What is density-functional theory

To get a first idea of what density-functional theory is about, it is useful to take a step back and recall some elementary quantum mechanics. In quantum mechanics we learn that all information we can possibly have about a given system is contained in the system s wave function, T. Here we will exclusively be concerned with the electronic structure of atoms, molecules and solids. The nuclear degrees of freedom (e.g., the crystal lattice in a solid) appear only in the form of a potential u(r) acting on the electrons, so that the wave function depends only on the electronic coordinates.2 Nonrelativistically, this wave function is calculated from Schrodinger s equation, which for a single electron moving in a potential v(r) reads 

If there is more than one electron (i.e., one has a many-body problem) 

2This is the so-called Born-Oppenheimer approximation. It is common to call v(r) a potential although it is, strictly speaking, a potential energy. 

Note that this is the same operator for any system of particles interacting via the Coulomb interaction, just as the kinetic energy operator 

3For materials containing atoms with large atomic number Z, accelerating the electrons to relativistic velocities, one must include relativistic effects by solving Dirac s equation or an approximation to it. In this case the kinetic energy operator takes a different form. 

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There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

What is obviously needed is a generally accepted recipe for how atomic states should be dealt with in approximate density functional theory and, indeed, a few empirical rules have been established in the past. Most importantly, due to the many ways atomic energies can be obtained, one should always explicitly specify how the calculations were performed to ensure reproducibility. From a technical point of view (after considerable discussions in the past among physicists) there is now a general consensus that open-shell atomic calculations should employ spin polarized densities, i. e. densities where not necessarily... 

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