# Onset of metallic transition in molecular liquid hydrogen

Jianqing Guo  School of Physics, Peking University, Beijing 100871, People’s
Republic of China  International Center for Quantum Materials, Peking
University, Beijing 100871, People’s Republic of China  Bingqing Cheng  The
Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg,
Austria  Limei Xu  School of Physics, Peking University, Beijing 100871,
People’s Republic of China  International Center for Quantum Materials, Peking
University, Beijing 100871, People’s Republic of China  Interdisciplinary
Institute of Light-Element Quantum Materials and Research Center for Light-
Element Advanced Materials, Peking University, Beijing, China.  Enge Wang
School of Physics, Peking University, Beijing 100871, People’s Republic of
China  International Center for Quantum Materials, Peking University, Beijing
100871, People’s Republic of China  Interdisciplinary Institute of Light-
Element Quantum Materials and Research Center for Light-Element Advanced
Materials, Peking University, Beijing, China.  Songshan Lake Materials Lab,
Institute of Physics, Chinese Academy of Sciences, Guangdong, China.  School
of Physics, Liaoning University, Shenyang, China  Ji Chen<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, People’s Republic of
China  Interdisciplinary Institute of Light-Element Quantum Materials and
Research Center for Light-Element Advanced Materials, Peking University,
Beijing, China.  Collaborative Innovation Center of Quantum Matter, Beijing
100871, People’s Republic of China

###### Abstract

Liquid-liquid phase transition of hydrogen is at the center of hydrogen phase
diagram as a promising route towards emergent properties such as the Wigner-
Huntington metallization, superconductivity, and superfluidity. Here we report
a study on the liquid-liquid phase transition of hydrogen using the state-of-
the-art diffusion quantum Monte Carlo and density functional theory
calculations. Our results suggest that the metallization process happens at
lower pressures and temperatures compared to the structural phase transition
of molecular to atomic hydrogen. The consequence is that metallized molecular
hydrogen is stable at a wide range of pressures and temperatures. Our study
breaks the conventional assumption that metallization coinciding with
dissociation of hydrogen molecule, and the molecular metallic hydrogen liquid
phase is likely to become the frontier of studying hydrogen phase transitions.

Hydrogen, the simplest and lightest element in nature, is famous for a complex
phase diagram and a rich variety of phase transitions. Among the many phases
of hydrogen, the liquid-liquid phase transition (LLPT) is of particular
interest ever since metallic hydrogen was predicted in 1935, and LLPT is
considered as a path to metallic hydrogen McMahon  _et al._ (2012); Nellis
(2021). Experimentally, the LLPT of hydrogen has been achieved under dynamic
and static compression in the fluid phase, and the evidence of metallization
was mostly provided by the change in optical reflectivity Dzyabura  _et al._
(2013); Zaghoo  _et al._ (2016, 2018); Jiang  _et al._ (2020); Weir  _et al._
(1996); Knudson  _et al._ (2015); Celliers  _et al._ (2018). Nevertheless,
challenges and controversies still exist in experiments especially at low
temperatures, including the uncertainty in determining the extreme pressure
and temperature conditions, the difficulties in structure characterizations,
and the lack of accurate transport measurements of conductivity McMahon  _et
al._ (2012); Fang  _et al._ (2019); Silvera and Dias (2021).

Theoretically, LLPT of hydrogen is often described as a transition from the
insulating molecular phase to the metallic atomic phase, supported by
computational simulations Scandolo (2003); Holst  _et al._ (2008); Morales
_et al._ (2010); Lorenzen  _et al._ (2010); Morales  _et al._ (2013); Geng
_et al._ (2019); Hinz  _et al._ (2020); Tian  _et al._ (2020); van de Bund
_et al._ (2021), indicating or assuming that the molecular-atomic transition
(MAT) and the insulating-metallic transition (IMT) occur simultaneously. In
recent years, new attention has been paid to the supercritical behavior and
the location of liquid-liquid critical point in the computational community,
involving studies using density functional theory (DFT), machine learning
potentials and quantum Monte Carlo (QMC) Li  _et al._ (2015); Gorelov  _et
al._ (2020); Cheng  _et al._ (2020); Karasiev  _et al._ (2021); Cheng  _et
al._ (2021); Tirelli  _et al._ (2021). Based on these studies, it is
understood that below the critical point, if the temperature is above the
melting line, LLPT is one phase transition where $\text{H}_{2}$ molecule
dissociates and metallizes simultaneously; but above the critical point, the
dissociation and metallization happen in a smooth and continuous manner. In
literature, both the molecular-atomic transition and the insulating-metallic
transition have been analysed in detail using the state-of-the-art
computational methods (see e.g. Refs. Pierleoni  _et al._ , 2018; Gorelov  _et
al._ , 2020). However, to the best of our knowledge, there is no theoretical
guarantee or experimental evidence that the metallization of molecular
hydrogen should exactly coincide with the dissociation, regardless if LLPT is
first order or not. On the contrary, experimental studies performed in the
solid regime of dense hydrogen has suggested that metallization can happen
without a molecular-atomic transition, motivating a re-examination of the
nature of LLPT Eremets  _et al._ (2019).

In this paper, we study the molecular-atomic and insulating-metallic phase
transition of liquid hydrogen, to answer the basic question of whether they
coincide with each other or one occurs before another. To provide microscopic
insights, we propose the molecular fraction of liquid hydrogen as the key
order parameter to compare the two kinds of phase transitions on the same
foot. We use the state-of-the-art diffusion quantum Monte Carlo (DMC) method
to benchmark DFT functionals and find that vdw-DF can accurately describe the
structural phase transition of dense liquid hydrogen. We also study the
electronic phase transition of liquid hydrogen by calculating the fundamental
gap as a function of molecular fraction using DMC and DFT methods. We find
that the insulating-metallic transition begins when around 20% hydrogen
molecules dissociate into hydrogen atoms, which requires much lower
dissociation than the half molecular-atomic phase transition. By mapping such
results onto the pressure-temperature phase diagram, we estimate that the
pressure of IMT is tens of GPa lower than structural MAT.

Figure 1:  Molecular and atomic liquid hydrogen from AIMD simulations. (a-b)
Snapshots of typical molecular liquid hydrogen (x = 0.96) and half-molecular-
half-atomic liquid hydrogen (x = 0.48) from the vdW-DF simulations. The
criterion for drawing the H-H bond is 0.9 Å. (c) Molecular fraction as a
function of temperature using different DFT functionals. The molecular
fraction order parameter x is defined in detail in the main text and the
Supplementary Material. Qualitatively the molecular faction x describes the
portion of $\text{H}_{2}$ molecules in liquid hydrogen. (d) HH pair
correlation functions $\text{G}_{\text{HH}}(\text{r})$ at different
temperatures calculated from vdw-DF AIMD simulations. Simulations were carried
out at the density of $\text{r}_{\text{s}}$ = 1.50 Bohr in the NVT ensemble
with 128 hydrogen atoms.

DFT-based ab initio molecular dynamics (AIMD) simulations were carried out
using the Quantum Espresso (QE) package Giannozzi  _et al._ (2009, 2017) with
the Perdew-Burke-Ernzerhof (PBE) Perdew  _et al._ (1996), BLYP-D2 Grimme
(2006), the vdW-DF Dion  _et al._ (2004), the vdW-DF2 Klimeš  _et al._ (2009)
and the SCAN Sun  _et al._ (2015) functionals. The interactions between the
valence electrons were treated with Hamann-Schlüter-Chiang-Vanderbilt
pseudopotentials Hamann  _et al._ (1979); Vanderbilt (1985). Each simulation
was performed in the canonical (NVT) ensemble, and the temperature was
controlled by a Nosé-Hoover thermostat Martyna  _et al._ (1992). DFT total
energy calculations with the PBE, the vdW-DF, the BLYP-D2, the SCAN, the HSE06
Heyd  _et al._ (2003) functionals were performed with both the QE package and
the VASP code Kresse and Furthmüller (1996). DMC calculations were performed
using the CASINO package Needs  _et al._ (2020), with the size-consistent DMC
algorithm ZSGMA Zen  _et al._ (2016). The recently developed energy consistent
correlated electron pseudopotentials (eCEPPs) Trail and Needs (2017) were
used. Further computational details can be found in the Supplemental Material
si .

To investigate the molecular-atomic phase transition we use an order
parameter, namely the molecular fraction. As suggested in previous studies, a
molecular fraction of 0.5 is a good criterion to characterize the structural
molecular-atomic phase transition Tamblyn and Bonev (2010); Geng  _et al._
(2019); Cheng  _et al._ (2020). Specifically, we follow the definition of
Cheng et al.Cheng  _et al._ (2020), where the molecular fraction x was defined
as the fraction of atoms with one neighbour within a smooth cutoff function
starts from 0.8 Å and decays to 0 at 1.1 Å (see Supplementary Material II for
more details and further validation using an agnostic classification scheme si
). There are other definitions of molecular fraction Tamblyn and Bonev (2010);
Geng  _et al._ (2019) and other order parameters, e.g. density/volume, heat
capacity, compressibility, and diffusion coefficient, that can be used to
describe the molecular-atomic transition on the temperature pressure phase
diagram Li  _et al._ (2015); Cheng  _et al._ (2020); Karasiev  _et al._
(2021). A detailed discussion on the thermodynamic properties involved in the
molecular-atomic LLPT is beyond the scope of this study. In this study we
focus on the molecular fraction, which provides the clearest microscopic
insight to the molecular-atomic transition. Fig. 1 (a) and (b) show snapshot
of two structures with a high molecular fraction of (x=0.96) and a low
molecular fraction (x=0.48), respectively.

In Fig. 1 (c) we plot the molecular fraction as a function of temperature by
performing systematic AIMD simulations using different DFT exchange
correlation functionals at the density of $\text{r}_{\text{s}}$ = 1.50 Bohr
with a system size of 128 H atoms. Tests on e.g. size effects and density
dependence are further presented and discussed in Fig. S2 and S3 si . The
simulations confirm that the molecular fraction can be used to monitor the
molecular-atomic LLPT as a function of temperature. The structural transition
is further illustrated by the pair correlation functions in Fig. 1 (d) and
Fig. S1 si , where the first peak is suppressed when the molecular fraction
decreases to $\sim$0.5. However, different functionals show quite different
transition temperatures at a constant density, which are consistent with
shifts of phase boundaries on a pressure-temperature phase diagram observed in
previous studies. Among the various functionals considered, we find that PBE
predict the lowest phase transition temperature and vdW-DF2 has the highest
phase transition temperature. The other three functionals, namely SCAN, vdW-
DF, and BLYP-D2, show behaviors that are in between the cases of PBE and vdW-
DF2. Overall, these computational observations of the qualitative difference
among different DFT functionals agree with the comparisons reported in
literature Clay  _et al._ (2014); Geng  _et al._ (2019).

Figure 2: Benchmark of DFT against DMC. 14 structures covering a wide range of
molecular fraction x were selected for the benchmark. The structures are
collected from vdW-DF AIMD and machine learning potential simulations of 128
hydrogen atoms with a density of rs=1.5 Bohr at different temperatures (see
Supplementary Material II for more details si ). The relative binding energy
is calculated as
$\text{E}_{\text{b}}^{\text{DFT}}-\text{E}_{\text{b}}^{\text{QMC}}$, where
$\text{E}_{\text{b}}=(\text{E}_{\text{liquid}}-\text{N}_{\text{H}}\times\text{E}_{\text{H}})/\text{N}_{\text{H}}$
is the binding energy.

The different results from AIMD simulations suggest that it is necessary to
further evaluate the accuracy of DFT functionals for liquid hydrogen.
Therefore, we calculate the binding energies of a set of structures via DMC
and use them as the benchmark to evaluate the performance of DFT functionals,
a strategy that has been adopted successfully for solid hydrogen Chen  _et
al._ (2014); Clay  _et al._ (2014); Drummond  _et al._ (2015). In Fig. 2, we
plot the relative binding energies of liquid hydrogen structures as a function
of the molecular fraction with respect to the DMC calculations
($\text{E}_{\text{b}}^{\text{DFT}}-\text{E}_{\text{b}}^{\text{QMC}}$). The
binding energy is defined as
$\text{E}_{\text{b}}=(\text{E}_{\text{liquid}}-\text{N}_{\text{H}}\times\text{E}_{\text{H}})/\text{N}_{\text{H}}$,
where $\text{E}_{\text{liquid}}$ is the total energy computed using DFT or
DMC, and $\text{E}_{\text{H}}$ is the energy of a single H atom. As one can
see, vdw-DF and BLYP+D2 describe the interactions in liquid hydrogen most
accurately among all the functionals. From the atomic side to the molecular
side, the relative binding energy differs by less than 10 meV per atom, and
the minor slope of the curve also benefits the prediction of phase boundary
between the molecular phase and the atomic phase. We also find that PBE and
SCAN tend to overestimate the stability of liquid hydrogen, and a large
positive slope means an severe overestimate of the relative stability of the
atomic phase. In contrast, vdw-DF2 underestimates the stability of liquid
hydrogen and the underestimation of the stability of the atomic phase is
stronger. Based on the benchmark, we conclude that vdw-DF and BLYP+D2 describe
the structural phase transition of liquid hydrogen most accurately, whereas
PBE and SCAN tend to facilitate the molecular-atomic transition and vdw-DF2
hinders the hydrogen molecules from dissociating into atoms. The benchmark
conclusions are in line with the AIMD results: the vdw-DF and BLYP+D2 results
coincide with each other within a small variation; PBE and SCAN predict lower
phase transition temperatures while vdw-DF2 has a higher phase transition
temperature. Note that the DFT methods that we used are different in both the
nature of the underlying exchange correlation functionals and the van der
Waals correction scheme. The importance of van der Waals interactions in dense
hydrogen have been mentioned in some studies, however further analyses show
that the differences in describing the molecular and atomic liquid hydrogen
are mainly dominated by the underlying exchange correlation functional while
the van der Waals interactions do not play a significant role (Fig. S4 si ).

Figure 3: Fundamental energy gap for liquid hydrogen as a function of
molecular fraction x calculated using different methods. Fundamental gap is
defined as
$\Delta=\text{E}(\text{N}+1)+\text{E}(\text{N}-1)-2\text{E}(\text{N})$, where
$\text{N}=128$ is the number of electrons, which equals the number of hydrogen
atoms in the system. Lines with different color show fundamental gap computed
using different methods. The structures are selected from those used in Fig.
2, and are detailed in the Supplementary Material. AMH, MMH and MIH indicates
regimes of atomic metallic hydrogen, molecular metallic hydrogen and molecular
insulating hydrogen, respectively. The background is colored with the
molecular fraction.

We now consider the electronic insulating-metallic phase transition of liquid
hydrogen. The fundamental gap of liquid hydrogen as a function of molecular
fraction calculated by DMC is shown in Fig. 3, computed according to
$\Delta=\text{E}(\text{N}+1)+\text{E}(\text{N}-1)-2\text{E}(\text{N})$. We
find that the fundamental gap $\Delta$ closes at molecular fraction $x=0.8$
(red dashed line), which is far from the half molecular-atomic phase
transition boundary $x=0.5$ (green dashed line). A fraction of $x=0.8$ means
liquid hydrogen becomes conductive when about 20% hydrogen molecules
dissociate into atomic hydrogen. Note that the structures used in computing
the fundamental gap are collected from simulations covering a wide range of
temperatures and pressures si . Therefore, despite that hydrogen metallizes
easier under higher pressures, the molecular fraction can be a robust order
parameter to characterize the metallization of hydrogen.

We can also compute the fundamental gap using DFT methods. As show in Fig. 3,
DFT calculations generally underestimate the fundamental gap compared with
DMC. Specifically, HSE06 and vdW-DF underestimates the gap by up to 30%,
whereas PBE may underestimate by more than 50%. However, an interesting fact
is that the molecular fraction where the fundamental gap closes is independent
of the calculation methods. All DFT functionals predict consistently a
threshold of molecular fraction $x=0.8$ for gap closing, therefore we can
directly calculate the fundamental gap using more economic methods to test the
size effects in fundamental gap calculations. So we select more configurations
from vdw-DF AIMD simulations with 512 hydrogen atoms at different densities
($\text{r}_{\text{s}}$ = 1.35 Bohr,$\text{r}_{\text{s}}$ = 1.50 Bohr and
$\text{r}_{\text{s}}$ = 1.65 Bohr respectively) and calculate their
fundamental gaps using PBE, as shown in Fig. S5 si . We find that the
insulating-metallic phase transition boundary is always around molecular
fraction $x=0.8$.

Figure 4: LLPT on the pressure-temperature phase diagram of hydrogen.
Insulating-metallic transition and molecular-atomic transition boundaries
determined in this work are shown as dark red and green solid lines,
respectively. NVT AIMD simulations of 512 hydrogen atoms were performed with
the vdW-DF functional at different temperatures and densities from which the
molecular fraction and pressure were calculated. The two transition boundaries
were estimated by the temperature and the pressure that reproduce the
molecular fraction x=0.8 (dark red line) and x=0.5 (green line), respectively.
Futher details can be found in Fig. S6 of the Supplemental Material si . MIH,
MMH and AMH indicates the molcular-insulating hydrogen, molecular-metallic
hydrogen and the atomic-metallic hydrogen in the liquid regime. The light red
line indicates the melting line adapted from Ref. Nellis, 2021. Symbols with
dashed lines (dark red) show several experimental measurements of insulating-
metallic phase transition. The dark blue dashed line shows the MAT phase
boundary obtained by Geng et al. using the vdW-DF functional Geng  _et al._
(2019). The light blue squares are IMT points calculated by Gorelov et al.
Gorelov  _et al._ (2020) using quantum Monte Carlo. Recent experimental
reports of the insulating-metallic transition boundary by Celliers et al.
Celliers  _et al._ (2018), Knudson et al. Knudson  _et al._ (2015) and Zaghoo
et al. Zaghoo  _et al._ (2016); Zaghoo and Silvera (2017) are presented by the
dark red points.

A clear prediction from the above results is that dense liquid hydrogen can be
metallized before the molecular-atomic phase transition, which implies that
experiments can find metallic hydrogen at lower temperatures/pressures than
atomic hydrogen. To provide a more quantitative prediction, we draw the
insulating-metallic ($x=0.8$) and molecular-atomic ($x=0.5$) phase transition
boundaries in the pressure-temperature phase diagram, using AIMD simulations
with the benchmarked vdW-DF functional (See Supplementary Material I and Fig.
S6 for more details si ). As shown by the pink shaded area in Fig. 4, at a
constant temperature the insulating-metallic transition pressure is shifted
downward 30-60 GPa from the molecular-atomic transition pressure. At a
constant pressure, the insulating-metallic transition temperature is lower
than the molecular-atomic transition temperature by a range of 300 to 1000 K.
The two phase boundaries mean that liquid hydrogen can be divided into three
phases, namely the molecular insulating hydrogen (MIH), the molecular metallic
hydrogen (MMH), and the atomic metallic hydrogen (AMH), as labelled in Fig. 3
and Fig. 4. Molecular semi-metallic phase has been observed in the low
temperature solid regime of dense hydrogenEremets  _et al._ (2019), and
partially dissociated molecular phases of solid hydrogen have also been
discussed experimentally and theoretically Dalladay-Simpson  _et al._ (2016);
Monserrat  _et al._ (2018). Here our prediction suggests that there is a wide
range of molecular metallic hydrogen in the liquid regime, where we can
further explore the intriguing electronic transitions of hydrogen. We note
that a different definition of molecular fraction was used in Ref. Geng  _et
al._ , 2019 to determine the MAT boundary, and our estimates agree quite well
with theirs, suggesting that the conclusions reached are insensitive to the
definition of the molecular fraction.

In experiments of liquid hydrogen, most of the measures are monitoring the
metallization instead of the structural phase transition. Therefore, the
separation of the two phase transition boundaries allows us to make further
quantitative comparison with experimental data, some of which are also plotted
in Fig. 4. Apart from the original results of Knudson et al., which was later
re-interpreted by Celliers et al.Celliers  _et al._ (2018), our new prediction
of the insulating-metallic transition boundary are in better agreements with
experimental data Zaghoo and Silvera (2017); Celliers  _et al._ (2018).
Vibrational measurements using e.g. Raman and Infrared spectroscopy have been
performed to identify structural transitions of dense solid hydrogen Howie
_et al._ (2015), and further extension of such techniques are desired to
determine the molecular-atomic LLPT. On the contrary, theoretical studies of
LLPT have mostly focused on structural phase transitions, and there is space
for further developments of theoretical methods, e.g. employing other
electronic structure calculations. Nuclear quantum effects can also lead to a
shift of both boundaries by a small amount Morales  _et al._ (2013); van de
Bund  _et al._ (2021), but it remains to be investigated whether nuclear
quantum effects on MAT and IMT are effectively the same, and how sensitive are
nuclear quantum effects to the choice of DFT functional.

To conclude, we have performed DMC calculations and AIMD simulations of liquid
hydrogen at a wide range of pressures and temperatures. Our main finding is
that the LLPT of hydrogen is in fact two separate transitions that do not
coincide with each other. Specifically, the insulting-metallic transition
occurs before the molecular-atomic phase transition as the temperature and
pressure increases, leading to a regime of molecular metallic hydrogen with a
width of 30-60 GPa and 300-1000 K. Our results provide an important addition
to the current understanding of the phase diagram of hydrogen. Last but not
the least, the onset of metallic transition in molecular liquid hydrogen is
encouraging for the experimental chasing of the holy grail of metallic
hydrogen and the seeking of high temperature superconductor in dense hydrides.

###### Acknowledgements.

The authors thank X.-Z. Li and A. Zen for helpful discussions. This work was
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant No. XDB33000000, and the National Natural Science
Foundation of China under Grant No. 11974024, 92165101, 11935002. We are
grateful for computational resources provided by TianHe-1A, the High
Performance Computing Platform of Peking University, Shanghai Supercomputer
Center, and the Platform for Data Driven Computational Materials Discovery of
the Songshan Lake Materials Lab.

## References

  * McMahon  _et al._ (2012) J. M. McMahon, M. A. Morales, C. Pierleoni, and D. M. Ceperley, Rev. Mod. Phys. 84, 1607 (2012).
  * Nellis (2021) W. J. Nellis, J. Phys. Chem. Lett. 12, 7972 (2021).
  * Dzyabura  _et al._ (2013) V. Dzyabura, M. Zaghoo, and I. F. Silvera, Proc. Natl. Acad. Sci. U.S.A. 110, 8040 (2013).
  * Zaghoo  _et al._ (2016) M. Zaghoo, A. Salamat, and I. F. Silvera, Phys. Rev. B 93, 155128 (2016).
  * Zaghoo  _et al._ (2018) M. Zaghoo, R. J. Husband, and I. F. Silvera, Phys. Rev. B 98, 104102 (2018).
  * Jiang  _et al._ (2020) S. Jiang, N. Holtgrewe, Z. M. Geballe, S. S. Lobanov, M. F. Mahmood, R. S. McWilliams, and A. F. Goncharov, Advanced Science 7, 1901668 (2020).
  * Weir  _et al._ (1996) S. T. Weir, A. C. Mitchell, and W. J. Nellis, Phys. Rev. Lett. 76, 1860 (1996).
  * Knudson  _et al._ (2015) M. D. Knudson, M. P. Desjarlais, A. Becker, R. W. Lemke, K. R. Cochrane, M. E. Savage, D. E. Bliss, T. R. Mattsson, and R. Redmer, Science 348, 1455 (2015).
  * Celliers  _et al._ (2018) P. M. Celliers, M. Millot, S. Brygoo, R. S. McWilliams, D. E. Fratanduono, J. R. Rygg, A. F. Goncharov, P. Loubeyre, J. H. Eggert, J. L. Peterson, N. B. Meezan, S. L. Pape, G. W. Collins, R. Jeanloz, and R. J. Hemley, Science 361, 677 (2018).
  * Fang  _et al._ (2019) W. Fang, J. Chen, Y. Feng, X.-Z. Li, and A. Michaelides, International Reviews in Physical Chemistry 38, 35 (2019).
  * Silvera and Dias (2021) I. F. Silvera and R. Dias, Advances in Physics: X 6, 1961607 (2021).
  * Scandolo (2003) S. Scandolo, Proc. Natl. Acad. Sci. U.S.A. 100, 3051 (2003).
  * Holst  _et al._ (2008) B. Holst, R. Redmer, and M. P. Desjarlais, Phys. Rev. B 77, 184201 (2008).
  * Morales  _et al._ (2010) M. A. Morales, C. Pierleoni, E. Schwegler, and D. M. Ceperley, Proc. Natl. Acad. Sci. U. S. A 107, 12799 (2010).
  * Lorenzen  _et al._ (2010) W. Lorenzen, B. Holst, and R. Redmer, Phys. Rev. B 82, 195107 (2010).
  * Morales  _et al._ (2013) M. A. Morales, J. M. McMahon, C. Pierleoni, and D. M. Ceperley, Phys. Rev. Lett. 110, 065702 (2013).
  * Geng  _et al._ (2019) H. Y. Geng, Q. Wu, M. Marqués, and G. J. Ackland, Phys. Rev. B 100, 134109 (2019).
  * Hinz  _et al._ (2020) J. Hinz, V. V. Karasiev, S. X. Hu, M. Zaghoo, D. Mejía-Rodríguez, S. B. Trickey, and L. Calderín, Phys. Rev. Research 2, 032065 (2020).
  * Tian  _et al._ (2020) C. Tian, F. Liu, H. Yuan, H. Chen, and Y. Gan, J. Phys. Condens. Matter 33, 015401 (2020).
  * van de Bund  _et al._ (2021) S. van de Bund, H. Wiebe, and G. J. Ackland, Phys. Rev. Lett. 126, 225701 (2021).
  * Li  _et al._ (2015) R. Li, J. Chen, X. Li, E. Wang, and L. Xu, New Journal of Physics 17, 063023 (2015).
  * Gorelov  _et al._ (2020) V. Gorelov, D. M. Ceperley, M. Holzmann, and C. Pierleoni, Phys. Rev. B 102, 195133 (2020).
  * Cheng  _et al._ (2020) B. Cheng, G. Mazzola, C. J. Pickard, and M. Ceriotti, Nature 585, 217 (2020).
  * Karasiev  _et al._ (2021) V. V. Karasiev, J. Hinz, S. X. Hu, and S. B. Trickey, Nature 600 (2021).
  * Cheng  _et al._ (2021) B. Cheng, G. Mazzola, C. J. Pickard, and M. Ceriotti, Nature 600, E15 (2021).
  * Tirelli  _et al._ (2021) A. Tirelli, G. Tenti, K. Nakano, and S. Sorella, arXiv:2112.11099 (2021).
  * Pierleoni  _et al._ (2018) C. Pierleoni, M. Holzmann, and D. M. Ceperley, Contributions to Plasma Physics 58, 99 (2018).
  * Eremets  _et al._ (2019) M. I. Eremets, A. P. Drozdov, P. P. Kong, and H. Wang, Nature Physics 15, 1246 (2019).
  * Giannozzi  _et al._ (2009) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009).
  * Giannozzi  _et al._ (2017) 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. Küçükbenli, 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, J. Phys.: Condens. Matter 29, 465901 (2017).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Grimme (2006) S. Grimme, J. Comput. Chem. 27, 1787 (2006).
  * Dion  _et al._ (2004) M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
  * Klimeš  _et al._ (2009) J. Klimeš, D. R. Bowler, and A. Michaelides, J. Phys. Condens. Matter 22, 022201 (2009).
  * Sun  _et al._ (2015) J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).
  * Hamann  _et al._ (1979) D. R. Hamann, M. Schlüter, and C. Chiang, Phys. Rev. Lett. 43, 1494 (1979).
  * Vanderbilt (1985) D. Vanderbilt, Phys. Rev. B 32, 8412 (1985).
  * Martyna  _et al._ (1992) G. J. Martyna, M. L. Klein, and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992).
  * Heyd  _et al._ (2003) J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 (2003).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Needs  _et al._ (2020) R. J. Needs, M. D. Towler, N. D. Drummond, P. López Ríos, and J. R. Trail, J. Chem. Phys. 152, 154106 (2020).
  * Zen  _et al._ (2016) A. Zen, S. Sorella, M. J. Gillan, A. Michaelides, and D. Alfè, Phys. Rev. B 93, 241118 (2016).
  * Trail and Needs (2017) J. R. Trail and R. J. Needs, J. Chem. Phys. 146, 204107 (2017).
  * (44) See Supplemental Material for more computational details, extended analyses, additional figures, and structure files.
  * Tamblyn and Bonev (2010) I. Tamblyn and S. A. Bonev, Phys. Rev. Lett. 104, 065702 (2010).
  * Clay  _et al._ (2014) R. C. Clay, J. Mcminis, J. M. McMahon, C. Pierleoni, D. M. Ceperley, and M. A. Morales, Phys. Rev. B 89, 184106 (2014).
  * Chen  _et al._ (2014) J. Chen, X. Ren, X.-Z. Li, D. Alfè, and E. Wang, J. Chem. Phys. 141, 024501 (2014).
  * Drummond  _et al._ (2015) N. D. Drummond, B. Monserrat, J. H. Lloyd-Williams, P. López Ríos, C. J. Pickard, and R. J. Needs, Nat. Commun. 6, 7794 (2015).
  * Zaghoo and Silvera (2017) M. Zaghoo and I. F. Silvera, Proc. Natl. Acad. Sci. U.S.A. 114, 11873 (2017).
  * Dalladay-Simpson  _et al._ (2016) P. Dalladay-Simpson, R. T. Howie, and E. Gregoryanz, Nature 529, 63 (2016).
  * Monserrat  _et al._ (2018) B. Monserrat, N. D. Drummond, P. Dalladay-Simpson, R. T. Howie, P. López Ríos, E. Gregoryanz, C. J. Pickard, and R. J. Needs, Phys. Rev. Lett. 120, 255701 (2018).
  * Howie  _et al._ (2015) R. T. Howie, P. Dalladay-Simpson, and E. Gregoryanz, Nature Materials 14 (2015).