Density-functional theory: Difference between revisions
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'''Density-functional theory''' is a set of theories in [[statistical mechanics]] that profit from the | |||
fact that 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 | the density. That is, the density (in its usual sense of particles | ||
per volume), which is a | per volume), which is a function of the position in inhomogeneous systems, | ||
uniquely defines the | uniquely defines the Helmholtz energy. By minimizing this Helmholtz energy one | ||
arrives at the true | arrives at the true Helmholtz energy of the system and the equilibrium | ||
density function. The situation | |||
parallels the better known electronic density functional theory, | parallels the better known electronic density functional theory, | ||
in which the energy of a quantum system is shown to be a functional | in which the energy of a quantum system is shown to be a functional | ||
of the electronic density (theorems by Hohenberg, Kohn, Sham, and Mermin. | of the electronic density (see the theorems by [[Hohenberg-Kohn-Mermin theorems |Hohenberg, Kohn, Sham, and Mermin]]). | ||
Starting from this fact, approximations are usually made in order | Starting from this fact, approximations are usually made in order | ||
| Line 15: | Line 15: | ||
In a local density theory the | In a local density theory the | ||
in which the dependence is local, as exemplified by the (exact) | in which the dependence is local, as exemplified by the (exact) | ||
Helmholtz energy of an ideal system: | |||
<math> | :<math>A_{id}=k_BT\int dr \rho(r) [\log \rho(r) -1 -U(r)],</math> | ||
where <math>U(r)</math> is an external potential. It is an easy exercise | where <math>U(r)</math> is an external potential. It is an easy exercise | ||
to show that Boltzmann's barometric law follows from minimization. | to show that [[Boltzmann's barometric law]] follows from minimization. | ||
An example of a weighed density theory would be the | An example of a weighed density theory would be the | ||
(also exact) excess | (also exact) excess Helmholtz energy for a system | ||
of | of [[1-dimensional hard rods]]: | ||
<math> | :<math>A_{ex}=-k_BT\int dz \rho(z) \log [1-t(z)],</math> | ||
where <math>t(z)=\int_{z-\sigma}^z dy \rho(y)</math>, | where <math>t(z)=\int_{z-\sigma}^z dy \rho(y)</math>, | ||
precisely an average of the density over the length of | precisely an average of the density over the length of | ||
the hard rods, <math>\sigma</math>. "Excess" means "over | the hard rods, <math>\sigma</math>. "Excess" means "over | ||
ideal", i.e., it is the total <math> | ideal", i.e., it is the total <math>A=A_{id}+A_{ex}</math> | ||
that is to be minimized. | that is to be minimized. | ||
==See also== | ==See also== | ||
*[[ | *[[van der Waals' density gradient theory]] | ||
*[[Ebner-Saam-Stroud]] | *[[Ebner-Saam-Stroud]] | ||
*[[Fundamental-measure theory]] | *[[Fundamental-measure theory]] | ||
*[[Hohenberg-Kohn-Mermin theorems]] | *[[Hohenberg-Kohn-Mermin theorems]] | ||
*[[Quantum density-functional theory]] | *[[Quantum density-functional theory]] | ||
*[[Ramakrishnan-Youssouff]] | *[[Ramakrishnan-Youssouff]] | ||
*[[Weighted density approximation]] | *[[Weighted density approximation]] | ||
**[[Kierlik and Rosinberg's weighted density approximation]] | |||
**[[Tarazona's weighted density approximation]] | |||
*[[Dynamical density-functional theory]] | |||
*[[Perdew-Burke-Ernzerhof functional]] | |||
*[[Becke-Lee-Yang-Parr functional]] (BLYP) | |||
==Interesting reading== | ==Interesting reading== | ||
*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 | |||
*[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)] | |||
*[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)] | |||
*[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)] | |||
[[category: Density-functional theory]] | [[category: Density-functional theory]] | ||
Latest revision as of 15:57, 14 June 2010
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:
![{\displaystyle A_{id}=k_{B}T\int dr\rho (r)[\log \rho (r)-1-U(r)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce17610f91bdd75afc6c39112119cec223be8dd0)
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:
![{\displaystyle A_{ex}=-k_{B}T\int dz\rho (z)\log[1-t(z)],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fecd984f021c42633fea5ed4df78638088adb303)
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)