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Teubner-Texte zur Physik

Thomas-Fermi Theory

H. Eschrig

Published Jan 1, 1996• 0 citations
Publisher

Abstract

After having started in Chapter 1 with the N -particle wavefunction, in the preceding chapter it was shown that the single-particle and two-particle density matrices suffice to calculate most groundstate properties of a many-body system described by the Hamiltonian (1.4). If one were able to classify the set of all admissible two-particle density matrices—the corresponding single-particle density matrices follow from them via (2.13)—then the groundstate energy, E , of the many-body system could be obtained as the absolute minimum of (2.46) over this set in a variational way, without reference to many-body wavefunctions as needed in (1.3). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-322-97620-8_4

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After having started in Chapter 1 with the N -particle wavefunction, in the preceding chapter it was shown that the single-particle and two-particle density matrices suffice to calculate most groundstate properties of a many-body system described by the Hamiltonian (1.4). If one were able to classify the set of all admissible two-particle density matrices—the corresponding single-particle density matrices follow from them via (2.13)—then the groundstate energy, E , of the many-body system could be obtained as the absolute minimum of (2.46) over this set in a variational way, without reference to many-body wavefunctions as needed in (1.3). These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The basic concern of density functional theory is the groundstate energy of an inhomogeneous interacting many-particle system as a function of the external potential acting on its density (and possibly as a function of the particle number, allowing for the consideration of ionization energies, affinities, and excitation gaps). This groundstate energy function is replaced by its Legendre transform as a function of the particle density, and the Legendre back transformation leads via insertion of an interaction-free reference system to an effective one-particle equation whose solution finds the ground-state energy and density to any given external potential. As was seen in Part I, this theory provides rather a reformulation than a solution of the many-body problem for the groundstate, because it does not explicitly provide the Legendre transform H [ n ] to E [ v, N ] (cf. (6.35, 6.31, and 6.36). The original many-body problem is now contained in H [ n ] which only can be guessed, although not without success. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .