# Kinetic energy functionals and the $N$-representability of the electron pair-density given by the classical map hyper-netted-chain (CHNC) method

## TL;DR

This paper demonstrates that the classical map hypernetted-chain (CHNC) method produces N-representable two-body densities for electrons, validated against quantum Monte Carlo results, and extends to classical calculations in quantum regimes.

## Contribution

The paper establishes the N-representability of CHNC-generated densities and applies the method to classical electron-ion systems in quantum regimes, validated by QMC, PIMC, and DFT comparisons.

## Key findings

- CHNC densities are N-representable and match QMC results.
- CHNC effectively models electron-ion systems at high temperature and density.
- CHNC is computationally efficient for quantum regimes where traditional methods are costly.

## Abstract

The classical map hypernetted-chain (CHNC) method for interacting electrons uses a kinetic energy functional in the form of a classical-fluid temperature. Here we show that the CHNC generated two-body densities and pair-distribution functions (PDFs) correspond to $N$-representable densities. Comparisons of results from CHNC with quantum Monte Carlo (QMC) and Path Integral Monte Carlo (PIMC) are used to validate the CHNC results. Since the PDFs are sufficient to obtain the equation of state or linear-response properties of electron-ion systems, we apply the CHNC method for fully classical calculations of electron-ion systems in the quantum regime, using hydrogen at 4000K and 350 times the solid density as an example since QMC comparisons are available. We also present neutral pseudo-atom (NPA) calculations which use rigorous density-functional theory (DFT) to reduces the many nuclear problem to an effective one-ion problem. The CHNC PDFs and NPA results agree well with the ion-ion, electron-ion and electron-electron PDFs from QMC, PIMC, or DFT coupled to molecular dynamics simulations where available. The PDFs of a 2D electron-hole system at 5K are given as an example of 2D `warm dense' matter where the electrons and the counter particles (holes) are all in the quantum regime. Basic methods like QMC, PIMC or even DFT become prohibitive while CHNC methods, being independent of the number of particles or the temperature, prove to be easily deployable.