In deriving the Kohn-Sham equation, it is a main step to extract the 'major' part of the true kinetic energy exactly by using the fictitious non-interacting system and calculating its kinetic energy. In the fictitious system the wavefunction is just a single determinant $\Psi_s$.

$$ T[\rho] = \langle{} \Psi[\rho]| \sum_{i=1}^N -\frac{1}{2}\nabla_i^2|\Psi[\rho]\rangle{} $$

$$ T_s[\rho] = \langle{} \Psi_s[\rho]| \sum_{i=1}^N -\frac{1}{2}\nabla_i^2|\Psi_s[\rho]\rangle{} $$

  • Why non-interacting kinetic energy is the major part of the true kinetic energy?
  • At what level of quantity it is different from the true kinetic energy?

I think the question is slightly equivalent to:

  • How different (quantitatively and intuitively) in kinetic energy of an electron state and its Slater approximating state.

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