2021

[1]Eva Zurek

[1]Department of Chemistry, State University of New York at Buffalo, Buffalo,
14260-3000, NY, USA

# Materials under high pressure: A chemical perspective

Katerina P. Hilleke<EMAIL_ADDRESS>Tiange Bi<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

At high pressure, the typical behavior of elements dictated by the periodic
table – including oxidation numbers, stoichiometries in compounds, and
reactivity, to name but a few – is altered dramatically. As pressure is
applied, the energetic ordering of atomic orbitals shifts, allowing core
orbitals to become chemically active, atypical electron configurations to
occur, and in some cases, non-atom-centered orbitals to form in the
interstices of solid structures. Strange stoichiometries, structures, and
bonding motifs result. Crystal structure prediction tools, not burdened by
preconceived notions about structural chemistry learned at atmospheric
pressure, have been applied to great success to explore phase diagrams at high
pressure, identifying novel structures in diverse chemical systems. Several of
these phases have been subsequently synthesized. Experimentally, access to
high-pressure regimes has been bolstered by advances in diamond anvil cell and
dynamic compression techniques. The joint efforts of experiment and theory
have led to startling successes stories in the realm of high-temperature
superconductivity, identifying many novel phases – some of which have been
synthesized – whose superconducting transition approaches room temperature.

###### keywords:

Pressure, crystal structure prediction, superconductivity, electronic
structure

##  1 Introduction

Our chemical knowledge and intuition, which has accumulated from our first
chemistry classes and experience living at atmospheric conditions, can be
completely knocked askew when the pressure variable is taken into
consideration. The range of pressures in our universe is wide, from 10-32 atm
in the vast spaces between galaxies all the way up to 10+32 atm in the core of
a neutron star. In the core of our own planet, Earth, pressures can reach up
to $\sim$3.5$\times 10^{6}$ atm (350 GPa), while many giant planets have core
pressures in the TPa regime. Understanding how chemistry changes under
pressure is key to uncovering how materials behave in these environments.
Experimental techniques based on diamond anvil cells (DACs) can access static
compression in the multi-megabar regime 1 ; Dubrovinsky:2015 ; Jenei:2018 ;
Dubrovinskaia:2016 ; Dewaele:2018 , while even higher pressures are accessible
via dynamic compression using shock wave or ramp techniques. 2 ; Hansen:2021 ;
Fratanduono:2021

However, producing these high pressures environments and analyzing the
resulting data is difficult. Zurek:2014i  Light elements tend to diffuse into
the diamonds of the DAC cell, weakening and breaking the apparatus, which is
financially expensive and wastes research time. Unintended side reactions
within the DAC can result in decompositions, unwanted product formation, and a
mixture of byproducts alongside the actual synthetic target. Even following a
successful synthesis, characterization is complex, since the capabilities of
X-ray diffraction (XRD) are limited for light elements. The pressure limit of
neutron diffraction is not high enough for many experiments, causing the
indirect-probe techniques of Raman and IR spectroscopy to step into the gap.
These difficulties have led to the development of a symbiotic relationship
between theoreticians and experimentalists, in which theoretical calculations
into promising novel compounds provide plausible synthetic targets, and aid in
structural characterization and analysis of properties once experimental
phases are observed. H3S and LaH10, two high-pressure phases with record-
breaking superconducting critical temperatures, $T_{c}$, have both been
investigated in such a symbiotic partnership. Strobel:2011a ; Duan:2014 ;
Li:2014 ; Drozdov:2015a ; geballe2018synthesis ; drozdov2019superconductivity
; Pers105 ; Peng:2017a ; Liu:2017-La-Y ; Kim:2008-Al

As is evident from the strange elemental ratios in the aforementioned iconic
superconductoring hydrides, pressure can have a profound effect on the
compositions, structures, properties, and stabilities of solid phases.
Hemley:2000 ; Song:2013a ; Grochala:2007a ; Goncharov:2013b ; Bhardwaj:2012a ;
Dubrovinsky:2013a ; McMillan:2013a ; Klug:2011a ; Naumov:2014a ; Zurek:2014i ;
Zurek:2016b ; Hermann-lip ; Zurek:2019k ; Zurek:2021k  Experimentally observed
compounds often could not have been predicted using the rules and bonding
schemes that stem from our ambient-pressure-taught intuition. For these
reasons, methods based on data mining 15  and chemical intuition 16 , tend to
break down when predicting solid phases under pressure. Therefore,
theoreticians have made use of crystal structure prediction (CSP) techniques,
which aim to pinpoint the global minimum as well as interesting local minimum
phases on the potential energy surface (PES) at a chosen pressure and
temperature. Once the unit cells and atomic configurations – often quite alien
to those seen at ambient pressure – are identified, theoreticians can
calculate the structural and electronic properties of the resulting phases,
either to compare with or to guide experimentalists. Theoretical calculations
can also probe pressures beyond those which are currently accessible
experimentally.

Herein, we first give an overview of several commonly used CSP algorithms,
followed by a survey of the unique behavior of matter compressed under extreme
pressures. We start by outlining the changes in electronic structure that are
induced by pressure, tracking how they manifest in strange stoichiometries,
structures, and bonding patterns in the phases that result. Finally, we
discuss the exciting superconducting phases that have emerged in high-pressure
experiments.

##  2 Crystal Structure Prediction

The problem tackled by CSP is one of global optimization. Within this
framework, the unit cell parameters and atomic positions – in other words, the
geometric coordinates of the crystalline structures – are the parameters that
are varied towards achieving the goal of minimizing the free energy. 18 ; 19 ;
20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29  These parameters include
three unit cell vectors, three unit cell angles, and 3$N-$3 degrees of freedom
in atomic positions, resulting in a total of 3$N+$3 parameters that need to be
determined to identify a global minimum structure in a PES for a phase with
$N$ atoms in the unit cell. The resulting problem is very difficult, for
several reasons. For one, it has been shown that the number of local minima in
the PES increase exponentially with the number of atoms in the unit cell. 30
Secondly, the computational cost of optimizing structures to the nearest local
minimum can be quite expensive, especially for the (typically) thousands of
structures that need to be considered in a single run. For this reason, early
CSP studies employed empirical potentials for small solid systems such as
Li3RuO4 31  and molecules including fullerene clusters. 32  Therefore, CSP is
an NP-hard (nondeterministic polynomial time hard) problem, since locating the
global minimum of homogeneous and heterogeneous systems both 55 ; 56  has no
algorithm that scales as a polynomial in the possible degrees of freedom.

Taking all of this into account, common meta-heuristic algorithms have been
designed to search for good (low-energy, stable) structures, but they are not
guaranteed to pinpoint the global minimum due to the sheer complexity of the
problem. However, it can be fruitful to analyze even the local minima that are
identified from such searches because they may possess promising properties,
and many are synthetically accessible. Various CSP methods have been designed
to sample the PES in different ways based on the situation at hand.

Some CSP methods, including random structure searches 23 ; 37 , evolutionary
(EA) and genetic algorithms (GA) 38 ; 39 ; 40 ; 41 ; 42 ; 43 ; 44 ; 45 , and
particle swarm optimization (PSO), 22 ; 46  search throughout the entire PES.
If minimal to no information is known regarding the target structure(s), this
type of method is preferred, because they start by sampling the entire PES to
identify promising regions and only then narrow their focus. Other methods,
such as metadynamics 35 , simulated annealing 36 , basin hopping 33 , and
minima hopping 34 , thoroughly explore only a particular, selected region of
the PES by overcoming energy barriers for a given starting structure, and
therefore multiple runs with various starting phases might be necessary.
Hybrid methods combining the aforementioned techniques also exist. 47 ; 48
Since interatomic potential parameters are often unreliable for squeezed
crystalline structures, a common practice for high pressure research is to
couple CSP with density functional theory (DFT) for local optimizations that
relax the crystal structures to the nearby minima. Some methods tackle this
problem by concurrently fitting DFT results produced during the search to
update parameters on-the-fly. 49 ; Tong:2018 ; Deringer:2018 ;
Podryabinkin:2019 ; Tong:2020

In order to acquaint the reader with various CSP techniques, we will next
describe the details of several popular methods that were mainly developed for
small inorganic crystals where each atom is treated independently. Although
the methods can be modified to include constraints on inter-molecular
interactions, 50  that would necessitate taking into consideration flexibilty
in conformation, nearly iso-energetic polymorphs of molecular crystals, and
large unit cells, 51  rendering it beyond the scope of the review. Other CSP
algorithms not included in the discussion herein are the ones for large
systems at ambient conditions involving empirical information, data mining or
machine learning. 15 ; 52 ; 53 ; 54

###  2.1 Potential Energy Surfaces

Potential energy surfaces or landscapes are highly dimensional, describing how
the energy of a system varies as a function of the 3$N+$3 degrees of freedom.
PESs are decorated with numerous local minima corresponding to potential
configurations of the $N$ atoms within some unit cell. The local minima
manifest as “valley”-like shapes in the PES, and they are separated by energy
barriers. Each structure defined or generated by a CSP algorithm typically
undergoes a process termed local optimization, where its energy is minimized
to reach the nearest local minimum. The PES may contain various features,
illustrated in Figure 1: 1) basins, within which all structures will optimize
to a single local minimum, 2) super basins, the domains in the potential
energy surface with multiple basins located next to each other, and 3)
funnels, super basins with multiple local minima that are separated by small
barriers.

Figure 1: Two-dimensional schematic of a potential energy surface, with red
dots indicating single structures at various locations. One dot sits at the
global minimum, the lowest energy point across the entire PES, while anoter
sits at a local minimum in the PES. A third sits atop a peak in the PES
representing a transition state between two funnels that each contain many
local minima. A fourth will, upon local optimization represented by a dotted
line, go to the local minimum of the basin in which it lies. Adapted with
permission from Zack, F.; Avery, P.; Wang, X.; Hilleke, K. P.; Zurek, E. _J.
Phys. Chem. C_ 2021, 125, 1601. XtalOpt-2020  Copyright 2021 American Chemical
Society.

Locating the global minimum of large and complex systems can be very
difficult. The “no free lunch” theorems conjecture that the success rates of
all algorithms will be the same when averaged over all PESs, 57  meaning that
varying the method employed will not necessarily help to improve the search
for a particular system. However, certain features of the PES can be leveraged
to the advantage of the researcher, even for a structure with many atoms in a
complex unit cell.

For example, there is no need to sample the entire PES, since the chemically
desired (physically reasonable) structures are typically located only in
certain areas, while the rest of the surface involves configurations with high
energy structures where atomic distances are too close or too far for them to
be energetically competitive. Removing these undesired regions from the
searches greatly accelerates the CSP run. Another property of the PES is that
low energy configurations occupy the largest amount of “hyperspace” in a PES,
58  so that randomly generated structures tend to fall into low energy basins
rather than regions with high energy. Finally, previous studies suggested that
low-energy basins are more likely to be located close to each other, with the
energy barriers between them being small. 59 ; 60  Thus, a configuration
falling in a random place in a funnel can be lightly perturbed, allowing it to
overcome the energetic barriers and continually move to lower energy until it
reaches the global minimum within the funnel. Overviews of the commonly used
CSP techniques that have been used to predict the structures of high pressure
materials are given below.

###  2.2 Following Imaginary Phonons

If the target compound resembles a known structure – a distortion of a highly
symmetric parent phase as in, for example, families of perovskites, one way to
carry out CSP is simply to follow imaginary phonon modes. Following local
optimization of a given starting configuration and calculations of its phonon
band structure, the presence of imaginary phonon modes indicate that a lower-
energy phase in the PES could be accessed by displacing atoms along the
eigenvectors of the soft phonon modes. The structure is modified with the
displaced atoms – typically resulting in a lower-symmetry structure – and a
subsequent local optimization is performed. The process needs to be repeated
until the final structure is located at a local minimum, with no imaginary
phonon modes. One special case that can occur is when the softest mode is
located at the $\Gamma$ point, in other words, the center of the Brillouin
zone. In this case, the perturbed structure would have the same unit cell as
the original but with the atoms moved along the displacement vectors.
Following these sorts of structural pathways allows us to enrich our chemical
knowledge, by getting a better of understanding of the relationships between
various structure types, and how symmetry breakings affect electronic
properties and energies.

Extensive predictions of complex systems have been done by following imaginary
phonon modes, and many of them show interesting properties. For example,
distortions along soft phonon modes from a rock-salt lattice in late
transition metal oxides lead to complex crystal structures 64  in which the
distortions are coupled either to magnetic ordering or Jahn-Teller effects.
Several structures explored for their promise as high-temperature
superconductors were identified by following soft phonon modes, including
silane 66 ; 65 ; 67  and a derivative of H3S intercalated wth methane CSH7 .
Distorted perovskite phases based on concerted tilting or rotations of
octahedra are frequently demonstrated to arise from atom displacements along
imaginary phonon modes in a high-symmetry cubic structure61 ; 62 ; 63 ;
Kaczkowski:2021 . High-pressure conversions between zircon, scheelite, and
fergusonite structures in MXO4 phases including CaMoO4Panchal:2017  and
YVO4Mittal:2019  can also be understood through this lens.

Although this method draws a clear route from inital structure to final
answer, there are limitations: the original structure should be close to the
target in the PES, in other words, they should have similar configurations.
Reorienting octahedra in a perovskite lattice requires little in terms of
atomic rearrangement; other high-pressure structures may be far more alien.
Another disadvantage is that the actual practice can be computationally
expensive even within DFT, since phonon calculations must be performed across
the entire Brillouin zone.

The method of following soft phonon modes can be combined with several of the
CSP techniques described below, however, by using their output to define the
initial starting guess. Because CSP techniques are typically limited to small
unit cell sizes, the most stable phases they find may show soft modes away
from the Zone center, which serve as a guide to a related, dynamically stable
compound.

###  2.3 Simulated Annealing

Another technique that probes local regions of the PES is simulated annealing,
which was inspired by the heating and quenching process of metallurgy that
recrystallizes materials 36 . To start, a random configuration of atoms is
generated and its energy assessed. From this starting point, a new structure
is produced by some combination of perturbation, random atomic displacement,
and atom permutation of the initial structure. The probability $P$ of the new
structure being accepted is based on the Metropolis Monte Carlo algorithm,
where $P=-\Delta E/k_{B}T$ ($\Delta E$ being the energy difference between the
initial and new structures and $k_{B}$ is the Boltzmann constant). The
variable $T$ is the “simulation temperature”, distinct from physical
temperature, that is employed in the mathematical expression controlling the
acceptance rate of new structures. After a period of “heating”, where values
of $T$ are large, a random number $\epsilon$ (0 $<\epsilon<$ 1) is used as a
cutoff value so that new structures will be accepted only if $\epsilon$ is
smaller than $P$. As the process iterates the $T$ variable is continually
decreased – analogous to the slow cooling of a solid material from the melt to
form crystals – to tighten the criteria of structure acceptance, as
illustrated schematically in Figure 2a. Once $T$ is reduced to 0, the process
will only accept structures with lower energies than the parent phase,
locating the neighboring local minima.

Figure 2: Schematic illustration of the ways in which CSP techniques traverse
a three-dimensional PES, including: (a) simulated annealing, (b) minima
hopping, (c) metadynamics, (d) random searches, (e) particle swarm
optimization, and (f) genetic algorithms. Adapted with permission from Zack,
F.; Avery, P.; Wang, X.; Hilleke, K. P.; Zurek, E. _J. Phys. Chem. C_ 2021,
125, 1601. XtalOpt-2020  Copyright 2021 American Chemical Society.

The success of this method relies significantly on choosing a good starting
structure. In addition, it is assumed that the high-temperature PES has a
similar landscape to the low-temperature PES, so that the local minima before
and after cooling are the same, but this does not always hold true. More
complicated search schemes, which can involve multiple heating/cooling cycles
and random mutations, have been proposed as techniques that explore broader
regions of the PES. However, random mutations in simulated annealing have to
be carefully handled for two reasons: 1) mutations that are too large can
cause a run to sample a large portion of the PES, effectively the same as a
random structure search, and 2) too-small mutations can fail to effectively
perturb the system. Molecular dynamics can also be used to perturb the
structures.

The application of the simulated annealing method has been fruitful, coupled
in many cases to Hartree-Fock and DFT methods. 20 ; 77  Many inorganic crystal
structures, including in the Pb-S systems, 78  Ca-C systems, 79  metal-N
systems, 82  B-N systems, 80  and Ge-F systems, 81  have been identified using
this method.

###  2.4 Basin and Minima Hopping

Similar to simulated annealing, the basin and minima hopping methods rely on a
judicious choice of starting structure. Basin hopping, developed mainly for
finite cluster systems 33 ; 83 ; 84 ; 85 ; 86 , relies on the Monte Carlo
algorithm to evaluate the fitness of proposed structures to be accepted or
rejected based on a carefully selected “temperature” cutoff. In this method,
all structures in a basin undergo relaxation to the same local minimum, so
that the PES is transformed into a series of connected step functions that are
each related to a local minimum. No penalty is imposed for sampling a region
of the PES that has already been searched.

Although the minima hopping method is very similar in design to basin hopping,
it uses molecular dynamics rather than Monte Carlo to traverse the PES. As
such, a temperature variable is not required. Instead, after relaxing a
structure into a nearby local minimum, the cutoff energy difference between
the original and optimized structures, labeled $E_{\text{diff}}$ in Figure 2b,
is set so that half of the new structures are accepted. Using soft phonon
modes to define inital trajectories for the molecular dynamics step can
increase the efficiency of the method, in-line with the Bell-Evans-Polanyi
principle regarding relative energies of local minima and the energetic
barriers surrounding them. Roy:2008 ; Sicher:2011  The algorithm can also
selectively favor exploring new regions of the PES by artificially raising the
kinetic energy employed in the molecular dynamics step. This mechanism is
invoked when already explored regions of the PES are revisited.

High pressure phases investigated by minima hopping include various polymorphs
of elemental carbon 88 ; 89 , as well as superconducting hydrides of
phosphorus, 90  sulfur, selenium, 91 , silicon 87  and ternary SxSe1-xH3
hydrides Amsler:2019 , as well as to identify new high-pressure structures of
SrTiO3Khajehpasha:2021 . Several binary systems that are known to be
immiscible at ambient pressures have been investigated with the minima hopping
method, including the intermetallic Fe-Bi Amsler:2017 ; Amsler:2018 , Cu-Bi
Amsler:2017b ; Clarke:2017 ; Amsler:2018 , and Ni-Bi Powderly:2017 ;
Amsler:2018  systems. Using a linear approximation to the enthalpy to predict
the stabilities of compounds found in 1 atm databases, the minima hopping
method identified stable phases in the immiscible-at-ambient-pressure systems
As-Pb, Al-Si, Sn-Bi, Fe-In, Hg-In, Hg-Sn, Re-Sn, Re-Br, and Re-Ga. Amsler:2018

###  2.5 Metadynamics

The metadynamics method is based on molecular dynamics simulations.
Exploration of the full PES is encouraged by filling already-visited areas
with Gaussian functions, represented by the dotted lines in Figure 2c, to
overcome barriers between known and unknown regions. 35  This so-called
“basin-flooding” is typically performed in a space defined by a particular
collection of variables, fewer than the 3$N+$3 needed to represent the full
crystalline lattice. Metadynamics has been frequently applied to studying
phase transitions 94 ; 95 , structural changes in solutions, and chemical
reactions, 35 , but has also been employed to predict several high pressure
structures, including phases of Ge 92 , Ca, 71 , N, Adeleke:2017  and carbon
dioxide. 93  One shortcoming of the metadynamics method is that it may flood a
transition basin separating multiple minima, thereby making it impossible to
find the global minimum. Thus, unless a good starting structure is known, it
is important to perform multiple searches to ensure adequate exploration of
the PES.

###  2.6 Random (Sensible) Structure Searches

The previously discussed CSP algorithms – following soft phonons, simulated
annealing, basin hopping, and metadynamics – all excel in the fine exploration
of particular regions of the PES and to some extent rely on a reasonable
structural guess as a starting point. The following three metaheuristics,
starting with random searches and proceeding to particle swarm optimization
methods and genetic/evolutionary algorithms specialize instead in surveying a
full PES to eventually narrow in on a final region.

Of these, the random search method is rooted in the most straightforward idea:
to generate crystal structures with randomized unit cell parameters and atomic
coordinates, and then to relax these structures to the closest local minima,
as illustrated in Figure 2d. This ensures wide coverage of the full PES, not
biased to any particular region. However, since randomly generated structures
might contain unrealistic interatomic distances, falling into unhelpfully high
energy regions of the PES, constraints can be applied to avoid atoms occupying
positions that are too near or too far from one another. If experimental
observations or chemical intuition confer knowledge regarding lattice
parameters, types, or space groups, such constraints can also be applied. For
structures containing molecules, quasi-molecules, or clusters, connectivity
constraints can further assist the procedure. Therefore, random structure
search methods can also be described as “sensible structure searches”. 23

The simplest random structure searches generate completely independent
structures, and therefore do not learn from their history. One algorithm that
can, in principle, learn during the course of its trajectory is the ab initio
random structure searching method (AIRSS). 23 ; 37  AIRSS allows users to
apply various constraints as outlined above. Users may also invoke a “shaking”
feature, which randomly mutates previously identified stable structures
through atomic displacement and unit cell deformation, thereby enabling the
search to overcome barriers between phases.

Random searches have been successfully used to predict many structures,
including elemental phases of hydrogen 69 ; 70 , lithium 71 ; 72 , nitrogen
Turnbull:2018 ; Sontising:2020 , and germanium Kelsall:2021 . Additionally,
high pressure phases of titanium dioxide 73 , silane Pickard:2006 , xenon
oxides Dewaele:2016 , MoBi2 Altman:2021 , as well as several high-temperature
superconducting hydrides Li:2019  have been explored through random searching
methods.

###  2.7 Particle Swarm Optimization

Inspired by the behavior of swarms of animal groups (i.e. flocks of birds,
schools of fish), 96  the particle swarm optimization algorithm, 46
implemented within the Crystal Structure AnaLYsis by Particle Swarm
Optimization (CALYPSO) code 22 , incorporates the positions and PES
trajectories of the collected group of putative structures to determine the
path a particular structure will take to traverse the PES.

The first applications of the particle swarm method focussed on predicting
cluster structures 97 . By now, the CALYPSO code is one of the most popular
methods for high pressure CSP, and has been employed to successfully predict
many novel materials, including multiple superconductors with high critical
temperatures 98 ; 99 ; 100 ; 101 ; Liu:2017-La-Y ; Li2MgH16 ; Sun:2020 ;
Chen:2021 , structures of elemental Li 148 , superhard B-N binary phases
Yang:2014 ; Zhang:2021 , Ce-F compounds, 102  and hydrides of Cl. 103

In the PSO algorithm, a set of random structures are created with constraints
on interatomic distances, volumes, and symmetry. These undergo local
relaxation to the nearest local minima. Each individual’s path through the PES
is defined both by its previous route and the traveling history of the other
individuals, expressed as a position plus a velocity. The velocity,
$\nu(t+1)$, is given as

$\nu(t+1)=\omega\nu(t)+c_{1}r_{1}(lbest(t)-x(t))\\\
+c_{2}r_{2}(gbest(t)-x(t))$ (1)

where $\omega$, whose range is between 0.4 and 0.9, is an inertia term that is
varied on-the-fly ; $c_{1}$ and $c_{2}$ are parameters comparing the current
position of the individual to the global minimum; $r_{1}$ and $r_{2}$ are
random numbers falling between 0 and 1; $x(t)$ is the current position of the
individual; $lbest$ is the current local minimum that the individual is
relaxed into; and $gbest$ is the best position, or current global minimum, of
the swarm. An example of this trajectory calculation is shown in Figure 2e.

The PSO approach has the dual benefit of continually learning from the search
history, while at the same time exploring the most promising regions of the
PES. To ensure the search does not become too narrow, new random structures
are continuously injected, thereby broadly sampling the rest of the PES.

###  2.8 Evolutionary Algorithms

In the past two decades, a plethora of CSP codes have been developed based on
evolutionary (or genetic) algorithms (EAs/GAs), including XtalOpt, 38 ;
XtalOpt-2020 ; XtalOpt-Original ; XtalOpt-r10 ; XtalOpt-r11 ; XtalOpt-r12 ;
104 ; 105  GASP, Tipton:2013  USPEX, 40 ; 106 ; 107  MAISE, 41  EVO, 42 , and
other codes developed by Trimarchi et al., 43 ; 108 , Abraham et al., 44
Fadda et al., 45 , Woodley and Catlow Woodley:2009 , Hammer et al. Hammer:2018
, and the “adaptive-Ga” of Wentzcovitch et al. 49 . These codes are coupled
with periodic first-principle simulation packages or interatomic potentials
for local optimizations. The evolutionary algorithm draws inspiration from the
concepts of evolutionary biology, wherein selection, mutation, and
reproduction operations cooperate to produce fit organisms – in the case of
CSP, global minima on a PES. The terms EA and GA are often used
interchangably, even though technically the two differ in the way that
offspring are created. In EAs the operators act on the structures in real
space, whereas in GAs the structural variables are first mapped onto a string,
akin to a chromosome, and the heredity operations are performed on these.

The first step in an EA search is to generate a set of random structures that
can be optionally constrained by a series of parameters including interatomic
distances, unit cell volumes and lattice parameters, space groups, and
molecular units. Optionally a search can be seeded by candidate structures, if
they are known. Following local optimization into the nearest local minimum,
the fitness of each structure is evaluated based on a thermodynamic variable
such as enthalpy or energy, which can in certain cases be weighted alongside
another variable, such as Vickers hardness XtalOpt-r12 . The fittest
structures serve as parents for the following generation, passing down
whichever structural features promote stability.

Offspring can be constructed via “breeding”, or crossover, operations in which
the unit cells of two parent structures are combined, or “mutation” operations
in which a single parent unit cell is perturbed in some way, as illustrated in
Figure 2f. The crossover operation, originally implemented for the prediction
of clusters 32 , involves rotating parent structures into a random
orientation, then cutting both and splicing them back into a single child.
Mutation operations range from permutation and exchange operators, which
switch atomic positions of different elements, to ripple operators which
displace atoms according to a periodic function, to strain operators which
alter the unit cell parameters entirely. They can also be combined with one
another - for example, the “stripple” operation implemented in XtalOpt 105
involves both strain and ripple processes.

The success of EA methods is demonstrated in the large number of high-pressure
crystal structures they have identified. These include hydrogen-rich phases
with “non-classical” stoichiometries 109 ; 110 ; 111 ; 112 ; 113 ; 114 ; 115 ;
116 ; 117 ; 118 ; 119 ; 120 ; 121 , elemental phases of boron Oganov:2009 ;
Hilleke:2021  and sodium 142 , binary systems ranging from Xe-O 122  to Sc-N
Aslam:2018  to Li-B 123 ; 126  and Be-B, 124 ; 125  (and the Li-Be-B ternary
system 127 ) alongside other ternary systems including Li-F-H LiFHn  and La-
B-H DiCataldo:2021 .

##  3 Chemistry Under Pressure

The application of external pressure has drastic consequences for materials
behavior. To a first approximation, an increase in pressure corresponds with a
decrease in the volume of condensed matter, driving up the importance of the
$PV$ contribution to system enthalpy in the expression $H=U+PV$. In such a
framework, structures that minimize the total volume, with high coordination
numbers and simple, highly symmetric atomic arrangements, should be the
natural endpoint of matter under extreme compression. These trends have been
encapsulated by various “rules” for materials at high pressure proposed by
Prewitt and Downs 128 , Grochala et al. Grochala:2007a  and Zhang et al.
Zhang:2017  Since our chemical intuition, trained at 1 atm, is hard-pressed to
generate meaningful insights at high pressures, these rules have been
developed from swathes of experimental data. Nevertheless, strange chemistry
is afoot, as supposedly simple metals such as Na and Li adopt open framework
structures 142 ; 147  and bizarre stoichiometries and structural motifs crop
up.

In addition to mechanical consequences, applied pressure influences the
electronic structure of compressed matter, altering the relative energies of
atomic or molecular orbitals. As a result, unfamiliar oxidation states,
reactive core electrons, and non-atom centered orbitals emerge in compressed
matter, manifesting in strange stoichiometries, structures, and bonding
arrangements. Elements that are immiscible or inert at ambient pressure can
form compounds. The resulting properties are diverse, ranging from
superconductors to topological materials.

In the following sections, we provide an overview of the many fascinating
results from the high pressure research community that demonstrate the
fabulous chemical and physical properties emerging under pressure. The
sections are organized according to different phenomena that arise at high
pressure – but as many stem from the common root of reordered orbital
energies, they are strongly coupled. Novel electronic structures result in
strange stoichiometries, which also involve bizarre bonding schemes. As such,
some examples will appear in multiple places, emphasizing different facets of
their novelty.

###  3.1 Electronic Structures

Pressure can have a profound impact on the electronic structure of matter.
Almost a century ago, Bernal proposed that all matter will become metallic
upon sufficient compression – including hydrogen. Around ten years later,
Wigner and Huntington further developed this idea, proffering 25 GPa as an
estimate of the pressure above which hydrogen would transform to a monoatomic
metallic solid phase. 130  At ambient pressure, hydrogen exists as a diatomic
molecule with a large energy gap between its highest occupied and lowest
unoccupied molecular orbitals. Under applied pressure, hydrogen molecules are
forced closer together, and bands are formed via orbital overlap. The
increased orbital overlap induced by applied pressure raises the energy of the
valence and lowers the energy of the conduction bands, eventually closing the
gap between them. Concurrently, the inter- and intra-molecular H-H distances
equilibrate to produce an equally spaced chain of hydrogen atoms – a state for
which the top of the valence band and the bottom of the conduction band are
degenerate in energy.

Experimental forays into metallizing hydrogen rapidly ascertained that the
original suggested applied pressure of 25 GPa was a gross underestimate 131 ;
132 ; 133 ; 134 ; 135 . At the same time theoretical studies sought to
ascertain the high-pressure structural transitions of elemental hydrogen
Naumov:2014a ; 70 ; 136 ; 69 ; 138 ; 139 , uncovering complex crystal
chemistry. Structures ranging from those that are layered, to ordered arrays
of H2 molecules to monatomic arrangements (Figure 3a) have been identified by
computations. The search for metallic hydrogen has become something of a “holy
grail” for the high-pressure research community, driving both theoretical and
experimental advances.

Figure 3: High pressure alters the electronic structure of atoms in materials.
Some of the fascinating compounds that have been predicted include: (a)
$I4_{1}/amd$ phase of metallic hydrogen, predicted ca. 500 GPa with monatomic
H (lines do not denote bonds, but serve as a guide to the eye), 70  (b)
electride $hP4$ structure of Na, with ISQs present in void spaces, 142  (c)
HgF4 where Hg assumes a +4 oxidation state, 155  and (d) Li3Cs, where anionic
Cs takes on a charge that is less than $-1$. 156

The metallization of hydrogen is one example of how pressure affects
electronic structure, and other remarkable consequences have been predicted or
observed. For instance, contrary to the expectation that under pressure all
materials should assume simple close-packed structures and become metallic,
several elements go rogue. Around 200 GPa, sodium transforms into an
insulating and transparent phase with a band gap of 1.3 eV. 142 ; 144 .
Theoretical predictions, later corroborated by experimental work, found a loss
of metallicity in Li above 70 GPa 145 ; 146 . What, then, has happened here?

As pressure increases, the electrons associated with an atom or ion occupy a
smaller and smaller space, therefore their quantized energy levels increase in
energy. An electron associated with a void or interstitial region in a
crystalline lattice would experience the same rise in energy – except that
such an electron, unlike a valence electron of an atom, would not have to
maintaining orthogonality to core electrons, thereby having less space in
which it could move. The decreased space results in a higher kinetic energy.
At some point, it would be energetically preferable for an electron to occupy
the interstitial spaces rather than the valence atom-centered orbitals. This
produces an _interstitial quasiatom_ (ISQ), which possesses quantized energy
levels for electrons Rousseau:2008 ; 149 ; Miao:2015 , and just like atom-
centered electrons the ISQ can participate in bonding and chemistry Miao:2017
; Miao:2015 . The ISQ-containing materials are electrides, which are well-
explored under ambient conditions. Dye:2009  In the case of Na, under
sufficient pressure its 2$p$ core orbitals start to overlap, squeezing its
valence 3$s$ electrons into the interstitial sites (Figure 3b) and rendering
the phase insulating, 142 ; 144  while in the case of Li it is the 2$s$
electrons that are localized into interstitial sites, with an array of
structures predicted. 71 ; 72 ; 147 ; 148 ; 145

A systematic investigation into the propensity towards ISQ formation used a
helium “compression chamber” comprised of a supercell fcc lattice of He atoms
whose unit cell parameters could be varied to simulate pressure. In this model
the central atom was either left empty (to model an ISQ) or replaced with
another element.149  The orbital energies of the ISQ model could then be
compared with those of the lone, compressed atom. Li and Na were found to
favor the formation of an ISQ at the lowest pressure of any element, followed
by Al and then Mg at much higher pressures. This study found that ISQ
formation is favored for elements with lower ionization potentials (reflecting
the potential facile loss of an electron to an interstitial region), with
relatively incompressible core electrons. Moreover, because the energies of
vacant $d$ orbitals do not increase as rapidly as occupied $s$ orbitals, ISQ
formation was stymied via $s\rightarrow d$ electronic transfer in elements
such as Cs. 149

This different rates of energy increase with pressure for atomic orbitals has
consequences beyond favoring or disfavoring electride formation. Under
pressure, $s\rightarrow p$, $s\rightarrow d$, $s\rightarrow f$, and
$d\rightarrow f$ electronic transitions drive strange behavior across the
periodic table.Rahm:2019  For example, cesium becomes a d1 metal by 15 GPa.
150 ; 151  With continuing pressure increases, the energetic reordering of Cs
atomic orbitals causes a sequence of phase transitions until reaching Cs IV, a
non-close-packed tetragonal structure, stable from 4.3 to 12 GPa. 152 .
Similar to what we saw for Li and Na, Cs IV is categorized as an electride,
with the maxima in the valence charge density found in the interstitial sites.
Above 70 GPa, a close-packed structure resurfaces with hybridized 5$p$ (semi-
core) and 6$d$ orbitals near the Fermi level. 154

The pressure-induced orbital reordering can also result in unusual oxidation
states. Typically at ambient pressure, the oxidation states assumed by each
element are dictated by their position on the periodic table relative to the
noble gases – gaining or losing only so many electrons as needed to obtain a
closed-shell configuration (or a half-filled d shell). Under pressure, though,
the oxidation states of mercury are predicted to diverge from the typical $+$2
to $+$3 and $+$4 in the mercury fluoride phases of HgF3 and HgF4 (Figure 3c)
at 40 and 75 GPa, respectively. 155  In HgF4, two Hg 5$d$ electrons are used
in forming bonds with four neighboring F atoms along with the usual 6$s$
valence electrons. Other predicted examples are the Li1-5Cs phases, in which
the Cs atoms assume an oxidation state beyond $-$1 due to electron transfer
from Li 2$s$ to the Cs 5$d$ orbitals. 156  The most negative charge on Cs is
seen in Li3Cs (Figure 3d). Moreover, a series of CsFn compounds wherein the Cs
5$p$ orbitals bond with the F 2$p$ orbitals, allowing Cs to access charges
larger than +1 have been studied. 102  The atypical electronic configurations
available under pressure result in curious stoichiometries and structural
motifs in these compressed phases.

###  3.2 Novel Stoichiometries

At atmospheric pressure, one can typically deduce the stoichiometry of a
binary compound using typical oxidation states to deduce their relative
ratios, e.g. Na+ and Cl- will form NaCl. However, under pressure the
stoichiometry of compounds can become very different from what is observed at
ambient conditions, and some elements that are immiscible or nearly so at 1
atm might undergo compound formation.

For example, pressure opens to binary combinations of sodium and chlorine a
range of stoichiometries beyond the simple 1:1: Na3Cl, Na2Cl, Na3Cl2, NaCl3,
and NaCl7 have all been predicted to form under pressure, and Na3Cl and NaCl3
have been synthesized. 171  As described in the previous section, cesium is a
component of multiple compounds with atypical stoichiometries. At atmospheric
pressure, CsF forms in the rock salt structure, with formal charges of $+$1
for Cs and $-$1 for F. Under pressure, computations have shown that the Cs
5$d$ orbitals become higher in energy than the F 2$p$, allowing Cs to attain
oxidation states up to $+$3 in CsF3 and $+$5 in CsF5.102 ; Miao:2017b  CsF2
and CsF4 stoichiometries with Cs in $+$2 and $+$4 oxidation states have also
been predicted. 164  With Li, Cs has been computed to form Li1-5Cs phases,
with Cs taking on the anionic role – in line with predicted changes in the
relative electronegativities of the alkali metals under pressure. 156 ;
Rahm:2019

A wide array of metal-hydrogen systems have been predicted to take on
stoichiometries under pressure that at first may sound absurd. In the Li-H
system, the LiH stoichiometry, which is the only one stable at 1 atm, is
joined by LiH2-8 109  and Li4-9H 117  phases under pressure. On the Li-rich
side, two nearly isoenthalpic Li5H structures with $Cmc2_{1}$ and $Abm2$
symmetry were predicted to be most enthalpically favorable. 117  Both are
built of Li8H building blocks that can be viewed as distorted bicapped
trigonal antiprisms with Li atoms on the vertices and H atoms at the center
(Figure 4a). The Li8H block can be thought of as a superalkali atom with one
more electron than a closed-shell number of eight, similar to building blocks
in suboxide compounds such as Rb9O2 and Cs11O3. 174  Since the
electronegativity difference between Li and H is so large, electron transfer
occurs from Li to H atoms, resulting in Li+ cations and hydridic hydrogen,
with the extra electrons donated by Li treated as anions to balance the
charge. The interactions within the Li8H cluster are treated as ionic, with
metallic bonding on the outside.

Figure 4: Examples of strange stoichiometries that arise in compressed
materials: (a) predicted Li5H ($Abm2$) built from Li8H subunits that behave as
superalkali atoms, 117  (b) synthesized MoBi2 in the CuAl2 type, with square
antiprisms of Bi containing Mo atoms, Altman:2021  (c) synthesized
$Pm\bar{3}m$ phase of XeFe3, 172 ; Stavrou:2018  and (d) synthesized
$Fm\bar{3}m$ Na2He, an insulating electride phase. Dong:2017

On the the hydrogen-rich side of the Li-H system, several phases were found to
be stable against decomposition into LiH and hydrogen, with LiH2 and LiH6
lying on the binary convex hull between 150 and 300 GPa. 37 ; 109  LiH2 is
semimetallic, with a lattice of Li$+$ cations and hydridic hydrogens combined
with 1D chains of H2 molecules. In LiH6, electron transfer occurs from Li to
H2 units, such that each H2 molecule carries $1/3$ of a negative charge. This
partially fills the H2 $\sigma^{*}$ antibonding bands, lengthening the
intramolecular H-H distances. Motivated by theoretical predictions of the
propensity of LiH6 to become superconducting, 175  experimentalists
successfully synthesized a lithium polyhydride compound above 130 GPa in a
DAC. 176  It was suggested that Li diffuses into the the DAC during
compression producing a layer of LiH6 between the diamond and the sample, and
another layer of LiH2 between LiH6 and LiH. The predicted metallicity was not
detected in experiments up to 215 GPa – however, later calculations using a
van der Waals including density functional suggested that LiH2, LiH7 and LiH9
may be candidate structures for the synthesized phases. Chen:2017a

Many other metal hydrides with unusual stoichiometries have been predicted
within DFT for MHn phases, with M = alkali metal and $n\geq 2$ or M = alkaline
or rare earth metal and $n>2$. 37 ; 98 ; 110 ; 111 ; 112 ; 113 ; 114 ; 115 ;
116 ; 177 ; 178 ; Liu:2017-La-Y ; 100 ; Peng:2017a ; ScH6-1 ; ScH6-2 ; ZrH6 ;
Li:2019 ; Sun:2020  A major motivation for these studies was the suggestion
that “chemical precompression”, achieved by doping hydrogen with other
elements, would lower hydrogen’s metallization pressure and the subsequent
onset of superconductivity. 66 ; 179

As with polyhydrides of Li, polyhydrides of Na were successfully synthesized
in a DAC above 40 GPa and 2000 K. 170  Experimental XRD patterns and Raman
spectral peaks matched well with those predicted for NaH3 and NaH7 phases,
which possessed H${}_{3}^{-}$ units. 110 ; 170  Several alkaline metal
hexahydrides (for example, MgH6 at 300 GPa, MgH6  CaH6 at 150 GPa, 98  SrH6 at
250 GPa, 116 ; 178  ScH6 at 285, 300 and 350 GPa, ScH6-1 ; ScH6-2 ; Peng:2017a
YH6 at 120 GPa, YH6-1 ; Peng:2017a ; 100  ZrH6 at 295 GPa, ZrH6  and LaH6 at
100 GPa Peng:2017a ) have been predicted to be good metals with almost free-
electron like DOS at the Fermi level. Therefore, it is not surprising that
they were predicted to be superconducting with relatively high estimated
$T_{c}$ values. Even higher $T_{c}$s have been predicted (and some have been
observed) for the decahydrides YH10 Liu:2017-La-Y ; Peng:2017a  and
LaH10,Liu:2017-La-Y ; Peng:2017a ; geballe2018synthesis ; Pers105 ;
drozdov2019superconductivity  as well as in CeH9 Li:2019  and YH9.Kong:2021

Several elemental systems that are immiscible at 1 atm can form alloys under
pressure. Li-Be is one of these, with no known binary alloys at ambient
pressure. However, theoretical studies have predicted that between 20-200 GPa,
several binary Li-Be phases including Li3Be, LiBe, LiBe2, and LiBe4 become
thermodynamically stable. 68  Their stabilization is proposed to occur through
a Hume-Rothery mechanism of Fermi sphere-Brillouin zone
intersection.Degtyareva:2006  A detailed analysis on the $P2_{1}/m$ LiBe phase
reveals a unique structure comprised of alternating layers of puckered Li
square nets and Be triangular nets. The two-dimensionality of the Be layers –
unexpected in a bimetallic alloy – is seen in a sharp, step-like peak in the
electronic DOS at 82 GPa. The 2$s$ electrons of Li join the Be layers as the
Li 1$s$ cores start to overlap, but the Be 1$s$ core, due to their smaller
size do not. It was proposed that the LiBe alloy, therefore, may be viewed as
a manifestation of a Zintl-type compound. 166

Bismuth is a component of several widely studied topological materials Bi2Te3
; Na3Bi  and superconductors Reynolds:1950 ; Matthias:1966 , but is immiscible
at ambient pressure with many transition metals – to the point that bismuth
has been used as a flux in ambient-pressure syntheses. Thompson:2011  Under
pressure, the situation changes substantially and binary phases of Bi with Fe
Walsh:2016 ; Amsler:2017 , Co Schwarz:2013 , Cu Clarke:2016 ; Clarke:2017 ;
Amsler:2017b , Mo Altman:2021 , and Sn Amsler:2018  have all been predicted
and in some cases synthesized. MoBi2,Altman:2021  FeBi2, Walsh:2016 ;
Amsler:2017  MnBi2, Walsh:2019  and NiBi2 Amsler:2018  (the Mn-Bi and Ni-Bi
systems display some ambient-pressure miscibility) all crystallize in the
CuAl2 structure type shown in Figure 4b that is adopted by several transition
metal pnictides. Several of these phases are predicted Amsler:2017 ;
Amsler:2017b  or measured Clarke:2016 ; Schwarz:2013  to be superconducting.

A number of phases containing noble gases, which are highly unreactive at 1
atm, can be stabilized under pressure. This may be key to understanding why
the abundance of xenon on earth is much lower than would be expected based on
its abundance in interstellar space, otherwise known as the “missing xenon
paradox”. 180  Many studies have been carried out to solve this paradox; 181
some hypothesizing that the“missing xenon” could react with other elements
under pressure.

Theoretical studies have predicted that Fe, Ni and O can form
thermodynamically stable structures with Xe under pressure. 122 ; 172 ; 173
Both Xe-Fe and Xe-Ni phases were predicted to be stable above 200 and 250 GPa,
respectively, with their high-temperature stabilities validated under the
quasiharmonic approximation. 172  Later experimental studies detected both
XeFe3 (Figure 3c) and XeNi3 alloys, as well as a mixed iron-nickel xenon
compound. Dewaele:2016b ; Stavrou:2018

Additionally, a plethora of Xe-O phases including Xe2O, Xe1,3,7O2, and XeO3
were computed to be stable against decomposition into the component elements
above 75 GPa. 122 ; 173  These studies suggest that the “missing Xe” might be
trapped by the minerals containing oxygen that exist in the Earth’s lower
mantle. 173  A recent theoretical study found that Xe can react with the newly
discovered iron peroxide compound (FeO2) to form ternary compounds of xenon,
iron, and oxygen with stoichiometries of Xe2FeO2 and XeFe3O6 predicted at
lower mantle conditions. Xe-Fe-O  Xe-O interactions were found to be strong in
these ternary compounds. No other noble gas elements were reactive with iron
peroxide, suggesting that it might specifically be “blamed” for the missing
xenon. Several other elements have been predicted to form compounds with xenon
under pressure, including magnesium, 182  nitrogen, Xe-N  cesium Zhang:2015 ,
and yttrium, Zhou:2019  while compound formation has been experimentally
observed when xenon is combined with water 183  and hydrogen 184 .

Other noble gases are known to become reactive under pressure – compounds have
been predicted for Ar with Ni Adeleke:2019 , Li Li:2015 , and Mg 182 .
Particularly exciting is a synthesized Na2He phase stable above 113 GPa in the
fluorite structure (Figure 3d) Dong:2017 . Subsequent theoretical studies
uncovered compounds of Fe and He Monserrat:2018 , He and ammonia Liu:2020 ,
and a ternary FeO2He compound Zhang:2018 . One study investigated the
stability of alkali metal oxides or sulfides intercalated with He and Ne.
Gao:2019  It has been proposed that helium can serve as something of an inert
“spacer” in ionic compounds with unequal numbers of cations and anions.
Liu:2018  Repulsions between the majority ions can prevent the formation of
close-packed structures – in which case an added He atom provides relief in
the $PV$ energetic term – and the added He keeps the majority ions separated
from one another. Through this lens, the Na2He phase, an electride, can be
treated as a compound of (Na+)2$\cdot 2e^{-}$ with He.

###  3.3 Structure and Bonding

Increased densification promotes electron density overlap between neighboring
atoms, which may react by adopting electron rich or electron poor multi-
centered bonding schemes. Grochala:2007a  In his seminal work on chemical
bonding, Pauling predicted that sufficiently large pressure would shorten the
intermolecular hydrogen bonds in water molecules and elongate the
intramolecular H-O bonds. 129  At some point, those two types of H-O bonds
would then become equivalent, using the lone pairs on an oxygen atom to form
bonds with the neighboring hydrogen atoms that are hydrogen-bound to oxygen at
ambient pressure. This phenomenon has been observed in experiments for the ice
X phase above 60 GPa, whose intra and intermolecular O-H bonds are symmetrized
and form three dimensional networks resembling the diamond structure, leaving
oxygen atoms four-coordinated. 158 ; 159  Several other hydrogen bond-
containing systems have been found both experimentally and theoretically to
show similar bond symmetrizing behavior under pressure, including HF, HCl,
HBr, 160  and the metastable phase of Li-F-H. LiFHn

Interesting structural motifs where multicentered bonds are formed are present
in several predicted high pressure hydride phases. 103 ; 111 ; 112 ; 115 ;
Duan:2015 ; LiFHn  For example, the trihydrogen cation (H${}_{3}^{+}$), a
dominant interstellar molecule, maximizes the overlap of hydrogen bonding
orbitals to form three-center two-electron bonds taking the shape of an
equilateral triangle. The symmetric H${}_{3}^{+}$ cluster occurs in the H5Cl
phase that has been theoretically predicted to be stable from 60-300 GPa. 103
; Duan:2015 . Another ternary hydride phase that is metastable at 300 GPa,
LiF4H4, contains a distorted version of this motif. LiFHn

With two more electrons than the trihydrogen cation, the trihydrogen anion
(H${}_{3}^{-}$) has a linear geometry with three-center four-electron bonds.
The ground state configuration of the trihydrogen anion, based on ab initio
calculations, shows a large difference between the two H-H bonds: 2.84 Å
versus 0.75 Å. 163  The PES of this cluster has two wells containing two
equivalent configurations, with the transition state at the peak of the energy
barrier corresponding to the bond-symmetrized arrangement where both H-H
distances are 1.06 Å. Hydrides of heavy group I (i.e. K, 112  Rb, 111  and Cs
113 ) or group II (i.e. Ba 115 ) metals display symmetric H${}_{3}^{-}$ ions.
For example, several nearly isoenthalpic CsH3 structures, all of which are
built from Cs+ and H${}_{3}^{-}$, become the lowest enthalpy points on the
convex hulls above 30 GPa. 113  Computational experiments have shown that the
softness of the cation is related to the formation of symmetric H${}_{3}^{-}$
anions under pressure. 112

The high pressure structures of metal hydrides present numerous curious
bonding arrangements for hydrogen atoms. In the pressure-stabilized alkaline
earth or rare metal tetrahydride MH4 phases, 98 ; 116 ; 178 ; Qian:Sc-2017 ;
ScH6-1 ; ScH6-2 ; Peng:2017a ; Liu:2017-La-Y ; Li:2019 ; Zurek:2018c  pressure
can also induce elongation or even dissociation of bonds. MH4  These
tetrahydrides assume the same $I4/mmm$ structure (ThCr2Si2-type) that occurs
in over 700 AB2X2 compounds and consists of three parts: metal cations,
hydridic hydrogen, and quasi-molecular hydrogen units (Figure 5a). Charge
transfer from trivalent or tetravalent metal atoms to the H2 $\sigma_{u^{*}}$
bands can elongate or break the H-H bonds in the quasi-molecular units. Any
still existing H-H bonds are further weakened by a Kubas-like interaction of
electron transfer from the H2 $\sigma_{g}$ to the metal $d$ orbitals, and
metal $d$ orbitals to the H2 $\sigma_{u^{*}}$ states. This means that the H-H
distance – which is related to the phases’ propensity towards
superconductivity – within the quasi-molecular orbitals can be tuned by the
choice of metal atoms and the applied pressure. MH4

Figure 5: Structural motifs in materials at high pressures: (a) predicted and
synthesized MH4 $I4/mmm$ phases adopted by several alkaline earth and rare
earth hydrides comprised of hydridic H (grey) and quasimolecular H2 (yellow)
units; MH4  (b) predicted CsF5 ($Fdd2$) phase with lightly distorted planar
pentagonal coordination environment of Cs reminiscent of the XeF5 molecule;
102  (c) predicted and synthesized BaGe3 ($I4/mmm$) phase adopted by several
polar intermetallic compounds with polyhedra built up of tetrel dumbbells. 168
; 169

Other wild and wonderful hydrogenic motifs are plentiful among the metal
hydrides; among the most iconic are the “clathrate-like” cages appearing in
superconducting phases such as CaH6. 98  Isostructural to YH6, Liu:2017-La-Y ;
Peng:2017a ; 100  MgH6, MgH6  ScH6, ScH6-1 ; ScH6-2  and more, the body-
centered lattice of CaH6 features Ca atoms at the centers of an H24 sodalite-
like framework of six square and eight hexagonal faces. Electron donation from
the Ca atom to the hydrogen sublattice has been postulated to populate a half-
filled degenerate nonbonding state for an H4 square unit, precluding
condensation into H2 molecular units but maintaining weak bonding character.
98  Similar weakly bound clathrate cages appear in MH9 phases such as CeH9
Li:2019  and MH10 compounds including LaH10. Liu:2017-La-Y  The Sc-H family of
high-pressure phases displays another strange structural motif: H5 pentagons.
ScH6-1 ; Xie:2020  These curious units appear joined together in strips in the
ScH9 and ScH12 structures, where plots of the Electron Localization Function
(ELF) confirms the presence of localized bonding. ScH6-1  In ScH10 at
$\sim$200 GPa two nearly isoenthalpic phases with $Immm$ and $P6_{3}/mmc$
symmetry feature H5 pentagons fused to one another in sets of three in a
“pentagraphene-like” motif. Xie:2020  As with many other metal superhydride
phases, the electrons donated from the metal atom (investigated for a hafnium
analogue HfH10) go into antibonding orbitals of the hydrogen framework,
weakening the H-H bonding and reducing H2 molecular character.

Pressure-induced changes in the energy ordering of atomic orbitals can cause
orbitals that would not normally participate in bonding to do so. One good
example is CsH3, where the Cs adopts a Cs+ configuration and the energy of the
Cs 5$p$ orbitals increases under pressure so they interact with the hydrogen
$s$ electrons. 113 . In the CsF2-6 phases found above 10 GPa that are stable
against decomposition into CsF and F2, the geometries and structural motifs
mirror the isoelectronic XeFn molecules. 102  For example, the building blocks
of CsF5 (Figure 5b) are pentagonal planar with Cs as the center atoms
coordinated by five F atoms, as seen in XeF5 molecules. 102  At 100 GPa and
high fluorine content (F:Cs ratio $\geq$ 2), the oxidation state of Cs can
increase almost linearly past $+1$ with increasing F content, according to
Bader analyses. In CsF2 and CsF3, covalent bonding was indicated between the
Cs semicore 5$p$ and F 2$p$ orbitals based on a crystal orbital Hamiltonian
population (COHP) analysis. 165  Thus, the pressure-reordering of orbital
energies has serious consequences on the possible bonding interactions between
cesium and fluorine atoms, allowing inner-shell electrons to actively
participate.

A series of polar intermetallic compounds MA3, where M is an alkaline earth or
rare earth metal and A is a group 14 tetrel element, adopt an $I4/mmm$
structure that only becomes stable under high pressure. In this structure,
found in CaGe3, 162  CaS3, YSi3, and LuSi3 167 , and predicted prior to its
synthesis for BaGe3 168 ; 169 , the tetrel atoms form a condensed lattice of
A-A dumbbells, which is layered with the metal atoms (Figure 5c).
Investigations using the electron localizability indictor or the COHP show
covalent bonding within the tetrel dumbbells and multicenter interactions
between the metal atom and the tetrel sublattice. 162 ; 167 ; 169  Several of
the phases were predicted to become Bardeen-Cooper-Schrieffer (BCS)
conventional superconductors below 10 K according to first-principles
calculations. 162 ; 167 ; 169  Furthermore, some of the phases could be
recovered at ambient conditions. The strong bonding networks formed under
pressure protect against decomposition as the pressure is released, since
large amounts of energy would be required to break the bonds. In this way,
high-pressure high-temperature techniques can be successfully used to
synthesize compounds with increased coordination components that do not align
with electron-precise counting schemes such as the Zintl concept. 166 .

###  3.4 Superconductivity

The propensity for superconductivity is one of the properties that is
significantly affected by the pressure variable. At least 23 elements undergo
superconducting transitions under pressure, 185  and the $T_{c}$ itself
depends greatly on pressure for superconducting elements and compounds alike.
The material with the highest 1 atm $T_{c}$, HgBa2Ca2Cu3O8, is an
unconventional superconductor whose $T_{c}$ of 133 K can be increased further
to 164 K at 31 GPa. 186  For over 20 years, this compound had the highest
critical temperature known, until a record-breaking 203 K was measured at 150
GPa in the sulfur-hydrogen system. 189

Hydrogen-rich compounds have become a target for pursuing high temperature
superconductivity under pressure as a proxy for metallic hydrogen itself. The
reason for this stems from Ashcroft’s proposal that the addition of another
element to hydrogen could reduce the required pressure for transitioning into
the superconducting state through “chemical pre-compression”. 136 ; 66 ;
DopingH  Herein, we overview several recent examples of high pressure hydrides
with remarkable $T_{c}$ values, and also illustrate how the successful synergy
between theoretical predictions and experimental synthesis led to their
discovery.

Previous theoretical studies 187  that estimated the superconducting
properties of compounds with the H2S stoichiometry at high pressure using the
Allen-Dynes-modified McMillan equation 188  identified a phase with an
estimated $T_{c}$ of 80 K at 160 GPa. Motivated by this prediction,
experimental work proceeded, resulting in a phase with measured $T_{c}$ below
100 K in good agreement with the expected values. 189  However, when the
sample was prepared at higher temperatures above 300 K, a material with a
record-breaking $T_{c}$ value of 203 K at 150 GPa was synthesized instead. 189

In a parallel set of studies, another group of researchers had previously
synthesized an (H2S)2H2 phase above 3.5 GPa. Strobel:2011a  This work inspired
a follow-up theoretical investigation that identified a series of structures
with the H3S stoichiometry at higher pressures, including an $R3m$ phase with
a $T_{c}$ of 155-166 K at 130 GPa. Duan:2014  At even higher pressures, this
structure was predicted to transform to a related $Im\overline{3}m$ phase
(Figure 6a) whose $T_{c}$ was estimated as 191-204 K at 200 GPa – in line with
the experimental results of Ref. 189 . Subsequent XRD analysis suggested that
the superconducting sample contained a mixture of the two predicted phases
along with the $\beta$-polonium phase of sulfur. 192  The high $T_{c}$ was
later confirmed by the Meissner effect, MeissnerH3S  and an optical
reflectivity experiment categorized the hydrogen sulfide material as a
conventional BCS superconductor whose superconductivity arises from electron-
phonon coupling. OptRefH3S

Figure 6: Selected high-temperature superconducting hydrides that have been
calculated to be stable or metastable at high pressures: (a) $Im\bar{3}m$ H3S,
the first high-temperature superconducting hydride that was synthesized, 189 ;
Duan:2014  (b) $R3m$ CSH7, one example of a predicted ternary superconducting
hydride, CSH7  (c) LaH10, the first clathrate-like superhydride to be made,
drozdov2019superconductivity ; Pers105  and (d) Li2MgH16, a predicted ternary
superhydride with an immensely high predicted $T_{c}$ of 473 K Li2MgH16

The exciting work on H3S prompted a slew of investigations into the
compositions and structures of the superconducting phases, the factors that
contribute to $T_{c}$, the isotope effect, the importance of anharmonicity,
and the role of hydrogen’s quantum nature. a324 ; a357 ; 91 ; 194 ; 195 ; 196
; 197 ; 198 ; a373 ; CSH38 ; a375 ; a376 ; a377 ; a378 ; a379 ; a380 ; a381 ;
a382 ; a383 ; Bianco:2018 ; Amsler:2019  For example, one theoretical
investigation concluded that the decomposition of the $Im\overline{3}m$ phase
into (SH-)(H3S+) perovskites is favorable. a384  First-principles molecular
dynamics (MD) simulations at 200 GPa and 200 K suggested that this perovskite
phase can be further separated into tetragonal and cubic regions, and the
resulting MD structure produces an XRD pattern in good agreement with the
experimental observations. a385

The high $T_{c}$ of H3S phases also inspired multiple studies on ternary
systems based on adding a third element to the S-H system. The electronic
structure of H3S features a maximum in the electronic DOS near the Fermi level
due to the presence of a pair of van Hove singularities on either side. CSH38
; a378  Tuning the electronic structure via doping with another element could
shift the Fermi level to lie directly on the maximum, strengthening the
electron-phonon coupling and raising the $T_{c}$ of the system.

In one study, currently in the spotlight, a carbonaceous sulfur hydride with a
measured $T_{c}$ of 288 K at 267 GPa was reported. CSH9  More recent XRD
analysis suggests that this phase is a derivative of the Al2Cu type. CSH10 ;
CSH11  Before the experimental discovery of the carbonaceous sulfur hydride,
structure predictions had actually been carried out on various carbon-doped
sulfur hydride stoichiometries: C1-2S1-2H1-8 up to 300 GPa, CSH7  and
C1-4S1-4H1-36 at 100 GPa. CSH35  A stoichiometry of CSH7 emerged as
particularly interesting in the calculations. A number of metastable phases
arising from the intercalation of methane molecules into the H3S framework
(Figure 6b) were found. The estimated $T_{c}$s for the predicted CSH7
structures fell well below that measured for the carbonaceous sulfur hydride
(194 K at 150 GPa and 181 K at 100 GPa), and their calculated equations of
state also did not match the experimental C-S-H data. A first-principles
investigation using the virtual crystal approximation (VCA), which employs
pseudoatoms based on weighted potentials of different elements to model
doping, found that a small amount of carbon incorporation into the H3S lattice
was sufficient to increase the estimated $T_{c}$ to 288 K. CSH36  However, the
VCA method disregards local chemical structure and bonding. Therefore, a more
recent theoretical study was carried out on supercells of H3S doped with
1.85-25% carbon (interpreted as SH${}_{3}\rightarrow$ CH3, producing six- or
four-coordinate C, or SH${}_{3}\rightarrow$ CH4, producing four-coordinate C).
CSHarXiv  Contrary to results obtained with the VCA, it was found that the
doping lowers the DOS at the Fermi level, but increases the logarithmic
average phonon frequency. The most promising phase for which $T_{c}$ could be
explicitly computed, CS3H13, was predicted to possess essentially the same
$T_{c}$ as $R3m$ H3S at 270 GPa. It was postulated that the most promising
stoichiometries for high-$T_{c}$ were CS15H49 or CS53H163, but explicit
calculations could not be performed due to the system size.

Besides sulfur, hydrides of another $p$-block element have inspired interest.
After loading phosphine into a DAC, researchers observed the onset of
superconductivity at 30 and 103 K at 83 and 207 GPa, respectively. 193
Follow-up theoretical studies appeared quickly, since the structural and
electronic properties of the superconducting phosphorus compounds were not
fully characterized. 90 ; 118 ; 201 ; 202 ; Bi-PH2 . Predictions suggested
that compression caused phosphine to decompose, with the measured
superconductivity arising from some combination of PH, PH2, PH3, and other
metastable phosphorus hydrides. 90 ; 118 ; 202 ; Bi-PH2  In one example, an
$I4/mmm$ PH2 phase had an estimated $T_{c}$ of 70 K at 200 GPa, 118 ,
increasing to 78 K at 220 GPa. 90  At lower pressures, PH2 phases were found
to be metastable, with enthalpies of formation and $T_{c}$ values in line with
the experimental observation at 83 GPa. Bi-PH2  A unique structure comprised
of cubic-like phosphorus layers capped by hydrogen atoms at the top and bottom
layers and further separated by layers of molecular hydrogen was observed.
Interestingly, the superconductivity was mainly derived from the P-H layers,
while the neutral and mobile H2 layers served to stabilize the structure by
separating the negatively charged H atoms.

Meanwhile, prior to the discovery of the carbonaceous sulfur hydride, the high
$T_{c}$ record of H3S was broken by another class of high pressure hydrides:
binary combinations of hydrogen with alkaline and rare earth metals with the
hydrogen atoms arranged in clathrate-like cages, as described in the previous
section. drozdov2019superconductivity ; Pers105  Experimental measurements on
$Fm\bar{3}m$ LaH10 detected $T_{c}$ values as high as 250 K
drozdov2019superconductivity  and 260-280 K Pers105 .

Theoretical investigations had previously predicted the stability of this
phase (Figure 6c) with La atoms occupying fcc positions on a cubic lattice
surrounded by an H32 clathrate-like cage. Liu:2017-La-Y ; Peng:2017a  An
isotypic yttrium analogue has also been predicted, with a $T_{c}$ of 305-325 K
at 250 GPa calculated via numerical solution of the Eliashberg equations,
Liu:2017-La-Y  but such a phase has not yet been experimentally obtained. A
ternary structure synthesized by mixing La and Y into the $Fm\bar{3}m$ lattice
has demonstrated $T_{c}$ up to 253 K at 183 GPa, where the addition of La is
thought to stabilize the YH10 framework. Semenok:2021b  In addition, other
ternary hydrides with impressive predicted $T_{c}$s and pressure stability
ranges have been identified, including a metastable Li2MgH16 phase (Figure 6d)
that is predicted to be a “hot” superconductor with a $T_{c}$ $\sim$473 K at
250 GPa. Li2MgH16  A family of phases based on adding a third element into
void spaces of the MH10 lattice, resulting in stoichiometries such as LaBH8
DiCataldo:2021 ; Liang:2021  and KB2H8 Gao:2021  show promise in maintaining
stability to pressures below a megabar. These discoveries are paving the way
to warm and light superconductivity at more readily accessible pressures.

##  4 Conclusion

For the vast majority of us who have never left Earth’s surface, the only way
in which we adjust our lives to account for the affects of pressure is to
adapt cooking recipes to account for water boiling at a lower temperature at
high altitudes. From the top of Mount Everest to the bottom of the Mariana
Trench the pressure difference is roughly only a tenth of a gigapascal. The
development of experimental techniques to access successively higher
pressures, coupled with a dramatic improvement in theoretical methods and
computational power has allowed researchers to explore materials at extreme
high pressure conditions. In many cases this has led to the discovery of a
profoundly different chemistry.

High pressure alters the relative orbital energies of atoms, driving
electronic transitions within atoms and ultimately resulting in unfamiliar,
let us call it extreme, chemistry. In some cases, valence electrons compressed
into void regions become interstitial quasiatoms, while in others the energies
of core orbitals increase enough to partake in bonding and chemistry. Elements
that are stubbornly immiscible with one another, or are just plain unreactive,
under atmospheric conditions form compounds at high pressures. Unfamiliar
structures can emerge, from extended 3D networks of hydrogen 98  to lattices
of tetrel dumbbells 162  to pentagonal planar CsF5 units 113  (with Cs taking
on a charge $>+1$ to boot!).

Experimental methods to synthesize and probe such strange phases have advanced
greatly, with improvements in diamond anvil cells including toroidal
Jenei:2018 ; Dewaele:2018  and double-stage diamond anvil cells Lobanov:2015 ,
as well as in dynamic compression experiments Fratanduono:2021 ; Hansen:2021
accessing higher and higher pressures. Theoretical methods remain closely
intertwined with experiment, providing guidance on promising systems to
investigate and elucidating the structural and electronic character of the
phases that are found.

Crystal structure prediction (CSP) techniques have proven immensely useful in
helping researchers begin to develop chemical intuition under pressure.
Ranging from methods that finely explore particular regions of a PES (e.g.
metadynamics and minima hopping) to those that promote broad exploration (e.g.
random searches, particle swarm optimization, and genetic algorithms), CSP has
been used to identify a plethora of new phases. The synergy between experiment
and theory is evident in the rapid progress made in the field of high-
temperature superconductivity in the hydrides, beginning with the discovery of
the record-breaking H3S and LaH10 phases. Experimental reports of room-
temperature superconductivitiy in a C-S-H phase has inspired numerous studies.
Complex phases combining three, four, or even more elements and strange
chemical behavior await to be uncovered as researchers further explore this
fascinating field.

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.

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