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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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CAuthor(s) 2021

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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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CAuthor(s) 2021

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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CAuthor(s) 2021

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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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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