A direct approach to computing the non-interacting kinetic energy functional

TL;DR

The paper tackles the long-standing challenge of explicitly formulating the non-interacting kinetic energy functional $T_{KS}(\rho)$ for general $N$-representable densities within DFT. It introduces a variational framework that uses an exact trigonometric (angle-field) parametrization of the density to enforce the density constraint without requiring an adjoint equation, enabling a direct computation of an extension $T^+$ of $T_{RKS}$ applicable beyond traditional $v$-representable densities. The authors prove conceptual equivalence with $T_{RKS}$ for non-interacting $v$-representable densities and provide a proof-of-concept in one dimension using KSDFT models, validating the approach against KS results with per-electron errors near 1 kcal/mol. They also develop a practical numerical strategy based on Radial Basis Function approximations and SLSQP optimization, discuss convexity properties of $T^+$, and outline pathways to higher-dimensional extensions and data-driven kinetic energy modeling in OFDFT. This framework offers a rigorous, scalable foundation for improving kinetic energy functionals and generating diverse density-kinetic energy datasets to advance orbital-free DFT and related applications.

Abstract

The non-interacting kinetic energy functional, $T_{KS}(ρ)$, plays a fundamental role in Density Functional Theory (DFT), but its explicit form remains unknown for arbitrary $N$-representable densities. Although it can, in principle, be evaluated by solving a constrained optimization problem, the associated adjoint problem is not always well-posed; moreover, even when it is, the corresponding adjoint operator may be singular. To the best of our knowledge, none of the existing approaches in the literature precisely determines the non-interacting kinetic energy functional for a given $N$-representable electron density, $ρ$. In this work, we present a variational framework for computing an extension of $T_{KS}(ρ)$ using an exact trigonometric reparametrization of the density that eliminates the need for an adjoint equation. We present a proof-of-concept numerical validation of the variational principle for the special case of one-dimensional Kohn-Sham systems. Our method, however, is general and provides a systematic foundation for computing $T_{KS}(ρ)$ in higher dimensions too, paving the way for improved kinetic energy functionals in DFT.

Figures (16)

• Figure 1: Plot of $g(x)$ over $[0,1]$.
• Figure 2: Histograms of non-interacting kinetic energy values, obtained from various initial conditions, for a given electron density, $\rho$, in each plot. Here, $\int_{\mathbb{R}} \rho = 2N = 6$.
• Figure 3: Histograms of non-interacting kinetic energy values, obtained from various initial conditions, for a given electron density, $\rho$, in each plot. Here, $\int_{\mathbb{R}} \rho = 2N = 6$ and $t_0^*$ is initial condition obtained from the orbitals of particle in a box. The value $T_s(t_0^*)$, , representing the non-interacting kinetic energy evaluated using $t_0^*$, indicates that this particular choice of initial condition yields a kinetic energy close to the minimum.
• Figure 4: Histograms showing the lowest ten percent of non-interacting kinetic energy values, obtained from various initial conditions, relative to the non-interacting kinetic energy of the Kohn Sham system for a given electron density, $\rho$, in each plot. Here, $\int_{\mathbb{R}} \rho = 2N = 6$, $t_0^*$ is initial condition obtained from the orbitals of particle in a box. The value $DT_s(t_0^*)$ is difference between non-interacting kinetic energy from the special initial condition and the non-interacting kinetic energy of the Kohn Sham system.
• Figure 5: Error between the Kohn Sham non-interacting kinetic energy and non-interacting kinetic energy obtained from our minimisation method with special initial condition, per electron, for a given electron density, $\rho$, such that $\int_{\mathbb{R}} \rho = 2N = 4$. Here, $n_e =1000$ and $n_{rbf} =15$ is used.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Remark
• Remark
• Theorem 3.2
• proof
• Lemma 3.3
• proof