I wonder if all DFT codes are based on the Kohn-Sham formalism?.
If other methods are available, what are their scopes?
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At least I was able to find three other forms of DFT listed as:
Orbital-Free Density Functional Theory (OFDFT) is, as the name suggests, an attempt to work with DFT without using the Kohn-Sham (KSDFT) approach of expressing the density as a sum of non-interacting orbital densities.
In the KS formulation, the kinetic energy functional explicitly depends on the choice of orbitals, whereas in principle the true energy functional should only depend on the overall density. The trade-off is that the KS kinetic energy functional has a well defined simple form, with all the many body effects describing the system being compressed into the exchange-correlation functional. For OFDFT, both the kinetic energy and exchange correlation functionals are undetermined and must be approximated in some way.
So why would one want to use OFDFT if its less easily defined and presumably less accurate? The main benefit is in the reduced scaling; while KSDFT generally scales cubically with the number of orbitals due to the need to perform a diagonalization, OFDFT scales linearly with system size. This obviously offers a huge potential benefit for studying large systems.