vitalys 发表于 2021-12-5 16:12
That's a pretty detailed wonderful explanation, thank you so much.
I assume that all the informat ...

You are correct.

Real space function means the variable corresponds to a point in three-dimensional space, unlike some functions defined in e.g. momentum space.

According to the first Hohenberg-Kohn theorem, in principle one can exactly evaluate electronic energy (E) of a system based on electron density distribution (ρ) of ground state. However, such an exact functional relationship E[ρ] has not been found in practice. So, you can at most using existing functionals (e.g. PBE, BLYP...) to *approximately* estimate E.

Notice that most quantum chemistry programs like Gaussian and ORCA implement Koha-Sham DFT (KS-DFT) rather than the DFT in original sense (which is often referred to as orbital-free DFT currently). Formally, the KS-DFT equation is an effective one-electron Schrodinger equation, by solving it you will obtain orbital wavefunctions, which will be used in evaluating E in the present SCF cycle and construct effective external potential utilized in the next SCF cycle. The main reason of introducing the orbital wavefunctions in the KS-DFT formalism is to obtain a relatively accurate electronic kinetic energy (T), which is a key component of E and can hardly be evaluated satisfactorily based on existing kinetic energy functional T[ρ] (except for certain metal systems. If you have interest, you may view J. Chem. Phys. 150, 204106 (2019) DOI: 10.1063/1.5095072, which collects almost all already proposed kinetic energy functionals).

Also note exchange-correlation energy Exc is an important part of E, it can be estimated via exchange-correlation (XC) functionals. Many popular XC functionals not only depend on ρ (and its derivatives), but also depend on orbital wavefunctions, these are known as hybrid functionals, such as B3LYP, PBE0, wB97XD, M06-2X.