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 differentconfined electronsand 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 © 2017 American Chemical Society 11570 DOI: 10.1021/acs.langmuir.7b01811 Langmuir 2017, 33, 11570−11573 Langmuir Article 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.