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# Density-functional theory

**Density-functional theory** is a set of theories in statistical mechanics that profit from the
fact that the Helmholtz energy function of a system can be cast as a functional of
the density. That is, the density (in its usual sense of particles
per volume), which is a function of the position in inhomogeneous systems,
uniquely defines the Helmholtz energy. By minimizing this Helmholtz energy one
arrives at the true Helmholtz energy of the system and the equilibrium
density function. The situation
parallels the better known electronic density functional theory,
in which the energy of a quantum system is shown to be a functional
of the electronic density (see the theorems by Hohenberg, Kohn, Sham, and Mermin).

Starting from this fact, approximations are usually made in order
to approach the true functional of a given system. An important
division is made between *local* and *weighed* theories.
In a local density theory the
in which the dependence is local, as exemplified by the (exact)
Helmholtz energy of an ideal system:

where is an external potential. It is an easy exercise to show that Boltzmann's barometric law follows from minimization. An example of a weighed density theory would be the (also exact) excess Helmholtz energy for a system of 1-dimensional hard rods:

where , precisely an average of the density over the length of the hard rods, . "Excess" means "over ideal", i.e., it is the total that is to be minimized.

## See also[edit]

- van der Waals' density gradient theory
- Ebner-Saam-Stroud
- Fundamental-measure theory
- Hohenberg-Kohn-Mermin theorems
- Quantum density-functional theory
- Ramakrishnan-Youssouff
- Weighted density approximation
- Dynamical density-functional theory
- Perdew-Burke-Ernzerhof functional
- Becke-Lee-Yang-Parr functional (BLYP)

## Interesting reading[edit]

- Robert Evans "Density Functionals in the Theory of Nonuniform Fluids", Chapter 3 pp. 85-176 in "Fundamentals of Inhomogeneous Fluids" (editor: Douglas Henderson) Marcel Dekker (1992) ISBN 978-0824787110
- Robert G. Parr "Density Functional Theory", Annual Review of Physical Chemistry
**34**pp. 631-656 (1983) - C. Ebner, H. R. Krishnamurthy and Rahul Pandit "Density-functional theory for classical fluids and solids", Physical Review A
**43**pp. 4355 - 4364 (1991) - Jianzhoung Wu "Density-functional theory for chemical engineering: from capillarity to soft materials", AIChE Journal
**52**pp. 1169 - 1193 (2005)