Why density functional theory favors a uniform electron density?

Why density functional theory (DFT) favors a uniform electron density when we use it to calculate ground state electronic structure self-consistently?

For example, when we use DFT for calculating the (ground state) electron density of two $$\text{H}$$ atoms with a separation somewhat larger then the bond length of an $$\text{H}_2$$ molecule, there will still be a non-negligible electron density between the two atoms as if there is still a covalent bond. As a result, the calculated electronic ground state is essentially an excited state of the system.

I cannot remember where did I read this, and I don't understand why DFT has such a problem. I think the problem resides in the fact that we use a local exchange-correlation functional $$E_{XC}$$ like LDA and GGA. But I don't understand why a local $$E_{XC}$$ will lead to a uniform electron density.

1 Answer

I think I can see the confusion, and I will do my best to explain.

As an initial attempt to use the electron density, the uniform electron gas (UEG) model was developed. This fictitious model assumes: an infinite number of electrons, a constant non-zero density, an infinite volume, and a uniformly distributed positive charge. IMPORTANT: The electron-electron interactions (correlation and exchange) are fundamentally ignored.

Two main theorems comprise modern DFT: Hohenberg-Kohn (HK) and Kohn-Sham (KS). In principle DFT is exact if we knew the universal HK functional (which contains electron-electron exchange-correlation terms); however, no mathematical description has been developed. As we do not know the exact form of the exchange-correlation term, a number of exchange-correlation functional have been developed. These can be derived from experimental thermochemical date or high-accuracy quantum calculations.

As exchange-correlation functionals are approximations, the electron density is not fully localised to a specific area. I hope this helps.