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Density functional theory

Density functional theory (DFT) is one of the most popular and successful quantum mechanical approaches to the many-body electronic structure calculations of molecular and condensed matter systems. Within the framework of DFT, the practically unsolvable many-body problem of interacting electrons is reduced to a solvable problem of a single electron moving in an averaged effective force field. This effective force field can be represented by a potential energy being created by all the other electrons as well as the atomic nuclei, which are seen as fixed in terms of the Born-Oppenheimer approximation.

Contents

1 Description of the Theory

2 Early Models

3 Derivation and Formalism

4 Approximations

5 Relativistic Generalization

6 Applications

7 References

8 Literature

Description of the Theory

In contrast to traditional methods like Hartree-Fock theory which are based on the complicated many-electron wavefunction DFT is written in terms of the electron density, giving this theory its name. DFT is an exact theory only for the free electron gas , while for the treatment of extended atomic systems various approximations have to be made. In many cases DFT gives quite satisfactory results in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum mechanical many-body problem.

DFT has been very popular for calculations in solid state physics since the 1970s. 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. DFT is now the leading method for electronic structure calculations in both fields.

However, there are still systems which are not described very well by DFT. One famous example is the false prediction of the band gap in semi-conductors. The method also fails to describe properly intermolecular interactions, especially van der Waals forces (dispersion).

Early Models

The first true density functional theory was developed by Thomas and Fermi in the 1920s. They calculated the energy of an atom by representing its kinetic energy as a functional of the electron density, combining this 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 was limited because it did not attempt to represent the exchange energy of an atom predicted by Hartree-Fock theory. An exchange energy functional was added by Dirac in 1928.

However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications because it is difficult to represent kinetic energy with a density functional, and it neglects electron correlation entirely.

Derivation and Formalism

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

$H \Psi = \left[{T}+{V}+{U}\right]\Psi = \left[\sum_i^N -\frac{\hbar^2}{2m}\nabla_i^2 + \sum_i^N V(\vec r_i) + \sum_{i<j}U(\vec r_i, \vec r_j)\right] \Psi = E \Psi$

where $\,\!N$ is the number of electrons and $\,\!U$ is the electron-electron interaction. The operators $\,\!T$ and $\,\!U$ are so-called universal operators as they are the same for any system, while $\,\!V$ is system dependent or non-universal. As one can see the actual difference between a single-particle problem and the much more complicated many-particle problem just arises from the interaction term $\,\!U$ . Now, there are many sophisticated methods for solving the many-body Schrödinger equation, e.g. there is diagrammatic perturbation theory in physics, while in quantum chemistry one often uses configuration interaction (CI) methods, based on the systematic expansion of the wave function in Slater determinants. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger 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 $\,\!U$ , onto a single-body problem without $\,\!U$ . In DFT the key variable is the particle density $n(\vec r)$ which is given by

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

Hohenberg and Kohn proved in 1964 [1] that the relation expressed above can be reversed, i.e. to a given ground state density $n_0(\vec r)$ it is in principle possible to calculate the corresponding ground state wave function $\Psi_0(\vec r_1,...,\vec r_N)$ . In other words, $\,\!\Psi_0$ is a unique functional of $\,\!n_0$ , i.e.

$\,\!\Psi_0 = \Psi_0[n_0]$

and consequently all other ground state observables $\,\!O$ are also functionals of $\,\!n_0$

$\left\langle O \right\rangle[n_0] = \left\langle \Psi_0[n_0] \left| O \right| \Psi_0[n_0] \right\rangle$

From this follows in particular, that also the ground state energy is a functional of $\,\!n_0$

$E_0 = E[n_0] = \left\langle \Psi_0[n_0] \left| T+V+U \right| \Psi_0[n_0] \right\rangle$ ,

where the contribution of the external potential $\left\langle \Psi_0[n_0] \left|V\right| \Psi_0[n_0] \right\rangle$ can be written explicitly in terms of the density

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

The functionals $\,\!T[n]$ and $\,\!U[n]$ are called universal functionals while $\,\!V[n]$ is obviously non-universal, as it depends on the system under study. Having specified a system, i.e. $\,\!V$ is known, one then has to minimise the functional

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

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

The variational problem of minimising the energy functional $\,\!E[n]$ can be solved by applying the Lagrangian method of undetermined multipliers, which was done by Kohn and Sham in 1965 [2]. 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

$E_s[n] = \left\langle \Psi_s[n] \left| T_s+V_s \right| \Psi_s[n] \right\rangle$ ,

where $\,\!T_s$ denotes the non-interacting kinetic energy and $\,\!V_s$ is an external effective potential in which the particles are moving. Obviously, $n_s(\vec r)\equiv n(\vec r)$ if $\,\!V_s$ is chosen to be

$V_s = V + U + \left(T_s - T\right)$ .

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

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

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

$n(\vec r )\equiv n_s(\vec r)= \sum_i^N \left|\phi_i(\vec r)\right|^2$ .

The effective single-particle potential $\,\!V_s$ can be written in more detail as

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

where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term $\,\!V_{\rm XC}$ is called exchange correlation potential. Here, $\,\!V_{\rm XC}$ includes all the many particle interactions. Since the Hartree term and $\,\!V_{\rm XC}$ depend on $n(\vec r )$ , 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 way. Usually one starts with an initial guess for $n(\vec r)$ , then one 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

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. 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. Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate. Using the latter (GGA) very good results for molecular geometries and ground state energies have been achieved. Many further incremental improvements have been made to DFT by developing better representations of the functionals.

Relativistic Generalization

The relativistic generalization of the DFT formalism leads to a current density functional theory.

Applications

In practice, Kohn-Sham theory can be applied in two distinct ways depending on what is being investigated. In the solid state, plane wave basis sets are used with periodic boundary conditions. Moreover, great emphasis is placed upon remaining consistent with the idealised model of a 'uniform electron gas', which exhibits similar behaviour to an infinite solid. In the gas and liquid phases, this emphasis is relaxed somewhat, as the uniform electron gas is a poor model for the behaviour of discrete atoms and molecules. Because of the relaxed constraints, a huge variety of exchange-correlation functionals have been developed for chemical applications. The most famous and popular of these is known as B3LYP [3-5]. The adjustable parameters of these functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually relatively 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 method). Hence, in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiment.

References

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864
[2] W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133
[3] A. D. Becke, J. Chem. Phys. 38 (1998) 3089
[4] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37 (1988) 785
[5] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98 (1994) 11623

Literature

Klaus Capelle, A bird's-eye view of density-functional theory

Last updated: 10-24-2004 05:10:45

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