Machine learning based nonlocal kinetic energy density functional for simple metals and alloys - Proposal and validation

Aug 5, 2026

Developing an accurate kinetic energy density functional (KEDF) is a key challenge in orbital-free density functional theory (OFDFT). This work proposes a machine learning based physical-constrained nonlocal (MPN) KEDF and implements it using bulk-derived local pseudopotentials and plane-wave basis sets in the ABACUS package. The model enforces three exact physical constraints: electron kinetic energy scaling, the free-electron-gas limit, and non-negativity of the Pauli energy density, and is tested on Li, Mg, Al and 59 alloys.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages16

Reading Time40 min

AccessFree to download

Introduction

Challenges in kinetic energy density functionals within OFDFT

Local/semilocal vs nonlocal KEDFs

Prior machine-learning approaches to KEDFs

Methodology: MPN KEDF Design

Physical constraints and enhancement-factor construction

Implementation in ABACUS with bulk-derived pseudopotentials

Benchmarking and Results

Test set: simple metals and alloys

Accuracy, transferability, and stability considerations

arXiv :2310 . 15591v2 [ cond-mat .mtrl-sci ] 3 Mar 2024 Machine learning based nonlocal kinetic energy density functional for simple metals and alloys Liang Sun 1 and Mohan Chen 1, 2, ∗ 1 HEDPS, CAPT, School of Physics and College of Engineering, Peking University, Beijing 100871, P. R. China 2 AI for Science Institute, Beijing 100080, P. R. China (Dated: March 5, 2024) Developing an accurate kinetic energy density functional (KEDF) remains a major hurdle in orbital-free density functional theory. We propose a machine learning based physical-constrained nonlocal (MPN) KEDF and implement it with the usage of the bulk-derived local pseudopotentialsand plane wave basis sets in the ABACUS package. The MPN KEDF is designed to satisfy three exact physical constraints: the scaling law of electron kinetic energy, the free electron gas limit, and the non-negativity of Pauli energy density. The MPN KEDF is systematically tested for simple metals, including Li, Mg, Al, and 59 alloys. We conclude that incorporating nonlocal information for designing new KEDFs and obeying exact physical constraints are essential to improve the accuracy, transferability, and stability of ML-based KEDF. These results shed new light on the construction of ML-based functionals. PACS numbers: 71.15.Mb, 07.05.Mh, 71.20.Gj I. INTRODUCTION Kohn-Sham density functional theory (KSDFT) is a widely-used ab initio method in materials science. [1, 2] However, its computational complexity of O(N3 ), where N is the number of atoms, poses significant challenges for large systems. Alternatively, orbital-free density functional theory (OFDFT) [3, 4] calculates the noninteracting electron kinetic energy Ts directly from the charge density instead of relying on the one-electron Kohn-Sham orbitals. As a result, OFDFT achieves amore affordable computational complexity of typically O (N lnN) or O(N) . [5–8] Given that Ts is comparable in magnitude to the total energy, the accuracy of OFDFT largely depends on the approximated form of the kinetic energy density functional (KEDF) . However, developing an accurate KEDF has been a major hurdle in the field of OFDFT for decades years. Over the past few decades, continuous efforts have been devoted to developing analytical KEDFs. [4, 9] In general, KEDFs can be classified into two categories. The first category comprises local and semilocal components in KEDFs, where the kinetic energy density is a function of the charge density, the charge density gradient, the Laplacian of charge density, or even higher-order derivatives of the charge density. [10–15] The second category consists of nonlocal forms of KEDFs, where the kinetic energy density is a functional of charge density, such that the kinetic energy density at each point in real space depends on the nonlocal charge density. [16–20] Typically, semilocal KEDFs are more computationally efficient, while nonlocal ones offer a higher accuracy. However, since most of the existing nonlocal KEDFs are ∗ [mohanchen@pku.edu.cn](mohanchen@pku.edu.cn) constructed based on the Lindhard response function, which is accurate for nearly free electron gas, they are mainly adequate for simple metals. [16, 17] Some KEDFs were proposed to describe semiconductor systems, but they cannot work well for simple metals. [18–20] As a result, a KEDF that works for both simple metal and semiconductor systems is still lacking, and it is still unclear how to construct it systematically. In recent years, machine learning (ML) techniques have been involved in the developments of computational physics. [21] In particular, the remarkable fitting ability of ML models has been demonstrated in various applications, including the fitting of potential energy surfaces in molecular dynamics [22, 23], as well as fitting exchangecorrelation functionals [24–27] and Hamiltonian matrices [28] within the framework of density functional theory (DFT) . [1, 2] Additionally, there have been endeavors to co

FAQs

What problem does this document address in OFDFT?

It targets the difficulty of developing an accurate kinetic energy density functional (KEDF) for orbital-free density functional theory.

How does the proposed MPN KEDF ensure physical correctness?

It is designed to satisfy three exact constraints: the scaling law for electron kinetic energy, the free electron gas limit, and non-negativity of the Pauli energy density.

Which materials and alloy systems are used to evaluate the method?

The document tests the MPN KEDF on simple metals including Li, Mg, and Al, and on 59 alloys.

Aug 5, 2026

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