A liquid-state theory for electron correlation functions and thermodynamics Shuangliang Zhao a,1 , Pingyun Feng b , Jianzhong Wu a,c,⇑ a Department of Chemical and Environmental Engineering, University of California, Riverside, CA 92521, USA b Department of Chemistry, University of California, Riverside, CA 92521, USA c Department of Mathematics, University of California, Riverside, CA 92521, USA article info Article history: Received 21 September 2012 In final form 7 November 2012 Available online 17 November 2012 abstract Correlation functions play an important role in determining the structural and energetic properties of electronic systems. However, exact results are available only for ideal systems free of electrostatic inter- actions. Here we present a liquid-state method that combines classical mapping of the Pauli exclusion principle with the universality ansatz of the bridge functional. We show excellent agreement of the new method with quantum Monte Carlo simulation for predicting the electronic correlation functions and thermodynamic properties of uniform electron systems. Ó 2012 Elsevier B.V. All rights reserved. Thermodynamic systems of non-interacting particles are dis- cussed in introductory texts of statistical mechanics [1,2]. For the ideal systems, the partition functions can be derived analyt- ically with either quantum or classical Hamiltonians and from which exact results are readily obtained for both the microscopic structure and the related thermodynamic properties. These ana- lytical expressions provide a starting point to study the proper- ties of multi-body systems with inter-particle energetics. Whereas exact results are no more attainable for most non-ideal thermodynamic systems due to multi-body interactions, myriad approximate methods have been developed in different contexts of quantum and statistical mechanics applications [3,4]. Common examples include random phase approximation, square-gradient approximation, self-consistent-field theories, integral-equation theories, cluster expansion methods, and various forms of density functional theory. Although similar mathematical procedures are routinely used for quantum and classical systems, their underly- ing connections are rarely explored. One exception is the suc- cessful application of the field-theory techniques, originally established for quantum many-body systems, to polymer physics [5]. Here we report a procedure in the opposite direction, i.e., uti- lization of a liquid-state method established for classical systems to account for multi-body correlation functions of a quantum- many-particles system. Specifically, we exploit the universality hypothesis of the bridge functional established in the classical li- quid-state theory to formulate excess Helmholtz energy of elec- tronic systems. We predict the pair correlation functions and energetics of uniform electronic systems in good agreement with simulation results. These correlation functions are difficult to cal- culate from conventional quantum-mechanical methods includ- ing the electronic density functional theory [6]. While here we demonstrate the numerical performance only for uniform elec- trons, the procedure is generally applicable to inhomogeneous systems. The concept of bridge functional was introduced about 20 years ago by Rosenfeld who first recognized that, for a wide variety of classical fluids, the free-energy functional due to the inter-particle interactions can be accurately described by a second-order density expansion (i.e., hypernetted chain approximation or HNC) plus a correction for high-order terms due to the short-range repulsion or the excluded-volume effects [7]. This high-order correction is referred to as the bridge functional, which is related but not iden- tical to the bridge function in the diagrammatic representation of the pair distribution function of a uniform fluid [8]. The bridge functional is a part of the Helmholtz energy arising from multi- body correlations while the bridge function is defined in terms of two-body correlations. Rosenfeld demonstrated that the second- order expansion conforms to the exact results for the long-range components of the inter-particle potential at both the ‘ideal gas’ and ‘ideal liquid’ limits, and that the bridge functional can be quan- titatively reproduced by that of a hard-sphere reference system [9]. The insensitivity of the bridge functional to the precise form of the inter-particle potential consists of a universality ansatz, which al- lows formulation of a generic expression for the Helmholtz energy due to inter-particle interactions. For a system of classical particles with pairwise additive poten- tial u ij (r), the excess Helmholtz energy at a given particle density profile q i (r) can be expressed in terms of that corresponds to a sys- tem of uniform density q 0 i , the first-order and the second-order expansions with respect to the local density deviations, and the bridge functional F B [q i (r)], viz.: 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.11.022 ⇑ Corresponding author. Tel.: +1 9518272413. E-mail address: jwu@engr.ucr.edu (J. Wu). 1 Current address: Department of Chemical Engineering, East China University of Science and Technology, Shanghai 200 237, China Chemical Physics Letters 556 (2013) 336–340 Contents lists available at SciVerse ScienceDirect Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett