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Density functional theory

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From Wikipedia, the free encyclopedia

Computational quantum mechanical modelling method to investigate electronic structure

Not to be confused withDiscrete Fourier transform.

Electronic structure methods
Valence bond theory
Coulson–Fischer theory Generalized valence bond Modern valence bond theory
Molecular orbital theory
Hartree–Fock method Semi-empirical quantum chemistry methods Møller–Plesset perturbation theory Configuration interaction Coupled cluster Multi-configurational self-consistent field Quantum chemistry composite methods Quantum Monte Carlo
Density functional theory
Time-dependent density functional theory Thomas–Fermi model Orbital-free density functional theory Adiabatic connection fluctuation dissipation theorem Görling–Levy perturbation theory Generalized Kohn Sham theory Optimized effective potential method Linearized augmented-plane-wave method Projector augmented wave method
Electronic band structure
Nearly free electron model Tight binding Muffin-tin approximation k·p perturbation theory Empty lattice approximation GW approximation Korringa–Kohn–Rostoker method
v t e

Density functional theory (DFT) is a computationalquantum mechanical modeling method used inphysics,chemistry andmaterials science to investigate theelectronic structure (ornuclear structure) (principally theground state) ofmany-body systems, in particular atoms, molecules, and thecondensed phases. Using this theory, the properties of a many-electron system can be determined by usingfunctionals - that is, functions that accept a function as input and output a single real number.^[1] In the case of DFT, these are functionals of the spatially dependentelectron density. DFT is among the most popular and versatile methods available incondensed-matter physics,computational physics, andcomputational chemistry.

DFT has been very popular for calculations insolid-state physics since the 1970s. However, DFT was not considered sufficiently accurate for calculations inquantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model theexchange andcorrelation interactions. Computational costs are relatively low when compared to traditional methods, such as exchange onlyHartree–Fock theory andits descendants that include electron correlation. Since, DFT has become an important tool for methods ofnuclear spectroscopy such asMössbauer spectroscopy orperturbed angular correlation, in order to understand the origin of specificelectric field gradients in crystals.

DFT sometimes does not properly describe:intermolecular interactions (of critical importance to understanding chemical reactions), especiallyvan der Waals forces (dispersion); charge transfer excitations;transition states, globalpotential energy surfaces, dopant interactions and somestrongly correlated systems; and in calculations of theband gap andferromagnetism insemiconductors.^[2] The incomplete treatment of dispersion can adversely affect the accuracy of DFT (at least when used alone and uncorrected) in the treatment of systems which are dominated by dispersion (e.g. interactingnoble gas atoms)^[3] or where dispersion competes significantly with other effects (e.g. inbiomolecules).^[4] The development of new DFT methods designed to overcome this problem, by alterations to the functional^[5] or by the inclusion of additive terms.^[6]^[7]^[8]^[9]^[10] Classical density functional theory uses a similar formalism to calculate the properties of non-uniform classical fluids.

Despite the current popularity of these alterations or of the inclusion of additional terms, they are reported^[11] to stray away from the search for the exact functional. Further, DFT potentials obtained with adjustable parameters are no longer true DFT potentials,^[12] given that they are not functional derivatives of the exchange correlation energy with respect to the charge density. Consequently, it is not clear if the second theorem of DFT holds^[12]^[13] in such conditions.

Overview of method

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In the context of computationalmaterials science,ab initio (from first principles) DFT calculations allow the prediction and calculation of material behavior on the basis of quantum mechanical considerations, without requiring higher-order parameters such as fundamental material properties. In contemporary DFT techniques the electronic structure is evaluated using a potential acting on the system's electrons. This DFT potential is constructed as the sum of external potentialsV_ext, which is determined solely by the structure and the elemental composition of the system, and an effective potentialV_eff, which represents interelectronic interactions. Thus, a problem for a representative supercell of a material withn electrons can be studied as a set ofn one-electronSchrödinger-like equations, which are also known asKohn–Sham equations.^[14]

Origins

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Although density functional theory has its roots in theThomas–Fermi model for the electronic structure of materials, DFT was first put on a firm theoretical footing byWalter Kohn andPierre Hohenberg in the framework of the twoHohenberg–Kohn theorems (HK).^[15] The original HK theorems held only for non-degenerate ground states in the absence of amagnetic field, although they have since been generalized to encompass these.^[16]^[17]

The first HK theorem demonstrates that theground-state properties of a many-electron system are uniquely determined by anelectron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem ofN electrons with3N spatial coordinates to three spatial coordinates, through the use offunctionals of the electron density. This theorem has since been extended to the time-dependent domain to developtime-dependent density functional theory (TDDFT), which can be used to describe excited states.

The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional.

In work that later won them theNobel prize in chemistry, the HK theorem was further developed byWalter Kohn andLu Jeu Sham to produceKohn–Sham DFT (KS DFT). Within this framework, the intractablemany-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effectivepotential. The effective potential includes the external potential and the effects of theCoulomb interactions between the electrons, e.g., theexchange andcorrelation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is thelocal-density approximation (LDA), which is based upon exact exchange energy for a uniformelectron gas, which can be obtained from theThomas–Fermi model, and from fits to the correlation energy for a uniform electron gas. Non-interacting systems are relatively easy to solve, as the wavefunction can be represented as aSlater determinant oforbitals. Further, thekinetic energy functional of such a system is known exactly. The exchange–correlation part of the total energy functional remains unknown and must be approximated.

Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original HK theorems, isorbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.

Derivation and formalism

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As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (theBorn–Oppenheimer approximation), generating a static external potentialV, in which the electrons are moving. Astationary electronic state is then described by awavefunctionΨ(r₁, …,r_N) satisfying the many-electron time-independentSchrödinger equation

{\hat {H}}\Psi =\left[{\hat {T}}+{\hat {V}}+{\hat {U}}\right]\Psi =\left[\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\right)+\sum _{i=1}^{N}V(\mathbf {r} _{i})+\sum _{i<j}^{N}U\left(\mathbf {r} _{i},\mathbf {r} _{j}\right)\right]\Psi =E\Psi ,

where, for theN-electron system,Ĥ is theHamiltonian,E is the total energy, ${\hat {T}}$ is the kinetic energy, ${\hat {V}}$ is the potential energy from the external field due to positively charged nuclei, andÛ is the electron–electron interaction energy. The operators ${\hat {T}}$ andÛ are called universal operators, as they are the same for anyN-electron system, while ${\hat {V}}$ is system-dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction termÛ.

There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction inSlater determinants. While the simplest one is theHartree–Fock method, more sophisticated approaches are usually categorized aspost-Hartree–Fock methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.

Here DFT provides an appealing alternative, being much more versatile, as it provides a way to systematically map the many-body problem, withÛ, onto a single-body problem withoutÛ. In DFT the key variable is the electron densityn(r), which for anormalizedΨ is given by

n(\mathbf {r} )=N\int {\mathrm {d} }^{3}\mathbf {r} _{2}\cdots \int {\mathrm {d} }^{3}\mathbf {r} _{N}\,\Psi ^{*}(\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{N})\Psi (\mathbf {r} ,\mathbf {r} _{2},\dots ,\mathbf {r} _{N}).

This relation can be reversed, i.e., for a given ground-state densityn₀(r) it is possible, in principle, to calculate the corresponding ground-state wavefunctionΨ₀(r₁, …,r_N). In other words,Ψ is a uniquefunctional ofn₀,^[15]

\Psi _{0}=\Psi [n_{0}],

and consequently the ground-stateexpectation value of an observableÔ is also a functional ofn₀:

O[n_{0}]={\big \langle }\Psi [n_{0}]{\big |}{\hat {O}}{\big |}\Psi [n_{0}]{\big \rangle }.

In particular, the ground-state energy is a functional ofn₀:

E_{0}=E[n_{0}]={\big \langle }\Psi [n_{0}]{\big |}{\hat {T}}+{\hat {V}}+{\hat {U}}{\big |}\Psi [n_{0}]{\big \rangle },

where the contribution of the external potential ${\big \langle }\Psi [n_{0}]{\big |}{\hat {V}}{\big |}\Psi [n_{0}]{\big \rangle }$ can be written explicitly in terms of the ground-state density $n_{0}$ :

V[n_{0}]=\int V(\mathbf {r} )n_{0}(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} .

More generally, the contribution of the external potential ${\big \langle }\Psi {\big |}{\hat {V}}{\big |}\Psi {\big \rangle }$ can be written explicitly in terms of the density $n {\displaystyle n}$ :

V[n]=\int V(\mathbf {r} )n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} .

The functionalsT[n] andU[n] are called universal functionals, whileV[n] is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified ${\hat {V}}$ , one then has to minimize the functional

E[n]=T[n]+U[n]+\int V(\mathbf {r} )n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r}

with respect ton(r), assuming one has reliable expressions forT[n] andU[n]. A successful minimization of the energy functional will yield the ground-state densityn₀ and thus all other ground-state observables.

The variational problems of minimizing the energy functionalE[n] can be solved by applying theLagrangian method of undetermined multipliers.^[18] First, one considers an energy functional that does not explicitly have an electron–electron interaction energy term,

E_{s}[n]={\big \langle }\Psi _{\text{s}}[n]{\big |}{\hat {T}}+{\hat {V}}_{\text{s}}{\big |}\Psi _{\text{s}}[n]{\big \rangle },

where ${\hat {T}}$ denotes the kinetic-energy operator, and ${\hat {V}}_{\text{s}}$ is an effective potential in which the particles are moving. Based on $E_{s}$ ,Kohn–Sham equations of this auxiliary noninteracting system can be derived:

\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{\text{s}}(\mathbf {r} )\right]\varphi _{i}(\mathbf {r} )=\varepsilon _{i}\varphi _{i}(\mathbf {r} ),

which yields theorbitalsφ_i that reproduce the densityn(r) of the original many-body system

n(\mathbf {r} )=\sum _{i=1}^{N}{\big |}\varphi _{i}(\mathbf {r} ){\big |}^{2}.

The effective single-particle potential can be written as

V_{\text{s}}(\mathbf {r} )=V(\mathbf {r} )+\int {\frac {n(\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} ^{3}\mathbf {r} '+V_{\text{XC}}[n(\mathbf {r} )],

where $V(\mathbf {r} )$ is the external potential, the second term is the Hartree term describing the electron–electronCoulomb repulsion, and the last termV_XC is the exchange–correlation potential. Here,V_XC includes all the many-particle interactions. Since the Hartree term andV_XC depend onn(r), which depends on theφ_i, which in turn depend onV_s, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e.,iterative) way. Usually one starts with an initial guess forn(r), then calculates the correspondingV_s and solves the Kohn–Sham equations for theφ_i. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation calledHarris functional DFT is an alternative approach to this.

Notes

The one-to-one correspondence between electron density and single-particle potential is not so smooth. It contains kinds of non-analytic structure.E_s[n] contains kinds of singularities, cuts and branches. This may indicate a limitation of our hope for representing exchange–correlation functional in a simple analytic form.
It is possible to extend the DFT idea to the case of theGreen functionG instead of the densityn. It is called asLuttinger–Ward functional (or kinds of similar functionals), written asE[G]. However,G is determined not as its minimum, but as its extremum. Thus we may have some theoretical and practical difficulties.
There is no one-to-one correspondence between one-bodydensity matrixn(r,r′) and the one-body potentialV(r,r′). (All the eigenvalues ofn(r,r′) are 1.) In other words, it ends up with a theory similar to the Hartree–Fock (or hybrid) theory.

Relativistic formulation (ab initio functional forms)

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The same theorems can be proven in the case of relativistic electrons, thereby providing generalization of DFT for the relativistic case. Unlike the nonrelativistic theory, in the relativistic case it is possible to derive a few exact and explicit formulas for the relativistic density functional.

Let one consider an electron in thehydrogen-like ion obeying the relativisticDirac equation. The HamiltonianH for a relativistic electron moving in the Coulomb potential can be chosen in the following form (atomic units are used):

H=c({\boldsymbol {\alpha }}\cdot \mathbf {p} )+eV+mc^{2}\beta ,

whereV = −eZ/r is the Coulomb potential of a pointlike nucleus,p is a momentum operator of the electron, ande,m andc are theelementary charge,electron mass and thespeed of light respectively, and finallyα andβ are a set ofDirac 2 × 2 matrices:

{\begin{aligned}{\boldsymbol {\alpha }}&={\begin{pmatrix}0&{\boldsymbol {\sigma }}\\{\boldsymbol {\sigma }}&0\end{pmatrix}},\\\beta &={\begin{pmatrix}I&0\\0&-I\end{pmatrix}}.\end{aligned}}

To find out theeigenfunctions and corresponding energies, one solves the eigenfunction equation

H\Psi =E\Psi ,

whereΨ = (Ψ(1), Ψ(2), Ψ(3), Ψ(4))^T is a four-componentwavefunction, andE is the associated eigenenergy. It is demonstrated in Brack (1983)^[19] that application of thevirial theorem to the eigenfunction equation produces the following formula for the eigenenergy of any bound state:

E=mc^{2}\langle \Psi |\beta |\Psi \rangle =mc^{2}\int {\big |}\Psi (1){\big |}^{2}+{\big |}\Psi (2){\big |}^{2}-{\big |}\Psi (3){\big |}^{2}-{\big |}\Psi (4){\big |}^{2}\,\mathrm {d} \tau ,

and analogously, the virial theorem applied to the eigenfunction equation with the square of the Hamiltonian yields

E^{2}=m^{2}c^{4}+emc^{2}\langle \Psi |V\beta |\Psi \rangle .

It is easy to see that both of the above formulae represent density functionals. The former formula can be easily generalized for the multi-electron case.

One may observe that both of the functionals written above do not have extremals, of course, if a reasonably wide set of functions is allowed for variation. Nevertheless, it is possible to design a density functional with desired extremal properties out of those ones. Let us make it in the following way:

F[n]={\frac {1}{mc^{2}}}\left(mc^{2}\int n\,d\tau -{\sqrt {m^{2}c^{4}+emc^{2}\int Vn\,d\tau }}\right)^{2}+\delta _{n,n_{e}}mc^{2}\int n\,d\tau ,

wheren_e inKronecker delta symbol of the second term denotes any extremal for the functional represented by the first term of the functionalF. The second term amounts to zero for any function that is not an extremal for the first term of functionalF. To proceed further we'd like to find Lagrange equation for this functional. In order to do this, we should allocate a linear part of functional increment when the argument function is altered:

F[n_{e}+\delta n]={\frac {1}{mc^{2}}}\left(mc^{2}\int (n_{e}+\delta n)\,d\tau -{\sqrt {m^{2}c^{4}+emc^{2}\int V(n_{e}+\delta n)\,d\tau }}\right)^{2}.

Deploying written above equation, it is easy to find the following formula for functional derivative:

{\frac {\delta F[n_{e}]}{\delta n}}=2A-{\frac {2B^{2}+AeV(\tau _{0})}{B}}+eV(\tau _{0}),

whereA =mc²∫n_e dτ, andB =√m²c⁴ +emc²∫Vn_e dτ, andV(τ₀) is a value of potential at some point, specified by support of variation functionδn, which is supposed to be infinitesimal. To advance toward Lagrange equation, we equate functional derivative to zero and after simple algebraic manipulations arrive to the following equation:

2B(A-B)=eV(\tau _{0})(A-B).

Apparently, this equation could have solution only ifA =B. This last condition provides us with Lagrangeequation for functionalF, which could be finally written down in the following form:

\left(mc^{2}\int n\,d\tau \right)^{2}=m^{2}c^{4}+emc^{2}\int Vn\,d\tau .

Solutions of this equation represent extremals for functionalF. It's easy to see that all real densities,that is, densities corresponding to the bound states of the system in question, are solutions of written above equation, which could be called the Kohn–Sham equation in this particular case. Looking back onto the definition of the functionalF, we clearly see that the functional produces energy of the system for appropriate density, because the first term amounts to zero for such density and the second one delivers the energy value.

Approximations (exchange–correlation functionals)

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The major problem with DFT is that the exact functionals for exchange and correlation are not known, except for thefree-electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately.^[20] One of the simplest approximations is thelocal-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

E_{\text{XC}}^{\text{LDA}}[n]=\int \varepsilon _{\text{XC}}(n)n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} .

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electronspin:

E_{\text{XC}}^{\text{LSDA}}[n_{\uparrow },n_{\downarrow }]=\int \varepsilon _{\text{XC}}(n_{\uparrow },n_{\downarrow })n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} .

In LDA, the exchange–correlation energy is typically separated into the exchange part and the correlation part:ε_XC =ε_X +ε_C. The exchange part is called the Dirac (or sometimes Slater)exchange, which takes the formε_X ∝n^1/3. There are, however, many mathematical forms for the correlation part. Highly accurate formulae for the correlation energy densityε_C(n_↑,n_↓) have been constructed fromquantum Monte Carlo simulations ofjellium.^[21] Although unrelated to the Monte Carlo simulation, the two variants provide comparable accuracy.^[22]

The LDA assumes that the density is the same everywhere. Because of this, the LDA has a tendency to underestimate the exchange energy and over-estimate the correlation energy.^[23] The errors due to the exchange and correlation parts tend to compensate each other to a certain degree. To correct for this tendency, it is common to expand in terms of the gradient of the density in order to account for the non-homogeneity of the true electron density. This allows corrections based on the changes in density away from the coordinate. These expansions are referred to as generalized gradient approximations (GGA)^[24]^[25]^[26] and have the following form:

E_{\text{XC}}^{\text{GGA}}[n_{\uparrow },n_{\downarrow }]=\int \varepsilon _{\text{XC}}(n_{\uparrow },n_{\downarrow },\nabla n_{\uparrow },\nabla n_{\downarrow })n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} .

Using the latter (GGA), very good results for molecular geometries and ground-state energies have been achieved.

Potentially more accurate than the GGA functionals are the meta-GGA functionals, a natural development after the GGA (generalized gradient approximation). Meta-GGA DFT functional in its original form includes the second derivative of the electron density (the Laplacian), whereas GGA includes only the density and its first derivative in the exchange–correlation potential.

Functionals of this type are, for example, TPSS and theMinnesota Functionals. These functionals include a further term in the expansion, depending on the density, the gradient of the density and theLaplacian (second derivative) of the density.

Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated fromHartree–Fock theory. Functionals of this type are known ashybrid functionals.

Generalizations to include magnetic fields

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The DFT formalism described above breaks down, to various degrees, in the presence of a vector potential, i.e. amagnetic field. In such a situation, the one-to-one mapping between the ground-state electron density and wavefunction is lost. Generalizations to include the effects of magnetic fields have led to two theories: current density functional theory (CDFT) and magnetic field density functional theory (BDFT). In both these theories, the functional used for the exchange and correlation must be generalized to include more than just the electron density. In current density functional theory, developed byVignale and Rasolt,^[17] the functionals become dependent on both the electron density and the paramagnetic current density. In magnetic field density functional theory, developed by Salsbury, Grayce and Harris,^[27] the functionals depend on the electron density and the magnetic field, and the functional form can depend on the form of the magnetic field. In both of these theories it has been difficult to develop functionals beyond their equivalent to LDA, which are also readily implementable computationally.

Applications

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C₆₀ withisosurface of ground-state electron density as calculated with DFT

In general, density functional theory finds increasingly broad application in chemistry and materials science for the interpretation and prediction of complex system behavior at an atomic scale. Specifically, DFT computational methods are applied for synthesis-related systems and processing parameters. In such systems, experimental studies are often encumbered by inconsistent results and non-equilibrium conditions. Examples of contemporary DFT applications include studying the effects of dopants on phase transformation behavior in oxides, magnetic behavior in dilute magnetic semiconductor materials, and the study of magnetic and electronic behavior in ferroelectrics anddilute magnetic semiconductors.^[2]^[28] It has also been shown that DFT gives good results in the prediction of sensitivity of some nanostructures to environmental pollutants likesulfur dioxide^[29] oracrolein,^[30] as well as prediction of mechanical properties.^[31]

In practice, Kohn–Sham theory can be applied in several distinct ways, depending on what is being investigated. In solid-state calculations, the local density approximations are still commonly used along withplane-wave basis sets, as an electron-gas approach is more appropriate for electrons delocalised through an infinite solid. In molecular calculations, however, more sophisticated functionals are needed, and a huge variety of exchange–correlation functionals have been developed for chemical applications. Some of these are inconsistent with the uniform electron-gas approximation; however, they must reduce to LDA in the electron-gas limit. Among physicists, one of the most widely used functionals is the revised Perdew–Burke–Ernzerhof exchange model (a direct generalized gradient parameterization of the free-electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gas-phase molecular calculations. In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part). Even more widely used is B3LYP, which is ahybrid functional in which the exchange energy, in this case from Becke's exchange functional, is combined with the exact energy from Hartree–Fock theory. Along with the component exchange and correlation functionals, three parameters define the hybrid functional, specifying how much of the exact exchange is mixed in. The adjustable parameters in hybrid functionals are generally fitted to a "training set" of molecules. Although the results obtained with these functionals are usually sufficiently accurate for most applications, there is no systematic way of improving them (in contrast to some of the traditionalwavefunction-based methods likeconfiguration interaction orcoupled cluster theory). In the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.

Density functional theory is generally highly accurate but highly computationally-expensive. DFT has been used with machine learning techniques - especially graph neural networks - to createmachine learning potentials. These graph neural networks approximate DFT, with the aim of achieving similar accuracies with much less computation, and are especially beneficial for large systems. They are trained using DFT-calculated properties of a known set of molecules.

Thomas–Fermi model

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The predecessor to density functional theory was theThomas–Fermi model, developed independently by bothLlewellyn Thomas andEnrico Fermi in 1927. They used a statistical model to approximate the distribution of electrons in an atom. The mathematical basis postulated that electrons are distributed uniformly in phase space with two electrons in every $h^{3}$ of volume.^[32] For each element of coordinate space volume $\mathrm {d} ^{3}\mathbf {r}$ we can fill out a sphere of momentum space up to theFermi momentum $p_{\text{F}}$ ^[33]

{\tfrac {4}{3}}\pi p_{\text{F}}^{3}(\mathbf {r} ).

Equating the number of electrons in coordinate space to that in phase space gives

n(\mathbf {r} )={\frac {8\pi }{3h^{3}}}p_{\text{F}}^{3}(\mathbf {r} ).

Solving forp_F and substituting into theclassical kinetic energy formula then leads directly to a kinetic energy represented as afunctional of the electron density:

{\begin{aligned}t_{\text{TF}}[n]&={\frac {p^{2}}{2m_{e}}}\propto {\frac {(n^{1/3})^{2}}{2m_{e}}}\propto n^{2/3}(\mathbf {r} ),\\T_{\text{TF}}[n]&=C_{\text{F}}\int n(\mathbf {r} )n^{2/3}(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} =C_{\text{F}}\int n^{5/3}(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r} ,\end{aligned}}

where

C_{\text{F}}={\frac {3h^{2}}{10m_{e}}}\left({\frac {3}{8\pi }}\right)^{2/3}.

As such, they were able to calculate theenergy of an atom using this kinetic-energy functional combined with the classical expressions for the nucleus–electron and electron–electron interactions (which can both also be represented in terms of the electron density).

Although this was an important first step, the Thomas–Fermi equation's accuracy is limited because the resulting kinetic-energy functional is only approximate, and because the method does not attempt to represent theexchange energy of an atom as a conclusion of thePauli principle. An exchange-energy functional was added byPaul Dirac in 1928.

However, the Thomas–Fermi–Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect ofelectron correlation.

Edward Teller (1962) showed that Thomas–Fermi theory cannot describe molecular bonding. This can be overcome by improving the kinetic-energy functional.

The kinetic-energy functional can be improved by adding thevon Weizsäcker (1935) correction:^[34]^[35]

T_{\text{W}}[n]={\frac {\hbar ^{2}}{8m}}\int {\frac {|\nabla n(\mathbf {r} )|^{2}}{n(\mathbf {r} )}}\,\mathrm {d} ^{3}\mathbf {r} .

Hohenberg–Kohn theorems

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The Hohenberg–Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential.

Theorem 1. The external potential (and hence the total energy), is a unique functional of the electron density.

If two systems of electrons, one trapped in a potential

v_{1}(\mathbf {r} )

and the other in

v_{2}(\mathbf {r} )

, have the same ground-state density

n(\mathbf {r} )

, then

v_{1}(\mathbf {r} )-v_{2}(\mathbf {r} )

is necessarily a constant.

Corollary 1: the ground-state density uniquely determines the potential and thus all properties of the system, including the many-body wavefunction. In particular, the HK functional, defined as

F[n]=T[n]+U[n]

, is a universal functional of the density (not depending explicitly on the external potential).

Corollary 2: In light of the fact that the sum of the occupied energies provides the energy content of the Hamiltonian, a unique functional of the ground state charge density, the spectrum of the Hamiltonian is also a unique functional of the ground state charge density.^[13]

Theorem 2. The functional that delivers the ground-state energy of the system gives the lowest energy if and only if the input density is the true ground-state density.

In other words, the energy content of the Hamiltonian reaches its absolute minimum, i.e., the ground state, when the charge density is that of the ground state.

For any positive integer

N {\displaystyle N}

and potential

v(\mathbf {r} )

, a density functional

F[n]

exists such that

E_{(v,N)}[n]=F[n]+\int v(\mathbf {r} )n(\mathbf {r} )\,\mathrm {d} ^{3}\mathbf {r}

reaches its minimal value at the ground-state density of

N {\displaystyle N}

electrons in the potential

v(\mathbf {r} )

. The minimal value of

E_{(v,N)}[n]

is then the ground-state energy of this system.

Pseudo-potentials

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The many-electronSchrödinger equation can be very much simplified if electrons are divided in two groups:valence electrons and inner coreelectrons. The electrons in the inner shells are strongly bound and do not play a significant role in the chemical binding ofatoms; they also partiallyscreen the nucleus, thus forming with thenucleus an almost inert core. Binding properties are almost completely due to the valence electrons, especially in metals and semiconductors. This separation suggests that inner electrons can be ignored in a large number of cases, thereby reducing the atom to an ionic core that interacts with the valence electrons. The use of an effective interaction, apseudopotential, that approximates the potential felt by the valence electrons, was first proposed by Fermi in 1934 and Hellmann in 1935. In spite of the simplification pseudo-potentials introduce in calculations, they remained forgotten until the late 1950s.

Pseudopotential representing the effective core charge. The physical image of the system with the accurate wavefunction and potential is replaced by a pseudo-wavefunction and a pseudopotential up to a cutoff value. In the image on the right, core electrons and atomic core are considered as the effective core in DFT calculations

Ab initio pseudo-potentials

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A crucial step toward more realistic pseudo-potentials was given by William C. Topp andJohn Hopfield,^[36] who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudo-potentials are obtained inverting the free-atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo-wavefunctions to coincide with the true valence wavefunctions beyond a certain distancer_l. The pseudo-wavefunctions are also forced to have the same norm (i.e., the so-called norm-conserving condition) as the true valence wavefunctions and can be written as

{\begin{aligned}R_{l}^{\text{PP}}(r)&=R_{nl}^{\text{AE}}(r),{\text{ for }}r>r_{l},\\\int _{0}^{r_{l}}{\big |}R_{l}^{\text{PP}}(r){\big |}^{2}r^{2}\,\mathrm {d} r&=\int _{0}^{r_{l}}{\big |}R_{nl}^{\text{AE}}(r){\big |}^{2}r^{2}\,\mathrm {d} r,\end{aligned}}

whereR_l(r) is the radial part of thewavefunction withangular momentuml, and PP and AE denote the pseudo-wavefunction and the true (all-electron) wavefunction respectively. The indexn in the true wavefunctions denotes thevalence level. The distancer_l beyond which the true and the pseudo-wavefunctions are equal is also dependent onl.

Electron smearing

[edit]

The electrons of a system will occupy the lowest Kohn–Sham eigenstates up to a given energy level according to theAufbau principle. This corresponds to the steplikeFermi–Dirac distribution at absolute zero. If there are several degenerate or close to degenerate eigenstates at theFermi level, it is possible to get convergence problems, since very small perturbations may change the electron occupation. One way of damping these oscillations is tosmear the electrons, i.e. allowing fractional occupancies.^[37] One approach of doing this is to assign a finite temperature to the electron Fermi–Dirac distribution. Other ways is to assign a cumulative Gaussian distribution of the electrons or using a Methfessel–Paxton method.^[38]^[39]

Classical density functional theory

[edit]

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.^[40]^[41]^[42]^[43] The classical non-relativistic method is correct forclassical fluids with particle velocities less than the speed of light andthermal de Broglie wavelength smaller than the distance between particles. The theory is based on thecalculus 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 fluidphase transitions, ordering in complex liquids, physical characteristics ofinterfaces andnanomaterials. Since the 1970s it has been applied to the fields ofmaterials science,biophysics,chemical engineering andcivil engineering.^[44] Computational costs are much lower than formolecular 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.^[45] 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 describingthermodynamic equilibrium states of many-particle systems with nonuniform density.^[46] Classical DFT has its roots in theories such as thevan der Waals theory for theequation of state and thevirial expansion method for the pressure. In order to account forcorrelation 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 byLeonard Ornstein andFrits Zernike in 1914.^[47] The connection to the densitypair distribution function was given by theOrnstein–Zernike equation. The importance of correlation for thermodynamic properties was explored through density distribution functions. Thefunctional 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 thePercus–Yevick equation linking the density distribution function and the direct correlation.^[48] Other closure relations were also proposed;theClassical-map hypernetted-chain method, theBBGKY hierarchy. In the late 1970s classical DFT was applied to the liquid–vapor interface and the calculation ofsurface tension. Other applications followed: thefreezing of simple fluids, formation of theglass phase, the crystal–melt interface anddislocation in crystals, properties ofpolymer systems, andliquid crystal ordering. Classical DFT was applied tocolloid dispersions, which were discovered to be good models for atomic systems.^[49] 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 andfluid 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 modelinteractions between particles. The spatially dependentdensity determines the local structure and composition of the material. It is determined as a function that optimizes the thermodynamic potential of thegrand canonical ensemble. Thegrand potential is evaluated as the sum of theideal-gas term with the contribution from external fields and an excessthermodynamic 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 functionalTaylor expansion. The excess free energy is then a sum of the contributions froms-body interactions with density-dependent effective potentials representing the interactions betweens 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 localchemical potential. The functional is not in general aconvex functional of the density; solutions may not be localminima. Limiting to low-order corrections in the local density is a well-known problem, although the results agree (reasonably) well on comparison to experiment.

Avariational principle is used to determine the equilibrium density. It can be shown that for constant temperature and volume the correct equilibrium density minimizes thegrand potential functional $\Omega$ of thegrand canonical ensemble over density functions $n(\mathbf {r} )$ . In the language of functional differentiation (Mermin theorem):

{\frac {\delta \Omega }{\delta n(\mathbf {r} )}}=0.

TheHelmholtz free energy functional $F {\displaystyle F}$ is defined as $F=\Omega +\int d^{3}\mathbf {r} \,n(\mathbf {r} )\mu (\mathbf {r} )$ .Thefunctional derivative in the density function determines the local chemical potential: $\mu (\mathbf {r} )=\delta F(\mathbf {r} )/\delta n(\mathbf {r} )$ .In classical statistical mechanics thepartition function is a sum over probability for a given microstate ofN classical particles as measured by the Boltzmann factor in theHamiltonian 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. Acorrelation function is introduced to describe the effective interaction between particles.

Thes-body density distribution function is defined as the statisticalensemble average $\langle \dots \rangle$ of particle positions. It measures the probability to finds particles at points in space $\mathbf {r} _{1},\dots ,\mathbf {r} _{s}$ :

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 }.

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:

{\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}).

Theradial distribution function withs = 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 $\Delta F$ from interactions between particles. In the grand ensemble the functional derivatives in the density yield the direct correlation functions $c_{s}$ :

{\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}).

The one-body direct correlation function plays the role of an effectivemean field. The functional derivative in density of the one-body direct correlation results in the direct correlation function between two particles $c_{2}$ . The direct correlation function is the correlation contribution to the change of local chemical potential at a point $\mathbf {r}$ for a density change at $\mathbf {r} '$ and is related to the work of creating density changes at various 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

\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} ').

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 ashard spheres and a term in a weak interaction such as the long rangeLondon 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 localizedGaussian function centered on crystal lattice points for the density in a solid, the hyperbolic function $\tanh(r)$ for interfacial density profiles.

Classical DFT has found many applications, for example:

developing new functional materials inmaterials science, in particularnanotechnology;
studying the properties of fluids atsurfaces and the phenomena ofwetting andadsorption;^[50]
understanding life processes inbiotechnology;
improvingfiltration methods for gases and fluids inchemical engineering;
fightingpollution of water and air in environmental science;
cell membranes by modelling complex systems withamphiphile compounds;
generating new procedures inmicrofluidics andnanofluidics.

The extension of classical DFT towards nonequilibrium systems is known as dynamical density functional theory (DDFT).^[51] DDFT allows to describe the time evolution of the one-body density $\rho ({\boldsymbol {r}},t)$ of a colloidal system, which is governed by the equation

{\frac {\partial \rho }{\partial t}}=\Gamma \nabla \cdot \left(\rho \nabla {\frac {\delta F}{\delta \rho }}\right)

with the mobility $\Gamma$ and the free energy $F {\displaystyle F}$ . 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.

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External links

[edit]

Walter Kohn, Nobel Laureate – Video interview with Walter on his work developing density functional theory by theVega Science Trust
Capelle, Klaus (2002). "A bird's-eye view of density-functional theory".arXiv:cond-mat/0211443.
Walter Kohn,Nobel Lecture
Argaman, Nathan; Makov, Guy (2000). "Density Functional Theory -- an introduction".American Journal of Physics.68 (2000):69–79.arXiv:physics/9806013.Bibcode:2000AmJPh..68...69A.doi:10.1119/1.19375.S2CID 119102923.
Electron Density Functional Theory – Lecture Notes
Density Functional Theory through Legendre Transformation Archived 2010-05-10 at theWayback Machine pdf
Burke, Kieron."The ABC of DFT"(PDF).
Modeling Materials Continuum, Atomistic and Multiscale Techniques, Book
NIST Jarvis-DFT
Clary, David C. (2024).Walter Kohn: From Kindertransport and Internment to DFT and the Nobel Prize. World Scientific Publishing.

Condensed matter physics

States of matter

Phase phenomena

Electrons in solids

Phenomena	Hall effect Quantum Hall effect Spin Hall effect Quantum spin Hall effect Berry phase Aharonov–Bohm effect Josephson effect Kondo effect
Theory	Drude model Free electron model Nearly free electron model Bloch's theorem Fermi liquid theory electronic band structure Anderson localization BCS theory tight binding model Hubbard model
Conduction	Insulator Mott insulator Semiconductor Semimetal Conductor Superconductor Topological insulator Spin gapless semiconductor
Couplings	Thermoelectricity Piezoelectricity Ferroelectricity Flexoelectricity Electrostriction