Non-interacting kinetic energy in Kohn-Sham approach is treated exactly via one-electron orbitals. In orbital free DFT kinetic energy should be approximated as an explicit functional of the electron density. Due to KE has the same order of magnitude as the total energy, errors of few percent in KE become large in absolute value. Exchange-correlation is another piece of the total energy which should be approximated, but it is the same in both, KS and orbital-free DFT. Standard orbital-free functionals such as Thomas-Fermi, second-order gradient approximation (SGA) and generalized gradient approximation (GGA) fail to predict binding in simple molecules and solids (no minima on energy surface). Recently developed non-empirical GGA KE functional predicts qualitatively and some times quantitatively correct results at low temperature, and it does very good job for equation of state (with errors in pressure less than 1%) in warm dense matter regime when KS calculation becomes intractable task due to elevated temperature. See attached paper and Refs. [18,19] therein.

Article Nonempirical Generalized Gradient Approximation Free Energy ...

Dear Mr. Menon:

Indeed, the Thomas-Fermi model is a good solution for warm dense matter. The orbital in Kohn-Sham DFT is not a true physical concept, but just a mean to calculate the kinetic energy. The TF model cannot treat the kinetic energy accurately, because it neglects the exchange-correlation energy which comes from Pauli principle.

Dear Dr. Kai Zhao

Thanks. I agree with you completely. The models of TF type are semi-classical. Though DFT attempts to generate a full quantum mechanical model of electrons in materials, its treatment of exchange effect is semi-empirical, though quite accurate.

Non-interacting kinetic energy in Kohn-Sham approach is treated exactly via one-electron orbitals. In orbital free DFT kinetic energy should be approximated as an explicit functional of the electron density. Due to KE has the same order of magnitude as the total energy, errors of few percent in KE become large in absolute value. Exchange-correlation is another piece of the total energy which should be approximated, but it is the same in both, KS and orbital-free DFT. Standard orbital-free functionals such as Thomas-Fermi, second-order gradient approximation (SGA) and generalized gradient approximation (GGA) fail to predict binding in simple molecules and solids (no minima on energy surface). Recently developed non-empirical GGA KE functional predicts qualitatively and some times quantitatively correct results at low temperature, and it does very good job for equation of state (with errors in pressure less than 1%) in warm dense matter regime when KS calculation becomes intractable task due to elevated temperature. See attached paper and Refs. [18,19] therein.

Article Nonempirical Generalized Gradient Approximation Free Energy ...

There is another way, the electron density is not set by the sum of the squares of orbitals, and as grid points. And operate electronic density given in tabular

In a simple picture, it is because the electron states assumed in TF model as well as in DFT are not consistent with the details of the functions which describe the realities of lectron states in a solid. In other words, these models are not consistent with the real state to an acceptable approximations. These models are valid closely to describe only high temperature (not at low temperatures) states of metals. Not specifically mentioning these models but situations like this have been analysed in the following paper. possibly it should help. This paper clearly demonstrates why certain many body theories based on single particle basis can not explain the properties of the system completely, accurately and in agreement with experiments in spite of their mathematical accuracies.

http://www.scribd.com/doc/110441679/Intrinsic-Problems-Superfluid-Theories

The question as posed is not really correct, although its intention certainly is. By definition of the Hohenberg-Kohn theorem (or Levy-Lieb to be more general) the TF kinetic energy functional can certainly be made to give the correct electron density if one knows the exact F[n]=min_{rho->n} Tr[rho(T+U)] functional. We can simply define E_{KHxc}[n] = F[n]-T_{TF}[n] and minimise E_{TF}+E_{KHxc}+E_{Exct} w.r.t. n. I'm sorry to be pedantic about this, but confusing between "density functional theory" and "approximations to density functional theory" can lead readers astray.

Putting the pedantry aside, density functional _approximations_ (DFAs) are much harder to make due to the aforementioned failure of the TF functional to reproduce the shell-structure. There has been work by (as I recall) the Burke group looking to make improved Kinetic energy approximations by machine learning (i.e. to find a functional T_{Learned}[n] that reproduces T_s[\{\phi[n]\}] on a variety of systems), but I'm not sure how useful these are in general. And as Valentin mentions, there are regimes where they work better or worse.

To complicated matters, there may very well be cases where T_s[\{\phi[n]\}] (the standard Kohn-Sham approximation to the kinetic energy) is actually quite a poor approximation to the true kinetic energy T[\Phi] even if one knows the exact one-body orbitals. I'm not aware of any actual cases, but it could appear

in situations where strong electron-repulsion correlation rears its ugly head.

Tim, T[\Phi] and T_s[\{\phi[n]\}] are different quantities and should not be compared. T[\Phi] describes the kinetic energy (KE) of interacting system. T_s[\{\phi[n]\}] is the non-interacting KE (i.e. the kinetic energy of non-interacting system), and that description is _exact_. The difference T_c=T[\Phi] - T_s[\{\phi[n]\}], according to the Kohn-Sham energy partition, is included in the correlation energy. Therefore one should compare between T_s[\{\phi[n]\}]+T_c[n] and T[\Phi]. If a DFT approximation for the total (interacting) KE fails it is due to poor correlation energy functional.

Hi Valentin, Yes T_s and T are different, but from my understanding generally they are quite similar in value and T_c

If I really drag my brain back to past lessons, I think we can write the correlation part of the kinetic energy via the adiabatic connection formula

T_c=\int_0^1 \d\lambda \int\d r\d r'/|r-r'| ( n_2^{\lambda} - \lambda n_2^1 )

i.e. it is the integrated difference between the linearly interpolated pair-density, and the nearly-linear actual pair-density at coupling constant \lambda. Mostly this is not very large.

Tim, the only point of my post was the following: T_s should not be viewed as an approximation to T. Though, as you mention, typically T_s is close to T, i.e. T_c

Valentin, I see your point. Although without E_c (and thus T_c) the Kohn-Sham T_s will not be correct as we won't be using the correct orbitals for it.

My argument is simply that one of the reasons approximations to KS-DFT tend to work better than approximations to TF-DFT is that T_c^{KS} is typically smaller than T_c^{TF}. Thus we are starting from a better approximation to the kinetic energy and approximating a smaller component of the total energy.

P.S. In your work on warm matter, do you find that T_{TF} (or e.g. T_{vW}) become closer to T?

For clarification: KS requires one approximate piece, exchange-correlation (XC) energy density functional (T_c is included in that term). Orbital-free DFT (OF-DFT) requires two approximate functionals, non-interacting KE (T_s[n]) and XC.

To access the accuracy of an orbital-free KE approximation one should compare between KS and OF-DFT results when both calculations are performed with the same XC functional (=> T_c in both cases are approximated on the same way). One can compare between KS and OF-DFT non-interacting KE energies, total energies and some other properties.

Orbital-free kinetic energy functionals which are more sophisticated as compared to the simplest Thomas-Fermi approximation, such as second-order gradient approximation (SGA), standard generalized gradient approximation (GGA), modified conjoint GGA, and non-local functionals usually provides better results for energies, lattice constants and equation of state.

https://www.southampton.ac.uk/assets/centresresearch/documents/compchem/DFT_L5.pdf

http://www.panstanford.com/pdf/9789814241236index.pdf