Density functional theory Thomas-Fermi model

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

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. 

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,... 

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 ... 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

Local-scaling transformations made their appearance in density functional theory (although in a disguised manner) in the works of Macke [58, 59]. Because the Thomas-Fermi theory corresponded to a free-electron gas model, and as such it was cast in terms of plane waves, any improvement on this theory required the introduction of deformed plane waves. Thus, initially local-scaling transformations were implicitly used when plane waves (defined in the volume V in ft3 and having uniform density p0 = N/V) ... 

Since the Thomas-Fermi functional is exact for the uniform electron gas, its failings must arise because the electron densities of chemical substances are far from uniform. This suggests that we construct the gradient expansion about the uniform electron gas limit such functionals will be exact for nearly uniform electron gases. An alternative perspective is to recall that the Thomas-Fermi theory is exact in the classical high-quantum number limit. The gradient expansion can be derived as a Maclaurin series in powers of ti it adds additional quantum effects to the Thomas-Fermi model. 

Another model potential can be obtained utilizing the statistical theory for the electron distribution in an atom due to the studies done by Thomas and Fermi [266, p. 145-156]. The Thomas-Fermi potential can be seen as the simplest potential possible within the framework of density functional theory in section 8.8. This close connection to DFT shows that exchange and correlation functionals can easily be introduced into the program code despite numerical... 

Density-functional theory has its conceptual roots in the Thomas-Fermi model of a uniform electron gas [325,326] and the Slater local exchange approximation [327]. A formalistic proof for the correctness of the Thomas-Fermi model was provided by Hohenberg-Kohn theorems, [328]. DFT has been very popular for calculations in sohd-state physics since the 1970s. In many cases DFT with the local-density approximation and plane waves as basis functions gives quite satisfactory results, for sohd-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum-mechanical many-body problem. 

In 1926 the physicist Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. No electron-shells are envisaged in this model which was independently rediscovered by Italian physicist Enrico Fermi two years later, and is now called the Thomas-Fermi method. For many years it was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals. The Thomas-Fermi method treats the electrons around the nucleus as a perfectly homogeneous electron gas. The mathematical solution for the Thomas-Fermi model is universal , which means that it can be solved once and for all. This should represent an improvement over the method that seeks to solve Schrodinger equation for every atom separately. Gradually the Thomas-Fermi method, or density functional theories, as its modem descendants are known, have become as powerful as methods based on orbitals and wavefunctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

---