Slater determinants density functional theory

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

In the second place, a quite useful characteristic of LS-DFT is that it renders possible to transform an arbitrary wavefunction, say, the Hartree-Fock single Slater determinant into a locally-scaled one associated with a given one-particle density such as the exact one. Thus, one can easily generate a locally-scaled Hartree-Fock wavefunction that yields the exact p. In this sense, one finds much common ground between LS-DFT and those constructive realizations of the constrained-search approach which reformulate the Hartree-Fock method as well as with those developments which pose the optimized potential method as a particular instance of density functional theory [42,43,57-61]. 

When first proposed, density functional theory was not widely accepted in the chemistry community. The theory is not rigorous in the sense that it is not clear how to improve the estimates for the ground-state energies. For wavefimction-based methods, one can include more Slater determinants as in a configuration interaction approach. As the wavefunctions improve via the variational theorem, the energy is lowered. In density functional theory, there is no... 

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