# Strong correlation between bonding network and critical temperature in
hydrogen-based superconductors

Francesco Belli  Centro de Física de Materiales (CSIC-UPV/EHU), Manuel de
Lardizabal Pasealekua 5, 20018 Donostia/San Sebastián, Spain  Fisika Aplikatua
1 Saila, Gipuzkoako Ingeniaritza Eskola, University of the Basque Country
(UPV/EHU), Europa Plaza 1, 20018 Donostia/San Sebastián, Spain  J. Contreras-
Garcia  Laboratoire de Chimie Théorique (LCT), Sorbonne Université CNRS, 75005
Paris (France)  Ion Errea  Centro de Física de Materiales (CSIC-UPV/EHU),
Manuel de Lardizabal Pasealekua 5, 20018 Donostia/San Sebastián, Spain  Fisika
Aplikatua 1 Saila, Gipuzkoako Ingeniaritza Eskola, University of the Basque
Country (UPV/EHU), Europa Plaza 1, 20018 Donostia/San Sebastián, Spain
Donostia International Physics Center (DIPC), Manuel de Lardizabal Pasealekua
4, 20018 Donostia/San Sebastián, Spain

###### Abstract

Recent experimental discoveries show that hydrogen-rich compounds can reach
room temperature superconductivity, at least at high pressures. Also that
there exist metallic hydrogen-abundant systems with critical temperatures of
few Kelvin, or even with no trace of superconductivity at all. By analyzing
through first-principles calculations the structural and electronic properties
of more than one hundred compounds predicted to be superconductors in the
literature, we determine that the capacity of creating a bonding network of
connected localized units is the key to enhance the critical temperature in
hydrogen-based superconductors, explaining the large variety of critical
temperatures of superconducting hydrogen-rich materials. We define a magnitude
named as the networking value, which correlates well with the predicted
critical temperature, much better than any other descriptor analyzed thus far.
This magnitude can be easily calculated for any compound by analyzing
isosurfaces of the electron localization function. By classifying the studied
compounds according to their bonding nature, we observe that the networking
value correlates with the critical temperature for all bonding types. Our
analysis also highlights that systems with weakened covalent bonds are the
most promising candidates for reaching high critical temperatures. The
discovery of the positive correlation between superconductivity and the
bonding network offers the possibility of screening easily hydrogen-based
compounds and, at the same time, sets clear paths for chemically engineering
better superconductors.

##  I Introduction

The field of hydrogen-based superconductivity has progressed enormously since
1968, when Ashcroft first proposed that pressurized hydrogen may become a high
temperature superconductor 1 . While the first discovered superconductors were
Th4H15 2  in 1970 and PdH 2B  in 1972, with not very promising critical
temperatures ($T_{c}$) of 7.6 K and 5 K respectively, the more recent
experimental discoveries at high pressures show that superconductivity on
hydrogen-based compounds can span from a few Kelvin to room temperature. Few
examples are the recently synthesized PrH94F , where $T_{c}=7$ K at 125 GPa,
AlH3 where no superconductivity was observed above 4 K despite the predictions
4E , and, on the other extreme, H3S DrozdovEremets_Nature2015 , YH9 4D ;
snider2020superconductivity , YH6 troyan2020anomalous , and LaH10 4 ; 4B  all
reaching critical temperatures well above 200 K at megabar pressures. In
addition, the recent observation of a $T_{c}$ of 288 K at 267 GPa in a
compound formed by sulfur, carbon, and hydrogen 4C  confirms that hydrogen-
based superconductors can be room-temperature superconductors. This
observation finally dismisses the maximal limits for $T_{c}$ initially
suggested by Cohen and Anderson for electron-phonon driven superconductivity 9
. Hydrogen-based compounds are thus the best currently available candidates to
reach ambient temperature and pressure superconductivity.

In order to discover new compounds with high $T_{c}$ at low pressures, a
simple physical-chemical understanding of the properties enhancing the
critical temperatures in hydrogen-based systems is necessary. In this regard,
the hundreds of compounds predicted to be superconductors with first
principles crystal structure prediction techniques constitute a rich working
dataset to extract conclusions 7 ; 15 ; doi:10.1146/annurev-
conmatphys-031218-013413 . Among these predictions, the highest $T_{c}$ values
are 300 K for pure metallic hydrogen 5 ; 6  and 326 K for the YH10 binary
compound 8 . Attempts have been made to increase $T_{c}$ further through
ternary compounds, for instance with H3S1-xPx 10  and Li2MgH16 11 . Aiming at
extracting useful information from this dataset, two main routes are being
explored: on the one hand, machine learning methods 12 ; 13 ; 14  are starting
to be employed to further increase the list of predicted systems, although the
obtained new compounds so far do not beat the already known; on the other
hand, additional efforts are being invested into classifying these
superconductors using simple footprints based on structural, chemical, and
electronic properties 13 ; 15 ; 17 . These studies suggest hydrogen rich
systems with highly symmetrical structures and high density of states (DOS) at
the Fermi level are the best candidates for high-temperature
superconductivity.

Even if these properties are able to suggest good trends, they serve necessary
but not sufficient conditions. This ultimately means there are not good
optimizers thus far: improving these parameters will not necessarily lead to
an improvement of the superconducting critical temperatures. In other words,
even if the footprints for a good superconductor are somewhat clear, we cannot
yet rely on simple variables to estimate the superconducting temperatures,
clarify the reason for such a broad spectrum of $T_{c}$ values, and,
ultimately, chemically engineer better superconductors.

In this work we investigate the chemical, structural, and electronic
properties through ab initio methods based on density functional theory (DFT)
for a set of 178 hydrogen-based superconductors previously predicted in the
literature 7 , including pure hydrogen and binary compounds. Our ultimate goal
is to provide a simple understanding of the origin of the high $T_{c}$ in
these compounds. We focus mainly on the electronic and structural properties
by means of chemical bonding descriptors, hydrogen-hydrogen distance,
electronic charge, and density of states at the Fermi level. We review the
impact on the predicted $T_{c}$ of many of these descriptors, which have been
already somewhat studied on a case to case basis in the literature but do not
reveal conclusive. We identify that the electron pairing and delocalization
play instead the final role. This takes us to propose a universal descriptor
based on the identification of electronic delocalization networks, identified
by means of the electron localization function (ELF) 18 ; 19 ; 20 ; 21 ; 22 ,
which quantifies the network of the electronic bonding. We define a simple
magnitude, the networking value, which is easily obtained from the calculation
of ELF isosurfaces. Such quantity reveals useful to have a first estimate of
the superconducting critical temperature without the need of performing
electron-phonon coupling calculations. To the best of our knowledge, it is the
first time that such a descriptor is proposed in the literature. We believe
the analysis of the networking value is the first step for a high-throughput
analysis of potential new high-$T_{c}$ systems. It also provides clear paths
for chemically engineering better hydrogen-based superconductors, guiding the
quest for high-$T_{c}$ compounds among the vast possibilities offered by
ternary compounds.

##  II Results

Chemical composition and bonding categories. In order to provide a
comprehensive understanding of the different type of hydrogen-based compounds
predicted to exist in the literature, it is convenient to categorize them in
families according to the nature of the chemical bonding of the hydrogen atoms
in the system. Our classification is guided by the study of the ELF 18 ; 19 ;
20 ; 21 ; 22  and the atomic charge distribution obtained through the Bader
analysis 23 ; 24 ; 25 ; 26  (see Methods for more details). The ultimate goal
of this investigation is to better understand what kind of chemical
interaction is the most beneficial for superconductivity among these
compounds.

After a thorough analysis of the ELF in our test set, we identify six
different families according to the nature of the chemical bonding, namely,
molecular systems, covalent systems, systems driven by weak covalent hydrogen-
hydrogen interactions, systems with electride behavior, ionic systems, and
isolated systems. In each case, the nature of the bonds is identified through
the analysis of the ELF saddle points between different atoms. Since bonding
properties are mainly local, each system can belong to more than just one
family. However, in order to simplify the analysis, we focus on the most
dominant feature for each compound. A representative for each family, together
with the distribution of the families through the groups of the periodic table
and the amount of hydrogen fraction in each case, is shown in Fig. 1. The
hydrogen fraction ($H_{f}$) takes the following form:

$H_{f}=\frac{N_{H}}{N_{H}+N_{X}},$ (1)

where $N_{H}$ and $N_{X}$ are the number of hydrogen and non-hydrogen atoms in
the primitive cell, respectively.

Figure 1: Classification of hydrogen-based superconductors according to their
bonding categories. The top panel shows representative systems for the
different categories with ELF isosurfaces at different values: TeH4 for
molecular systems (magenta), PH2 for covalent systems (purple), ScH4 for
systems with weak covalent hydrogen-hydrogen interactions (dark green), CrH
for elecrtides (yellow), PdH for isolated systems (orange), and ScH2 for ionic
systems (light green). The lower panel shows respectively the critical
temperature as a function of the host atom periodic group (left panel) and the
hydrogen fraction in the compounds (right panel). In the left panel, the name
of some compounds is explicitly given together with the index (in parenthesis)
given to each compound in Extended Data Tables I - III. In the right panel,
and also in Figs. 2 and 4, just the index is given for these compounds for the
sake of brevity.

The molecular family describes all systems having at least one pair of
hydrogen atoms forming a molecule. The latter can be identified through the
ELF analysis by locating an isosurface surrounding an isolated molecule at
very high values of ELF (see the magenta surface around the hydrogen molecules
in TeH4 in Fig. 1). A system is chosen to be molecular if an isolated pair of
hydrogen atoms appears connected at a value of ELF higher than 0.85, i.e., if
the minimum ELF value in between the hydrogen atoms is above 0.85. Systems
with molecular behavior appear between groups 1 to 4 and 13 to 17. All of them
have a value of $H_{f}>0.\overline{6}$, meaning that in the unit cells there
are at least two hydrogen atoms per host. We note that molecular systems tend
to have very high values of $H_{f}$, which reflects that in most cases several
molecules exist per host atom. The highest critical temperatures for these
molecular compounds have been predicted for ScH12(33) and TeH4(150), with
$T_{c}$ values around 150 K (the number in parenthesis after a given compound
corresponds to the index given for each compound in Supplementary Data Tables
I - III). The critical temperatures of the molecular compounds span from few
Kelvin to these high values.

The covalent family is composed of systems where covalent bonds between
hydrogen and the host atoms are dominant. Throughout our discussion we shall
label the host atom as X, with X$\neq$H. For H-X bonds, the covalent character
can be identified by an elongation toward the host atom of the ELF isosurface
surrounding the hydrogen (i.e. a polarized covalent bond). This is exemplified
by the purple surfaces pointing from the H (small spheres) to the P atoms (big
spheres) in PH2 in Fig. 1. Covalent systems appear for groups 13 to 17, and
are mostly related to the host atom’s $p$ type orbital character. Some of the
highest critical temperature for these systems have been predicted for
H3S(138) at 200 K, BH3(77) at 125 K, and H3Se(148) at 110 K.

The weak covalent hydrogen-hydrogen interaction family is dominated by
compounds with predominant weak H-H covalent interactions. The difference with
the molecular case is that hydrogen molecules or clusters appear elongated or
quasi dissociated. This is illustrated with the dark green surfaces in ScH4 in
Fig. 1, where bonds between hydrogen atoms appear at much lower values of ELF.
From a quantitative point of view we assume that a group of hydrogen atoms is
weakly bonded if the ELF at the bond point is within the range [0.4 - 0.85].
These systems mostly appear between groups 1 to 5. In this bonding type,
interactions seem to be purely related to hydrogen atoms whilst the host atoms
appear as inert or acting as a chemical pre-compressor or electron donor.
Compounds with this bonding characteristic tend to contain many hydrogen atoms
per host, as it happens for the molecular compounds. For compounds with the
host atom in a low group of the periodic table, the host atoms valence
electrons are donated to the hydrogen atoms resulting in a weakening of the
H-H bonds, which translates into an increment of the H-H distance. This family
shows the highest predicted critical temperatures, the highest being 326 K for
YH10(27) and 300 K for metallic hydrogen H(76).

The ionic family is formed by those systems whose hydrogen atoms show an ionic
character. This is identified by an isolated proto-spherical ELF isosurfaces
surrounding the hydrogen atoms. In addition, for a system to be hereby
considered as ionic the mean extra charge per hydrogen atom must be more than
0.5 electrons. This is illustrated by the spherical light green surfaces
around hydrogen in ScH2 in Fig. 1, where the charge on H is 0.67 electrons.
Ionic behavior was observed between groups 2 to 5 of the periodic table, in
all cases with low values of $H_{f}$. The ionic bonding originates from the
strong difference in electronegativity between host and hydrogen atoms, which
is always increased under pressure 31 . This family transforms into the weak
covalent hydrogen-hydrogen interaction class when $H_{f}$ increases, since the
donated charge becomes small in respect to the number of recipient hydrogen
atoms. Critical temperatures for these systems are low, with the highest being
45 K for BeH2(4).

The electride family contains systems featuring electride behavior, i.e.
compounds with electrons localized in the voids. The latter can be identified
by isolated pockets of localized electrons in empty space of the crystal as
the ones shown in yellow for CrH in Fig. 1. From a quantitative viewpoint,
electride behavior is characterized in terms of isolated isosurfaces not
surrounding any nuclei with ELF maximum values in between 0.35 to 0.7. Note
that metallic compounds are also included in this family. Metallic cases also
show isolated bubbles of ELF occupying the voids, but their profile is
flatter. Given the difficulties to set a quantitative barrier, we have merged
them in a unique family. Electrides and metals only appear mainly between
groups 5 and 10, and are among the systems with the lowest value of $H_{f}$
reaching a maximum of 3 hydrogen atoms per host ($H_{f}=0.75$). Critical
temperatures for these systems are low, not reaching above 50K, exception made
for the Si2H6(94) with a $T_{c}$ of 153 K, where the electride-metallic
behavior seems to be arising from H-Si covalent bonds (not justifying a
further differentiation between electrides and metals).

The remaining family includes all the materials featuring extremely weak bonds
between hydrogen and host atoms. These systems have been named as isolated and
are identified by the lack of any kind of connection of the ELF isosurfaces
above 0.25. These systems have low critical temperatures not reaching above 40
K and appear mainly between groups 5 and 12 of the periodic table. They also
show a weak capacity of hosting a large number of H atoms per X atom, as they
have the lowest values of $H_{f}$.

Overall our results highlight that characterizing the bonding type of a solid
thanks to these families enables to discard a great number of compounds as
potential high-$T_{c}$ compounds. Covalent interactions, be it weak H-H or X-H
are the most favorable for high-temperature superconductivity. This allows to
identify the potential interesting combination of elements, especially with
respect to the increasing search among ternary compounds. The lowest $T_{c}$
values appear for electrides and isolated compounds, mostly appearing between
groups 5 and 12, which do not show lots of potential as high-$T_{c}$
compounds. These families also show the lowest values of $H_{f}$.

Figure 2: Structural and electronic properties. Panel a) shows $T_{c}$ as a
function of the shortest hydrogen-hydrogen distance for all the compounds.
Panel b) shows $T_{c}$ as a function of the mean extra electrons per hydrogen
atom. Panels c), d), and e), respectively, show $T_{c}$ as a function of the
total DOS at the Fermi level, the DOS at the Fermi level projected on the
hydrogen $s$ orbitals per hydrogen, and the fraction of the total DOS at the
Fermi level coming from the hydrogen orbitals.

Hydrogen-hydrogen distance and electronic properties. After categorizing the
different bonding families of hydrogen-based superconductors, we focus on
understanding the trends of the predicted $T_{c}$ with structural and
electronic properties. The results are summarized in Fig. 2. The analysis
focuses on the shortest hydrogen-hydrogen distance for each compound, the
charge distribution on hydrogen atoms, and the density of states at the Fermi
level with the ultimate goal of finding correlations between such descriptors
and the critical temperature.

Even if no general trend is observed when plotting $T_{c}$ as a function of
the H-H shortest distance, several conclusions can be drawn. The structural
analysis highlights an increment of the superconducting critical temperature
with the increase of the shortest hydrogen-hydrogen distance for those systems
where the bonding is driven by pure hydrogen interactions, meaning the
molecular and weak covalent hydrogen-hydrogen interaction families. For these
two families the H-H distance spans from 0.74 Å for systems with $T_{c}$ below
1 K to a maximum of around 1.35 Å for the compounds with highest critical
temperatures. In the region between 0.9 and 1.35 Å lie the currently predicted
compounds with the highest superconducting temperatures reaching values as
high as 300 K. In other words, elongated H-H interactions promote $T_{c}$.
Interestingly, our analysis highlights that such H-H distance variation is not
related to a variation of pressure, meaning that on a broad level, increasing
the pressure does not necessarily yield an increment of the bonding distance,
so composition rather than pressure would be a more relevant variable to tweak
(see Supplementary Data Table I to see the pressure at which each compound is
studied).

The shortest hydrogen-hydrogen distance for the covalent family spans between
1 and 2.5 Å. Low symmetry covalent systems appear between 1 to 1.45 Å where
the short H-H distance is due to inter hydrogen bonds appearing beside the
dominant hydrogen-host bonds. The highest reported superconducting temperature
for these systems is about 135 K for BH3(77). Interestingly, at around 1.55 Å
the covalent family shows a sharp spike in the predicted $T_{c}$ through
systems sharing linear $X=H=X$ bonds originating through the host $p$
orbitals, a $\bar{3}$m point group, a value of $H_{f}$ equal to 0.75, and a
lack of direct hydrogen-hydrogen bonds. Here lay systems as H3S(138),
H3Se(148), GaH3(80), and GeH3(106). Interestingly, the Si2H6(94) electride
also shares these features even if the bonding nature is slightly different.
All these compounds are shown in Fig. 3 with a representative ELF isosurface.
For H-H distances above approximately 1.55 Å a sharp drop in $T_{c}$ appears,
with systems not reaching above 50 K. This zone lacks direct hydrogen-hydrogen
bonds and features systems with a low percentage of hydrogen, with mostly
isolated, electride, and ionic behavior. Hence, covalent elongated bonds, H-H
or H-X, with high $H_{f}$ seem to be the best candidates to increase $T_{c}$.

Figure 3: Highly symmetric high-$T_{c}$ systems . This figure shows five
compounds related to the $T_{c}$ spike around a hydrogen-hydrogen distance of
1.55 Å. The pink spheres refer to hydrogen atoms, while the rest refer to the
respective host atoms. An ELF isosurface is depicted in each case. Also the
shortest H-H ($D_{HH}$) and H-X ($D_{HX}$) distances are noted. Most of these
compounds belong to the covalent family and show a purple ELF isosurface. The
Si2H6 compound is an electride (note the small ELF bubbles in empty sites) and
its ELF isosurface is shown in orange.

Panel b of Fig. 2 shows the mean extra electron per hydrogen atom obtained
through the analysis of the Bader charge (see Methods). The covalent family
has been excluded from this panel due to the unreliability of the Bader
anlysis, which arises from the difficulties in assigning shared electrons in
the H-X bonds to each atom. We estimate the mean extra electron per hydrogen
atom ($\bar{\rho}$) as

$\bar{\rho}=\frac{\sum_{i=1}^{N_{H}}Q_{i}-N_{H}}{N_{H}},$ (2)

where $Q_{i}$ is the number of electrons assigned to the $i$th hydrogen atom
by the Bader analysis, while $N_{H}$ is the total number of hydrogen atoms in
the primitive cell.

Overall hydrogen atoms tend to gain electrons due to their higher
electronegativity with respect to other atoms in the cell. We observe that the
highest $T_{c}$ values are associated with small charge transfers, i.e. from 0
to 0.25 extra electrons per hydrogen atom. As the extra charge increases,
$T_{c}$ drops sharply below 50 K, exception made for the BeH2(5), compound
belonging to the ionic family with $T_{c}=97$ K, and Si2H6(94), with
$T_{c}=153$ K and part of the elecrtide family.

It is often mentioned in the literature that an increment of (negative) charge
on the hydrogen atom leads to a weakening of the hydrogen-hydrogen bonds 11 ;
104 ; 112 ; https://doi.org/10.1002/wcms.1330 . Our results show that both
molecular and weak covalent hydrogen-hydrogen interaction families show non-
negligible extra electrons on the hydrogen atoms. We also observe that the
extra charge on the hydrogen is responsible for a slight increment of the H-H
distance for the weak covalent hydrogen-hydrogen interaction family. In fact,
compounds within this family with a shorter H-H distance tend to have less
extra electrons per hydrogen atom. The maximum extra electron per hydrogen
among this family is around 0.5, a value that increases up to 1 in the
molecular systems.

Panels c, d, and e in Fig. 2 report the results for the DOS analysis. A
simplified equation for the electron-phonon coupling constant $(\lambda)$ may
be given by PhysRev.167.331

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

where $N(E_{F})$ is the DOS at the Fermi energy, $\langle I^{2}\rangle$ is a
term that is related to the electron-phonon matrix elements, $M$ is the mean
mass of the primitive cell, and $\langle\omega^{2}\rangle$ is the mean
quadratic phonon frequency. This suggests that a higher $N(E_{F})$ should give
rise to a higher electron-phonon coupling and, thus, also a higher
superconducting critical temperature. Although this is a favorable condition
for good superconductors, the results for the DOS prove this condition not to
be univocally sufficient 16 . The total DOS per electron at the Fermi level
shown in panel c of Fig. 2 shows a sharp increment in the highest
superconducting temperatures for a DOS value of around 0.015 $eV^{-1}$.
However, for such values of the DOS, compounds with very low $T_{c}$ can still
be found, making the total DOS at the Fermi level not a good descriptor of
high-$T_{c}$ compounds.

Considering that H atoms due to their light mass are responsible for the large
values of $T_{c}$ in these compounds, we analyze the projection of the DOS at
the Fermi level onto hydrogen atoms (see panel d of Fig. 2). However, the
trends are similar to those obtained for the total DOS. In this case, the
chance to have a higher superconducting critical temperature is related to an
increment of DOS up to a value of 0.06 $eV^{-1}N_{H}^{-1}$. In particular, our
results suggest that the key for high superconductivity is not strictly
related to the value of the DOS. Instead, the fraction of active hydrogen
atoms at the Fermi energy (see panel e) seems to be more relevant. In
agreement with our previous findings, only molecular and weak covalent
hydrogen-hydrogen interaction families are able to reach high amounts of DOS
coming from the hydrogen active atoms. This is directly related to the fact
that the H-H bonds in those systems have a great contribution to the HOMO and
LUMO. Nevertheless, false positives appear where the $T_{c}$ appears very low
despite the large contribution of hydrogen states to the Fermi surface. In
fact, the DOS misses the information on how electrons are coupled with the
lattice vibrations. Systems with very high values of the DOS at the Fermi
energy but with very low electron-phonon coupling will not exhibit high
superconducting critical temperatures.

Networking through the ELF. In order to screen new superconductors and guide
the quest for new superconducting compounds, we need to look for an easily
computable variable that will tell us not only about the interatomic bonding
properties, but also how prone the system is to electron-phonon coupling. Even
if some trends can be observed as discussed above, the descriptors analyzed so
far in Figs. 1 and 2 are unable to capture when electrons couple more strongly
to lattice vibrations and, thus, do not correlate with $T_{c}$. In this
section we propose a new observable based on the study of electron
(de)localization that, instead, correlates well with the predicted
superconducting critical temperature.

The ELF is a proper function to analyze the degree of electronic localization
as high value isosurfaces of the ELF reveal regions in space where electrons
localize. In fact, for isosurfaces with values close to 1 the electrons are
localized generally on atomic sites, and they start to delocalize toward
neighbors and form bonds as the ELF value decreases (see Methods). In order to
analyze this delocalization on a crystal size scale, we define the networking
value $\phi$ as the highest value of the ELF that creates an isosurface
spanning through the whole crystal in all three cartesian directions. The
$\phi$ value can thus be easily extracted by calculating the ELF and
determining at which value a crystal sized isosurface is created when lowering
the ELF value from 1. This isosurface encloses most of the atoms in the
crystal, but not necessarily all. The ELF saddle points reveal crucial for the
determination of the networking value, especially for hydrogen based
compounds. For these systems, where hydrogen-hydrogen bonds are dominant, the
ELF saddle points identify the interatomic bonds that pave the crystal sized
network. Thus, for the determination of the networking value it is sufficient
to identify the ensemble of ELF saddle points at the highest value of ELF able
to bridge the gap between different atoms and create the 3D network.

In Fig. 4 we provide the ELF isosurface related to the $\phi$ value as well as
the network created by the saddle points for this ELF value for PdH(70),
YH4(15), and H3S(138). These three cases are related to the most common types
of networks identified during the analysis: isolated, weak covalent hydrogen-
hydrogen interaction, and covalent families, respectively. In PdH(70) the
network includes both Pd and H atoms. The isolated behavior of the atoms leads
to the ELF bubbles around the atoms getting in contact at very low values of
ELF, 0.19. For YH4(15) the highest 3D connecting network appears at ELF=0.43,
and is constructed only by hydrogen atoms showing a weak covalent interaction.
This is one of the cases in which the 3D network subsisting at the highest
value of ELF does not include all the atoms in the unit cell. For the case of
H3S(138), two interlaced networks appear at ELF=0.68 due to its high symmetry,
which is supported by the weak $H=S$ covalent bonds. Another type of network
is the one arising from the electride systems, where the connection appears
through the isolated pockets of charge in the empty zones of the unit cell.

Interestingly, the networking value correlates rather well with $T_{c}$ as
shown in Fig. 4, clearly much better than any other descriptor based on the
structure or the electronic properties studied so far in the literature and in
Figs. 1 and 2. We attribute this positive correlation to the capacity of this
descriptor to somehow measure how the lattice vibrations affect the electronic
cloud. This is not surprising, because if atoms are connected among them with
highly localized electrons, phonon vibrations are prone to affect more
electrons. The fact that the network is required to span through all the
crystal seems also reasonable, as all, or at least many, phonon modes are
expected to affect the electronic cloud.

Few attempts were made to improve the correlation between $\phi$ and $T_{c}$
by introducing additional conditions for the network identification, such as
having all the hydrogen atoms be part of the network, connecting the atoms
through short direct paths, and creating connections between close hydrogen
atoms up to a given distance. The introduction of these restrictions however
produced only minor improvements at the prize of complicating the definition
of $\phi$. On the opposite side, relaxing the condition about the 3D nature of
the network to simply two dimensional sheets or one dimensional worms worsened
the correlation. This suggests that forming a 3D bonding network is crucial,
which we believe it is related to the fact that in this case the networking
value better captures the overall coupling of the phonon modes with the
electrons as argued above.

As seen in Fig. 4, an improvement in the correlation is obtained by
multiplying $\phi$ by the hydrogen fraction $H_{f}$:

$\Phi=\phi H_{f}.$ (4)

A reasonable explanation for such improvement is that the hydrogen fraction is
a rough estimation of the multiplicity of hydrogen bonds. Systems with few H
atoms will tend to form less bonds in which H atoms participate. This is in
contrast with hydrogen rich systems with an incredible number of bonds in
which H atoms participate. An example is HSe(148) with just 7 bonds per
formula unit and $H_{f}=0.5$, in contrast with LaH10(30) with 20 bonds per
formula unit and $H_{f}$ = 0.90.

We have also investigated the possible correlation between the networking
value and the DOS which was found missing (see Fig. 5). The fact the
networking value does not correlate at all with the DOS at the Fermi level ,
not even with the DOS coming from H atoms, underlines that these two
descriptors are measuring different things. While $\phi$ is able to capture
the effect of the electron-phonon interaction, the DOS cannot, and
consequently does not correlate well with $T_{c}$. On the contrary, $\phi$, or
its product with the hydrogen fraction $\Phi$, determine for the first time a
magnitude that shows a striking correlation with $T_{c}$, and constitutes a
first positive and cheap identification of hydrogen-based superconductors.

Figure 4: Networking value. The upper panel shows the ELF isosurface and the
three dimensional network spanning through all the crystal, which is formed by
the ELF saddle points and the atoms, associated with the $\phi$ value for
PdH(70), YH4(15), and H3S(138). The networking value is given for each case.
The bottom panels show the critical temperature $T_{c}$ as a function of the
networking value $\phi$ (left) and the networking value multiplied by $H_{f}$,
$\Phi=\phi H_{f}$ (right). In both cases there is a linear correlation with
the $T_{c}$, although $\Phi$ correlates slightly better. Figure 5: Networking
value with respect to the DOS. The figure shows the networking value $\phi$ as
a function of the hydrogen contribution to the DOS at the Fermi energy per
hydrogen atom.

##  III Discussion

After the large analysis presented in this work, we can conclude that the
highest critical temperatures are achieved in the molecular, covalent and weak
covalent hydrogen-hydrogen interaction families. These three families are
different expressions of covalent bonds, where electrons are strongly
localized. The highest critical temperatures appear between groups 1 and 5 of
the periodic table, where bonds are mainly driven by covalent hydrogen-
hydrogen interactions, and 13 and 16, where covalent bonds are predominantly
between hydrogen and host atoms. It seems that these covalent compounds reach
their highest $T_{c}$ values for systems with large symmetry and hydrogen-host
bonds at a distance of about 1.55 Å, without direct hydrogen-hydrogen bonding.

It is important to remark that our work shows how the molecular family
transitions towards the weak covalent hydrogen-hydrogen interaction one, with
an associated increase in $T_{c}$. The transition is smooth, starting from low
$T_{c}$ systems with only hydrogen molecules, such as H4I(164), going then
through a mixed phase where molecules expand and intermolecular H-H
interactions start to be present as for ScH9(31) and ScH7(34), to finally
transition towards a full weak interacting behavior with no molecules as found
for YH10(27) and LaH10(30), which have the largest networking value and
$T_{c}$. This suggests that stretching hydrogen molecules is beneficial for
superconductivity. The same conclusion is reached looking at the two pure
hydrogen phases studied here (systems 176 and 177). The hydrogen phase
ascribed to the weak covalent hydrogen-hydrogen interaction behavior (176)
shows a far higher $T_{c}$ of around 300 K compared to the purely molecular
phase (177), with $T_{c}=109$ K. Therefore, stretching hydrogen molecules by
chemical or mechanical means in systems containing many H2 units seems a very
promising path to discovering new high-$T_{c}$ compounds. This seems to put in
context the extraordinary prediction of a critical temperature of 473 K in
Li2MgH16 11 , where doping a molecular MgH16 compound with Li brakes the
molecular units, transforming the system into one with weak covalent hydrogen-
hydrogen interactions.

The networking value $\phi$ defined here is able to capture effectively how
sensitive the electronic cloud is on average to lattice vibrations, and,
consequently, correlates well with $T_{c}$. The networking value improves all
other structural or electronic descriptors previously studied 13 ; 15 ; 17 .
As extracting $\phi$ simply requires the analysis of ELF isosurfaces, which
can be easily obtained post-processing DFT ground state calculations, it
offers a simple way of screening hydrogen-based superconductors, as well as
showing the correct directions to chemically engineering better hydrogen-based
superconductors. Interestingly, as the definition of $\phi$ is completely
general, not limited to the presence of hydrogen in the system, it could
potentially be used to estimate the $T_{c}$ of all phonon-mediated
superconductors. Eventually, it may be also worth studying it in
unconventional superconductors, as the $\phi$ is only based on the analysis of
the electronic cloud and correlations could also appear.

As a final word, we would like to underline that the superconducting critical
temperatures used to find correlations are extracted directly from the
literature, without being recalculated (see Supplementary Data Table I to
check the reference from which the $T_{c}$ value was taken for each case). All
the $T_{c}$ values were obtained by first principles DFT calculations, but at
different levels of theory, for instance, for the estimation of the critical
temperature. The wide grey area in Fig. 4 could be the result of such
inconsistencies. In addition, most of these $T_{c}$ values have been obtained
assuming that the ground state structure is the one given by the minimum of
the Born-Oppenheimer energy surface (classical approximation) and that lattice
vibrations can be described within the harmonic approximation around these
positions. However, in hydrogen-based superconductors, recent calculations
have shown that the crystal structure can be largely modified by ionic quantum
effects and that anharmonicity strongly renormalizes the obtained harmonic
phonon spectra, which can strongly impact the predicted
$T_{c}$PhysRevLett.111.177002 ; PhysRevB.89.064302 ; 33 ; 34 ; Errea81 ; 6 ; 8
; Bianco2018 ; hou2021strong . These effects could produce a narrower
correlation between $T_{c}$ and $\phi$ by introducing variations due to
corrections on the structure and $T_{c}$.

##  IV Methods

DFT Calculations. We carry out our analysis on a sample of 178 compounds
containing hydrogen that had been previously predicted to be superconductors
in the literature. Most of the chosen compounds were those summarized in Ref.
7 , which had been analyzed before in the literature scList ; 100 ; 101 ; 102
; 103 ; 104 ; 105 ; 106 ; 107 ; 108 ; 109 ; 110 ; 111 ; 112 ; 113 ; 115 ; 116
; 117 ; 118 ; 119 ; 120 ; 121 ; 122 ; 123 ; 124 ; 125 ; 126 ; 127 ; 128 ; 129
; 130 ; 131 ; 132 ; 133 ; 134 ; 135 ; 136 ; 137 ; 138 ; 139 ; 140 ; 141 ; 142
; 143 ; 144 ; 145 ; 146 ; 147 ; 148 ; 149 ; 150 ; 151 ; 152 ; 153 ; 154 ; 155
; 156 ; 157 ; 158 ; 159 ; 160 ; 161 ; 162 ; 163 ; 164 ; 165 ; 166 ; 167 ; 168
; 169 ; 170 ; 171 ; 172 ; 173 ; 174 ; 175 ; 176 ; 177 ; 178 ; 179 ; 180 ; 181
; 182 ; 183 ; 184 ; 185 ; 186 ; 187 ; 188 ; 189 ; 190 . We took each of these
compounds, relax them classically at a given pressure with DFT at the Born-
Oppenheimer minimum position, and calculated electronic properties for them.
Due to the immense work required, we did not perform the $T_{c}$ calculations,
but took the values predicted in the literature. Among the 178 compounds, 43
were discarded due to lack of information on the atomic structure, which made
impossible their analysis. Supplementary Data Tables I-III summarize all the
calculated data presented here, as well as from which reference the $T_{c}$
value was taken.

All DFT calculations were performed with the plane-wave Quantum ESPRESSO (QE)
package 27 ; 28 . The exchange correlation potential was approximated with the
Perdew-Burke-Ernzerhof parametrization GGA-PBE . At least the first few upper
core orbitals were included in the pseudopotential for the host atom. The
cutoff for the wave-functions and the density were respectively 70 Ry and 700
Ry. Integrations over the Brillouin zone were performed with the Methfessel-
Paxton smearing technique PhysRevB.40.3616 , with a 0.02 Ry broadening. These
integrations were performed with dense $\mathbf{k}$ point grids, where a
volume of 0.001 Å-3 was occupied per $\mathbf{k}$ point in the Brillouin zone
for the self-consistent calculation and a volume of 0.0002 Å-3 for the non
self-consistent calculation. The electronic properties such as the ELF, the
DOS, and the charge distribution were calculated for each system using the QE
post-processing tools through the results obtained for the non self-consistent
calculations.

Bader charges.  There have been numerous approaches in order to determine the
charge associated to an atom in a molecule. Probably, one of the most useful
in solid state is that derived from the electron density, introduced by Bader
and coworkers in what is called the Quantum Theory of Atoms in Molecules BA90
. In an ordinary solid, the electron density has its maxima (cusps) at the
nuclei and decays exponentially as the electron density moves away from the
nuclei. The resulting topology looks like an assemblage of mountains, each of
which is identified as an atom. The zero gradient surface around these maxima
are well defined surfaces that lead to atoms as non-overlapping units. This
allows determining their charge by mere integration of the electron density
within their associated region of space. Since these regions are non-
overlapping these charges have the advantageous properties of being additive.

The Electron Localization Function.  The electron localization function (ELF)
was developed by Becke and Edgecombe in 1990 BE90  for the analysis in real
space of electron localization, and later on reinterpreted by Savin KS96  in
terms of the Pauli kinetic energy density ($t_{p}$) corrected by the
homogeneous electron gas kinetic energy density ($t_{w}$):

$\displaystyle\chi=\frac{1}{1+(t_{p}/t_{w})^{2}}.$ (5)

$\chi$ is then mapped to run from 0 to a maximum value of 1:

$\displaystyle ELF=\frac{1}{1+\chi^{2}}.$ (6)

Values close to 1 appear in those places where electrons are localized. Hence,
maxima appear in the bonds (as well as in cores and lone pairs). It should be
noted that hydrogen constitutes a peculiar case. A molecule such as N2 will
feature a maximum in the middle of the inter-nitrogen distance associated with
the N-N bond and separated from the N cores. Since a hydrogen molecule only
has 2 electrons, a maximum does not appear for the H-H bond, but rather a
surface with very high ELF that encapsulates the H2 molecule.

It is easy to see from Eq. 6 that the value $ELF=$0.5 is associated with the
distribution in a homogeneous electron gas of the same density as the point of
study. Indeed, metals are characterized by very flat ELF profiles slightly
deviating from 0.5.

Whereas the maxima of ELF provide a measure of how localized electrons are,
its value in between these maxima characterize delocalization in between these
regions, i.e. how easy it is for electrons to go from one localized unit to
another deloc . When the “easiest pathway” is analyzed in the full crystalline
cell, we end up with a picture of how easy it is for electrons to move across
the crystal.

##  V Data availability

All the results presented in this work are summarized in Supplementary Data
Tables I-III. Further data or details are available from the corresponding
author upon reasonable request.

##  VI Code availability

Quantum ESPRESSO is an open-source suite of computational tools available at
https://www.quantum-espresso.org.

## References

  * (1) N. W. Ashcroft.  Metallic hydrogen: A high-temperature superconductor?  Phys. Rev. Lett., 21:1748–1749, Dec 1968.
  * (2) C. B. Satterthwaite and I. L. Toepke.  Superconductivity of hydrides and deuterides of thorium.  Phys. Rev. Lett., 25:741–743, Sep 1970.
  * (3) T. Skoskiewicz.  Superconductivity in the palladium-hydrogen and palladium-nickel-hydrogen systems.  physica status solidi (a), 11(2):K123–K126, 1972.
  * (4) Di Zhou, Dmitrii V. Semenok, Defang Duan, Hui Xie, Wuhao Chen, Xiaoli Huang, Xin Li, Bingbing Liu, Artem R. Oganov, and Tian Cui.  Superconducting praseodymium superhydrides.  Science Advances, 6(9), 2020.
  * (5) Igor Goncharenko, M. I. Eremets, M. Hanfland, J. S. Tse, M. Amboage, Y. Yao, and I. A. Trojan.  Pressure-induced hydrogen-dominant metallic state in aluminum hydride.  Phys. Rev. Lett., 100:045504, Jan 2008.
  * (6) A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, and S. I. Shylin.  Conventional superconductivity at 203 kelvin at high pressures in the sulfur hydride system.  Nature, 525:73, ags 2015.
  * (7) PP Kong, VS Minkov, MA Kuzovnikov, SP Besedin, AP Drozdov, S Mozaffari, L Balicas, FF Balakirev, VB Prakapenka, E Greenberg, et al.  Superconductivity up to 243 k in yttrium hydrides under high pressure.  arXiv preprint arXiv:1909.10482, 2019.
  * (8) Elliot Snider, Nathan Dasenbrock-Gammon, Raymond McBride, Xiaoyu Wang, Noah Meyers, Keith V. Lawler, Eva Zurek, Ashkan Salamat, and Ranga Dias.  Superconductivity to 262 kelvin via catalyzed hydrogenation of yttrium at high pressures, 2020.
  * (9) Ivan A. Troyan, Dmitrii V. Semenok, Alexander G. Kvashnin, Andrey V. Sadakov, Oleg A. Sobolevskiy, Vladimir M. Pudalov, Anna G. Ivanova, Vitali B. Prakapenka, Eran Greenberg, Alexander G. Gavriliuk, Viktor V. Struzhkin, Aitor Bergara, Ion Errea, Raffaello Bianco, Matteo Calandra, Francesco Mauri, Lorenzo Monacelli, Ryosuke Akashi, and Artem R. Oganov.  Anomalous high-temperature superconductivity in yh6, 2020.
  * (10) Maddury Somayazulu, Muhtar Ahart, Ajay K. Mishra, Zachary M. Geballe, Maria Baldini, Yue Meng, Viktor V. Struzhkin, and Russell J. Hemley.  Evidence for superconductivity above 260 k in lanthanum superhydride at megabar pressures.  Phys. Rev. Lett., 122:027001, Jan 2019.
  * (11) AP Drozdov, PP Kong, VS Minkov, SP Besedin, MA Kuzovnikov, S Mozaffari, L Balicas, FF Balakirev, DE Graf, VB Prakapenka, et al.  Superconductivity at 250 k in lanthanum hydride under high pressures.  Nature, 569(7757):528–531, 2019.
  * (12) Elliot Snider, Nathan Dasenbrock-Gammon, Raymond McBride, Mathew Debessai, Hiranya Vindana, Kevin Vencatasamy, Keith V. Lawler, Ashkan Salamat, and Ranga P. Dias.  Room-temperature superconductivity in a carbonaceous sulfur hydride.  Nature, 586(7829):373–377, 2020.
  * (13) Marvin L. Cohen and P. W. Anderson.  Comments on the maximum superconducting transition temperature.  AIP Conference Proceedings, 4(1):17–27, 1972.
  * (14) Tiange Bi, Niloofar Zarifi, Tyson Terpstra, and Eva Zurek.  The search for superconductivity in high pressure hydrides.  In Reference Module in Chemistry, Molecular Sciences and Chemical Engineering. Elsevier, 2019.
  * (15) JosÃ© A. Flores-Livas, Lilia Boeri, Antonio Sanna, Gianni Profeta, Ryotaro Arita, and Mikhail Eremets.  A perspective on conventional high-temperature superconductors at high pressure: Methods and materials.  Physics Reports, 856:1 – 78, 2020.  A perspective on conventional high-temperature superconductors at high pressure: Methods and materials.
  * (16) Chris J. Pickard, Ion Errea, and Mikhail I. Eremets.  Superconducting hydrides under pressure.  Annual Review of Condensed Matter Physics, 11(1):57–76, 2020.
  * (17) Christoph Heil, Simone di Cataldo, Giovanni B. Bachelet, and Lilia Boeri.  Superconductivity in sodalite-like yttrium hydride clathrates.  Phys. Rev. B, 99:220502, Jun 2019.
  * (18) Miguel Borinaga, Ion Errea, Matteo Calandra, Francesco Mauri, and Aitor Bergara.  Anharmonic effects in atomic hydrogen: Superconductivity and lattice dynamical stability.  Phys. Rev. B, 93:174308, May 2016.
  * (19) Miguel Borinaga, P Riego, A Leonardo, Matteo Calandra, Francesco Mauri, Aitor Bergara, and Ion Errea.  Anharmonic enhancement of superconductivity in metallic molecularCmca - 4 hydrogen at high pressure: a first-principles study.  Journal of Physics: Condensed Matter, 28(49):494001, oct 2016.
  * (20) Yanfeng Ge, Fan Zhang, and Yugui Yao.  First-principles demonstration of superconductivity at 280 k in hydrogen sulfide with low phosphorus substitution.  Phys. Rev. B, 93:224513, Jun 2016.
  * (21) Ying Sun, Jian Lv, Yu Xie, Hanyu Liu, and Yanming Ma.  Route to a superconducting phase above room temperature in electron-doped hydride compounds under high pressure.  Phys. Rev. Lett., 123:097001, Aug 2019.
  * (22) Michael J. Hutcheon, Alice M. Shipley, and Richard J. Needs.  Predicting novel superconducting hydrides using machine learning approaches.  Phys. Rev. B, 101:144505, Apr 2020.
  * (23) Takahiro Ishikawa, Takashi Miyake, and Katsuya Shimizu.  Materials informatics based on evolutionary algorithms: Application to search for superconducting hydrogen compounds.  Phys. Rev. B, 100:174506, Nov 2019.
  * (24) Valentin Stanev, Corey Oses, A. Gilad Kusne, Efrain Rodriguez, Johnpierre Paglione, Stefano Curtarolo, and Ichiro Takeuchi.  Machine learning modeling of superconducting critical temperature.  npj Computational Materials, 4, June 2018.
  * (25) Dmitrii V. Semenok, Ivan A. Kruglov, Igor A. Savkin, Alexander G. Kvashnin, and Artem R. Oganov.  On distribution of superconductivity in metal hydrides.  Current Opinion in Solid State and Materials Science, 24(2):100808, 2020.
  * (26) A. D. Becke and K. E. Edgecombe.  A simple measure of electron localization in atomic and molecular systems.  The Journal of Chemical Physics, 92(9):5397–5403, 1990.
  * (27) Andreas Savin, Ove Jepsen, JÃŒrgen Flad, Ole Krogh Andersen, Heinzwerner Preuss, and Hans Georg von Schnering.  Electron localization in solid-state structures of the elements: the diamond structure.  Angewandte Chemie International Edition in English, 31(2):187–188, 1992.
  * (28) Yuri Grin, Andreas Savin, and Bernard Silvi.  The ELF Perspective of chemical bonding, chapter 10, pages 345–382.  John Wiley and Sons, Ltd, 2014.
  * (29) Patricio Fuentealba, E. Chamorro, and Juan C. Santos.  Chapter 5 understanding and using the electron localization function.  In Alejandro Toro-LabbÃ©, editor, Theoretical Aspects of Chemical Reactivity, volume 19 of Theoretical and Computational Chemistry, pages 57 – 85. Elsevier, 2007.
  * (30) Konstantinos Koumpouras and J Andreas Larsson.  Distinguishing between chemical bonding and physical binding using electron localization function (ELF).  Journal of Physics: Condensed Matter, 32(31):315502, may 2020.
  * (31) W Tang, E Sanville, and G Henkelman.  A grid-based bader analysis algorithm without lattice bias.  Journal of Physics: Condensed Matter, 21(8):084204, jan 2009.
  * (32) Edward Sanville, Steven D. Kenny, Roger Smith, and Graeme Henkelman.  Improved grid-based algorithm for bader charge allocation.  Journal of Computational Chemistry, 28(5):899–908, 2007.
  * (33) Graeme Henkelman, Andri Arnaldsson, and Hannes Jonsson.  A fast and robust algorithm for bader decomposition of charge density.  Computational Materials Science, 36(3):354 – 360, 2006.
  * (34) Min Yu and Dallas R. Trinkle.  Accurate and efficient algorithm for bader charge integration.  The Journal of Chemical Physics, 134(6):064111, 2011.
  * (35) Martin Rahm, Roberto Cammi, NW Ashcroft, and Roald Hoffmann.  Squeezing all elements in the periodic table: electron configuration and electronegativity of the atoms under compression.  Journal of the American Chemical Society, 141(26):10253–10271, 2019\.
  * (36) Hui Wang, John S. Tse, Kaori Tanaka, Toshiaki Iitaka, and Yanming Ma.  Superconductive sodalite-like clathrate calcium hydride at high pressures.  Proceedings of the National Academy of Sciences, 109(17):6463–6466, 2012.
  * (37) Feng Peng, Ying Sun, Chris J. Pickard, Richard J. Needs, Qiang Wu, and Yanming Ma.  Hydrogen clathrate structures in rare earth hydrides at high pressures: Possible route to room-temperature superconductivity.  Phys. Rev. Lett., 119:107001, Sep 2017.
  * (38) Hui Wang, Xue Li, Guoying Gao, Yinwei Li, and Yanming Ma.  Hydrogen-rich superconductors at high pressures.  WIREs Computational Molecular Science, 8(1):e1330, 2018.
  * (39) W. L. McMillan.  Transition temperature of strong-coupled superconductors.  Phys. Rev., 167:331–344, Mar 1968.
  * (40) Lilia Boeri.  Understanding Novel Superconductors with Ab Initio Calculations, pages 73–112.  Springer International Publishing, Cham, 2020.
  * (41) Ion Errea, Matteo Calandra, and Francesco Mauri.  First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds.  Phys. Rev. Lett., 111:177002, Oct 2013.
  * (42) Ion Errea, Matteo Calandra, and Francesco Mauri.  Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides.  Phys. Rev. B, 89:064302, Feb 2014.
  * (43) Ion Errea, Francesco Belli, Lorenzo Monacelli, Antonio Sanna, Takashi Koretsune, Terumasa Tadano, Raffaello Bianco, Matteo Calandra, Ryotaro Arita, Francesco Mauri, and JosÃ© A. Flores-Livas.  Quantum crystal structure in the 250-kelvin superconducting lanthanum hydride.  Nature, 578:66 – 69, 2020.
  * (44) Ion Errea, Matteo Calandra, and Francesco Mauri.  First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds.  Phys. Rev. Lett., 111:177002, Oct 2013.
  * (45) Ion Errea, Matteo Calandra, Chris J. Pickard, Joseph R. Nelson, Richard J. Needs, Yinwei Li, Hanyu Liu, Yunwei Zhang, Yanming Ma, and Francesco Mauri.  Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system.  Nature, 532(7597):81–84, Apr 2016.  Letter.
  * (46) Raffaello Bianco, Ion Errea, Matteo Calandra, and Francesco Mauri.  High-pressure phase diagram of hydrogen and deuterium sulfides from first principles: Structural and vibrational properties including quantum and anharmonic effects.  Physical Review B, 97(21), June 2018.
  * (47) Pugeng Hou, Francesco Belli, Raffaello Bianco, and Ion Errea.  Strong anharmonic and quantum effects in pm-3n alh3 under high pressure: A first-principles study, 2021.
  * (48) Tiange Bi, Niloofar Zarifi, Tyson Terpstra, and Eva Zurek.  The search for superconductivity in high pressure hydrides.  In Reference Module in Chemistry, Molecular Sciences and Chemical Engineering. Elsevier, 2019.
  * (49) Yu Xie, Quan Li, Artem R. Oganov, and Hui Wang.  Superconductivity of lithium-doped hydrogen under high pressure.  Acta Crystallographica Section C, 70(2):104–111, Feb 2014.
  * (50) Dawei Zhou, Xilian Jin, Xing Meng, Gang Bao, Yanming Ma, Bingbing Liu, and Tian Cui.  Ab initio study revealing a layered structure in hydrogen-rich kh6 under high pressure.  Phys. Rev. B, 86:014118, Jul 2012.
  * (51) Shuyin Yu, Qingfeng Zeng, Artem R. Oganov, Chaohao Hu, Gilles Frapper, and Litong Zhang.  Exploration of stable compounds, crystal structures, and superconductivity in the be-h system.  AIP Advances, 4(10):107118, 2014.
  * (52) Xiaolei Feng, Jurong Zhang, Guoying Gao, Hanyu Liu, and Hui Wang.  Compressed sodalite-like mgh6 as a potential high-temperature superconductor.  RSC Adv., 5:59292–59296, 2015.
  * (53) Yanchao Wang, Hui Wang, John S. Tse, Toshiaki Iitaka, and Yanming Ma.  Structural morphologies of high-pressure polymorphs of strontium hydrides.  Phys. Chem. Chem. Phys., 17:19379–19385, 2015.
  * (54) James Hooper, Bahadir Altintas, Andrew Shamp, and Eva Zurek.  Polyhydrides of the alkaline earth metals: A look at the extremes under pressure.  The Journal of Physical Chemistry C, 117(6):2982–2992, 2013.
  * (55) Duck Young Kim, Ralph H. Scheicher, Ho-kwang Mao, Tae W. Kang, and Rajeev Ahuja.  General trend for pressurized superconducting hydrogen-dense materials.  Proceedings of the National Academy of Sciences, 107(7):2793–2796, 2010.
  * (56) Yong-Kai Wei, Jiao-Nan Yuan, Faez Iqbal Khan, Guang-Fu Ji, Zhuo-Wei Gu, and Dong-Qing Wei.  Pressure induced superconductivity and electronic structure properties of scandium hydrides using first principles calculations.  RSC Adv., 6:81534–81541, 2016.
  * (57) Yinwei Li, Jian Hao, Hanyu Liu, S Tse John, Yanchao Wang, and Yanming Ma.  Pressure-stabilized superconductive yttrium hydrides.  Scientific Reports, 5:9948, 2015.
  * (58) Shifeng Qian, Xiaowei Sheng, Xiaozhen Yan, Yangmei Chen, and Bo Song.  Theoretical study of stability and superconductivity of ${\mathrm{sch}}_{n}$ ($n=4--8$) at high pressure.  Phys. Rev. B, 96:094513, Sep 2017.
  * (59) Xiaoqiu Ye, Niloofar Zarifi, Eva Zurek, Roald Hoffmann, and NW Ashcroft.  High hydrides of scandium under pressure: potential superconductors.  The Journal of Physical Chemistry C, 122(11):6298–6309, 2018.
  * (60) Bin Li, Zilong Miao, Lei Ti, Shengli Liu, Jie Chen, Zhixiang Shi, and Eugene Gregoryanz.  Predicted high-temperature superconductivity in cerium hydrides at high pressures.  Journal of Applied Physics, 126(23):235901, 2019.
  * (61) Hanyu Liu, Ivan I. Naumov, Roald Hoffmann, N. W. Ashcroft, and Russell J. Hemley.  Potential high-tc superconducting lanthanum and yttrium hydrides at high pressure.  Proceedings of the National Academy of Sciences, 114(27):6990–6995, 2017.
  * (62) IO Bashkin, MV Nefedova, VG Tissen, and EG Ponyatovskii.  Superconductivity in the ti-d system under pressure.  Physics of the Solid State, 40(12):1950–1952, 1998.
  * (63) Kavungal Veedu Shanavas, L Lindsay, and David S Parker.  Electronic structure and electron-phonon coupling in tih 2.  Scientific Reports, 6:28102, 2016.
  * (64) Xiao-Feng Li, Zi-Yu Hu, and Bing Huang.  Phase diagram and superconductivity of compressed zirconium hydrides.  Phys. Chem. Chem. Phys., 19:3538–3543, 2017.
  * (65) Yunxian Liu, Xiaoli Huang, Defang Duan, Fubo Tian, Hanyu Liu, Da Li, Zhonglong Zhao, Xiaojing Sha, Hongyu Yu, Huadi Zhang, et al.  First-principles study on the structural and electronic properties of metallic hfh 2 under pressure.  Scientific reports, 5:11381, 2015.
  * (66) Changbo Chen, Fubo Tian, Defang Duan, Kuo Bao, Xilian Jin, Bingbing Liu, and Tian Cui.  Pressure induced phase transition in mh2 (m = v, nb).  The Journal of Chemical Physics, 140(11):114703, 2014.
  * (67) Guoying Gao, Roald Hoffmann, N. W. Ashcroft, Hanyu Liu, Aitor Bergara, and Yanming Ma.  Theoretical study of the ground-state structures and properties of niobium hydrides under pressure.  Phys. Rev. B, 88:184104, Nov 2013.
  * (68) Quan Zhuang, Xilian Jin, Tian Cui, Yanbin Ma, Qianqian Lv, Ying Li, Huadi Zhang, Xing Meng, and Kuo Bao.  Pressure-stabilized superconductive ionic tantalum hydrides.  Inorganic chemistry, 56(7):3901–3908, 2017.
  * (69) Shuyin Yu, Xiaojing Jia, Gilles Frapper, Duan Li, Artem R Oganov, Qingfeng Zeng, and Litong Zhang.  Pressure-driven formation and stabilization of superconductive chromium hydrides.  Scientific reports, 5:17764, 2015.
  * (70) Xiaofeng Li, Hanyu Liu, and Feng Peng.  Crystal structures and superconductivity of technetium hydrides under pressure.  Phys. Chem. Chem. Phys., 18:28791–28796, 2016.
  * (71) Alexander G Kvashnin, Ivan A Kruglov, Dmitrii V Semenok, and Artem R Oganov.  Iron superhydrides feh5 and feh6: stability, electronic properties, and superconductivity.  The Journal of Physical Chemistry C, 122(8):4731–4736, 2018.
  * (72) Arnab Majumdar, John S. Tse, Min Wu, and Yansun Yao.  Superconductivity in ${\mathrm{feh}}_{5}$.  Phys. Rev. B, 96:201107, Nov 2017.
  * (73) Yunxian Liu, Defang Duan, Fubo Tian, Chao Wang, Yanbin Ma, Da Li, Xiaoli Huang, Bingbing Liu, and Tian Cui.  Stability and properties of the ru-h system at high pressure.  Phys. Chem. Chem. Phys., 18:1516–1520, 2016.
  * (74) Yunxian Liu, Defang Duan, Xiaoli Huang, Fubo Tian, Da Li, Xiaojing Sha, Chao Wang, Huadi Zhang, Ting Yang, Bingbing Liu, et al.  Structures and properties of osmium hydrides under pressure from first principle calculation.  The Journal of Physical Chemistry C, 119(28):15905–15911, 2015\.
  * (75) Liyuan Wang, Defang Duan, Hongyu Yu, Hui Xie, Xiaoli Huang, Yanbin Ma, Fubo Tian, Da Li, Bingbing Liu, and Tian Cui.  High-pressure formation of cobalt polyhydrides: A first-principle study.  Inorganic Chemistry, 57(1):181–186, 2018.
  * (76) B Stritzker and W Buckel.  Superconductivity in the palladium-hydrogen and the palladium-deuterium systems.  Zeitschrift für Physik A Hadrons and nuclei, 257(1):1–8, 1972\.
  * (77) Ion Errea, Matteo Calandra, and Francesco Mauri.  First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds.  Physical review letters, 111(17):177002, 2013.
  * (78) Chao-Hao Hu, Artem R Oganov, Qiang Zhu, Guang-Rui Qian, Gilles Frapper, Andriy O Lyakhov, and Huai-Ying Zhou.  Pressure-induced stabilization and insulator-superconductor transition of bh.  Physical review letters, 110(16):165504, 2013.
  * (79) Kazutaka Abe and NW Ashcroft.  Crystalline diborane at high pressures.  Physical Review B, 84(10):104118, 2011.
  * (80) Yong-Kai Wei, Ni-Na Ge, Guang-Fu Ji, Xiang-Rong Chen, Ling-Cang Cai, Su-Qin Zhou, and Dong-Qing Wei.  Elastic, superconducting, and thermodynamic properties of the cubic metallic phase of alh3 via first-principles calculations.  Journal of Applied Physics, 114(11):114905, 2013.
  * (81) Igor Goncharenko, MI Eremets, M Hanfland, JS Tse, M Amboage, Y Yao, and IA Trojan.  Pressure-induced hydrogen-dominant metallic state in aluminum hydride.  Physical Review Letters, 100(4):045504, 2008.
  * (82) Pugeng Hou, Xiusong Zhao, Fubo Tian, Da Li, Defang Duan, Zhonglong Zhao, Binhua Chu, Bingbing Liu, and Tian Cui.  High pressure structures and superconductivity of alh3(h2) predicted by first principles.  RSC Adv., 5:5096–5101, 2015.
  * (83) Guoying Gao, Hui Wang, Aitor Bergara, Yinwei Li, Guangtao Liu, and Yanming Ma.  Metallic and superconducting gallane under high pressure.  Physical Review B, 84(6):064118, 2011.
  * (84) Yunxian Liu, Defang Duan, Fubo Tian, Hanyu Liu, Chao Wang, Xiaoli Huang, Da Li, Yanbin Ma, Bingbing Liu, and Tian Cui.  Pressure-induced structures and properties in indium hydrides.  Inorganic chemistry, 54(20):9924–9928, 2015.
  * (85) Y Yao, J. S Tse, Y Ma, and K Tanaka.  Superconductivity in high-pressure SiH 4.  Europhysics Letters (EPL), 78(3):37003, apr 2007.
  * (86) JS Tse, Y Yao, and K Tanaka.  Novel superconductivity in metallic snh 4 under high pressure.  Physical Review Letters, 98(11):117004, 2007.
  * (87) Ji Feng, Wojciech Grochala, Tomasz Jaroń, Roald Hoffmann, Aitor Bergara, and NW Ashcroft.  Structures and potential superconductivity in sih 4 at high pressure: En route to "metallic hydrogen".  Physical Review Letters, 96(1):017006, 2006.
  * (88) Huadi Zhang, Xilian Jin, Yunzhou Lv, Quan Zhuang, Yunxian Liu, Qianqian Lv, Kuo Bao, Da Li, Bingbing Liu, and Tian Cui.  High-temperature superconductivity in compressed solid silane.  Scientific reports, 5:8845, 2015.
  * (89) Xiao-Jia Chen, Jiang-Long Wang, Viktor V Struzhkin, Ho-kwang Mao, Russell J Hemley, and Hai-Qing Lin.  Superconducting behavior in compressed solid sih 4 with a layered structure.  Physical review letters, 101(7):077002, 2008.
  * (90) Miguel Martinez-Canales, Artem R Oganov, Yanming Ma, Yan Yan, Andriy O Lyakhov, and Aitor Bergara.  Novel structures and superconductivity of silane under pressure.  Physical review letters, 102(8):087005, 2009.
  * (91) Yinwei Li, Guoying Gao, Yu Xie, Yanming Ma, Tian Cui, and Guangtian Zou.  Superconductivity at 100 k in dense sih4 (h2) 2 predicted by first principles.  Proceedings of the National Academy of Sciences, 107(36):15708–15711, 2010.
  * (92) Xilian Jin, Xing Meng, Zhi He, Yanming Ma, Bingbing Liu, Tian Cui, Guangtian Zou, and Ho-kwang Mao.  Superconducting high-pressure phases of disilane.  Proceedings of the National Academy of Sciences, 107(22):9969–9973, 2010.
  * (93) Guoying Gao, Artem R Oganov, Aitor Bergara, Miguel Martinez-Canales, Tian Cui, Toshiaki Iitaka, Yanming Ma, and Guangtian Zou.  Superconducting high pressure phase of germane.  Physical review letters, 101(10):107002, 2008.
  * (94) Chao Zhang, Xiao-Jia Chen, Yan-Ling Li, Viktor V Struzhkin, Russell J Hemley, Ho-Kwang Mao, Rui-Qin Zhang, and Hai-Qing Lin.  Superconductivity in hydrogen-rich material: Geh 4.  Journal of superconductivity and novel magnetism, 23(5):717–719, 2010.
  * (95) Huadi Zhang, Xilian Jin, Yunzhou Lv, Quan Zhuang, Qianqian Lv, Yunxian Liu, Kuo Bao, Da Li, Bingbing Liu, and Tian Cui.  Investigation of stable germane structures under high-pressure.  Physical Chemistry Chemical Physics, 17(41):27630–27635, 2015.
  * (96) Guohua Zhong, Chao Zhang, Xiaojia Chen, Yanling Li, Ruiqin Zhang, and Haiqing Lin.  Structural, electronic, dynamical, and superconducting properties in dense geh4 (h2) 2.  The Journal of Physical Chemistry C, 116(8):5225–5234, 2012.
  * (97) PuGeng Hou, FuBo Tian, Da Li, ZhongLong Zhao, DeFang Duan, HuaDi Zhang, XiaoJing Sha, BingBing Liu, and Tian Cui.  Ab initio study of germanium-hydride compounds under high pressure.  RSC advances, 5(25):19432–19438, 2015.
  * (98) M Mahdi Davari Esfahani, Artem R Oganov, Haiyang Niu, and Jin Zhang.  Superconductivity and unexpected chemistry of germanium hydrides under pressure.  Physical Review B, 95(13):134506, 2017.
  * (99) Kazutaka Abe and NW Ashcroft.  Quantum disproportionation: The high hydrides at elevated pressures.  Physical Review B, 88(17):174110, 2013.
  * (100) Huadi Zhang, Xilian Jin, Yunzhou Lv, Quan Zhuang, Yunxian Liu, Qianqian Lv, Da Li, Kuo Bao, Bingbing Liu, and Tian Cui.  A novel stable hydrogen-rich snh 8 under high pressure.  RSC advances, 5(130):107637–107641, 2015.
  * (101) Guoying Gao, Artem R Oganov, Peifang Li, Zhenwei Li, Hui Wang, Tian Cui, Yanming Ma, Aitor Bergara, Andriy O Lyakhov, Toshiaki Iitaka, et al.  High-pressure crystal structures and superconductivity of stannane (snh4).  Proceedings of the National Academy of Sciences, 107(4):1317–1320, 2010.
  * (102) JS Tse, Y Yao, and K Tanaka.  Novel superconductivity in metallic snh 4 under high pressure.  Physical Review Letters, 98(11):117004, 2007.
  * (103) Huadi Zhang, Xilian Jin, Yunzhou Lv, Quan Zhuang, Ying Li, Kuo Bao, Da Li, Bingbing Liu, and Tian Cui.  Pressure-induced phase transition of snh4: a new layered structure.  RSC Adv., 6:10456–10461, 2016.
  * (104) M Mahdi Davari Esfahani, Zhenhai Wang, Artem R Oganov, Huafeng Dong, Qiang Zhu, Shengnan Wang, Maksim S Rakitin, and Xiang-Feng Zhou.  Superconductivity of novel tin hydrides (sn n h m) under pressure.  Scientific reports, 6:22873, 2016.
  * (105) I. A. Troyan A.P. Drozdov, M. I. Eremets.  Superconductivity above 100 k in ph3 at high pressures.  arXiv:1508.06224.
  * (106) José A Flores-Livas, Maximilian Amsler, Christoph Heil, Antonio Sanna, Lilia Boeri, Gianni Profeta, Chris Wolverton, Stefan Goedecker, and EKU Gross.  Superconductivity in metastable phases of phosphorus-hydride compounds under high pressure.  Physical Review B, 93(2):020508, 2016.
  * (107) Hanyu Liu, Yinwei Li, Guoying Gao, John S Tse, and Ivan I Naumov.  Crystal structure and superconductivity of ph3 at high pressures.  The Journal of Physical Chemistry C, 120(6):3458–3461, 2016.
  * (108) Tiange Bi, Daniel P Miller, Andrew Shamp, and Eva Zurek.  Superconducting phases of phosphorus hydride under pressure: stabilization by mobile molecular hydrogen.  Angewandte Chemie, 129(34):10326–10329, 2017.
  * (109) Tiange Bi, Daniel P Miller, Andrew Shamp, and Eva Zurek.  Superconducting phases of phosphorus hydride under pressure: stabilization by mobile molecular hydrogen.  Angewandte Chemie, 129(34):10326–10329, 2017.
  * (110) Yuhao Fu, Xiangpo Du, Lijun Zhang, Feng Peng, Miao Zhang, Chris J Pickard, Richard J Needs, David J Singh, Weitao Zheng, and Yanming Ma.  High-pressure phase stability and superconductivity of pnictogen hydrides and chemical trends for compressed hydrides.  Chemistry of Materials, 28(6):1746–1755, 2016.
  * (111) Kazutaka Abe and NW Ashcroft.  Stabilization and highly metallic properties of heavy group-v hydrides at high pressures.  Physical Review B, 92(22):224109, 2015.
  * (112) Yanbin Ma, Defang Duan, Da Li, Yunxian Liu, Fubo Tian, Hongyu Yu, Chunhong Xu, Ziji Shao, Bingbing Liu, and Tian Cui.  High-pressure structures and superconductivity of bismuth hydrides.  arXiv preprint arXiv:1511.05291, 2015.
  * (113) Yinwei Li, Jian Hao, Hanyu Liu, Yanling Li, and Yanming Ma.  The metallization and superconductivity of dense hydrogen sulfide.  The Journal of chemical physics, 140(17):174712, 2014.
  * (114) Defang Duan, Yunxian Liu, Fubo Tian, Da Li, Xiaoli Huang, Zhonglong Zhao, Hongyu Yu, Bingbing Liu, Wenjing Tian, and Tian Cui.  Pressure-induced metallization of dense (h 2 s) 2 h 2 with high-t c superconductivity.  Scientific reports, 4:6968, 2014.
  * (115) Ion Errea, Matteo Calandra, Chris J Pickard, Joseph R Nelson, Richard J Needs, Yinwei Li, Hanyu Liu, Yunwei Zhang, Yanming Ma, and Francesco Mauri.  Quantum hydrogen-bond symmetrization in the superconducting hydrogen sulfide system.  Nature, 532(7597):81–84, 2016.
  * (116) Ryosuke Akashi, Mitsuaki Kawamura, Shinji Tsuneyuki, Yusuke Nomura, and Ryotaro Arita.  First-principles study of the pressure and crystal-structure dependences of the superconducting transition temperature in compressed sulfur hydrides.  Physical Review B, 91(22):224513, 2015.
  * (117) Yinwei Li, Lin Wang, Hanyu Liu, Yunwei Zhang, Jian Hao, Chris J Pickard, Joseph R Nelson, Richard J Needs, Wentao Li, Yanwei Huang, et al.  Dissociation products and structures of solid h 2 s at strong compression.  Physical Review B, 93(2):020103, 2016.
  * (118) Takahiro Ishikawa, Akitaka Nakanishi, Katsuya Shimizu, Hiroshi Katayama-Yoshida, Tatsuki Oda, and Naoshi Suzuki.  Superconducting h 5 s 2 phase in sulfur-hydrogen system under high-pressure.  Scientific reports, 6:23160, 2016.
  * (119) José A Flores-Livas, Antonio Sanna, and EKU Gross.  High temperature superconductivity in sulfur and selenium hydrides at high pressure.  The European Physical Journal B, 89(3):63, 2016.
  * (120) Yanfeng Ge, Fan Zhang, and Yugui Yao.  First-principles demonstration of superconductivity at 280 k in hydrogen sulfide with low phosphorus substitution.  Physical Review B, 93(22):224513, 2016.
  * (121) Shoutao Zhang, Yanchao Wang, Jurong Zhang, Hanyu Liu, Xin Zhong, Hai-Feng Song, Guochun Yang, Lijun Zhang, and Yanming Ma.  Phase diagram and high-temperature superconductivity of compressed selenium hydrides.  Scientific reports, 5(1):1–8, 2015.
  * (122) Xin Zhong, Hui Wang, Jurong Zhang, Hanyu Liu, Shoutao Zhang, Hai-Feng Song, Guochun Yang, Lijun Zhang, and Yanming Ma.  Tellurium hydrides at high pressures: High-temperature superconductors.  Physical Review Letters, 116(5):057002, 2016.
  * (123) Yunxian Liu, Defang Duan, Fubo Tian, Chao Wang, Gang Wu, Yanbin Ma, Hongyu Yu, Da Li, Bingbing Liu, and Tian Cui.  Prediction of stoichiometric poh n compounds: crystal structures and properties.  RSC advances, 5(125):103445–103450, 2015.
  * (124) Defang Duan, Fubo Tian, Zhi He, Xing Meng, Liancheng Wang, Changbo Chen, Xiusong Zhao, Bingbing Liu, and Tian Cui.  Hydrogen bond symmetrization and superconducting phase of hbr and hcl under high pressure: An ab initio study.  The Journal of chemical physics, 133(7):074509, 2010.
  * (125) Defang Duan, Fubo Tian, Zhi He, Xing Meng, Liancheng Wang, Changbo Chen, Xiusong Zhao, Bingbing Liu, and Tian Cui.  Hydrogen bond symmetrization and superconducting phase of hbr and hcl under high pressure: An ab initio study.  The Journal of chemical physics, 133(7):074509, 2010.
  * (126) Siyu Lu, Min Wu, Hanyu Liu, S Tse John, and Bai Yang.  Prediction of novel crystal structures and superconductivity of compressed hbr.  RSC Advances, 5(57):45812–45816, 2015.
  * (127) Changbo Chen, Ying Xu, Xiuping Sun, and Sihan Wang.  Novel superconducting phases of hcl and hbr under high pressure: an ab initio study.  The Journal of Physical Chemistry C, 119(30):17039–17043, 2015\.
  * (128) Andrew Shamp and Eva Zurek.  Superconducting high-pressure phases composed of hydrogen and iodine.  The Journal of Physical Chemistry Letters, 6(20):4067–4072, 2015\.
  * (129) Defang Duan, Fubo Tian, Yunxian Liu, Xiaoli Huang, Da Li, Hongyu Yu, Yanbin Ma, Bingbing Liu, and Tian Cui.  Enhancement of t c in the atomic phase of iodine-doped hydrogen at high pressures.  Physical Chemistry Chemical Physics, 17(48):32335–32340, 2015.
  * (130) Xiaozhen Yan, Yangmei Chen, Xiaoyu Kuang, and Shikai Xiang.  Structure, stability, and superconductivity of new xe–h compounds under high pressure.  The Journal of chemical physics, 143(12):124310, 2015.
  * (131) David C Lonie, James Hooper, Bahadir Altintas, and Eva Zurek.  Metallization of magnesium polyhydrides under pressure.  Physical Review B, 87(5):054107, 2013.
  * (132) José A Flores-Livas, Maximilian Amsler, Christoph Heil, Antonio Sanna, Lilia Boeri, Gianni Profeta, Chris Wolverton, Stefan Goedecker, and EKU Gross.  Superconductivity in metastable phases of phosphorus-hydride compounds under high pressure.  Physical Review B, 93(2):020508, 2016.
  * (133) Andrew Shamp, Tyson Terpstra, Tiange Bi, Zackary Falls, Patrick Avery, and Eva Zurek.  Decomposition products of phosphine under pressure: Ph2 stable and superconducting?  Journal of the American Chemical Society, 138(6):1884–1892, 2016\.
  * (134) Defang Duan, Fubo Tian, Xiaoli Huang, Da Li, Hongyu Yu, Yunxian Liu, Yanbin Ma, Bingbing Liu, and Tian Cui.  Decomposition of solid hydrogen bromide at high pressure.  arXiv preprint arXiv:1504.01196, 2015.
  * (135) Miguel Borinaga, Ion Errea, Matteo Calandra, Francesco Mauri, and Aitor Bergara.  Anharmonic effects in atomic hydrogen: Superconductivity and lattice dynamical stability.  Physical Review B, 93(17):174308, 2016.
  * (136) Miguel Borinaga, P Riego, A Leonardo, Matteo Calandra, Francesco Mauri, Aitor Bergara, and Ion Errea.  Anharmonic enhancement of superconductivity in metallic molecular cmca- 4 hydrogen at high pressure: a first-principles study.  Journal of Physics: Condensed Matter, 28(49):494001, 2016.
  * (137) Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Matteo Calandra, Roberto Car, Carlo Cavazzoni, Davide Ceresoli, Guido L Chiarotti, Matteo Cococcioni, Ismaila Dabo, Andrea Dal Corso, Stefano de Gironcoli, Stefano Fabris, Guido Fratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco Mauri, Riccardo Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbraccia, Sandro Scandolo, Gabriele Sclauzero, Ari P Seitsonen, Alexander Smogunov, Paolo Umari, and Renata M Wentzcovitch.  QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials.  Journal of Physics: Condensed Matter, 21(39):395502, sep 2009.
  * (138) P Giannozzi, O Andreussi, T Brumme, O Bunau, M Buongiorno Nardelli, M Calandra, R Car, C Cavazzoni, D Ceresoli, M Cococcioni, N Colonna, I Carnimeo, A Dal Corso, S de Gironcoli, P Delugas, R A DiStasio, A Ferretti, A Floris, G Fratesi, G Fugallo, R Gebauer, U Gerstmann, F Giustino, T Gorni, J Jia, M Kawamura, H-Y Ko, A Kokalj, E KÃŒçÃŒkbenli, M Lazzeri, M Marsili, N Marzari, F Mauri, N L Nguyen, H-V Nguyen, A Otero de-la Roza, L Paulatto, S Poncé, D Rocca, R Sabatini, B Santra, M Schlipf, A P Seitsonen, A Smogunov, I Timrov, T Thonhauser, P Umari, N Vast, X Wu, and S Baroni.  Advanced capabilities for materials modelling with quantum ESPRESSO.  Journal of Physics: Condensed Matter, 29(46):465901, oct 2017.
  * (139) John P. Perdew, Kieron Burke, and Matthias Ernzerhof.  Generalized Gradient Approximation Made Simple.  Phys. Rev. Lett., 77:3865–3868, Oct 1996.
  * (140) M. Methfessel and A. T. Paxton.  High-precision sampling for brillouin-zone integration in metals.  Phys. Rev. B, 40:3616–3621, Aug 1989.
  * (141) Bader R. F. W.  Oxford University Press, Oxford, 1990.
  * (142) A.D. Becke and K.E. Edgecombe.  J. Chem. Phys, 92:5397, 1990.
  * (143) M. Kohout and A. Savin.  Int. J. Quant. Chem., 60:875, 1996.
  * (144) J. Contreras-Garcia and J. M. Recio.  Electron delocalization and bond formation under the elf framework.  Theor. Chem. Acc., 128:411, 2011.

##  VII Acknowledgements

This research was supported by the European Research Council (ERC) under the
European Unions Horizon 2020 research and innovation programme (grant
agreement No. 802533).

##  VIII Author contributions

F.B. performed all the DFT calculations. I.E. conceived the project. All
authors contributed to the analysis of the data and to the writing of the
manuscript.

##  IX Competing interests

The authors declare no competing interests.