performed structure searches at $300$, $400$, $500$ and $750$ GPa, using randomly generated structures containing between $1$ and $15$ atoms and possessing between $1$ and $48$ symmetry operations. We then performed searches at these same pressures with $16$, $21$, $24$ and $32$ atoms. Finally, we performed much smaller searches in peturbed supercells of our various host-guest approximants, which involved up to $288$ atoms. In total, we considered around $7500$ fully relaxed structures.

\section{\label{results}Results and Discussion}
\subsection{\label{relative_enths}Relative Enthalpies}

\begin{figure}[h!]
    \hspace{-0.22cm}
    \includegraphics[width=0.49\textwidth]{NOCOLOUR_phase_diagram.pdf}

    \vspace{0.25cm}

    \includegraphics[width=0.22\textwidth]{R-3m_Fdd2_temps.pdf}
    \hspace{0.35cm}
    \includegraphics[width=0.22\textwidth]{HG_bcc_temps.pdf}

    \caption{\textbf{Top:} Static-lattice phase diagram of S calculated with the PBE functional. `HG' denotes the host-guest phases, and for clarity, they have been plotted as a single curve. The inset shows the small enthalpy differences between the $Fdd2$ structure and HG phase. The simple-cubic $Pm\bar{3}m$ structure considered in previous studies \cite{Whaley_Baldwin,USPEX,sulfur_no_SC,Rudin,sulfur_and_Se_Rudin} is shown for comparison. \textbf{Bottom:} Phase diagrams with vibrational effects included (at the level of the harmonic approximation) in the vicinity of the $R\bar{3}m \rightarrow Fdd2$ transition (left) and HG $\rightarrow Im\bar{3}m$ transition (right). The black dashed line on the left diagram marks the boundary of the second order $Fdd2 \rightarrow Fddd$ phase transition.}

    \label{phase_diagram}
\end{figure}

\begin{figure*}
    \centering
    \begin{subfigure}[b]{0.3\textwidth}
        \hspace{-1cm}\includegraphics[width=0.2\textwidth]{yellow_axes.png}
        \includegraphics[width=0.75\textwidth]{all_yellow_Fdd2.png}
        \caption{Looking down the pseudo-guest chain axes. From top-to-bottom, the pseudo-guest chains are arranged in an AB stacking.\newline\newline}
        \end{subfigure}
        %
        \hspace{0.5cm}
        %
        \begin{subfigure}[b]{0.3\textwidth}
        \hspace{-1cm}\includegraphics[width=0.2\textwidth]{yellow_axes.png}
        \includegraphics[width=0.75\textwidth]{red_guests_Fdd2.png}
        \caption{The same structure, but with the two different types of pseudo-guest chains artificially coloured red and blue, shown looking down the axis of the red guests. \newline}
        \end{subfigure}
        %
        \hspace{0.5cm}
        %
        \begin{subfigure}[b]{0.3\textwidth}
        \hspace{-0.8cm}\includegraphics[width=0.164\textwidth]{blue_axes.png}
        \includegraphics[width=0.75\textwidth]{blue_guests_Fdd2.png}
        \caption{The same as the image on the immediate left, but rotated $45 \degree$ anticlockwise around an axis running top-to-bottom of the page (i.e. perpendicular to the chains). The result is an equivalent structure.}
        \end{subfigure}

    \caption{The commensurate $Fdd2$ structure, viewed down the `pseudo-guest' chains. The relation of this view to the lattice vectors of the conventional (64-atom) orthorhombic cell is shown in Fig. \ref{Fdd2_birds_eye_view}.}
    \label{Fdd2_detail}
    \vspace{0.25cm}
\end{figure*}

\begin{figure}[h!]
    \hspace{-0.70cm}
    \includegraphics[width=0.075\textwidth]{axes_down_c.png}
    \hspace{+0.15cm}
    \includegraphics[width=0.30\textwidth]{Fdd2_chains_birds_eye.pdf}

    \vspace{0.325cm}

    \hspace{-0.70cm}
    \includegraphics[width=0.075\textwidth]{axes_down_a.png}
    \hspace{+0.15cm}
    \includegraphics[width=0.32\textwidth]{down_a.png}

    \vspace{0.325cm}

    \hspace{-0.70cm}
    \includegraphics[width=0.075\textwidth]{axes_down_b.png}
    \hspace{+0.15cm}
    \includegraphics[width=0.125\textwidth]{down_b.png}

    \caption{The $Fdd2$ structure, in the same colour scheme as Fig. \ref{Fdd2_detail}. Two example chains (like those shown in Fig. \ref{Fdd2_detail}) have been highlighted with red and blue lines (top figure only). The conventional (64-atom) orthorhombic unit cell is indicated in black.}
    \label{Fdd2_birds_eye_view}
\end{figure}

Fig. \ref{phase_diagram} shows our phase diagram of S calculated using the PBE functional. In addition to PBE, table \ref{transitions_table} shows the (static-lattice) transition pressures according to the LDA and PBEsol; the order of phase transitions is unaffected by the choice of functional. Using PBE, we determine that at $266$ GPa, S transitions from the $R\bar{3}m$ ($\beta$-Po) phase to a structure of $Fdd2$ symmetry with $16$ ($64$) atoms in its primitive (conventional) cell, which visually resembles a strongly distorted version of the Ba-IVa HG phase.

\addtolength{\tabcolsep}{+1.25pt}
\begin{table}[!htbp]
\begin{tabular}{c c c c}
\hline\hline
Functional & $R\Bar{3}m \rightarrow$ $Fdd2$ & $Fdd2 \rightarrow$ HG & HG $\rightarrow Im\bar{3}m$ \\ \hline
LDA & $249$  & $378$ & $700$ \\
PBE & $266$  & $386$ & $679$ \\
PBEsol & $245$  & $366$ & $644$ \\
\hline\hline
\end{tabular}
\caption{Static-lattice transition pressures in GPa according to LDA, PBE and PBEsol.}
\label{transitions_table}
\end{table}
\addtolength{\tabcolsep}{-1.25pt}

This $Fdd2$ structure is detailed in Figs. \ref{Fdd2_detail} and \ref{Fdd2_birds_eye_view}. Whilst it clearly resembles the Ba-IVa HG structure, it has the property of being self-similar when rotated $45 \degree$ about an axis perpendicular to to the pseudo-guest chains. We use the term pseudo-guest, as this property implies that there is no longer a unique chain axis, and therefore it cannot be equivalent to the `standard' Ba-IVa type HG structure (although, as we will show, S does eventually adopt the Ba-IVa HG structure at higher pressures). The equivalence through a $45 \degree$ rotation means that there are two nonintersecting pseudo-guest chains, running in different directions in offset planes, as detailed in Fig. \ref{Fdd2_birds_eye_view}. Each pseudo-guest chain forms part of the host structure when looking down the other chain's axis (and vice versa), and thus it is no longer possible to decompose the structure into two distinct host and guest lattices (see Figs. \ref{Fdd2_detail}(b) and \ref{Fdd2_detail}(c)). Whilst the pseudo-guest chains in the same layer are in alignment, those in the layer below are shifted by half of the intra-chain atomic separation, thus giving the chains an AB ordering when viewed from the side.
\par
We tested the stability of this structure for a few other elements that possess a high-pressure HG phase (such as Al and As), but in each case it was never the lowest-enthalpy phase. The `two-chain' motif is reminiscent of the structure of the ternary incommensurate compound Hg$_{3-\delta}$AsF$_6$ \cite{incommensurate_mercury_compound_axe,incommensurate_mercury_compound_buiting,incommensurate_mercury_compound_degroot}, in which chains of mercury atoms, running at $90 \degree$ to one another, sit within an AsF$_6$ framework. However, unlike this compound, and the incommensurate Ba-IVa phase, our calculations suggest that the $Fdd2$ structure is in fact a commensurate crystal and is not \textit{quasi}-periodic. To demonstrate that this is the case, we first show that at higher pressures, S \textit{does} exhibit a truly incommensurate Ba-IVa type HG structure, and then use these findings to show that the $Fdd2$ phase cannot be incommensurate.
\par
The Ba-IVa HG structure, which occurs (with minor variations) in multiple elements across the periodic table \cite{HG_Barium,HG_alkali_metals,Rb-IV,HG_Aluminium,HG_group_V,HG_Bismuth_SC,Haussermann_2002,HG_Strontium,HG_scandium,HG_Antimony,HG_Calcium}, comprises a tetragonal `host' structure of $I4/mcm$ symmetry with $8$ atoms in the (conventional) unit cell, and one of two possible coexisting `guest' structures; either a $C$-face centered tetragonal structure, or an $M$-face centered monoclinic structure, with atoms at $(0,0,0)$ and $(1/2,1/2,0)$ in both cases. The host and guest structures are incommensurate along their $c$-axes with an ideal ratio $\gamma_0$ that is remarkably close to $4/3$. In some cases, such as Bi-III and Sb-II \cite{modulated_Bi_and_Sb,Kartoon_Bismuth} or Ca-VII \cite{chain_ordered_calcium}, the positions of both the host and guest atoms in the HG structure are distorted/modulated relative to their `ideal' HG positions, giving both the channels and chains a `wiggly' or worm-like appearance.
\par
We find that above $386$ GPa, a Ba-IVa type HG approximant of $C2/c$ symmetry ($\gamma=4/3$), with $16$ (32) atoms in its primitive (conventional) cell, becomes lower in enthalpy than the $Fdd2$ structure. Similar to the $Fdd2$ structure, the atoms in the $C2/c$ approximant are distorted from their ideal positions; however, a unique chain axis for the guest atoms now exists, and thus the $C2/c$ approximant is of the `standard' Ba-IVa type. Below $386$ GPa, the $C2/c$ approximant is dynamically unstable, and it develops an imaginary optical $\Gamma$-point phonon mode that leads to the $Fdd2$ structure. The structural and enthalpic differences between the $C2/c$ approximant and the $Fdd2$ structure are small, and over the stability range of the $Fdd2$, they differ in enthalpy by at most $1.5$ meV per atom (see inset in Fig. \ref{phase_diagram}).

In order to calculate the ideal incommensurate host-guest ratio $\gamma_0$ at which the enthalpy is a minimum, we fitted the enthalpy of various commensurate approximants to a quadratic curve (see Supplemental Fig. 3 \cite{supp}), from which the ideal incommensurate ratio can be read off as the minimum. Fig. \ref{ratio_with_pressure} shows that $\gamma_0$ slowly decreases monotonically with increasing pressure, and that the choice of exchange-correlation functional does not affect the results significantly. This slow monotonic decrease in $\gamma_0$ with pressure is also seen in the HG phase of sodium \cite{HG_alkali_metals}, but is opposite to the cases of aluminium \cite{HG_Aluminium} and bismuth \cite{Kartoon_Bismuth}. In potassium and rubidium $\gamma_0$ initially decreases with pressure, before this behaviour reverses and $\gamma_0$ instead increases with pressure \cite{HG_alkali_metals}.

\begin{figure}[h!]
    \hspace{-1.5cm}
    \includegraphics[width=0.495\textwidth]{ratio_with_pressure.pdf}
    \caption{Dependence of the ideal incommensurate ratio with pressure. Data was taken in $100$ GPa intervals and fitted to a cubic spline.}
    \label{ratio_with_pressure}
\end{figure}

The procedure used to calculate $\gamma_0(p)$ allows us to propose that the $Fdd2$ structure is \textit{not} a commensurate approximant to an incommensurate phase, despite its visual similarity to the Ba-IVa structure; of all the approximants used to construct our $H(\gamma)$ quadratic fits ($9$ in total), only the $16$-atom approximant collapsed into the two-chain pattern at low pressures; approximants with other $\gamma$ values simply remained in the `single chain' Ba-IVa type structure. This suggests that the $Fdd2$ structure does not belong to a family of HG approximants, but rather is a $16$-atom structure in its own right. Additionally, we re-emphasise the fact that since one cannot decompose the $Fdd2$ into a unique host and a unique guest structure, it is not possible to even define a $\gamma$ value for the $Fdd2$.
\par
Relative to the more densely packed bcc ($Im\bar{3}m$) and trigonal ($R\bar{3}m$) phases, the HG structures are stabilised by a relatively large reduction in the total electronic energy. At 500 GPa, despite being ${\sim} 0.719\%$ larger in volume, the HG phase has a total electronic energy $0.190$ eV per atom lower than the bcc structure, which is more than enough to offset the increased $pV$ term and make it a favourable structure. The idea that S becomes more open-packed at these pressures had previously been considered by Rudin \textit{et. al.} \cite{Rudin,sulfur_and_Se_Rudin} who proposed an $R\bar{3}m \rightarrow$ simple-cubic ($Pm\bar{3}m$) transition, but neither that work, nor two previous high-pressure structure searching studies \cite{USPEX,Whaley_Baldwin}, located the $Fdd2$ and HG phases. We further note the significance of our findings on the construction of convex hulls for sulfur compounds (e.g. H$_2$S and H$_3$S) at these pressures; the HG phase of S is lower in enthalpy than the previously-assumed ground state ($R\bar{3}m$) by 11 meV per atom at 300 GPa, and up to 57 meV at 500 GPa.
\par
The inclusion of nuclear vibrational effects does not change the phase transition sequence, and at low temperatures, vibrational Zero-Point Energies (ZPEs) cause only small deviations from the static-lattice pressure boundaries (see Fig. \ref{phase_diagram}). For example, at the $R\bar{3}m \rightarrow Fdd2$ transition, we find the ZPE of the $R\bar{3}m$ phase to be  $1.4$ meV per atom higher than that of the $Fdd2$ structure, which lowers the transition pressure by ${\sim}4$ GPa from the static-lattice value in Table \ref{transitions_table}. The ZPEs of the $Fdd2$ structure and the $C2/c$ approximant are identical to within our convergence limits. At the HG $\rightarrow Im\bar{3}m$ transition, we find the HG ZPE to be higher than that of the $Im\bar{3}m$ phase by $4.8$ meV per atom, and the transition pressure reduces by ${\sim}22$ GPa compared to the value in Table \ref{transitions_table}. At higher temperatures ($\gtrsim 300$ K), vibrational effects reduce the transition pressures of both the $R\bar{3}m \rightarrow Fdd2$ and HG $\rightarrow Im\bar{3}m$ transitions. The lowering of the transition pressure for the $R\bar{3}m \rightarrow Fdd2$ transition is particularly interesting for two reasons. Firstly, it provides an experimental route to access the $Fdd2$ structure at slightly lower pressures, somewhat reducing the difficulty of any diamond anvil experiments. Secondly, it allows for observation of a new structure of $Fddd$ symmetry, also with 16 (64) atoms in its primitive (conventional) unit cell. This $Fddd$ structure is derived from the $Fdd2$ structure, which continuously develops an imaginary $\Gamma$-point phonon mode below $215$ GPa, pushing it into the $Fddd$ symmetry - the resulting transition between the phases is thus second order. We note that this $Fddd$ structure is distinct from the well-known ambient-pressure orthorhombic ground state of sulfur (also of $Fddd$ symmetry) which comprises S8 rings. Our calculations show that at temperatures above $\sim 1260$ K, the $Fddd$ phase should become the ground state at $215$ GPa (again, see Fig. \ref{phase_diagram}). We note that a recent publication \cite{sulfur_melting_curve} on the high-pressure melting curve of S achieved temperatures of $\sim 1700$ K at $51$ GPa using a laser-heated diamond anvil cell (LHDAC), and extrapolation of their data using a Simons-Glatzel fit \cite{simons-glatzel} suggests that the melting temperature of S should be at least $\sim 3850$ K at $200$ GPa.

\subsection{\label{chain_ordering}Chain Ordering}

After establishing the $Fdd2$ structure and $C2/c$ approximant as competitive phases, we conducted auxiliary searches with peturbed supercells of the $C2/c$ approximant containing $32$, $64$, $128$ and $288$ atoms. These searches produced two new HG approximants of $P4/ncc$ symmetry ($32$ atoms) and $Pcca$ symmetry ($64$ atoms) with slightly lower enthalpies. These approximants have the same commensurate ratio as the $C2/c$ approximant ($\gamma = 4/3$) and feature the same host structure, but possess different guest structures. The bottom of Fig. \ref{stacking_comparisons} shows the relative enthalpies of these different host-guest approximants. It can be seen that the $16$-atom $C2/c$ approximant is actually favoured only in a narrow pressure range ($384$-$400$ GPa), before the $64$-atom $Pcca$ approximant becomes favourable between $400$-$590$ GPa. The $32$-atom $P4/ncc$ approximant is then the ground state between $590$-$679$ GPa, with the bcc $Im\bar{3}m$ phase becoming stable thereafter.

Each of the $C2/c$, $Pcca$ and $P4/ncc$ approximants represent a different ordering of the guest atom chains, in which adjacent chains are displaced by a different amount parallel to the direction of the chain axes (i.e. have different heights in the $z$ direction). We found that this displacement was always either $1/3$ or $2/3$ of the guest lattice $c$ axis length ($c_{G}$). Fig. \ref{stacking_comparisons} shows the different orderings as viewed along, and perpendicular to, the axes of the chains. In each case, the chains can clearly be described as having been stacked in a certain order, with the $C2/c$, $Pcca$ and $P4/ncc$ approximants possessing a stacking order of ABC, ABAC and AB respectively. Fig. \ref{stacking_comparisons} also shows that the aforementioned distortions/modulations of the host and guest structures are slightly different in each case, being largest for the $C2/c$ structure and smallest for the $P4/ncc$ structure. This suggests a coupling between the chain orderings and the distortions; an observation that we shall return to later.

\begin{figure}[h!]

    \begin{subfigure}[t]{0.2\textwidth}
        \includegraphics[width=\linewidth]{C2c_stacking_labelled.png}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.21\textwidth}
        \hspace{-1.1cm}
        \includegraphics[width=\linewidth]{C2c_face_labelled.png}
    \end{subfigure}

    \vspace{0.521cm}

    \begin{subfigure}[t]{0.2\textwidth}
        \includegraphics[width=\linewidth]{Pcca_stacking_labelled.png}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.21\textwidth}
        \hspace{-1.1cm}
        \includegraphics[width=\linewidth]{Pcca_face_labelled.png}
    \end{subfigure}

    \vspace{0.521cm}

    \begin{subfigure}[t]{0.2\textwidth}
        \includegraphics[width=\linewidth]{P4ncc_stacking_labelled.png}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.22\textwidth}
        \hspace{-1.05cm}
        \includegraphics[width=\linewidth]{P4ncc_face_labelled.png}
    \end{subfigure}

    \vspace{0.31cm}

    \hspace{-0.5cm}\includegraphics[width=0.082\textwidth]{chains_down_b.png}
    \hspace{+4.5cm}\includegraphics[width=0.0794\textwidth]{chains_down_c.png}

    \vspace{0.298cm}

    \hspace{-0.6cm}\includegraphics[width=0.475\textwidth]{NOCOLOUR_chain_enthalpies.pdf}

    \caption{\textbf{Top:} ABC $(0,\frac{1}{3},\frac{2}{3})$ stacking of the $16$-atom $C2/c$ approximant. \textbf{Middle:} ABAC $(0,\frac{1}{3},0,\frac{2}{3})$ stacking of the $64$-atom $Pcca$ approximant. \textbf{Bottom:} AB $(0,\frac{2}{3})$ stacking of the $32$-atom $P4/ncc$ approximant. The relative stability of the stackings is shown beneath the structures; the $Fdd2$ and bcc phases become stable immediately off to the left and right of the plot, respectively.}
    \label{stacking_comparisons}
\end{figure}

The lowest-enthalpy stacking ordering at a given pressure is a trade off between minimisation of the total electronic energy and the $pV$ term. The AB ordering of the $P4/ncc$ approximant stacks most efficiently to give the lowest $pV$ term at all pressures, but has the largest total electronic energy of all approximants. This situation is reversed in the ABC stacking of the $C2/c$ approximant, which has the lowest electronic energy but the largest $pV$ term. The $Pcca$ approximant of ABAC stacking is a compromise between these two motifs and explains why it is favoured over a comparatively large pressure range. At lower pressures ($<400$ GPa), the $pV$ term is less important than the electronic term, and the $C2/c$ approximant becomes the ground state. At much larger pressures ($>590$ GPa), the $pV$ term becomes dominant, and the $P4/ncc$ approximant is the lowest enthalpy phase.
\par
Chain orderings that differ from the `ideal' all-aligned case have been discovered in other host-guest phases. The AB-ordered HG phase (our $P4/ncc$ approximant) that S adopts above $590$ GPa is also the ground state of the group-V elements As, Sb and Bi at much lower pressures \cite{HG_group_V,Haussermann_2002}. The Ca-VII phase of Calcium \cite{chain_ordered_calcium} is well-described by a $128$-atom approximant in which the guest chains have different $z$-heights in both the $a$ and $b$ directions (as opposed to S, where the $z$-heights change only along the $a$ direction). In the case of Ba-IVc, the structure adopted by Barium at pressures a few GPa higher than the Ba-IVa phase, the chain displacements are arranged in such a way as to produce an extremely complicated, mesoscopic-scale, interlocking S-shape patterning, which is well-represented by a $768$-atom cell \cite{complicated_barium}.
\par
Whilst the enthalpy differences between the ABC, ABAC and AB stackings are mostly ${\sim}1$ meV per atom, we note that the `ideal' stacking (of $I4/mcm$ symmetry), in which all guest chains are perfectly aligned (i.e., AAA...), is higher in enthalpy at all pressures by at least $20$ meV per atom and up to $45$ meV, indicating that perfect chain alignment is strongly disfavoured in S. This raises the question of why any stacking that differs from the all-aligned case has such a drastically reduced enthalpy. Surprisingly, we find that explicitly offsetting the chains causes a only a very small enthalpy reduction in itself. Instead, we show that the answer lies in the distortions/modulations of the host and guest atoms, which cannot occur unless the chains are first offset.
\par
Fig. \ref{enthalpy_evolution} details the situation for the ABC case at $387$ GPa, although the ABAC and AB cases are analogous. Starting from the all-aligned $I4/mcm$ approximant, we manually offset the chains such as to have an ABC ordering, keeping all else fixed. This gives rise to an enthalpy reduction of $4$ meV per atom, which can be attributed to the opening of several directional band gaps at the Fermi level (see Supplemental Fig. 5 \cite{supp}). We subsequently relax the cell vectors (with the atomic positions fixed), which gives rise to a further enhalpy reduction of $1$ meV. Finally, we add in the modulation of the host and guest atoms, which further reduces the enthalpy by $34$ meV; an order of magnitude larger than the reductions caused by the chain offsets and the cell vector relaxation. The bottom of Fig. \ref{enthalpy_evolution} shows that these energetic reductions arise as a result of shifting eDOS weight to lower energies, whilst little changes at the Fermi level itself.

\begin{figure}[!htbp]

    \hspace{-0.69cm}
    \includegraphics[width=0.49\textwidth]{enthalpy_evolution_with_figures.pdf}

    \hspace{-0.35cm}
    \includegraphics[width=0.49\textwidth]{DOS_changes.pdf}

    \caption{\textbf{Top:} Schematic decomposing the $39$ meV enthalpy reduction that occurs going from the all-aligned approximant to the ABC ordered approximant at $387$ GPa. The enthalpy axis is logarithmic. \textbf{Bottom:} The corresponding changes in the eDOS. The Fermi level is set at $0$ eV.}
    \label{enthalpy_evolution}
\end{figure}

Since the modulations provide the greatest contribution by far to the total enthalpy reduction, it is natural to ask why offsetting the guest chains - a change that causes only a very small enthalpy reduction in itself - is necessary to see these modulations appear. When the guest chains are offset, the host atoms are exposed to an asymmetric environment which creates forces that cause the host atoms to distort, which in turn distorts the guest atoms. If the chains are all-aligned, then the environment on either side of the host atoms is the same; symmetry implies that these forces do not arise, and there are therefore no distortions. In the ABC and ABAC stackings, both the host and guest atoms are distorted in all of the $x$, $y$ and $z$ directions (see Fig. \ref{stacking_comparisons}). However, in the case of AB stacking, symmetry dictates that the guests are displaced only in the $z$ direction. These $z$ distortions longitudinally pinch together pairs of guest atoms along the chain axes and dimerize them such that the covalent bonds alternate in a short-long-short-long pattern, with the short bond ${\sim}3.2 \%$ smaller than the long bond. This 1D dimerization is referred to as \textit{quasi}-pairing in the literature \cite{modulated_Bi_and_Sb}, and the same behaviour has recently been predicted to occur in the CDW phase of sulfur at lower pressures \cite{Whaley_Baldwin}. Detailed experimental \cite{modulated_Bi_and_Sb} and theoretical \cite{Kartoon_Bismuth} analyses of both the host and guest distortions in the Bi-III structure confirm that the modulations play an indispensable role in the stability of HG phases.
\par
In addition to the structures found in our structure search, we calculated the enthalpy of various complex structures that appear in the vicinity of HG phases in other elements, such as oC52 in rubidium \cite{oC52_rubidium}, oC84 in caesium \cite{oC84_caesium}, oP8 in potassium \cite{oP8_potassium}, and A7 in the group-V elements \cite{Haussermann_2002}. These structures were found to be at least $0.1$ eV per atom above the ground state at all pressures considered here. We also constructed some $\gamma=4/3$ approximants with longer-period chain stackings such as ABAAC, ABCAB, ABCACB and permutations thereof, as well as configurations where the chain heights changed in the $b$- direction also (see Supplemental Material for a table of all the stackings that were tested \cite{supp}). Most of these structures collapsed to the 64-atom $Pcca$ ABAC stacking, which seems to have a particularly large basin of attraction in the configuration space of chain orderings. Those that did not collapse to the ABAC stacking simply had a higher enthalpy at all pressures, although we note that an ABCACB stacking was particularly close to becoming the ground state ($+0.2$ meV above the ABAC stacking at $550$ GPa). We further considered the effect of spin polarisation in our calculations, and seeded our structures with varying ferromagnetic and anti-ferromagnetic orderings, but in all cases, the density relaxed to a spin-unpolarised configuration.

\subsection{\label{charge_density}Charge Density}

We compare charge density isosurfaces of HG sulfur in Fig. \ref{voids_and_covalents} at moderate ($35\%$ of maximum) and very small ($0.3\%$ of maximum) values, alongside several 2D slices of the charge density taken perpendicular to the guest chains. The charge density of S is peaked near atomic sites and along lines connecting neighbouring atoms, which is representative of a covalently bonded solid. S exhibits no significant interstitial localisation of charge and therefore does not constitute a conventional electride, unlike the HG phases of the alkali metals \cite{HG_alkali_metals} or aluminium \cite{HG_Aluminium}. However, S displays curious behaviour when studied at very small isosurface values, which correspond to charge-depleted regions. The middle of Fig. \ref{voids_and_covalents} shows that these charge-depleted regions form well localised lobes, which we refer to as charge `voids', and co-incide with minima in the Electron Localisation Function (ELF) \cite{ELF_paper}. These voids form a structure equivalent to the host lattice of the alkali metal HG phases \cite{HG_alkali_metals}, with one host unit situated at the midpoint of every `short' intra-chain covalent bond. The bottom of Fig. \ref{voids_and_covalents} shows the 2D charge density taken at the midpoints of these `short' bonds for the AB case; when these slices are superimposed, the full set of voids (as seen in the middle of Fig. \ref{voids_and_covalents}) is recovered.

\begin{figure}[!htbp]
    \includegraphics[width=0.368\textwidth]{red_covalent_bonds.png}

    \vspace{0.125cm}

    \includegraphics[width=0.368\textwidth]{blue_voids.png}

    \vspace{0.15cm}

    \includegraphics[width=0.46\textwidth]{2D_slices_LABELLED.pdf}

    \caption{Charge density in HG sulfur. \textbf{Top:} Isosurface at $35\%$ of maximum, displaying clear covalent bonds. \textbf{Middle:} Isosurface at $0.3\%$ of maximum, showing the charge-depleted `voids'. \textbf{Bottom:} 2D slices of the charge density taken perpendicular to the guest chains for the AB ordering (guest atoms highlighted). The ABC and ABAC stackings give similar charge distributions, but the voids have different $z$ coordinates in each case (as they are tied to the guest structure).}
    \label{voids_and_covalents}
\end{figure}

Remarkably, these voids are the exact locations of charge density \textit{maxima} in the HG phase of aluminium \cite{HG_Aluminium}. The duality between the charge densities of S and Al can be explained in terms of the intra-chain bonding between the guest atoms, which is present in S but absent in Al. In S, the short intra-chain covalent bonds contain significantly more charge than the weaker long bonds; this excess charge is drawn from the charge voids, and explains why there is precisely one set of voids per short bond.
\par
We note that in S there is no opening of a pseudogap in the electronic Density of States (eDOS) at the Fermi level (see Fig. \ref{enthalpy_evolution} and Supplemental Fig. 6 \cite{supp}). This is in contrast to the behaviour of high-pressure electrides \cite{HG_alkali_metals,HG_Aluminium}, in which there is often a dramatic reduction in the eDOS at the Fermi level (in Al, for example, the eDOS reduces by a factor of ${\sim}4$). Interestingly, we find that there is still significant $s \rightarrow d$ charge transfer in S; for the HG phase at $500$ GPa, the Fermi-level eDOS comprises $11\%$ $s$ character, $42\%$ $p$ character and $47\%$ $d$ character.

\subsection{\label{TCs}Superconductivity and Phason Modes}

We have calculated the superconducting critical temperatures of our high-pressure sulfur phases using Migdal-Eliashberg theory within Density Functional Perturbation Theory (DFPT) using the \verb|QUANTUM ESPRESSO| code \cite{QE}. We used the McMillan-Allen-Dynes formula \cite{McMillan_Allen_Dynes} to calculate Tc:

\begin{equation}
    T_c = \frac{\omega_{_\mathrm{log}}}{1.2} \mathrm{exp} \bigg( \frac{-1.04(1+\lambda)}{\lambda - \mu^{*} (1+0.62 \lambda) } \bigg)
\end{equation}

Where $\omega_{_\mathrm{log}}$ is the logarithmically averaged phonon frequency, $\lambda$ is the electron-phonon coupling constant, and $\mu^{*}$ is the Coulomb pseudopotential parameter, rescaled from the bare value $\mu$ to account for retardation \cite{Morel_Anderson}:

\begin{equation}
    \mu^{*} = \frac{\mu}{1 + \mu \ \mathrm{ln}\big( \frac{\epsilon_F}{\Theta_D} \big)}
\label{renormalised_mu}
\end{equation}

Where $\epsilon_F$ is the Fermi energy, and $\Theta_D$ is the phonon Debye frequency. We note that a relatively recent publication \cite{SCDFT_sulfur} has employed Superconducting DFT (SCDFT) \cite{SCDFT_1,SCDFT_2} to calculate the $T_c$ of the $R\bar{3}m$ phase at pressures up to $200$ GPa, and shows that the value of $\mu$ for S in this regime remains nearly constant at $0.20$. Rescaling this using Eqn. \ref{renormalised_mu}, we obtain a $\mu^{*}$ value of $0.09$ that is essentially constant across all the phases, since the ratio $\epsilon_F / \Theta_D$ changes very little between the phases and with increasing pressure. Further, a previous publication \cite{zakharov_and_cohen_sulfur} derived a $\mu^{*}$ value of $0.08$ for the bcc phase at $500$ GPa using the static Thomas-Fermi dielectric function. We therefore choose $\mu^{*} \in [0.08,0.10]$ as a reasonable range of representative $\mu^{*}$ values.

\begin{figure}[h!]

    \hspace{-0.5cm}
    \includegraphics[width=0.45\textwidth]{Tc_vs_pressure.pdf}

    \caption{Superconducting critical temperatures of the sulfur phases as a function of pressure. Dotted vertical lines denote the structural phase transitions discussed in this work. The shading corresponds to $\mu^{*} \in [0.08,0.10]$; the grey region connects the lowest-enthalpy structures at each pressure, and the purple region in the top left is a possible metastable path if S can be trapped in the $Fdd2$ symmetry below $270$ GPa. The data has been linearly interpolated.}
    \label{Tc_vs_pressure}
\end{figure}

Fig. \ref{Tc_vs_pressure} shows our calculated $T_c$ values as a function of pressure. It can be seen that $T_c$ is expected to peak shortly after the transition to the $Fdd2$ phase, where we predict $T_c$ to be $24.8$ - $28.2$ K at $271$ GPa. For a purely elemental solid, this is a high critical temperature and is close to the record among the elements (excluding hydrogen) of $29$ K in high-pressure Ca-VII \cite{SC_elements_review,Calcium_experimental,chain_ordered_calcium}. Phonon calculations show that although it is not the lowest enthalpy phase at such pressures, the $Fdd2$ structure is dynamically stable down to pressures as low as $215$ GPa (below here, it develops a soft mode that pushes it into the $Fddd$ structure, as previously mentioned). Since we find the $T_c$ of the $Fdd2$ phase to be nearly constant in this region, it is possible that upon reducing pressure from $>270$ GPa, hysteresis effects may allow $T_c$ to take a path similar to that indicated in purple in the top left of Fig. \ref{Tc_vs_pressure}. We note that whilst $T_c$ falls monotonically with pressure after the transition to the $Fdd2$ phase, we predict that S will remain superconducting up to at least $700$ GPa.
\par
Whilst we used \textit{isotropic} Migdal-Eliashberg theory to calculate $T_c$, we note that the aforementioned SCDFT calculations in Ref. \cite{SCDFT_sulfur} account fully for the Fermi surface anisotropy. Our $T_c$ results for the $R\bar{3}m$ phase at $250$ GPa are slightly higher (by ${\sim}3$ K) than the $T_c$ obtained from the use of SCDFT in Ref. \cite{SCDFT_sulfur}, although our Eliashberg function $\alpha^2F(\omega)$ is very similar (see Fig. \ref{eliashberg_function}). Conversely, our $T_c$ value for the $R\bar{3}m$ structure is actually slightly lower (by ${\sim}4$ K) than the experimental value obtained by Drozdov et. al. \cite{drozdov_hydrogen_sulfide_experiment}.

\begin{figure}[h!]

    \vspace{-0.25cm}
    \hspace{-0.4cm}
    \includegraphics[width=0.48\textwidth]{phason_mode.pdf}

    \caption{Phonon dispersion relation for the AB-ordered $P4/ncc$ approximant at $375$ GPa (commensurate ratio $\gamma = 4/3$). The $\Gamma \rightarrow$ X direction runs across the tetragonal face of the HG structure (perpendicular to the guest chains). Two low-energy phason modes, which correspond to relative movement of the host and guest lattices, have been highlighted in green and red. In the limit of a truly incommensurate structure, the frequencies of these modes vanish.}
    \label{phason_mode}
\end{figure}

In the absence of complications such as the formation of finite-sized domains, the energy of a \textit{truly incommensurate} HG structure must necessarily be invariant with respect to relative translation of the host and guest lattices. Such a relative translation can be achieved with an optical phonon mode running perpendicular to the guest chains, which displaces the host and guest structures in opposite directions. The energy of such a phonon mode should be exactly zero for a true \textit{quasi}-crystal, although the use of commensurate approximants in \textit{ab}-\textit{initio} studies such as this work means that they appear to have a small positive frequency in a phonon calculation. Fig. \ref{phason_mode} shows the phonon dispersion relation for the AB-ordered $P4/ncc$ approximant, in which two low-energy optical modes can be seen. These `phason' modes naturally lead to large values of the mode-specific electron-phonon coupling constant $\lambda_{\mathrm{\textbf{q}},\nu}$, which varies with phonon frequency as $\sim \omega^{-1}_{\mathrm{\textbf{q}},\nu}$, and give rise to strong-coupling superconductivity in HG phases \cite{HG_Bismuth_SC}; however, as mentioned, our use of a commensurate approximant forces the phason modes in Fig. \ref{phason_mode} to have a nonzero frequency. Consequently in Table \ref{superconducting_properties}, which summarises relevant superconducting properties of interest for several of the structures, the value of the electron-phonon coupling constant $\lambda$ for the HG phase is \textit{not} in the strong-coupling regime where $\lambda >> 1$ \cite{McMillan_Allen_Dynes}. It is probable that the use of a larger approximant (with accompanying lower phason frequencies) would give rise to a larger value of  $\lambda$, although such a calculation is beyond the scope of this work and would be computationally expensive. We also note that solving the Eliashberg equations (necessary for the true strong coupling case) gave very similar $T_c$ values to those obtained using the McMillan-Allen-Dynes formula for all of the structures discussed in this work. Interestingly, we find that $\lambda$ peaks in the $Fdd2$ phase, which by virtue of being a commensurate structure, cannot possess a phason mode. Nonetheless, we still predict a relatively high $T_c$ of $16.4$ - $19.8$ K for the HG phase at $425$ GPa, which we partially attribute to the lack of a significant Fermi-level pseudogap in the eDOS (see Table \ref{superconducting_properties}).

\addtolength{\tabcolsep}{-0.85pt}
\begin{table}[!htbp]
\begin{tabular}{c c c c c c}
\hline\hline
\centering
Structure & Pressure (GPa) & $T_c$ (K) & $\lambda$ & $\langle \omega_{log} \rangle$ (K) & $\mathcal{N}(\epsilon_F)$ \\ \hline
$R\bar{3}m$ & $250$ & $17.6$ & $0.689$  & $469$ & $0.0390$ \\
$Fdd2$ & $271$ & $26.5$ & $0.847$  & $439$ & $0.0414$ \\
HG $(\gamma=\frac{4}{3})$ & $425$ & $18.1$ & $0.738$  & $417$ & $0.0445$ \\
$Im\bar{3}m$ & $700$ & 12.6 & $0.543$  & $660$ & $0.0470$ \\
\hline\hline
\end{tabular}
\caption{Superconducting critical temperature $T_c$, electron-phonon coupling constant $\lambda$, logarithmic average frequency $\langle \omega_{log} \rangle$ and Fermi-level electronic density of states $\mathcal{N}(\epsilon_F)$ (in units of electrons per eV per \AA $^{3}$) for several of the structures discussed in this work at various pressures.}
\label{superconducting_properties}
\end{table}
\addtolength{\tabcolsep}{+0.85pt}

\begin{figure}[h!]

    \begin{subfigure}[t]{0.236\textwidth}
        \hspace{-0.2cm}
        \includegraphics[width=\linewidth]{a2F_R-3m_250GPa_with_SG_label.pdf}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.236\textwidth}
        \hspace{-0.3cm}
        \includegraphics[width=\linewidth]{a2F_Fdd2_271GPa_with_SG_label.pdf}
    \end{subfigure}

    \vspace{0.4cm}

    \begin{subfigure}[t]{0.236\textwidth}
        \hspace{-0.2cm}
        \includegraphics[width=\linewidth]{a2F_C2c_425GPa_with_SG_label.pdf}
    \end{subfigure}
    \hfill
    \begin{subfigure}[t]{0.236\textwidth}
        \hspace{-0.3cm}
        \includegraphics[width=\linewidth]{a2F_Im-3m_700GPa_with_SG_label.pdf}
    \end{subfigure}

    \caption{Eliashberg function $\alpha^2F(\omega)$ for selected structures at given pressures. The contribution of each phonon branch to $\alpha^2F(\omega)$ has been shaded with a different colour. The (cumulative) electron-phonon coupling constant $\lambda(\omega)$ has been plotted with a dashed blue line.}
    \label{eliashberg_function}
\end{figure}

\section{\label{conclusion}Conclusions}

We have shown that upon increasing pressure beyond the experimentally-known $R\bar{3}m$ ($\beta$-Po) phase, sulfur adopts a commensurate structure of $Fdd2$ symmetry in the range $266$-$386$ GPa that resembles a strongly distorted Ba-IVa phase. Between $386$ and $679$ GPa, S possesses a truly incommensurate HG structure of the Ba-IVa type. Above $679$ GPa, we find that S adopts a more closely packed bcc structure. We have accounted for the effects of finite temperatures on the phase boundaries, and have shown that all of the transition pressures can be lowered with an increase in temperature, as well as predicting the existence of an $Fddd$ phase below $215$ GPa and at temperatures $\gtrsim 1260$ K.
\par
We have studied the properties of the incommensurate HG phase in detail, and have shown that the ideal incommensurate ratio $\gamma_{0}$ decreases monotonically with increasing pressure. Over the stability range of the HG phase, S transforms between several competing chain-ordered variants of the incommensurate HG structure, with an ABAC chain stacking stable over most of this pressure range. These different chain orderings are intimately coupled to modulations of the host and guest atoms, which contribute substantially to the stability of the HG phases. We find that there is no significant interstitial localisation of charge in S, instead showing that there are notable localised \textit{absences} of charge (`voids'). We have calculated the superconducting critical temperature of S for several of the structures, and expect $T_c$ to peak between $24.8$ and $28.2$ K in the $Fdd2$ phase at $271$ GPa. We predict that S remains a superconductor up to at least $700$ GPa.

\section{\label{acknowledgements}Acknowledgements}

We wish to thank Malte Grosche for useful discussions. The computational resources for this project were provided by the Cambridge Service for Data Driven Discovery (CSD3), and we are further grateful for computational support from the UK national high performance computing service, ARCHER, for which access was obtained via the UKCP consortium and funded by EPSRC grant ref EP/P022561/1.

\nocite{*}

\section{\label{pseudo}Pseudopotentials}
We used a sulfur pseudopotential designed using the OTFG code bundled with \verb|CASTEP|. Our pseudopotential had a cutoff radius of $0.74$ \AA $ \ =1.4$ Bohr (far smaller than half the smallest interatomic separation encountered in this study) and included the $n=3$ valence shell only. Our tests show that inclusion of the core $n=2$ states has no effect on resolving enthalpy differences at these pressures. The local channel was chosen as $l_{loc}=3$. This pseudopotential was used for all \verb|CASTEP| calculations in the manuscript.
\\\\
The OTFG code string used the generate the pseudopotential was:

\begin{verbatim}
    S 3|1.4|4|10|12|30:31:32(qc=7)
\end{verbatim}

For the \verb|QUANTUM ESPRESSO| (QE) superconducting critical temperature calculations, the default scalar relativistic, ultrasoft PBE pseudopotential for S from  \href{https://dalcorso.github.io/pslibrary/}{pslibrary} were found to be sufficient in all cases, except for the $C2/c$ approximant at $425$ GPa, which required a smaller $R_c$. For this calculation, a custom pseudopotential with cutoff radius $R_c=1.6$ Bohr was generated via the \verb|ld1.x| code bundled with QE, using the following input file:

\begin{verbatim}
&input
   title='S',
   zed=16.,
   rel=1,
   config='[Ne] 3s2 3p4 3d-1',
   iswitch=3,
   dft='pbe'
 /
 &inputp
   lpaw=.false.,
   pseudotype=3,
   file_pseudopw='S.UPF',
   author='Jack Whaley-Baldwin',
   lloc=-1,
   rcloc=1.6,
   which_augfun='PSQ',
   rmatch_augfun_nc=.true.,
   nlcc=.true.,
   new_core_ps=.true.,
   rcore=1.4,
   tm=.true.
 /
6
3S  1  0  2.00  0.00  1.40  1.60  0.0
3S  1  0  0.00  6.00  1.40  1.60  0.0
3P  2  1  4.00  0.00  1.40  1.60  0.0
3P  2  1  0.00  3.00  1.40  1.60  0.0
3D  3  2  0.00  0.10  1.40  1.60  0.0
3D  3  2  0.00  0.30  1.40  1.60  0.0
\end{verbatim}

This custom pseudopotential was tested against its respective default pslibrary pseudopotential for a few structures, and was found to agree in volume to within $0.1 \%$, and relative enthalpies agreed to $\pm 0.1$ meV.

\section{\label{conv_calcs}Convergence Calculations}

Here, we present convergence calculations at $400$ and $800$ GPa, which span the majority of the pressure range covered. We consider as representative structures the $R\bar{3}m$ stucture at $400$ GPa and the $Im\bar{3}m$ structure at $800$ GPa. We show only the convergence results for the PBE pseudopotential here, because the converged cutoffs and \textbf{k}-point spacings for the the LDA and PBEsol pseudopotentials are nearly identical.
\\\\
Supplemental Figs. \ref{400gpa_convergence} and \ref{800gpa_convergence} show that our chosen parameters of a plane-wave cutoff of $850$ eV, a \textbf{k}-point spacing of $2\pi \times 0.01 \ $\AA$^{-1}$, and the \verb|CASTEP| default electronic smearing temperature ($\approx 1200$ K) were sufficient to achieve absolute convergence of at least $\pm 0.2$ meV throughout the pressure range of interest; the relative energies of the phases are converged to better than this.
\\\\
Such stringent convergence was necessary as the different chain-ordered host-guest phases are separated by enthalpy differences of order $\approx 0.5$ meV.

\begin{figure}[h!]
    \hspace{-0.5cm}
    \includegraphics[width=0.45\textwidth]{supplemental/R-3m_cutoff_400gpa.pdf}

    \vspace{0.25cm}

    \hspace{-0.75cm}
    \includegraphics[width=0.43\textwidth]{supplemental/R-3m_kpoints_400gpa.pdf}

    \vspace{0.25cm}

    \hspace{-0.95cm}
    \includegraphics[width=0.45\textwidth]{supplemental/R-3m_smearing_400gpa.pdf}
    \caption{Absolute convergence of the total electronic energy of the $R\bar{3}m$ structure at $400$ GPa, for the plane-wave cutoff (top), \textbf{k}-point spacing (middle) and electronic smearing temperature (bottom). $\pm 1$ meV and $\pm 0.5$ meV lines are shown relative to the most expensive result.}
    \label{400gpa_convergence}
\end{figure}

\begin{figure}[h!]
    \hspace{-0.5cm}
    \includegraphics[width=0.45\textwidth]{supplemental/Im-3m_cutoff_800gpa.pdf}

    \vspace{0.25cm}

    \hspace{-0.7cm}
    \includegraphics[width=0.43\textwidth]{supplemental/Im-3m_kpoints_800gpa.pdf}

    \vspace{0.25cm}

    \hspace{-0.7cm}
    \includegraphics[width=0.45\textwidth]{supplemental/Im-3m_smearing_800gpa.pdf}
    \caption{Absolute convergence of the total electronic energy of the $Im\bar{3}m$ structure at $800$ GPa, for the plane-wave cutoff (top), \textbf{k}-point spacing (middle) and electronic smearing temperature (bottom). $\pm 1$ meV and $\pm 0.5$ meV lines are shown relative to the most expensive result.}
    \label{800gpa_convergence}
\end{figure}

\clearpage

\section{\label{ratio_fitting}Ideal Host-Guest Ratio Fitting Procedure}

Supplemental Fig. \ref{gamma_parabola} details how the `ideal' host-guest $\gamma$-value was obtained for a given pressure.
\\\\
In addition to those approximants found in the search, several approximants with different $\gamma$-values were constructed `by hand', and the resulting enthalpies were fitted to a quadratic curve. The ideal value $\gamma_0$ is simply read off from the minimum of this curve.

\begin{figure}[h!]
    \hspace{-0.7cm}
    \includegraphics[width=0.475\textwidth]{supplemental/HG_xplot_HULL.pdf}
    \caption{Dependence of the enthalpy of the HG phase on the $c_H / c_G$ ratio at $400$ GPa using the PBE functional. The black line is a least-squares fit to a quadratic curve, and the `ideal' value $x_0$ is taken from the minimum of the curve. Each point represents a particular host-guest approximant, and the label shows the number of atoms in that approximant. By constructing similar curves at different pressures, we can obtain $\gamma_0$ at each pressure and thus plot Fig. 4 in the manuscript.}
    \label{gamma_parabola}
\end{figure}

\newpage
\section{\label{C2c_and_Fdd2}Enthalpies of C2/c and Fdd2 Structures}

The $C2/c$ HG approximant exhibits an imaginary phonon mode at the $\Gamma$-point below $380$ GPa, which distorts it into the $Fdd2$ structure.
\\\\
The $C2/c$ and $Fdd2$ structures are very nearly identical, both structurally and in enthalpy. The enthalpy difference between them is at most $\approx 1.5$ meV at $275$ GPa. Supplemental Fig. \ref{C2_vs_C2c_vs_Fdd2} compares the enthalpies of the two structures, in addition to another enthalpy curve that detects only the $C2$ symmetry group, which is a subgroup of both $C2/c$ and $Fdd2$. This shows that sulfur `selects' the $Fdd2$ symmetry below $380$ GPa and the $C2/c$ HG approximant above this pressure.

\begin{figure}[h!]
    \hspace{-0.275cm}
    \includegraphics[width=0.46\textwidth]{supplemental/C2_C2c_Fdd2_comparison.pdf}
    \caption{Relative enthalpies of the $C2$, $C2/c$ and $Fdd2$ structures. $C2$ is a symmetry subgroup of both $C2/c$ and $Fdd2$.}
    \label{C2_vs_C2c_vs_Fdd2}
\end{figure}

\newpage
\section{\label{other_chain_orderings}Enthalpies of Other Stacking Orderings}

In the manuscript, we state that we tested various other stacking orderings that turned out to either be not energetically competitive, or simply collapsed into another chain ordering. Supplemental Table \ref{transitions_table} details these other stacking orderings.

\addtolength{\tabcolsep}{+5pt}
\begin{table}[!htbp]
\begin{tabular}{l l l l}
\hline\hline
Stacking Sequence & Chain Heights & Smallest Enthalpy (meV/atom) & Pressure (GPa) \\ \hline
ABCACB & $(0,1,2,0,2,1)$  & $+0.3$ & 550 \\
ABACBCBC$^{\dagger}$ & $(0,1,0,2,1,2,1,2)$  & $+0.8$ & $500$ \\
AB$^{\dagger}$ & $(0,1)$  & $+1.3$ & $450$ \\
ABB & $(0,1,1)$  & $+16.9$ & $480$ \\
ABACB & $(0,1,0,2,1)$  & collapsed to ABC : $(0,1,2)$ & N/A \\
ABB & $(0,2,2)$  & collapsed to ABB : $(0,1,1)$ & N/A \\
ABBB & $(0,1,1,1)$  & collapsed to ABAC : $(0,1,0,2)$ & N/A \\
ABBC & $(0,1,1,2)$  & collapsed to ABAC : $(0,1,0,2)$ & N/A \\
ABBC & $(0,2,2,1)$  & collapsed to ABAC : $(0,1,0,2)$ & N/A \\
\hline\hline
\end{tabular}
\caption{Explicitly tested stacking sequences that were constructed by hand. The `Chain Heights' ($c$-axis displacement as one moves along the $a$-direction) are given in units of a third of the intra-chain atomic spacing, e.g. `2' corresponds to $\frac{2}{3}$. The `Smallest Enthalpy' is measured relative to the lowest-enthalpy stacking sequence as given in the manuscript, which is ABC in the range $384$-$400$ GPa, ABAC in the range $400$-$590$ GPa, and AB in the range $590$-$679$ GPa. The `Pressure' column in this table is the pressure at which the enthalpy of that sequence was lowest. Where the stacking sequence collapsed into another ordering, `N/A' (Not Applicable) has been placed in the Pressure column. Sequences marked with a dagger ($^{\dagger}$) developed (after relaxation) $c$-axis displacements that changed when moving along the $b$-axis also.}
\label{transitions_table}
\end{table}
\addtolength{\tabcolsep}{-5pt}

\clearpage

\section{\label{band_gap_chain_ordering}Band Gaps Opened by Chain Ordering}

Supplemental Fig. \ref{bandstructures} below shows the bandstructures of the $\gamma=4/3$ approximant with all chains aligned (i.e. AAAAAA...) and then with  an ABC ordering. It can be seen that adding the ABC ordering opens up several band gaps at the Fermi level, and this contributes to the lowering of electronic energy as described in Fig. 6 of the main manuscript.

\begin{figure}[!htbp]

    \hspace{-0.4cm}
    \includegraphics[width=0.435\textwidth]{I4mcm_GX_GS_only_HIGHLIGHTED.pdf}

    \vspace{0.4cm}

    \hspace{-0.4cm}
    \includegraphics[width=0.435\textwidth]{C2c_GX_GS_only_HIGHLIGHTED.pdf}

    \caption{Electronic bandstructures at $387$ GPa along $\Gamma \rightarrow$ X:$(0.5,0,0)$ and $\Gamma \rightarrow$ S:$(0.5,0.5,0)$ high symmetry directions for a ($96$-atom) $3\times1\times1$ tiling of the ideal $I4/mcm$ structure. Band gap openings are highlighted with cyan boxes. \textbf{Top:} With all chains aligned. \textbf{Bottom:} With the chains offset `by hand' as to have an ABC ordering along the $a$-axis.}
    \label{bandstructures}
\end{figure}

\clearpage

\section{\label{eDOS}Electronic Density of States}

The top of Supplemental Fig. \ref{eDOSs} compares the electronic Densities of States (eDOS) of the $Im\bar{3}m$ phase and 64-atom $Pcca$ HG approximant. The HG phase features a relative reduction in total electronic energy as a result of transferring its eDOS weight to lower energies. A small reduction in the Fermi level eDOS is also visible.
\\\\
For comparison, the bottom of Supplemental Fig. \ref{eDOSs} compares the eDOSs of HG aluminium at $5.20$ TPa and HG sulfur at $500$ GPa (where the 64-atom $Pcca$ approximant was used to model the HG phase). It can clearly be seen that a drastic drop in the Fermi level eDOS that is present in Al does not occur for S, where the eDOS at $E_F$ is in fact relatively flat.

\begin{figure}[h!]
    \hspace{-0.3cm}
    \includegraphics[width=0.45\textwidth]{supplemental/eDOS_Im-3m_vs_Pcca.pdf}

    \vspace{0.35cm}

    \includegraphics[width=0.45\textwidth]{supplemental/eDOS_Al16_vs_Pcca.pdf}
    \caption{Electronic Densities of States (eDOS). \textbf{Top:} For the 64-atom $Pcca$ approximant and $Im\bar{3}m$ phase at $500$ GPa. \textbf{Bottom:} For HG Aluminium at $5.20$ TPa and HG sulfur at $500$ GPa (64-atom $Pcca$ approximant) - These pressures roughly correspond to the midpoint of the respective HG stability ranges.}
    \label{eDOSs}
\end{figure}

\section{\label{eos_discos}Volume Discontinuity}

Supplemental Fig. \ref{volume_discontinuities} shows that the $R\bar{3}m \rightarrow Fdd2$ and HG $\rightarrow Im\bar{3}m$ transitions introduce volume discontinuities into the equation of state of high-pressure sulfur. The fractional volume change $\frac{|\Delta V|}{V_0}$ is $ \approx 0.71 \%$ upon transition to the $Fdd2$ phase, and $ \approx 0.99 \%$ upon transition to the $Im\bar{3}m$ phase. The magnitude of the volume change at the $Fdd2 \rightarrow$ HG transition (not shown here) was less than $0.01 \%$.

\begin{figure}[h!]
    \includegraphics[width=0.45\textwidth]{supplemental/R-3m_to_Fdd2_vol_disco.pdf}

    \vspace{0.5cm}

    \hspace{0.05cm}
    \includegraphics[width=0.45\textwidth]{supplemental/P4ncc_to_Im-3m_vol_disco.pdf}

    \caption{Volume discontinuities with pressure for the $R\bar{3}m \rightarrow Fdd2$ transition (top) and the HG $\rightarrow Im\bar{3}m$ transition (bottom).}
    \label{volume_discontinuities}
\end{figure}

\clearpage

\section{\label{diffraction} Powder Diffraction Patterns}

\begin{figure}[h!]
    \includegraphics[width=0.43\textwidth]{supplemental/Fdd2_275GPa_powder_diffraction.png}

    \vspace{0.4cm}

    \includegraphics[width=0.43\textwidth]{supplemental/C2c_275GPa_powder_diffraction.png}

    \vspace{0.4cm}

    \includegraphics[width=0.43\textwidth]{supplemental/R-3m_250GPa_powder_diffraction.png}

    \caption{Simulated powder diffraction patterns for the $Fdd2$ phase at $275$ GPa (top), the $C2/c$ ABC-ordered HG approximant at $275$ GPa (middle) and $R\bar{3}m$ phase at 250 GPa (bottom). The incident radiation wavelength was set to $0.4$ \AA.}
    \label{powder_diffraction_patterns}
\end{figure}

Supplemental Fig. \ref{powder_diffraction_patterns} shows simulated powder diffraction patterns for the $Fdd2$, $C2/c$ approximant ($\gamma = 4/3$) and $R\bar{3}m$ ($\beta$-Po) phases. At $275$ GPa, the $Fdd2$ phase should be the ground state structure. However, we present diffraction patterns for both the $C2/c$ approximant \textbf{\textit{and}} the $Fdd2$ structure at $275$ GPa, in order to assist experimentalists in identifying any difference between the diffraction patterns of these two (very) similar structures.
\\\\
We note that the most significant difference is the absence of a peak at $2 \theta \sim 14.1 ^{\circ}$ in the pattern of the $C2/c$ approximant, which is present in the $Fdd2$ pattern. In fact, there are several peaks present in the $Fdd2$ pattern that are missing in the $C2/c$. We also note that the relative heights of a few peaks are changed between the two patterns. We expect the real host-guest diffraction pattern (to which the $C2/c$ is an approximation) to possess satellite peaks, which will not be present on the pattern presented here.

\clearpage

\section{\label{Al_and_As}Aluminium and Arsenic Phase Diagrams}

In the manuscript, we mention that the $Fdd2$ structure does not become the ground state in other elements that possess high-pressure HG phases. In Supplemental Fig. \ref{Al_and_As_phase_diagram} below, we present, as examples, the phase diagrams of Al and As with the $Fdd2$ structure added in.

\begin{figure}[h!]
    \hspace{-0.45cm}
    \includegraphics[width=0.46\textwidth]{supplemental/aluminium_phase_diagram.pdf}

    \vspace{0.5cm}

    \hspace{-0.45cm}
    \includegraphics[width=0.46\textwidth]{supplemental/arsenic_phase_diagram.pdf}

    \caption{Phase diagrams of aluminium in the terapascal regime (top) and arsenic between $20$-$100$ GPa (bottom). For Al, the $C2/c$ approximant was used to represent the HG phase (as per Ref. \cite{HG_Aluminium}), and for As, the 32-atom $P4/ncc$ approximant was used (as per Ref. \cite{Haussermann_2002}).}
    \label{Al_and_As_phase_diagram}
\end{figure}

Whilst certainly competitive, the $Fdd2$ structure does not become the ground state in either of these cases.

\newpage

\section{\label{TC} Superconducting Critical Temperature Calculations}

We used the plane-wave DFT code \verb|QUANTUM ESPRESSO| \cite{QE} for electron-phonon coupling calculations and subsequent calculation of the superconducting critical temperature $T_c$ within Migdal-Eliashberg theory and using the McMillan-Allen-Dynes formula \cite{McMillan_Allen_Dynes}. Density Functional Perturbation Theory (DFPT) was used for all phonon calculations.
\\\\
A crucial consideration in accurate $T_c$ calculations is the choice of the double-delta smearing parameter $\sigma$ in the calculation of the Eliashberg function:

\begin{align}
    \alpha^2 F(\omega) = \frac{1}{g(\epsilon_F)} \sum_{m,n} \sum_{\vec{q},\nu} \delta(\omega - \omega_{\vec{q},\nu}) \sum_{\vec{k}} |g^{\vec{q},\nu,m,n}_{\vec{k}+\vec{q},\vec{k}}|^2
    \notag\\
    \times \ \delta(\epsilon_{\vec{k}+\vec{q},m}-\epsilon_F) \  \delta(\epsilon_{\vec{k},n}-\epsilon_F)
\label{eliashberg}
\end{align}

Where the delta functions are replaced by Gaussians of width $\sigma$:

\begin{align}
    \delta(\epsilon_{\vec{q},\nu}-\epsilon_F) \rightarrow \mathrm{exp} \bigg( \frac{(\epsilon_{\vec{q},\nu} - \epsilon_F)^2}{\sigma^2} \bigg)
\label{gaussian}
\end{align}

This replacement is necessary, as Eqn. \ref{eliashberg} can only be used as-is in the infinite \textbf{q}- and \textbf{k}- kpoint sampling limit. The use of finite \textbf{q}- and \textbf{k}- point grids in electronic structure calculations necessitates the replacement of the delta functions by Gaussians.
\\\\
Careful selection of the $\sigma$ value is very important in elemental systems, where $T_c$ values are generally of order $10$ K, and thus an error of a few Kelvin represents a large relative error. The optimal value of $\sigma$ can be determined by calculating $T_c(\sigma)$ for two similarly-sized \textbf{k}-point grids and finding the point at which the critical temperatures of the two grids start to diverge. It is a necessary requirement that the \textbf{k}-point grid be an integer multiple of the \textbf{q}-point grid. As an example, if we use a 6x6x6 \textbf{q}-point grid, we might calculate $T_c(\sigma)$ for a 24x24x24 and a 30x30x30 $k$-point grid, corresponding to $kpq$ (`\textbf{k}-points per \textbf{q}-point') values of 4 and 5, respectively. This procedure is described in detail in  ref. \cite{niobium_Tc}, and we show an example of such a $T_c$ calculation for the $Fdd2$ structure in Supplemental Fig. \ref{tc_method} below.

\begin{figure}[h!]
    \includegraphics[width=0.45\textwidth]{supplemental/tc_method_smearing.pdf}

    \caption{$T_c$ of the $Fdd2$ structure at $271$ GPa as a function of smearing width $\sigma$ for two \textbf{k}-point grids - one with $4$ \textbf{k}-points per \textbf{q}-point, and one with $5$ \textbf{k}-points per \textbf{q}-point. A green arrow indicates the point where the two curves start to diverge, and the green dotted line shows that the ideal smearing width is $\sim{0.06}$ Rydberg in this case.}
    \label{tc_method}
\end{figure}

\newpage

\section{\label{structures} Structural Information}

We present (primitive) \verb|.cell| files for various structures below.
\\\\

\color{red}
$Fddd \ @ \ 200$ GPa
\color{black}
\begin{verbatim}

    ang
      0.0000000   3.0716500   7.5560978
      2.8136036   0.0000000   7.5560978
      2.8136036   3.0716500   0.0000000

      S  0.581656103  0.791591252  0.050624316
      S  0.071197879  0.428802121  0.071197879
      S  0.397001150  0.397001150  0.102998850
      S  0.852998850  0.852998850  0.147001150
      S  0.178802121  0.821197879  0.178802121
      S  0.668343897  0.458408748  0.199375684
      S  0.102998850  0.102998850  0.397001150
      S  0.428802121  0.071197879  0.428802121
      S  0.673871671  0.199375684  0.458408748
      S  0.791591252  0.581656103  0.576128329
      S  0.050624316  0.576128329  0.581656103
      S  0.199375684  0.673871671  0.668343897
      S  0.458408748  0.668343897  0.673871671
      S  0.576128329  0.050624316  0.791591252
      S  0.821197879  0.178802121  0.821197879
      S  0.147001150  0.147001150  0.852998850

\end{verbatim}

\newpage

\color{red}
$R\bar{3}m \ @ \ 250$ GPa
\color{black}
\begin{verbatim}

    ang
    1.6211882   0.9359935   0.8405054
    -1.6211882   0.9359935   0.8405054
    0.0000000  -1.8719869   0.8405054

    S  0.500000000  0.500000000  0.500000000

\end{verbatim}

\color{red}
$Fdd2 \ @ \ 300$ GPa
\color{black}
\begin{verbatim}

    ang
      0.0000000   7.2469132   2.6760045
      2.9544756   0.0000000   2.6760045
      2.9544756   7.2469132   0.0000000

      S  0.460350598  0.792807744  0.055443387
      S  0.731418288  0.482390943  0.064087521
      S  0.058060264  0.051661061  0.071035871
      S  0.178964129  0.430757196  0.191939736
      S  0.430757196  0.178964129  0.198338939
      S  0.850284770  0.573046329  0.274747620
      S  0.573046329  0.850284770  0.301921281
      S  0.975252380  0.948078719  0.399715230
      S  0.558601729  0.194556613  0.457192256
      S  0.185912479  0.527896752  0.518581712
      S  0.948078719  0.975252380  0.676953671
      S  0.792807744  0.460350598  0.691398271
      S  0.482390943  0.731418288  0.722103248
      S  0.527896752  0.185912479  0.767609057
      S  0.194556613  0.558601729  0.789649402
      S  0.051661061  0.058060264  0.819242804

\end{verbatim}

\color{red}
$C2/c \ @ \ 400$ GPa
\color{black}
\begin{verbatim}

    ang
      2.8740970  -2.5844681   0.0000000
      2.8740970   2.5844681   0.0000000
     -2.5688715   0.0000000   7.0105305

      S  0.257215059  0.026326505  0.048178790
      S  0.578217142  0.431995613  0.119203117
      S  0.095583293  0.639743506  0.124195854
      S  0.709544949  0.927654278  0.230208302
      S  0.072345722  0.290455051  0.269791698
      S  0.360256494  0.904416707  0.375804146
      S  0.568004387  0.421782858  0.380796883
      S  0.973673495  0.742784941  0.451821210
      S  0.026326505  0.257215059  0.548178790
      S  0.431995613  0.578217142  0.619203117
      S  0.639743506  0.095583293  0.624195854
      S  0.927654278  0.709544949  0.730208302
      S  0.290455051  0.072345722  0.769791698
      S  0.904416707  0.360256494  0.875804146
      S  0.421782858  0.568004387  0.880796883
      S  0.742784941  0.973673495  0.951821210

\end{verbatim}

\color{red}
$Pcca \ @ \ 500$ GPa
\color{black}
\begin{verbatim}

    ang
     10.1310268   0.0000000   0.0000000
      0.0000000   5.0612578   0.0000000
      0.0000000   0.0000000   7.5490969

      S  0.921154366  0.141917384  0.001201932
      S  0.578845634  0.858082616  0.001201932
      S  0.178845563  0.342070917  0.004277600
      S  0.321154437  0.657929083  0.004277600
      S  0.750000000  0.000000000  0.027404472
      S  0.003489518  0.491437926  0.122581360
      S  0.496510482  0.508562074  0.122581360
      S  0.570435274  0.163949693  0.162606534
      S  0.929564726  0.836050307  0.162606534
      S  0.828512529  0.359431377  0.164791163
      S  0.671487471  0.640568623  0.164791163
      S  0.326611856  0.346111455  0.168767685
      S  0.173388144  0.653888545  0.168767685
      S  0.076817651  0.146729300  0.171665485
      S  0.423182349  0.853270700  0.171665485
      S  0.250000000  0.000000000  0.227082257
      S  0.750000000  0.000000000  0.272917743
      S  0.923182349  0.146729300  0.328334515
      S  0.576817651  0.853270700  0.328334515
      S  0.673388144  0.346111455  0.331232315
      S  0.826611856  0.653888545  0.331232315
      S  0.171487471  0.359431377  0.335208837
      S  0.328512529  0.640568623  0.335208837
      S  0.429564726  0.163949693  0.337393466
      S  0.070435274  0.836050307  0.337393466
      S  0.996510482  0.491437926  0.377418640
      S  0.503489518  0.508562074  0.377418640
      S  0.250000000  0.000000000  0.472595528
      S  0.821154437  0.342070917  0.495722400
      S  0.678845563  0.657929083  0.495722400
      S  0.078845634  0.141917384  0.498798068
      S  0.421154366  0.858082616  0.498798068
      S  0.578845634  0.141917384  0.501201932
      S  0.921154366  0.858082616  0.501201932
      S  0.321154437  0.342070917  0.504277600
      S  0.178845563  0.657929083  0.504277600
      S  0.750000000  0.000000000  0.527404472
      S  0.496510482  0.491437926  0.622581360
      S  0.003489518  0.508562074  0.622581360
      S  0.929564726  0.163949693  0.662606534
      S  0.570435274  0.836050307  0.662606534
      S  0.671487471  0.359431377  0.664791163
      S  0.828512529  0.640568623  0.664791163
      S  0.173388144  0.346111455  0.668767685
      S  0.326611856  0.653888545  0.668767685
      S  0.423182349  0.146729300  0.671665485
      S  0.076817651  0.853270700  0.671665485
      S  0.250000000  0.000000000  0.727082257
      S  0.750000000  0.000000000  0.772917743
      S  0.576817651  0.146729300  0.828334515
      S  0.923182349  0.853270700  0.828334515
      S  0.826611856  0.346111455  0.831232315
      S  0.673388144  0.653888545  0.831232315
      S  0.328512529  0.359431377  0.835208837
      S  0.171487471  0.640568623  0.835208837
      S  0.070435274  0.163949693  0.837393466
      S  0.429564726  0.836050307  0.837393466
      S  0.503489518  0.491437926  0.877418640
      S  0.996510482  0.508562074  0.877418640
      S  0.250000000  0.000000000  0.972595528
      S  0.678845563  0.342070917  0.995722400
      S  0.821154437  0.657929083  0.995722400
      S  0.421154366  0.141917384  0.998798068
      S  0.078845634  0.858082616  0.998798068

\end{verbatim}

\color{red}
$P4/ncc \ @ \ 650$ GPa
\color{black}
\begin{verbatim}

    ang
      4.9149353   0.0000000   0.0000000
      0.0000000   4.9149353   0.0000000
      0.0000000   0.0000000   7.3482019

      S  0.500000000  0.000000000  0.052805013
      S  0.154488689  0.145587719  0.083832564
      S  0.645587719  0.345511311  0.083832564
      S  0.354412281  0.654488689  0.083832564
      S  0.845511311  0.854412281  0.083832564
      S  0.000000000  0.500000000  0.193048082
      S  0.848729494  0.151270506  0.250000000
      S  0.348729494  0.348729494  0.250000000
      S  0.651270506  0.651270506  0.250000000
      S  0.151270506  0.848729494  0.250000000
      S  0.500000000  0.000000000  0.306951918
      S  0.145587719  0.154488689  0.416167436
      S  0.654488689  0.354412281  0.416167436
      S  0.345511311  0.645587719  0.416167436
      S  0.854412281  0.845511311  0.416167436
      S  0.000000000  0.500000000  0.447194987
      S  0.500000000  0.000000000  0.552805013
      S  0.845511311  0.145587719  0.583832564
      S  0.354412281  0.345511311  0.583832564
      S  0.645587719  0.654488689  0.583832564
      S  0.154488689  0.854412281  0.583832564
      S  0.000000000  0.500000000  0.693048082
      S  0.151270506  0.151270506  0.750000000
      S  0.651270506  0.348729494  0.750000000
      S  0.348729494  0.651270506  0.750000000
      S  0.848729494  0.848729494  0.750000000
      S  0.500000000  0.000000000  0.806951918
      S  0.854412281  0.154488689  0.916167436
      S  0.345511311  0.354412281  0.916167436
      S  0.654488689  0.645587719  0.916167436
      S  0.145587719  0.845511311  0.916167436
      S  0.000000000  0.500000000  0.947194987

\end{verbatim}

\newpage

\color{red}
$Im\bar{3}m \ @ \ 700$ GPa
\color{black}
\begin{verbatim}

    ang
     -1.1027680   1.1027680   1.1027680
      1.1027680  -1.1027680   1.1027680
      1.1027680   1.1027680  -1.1027680

      S  0.000000000  0.000000000  0.000000000

\end{verbatim}

redpajama_set_name: RedPajamaArXiv


text:
John Baird 'deeply skeptical' of Iran's intentions in nuclear deal
Jessica Barrett
The following is an excerpt. To view the full article please visit the link below.
Canada's "hawkish" reaction to Iran's historic deal with six world powers is out of step with the international community, experts say.
University of Waterloo political science professor Dr. Bessma Momani and former Canadian ambassador to Iran, John Mundy, said Sunday that Canada's hard-line stance on Iran shows we are lagging behind our allies.
"We really stand outside the consensus of the G7 on this issue," said Momani, also a professor at the Balsillie School of International Affairs.
Meanwhile Mundy, Canada's last ambassador in Iran, said the agreement is a "defeat" for Canada's foreign policy, which favours sanctions over negotiation.
The deal will see Iran...

redpajama_set_name: RedPajamaCommonCrawl


I've created a "reference" category for our Jnews site and marked 3 published posts as reference posts, however when I go the site and click on the reference category, only one of the three posts shows up. All 3 are marked as "reference" in the Jnews primary category section, and the "reference" category has been marked as the primary category in the WordPress category section as well (along with two other categories). All 3 articles look identically set up/tagged/categorised in the WordPress post editor, yet only one of them shows up in the reference category.
Could someone please advise why this is the case?
In this case, can you provide me your login access to check directly on our side? You can send it here and don't forget to check Set as private reply option below.
Thanks for getting back to me Ferdy.
This issue has (magically) resolved itself, however I don't think the root of the problem is gone. I don't think the site is updating to changes we make fast enough – it takes several hours, sometimes a day, until some changes come through, such as the above categories problem and more. As another example, if you look at my other recent post, you told me to set the post date setting from modified to published. While I did this, it hasn't updated on the site yet – I think it will eventually, but it's odd that it's not doing it immediately.
Is there some kind of problem or setting that would be causing the site to update this slowly? Maybe something to do with caching, especially server side or on the theme's part???
Just to add to this – the date will update on the post itself, but it won't be reflected on the homepage. Eg. If I change a post from February to January, the post itself shows the update January date but that's not reflected on the homepage either on the text of the date or the ordering of the posts.
It also did eventually update the posts in about an hour, but why did it take so long?? Shouldn't it be immediate.
I really need an update on this. I added 3 new posts last night and none of them have shown up on the site since then AT ALL. Why is it taking hours for the site to update settings and display posts? Is it a caching problem. I know it's to do with the JNews theme because if I live preview another theme that's not Jnews, everything is there.
In this case, can you provide me with login access to your website? it seems we need to do further investigation on your website. You can send it here and don't forget to check Set as private reply option below.
So wordpress support identified this error in the PHP of the Jnews theme. Is there any way to fix this without uninstalling and reinstalling the theme, as we would lose all the customisation we've done to the theme.
This reply was modified 2 months ago by digitalvix3n.
We have tried that with a fresh download of the theme zip file. Still have the same problem.
Regarding this issue, I will need to do further checking on your website. Can you provide me login access into your website and FTP access? you can send it here and don't forget to check Set as private reply option below.
It's hosted by wordpress so there's no access to FTP or no access to logs.


Eleazar, one of the scribes in high position, a man now advanced in age and of noble presence, was being forced to open his mouth to eat swine's flesh. But he, welcoming death with honor rather than life with pollution, went up to the rack of his own accord, spitting out the flesh, as all ought to go who have the courage to refuse things that it is not right to taste, even for the natural love of life.
Those who were in charge of that unlawful sacrifice took the man aside because of their long acquaintance with him, and privately urged him to bring meat of his own providing, proper for him to use, and to pretend that he was eating the flesh of the sacrificial meal that had been commanded by the king, so that by doing this he might be saved from death, and be treated kindly on account of his old friendship with them. But making a high resolve, worthy of his years and the dignity of his old age and the gray hairs that he had reached with distinction and his excellent life even from childhood, and moreover according to the holy God-given law, he declared himself quickly, telling them to send him to Hades.
So in this way he died, leaving in his death an example of nobility and a memorial of courage, not only to the young but to the great body of his nation.


text:
Sony will make a 5G phone, but only when the tech rolls out
By sysadmin on February 25, 2019 in Reviews
Think that Sony isn't interested in 5G just because the Sony Xperia 1 doesn't support it? Think again.
Sony is showing off a new prototype 5G device at MWC 2019, and the company has confirmed to TechRadar it won't be pursuing 5G in the consumer space until 5G is fully offered by mobile networks.
Instead of packing it inside a commercially available phone – think LG V50, Huawei Mate X or Samsung Galaxy S10 5G – the company is showing off a prototype that's never intended to come to market.
Patience at Sony
The device is being shown on both Sony and Qualcomm's booths at the show. The prototype is sporting Qualcomm's Snapdragon 855, which is the same chipset as inside the Xperia 1.
The Snapdragon 855 is capable of supporting 5G – that's the chipset inside the LG V50 – so why won't Sony's latest flagship support 5G?
Mitsuya Kishida, President of Sony Mobile told TechRadar, "Obviously, we are heavily cooperating with our partners to develop for 5G, and depending on the country and the customers need we will introduce the 5G model accordingly.
"It will be a separate model and depend on the country, again. How to introduce the solution is going to be different for our customers."
However, he did confirm that Sony was intending to have a device on the market when the 5G wave of phones fully begins.
Whether that means we'll see a variant of the Xperia 1 – much like how Xiaomi has plans to rereleased the Mi Mix 3 with a new 5G version – or a completely new phone remains to be seen.
It's also a benefit of Sony's schedule of releasing a new flagship phone every six months too.
"Being the first in 5G… that's not what we're trying to achieve."
Mitsuya Kishida, President Sony Mobile
By the time 5G begins its rollout later this year, Sony may be ready to introduce the Xperia 2 that is likely to still sport the Snapdragon 855 but may be ready with 5G compatibility.
Kishida continued, "Being the first in 5G… that's not what we're trying to achieve. I'm really happy that our engineering skill and the speed of development towards 5G, I'm very comfortable with that."
One of the other reasons for Sony avoiding 5G at this stage is also down to the effect it can have on the design of the phone too.
Silke Schild, head of external communications at Sony Mobile told TechRadar, "We don't want to compromise on the design [of the Xperia 1] either, integrating additional antennas etcetera."
That may mean the next phone from Sony is set to be thicker and heavier so it can support 5G, but we likely won't know that for quite a while now and it may be IFA 2019 at the earliest for us to next hear about a flagship Sony handset.
MWC (Mobile World Congress) is the world's largest showcase for the mobile industry, stuffed full of the newest phones, tablets, wearables and more. TechRadar is reporting live from Barcelona all week to bring you the very latest from the show floor. Head to our dedicated MWC 2019 hub to see all the new releases, along with TechRadar's world-class analysis and buying advice about your next phone.
Mobile phones are too fragile for European businesses
How to watch This Time with Alan Partridge online: stream from the UK or abroad


text:
Texas Legislature 2019
With major policy differences emerging, state leaders continue to project unity
Halfway through the legislative session, lawmakers say they're getting along. But they have a long way to go to achieve their goals this year.
by Emma Platoff and Cassandra Pollock March 18, 2019 12 AM Central
Gov. Greg Abbott, Lt. Gov. Dan Patrick and House Speaker Dennis Bonnen held a joint press conference on the lawn of the Governor's Mansion on Jan. 9. Credit: Miguel Gutierrez Jr. / The Texas Tribune
The three Republicans who lead Texas state government have made it clear: They know what the state's top problems are and they're working together to fix them now, during the brief window they get every two years. But resolutions on those consensus issues have — predictably — proved harder to settle on.
As the 140-day session marks its halfway point this week, must-do reforms to the state's property tax and school finance systems remain only partially baked. A property tax bill originally filed with identical language in both chambers remains in purgatory, with approval from the Senate's property tax committee but without a vote from the full body; in the House, it's yet to advance past the panel.
And the chambers are perhaps further apart on the second priority issue, school finance. The House is charging ahead on its version, but an as-yet unfinished Senate proposal takes a different tack.
But even with the different approaches the Texas Senate and House have adopted on those two issues — and despite the pace that's causing some at the Capitol to worry about getting it all done in time — Lt. Gov. Dan Patrick and House Speaker Dennis Bonnen, along with Gov. Greg Abbott, appear determined to keep the negotiations civil, at least in public. In the Legislature's recent past, personal feuds stymied policy priorities; this year, even with public displays of cooperation, the challenges remain steep.
Of course, for some seasoned political observers, the current dynamic at the Capitol is to be expected — even when lawmakers agree on the focus, the devil is always in the details. Still, others wonder: In a session billed as one big campfire sing-along, what's taking so long?
"Even though it sounds like we're halfway through, it just kind of shifts into a second gear," said Sen. Paul Bettencourt, the Houston Republican shepherding property tax reforms through the upper chamber. "The halfway point really isn't the halfway point. It's really when things start picking up."
The chambers have hardly stalled. A House committee is expected to vote on its budget proposal Monday. That budget bill, House Bill 1, could then hit the floor of the lower chamber by the end of March — the earliest House members have debated the only piece of legislation that's constitutionally required to pass in several years. The Senate, meanwhile, has already unanimously passed its supplemental budget, which would put roughly $6 billion toward leftover state expenses, such as Hurricane Harvey recovery, and work continues on its own budget proposal.
Still, the pace is notably more sluggish than the most recent legislative session, particularly in the Senate. By March 1, 2017, the upper chamber had passed all four of the governor's legislative priorities — along with a controversial anti-"sanctuary cities" bill that would prove among the most divisive measures of the legislative year. This year, that same milestone came and went without even a public unveiling of the lieutenant governor's priority bills. Those were announced a week later, in a Friday evening press release that included a host of priorities Patrick has made little mention of in recent months.
Patrick has said the upper chamber's slower start this year is by design — a chance for new leadership in the House to keep pace — but skeptics wonder if some of his priority measures simply don't have the support to move.
"We moved very fast in the Senate the last two sessions, pushing a lot of strong conservative legislation to the House because the former speaker had a different view on things," Patrick told Lubbock radio host Chad Hasty last month. "Now, it's not as important. ... It's more important that we try to work together. The House will lead on some issues; we'll lead on some issues. It's a different pace."
Still, with that cooperation ongoing, interchamber policy differences continue to dog Republican leaders, particularly on school finance — an issue that state leaders have long been promising to fix. After proposed overhauls ended in stalemate during the 2017 legislative sessions, a group of legislators appointed by Abbott, Patrick and former House Speaker Joe Straus spent months studying ways to improve the system as part of a school finance commission.
That panel — including House Public Education Chairman Dan Huberty, R-Houston, and Senate Education Chairman Larry Taylor, R-Friendswood — released more than 30 recommendations aimed at addressing deep flaws in the state's public education system. Now, the chambers have unveiled proposals that are at odds in significant ways.
Surrounded by dozens of his House colleagues, Huberty unveiled a roughly 200-page bill earlier this month that would put $9 billion toward public schools and property tax relief. Days later, amid a Friday evening filing deadline, his Senate counterpart, Taylor, filed an incomplete version of his own bill. Unlike the House proposal, the Senate bill would institute some type of outcomes-based funding for school districts, paying them more based on how well third-graders perform academically. The House has proposed compressing property tax rates by 4 cents, meaning lower bills for homeowners; such a proposal doesn't appear in the Senate version, at least in its current form.
Beyond that, the Senate's school finance proposal is difficult to scrutinize, as it's not finished. Bettencourt said he and Taylor are still working on the meat of the proposal, a pitch for how to overhaul school district property taxes, which constitute a significant chunk of homeowners' property tax bills. Until that takes shape, the tax chairman added, he's not whipping the Senate floor for a vote on the property tax bill he filed in January.
Perhaps the most notable break between the chambers is over teacher pay. All three state leaders have made clear that it's a priority, but their approaches have differed: While Patrick has pushed heavily for a $5,000 across-the-board teacher pay raise — a measure that passed the Senate unanimously — Bonnen has suggested that he would prefer to let school districts decide how to best use their education dollars. Those differences sharpened earlier this month, when the House unveiled its school finance bill that instead proposed tackling the issue by raising minimum salaries for educators and increasing health and pension benefits, among other things.
Asked about the Senate's teacher raise proposal, Bonnen said, "I don't know how you call a $5,000, across-the-board teacher pay raise" a "plan."
"What we have is a plan," Bonnen continued. "I think teachers are some of the smartest people in Texas, and they are going to figure out that the Texas House has a winning plan for the teachers and students in Texas."
Through a spokesperson, Patrick acknowledged at the time that the two chambers "have taken different approaches" on the issue but emphasized that both remain focused on property tax and school finance reform.
Last session, such stark policy differences might have devolved into interchamber finger-pointing. The 85th Legislature was defined by a showdown between Patrick and Straus, the former House speaker whom Patrick blasted as uncooperative and insufficiently conservative. Amid a policy stalemate, Patrick ended the legislative year by slamming Straus in the most insulting Texas terms.
"Thank goodness Travis didn't have the speaker at the Alamo," Patrick said. "He might have been the first one over the wall."
But this year, with a new speaker whom those on the right and left are still hesitant to criticize, leaders have taken care to give off the impression that negotiations remain on the rails. At least publicly, the "Big Three" have maintained the unified front that kicked off the session with joint press conferences hailed as historic.
Bonnen, in an effort to keep negotiations running smoothly, told his top House lieutenants at a weekly lunch meeting last Monday to avoid publicly disparaging the upper chamber over differences on school finance. Bonnen's message during that meeting, according to several people who heard the speaker's comments, was clear: Let's not poke the bear in the Senate.
Patrick, meanwhile, told conservative activists in North Texas later that day that his "No. 1 goal this session" is to ensure Bonnen is re-elected as speaker in 2021.
"I love it," Bonnen told Hasty, the Lubbock radio host, in February. "We will disagree at some point, but we're not going to get on your radio show … and talk about those disagreements."
In a separate interview with Hasty, Patrick had made much the same point: "If we disagree, we're gonna disagree without being disagreeable," he said.
But even if everyone's playing nice, the thorny issues that have stymied lawmakers in sessions past aren't getting easier to resolve. And the pressure is only mounting as they near the finish line — the official end, or "sine die," to the session on May 27.
The pace beneath the Capitol dome has some insiders quietly questioning whether the 140-day session will be long enough or whether the governor might need to call lawmakers back to Austin after May for additional work. For their part, state leaders remain confident the issues will be addressed this legislative year — no matter how long it takes.
"They have to be passed. If they have to be passed in a special [session], they have to be passed in a special," Bettencourt said. "I'm optimistic that we won't have to do that. But I also told my landlord I need the place."


Android Wear, Google's slowly-but-steadily improving wearables platform, will be receiving an update in the near future with a pretty useful feature. On Thursday, Google announced a new Android Wear SDK version as well as the newest latest version of Android Wear, Android Wear 2.9.0.
Android Wear 2.9.0 adds an Android Oreo-like notification dot, which was optional before, to the bottom of first- and third-party watch faces. The dot-shaped icon is white by default, but watch face creators and developers can customize its color and the color of the ring around it via WatchFaceStyle.setAccentColor.
Alternatively, they can switch off the notification dot and replace it with a counter showing the total number of unread notifications, which will appear at top of the watch face in the system tray. Or if they're feeling adventurous, they can grab the number of unread notifications using WatchFaceStyle.getUnreadCount and tinker with the font and style.
In addition to the new unread notification indicators, the latest Android Wear SDK improves the ComplicationDrawable class—the styleable drawable object that governs the behavior of watch face complications—that was introduced at Google I/O 2017.
Starting with Wear SDK 2.2.0, complications that lack the necessary permissions to display content (for example, a weather widget that doesn't request access to the internet) automatically launch permission requests in the onTap method.
The other big change has to do with screen redraws. Previously, Google recommended that watch faces with complications containing images or icons redraw the screen every second so that users aren't left waiting while they loaded. But that doesn't make a lot of sense for watch faces that update once per minute. Google's solution is a new callback in ComplicationDrawable, Drawable.Callback, that redraws the watch face when images have loaded.
You can preview Android Wear 2.9.0 by using the new Android Oreo-based emulator, but you should beware of a few bugs. Google says that activating Theater Mode in the emulator can cause it to remain stuck until you clear the emulator data, and within the window of emulators for API levels 25 and 26, the power button doesn't work. In addition, the China version of the emulator sometimes begins flashing if you use the handwriting input method, and if you click the button for the on-screen keyboard, the keyboard blocks half the screen.


text:
id: 238
title: 'A ConfigMgr Admin's Guide to WOL Magic Packet: Part Three'
date: 2013-12-15T19:22:12+00:00
author: Ryan Ephgrave
layout: post
guid: https://www.ephingadmin.com/?p=238
permalink: /a-configmgr-admins-guide-to-wol-magic-packet-part-three/
categories:
  - ConfigMgr
  - WOL
tags:
  - BIOS
  - ConfigMgr
  - Magic Packet
  - WOL
---
If you follow Part One and Part Two of this guide, you should have a very high success rate with WOL in your environment. If you don't, this guide will go through a number of gotchas and troubleshooting steps you can follow to get a better success rate.

Create a test package

I suggest making a collection with your computers that refuse to wake up and a test package to deploy to this collection. For the test package, I suggest making a standard program package and the command line simply should be cmd /c. This way you can deploy the package to these computers as required and not have to worry about changes being made. To re-send packets, you can either delete the deployment and re-create it, or change the deadline of the deployment to a minute into the future.

Maintenance Windows

WOL Packets are sent when a deployment deadline is reached, regardless of maintenance windows. So if you want to use WOL with your deployments, make sure the deadline is set inside your maintenance window or your computers might go back to sleep before the maintenance window hits.

Log Files

If you want to make sure the ConfigMgr server is sending those packets, you will want to check the log files. WolCmgr.log will tell you which clients need to be sent packets and the number of packets sent / retried. Wolmgr.log will tell you when it will wake up the computers.

Are The Packets Sent:

If you really want to be sure the packets are being sent, and want to make sure they are going to the right MAC address, use a WOL sniffer on the server. The wol packet sniffer I use is by ProfShutdown and you can get it here (last link on page).  Run this program as admin, and you will see this screen:

Whenever a magic packet is sent, you will see all the information on the packet in this screen:

If you know which computer isn't waking up, you can copy and paste this information into notepad and do a ctrl+f for that MAC address. If the MAC address is listed, you know the packet is being sent. If it isn't listed, make sure there is a Hardware Inventory on that computer with the correct MAC address.

Are The Packets Received:

Sometimes the packets are not being received by the computer, and this is the reason why it isn't waking up. Go to the computer with the problem, turn it on, and log in. Open the WOL Sniffer program I had you download earlier as an admin, and then send packets to that computer. If you see the packet on the computer, you know the packets can get to the computer, so you can generally rule out a network problem. If you are using Unicast, remember the packet won't be sent to the computer if the router's ARP cache is cleared. These are generally cleared on a schedule, and the default is usually 30 minutes. Talk to your networking guys about increasing the time between cache clears, or switch to subnet directed broadcasts.

Is the NIC On:

If there are no link lights on the back of the computer, where the network cord is plugged in, then your NIC is probably not getting any power. Because of this, it won't be able to turn on the computer when packets are sent. Check BIOS and / or Windows to see which is turning off the NIC. The next trick will tell you how to narrow down which is causing the problem.

Is This A Windows Problem or BIOS Problem:

One interesting trick you can do is power off the computer, and then power it back on going into BIOS. Sometimes, Windows turns the NIC off when the computer is powered down, and this action will turn back on the NIC. Now, power off the computer (make sure it doesn't boot into Windows) and try to wake up the computer. If it wakes up, and it never woke up before, you are probably looking at a Windows setting problem or a driver bug. If it doesn't wake up, check the BIOS. make sure all the settings are correct, and if they are look into updating or downgrading the BIOS.

If you have any other troubleshooting tips, please let me know in the comments!

redpajama_set_name: RedPajamaGithub


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