Density functional theory
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.
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.
[4] The development of new DFT methods designed to overcome this problem, by alterations to the functional[5] or by the inclusion of additive terms,[6][7][8][9][10] is a current research topic.
[14] 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).
[15] 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.
[16][17] 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.
A stationary electronic state is then described by a wavefunction Ψ(r1, …, rN) satisfying the many-electron time-independent Schrödinger equation where, for the N-electron system, Ĥ is the Hamiltonian, E is the total energy,
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.
The variational problems of minimizing the energy functional E[n] can be solved by applying the Lagrangian method of undetermined multipliers.
It is demonstrated in Brack (1983)[19] that application of the virial theorem to the eigenfunction equation produces the following formula for the eigenenergy of any bound state: and analogously, the virial theorem applied to the eigenfunction equation with the square of the Hamiltonian yields It is easy to see that both of the above formulae represent density functionals.
Highly accurate formulae for the correlation energy density εC(n↑, n↓) have been constructed from quantum Monte Carlo simulations of jellium.
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.
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 for electrons delocalised through an infinite solid.
Among physicists, one of the most widely used functionals is the revised Perdew–Burke–Ernzerhof exchange model (a direct generalized gradient parameterization of the free-electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gas-phase molecular calculations.
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).
These graph neural networks approximate DFT, with the aim of achieving similar accuracies with much less computation, and are especially beneficial for large systems.
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.
The kinetic-energy functional can be improved by adding the von Weizsäcker (1935) correction:[41][42] The Hohenberg–Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential.
The use of an effective interaction, a pseudopotential, that approximates the potential felt by the valence electrons, was first proposed by Fermi in 1934 and Hellmann in 1935.
Based on that idea, modern pseudo-potentials are obtained inverting the free-atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo-wavefunctions to coincide with the true valence wavefunctions beyond a certain distance rl.
The distance rl beyond which the true and the pseudo-wavefunctions are equal is also dependent on l. The electrons of a system will occupy the lowest Kohn–Sham eigenstates up to a given energy level according to the Aufbau principle.
Classical DFT is a popular and useful method to study fluid phase transitions, ordering in complex liquids, physical characteristics of interfaces and nanomaterials.
[51] Computational costs are much lower than for molecular dynamics simulations, which provide similar data and a more detailed description but are limited to small systems and short time scales.
[52] As in electronic systems, there are fundamental and numerical difficulties in using DFT to quantitatively describe the effect of intermolecular interaction on structure, correlations and thermodynamic properties.
Classical DFT addresses the difficulty of describing thermodynamic equilibrium states of many-particle systems with nonuniform density.
Theories were developed for simple and complex liquids using the ideal gas as a basis for the free energy and adding molecular forces as a second-order perturbation.
The grand potential is evaluated as the sum of the ideal-gas term with the contribution from external fields and an excess thermodynamic free energy arising from interparticle interactions.
In the simplest approach the excess free-energy term is expanded on a system of uniform density using a functional Taylor expansion.
When the structure of the system to be studied is not well approximated by a low-order perturbation expansion with a uniform phase as the zero-order term, non-perturbative free-energy functionals have also been developed.
