# Derivation of the Weizs\"acker Density Functional from Probability Theory

## TL;DR

This paper proves that the Weizsäcker potential is an exact component of the universal density functional in ground-state density functional theory, using probability theory and kinetic energy analysis without approximations.

## Contribution

It provides a rigorous, approximation-free derivation of the Weizsäcker potential as an exact term in DFT's universal functional.

## Key findings

- Weizsäcker potential is exact in the universal functional.
- The proof relies solely on probability theory and kinetic energy form.
- Other kinetic energy terms are also examined.

## Abstract

We demonstrate that the Weizs\"acker potential is an exact term in the universal functional in density functional theory (DFT) for the ground state of a system with $N$ electrons. This proof uses no approximations or physical arguments, and follows from the form of kinetic energy of the ground state and probability theory. We also examine the form of the other terms in the kinetic energy.