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-cl16 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)

Goddard, William A., III

Kim, Hyungjun

Su, Julius T.

English

Caltech Authors - Main

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 compres-
sibility 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 reinterpre-
tation 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 octahedral site
having six Li nuclei neighbors (four nuclei are
ﬃﬃﬃ
2
p
∕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 mechan-
ical 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 loca-
lized electrons interacting with themselves, which the exchange-
correlation 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, be-
cause 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.
To assess how the structural distortion for the eFF description
of bcc can be resolved by using a multiconfiguration wave-
function, we considered 500 different eFF electronic configura-
tions. 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 config-
urations. 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 configura-
tions (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 func-
tional theory studies show that the equilibrium density, equili-
brium 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-to-
cI16 solid-solid transition curve during the isothermal compres-
sion (Lower). In the Inset, the EOS of fcc Li in the low pressure
regime (ρ ≤ 0.9 g∕cm3) is compared with the experimental re-
sults (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 func-
tion of the scaled distance, r∕rs. 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,
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108 1 of 7
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.
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∕rs 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
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 ap-
proximately nine suggesting that the compression induces more
significant IE–Li interaction while sacrificing the Li–Li interac-
tion (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).
Diffusion of Electrons. Fig. S11 shows the change of MSD of in-
terstitial 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 reverbera-
tion 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. KnudsonMD, 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.
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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 2 of 7
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.
 10  15  20
Diffraction angle 2θ (deg)
B  Predicted x-ray diffraction pattern
eFF (300K dynamics)
xtal
110
200 211 220 310
2 3 4 5 6 7
Distance (r)
A  Pair correlation function, g(r)
eFF (300K dynamics) xtal
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).
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108 3 of 7
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.
0
20
40
60
80
100
0.5 1.0 1.5
Pr
es
su
re
 (G
Pa
)
Density (g/cm3)
Calculated Shock Hugoniot Curves
initial phase:
bcc
initial phase:
fcc
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 4 of 7
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 5 of 7
00.5
1
0.5 1.0 1.5
2n
d 
m
in
/3
rd
 m
ax
Density (g/cm3)
C
fcc
cI16
amor
y=0.5
1.34
8
10
12
14
Fi
rs
t C
N
B
fcc
cI16
amor
0 1 2 3 4 5
Pa
ir 
co
rre
la
tio
n 
fu
nc
tio
n,
 g
(r)
Scaled distance (r/rs)
A
ρ=0.53
(rs=1.727)
ρ=0.9
(rs=1.452)
ρ=1.2
(rs=1.319)
ρ=1.3
(rs=1.284)
ρ=1.4
(rs=1.253)
ρ=1.5
(rs=1.224)
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∕rs. (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 1.0g∕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∕rs 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 6 of 7
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.
Kim et al. www.pnas.org/cgi/doi/10.1073/pnas.1110322108 7 of 7


High-temperature high-pressure phases of lithium from electron force field (eFF) quantum electron dynamics simulations

http://authors.library.caltech.edu/25434/1/Kim2011p15868P_Natl_Acad_Sci_Usa.pdf

High-temperature high-pressure phases of lithium from electron force field (eFF) quantum electron dynamics simulations

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