Local Density Approximation for the Short-Range Exchange Free Energy Functional

Local Density Approximation for the Short-Range Exchange Free Energy Functional

Fengyuan Xuan,

^†

Jeng-Da Chai,*

^,‡,§,∥

and Haibin Su*

^,⊥

†Centre for Advanced 2D Materials, National University of Singapore, Block S16, Level 6, 6 Science Drive 2, Singapore 117546, Singapore

‡Department of Physics,^§Center for Theoretical Physics, and^∥Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

⊥Department of Chemistry, The Hong Kong University of Science and Technology, Kowloon, Hong Kong 999077, China

ABSTRACT: Analytical expressions for the exchange free energy per particle of the uniform electron gas (UEG) associated with the short-range (SR) interelectronic interaction at the low- and high-temperature limits are examined, yielding an accurate analytical parametrization for the SR exchange free energy per particle of the UEG as a function of the uniform electron density, temperature, and range- separation parameter. This parametrization constitutes the local density approximation for the SR exchange free energy functional, which can be thefirst step towardfinding generally accurate range-separated hybrid functionals in both finite-temperature density functional theory and thermally assisted-occupation density functional theory.

■

INTRODUCTION

Over the past decades, Kohn−Sham density functional theory (KS-DFT)^1,2 has been one of the most powerful quantum- mechanical methods for studying the ground-state properties of atoms, molecules, and bulk materials.^3,4However, since the exact exchange−correlation (XC) energy functional Exc[ρ] in KS-DFT remains unknown, numerous efforts have been devoted to improving the accuracy of approximate XC energy functionals.

The uniform electron gas (UEG) is an important system from which the local density approximation (LDA) for the XC energy functional, E_xc^LDA, can be developed. However, a major shortcoming associated with the LDA is the self-interaction error (SIE),⁵which can be efficiently reduced by incorporating the Hartree−Fock (HF) exchange energy ExHF into the LDA XC energy functional E_xc^LDA. A global hybrid (GH) XC energy functional^6,7 may generally achieve this. However, to reduce the SIE problem more effectively, the range-separated hybrid (RSH) scheme can be adopted.⁸⁻¹⁹Particularly, in the long- range corrected hybrid (LCH) scheme, the Coulomb operator is partitioned into the long-range (LR) and short-range (SR) operators, commonly achieved with the use of the standard error function (erf) and the complementary error function (erfc)

ω ω

| − ′| = | − ′|

| − ′| + | − ′|

| − ′|

r r

r r r r

r r r r

1 erf( ) erfc( )

(1) On the right-hand side ofeq 1, thefirst term is the LR operator (i.e., the erf operator), the second term is the SR operator (i.e., the erfc operator), and ω is the range-separation parameter

controlling the range of the SR operator (atomic units are used throughout the paper). The LCH XC energy functional can be commonly expressed as E_xc^LCH= E_x^LR‑HF+ E_x^SR‑LDA+ E_c^LDA, where E_x^LR‑HF is the HF exchange energy associated with the LR operator, E_x^SR‑LDA is the LDA exchange energy functional associated with the SR operator, and E_c^LDA is the LDA correlation energy functional (i.e., associated with the Coulomb operator). Accordingly, the SIE associated with an LCH XC energy functional can be greatly reduced, and the corresponding XC potential possesses the correct −1/r asymptote in the asymptotic regions of atomic and molecular systems. At ω = 0 (i.e., the SR interelectronic interaction reduces to the Coulomb interelectronic interaction), E_xc^LCH reduces to E_xc^LDA. Note also that in the LCH scheme, E_x^SR‑LDA and E_c^LDA, which are the functionals based on the LDA (i.e., the simplest density functional approximation), may be replaced with those based on more sophisticated density functional approximations [e.g., the generalized-gradient approximations (GGAs)] for improved accuracy.

Finite-temperature density functional theory (FT-DFT) was first proposed by Mermin,²⁰ wherein the grand canonical ensemble is adopted to study the thermodynamic properties of a physical system at temperature T. To obtain the thermal equilibrium density in FT-DFT, one needs to solve the following self-consistent equations

Received: February 1, 2019 Accepted: April 17, 2019 Published: April 26, 2019

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lmoo

noo |

}oo~

ooψ ψ

−∇

+v r r = ϵ r

2 ( ) _i( ) _{i i}( )

2

s (2)

where

∫

^{ρ μ}

= + ′

| − ′| ′ +

v r v r r

r r r r

( ) ( ) ( )

d ( )

s ext xc (3)

is the effective potential, vext(r) is the external potential, and μxc(r) =δFxc[ρ]/δρ(r) is the XC potential (i.e., the functional derivative of the XC free energy functional F_xc[ρ]). The thermal equilibrium density is given by

∑

ρ = |ψ | + ^μ

=

∞

r r ϵ −

( ) ( ) /(1 e )

i i

k T 1

2 (i )/(B )

(4) where μ is the chemical potential chosen to conserve the number of electrons and k_Bis the Boltzmann constant. At T = 0, FT-DFT reduces to KS-DFT and the XC free energy functional F_xc[ρ] reduces to the XC energy functional Exc[ρ].

Although FT-DFT has been proposed for several decades, only KS-DFT has received massive applications in solid-state physics and quantum chemistry. The main reason may be that the“thermal effect” is commonly regarded as an effect only on nuclei. According to the Born−Oppenheimer approximation, the Hamiltonian of electrons can be separated from that of nuclei. The coupling between the vibrational modes of nuclei and the motion of electrons can be properly described by some model Hamiltonian and treated as perturbation.²¹ In many other cases, electrons are not in the extreme conditions and hence the thermal effect can simply be neglected. Therefore, the properties of atoms, molecules, and bulk materials are usually studied using KS-DFT (i.e., FT-DFT with T = 0).

Although several efforts have been devoted to understanding the properties of thermal density functionals,²²⁻²⁶ the applications of FT-DFT are only limited to hot plasmas and warm dense matter.²⁷⁻²⁹ Besides, in conventional FT-DFT calculations, for simplicity, the XC free energy functional F_xc[ρ] has been commonly approximated by the “zero- temperature” XC energy functional Exc[ρ] evaluated with the thermal equilibrium densityρ(r) at temperature T.

In 1984, an analytical parametrization for the LDA XC free energy functional F_xc^LDA[ρ] was proposed by Perrot and Dharma-wardana,³⁰ based on the low-temperature limit for the exchange free energy derived by Horovitz and Thieberger.³¹ In 2000, a classical mapping method³² was adopted to study finite-temperature electron liquid.³³ Later, the classical representation of quantum systems at equilibrium was applied to study two special quantum systems.³⁴⁻³⁷ Recently, the restricted path integral Monte Carlo (PIMC) approach was adopted to numerically evaluate the XC free energy per particle of the warm dense UEG.³⁸The numerical data were soon adopted by Karasiev and co-workers for the development of an accurate parametrization for F_xc^LDA[ρ].³⁹ Very recently, the most advanced density matrix quantum Monte Carlo method was adopted to resolve the discrepancy between the configuration and restricted PIMC results.⁴⁰Since these simulations are only limited to small systems, thefinite- size errors were further corrected from the ab initio finite-N quantum Monte Carlo calculations and compared with the previous results where the significant difference was found.⁴¹In addition, a careful study on the free energy functional for a noninteracting electron system at temperature T was recently performed by Dufty and Trickey.⁴²

Nevertheless, owing to the local approximation, the LDA XC free energy functional F_xc^LDA[ρ] should still suffer from the SIE problem. To reduce this, the aforementioned LCH scheme can be extended to FT-DFT, i.e., incorporating the HF exchange free energy F_x^HF into the LDA XC free energy functional F_xc^LDA. The resulting LCH XC free energy functional can be expressed as F_xc^LCH = F_x^LR‑HF + F_x^SR‑LDA + F_c^LDA, where F_x^LR‑HFis the HF exchange free energy associated with the LR operator (e.g., the erf operator), F_x^SR‑LDAis the LDA exchange free energy functional associated with the SR operator (e.g., the erfc operator), and F_c^LDAis the LDA correlation free energy functional (i.e., associated with the Coulomb operator). Atω = 0 (i.e., the SR interelectronic interaction reduces to the Coulomb interelectronic interaction), F_x^SR‑LDA reduces to the LDA exchange free energy functional F_x^LDA and hence F_xc^LCH reduces to F_xc^LDA. Note that in FT-DFT, F_x^LR‑HFis well-defined and F_c^LDA is readily available in the literature. Therefore, the focus of the present work is a reliably accurate parametrization for F_x^SR‑LDA. For brevity, hereafter, we adopt “the SR LDA exchange free energy functional” (FxSR‑LDA) for “the LDA exchange free energy functional associated with the SR operator” and adopt “the SR LDA exchange energy functional”

(E_x^SR‑LDAor F_x^SR‑LDAwith T = 0) for“the LDA exchange energy functional associated with the SR operator”.

On the other hand, in 2012, Chai developed thermally assisted-occupation density functional theory (TAO-DFT)⁴³ for the study of the ground-state properties of nanoscale systems with strong static correlation effects (which are

“challenging systems” for traditional electronic structure methods).⁴⁴⁻⁵¹ In contrast to KS-DFT, an auxiliary system of N noninteracting electrons at some“fictitious temperature”

is employed in TAO-DFT, wherein strong static correlation is explicitly approximated by the entropy contribution (e.g., see eq 26 of ref43). A self-consistent scheme for determining the fictitious temperature in TAO-DFT was recently proposed.⁵² In 2017, the GH and RSH schemes in TAO-DFT were also developed for a wide range of applications.⁵³Relative to local and semilocal density functionals in TAO-DFT, GH func- tionals in TAO-DFT were shown to possess reduced SIEs.

However, to employ the RSH scheme in TAO-DFT, the SR (or LR) exchange free energy functional (at the fictitious temperature) should be further developed. Therefore, the development of SR LDA exchange free energy functional can also be the first step toward finding generally accurate RSH functionals in TAO-DFT, highlighting the value of the present work.

For the reasons given above, there is a strong need for developing the SR LDA exchange free energy functional for the RSH schemes in both FT-DFT and TAO-DFT. InResults and Discussion, wefirst examine analytical expressions for the SR exchange free energy per particle of the UEG at the low- and high-temperature limits. For the low-temperature limit, the first-order dependence on temperature is found to be absent, similar to that found for the exchange free energy per particle of the UEG. Moreover, the zero-temperature limit agrees exactly with that reported in the literature.⁵⁴ Based on these limits and our findings, we develop a reliably accurate analytical parametrization for the SR exchange free energy of the UEG as a function of the uniform electron density, temperature, and range-separation parameter, retaining the correct zero-temperature and high-temperature limits. Besides, the SR exchange potential of the UEG, obtained directly from the functional derivative of the parametrized SR exchange free

energy of the UEG, is reliably accurate relative to the corresponding numerical value. For a general system, the SR LDA exchange free energy functional is developed. In the last section, we give our conclusions.

■

RESULTS AND DISCUSSION

SR Exchange Free Energy Per Particle of the UEG.

Consider a spin-unpolarized system containing N electrons associated with the SR interelectronic interaction (i.e., the erfc operator given byeq 1with the range-separation parameterω) in a volume V at temperature T, with a positive background charge keeping the system neutral. Here, N and V are taken to infinity in the manner that keeps the electron density (ρ = N/

V)finite. The SR exchange free energy of the UEG, FxSR, can be expressed in both momentum space and coordinate space^30,31

∬

∬

π

π

ω

= − | − ′| −

′

=− | − ′|

| − ′| [ | − ′| ] ′

−| − ′| ω

F V ′

n n

G

k k k

k r r

r r r r r r

(2 )

4 (1 e ) d

d

erfc( )

( ) d d

k k

k k x

SR

6 2

/4

2

2 2

(5) where nk is given by the Fermi−Dirac distribution function (withβ = 1/(kBT) and k = |k|)

=

+ ^{β −μ}

n 1

1 e

k (^k2₂ ) (6)

and G(x) (with x≡ |x|) is defined as

∫

= π

+ ^{β μ}

∞

−

G x x

p px p

( ) 1

2

sin( )

1 e

2 d

0

(^p2₂ )

(7) On the basis ofeq 5, one can replace the chemical potentialμ with the uniform electron densityρ (i.e., the inversion of the equation below can be performed, as shown in refs30and55)

∑ ∑

ρ = = =

+ ^{β μ}

− −

−

N

V 2V n 2V 1

1 e

k k

k

1 1

(^k2₂ ) (8)

and numerically evaluate the SR exchange free energy per particle of the UEG

λ = = ρ

f k t F

N F ( , , ) V

x SR

F xSR

xSR

(9) as a function of the Fermi wave vector k_F= (3π²ρ)^1/3and two dimensionless variables (t≡ kBT/E_Fandλ ≡ ω/kF). Here, E_F= k_F²/2 is the Fermi energy. It is convenient to define the scaled SR exchange free energy per particle of the UEG

λ

μ ≡ ̃ λ

f k t

k f t

( , , )

( , 0) ( , )

x SR

F

x F x

SR

(10) which is a function of the dimensionless variables t andλ only.

Here,μx(k_F, 0) =−kF/π is the exchange potential of the UEG at the Fermi wave vector k_Fand t = 0.

In the following discussion, for the given values ofρ and ω, we examine the SR exchange free energy per particle of the UEG at the low-t and high-t limits. Therefore, the low-t and high-t limits discussed are equivalent to the low-temperature (low-T) and high-temperature (high-T) limits, respectively.

Low-Temperature Limit. Similar to the procedures of Horovitz and Thieberger,³¹wefirst express FxSR [given by eq 5] as a function of z ≡ e^βμ

∫

= ∂

∂ ′ ′

F F

z dz

z xSR

0 x SR

(11)

where the lower limit of integration is determined by direct calculation ofeq 5in the limit z→ 0

i kjjjjj jj

y {zzzzz zz β ω

βω πβ

= − −

−

=

→ →

F Vz

lim lim 1

1 (2 ) 0

z 0 xSR z

0 2

2 2

(12)

The derivative of F_x^SRwith respect to z is given by lmo

no

|}o

∫

~o π β

ω π

∂

∂ = − [ ] − ^ω [ ]

∞ −

F z

V

z g(0) 2 g x x

e ^x ( ) d

xSR

3

2

0

2 2 2

(13) where g(x) is defined as

∫

=

+ ^{β μ}

∞

−

g x px

p

( ) cos( )

1 e

d

0

(^p2₂ )

(14)

We now proceed to evaluate F_x^SR[given byeq 11] at the large-z limit, which corresponds to the low-t limit (e.g., see eq 5 of ref 55). It can be shown that

ω

∫

π ^ω [ ] ≤ [ ] = β

∞ −

g x x g h z

2 e _x ( ) d (0) ( )

0

2 2

2 2

(15)

where h(z) is a function of z only. Therefore, the integration of eq 13with respect to z from 0 to a constant z₁(see ref31for the reason that one can choose such a constant) yields at most a second-order temperature-dependent term, which can be expressed as δ/β². Accordingly, one can separate the integration in eq 11 with respect to z′ into two parts: (i) from 0 to z₁ and (ii) from z₁ to z. Using the generalized Sommerfeld’s lemma,⁵⁶in the range of z > z₁, one obtains the following low-temperature expansion

lm oo noo

Ä ÇÅÅÅÅÅ ÅÅÅÅÅ

É ÖÑÑÑÑÑ ÑÑÑÑÑ Ä

ÇÅÅÅÅÅ ÅÅÅÅÅ i

kjjjj i

kjjj y{zzzy {zzzz i

kjjjj i

kjjj y{zzzy {zzzz É

ÖÑÑÑÑÑ ÑÑÑÑÑ|

}oo

~oo

π β β β π

β ω

π

π ω π

ω ω

π π

β ω

π ω ω

∂

∂ = − − +

− − +

− − + +

ω

ω

−

−

F z

V

z k

k

k k

k k

k 1

2 2 2

3 ...

2

2 e 1 erf

12 1 e erf ...

k

k x

SR

3

2 2

2

/

2 2 2

/

2 2

2 2

(16) where the variable is rescaled to k= 2 ln z/ β. The integration in eq 11 with respect to z′ from z1 to z can be transformed into integration with respect to k from

= β

k₁ 2 lnz₁/ to 2μ, which yields the SR exchange free energy per volume of the UEG at the low-temperature limit

Ä ÇÅÅÅÅÅ ÅÅÅÅ

É Ö ÑÑÑÑÑ ÑÑÑÑÑ δ

∫

β β

δ

β π β β μ π

βμ π

β π

= + ∂

∂

≈ − + + [ − ]

+ [ − ]

F μ

V

F

z z k k

I k I k J k J k

d

1

2 2

3ln( ) 1

2 ( ) ( )

1

12 ( ) ( )

k x

SR 2

2 xSR

2 3 2

2 2 2

3 1

2 1

1

(17)

here Ä

ÇÅÅÅÅÅ

ÅÅÅÅ i

kjjj y{zzzÉ ÖÑÑÑÑÑ ω ÑÑÑÑ

ω π

ω ω

= − + − ^{− ω} +

I k k k k k

( ) 3 3 (2 )e 2

k erf

2

2 2 2 ₂/ ₂ 3

(18) and

i kjjjj y

{zzzz i kjjjj y

{zzzz i

kjjj y{zzz

ω ω

π ω

= − + − ω

J k k k

k

( ) Ei 2 ln 2 k

erf

2 2

2

2 (19)

where the exponential integral Ei(x) is defined as

∫

= −

−

∞ −

x s s

Ei( ) e

d

x s

(20) The first two terms of eq 17 are ω-independent and are the same as the results of Horovitz and Thieberger.³¹The last two terms (i.e., those with I(k) and J(k)) areω-dependent, which are the new results brought by the SR interelectronic interaction. From eq 17, we find that the first-order temperature-dependent term vanishes, which is similar to that found from the exchange free energy per volume of the UEG at the low-temperature limit.³¹Note also that in the last term of eq 17, J(k₁)/β², besides an analogue term with the logarithmic term of T²ln T in ref31we arrive at an additional term of T²Ei(−T), which originates from the SR interelec- tronic interaction. For an analysis of the divergent term, one may refer to ref57. The dependence on k (i.e., (2μ)^1/2) can be replaced by the uniform electron density ρ (see eq 12 of ref 31), as T approaches zero.

In addition, asω → 0 (i.e., the SR interelectronic interaction reduces to the Coulomb interelectronic interaction), eq 17 correctly reduces to the exchange free energy per volume of the UEG at the low-temperature limit

Ä ÇÅÅÅÅÅ ÅÅÅ

É ÖÑÑÑÑÑ ÑÑÑ

π π β

π π

β δ β

= − + + − +

F V

E 1 E O

2 3 3 ln( ) ( )

x F2

3 2 2 F

4

(21) which was previously reported by Horovitz and Thieberger.³¹ Note that thefirst-order temperature-dependent term ineq 21 vanishes, which later guided the parametrization function for the LDA exchange free energy functional of Perrot and Dharma-wardana.³⁰

Besides, on the basis of eq 17, as 1/β → 0 (i.e., as T or t reduces to zero), the scaled SR exchange free energy per particle of the UEG at the zero-temperature limit (first derived by Gill, Adamson, and Pople⁵⁴) can be correctly obtained

Ä ÇÅÅÅÅÅ

ÅÅÅÅ i

kjjj y{zzz É

ÖÑÑÑÑÑ ÑÑÑÑ λ

μ λ

λ π

λ λ λ λ λ

= = ̃ =

= − − + + − ^{− λ}

f k t

k f t

( , 0, )

( , 0) ( 0, )

3

4 2 2 erf 1

3 (2 )e

x SR

F

x F x

SR

3 3 1/ ²

(22)

High-Temperature Limit. Here, we examine the SR exchange free energy per particle of the UEG at the high- temperature limit. The Taylor series expansion of g(x) [seeeq 14] with respect to z≡ e^βμ yields

∑

^π

= − β

+

β

=

∞

+ − +

g x z

( ) ( 1) m

2 ( 1) e

m

m m x m

0

1 ²/2 ( 1)

(23) At the small-z limit, which corresponds to the high-t limit (e.g., see eq 5 of ref55), one can keep only thefirst term and obtain

π β π β

∂

∂ ≈ − [ ] +

+ _βω F

z

V

z g V z

(0) 2 1

x SR

3

2

2 2 1

2 (24)

Consequently, the resulting F_x^SRcan be expressed as

∫

^{πβρ π β}

= ∂

∂ ′ ′ ≈ − +

+ _βω

F F

z z V V z

d 2 4 1

z

xSR 0

x

SR 2

2 2 2

1

2

(25) Using eq 2.3 of ref 30 to replace z with ρ at the high- temperature limit, the scaled SR exchange free energy per particle of the UEG at the high-temperature limit⁵⁸ can be obtained

i kjjjjj j

y {zzzzz z λ

μ λ

λ λ

= = ̃ =

= −

+

f k t h

k f t h

t t

( , , )

( , 0) ( , )

1

3 1 2

4 2

x SR

F

x F x

SR

2 (26)

where t = h stands for the high-temperature limit.

Note that asω → 0 or λ → 0 (i.e., the SR interelectronic interaction reduces to the Coulomb interelectronic inter- action), eq 26 correctly reduces to the scaled exchange free energy per particle of the UEG at the high-temperature limit, i.e., 1/(3t).³⁰

Parametrization for the SR Exchange Free Energy Per Particle of the UEG. In the work of Perrot and Dharma- wardana,³⁰afitting function was proposed to parametrize the numerical data of the exchange free energy per particle of the UEG. As thefirst-order temperature-dependent term vanishes at the low-temperature limit,³¹ the temperature-dependent term in the fitting function of Perrot and Dharma-wardana starts from the t²term (see eq 3.2 of ref30).

In the present work, to incorporate the correct zero- temperature limit f̃_x^SR(t = 0, λ) [see eq 22] and high- temperature limit f̃_x^SR(t = h, λ) [see eq 26], and also our findings that the first-order temperature-dependent term vanishes (i.e., the temperature-dependent term starts from the t² term) at the low-temperature limit, we propose the followingfitting function

λ

μ λ λ λ

λ λ λ

λ

= ̃ = ̃ = [

+ − ] + ̃ = [ −

− − ]

λ

λ λ

λ

−

− −

−

f k t

k f t f t y

y f t h y

y ( , , )

( , 0) ( , ) ( 0, ) ( )e

(1 ( ))e ( , ) 1 ( )e

(1 ( ))e

x t

x t x t

x t

x SR

F

x F x

SR

x SR

1

( )

1

( ) x

SR

2

( )

2

( )

1 3

2 2

3 3

4 4

(27)

to parametrize the numerical data of the scaled SR exchange free energy per particle of the UEG f̃_x^SR(t,λ) [given byeq 10], where x_i(λ) (i = 1, 2, 3, and 4) is defined as

λ ≡ + ^{− λ}+ − − ^{− λ}

x_i( ) c_i1 c_i2e ^ci3 (c_i0 c_i1 c_i2)e ^ci4 (28) and y_i(λ) (i = 1 and 2) is defined as

λ ≡ + ^{− λ}+ − − ^{− λ}

y_i( ) d_i1 d_i2e ^di3 (d_i0 d_i1 d_i2)e ^di42 ₍₂₉₎ Thefitting to the numerical data is performed in the range of 0

< t < 12 and 0 <λ < 20. The optimized parameters for xi(λ) and y_i(λ) are shown inTables 1and2, respectively.

Figure 1shows the surface plot for the parametrization of f̃_x^SR(t, λ) [given by eq 27] and the scattered circles for the

numerical data of f̃_x^SR(t,λ) [given byeq 10].Figure 2shows the fitting curves for the parametrization of f̃xSR(t, λ) [given byeq 27] and the scattered circles for the numerical data of f̃_x^SR(t,λ) [given byeq 10] at variousλ values.

To examine the accuracy of our parametrization, Figure 3 shows the relative error of the parametrization of f̃_x^SR(t, λ) [given by eq 27], where the relative error is defined as the absolute value of ((parametrization [given by eq 27] − numerical data [given byeq 10])/(numerical data [given byeq 10])). Relative to the numerical data, our parametrization is reliably accurate. The relative error is vanishingly small in the low-t and high-t regions. However, in the intermediate-t region, Table 1. Optimized Parameters forxi(λ) (i = 1, 2, 3, and 4)

[See Equation 28]

i 1 2 3 4

c_i0 0.3729 5.6674 0.2127 16.0023

c_i1 0.0051 0.1777 0.0036 0.0894

c_i2 0.0438 0.7474 0.0258 7.1526

c_i3 0.3485 0.3471 0.2023 14.8795

c_i4 2.6256 2.7513 1.9715 2.0447

Table 2. Optimized Parameters foryi(λ) (i = 1 and 2) [See Equation 29]

i 1 2

d_i0 0.2839 0.7331

d_i1 0.9912 0.9973

d_i2 −0.4510 −0.1511

d_i3 0.4941 0.6022

d_i4 1.0075 1.7642

Figure 1.Scaled SR exchange free energy per particle of the UEG, f̃_x^SR(t,λ), as a function of t and λ. Surface: parametrization [given byeq 27].

Circles: numerical data [given byeq 10].

Figure 2.Scaled SR exchange free energy per particle of the UEG, f̃_x^SR(t,λ), as a function of t and λ. Lines: parametrization [given byeq 27]. Circles: numerical data [given by eq 10]. Here, magenta, red, yellow, cyan, green, and blue correspond toλ = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.

the relative error is slightly larger, especially for the larger λ.

The maximum relative error is 0.087 (i.e., the maximum percentage error = 8.7%) at t = 6 and λ = 20. Therefore, further investigation on the expression of f̃_x^SR(t,λ) at the large-λ limit can be essential for improved parametrization.

SR Exchange Potential of the UEG. In the work of Perrot and Dharma-wardana,³⁰ the exchange potential of the UEG was parametrized separately, which can, however, be inconsistent with the functional derivative of their para- metrized exchange free energy functional of the UEG. It is worth mentioning that Karasiev and co-workers⁵⁵ recently proposed a more accurate parametrization (compared to the one from ref30) for the UEG exchange free energy functional with exchange potential calculated as the corresponding functional derivative.

For consistency, in the present work, the SR exchange potential of the UEG, μxSR, is obtained directly from the functional derivative of F_x^SR

μ δ

δρ ρ

= F = + ρ

f df

x d

SR x

SR x

SR x

SR

(30) Substituting f_x^SR= f_x^SR(k_F, t,λ) = μx(k_F, 0) f̃_x^SR(t,λ) = (−kF/π) f̃_x^SR(t,λ) [given by eq 27] intoeq 30,μxSRcan be expressed as

lm ooo nooo

|} ooo

~ooo

μ μ λ

λ ρ λ

ρ λ

ρ

λ λ

λ ρ

λ λ λ λ

λ

=

= + ∂

∂ + ∂

∂ + ∂

∂

= − ∂

∂ − ∂

∂ k t

f k t f k t

k

k

f k t t

t f k t

f k t t f k t

t

f k t ( , , )

( , , ) ( , , ) d

d ( , , ) d

d

( , , ) d d 4 ( , , )

3

2 3

( , , ) 3

( , , )

x SR

x SR

F

x SR

F

x SR

F F

F

x SR

F x

SR F

x SR

F x

SR

F x

SR F

(31) On the basis ofeq 31, it is convenient to define the scaled SR exchange potential of the UEG

μ λ

μ μ λ

λ λ

λ λ

λ

≡ ̃

=

̃

− ∂ ̃

∂

− ∂ ̃

∂ k t

k t

f t t f t

t f t

( , , )

( , 0) ( , )

4 ( , ) 3

2 3

( , )

3

( , )

x SR

F

x F x

SR

x SR

x SR

x SR

(32) Figure 3. Relative error of the parametrization of the scaled SR

exchange free energy per particle of the UEG, f̃xSR(t,λ), as a function of t andλ. Here, the relative error is defined as the absolute value of ((parametrization [given by eq 27]− numerical data [given by eq 10])/(numerical data [given byeq 10])).

Figure 4.Scaled SR exchange potential of the UEG,μ̃xSR(t,λ), as a function of t and λ. Surface: parametrization [given byeq 32]. Circles: numerical data [given by differentiatingeq 10].

which is a function of the dimensionless variables t andλ only.

As f̃_x^SR(t,λ) [given byeq 27] is parametrized,μ̃xSR(t,λ) [given byeq 32] can be evaluated analytically.

Figure 4shows the surface plot forμ̃xSR(t,λ) [given byeq 32]

and the scattered circles for the corresponding numerical data [given by differentiating eq 10]. Figure 5 shows the fitting curves forμ̃xSR(t,λ) [given by eq 32] and the scattered circles for the corresponding numerical data [given by differentiating eq 10].

To assess the accuracy of our parametrization, Figure 6 shows the relative error of the parametrization of μ̃xSR(t, λ)

[given by eq 32], where the relative error is defined as the absolute value of ((parametrization [given by eq 32] − numerical data [given by differentiating eq 10])/(numerical data [given by differentiating eq 10])). A good agreement between our parametrization and the corresponding numerical data can be clearly seen from thefigure. The relative error is vanishingly small in the low-t and high-t regions. Nonetheless, in the intermediate-t region, the relative error is slightly larger, especially for the largerλ. The maximum relative error is 0.108 (i.e., the maximum percentage error = 10.8%) at t = 7.7 andλ

= 20. As mentioned previously, the accuracy of the parametrization may be further improved by investigating the expression of f̃_x^SR(t, λ) (and hence the corresponding expression ofμ̃xSR(t,λ) given byeq 32) at the large-λ limit.

LDA for the SR Exchange Free Energy Functional. In the previous section, the SR exchange free energy per particle of the UEG f_x^SR(t, λ) has been discussed and parametrized, which can now be extended to a general system.

Consider a spin-unpolarized system containing N electrons associated with the SR interelectronic interaction (i.e., the erfc operator given byeq 1with the range-separation parameterω) at temperature T, in the presence of an external potential v_ext(r). The LDA for the SR exchange free energy per particle can be obtained by replacing the uniform electron densityρ in eq 27with the local electron densityρ(r). Accordingly, kF, E_F, t,λ, and μx(k_F, 0) are replaced with k_F(r) = [3π²ρ(r)]^1/3, E_F(r)

= [k_F(r)]²/2, t(r) = k_BT/E_F(r),λ(r) = ω/kF(r), andμx(k_F(r), 0) = −kF(r)/π, respectively. Consequently, the SR LDA exchange free energy functional can be expressed as

∫

∫

∫

∫

ρ ρ λ

ρ μ λ

π ρ λ

ρ λ

[ ] =

= ̃

=− ̃

= [ ] ̃

F ‐ f k t

k f t

k f t

C f t

r r r r r

r r r r r

r r r r r

r r r r

( ) ( ( ), ( ), ( ))d ( ) ( ( ), 0) ( ( ), ( ))d 1 ( ) ( ) ( ( ), ( ))d

( ) ( ( ), ( ))d

x SR LDA

x SR

F

x F x

SR

F x

SR

x 4/3

x SR

(33) where C_x= −(3/π)^1/3and f̃_x^SR(t(r),λ(r)) [given byeq 27] is the scaled SR exchange free energy per particle of the UEG at the local electron density ρ(r). The SR LDA exchange potential is given by the functional derivative of F_x^SR‑LDA[ρ]

μ δ ρ

= δρ [ ]

‐

F ‐

r r

( ) ( )

x

SR LDA xSR LDA

(34) Owing to the spin-scaling relation,⁵⁹the extension of the SR LDA exchange free energy functional to a spin-polarized system (i.e., with theα-spin density ρα(r),β-spin density ρβ(r), temperature T, and range-separation parameter ω) is straightforward

ρ ρ ρ ρ

[_{α β}] = [ _α] + [ _β]

‐ ‐ ‐

F , 1 F F

2( 2 2 )

x SR LDA

x SR LDA

x SR LDA

(35) where the spin-polarized functional F_x^SR‑LDA[ρα,ρβ] [seeeq 35]

can be conveniently expressed by the spin-unpolarized functional F_x^SR‑LDA[ρ] [seeeq 33].

■

CONCLUSIONS

In summary, we have examined analytical expressions for the SR exchange free energy per particle of the UEG at the low- and high-temperature limits. The SR interelectronic interaction brings extra terms in the two limiting forms when compared with those for the Coulomb interelectronic interaction. At the low-temperature limit, the temperature-dependent term starts from the t² term, which is similar to that found for the exchange free energy per particle of the UEG. An analytical fitting function has been proposed for the SR exchange free energy per particle of the UEG. Accordingly, the SR LDA exchange free energy functional for a general system has been developed, with which RSH functionals can be readily devised in both FT-DFT and TAO-DFT.

Figure 5.Scaled SR exchange potential of the UEG,μ̃xSR(t,λ), as a function of t andλ. Lines: parametrization [given byeq 32]. Circles:

numerical data [given by differentiatingeq 10]. Here, magenta, red, yellow, cyan, green, and blue correspond toλ = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0, respectively.

Figure 6. Relative error of the parametrization of the scaled SR exchange potential of the UEG, μ̃xSR(t,λ), as a function of t and λ.

Here, the relative error is defined as the absolute value of ((parametrization [given by eq 32] − numerical data [given by differentiating eq 10])/(numerical data [given by differentiatingeq 10])).

In the future, we plan to develop the SR exchange free energy functionals based on more sophisticated density functional approximations (e.g., GGAs) to further improve the accuracy of RSH functionals in both FT-DFT and TAO- DFT. Note that an accurate GGA XC free energy functional (i.e., associated with the Coulomb operator) has been recently developed.⁶⁰

■

AUTHOR INFORMATION Corresponding Authors

*E-mail:[email protected](J.-D.C.).

*E-mail:[email protected](H.S.).

ORCID

Jeng-Da Chai: 0000-0002-3994-2279

Haibin Su: 0000-0001-9760-6567 Notes

The authors declare no competingfinancial interest.

■

ACKNOWLEDGMENTS

Work at NTU (Taiwan) was supported by the Ministry of Science and Technology of Taiwan (Grant No. MOST107- 2628-M-002-005-MY3), the National Taiwan University ( G r a n t N o s . N T U - C C - 1 0 7 L 8 9 2 9 0 6 ; N T U - C C P - 106R891706; NTU-CDP-105R7818), and the National Center for Theoretical Sciences of Taiwan. The work is supported in part by the Society of Interdisciplinary Research (SOIREE), HKUST Grants (IGN17SC04; R9418). The authors are grateful for Mark Casida, Peter Gill, Andreas Savin, and Debashis Mukherjee for stimulating discussions.

The authors would also like to thank Tsung-Jen Liao for useful discussions.

■

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