Density-functional theory: Difference between revisions

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This is a set of theories in statistical mechanics that profit from the
'''Density-functional theory''' is a set of theories in [[statistical mechanics]] that profit from the
fact that the free energy of a system can be cast as a functional of
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 funtion of the position in inhomogeneous systems,
per volume), which is a function of the position in inhomogeneous systems,
uniquely defines the free energy. By minimizing this free energy one
uniquely defines the Helmholtz energy. By minimizing this Helmholtz energy one
arrives at the true free energy of the system and the equilibrium
arrives at the true Helmholtz energy of the system and the equilibrium
densify function. The situation
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)
free energy of an ideal system:
Helmholtz energy of an ideal system:


<math>F_{id}=kT\int dr \rho(r) [\log \rho(r) -1 -U(r)],</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 free energy for a system
(also exact) excess Helmholtz energy for a system
of 1D hard rods:
of [[1-dimensional hard rods]]:


<math>F_{ex}=-kT\int dz \rho(z) \log [1-t(z)],</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>F=F_{id}+F_{ex}</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==
*[[Dynamical density-functional theory]]
*[[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]]
*[[Kierlik and Rosinberg's weighted density approximation]]
*[[Quantum density-functional theory]]
*[[Quantum density-functional theory]]
*[[Ramakrishnan-Youssouff]]
*[[Ramakrishnan-Youssouff]]
*[[Tarazona's weighted density approximation]]
*[[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", in "Fundamentals of Inhomogeneous Fluids" (ed. D. Henderson). Marcel Dekker.
*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/10.1146/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/10.1103/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:

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]

Interesting reading[edit]