Density functional theory (DFT) is in principle an exact theory widely used to explore many-electron systems, from single atoms and small molecules to crystalline solids and large bio-complexes.

Fundamentally, a many-electron problem is addressed by the Schrödinger equation (or the Dirac equation, when relativity is important). The main quantity in this equation is the wavefunction, \( \Psi (\vec r_1, \vec r_2, ... , \vec r_N)\), which depends on the positions of all the electons in the system: \(\vec r_1\) through \(\vec r_N\). When the system possesses more than a few electons, even storing the wavefunction (not to mention exactly calculating it) becomes formidable. Aiming at systems with thousands of electons (or even millions), alternative approaches ought to be considered.

Density functional theory is such an approach. As has been shown by its pioneers, Pierre Hohenberg and Walter Kohn [1], the ground-state density of the system, \(n(\vec r)\), includes all the information about it, and all the other physical or chemical quantities are functionals of the density; hence the name density functional theory. Irrespectively of the number of electrons, the density depends only on one vector coordinate, \(\vec r\). Therefore, it can be easily stored and addressed even for very large N. Last but not least,  \(n(\vec r)\) seems to be a more intuitive quantity to describe the system.

But how to obtain the density of a given system and devise the desired physical quantities from it? Usually this is done following the Kohn-Sham approach [2], where the density is reconstructed using non-interacting electrons that are subject to a special, effective potential. This potential includes, in addition to the electron-nuclei attraction and the classical Hartree repulsion of the "electron cloud", the exchange and correlation potential, which is responsible to all the quantum electron-electron interactions in the system, and therefore crucial in chemistry. This last term is almost always unknown and has to be approximated. In the last 50 years a major scientific effort has been invested in development of highly-accurate approximations useful in chemistry, physics and materials science.

References

1. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964)

2. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965) 

3. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, 1989).

4. C. Fiolhais, F. Nogueira and M. A. L. Marques (eds.), A Primer in Density Functional Theory, Lectures in Physics Vol. 620 (Springer, 2003)

5. E. Kaxiras, Atomic and Electronic Structure of Solids (Cambridge University Press, 2003)

6. C. Cramer, Essentials Of Computational Chemistry: Theories And Models (Wiley, 2004)

7. R. Martin, Electronic Structure (Cambridge Unviersity Press, 2004)

8. R. O. Jones, Density functional theory: Its origins, rise to prominence, and future, Rev. Mod. Phys. 87, 897 (2015)

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