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Revista de la Sociedad Química de México

The foundations of density functional theory and wave quantum mechanics

Jaime Keller

Published Jan 1, 2000• 4 citations
OpenAlex

Abstract

We analyze the notion of Density Functional Theory from the definition of a density of action over spacetime. We derive a theory to obtain the energy as a function of particle density, which contains the standard Density Functional Theory (DFT) and enlarges it by considering in one functional both the density of particles and the energy density per particle. The formulation presents a mathematical structure where the particle density is a factor of the energy per particle. The starting point is the definition of a global action density K(x) from which, by successive projections, the density Kj for a given type j of particles is projected. This density is then factorized into gauge dependent factors, which are shown to generate known relations and structures of quantum mechanics. Within DFT the minimization of the energy density functional, with respect to changes in the density, contains as a consequence two terms: the first corresponds to the standard density functional theory for non-interacting particles, and the second to the optimization of the kinetic and the interparticle interaction energy, terms that go beyond the standard DFT. We show explicitly the relation of the present approach to standard Wave Quantum Mechanics and show explicitly the reasons within spacetime- action of several basic postulates of Quantum Mechanics.

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arXiv:2206.10097•Lecture notes series, Institute For Mathematical Sciences•Feb 26, 2023•1 citation
We analyze the notion of Density Functional Theory from the definition of a density of action over spacetime. We derive a theory to obtain the energy as a function of particle density, which contains the standard Density Functional Theory (DFT) and enlarges it by considering in one functional both the density of particles and the energy density per particle. The formulation presents a mathematical structure where the particle density is a factor of the energy per particle. The starting point is the definition of a global action density K(x) from which, by successive projections, the density Kj for a given type j of particles is projected. This density is then factorized into gauge dependent factors, which are shown to generate known relations and structures of quantum mechanics. Within DFT the minimization of the energy density functional, with respect to changes in the density, contains as a consequence two terms: the first corresponds to the standard density functional theory for non-interacting particles, and the second to the optimization of the kinetic and the interparticle interaction energy, terms that go beyond the standard DFT. We show explicitly the relation of the present approach to standard Wave Quantum Mechanics and show explicitly the reasons within spacetime- action of several basic postulates of Quantum Mechanics.
Quantum Mechanics is a successful model to describe physical phenomena. We show that a simple set of postulates about the description of objects in Nature allows the derivation of the principles of QM, transforming this theory into a deductive approach to Physics. The main postulate is: Objects are described as energy distributions in Space–Time. 1 Quantum Mechanics from START It is universally accepted that Quantum Mechanics is a successful model to describe physical phenomena, in particular of the electronic structure of atoms, molecules and solids. As such it has been reformulated in several different forms (see for example: Nine formulations of quantum mechanics, Daniel F. Styer et al. [15]). The aim of the present paper is to show that a simple set of postulates about the description of objects in Nature allows the derivation of the principles of QM, transforming this theory into a deductive approach to Physics. In our approach the main postulate is: Objects are described as energy distributions in Space–Time. The theory is in fact a systematic mathematical analysis of this postulate and of other general considerations about action, energy–momentum and the role of the theory as a useful description of Nature. The paper also shows the fundamental and basic status of Density Functional Theory, also contained in our approach. The initial mathematical structure is a geometric (quadratic form based) union of space, time and action. In physics bundles of trajectories are used to describe the objects of nature: distribution of heights over a surface, a real object is described as a bundle of space-time trajectories, here we use a description as a bundle of action valves in space-time (energy densities in space following light-like lines). The author is fully aware that the presentation is strongly guided by the knowledge of the usual computational approach to study atoms, molecules and condensed matter electronic structure.
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, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
In these notes I have given a personally flavored exposé of static density-functional theory (DFT). I have started from standard many-body physics at a very elementary level and then gradually introduced the basic concepts of DFT. Successively more advanced topics are added and at the end I even discuss a few not yet published theories. The discussion represents many of the personal views of the author and there is no attempt at being comprehensive. I fully realize that I am often 'unfair' in treating the achievements of other researchers. Many topics of standard DFT are deliberately left out like, e.g., time-dependence, excitations, and magnetic or relativistic effects. These notes represent a compilation of a series of lectures given at at the EXC!TING Summer School DFT beyond the ground state at Riksgränsen, Sweden in June of 2003.
Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.