Thomas-Fermi method

In the field of quantum chemistry, theories on electronic motion states commencing with the Hartree-Fock method have intended to incorporate electron correlation more efficiently. However, due to the long computational times required, quantum chemistry calculations had been mostly restricted to trial applications by theoreticians until the 1980s. In the 1990s, density functional theory (DFT) appeared in quantum chemistry to resolve this situation. After this, DFT has become widespread, so that is now the main theory, which is used in more than 80% of quantum chemistry papers in 2014. 

Tsuneda, Density Functional Theory in Quantum Chemistry, DOI 10.1007/978-4-431-54825-6 4, Springer Japan 2014 

This kinetic energy functional was the first local density approximation (LDA). In the next year (1928), Fermi independently derived the same kinetic energy functional as Thomas s functional using Fermi statistics at the absolute zero point, completing what is now known as the Thomas-Fermi method, on the basis of the Hartree method (see Sect. 2.1) (Fermi 1928). 

Although the Thomas-Fermi method is an interesting theory representing the Hamiltonian operator as the functional only of the electron density, even qualitative discussions cannot be contemplated based on this method in actual electronic state calculations. Dirac considered that this problem may be attributed to the lack of exchange energy (see Sect. 2.4), which was proposed in the same year (Fock 1930), and proposed the first exchange functional of electron density p (Dirac 1930), 

This functional is found to be the exact LDA exchange functional. Furthermore, von Weizsacker proposed a correction term using the gradient of electron density for the Thomas-Fermi kinetic energy functional (von Weizsacker 1935), 

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Confusion is created by the often-quoted results of calculations by Latter that did predict some of the above ordering on the badis of the rather crude Thomas-Fermi method of approximation 20). More recent Hartree-Fock calculations on atoms show, for example, that the 3d level is definitely of lower energy than that of 4s (21). 

In 1926 Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. Electron-shells are not envisaged in this model, which was independently rediscovered by Enrico Fermi two years later. For many years the Thomas-Fermi method 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.17... 

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

But alas most of what has been described so far concerning density theory applies in theory rather than in practice. The fact that the Thomas-Fermi method is capable of yielding a universal solution for all atoms in the periodic table is a potentially attractive feature but is generally not realized in practice. The attempts to implement the ideas originally due to Thomas and Fermi have not quite materialized. This has meant a return to the need to solve a number of equations separately for each individual atom as one does in the Hartree-Fock method and other ab initio methods using atomic orbitals. 

Edward Teller showed that the Thomas-Fermi method cannot predict binding in atoms. 

These F-values are not so reliable as those calculated by Hartree s method. On the other hand, they are obtained with much less labor, Hartree s calculations having so far been carried out for only a small number of atoms. In figure 10 F-curves are shown for Li+, Na+, K+, and Rb+ as obtained by the method described in this paper, by Hartree s method and by the Thomas-Fermi method. It is seen that for all... 

An intuitive extension of the quantum-mechanical discription to more complicated atoms is by the Thomas-Fermi method [62]. 

I do not recall when I first heard of the Hohenberg-Kohn-Sham papers, but I do know that the quantum chemistry community at first paid little attention to them. In June of 1966 Lu Sham spoke about DFT at a Gordon Conference. But in those days, there was more discussion about another prescription that had been on the scene since 1951, the Slater Xa method. The Xa method was a well-defined, substantial improvement over the Thomas-Fermi method, a sensible approximation to exact Kohn-Sham. Debate over Xa went on for a number of years. Slater may never have recognized DFT as the major contribution to physics that it was. [When I asked John Connolly five or six years ago how he thought Slater had viewed DFT, he replied that he felt that Slater regarded it as obvious. ]... 

The actual calculation of the repulsive forces needs of course a very exact knowledge of the charge distribution on the surface of the molecules, and therefore presents considerable difficulties hitherto, a detailed calculation could only be carried out for the very simplest case of He. The most successful attempts in this direction so far have applied the ingenious Thomas-Fermi method which takes the Pauli Principle directly as a basis and is accordingly able, neglecting many unessential details, to account for just that effect which is characteristic of this penetration mechanism. 

The Thomas-Fermi method and the Xa scheme were at the time of their inceptions considered as useful models based on the notion that the energy of an electronic system can be expressed in terms of its density. A formal proof of this notion came in 1964 when it was shown by Hohenberg and Kohn [38] that there is a unique relation between density and energy. The year after Kohn and Sham put forward a practical variational DFT approach in which they replaced E of (2) with a combined exchange and correlation term... 

Using the spherical jellium model and the extended Thomas-Fermi method described in Sect. 2.2 above, we have calculated the critical number Nq for q = 1, 2, 3. In our calculation these give the cluster sizes at which the successive electron affinities Aj (designated simply by A in previous sections), A2 and A3 become positive. These quantities are given by... 

Table 4. Critical cluster size at which the singly, doubly and triply charged sodium cluster anions become stable against electron detachment. The results of the extended Thomas-Fermi method have been obtained with X = 0.5. The entry labelled NL-WDA corresponds to the calculations [18] with the non local kinetic and exchange energy functionals mentioned in Sect. 2.2...

Although various attempts have been made to modify the Thomas-Fermi method, all of these attempts have failed to make the method reliable, because it has no physical background establishing the uniqueness of solutions and the existence of density functionals, and it also cannot reproduce even chemical bonds qualitatively. As a consequence, this method had been forgotten until the mid-1960s. 

In 1964, the concept of the Thomas-Fermi method was revived by a theorem called the Hohenberg-Kohn theorem (Hohenberg and Kohn 1964). This theorem consists of the following two subsidiary theorems for nondegenerate ground electronic states ... 

In Chap. 4, the Kohn-Sham equation, which is the fundamental equation of DFT, and the Kohn-Sham method using this equation are described for the basic formalisms and application methods. This chapter first introduces the Thomas-Fermi method, which is conceptually the first DFT method. Then, the Hohenberg-Kohn theorem, which is the fundamental theorem of the Kohn-Sham method, is clarified in terms of its basics, problems, and solutions, including the constrained-search method. The Kohn-Sham method and its expansion to more general cases are explained on the basis of this theorem. This chapter also reviews the constrained-search-based method of exchange-correlation potentials from electron densities and... 

The Thomas-Fermi method incorporates quantum effects only to the extent that the electrons obey the exclusion principle. A natural goal of a... 

Gradually, the Thomas-Fermi method or its modem descendants, which are known as density functional theories, have become equally powerful compared to methods based on orbitals and wavefimctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. The solution is expressed in terms of the variable Z, which represents atomic number, the crucial feature that distinguishes one kind of atom from any other element. One does not need to repeat the calculation separately for each atom, but this advantage applies only in principle, as discussed below. 

Marconi and March [45], in early work, studied relativistic generalizations of the l/Z expansion (1) by invoking the Vallarta-Rosen [46] generalization of the original Thomas-Fermi method [4] [5]. This generalization replaces the constant chemical potential unr of non-relativistic density-functional... 

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. 

The density functional theoiy (DFT) was mentioned previously in Chapter 4, and the Thomas-Fermi method can also be viewed as a special case of DFT. DFT has become very popular over the last 20 years or so. The reasons are obvious — it scales as where N is the number of electrons—by contrast to standard ab initio Hartree-Fock theory, which scales as N, and the results are more accurate (not to mention accurate ab initio theories, which scale as N ). Furthermore the DFT theory operates in three dimensions (x, y, z) in which the electron density is defined—no matter how many electrons are involved. DFT theory has been reviewed and described in numerous books and review articles see, for instance, [212]. We shall therefore only give a brief presentation of it here. 

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