2022

[1]Eva Zurek

1]Department of Chemistry, State University of New York at Buffalo, 777
Natural Sciences Complex, Buffalo, 14260-3000, NY, USA

# Crystal Chemistry at High Pressure

Katerina P. Hilleke<EMAIL_ADDRESS><EMAIL_ADDRESS>[

##  1 Introduction

Jupiter, the largest planet in our solar system, is a gas giant comprised
mostly of hydrogen and helium. After penetrating its atmosphere, a mixture of
hydrogen and helium, methane and ammonia, one would find a massive sea of
hydrogen – behaving very differently than one might expect from general
chemistry classes Guillot:2005 . The hydrogen layer surrounding Jupiter’s
core, under immense pressure and at high temperature, is metallic. This is
just one example of how the pressure variable is so important in determining
chemical and physical behavior, bearing consequences for the evolution and
dynamics of planetary interiors Zaghoo:2017a .

Under pressure, the behavior of the elements as well as the compounds they
form can change drastically from what we, as beings living and learning at 1
atmosphere, are used to. Hydrogen, as described above, can become metallic at
high pressure, joining its Group I cousins the alkali metals. It may be
perplexing then to learn that the alkali metals lithium Matsuoka:2009  and
sodium Ma:2009 ; Polsin:2022  become semiconducting and insulating,
respectively, when squeezed. Potassium starts to behave more like a transition
metal Bukowinski:1976 ; Atou:1996 ; Parker:1996 . When combined with one
another, the unexpected behavior of the elements under pressure makes for
correspondingly curious compounds. Sodium chlorides with vastly divergent
stoichiometries from the typical 1:1 (Figure 1) have been predicted and
synthesized Zhang:2013 , stubbornly inert helium atoms form a compound with
sodium Dong:2017 , and metal superhydrides wherein the hydrogen atoms coalesce
into clathrate-like networks have been reported geballe2018synthesis . Various
“rules” for the behavior of materials at high pressure have been proposed by
Prewitt and Downs Prewitt:1998 , Grochala and co-workers Grochala:2007a , and
Zhang _et. al._ Zhang:2017 , to name a few – and have undergone progressive
revision as more is uncovered about the structures and materials far below us
in the Earth’s core and far away in the planets and stars.

Figure 1: The familiar crystal structure of NaCl, table salt, (a) is the only
known stable compound of sodium and chlorine at ambient pressure. Under
pressures exerted by a diamond anvil cell (b), a variety of additional
stoichiometries and crystal structures decorate the sodium-chlorine phase
diagram, including the $P4/mmm$ Na3Cl, $P4/m$ Na3Cl2, $Imma$ NaCl2, and
$Pm\bar{3}$ NaCl7 Dong:2017  structures shown here.

Over the past century, experimental methods have evolved to create
progressively higher pressures in a laboratory setting, allowing us to
directly probe the behavior of materials under extreme conditions. Pressures
in the megabar range can now be routinely achieved, albeit still requiring
delicate setups Jayaraman:1983 ; Bassett:2009 ; Li:2018c . Diamond anvil cells
(DACs) combine the superior hardness of this desired polymorph of carbon with
its optical transparency, enabling the creation of the highest static
pressures and interrogation of the sample via visual and spectroscopic means.
Engineering advances from multistage compression apparatuses Dubrovinsky:2012
; Dubrovinskaia:2016  to toroidal DACs Dewaele:2018 ; Jenei:2018  have driven
the experimental ceiling for static pressures ever higher. Dynamic compression
experiments under ramp or shock conditions, driven by gas guns, lases pulses,
or magnetic fields can reach well into the terapascal regime Duffy:2019 ;
Hansen:2021 ; Fratanduono:2021 , allowing us to study the behavior of diamond
at 5 TPa Smith:2014  and iron at conditions thought to be in super-Earths
Kraus:2022a . Diagnostic techniques for characterizing the resulting
substances can be difficult to implement in both dynamic and static
compression, often requiring theoretical support for their interpretation
Zurek:2014i .

Far cheaper than high-pressure experiments are computations modeling high-
pressure systems. Band structures, phonons, Raman or infra-red spectra and
numerous material properties can all be calculated without so much as stepping
foot into a laboratory. Crystal structure prediction (CSP) techniques, not
weighted down by preconceived, atmospheric pressure-based, notions of how
atoms might appropriately arrange themselves in a unit cell, can identify
which structures can exist under high-pressure conditions Pickard:2009 ;
Oganov:2011 ; Wang:2014 . The computational exploration of potential energy
landscapes will not be discussed in our contribution; to learn about methods
that can be employed to identify the global, as well as important local,
minima we point the reader to an excellent chapter in this book “Crystal
Structure Prediction” by Andreas Hermann, Lewis J. Conway and Chris J.
Pickard. Exploratory calculations highlight promising phases for further
experimental investigation, but computation can just as well follow
experimental results, elucidating behavior and filling in the gaps. The
resulting feedback loop of experiment and theory has driven the discovery and
characterization of a plethora of phases ranging from the superhard
Oganov:2009 ; Zhang:2015b ; Avery:2019  to the superconducting Liu:2017-La-Y ;
Peng:2017 ; Kruglov:2018 ; geballe2018synthesis ; drozdov2019superconductivity
; Semenok:2021b .

In the following sections we build a framework for understanding the behavior
of materials at high pressure, starting from the effects of pressure on the
atoms themselves, driving electronic transitions and altering periodic trends.
From there, the various manifestations of high pressure in the solid state are
sorted into categories (which are not mutually exclusive, but rather
illustrative), starting with exotic electronic structures and electrides. We
discuss compounds of the noble gases and those containing elements that are
immiscible at ambient pressure, as well as crystal lattices that contain
bizarre geometrical motifs and bonding configurations. Finally, we survey the
effects of high pressure on superconductivity, a field that has recently
undergone a veritable explosion as high pressure phases toe the line of room-
temperature superconductivity Zurek:2020k .

##  2 The atom under pressure

Chemistry describes the interactions between the 118 distinct elements that
are organized in the periodic table, proposed by Mendeleev while classifying
elements according to their chemical properties observed at atmospheric
conditions. The trends found within the periodic table can be used to compare
atoms according to their size, the number of electrons surrounding their
nuclei, and to make predictions as to whether those electrons are held tightly
or loosely. Moreover, the periodic table allows students and researchers to
predict how different elements will interact with one another: will they form
compounds, emulsions or alloys, or will they be unreactive? If reactivity is
suspected, the periodic table can be used to guess if the bonds are covalent,
ionic, or (usually) somewhere in between. Across the periodic table, trends in
properties such as atomic radii, electronegativity, and oxidation state can be
mapped, leading one to conclude that fluorine, the most electronegative
element, will gain electrons in a binary phase thereby achieving an F-
configuration with a filled valence shell. On the other hand, cesium, as the
least electronegative element (neglecting francium, whose miniscule half-life
renders it basically experimentally irrelevant), typically assumes an
oxidation state of 0 or +1.111When dissolved in amines or ethers, Na, K, Rb,
and Cs can assume the unusual oxidation state of -1. Dye:2009  These so-called
_alkalides_ are thought to form ion-pairs with solvated metal cations.
Zurek:2011c ; Zurek:2020h ; Abella:2021  Yet at high pressure, several LinCs
phases have been predicted Botana:2014  where Cs attains unusual formal
oxidation states thought to be in excess of -2 due to substantial electron
transfer from lithium to cesium. How can the stability of these unintuitive
stoichiometries and their resulting electronic structures be rationalized?

Let us consider electronegativity a little more. Although Pauling’s
Pauling:1932  is the most widely adopted, a number of metrics have been used
to produce scales of electronegativity for the elements. In Pauling’s
formulation, electronegativity differences between pairs of atoms A and B are
calculated from the homo- and heteronuclear bond dissociation energies, then
referenced against the electronegativity of H being set to 2.1 (later adjusted
to 2.2). In this regime, the electronegativities of fluorine, lithium, and
cesium are 3.98, 0.98, and 0.79, respectively. Several other scales have been
proposed, including Mulliken’s “absolute electronegativity” Pearson:1985
being the average of the first ionization energy and electron affinity of an
atom Mulliken:1934 . Dong _et al._ modify the Mulliken definition for elements
under high pressure taking as a reference the homogeneous electron gas rather
than the vacuum Dong:2022 . Allen’s electronegativities are derived from the
average energies of the valence electrons in the atom Allen:1989 , and a
closely related scheme has recently been proposed by Rahm et al Rahm:2019 ;
Rahm:2021 ; Racioppi:2021 , where electronegativity is calculated as the
average of the electron binding energies of ground state valence electrons –
approximately translated to the Allen scale if averaging over valence
electrons alone. Broadly speaking, the common factor of importance in all of
these is the collection of atomic orbital energies and the differences between
them. Under pressure, those change.

The prototypical model for understanding the quantized behavior of energy
levels in a confined system is the particle in a box. The resulting energy
levels for a particle of mass $m$ in a box of width $L$ are given by
$E_{n}=\frac{\hbar^{2}n^{2}\pi^{2}}{2mL^{2}}$, where $n$ is the principal
quantum number. To consider the effects of pressure on this model, we could
simply make the box smaller by reducing the width $L$, which has the effect of
increasing the $E_{n}$. Thus, energy levels (orbital energies) will increase
under pressure. The complicating factor is that the rate of this increase is
not the same for each orbital because the number of radial nodes plays a role.
The peak density of a 4s orbital is further from the nucleus than a 4p
orbital, since the electrons occupying the 4s orbital must maintain
orthogonality to those in the 1s, 2s, and 3s orbitals, while those in the 4p
orbital must only contend with the 3p and 2p orbitals (this analogy may be
extended to the 4d and 4f shells). With more electron density being further
from the nucleus, the electrons in the 4s orbital will feel the effects of
pressure more strongly than those in the 4p orbital. This is illustrated
schematically in Figure 2a and 2b. At certain levels of confinement,
electronic $n\textrm{s}\rightarrow n\textrm{p}$,
$n\textrm{s}\xrightarrow{}(n-1)\textrm{d}$, and
$(n-1)\textrm{d}\xrightarrow{}(n-2)\textrm{f}$ transitions can become
favorable. This is the reason why pressure drives rearrangements of the
orbital energies of an atom, with ensuing electronic transitions. (In section
3.2, we will explore the opportunities presented by another sort of
destination orbital, this time one not centered on an atom.)

In compressed lithium, the ground state electronic configuration transitions
from $1\textrm{s}^{2}2\textrm{s}^{1}\rightarrow
1\textrm{s}^{2}2\textrm{p}^{1}$, while cesium undergoes a
$6\textrm{s}^{1}\rightarrow 5\textrm{d}^{1}$ pressure induced transformation.
The exact pressures at which these electronic changes occur depend, of course,
on the chemical environment of the lithium or cesium atoms – and in
calculations, on the theoretical methodology. In Cs, the
$\textrm{s}\rightarrow\textrm{d}$ transition is thought to drive the
transformation to the complex Cs-III structure with 84 atoms in the unit cell
that is stable near 4.2 GPa Sternheimer:1950 ; Louie:1974 ; Takemura:1982 ;
Connerade:2000 ; Rahm:2019b ; Zurek:2005a , while in lithium the pressure at
which the s-p mixing occurs is believed to be somewhat higher, in the megabar
to multimegabar range, dependent on the chemical environment and level of
theory Neaton:1999 ; Hanfland:2000 ; Naumov:2015 ; Rahm:2019b . Another
example includes potassium, whose complex phase diagram under pressure (see,
_e.g._ Figure 2c,d), has been in part attributed to this s $\rightarrow$ d
electronic transition. Degtyareva:2014  The energies of core orbitals can also
become relevant; for example, in Cs-VI the 5p bands hybridize with the 6s
Takemura:2000 ; McMahan:1984 , making them accessible for chemical
interactions, and the core orbitals of K have also been proposed to be key to
its structural diversity. Degtyareva:2014  Overall, one effect of pressure on
the alkali metals is that their electronegativities undergo quite a
rearrangement, ending up with cesium being more electronegative than lithium
Rahm:2019b ; Dong:2022 . From this perspective, the formation of cesium anions
in the LinCs phases begins to make sense.

Figure 2: External pressure on an atom raises the energy of atomic orbitals as
they are constrained to a smaller space, but the rate of this increase differs
between s, p, and d orbitals (a), (b), favoring s $\rightarrow$ d transitions
for many of the alkali metals, including potassium. Potassium adopts the $bcc$
structure at 0 GPa (c) but at higher pressures takes on a series of complex
structures including the incommensurately modulated host-guest _t_ I19 phase
McMahon:2006b  (d).

Among the $d$-block similar reorderings are predicted to take place, with the
group 10 metals Ni, Pd, and Pt preferring $d^{10}$ closed-shell
configurations, as compared to the $s^{2}d^{8}$ favored at ambient conditions,
while the group 11 and 12 metals become electron donors Dong:2022 . The former
transition is associated with a spike in the estimated chemical hardness –
obtained as half of the HOMO-LUMO gap – of Ni, Pd, and Pt, reaching values
comparable to some of the noble gases. This is contrary to the general trend
where the hardness of most of the elements in the periodic table decreases
with pressure as energy levels become closer to one another Dong:2022 . The
resulting changes in the relative hardness of pairs of elements can lead to
changes in compound stability arising from HSAB (hard-soft acid-base)
arguments, and the appearance of strange multicenter bonding manifolds in
certain high pressure phases have been linked to general increases in softness
Hooper:2012 .

From transition-metal-like behavior in the _s_ -block to relative inertness in
the formerly-$d^{8}$-transition metals, atoms under compression can behave
very differently from their ambient-pressure selves, and the consequences for
materials under pressure are far-reaching. We will now explore some of the
wild and wonderful structures and phenomena that result.

##  3 The crystal under pressure

At ambient pressure a majority of the solid, metallic elements of the periodic
table adopt very simple, symmetric, structures that are close-packed. The most
stable geometries are those that minimize the free energy. However, at low
temperatures the entropic contributions between different solid phases are
typically negligible, so computational studies often employ the enthalpy to
determine the structures that are preferred. With increasing pressure the
enthalpy, consisting of the internal energy and pressure-volume terms
($H=U+PV$), becomes dominated by the $PV$ term. It would be natural,
therefore, to imagine that close-packed structures with increased coordination
numbers become preferred at high pressures. The reality, however, does not
coincide with our expectations. For example, within cesium the nearest
neighbor coordination number first increases from 8 (bcc) to 12 (fcc), then
decreases to about 10 (Cs-III), 8 (Cs-IV) and finally increases again to 10/11
(Cs-V) and 12 (Cs-VI). These structural transitions are believed to be driven
by the previously discussed pressure-induced $s\rightarrow d$ valence
electronic transition within the constituent atoms, which causes the
interatomic distances to become smaller compared to the ranges of the
wavefunction Sternheimer:1950 .

Moving beyond elemental crystal structures, several compounds with seemingly
bizarre stoichiometries (at least from the perspective of minds that
experience a 1 atmosphere reality) have been predicted and/or synthesized
under pressure. The familiar combination of sodium and chlorine, table salt
and prototypical ionic compound with a 1:1 ratio, is not the only stable
crystalline structure in the Na-Cl phase diagram. At least two unique
stoichiometries, Na3Cl and NaCl3, were synthesized, and several others were
predicted to become stable when squeezed Zhang:2013 . Noble gases xenon,
argon, and helium are active components of solid compounds that have been
synthesized Dewaele:2016b ; Stavrou:2018 ; Dong:2017 , and _very_ hydrogen-
rich compounds such as YH9, LaH10, and CaH6 that are high-temperature
superconductors with superconducting critical temperatures ($T_{c}$s)
approaching room temperature have been made Snider:2021 ; geballe2018synthesis
; drozdov2019superconductivity ; Somayazulu:2019-La ; Ma:2022 ; Li:2022 .
These metal superhydrides are materials-by-design success stories inspired by
theoretical predictions Wang:2012 ; Liu:2017-La-Y ; Peng:2017 .

In the following sections, we explore the plethora of exciting materials that
can be created using high pressure, with all their intriguing structural and
behavioral phenomena. While their stability and existence can be traced to the
pressure driven electronic rearrangements of the constituent atoms described
in Section 2, the manifestations of these rearrangements can take many forms.
Below, we describe a variety of illustrative phases sorted into a series of
categories, but by necessity these categories will, at times, overlap.
Nevertheless, all exemplify the ramifications of high pressure on solid-state
chemistry.

###  3.1 Electronic structure

At atmospheric pressure, the stoichiometries of many inorganic solid-state
compounds can be predicted from the most common oxidation states of their
constituent elements. Usually, alkali metals and alkaline earth metals possess
oxidation states of +1 and +2 respectively, so when combined with the O2- ion
one would expect Na2O and MgO, as well as K2O and SrO to form. The noble gases
are mostly inert, the p-block is amenable to forming covalently bonded
networks, and while a wide variety of oxidation states, which correspond to
various filled or half-filled orbitals, are available to several transition
metals, Zn, Cd, and Hg steadily persist in maintaining their $d^{10}$
configurations.

The consequences of the orbital energy shifts discussed in Section 2 mean that
several of these “rules” no longer apply at high pressures, and elements can
adopt unusual oxidation states in compounds with unexpected stoichiometries.
The LinCs phases Botana:2014  used to illustrate the effects of pressure on
electronegativity provide one such example. Above 70 GPa, the very lithium
rich Li5Cs phase is predicted to become stable, joined at higher pressures by
Li3Cs, Li4Cs, and LiCs Botana:2014 . Remarkably, the calculated Bader charges
on the Cs atoms in these stoichiometries are all more negative than -1, and
since calculated Bader charges frequently underestimate formal oxidation
states, in these compounds Cs may attain a formal oxidation state that is
potentially lower than -2. While alkali metal anions (alkalides) have
previously been captured at ambient pressures with cryptands Dye:1979 , they
achieve charges only up to -1 with the additional electron going into the $n$s
orbital. In the case of the LinCs phases, however, pressure induces
significant Li 2s $\rightarrow$ 2p and Cs 6s $\rightarrow$ 5d electronic
transitions, with the latter increasing progressively with higher Li content,
thereby facilitating the acceptance of electron density by the typically
unoccupied Cs 5d orbitals.

For K, Cs, and Rb the pressure-driven $n\textrm{s}\rightarrow(n-1)\textrm{d}$
transitions led to predictions of transition-metal like behavior
Bukowinski:1976 ; Pickard:2011  in the formation of intermetallic compounds
with actual transition metals Parker:1996 ; Hasegawa:1997 ; Lee:2003 ;
Adeleke:2020 . Within some of these compounds, the transition metal elements
assume exotic electronic configurations, as in the case of the predicted
potassium iridide K3Ir (Figure 3a) containing the Ir3- anion Brgoch:2016 . The
Ir 5$d$ orbital becomes fully occupied as a result of electron transfer from
K, echoed in the later predicted Rb3Ir and Cs3Ir phases Lotfi:2019 . K3Ir and
Rb3Ir share the $Pmnm$ Cu3Ti structure type, while Cs3Ir adopts the $P2_{1}/m$
Ni3Ta type, which consist of Ir@M8 and M@M8 distorted cubes, and Ir@M12
distorted cuboctahedra respectively. In combination with Li under high
pressure, Au displays a similar ability to adopt a significantly negative
formal charge of less than -3 in the predicted phases Li4Au and Li5Au – where
electrons donated from Li are placed into the empty Au 6p orbitals Yang:2016 ,
which are less destabilized than the Li 2s or 2p under pressure.

Figure 3: High pressure compounds where the transition metal atoms adopt
curious electronic configurations in (a) K3Ir Brgoch:2016  (Cu3Ti type), with
Ir@K8 and K@K8 distorted cubes and iridide I3- anions, and (b) HgF4
Botana:2015 , in which d8 configurations on the Hg lead to square planar
geometries.

Pressure can also promote chemical interactions with core or semi-core
orbitals, as in the case of HgF3 and HgF4 Botana:2015 . These stoichiometries
are predicted to become stable above 73 and 38 GPa, respectively, and above
200 GPa HgF4 is computed to decompose into HgF3 and F2. The $I4m$ symmetry
HgF4 crystal possesses square planar HgF4 units typical of a d8 organometallic
complex, with the Electron Localization Function (ELF) Savin:2005  confirming
covalent Hg-F interactions. To form the four Hg-F bonds, not only the Hg 6$s$
but also two of the semicore 5$d$ electrons are required. In HgF3, the
$Fm\bar{3}m$ structure (which distorts below 100 GPa to $C2/m$ symmetry)
involves a fluorite-type HgF${}_{2}^{+}$ lattice stuffed with F- ions, leaving
Hg with a d9 configuration. A series of predicted CsFn phases, in which the Cs
5p electrons participate in Cs-F covalent bonds, are another example of the
pressure-induced activation of core electrons Miao:2013 . Their crystal
structures display motifs resembling the isoelectronic [XeFn]- molecules – for
example, $Fdd2$ CsF5 contains planar pentagonal CsF5 units, similar to the
[XeF5]- anion. With increasing fluorine content, the formal oxidation state on
cesium reaches values greater than +1.

###  3.2 High Pressure Electrides

Electrides are solids where the electrons, localized on non nuclear-centric
sites, behave as anions Liu:2020b . They are conceptually related to solvated
electrons, in which the excess electrons can be thought to occupy cavities in
the fluid Zurek:2009 , as well as alkalide liquids Zurek:2011c ; Zurek:2020h
or alkalide solids where alkali metal anions fill the interstitial voids
Dye:1979 . Although many types of electride families are known at atmospheric
conditions, including those that are organic, inorganic, intermetallic and
those where the electron localization is restricted to various dimensions or
possesses topological properties, herein, we restrict the discussion to high
pressure electrides: systems where the electron localization occurs as a
response to compression Miao:2015 .

Though the formation of high pressure electrides has been rationalized in many
ways, including pressure induced orbital rehybridizations Neaton:2001 ;
Ma:2009 ; Li:2010 , and multicenter bond formation Dong_Oganov_Chapter ;
Modak:2019 , herein we focus on a simple model proposed by Miao and Hoffmann
Miao:2014 ; Miao:2015 . As atoms in a solid compound are compressed, raising
their orbital energies, the electrons in the highest-energy orbitals can
vacate the atom entirely and occupy the interstices of the crystal lattice
instead. To understand why this might occur, Miao and Hoffmann pointed out
that orbitals can be ascribed to these voids, which can be thought of as
interstitial quasiatoms (ISQs). At ambient pressure, the ISQ energies are
higher than the atom-centered ones. However, unlike the atom-centered
orbitals, those of the ISQ do not experience the repulsive effect caused by
the core electrons, and their increase in energy with pressure will be less
than the atom-centered orbitals. When the ISQ orbital energies fall below the
atom-centered ones they will be occupied, thereby localizing the valence
electrons in the interstitial regions. These electrons, detached from the
nuclei, serve as anions and the corresponding compounds are called
_electrides_.

For several simple metals, high-pressure phases identified as electrides via
calculations have been subsequently studied experimentally. In sodium, Neaton
and Ashcroft posited that under pressures high enough to induce overlap of the
$2p$ orbitals, a combination of Pauli repulsion and core orthogonality
constraints would drive the valence electrons away from the ionic cores, to
localize in the crystalline interstices instead. This electronic
redistribution would result in a metal-to-insulator transition Neaton:2001 .
Later CSP calculations – with experimental confirmation of an optically
transparent, wide-bandgap insulating phase in the same publication – proposed
an insulating $hP$4 phase with $P6_{3}/mmc$ symmetry to become stable above
260 GPa. Ma:2009  The new $hP$4 phase was in fact experimentally observed at
pressures as low as 200 GPa, but the discrepancy was ascribed to a combination
of thermal effects as well as the preferential stabilization of metallic
states by the computational method employed. Moving to higher pressures and
temperatures, evidence for the $hP$4 phase has been obtained in shock-ramp
experiments Polsin:2022 . However, _in situ_ X-ray diffraction (XRD) revealed
peaks that could not be attributed to $hP$4 between 240-325 GPa at
temperatures in the thousands-of-degrees Kelvin. Consistent with these dynamic
compression experiments, calculations showed that the free energy of a
$P6_{3}/m$ symmetry phase that is a topological electride was lower than that
of $hP$4 at these pressures and temperatures Wang:2022b . At higher pressures
yet, ca. 15.5 TPa, sodium is predicted to adopt a curious, metallic _cI_ 24
electride phase consisting of Na12 icosahedra Li:2015 . In the insulating
$hP$4 structure that dominates much of the high-pressure landscape in sodium,
highlighted in Figure 4a, the atoms occupy the Ni sites of the Ni2In structure
type with the ISQs on the In sites, in line with the treatment of this phase
as (Na+)2E2- (where E2- denotes a doubly-occupied ISQ). In fact, several A2X
alkali metal chalcogenides adopt the antifluorite $Fm\bar{3}m$ crystal
structure at ambient conditions, but eventually transition to the Ni2In
structure type under pressure Vegas:2002 ; Santamaria-Perez:2011 . Potassium
also adopts the $hP$4 phase when squeezed Marques:2009 ; Degtyareva:2014 .

Lithium presents another example of complex structural and electronic behavior
under pressure, as first postulated by Neaton and Ashcroft who suggested it
may adopt an insulating, paired ground state Neaton:1999 . Subsequent
experiments revealed that Li assumes the same semimetallic $cI$16
Hanfland:2000  structure found in Na McMahon:2007 . At higher pressures, Li
transitions to a number of unique phases, such as those with orthorhombic
C-centered lattices and 88, 40, and 24 atoms in their unit cells which have
been observed Guillaume:2011 . One of these, $oC$40 with the $Aba2$ space
group, is an electride displaying especially interesting behavior Miao:2017 .
In this phase, ISQs occupy three separate symmetry-distinct sites, two that
are doubly occupied (E${}^{\textrm{II}}$) and the third singly occupied
(E${}^{\textrm{I}}$), so its primitive cell can be considered as
Li20E${}^{\textrm{II}}_{8}$E${}^{\textrm{I}}_{4}$. The
E${}^{\textrm{I}}$-E${}^{\textrm{I}}$ distance remains roughly constant at a
short 1.3 Å from 50-80 GPa, with an elevated electron density found between
the ISQs. Crystal Orbital Hamilton Population (COHP) Dronskowski:1993 , ELF,
and projected density of states (PDOS) analyses all indicate bonding character
between the E${}^{\textrm{I}}$ sites, and examination of the $\Gamma$-point
band-decomposed charge densities revealed bonding and antibonding states
analogous to the $\sigma_{g}$ and $\sigma_{u}^{*}$ orbitals in H2. In fact,
computations have shown that ISQs can form covalent, ionic and metallic bonds
with atoms as well as with other ISQs Miao:2015 ; Miao:2015b ; Li:2010 ;
Botana:2016 ; Miao:2017 . For this reason, Miao has espoused the idea that
ISQs may be thought of as a chemical element and placed above helium in the
periodic table under pressure Zurek:2019k .

Figure 4: High pressure electride formation is observed in a number of complex
high-pressure phases of the alkali metals, including both the insulating (a)
_h_ P4 phase of Na Ma:2009  (Ni2In structure type) where ISQs containing two
electrons occupy the In sites, and (b) semiconducting _oC_ 40 Li, containing
three inequivalent ISQ sites of which two (black) are doubly occupied, and the
third (pink) is singly occupied so a bond is formed between nearest neighbors,
as highlighted by the lines that join them.

The proclivity of the elements to ISQ formation has been investigated by
comparing the energies of their orbitals calculated at different pressures
using a He confinement model with those of an ISQ 1s orbital Miao:2014 .
Unsurprisingly, Li and Na were found to favor ISQ formation at relatively low
pressures, with Mg, Al, In, and Tl predicted to follow suit at higher
pressures. Among the heavier alkali metals, the energies of the valence s
orbitals were found to rapidly increase in energy relative to that of the ISQ
1s – but as previously discussed, these elements are also susceptible to a
pressure induced electronic $n\textrm{s}\rightarrow(n-1)\textrm{d}$
transition. Within cesium, for example, the increased d occupation, already
noted by Sternheimer in 1950 Sternheimer:1950 , was invoked to explain the
curious structure of Cs-IV adopted at 4.3 GPa, where the Cs atoms possess a
coordination number of 8 Takemura:1982 . This decrease in coordination number
with increasing pressure appeared so counterintuitive to Linus Pauling that he
presented an alternative structure solution assuming cubic symmetry and
invoking icosahedral clusters Pauling:1989 . Pauling’s hypothesis turned out
to be incorrect. Importantly, von Schnering and Nesper vonSchnering:1987
recognized that the Cs atoms in the Cs-IV structure occupied the Th sites of
the ThSi2 lattice – and that the valence electron density exhibited maxima not
near the Cs atoms but at the Si positions of ThSi2 and between the Si-Si
bonds, dubbing this phase an electride. This $I4_{1}/amd$ structure is also
assumed by K and Rb under pressure Olijnyk:1983 ; Ma:2008 ; Lundegaard:2009 .

If the high-pressure phase behavior of the alkali metals were not complex
enough, Na, K, and Rb all adopt different versions of an incommensurately
modulated host-guest lattice (similar to the W5Si3 type), often referred to as
the $tI$19 structure, a model of which is illustrated for K in Figure 2d.
Olijnyk:1983 ; Schwarz:1999 ; McMahon:2001 ; McMahon:2006 ; McMahon:2006b ;
Gregoryanz:2008 ; Ma:2009 ; Woolman:2018  All three share the same host
lattice, but display different periodicity in the guest lattice. In a study
using commensurate approximants to model the electronic structure, electron
localization in the interstitial spaces was found, with some highly localized
basins as well as a 1D channel of electron density lying in channels of the
host structure Woolman:2018 . Another study proposed that electrides in the
heavy alkali metals could be stabilized via ferromagnetic ordering
Pickard:2011 .

Numerous predicted binary and ternary systems under pressure also behave as
electrides, including two Li3Fe phases (with $P6/mmm$ and $P4/mbm$ symmetries)
Zhou:2016 , $P4/mbm$ Na3Fe Zhou:2016b , and a superconducting Y3Si phase
Zhang:2021 . In a range of LinI stoichiometry phases predicted above 50 GPa,
ISQs form within the interstices between I-centered Li polyhedra – but higher
pressures drive electron density back from the ISQs to the iodine atoms,
filling the 5p and eventually the 5d orbitals, skipping the 6s, in line with
disfavoring s orbital occupation under pressure Botana:2016 . Finally, several
electride phases have been identified involving the famously unreactive noble
gases - including a particularly surprising case Dong:2017  where the noble
gas is crucial to the stability of the synthesized structure.

###  3.3 Compounds of noble gases

Where did all the xenon go?

Relative to the abundance of Ar and Kr in the Earth’s atmosphere, the amount
of Xe is strikingly lower than it should be, a problem known as the “missing
xenon paradox”. Geoscientists have explained this discrepancy in many
different ways Pepin:2002 ; Shcheka:2012 ; Marty:2012 , but a growing body of
evidence suggests that the Xe did not escape, and instead it has been
incorporated into the minerals found within the Earth. At Earth’s core
pressures, both the atomic orbital energies and the relative electronegativies
of the elements, including Xe and the Fe and Ni that comprise the majority of
the core, are significantly perturbed Rahm:2019 ; Dong:2022 . Therefore, it
should not be a surprise that their reactivity differs from our 1 atmosphere
expectations.

Prior to the advent of widespread CSP, computational investigations concluded
that Xe incorporation into Fe and Ni would not occur, at least not to a large
extent Caldwell:1997 ; Lee:2006 . These studies, which relied on the
assumption that the Xe-metal alloys adopted crystal lattices similar to those
of the elemental metals, turned out to be incorrect. Later CSP-based studies
predicted the emergence of stable Xe-Fe and Xe-Ni compounds above 250 and 200
GPa, respectively Zhu:2014 , with $Pm\bar{3}m$ XeFe3 (AuCu3-type) and $Pmmn$
XeNi3 (based on Xe@Ni12 cuboctahedra) having the lowest enthalpies of
formation, although $P\bar{6}2m$ XeFe5 and XeNi5, as well as $P2_{1}/m$ XeNi6
also appeared on the convex hull. Confirming these predictions, experimental
studies have synthesized XeNi3 and Xe(Fe/Ni)3 phases at high pressure,
although with slightly different structures than those predicted. This
includes $Pm\bar{3}m$ for XeNi3, either as an ordered AuCu3 Dewaele:2016b  or
disordered CrNi3 alloy Stavrou:2018 , and a mixture of $fcc$ and
$Pmmn$-symmetry phases for XeFe3 Stavrou:2018 . In these systems, the Fe and
Ni atoms behave as oxidants, accepting 5p electron density from Xe Zhu:2014 ;
Stavrou:2018 , in agreement with the predicted increase in electronegativity
differences between Xe and the transition metals at high pressure Rahm:2019 .
Xe2FeO2 and XeFe3O6, both involving substantial Fe-O and Xe-O bonding, have
also been computed to be stable at pressures relevant to the Earth’s core
Peng:2020 . The high-pressure ArNi phase, in which some Ni 3d electron density
is transferred to the Ar 4s, inducing a magnetic moment on the Ni, has been
synthesized Adeleke:2019 .

CSP calculations have also predicted stable compounds containing Xe, or other
noble gas (NG) elements, and Mg above 125-250 GPa Miao:2015b . This includes
Mg-Xe and Mg-Kr phases, which adopt structures based on stacked square
lattices of Mg and the NGs in different patterns, ranging from the CsCl type
($Pm\bar{3}m$) to more complex $P4/nmm$ or $I4/mmm$ arrangements for MgNG and
Mg2NG stoichiometries. Compounds of Mg with Ar, on the other hand, were found
to favor hexagonal arrangements such as anti-NiAs type MgAr ($P6_{3}/mmc$). In
these compounds the energies of the metal 3s orbitals increase precipitously
in comparison to the outer shell d orbitals of the noble gases inducing Mg 3s
$\rightarrow$ NG d orbital transfer. The ELF of Mg2NG (NG=Xe, Kr, and Ar)
phases shows an additional interesting feature: ISQ formation. This occurs far
below the pressures at which elemental Mg is predicted to form an electride
Li:2010 ; Miao:2014 . Two reasons have been used to explain this phenomenon
Miao:2015b . First, far fewer ISQ sites – concomitantly occupying less space –
relative to the elemental Mg electride are necessary to accept the displaced
valence electrons of Mg, as many of them are transferred to the NG atoms
instead. In addition, the NG atoms promote the formation of larger
interstitial spaces in the structure, stabilizing the ISQ at lower pressures.
Under moderate pressures of ca. 50-300 GPa, the energies of the outer shell Xe
d orbitals are similar to those of the ISQ 1s, although with higher pressure
they become lower in energy, congruent with the gradual ISQ 1s $\rightarrow$
NG d electron transfer with increasing pressure up to 600 GPa Miao:2015b .

Several other stable compounds of Xe have been predicted at high pressures,
including XeO, XeO2, and XeO3 Zhu:2013 , as well as Xe3O2, Xe2O, and Xe7O2
Hermann:2014 , while Xe2O5 and Xe3O2 have both been experimentally observed at
pressures lower than 100 GPa. Dewaele:2016  Krypton oxide, KrO, has been
predicted as well Zaleski-Ejgierd:2016 , as have xenon nitrides Peng:2015 ;
Howie:2016  and carbides Bovornratanaraks:2019 . Fluorides of argon
Kurzydlowski:2015 , krypton Kurzydlowski:2017 , and xenon Peng:2016  – with
Xe-Xe dimers cropping up in Xe2F and XeF – have all been predicted. Several
xenon chlorides including XeCl, XeCl2, and XeCl4, with the former two being
metastable by 10 GPa and reaching the convex hull by 60 GPa have been
computationally studied Zarifi:2018 . When they are combined with Li, the
noble gases Ar Li:2015b  and Xe Liu:2017  are predicted to behave as anions,
with the Li 2s orbital rising above the Xe and Kr outer shell d orbitals.
Several cesium xenides are predicted to be stable at high pressures, many
adopting alternate colorings of a distorted bcc lattice Zhang:2015 . There is
experimental evidence for the formation of a phase mixing Xe with water ice at
conditions expected for planets such as Uranus and Neptune Sanloup:2013 .

Figure 5: Conventional unit cell of helium-containing Na2He Dong:2017  in the
$Fm\bar{3}m$ space group (a), an electride with localized electron pairs
occupying octahedral vacancies (b). The mechanism by which He incorporation
serves to stabilize the structure is illustrated in (c), which shows
schematically how He insertion reduces electrostatic repulsions involved in
the A2B (Na2E) stoichiometry.

Helium is famously the most inert element at ambient pressure by virtue of its
closed-shell electronic configuration, zero electron affinity and large
ionization potential. Nonetheless, a number of stable helium-containing
compounds have recently been predicted at high pressure, including those with
iron Monserrat:2018 , ammonia Liu:2020c , water Liu:2015b ; Liu:2019 ,
nitrogen Vos:1992 ; Li:2018d , and even with _other noble gases_ Wang:2015c ;
Cazorla:2009  (the van der Waals compound NeHe2, a Laves phase in the MgZn2
structure has been experimentally observed Loubeyre:1993 ).

A particularly noteworthy example is provided by Na2He Dong:2017 , an
electride phase with a fluorite-like lattice (Figure 5a) in which every Na8
cube that does not contain a He atom is instead occupied by an electron pair
(Figure 5b), so that the phase can be expressed as (Na+)2(E2-)He. Although He
does not participate in any bond formation, its presence is nevertheless a
crucial stabilizing force in this phase, which has been successfully
synthesized above 113 GPa Dong:2017 . A subsequent computational study showed
that the He atoms act as inert “spacers” to reduce the Madelung repulsion
resulting from the unequal amounts of cations and anions in the parent
(Na+)2(E2-) phase Liu:2018 . Reaction enthalpies for helium in combination
with ionic AB, A2B and AB2 compounds revealed that He incorporation was
generally favored when the cation:anion ratios were unequal such as in MgF2
and Li2O, but not for AB-type phases such as LiF or MgO, in line with the
prediction of successful He incorporation into certain alkali oxides and
sulfides Gao:2019b . Helium placement in the ionic lattices tends to separate
ions of similar charge as shown schematically in Figure 5c. An FeO2He phase in
which the Fe and O atoms form a fluorite lattice and the He atoms occupy the
remaining Fe8 cubes (isopointal to Na2EHe) was predicted to be stable above
120 GPa Zhang:2018 , with He appearing to play the same role of spacing agent.
This mechanism allows even the most inert noble gases to play an active role
in stabilizing compounds at high pressure, all without forming a single
chemical bond.

###  3.4 Miscibility under pressure

The noble gases obtained their names due to their general lack of reactivity,
but under ambient conditions, numerous combinations of elements resist mixing
to form alloys or stoichiometric compounds. The proclivity or reluctance of a
pair of elements towards mixing has been explained in a variety of ways,
resulting in predictive rules including those of Hume-Rothery and co-workers
Hume-Rothery:1969 , Miedema’s model Miedema:1973 ; Miedema:1973b , Darken and
Gurry’s maps Darken:1953 , and more Chelikowsky:1979 ; Alonso:1980 ; Wang:2015
. As we will soon see, pressure turns out to be a useful variable that can
alter the (im)miscibility of two or more elements.

Consider, for example, magnesium and iron, whose large size and small
electronegativity differential at ambient pressure makes compound formation
intractable Dubrovinskaia:2004 . According to Miedema’s rules, compound
formation is favored by large electronegativity differences and similar charge
densities Miedema:1973 . The electronegativity difference between the two
elements greatly increases under pressure Rahm:2019b ; Dong:2022  – and
because Mg is more compressible, its radius approaches that of Fe when
squeezed, thereby increasing the miscibility of the two elements. As a result,
stable Mg-Fe compounds have been computationally Gao:2019 ; Fang:2021  and
experimentally Dubrovinskaia:2005  studied under pressure. The higher
compressibility of K has also been found to favor compound formation with
transition metals such as Ag, even at pressures below which K is anticipated
to undergo an $n$s $\rightarrow$ ($n-1$)d transition Atou:1996 .

Another case-in-point of pressure induced reactivity are the Li-Be alloys
predicted to be stable in the megabar regime Feng:2008 . By 20 GPa LiBe2
reaches the Li-Be convex hull, where it is joined by LiBe4 (shown in Figure
6a) and LiBe by 80 GPa. At 100 GPa the latter trades its place on the convex
hull with Li3Be. Alignment of strong diffraction peaks with 2$k_{F}$ (twice
the free-electron Fermi wavevector) is suggestive of stabilization through a
Fermi surface-Brillouin zone interaction mechanism, which has been used to
explain the particular stabilities (and electron counts which make them so) of
Hume-Rothery electron phases Degtyareva:2006 ; Berger:2011 . Furthermore, at
around 82 GPa, an odd feature emerges in the DOS curve of $P2_{1}/m$ LiBe: the
base of the valence band appears as a step-like function, remains flat for
$\sim$4 eV, and sharply increases once more before more complex features take
over. This is linked to a distinct separation – made possible by the pressure-
induced increase in the electronegativity difference between Li and Be –
between high- and low-electron density planes associated with the Be and Li
atoms, respectively, leading to quasi-2D-like behavior in a geometrically 3D
structure. Stabilization through Fermi sphere interaction with higher zones
(referred to as Jones zones from the Mott-Jones formulation of this mechanism)
was invoked to propose a high-pressure NaAl phase in the NaTl structure type
Feng:2010  just above 12 GPa.

Another element that does not undergo compound formation with Li at ambient
conditions is Fe. Just above 40 GPa, however, Li3Fe ($P6/mmm$) and LiFe
($Fd\bar{3}m$, NaTl-type) phases are computed to lie on the convex hull
Zhou:2016 . Some interstitial electron localization appears in Li3Fe, both in
$P6/mmm$ symmetry as well as the $P4/mbm$ symmetry computed to prevail just
above 60 GPa which is shown in Figure 6b. Both Li3Fe phases are host-guest
lattices, with the Fe atoms lying in larger hexagonal ($P6/mmm$) and
heptagonal ($P4/mbm$) channels, and the electron localization is found within
the smaller triangular and square channels. Na3Fe is predicted to adopt this
same $P4/mbm$ phase between 120 and 300 GPa Zhou:2016b . An additional Li3Fe2
phase with $C2/m$ symmetry appears on the convex hull by 80 GPa, involving Fe
zigzag chains in combination with alternating Li linear or armchair chains.

Figure 6: High pressure enables the mixing of elements which do not form
compounds or alloys otherwise. These include (a) LiBe4, a layered compound of
Li and Be Feng:2008 , (b) Li3Fe, a host-guest compound with Fe atoms as the
guests to a Li-based host lattice Zhou:2016 , and (c) CuBi Guo:2017 ;
Clarke:2017 , one representative of the many bismuth-containing compounds
which have been found at high pressure. CuBi has been experimentally observed.

Recently, a host of bismuth-containing phases have been found to be stabilized
at high pressure. Bismuth is a component in numerous topologically nontrivial
materials Chen:2009 ; Liu:2014  and superconducting systems at ambient
pressure Reynolds:1950 ; Matthias:1966 ; Hor:2010 , yet under these conditions
does not form stable compounds with many of the transition metals. In
combination with the exotic bulk properties of bismuth, the various magnetic
or otherwise electronically nontrivial properties of the transition metals
make for a tantalizing combination in high pressure experiments. Under
pressure, the estimated electronegativity of Bi drops precipitously in
comparison to most of the d-block, rendering it more reactive Rahm:2019b ;
Dong:2022 . Indeed, above 32 GPa, FeBi2 Walsh:2016 ; Amsler:2017  was observed
to form in DAC experiments in the $I4/mcm$ Al2Cu structure type shared by
NiBi2 Amsler:2018  and MnBi2 Walsh:2019  (both are stabilized by high pressure
although other Ni-Bi and Mn-Bi phases are accessible at atmospheric pressure),
as well as certain high-pressure transition metal pnictides Takizawa:1999 ;
Wu:2009 ; Poffo:2012 . A second phase, FeBi3 with $Cmcm$ symmetry, has also
been predicted to lie on the convex hull between 36 and 39 GPa, but this
narrow stability range likely hinders synthetic accessibility Amsler:2017 .

In fact, high-pressure high-temperature methods have been used to synthesize a
wide variety of bismuth-containing compounds including CoBi3 Schwarz:2013 ;
Tence:2014 , which adopts the $Pnma$ NiBi3 structure type, becomes stable by 5
GPa, and is a superconductor with a $T_{c}$ just below 0.5 K. Synthesized
binaries of Cu and Bi include Cu11Bi7 at 3 GPa Clarke:2016  and CuBi (Figure
6c) at 6 GPa Guo:2017 ; Clarke:2017 , as well as a $I4/mmm$ Cu2Bi phase above
50 GPa Amsler:2017b , which is possibly overtaken by a Cu7Bi2 phase
Amsler:2018 . Of the second-row transition metals, MoBi2, also in the Al2Cu
structure type, has been synthesized above 35 GPa, while evidence for a Mo-Bi
bcc-type alloy appeared above 5 GPa Altman:2021 .

The Linear Approximation to Enthalpy (LAE), a tool for rapid and
computationally cheap evaluation of formation enthalpies, was used to explore
the high-pressure stability of structures in binary ambient-immiscible systems
Amsler:2018 . In concert with the minima hopping CSP method, several new
phases were found to be stabilized by 50 GPa: PbAs, Si3Al, SiAl, SiAl3, BiSn3
– yet another bismuthide – In3Fe, Hg3In, HgIn, HgIn3, Hg3Sn, ReSn3, ReBr3,
ReGa, and ReGa3. Only a limited range of stoichiometries (A3B, AB, and AB3)
was sampled, encouraging further investigation into each of these systems –
but now there is preliminary data to suggest fertile ground.

###  3.5 Geometries and bonding

In the previous sections, we have explored curious electronic interactions
made possible by external pressure and compound formation between unexpected
species. Here, we shift our focus to the particular geometrical arrangements
that emerge in materials under pressure. With higher density, atoms are forced
into closer proximity – the possibility of electride formation notwithstanding
– which can promote multicenter bonding in both electron-poor and electron-
rich contexts as coordination numbers increase Grochala:2007a . When electron-
precise species are closely bunched under compression, electron-deficient
multicenter bonding can emerge as the constituent electrons are needed to span
more bonding interactions.

Bond symmetrization is a frequent secondary consequence of compression: an
asymmetric fragment forced to occupy a progressively smaller space has less
room for asymmetry. Often, this leads to a collapse into a symmetric and
multicentered bonding regime, as was predicted for water ice under pressure by
Pauling Pauling:1960 . He suggested that the intermolecular hydrogen bonds
between adjacent water molecules would shorten with pressure Pauling:1960 ,
eventually becoming equivalent with the intramolecular O-H bonds, as
illustrated in Figure 7a. This prediction was verified experimentally upon the
discovery of ice X, where the lone pairs of the oxygen atoms are used to form
additional covalent bonds, rendering the oxygen atoms tetrahedrally
coordinated by hydrogens in a diamond-like network Goncharov:1996 ;
Bernasconi:1998 . Pressure-induced hydrogen bond symmetrization has also been
observed in computations on hydrogen halide systems such as HF, HCl and HBr
Zhang:2010 , as well as in the record-breaking superconductor $Im\bar{3}m$ H3S
Errea:2016 .

Small homoatomic clusters alien to the 1 atmosphere pressure-trained mind are
found or predicted for other elements in high pressure crystal structures. The
wide structural variety has been explained by the increased stabilization of
homonuclear bonds as compared to more polar or ionic bonds under pressure
Zurek:2019k , favoring single-element clustering. An example of a compound
containing novel homonuclear motifs is $Pnma$ NaCl3 (Figure 7b), computed to
be stable from 20 to 48 GPa, featuring a linear Cl${}_{3}^{-}$ anion
reminiscent of the more familiar triiodide I${}_{3}^{-}$ Zhang:2013 . Another
such motif is the pentazolate N${}_{5}^{-}$ ring, which can store more energy
than the related azide anion N${}_{3}^{-}$, but is challenging to synthesize
at ambient pressure Vij:2002 ; Wang:2018 . This species, ubiquitous in high-
pressure phases, is predicted to be a constituent of LiN5 Shen:2015 ;
Peng:2015a ; Laniel:2018b ; Zhou:2020  – a phase that has been successfully
quenched to ambient conditions after synthesis at 45 GPa Laniel:2018  – to
sodium pentazolates NaN5 and Na2N5 Steele:2015 , CsN5 Peng:2015b , CuN5
Li:2018 , MgN10 and BeN10 Xia:2019 , ZnN10 Liu:2020 , BaN5 and BaN10
Huang:2018 , SnN20 Wang:2020 , and IrN7 Du:2021 . Polynitrogen chains feature
in many proposed high-pressure compounds of Cs Peng:2015b , Fe Wu:2018 , Zn
Liu:2020 , Ba Huang:2018 , Sn Wang:2020 , Cd Niu:2021 , Gd Liu:2021 , and Ta
Bykov:2021 , the last of which has been experimentally realized. High pressure
also facilitates the formation of silicon clusters in predicted phases
including Si4 squares in CaSi Gao:2014 , extended networks and clathrate-like
cages in silicides such as CsSi6 Li:2018b , MgSi5 Hubner:2019 , and several
lithium silicide compounds Zhang:2016 .

Figure 7: Geometrical and bonding adaptations at high pressures. With pressure
(a), covalent O-H and intermolecular hydrogen bonds between separate water
molecules in the ice VIII phase equilibrate to yield the symmetric ice X phase
Goncharov:1996 ; Bernasconi:1998 . Pressure coincides with the appearance of
strange motifs including (b) linear trichloride anions in $Pnma$ NaCl3
Zhang:2013 , (c) clusters of Ge dumbbells (inset) in $I4/mmm$ BaGe3 Zurek:2015
; Castillo:2016 , and (d) Li8H “superatom-like” building blocks (inset) in
$Abm2$ Li5H Hooper:2012c . Except for Li5H, all have been experimentally
realized.

Tetrel clusters comprise a family of polar intermetallic $I4/mmm$ symmetry
compounds formed from alkaline earth or rare earth metals and group 14 tetrels
in a 1:3 ratio. An example of these isotypic compounds, BaGe3, is shown in
Figure 7c. The clusters within it may be described as tetrel dumbbells
condensed into cubes, which are capped on four equatorial faces by additional
dumbbells shared with a neighboring cube, forming a loose three-dimensional
network. This structure has been experimentally observed in Ca, Y, and Lu
silicides Schwarz:2012  (and later identified in a CSP investigation of the
Y-Si system Zhang:2021 ), and a related distorted $I\bar{4}2m$ BaSi3 phase has
also been synthesized Hubner:2019b . Alkaline earth trigermanides CaGe3
Schnelle:2012 , SrGe3 Nishikawa:2015 , and BaGe3 Zurek:2015 ; Castillo:2016
have also been found to adopt this structure. The distribution of electrons
within the clusters aligns with two-center two-electron bonds along the tetrel
dumbbells, with multicenter interactions between the tetrel and rare
earth/alkaline earth Schwarz:2012 ; Zurek:2015 ; Castillo:2016b .
Superconductivity has been measured in some of these compounds, albeit at low
temperatures Schwarz:2012 ; Castillo:2016 , augmented by predictions from
first principles calculations Zurek:2019k ; Castillo:2016b ; Zhang:2021 .
Additionally, the stability conferred by the strong covalently bonded networks
permits some of these phases, synthesized at high-temperatures and high-
pressures, to be recovered at ambient conditions Castillo:2016b ; Hubner:2019b
. A similar example is presented by elemental carbon – diamond is its ground
state at high-pressures, but due to the immense strength of its sp3 covalently
bonded network, the energetic barrier for its transition to the lower-enthalpy
allotrope graphite is too high and it persists “forever” under ambient
conditions. Furthermore, laser-driven ramp compression studies of carbon to 2
TPa have found that carbon stubbornly maintains the diamond structure well
beyond its predicted high-pressure stability limits Lazicki:2021 , as the
barriers to breaking the sp3 bonds remain large under pressure.

Another example of unique clusters predicted to form only at high pressures
are found within a family of lithium subhydrides Hooper:2012c . Computations
uncovered two nearly isoenthalpic Li5H phases that had the most negative
enthalpies of formation. Both were built of Li8H units that behaved as
superatoms analogous to similar units in synthesized Rb9O2 and Cs11O3
suboxides Simon:2010 . One of these, with $Abm2$ symmetry, is shown in Figure
7d. The Li8H cluster, a distorted bicapped trigonal antiprism of Li
encapsulating a single H atom, has one electron in excess of the closed-shell
octet and thus behaves as a superalkali atom.

In addition to the well-known H2 molecular units and H- hydridic species,
hydrogen atoms can form other distinct clusters. One of these, the trihydrogen
cation, H${}_{3}^{+}$, is in fact one of the most abundant species in the
universe – but it is also largely relegated to interstellar space and the
atmospheres of gas giant planets Oka:2013 . High pressure crystal chemistry
offers another opportunity. The halogen polyhydride $Cc$ H5Cl Wang:2015b ;
Duan:2015 ; Zeng:2017 , predicted to become stable above 100 GPa, contains
slightly distorted H${}_{3}^{+}$ clusters with H-H distances of 0.74, 0.97 and
1.01 Å Wang:2015b . By 300 GPa the three distances converge to 0.87-0.88 Å,
yielding a nearly-perfect equilateral triangle. With sufficient pressure, this
H${}_{3}^{+}$ unit interacts with a neighboring H2 molecule forming a twisted
bowtie-like loosely interacting H${}_{5}^{+}$ motif Wang:2015b ; Zeng:2017 .
Metastable predicted H2F, H3F, and H5F species, as well as H5Br also contain
this triangular H${}_{3}^{+}$ cation Duan:2015 , as does the metastable $P1$
LiF4H4 Bi:2021 .

With two extra electrons, the trihydride anion, H${}_{3}^{-}$, prefers a
linear arrangement involving a three-center four-electron bond. Quantum
chemical calculations have shown that the ground state geometry of the
isolated trihydride anion possesses one H-H bond that is substantially longer
than the other (2.84 vs. 0.75 Å), with the transition state between the
H-H$\cdot\cdot\cdot$H and H$\cdot\cdot\cdot$H-H configurations corresponding
to the symmetric case Ayouz:2010 . Nevertheless, certain predicted high-
pressure hydrides of the heavy alkali metals K Hooper:2012 , Rb Hooper:2012a ,
and Cs Shamp:2012  (as well as the alkaline earth metal Ba Hooper:2013 )
feature an H${}_{3}^{-}$ anion symmetrized via pressure. Synthesized NaH7 is
thought to contain an asymmetric linear H${}_{3}^{-}$ motif Struzhkin:2016 .
Linear H${}_{3}^{-}$ units are also predicted to appear in various indium
Liu:2015  and lithium polyhydrides Chen:2017 .

Scandium polyhydrides, meanwhile, are predicted to feature five-membered rings
of hydrogen atoms in various arrangements. In $I4_{1}/md$ ScH9, which lies on
the convex hull around 300 GPa Zurek:2018b , strips of edge-sharing H5
pentagons are stacked perpendicular to one another along the _c_ axis, linked
by vertices. The strips are separated by additional H atoms in molecular H2
units. Around 250 GPa, ScH10 adopts a $Cmcm$ structure in which H5 pentagons
are grouped into sets of three, sharing edges and a single common vertex. This
phase is nearly isoenthalpic with another ScH10 structure with the same (H5)3
“pentagraphenelike” clusters but arranged in $P6_{3}/mmc$ symmetry Xie:2020 .
ELF demonstrates bonding within the H5 units Zurek:2018b ; Xie:2020 . This
same pentagraphenelike structure is expected to lie on the Lu-H convex hull at
300 GPa and to be very near the Hf-H and Zr-H hulls Xie:2020 . For higher
hydrogen contents yet, ScH12 is predicted to be built of stacked strips of
edge-sharing H5 pentagons spaced by Sc Zurek:2018b .

Indeed, the first, most simple, element does not like to be outdone! One more
class of high-pressure hydrogen rich materials whose prediction sparked
tremendous experimental synthetic efforts are the so-called “metal
superhydrides”. The reason why scientists have pursued them in earnest is the
prediction, verified by recent experiments, of conventional superconductivity
at temperatures approaching those experienced in a cold room (the $T_{c}$ of
LaH10 is about 10 ∘F – January in Siberia), or a crisp fall day (the $T_{c}$
of C-S-H is about 60 ∘F), albeit still at very high pressures! CSP
investigations into the high-pressure Ca-H system revealed a curious
$Im\bar{3}m$ CaH6 phase (Figure 8a) stable above 150 GPa in which the Ca atoms
were arranged in a bcc lattice and the H atoms condensed into a sodalite-like
H24 framework Wang:2012 . All H-H distances were equivalent, 1.24 Å at 150
GPa, and ELF analysis confirmed weak covalent bonding between the H atoms.
This phase, a good metal, was predicted to exhibit large electron-phonon
coupling, and indeed first principles calculations estimated a superconducting
transition temperature $T_{c}$ of 220-235 K at 150 GPa. Subsequent synthetic
exploration led to measurement of $T_{c}$ over 200 K at 160-170 GPa for CaH6
Ma:2022 ; Li:2022 .

Figure 8: Clathrate-like metal hydrides predicted then synthesized at high
pressures. Their remarkable superconducting properties are tied to strong
electron-phonon coupling. Several metal-hydrogen stoichiometries adopt these
so-called “superhydride” motifs, including $Im\bar{3}m$ MH6, exemplified by
CaH6 Wang:2012 ; Li:2022 ; Ma:2022 , (b) $P6_{3}/mmc$ MH9, exemplified by YH9
Peng:2017 ; Snider:2021 , and (c) $Fm\bar{3}m$ MH10, exemplified by LaH10
Liu:2017-La-Y ; Peng:2017 ; drozdov2019superconductivity ; Somayazulu:2019-La
.

The computational discovery of CaH6 was shortly followed up by fruitful
theoretical investigations into related metal-hydrogen systems, turning up
isostructural phases for Mg Feng:2015a , Sc Qian:Sc-2017 ; Abe:Sc-2017 ;
Zurek:2018b , Y Li:2015a ; Liu:2017-La-Y ; Peng:2017 , Pu Zhao:2020 , Tb
Hai:2021 , Eu Semenok:2021 , and Pm-Lu Sun:2020b . Structures that are
distortions of this high symmetry phase have been predicted as well. This
includes a tetragonally-distorted $I4/mmm$ ZrH6 variant Abe:2018-Zr , along
with an $R\bar{3}m$ phase in which opposite hexagonal faces of the H24 cubic
sodalite framework are opened for SrH6 Hooper:2013b ; Wang:2015a ; Zurek:2018d
and LaH6 Liu:2017-La-Y ; Peng:2017 , although the latter may not lie on the
convex hull. $Imm2$ BaH6 Hooper:2013 , which contains some of the
H${}_{3}^{-}$ trihydride anions explored above, can be thought of as a highly
fragmented version of the sodalite framework. The role of distortions of the
high-symmetry $Im\bar{3}m$ structure adopted by CaH6 tends to reduce the
density of states at the Fermi level, $E_{F}$, thereby also lowering $T_{c}$.
The origin of such distortions has recently been investigated using the lens
of DFT-Chemical Pressure Hilleke:2022 .

Higher hydrogen content allows for other clathrate-like arrangements of
hydrogen, from $P6_{3}/mmc$ YH9 Peng:2017  (Figure 8b) to $Fm\bar{3}m$ LaH10
Liu:2017-La-Y ; Peng:2017  (Figure 8c). Distorted versions of these two
structures have also been predicted, including $C2/m$ CaH9 Shao:2019b  and
$P1$ Eu4H36 Semenok:2021 , as well as $R\bar{3}m$ CaH10 Shao:2019b  and AcH10
Semenok-2018 . In the case of LaH10, quantum anharmonic effects were found to
be key in stabilizing the $Fm\bar{3}m$ structure over less symmetric variants
Errea:2020 . Other more complex clathrate-like hydrogenic frameworks have been
predicted as well. One example is a diamond-like lattice of Mg@H28 clusters
intercalated with Li@H18 units, which comprise the $Fd\bar{3}m$ Li2MgH16
phase. This compound is an example of a “hot” superconductor whose estimated
$T_{c}$, 351 K at 300 GPa, is well above room temperature Sun:2019 . Such
materials are under intense speculation and investigation for their promise
towards achieving room-temperature superconductivity, as described in Section
4.3 below.

##  4 Superconductivity

The 1911 discovery of a phenomenon in which a substance’s resistivity can
plummet to zero Onnes:1911  sparked countless investigations and resulted in a
Nobel prize for Heike Kamerlingh Onnes, as well as a number of future Nobel
prizes (directly or indirectly). The mechanism of superconductivity, and the
search for new superconducting materials, has fascinated scientists for over a
century. A key parameter for superconductors is the critical temperature,
$T_{c}$, below which a material becomes superconducting. For a number of
illustrative superconducting materials, $T_{c}$ is plotted against the
pressure at which they are stable in Figure 9. Of course, practical
applications of superconductivity are limited if temperatures very near 0 K
are required, and for some decades the highest known $T_{c}$ values lingered
in the low twenties Matthias:1954 ; Gavaler:1973 . This sparked debate
regarding a possible natural “cap” on superconductivity around these
temperatures AndersonMatthias:1964 . However, a family of cuprates whose
superconducting mechanism has yet to be explained were the first to break the
liquid nitrogen barrier Wu:1987 , achieving $T_{c}$s over 160 K in the case of
pressurized HgBa2Cam-1CumO2m+2+δ Gao:1994 .

Figure 9: Critical superconducting temperature, $T_{c}$, plotted against
pressure for selected materials. Data points based on experimental
measurements of $T_{c}$ are plotted with filled circles, while those estimated
via theoretical calculations are plotted with empty circles. Elemental
$T_{c}$s are given in brown, intermetallics in green, cuprates (belonging to
the non-BCS unconventional family) in orange, and hydrides in blue. The
boiling point of liquid nitrogen and room temperature are provided to guide
the eye, with a star marking the “holy grail” of room-temperature
superconductivity at ambient pressure. Data was collected from references
Onnes:1911 ; Finnemore:1966 ; Gavaler:1973 ; Wu:1987 ; Gao:1994 ;
Nagamatsu:2001 ; Sakata:2011 ; Drozdov:2015 ; Drozdov:2015a ; Liu:2017-La-Y ;
Peng:2017 ; Somayazulu:2019-La ; drozdov2019superconductivity ; Salke:2019 ;
Sun:2019 ; Snider:2020 ; diCataldo:2021 ; Gao:2021 ; Zhang:2022 .

The 2001 discovery of superconductivity up to 39 K in MgB2 Nagamatsu:2001
provided experimental evidence that non-cuprate phases could be promising
superconductors, despite the fact that they deviated from the collection of
empirical rules enumerated by B. T. Matthias in the 1950s and 1960s
Matthias:1957 . These ranged from favorable valence electron concentrations to
a general distrust for theorists. MgB2 belongs, with early and long-term
record holders Nb3Sn and Nb3Ge, as well as the clathrate-like hydrides
discussed in Section 3.5, to the family of “conventional” superconductors
whose mechanism is thought to follow the theory propounded by Bardeen, Cooper,
and Schrieffer in 1957 bcs1 ; bcs2 . This discovery revitalized interest in
conventional superconductors Pickett:2007 , leading researchers to wonder what
the trajectory of superconductivity research would have looked like had the
1957 measurements of the heat capacity of MgB2 Swift:1957  captured the
discontinuity that appears upon the superconducting transition.

Within BCS theory, two electrons of opposite momentum and spin that are within
$\pm\hbar\omega_{\text{cut}}$ (an energy in line with the phonon energies) of
the Fermi surface may, at long distances, overcome Coulombic repulsion and
experience a net attractive potential when the lattice is polarized through
phonon vibrations. This forms a Cooper pair, a composite boson of weakly
interacting species. Thermal energy can break the Cooper pairs, with $T_{c}$
describing the temperature at which this occurs and the superconducting state
is destroyed. From this construction, the $T_{c}$ for a material can be
estimated by

$k_{B}T_{c}=1.14\hbar\omega\exp\bigg{[}{\frac{-1}{N_{F}V}}\bigg{]}$ (1)

where $\omega$ is the average phonon energy, $N_{F}$ is the single spin
electronic density of states at $E_{F}$, and $V$ is the pairing potential
between two electrons resulting from the electron-phonon interaction. This
suggests that high $T_{c}$ is correlated with a large $N_{F}$ (a feature
potentially tunable via judicious doping), strong coupling between electrons
and phonons, and high phonon frequencies. A frequently used semiempirical
formula to estimate the $T_{c}$ of a conventional superconductor is the Allen-
Dynes modified McMillan equation McMillan:1968 ; Dynes:1972 ; Allen:1975 :

$T{{}_{c}}=\frac{\omega_{\text{ln}}}{1.2}\text{exp}\bigg{[}-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\bigg{]}.$
(2)

where $\lambda$ is the electron-phonon coupling constant, $\omega_{\text{ln}}$
is the logarithmic average phonon frequency and $\mu^{*}$ is the Coulomb
repulsion parameter, which is typically treated semiempirically. An
approximate – and illustrative – formula to estimate $\lambda$ was proposed by
Hopfield as Hopfield:1969

$\lambda=\frac{N_{F}\langle I^{2}\rangle}{M\langle\omega^{2}\rangle}$ (3)

where $\langle I^{2}\rangle$ are the electron-phonon matrix elements averaged
over the Fermi surface, $M$ is the atomic mass, and $\langle\omega^{2}\rangle$
the mean phonon frequency. To increase $\lambda$, then, $N_{F}$ and the
electron-phonon matrix elements should be increased. Converse to expectations
from Equation 1 and Equation 2, where the average phonon energy was directly
proportional to $T_{c}$, here an increase in $\langle\omega^{2}\rangle$ serves
to decrease $\lambda$. In the denominator of Equation 3,
$\langle\omega^{2}\rangle$ and $M$ will naturally tend to counteract one
another, as increases in atomic mass lead to softer phonon frequencies. In
fact, evidence suggests that $T_{c}$ frequently increases at the edge of
dynamic instability, where soft phonons promote strong coupling Quan:2019 .

In the following sections, we describe the effect of pressure on the
propensity for superconductivity of the elements, hydrogen in particular. We
end by discussing families of hydrogen-rich phases that are extremely
promising towards achieving the once-distant goal of room-temperature
superconductivity (albeit potentially only at very high pressures).

###  4.1 The elements

Most of the elements in the periodic table can be superconducting given the
right conditions. This includes, so far without exception, rather low
temperatures. However, external pressure can affect the $T_{c}$, or even
induce superconductivity in some elements Buzea:2004 ; Schilling:2006 ;
Hamlin:2015 ; Shimizu:2015 . In fact, calcium at ambient pressure is not a
superconductor and achieves $T_{c}=$ 29 K at 216 GPa Sakata:2011 . In
comparison, the highest elemental $T_{c}$ at ambient pressure is 9.2 K for
niobium Finnemore:1966 , while only lithium among the alkali metals
Tuoriniemi:2007 , and only beryllium Falge:1967  among the alkaline earth
metals are known to superconduct at ambient pressure, both well under 0.1 K.
Of the fifty-four known superconducting elements of the periodic table, only
thirty-one are superconductors at ambient pressure.

Early indications regarding the ability of pressure to either enhance or
suppress superconductivity Onnes:1925 ; Jennings:1958 ; Smith:1967  were less
than encouraging. For simple metals whose electronic structure aligns with a
mostly free-electron model, such as Zn, Cd, Hg, and the group 13 metals,
applied pressure serves to suppress what superconductivity is present at
ambient pressure Jennings:1958 ; Smith:1967 . In such free-electron-like
metals, the effect of pressure is to broaden electronic bands and increase
phonon frequencies due to the stiffer lattice. Band broadening reduces the
electronic density of states at $E_{F}$, while a stiffer lattice is correlated
with weaker coupling between electrons and phonons, both effects being
detrimental to the superconductivity of a system. At ambient pressure, the
alkali metals behave as free-electron metals, but as we have seen in Section
3.2, with pressure their electronic structure rapidly diverges from these
expectations.

For Li, $T_{c}$ is highly pressure-dependent and reliant on complex crystal
chemistry Struzhkin:2002 ; Shimizu:2002 ; Deemyad:2003 . Like all of the
alkali metals, it adopts a bcc crystal structure at ambient pressure, which in
short order transitions to the fcc structure. Up to 8 GPa, the increase in
pressure is reflected in an increase in phonon frequencies, typical for a
stiffening lattice, but with further pressure the phonons become softer and
just above 30-40 GPa, imaginary modes related to a structural instability
appear. This pressure corresponds to another phase transition, this time to
the _h_ R1 ($R\bar{3}m$) structure, and shortly thereafter to the _c_ I16.
Maxima in $T_{c}$ are associated with the onset of dynamical instability, as
the very soft phonon motions strongly bolster the electron-phonon coupling
Quan:2019 ; Chen:2020 . From its mostly spherical character at 0 GPa, where
lithium adopts a bcc crystal structure, the pressure-induced 2s $\rightarrow$
2p electronic transition drives an increasingly anisotropic Fermi surface
featuring _hot spots_ of especially strong coupling Kasinathan:2006 , losing
free-electron-like behavior and leading to Fermi surface nesting (FSN)
Profeta:2006 ; Rodriguez-Prieto:2006 ; Xie:2007 . Phonon softening, in
particular along the $\Gamma\rightarrow\textrm{K}$ path, is accompanied by
enhancement of the electron-phonon coupling Yao:2009b ; Bazhirov:2010 , with
the result that $T_{c}$ grows from practically zero to a maximum of $\sim$20
K. This value is among the higher elemental $T_{c}$s, as a result of the
pressure-induced electronic transitions in lithium. Following the structural
transition to the _c_ I16 phase, lithium remains superconducting but $T_{c}$
decreases due to a reduction of the FSN and concomitant smaller electron-
phonon coupling Yao:2009b ; Rousseau:2011 ; Yue:2018 . At high enough
pressures, lithium undergoes a metal-semiconductor transition – and eventually
goes back to being a (poor) metal upon transitioning to the _o_ C24 ($Cmca$)
phase Matsuoka:2014 , but one nonetheless predicted to be superconducting with
an estimated $T_{c}$ of 14 K at 200 GPa Yan:2016 .

In cesium Wittig:1970 ; Schwarz:1998  and rubidium Deng:2019 ; Schwarz:1999 ,
the onset of superconductivity with pressure is associated with the _o_ C16
(Cs-V and Rb-IV) structures, alongside the
$\textrm{n}s\rightarrow(\textrm{n}-1)d$ transition Deng:2019 . An increase in
_d_ -character in the electronic states at $E_{F}$ is correlated with higher
$T_{c}$ in the transition metals Debessai:2008 ; Slocombe:2015  and applies
here as well to the heavier alkali metals – which, as we have seen, behave
under pressure as transition metals themselves.

###  4.2 Hydrogen

Vitaly Ginzburg, awarded the Nobel Prize in Physics in 2003 for his work in
superconductivity and superfluidity, formulated a list of, in his view, the
thirty most pressing problems for physics in the 21st century Ginzburg:1999 ;
Ginzburg:2003 . Following controlled nuclear fusion, the second and third
items on this list were high-temperature (room-temperature) superconductivity
and metallic hydrogen. These problems are not unrelated.

In 1926, J. D. Bernal proposed that under sufficient pressure hydrogen would
transition to a metallic state, but it took nearly a decade for pen to be put
to paper by Wigner and Huntington Wigner:1935 . Their 1935 suggestion that
hydrogen could be metallized by 25 GPa, estimated using a series of
assumptions regarding crystal structure and compressibility, proved to be an
immense underestimate. In 1968 Neil Ashcroft explicitly linked the quest for
metallic hydrogen with the quest for superconductors with higher $T_{c}$s,
with the suggestion that metallic hydrogen itself would be quite a fantastic
superconductor Ashcroft:1968 . Hydrogen, being the lightest element, can
possess the highest frequencies (as a diatomic molecule) and experience a
large electron-phonon coupling due to the lack of screening by core electrons.
Moreover, in the metallic state its DOS at $E_{F}$ is thought to be quite
high, making for a very attractive material.

The pressure required to metallize hydrogen, however, is in the multimegabar
range. Claims of metallic or semimetallic hydrogen have been made for DAC
experiments at very high pressures and low temperatures, toeing the line of
the practical limits of these techniques Dias:2017 ; Eremets:2019 ;
Loubeyre:2020 . Complicating the picture, different experiments used different
measures to characterize hydrogen’s transition to metallicity, from
vibrational spectroscopic techniques such as IR Loubeyre:2020 , optical
measurements such as reflectance and opacity Dias:2017 , and resistivity
measurements Eremets:2019 , as well as different scales to calibrate pressure.
At times, this led to seemingly contradictory results, prompting questions
regarding experimental accuracy and reproducibility Liu:2017b ; Goncharov:2017
; Gregoryanz:2020 . _Ab initio_ calculations taking into account the quantum
fluctuations of the hydrogen nuclei, however, can reconcile some of these
differences, finding in the $C2/c$-24 high-pressure phase closing of the
electronic gap (and transition to metallicity) before the closing of the
optical gap (and transition from transparency to reflectivity) Monacelli:2021
.

Additionally, the impractically low $T_{c}$s of ambient pressure materials are
then traded for a much higher $T_{c}$ in pure metallic hydrogen, but at
impractically high pressures! In fact, _ab initio_ modeling suggests
progressive jumps in $T_{c}$ with a transition from the molecular (estimated
$T_{c}$ = 356 K near 500 GPa) to the atomic phase ($T_{c}$ increasing to 481 K
ca. 700 GPa), and with an atomic-atomic phase transition at $\sim$1-1.5 TPa
driving up $\lambda$ and resulting in an immense estimated $T_{c}$= 764 K
McMahon:2011 . To address the hydrogen metallization problem, Ashcroft
proposed another strategy – instead of pure hydrogen, hydrogen-rich metallic
alloys could be targeted as putative superconductors Ashcroft:2004 . The
presence of additional atoms in the hydrogen matrix would confer a _chemical
precompression_ , thereby lowering the external pressure required to reach the
metallic state. Under ambient conditions, the crystal structures adopted by
metal hydrides are largely subject to the dictates of balanced oxidation
states, hence alkali metal hydrides assume the rock salt structure, and
hydrides of +2 metals favor fluorite or Co2Si structures, while trivalent
metal hydrides tend towards the BiF3 structure, and so on Smithson:2002 ;
Bourgeois:2017 . Much higher hydrogen content would be needed for such
hydrogen-rich alloys, as suggested by Ashcroft, and furthermore many of the
resulting structures might differ greatly from anything observed at ambient
conditions. Defying the recommendations of Matthias, the simultaneous and
serendipitous advances in CSP methods meant that theoreticians were well
poised to answer this call!

###  4.3 Clathrate-like hydrides

They were successful. As described above, calculations on the high-pressure
Ca-H system located the CaH6 phase described in Section 3.5 Wang:2012 , in
which Ca atoms are embedded into a hydrogenic sodalite-like clathrate
framework. The strong electron-phonon coupling predicted for $Im\bar{3}m$ CaH6
can be traced to breathing and rocking modes of the square H4 units of the
sodalite framework Wang:2012 . The molecular orbital diagram for such an H4
square has, above a filled bonding state, a half-occupied degenerate non-
bonding state. Assuming full ionization (integrated charges within atomic
basins according to the Quantum Theory of Atoms in Molecules indicate roughly
1 electron per Ca is transferred to the hydrogen network), these orbitals
accept the roughly 1/3 electron per H transferred from the Ca atom. This
favors symmetry-breaking Jahn-Teller distortions – key contributions to the
electron-phonon coupling – that lift the degeneracy.

Similar clathrate-like hydrides, as outlined in Section 3.5, were rapidly
predicted in several other systems. $Im\bar{3}m$ YH6 has been synthesized with
a measured $T_{c}$ of 224 K at 160 GPa Troyan:2021a . Hydrides with even
higher hydrogen content have been synthesized as well, including a
$P6_{3}/mmc$ YH9 phase with a $T_{c}$ of 262 K at ca. 180 GPa Snider:2021  (a
subsequent study reported a slightly lower $T_{c}$ of 243 K at 200 GPa
Kong:2021a ), and an isotypic CeH9 phase whose $T_{c}$ has only been predicted
(57 K at 88 GPa and up to 100 K at 130 GPa Peng:2017 , or 105-117 K by 200 K
Salke:2019 ). The relatively low pressure required to stabilize CeH9 has been
ascribed to strong chemical precompression from the delocalized Ce 4 _f_
electrons Jeon:2020 . The reported $T_{c}$s for $Fm\bar{3}m$ LaH10 (250 and
260 K at 170 and 185 GPa, respectively drozdov2019superconductivity ;
Somayazulu:2019-La ), are in line with theoretical predictions of 257-274 K at
250 GPa Liu:2017-La-Y . Isotypic $Fm\bar{3}m$ YH10 is computed to be a room
temperature superconductor with a $T_{c}$ of 305-327 K at 250 GPa
Liu:2017-La-Y . However, YH10 has thus far eluded synthetic efforts. Partial
doping with lanthanum appears to be one strategy to stabilize YH10: a series
of ternary (La/Y)H10 phases have been experimentally observed, with measured
$T_{c}=$ 253 K Semenok:2021b .

Although the $T_{c}$s of many clathrate-like hydrides are stunning, these
phases will surely decompose well above atmospheric pressures! CeH9 is
remarkable for the comparatively low, sub-megabar, pressures at which it
maintains dynamic stability Salke:2019 . In an attempt to preserve the
loosely-bound hydrogenic clathrate frameworks that are associated with such
strong electron-phonon coupling to lower pressures, one promising strategy
involves the addition of a third element in an attempt to further chemically
precompress the hydrogenic lattices. The $Fm\bar{3}m$ LaBH8 phase, which is
predicted to maintain dynamic stability down to 40 GPa diCataldo:2021 , can be
derived from LaH10 by removing two hydrogen atoms per formula unit and placing
boron atoms into the center of H8 cubes that are empty in LaH10 Liang:2021 ;
Zhang:2022 . LaBH8 has an estimated $T_{c}$ of 126 K at 50 GPa diCataldo:2021
– and the isostructural LaBeH8 phase is predicted to achieve a $T_{c}$ of 183
K at 20 GPa Zhang:2022 . Other XYH8 phases have been proposed, with a variety
of possible elemental combinations ripe for tuning stability and properties
Zhang:2022 . A second possibility is afforded by the XY2H8 phases that can be
constructed by leaving the H8 cubes empty, but stuffing the center of H4
tetrahedra instead. KB2H8 (dynamically stable to 12 GPa Gao:2021 ) and LaC2H8
(dynamically stable down to 70 GPa Durajski:2021 ) are two representatives of
this structural arrangement, with estimated $T_{c}$s of 134-146 and 69 K,
respectively.

Key to the success of the clathrate-like hydrides is the maintenance of
loosely-coordinated networks of hydrogen, rather than condensation into H2
molecules. The effect of H-H interatomic distances on superconductivity can be
seen in the MH4 hydrides, which adopt the $I4/mmm$ structure shared with
ThCr2Si2 Shatruk:2019 ; Bi:2021b . Hydrogen occupies two inequivalent sites in
the ThCr2Si2 structure – the apical Ha (Wyckoff position 4 _e_ , Si) and basal
Hb (4 _d_ , Cr). A plethora of metal hydrides have been predicted or
synthesized in this structure type under pressure, including Ca Wang:2012 ;
Zurek:2018c ; Shao:2019b  and Sr Hooper:2013b ; Wang:2015a , Sc Abe:Sc-2017 ;
Qian:Sc-2017 ; Zurek:2018b , Y Li:2015a ; Liu:2017-La-Y , and Zr Abe:2018-Zr
and rare earths La Liu:2017-La-Y , Ce Peng:2017 ; Li:2019 ; Salke:2019 , Pr
Peng:2017 ; Zhou:2020b ; PenaAlvarez:2020 , Pu Zhao:2020 , Tb Hai:2021 , Eu
Semenok:2021 , Nd Peng:2017 ; Zhou:2020c , and Th Kvashnin:2018 ;
Semenok:2020c , making systematic study enticing and useful Bi:2021b . With
metal oxidation states ranging from +2 to +4, the formulas of these compounds
can be written as Mx+(H${}_{b}^{-}$)2(Ha)${}_{2}^{(x-2)-}$, with hydridic Hb
and a range of charges possible on the Ha atoms. The $T_{c}$s of these phases
are correlated with the length of the Ha-Ha contacts, which can behave
anywhere from covalently bound H2 units to fully dissociated hydridic anions
depending on the metal atom – similar to the behavior of the X-X bond in
ThCr2Si2-type AB2X2 phases Hoffmann:1985 . The size of the metal atom can be
relevant, as larger atoms will stretch the Ha-Ha contact through purely steric
interactions, but more important is the valency of the metal atom. Electron
transfer from the electropositive metal into the Ha-Ha motif directly
populates the H2 $\sigma_{u}^{*}$ antibonding orbitals, but is also driven by
a Kubas-like two-pronged mechanism of H2 $\sigma_{g}\rightarrow$ M $d$
donation, and M $d\rightarrow$ H2 $\sigma_{u}^{*}$ back-donation. With enough
H2 $\sigma_{u}^{*}$ population, the Ha atoms behave in a hydridic fashion
lowering the $T_{c}$, as seen in ZrH4 Abe:2018-Zr  and ThH4 Kvashnin:2018 .
Donation of sufficient electron density to weaken, but not fully break, the
Ha-Ha bonding interaction results in a much higher DOS at $E_{F}$ and enhanced
$T_{c}$, as in YH4 Li:2015a .

###  4.4 Covalent hydrides

The first hydride to top the charts, as it were, was not of the metal
clathrate-like family, but instead came from attempts to metallize H2S. Theory
identified an H2S compound that was computed to possess a $T_{c}$ of 80 K at
160 GPa Li:2014 . Experimental confirmation followed shortly thereafter,
finding a phase with $T_{c}$ $<$ 100 K, but in the process a higher-
temperature preparation method yielded a sample with a $T_{c}$ of 203 K at 150
GPa Drozdov:2015a . A few years before, synthetic exploration into the
(H2S)2H2 stoichiometry found a phase stabilized by pressure-induced hydrogen
bonding above 3.5 GPa Strobel:2011 . This inspired CSP investigations of the
H3S stoichiometry, which found an $R{3}m$ phase with $T_{c}=$ 155-166 K at 130
GPa Duan:2014 . By 180 GPa this structure transitioned to one with
$Im\bar{3}m$ symmetry (Figure 10a) for which the estimated $T_{c}$ was 191-204
K at 200 GPa. Serendipitously the experimental Drozdov:2015a  and theoretical
Duan:2014  manuscripts appeared at nearly the same time. Subsequent XRD
studies supported the identification of the 203 K superconductor as
$Im\bar{3}m$ H3S Einaga:2016 , though other structures have been proposed
Li:2016 ; Gordon:2016 ; Ishikawa:2016 ; Akashi:2016 ; Yao:2018 ; Laniel:2020 .

A host of studies on the H3S superconductor have followed, exploring the
isotope effect, role of anharmonicity, and possible quantum effects Errea:2016
; Ortenzi:2016 ; Durajski:2015 ; Durajski:2016 ; Gorkov:2016 ; Goncharov:2016
; Jarlborg:2016 ; Bussmann-Holder:2017 ; Szczesniak:2017 ; Azadi:2017 ;
Arita:2017 . The inclusion of quantum nuclear motions lowered the pressure
where the less symmetric $R3m$ phase was predicted to transition to the
$Im\bar{3}m$ structure with symmetric H-S bonds into the range of pressures
where high $T_{c}$s had been measured Errea:2016 . One of the most striking
features of the electronic structure of H3S are a pair of van Hove
singularities bracketing $E_{F}$ Quan:2016 ; Sano:2016 . Shifting the position
of $E_{F}$, potentially by doping, could increase the number of states that
can participate in the electron-phonon coupling mechanism, and therefore
increase the $T_{c}$ of the system.

Doping is a common strategy used for precise tuning of $E_{F}$, and
computations using the virtual crystal approximation (VCA) suggested that the
addition of a little bit of phosphorus, carbon, or silicon could raise the
$T_{c}$ into the room-temperature regime $>$ 280 K Heil:2015 ; Ge:2016 ;
Ge:2020 . In the VCA, alchemical pseudoatoms are constructed from weighted
averages of the component atom potentials. The resulting chemical chimeras,
however, cannot accurately model the local structural and electronic effects
that arise when one atom is replaced with an entirely different element. This
throws off, in particular, the very dynamical response properties that one
must calculate carefully to obtain reasonable estimates of $T_{c}$. Additional
studies based on actual doped H3S models constructed as supercells have sought
to explore the local effects of doping Liu:2018c ; Amsler:2019 ; Guan:2021 ;
Wang:2022 , although the calculation of dynamical properties of the requisite
large unit cells can be prohibitively expensive.

Figure 10: The covalent hydride $Im\bar{3}m$ H3S Drozdov:2015a  (a) represents
a breakthrough in high-$T_{c}$ conventional superconductivity. Peaks in the
electronic DOS near $E_{F}$ have prompted numerous investigations on doped
versions of H3S, discovering phases such as the methane-intercalated (CH4)H3S
= CSH7 Cui:2020  (b), among many others. A recent synthesis of a room-
temperature superconductor consisting of carbon, sulfur, and hydrogen
generated even more momentum, with diffraction analysis performed at pressures
below those where the room-temperature superconducting transition were
observed, revealing the presence of a phase based on the $I4/mcm$ Al2Cu-like
structure adopted by the van der Waals (H2S)H2 phase Strobel:2011 ;
Lamichhane:2021 ; Goncharov:2022  (c).

One particularly promising system involved the addition of carbon to the H3S
lattice by way of methane intercalation Cui:2020 ; Sun:2020 ; Harshman:2022 .
Stoichiometries that are a linear combination of CH4 and H3S (the most simple
of which is CSH7) proved especially interesting. They yielded a variety of
dynamically stable (although energetically metastable) structures, which
differed in the orientation of the methane molecules encapsulated in the H3S
lattice. Some of the highest $T_{c}$s predicted for these phases were 181 K at
100 GPa for $I\bar{4}3m$ Sun:2020 , and 181-194 K for the $R3m$ symmetry
structure Cui:2020  shown in Figure 10b.

Independently, photochemical synthesis in the C-S-H system yielded the first
report of room-temperature superconductivity, achieving a $T_{c}$ of 288 K at
267 GPa Snider:2020 . This report has inspired a slew of follow-up work and
much debate Hirsch:2021 ; Gubler:2022 ; Goncharov:2022 . XRD analysis
performed at pressures below the purported room-temperature superconducting
transition Lamichhane:2021 ; Bykova:2021  is consistent with the Al2Cu
geometry (as well as an orthorhombic $Pnma$ variant) associated with CH4-H2
Somayazulu:1996  and H2S-H2 Strobel:2011  (Figure 10c). This may suggest an
overall stoichiometry of [(CH4)2H2]x[(H2S)2H2]y for the room-temperature C-S-H
superconductor, although subsequent phase transitions at higher pressure to
the high-$T_{c}$ superconducting phase cannot be ruled out. In fact,
additional studies indicate just such a structural transition occurs to form
the room-temperature superconducting phase, with indications of methane
signatures in the Raman spectra Goncharov:2022 . As was the case for the
binary H3S system, it appears that a panoply of metastable phases may be
accessible by slight variations on synthetic procedure, in particular on
carbon content Smith:2021 , offering plenty of space for further experimental
and theoretical discoveries.

In addition to sulfur-based covalent hydrides, phosphorus hydrides have
sparked interest after compression of a phosphine (PH3) sample yielded a
material that became superconducting at 30 K at 83 GPa, increasing to 103 K at
207 GPa Drozdov:2015 . The structure and composition of the responsible phase
or phases was unclear, prompting an array of follow-up studies levying CSP
techniques to identify plausible compounds Fu:2016 ; Liu:2016 ; Shamp:2016 ;
Flores-Livas:2016 ; Bi:2017 . Pressure was found to drive the decomposition of
phosphine into a variety of products with stoichiometries including PH, PH2,
PH3, and more. A predicted $C2/m$ PH3 phase Liu:2016  featuring P-P bonds (in
contrast to the H3S superconductor, which has no S-S bonding) was estimated to
be superconducting below 83 K at 200 GPa, in line with the experimental
values. Another study suggested that multiple metastable decomposition
products of phosphine, including those with PH and PH2 stoichiometries, might
in combination be responsible for the observed superconductivity Flores-
Livas:2016 . PH2 phases with $C2/m$ and $I4/mmm$ symmetries, differing by a
tilt in the component H-P-H moieties, had estimated $T_{c}$s of 76 and 70 K,
respectively Shamp:2016 . Later, another set of PH2 phases were proposed
consisting of simple cubic layers of phosphorus capped with hydrogen atoms and
further intercalated with H2 molecules acting as Coulombic spacing agents
Bi:2017 . At 80 GPa, these structures had estimated $T_{c}$s ca. 30 K, similar
to the values that were measured. Raman spectroscopic measurements provided
evidence for phosphine dimerization coupled with dehydrogenation under
pressure, yielding compositions such as P2H4 and P3H6 Liu:2018b ; Yuan:2019 .
In these phases, low temperatures were required to maintain stability at
multimegabar pressures. The $T_{c}$ of a predicted $C2/m$ P4H6 phase was
estimated to be 67 K at 200 GPa Yuan:2019 .

The plethora of metastable P-H compounds under pressure has prompted
computational investigations into ternary systems containing phosphorus and
hydrogen. Above 250 GPa, an $R\bar{3}$ LiP2H14 phase, consisting of P@H9
clusters spaced by Li atoms as well as isolated H atoms, achieves an estimated
$T_{c}$ of 169 K at 230 GPa (where it is metastable) Li:2020 . $Pm\bar{3}$
LiPH6, a colored variant of the A15 crystal structure adopted by intermetallic
superconductors Nb3Ge Gavaler:1973  and Nb3Sn Matthias:1954 , has an estimated
$T_{c}$ of 150-167 K at 200 GPa (where it is metastable) Shao:2019 . In the
S-P-H system, obviously tantalizing for its connection to the H3S
superconductor as well as to phosphine derivatives, relatively low $T_{c}$s
were predicted for phases on the high-pressure convex hulls, but low-lying
metastable structures based on phosphorus substitution into the $Im\bar{3}m$
H3S lattice were promising, including $Im\bar{3}m$ S7PH24, which had an
estimated $T_{c}$ of 183 K at 200 GPa Geng:2022 .

##  5 Conclusion

Although the entirety of the lived human experience resides within a vastly
narrow pressure range, the universe is not so simple. The chemistry we know at
1 atmosphere is not the chemistry of Jupiter, Saturn, or even the center of
our own planet Earth. Starting from the periodic table itself, the
ramifications of pressure are rapidly found to alter elemental behavior and,
consequently, how the elements interact with one another to form new and
bizarre phases. Potassium, in its guise as a “transition metal”, enjoys all
manner of new chemical interactions – in compound formation and in the wildly
complex electride elemental structures it adopts. Cesium can become anionic,
and helium takes an active role in stabilizing a network of sodium and
interstitial quasiatoms. Strange geometrical and bonding motifs from clusters
to networks abound.

Yet not only are the structures of phases – electronic and crystalline –
molded by high pressure, but high-pressure studies have revolutionized the
search for high-temperature superconductivity. Pressure induces
superconductivity in a plethora of elements, and drives the formation of
phases containing structural motifs whose atomic vibrations can be strongly
coupled to the underlying electronic structure. From the clathrate-like LaH10
to the covalent H3S – and the intensely-discussed CSH room-temperature
superconductor – the playing field of high-pressure materials is a promising
one for the future.

## Acknowledgments

We acknowledge the NSF (DMR-1827815) for financial support. This material is
based upon work supported by the U.S. Department of Energy, Office of Science,
Fusion Energy Sciences under Award Number DE-SC0020340 to E.Z. K.P.H. thanks
the US Department of Energy, National Nuclear Security Administration, through
the Chicago-DOE Alliance Center under Cooperative Agreement Grant No. DE-
NA0003975 for financial support. We thank Giacomo Scilla for his help in
editing and preparing the manuscript.

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