Density Functional Theory for Electron Gas and for Jellium J

Article

Cite This: Langmuir 2017, 33, 11570-11573 pubs.acs.org/Langmuir

Density Functional Theory for Electron Gas and for Jellium J. W. Dufty

Department of Physics, University of Florida, Gainesville, Florida 32611, United States

ABSTRACT: Density functional theory relies on universal functionals characteristic of a given system. Those functionals in general are diﬀerent for electron gas and for jellium (electron gas with a uniform background). However, jellium is frequently used to construct approximate functionals for an electron gas (e.g., local density approximation and gradient expansions). The precise relationship of the exact functionals for the two systems is addressed here. In particular, it is shown that the exchange- correlation functionals for the inhomogeneous electron gas and inhomogeneous jellium are the same. This justiﬁes theoretical and quantum Monte Carlo simulation studies of jellium to guide the construction of functionals for an electron gas. Related issues of the thermodynamic limit are noted as well.

I. INTRODUCTION Their relationship was established by Mermin, who extended the original ground-state theorems of Hohenberg and Kohn to Keith Gubbins has been an inspiration to students and fi 8,9 colleagues for more than 50 years, through direct mentoring nite temperatures for quantum systems described by the Gibbs ensemble of statistical mechanics. This early history is and collaborations and indirectly through his extensive fl innovative publications. My own collaboration was very early brie y reviewed in refs 5 and 6. Recently, the two paths have rejoined in an effort to describe conditions of warm, dense in my career as a new faculty member in the Physics 10 Department at the University of Florida. Keith had just finished matter. These are solid density conditions, but temperatures his book with Tim Reed showing the relevance of statistical for electrons range from the zero temperature ground state to mechanics to applications in chemical and other engineering well above the Fermi temperature. They include domains with fields. He continued to demonstrate the importance of atomic and molecular coexistence and with relevant association formulating practical approximations to complex systems and dissociation chemistry. A central ingredient is the through their firm foundation in basic statistical mechanics. (intrinsic) free energy functional F, consisting of a non- He also demonstrated by example the need to ignore the interacting contribution F0, a Hartree contribution, FH, and a artificial interfaces between applied and basic disciplines, taking remainder called the exchange-correlation free energy func- recent developments in the mathematics and physics literature tional, Fxc. The exact functional form for the Hartree to formulate innovative and accurate descriptions of complex contribution is known, and the contribution from F0 is treated systems via the reinforcement of theory, simulation, and exactly (numerically) within the Kohn−Sham formulation of experiment. An example is his early application of nonlocal DFT11 extended to finite temperatures. Hence, the primary classical density functional theory (DFT) for adsorption in challenge for applications is the construction of the exchange- carbon slit pores, verified by simulation and including a correlation density functional. discussion of problems of extracting experimental parameters An important constraint is its equivalence with the needed for the calculation.1 The presentation here has in corresponding functional for jellium (electrons in a uniform common the tool of density functional theory for application to neutralizing background12) when evaluated at a uniform a complex system (warm dense matter). But the system is quite density. Although still a difficult quantity to determine, the differentconfined electronsand hence quantum effects can latter has been studied widely by approximate theoretical dominate except at the highest temperatures. The emphasis is methods12,13 andmorerecentlybyquantumsimulation on formal relationships to guide practical applications, so it is methods across the temperature−density plane.14 An accurate hoped that Keith will appreciate some of his spirit in the fitting function for practical applications now exists.15 Its utility following. for DFT is within the “local density approximation”, first The historical development of DFT has followed two quite different paths, one for quantum systems at zero temperature − Special Issue: Tribute to Keith Gubbins, Pioneer in the Theory of (e.g., electrons in atoms, molecules, and solids)2 5 and one for Liquids temperature-dependent classical nonuniform fluids (e.g., two- phase liquids and porous media).6,7 The former has focused on Received: May 31, 2017 the determination of the ground-state energy whereas the latter Revised: July 17, 2017 has focused on the classical free energy of thermodynamics. Published: July 24, 2017

N 2 proposed in ref 11 for the zero temperature energy functional p̂ 1 (()nnnnn̂ rr− )(()̂ ′− ) − ()̂ rrrδ (−′ ) Ĥ = ∑ α +′e2 ddrr bb and extended to finite temperatures for the free energy j ∫ |−′| α=1 2m 2 rr functional. It assumes that the nonuniform system Fxc can be (3) represented at each point by the uniform jellium Fxc evaluated at the density for that point. The constant nb denotes the density of a uniform neutralizing The objective here is to clarify the precise relationship of the background for the electrons. When the grand canonical DFT functionals for the two different systems, electron gas and ensemble is considered, it is given by nb = N/V, where N is the jellium, for general nonuniform densities. The primary new average particle number. The two Hamiltonians are seen to be result is that the exchange-correlation functionals for the two related by systems are equivalent and the total free energies (intrinsic plus Ĥ = HvnÊ + d()()rrr̂ + external potential) are the same. In the analysis, it noted that je∫ b b (4) jellium is usually considered in the thermodynamic limit, N → with ∞, V → ∞, N /V = constant, where N and V are the average particle number and volume, respectively. It has been proven 2 nb 16 vb()rr=−e ∫ d ′ that the jellium free energy is well defined in this limit, |−′|rr (5) whereas that for the electron gas is not (it does not scale as the volume for large system size due in part to a lack of charge The second term on the right side of eq 4 is the potential of interaction between the electrons and the background, and the neutrality). Nevertheless, the exchange-correlation component third term is the background self-energy of the free energy for the electrons does have a proper thermodynamic limit as a consequence of the equivalence 2 1 ()neb 1 demonstrated here. In the next section, Hamiltonians for the Eb =′∫∫ddrr =− d()rrnvbb 2 |−′|rr 2 (6) isolated electron gas and jellium are defined. Next, the statistical mechanical basis for DFT is described for the grand canonical Equation 3 is the usual definition of jellium as the electron ensemble. The grand potential (proportional to the pressure) is system plus a uniform neutralizing background. defined as a functional of a given external potential v (r) A. Grand Potential Functionals. Now consider the ex r (occurring through a local chemical potential μ(r)=μ − addition of an external single-particle potential vex( ) to the r vex( )), and the corresponding functionals for the inhomoge- electron and jellium Hamiltonians. The equilibrium properties neous electron gas and inhomogeneous jellium are related. for the corresponding inhomogeneous systems are defined by μ r r r Associated with ( ) are the densities ne( ) and nj( ) for the the grand canonical potentials two systems. Their relationship is established. Next, the strict ∞ ̂ (NH )−βμ (e−∫ drr () n ())̂ r concavity of the grand potential functionals assures a one-to- βΩ|=−e(,βμVTre ) ln∑ one relationship of the densities to μ(r) so that a change in N=0 (7) variables is possible. This is accomplished by Legendre ∞ fi ̂ transformations which de ne the free energy density func- (NH )−βμ (j−∫ drr () n ())̂ r βΩ|=−j(,βμVTre ) ln∑ tionals. It is noted that the free energy density functional (8) obtained in this way is precisely that of Mermin’s DFT. The N=0 free energies differ by the potential energy of the background Here, the local chemical potential is defined by charge. It is shown that this cancels the differences between the μ()rr=−μ v () corresponding Hartree free energy contributions, resulting in ex (9) the desired equivalence of the exchange-correlation functionals. These grand potentials are functions of the inverse temperature β and the volume V and functionals of the local chemical μ r Ω β |· Ω β II. INHOMOGENEOUS ELECTRON AND JELLIUM potential ( ). The functionals themselves, e( , V ) and j( , |· ̂ ̂ FUNCTIONALS V ), are characterized by He and Hj, respectively (the notation Ω(β, V|·) is used to distinguish the functional from its value The Hamiltonian for an isolated system of N electrons in a Ω(β, V|μ) when evaluated at μ). Because the latter two are volume V is given by different, the functionals are different. However, from eq 4 they have the simple relationship N p2̂ ̂ α 1 2 nn()̂ rr (̂ ′− ) n ()̂ rrrδ (−′ ) He = ∑ +′e ddrr βΩ|=+Ω|−(,βμVE ) β ββμ (, Vv ) ∫ |−′| jbeb(10) α=1 2m 2 rr (1) In fact, all average properties in the corresponding grand ensembles have a similar relationship. For example, a property where m and e are the electron mass and charge, respectively, represented by the operator X̂has the averages and δ(r) is the Dirac delta function. The number density ∞ operator for position r is ̂ (NHn )ββΩ−ee (−∫ drr μ () ())̂ r ̂ Xe(,βμVTree|≡ ) ∑ X N N=0 (11)

n()̂ rrq=−∑ δ (̂α ) ∞ ̂ α=1 (2) (NHn )ββΩ−j (j−∫ drr μ () ())̂ r ̂ Xj(,βμVTree|≡ ) ∑ X N=0 (12) The position and momentum operators for electron α are q̂α and p̂α, respectively. A caret over a symbol denotes the operator so with eqs 4 and 10 they are related by corresponding to that variable. The related jellium Hamiltonian X (,βμVXVv|= ) (, βμ |− ) is jeb (13)

11571 DOI: 10.1021/acs.langmuir.7b01811 Langmuir 2017, 33, 11570−11573 Langmuir Article

B. Free Energy Density Functionals. The local number 1 2 (()nnnnrr−′−bb )(() ) FVnejH(,β |= ) ∫ ddrr ′ densities are 2 |−′|rr (23) δβΩ|(, βV μ ) δβΩ|j(, βV μ ) nV(,r βμ ,|=− ) e ,(,,nVr βμ|=− ) e δβμ()r j δβμ()r In particular, their system size dependence is quite different. (14) For example, in the uniform density limit (assuming a spherical β | → 2 5/3 π 2 Ω β |μ volume) FeH( , V n) C(nee) V (with C =(4 ) (1/15) (3/ The derivative of j( , V ) is taken at constant nb. Using eq 10 π 5/3 ≃ β | (4 )) 0.967), whereas FjH( , V n) vanishes in this limit. or 13, it is seen that the number densities are related by This difference is the reason (along with charge neutrality) that F (β, V|n) has a proper thermodynamic limit16 while F (β, V|n) nj(,rrβμ ,Vn|= )eb (, βμ , Vv |− ) (15) j e does not (because it does not scale linearly with the volume). Ω β |· Ω β |· fi It can be shown that e( , V ) and j( , V ) (again with At rst sight, this seems at odds with eq 20 because the left constant nb) are strictly concave functionals so that eq 14 side has a thermodynamic limit whereas each term on the right fi ⇔ μ ⇔ de nes the one-to-one invertible relationships ne and nj separately does not. However, the second and third terms μ β μ → β . Consequently, a change in variables , V, , V, ne for the cancel the singular volume dependence of FeH so that β μ → β inhomogeneous electron system and , V, , V, nj for inhomogeneous jellium is possible. The corresponding FVnvneH(βββ ,|+ )∫ drrr b () () + EFVn b = jH ( , | ) fi Legendre transformations then de ne the free energy func- (24) tionals of these densities This observation, together with the equavalence of the FVn(,ββμμβμ|=Ω|+ ) (, V ) drr () n (, r , V | ) noninteracting contributions, leads to the equivalence of the eee∫ e exchange-correlation contributions for the electron gas and (16) jellium FVnFVn(,ββ|= ) (, | ) FVn(,ββμμβμ|=Ω|+ ) (, V ) drr ()(, n r , V | ) jxc exc (25) jjj∫ j (17) Note that this equivalence applies for general inhomogeneous These are precisely the density functionals of DFT (e.g., densities, extending the familiar relationship for uniform defined by Mermin8). systems. Their relationship follows using eqs 4 and 15 These are two different inhomogeneous systems, yet their correlations are the same for every admissable density. The βββμ|= +Ω |− FVnEjjbe(, ) (, V v b ) functionals are universal in the sense that their forms are +−∫ drr (μβμ ()vnbe ())(, r r , V |− v b ) independent of the underlying external potential. However, the potential associated with the chosen density of their argument +|−∫ d()(,,rrrvnbeβμ V v b ) is different in each case. To see this, consider the functional derivative of eq 20 with respect to the density =||−+FVnveeb(,βμ ( ))∫ drr vn be () (, r βμβ , V |−+ v bb ) E (18) δβ | FVnj(, ) δβFVne(,| ) ==μ()r + vb()r and using eq 15 again gives the desired result. δn()r δn()r (26) Hence, the density n is determined from the jellium functional FVnFVnjejbjb(,ββ|=j ) (, |+ )∫ drr v ()( n r |+ μβ ) E by (19) δβFVn(,| ) Because μ(r) is arbitrary, so also is n (r|μ) and eq 19 can be j j ==−μμ()rrvex () written more simply as δn()r (27) Alternatively, the same density is determined from the electron FVnFVn(,ββ|= ) (, |+ ) drrr v ()() n + β E je∫ b b (20) functional by δβFVn(,| ) e =−μμ()rrvvv () =−+ ( () rr ()) III. EXCHANGE-CORRELATION FUNCTIONAL δn()r bexb EQUIVALENCE (28) Traditionally, the free energy density functional is separated r For jellium, the potential is vex( ), and for the electron gas, it is into a noninteraction contribution, F , a mean-field Hartree r r 0 vex( )+vb( ). The two solutions to eqs 27 and 28 have the contribution, FH, and the remaining exchange-correlation same relationship as expressed in eq 15. contribution, Fxc IV. DISCUSSION FF= 0Hx++ F Fc (21) Practical approximations for the electron gas exchange- Clearly, F0 is the same for the electron and jellium systems, as correlation functional are typically introduced at the level of r follows from eq 20 because vb( ) = 0 in this case. However, the its density, defined by Hartree terms (defined as the average intrinsic internal energy with the pair correlation function equal to unity) are different FVnf(,ββ|= ) drr (, , Vn | ) exc ∫ exc (29) 1 2 nn()rr (′ ) A formal functional expansion about the density at point r can FVneeH(,β |= ) ∫ ddrr ′ 2 |−′|rr (22) be performed

11572 DOI: 10.1021/acs.langmuir.7b01811 Langmuir 2017, 33, 11570−11573 Langmuir Article

fVnfVn(,rrββ ,|= ) (, , || ) (7) Lutsko, J. Recent Developments in Classical DensityFunctional exc exc n()r 2010 − δβfVn(,r ,| ) Theory. Adv. Chem. Phys. , 144,1 93. +′∫ dr exc ((nnrr′− ) ()) (8) Mermin, D. Thermal Properties of the InhomogeneousElectron δn()r′ n()r Gas. Phys. Rev. 1965, 137, 1441−1443. (9) Eschrig, H. T > 0 Ensemble-state Density Functional Theory via 2010 δβfVn(,r ,| ) Legendre Transform. Phys. Rev. B: Condens. Matter Mater. Phys. , +′″∫ ddrrexc (nnnn (rrrr′− ) ( ))( ( ″− ) ( )) + ... 82, 205120. δδnn()rr′″ ( ) n()r (10) Frontiers and Challenges in Warm Dense Matter; Graziani, F. (30) et al., Eds.; Spinger Verlag: Heidelberg, 2014. (11) Kohn, W.; Sham, L. J. Self-Consistent Equations IncludingEx- The coefficients are evaluated at the uniform density n(r); i.e., change and Correlation Effects. Phys. Rev. 1965, 140, 1133−1138. all functional density dependence is evaluated at the same value. (12) Giuliani, G.; Vignale, G.; Quantum Theory of the Electron Liquid; Consequently, the lead term is just the uniform electron gas Cambridge University Press: New York, 2005. exchange-correlation free energy per unit volume (13) Sjostrom, T.; Dufty, J. Uniform electron gas atfinite temperatures. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 115123. 1 and references therein. f (,r ββ ,Vn|| ) =FVn(, , ) |= exc nnn()rrV exc ee () (31) (14) Dornheim, T.; Groth, S.; Sjostrom, T.; Malone, F.; Foulkes, W.; Bonitz, M. Ab initio Quantum Monte Carlo simulation of the This is known as the local density approximation. Similarly, warmdense electron gas in the thermodynamic limit. Phys. Rev. Lett. subsequent terms in eq 30 are the response functions for the 2016, 117, 156403. and references therein. uniform electron gas. From the above analysis, all of these can (15) Karasiev, V.; Sjostrom, T.; Dufty, J.; Trickey, S. B. Accurate now be identifed with those for jellium, which has been studied Homogeneous Electron Gas Exchange-Correlation Free Energy for extensively. As noted in the Introduction, an accurate analytic LocalSpin-Density Calculations. Phys. Rev. Lett. 2014, 112, 076403. fit for F (β, V, n ) is now available across the entire β, n (16) Lieb, E.; Narnhofer, H. The thermodynamic limit for jellium. J. jxc e e 1975 − plane,15 so the local density approximation is known explicitly. Stat. Phys. , 12, 291 310. Similarly, the first few response functions for jellium are known as well.12 More generally, approximations for exchange correlations away from the uniform limit (e.g., generalized gradient approximations) can be addressed for inhomogeneous jellium as well. Although this is a difficult problem, it is placed in a more controlled thermodynamic context due to charge neutrality and extensivity. ■ AUTHOR INFORMATION ORCID J. W. Dufty: 0000-0002-2145-3733 Notes The author declares no competing financial interest. ■ ACKNOWLEDGMENTS This research was supported by U.S. DOE grant DE- SC0002139. ■ REFERENCES (1) Lastoskie, C.; Gubbins, K.; Quirke, N. Pore sizeheterogeneity and the carbon slit pore: a density functional theory model. Langmuir 1993, 9, 2693−2702. (2) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, 864−871. (3) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford: New York, 1989. (4) Dreizler, R. M.; Gross, E. K. U. DensityFunctional Theory: An Approach to the Quantum Many-Body Problem; Springer Verlag: Berlin, 1990. (5) Jones, R. O. Density functional theory: Itsorigins, rise to prominence, and future. Rev. Mod. Phys. 2015, 87, 897−923. (6) Evans, R. Density Functionals in the Theory of Non-Uniform Fluids. In Fundamentals of Inhomogeneous Fluids, Henderson, D., Ed.; Marcell Dekker: New York, 1992; pp 85−176. The nature of theliquid-vapour interface and other topics in the statistical mechanics ofnon-uniform, classical fluids. Adv. Phys. 1979, 28, 143−200. Density Functional Theory for Inhomogenous Fluids: Statics, Dynamics, and Applications. In 3rd Warsaw School of Statistical Physics; Cichocki, B., Napiowski,́ M., Piasecki, J., Eds.; Warsaw University Press: Warsaw, 2010; pp 43−85.

11573 DOI: 10.1021/acs.langmuir.7b01811 Langmuir 2017, 33, 11570−11573