Ab initio calculations using Kohn-Sham (KS) density functional theory (DFT)1,2 can accurately describe the fundamental properties of various materials. However, its computational cost scales with the cube of the number of electrons in the simulation cell, which poses a major challenge to large-scale simulations. In contrast, orbital-free (OF) DFT is inherent of the lower computational cost that scales linearly with the number of atoms in the system, as it relies only on the electron density and the use of KS orbitals is avoided. As a result, OF-DFT is successfully applied to large-scale simulations of systems with up to millions of atoms3,4,5,6.
where fi, \(V_nl(\bfr^\prime,\bfr)=\langle \bfr^\prime \hatV_nl \bfr\rangle\), and \(\gamma _s(\bfr,\bfr^\prime)=\sum _if_i\psi _i(\bfr)\psi _i^* (\bfr^\prime)\) represent the occupation number of the ith KS orbital ψi, the real-space representation of the nonlocal part pseudopotential, and the non-interacting density matrix, respectively. Considering that the density matrices \(\gamma _s[\rho ](\bfr,\bfr^\prime)\) can be used to approximate the KEDFs7,56,57, an NLPP energy density functional (NLPPF) relying directly on the density matrix is proposed to evaluate the nonlocal electron-ion interaction energy. The nonlocal electron-ion interaction energy is then rewritten as a function of electron density
Recent Advances In Orbital Free Density Functional Theory Pdf
It proves useful to incorporate thecondition of eq 5 intoapproximate density functionals,particularly for nearly free-electron metals. This information issufficient for determining the weight function in eq 4:
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.
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 properties of non-uniform classical fluids.
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.
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.
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 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]
A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield[37] and more recently Cronin[citation needed], who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. 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 pseudo-wavefunctions are also forced to have the same norm (i.e., the so-called norm-conserving condition) as the true valence wavefunctions and can be written as
Classical density functional theory is a classical statistical method to investigate the properties of many-body systems consisting of interacting molecules, macromolecules, nanoparticles or microparticles.[41][42][43][44] The classical non-relativistic method is correct for classical fluids with particle velocities less than the speed of light and thermal de Broglie wavelength smaller than the distance between particles. The theory is based on the calculus of variations of a thermodynamic functional, which is a function of the spatially dependent density function of particles, thus the name. The same name is used for quantum DFT, which is the theory to calculate the electronic structure of electrons based on spatially dependent electron density with quantum and relativistic effects. Classical DFT is a popular and useful method to study fluid phase transitions, ordering in complex liquids, physical characteristics of interfaces and nanomaterials. Since the 1970s it has been applied to the fields of materials science, biophysics, chemical engineering and civil engineering.[45] 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. Classical DFT is valuable to interpret and test numerical results and to define trends although details of the precise motion of the particles are lost due to averaging over all possible particle trajectories.[46] 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 allows the calculation of the equilibrium particle density and prediction of thermodynamic properties and behavior of a many-body system on the basis of model interactions between particles. The spatially dependent density determines the local structure and composition of the material. It is determined as a function that optimizes the thermodynamic potential of the grand canonical ensemble. 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. The excess free energy is then a sum of the contributions from s-body interactions with density-dependent effective potentials representing the interactions between s particles. In most calculations the terms in the interactions of three or more particles are neglected (second-order DFT). 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. The minimization of the grand potential functional in arbitrary local density functions for fixed chemical potential, volume and temperature provides self-consistent thermodynamic equilibrium conditions, in particular, for the local chemical potential. The functional is not in general a convex functional of the density; solutions may not be local minima. Limiting to low-order corrections in the local density is a well-known problem, although the results agree (reasonably) well on comparison to experiment. 2ff7e9595c
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