An improved classical mapping method for homogeneous electron gases at finitetemperatureYu Liu and Jianzhong Wu Citation: The Journal of Chemical Physics 141, 064115 (2014); doi: 10.1063/1.4892587 View online: http://dx.doi.org/10.1063/1.4892587 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A bridge-functional-based classical mapping method for predicting the correlation functions of uniform electrongases at finite temperature J. Chem. Phys. 140, 084103 (2014); 10.1063/1.4865935 Uniform electron gases. I. Electrons on a ring J. Chem. Phys. 138, 164124 (2013); 10.1063/1.4802589 Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gasmodels J. Chem. Phys. 136, 244101 (2012); 10.1063/1.4720076 Transcorrelated calculations of homogeneous electron gases J. Chem. Phys. 136, 224111 (2012); 10.1063/1.4727852 Stopping power for a charged particle moving through three-dimensional nonideal finite-temperature electrongases Phys. Plasmas 18, 072701 (2011); 10.1063/1.3600533

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THE JOURNAL OF CHEMICAL PHYSICS 141, 064115 (2014)

An improved classical mapping method for homogeneous electron gasesat finite temperature

Yu Liu and Jianzhong Wua)

Department of Chemical and Environmental Engineering and Department of Mathematics,University of California, Riverside, California 92521, USA

(Received 16 June 2014; accepted 28 July 2014; published online 14 August 2014)

We introduce a modified classical mapping method to predict the exchange-correlation free energyand the structure of homogeneous electron gases (HEG) at finite temperature. With the classicalmap temperature parameterized on the basis of the quantum Monte Carlo simulation data for thecorrelation energy and exact results at high and low temperature limits, the new theoretical proceduregreatly improves the classical mapping method for correlating the energetic properties HEG overa broad range of thermodynamic conditions. Improvement can also be identified in predicting thelong-range components of the spin-averaged pair correlation functions. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4892587]

I. INTRODUCTION

Properties of electronic systems at finite temperature areof great importance in practical applications and have at-tracted much attention in modern chemistry and physics.1–4

At finite temperature, homogeneous electron gases (HEG)provides a good reference for studying the properties of warmdense matter (WDM)1 and inertial confinement fusion (ICF).3

Because the typical temperature for these systems is of theorder of 105 K, electrons behave totally different from whenthey are at low temperature states. The predominance of ex-cited states and the entropic contributions makes theoreti-cal descriptions of finite-temperature electronic systems morecomplicated in comparison to those for electronic systems at0 K.

One of the most important applications dependent onknowing the properties of HEG is finite-temperature densityfunctional theory (FT-DFT), a generalization of electronicDFT at 0 K to include the entropic effects.5–9 FT-DFT cal-culations hinge on analytical expressions for the free energy,which is a functional of the electronic density profile ρ(r):5, 10

F [ρ(r)] = Ks[ρ(r)] − T Ss[ρ(r)] + Fee[ρ(r)] + Fne[ρ(r)]

+Fxc[ρ(r)], (1)

where Ks[ρ(r)] and Ss[ρ(r)] are, respectively, the kinetic en-ergy and the entropy functionals of a non-interacting refer-ence system; T is the absolute temperature; Fee[ρ(r)] andFne[ρ(r)] are the classical electrostatic energies due to theelectron-electron and the nuclear-electron Coulomb interac-tions; and Fxc[ρ(r)] represents the exchange-correlation freeenergy. The energetic properties are attainable for a system ofnon-interacting (or ideal) electrons on the basis of the Fermi-Dirac statistics.11 The classical electrostatic energies can becalculated directly from the ionic density profiles based onthe Coulomb law. In general, the exchange-correlation freeenergy cannot be formulated exactly. As for the exchange-

a)Electronic mail: jwu@engr.ucr.edu.

correlation energy in applications of the conventional DFTto electronic systems at 0 K, the local density approximation(LDA) provides a convenient starting point to estimate Fxc[ρ]:

F LDAxc [ρ] =

∫ρ(r)fxc[ρ(r)]dr, (2)

where fxc(ρ) stands for the exchange-correlation free energyper particle in a homogeneous electron gas. Like the con-ventional DFT, improvements over LDA may be achievedwith the generalized gradient approximation (GGA) or meta-GGA.12–16

For a homogeneous electron gas at finite temperature T,the exchange correlation energy and free energy are relatedthrough the Gibbs-Helmholtz equation:

εxc = fxc − T∂fxc

∂T, (3)

where εxc(ρ) stands for the exchange-correlation energy perparticle. At T = 0 K, fxc(ρ) and εxc(ρ) are identical andcan be directly obtained from quantum Monte Carlo (QMC)simulation.17, 18 The situation is more complicated at finitetemperature because QMC provides information only forεxc(ρ).4 Because of the integration constant, Eq. (3) alone isinsufficient to determine fxc(ρ) from εxc(ρ). For example, it iseasy to show that fxc(T) = f0(T) and fxc(T) = f0(T) + aT wouldlead to the same εxc(T) for arbitrary a independent of temper-ature. By incorporating asymptotic results at large electrondensity and at the zero and high-T limits, Karasiev et al.10 in-troduced a semi-empirical expression for fxc that contains a setof parameters fitted with εxc from QMC. These correlationsare convenient to calculate the thermodynamic properties ofHEG over broad temperature and density ranges.

Among a number of alternative methods to calcu-late the exchange-correlation free energy the classical map-ping method introduced by Dharma-Wardana et al. hasthe advantages of theoretical simplicity and computationaleffectiveness.19, 20 The key idea in classical mapping is to re-place the multi-body effects due to the Pauli exclusion prin-ciple by using a reference classical system that reproduces

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064115-2 Y. Liu and J. Wu J. Chem. Phys. 141, 064115 (2014)

the pair correlation functions of the quantum system. In otherwords, the quantum effects are taken into account by an effec-tive pairwise additive potential of a classical system at a refer-ence temperature. Because we can readily calculate the prop-erties of the classical system from conventional liquid-statetheories (or molecular simulation methods),19–22 the classicalmap avoids direct calculation of the multi-body wave func-tions and, thus, is computationally much more efficient thanQMC. Moreover, the two-body correlation functions obtainedfrom the liquid-state theories allow for a straightforward pre-diction of the exchange-correlation free energy, which makesthe classical mapping method more appealing for practical us-age.

In our previous publications,22, 23 we used the classicalmapping method to predict the pair correlation functions andthe energetic properties of uniform electron gases at 0 K andthe electronic structure at finite temperature. We demonstratedthat classical mapping yields structural information in excel-lent agreement with QMC data and that a new exchange-correlation functional can be formulated on the basis of theHEG pair correlation functions to avoid self-interaction andstatic correlation errors.24 In this work, we employ the clas-sical mapping method to predict the exchange-correlationfree energy of homogeneous electron gases at finite temper-ature by using a new procedure to obtain the reference clas-sical temperature. The modified classical mapping methodimproves the theoretical predictions for both the correlationenergy and the microscopic structure of uniform electrongases. The remainder of this article is organized as follows:Section II introduces the theoretical framework for the clas-sical mapping method and the related liquid-state theory.Section III provides the numerical results and discussions. Fi-nally, main conclusions will be drawn in Sec. IV.

II. THEORY

We consider a spin-unpolarized HEG with average num-ber density ρ and absolute temperature T. The quantum sys-tem is “mapped” into a reference classical system with thesame particle density ρ and an effective temperature Tcf. Toensure that the quantum and classical systems have the samethe structure as measured by the spin-resolved pair correlationfunctions, the classical system consists of two types of parti-cles with an effective pairwise additive potential uij(r). Ac-cording to Dharma-Wardana et al.,19 uij(r) includes an effec-tive Pauli potential P(r) and a diffraction-corrected Coulombenergy:

uij (r) = P (r)δij + 1

r[1 − exp(r

√πm∗Tcf)], (4)

where r stands for the center-to-center distance between a pairof particles, subscript i and j denote the spin index of eachparticle, δij is the Kronecker delta function reflecting that thePauli exclusion principle applies only to electrons of the samespin, and m∗ is the effective mass of electrons, which is set as1 in Hartree atomic units (a.u.). Throughout this work, weuse a.u. for all physical variables. The numerical values fortemperature are dimensionless, defined as kBT/εF, where kB isthe Boltzmann constant and εF is the Fermi energy.

For a classical system with a pairwise additive potential,we can readily calculate the pair correlation functions gij(r),or equivalently, the total correlation functions hij(r) = gij(r)− 1, from the Ornstein-Zernike integral-equation theory withthe hypernetted-chain (HNC) approximation:25

hij (r) = cij (r) +∑

k

ρk

∫cik(r ′)hkj (|r − r′|)dr′, (5)

hij (r) = exp

[−uij (r)

kBTcf

+ hij (r) − cij (r)

]− 1, (6)

where ρk is the number density of component k and cij(r)represents the direct correlation function (DCF). It should benoted that the DCFs of the quantum and classical systems aredifferent because of extra quantum correlations; Eqs. (5) and(6) are valid only for the classical reference system. Besides,the HNC approximation discussed in this work has no con-nection with quantum HNC26 or Fermi-HNC theory.27 As dis-cussed in our previous work,22, 23 the bridge functions makenegligible contributions to the properties of uniform electrongases.

One key conjecture of the classical mapping method isthat the classical and quantum systems yield the same paircorrelation functions with and without the Coulomb interac-tion. Because hij(r) can be solved from the classical method,via Eqs. (5) and (6), we can readily calculate the exchangecorrelation free energy of the electronic system according tothe adiabatic connection:28

fxc = 1

4

∫ 1

0dα

∫h11(r, α) + h12(r, α)

rdr, (7)

where hij(r,α) denotes the pair correlation function of a clas-sical reference system with a reduced Coulomb potential:

uij (r, α) = P (r)δij + α

r[1 − exp(r

√πm∗Tcf)]. (8)

The numerical procedure for solving hij(r,α) is the same asthat for hij(r), i.e., from Eqs. (5) and (6).

For the application of the integral-equation theory to thereference classical system, we need to define the effectivePauli potential P(r) and the classical reference temperatureTcf. The Pauli potential can be obtained by using the classicalmapping method for a non-interacting reference quantum sys-tem of the same density. In that case, the Coulomb potentialhas been taken out and the corresponding pair potential in theclassical reference system is represented by

u0ij (r) = P (r)δij . (9)

Plugging Eq. (9) into (5) and (6) allows us to solve forP(r) using the corresponding pair correlation functions of thequantum system as the input. For HEG at finite tempera-ture, the latter can be exactly solved by a combination of theSchrödinger equation for free electrons, the Slater determi-nant, and the Fermi-Dirac distribution:11

h0ij (r) =

⎧⎨⎩

−[

zπ2�2rρ

∫ ∞0

t sin(tr/�)z+et2

dt]2

i = j

0 i �= j, (10)

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064115-3 Y. Liu and J. Wu J. Chem. Phys. 141, 064115 (2014)

where � = 1/√

πm∗T is the electron thermal wave lengthand z is known as fugacity, which can be determined from

z

π2�3

∫ ∞

0

t2

z + et2 dt = ρ. (11)

In the original classical mapping method proposed byDharma-Wardana et al.,19 a simple expression is used to relatethe classical temperature and the absolute temperature:

Tcf =√

T 2 + T 2q , (12)

where the “quantum temperature” is empirically correlatedwith the electron density

Tq = 1

a + br1/2s + crs

. (13)

In above equations, rs = (3/4πρ)1/3 stands for the Wigner-Seitz radius, and parameters a = 1.594, b = −0.316, andc = 0.024 were fitted with the correlation energy from QMCat 0 K. While Eq. (12) satisfies the asymptotic limit Tcf → Twhen T → ∞, it has been shown that the classical temperaturepredicted by Eqs. (12) and (13) yields unsatisfactory correla-tion energy at finite temperature.29–31 To remediate the defi-ciencies of these correlations, here we propose a new proce-dure to estimate the classical temperature. Instead of invokingthe unphysical quantum temperature, we simply use the func-tional form to empirically correlate the classical temperaturedirectly in terms of the absolute temperature and the electrondensity:

Tcf = 1

a(T ) + b(T )r1/2s + c(T )rs

. (14)

To identify suitable analytical expressions for thetemperature-dependent parameters a(T), b(T), and c(T),we impose the following constraints:

1) Tcf should reproduce the correlation energy at 0 K, i.e.,a(0) = 1.594, b(0) = −0.316, and c(0) = 0.024.

2) When T → ∞, Tcf → T for any electron density. Equa-tion (14) suggests that a(T ) ∼ T −1, b(T) and c(T) shouldbe negligible in comparison to a(T) in the high tempera-ture limit.

Within these constraints, we tested several functional formsfor a(T), b(T), and c(T) and found that the following relationsare most accurate for reproducing the QMC data for the cor-relation energy:4

⎧⎪⎪⎨⎪⎪⎩

a(T ) = 4772.126T 2e−5.13√

T + 1T +0.627

b(T ) = (−201.552T 2 − 0.316)e−2.687T

c(T ) = (21.96T 2 + 0.024)e−2.7T

. (15)

In obtaining the coefficients in Eq. (15), we first calcu-lated the classical temperature from the correlation energygiven by QMC for each thermodynamic state; parametersa(T), b(T), and c(T) were then determined by fitting the classi-

cal temperature with Eq. (14) for each temperature. In classi-cal mapping, the correlation energy was calculated from a nu-merical derivative of the correlation free energy with respectto temperature via Eqs. (17) and (3).

III. RESULTS AND DISCUSSION

Figure 1 shows the thermodynamic temperature T and thecorresponding classical reference temperature Tcf for HEG attwo representative densities, rs = 1 and 40 in a.u. Interest-ingly, the relationship is nonmonotonic at both high and lowelectronic densities. The classical temperature exhibits a sin-gle minimum value at high density (rs = 1) yet double min-ima at low density (rs = 40). The trough at low temperature(T < 4 a.u.) cannot be captured by the simple equation pro-posed by Dharma-Wardana et al. (viz., Eq. (12)). In mostcases, Tcf is lower than T, which contradicts to Eq. (12) andimplies that a meaningful quantum temperature cannot be de-fined. At high density (rs = 1), there is a minimum classicaltemperature at T = 0.5 a.u. consistent with previous inves-tigations by Karasiev et al.10 and Brown et al.31 The mini-mum classical temperature reflects a trough in the exchange-correlation energy at the same region as identified by QMC.At low density (rs = 40), the irregular peak is originated fromthe testing function of Tcf, viz., Eq. (15). One should noticethat the definitions of the kinetic energy are different in quan-tum and classical methods. While Tcf is proportional to thekinetic energy of the classical reference system, a similar cor-relation cannot be established for the quantum system. Be-cause Tcf is obtained from the QMC data for the correlationenergy, it provides no information on the quantum kinetic en-ergy. Due to the lack of constraints between T and Tcf, weconjecture that Tcf < T is physically acceptable.

One of the most important usages of the HEG model liesin its applications to DFT calculations. At finite temperature,the local density approximation (and its improvements) re-quires an expression for the exchange-correlation free energyas a function of the electron density. In conventional DFT cal-culations, the exchange correlation free energy is separatedinto two parts, the exchange contribution and the correlationcontribution. The exchange free energy is defined in terms of

0 2 4 6 8 100.01

0.1

1

10

Tcf

rs = 1

rs =40

FIG. 1. The dependence of the classical map temperature Tcf on the thermo-dynamic temperature T of spin-unpolarized electron gases at two densities.Both temperatures are dimensionless, and electron density is given in termsof Wigner-Seitz radius rs with atomic units.

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064115-4 Y. Liu and J. Wu J. Chem. Phys. 141, 064115 (2014)

0 10 20 30 40

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Cor

rela

tion

Ene

rgy

(a.u

.)rs (a.u.)

T=0.0625 This work T=0.0625 QMC T=0.0625 DW approximation T=0.5 This work T=0.5 QMC T=0.5 DW approximation

(a)

0 10 20 30 40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00 (b)

Cor

rela

tion

Ene

rgy

(a.u

.)

rs (a.u.)

T=1 This work T=1 QMC T=1 HNC Prediction

0 10 20 30 40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00(c)

Cor

rela

tion

Ene

rgy

(a.u

.)

rs (a.u.)

T=2 This work T=2 QMC T=2 DW approximation

0 10 20 30 40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00 (d)

Cor

rela

tion

Ene

rgy

(a.u

.)

rs (a.u.)

T=4 This work T=4 QMC T=4 DW approximation

FIG. 2. Correlation energy versus density for homogeneous electron gases at different temperatures. Here, DW approximation means classical mapping methodusing the classical reference temperature suggested by Dharma-Wardana et al., i.e., Eqs. (12) and (13). All values are in atomic units.

that of a reference non-interacting quantum system that has adensity the same as that of the real system:

fx = 1

4

∫h11(r, 0) + h12(r, 0)

rdr. (16)

The correlation part represents the difference between theexchange-correlation free energy of the real system and thatof the non-interacting reference system:

fc = fxc − fx. (17)

At T = 0 K, the exchange-correlation energy and free en-ergy are identical, and both can be calculated from QMC andclassical mapping directly. At finite temperature, however,QMC gives explicit results only for the exchange-correlationenergy, while the classical mapping method gives the freeenergy. For a direct comparison of QMC and the classicalmapping method, we calculate the correlation energy fromthe correlation free energy using the Gibbs-Helmholtz equa-tion given by Eq. (3). Toward that end, the partial derivativewith respect to temperature can be evaluated with the finite-difference method.

For a comparison of the classical mapping predictionswith the simulation data, we consider first the correlationenergy of HEG as a function of the electron density at dif-ferent temperatures. Figure 2 presents the correlation energyobtained from QMC and that from the modified classical map-ping method. Without any correction for the classical temper-ature, the original classical mapping method overpredicts thecorrelation energy by almost 50% at high density. The poorperformance can be attributed to the large classical referencetemperature used in mapping the quantum system. Becausethe higher the temperature, the less important the particle in-teractions in classical systems, the pair correlation functionh(r) resembles more like that of an ideal system at high T (i.e.,it becomes closer to 0). According to Eq. (7), the exchange-correlation free energy approaches 0 as h(r) vanishes, explain-ing the large correlation energy predicted by classical map-ping. By using a modified classical reference temperature, weare able to obtain the correlation energy in better agreementwith the QMC data.

Figure 3 shows the exchange-correlation free energy asa function of the electron density (a) and temperature (b)

0 10 20 30 40-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

f xc

rs

T=0.0625 This workT=0.0625 Karasiev et alT=8 This workT=8 Karasiev et al

(a)

0.1 1 10-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

f xc

T

This work Karaseiv et al(b)

rs = 1

FIG. 3. A comparison of the exchange correlation free energy from this work and the results by Karaseiv et al.10

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064115-5 Y. Liu and J. Wu J. Chem. Phys. 141, 064115 (2014)

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

g(r)

r/rs

rs = 1 This work

rs = 1 QMC

rs = 1 DW approximation

rs = 40 This work

rs = 40 QMC

rs = 40 DW approximation

(a)

0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

g(r)

r/rs

rs = 1 This work

rs = 1 QMC

rs = 1 DW approximation

rs = 40 This work

rs = 40 QMC

rs = 40 DW approximation

(b)

FIG. 4. Spin-averaged radial distribution functions of homogeneous electron gases at two temperatures T = 0.0625 (a) and 1 (b).

according to Eq. (7). Because QMC data are not avail-able, here we compare the numerical results from this workwith those from the semi-empirical correlations proposed byKarasiev et al.10 The two methods give similar dependenceof the exchange-correlation energy as a function of the elec-tron density, i.e., the fxc ∼ rs curve. On the effect of tempera-ture (the fxc ∼ T curve), however, good agreements are foundonly at the low temperature region (T < 0.5). The classicalmapping method predicts lower free energy and much largercurvature at high temperature. According to Karasiev et al.10

(Figure 1 of the reference), our results lie between their pre-dictions of fxc and εxc obtained from different methods, sug-gesting that our prediction should be at least qualitatively cor-rect. Because the free energy is not calculated in QMC explic-itly, the discrepancy between these two theoretical predictionsis calling for further investigations. From a practical perspec-tive, the accuracy of these methods may be tested by theirimplementation into FT-DFT calculations.

The classical mapping method provides information notonly about energy and free energy but also the pair correlationfunctions of uniform electron systems. The latter were not re-ported by Karasiev et al.10 Figure 4 compares the theoreticaland simulation results for the spin-averaged radial distributionfunction (RDF), defined as

g(r) = 1

2[h11(r) + h12(r)] + 1. (18)

As discussed above, the classical temperature plays an im-portant role in predicting the pair correlation functions. Ourmodified classical mapping gives the RDF closer to QMCdata because it employs lower temperature, which makes theinter-participle interactions more important. In particular, thenew method captures the peak value of the radial distributionfunction at low density rs = 40, which reflects strong elec-tronic correlations. Figure 4 suggests that simple approxima-tions on the classical reference temperature would not lead tocorrect long-range correlations in the quantum system. Whilethe Dharma-Wardana approximation yields better results atshort range, the integrand in Eq. (7), dr/r = 4πrdr, meansthat the long-range contribution dominates in calculation ofthe exchange correlation free energy. As a result, theoreticalmethods that give a more accurate description of the long-range components of the correlation functions also provide amore accurate prediction of the energetic properties.

IV. CONCLUSIONS

We have revised the classical mapping method for pre-dicting the properties of uniform electron gases (HEG) at fi-nite temperature by introducing a set of new correlations forthe classical reference temperature. The semi-empirical cor-relations were parameterized with recent QMC data for thecorrelation energies and the asymptotic results at low and hightemperatures. Although the simulation data for the correlationenergy have been used as an input for fitting the temperature-dependent parameters, the classical mapping method pro-vides additional entropic information not directly accessiblethrough QMC.

The new theoretical procedure was calibrated with thecorrelation energy, exchange-correlation free energy, and paircorrelation functions of HEG at finite temperature from QMCand/or previous theoretical predictions. Specifically, the cor-relation energy and pair correlation functions were com-pared with those based on the classical temperature proposedby Dharma-Wardana et al.19 and with QMC results,4 whilethe exchange-correlation free energy was compared withthe semi-empirical correlations proposed earlier by Karasievet al.10 The new theoretical procedure yields the correlationenergy in excellent agreement with QMC and shows signif-icant improvement over the classical mapping method orig-inally proposed by Dharma-Wardana et al. In particular, thenew correlations for the classical temperature show improve-ments on the classical mapping method for predicting the paircorrelation functions at the long-range region.

To a certain degree, the classical mapping method pro-posed in this work complements the earlier publication byKarasiev et al.10 for correlating the energetic properties ofuniform electron gases at finite temperature. While, in gen-eral, the classical mapping method yields the exchange-correlation free energy in good agreement with that fromKarasiev et al.,10 we have noticed significant disagreementat high temperature. Because QMC data are not available forthe exchange-correlation free energy, the difference may betested in terms of their usefulness, for example, in future ap-plications to FT-DFT calculations.

ACKNOWLEDGMENTS

J.W. is grateful to the U.S. Department of Energy (DOE)(DE-FG02-06ER46296) for the financial support of this

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064115-6 Y. Liu and J. Wu J. Chem. Phys. 141, 064115 (2014)

research. The authors are also grateful to the National En-ergy Research Scientific Computing Center (NERSC), whichis supported by the Office of Science of the U.S. Departmentof Energy under Contract No. DE-AC03-76SF0009.

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