Density functional theory
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Density functional theory (DFT) is one of the most popular approaches to quantum mechanical many-body electronic structure calculations of molecular and condensed matter systems.
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Description of the theory
Traditional methods in electronic structure, like Hartree-Fock theory 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 <math>3N<math> variables, three spatial variables for each of the <math>N<math> electrons, the density is only a function of three variables and is a simpler quantity to deal with both conceptually and practically.
Although density functional theory has its conceptual roots in the Thomas-Fermi model, DFT was not put on a firm theoretical footing until the Hohenberg-Kohn (HK) theorem which 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. Moreover, the HK theorem proves that the ground state density minimizes the total electronic energy of the system. Since the HK theorem holds only for the ground state, DFT is also a ground state theorem.
The Hohenberg-Kohn theorem is only an existence theorem, stating that the mapping exists, but does not provide any such exact mapping. It is in these mappings that approximations are made. The most popular such mapping is local-density approximation (LDA) which gives an approximate mapping from the density of the system to the total energy. The LDA is exact for the uniform electron gas, also known as jellium.
In practise, the HK theorem is not often used to directly make calculations. Instead, the most common present-day implementation of density functional theory is through the Kohn-Sham method. 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.
In many cases DFT with the local-density approximation 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 a leading method for electronic structure calculations in both fields.
However, there are still systems which are not described very well by LDA. The LDA fails to describe properly intermolecular interactions, especially van der Waals forces (dispersion). Another popularly cited result is the underestimation of the band gap in semiconductors, but this strictly, does not demonstrate a failure since DFT is a ground state theory, and the band gap is an excited state property.
Early models: Thomas-Fermi model
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 <math>\,\!V<math> in which the electrons are moving. A stationary electronic state is then described by a wave function <math>\Psi(\vec r_1,\dots,\vec r_N)<math> fulfilling the many-electron Schrödinger equation
- <math> 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 where <math>\,\!N<math> is the number of electrons and <math>\,\!U<math>
is the electron-electron interaction. The operators <math>\,\!T<math>
and <math>\,\!U<math> are so-called universal operators as they are the same
for any system, while <math>\,\!V<math> 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 <math>\,\!U<math>. 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 <math>\,\!U<math>, onto a single-body problem
without <math>\,\!U<math>. In DFT the key variable is the particle density
<math>n(\vec r)<math> which is given by
\Psi^*(\vec r,\vec r_2,\dots,\vec r_N) \Psi(\vec r,\vec r_2,\dots,\vec r_N).<math>
Hohenberg and Kohn proved in 1964 [1] that
the relation expressed above can be reversed,
i.e. to a given ground state density <math>n_0(\vec r)<math> it is
in principle possible to calculate the corresponding ground state
wave function <math>\Psi_0(\vec r_1,\dots,\vec r_N)<math>. In other words,
<math>\,\!\Psi_0<math> is a unique functional of <math>\,\!n_0<math>, i.e.
and consequently all other ground state observables <math>\,\!O<math> are also
functionals of <math>\,\!n_0<math>
\left\langle \Psi_0[n_0] \left| O \right| \Psi_0[n_0] \right\rangle.<math>
From this follows in particular, that also the ground state energy is a
functional of <math>\,\!n_0<math>
\left\langle \Psi_0[n_0] \left| T+V+U \right| \Psi_0[n_0] \right\rangle<math>,
where the contribution of the external potential
<math>\left\langle \Psi_0[n_0] \left|V\right| \Psi_0[n_0] \right\rangle<math>
can be written explicitly in terms of the density
The functionals <math>\,\!T[n]<math> and <math>\,\!U[n]<math>
are called universal functionals
while <math>\,\!V[n]<math> is obviously non-universal, as it depends on the
system under study. Having specified a system, i.e. <math>\,\!V<math> is known,
one then has to minimise the functional
with respect to <math>n(\vec r)<math>, assuming one has got reliable
expressions for <math>\,\!T[n]<math> and <math>\,\!U[n]<math>.
A successful minimisation of
the energy functional will yield the ground
state density <math>\,\!n_0<math> and thus all other ground state observables.
The variational problem of minimising the energy functional <math>\,\!E[n]<math>
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
\left\langle \Psi_s[n] \left| T_s+V_s \right| \Psi_s[n] \right\rangle,<math>
where <math>\,\!T_s<math> denotes the non-interacting kinetic energy and <math>\,\!V_s<math>
is an external effective potential in which the particles are moving.
Obviously, <math>n_s(\vec r)\equiv n(\vec r)<math> if <math>\,\!V_s<math>
is chosen to be
Thus, one can solve the so-called Kohn-Sham equations of
this auxiliary non-interacting system
= \epsilon_i \phi(\vec r), <math>
which yields the orbitals <math>\,\!\phi_i<math> that reproduce the density
<math>n(\vec r)<math> of the original many-body system
\sum_i^N \left|\phi_i(\vec r)\right|^2. <math>
The effective single-particle potential <math>\,\!V_s<math>
can be written in more detail as
{\rm d}^3r' + V_{\rm XC}[n_s(\vec r)],<math>
where the second term denotes the so-called Hartree term
describing the electron-electron Coulomb repulsion, while
the last term <math>\,\!V_{\rm XC}<math> is called exchange correlation potential.
Here, <math>\,\!V_{\rm XC}<math> includes all the many particle interactions.
Since the Hartree term and <math>\,\!V_{\rm XC}<math> depend on
<math>n(\vec r )<math>, which
depends on the <math>\,\!\phi_i<math>, which in turn depend on
<math>\,\!V_s<math>, 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 <math>n(\vec r)<math>, then
one calculates the corresponding <math>\,\!V_s<math> and solves the
Kohn-Sham equations
for the <math>\,\!\phi_i<math>. From these one calculates a new
density and starts again. This procedure is then repeated until
convergence is reached.
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:
The local spin-density approximation (LSDA) is a straightforward
generalization of the LDA to include electron spin:
Highly accurate formulae for the exchange-correlation energy density
<math>\epsilon_{XC}(n_\uparrow,n_\downarrow)<math> have been constructed
from simulations of a free-electron gas.
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.
The relativistic generalization of the DFT formalism leads to a current
density functional theory.
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.
[1] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864 es:Teoría del Funcional de la Densidad
fr:Théorie de la fonctionnelle de la densité
ko:밀도범함수이론
ja:密度汎関数理論
pl:Teoria funkcjonałów gęstości
Approximations
Relativistic generalization
Applications
References
[2] W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133
[3] A. D. Becke, J. Chem. Phys. 98 (1993) 5648
[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