Density functional formalism

Jones R O 1987 Molecular calculations with the density functional formalism Advances in Chemicai Physics vol LXVIl, ed K P Lawley (New York Wiley-Interscience) pp 413-37... 

Svane A and Gunnarsson Q 1990 Transition-metal oxides in the self-interaction-corrected density-functional formalism Phys. Rev. Lett. 65 1148... 

Gunnarsson O and B I Lundqvist 1976. Exchange and Correlation in Atoms, Molecules, and Solids by the Spin-density-functional Formalism. Physical Review B13.-4274-4298. 

Jones, R. O., Gunnarsson, 1989, The Density Functional Formalism, its Applications and Prospects , Rev. Mod. 

Theophilou, A., 1979, The Energy Density Functional Formalism for Excited States , J. Phys. C, 12, 5419. 

Gunnarson, O., and B. I. Lundqvist. 1976. Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B 13, 4274. 

This energy functional attains its minimum for the true electronic density profile. This offers an attractive scheme of performing calculations, the density functional formalism. Instead of solving the Schrodinger equation for each electron, one can use the electronic density n(r) as the basic variable, and exploit the minimal properties of Eq. (17.8). Further, one can obtain approximate solutions for n(r) by choosing a suitable family of trial functions, and minimizing E[n(r)] within this family we will explore this variational method in the following. 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

To make practical calculations using the density-functional formalism, Kohn and Sham (1965) show that the condition of minimizing the energy is equivalent to a set of ordinary differential equations that can be solved by a self-... 

Lang, N. D. (1973). The density-functional formalism and the electronic structure of metal surfaces. In Solid State Physics, edited by H. Ehrenreich, F. Seitz, and D. Turnbull, Vol. 28, Academic, New York. 

In addition to the cluster calculations, we report details of recent first-principles calculations based on the density functional formalism. These calculations employ periodic boundary conditions to allow investigation of the entire zeolite lattice, and therefore the use of a plane-wave basis set is applicable. This has a number of advantages, most notably that the absence of atom-centered basis functions results in no basis set superposition error (BSSE) (272), which arises as a result of the finite nature of atom-centered basis sets. Nonlocal, or gradient, corrections are applicable also, just as they are in the cluster calculations. 

Modern density functional theory (DFT) provides an enormous simplification of the many-body problem [1-7], It enables one to replace the complicated conventional wavefiinction approach with the simpler density functional formalism. The ground-state properties of the system under investigation are obtained through a minimization over densities rather than a minimization over wavefunctions. The electron density is especially attractive for calculations involving large systems, because it contains only three dimensions regardless of the size of the system. In addition, even for relatively small systems, it has been found that density-functional methods, for certain situations, often yield results competitive with, or superior to those obtained from various traditional wavefiinction approaches. 

Focusing on the variational principle present at the heart of the Density Functional formalism (actually a minimum principle), Eq.(4) must be minimised with respect to the variations of the wavefunctions, subject to the following orthonormalisation constraints ... 

In this work we study a number of isolated clusters which may be relevant for understanding the clustering in the liquid alloys. Of course, the behaviour of those clusters in the alloy may be more complicated due to the interaction with the condensed medium, but by studying free clusters we expect to obtain useful information about the tendency of the atoms to cluster in the alloy. A preliminary calculation [9] using the Density Functional Formalism (DFT) [10, 11] and a simplified model for the cluster structure [12] has confirmed the high stability of the A4Pb and A4Pb4 species (with A = Li, Na, K, Rb, Cs). However, the drastic simplification of the cluster structure used in that model calls for more accurate calculations. Consequently, in this work we report the results of ab initio molecular-dynamics DFT calculations. 

Before preceding, it is useful to consider the form of the force-force correlation function, which is given in Equation (21), with Equations (22), (24), (25), (26) and (27). The form of the force-force correlation function, derived using density functional formalism, is employed because it permits the use of very accurate equations of state for solvents like ethane and CO2 to describe the density dependence and temperature dependence of the solvent properties. These equations of state hold near the critical point as well as away from it. Using the formalism presented above, we are able to build the known density and temperature-dependent properties of the... 

Iron-series Dimers.—Harris and Jones96 have in addition calculated binding energy curves for low-lying states of the 3local spin-density approximation for the exchange and correlation energy. 

This is our new equation for the single-site density distributions which generalizes the LMBW equation (2) to polyatomic fluids, below called a site-site LMBW (SS-LMBW) equation [47]. As distinct from the site-site DFT approaches [20-24], the SS-LMBW equation properly treats the short- and long-range correlations coupled in the site-site direct correlation function Cy (r2, rs). The SS-LMBW theory also differs from the RISM approach to inhomogeneous polyatomic fluids derived within the density functional formalism by Chandler et al. [25,26]. In Equations (3.7) and (3.13) of Ref. [26] for the site density profiles p (r), the orientational averaging with the intramolecular matrix (r 12)... 

Grimme extended the double hybrid density functional formalism to be able to compute BCD. For a test suite of six molecules, performance of B2PLYP is notably better than for B3LYP. 

Self-consistent LCAO results within the density-functional formalism have also recently been obtained for a-Si02 (Xu and Ching, (1988a,b). 

Here we will not discuss the density functional formalism in depth, but we refer the reader to Ref.(3) for access to the extensive and growing literature on its fundamental theoretical... 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

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