Generalized Gradient Approximation

Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT 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. Computational costs are relatively low when compared to traditional methods, such as exchange only Hartree–Fock theory and its descendants that include electron correlation. Since, DFT has become an important tool for methods of nuclear spectroscopy such as Mössbauer spectroscopy or perturbed angular correlation, in order to understand the origin of specific electric field gradients in crystals.

Despite recent improvements, there are still difficulties in using density functional theory to properly describe: intermolecular interactions (of critical importance to understanding chemical reactions), especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces, dopant interactions and some strongly correlated systems; and in calculations of the band gap and ferromagnetism in semiconductors. 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. 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. Classical density functional theory uses a similar formalism to calculate properties of non-uniform classical fluids.

Despite the current popularity of these alterations or of the inclusion of additional terms, they are reported to stray away from the search for the exact functional. Further, DFT potentials obtained with adjustable parameters are no longer true DFT potentials, 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 in such conditions.

Overview of method

In the context of computational materials 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 potentials , which is determined solely by the structure and the elemental composition of the system, and an effective potential , which represents interelectronic interactions. Thus, a problem for a representative supercell of a material with electrons can be studied as a set of one-electron Schrödinger-like equations, which are also known as Kohn–Sham equations.

Origins

Although density functional theory has its roots in the Thomas–Fermi model for the electronic structure of materials, DFT was first put on a firm theoretical footing by Walter Kohn and Pierre Hohenberg in the framework of the two Hohenberg–Kohn theorems (HK). The original HK theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.

The first HK theorem demonstrates that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of electrons with spatial coordinates to three spatial coordinates, through the use of functionals of the electron density. This theorem has since been extended to the time-dependent domain to develop time-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 the Nobel prize in chemistry, the HK theorem was further developed by Walter Kohn and Lu Jeu Sham to produce Kohn–Sham DFT (KS DFT). Within this framework, the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting 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. Non-interacting systems are relatively easy to solve, as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic 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, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the noninteracting system.

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 , in which the electrons are moving. A stationary electronic state is then described by a wavefunction satisfying the many-electron time-independent Schrödinger equation

: \hat H \Psi = \left hat T + \hat V + \hat U\rightPsi = \left sum_^N \left(-\frac \nabla_i^2\right) + \sum_^N V(\mathbf r_i) + \sum_^N U\left(\mathbf r_i, \mathbf r_j\right)\right\Psi = E \Psi,

where, for the -electron system, is the Hamiltonian, 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, and is the electron–electron interaction energy. The operators $\hat T$ and are called universal operators, as they are the same for any -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 term .

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 , onto a single-body problem without . In DFT the key variable is the electron density , which for a normalized is given by

: $n(\mathbf r) = N \int^3 \mathbf r_2 \cdots \int^3 \mathbf r_N \, \Psi^*(\mathbf r, \mathbf r_2, \dots, \mathbf r_N) \Psi(\mathbf r, \mathbf r_2, \dots, \mathbf r_N).$

This relation can be reversed, i.e., for a given ground-state density it is possible, in principle, to calculate the corresponding ground-state wavefunction . In other words, is a unique functional of ,

: \Psi_0 = \Psi _0

and consequently the ground-state expectation value of an observable is also a functional of :

: O _0= \big\langle \Psi _0\big, \hat O \big, \Psi _0\big\rangle.

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

: E_0 = E _0= \big\langle \Psi _0\big, \hat T + \hat V + \hat U \big, \Psi _0\big\rangle,

where the contribution of the external potential \big\langle \Psi _0\big, \hat V \big, \Psi _0\big\rangle can be written explicitly in terms of the ground-state density $n_0$:

: V _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 $n$:

: V = \int V(\mathbf r) n(\mathbf r) \,\mathrm d^3 \mathbf r.

The functionals and are called universal functionals, while 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 = T + U + \int V(\mathbf r) n(\mathbf r) \,\mathrm d^3 \mathbf r

with respect to , assuming one has reliable expressions for and . A successful minimization of the energy functional will yield the ground-state density and thus all other ground-state observables.

The variational problems of minimizing the energy functional can be solved by applying the Lagrangian method of undetermined multipliers. First, one considers an energy functional that does not explicitly have an electron–electron interaction energy term,

: E_s = \big\langle \Psi_\text \big, \hat T + \hat V_\text \big, \Psi_\text \big\rangle,

where $\hat$ denotes the kinetic-energy operator, and $\hat_\text$ 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 \nabla^2 + V_\text(\mathbf r)\right\varphi_i(\mathbf r) = \varepsilon_i \varphi_i(\mathbf r),

which yields the orbitals that reproduce the density of the original many-body system

: $n(\mathbf r ) = \sum_^N \big, \varphi_i(\mathbf r)\big, ^2.$

The effective single-particle potential can be written as

: V_\text(\mathbf r) = V(\mathbf r) + \int \frac \,\mathrm d^3 \mathbf r' + V_\text (\mathbf r)

where $V(\mathbf r)$ is the external potential, the second term is the Hartree term describing the electron–electron Coulomb repulsion, and the last term is the exchange–correlation potential. Here, includes all the many-particle interactions. Since the Hartree term and depend on , which depends on the , which in turn depend on , 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 , then calculates the corresponding and solves the Kohn–Sham equations for the . From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation called Harris 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. 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 the Green function instead of the density . It is called as Luttinger–Ward functional (or kinds of similar functionals), written as . However, 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-body density matrix and the one-body potential . (Remember that all the eigenvalues of 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)

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 a hydrogen-like ion obeying the relativistic Dirac equation. The Hamiltonian for a relativistic electron moving in the Coulomb potential can be chosen in the following form (atomic units are used):

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

where is the Coulomb potential of a pointlike nucleus, is a momentum operator of the electron, and , and are the elementary charge, electron mass and the speed of light respectively, and finally and are a set of Dirac 2 × 2 matrices:

:$\begin \boldsymbol\alpha &= \begin 0 & \boldsymbol\sigma \\ \boldsymbol\sigma & 0 \end, \\ \beta &= \begin I & 0 \\ 0 & -I \end. \end$

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

: $H\Psi = E\Psi,$

where is a four-component wavefunction, and is the associated eigenenergy. It is demonstrated in Brack (1983) that application of the virial theorem to the eigenfunction equation produces the following formula for the eigenenergy of any bound state:

: $E = mc^2 \langle \Psi , \beta , \Psi \rangle = mc^2 \int \big, \Psi(1)\big, ^2 + \big, \Psi(2)\big, ^2 - \big, \Psi(3)\big, ^2 - \big, \Psi(4)\big, ^2 \,\mathrm\tau,$

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 = \frac \left(mc^2 \int n \,d\tau - \sqrt \right)^2 + \delta_ mc^2 \int n \,d\tau,

where in Kronecker delta symbol of the second term denotes any extremal for the functional represented by the first term of the functional . The second term amounts to zero for any function that is not an extremal for the first term of functional . 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 _e + \delta n= \frac \left(mc^2 \int (n_e + \delta n) \,d\tau - \sqrt \right)^2.

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

: $\frac = 2A - \frac + eV(\tau_0),$

where , and , and is a value of potential at some point, specified by support of variation function , 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 if . This last condition provides us with Lagrange
equation for functional , 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 functional . 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 functional , 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)

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. One of the simplest approximations is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

: E_\text^\text = \int \varepsilon_\text(n) n(\mathbf r) \,\mathrm d^3 \mathbf r.

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

: E_\text^\text _\uparrow, n_\downarrow= \int \varepsilon_\text(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: . The exchange part is called the Dirac (or sometimes Slater) exchange, which takes the form . There are, however, many mathematical forms for the correlation part. Highly accurate formulae for the correlation energy density have been constructed from quantum Monte Carlo simulations of jellium. A simple first-principles correlation functional has been recently proposed as well. Although unrelated to the Monte Carlo simulation, the two variants provide comparable accuracy.

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. 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) and have the following form:

: E_\text^\text _\uparrow, n_\downarrow= \int \varepsilon_\text(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 the Minnesota Functionals. 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 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 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 by Vignale

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