High-temperature high-pressure phases of lithium
from electron force field (eFF) quantum
electron dynamics simulations
Hyungjun Kima, Julius T. Sub, and William A. Goddard IIIa,b,1
a
Center for Materials Simulations and Design, Graduate School of Energy, Environment, Water, and Sustainability (World Class University), Korea
Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea; and bMaterials and Process Simulation Center, California Institute of
Technology, Pasadena, CA 91125
Contributed by William A. Goddard III, July 11, 2011 (sent for review November 3, 2010)
wavepacket dynamics ∣ interstitial electron model ∣ symmetry breaking
T
here are great uncertainties in the properties of matter at the
high compression (several times ambient densities) and high
temperatures (20,000–2,000,000 K) characteristic of deep interiors of giant planets (1), conditions of thermonuclear fusion, and
phenomena generated by shocks from planetary impact (2). New
methods for experimental study of these regimes are being developed (National Ignition Facility at Lawrence Livermore National
Laboratory, Z-Pinch at Sandia National Laboratory) that generate data about materials under extreme conditions. However,
theoretical and computational methods used to predict properties of warm dense matter [high temperatures (T), high pressures
(P), and under rapidly changing conditions] have serious shortcomings leading to considerable uncertainties due to high degrees
of electronic excitation, structural and electronic heterogeneity,
and complex transient dynamics. This contrasts with the situation
at room temperature (RT), where a wealth of structural and
energetic data on compressed phases is available via experiments
in diamond anvil cells and gas guns (3–5), and from theoretical
studies such as density functional theory (DFT).
To provide fresh insight into the dynamical properties of warm
dense matter, we developed the eFF (electron force field) methodology that explicitly solves the time-dependent Schrödinger
equation including all two-body interactions, with the restrictions
that the electrons are described as Gaussian wave packets and
that the wavefunction is described as a Hartree product (with
exchange terms replaced by a spin dependent Hamiltonian). The
eFF makes it practical to describe the nonadiabatic quantum
dynamics of extended systems containing highly excited electrons
at a computational cost comparable to that of classical molecular
www.pnas.org/cgi/doi/10.1073/pnas.1110322108
dynamics (6). This allows the dynamics of electrons and nuclei to
be treated on an equal footing, without the adiabatic assumption
of first solving for stationary states of the electrons. This nonadiabatic approach is very important for the systems at high temperature range (>10;000 K) because the substantial kinetic
energy of the electrons makes their dynamics no longer subordinate to the dynamics of nuclei. Thus the eFF method provides a
useful complement to current DFT methods, which are based on
ground state electronic theory, and hence may not properly handle electronic dynamics for rapidly evolving systems. It is also a
complement to classical force field methods, which contain no
explicit description of electron dynamics.
Previously, we reported the use of eFF to simulate the shock
thermodynamics of hydrogen, including the transformation from
molecules to atoms to plasma under high temperatures and
compressions (see Figs. S1 and S2) (7). We also reported the
application of eFF to describing the electron dynamics of the
highly excited states in diamonoid nanoparticles formed by relaxation of valence electrons into core holes (Auger process)
(8). These studies agree with available experimental data.
Here we extend the eFF into systems containing both tightly
bound core electrons and loosely bound metallic electrons, where
previous wave packet methods have had considerably difficultly.
The alkali metals were long thought to be rather simple nearly
free-electron metals, but in the last decade have been shown
to be rather complex, with localization transitions at high P
(9–17). The spherical shape of the ground state orbital led initially to the belief that the solid would not exhibit phase transitions
beyond the highly coordinated phases; i.e., face-centered cubic
(fcc) phase after body-centered cubic (bcc) under high pressure
(Figs. S3–S5) (18). This description of alkali metals in terms of
nuclear-centered s-like valence electrons was shown to be incorrect from generalized valence bond calculations on Li clusters
(19) and later from DFT calculations on bulk Li (20), which predicted that Li would undergo a symmetry breaking solid-solid
phase transition. The driving force for these structural distortions
is the shifting of the valence electrons to interstitial positions
between the nuclei, leading to strong bonds of a single electron
to several nuclei. The increased stability of these interstitialcentered electrons provides the driving force toward decreased
nuclear coordination number (CN) of the solid at high pressures.
Such predictions stimulated X-ray diffraction experiments of
compressed Li (10), that found a transition from fcc (CN 12)
to the cI16 (CN 11) structure in which each electron is close to
eight nuclei. This was followed by further DFT studies showing
the cI16 (9, 21–24) structure to be stable at high P and low T
Author contributions: H.K., J.T.S., and W.A.G. designed research; H.K. performed research;
H.K., J.T.S., and W.A.G. analyzed data; and H.K., J.T.S., and W.A.G. wrote the paper.
The authors declare no conflict of interest.
1
To whom correspondence should be addressed. E-mail: wag@wag.caltech.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1110322108/-/DCSupplemental.
PNAS ∣ September 13, 2011 ∣ vol. 108 ∣ no. 37 ∣ 15101–15105
PHYSICS
We recently developed the electron force field (eFF) method for
practical nonadiabatic electron dynamics simulations of materials
under extreme conditions and showed that it gave an excellent
description of the shock thermodynamics of hydrogen from molecules to atoms to plasma, as well as the electron dynamics of the
Auger decay in diamondoids following core electron ionization.
Here we apply eFF to the shock thermodynamics of lithium metal,
where we find two distinct consecutive phase changes that manifest themselves as a kink in the shock Hugoniot, previously
observed experimentally, but not explained. Analyzing the atomic
distribution functions, we establish that the first phase transition
corresponds to (i) an fcc-to-cI16 phase transition that was observed
previously in diamond anvil cell experiments at low temperature
and (ii) a second phase transition that corresponds to the formation of a new amorphous phase (amor) of lithium that is distinct
from normal molten lithium. The amorphous phase has enhanced
valence electron-nucleus interactions due to localization of electrons into interstitial locations, along with a random connectivity
distribution function. This indicates that eFF can characterize and
compute the relative stability of states of matter under extreme
conditions (e.g., warm dense matter).
(11, 25). However no attempts have been made to understand
what happens at high pressures and temperatures.
Here we apply eFF to compute the electronic and structural
properties of phases of lithium over a wide range of temperatures
and pressures (up to 20,000 K and 100 GPa), with comparisons to
experiments capable of accessing those extreme conditions.
Results and Discussion
To validate that eFF leads to a correct description of the fcc-tocI16 transition under ambient conditions, we performed isothermal dynamics at 300 K on fcc and cI16 lattices at various densities
[ρ ¼ 0.53 (normal) to 1.5 g∕cm3 ]. We find that eFF describes
both structures well, with predicted X-ray diffraction patterns
exactly matching those from the experiment of (10) (see Fig. S6)
including the transition at P ¼ 40 GPa (see Fig. S7).
The locations and sizes of the valence electrons, as well as the
positions of the nuclei, change over the course of eFF dynamics
(see Fig. 1). For both the fcc and cI16 structures, eFF leads to
valence electrons located in interstitial positions between several
nuclei, in agreement with the interstitial electron (IE) model of
Li (19, 26). For the metallic fcc phase, we find IEs located at
octahedral interstices between Liþ ions (six coordinate), whereas
in the semimetallic cI16, half of the IEs are located inside trapezoidal zigzag chains in the (010) direction, with the other half
sitting in between ions bridging the chains (eight coordinate
but with four shorter bonds). These locations are identical to the
electron density maxima observed in DFT calculations (10).
We then proceeded to compute the shock behavior of Li
solid, assuming that a thermodynamic equilibrium is established
in the bulk solid as the shock front passes through (27). Under
those conditions, the pressure P, volume V , and internal energy E
satisfy the Rankine–Hugoniot equation: E-E0 þ ð1∕2ÞðV -V 0 Þ
ðP þ P0 Þ ¼ 0, where the subscript zero denotes the initial state
before the compression (taken to be ρ0 ¼ 0.53 g∕cm3 and
T 0 ¼ 291 K). We performed eFF dynamics on systems of varying
ρ and T to determine P, V , and E satisfying the above equation.
The initial structure in each case was taken to be either fcc or
cI16, depending on which structure had a lower internal energy
at the given ρ. The detailed procedure is described in the method
section.
Fig. 2 shows that the resulting eFF Hugoniot curve matches
experiments (28–31), including a kink at 1.2 to 1.3 g∕cm3 and
50 GPa. To understand and characterize the microscopic origin
of this effect, we show in Fig. 3A the nuclear pair distribution
functions, g(r), for various points on the Hugoniot curve. Here,
the axis is scaled by r s ¼ ð3∕4πnÞ1∕3 , where n is the number den-
A fcc
B cI16
Fig. 1. The eFF finds as global minima valence electrons in Li that are centered at interstitial sites between Liþ ions. (A) The eFF electron density map of
valence electrons for fcc Li solid at ambient density (ρ ¼ 0.53 g∕cm3 ). Shown
is the density sampled in a (100) plane through the solid, which ranges from
black (low density) to yellow (high density); with the positions of the nuclei
denoted by white circles. Here the valence electrons occupy octahedral interstitial sites. (B) Same plot for cI16 Li solid at high density ρ ¼ 1.3 g∕cm3. Here
the electrons occupy positions within trapezoids of nuclei, and between
neighboring atoms bridging the trapezoidal chains.
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www.pnas.org/cgi/doi/10.1073/pnas.1110322108
Fig. 2. Shock Hugoniot curve for solid Li computed from eFF dynamics (black
circles and red dashed line) agrees well with experiment (violet squares from
ref. 28, brown triangles from ref. 29, orange reverse triangles from ref. 30,
and gray diamonds from ref. 31). The computed T at each ρ is shown with
text on the plot. Both the eFF and experimental shock Hugoniot curves show
a kink at 1.2–1.3 g∕cm3 and approximately 50 GPa. The eFF structural analyses show that this kink results from two closely spaced phase transitions: a
solid-solid transition at 1.21 g∕cm3 and 47 GPa (close to the values of
ρ ¼ 1.20 g∕cm3 and P ¼ 40 GPa calculated with eFF at 300 K in isothermal
compression and the values of P ¼ 45 GPa from experiment; Fig. S7); and
a second, a cI16-to-amor phase transition at 1.33 g∕cm3 and 52 GPa. To access
the high pressure and high temperature phase of amor, we further investigated two other thermodynamic paths: Case 1: Heating up the compressed
system at ρ ¼ 1.5 g∕cm3 a from 300 to 10,000 K, and, Case 2: Compressing
the system from the ambient density of 0.53 g∕cm3 to 1.5 g∕cm3 under high
temperature of 10,000 K. The results are analyzed in Fig. 4.
sity. Integrating g(r) until the first minimum leads to the CN for
each atom (see Fig. S8B).
For ρ ≤ 1.2 g∕cm3 , the g(r) exhibits a clear fcc pattern with
CN ¼ 12. Near 1.3 g∕cm3 , the g(r) pattern changes dramatically,
while still showing long range order. This pattern matches that of
the cI16 solid, leading to the same CN (10.9 at 4,009 K, compared
to 11 for the ideal cI16 solid) although the peak positions are
severely broadened due to the elevated T of the system (4,009 K).
Above 1.4 g∕cm3 , the pattern of g(r) changes dramatically
from that of cI16 phase. The long range order in g(r) disappears
(the third peak of g(r) is nearly gone) and the CN drops to 9.48 at
1.4 g∕cm3 and 9.29 at 1.5 g∕cm3 . This phase, which we denote as
amor, represents a state of lithium whose structural and electronic properties have not been described previously.
To quantify the transition to the amor phase, we compare the
ratio of peak height at the second minimum and peak height at
the third maximum, which shows a large increase at 1.32 g∕cm3
(see Fig. S8C). We find that the combination of the fcc-to-cI16
transition plus the cI16-to-amor transition is responsible for the
kink observed in both the experimental and eFF Hugoniot curves.
To further characterize the amor phase, we define a topological connectivity distribution function, P(n), where one by one we
examine each bond pair (IJ) and ask how many other atoms, n,
are bonded to both I and J. [The bond length is based on the
distance to the first minimum in the g(r) after the first maximum.]
This connectivity distribution function, P(n), quantifies how the
connectivity is locally clustered among the nearest neighbors of
a specific bond. For fcc crystal, Pf cc ð4Þ ¼ 1 whereas all other
Pf cc ðnÞ ¼ 0; for cI16 crystal, PcI16 ð3Þ ¼ 8∕11, PcI16 ð4Þ ¼ 3∕11,
Kim et al.
and PcI16 ðnÞ ¼ 0 otherwise. This connectivity distribution function is related to the Honeycutt–Anderson (HA) analysis (32).
To obtain P(n) of the amor phase, we analyzed six different
simulation trajectories at T ¼ 10;000–20;000 K and ρ ¼ 1.4–1.5
g∕cm3 (one is from the 1.5 g∕cm3 case is on the Hugoniot curve).
All lead to the same form for Pamor ðnÞ as shown in Fig. 3B,
indicating that P(n) is a good order parameter for characterizing
the amor phase. Interestingly, we find that Pamor ðnÞ is fitted
well by a Gaussian distribution function (R2 ¼ 0.98); with mean
value of n ¼ 2.6 and a standard deviation is 0.9. Because P(n) for
a random graph is a binomial distribution (whose large N limit
is a Gaussian distribution; N ¼ number of samples), this implies
that the amor phase has a nearly random clustering with an
average of 2.6 neighbors per bond pair.
To describe the phase transition quantitatively, we write the
P(n) of each system as a linear combination of Pfcc ðnÞ, PcI16 (n),
and Pamor (n):
PðnÞ ¼ f fcc Pfcc ðnÞ þ f cI16 PcI16 ðnÞ þ f amor Pamor ðnÞ þ E;
[1]
where f fcc , f cI16 , and f amor are the composition fractions of the
fcc, cI16, and amor phases, respectively. We used a least squares
procedure to fit P(n) subject to the constraints that all coefficients
are nonnegative. The magnitude of the residual function E, when
squared, never exceeded 0.05, validating our assumption that
Eq. 1 holds.
*Liquid phase under low pressure is generated by the eFF simulation at 0.53 g∕cm3
and 9,388 K.
Kim et al.
Fig. 4. (A) Change of pressure while heating up the system under high compression (Case 1 of Fig. 2). This shows that the slope of amor phase is 2.3 times
larger than that of cI16 phase, indicating that the amor phase has a much
larger entropy than the crystalline phase of cI16. (B) Change in the number
of nuclei coordinated to each interstitial electron during the isothermal
compression at 10,000 K (Case 2 of Fig. 2). Under low compression, valence
electron of liqd phase is coordinated by approximately 6 Liþ nuclei. Under
high compression, the Li CN increases until approximately nine suggesting
that the compression induces more significant IE–Li interactions while sacrificing the Li–Li interaction (resulting from inner-shell formation). This is very
similar to the phase transition from fcc to cI16 (compare Li CN of fcc and cI16
are six and eight, respectively).
The coefficients obtained from this fit along the primary
Hugoniot curve are shown in Fig. 3C. We find that along the
Hugoniot, the system is 100% fcc solid for ρ ≤ 1.1 g∕cm3 , with
f fcc then dropping rapidly to 0 by 1.3 g∕cm3 , with the transition
at 1.21 g∕cm3 . Simultaneous with this decrease of f fcc , f cI16 dramatically increases attaining 84% cI16 character at 1.25 g∕cm3 ,
along with 11% of the amor phase. Then, the cI16 phase disappears by ρ ¼ 1.5 g∕cm3 with the second phase transition at
1.33 g∕cm3 . We note that this transition at ρ of 1.33 g∕cm3
matches the transition density of ρ ¼ 1.32 g∕cm3 deduced from
the disappearance of the third peak in g(r) in Fig. 3C.
We also examined the mean squared displacement (MSD) to
provide a measure of the diffusion rate. The MSD of the amor
phase shows fluidity three times that of the solid cI16 (or fcc)
phases (see Fig. S9).
Next we compare the amor phase with the liquid phase (liqd)
found at high T but low P. Fig. 3B shows that the topological indices of Li–Li bonding, Pliqd ðnÞ vs. Pamor ðnÞ, are quite distinct.
Thus liqd has a P(n) of four to seven neighbors to each bond,
and deviates significantly from the Gaussian distribution for amor
as shown in Fig. 3B.
To further clarify the comparison of the amor phase to other
phases, we investigated the equation of states for two cases.
PNAS ∣ September 13, 2011 ∣
vol. 108 ∣
no. 37 ∣
15103
PHYSICS
Fig. 3. (A) Nuclear-nuclear pair correlation function g(r). This shows a loss of
order, with well-defined solid peaks vanishing as ρ increases from 1.2 g∕cm3
to 1.4 g∕cm3 . For consistency, g(r) is plotted as a function of the scaled
distance, r∕r s . (B) Comparison of the topological index P(n) (number of atoms
bonded to both atoms of each bond) for the various phases from eFF trajectories. The blue and green lines indicate P(n) for fcc and cI16, respectively.
For the amor phase, we examined six slightly different conditions (listed below) all of which exhibit a similar P(n) behavior. Indeed this composite P(n)
function is well fitted by a Gaussian function G(n) centered at 2.6 with a
width of 0.9. We interpret this to indicate that all these cases correspond to
the same amor phase. In contrast the normal liqd phase of Li (orange line*)
shows higher CN (16 vs. 9) and a much broader P(n) distribution. The six
amor cases are: (i) fρ;T g ¼ ð1; the point on the Hugoniot curveÞf1.5;9347g,
(ii)f1.5;10580g, (iii)f1.5;19910g, (iv)f1.4;10050g, (v)f1.4;15260g, and
(vi)f1.4;19720g. (C) Fractional compositions of the Li phases along the Hugoniot curve based on P(n). The f f cc and f amor fitted by a hyperbolic tangent
function of the form 0.5½1 − tanh½aðx − x c Þ, where fa;x c g ¼ f134.83;1.21g
for fcc and fa;x c g ¼ f−17.00;1.33g for amor. This leads to a fcc-to-cI16
transition at 1.21 g∕cm3 and the cI16-to-amor transition at 1.33 g∕cm3 .
1. Case 1: Constant ρ ¼ 1.5 g∕cm3 with T increasing from 300
to 10,000 K (cI16-to-amor phase transition in Fig. 2), and
2. Case 2: Isothermal compression at T ¼ 10;000 K from a
density of 0.53 to 1.5 g∕cc (liqd-to-amor phase transition;
in Fig. 2).
Case 1 leads to the P vs. T at 1.5 g∕cm3 shown in Fig. 4A.
Here ð∂P∕∂TÞV ¼ 0.0020 GPa∕K for cI16 whereas ð∂P∕∂TÞV ¼
0.0046 GPa∕K for amor. From Maxwell’s relations ð∂P∕∂TÞV ¼
ð∂S∕∂V ÞT , the entropy change per volume change at constant
temperature. This indicates that the amor phase has 2.3 times
larger entropy than the cI16 phase. This implies that the “increased T” at the shock wave front is more responsible than the
“increased P” for the transition from the ordered crystalline
phase (energetically favored phase) to the amor phase (entropically favored phase).
Then, we analyzed the change of g(r) during the isothermal
compression at 10,000 K (Case 2; refer Fig. S10A). The CN of
liqd phase at ρ ¼ 0.53 g∕cm3 is nearly 16, much larger than
the CN of amor phase, approximately nine. Until 1.0 g∕cm3 ,
the compression increases the CN of Li monotonically, a pattern
typical for fluids (33). At 1.0 g∕cm3 , we observe that an innershell structure develops [appearing as a new second peak, which
decreases the CN (see Fig. S10B)] by changing the bond distance
The physical origin of the formation of the inner-shell structure
is suggested from the IE-Liþ coordination number in Fig. 4B.
At 0.53 g∕cm3 , the IE of liqd phase shows that each electron
is surrounded by approximately six Liþ ions (as in the fcc phase),
but the compression increases the number of Liþ ions coordinating an IE until approximately nine at 1.5 g∕cm3 (versus the lower
IE-Liþ coordinations of six and eight for fcc solids and cI16 solids,
respectively). Fitting to a hyperbolic tangent function leads a
transition point at 0.95 g∕cm3 , at which the inner shell forms.
This indicates that the driving force of liqd-to-amor phase transition is rooted on the same physical origin that governs the fccto-cI16 phase transition; extremely high pressure induces more
IE–Li interactions while sacrificing Li–Li interactions, leading
to an increasing IE–Li CN and decreasing Li–Li CN. Based on
the Li CN of IE, we conclude that the fluidic phases under high
T, liqd and amor, have local IE–Li interactions similar to that of
the solid phases under low T, fcc and cI16, respectively.
We expect that the amor phase will have a lower electrical
conductivity than the liqd phase because of increased interaction
between valence electrons and nuclei, just as in the comparison
of cI16 and fcc cases. This lower fluidity of valence electrons (see
Fig. S11) is related to the actual diffusion of orbitals. Experiments
on the changes in the electrical conductivity along the Hugoniot
might provide an experimental measure for observing the liqdto-amor phase transition.
Summarizing, we find that the simple eFF method (using
the same functional form, parameters, and potentials as for the
applications to the hydrogen Hugoniot and the Auger dynamics
of diamondoids) captures accurately complex phase transitions
exhibited by Li up to 20,000 K and 100 GPa, indicating that
eFF can describe systems with both localized core elections and
diffuse metallic bonding electrons. The eFF leads to an excellent
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relative stability of many different phases at high P and T. This
suggests that eFF will be useful for describing the structural,
energetic, and electronic properties of warm, dense matter containing complex combinations of covalent, ionic, and metallic
bonding.
Methods
The eFF Dynamics. We performed the eFF dynamics for total simulation times
of 1–2 ps with 1– 2 ps of preequilibration and a time integration step of
dt ¼ 5 attoseconds. With eFF the time-dependent Schrödinger equation uses
an empirical exchange term with three universal parameters that are exactly
the same as in previous publications (6–8). Individual trajectories were integrated in the microcanonical ensemble. We equilibrated the Li system at
each temperature by starting with an energy-minimized geometry, giving
the nuclei a Maxwell–Boltzmann distribution of initial velocities. By varying
the initial velocity distributions temperature, we obtained a correct final
temperature with energy properly partitioned between all degrees of freedom. Either periodic fcc (3 × 3 × 3 supercell, 108 atoms) or cI16 (2 × 2 × 2
supercell, 128 atoms) lattices were employed as an initial structure for the
simulations.
Theoretical Shock Hugoniot Curve. For each density, we started the simulation
with two different crystal structures of fcc and cI16. Then, we determined
the P, T , and E conditions satisfying Rankine–Hugoniot conditions with
both initial structures. Between two sets of P, T , and E determined from
two independent initial structures, we chose the set having the lower internal energy E. This procedure yields initial structures of (i) fcc structures, when
0.53 ≤ ρ < 1.2 g∕cm3 , and (ii) cI16 structures, otherwise. These initial conditions are identical to the conditions having the lower internal energies
during the 300 K isothermal compression (see Fig. S7).
ACKNOWLEDGMENTS. This work was supported by Defense Advanced
Research Planning Agency–Office of Naval Research (PROM), the US Department of Energy (Predictive Science Academic Alliance Program), and WCU
[through the National Research Foundation of Korea funded by the Ministry
of Education, Science, and Technology (R31-2008-000-10055-0)].
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Kim et al.
PNAS ∣ September 13, 2011 ∣
vol. 108 ∣
no. 37 ∣
15105
Supporting Information
Kim et al. 10.1073/pnas.1110322108
SI Results and Discussion
Discussion of H2 Shock Hugoniot. At the time of our publication
of the eFF description of the H2 shock Hugoniot in 2007, eFF
agreed with experiments (2–4) and path integral Monte Carlo
(PIMC) theory (5–7) up to 100 GPa, but eFF led to a compressibility 20% higher than PIMC for 100–200 GPa. More recent
experiments in 2009 (8), shown as blue dots in Fig. S1 agreed with
the eFF result that these is a large increase in the compressibility
between 80 and 150 GPa, confirming the accuracy of eFF for
extreme conditions. The initial Lawrence Livermore National
Laboratory (LLNL) report (8) showed amazing agreement with
the absolute compressibility predicted above 100 GPa, as shown
in Fig. S1.
However more recently the Sandia National Laboratory (SNL)
group Knudson and Desjarlais (9) reinterpreted the LLNL data,
using a more accurate pressure standard based on a recalibration
of the shock compression of quartz to 1.6 TPa: This ends up
with the Hugoniot in Fig. S2, which retains the large increase
in compressibility found in the eFF calculations, but now leads
to disagreement with previous experiments in the region below
100 GPa. We are not aware that there has been any resolution
of this disagreement in the experiments.
This increase in compressibility observed experimentally above
100 GPa was found in eFF to arise from the transition from
molecular H2 to atomic H that occurs above 100 GPa in the
Hugoniot. This LLNL experiments even with the SNL reinterpretation strongly supports the accuracy of eFF in predicting the
phase transitions, but leaves open the question of the absolute
accuracy of the eFF compressibility at high temperatures.
Comparison of Face-Centered Cubic (fcc) and Body-Centered Cubic
(bcc) Phases. For the bcc structure, the electron force field
(eFF) prefers to have each electron in a squashed
pffiffiffi octahedral site
having six Li nuclei neighbors (four nuclei are 2∕2a distant and
two nuclei are 1∕2a distant; a is the unit cell parameter), which is
similar to the fcc phase where eFF leads to an electron in each
octahedral site, with six Li nuclei neighbors at equal distances.
For the fcc phase there is one such octahedral site per atom.
However, in the bcc phase, there are three equivalent sites for
each electron. Considering a (4 × 4 × 4) bcc cell, there are 128
valence electrons to distribute among 384 octahedra, leading
to a huge number of combinations. The exact quantum mechanical ground state would consider the optimum superposition
(resonance) of these combinations, with correlations between
them to prevent double occupation. The density functional theory
(DFT) approximation would have approximately 64 occupied
molecular orbitals per cell, leading to some problems with localized electrons interacting with themselves, which the exchangecorrelation terms would aim to correct. The current eFF method
keeps whole electrons in each localized orbital and must describe
the resonance by hopping these orbitals from site to site. This
will lead to some coupling of the electrons with the lattice, because each single configuration would like to distort the ions
about the current positions of the electrons. Consequently, the
current eFF leads to a minimized structure for finite supercells
(e.g., 128 Li atoms) that distorts slightly. However the dynamical
structure at 300 K for eFF leads to a pair correlation function
and an X-ray diffraction pattern quite close to the bcc crystal
structure but slightly distorted (Fig. S3). We anticipate that longer
dynamics allowing cell dynamics with much longer cell damping
constants would lead to an optimum structure with exactly the bcc
overall cell size.
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108
To assess how the structural distortion for the eFF description
of bcc can be resolved by using a multiconfiguration wavefunction, we considered 500 different eFF electronic configurations. Here we fixed the nuclei positions at the bcc lattice sites,
and carried out eFF minimizations only for the electrons using
500 randomly chosen initial electronic configurations. We then
obtained the forces applied on the nuclei, which is the origin of
structural distortions. We compared these averaged forces to
those from a single electronic configuration.
Fig. S4 shows the resolved forces on the 128 nuclei along
x-direction (A), y-direction (B), and z-direction (C) for the single
eFF configuration compared to the average over the 500 configurations. The forces from a single wavefunction fluctuate up to
approximately 30 kcal∕mol∕Å, which would tend to distort the
bcc structure. However, the consideration of multiple configurations (500 among 6128 ) dramatically reduces the applied forces to
become almost zero, leading to a stable bcc structure.
For the fcc and cI16 phases more relevant to the high pressure
phase, there is one available interstitial site per valence electrons,
so that issues involving multiconfigurational wavefunction are
negligible and eFF preserves the structures without distortions.
This lack of distortion for model supercell sizes is the reason
why we chose the fcc structure as our reference state for the
Hugoniot calculation (but using the bcc density of 0.53 g∕cm3 ).
We consider that this choice is reasonable because density functional theory studies show that the equilibrium density, equilibrium energy, bulk modulus, and its derivative with respect to
pressure of bcc and fcc lithium are almost identical (1)
To clarify how different choices for the initial state affect the
total shock Hugoniot curve, we show below the Hugoniot curve
resulting from using the bcc structure as the initial structure.
The bcc initial structure leads to a slightly larger pressure than
for the fcc initial structure (Fig. S5). However, the differences
are marginal (<8%) leading to a nearly identical shock Hugoniot
curves.
Comparison of eFF Diffraction Pattern and Structure with Experiment.
Fig. S6 compares the eFF predicted structures and calculated
synchrotron X-ray diffraction patterns with experiment (10). We
see excellent agreement.
eFF Isothermal Compression at Room Temperature Results. Fig. S7
shows the isothermal compression at room temperature. The
black line in the top figure shows the equation of states (EOS)
of more stable phase, which guides the eye to track the fcc-tocI16 solid-solid transition curve during the isothermal compression (Lower). In the Inset, the EOS of fcc Li in the low pressure
regime (ρ ≤ 0.9 g∕cm3 ) is compared with the experimental results (11–13). At the bottom, we show the total energy per lithium
atom during the 300 K isothermal compression is calculated
using the initial structures of fcc Li and cI16 Li.
eFF Isothermal Compression at Room Temperature Results. The Pair
correlation function g(r) in Fig. S8A shows a loss of order, with
well-defined solid peaks vanishing as ρ is increased from
1.2 g∕cm3 to 1.4 g∕cm3 . For consistency, g(r) is plotted as a function of the scaled distance, r∕r s . The CN decreases from 12 (fcc)
to 11 (cI16) to 9.29 (amor), showing the transition from fcc
to cI16 to the amor phase. In contrast the CN of the liqd phase
under low P is 16.1 indicating that the newly formed amor phase
is distinct from the liqd phase. Fig. S8C shows that the third peak
of g(r) disappears as ρ increases from 1.2 g∕cm3 to 1.5 g∕cm3 ,
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which is quantified using the ratio of g(r) at the second minimum
point to g(r) at the third maximum point. A rapid transition is
observed at 1.32 g∕cm3 , which we consider as the transition point
from cI16 to amor (intercept with the line y ¼ 0.5).
Diffusion. Fig. S9 shows the mean square displacements (MSD) of
nuclei of four different phases; The fluidity of amor phase is
about 2∕3 that of liqd phase, but 2.5 times larger than that of
the solid fcc or cI16 phases.
compression at 10,000 K, which drops near ρ ¼ 1.0 g∕cm3 due
to the formation of the inner. Under low compression, valence
electron of liqd phase is coordinated by approximately 6 Liþ
nuclei. Under high compression, the Li CN increases until approximately nine suggesting that the compression induces more
significant IE–Li interaction while sacrificing the Li–Li interaction (which results in inner shell formation). This is very similar
to the phase transition from fcc to cI16 (compare Li CN of fcc and
cI16 are six and eight, respectively).
Isothermal Compression at 10,000 K, Case 2. Fig. S10 shows the
results of isothermal compression at 10,000 K (Case 2 of Fig. 2).
The new distinctive peak appears at r∕r s ¼ 2.56, becomes more
significant at higher compression. This indicates the formation of
an inner shell structure during the liqd-to-amor phase transition.
This is supported by the change of CN during the isothermal
Diffusion of Electrons. Fig. S11 shows the change of MSD of interstitial electrons (IEs) during liqd-to-amor phase transition
at 10,000 K. The substantial suppression of MSD at the phase
transition from liqd-to-amor suggests that the IEs of amor phase
is more localized than that of liqd phase.
1. Dacorogna MM, Cohen ML (1986) First-principles study of the structural properties of
alkali metals. Phys Rev B 34:4996–5002.
2. Nellis WJ, et al. (1983) Equation-of-state data for molecular-hydrogen and deuterium
at shock pressures in the range 2–76 GPa (20–760 KBar). J Chem Phys 79:1480–1486.
3. Knudson MD, Hanson DL, Bailey JE, Hall CA, Asay JR (2003) Use of a wave reverberation technique to infer the density compression of shocked liquid deuterium to
75 GPa. Phys Rev Lett 90:035505.
4. Boriskov GV, et al. (2005) Shock compression of liquid deuterium up to 109 GPa. Phys
Rev B 71:092104.
5. Magro WR, Ceperley DM, Pierleoni C, Bernu B (1996) Molecular dissociation in hot,
dense hydrogen. Phys Rev Lett 76:1240.
6. Militzer B (2002) Path integral Monte Carlo simulations of hot dense hydrogen. PhD
dissertation (University of Illinois, Urbana, IL).
7. Militzer B, Ceperley DM (2000) Path integral Monte Carlo calculation of the deuterium
Hugoniot. Phys Rev Lett 85:1890–1893.
8. Hicks DG, et al. (2009) Laser-driven single shock compression of fluid deuterium from
45 to 220 GPa. Phys Rev B 79:014112.
9. Knudson MD, Desjarlais MP (2009) Shock compression of quartz to 1.6 TPa: Redefining
a pressure standard. Phys Rev Lett 103:225501
10. Hanfland M, Syassen K, Christensen NE, Novikov DL (2000) New high-pressure phases
of lithium. Nature 408:174–178).
11. Olinger B, Shaner JW (1983) Lithium, compression and high-pressure structure. Science
219:1071–1072.
12. Vaidya SN, et al. (1971) The compression of the alkali metals to 45 kbar. J Phys Chem
Solids 32:2454–2556.
13. Grover R, et al. (1969) On the compressibility of the alkali metals. J Phys Chem
30:2091–2103.
Fig. S1. The Shock Hugoniot from the 2007 eFF paper showing the experiments available then plus the new data from LLNL (8) published in 2009. These new
experiments in 2009 (shown as blue dots) agree with the eFF results that these is a large increase in the compressibility between 80 and 150 GPa, confirming the
accuracy of eFF for extreme conditions. This increase in compressibility observed experimentally above 100 GPa was found in eFF to arise from the transition
from molecular H2 to atomic H that occurs above 100 GPa in the Hugoniot.
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108
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Fig. S2. Reinterpretation of the LLNL data (8) by SNL (9). At the time of the report in 2007, the eFF agreed with experiments (2–4) and PIMC theory (5–7) up to
100 GPa, but eFF led to a compressibility 20% higher than PIMC for 100–200 GPa. More recent experiments in 2009 (shown as blue dots) (8), however, agree
with the eFF results that these is a large increase in the compressibility between 80 and 150 GPa, confirming the accuracy of eFF for extreme conditions. This
increase in compressibility observed experimentally above 100 GPa was found in eFF to arise from the transition from molecular H2 to atomic H that occurs
above 100 GPa in the Hugoniot.
A Pair correlation function, g(r)
eFF (300K dynamics)
2
3
B
xtal
4
5
Distance (r)
6
7
Predicted x-ray diffraction pattern
110
200
211
xtal
220
310
eFF (300K dynamics)
10
15
Diffraction angle 2θ (deg)
20
Fig. S3. (Upper) Pair correlation function, g(r) is computed from the 0.5 ps eFF dynamics of 0.53 g∕cm3 bcc structure (red line). For the comparison, g(r) of the
bcc crystal is shown simultaneously (black line). (Lower) Calculated synchrotron X-ray diffraction patterns (λ ¼ 0.4124 Å) of 0.53 g∕cm3 bcc structure at 300 K.
One hundred diffraction patterns from 100 snapshots generated from 0.5 ps eFF dynamics are averaged (red line). For the comparison, X-ray diffraction pattern
of the bcc crystal is shown simultaneously (black line).
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Fig. S4. Comparison of (A) x-directional, (B) y-directional, and (C) z-directional forces applied on the nuclei from the single wavefunction description and from
the multiconfigurational wavefunction description. To generate multiconfigurational wavefunction, 500 randomly chosen initial electronic configurations
were minimized using eFF, then, the averaged forces were calculated.
Calculated Shock Hugoniot Curves
100
Pressure (GPa)
80
initial phase:
bcc
60
40
initial phase:
fcc
20
0
0.5
1.0
3
Density (g/cm )
1.5
Fig. S5. Shock Hugoniot curve for solid Li computed from eFF dynamics when bcc is used as an initial phase (red circles and line) and fcc is used as an initial
phase (black circles and line). Regardless of the initial phases, the shock Hugoniot curves are almost identical.
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108
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Fig. S6. The eFF predicted structures and calculated synchrotron X-ray diffraction patterns: (A) 0.53 g∕cm3 of fcc Li at 300 K; (B) 1.3 g∕cm3 of cI16 Li at 300 K.
For each diagram, 100 diffraction patterns from 100 snapshots generated from 0.5 ps eFF dynamics are averaged. The wavelength is set by 0.4124 Å for the
direct comparison to the experimental data from ref. 10.
Fig. S7. Isothermal compression at room temperature. (Upper) Black line shows the EOS of more stable phase, which guides the eye to track the fcc-to-cI16
solid-solid transition curve during the isothermal compression (Lower). In the Inset, the EOS of fcc Li in the low pressure regime (ρ ≤ 0.9 g∕cm3 ) is compared
with the experimental results (11–13) (Lower) Total energy per lithium atom during the 300 K isothermal compression is calculated using the initial structures of
fcc Li and cI16 Li.
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108
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14
Pair correlation function, g(r)
ρ=1.5
(rs=1.224)
ρ=1.4
(rs=1.253)
First CN
A
ρ=1.3
(rs=1.284)
12
fcc
10
ρ=0.53
(rs=1.727)
0
1
2
3
4
5
Scaled distance (r/rs)
2nd min/3rd max
ρ=0.9
(rs=1.452)
cI16
amor
8
1
ρ=1.2
(rs=1.319)
B
C
amor
y=0.5
0.5
cI16
fcc
0
0.5
1.0
1.34 1.5
3
Density (g/cm )
Fig. S8. (A) Pair correlation function g(r). This shows a loss of order, with well-defined solid peaks vanishing as ρ is increased from 1.2 g∕cm3 to 1.4 g∕cm3 . For
consistency, g(r) is plotted as a function of the scaled distance, r∕r s . (B) The CN decreases from 12 (fcc) to 11 (cI16) to 9.29 (amor), showing the transition from fcc
to cI16 to the amor phase. In contrast the CN of the liqd phase under low P is 16.1 indicating that the newly formed amor phase is distinct from the liqd phase.
(C) The third peak of g(r) disappears as ρ increases from 1.2 g∕cm3 to 1.5 g∕cm3 , which is quantified using the ratio of g(r) at the second minimum point to g(r)
at the third maximum point. A rapid transition is observed at 1.32 g∕cm3 , which we consider as the transition point from cI16 to amor (intercept with the line
y ¼ 0.5).
Fig. S9. MSD of nuclei of four different phases. (i) Blue solid line: fcc phase (ρ ¼ 1.0 g∕cm3 , T ¼ 1;718 K). (ii) Green solid line: cI16 phase (ρ ¼ 1.3 g∕cm3 ,
T ¼ 4;009 K). (iii) Red solid line: amor phase (ρ ¼ 1.5 g∕cm3 , T ¼ 9;347 K). (iv) Orange dashed-double dotted line: liqd phase (ρ ¼ 0.53 g∕cm3 ,
T ¼ 9;388 K). The fluidity of amor phase is about 2∕3 that of liqd phase, but 2.5 times larger than that of the solid fcc or cI16 phases.
Fig. S10. Isothermal compression at 10,000 K (Case 2 of Fig. 2). (A) Change of pair correlation function during the isothermal compression at 10,000 K. At
ρ ¼ 1.0 g∕cm3 , a new distinctive peak appears at r∕r s ¼ 2.56, which becomes more significant at higher compression. This indicates the formation of an inner
shell structure during the liqd-to-amor phase transition. (B) Change of CN during the isothermal compression at 10,000 K. Due to the formation of the inner
shell a large drop of the CN is observed near ρ ¼ 1.0 g∕cm3. (C) Change of nuclei coordination number (CN) of the interstitial electron during the isothermal
compression at 10,000 K. Under low compression, valence electron of liqd phase is coordinated by approximately 6 Liþ nuclei. Under high compression, the Li
CN increases until approximately nine suggesting that the compression induces more significant IE–Li interaction while sacrificing the Li–Li interaction (which
results in inner shell formation). This is very similar to the phase transition from fcc to cI16 (compare Li CN of fcc and cI16 are six and eight, respectively).
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108
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Fig. S11. Change of MSD of IEs during liqd-to-amor phase transition at 10,000 K. The phase transition from liqd-to-amor leads a substantial suppression of
MSD, which infers that the IEs of amor phase is more localized than that of liqd phase.
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