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I am writing my own (orbital-free, for now) 3D density functional theory code from scratch. It currently works correctly for the simple harmonic oscillator potential $V_\text{ext}(\mathbf r) = \frac12\omega^2 |\mathbf r|^2$, yielding a ground state energy of $\frac32\omega$ (atomic units). I am trying to extend this to find the GS energy of a hydrogen atom.

The SHM code uses the von-Weizsäcker KE functional and the external potential term $\int d\mathbf r\ n(\mathbf r)V_\text{ext}(\mathbf r)$. However substituting $V_\text{ext}$ for $-1/r$ (the potential generated by a proton at the origin) yields an energy of around -12 eV, which is too high. Moreover, unlike for the SHM, the obtained GS energy seems to vary depending on the grid size and spacing, suggesting that I am not including the correct functionals in the energy calculation. Further evidence for this is the energy of the explicit 1s orbital density also being slightly too high (~11 eV).

So my question is: am I using the wrong kinetic energy functional? Should I be adding other terms to the energy expression? What is the setup to correctly obtain the correct GS for the H atom?

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As far as I know, the issue of the kinetic energy functional is how people first came to use orbital based functionals in any capacity. Meaning the von Weizsäcker functional, albeit the best orbital free approximation for the kinetic energy there is, is not very accurate.

Orbital free DFT is an active field of research that only moves very slowly. I recommend this review from 2015: Frank Discussion of the Status of Ground-State Orbital-Free DFT

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    $\begingroup$ For what it's worth, there have been a few promising papers using ML methods to improve orbital-free calculations (e.g., using a grid ML or attempting to learn the functional). I don't know of any reviews yet, but Hauser 2020 is one example. $\endgroup$ Aug 16 at 14:01

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