Thomas-Fermi Theorems

3 FORMALIZATION OF THOMAS-FERMI THEORY 5.3.1 THOMAS-FERMI THEOREMS 

The TF theory was independently developed by Thomas (1927) and Fermi (1927), and proposes for the total energy as the density functional, the expression  

The correction through the exchange term was introduced by Dirac (1930) through which the TF theory becomes the DFT theory and is amended as follows  

Another correction this time on the kinetic term, brought by von Weizsaecker (1935) transforms the TF theory in TFW theory and the last relation will be properly modified the present discussion follows (Putz, 2012)  

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

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In summary, the original Thomas-Fermi-Dirac DFT was unable to give binding in molecules. This was corrected by Kohn-Sham, [11] who chose to use an orbital rather than density evaluation of the kinetic energy. By the virial theorem, = —E, so this was a necessity to obtain realistic results for energies. Next, it was shown that the exact exchange requires an orbital-dependent form, too. [47,48] The future seems to demand an orbital-dependent form for the correlation. 

Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

Note that -y and y usually approach the Thomas-Fermi limit ( ) (Table 4.4) except for hydrogen, where y=2 because of the virial theorem, E=T + Vne, with E= - T. 

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 HK theorem tells us that it should be possible to calculate all observables using the density alone, since the HK theorem guarantees that they are all functionals of no(r). In practice, however, one does not know how to do this explicitly. In particular, to perform such calculations one needs reliable approximations for T[n] and U n] to begin with. Prior to discussing a practical way of attacking these problems, we recall an older, but still occasionally useful, alternative the Thomas-Fermi (TF) approximation [257]. 

Thomas-Fermi model [18], or with embellishments by Dirac, the Thomas-Fermi-Dirac model [18], 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). 

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 model does not suffice for chanistry. In the Thomas-Fermi theory, all atomic anions are predicted to be unstable. Also, the energy of a molecule is always greater than the energy of isolated atoms, so the Thomas-Fermi theory predicts that no molecule is stable. This result is called the Teller nonbinding theorem." ... 

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. 

Kohn-Sham DFT is widely used in electronic structure calculations of the ground state properties of atoms, molecules, and solids. DFT was introduced by Hohenberg and Kohn (1964). Based on the Thomas-Fermi model (Fermi, 1927 Thomas, 1927), they laid out the fundamental theorem which stated that the electron density determines the external potential and as a result, immediately, it also uniquely determines the Hamiltonian operator (Hohenberg and Kohn, 1964). One year later, in 1965, Kohn and Sham declared another theorem resembling the variational principle which stated that the true ground state density would deliver the ground state energy of the system (Kohn and Sham, 1965). 

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