Orbitals and the Non-Interacting Reference System

Let us recall that the Hohenberg-Kohn theorems allow us to construct a rigorous many-body theory using the electron density as the fundamental quantity. We showed in the previous chapter that in this framework the ground state energy of an atomic or molecular system can be written as 

Of these, only J[p] is known, while the explicit forms of the other two contributions remain a mystery. The Thomas-Fermi and Thomas-Fermi-Dirac approximations that we briefly touched upon in Chapter 3 are actually realizations of this very concept. All terms present in these models, i. e., the kinetic eneigy, the potential due to the nuclei, the classical 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant Osd constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the tme N-electron wave function, we showed in Section 1.3 that SD can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the effective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as 

The HF spin orbitals x, that appear in this expression are chosen such that the expectation value Ejjp attains its minimum (under the usual constraint that the remain orthonormal) 

Of course, all this is not new but only a recapitulation of results from Chapter 1. The important connection to density functional theory is that we now go on to exploit the above kinetic energy expression, which is valid for non-interacting fermions, in order to compute the major fraction of the kinetic energy of our interacting system at hand. 

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