The density-functional formalism

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. 

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). 

The total energy E of the system is also a functional of the density distribution, E = [] (r)]. Therefore, if the form of this functional is known, the ground-state electron density distribution n t) can be determined by its Euler-Lagrange equation. However, except for the electron gas of almost constant density, the form of the functional [ (r)] cannot be determined a priori. 

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- 

The N solutions with the lowest energy values are used to calculate the 

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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... 

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

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. 

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. 

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. 

Iron-series Dimers.—Harris and Jones96 have in addition calculated binding energy curves for low-lying states of the 3density functional formalism with a local 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)... 

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... 

In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation elfects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations,... 

Petrosyan et have presented a new method for describing solvated systems. It is based on the density-functional formalism and related to the approaches of section IIG. It combines a quantum-mechanical density-functional treatment of the solute... 

In their simulations, they use a supercell approach with the cell of Fig. 22 repeated periodically in all three dimensions (with, however, some further layers of Pt atoms). All atoms were treated quantum-mechanically within the density-functional formalism. 

We consider a cluster of AT He atoms of mass m and radius ro, together with a single excess electron. The subsystem of the helium atoms will be treated by the density functional formalism [113, 247]. The excess electron will be treated quantum mechanically. The energetics and charge distribution of the electron were calculated within the framework of the adiabatic approximation for each fixed nuclear configuration. 

At present the most satisfactory foundation of the one-electron picture for metals is provided by the local approximation to the density-functional formalism of Hohenberg and Kohn [1.5] and Kohn and Sham [1.6]. This one-electron theory is presented in Chap.7 when ground-state properties of crystalline solids are discussed. Here we note that the local-density (LD) formalism, like the Xa method, leads to an effective one-electron potential which is a function of the local electron density as expressed by (1.2). 

The approach we have discussed here addresses both problems with comparable emphasis. The density functional formalism, with the LSD approximation for the exchange-correlation energy, provides us with an approximate method of calculating energy surfaces, and the results have predictive value in many contexts. DF can also be carried out with comparable ease for all elements. When coupled with MD at elevated temperatures (simulated annealing), it is possible to study cases where the most stable isomers are unknown, or where the energy surfat have many local minima. 

Several authors [8,9,15-18], have recently attempted the calculation of c. All those calculations have several points in common a) The use of the density functional formalism [14] with approximate functionals for the kinetic energy of the electrons, like that of Eq. (14), or including higher order gradient corrections, b) A local density (LDA) description of exchange and correlation effects. The LDA exchange energy is... 

Clemenger s model is, however, non-selfconsistent. Ekardt and Penzar [41,4.3] have extended the jellium mode to account for spheroidal deformations. In this model the ionic background is represented by a distribution of positive charge with constant density and a distorted, spheroidal, shape. The advantage with respect to Clemenger s model is that the spheroidal jellium model is parameter-free and that the calculation of the electronic wave functions is performed self-consistently using the density functional formalism. The distortion parameter is determined by solving the Kohn-Sham equations for different... 

How does the surface electronic profile deform with charging Although there was no direct observation of the charge-induced profile modulation, a lot is known from the theory and indirect manifestation of this effect in field emission. For the jellium model, after the pioneering papers [110, 113, 114], the problem was studied in detail in Ref. [115] within the trial function version of the density functional formalism [116] and in Ref. [117, 118] within the more general Kohn-Sham scheme [119]. We summarize here the properties necessary for our further discussion. 

During the last few years, the latter view has received more supporting evidence. Already the early experimental work of Giraultand Schiffrin [71], who determined the surface excess of water at the interface with 1,2-dichloroethane, had indicated the existence of a mixed boundary layer. Recent X-ray scattering experiments [72] indicate an average interfacial width of the order of 3 to 6 A. These experiments are in line both with model calculations based on the density functional formalism [73] and with computer simulations [74, 75]. Accordingly, the interface is best visualized as rough on a molecular scale as indicated in Fig. 13. 

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