Noninteracting reference system

The next step is crucial. We have shown above that the exact wave functions of noninteracting fermions are Slater determinants.12 Thus, it will be possible to set up a noninteracting reference system, with a Hamiltonian in which we have introduced an effective, local potential Vs(r) ... 

With this construction of the noninteracting reference system one finds for the g-dependent interacting ground state energy E g),... 

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r) as a sum over densities of the occupied orbitals, that is, p(r) = Xl<)>,l2, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential v(r) of the nuclei and the classical Coulomb repulsion VCoul(r) within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been folded into a local potential vxc r) ... 

In this equation, T [p] is the kinetic energy of the noninteracting reference system, Vg ([p] the interaction energy with some external potential, J[p] the Coulomb interaction of the electrons, and represents the exchange-correlation functional... 

The additional information of the spin density p (r) can then be directly exploited in the exchange-correlation functionals. For open-shell systems, two different restrictions are possible when introducing the noninteracting reference system [109, 111]. We can require (i) that only the total electron density of the fully interacting and of the reference system agree or (ii) that, in addition, the spin densities of the two systems are exactly the same. The first condition leads to a spin-restricted Kohn-Sham DFT formulation, while for the latter a spin-unrestricted Kohn-Sham DFT framework is required [109]. 

The utility of KS-DFT arises because the properties of the noninteracting reference system often closely resemble the properties of the target interacting electron system. 

Higuchi, M. Higuchi, K. Pair density functional theory utilizing the noninteracting reference system an effective initial theory. Phys. Rev. B 2008, 78, 125101. 

In relativistic current-density funtional theory discussed above, the noninteracting reference system is chosen such that it possesses the same 4-current... 

Hence, it is possible to develop a relativistic density-only d ) KS-DFT, which will resemble the nonrelativistic spin-restricted KS-DFT formalism, by only requiring that the noninteracting reference system possesses the same density as the interacting system. Then, the total (relativistic) energy functional can be decomposed as... 

Here, ipi r) denotes one of the N 4-spinors of Eq. (8.213). The noninteracting reference system can then be set up in such a way that, in addition to the electron density, some parts of the magnetization agree with those of the fully interacting system. 

These different choices for the noninteracting reference system imply dif-ferent definitions of the noninteracting kinetic energy functionals, Tg p, mz] and m ], and accordingly also of the exchange-correlation energies,... 

The first term arises from the interaction of the dectrons with an external potential (typically due to the dectrostatic interaction with the nudd) the second term is the kinetic energy of a noninteracting reference system (electronic gas),... 

Insertion of (2.72) into the coupling constant integral (2.71) leads to the standard energy correction which results from switching on some perturbation to a noninteracting reference system. 

To facilitate the discussion, we couch DFT in the language of p, the first-order reduced density operator of the noninteracting reference system. Consider an N electron system in a spin-compensated state and in an external potential Wext(r) (extension to spin-polarized state is trivial). The real space representation of p is the density matrix... 

The last term in Eq. (2) represents the main problem of the DFT. In DFT methodologies the optimal electron density (p ) is computed following the variational principle. Kohn and Sham proposed that a real electron density can be represented by a fictitious noninteracting reference system. Electrons in the latter system do not interact, but its ground-state electron density distribution is exactly the same as corresponding to the real system under consideration. The deviation in the behavior of noninteracting electrons from that of the real ones is then taken into account by the unknown XC functional that is included in DFT methods in an approximate form. The development of such approximate functional is a very active research topic in theoretical chemistry. 

---