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Phys. Plasmas 28, 032706 (2021); https://doi.org/10.1063/5.0040062 28, 032706
© 2021 Author(s).
Efficacy of the radial pair potential
approximation for molecular dynamics
simulations of dense plasmas
Cite as: Phys. Plasmas 28, 032706 (2021); https://doi.org/10.1063/5.0040062
Submitted: 09 December 2020 . Accepted: 05 February 2021 . Published Online: 11 March 2021
Lucas J. Stanek, Raymond C. Clay, M. W. C. Dharma-wardana, Mitchell A. Wood, Kristian R. C. Beckwith,
and Michael S. Murillo
COLLECTIONS
This paper was selected as Featured
Efficacy of the radial pair potential approximation
for molecular dynamics simulations of dense
plasmas
Cite as: Phys. Plasmas 28, 032706 (2021); doi: 10.1063/5.0040062
Submitted: 9 December 2020 .Accepted: 5 February 2021 .
Published Online: 11 March 2021
Lucas J. Stanek,
1,2,a)
Raymond C. Clay III,
2,b)
M. W. C. Dharma-wardana,
3
Mitchell A. Wood,
2
Kristian R. C. Beckwith,
2
and Michael S. Murillo
1,c)
AFFILIATIONS
1
Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing,
Michigan 48824, USA
2
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
3
National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada
a)
Author to whom correspondence should be addressed: staneklu@msu.edu
b)
Electronic mail: rclay@sandia.gov
c)
Electronic mail: murillom@msu.edu
ABSTRACT
Macroscopic simulations of dense plasmas rely on detailed microscopic information that can be computationally expensive and is difficult to
verify experimentally. In this work, we delineate the accuracy boundary between microscale simulation methods by comparing Kohn–Sham
density functional theory molecular dynamics (KS-MD) and radial pair potential molecular dynamics (RPP-MD) for a range of elements,
temperature, and density. By extracting the optimal RPP from KS-MD data using force matching, we constrain its functional form and dis-
miss classes of potentials that assume a constant power law for small interparticle distances. Our results show excellent agreement between
RPP-MD and KS-MD for multiple metrics of accuracy at temperatures of only a few electron volts. The use of RPPs offers orders of magni-
tude decrease in computational cost and indicates that three-body potentials are not required beyond temperatures of a few eV. Due to its
efficiency, the validated RPP-MD provides an avenue for reducing errors due to finite-size effects that can be on the order of 20%.
V
C2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://
creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/5.0040062
I. INTRODUCTION
High energy-density science relies heavily on computational models
to provide information not accessible experimentally due to the high
pressure and transient environments. The plasmas in these experiments
typically contain strongly coupled ions and partially degenerate electrons,
which constrains our microscopic modeling choices to molecular
dynamics (MD) and Monte Carlo approaches. Interfacial mixing in
warm dense matter, for example, requires costly, large-scale MD simula-
tions; but such simulations reveal previously unknown transport mecha-
nisms.
1
It is therefore crucial to quantify the efficacy of computational
models in different regions of species-temperature-density space so that
the cheapest accurate model can be exploited to address such prob-
lems.
2–4
While it is desirable to use short-range, radial, pair potentials
(RPPs) to maximize the length and time scales, N-body energies may be
required in some cases. Few studies have been carried out that compre-
hensively assess the limitations of RPPs and the regimes of utility for the
extant forms; given that the force law is the primary input into MD mod-
els, it essential to have quantitative information about these force models.
A wide variety of RPPs have been developed for modeling dense
plasmas. In some cases the accuracy of the model can be inferred from
its theoretical underpinnings; in other cases, comparison to higher-
fidelity approaches or experiments is needed. Limitations of the RPP
approximation are generally unknown unless compared to an N-body
potential simulation result such as Kohn–Sham density functional theory
molecular dynamics (KS-MD). Both KS-MD simulations and this com-
parison are time-consuming processes that are limited to the temperature
regime in which the pseudopotentials necessary for KS-MD are valid.
5,6
Moreover, comparisons between RPP-MD and KS-MD are limited in
the literature, have not been carried out for a range of elements and tem-
peratures, and are often validated with integrated quantities where indi-
vidual particle dynamics have been averaged and results are subject to
cancelation of errors.
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In this work, we carry out KS-MD simulations for a range of ele-
ments, temperatures, and densities, allowing for a systematic compari-
son of three RPP models. While multiple RPP models can be
selected,
7–11
we choose to compare the widely used Yukawa potential,
which accounts for screening by linearly perturbing around a uniform
density in the long-wavelength (Thomas–Fermi) limit, a potential con-
structed from a neutral pseudo-atom (NPA) approach,
12–15
and the
optimal force-matched RPP that is constructed directly from KS-MD
simulation data.
Each of the models we chose impacts our physics understanding
and has clear computational consequences. For example, success of
the Yukawa model reveals the insensitivity to choices in the pseudopo-
tential and screening function and allows for the largest-scale simula-
tions. Large improvements are expected from the NPA model, which
makes many fewer assumptions with a modest cost of pre-computing
and tabulating forces. (See the Appendix for more details on the NPA
model.) The force-matched RPP requires KS-MD data and is therefore
the most expensive to produce, but it reveals the limitations of RPPs
themselves since they are by definition the optimal RPP.
Using multiple metrics of comparison between RPP-MD and
KS-MD including the relative force error, ion–ion equilibrium radial
distribution function g(r), Einstein frequency, power spectrum, and
the self-diffusion transport coefficient, the accuracy of each RPP model
is analyzed. By simulating disparate elements, namely, an alkali metal,
multiple transition metals, a halogen, a nonmetal, and a noble gas, we
see that force-matched RPPs are valid for simulating dense plasmas at
temperatures above fractions of an eV and beyond. We find that for
all cases except for low temperature carbon, force-matched RPPs accu-
rately describe the results obtained from KS-MD to within a few per-
cent. By contrast, the Yukawa model appears to systematically fail at
describing results from KS-MD at low temperatures for the conditions
studied here validating the need for alternate models such as force-
matching and NPA approaches at these conditions.
In Sec. II, we discuss how RPPs arise from second order perturba-
tion theory and how their representation influences the shape of g(r)
due to particle crowding and/or attraction. Comparisons between
RPPs and KS-MD are done in Sec. III, where we begin by comparing
interparticle forces illustrating how an increase in temperature indi-
cates an increase in accuracy. In addition, the microfield distribution
of forces, Einstein frequency, power spectrum, self-diffusion coeffi-
cient, and g(r) are compared, highlighting how an approximately accu-
rate g(r) does not ensure similar accuracy in time correlation functions
and transport coefficients. A description of how we accurately com-
pute the self-diffusion coefficient and its uncertainty when finite-size
errorsarenon-negligibleisgiveninSec.III C. This further emphasizes
theneedforRPPs,asweminimizefinite-sizeerrorsinKS-MDsimula-
tions by making the necessary corrections as shown in Sec. III E.We
conclude by comparing fully converged (in particle number and simu-
lation time) self-diffusion coefficients to an analytic transport theory;
benchmarking its accuracy and providing an effective interaction cor-
rection to extend the range of applicability.
II. MODELS FOR THE INTERACTION POTENTIAL
The theoretical foundations of the models we will compare are
described in this section; their connections are shown in Fig. 1.We
compare three classes of interactions that are based on the ionic
N-body energy, shown in the top box, pair interactions that are pre-
computed and are analytic or tabulated, shown in the lower-left box,
and optimal pair interactions extracted from the N-body results,
shown in the lower-right box. By comparing these three approaches,
we aim to answer several specific questions. First, given the nuclear
charge Z, ionic number density n
i
, and temperature T, what ranges in
fZ;ni;Tgspace are the fast, pre-computed interactions valid and
therefore allow for large-scale heterogeneous simulations? Second,
how accurate is the “optimal” pair interaction, and what do its limita-
tions reveal about the need for three-body interactions (and perhaps
beyond)? Can these interactions be used to test and correct for finite-
size errors? Third, can the optimal interactions guide the development
of pre-computed interactions? To simplify the discussion we will con-
sider single species matter with a range of Z, each species at its normal
solid ionic mass density q
i
, or in some cases half of that, and in ther-
modynamic equilibrium at temperature T. While we do not consider
mixtures in this work, the framework is general and can be straightfor-
wardly applied to them.
FIG. 1. Connections between different
portions of this work. N-body potentials,
shown in the top box, are used to validate
pair potential models (lower left) and
produce optimal tabulated potentials
(lower right). Both pre-computed RPPs
and tabulated force-matched RPPs pro-
vide finite-size corrections to KS-MD data
assuming they accurately reproduce the
Kohn–Sham potential energy surface. The
tabulated force-matched RPPs highlight
the appropriate RRP representation (e.g.,
oscillations). The pre-computed RPPs
give physical intuition to the representation
determined by the KS-MD data.
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Assuming the Born–Oppenheimer approximation holds, we
define a potential energy surface for the ions as
Utot ¼UNðr1;r2;…;rNÞ:(1)
Physically, the ionic potential energy surface is determined by the elec-
tronic charge distribution arising from ions at a particular set of coordi-
nates; in general, (1) does not simplify into sums over pairwise terms.
There are two major approaches to obtaining (1) in practice. The
approach represented by the top box in Fig. 1 computes the electronic
charge distribution for each ionic distribution. This is achieved computa-
tionally in Kohn–Sham approaches by reducing the electron many-body
problem to a single-electron problem in which the Kohn–Sham electron
moves in the external field of N-ionic centers. The dominant computa-
tional cost comes from solving an NoNoset of eigenvalue equations,
where N
o
is the number of single-particle orbitals. Even though the elec-
tron many-body problem has been simplified to a one-body problem,
matrix diagonalization incurs a cost of OðN3
oÞ, and at high temperatures
the smearing of the Fermi–Dirac distribution requires an increasing
number of orbitals leading to significant increases in computational cost.
The complexity of the electron charge distribution also demands the use
of an advanced “Jacob’s ladder” of exchange-correlation (XC) functions
to address the electron many body problem.
This approach yields an intrinsically ionic N-body potential
energy surface; the electronic density is computed using a description
appropriate to the choice of fZ;ni;Tg. The second approach to calcu-
lating the potential energy surface is to use a cluster-type expansion,
which takes the form
Utot ¼X
N
i
U1ðriÞþX
N
i;j
U2ðri;rjÞþX
N
i;j;k
U3ðri;rj;rkÞþ:(2)
When this expansion can be truncated with only a few terms, interac-
tions can be pre-computed, and fast neighbor algorithms allow for a
very rapid evaluation of forces, typically many orders of magnitude
faster than through use of (1). This allows, for example, for simulations
with trillions of particles.
16–18
However, the disadvantages are that the
computational cost increases rapidly as more terms are included, and
the accuracy of a specific truncation and choice of functional forms
with that truncation are not usually known; part of our goal is to assess
how accurate the potential energy surface in (1) can be represented by
the first two terms of (2).
A. N-body interaction potentials
The most accurate forces are obtained from the gradient of the
total energy in (1), which requires the entire ionic configuration.
Although machine learning approaches are enabling the ability to pre-
learn that relationship,
19–21
it remains more common to compute the
forces for each ionic configuration during the simulation (“on-the-
fly”). We obtain the electronic number density for each ionic configu-
ration in the Kohn–Sham–Mermin formulation of the density
neðrÞ¼X
i
fiðTÞj/iðrÞj2;(3)
where Tis the temperature of the system in energy units, the Fermi
occupations are given by fiðTÞ¼ð1þebðEilÞÞ1, and the
Kohn–Sham–Mermin orbitals /iðrÞsatisfy
1
2r2þveff ðrÞ
/iðrÞ¼i/iðrÞ;(4)
where
veff ðrÞ¼Vext ðrÞþðdr0neðr0Þ
jrr0jþdExc q
½
dqðrÞ
"#
(5)
is a sum of the external (Nion–electron), Hartree, and exchange-
correlation energies. Our KS-MD simulations were done using the
Vienna Ab initio simulation package (VASP).
22–25
The finite tempera-
ture electronic structure was treated with the Mermin free-energy
functional, and we used the Perdew–Burke–Ernzerhof (PBE) func-
tional for the exchange correlation energy.
26
To improve computa-
tional efficiency, we eliminated the chemically inactive core electrons
with the projector augmented-wave
27
pseudopotential. Due to the
anticipated high temperatures and small interionic separations, we
used the smallest core “GW” pseudopotentials available in VASP.
Sixty-four atoms (N¼64) were used in these simulations, with an
energy cutoff of 800eV and at the Baldereschi mean-value k-point
28
for all temperatures ranging from T¼0.5 to 15 eV. A simulation time
step of 0.1 fs was used, and the total simulation lengths for each case
vary and are on the order of a few picoseconds. All KS-MD simula-
tions were first equilibrated in the NVT ensemble and then carried out
in the NVE ensemble where data were collected.
B. Force matching
After the Kohn–Sham potential energy surface has been com-
puted, we aim to construct a compact representation of (1) with (2).
By assuming a parameterized functional form for (2), the force-
matching procedure
29–34
was used to generate the optimal RPP model
based on the KS-MD force data. From each KS-MD simulation, a
dataset of K3NM forces (3 force components, Natoms, and M
atomic configurations) is obtained. Atomic simulation data at nearby
time points are highly correlated; thus, a stride between atomic config-
urations was used to generate 100–200 independent configurations.
With each KS-MD dataset, we determine the optimal RPP for
that system by minimizing the loss function
LðfÞ¼X
K
k¼1
wkðFkðfÞF0
kÞ2:(6)
Here, fis a set of optimizable parameters, FkðfÞis the kth force for the
parameterized model with parameters f,F0
kis the kth force from KS-
MD reference dataset, and w
k
is a weight factor. The weight factor
wk¼1=ðF0
kþeÞ2ensures that both large and small forces contribute
equally to (6). The parameter eshould be varied for each temperature
and element but in most cases here, e1.
Thechoiceofparameterizationcaneitherhaveapre-computed
functional form such as (8) or be determined completely from the data
as is the case for a tabulated potential
35
with spline interpolation—the
choice in this work. For each system, we begin by sampling a TFY
RPP at 15 locations in rand use that as the initial condition for the
force-matching procedure. The TFY RPP is sampled such that
rmin <r<8˚
Awherer
min
is the minimum ionic separation in the KS-
MD dataset. To ensure that the core repulsion and/or attractive oscilla-
tion regions are sampled sufficiently, 10 points are placed in the region
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where rmin <r<4˚
A, leaving the 5 remaining points to be placed
where r>4˚
A. To test for convergence of the optimal force-matched
RPP, two optimization methods were used (specifically simulated
annealing and differential evolution). By choosing a tabulated potential
form for the RPP, the explicit form of the model is entirely determined
from the KS-MD force data andnot limited to a fixed functional form.
While the force-matched RPP yields the best RPP to reproduce
the KS-MD force data, it could be the case three-body and higher
interactions are non-negligible. To check this, we selectively employ
the spectral neighborhood analysis potential (SNAP) which constructs
a potential energy surface from a set of four-body descriptors (bispec-
trum components), where each descriptor is independently weighted,
and these weights are determined by regressing against KS-MD data
of energies and forces. A descriptor captures the strength of density
correlations between neighboring atoms and the central atom within a
given cutoff distance; details can be found in Refs. 36 and 37.The
parameterization of the SNAP uses descriptors of the local atomic
environment capturing up to four-body interactions when represented
in the form of (2), so lower errors associated with SNAP compared to
an optimal RPP are entirely due to three- and four-body interactions.
While higher bodied inter-atomic potentials exist in the literature,
38
it
can be expected there are diminishing accuracy returns with higher
interaction moments; thus, SNAP offers a leading order check on the
RPP compared here.
SNAP potentials utilizing 56 bispectrum component descriptors
were trained on 10% of the KS-MD dataset and additionally tested
against an additional 10% to ensure regression errors were properly
minimized and avoided over-fitting of the KS-MD data.
C. Radial pair potentials
As the computational cost of using on-the-fly N-body interac-
tions is often prohibitive, the least expensive approach utilizes pre-
computed RPPs ignoring most of the terms in (2). Many functional
forms for the RPP have been proposed for application to warm dense
matter often using the second-order perturbation-theory interaction
energy
uðkÞ¼hZi2uCðkÞþjueiðkÞj2vðkÞ;(7)
which is the standard Fourier-space result
39
writtenintermsofthemean
ionization state hZi, the bare Coulomb potential uCðkÞ¼4pe2=k2,the
electron–ion pseudopotential uei ðkÞ, and the susceptibility vðkÞ.
In practice, pair interactions are constructed using nearly the
same steps as for the N-body interactions, with the primary difference
being that each ion is replaced with a single “average atom” (AA),
which is an all-electron, non-linear, finite-temperature density func-
tional theory calculation;
40
such calculations can also be relativis-
tic.
41,42
From the AA, a pseudopotential uei ðkÞand an accurate free/
valence electron response function vðkÞare constructed and (7) is
formed. This approach has three strengths: (1) typical AA models are
not limited to low temperatures, (2) the interaction (7) can be pre-
computed for use in MD, and (3) pair interactions with a fast nearest
neighbor algorithm are very computationally efficient. As we alluded
to above, the accuracy loss attendant to these strengths is what we
wish to determine in this work. The AA itself is aware of the ionic
number density n
i
, which sets the ion-sphere radius ai¼ð3=4pniÞ1=3,
and includes the fact that there is only one ion in the ion sphere, which
implies a g(r); this indirect inclusion of higher-order terms in (2) is
true for all AA-based interactions.
Among the simplest variants of (7), one approximates the pseu-
dopotential as ueiðkÞ4phZie2=k2, where the mean ionization state
hZiresults from a AA calculation,
40
and vðkÞin its long-wavelength
(Thomas–Fermi) limit vTF ðkÞ; this is known as the “Yukawa” interac-
tion.
10,43
Here, we employ a Yukawa interaction with inputs from a
Thomas–Fermi AA,
1
which we will refer to as “TFY.” This procedure
yields an analytic potential in real space of the form
uTFY ðrÞ¼hZi2e2
rexp r=kTF
ðÞ;(8)
where the electron screening is approximated by the Thomas–Fermi
screening length
k2
TF ¼ffiffiffiffiffiffi
8T
p
pF1=2ðbleÞ;(9)
where F1=2is the Fermi–Dirac integral of order 1=2;b¼1=T,
and l
e
is the electron chemical potential. Pad
e fits of Fermi–Dirac
integrals and their inverses are carried out in Refs. 44 and 45.An
approximation to these fits
46
yields
k2
TF 4pnee2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T2þ2
3EF2
s;(10)
where the Fermi energy EF¼h2ð3p2neÞ2=3=2me.NotethattheTFY
interaction is monotonically decreasing (purely repulsive).
Computationally, the TFY model is highly desirable because of its radial,
pair, analytic form with an exponentially damped short range. Its weak-
nesses are the relatively approximate treatments of ueiðkÞand vðkÞ.The
TFY model can be extended by including the gradient corrections to
vTF ðkÞ, but otherwise retaining the other approximations. This improve-
ment yields the Stanton–Murillo potential;
10
the gradient correction to
vTF ðkÞintroduces oscillations in the potential in some plasma regimes
that are absent in the monotonic TFY model. Moreover, gradient correc-
tions add improvements to the cusp at the origin and the large-rasymp-
totic behavior. Here, however, we will only employ the simpler TFY
model.
A great deal of accuracy can be gained by abandoning analytic
inputs to (7). In this case, self-consistent numerical calculations of
each of the terms can be carried out, still allowing for pre-computed
interactions; there is essentially no computational overhead for tabu-
lated interactions.
35
Here, we employ a NPA model that yields both
the mean ionization state and its pseudopotential using a
Kohn–Sham–Mermin approach, as described above, but with a finite-
temperature exchange-correlation potential; the susceptibility is deter-
mined by the Lindhard function with local field corrections.
13
Note
that the electron–ion pseudopotential ueiðkÞintroduces additional
oscillations on length scales different from vðkÞalthough the Friedel
oscillations in vðkÞcontribute much more to the pair interaction. Note
that the name “NPA” has been used by many authors to several differ-
ent average-atom models, and many of them involve approximations
that limit those models to higher temperatures, e.g., T>EF;however,
here we use the one-center density functional theory model developed
by Dharma-wardana and Perrot as this model has been tested at high
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temperatures as well as at very low temperatures, and found to agree
closely with more detailed N-center density functional theory simula-
tions and path-integral quantum calculations where available.
It is worth comparing predictions based on (7) with other forms
suggested previously. A popular RPP for warm dense matter studies is
the short-range repulsion interaction, which adds a long-range,
power-law correction to the TFY model of the form A=r4;
9,47–52
for
A>0, this is also a monotonic interaction, with the goal of increasing
the strength of the TFY model, which underestimates the peak height
of g(r). In Fig. 2, we examine this ansatz by computing a NPA interac-
tion for Al at solid density and T¼1 eV. To find the “best” power law,
we multiply the NPA interaction by various powers r
a
to find regions
where the interaction is flat; a flat region with a¼4wouldrecoverthe
short-range repulsion interaction. It is clear that the A=r4is only valid
over a very small range of rvalues; importantly, the NPA interaction
shows that the exponent aincreases as rbecomes large, which is a true
short-ranged interaction—the empirical correction the short-range
repulsion model adds greatly overestimates the strength of the interac-
tion at large interparticle separations.
12
Worse, the short-range repul-
sion model potentially gets an accurate answer for the wrong reason,
as we explore in Fig. 3.
Because the form (7) generally has oscillations, the enhanced
peak height of g(r) from the NPA model over the TFY model occurs
for two, independent reasons. Attractive regions of the interaction, as
shown in the top panel of Fig. 3, can produce very strong peaks in g(r).
Conversely, stronger overall repulsion at intermediate rcan lead to a
similar g(r) behavior, as shown in the bottom panel of Fig. 3,butwith
rapid decay of the interaction at larger r. The functional form (7) natu-
rally contains both the “crowding” and “attraction” behaviors as spe-
cial cases. Figure 4 shows a comparison of the RPPs for C, Al, V, and
Au at T¼0.5and5eV.TheTFYmodelispurelymonotonic,whereas
the force-matched and NPA RPPs have attractive and repulsive
regions in their oscillations. Below, we will explore the consequences
of these features of the interaction on ionic transport.
Once the RPPs have been constructed, MD simulations were car-
ried out using in the large-scale atomic/molecular massively parallel
simulator (LAMMPS).
53
For the tabulated RPPs (force-matched and
NPA) a linear interpolation was needed to determine the force value
between tabulation points. To make a direct comparison between the
RPP-MD and KS-MD results, all simulations were carried out in a
three dimensional periodic box with 64 atoms and a time step of 0.1 fs.
The length of each simulation is identical to the corresponding simula-
tion performed with KS-MD. Keeping these conditions identical
avoids the unintentional reduction in statistical errors between KS-
MD and RPP-MD. All simulations were first equilibrated in the NVT
ensemble so that the average temperature for each simulation during
the data collection phase is within 1% of the reported temperature in
Table I. The data collection phase was carried out in the NVE ensem-
ble. In Sec. III E, a finite-size effect study was done for the cases of C at
2.267 g/cm
3
and V at 6.11 g/cm
3
where the total simulation length was
increased by 10 times and the number of atoms Nincreases from 64 to
256, 3375, and 8000.
III. RESULTS
A. Force error analysis
One metric for establishing the accuracy of approximations to
the Kohn–Sham potential energy surface is to compute relative force
FIG. 2. NPA RPP for Al at 2.7 g/cm
3
and T¼1 eV. Various power laws are valid at
different values of r. The appropriate power law for a given range of ris shaded
and denoted with a “2,” “4,” or “6.”
FIG. 3. Comparison of TFY and NPA RPPs for C and Al with corresponding g(r)
computed from MD simulation: (a) C at 2.267 g/cm
3
and T¼0.5 eV. The increase
in magnitude of the first g(r) peak results, in this case, from particle attraction. (b) Al
at 2.7 g/cm
3
and T¼1 eV. In this case, it is particle crowding increases the magni-
tude of the first g(r) peak.
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errors between Kohn–Sham force data and a parameterized model
(RPP or many-body potential) for Mparticle coordinate configura-
tions. For this, we compute the mean-absolute force error
MAE ¼1
3MN X
a;i;mjFðPARÞ
a;i;mFðKSMDÞ
a;i;mj;(11)
where FðPARÞ
a;i;mand FðKSMDÞ
a;i;mare the ath force components (x,y,orz)
on the ith atom in particle coordinate configuration number mfor the
parameterized model and the KS-MD force data, respectively.
Note that a direct comparison of the mean absolute error
between different elements, temperatures, and densities cannot be
done as the distributions of forces associated with systems of differ-
ent elements at different thermodynamic conditions are in general
quite different. This can be observed in Fig. 5 where a microfield
distribution of the force magnitudes is shown. In all cases but C at
2.267 g/cm
3
and T¼5 eV, the TFY model peaks at a smal ler field
value than KS-MD. In contrast, for C, V, and Au at T¼0.5 eV, the
NPA RPP peaks at a higher field value than KS-MD. These trends
can be connected back to (7) where the choice of hZi;ueiðkÞ,and
vðkÞall contribute to the construction of a RPP model and hence
the force magnitudes. More work needs to be done to determine
how each term influences the RPP model, the predicted forces, and
observables.
As the microfield force distributions vary for different elements
and temperatures, the mean absolute error will also vary. To this end,
we seek a scale factor for (11) to normalize the results across the differ-
ent elements, temperatures, and densities studied here. Such a scale
factor is the “mean absolute force” defined as
MAF ¼1
3MN X
a;i;mjFðKSMDÞ
a;i;mj:(12)
Using (11) and (12), we define the relative force error as
RFE ¼X
i;a;mjFðPARÞ
a;i;mFðKSMDÞ
a;i;mj
X
a;i;mjFðKSMDÞ
a;i;mj:(13)
This metric has the following desirable property: if the mean absolute
error changes with density or temperature in the same way as the
underlying force distribution, the relative force error will maintain
roughly the same value. Therefore, as we change the thermodynamic
conditions for a given element, (13) provides a temperature indepen-
dent metric as measured with respect to a KS-MD force data
“baseline.” Intuitively, when (13) evaluates to 1, the mean absolute
error is the same order of magnitude as the mean absolute force and
when ð13Þis zero, the parameterized model is exactly reproducing the
per-component KS-MD force data.
Figure 6 displays (13) as a function of temperature for C, Al, V,
and Au where general trends can be observed. One trend is that for
most RPPs, the relative force error decreases toward higher tempera-
tures, which confirms an intuition long held for the validity of the
NPA and TFY models. However, for all systems pictured except C,
force matching drastically reduces the relative force error compared to
the NPA and TFY results. Specifically, the force-matched RPPs rou-
tinely achieve a relative force error of roughly 0.05 above T¼5eV.
Except for the case of the NPA RPP for Al, the NPA and TFY RPPs
maintain an error of around 0.2 across the entire the temperature
range.
The second major observation from Fig. 6 is that while force-
matched RPPs drastically lower the observed relative force errors
across temperatures compared against other RPPs, we immediately see
where a RPP approximation is likely invalid. For example, the relative
force error for C using the force-matched RPP is uncharacteristically
FIG. 4. The RPP models normalized by temperature vs distance for C, Al, V, and Au. Top row, T¼0.5 eV: the representation of the RPP is element dependent with strong
agreement for aluminum. Bottom row, T¼5 eV: The agreement between models improves significantly. The differences in the representation can be connected back to (7)
where the treatment of the mean ionization, electron–ion pseudopotential, and susceptibility define the RPP.
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TABLE I. The self-diffusion coefficient for all systems. For each RPP model, the number of particles, time step, and simulation length were kept identical for each element, den-
sity, and temperature. Finite-size corrections are carried out in Sec. III E.
Element q
i
(g/cm
3
)T(eV) DKSMD ð103˚
A2=fsÞDFM ð103˚
A2=fsÞDTFY ð103˚
A2=fsÞDNPA ð103˚
A2=fsÞ
Li 0.513 0.054 1.4 60.13 1.27 60.054 5.6 60.39 1.26 60.077
C 2.267 0.47 2.4 60.12 9.3 60.20 11.0 60.60 1.69 60.060
1.0 18.6 60.7 27.2 60.77 25 61.64 11.0 60.45
2.0 46 61.68 49 61.0 42 63.70 43 63.86
4.9 85 659465.82 106 65.37 92 63.45
10.0 215 64.49 151 64.63
15 266 63.46 198 65.10
20 349 67.60 249 62.35
28 423 613.28 324 612.17
Al 2.7 0.1 1.6 60.14 0.35 60.021 7
0.50 3.8 60.16 4.1 60.13 9.17 60.099 3.9 60.11
1.1 9.8 60.30 9.4 60.11 17.5 60.70 8.5 60.44
2.0 18.7 60.50 18.8 60.68 34.8 60.52 18.8 60.40
4.9 48 63.56 49 63.17 72 63.03 54 62.7
9.2 83 61.63 84 65.67 122 65.83 94 68.80
10.0 131 66.77 105 613.53
15.4 134 63.68 129 63.37 169 65.26 142 66.17
20.0 197 65.21 151 62.55
30.0 252 64.74 203 66.44
Ar 1.395 0.48 10.7 60.43 12 61.03 19 61.10
1.0 20.1 60.89 26 63.0 39 62.22
2.0 48 61.75 45 62.84 85 68.75
5 143 66.55 171 64.09
10.0 210 614.53 179 66.21
15.0 235 613.34 193 611.95
20.0 255 66.73 209 68.91
30.0 268 62.73 228 68.26
V 6.11 0.49 2.25 60.050 2.86 60.079 3.9 60.18 0.91 60.027
1.0 5.5 60.21 6.5 60.16 7.9 60.36 6.6 60.15
2.1 11.6 60.78 12.5 60.68 17.8 60.74 14.8 60.50
4.8 24.2 60.63 24.7 60.88 41 62.76 27.7 60.90
9.5 46 63.41 42 62.65 68 62.10 47.6 60.93
14.6 53 61.81 57 63.25 84 64.83 63 61.19
20.0 103 66.10 82.7 60.78
30.0 134 68.57 96 61.86
3.055 0.5 9.0 60.81 11.3 60.29 8.7 60.23
0.97 14.7 60.47 15.4 60.43 19 61.39
2.0 23 61.13 24 61.84 31 61.26 27 61.84
4.9 47 64.38 44 62.52 66 67.30 48 62.02
Fe 7.874 0.51 2.13 60.047 2.34 60.042 2.84 60.030
1.1 5.27 60.098 5.5 60.16 5.9 60.39
2.1 10.4 60.72 10.4 60.73 14.8 60.46 9.2 60.47
5.0 20.4 60.61 22.0 60.97 32 61.41 27.1 60.19
10.4 35 61.14 38 61.40 54 61.25 49 62.90
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high (roughly 0.6) until T¼5 eV. A similar situation appears for the
case of V at T¼0.5 eV where the relative force error for the force-
matched RPP is roughly 0.25. We can demonstrate explicitly that these
discrepancies come from the neglect of three-body and higher interac-
tions by showing relative force errors using a SNAP model. For C, the
relative force error drops from roughly 0.6 using a force-matched RPP
to 0.2 using a SNAP model at T¼0.5eV. Likewise for V, the relative
force error drops from roughly 0.25 using a force-matched RPP to
0.07 using a SNAP model at the same temperature.
Ultimately, it is not the component-wise force or the interaction
potential we care about generating, but rather observables such as g(r)
and the self-diffusion coefficient. To address this connection, we
examine correlations between the force error and the self-diffusion
coefficient error, as shown in Fig. 7. While there is a general trend with
increasing errors in both quantities (shown with a linear fit), there are
also some clear outliers. For the case of C at 2.267g/cm
3
and
T¼0.5 eV, we find that the NPA and TFY RPPs produce a self-
diffusion coefficient that differs from the KS-MD result by many
TABLE I. (Continued.)
Element q
i
(g/cm
3
)T(eV) DKSMD ð103˚
A2=fsÞDFM ð103˚
A2=fsÞDTFY ð103˚
A2=fsÞDNPA ð103˚
A2=fsÞ
15.0 83 65.18 60 62.60
20.0 97 62.93 70 61.04
30.0 103 64.69 83.0 60.94
3.937 0.51 6.0 60.39 8.5 60.94 6.2 60.21
1.1 15.8 60.70 15.6 60.67 14.4 60.33
2.1 20 61.18 22 62.07 27 61.28
Au 19.30 0.52 0.92 60.028 0.71 60.084 1.67 60.12 0.51 60.042
1.1 2.0 60.11 1.92 60.088 3.9 60.42 1.66 60.069
1.9 4.0 60.14 3.4 60.15 6.6 60.16 3.52 60.05
5.0 7.8 60.40 8.2 60.21 15.3 60.63 10.7 60.50
9.7 14.4 60.64 15 61.19 25 61.94 16.6 60.86
15.0 19.82 60.80 22 62.56 30 61.95 25 62.79
20.0 39 63.49 28.0 60.97
30.0 56 62.23 33 61.26
FIG. 5. Microfield distributions for C, Al, V, and Au. The observed trends of the microfields agree with the trends of the self-diffusion coefficients in Table I. In general, when the
microfields are similar to that of KS-MD, the agreement between the self-diffusion coefficient increases. To assess the importance of three-body or higher interactions, SNAP
results are reported for C and V at both T¼0.5 and 5 eV.
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factors. However, C under these conditions exists in several charge
states with transient bonding; only the NPA accounts for this. (More
details are found in the Appendix.) This case is marked with arrows in
Fig. 7. Conversely, for V at T¼1 eV, the relative self-diffusion percent
error is low, yet the relative force error is high. The imperfect mapping
of relative self-diffusion error vs relative force error suggests that phys-
ics beyond a RPP is needed, possibly at least a three-body angular
dependence, but further work is needed.
B. Radial distribution function and the Einstein
frequency
The radial distribution function
30
is a measure of spatial correla-
tions normalized by the ideal gas. It has been shown that, in general,
there always exists a RPP that can reproduce g(r)fromaN-body simu-
lation,
54
and the force-matching procedure provides an avenue for
obtaining this RPP. Figure 8 compares g(r) computed from MD
simulations for all RPP models for C, Al, V, and Au. Each row corre-
sponds to a different temperature, and clear trends can be observed,
such as the improvement in agreement between models as the temper-
ature increases. We note that the force-matched RPP always obtains
the correct g(r), and the NPA model generally predicts the location of
the first peak but sometimes over-predicts the magnitude or misses
the location of the first peak altogether as observed in the case of V at
6.11 g/cm
3
for T¼0.5 eV. The TFY model always underestimates the
magnitude of the first peak height, and the location is usually shifted.
Insight into the connection between the g(r) peak height and the
self-diffusion coefficient can be obtained from the normalized velocity
autocorrelation function
55
ZðtÞ¼hvðtÞvð0Þi
hvð0Þvð0Þi;(14)
where vðtÞis the velocity of a particle at time tand hi is an ensem-
ble average over particles and time. A short time expansion of (14)
yields
ZðtÞ¼1X2
0
t2
2!þ;(15)
where X0is the Einstein frequency
X2
0¼4pqi
3mið1
0
dr r2gðrÞr2uðrÞ;(16)
where m
i
is the ion mass in grams. The Einstein frequency gives
insight into the relationship between u(r)andg(r), highlighting how
different regions are weighted more or less depending on the curvature
of u(r). In Fig. 9,theintegrandof(16) is shown. For the TFY model,
the integrand is always smaller than those predicted by force-matched
and NPA RPPs. The area under each curve in Fig. 9 can be directly
connected to the self-diffusion coefficient through the Green–Kubo
relation (in three dimensions)
D¼T
mð1
0
dt ZðtÞ;(17)
substituting (15) into (17). Doing so shows that the TFY model will
always predict a larger self-diffusion coefficient than the force-
matched or NPA model as the area under these curves is larger. This is
FIG. 6. Relative force error vs temperature computed from (13) for C, Al, V, and Au. The red shaded region indicates force accuracy of 0:1 and the blue shaded region indi-
cates force accuracy of 0:05. SNAP and force-matched RPP yields the lowest relative force error and decreases or remains constant as temperature increases. This indi-
cates an increases in accuracy of the RPP models as temperature increases.
FIG. 7. Relative (to KS-MD) self-diffusion error vs the relative force error for C, Al,
V, and Au. The size of each point corresponds to the atomic number. The gray
dashed line is a linear fit to the points showing a positive correlation between self-
diffusion error and force error.
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confirmed later when the self-diffusion coefficients are explicitly calcu-
lated as discussed in Sec. III C.
C. Self-diffusion
Another approach to compute the self-diffusion coefficient is via
the slope of the mean-squared displacement from the Einstein relation
D¼lim
t!1 hjrðtÞrð0Þj2i
6t:(18)
Both (17) and (18) can be used to compute the self-diffusion coeffi-
cient and have been shown to be equivalent.
56
In this work, the self-
diffusion coefficient has been calculated from a linear fit to the mean-
squared displacement, hjrðtÞrð0Þj2i.
Due to finite-size effects, two problems arise when computing the
slope and uncertainty of the linear fit. First, we must ensure that the
linear fit is carried out in the late-time linear regime of the mean-
squared displacement. Second, we dismiss statistically unconverged
late-time behavior of the mean-squared displacement where the
ensemble average contains sparse amounts of data. To remedy both of
these concerns, we uniformly randomly sub-sample the mean-squared
displacement 100 times with 10 points along each sub-sample. Next, a
linear fit is determined for each sub-sample, and the standard devia-
tion of the sub-sample slopes is computed. Once the standard devia-
tion is known, a cutoff time is calculated by determining the point in
time that the standard deviation of the sub-sample fits is less than half
of the standard deviation computed from sub-sample fits to the entire
mean-squared displacement. The simulation data for the mean-
squared displacement after the cutoff time is discarded, and the fitting
procedure described above is repeated. The average, and standard
deviation of the fits to the reduced dataset yield self-diffusion coeffi-
cient and the uncertainty, respectively, and are reported in Table I and
displayed in Fig. 10.
Given the values for the self-diffusion coefficient reported in
Table I, we can answer the following question: at what temperature are
computationally inexpensive models adequate? To do this, we com-
pute the relative self-diffusion coefficients DNPA=DKSMD and
DTFY =DNPA . For example, the top panel in Fig. 11 suggests that NPA
models may be accurate from T¼1 eV and above if the target error
tolerance is 50% of the self-diffusion coefficient computed from KS-
MD. Similarly in the bottom figure, the TFY model is generally accu-
rate to within 50% of the NPA model from T¼5 eV and beyond.
Two important observations can be made from the trends in Fig. 11.
The top panel illustrates temperatures at which an N-body potential is
needed and when NPA is adequate. The bottom panel shows a compari-
son with TFY, which has the simplest ueiðkÞand vðkÞ,andweseetem-
peratures at which TFY becomes comparable to NPA, suggesting when
we can exploit simpler approximations for those inputs.
D. Power spectrum
The self-diffusion coefficient is useful for comparing and quanti-
fying the accuracy of RPP models and transport theories, but in order
FIG. 8. The radial distribution functions for C, Al, V, and Au are shown. The top row corresponds to T¼0.5eV, the middle row T¼2 eV, and the bottom row T¼5 eV. The
force-matched RPP always reproduces the g(r) obtained from KS-MD.
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to assess how accurately the particle dynamics are reproduced, we look
at the power spectrum of the velocity autocorrelation function Z(t)
~
ZðÞ¼ð1
0
dt cos 2pt
ðÞ
ZðtÞ:(19)
In Fig. 12,wecompare~
ZðÞcalculated using TFY, force-matched,
and NPA RPPs against results obtained from KS-MD. We find that
with the exception of low temperature C and V, force-matched RPPs
agree with the KS-MD results across the entire frequency range. This,
combined with the low relative force errors and accurate reproduction
of static properties discussed previously, indicates that the force-
matched RPPs accurately approximate the Kohn–Sham potential
energy surface. For higher temperatures, the NPA RPP is very similar
to the force-matched RPP for low and high frequencies for all
FIG. 9. The integrand of the Einstein frequency (16). All integrands are consistent with values reported in Table I as the self-diffusion coefficient decrease as the integral of the
Einstein frequency increases. This allows for a “by eye” comparison of the self-diffusion coefficient from different RPP models.
FIG. 10. Self-diffusion coefficients for different elements and densities vs temperature. The numerical values are reported in Table I. For all cases all models predict values
that have roughly the same order of magnitude. The only case where the force-matched RPP fails to reproduce the KS-MD self-diffusion coefficient is for C at 2.267 g/cm
3
and
T¼0.5 eV. The TFY RPP model generally predicts larger self-diffusion coefficients, which is consistent with the Einstein frequency in Fig. 9. Note that the NPA RPP model, in
contrast, agrees with results obtained from the force-matched RPP and KS-MD models very well.
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elements. For T¼0.5 eV, the dynamics predicted from the NPA model
are noticeably less similar to those from KS-MD where NPA underesti-
mates the prevalence of low-frequency modes in Au and both low and
high-frequency modes in V. Interestingly, the NPA RPP captures the
single-particle dynamics of low temperature C very well, but Figs. 4 and
8indicate that this agreement comes at the expense of sacrificing the
accuracy of static properties. Finally, the TFY RPP exhibits roughly the
same trends across all elements and temperatures—overestimation of
the low frequency modes and underestimation of the high-frequency
modes except for the case of C at 2.267 g/cm
3
and T¼5eVwhereexcel-
lent agreement with KS-MD is observed.
E. Finite-size corrections
Generally, thousands or even millions of atoms are needed to
approximate the thermodynamic limit.
1,16
While the KS-MD frame-
work provides an accurate description of the electronic structure and
the N-body potential is determined on-the-fly, corrections for finite-
size effects must be considered. When the shear viscosity gof the sys-
tem is known, finite-size corrections can be determined from Ref. 57
D1¼DNþnT
6pgL;(20)
where D1is the self-diffusion coefficient in the thermodynamic limit,
D
N
is the self-diffusion coefficient computed from a system of finite
number of particles N,andn¼2:837 297 for cubic simulation boxes of
lengthLwithperiodicboundaryconditions.Whengis unknown, mul-
tiple simulations of increasing particle number are carried out, and a lin-
earfitisusedtodetermineD1. Results from this procedure are shown
in Fig. 13 where D1is determined via linear extrapolation to 1=L¼0.
By finding the percent difference in D1and D
N
,weapproximate
the errors from finite-size effects in the KS-MD self-diffusion coeffi-
cient at these conditions. The approximate error in KS-MD for the case
FIG. 12. The normalized power spectrum for C, Al, V, and Au. For C at T¼0.5 eV, the single particle dynamics are poorly described by the TFY and force-matched models
but more accurately described with the NPA model. As the temperature increases from T¼0.5 to 5 eV, all models more accurately reproduce small and high frequency dynam-
ics with the most notable improvement for C.
FIG. 11. Relative self-diffusion coefficients. The shaded region brackets the range
of 0.5 and 0.5. (a) For all cases except V at T¼0.5 eV, the points fall within the
bounds of the bracketed region. The NPA RPP fails to reproduce the KS-MD results
at T¼0.5 eV, thereby revealing a temperature boundary below which KS-DFT is
needed. In (b), the points are within 50% of the NPA value from T¼5 eV and
above for most cases. The orange point marked with an arrow has been reduced
by a factor of 1/2 to improve clarity of the banded region.
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shown in Fig. 13,is20%. While the error will vary with {Z,n,T}, the
impact of finite-size effects is significant. From this study, the most
promising approach is to fully converge the NPA MD results, using
force-matched RPPs when necessary (for low temperatures Tⱗ1eV).
Finite-size corrections allow for a direct comparison to analytic
transport theories, namely, the Stanton–Murillo model.
58
The
Stanton–Murillo model provides a closed form solution for ionic self-
diffusion by using an effective interaction potential in a Boltzmann kinetic
theory framework. The major benefit of this model is that the computa-
tion of ionic transport is nearly instantaneous. However, its applicability
in the cold dense matter and warm dense matter regimes is unknown.
The results in Table II show that the effective interaction
approach of the Stanton–Murillo model captures much of the many-
body physics included in the TFY RPP results. The main weakness of
the model, and also TFY, is therefore the functional form of the inter-
action they employ as the differences with the force-matched and
NPA columns reveal. Because self-diffusion is a relatively simple trans-
port coefficient,
58
more work is needed to quantify these trends for
other transport properties.
With the converged self-diffusion data, we generate an effective
interaction correction to the Stanton–Murillo model. The effective
interaction corrected Stanton–Murillo model is
DCSM ¼aðZ;TÞDSM;(21)
where aðZ;TÞis determined by fitting the ratio of the self-diffusion
coefficient from the best performing RPP model and the self-diffusion
coefficient computed from the Stanton–Murillo model D
SM
to the
functional form
aðZ;TÞ¼aerfðbTÞ
bT þ1;(22)
which asymptotes to D
SM
as Tincreases. Here the “best performing
RPP model” refers to the RPP model that most accurately reproduced
the self-diffusion coefficient computed from 64 particle KS-MD
TABLE II. Self-diffusion coefficient in the thermodynamic limit. Both elements are at solid density (2.267 g/cm
3
for C, and 6.11g/cm
3
for V).
Element T(eV) DFM ð103˚
A2=fsÞDNPA ð103˚
A2=fsÞDTFY ð103˚
A2=fsÞDSM ð103˚
A2=fsÞDCSM ð103˚
A2=fsÞ
C 0.47 10.55 2.14 12.66 13.08 2.14
1.0 32.44 14.21 25.57 26.11 13.87
2.0 56.70 43.12 51.14 50.53 39.23
4.9 99.51 109.55 117.88 118.34 106.76
10.0 169.91 210.76 217.34 206.71
15.0 219.99 296.21 293.54 284.10
20.0 256.33 342.15 356.41 348.04
28.0 327.32 470.10 439.44 432.07
V 0.49 4.14 1.01 5.42 6.76 4.39
1.0 8.54 8.53 11.53 12.26 7.96
2.1 15.67 18.87 22.56 23.14 15.03
4.8 28.72 31.34 42.42 46.25 30.18
9.5 49.49 54.90 73.76 77.10 51.16
14.6 66.98 74.44 99.82 99.78 67.97
20.0 87.90 118.24 117.8 83.15
30.0 105.60 143.63 141.66 106.94
50.0 131.84 175.35 171.07 143.02
75.0 178.34 202.93 194.31 172.91
100.0 209.50 207.20 211.86 194.36
FIG. 13. Finite-size effect study for V at 6.11 g/cm
3
and T¼2 eV. Identical MD sim-
ulations were carried out with increasing particle number. Extrapolating with a linear
fit (gray dashed line) to 1=L¼0 approximates the thermodynamic limit, correcting
the values in Table I.
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simulations. The parameters aand bare reported in Table III for C at
2.267 g/cm
3
and V at 6.11 g/cm
3
, and their values vary considerably
between both cases emphasizing the need for a comprehensive finite
size effect study to produce correction factors for additional elements
and conditions. This correction factor allows for the use of the
Stanton–Murillo model in regions of previously unknown accuracy.
The finite-size corrections along with the corrected Stanton–Murillo
model results are shown Fig. 14 with the numerical values given in
Table II. Note that for low temperature C at 2.267 g/cm
3
,thebestper-
forming RPP model was NPA (as reported in Fig. 10 and Table I)
explaining why the corrected Stanton–Murillo model tends toward the
NPA RPP at low temperatures. For V at 6.11 g/cm
3
, the best perform-
ing RPP model was the force-matched RPP again explaining the low
temperature trend.
In an attempt to summarize our work in a single figure, Fig. 15
shows our suggested use cases for all RPPs studied here for two relative
self-diffusion accuracies computed from Table I. When points (the
average value or its uncertainty) for a given model are within the
appropriate tolerance (30% for the top panel and 15% for the bottom
panel), we consider the model as being accurate for that temperature
and element and is denoted with a colored bar or arrow. We rank the
computational expense from lowest to highest as TFY, NPA, force
matching, and KS-MD. When a computationally cheaper model is
accurate, it replaces the more computationally expensive model in
Fig. 15. Based on trends observed in Figs. 6,11,and14, we assume
that the models remain accurate for higher temperatures and illustrate
this by upward pointing colored arrows. For example, consider the
case of Fe in the top panel of Fig. 15. The force-matched RPP is
TABLE III. Coefficients a, and bfor the effective interaction correction (22). Note that
the values of aand bvary considerably for each element.
Element ab
C (2.267 g/cm
3
) 2.198 1.032
V (6.11 g/cm
3
) 0.037 67 0.311 2
FIG. 14. Self-diffusion coefficient vs temperature in the thermodynamic limit. The
points displayed here are taken from Table II. (a) Self-diffusion coefficient for C at
2.267 g/cm
3
. (b) Self-diffusion coefficient for V at 6.11 g/cm
3
. The Stanton–Murillo
model (denoted SM) fails for low temperature C. For V, the Stanton–Murillo model
shows excellent agreement with the force-matched RPP even at low temperatures.
The validity of the Stanton–Murillo model is extended to low temperatures with an
effective interaction correction (denoted CSM).
FIG. 15. Suggested use cases for each RPP model based on the relative self-
diffusion coefficient error (between RPP-MD and KS-MD) and cheapest computa-
tion cost. The top and bottom panels correspond to a 30% and 15% relative error,
respectively. The elements denoted with a subscript of “1/2” corresponds to half
solid density (V at 3.055 g/cm
3
and Fe at 3.937 g/cm
3
). The colored bars indicate
the computationally cheapest RPP that generates a self-diffusion coefficient to
within the specified error tolerance available for that system based on Table I. The
empty space under each bar indicates regions where no KS-MD data were col-
lected, so no assessment on a RPPs accuracy can be made.
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accurate to within 30% of the KS-MD result from T ¼0.5 eV and up.
The NPA model, which is computationally cheaper than the force-
matched RPP, becomes accurate (within 30% of KS-MD) at T¼2eV
and up, hence the transition between the force-matched and NPA
models. For Al, the NPA RPP is within 15% of KS-MD at all tempera-
tures. However, at T¼15 eV the TFY model becomes accurate there-
fore replacing the NPA RPP.
IV. CONCLUSIONS AND OUTLOOK
A systematic study of various RPPs for molecular dynamics sim-
ulations of dense plasmas was performed for a wide range of elements
vs temperature for solid and half-solid density cases. Of the RPPs
studied in this work, RPPs constructed from a NPA approach come
closest to accurately reproducing the transport and structural proper-
ties predicted by KS-MD. The failures of NPA for metals near
T¼0.5 eV are expected: V is a polyvalent metal and s-d hybridization
occurs in Au, which is not treated at all in our variant of the NPA
model. Thus, it is unclear if inaccuracies in NPA reveal the need for
N-body interactions or an improved NPA treatment. Moreover, finite-
size corrections to KS-MD are seen to be significant; prior work on Si
suggests that at least 108 particles are needed to accurately treat ele-
ments like C at low temperatures.
59
Studies on C and Si where there
are transient covalent bonding at low temperatures have raised
the inadequacy of the PBE XC-functional that has been used here. In
Ref. 59, the SCAN functional was used showing remarkable agreement
between VASP calculations and NPA results for supercooled high-
density Si. This implies that VASP calculations for systems in the low
temperature warm dense matter regime become sensitive to the choice
of the XC-functional. Similarly, the XC-functionals for transition met-
als like V, Fe, etc., are known to need Hubbard-type corrections that
are not included in our studies. Although this work does not fully
resolve these issues, the trends seen for the lowest temperature for C,
V, and Au should be examined in detail in future work. Additionally,
the NPA model is exceptionally accurate for Al. As Al is a free electron
metal, its electronic structure is well described as a Fermi-liquid, the
precise physical model in which NPA performs well. In the cases
where the electronic structure of the system is not well described as a
Fermi-liquid, the performance of the NPA model decreases at low
temperature, further emphasizing the need for a comprehensive study
over a range of elements and conditions.
As in previous works,
9,60
the TFY model predicts the least struc-
tured g(r). Notionally, the accuracy of the TFY model appears to fol-
low the machine learning trend of hZi=Z>0:35
61
although it was not
possible to use all models in this work at high enough temperatures to
be quantitative. In contrast, the NPA model with its improved
Kohn–Sham treatment and use of a pseudopotential in (7) eliminates
most of these errors except for C and V at T¼0.5 eV, elements for
which we would recommend NPA for T>2 eV. Because we examined
seven diverse elements over the warm dense matter regime, the accu-
racy of NPA (and for moderate temperature, even TFY) suggests that
no additional “short-range repulsion”
9,47–52
is needed beyond (7);as
(7) does not contain core–core repulsion, the structure of the interac-
tion is more likely to be effective core–valence repulsion captured by
ueiðkÞ, as well as structure in vðkÞbeyond vTF ðkÞ.However,wenote
(see the Appendix) that in treating weakly ionized systems like warm-
dense Ar with a mean ionization of hZi¼0:3, some 70% of the Ar
atoms are neutral, while about 30% of the atoms are singly ionized.
Thus, the neutrals interact via a core–core interaction screened by the
free electrons. In such cases the use of (7) alone is inadequate. The
NPA model treats such a two-component mixture using three pair
potentials. In general, core–core interactions are important for weakly
ionized atoms with a large core. These core–core interactions can be
readily calculated using the core–electron density obtained from the
NPA Kohn–Sham calculation.
As expected, the force-matched RPP reproduced the g(r)
computed from KS-MD for all cases. In only one case, again C at
2.267 g/cm
3
and T¼0.5 eV, the force-matched RPP overestimated the
self-diffusion coefficient; this suggests that the spherical pair interac-
tion is not applicable, and non-spherical corrections, which could
include three-body contributions, are needed as suggested by the near-
perfect agreement of the SNAP and KS-MD microfield of force magni-
tudes in Fig. 5. However, for all cases considered with T>1 eV, the
g(r) and self-diffusion coefficient are adequately described by a RPP.
With the force-matched-validated NPA interaction, pre-computing
the interaction allows for much larger pair-potential simulations.
As fast analytic expressions for transport coefficients are needed
for hydrodynamic modeling, we compared our self-diffusion results
from all models to the Stanton–Murillo model for both C and V. In
both cases, the Stanton–Murillo model was consistent with the TFY
model (on which it is based) and both have agreement with force-
matched-based results. The error between the Stanton–Murillo model
and the force-matched results is <65% below T¼10 eV for V and
<25% below T¼5 eV for C, adding confidence to the use of this
model in hydrodynamics models above that temperature. For experi-
ments that are rapidly heated above a few eV, little time is spent where
the errors are large; because the transport coefficients are numerically
very small during this transient heating, negligible transport can occur
during that time. For example, note that the V diffusion coefficient
varies by a factor of about 30 in the range T¼0.5–100 eV. Conversely,
for experiments that dwell at lower temperatures, we provide a RPP-
based correction factor to the Stanton–Murillo model with an error of
less than 1% for C at T¼0.5 eV and 6% for V at T¼0.5 eV.
Our results suggest several new avenues of investigation. From a
data science perspective, larger collections of systematically obtained
simulation results would aid in better defining accuracy boundaries. In
particular, more elements that produce more material types should be
studied. For mixtures, N-body potentials could be explored; here, we
cast all of the pair potentials as heteronuclear. Additionally, our con-
clusions are based on studies of the microfield distribution of forces,
Einstein frequency, power spectrum, self-diffusion coefficient, and
g(r), which could be extended to include other properties such as vis-
cosities and interdiffusion in mixtures, electrical conductivity, thermal
conductivity, and ion-dynamical properties like the speed of sound.
15
While in this work, we focused primarily on force matching, effective
interaction potentials can be obtained through “structure
matching.”
30,62,63
Finally, as very large scale simulations become more
common, spatially heterogeneous plasmas can be modeled; much less
is known about potentials in such environments, although recent work
has explored non-spherical potentials.
1
ACKNOWLEDGMENTS
Sandia National Laboratories is a multimission laboratory
managed and operated by National Technology and Engineering
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CAuthor(s) 2021
Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. Department of Energy’s National
Nuclear Security Administration under Contract No. DE-
NA0003525. This paper describes objective technical results and
analysis. Any subjective views or opinions that might be expressed
in the paper do not necessarily represent the views of the U.S.
Department of Energy or the United States Government. The
authors would like to thank Jeffrey Haack (LANL), Liam Stanton
(SJSU), Patrick Knapp (SNL), and Stephanie Hansen (SNL) for
insightful discussions, and Josh Townsend (SNL) for his library of
VASP post-processing tools. LAMMPS can be accessed at http://
lammps.sandia.gov.
APPENDIX: NPA DETAILS AND EXTENSIONS
In this appendix, we provide a brief background on NPA cal-
culations, and specific extensions needed for the argon, iron and
vanadium cases at low temperatures. While we believe the results
are formally correct for these cases, we address the nature of the
questions being asked and the extensions that are needed.
1. NPA formulation
The NPA model applies primarily to warm-dense fluids with
spherical symmetry although the NPA method can be applied
equally well to crystals.
64,65
Importantly, there is no unique NPA
model;
40,42,66,67
here, we describe a specific set of choices
14,65,68,69
based around a formal statement of the theory.
69
A key difference
between many average-atom models and the NPA model used here
is that the free electrons are not confined to the Wigner–Seitz
sphere, but move in all of space as approximated by a very large
correlation sphere, of radius R
c
which is ten to twenty times the
Wigner–Seitz radius.
40
Our NPA model begins with the variational property of the
grand potential X½ne;nias a functional of the one-body densities
ne¼neðrÞfor electrons, and ni¼niðrÞfor ions. Only a single
nucleus of the material is used and taken as the center of the coordi-
nate system. The other ions (“field ions”) are replaced by their one-
body density distribution qðrÞ: DFT asserts that the physics is solely
given by the one-body distribution; i.e., we do not need two-body,
three-body, and such information as they get included via
exchange-correlation (XC)-functionals. That is, there is no cluster
expansion of the total potential (1) as in (2) of the main text. The
terms beyond the pair-interaction are not neglected, but included in
the ion–ion XC potential which is not used in VASP-type calcula-
tions. Note that this formulation differs from N-center codes
24,70
like the VASP or ABINIT. Moreover, there can be other differences;
for example, we have used the finite-Telectron XC-functional by
Perrot and Dharma-wardana,
71
while the PBE implemented on
VASP is a T¼0 XC-functional. The finite-Tfunctional used is in
good agreement with quantum Monte Carlo XC-data
68
in the den-
sity and temperature regimes of interest.
The artifice of using a nucleus at the origin converts the one
body ion density qðrÞand the electron density n(r) into effective
two body densities in the sense that
niðrÞ¼
nigiiðrÞ;neðrÞ¼
negeiðrÞ:(A1)
The origin need not be at rest; however, most ions are heavy enough
that the Born–Oppenheimer approximation is applicable. Here
ni;
neare the mean ion density and the mean free electron density,
respectively. Bound electrons are assumed to be firmly associated
with each ionic nucleus and contained in their “ion cores” of radius
r
c
such that
rc<ai:(A2)
In some cases, e.g., some transition metals, and for continuum reso-
nances etc., this condition for a compact core may not be met,
and additional steps are needed. We assume a compact core as a
working hypothesis. The DFT variational equations used here are
dXne;ni
½
dne¼0;dXne;ni
½
dni¼0:(A3)
These directly lead to two coupled Kohn–Sham equations where
the unknown quantities are the XC-functional for the electrons, and
the ion-correlation functional for the ions.
72
If the Born–
Oppenheimer approximation is imposed, the ion–electron
XC-functional may also be neglected. Approximations arise in
modeling these functionals and in decoupling the two Kohn–Sham
equations
14,73
to some extent, for easier numerical work. The first
equation gives the usual Kohn–Sham equation for electrons moving
in the external potential of the ions. This is the only DFT equation
used in N-center codes in which ions define a periodic structure
evolved by MD, followed by a Kohn–Sham solution at each step. In
contrast, NPA employs the one-body ion density niðrÞ; it was
shown in Ref. 69 that the ionic DFT equation can be identified as a
Boltzmann-like distribution of field ions around the central ion, dis-
tributed according to the “potential of mean force” well known in
the theory of fluids. In such a formulation, the ion–ion correlation
functional Fii
xc was identified to be the sum of hypernetted-chain
diagrams plus the bridge diagrams as an exact result formally
although the bridge diagrams cannot be evaluated exactly.
The mean electron density
necan also be specified as the num-
ber of free electrons per ion, viz., the mean ionization state hZi.
Although the material density
niis specified, the mean free electron
density
neis unknown at any given temperature, as it depends on
the ionization balance which is controlled by the free energy mini-
mization given in (A3). Hence, a trial value for
ne(i.e., equivalently,
a trial value for hZi) is assumed and the thermodynamically consis-
tent niðrÞis determined. This is repeated until the target mean ion
density
niis obtained.
This means that the Kohn–Sham equation has to be solved for
a single electron moving in the field of the central ion; its ion distri-
bution
qgðrÞis modified at each iteration with modification of the
trial
Zuntil the target material density is found. However, it was
noticed very early
14,65
that the Kohn–Sham solution was quite
insensitive to the details of the g(r) and hence a simplification was
possible. The simplification was to replace the trial g(r) at the trial
Zby a cavity-like distribution
gcavðrÞ¼0;rai;gcav ðrÞ¼1;r>ai:(A4)
Here the a
i
is the trial value of a
i
, based on the trial
ne. Hence,
adjusting the gcavðrÞat each iteration requires only adjusting the
trail a
i
to achieve self-consistency. The self-consistency in the ion
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distribution is rigidly controlled by the Friedel sum rule for the
phase shifts of the Kohn–Sham-electrons.
69
This ensures that
q¼
n=Z. Thus, a valuable result of the calculation using the cavity
model of g(r) is the self-consistent value of the mean ionization
state Z, which is both an atomic quantity and a thermodynamic
quantity.
Here we note crucial simplifications used in implementing the
NPA. Given that the electron distribution n(r) obtained self-
consistently can be written as a bound-electron term and a free-
electron term because of the condition specified by (A2), we have
neðrÞ¼cðrÞþnfðrÞ;DneðrÞ¼nfðrÞþ
ne:(A5)
The core–electron density (made up of “bound electrons”) is
denoted by c(r). The free electron density nfðrÞis the response of
an electron fluid containing a cavity that mimics niðrÞ. It contrib-
utes to the potential on the electrons. The response of a uniform
electron gas to the central ion can be obtained by subtracting the
effect of the cavity using the known static interacting linear
response function vðk;
ne;TÞof the electron fluid. That is, from
now on we take it that the charge density nfðrÞand the charge
pileup DnfðrÞare both corrected for the presence of the cavity, but
we use the same symbols.
2. Pair-potentials for NPA mixtures: Ionic contributions
With the basic NPA formulated, we turn to the construction of
pair potentials, with a focus on the more challenging cases we con-
sider in the main text. It is very common to treat WDM and liquid
metals through purely ionic interactions, which are adequately eval-
uated in second-order perturbation theory unless the free electron
density and the temperature (T=EF) are very low. Such interactions,
which generalizes (7) to mixtures, so are written in k-space as
UabðkÞ¼ZaZbVk(A6)
þfUei
aðkÞDnf
bðkÞþUei
bðkÞDnf
aðkÞg=2;(A7)
Vk¼4p=k2:(A8)
In the NPA theory for mixtures, Z
a
and Z
b
are integers, while in the
simple (average-atom) NPA, the Zs¼RsxsZsis used. The electron
density pileup is calculated using the linear-response property of
the pseudopotential
DnfðkÞ¼UeiðkÞvðkÞ:(A9)
Here, since nfðkÞhas been calculated via Kohn–Sham, it has all the
non-linear effects included by the construction of UeiðkÞ. The extent
of the validity of such a quasi-linear pseudopotential is discussed in
Ref. 13. Unlike in the average atom NPA, the U
ei
used in mixture-
theory is the pseudoptential of the ion with the appropriate integer
ionization. The interacting electron gas response function used in
these calculations includes a local-field factor chosen to satisfy the
finite temperature electron-gas compressibility sum rule, and is
given explicitly by
vðk;TeÞ¼ v0ðk;TeÞ
1Vkð1GkÞv0ðk;TeÞ;(A10)
Gk¼ð1j0=jÞðk=kTFÞ;Vk¼4p=k2;(A11)
kTF ¼f4=ðparsÞg1=2;a¼ð4=9pÞ1=3:(A12)
Here v
0
is the finite-TLindhard function, V
k
is the bare Coulomb
potential, and G
k
is a local-field correction (LFC). The finite-Tcom-
pressibility sum rule for electrons is satisfied since j
0
and jare the
non-interacting and interacting electron compressibilities, respec-
tively, with jmatched to the FxcðTÞused in the Kohn–Sham calcu-
lation. In (A12),kTF appearing in the LFC is the Thomas–Fermi
wave vector. We use a G
k
evaluated at k!0 for all kinstead of the
more general k-dependent form [e.g., Eq. (50) in Ref. 71] since the
k-dispersion in G
k
has negligible effect for the WDMs of this study.
3. Pair-potentials for NPA mixtures: Core interactions
We now consider extensions to this formulation that includes
the effect of core electrons (see Appendix B of Ref. 14). Core–core
interactions are important for atoms like argon, sodium, potassium,
gold, etc., with large cores and zero or low hZi. Here we use a sim-
plified approach in discussing the case of low-Targon rather than
the more exact approach given in Ref. 14. The total pair-interaction
w
ab
is of the form
wab ¼Uðca;cbÞþfUðca;fbÞþUðfa;cbÞg þ Uðfa;fbÞ;
¼UccðrÞþUcf ðrÞþUff ðrÞ:(A13)
The first term, Uc¼Uðca;cbÞis the interaction between the two
atomic cores, and it is the only term found in a neutral gas of pure
argon atoms. The second term is the interaction of the core elec-
trons of the neutral atom awith the pseudo-potential of the ion b
with integer charge Z
b
, while the indices are interchanged in the
third term. We need to evaluate the first two terms involving core
electrons, while the last term is given by (7) of the main text. How
to evaluate these in the NPA at any temperature and density is
given in Appendix B of Ref. 14 to the same level of approximation
as (A6), i.e., to second-order in perturbation theory in the screened
interactions. It is found that such evaluations for argon give results
close to parameterized potentials similar to the Lennard–Jones (LJ)
or more sophisticated potentials, but inclusive of a screening correc-
tion. Thus, at the LJ-level of approximation, UcðkÞfor two neutrals
immersed in the electron gas is approximately given by
ULJ ðkÞf1þVkvðkÞg. For two charged ions, a correction factor of
fðZnZintÞ=Zng2, where Z
n
is the nuclear charge, and Z
int
is the
integer-ionization of the ion is included because the ion cores have
less electrons than the neutral cores.
4. Argon in low-T WDM states
The NPA calculation at density q’1.4 g/cm
3
and tempera-
ture T¼2 eV yields a mean charge of hZi’0:3; however, this
implies a mixture where 30% of the argon atoms are in the Ar
þ
state, while 70% of the atoms are neutral atoms. We ignore the
Ar
2þ
ionization state as the second ionization energy is about 27 eV
(ignoring plasma corrections). So, while the NPA tries to assign a
mean ionization state, this argon system is better treated as a two
component mixture with x
a
,x
b
, where a¼Ar;b¼Arþ.
Thus, we need the three pair-potentials U
ab
, i.e., for Ar–Ar,
Ar–Ar
þ
, and Ar
þ
–Ar
þ
interactions. Here the atomic species are
immersed in the electron gas resulting from the ionization. These
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CAuthor(s) 2021
pair-interactions can be rigorously calculated from first-principles
using the atomic and ionic electron densities obtained in the NPA,
as discussed in Appendix B of Ref. 14. Here we follow a more sim-
plified calculation of the pair potentials exploiting known models
for argon, suitable for this mixture with 70% neutral argon.
The Ar
þ
ion interacts with the neutral Ar atom by polarizing
the core–electron distribution. This distortion is usually described
in terms of the dipole polarizability aand quadrupole polarizability
bof the Ar atom, viz.,
Ucf ðrÞ¼Uðca;fbÞþUðcb;faÞ;(A14)
Uðca;fbÞðrÞ¼C4=r4þC6=r6þ;(A15)
C4¼aðZintÞ2=2;C6¼bðZint Þ2=2:(A16)
This interaction will be screened by the polarization processes of
the background electron gas. Instead of using the truncated multi-
pole expansion, an improved calculation may be done as in
Appendix B of Ref. 14. The ion–atom contribution from (A14) con-
tains only the first term, as atom ais neutral and has no free elec-
trons. In the regime T¼2 eV for Ar, we have simply approximated
the core–core interaction of the atom–ion interaction by the mean
value of atom–atom and ion–ion core–core interaction.
In Fig. 16, we display the pair potentials for Ar–Ar, Ar
þ
–Ar,
and Ar
þ
–Ar
þ
at T¼2 eV at a density of 1.395 g/cm
3
. The composi-
tions 0.7 for Ar, and 0.3 for Ar
þ
are obtained from the simple NPA
calculation. We have not attempted to optimize these composition
fractions using steps indicated in Ref. 14.
5. Iron and Vanadium in low-T WDM states
The NPA uses an “isotropic” atomic model even for iron,
vanadium, and other transition metals, and ignores the s-dhybridi-
zation energy E
hyb
that re-arranges the electron distribution among
the dand selectron states near the Fermi energy. The need for s-d
hybridization is best seen by looking at the low temperature band
structure of such a transition metal. The unhybridized bands of an
“isotropic model” for vanadium are such that the free-electron like
band crosses the d-bands, and the s-dinteraction redistributes the
electrons. Furthermore, instead of the s-wave local pseudopotential
(as employed here), an angular-momentum dependent form is
more appropriate at low-T. Hence, the calculation of the mean ioni-
zation has to be accordingly modified. However, the simpler picture
reemerges when the temperature exceeds the s-dhybridization
energy.
Furthermore, charge polarization fluctuations of the d-elec-
trons can couple with those of the electron gas (e.g., as discussed by
Maggs and Ashcroft
74
). Such effects, as well as “on-site Hubbard U”
effects need to be included in the electron XC-functionals to prop-
erly treat transition metals. The usual XC-functionals made avail-
able with standard codes do not include these effects. However, at
sufficiently high temperatures, the ionization becomes high enough
to screen these effects and the theory simplifies. The pair-potentials
for iron provided by the isotropic model with no s-dhybridization
and other d-band effects lead to cluster formation at low-T. Here,
unlike in liquid carbon or silicon, the bonding is not transient.
Hence low-TMD simulations will show no movement of the ions
and no diffusion. In the case of silicon and carbon, both of which
are known to show varying degrees of transient bonding behavior,
NPA was shown to produce an accurate description of the high
density liquid phase, with structure factors and PDFS obtained
from the NPA-HNC procedure agreeing closely
75
with those from
218-atom simulations with DFT.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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FIG. 16. The pair potentials among the Ar and Ar
þ
species of the two component
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