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Classical Model for Jellium

A simple, practical model for computing the equilibrium thermodynamics and structure of jellium by classical strong coupling methods is proposed. An effective pair potential and coupling constant are introduced, incorporating the ideal gas, low density, and weak coupling quantum limits. The resulting parameter free, analytic model is illustrated by the calculation of the pair correlation function over a wide range of temperatures and densities via strong coupling classical liquid state theory. The results compare favorably with the first finite temperature restricted path integral Monte Carlo simulations reported recently.

The objective here is to provide an effective classical system representing the thermodynamics of the real quantum jellium, using an effective pair potential and an effective coupling constant. Simple analytic expressions are given, based on the more complete but complex results of reference [9]. Application of this model is illustrated using the hypernetted chain (HNC) integral equation of classical liquid state theory to calculate the pair correlation function. Comparison of these calculations with the first finite temperature restricted path integral Monte Carlo (PIMC) simulation results reported recently [10] show good agreement over a wide range of densities and temperatures.
Jellium has a broader current interest than its historical role as a test bed for quantum many body theories. Its thermodynamic properties, particularly the density dependence of the free energy, provide the basis for local density approximations (LDA) in electron density functional theory (DFT) [11]. Fits to zero temperature PIMC simulations have been the basis for virtually all LDA DFT for the past thirty years. Until now there has been no corresponding basis for an LDA at finite temperatures from either theory or simulation. Such conditions of solid densities at temperatures comparable to the Fermi temperature are of central interest to the new studies of "warm, dense, matter" [12].
The system of interest is a collection of N charges with Coulomb pair interactions φ(r) in a uniform neutralizing background, at equilibrium [13]. The thermodynamic variables are the temperature and density, T ≡ 1/k B β and n. A corresponding effective classical system is considered with pair interactions φ c (r) in a uniform neutralizing background, at equilibrium with temperature and density T c and n c . The correspondence of the classical and quantum systems is established by defining φ c (r), T c , n c as functions or functionals of φ(r), T, n in such a way as to assure the equivalence of selected equilibrium properties. Three such conditions are chosen [8]. The first two are equivalence of the densities and pair correlation functions The remaining condition fixing T c is replaced by a corresponding condition for an effective coupling constant, as discussed below.
To be useful, the condition equating pair correlation functions must be invertible, φ c (r) = g −1 c (r, β c , n c | g), which entails solution to the classical many-body problem. (this inversion does not need to be unique; see final comments below). In the special case of the ideal gas limit the result is known as the Pauli potential, arXiv:1302.4507v1 [cond-mat.stat-mech] 19 Feb 2013 denoted φ (0) c (r). Even in this case the inversion cannot be accomplished exactly, but good approximations are known [5,9]. The relevant dimensionless thermodynamic parameters for the quantum system are the temperature relative to the Fermi temperature t = 1/β F and r s = r 0 /a B , the mean distance between particles r 0 (defined by 4πnr 3 0 /3 = 1) relative to the Bohr radius a B . Hence the dimensionless effective potential where x ≡ r/r 0 . It has been recognized that the Pauli potential depends only on t. Two exact limits for ∆ * (x, t, r s ) are important for the discussion here. The first is the weak coupling limit where c c (x, t, r s ) is the direct correlation function. It is related to the pair correlation function g c (x, t, r s ) by the exact Ornstein-Zernicke equation [2]. Using the correspondence conditions (1) the Ornstein-Zernicke equation defines the direct correlation function in terms of the quantum pair correlation function A sufficient condition for weak coupling is large x, for which the behavior of c c (x, t, r s ) is determined from the perfect screening sum rule for g(x, t, r s ) [18], giving This is the same form as for the classical one component plasma, except with the classical Coulomb coupling constant Γ = βq 2 /r 0 = 4 2/3π 2 1/3 r s /3t replaced by the effective coupling constant Here At low temperatures and fixed density Γ becomes divergent whereas the effective coupling constant remains finite Γ c (0, r s ) 1. 155 r The second exact limit is that for low density and weak coupling. The condition of low density means that g(x, , t, r s ) is determined by the two electron Slater sum. The weak coupling ∆ * (x, t, r s ) in that case is known as the Kelbg potential [3,4,14] where a = (r s /Γ) 1/2 . This weak coupling result at low density can be improved by imposing the exact behavior of the two particle Slater sum at x = 0, to include some strong coupling effects [3,4,15,16]. The modified form is with γ (Γr s ) = − (πΓr s ) where S(Γr s ) is the two electron relative coordinate Slater sum at x = 0 S(Γr s ) = −4 (πΓr s ) The proposal here is to further extend this Kelbg form to apply broadly across a wide range of values t, r s by imposing the exact asymptotic limit (5). This is accomplished by replacing Γ with the effective coupling constant Γ c given by (6). The approximate effective pair potential is thus Since ∆ * K (x, Γ c , r s ) is an analytic, parameter free form this potential is suitable for practical applications in classical many-body theory, classical Monte Carlo calculations, and molecular dynamics simulations.
To illustrate the utility of this model potential the pair correlation function g(x, t, r s ) for jellium is calculated here using the classical liquid state HNC integral equation [2]. The first step is a determination of φ (0) * c (x, t) for the ideal Fermi gas. Since the pair correlation function g (0) (x, t, r s ) is known exactly, the HNC equations can be inverted to determine φ (0) * c (x, t). These equations are solved numerically using the method of Ng [19]. Next, with φ (0) * c (x, t) known the pair correlation function for jellium can be determined from the HNC equations using (11).
Very recently restricted path integral Monte Carlo (PIMC) simulations have been reported for the pair correlation function spanning conditions ranging from extreme quantum to semi-classical. These results provide important benchmarks for existing quantum many-body methods, as well as the approach proposed here. Consider first the relatively high temperature t = 8. Figure  (1a) shows good agreement with PIMC for all densities, 1 ≤ r s ≤ 40. Figure (1b) shows the same comparison for t = 1. Again the agreement is good, except at the extreme condition r s = 40. In this case a strong correlation peak has formed that is badly under estimated by the theory, although its location is adequately described. Generally, it is found for t ≤ 1 the theory is quite good for 1 ≤ r s ≤ 10. This is illustrated in Figures (2a) and (2b) at t = 0.5 and 0.0625 (the latter is essentially the same as t = 0, as confirmed by a comparison with diffusion Monte Carlo simulations at t = 0 [20]). Some trends are evident even from this limited data. For example, the temperature dependence for t ≤ 0.5 is quite weak for r s > 1. However, for r s = 1 a significant temperature dependence is seen for 0.5 ≤ t ≤ 8. In summary, the model potential (11) provides a practical form for the analysis of jellium using classical methods under conditions that are difficult to access by existing quantum methods (e.g., r s > 1 and t < 10).
The thermodynamic properties of jellium can be calculated from the pair correlation function. For example, the pressure can be obtained from a coupling constant integration. Let p(t, r s , q) be the exact quantum pressure and g (r, t, r s , q) the exact quantum pair correlation function where now the dependence on the charge q has been made explicit. Then Here φ(r) is the Coulomb pair potential. Therefore, approximating g (r, t, r s , y) by the corresponding classical result obtained using the model potential (11) determines the pressure for arbitrary t, r s . A more direct approach would be classical Monte Carlo integration of the Gibbs distribution for the free energy As noted in the introduction, the idea of an effective classical pair potential with an effective classical temperature was already introduced more than ten years ago by Perrot and Dharma-wardana [6]. Instead of the Kelbg potential they chose the Deutsch potential [17], originally introduced as a simpler representation of the Kelbg potential. The PDW effective classical potential is similar to (11), but with ∆ * K (x, Γ c , r s ) replaced by ∆ * P DW (x, Γ P DW , r s ) ≡ Γ P DW x −1 (1 − exp(−bx)) (14) Here, b = (πr s /Γ P DW ) 1/2 and the effective coupling constant is This follows from their phenomenological form for the classical temperature interpolating between the real temperature T and a finite value T 0 at T = 0. The single parameter T 0 /T is determined by fitting the classical correlation energy calculated with this potential to the quantum exchange/correlation energy determined from PIMC at T=0. It is given as an explicit fitting function of r s in reference [6]. Although the dependence of Γ P DW on t, r s is quite different from that derived here, and the shape of the resulting effective pair potential can be quite different, nevertheless the HNC pair correlation function calculated from the PDW potential has a similar accuracy to that reported here. This indicates that an effective pair potential has no inherent physical interpretation, but rather is a non-unique tool for generating physical properties of interest through classical manybody methods. Here that potential has been constructed by imposing three exact constraints: the ideal gas limit, low density limit, and large distance limit. Consequently no fitting parameters are required. The result provides theoretical support for the ideas of reference [6] and provides insight into the relevant physical mechanisms. For example, the exact screening sum rule that determines the form of Γ c here appears to incorporate quantum effects as significant as those of Γ P DW imposed by empirical T = 0 exchange/correlation energy data. This research has been supported by NSF/DOE Partnership in Basic Plasma Science and Engineering award DE-FG02-07ER54946 and by US DOE Grant de-sc0002139.