Editing Density functional theory (section)

==Classical density functional theory==
Classical density functional theory is a classical statistical method to investigate the properties of many-body systems consisting of interacting molecules, macromolecules, nanoparticles or microparticles.<ref>{{cite journal|last1=Evans|first1=Robert|date=1979|title=The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids|journal=Advances in Physics|volume=281|issue=2|pages=143–200|bibcode=1979AdPhy..28..143E|doi=10.1080/00018737900101365}}</ref><ref>{{Cite journal| doi = 10.1088/0953-8984/28/24/240401| issn = 0953-8984| volume = 28| issue = 24| pages = 240401| last1 = Evans| first1 = Robert| last2 = Oettel| first2 = Martin| last3 = Roth| first3 = Roland| last4 = Kahl| first4 = Gerhard| title = New developments in classical density functional theory| journal = Journal of Physics: Condensed Matter| date=2016| pmid = 27115564| bibcode = 2016JPCM...28x0401E| doi-access = free}}</ref><ref>{{cite journal|last1=Singh|first1=Yaswant|date=1991|title=Density Functional Theory of Freezing and Properties of the ordered Phase|journal=Physics Reports|volume=207|issue=6|pages=351–444|bibcode=1991PhR...207..351S|doi=10.1016/0370-1573(91)90097-6}}</ref><ref>{{cite book|last1=ten Bosch|first1=Alexandra|url=http://atomstooceans.com|title=Analytical Molecular Dynamics:from Atoms to Oceans|date=2019|isbn=978-1091719392}}</ref> The classical non-relativistic method is correct for [[classical fluids]] with particle velocities less than the speed of light and [[thermal de Broglie wavelength]] smaller than the distance between particles. The theory is based on the [[calculus of variations]] of a thermodynamic functional, which is a function of the spatially dependent density function of particles, thus the name. The same name is used for quantum DFT, which is the theory to calculate the electronic structure of electrons based on spatially dependent electron density with quantum and relativistic effects. Classical DFT is a popular and useful method to study fluid [[phase transitions]], ordering in complex liquids, physical characteristics of [[interface (matter)|interfaces]] and [[nanomaterials]]. Since the 1970s it has been applied to the fields of [[materials science]], [[biophysics]], [[chemical engineering]] and [[civil engineering]].<ref>{{cite journal|last1=Wu|first1=Jianzhong|date=2006|title=Density Functional Theory for chemical engineering:from capillarity to soft materials|journal=AIChE Journal|volume=52|issue=3|pages=1169–1193|doi=10.1002/aic.10713|bibcode=2006AIChE..52.1169W }}</ref> Computational costs are much lower than for [[molecular dynamics]] simulations, which provide similar data and a more detailed description but are limited to small systems and short time scales. Classical DFT is valuable to interpret and test numerical results and to define trends although details of the precise motion of the particles are lost due to averaging over all possible particle trajectories.<ref>{{cite journal|last1=Gelb|first1=Lev D.|last2=Gubbins|first2=K. E.|last3=Radhakrishnan|first3=R.|last4=Sliwinska-Bartkowiak|first4=M.|date=1999|title=Phase separation in confined systems|journal=Reports on Progress in Physics|volume=62|issue=12|pages=1573–1659|bibcode=1999RPPh...62.1573G|doi=10.1088/0034-4885/62/12/201|s2cid=9282112}}</ref> As in electronic systems, there are fundamental and numerical difficulties in using DFT to quantitatively describe the effect of intermolecular interaction on structure, correlations and thermodynamic properties.

Classical DFT addresses the difficulty of describing [[thermodynamic equilibrium]] states of many-particle systems with nonuniform density.<ref>{{cite book|last1=Frisch|first1=Harry|title=The equilibrium theory of classical fluids|last2=Lebowitz|first2=Joel|date=1964|publisher=W. A. Benjamin|location=New York}}</ref> Classical DFT has its roots in theories such as the [[Van der Waals equation|van der Waals theory]] for the [[equation of state]] and the [[virial expansion]] method for the pressure. In order to account for [[correlation]] in the positions of particles the direct correlation function was introduced as the effective interaction between two particles in the presence of a number of surrounding particles by [[Leonard Ornstein]] and [[Frits Zernike]] in 1914.<ref>{{cite journal|last1=Ornstein|first1=L. S.|last2=Zernike|first2=F.|date=1914|title=Accidental deviations of density and opalescence at the critical point of a single substance|url=http://www.dwc.knaw.nl/DL/publications/PU00012727.pdf|journal=Royal Netherlands Academy of Arts and Sciences|series=Proceedings|volume=17|pages=793–806|bibcode=1914KNAB...17..793.}}</ref> The connection to the density [[pair distribution function]] was given by the [[Ornstein–Zernike equation]]. The importance of correlation for thermodynamic properties was explored through density distribution functions. The [[functional derivative]] was introduced to define the distribution functions of classical mechanical systems. Theories were developed for simple and complex liquids using the ideal gas as a basis for the free energy and adding molecular forces as a second-order perturbation. A term in the gradient of the density was added to account for non-uniformity in density in the presence of external fields or surfaces. These theories can be considered precursors of DFT.

To develop a formalism for the statistical thermodynamics of non-uniform fluids functional differentiation was used extensively by Percus and Lebowitz (1961), which led to the [[Percus–Yevick approximation|Percus–Yevick equation]] linking the density distribution function and the direct correlation.<ref>{{cite journal|last1=Lebowitz|first1=J. L.|last2=Percus|first2=J. K.|date=1963|title=Statistical Thermodynamics of Non-uniform Fluids|journal=Journal of Mathematical Physics|volume=4|issue=1|pages=116–123|bibcode=1963JMP.....4..116L|doi=10.1063/1.1703877}}</ref> Other closure relations were also proposed;the [[Classical-map hypernetted-chain method]], the [[BBGKY hierarchy]]. In the late 1970s classical DFT was applied to the liquid–vapor interface and the calculation of [[surface tension]]. Other applications followed: the [[freezing]] of simple fluids, formation of the [[glass]] phase, the crystal–melt interface and [[dislocation]] in crystals, properties of [[polymer]] systems, and [[liquid crystal]] ordering. Classical DFT was applied to [[colloid]] dispersions, which were discovered to be good models for atomic systems.<ref>{{cite journal|last1=Löwen|first1=Hartmut|date=1994|title=Melting, freezing and colloidal suspensions|journal=Physics Reports|volume=237|issue=5|pages=241–324|bibcode=1994PhR...237..249L|doi=10.1016/0370-1573(94)90017-5}}</ref> By assuming local chemical equilibrium and using the local chemical potential of the fluid from DFT as the driving force in fluid transport equations, equilibrium DFT is extended to describe non-equilibrium phenomena and [[fluid dynamics]] on small scales.

Classical DFT allows the calculation of the equilibrium particle density and prediction of thermodynamic properties and behavior of a many-body system on the basis of model [[Fundamental interaction|interactions]] between particles. The spatially dependent [[density]] determines the local structure and composition of the material. It is determined as a function that optimizes the thermodynamic potential of the [[grand canonical ensemble]]. The [[grand potential]] is evaluated as the sum of the [[ideal-gas]] term with the contribution from external fields and an excess [[thermodynamic free energy]] arising from interparticle interactions. In the simplest approach the excess free-energy term is expanded on a system of uniform density using a functional [[Taylor expansion]]. The excess free energy is then a sum of the contributions from ''s''-body interactions with density-dependent effective potentials representing the interactions between ''s'' particles. In most calculations the terms in the interactions of three or more particles are neglected (second-order DFT). When the structure of the system to be studied is not well approximated by a low-order perturbation expansion with a uniform phase as the zero-order term, non-perturbative free-energy functionals have also been developed. The minimization of the grand potential functional in arbitrary local density functions for fixed chemical potential, volume and temperature provides self-consistent thermodynamic equilibrium conditions, in particular, for the local [[chemical potential]]. The functional is not in general a [[convex function]]al of the density; solutions may not be local [[minima]]. Limiting to low-order corrections in the local density is a well-known problem, although the results agree (reasonably) well on comparison to experiment.

A [[variational principle]] is used to determine the equilibrium density. It can be shown that for constant temperature and volume the correct equilibrium density minimizes the [[grand potential]] functional <math>\Omega</math> of the [[grand canonical ensemble]] over density functions <math>n(\mathbf r)</math>. In the language of functional differentiation (Mermin theorem):

:<math>\frac{\delta \Omega}{\delta n(\mathbf r)} = 0.</math>

The [[Helmholtz free energy]] functional <math>F</math> is defined as <math>F = \Omega + \int d^3 \mathbf r\, n(\mathbf r) \mu(\mathbf r)</math>.
The [[functional derivative]] in the density function determines the local chemical potential: <math>\mu(\mathbf r) = \delta F(\mathbf r) / \delta n(\mathbf r)</math>.
In classical statistical mechanics the [[partition function (statistical mechanics)|partition function]] is a sum over probability for a given microstate of {{mvar|N}} classical particles as measured by the Boltzmann factor in the [[Hamiltonian function|Hamiltonian]] of the system. The Hamiltonian splits into kinetic and potential energy, which includes interactions between particles, as well as external potentials. The partition function of the grand canonical ensemble defines the grand potential. A [[Correlation function (statistical mechanics)|correlation function]] is introduced to describe the effective interaction between particles.

The ''s''-body density distribution function is defined as the statistical [[ensemble average]] <math>\langle\dots\rangle</math> of particle positions. It measures the probability to find ''s'' particles at points in space <math>\mathbf r_1, \dots, \mathbf r_s</math>:

:<math>n_s(\mathbf r_1, \dots, \mathbf r_s) = \frac{N!}{(N - s)!} \big\langle \delta(\mathbf r_1 - \mathbf r'_1) \dots \delta(\mathbf r_s - \mathbf r'_s) \big\rangle.</math>

From the definition of the grand potential, the functional derivative with respect to the local chemical potential is the density; higher-order density correlations for two, three, four or more particles are found from higher-order derivatives:

:<math>\frac{\delta^s \Omega}{\delta \mu(\mathbf r_1) \dots \delta\mu(\mathbf r_s)} = (-1)^s n_s(\mathbf r_1, \dots, \mathbf r_s).</math>

The [[radial distribution function]] with ''s''&nbsp;=&nbsp;2 measures the change in the density at a given point for a change of the local chemical interaction at a distant point.

In a fluid the free energy is a sum of the ideal free energy and the excess free-energy contribution <math>\Delta F</math> from interactions between particles. In the grand ensemble the functional derivatives in the density yield the direct correlation functions <math>c_s</math>:

:<math>\frac{1}{kT} \frac{\delta^s \Delta F}{\delta n(\mathbf r_1) \dots \delta n(\mathbf r_s)} = c_s(\mathbf r_1, \dots, \mathbf r_s).</math>

The one-body direct correlation function plays the role of an effective [[mean field]]. The functional derivative in density of the one-body direct correlation results in the direct correlation function between two particles <math>c_2</math>. The direct correlation function is the correlation contribution to the change of local chemical potential at a point <math>\mathbf r</math> for a density change at <math>\mathbf r'</math> and is related to the work of creating density changes at different positions. In dilute gases the direct correlation function is simply the pair-wise interaction between particles ([[Debye–Huckel equation]]). The Ornstein–Zernike equation between the pair and the direct correlation functions is derived from the equation

:<math>\int d^3 \mathbf r''\, \frac{\delta \mu(\mathbf r)}{\delta n(\mathbf r'')} \frac{\delta n(\mathbf r'')}{\delta\mu(\mathbf r')} = \delta(\mathbf r - \mathbf r').</math>

Various assumptions and approximations adapted to the system under study lead to expressions for the free energy. Correlation functions are used to calculate the free-energy functional as an expansion on a known reference system. If the non-uniform fluid can be described by a density distribution that is not far from uniform density a functional Taylor expansion of the free energy in density increments leads to an expression for the thermodynamic potential using known correlation functions of the uniform system. In the square gradient approximation a strong non-uniform density contributes a term in the gradient of the density. In a perturbation theory approach the direct correlation function is given by the sum of the direct correlation in a known system such as [[hard spheres]] and a term in a weak interaction such as the long range [[London dispersion force]]. In a local density approximation the local excess free energy is calculated from the effective interactions with particles distributed at uniform density of the fluid in a cell surrounding a particle. Other improvements have been suggested such as the weighted density approximation for a direct correlation function of a uniform system which distributes the neighboring particles with an effective weighted density calculated from a self-consistent condition on the direct correlation function.

The variational Mermin principle leads to an equation for the equilibrium density and system properties are calculated from the solution for the density. The equation is a non-linear integro-differential equation and finding a solution is not trivial, requiring numerical methods, except for the simplest models. Classical DFT is supported by standard software packages, and specific software is currently under development. Assumptions can be made to propose trial functions as solutions, and the free energy is expressed in the trial functions and optimized with respect to parameters of the trial functions. Examples are a localized [[Gaussian function]] centered on crystal lattice points for the density in a solid, the hyperbolic function <math>\tanh(r)</math> for interfacial density profiles.

Classical DFT has found many applications, for example: 
* developing new functional materials in [[materials science]], in particular [[nanotechnology]];
* studying the properties of fluids at [[surface]]s and the phenomena of [[wetting]] and [[adsorption]];<ref>Hydrophobicity of ceria, [https://hal.archives-ouvertes.fr/hal-02308396/document/#page=5 Applied Surface Science, 2019, 478, pp.68-74.] in HAL archives ouvertes</ref>
* understanding life processes in [[biotechnology]];
* improving [[filtration]] methods for gases and fluids in [[chemical engineering]];
* fighting [[pollution]] of water and air in environmental science;
* cell membranes by modelling complex systems with [[amphiphile]] compounds;
* generating new procedures in [[microfluidics]] and [[nanofluidics]].

The extension of classical DFT towards nonequilibrium systems is known as dynamical density functional theory (DDFT).<ref>{{cite journal|last1=te Vrugt|first1=Michael|author2-link=Hartmut Löwen|last2=Löwen|first2=Hartmut|last3=Wittkowski|first3=Raphael|date=2020|title=Classical dynamical density functional theory: from fundamentals to applications|journal=Advances in Physics|volume=69|issue=2|pages=121–247|doi=10.1080/00018732.2020.1854965|arxiv=2009.07977|bibcode=2020AdPhy..69..121T|s2cid=221761300}}</ref> DDFT allows to describe the time evolution of the one-body density <math>\rho(\boldsymbol{r},t)</math> of a colloidal system, which is governed by the equation

:<math>\frac{\partial \rho}{\partial t} = \Gamma \nabla \cdot \left(\rho\nabla \frac{\delta F}{\delta \rho} \right)</math>

with the mobility <math>\Gamma </math> and the free energy <math> F </math>. DDFT can be derived from the microscopic equations of motion for a colloidal system (Langevin equations or Smoluchowski equation) based on the adiabatic approximation, which corresponds to the assumption that the two-body distribution in a nonequilibrium system is identical to that in an equilibrium system with the same one-body density. For a system of noninteracting particles, DDFT reduces to the standard diffusion equation.