Adf Density Functional

[Nilsa Housman]

Densityfunctional 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.

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.[1] 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)[2] or where dispersion competes significantly with other effects (e.g. in biomolecules).[3] The development of new DFT methods designed to overcome this problem, by alterations to the functional[4] or by the inclusion of additive terms,[5][6][7][8][9] is a current research topic. 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[10] to stray away from the search for the exact functional. Further, DFT potentials obtained with adjustable parameters are no longer true DFT potentials,[11] 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[11][12] in such conditions.

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 N electrons with 3N 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.

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.

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 n(r), which for a normalized Ψ is given by

The functionals T[n] and U[n] are called universal functionals, while V[n] is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified V ^ \displaystyle \hat V , one then has to minimize the functional

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 H for a relativistic electron moving in the Coulomb potential can be chosen in the following form (atomic units are used):

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:

where ne in Kronecker delta symbol of the second term denotes any extremal for the functional represented by the first term of the functional F. The second term amounts to zero for any function that is not an extremal for the first term of functional F. 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:

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

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.[19] 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:

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.[24] 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)[25][26][27] and have the following form:

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.

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 and Rasolt,[16] 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,[28] 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.

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 and dilute magnetic semiconductors.[1][29] It has also been shown that DFT gives good results in the prediction of sensitivity of some nanostructures to environmental pollutants like sulfur dioxide[30] or acrolein,[31] as well as prediction of mechanical properties.[32]

Density functional theory is generally highly accurate but highly computationally-expensive. In recent years, DFT has been used with machine learning techniques - especially graph neural networks - to create machine 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. Researchers have been trying to approximate DFT with machine learning for decades, but have only recently made good estimators. Breakthroughs in model architecture and data preprocessing that more heavily encoded theoretical knowledge, especially regarding symmetries and invariances, have enabled huge leaps in model performance. Using backpropagation, the process by which neural networks learn from training errors, to extract meaningful information about forces and densities, has similarly improved machine learning potentials accuracy. By 2023, for example, the DFT approximator Matlantis could simulate 72 elements, handle up to 20,000 atoms at a time, and execute calculations up to 20,000,000 times faster than DFT with similar accuracy, showcasing the power of DFT approximators in the artificial intelligence age. ML approximations of DFT have historically faced substantial transferability issues, with models failing to generalize potentials from some types of elements and compounds to others; improvements in architecture and data have slowly mitigated, but not eliminated, this issue. For very large systems, electrically nonneutral simulations, and intricate reaction pathways, DFT approximators often remain insufficiently computationally-lightweight or insufficiently accurate.[33][34][35][36][37]

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