# Nonlocal Kinetic Energy Functionals By Functional Integration

## TL;DR

This paper introduces a new nonlocal kinetic energy functional, MGP, constructed via functional integration, which accurately models electron densities and energies in various materials with low computational cost.

## Contribution

The work presents a novel nonlocal kinetic energy functional derived from the second derivative, satisfying key physical conditions and enabling efficient orbital-free DFT calculations.

## Key findings

- MGP accurately predicts equilibrium properties of silicon and III-V semiconductors.
- MGP is numerically stable and converges within 12 iterations.
- Computational cost of MGP is comparable to existing GGA functionals.

## Abstract

Since the seminal works of Thomas and Fermi, researchers in the Density-Functional Theory (DFT) community are searching for accurate electron density functionals. Arguably, the toughest functional to approximate is the noninteracting Kinetic Energy, $T_s[\rho]$, the subject of this work. The typical paradigm is to first approximate the energy functional, and then take its functional derivative, $\frac{\delta T_s[\rho]}{\delta \rho(r)}$, yielding a potential that can be used in orbital-free DFT, or subsystem DFT simulations. Here, this paradigm is challenged by constructing the potential from the second-functional derivative via functional integration. A new nonlocal functional for $T_s[\rho]$ is prescribed (which we dub MGP) having a density independent kernel. MGP is constructed to satisfy three exact conditions: (1) a nonzero "Kinetic electron" arising from a nonzero exchange hole; (2) the second functional derivative must reduce to the inverse Lindhard function in the limit of homogenous densities; (3) the potential derives from functional integration of the second functional derivative. Pilot calculations show that MGP is capable of reproducing accurate equilibrium volumes, bulk moduli, total energy, and electron densities for metallic (BCC, FCC) and semiconducting (CD) phases of Silicon as well as of III-V semiconductors. MGP functional is found to be numerically stable typically reaching selfconsistency within 12 iteration of a truncated Newton minimization algorithm. MGP's computational cost and memory requirements are low and comparable to the Wang-Teter (WT) nonlocal functional or any GGA functional.