Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.
Although density functional theory has its conceptual roots in the Thomas-Fermi model described below, DFT was put on a firm theoretical footing by the Hohenberg-Kohn theorems (H-K). The first of these demonstrates the existence of a one-to-one mapping between the ground-state electron density and the ground-state wavefunction of a many-particle system. Further, the second H-K theorem proves that the ground-state density minimizes the total electronic energy of the system. The original H-K theorems held only for the ground state in the absence of magnetic field, although they have since been generalized.
The theorems can be extended to the time-dependent domain to derive time-dependent density functional theory (TDDFT), which can be also used to describe excited states.
The first Hohenberg-Kohn theorem is an existence theorem, stating that the mapping exists. That is, the H-K theorems tell us that the electron density that minimizes the energy according to the true total energy functional describes all that can be known about the electronic structure. The H-K theorems do not tell us what the true total-energy functional is, only that it exists.
The most common implementation of density functional theory is through the Kohn-Sham method, which maps the properties of the system onto the properties of a system containing non-interacting electrons under a different potential. The kinetic-energy functional of such a system of non-interacting electrons is known exactly. The exchange-correlation part of the total-energy functional remains unknown and must be approximated. Another approach, less popular than Kohn-Sham DFT (KS-DFT) but arguably more closely related to the spirit of the original H-K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the interacting system.
Description of the theory
Traditional methods in electronic structure theory, in particular Hartree-Fock theory and its descendants, are based on the complicated many-electron wavefunction. The main objective of density functional theory is to replace the many-body electronic wavefunction with the electronic density as the basic quantity. Whereas the many-body wavefunction is dependent on variables, three spatial variables for each of the electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically.
Within the framework of Kohn-Sham DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas-Fermi model, and from fits to the correlation energy for a uniform electron gas.
DFT has been very popular for calculations in solid state physics since the 1970s. In many cases DFT with the local-density approximation gives quite satisfactory results, for solid-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum mechanical many-body problem. However, it was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. DFT is now a leading method for electronic structure calculations in both fields. Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe intermolecular interactions,
especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other strongly correlated systems; and in calculations of the band gap in semiconductors. Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g., interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.
Derivation and formalism
As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born-Oppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction satisfying the many-electron Schrödinger equation
where is the electronic molecular Hamiltonian, is the number of electrons, is the -electron kinetic energy, is the -electron potential energy from the external field, and is the electron-electron interaction energy for the -electron system. The operators and are so-called universal operators as they are the same for any system, while is
system dependent or non-universal. The difference between having separable single-particle problems and the much more complicated many-particle problem arises from the interaction term .
There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the Hartree-Fock method, more sophisticated approaches are usually categorized as post-Hartree-Fock methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.
Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with , onto a single-body problem without . In DFT the key variable is the particle density , which for a normalized is given by
This relation can be reversed; i.e., for a given ground-state density it is possible, in principle, to calculate the corresponding ground-state wavefunction . In other words, is a unique functional of ,
and consequently the ground-state expectation value of an observable is also a functional of
In particular, the ground-state energy is a functional of
where the contribution of the external potential can be written explicitly in terms of the ground-state density
More generally, the contribution of the external potential can be written explicitly in terms of the density ,
The functionals and are called universal functionals, while is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified , one then has to minimize the functional
with respect to , assuming one has got reliable expressions for and . A successful minimization of the energy functional will yield the ground-state density and thus all other ground-state observables.
The variational problem of minimizing the energy functional can be solved by applying the Lagrangian method of undetermined multipliers. Hereby, one uses the fact that the functional in the equation above can be written as a fictitious density functional of a non-interacting system
where denotes the non-interacting kinetic energy and is an external effective potential in which the particles are moving. Obviously, if is chosen to be
Thus, one can solve the so-called Kohn-Sham equations of this auxiliary non-interacting system,
which yields the orbitals that reproduce the density of the original many-body system
The effective single-particle potential can be written in more detail as