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# Density functional theory

Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

## Hohenberg-Kohn theorems

Although density functional theory has its conceptual roots in the Thomas-Fermi model described below, DFT was put on a firm theoretical footing by the Hohenberg-Kohn theorems (H-K). The first of these demonstrates the existence of a one-to-one mapping between the ground-state electron density and the ground-state wavefunction of a many-particle system. Further, the second H-K theorem proves that the ground-state density minimizes the total electronic energy of the system. The original H-K theorems held only for the ground state in the absence of magnetic field, although they have since been generalized.

The theorems can be extended to the time-dependent domain to derive time-dependent density functional theory (TDDFT), which can be also used to describe excited states.

The first Hohenberg-Kohn theorem is an existence theorem, stating that the mapping exists. That is, the H-K theorems tell us that the electron density that minimizes the energy according to the true total energy functional describes all that can be known about the electronic structure. The H-K theorems do not tell us what the true total-energy functional is, only that it exists.

The most common implementation of density functional theory is through the Kohn-Sham method, which maps the properties of the system onto the properties of a system containing non-interacting electrons under a different potential. The kinetic-energy functional of such a system of non-interacting electrons is known exactly. The exchange-correlation part of the total-energy functional remains unknown and must be approximated. Another approach, less popular than Kohn-Sham DFT (KS-DFT) but arguably more closely related to the spirit of the original H-K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the interacting system.

## Description of the theory

Traditional methods in electronic structure theory, in particular Hartree-Fock theory and its descendants, are based on the complicated many-electron wavefunction. The main objective of density functional theory is to replace the many-body electronic wavefunction with the electronic density as the basic quantity. Whereas the many-body wavefunction is dependent on $3N$ variables, three spatial variables for each of the $N$ electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically.

Within the framework of Kohn-Sham DFT, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas-Fermi model, and from fits to the correlation energy for a uniform electron gas.

DFT has been very popular for calculations in solid state physics since the 1970s. In many cases DFT with the local-density approximation gives quite satisfactory results, for solid-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum mechanical many-body problem. However, it was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions. DFT is now a leading method for electronic structure calculations in both fields. Despite the improvements in DFT, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces and some other strongly correlated systems; and in calculations of the band gap in semiconductors. Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g., interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.

## Derivation and formalism

As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born-Oppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction $Psi\left(vec r_1,dots,vec r_N\right)$ satisfying the many-electron Schrödinger equation

where $hat H$ is the electronic molecular Hamiltonian, $N$ is the number of electrons, $hat T$ is the $N$-electron kinetic energy, $hat V$ is the $N$-electron potential energy from the external field, and $hat U$ is the electron-electron interaction energy for the $N$-electron system. The operators $hat T$ and $hat U$ are so-called universal operators as they are the same for any system, while $hat V$ is system dependent or non-universal. The difference between having separable single-particle problems and the much more complicated many-particle problem arises from the interaction term $hat U$.

There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the Hartree-Fock method, more sophisticated approaches are usually categorized as post-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, with $hat U$, onto a single-body problem without $hat U$. In DFT the key variable is the particle density $n\left(vec r\right)$, which for a normalized $,!Psi$ is given by

$n\left(vec r\right) = N int\left\{rm d\right\}^3r_2 int\left\{rm d\right\}^3r_3 cdots int\left\{rm d\right\}^3r_N Psi^*\left(vec r,vec r_2,dots,vec r_N\right) Psi\left(vec r,vec r_2,dots,vec r_N\right)$

This relation can be reversed; i.e., for a given ground-state density $n_0\left(vec r\right)$ it is possible, in principle, to calculate the corresponding ground-state wavefunction $Psi_0\left(vec r_1,dots,vec r_N\right)$. In other words, $,!Psi_0$ is a unique functional of $,!n_0$,

$,!Psi_0 = Psi\left[n_0\right]$

and consequently the ground-state expectation value of an observable $,hat O$ is also a functional of $,!n_0$

$O\left[n_0\right] = leftlangle Psi\left[n_0\right] left| hat O right| Psi\left[n_0\right] rightrangle$

In particular, the ground-state energy is a functional of $,!n_0$

$E_0 = E\left[n_0\right] = leftlangle Psi\left[n_0\right] left| hat T + hat V + hat U right| Psi\left[n_0\right] rightrangle$

where the contribution of the external potential $leftlangle Psi\left[n_0\right] left|hat V right| Psi\left[n_0\right] rightrangle$ can be written explicitly in terms of the ground-state density $,!n_0$

$V\left[n_0\right] = int V\left(vec r\right) n_0\left(vec r\right)\left\{rm d\right\}^3r$

More generally, the contribution of the external potential $leftlangle Psi left|hat V right| Psi rightrangle$ can be written explicitly in terms of the density $,!n$,

$V\left[n\right] = int V\left(vec r\right) n\left(vec r\right)\left\{rm d\right\}^3r$

The functionals $,!T\left[n\right]$ and $,!U\left[n\right]$ are called universal functionals, while $,!V\left[n\right]$ 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

$E\left[n\right] = T\left[n\right]+ U\left[n\right] + int V\left(vec r\right) n\left(vec r\right)\left\{rm d\right\}^3r$

with respect to $n\left(vec r\right)$, assuming one has got reliable expressions for $,!T\left[n\right]$ and $,!U\left[n\right]$. A successful minimization of the energy functional will yield the ground-state density $,!n_0$ and thus all other ground-state observables.

The variational problem of minimizing the energy functional $,!E\left[n\right]$ can be solved by applying the Lagrangian method of undetermined multipliers. Hereby, one uses the fact that the functional in the equation above can be written as a fictitious density functional of a non-interacting system

$E_s\left[n\right] = leftlangle Psi_s\left[n\right] left| hat T_s + hat V_s right| Psi_s\left[n\right] rightrangle$

where $hat T_s$ denotes the non-interacting kinetic energy and $hat V_s$ is an external effective potential in which the particles are moving. Obviously, $n_s\left(vec r\right) stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} n\left(vec r\right)$ if $hat V_s$ is chosen to be

$hat V_s = hat V + hat U + left\left(hat T - hat T_sright\right)$

Thus, one can solve the so-called Kohn-Sham equations of this auxiliary non-interacting system,

$left\left[-frac\left\{hbar^2\right\}\left\{2m\right\}nabla^2+V_s\left(vec r\right)right\right] phi_i\left(vec r\right) = epsilon_i phi_i\left(vec r\right)$

which yields the orbitals $,!phi_i$ that reproduce the density $n\left(vec r\right)$ of the original many-body system

$n\left(vec r \right) stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} n_s\left(vec r\right)= sum_i^N left|phi_i\left(vec r\right)right|^2$

The effective single-particle potential can be written in more detail as

$V_s\left(vec r\right) = V\left(vec r\right) + int frac\left\{e^2n_s\left(vec r,\text{'}\right)\right\}$
> {rm d}^3r' + V_{rm XC}[n_s(vec r)]

where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term $,!V_\left\{rm XC\right\}$ is called the exchange-correlation potential. Here, $,!V_\left\{rm XC\right\}$ includes all the many-particle interactions. Since the Hartree term and $,!V_\left\{rm XC\right\}$ depend on $n\left(vec r \right)$, which depends on the $,!phi_i$, which in turn depend on $,!V_s$, 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 for $n\left(vec r\right)$, then calculates the corresponding $,!V_s$ and solves the Kohn-Sham equations for the $,!phi_i$. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached.

## Approximations (Exchange-correlation functionals)

The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

$E_\left\{rm XC\right\}\left[n\right]=intepsilon_\left\{rm XC\right\}\left(n\right)n \left(vec\left\{r\right\}\right) \left\{rm d\right\}^3r$

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

$E_\left\{rm XC\right\}\left[n_uparrow,n_downarrow\right]=intepsilon_\left\{rm XC\right\}\left(n_uparrow,n_downarrow\right)n \left(vec\left\{r\right\}\right)\left\{rm d\right\}^3r$

Highly accurate formulae for the exchange-correlation energy density $epsilon_\left\{rm XC\right\}\left(n_uparrow,n_downarrow\right)$ have been constructed from quantum Monte Carlo simulations of a free electron model.

Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate:

$E_\left\{XC\right\}\left[n_uparrow,n_downarrow\right]=intepsilon_\left\{XC\right\}\left(n_uparrow,n_downarrow,vec\left\{nabla\right\}n_uparrow,vec\left\{nabla\right\}n_downarrow\right)$
n (vec{r}) {rm d}^3r

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 meta-GGA functions. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the Laplacian (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 from Hartree-Fock theory. Functionals of this type are known as hybrid functionals.

## Generalizations to include magnetic fields

The DFT formalism described above above breaks down, to various degrees, in the presence of a vector potential, i.e. a magnetic 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 different theories: current density functional theory (CDFT) and magnetic field 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 by Vignale and Rasolt, 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, 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

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 with plane wave basis sets, as an electron gas approach is more appropriate 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, probably the most widely used functional is the revised Perdew-Burke-Ernzerhof exchange model (a direct generalized-gradient parametrization 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 a hybrid 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 funсtionals, 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. Unfortunately, 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 traditional wavefunction-based methods like configuration interaction or coupled cluster theory). Hence in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.

For molecular applications, in particular for hybrid functionals, Kohn-Sham DFT methods are usually implemented just like Hartree-Fock itself.

## Thomas-Fermi model

The predecessor to density functional theory was the Thomas-Fermi model, developed by Thomas and 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 h3 of volume. For each element of coordinate space volume d3r we can fill out a sphere of momentum space up to the Fermi momentum pf

$\left(4/3\right)pi p_f^3\left(vec\left\{r\right\}\right)$

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

$n\left(vec\left\{r\right\}\right)=frac\left\{8pi\right\}\left\{3h^3\right\}p_f^3\left(vec\left\{r\right\}\right)$

Solving for pf and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:

$T_\left\{TF\right\}\left[n\right]=C_Fint n^\left\{5/3\right\}\left(vec\left\{r\right\}\right) d^3r$
where   $C_F=frac\left\{3h^2\right\}\left\{10m\right\}left\left(frac\left\{3\right\}\left\{8pi\right\}right\right)^\left\{2/3\right\}$
As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nuclear-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 the exchange energy of an atom as a conclusion of the Pauli principle. An exchange energy functional was added by 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 of electron correlation.

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 the Weizsäcker (1935) correction:

$T_W\left[n\right]=frac\left\{1\right\}\left\{8\right\}frac\left\{hbar^2\right\}\left\{m\right\}intfrac\left\{|nabla n\left(vec\left\{r\right\}\right)|^2\right\}\left\{n\left(vec\left\{r\right\}\right)\right\}dr$

## Books on DFT

• R. Dreizler, E. Gross, Density Functional Theory (Plenum Press, New York, 1995).
• C. Fiolhais, F. Nogueira, M. Marques (eds.), A Primer in Density Functional Theory (Springer-Verlag, 2003).
• Kohanoff, J., Electronic Structure Calculations for Solids and Molecules: Theory and Computational Methods (Cambridge University Press, 2006).
• W. Koch, M. C. Holthausen, A Chemist's Guide to Density Functional Theory (Wiley-VCH, Weinheim, ed. 2, 2002).
• R. G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989), ISBN 0-19-504279-4, ISBN 0-19-509276-7 (pbk.).
• N.H. March, Electron Density Theory of Atoms and Molecules (Academic Press, 1992), ISBN 0-12-470525-1.