Microscopic Modelling of Correlated Low-dimensional Systems

Chapter 2: Method 9
system. A quantity with less degrees of freedom can provide all the useful information and
be chosen as the central variable. This is indeed the case and the main subject of density
functional theory.
2.2 Density Functional Theory (DFT)
Since its introduction by Hohenberg-Kohn and Kohn-Sham in the 1960s [47], [64] density
functional theory (DFT) has evolved into a powerful tool that is widely used in condensed
matter theory, computational materials science and quantum chemistry for the calculation
of electronic, magnetic and structural properties of molecules and solids. The method has
been remarkably successful in predicting, reproducing and explaining a wide variety of
physical and chemical properties materials.
The formulation of the two theorems of Hohenberg and Kohn is as follows:
• Theorem I: For any system of interacting particles in an external potential Vext(r),
the potential Vext(r) is determined uniquely, except for a constant, by the ground state
particle density ρ0(r).
An immediate consequence is that the ground state expectation value of any obser-
vable is a unique functional of the exact ground state electron density. Therefore all
properties of the system are completely determined given only the ground state density
ρ0(r).
• Theorem II: A functional for the energy E[ρ] in terms of the density ρ(r) is defined:
where,
�
E[ρ(r)] =
υext(r)ρ(r) dr + F [ρ(r)] (2.7)
F [ρ(r)] =< φ|T + Vee|φ > (2.8)
the Hohenberg-Kohn density functional F [ρ(r)] is universal for any many-electron
systems. E[ρ(r)] reaches its minimal value for the ground state density corresponding
to Vext.