Non-interacting reference system

So far so good, but before we are in business with this concept we need to find a prescription for how we can uniquely determine the orbitals in our non-interacting reference system. In other words, we ask how can we define Vs such that it really provides us with a Slater determinant which is characterized by exactly the same density as our real system To solve this problem, we write down the expression for the energy of our interacting, real... 

If we compare this equation with the one-particle equations from the non-interacting reference system, we see immediately that the expression in square brackets, i. e. Veff, is identical to Vs of equation (5-8) above... 

For each X the effective external potential A t is adapted such that the density always equals the density of the fully interacting system. Hence p(r) is independent of the actual value of X. Clearly, for X = 0, we recover the Hamilton operator for the non-interacting reference system, and V ° = Vs, while X = 1 is the regular expression for the real system... 

The expectation values in (6) and (7) are with respect to the KS determinant which represents the state of the non-interacting reference system... 

The exact linear density response can be related to the linear density response of a non-interacting reference system of electrons fts using the Dyson equation ... 

Here p r) is the electron density, T[p] is the kinetic energy, and Vee[p] collects aU electron-electron interaction energies. A major technical problem is the accurate description of the kinetic energy functional T[p], In the Kohn-Sham approximation this problem is avoided by introducing orbitals of a non-interacting reference system. Levy and Perdew (Levy 1979 Levy and Perdew 1985) have shown that these orbitals are delivered by the following minimization procedure ... 

Instead of calculating the electron density from the wavefunction of the interacting system, in the Kohn-Sham method, it is determined using a non-interacting reference system, which has the same electron density as the interacting system ... 

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