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| '''Density-functional theory''' is a set of theories in [[statistical mechanics]] that profit from the
| | *[[Classical DFT]] |
| fact that the [[Helmholtz energy function]] of a system can be cast as a functional of
| | *[[Quantum DFT]] |
| the density. That is, the density (in its usual sense of particles
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| per volume), which is a function of the position in inhomogeneous systems,
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| uniquely defines the Helmholtz energy. By minimizing this Helmholtz energy one
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| arrives at the true Helmholtz energy of the system and the equilibrium
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| density function. The situation
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| parallels the better known electronic density functional theory,
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| in which the energy of a quantum system is shown to be a functional
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| of the electronic density (see the theorems by [[Hohenberg-Kohn-Mermin theorems |Hohenberg, Kohn, Sham, and Mermin]]).
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| Starting from this fact, approximations are usually made in order
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| to approach the true functional of a given system. An important
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| division is made between ''local'' and ''weighed'' theories.
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| In a local density theory the
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| in which the dependence is local, as exemplified by the (exact)
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| Helmholtz energy of an ideal system:
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| :<math>A_{id}=k_BT\int dr \rho(r) [\log \rho(r) -1 -U(r)],</math>
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| where <math>U(r)</math> is an external potential. It is an easy exercise
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| to show that [[Boltzmann's barometric law]] follows from minimization.
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| An example of a weighed density theory would be the
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| (also exact) excess Helmholtz energy for a system
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| of [[1-dimensional hard rods]]:
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| :<math>A_{ex}=-k_BT\int dz \rho(z) \log [1-t(z)],</math>
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| where <math>t(z)=\int_{z-\sigma}^z dy \rho(y)</math>,
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| precisely an average of the density over the length of
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| the hard rods, <math>\sigma</math>. "Excess" means "over
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| ideal", i.e., it is the total <math>A=A_{id}+A_{ex}</math>
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| that is to be minimized.
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| ==See also==
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| *[[van der Waals' density gradient theory]] | |
| *[[Ebner-Saam-Stroud]]
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| *[[Fundamental-measure theory]]
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| *[[Hohenberg-Kohn-Mermin theorems]]
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| *[[Quantum density-functional theory]] | |
| *[[Ramakrishnan-Youssouff]]
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| *[[Weighted density approximation]]
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| **[[Kierlik and Rosinberg's weighted density approximation]]
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| **[[Tarazona's weighted density approximation]]
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| *[[Dynamical density-functional theory]]
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| *[[Perdew-Burke-Ernzerhof functional]]
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| *[[Becke-Lee-Yang-Parr functional]] (BLYP)
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| ==Interesting reading==
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| *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
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| *[http://dx.doi.org/10.1146/annurev.pc.34.100183.003215 Robert G. Parr "Density Functional Theory", Annual Review of Physical Chemistry '''34''' pp. 631-656 (1983)]
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| *[http://dx.doi.org/10.1103/PhysRevA.43.4355 C. Ebner, H. R. Krishnamurthy and Rahul Pandit "Density-functional theory for classical fluids and solids", Physical Review A '''43''' pp. 4355 - 4364 (1991)]
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| *[http://dx.doi.org/10.1002/aic.10713 Jianzhoung Wu "Density-functional theory for chemical engineering: from capillarity to soft materials", AIChE Journal '''52''' pp. 1169 - 1193 (2005)]
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| [[category: Density-functional theory]]
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