Electrons Thomas-Fermi model

Our treatment so far has dealt with non-interacting electrons, yet we know for sure that electrons do interact with each other. Dirac (1930b) studied the effects of exchange interactions on the Thomas-Fermi model, and he soon discovered that this effect could be modelled by adding an extra term... 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

Actually, the first attempts to use the electron density rather than the wave function for obtaining information about atomic and molecular systems are almost as old as is quantum mechanics itself and date back to the early work of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of only historical interest. We therefore refrain from an in-depth discussion of the Thomas-Fermi model and restrict ourselves to a brief summary of the conclusions important to the general discussion of DFT. The reader interested in learning more about this approach is encouraged to consult the rich review literature on this subject, for example by March, 1975, 1992 or by Parr and Yang, 1989. 

In the Thomas-Fermi model,49 the kinetic energy density of the electron gas is written as... 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

The important features of ael are represented by the Thomas-Fermi model of the atom which assumes that orbital electrons screen exponentially the nuclear charge 7. On this model we have,... 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

The hardness kernels in Equation 24.110 depend on the kinetic energy functional as well as on the electron-electron interactions. Thomas-Fermi models can be used to evaluate the kinetic part of these hardness kernels and can be combined with a band structure calculation of the linear response X -... 

The p ) moments for all the neutral atoms can be fit to simple functions of the number of electrons [232,263]. These functions are heuristic extensions of expressions [264-266] derived on the basis of the Thomas-Fermi model [267-271], with the Scott-Schwinger correction [272,273] for strongly bound electrons. Thus, the Hartree-Fock (p ) for the neutral atoms from hydrogen (A = 1) through lawrencium (A = 103) can be fit [232] as follows ... 

During this period, accurate solutions for the electronic structure of helium (1) and the hydrogen molecule (2) were obtained in order to verify that the Schrodinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree (3) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model (4) based on total charge density rather than individual orbitals was used. 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

The radiation-hydrodynamic simulation includes the Quotidien EOS [29] and Ion EOS based on the Cowan model [30], For the electron component, a set of fitting formulae derived from the numerical results from the Thomas-Fermi model and a semi-empirical bonding correction [31] are adopted. The effective Z-number of the partially ionized plasma is obtained from the average atom model. Radiation transport is treated by multigroup diffusion. 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

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