Nonlocal Functionals Inspired by the Strongly Interacting Limit of DFT: Exact Constraints and Implementation

Capturing strong correlation effects remains a key challenge for the development of improved exchange–correlation (XC) functionals in density functional theory. The recently proposed multiple radii functional (MRF) [J. Phys. Chem. Lett. 2017, 8, 2799; J. Chem. Theory Comput. 2019, 15, 3580] was designed to capture strong correlation effects seamlessly, as its mathematical structure draws from that of the exact XC functional in the limit of infinite correlations. The MRF functional provides a framework for building approximations along the density-fixed adiabatic connection, delivers accurate XC energy densities in the standard DFT gauge (same as that of the exact exchange energy density), and is free of one-electron self-interaction errors. To facilitate the development of XC functionals based on the MRF, we examine the behavior of the MRF functional when applied to uniform and scaled densities and consider how it can be made exact for the uniform electron gas. These theoretical insights are then used to build improved forms for the fluctuation function, an object that defines XC energy densities within the MRF framework. We also show how the MRF fluctuation function for physical correlation can be easily readjusted to accurately capture the XC functional in the limit of infinite correlations, demonstrating the versatility of MRF for building approximations for different correlation regimes. We describe the implementation of MRF using densities expanded on Gaussian basis sets, which improves the efficiency of previous grid-based MRF implementations.

在密度泛函理论（density functional theory, DFT）中，精准捕捉强关联效应仍是开发更优异的交换-关联（exchange–correlation, XC）泛函的核心挑战。近期提出的多半径泛函（multiple radii functional, MRF）[J. Phys. Chem. Lett. 2017, 8, 2799; J. Chem. Theory Comput. 2019, 15, 3580] 旨在无缝捕捉强关联效应，因其数学结构源自无限关联极限下精确XC泛函的形式。MRF泛函提供了沿固定密度绝热连接构建近似的理论框架，可在标准DFT规范（与精确交换能密度的规范一致）下生成精准的XC能密度，且无单电子自相互作用误差。为推动基于MRF的XC泛函开发工作，我们研究了MRF泛函在均匀密度与缩放密度下的表现，并探讨了如何使其对均匀电子气严格精确。随后，我们将这些理论洞察用于改进涨落泛函（fluctuation function）的形式——该对象是MRF框架内定义XC能密度的核心载体。我们还展示了如何轻松调整物理关联下的MRF涨落泛函，以在无限关联极限下精准匹配XC泛函，验证了MRF在构建不同关联区域近似方案时的通用性。我们还介绍了基于高斯基组（Gaussian basis sets）展开密度的MRF实现方法，该方法相较于此前基于网格的MRF实现方案，有效提升了计算效率。